APPLIED THERMODYNAMICS FOR ENGINEERS BY WILLIAM D. ENNIS, M.E. MEMBER OF AMERICAN SOCIETY OF MECHANICAL ENGINEERS PROFESSOR OF MECHANICAL ENGINEERING IN THE POLYTECHNIC INSTITUTE OF BROOKLYN 316 ILLUSTRATIONS FOURTH EDITION, CORRECTED NEW YORK D. VAN NOSTRAND COMPANY 25 PARK PLACE 1915 COPYRIGHT, 1910, BY D. VAN NOSTRAND COMPANY COPYRIGHT, 1913, BY D. VAN NOSTRAND COMPANY COPYRIGHT, 1915, BY D. VAN NOSTRAND COMPANY THE SCIENTIFIC PRESS ROBERT DRUMMOND AND COMPANY BROOKLYN, N, V. PREFACE TO THE THIRD EDITION THIS book was published in the fall of 1910. It was the first new American book in its field that had appeared in twenty years. It was not only new in time, it was new in plan. The present edition, which represents a third printing, thus demands careful revision. The revision has been comprehensive and has unfortunately somewhat increased the size of the book a defect which further time may, however, permit to be overcome. Such errors in statement or typography as have been discovered have been eliminated. Improved methods of presentation have been adopted wherever such action was possible. Answers to many of the numerical prob- lems have now been incorporated, and additional problems set. Expanded treatment has been given the kinetic theory of gases and the flow of gases; and results of recent studies of the properties of steam have been discussed. There will be found a brief study of gas and vapor mixtures, undertaken with special reference to the use of mixtures in heat engines. The gas engine cycle has been subjected to an analysis which takes account of the varying specific heats of the gases. The section on pressure turbines has been rewritten, as has also the whole of Chapter XV, on results of engine tests the latter after an entirely new plan. A new method of design of com- pound engines has been introduced. Some developments from the engineering practice of the past three years are discussed such as Orrok's condenser constants; Clayton's studies of cylinder action (with application to the Hirn analysis and the entropy diagram), the Humphrey internal combustion pump, the Stumpf uniflow engine and various gas-engine cycles. The section on absorption systems of refrigeration has been extended to include the method of computing a heat balance. Brief additional sections on applica- tions of the laws of gases to ordnance and to balloon construction are submitted. A table of symbols have been prefixed to the text, and a " reminder " page on the forms of logarithmic transformation iii iv PREFACE may be found useful. The Tyler method of solving exponential equations by hyperbolic functions will certainly be found new. In spite of these changes ; the inductive method is retained to the largest extent that has seemed practicable. The function of the book is to lead the student from what is the simple and obvious fact of daily experience to the comprehensive generalization. This seems more useful than the reverse procedure. POLYTECHNIC INSTITUTE OF BROOKLYN, NEW YORK, 1913. PREFACE TO THE FIRST EDITION " APPLIED THERMODYNAMICS " is a pretty broad title ; but it is intended to describe a method of treatment rather than unusual scope. The writer's aim has been to present those fundamental principles which concern the designer no less than the technical student in such a way as to convince of their importance. The vital problem of the day in mechanical engineering is that of the prime mover. Is the steam engine, the gas engine, or the turbine to survive? The internal combustion engine works with the wide range of temperature shown by Carnot to be desirable; but practically its superiority in efficiency is less marked than its temperature range should warrant. In most forms, its entire charge, and in all forms, the greater part of its charge, must be compressed by a separate and thermally wasteful operation. By using liquid or solid fuel, this complication may be limited so as to apply to the air supply only ; but as this air supply constitutes the greater part of the combustible mixture, the difficulties remain serious, and there is no present means available for supplying oxygen in liquid or solid form so as to wholly avoid the necessity for compression. The turbine, with superheat and high vacuum, has not yet surpassed the best efficiency records of the reciprocating engine, although commercially its superior in many applications. Like the internal combustion engine, the turbine, with its wide temperature range, has gone far toward offsetting its low efficiency ratio ; where the temperature range has been narrow the economy has been low, and when running non-condensing the efficiency of the turbine has compared unfavorably with that of the engine. There is promise of development along the line of attack on the energy losses in the turbine; there seems little to be accomplished in reducing these losses in the engine. The two motors may at any moment reach a parity. vi PREFACE These are the questions which should be kept in mind by the reader. Thermodynamics is physics, not mathematics or logic. This book takes a middle ground between those text-books which replace all theory by empiricism and that other class of treatises which are too apt to ignore the engineering significance of their vocabulary of differential equations. We here aim to present ideal operations, to show how they are modified in practice, to amplify underlying principles, and to stop when the further application of those principles becomes a matter of machine design. Thermo- dynamics has its own distinct and by no means narrow scope, and the intellectual training arising from its study is not to be ignored. We here deal only with a few of its engineering aspects ; but these, with all others, hark back invariably to a few fundamental princi- ples, and these principles are the matters for insistent emphasis. Too much anxiety is sometimes shown to quickly reach rules of practice. This, perhaps, has made our subject too often the barren science. Rules of practice eternally change ; for they depend not alone on underlying theory, but on conditions current. Our theory should be so sound, and our grasp of underlying principles so just, that we may successfully attack new problems as they arise and evolve those rules of practice which at any moment may be best for the conditions existing at that moment. But if Thermodynamics is not differential equations, neither should too much trouble be taken to avoid the use of mathematics which every engineer is supposed to have mastered. The calculus is accordingly employed where it saves time and trouble, not else- where. The so-called general mathematical method has been used in the one application where it is still necessary ; elsewhere, special methods, which give more physical significance to the things de- scribed, have been employed in preference. Formulas are useful to the busy engineer, but destructive to the student; and after weighing the matter the writer has chosen to avoid formal definitions and too binding symbols, preferring to compel the occasionally reluctant reader to grub out roots for himself an excellent exor- cise which becomes play by practice. The subject of compressed air is perhaps not Thermodynamics, but it illustrates in a simple way many of the principles of gases PREFACE vii and has therefore been included. Some other topics may convey an impression of novelty; the gas engine is treated before the steam engine, because if the order is reversed the reader will usually be rusty on the theory of gases after spending some weeks with vapor phenomena ; a brief exposition of multiple-effect distillation is pre- sented; a limit is suggested for the efficiency of the power gas producer ; and, carrying out the general use of the entropy diagram for illustrative purposes, new entropy charts have been prepared for ammonia, ether, and carbon dioxide. A large number of prob- lems has been incorporated. Most of these should be worked with the aid of the slide rule. Further originality is not claimed. The subject has been written, and may now be only re-presented. All standard works have been consulted, and an effort has been made to give credit for methods as well as data. Yet it would be impossible in this way*to fully acknowledge the beneficial influence of the writer's former teachers, the late Professor Wood, Professor J. E. Denton, and Dr. D. S. Jacobus. It may be sufficient to say that if there is anything good in the book they have contributed to it ; and for what is not good, they are not responsible. POLYTECHNIC INSTITUTE OP BROOKLYN", NEW YORK, August, 1910. CONTENTS CHAPTER PAGE TABLE OF SYMBOLS xiii I. THE NATURE AND EFFECTS OF HEAT 1 II. Ta^ HEAT UNIT: SPECIFIC HEAT: FIRST LAW OF THERMODYNAMICS 11 III. LAWS OF GASES: ABSOLUTE TEMPERATURE: THE PERFECT GAS . 19 IV. THERMAL CAPACITIES- SPECIFIC HEATS OF GASES: JOULE'S LAW . 32 V. GRAPHICAL REPRESENTATIONS: PRESSURE-VOLUME PATHS OF PER- FECT GASES 43 VI. THE CARNOT CYCLE 76 VII. THE SECOND LAW OF THERMODYNAMICS 84 VIII. ENTROPY 92 IX. COMPRESSED AIR 106 The cold air engine: cycle, temperature fall, preheaters, design of engine: the compressor: cycle, form of compression curve, jackets, multi-stage compression, intercooling, power consump- tion: engine and compressor relations: losses, efficiencies, en- tropy diagram, compressor capacity, volumetric efficiency, design of compressor, commercial types: compressed air trans- mission. X. HOT-AIR ENGINES 145 XI. GAS POWER . 162 The producer: limit of efficiency: gas engine cycles: Otto, Car- not, Atkinson, Lenoir, Brayton, Clerk, Diesel, Sargent, Frith, Humphrey: practical modifications of the Otto cycle: mixture, compression, ignition, dissociation, clearance, expansion, scav- enging, diagram factor: analysis with variable specific heats considered: principles of design and efficiency: commercial gas engines: results of tests: gas engine regulation. XII. THEORY OF VAPORS 230 Formation at constant pressure: saturated steam: mixtures: superheated steam: paths of vapors: vapors in general: steam cycles: steam tables. is x CONTENTS CHAPTER PAOB XIII. THE STEAM ENGINE ......... 298 Practical modifications of the Rankino cycle: complete and incom- plete expansion, wiredrawing, cylinder condensation, ratio of expansion, the steam jacket, use of superheated steam, actual expansion curve, mean effective pressure, back pressure, clear- ance, compression, valve action: the entropy diagram: cylinder feed and cushion steam, Boulvin's method, preferred method: multiple expansion: desirability of complete expansion, conden- sation losses in compound cylinders, Woolf engine, receiver engine, tandem and cross compounds, combined diagrams, design of compound engines, governing, drop, binary vapor engine- engine tests: indicators, calorimeters, heat supplied, heat rejected, heat transfers: regulation: types of steam engine. XIV. THE STEAM TURBINE ........ 363 Conversion of heat into velocity: the turbine cycle, effects of fric- tion, rate of flow, efficiency in directing velocities: velocity compounding, pressure compounding: efficiency of the turbine: design of impulse and pressure turbines: commercial types and applications. XV. RESULTS OF TRIALS OF STEAM ENGINES AND STEAM TURBINES 397 "Economy, condensing and non-condensing, of various commercial forms with saturated and superheated steam: mechanical effi- ciencies. XVI. THE STEAM POWER PLANT ....... 415 Fuels, combustion economy, air supply, boilers, theory of draft, fans, chimneys, stokers, heaters, superheaters, economizers, condensers, pumps, injectors, XVII. DISTILLATION .......... 430 The still, evaporation in vacuo, multiple-effect evaporation. FUSION : Change of volume during change of state, pressure-temperature relation, latent* hoat of fusion of ice. LiQUKKAOTION OF Preswure and cooling, critical temperature, cascade system, regen- erative apparatus. XVTII. MECHANICAL RKFIUCIKUATION ....... 454 Air machines: reversed cycle, Bell-Coleman machine, deiw air apparatus, coefficient of performance, Kelvin warming machine: vapor-compression machines: the cycle, choice of fluid, ton- nage rating, ice-melting effect, design of compressor: the absorp- tion system, heat balance: methods and fiolde of application: ice- making; commercial efficiencies. CHARACTERISTIC SYMBOLS F = Fahrenheit; C = Centigrade; R = R6aumur; - Radiation (Art. 25); = gas constant for air = 53,36 fUb. =00686B.t.u.; = ratio of expansion; P,p = pressure: usually Ib. per sq, in. absolute; V, v = volume, cu. ft: usually of 1 Ib. ; = velocity (Chapter XIV); Tj Z = temperature, usually absolute; !T=heat to produce change of tem- perature (Art. 12); E = change of internal energy; 7 = disgregation work; Q,#=heat absorbed or emitted; = total heat above 32 of 1 Ib. of dry vapor; h = heat emitted; ^heat of liquid above 32 F; =head of liquid; c= constant; ^specific heat; s= specific heat; r=gas constant (Art. 52); = internal heat of vaporization; -ratio of expansion; jrj -7= -specific heat; m dT 'dH T -=-= entropy; 34,5 Ibs. water per hour from and at 212 F.=l boiler E.P.; 42 42 B.t.u. per min. = 1 H.P.; 2545 B.t.u. per hour = 1 H.P,; 17.59 B.t.u. per minute =1 watt; W) w= weight (Ib.); W - external mechanical work; S= piston speed, feet per minute; A =piston area, square inch; k- specific heat at constant pres- sure; I - specific heat at constant volume; k y=r> n=polytropic exponent; N,n= entropy; e- coefficient of elasticity; ^external work of vaporization; pro =mean effective pressure; J> = piston displacement (Art, 190); r.p.m. revolutions per minute; H.P, =horse-power; d= density; gf-32.2; 778= mechanical equivalent of heat; 459.6(460)= absolute temperate at Fahrenheit zero; L=heat of vaporization; x - dryness fraction ; 7 =factor of evaporation; 7fc= entropy of dry steam; Tie = entropy of vaporization; nw = entropy of liquid. CHAPTER I THE NATURE AND EFFECTS OF HEAT 1. Heat as Motive Power. All artificial motive powers derive their origin from heat. Muscular effort, the forces of the waterfall, the wind, tides and waves, and the energy developed by the combustion of fuel, may all be traced back to reactions induced by heat. Our solid, liquid, and gaseous fuels are stored-up solar heat in the forms of hydrogen and carbon. 2. Nature of Heat. We speak of bodies as "hot" or "cold," referring to certain impressions which they produce upon our senses. Common experimental knowledge regarding heat is limited to sensations of temper- ature. Is heat matter, force, motion, or position ? The old " caloric " theory was that "heat was that substance whose entrance into our bodies causes the sensation of warmth, and whose egress the sensation of cold." But heat is not a " substance " similar to those with which we are familiar, for a hot body weighs no more than one which is cold. The calorists avoided this difficulty by assuming the existence of a weightless material fluid, caloric. This substance, present in the interstices of bodies, it was contended, produced the effects of heat; it had the property of passing between bodies over any intervening distance, Friction, for example, de- creased the capacity for caloric; and consequently some of the latter " flowed out," as to the hand of the observer, producing the sensatiou of heat. Davy, however, in 1799, proved that friction does not diminish the capacity of bodies for containing heat, by rubbing together two pieces of ice until they melted. According to the caloric theory, the resulting water should have had less capacity for heat than the original ice : but the fact is that water has actually about twice the capacity for heat that ice has ; or, in other words, the specific heat of water is about 1.0, while that of ice is 0,504. The caloric theory was further assailed by Rumford, who showed that the supply of heat from a body put under appropriate conditions was so nearly inexhaustible that the source thereof could not be conceived as being even an " imponderable " substance. The notion of the calorists was that the different specific heats of bodies were due to a varying capac- ity for caloric ; that caloric might be squeezed out of a body like water from a sponge. Kumford measured the heat generated by the boring of cannon in the arsenal at Munich. In one experiment, a gun weighing 2 APPLIED THERMODYNAMICS 113,13 Ib. was heated 70 E., although the total weight of borings produced was only 837 grains troy. In a later experiment, Rtimford succeeded in boiling water by the heat thus generated. He argued that "anything which any insulated body or system of bodies may continue to furnish tuithout limitation cannot possibly be a material substance." The evolution of heat, it was contended, might continue as indefinitely as the generation of sound following the repeated striking of a bell (1).* Joule, about 1845, showed conclusively that mechanical energy alone sufficed for the production of heat, and that the amount of heat generated was always proportionate to the energy expended. A view of his apparatus is given in Fig. 1, v and h being the verti- cal and horizontal sections, respectively, of the container shown at <?. Water being placed in 0, a rotary motion of the contained brass paddle wheel was caused by the de- scent of two leaden weights suspended by cords. The rise in temperature of the FIG. 1. Arts. 2, 30. Joule's Apparatus, water was noted, the expended work (by the falling weights) com- puted, and a proper correction made for radiation. Similar experi- ments were made with mercury instead of water. As a result of his experiments, Joule reached conclusions which served to finally overthrow the caloric theory* 3, Mechanical Theory of Heat. Various ancient and modern philosophers had conceded that heat was a motion of the minute particles of the body, some of them suggesting that such motion * Figures in parentheses signify references grouped at the ends ot the chapters. THE NATURE AND EFFECTS OF HEAT 3 was produced by an "igneous matter/' Locke denned heat as "a very brisk agitation of the insensible parts of the object, which pro- duces in us that sensation from which we denominate the object hot ; so [that] what in our sensation is heat, in the object is nothing but motion." Young argued, "If heat be not a substance, it must be a quality; and this quality can only be a motion." This is the modern conception. Heat is energy : it can perform work, or pro- duce certain sensations ; it can be measured by its various effects. It is regarded as " energy stored in a substance by virtue of the state of its molecular motion" (2). Conceding that heat is energy, and remembering the expression for energy, I mv z , it follows that if the mass of the particle does not change, its velocity (molec- ular velocity) must change; or if heat is to include potential energy, then the molecular configuration must change. The molecular vibrations are invisible, and their precise nature unknown. Rankine's theory of molecular vortices assumes a law of vibration which has led to some useful results. Since heat is energy, its laws are those generally applicable to energy, as laid down by Newton : it must have a commensurable value ; it must be convertible into other forms of energy, and they to heat; and the equivalent of heat energy, expressed in mechanical energy units, must be constant and determinable by experiment. 4. Subdivisions of the Subject. The evolutions and absorptions of heat accompanying atomic combinations and molecular decompo- sitions are the subjects of thermochemistry. The mutual relations of heat phenomena, with the consideration of the laws of heat trans- mission, are dealt with in general physics. The relations between heat and mechanical energy are included in the scope o applied engi- neering thermodynamics, which may be defined as the science of the mechanical theory of heat. While thermodynamics is thus apparently only a subdivision of that branch of physics which treats of heat, the relations which it considers are so important that it may be regarded as one of the two fundamental divisions of physics, which from this standpoint includes mechanics dealing with the phenomena of ordinary masses and thermodynamics treating of the phenomena of molecules. Thermodynamics is the science of energy. 5. Applications of Thermodynamics. The subject has far-reaching applications in physios and chemistry. In its mechanical aspects, it deals 4 APPLIED THERMODYNAMICS with matters fundamental to the engineer. After developing the general laws and dwelling briefly upon ideal processes, we are to study the condi- tions affecting the efficiency and capacity of air, gas, and steam engines and the steam turbine; together with the economics of air compression, distillation, refrigeration, and gaseous liquefaction. The ultimate engi- neering application of thermodynamics is in the saving of heat, an appli- cation which becomes attractive when viewed in its just aspect as a saving of money and a mode of conservation of our material wealth. 6. Temperature. A hot body, in common language, IB one whose temperature is high, while a cold body is one low in temperature. Tem- perature, then, is a measure of the hottwss of bodies. From a riso in tem- perature, we infer an accession, of heat; or from a fall in temperature, a loss of heat.* Temperature is not, however, a satisfactory measure of quantities of heat. A pound of water at 200 contains very much more hieat than a pound of lead at the same temperature ; this may be demon- strated by successively ooolmg the bodies in a bath to the same final tem- perature, and noting the gain of heat by the bath. Furthermore, immense quantities of heat are absorbed by bodies in passing from the solid to the liquid or from the liquid to the vaporous conditions, without any change* in temperature whatever. Temperature defines a condition of heat only. It is a measure of t7ie capacity of the body for coni'iniinimting heat to otlwr bodies. Heat always passes from a body of relatively high temperature j it never passes of itself from a cold body to a hot one. Wherever two bodies of different temperatures are in thermal juxtaposition, an inter- change of heat takes place ; the cooler body absorbs heat from the hotter body, no matter which contains initially the greater quantity of heat, until the two are at the saine temperature, or in thermal (tquflibrhwH,. Two bodies are at the same temperature when there is no tendency toward a transfer of heat between them. Measurements of temperature ai'o in gen- eral based upon arbitrary scales, standardized by comparison with some physically established " fixed " point. One of these fixed temperatures is that minimum at which pure water boils when under normal atmospheric pressure of 14.697 Ib. per square inch; viz. 212 F. Another is the maximum temperature of melting ice at atmospheric pressure*, which is 32 F. Our arbitrary scales of temperature cannot be expressed iu terms of the fundamental physical units of length and weight 7- Measurement of Temperature. Temperatures are measured by thermome- ters. The common type of instrument consists of a connected bulb and vertical tube, of glass, in which is contained a liquid. Any change in temperature affects * "... the change in temperature is the thing observed and ... the idea of heat is introduced to account for the change. , ." Gtoodimough. THE NATURE AND EFFECTS OF HEAT 5 the volume of the liquid, and the portion in the tube consequently rises or falls. The expansion of solids or of gases is sometimes utilized m the design of thermom- eters, Mercury and alcohol are the liquids commonly used. The former freezes at -38 F. and boils at 675 F. The latter freezes at -203 F. and boils at 173 F. The mercury thermometer is, therefore, more commonly used for high tempera- tures, and the alcohol for low (2a). 8. Thermometric Scales. The Fahrenheit thermometer, generally employed by engineers in the United States and Great Britain, divides the space between the "fixed points" (Art. 6) into 180 equal degrees, freezing being at 32 and boiling at 212. The Centigrade scale, employed by chemists and physicists (sometimes described as the Celsius scale), calls the freezing point and the boiling point 100. On the Reaumur scale, used in Russia and a few other countries, water freezes at and boils at 80. One de- gree on the Fahrenheit scale is, therefore, equal to | C., or to R. In making transformations, care must be taken to regard the differ- ent zero point of the Fahrenheit thermometer. On all scales, tem- peratures below zero are distinguished by the minus ( ) prefix. The Centigrade scale is unquestionably superior in facilitating arithmetical calculations; but as most English papers and tables are published in Fahrenheit units, we must, for the present at least, use that scale of temperatures. 9. High. Temperature Measurements. For measuring temperatures above 800 ' F., some form of pyrometer must be employed. The simplest of these is the metallic pyrometer, exemplifying the principle that different metals expand to dif- ferent extents when heated through the same range of temperature. Bars of irou and brass are firmly connected at one end, the other ends being free. At some standard temperature the two bars are of the same length, and the indicator, con- trolled jointly by the two free ends of the bars, registers that temperature. When the temperature changes, the indicator is moved to a new position by the relative distortion of the free ends. In the Le Chatelier electric pyrometer, a thermoelectric couple is employed. For temperatures ranging from 300 C. to 1500 C., one element is made of platinum, the other of a 10 per cent, alloy of platinum with rhodium. Any rise in tempera- ture at the junction of the elements induces a flow of electric current, which is con- ducted by wires to a galvanometer, located in any convenient position. The ex- pensive metallic elements are protected from oxidation by enclosing porcelain tubes. In the Bristol thermoelectric instrument, one element is of a platinum- rhodium alloy, the other of a cheaper metal. The electromotive force is indicated by a Weston millivoltmeter, graduated to read temperatures directly. The in- strument is accurate up to 2000 F. The electrical resistance pyrometer is based on the law of increase of electrical resistance with increase of temperature. In Cal- lendar's form, a coil of fine platinum wiie i wound on a serrated mica fram*. The instrument is enclosed in porcelain, and placed in the space the temperature 6 APPLIED THERMODYNAMICS of which is to be ascertained. The resistance is measured "by a Wheatstone bridge, a galvanometer, or a potentiometer, calibrated to read temperatures directly. Each instrument must be separately calibrated. Optical pyrometers are based oil the principle that the colors of bodies vary with their temperatures (26). In the Morse thermogage, of this type, an incandescent lamp is wired in circuit with a rheostat and a millivoltmeter. The lamp is located between the eye and the object, and the current is regulated until the lamp be- comes invisible. The temperature is then read directly from the calibrated milli- voltmeter. The device is extensively used in hardening steel tools, and has been employed to measure the temperatures in steam boiler furnaces. 10. Cardinal Properties. A cardinal or integral property of a substance is any property which is fully defined by the immediate state of the substance. Thus, weight, length, specific gravity, are cardinal properties. On the other hand, cost is a non-cardinal prop- erty ; the cost of a substance cannot be determined by examination of that substance; it depends upon the previous history of the sub- stance. Any two or three cardinal properties of a substance may be used as coordinates in a graphic representation of the state of the sub- stance. Properties not cardinal may not be so used, because such properties do not determine, nor are they determinable by, the pres- ent state of the substance. The cardinal properties employed in thermodynamics are five or six in number.* Three of these are pres- sure, volume, and temperature ; pressure being understood to mean specific pressure, or uniform pressure per unit of surface, exerted by or upon the body, and volume to mean volume per unit of weight. The location of any point in space is fully determined by its three coordi- nates. Similarly, any three cardinal properties may serve to fix the thermal condition of a substance. The first general principle of thermodynamics is that if two of the three named cardinal properties are known, these two enable us to calcu- late the third. This principle cannot be proved d priori ; it is to be justi- fied by its results in practice. Other thermodynamic properties than pressure, volume, and temperature conform to the same general principle (Art. 169) ; with these properties we are as yet unacquainted. A correlated principle is, then, that any two of the cardinal properties suffice to fully determine the state of the substance, For certain gases, the general prin- ciple may be expressed; PV= (f}T *For gases, pressure, volume, temperature, internal energy, entropy; for wet vapors, dryness is another* THE NATURE AND EFFECTS OF HEAT 7 while for other gaseous fluids more complex equations (Art. 363) must be used. In general, these equations are, in the language of analytical geometry, equations to a surface. Certain vapors cannot be represented, as yet, by any single equation between P, F, and T, although correspond- ing values of these properties may have been ascertained by experiment. With other vapors, the pressure may be expressed as a function of the temperature, while the volume depends both upon the temperature an<l upon the proportion of liquid mingled with the vapor. 11. Preliminary Assumptions. The greater part of the subject deals with substances assumed to be in a state of mechanical equilibrium, all changes being made with infinite slowness. A second assumption is that no chemical actions occur during the thermodynamic trans- formation. In the third place, the substances dealt with are assumed to be so homogeneous, as to be in uniform thermal condition through- out : for example, the pressure property must involve equality of pressure in all directions ; and this limits the consideration to the properties of liquids and gases. The thermodynamics of solids is extremely complex, because of the obscure stresses accompanying their deformation (3), Kelvin (4) has presented a general analysis of the action of any homogeneous solid body homogeneously strained. 12. The Three Effects of Heat. Setting aside the obvious un- classified changes in pressure, volume, and temperature accompanying manifestations of heat energy, there are three known, ways in which heat may be expended. They are : (#) In a change of temperature of the substance. (6) In a change of physical state of the substance. (<?) In the performance of external work by or upon the substance. Denoting these effects by T, I, and W, then, for any transfer of heat JJ", we have the relation H= T + I + W, any of the terms of which expression may be negative. It should be quite obvious, therefore, that changes of temperature alone are in- sufficient to measure expenditures of heat. Items (#) and (6) are sometimes grouped together as indications of a change in the INTERNAL ENERGY (symbol E) of the heated substance, the term being one of the first importance-, which it is 8 APPLIED THERMODYNAMICS essential to clearly apprehend. Items (5) and (c) are similarly some- times combined as representing the total work. 13. The Temperature Effect. Temperature indications of heat activity are sometimes refened to as " sensible heat." The addition of heat to a substance may either raise or lower its temperature, in accordance \v ith the fundamental equation of Art. 12. The temperature effect of heat, from the standpoint of the mechanical theory, is due to a change in the velocity of molecular motion, in conse- quence of which the kinetic energy of that motion changes. This effect is therefore sometimes referred to as vibration work. Clausiua called it actual energy. 14. External Work Effect. The expansion of solids and fluids, due to the supply of heat, is a familiar phenomenon. Heat may cause either expansion or contraction, which, if exerted against a resistance, may suffice to perform mechanical work. 15. Changes of Physical State. Broadly speaking, such effects include all changes, other than those of temperature, within the sub- stance itself. The most familiar examples are the change between the solid and the liquid condition, when the substance melts or freezes, and that between the liquid and the vaporous, when it boils or condenses ; but there are intermediate changes of molecular aggrega- tion in all material bodies which are to be classed with these effects under the general description, disgregation work. The mechanical theory assumes that in such changes the molecules are moved into new positions, with or against the lines of mutual attraction. These movements are analogous to the "partial raising or lowering of a weight which is later to be caused to perform work by its own descent. The potential energy of the substance is thus changed, and positive or negative work is performed against internal resisting forces." When a substance changes its physical state, as from water to steam, it can be shown that a very considerable amount of external work is done, iu consequence of the increase in volume which occurs, and which may be made to occur against a heavy pressure. This external work is, however, equivalent only to a very small proportion of the total heat supplied to produce evaporation, the balance of the heat having been expended in the performance of disgregation work. The molecular displacements constituting disgregation work are exemplified in THE NATURE AND EFFECTS OF HEAT .9 16. Solid, Liquid, Vapor, Gas. Solid bodies are those which resist tendencies to change their form or volume. Liquids are those bodies which in all of their parts tend to preserve definite volume, and which are practically unresistant to influences tending to slowly change their figure. Gases are unresistant to slow changes in figure or to increases in volume. They tend to expand indefinitely so as to completely fill any space in which they are contained, no matter what the shape or the size of that space may be. Most substances have been observed in all three forms, under appropriate conditions ; and all substances can exist in any of the forms. At this stage of the discussion, no essential difference need be drawn between a vapor and a gas. Formeily, the name vapor was applied to those gaseous substances which at ordinary temperatures were liquid, while a " gas " was a substance never observed in the liquid condition. Since all of the so-called "permanent" gases have been liquefied, this distinction has lost its force. A useful definition of a vapor as distinct from a true gas will be given later (Art. 380). Under normal atmospheric pressure, there exist well-defined tempera- tares at which various substances pass from the solid to the liquid and from the liquid to the gaseous conditions. The temperature at which the former change occurs is called the melting point or freezing point; that of the latter is known as the boiling point or temperature of condensation. 17. Other Changes of State. Although the operation described as boiling occurs, for each liquid, at some definite temperature, there is an almost continual evolution of vapor from nearly all liquids at temperatures below their boiling points. Such "insensible" evaporation is with some substances non-existent, or at least too small in amount to permit of measurement: as in the instances of mercury at 32 F. or of sulphuric acid at any ordinary temperature. Ordinarily, a liquid at a given temperature continues to evaporate so long as its partial vapor pressure is less than the maximum pressure corresponding to its temperature. The inter- esting phenomenon of sublimation consists in the direct passage from the solid to the gaseous state. Such substances as camphor and iodine manifest this property. Ice and snow also pass directly to a state of vapor at temperatures far below the freezing point. There seem to be no quantitative data on the heat relations accom- panying this change of state (see Art. 382 6). 18. Variations in " Fixed] Points." Aside from the influence of pressure (Arts. 358, 603), various causes may modify the positions of the "fixed points" of the thermometric scale. Water may be cooled below 32 F. without freezing, if kept perfectly still. If free from air, water boils at 270-290 F. Minute particles of air are necessary to start evaporation sooner; their function is probably to aid in the diffusion of heat. (1) Tyndall: Heat as a Mode of Motion. (2) Nichols and Franklin: The F,le- ments of Physics, I, 161. (2o) Heat Treatment of High Temperature Mercurial Thermometers, by Dickinson; Bulletin of the Bureau of Standards, 2, 2. (2&) See the paper, Optical Pyrometry, by Waidner and Burgess, Bulletin of the Bureau of Standards, 1, 2. (3) See paper by J. E. Siebel: The Molecular Constitution of 10 APPLIED THERMODYNAMICS Solids, in Science, Nov. 5, 1909, p. 654. (4) Quarterly Mathematical Journal, April, 1855. (5) Darling: Heat for Engineers, 208. SYNOPSIS OF CHAPTER I Heat is the universal source of motive power. Theories of heat : the caloric theory heat is matter; the mechanical theory heat is molecular motion, mutually conveitible with mechanical energy. THEBMOOHEMISTRY, THERMODYNAMICS. Thermodynamics : the mechanical theory of heat ; in its engineering applications, the science of heat-motor efficiency. Heat intensity, temperature : definition of, measurement of ; pyrometers. Thermometric scales: Fahrenheit, Centigrade, Reaumur; fixed points and their variations Cardinal properties : pressure, volume, temperature; PF=(/)!T. Assumptions: uniform thermal condition ; no chemical action ; mechanical equilibrium, Effects of heat : Bf- T+I+ W\ T+I= E= "internal energy " ; J7= external work. Changes of physical stale, perceptible and imperceptible: I=disgregation work. Solid, liquid, vapor , gas: melting point, boiling point; insensible evaporation; sublimation. PROBLEMS 1. Compute the freezing points, on the Centigrade scale, of mercury and alcohol. (Ans., mercury, 38.9: alcohol, 130.6,) 2. At what temperatures, RSaumur, do alcohol and mercury boil? (Ans., mer- cury, 285.8: alcohol, 62.7.) 3. The normal temperature of the human body is 98.6 F. Express in Centigrade degrees, (Ana., 37 C.) 4. At what temperatures do the Fahrenheit and Centigrade thermometers read alike? (Ans., -40.) 5. At what temperatures do the Fahrenheit and Rgaumur thermometers read alike? (Ans., -25.6.) a. Express the temperature 273 C. on the Fahrenheit and Reaumur scales. 3., -459.4 F.: -218.4 R.) CHAPTER II THE HEAT UNIT: SPECIFIC HEAT: FIRST LAW OF THERMODYNAMICS 19. Temperature Waterfall Analogy. The difference between temperature and quantity of heat may be apprehended from the analogy of a waterfall. Tem- perature is like the head of water ; the energy of the fall depends upon the head, but cannot be computed without knowing at the same time the quantity of water. As waterfalls of equal height may differ in power, while those of equal power may differ in fall, so bodies at like temperatures may contain different quantities of heat, and those at unequal temperatures may be equal in heat contents. 20. Temperatures and Heat Quantities. If we mix equal weights of water at different temperatures, the resulting temperature of the mix- ture will be very nearly a mean between the two initial temperatures. If the original weights are unequal, then the final temperature will be nearer that initially held by the greater weight. The general principle of transfer is that The loss of heat by the hotter water will equal the gain of heat by the colder. Thus, 5 Ib. of water at 200 mixed with 1 Ib. at 104 gives 6 Ib. at 184; the hotter water having lost 80 " pound-degrees," and the colder water having gained the same amount of heat. If, however, we mix the 5 Ib. of hot water with 1 Ib. of some other substance say linseed oil the resulting temperature will not be 184, but 194.6, if the initial tem- perature of the oil is 104. 21. General Principles. Before proceeding, we may note, in addition to the principle just laid down, the following laws which are made apparent by the ex- periments described and others of a similar nature : (a) In a homogeneous substance, the movement of heat accom- panying a given change of temperature * is proportional to the weight of the substance. (J) The movement of heat corresponding to a given change of * Not only the amount, but the method^ of changing the temperature must be fixed (Art. 57). 11 12 APPLIED THERMODYNAMICS temperature is not necessarily the same for equal intervals at all parts of the thermoinetric scale ; thus, water cooling from 200 to 195 does not give out exactly the same quantity of heat as in cool- ing from 100 to 95. <Y) The loss of heat during cooling through a stated range of temperature is exactly equal to the gain of heat during warming through the same range. 22. The Heat Unit. Changes of temperature alone do not measure heat quan- tities, because heat produces other effects than that of temperature change. If, however, we place a body under " standard" conditions, at which these other effects, if not known, are at least constant, then we may define a unit of quantity of heat by reference to the change m temperature which at produces, understand- ing that there may be included perceptible or imperceptible changes of other kinds, not affecting the constancy of value of the unit. The British Thermal Unit is that quantity of heat which is expended in raising the temperature of one pound of water (or in producing other effects during this change in temperature) from 62 to 63 F.* To heat water over this range of temperature requires very nearly the same expenditure of heat as is necessary to warm it 1 at any point on the thermometric scale. In fact, some writers define the heat unit as thab quantity of heat necessary to change the temperature front 39.1 (the temperature of maximum density) to 40.1. Others use the ranges 32 to 33, 59 to 60, or 39 to 40. The range first given is that most recently adopted. 23. French TTnits. The French or C. G. S. unit of heat is the calorie, the amount of heat necessary to raise the temperature of one kilogram of water 1 C. Its value is 2.2046 X f = 3.96832 B. t. u., and 1 B. t. u. = 0.251996 cal. The calorie is variously measured from 4 to 5 and from 14.5 to 15.5 (J. The gram-calorie is the heat required to raise the temperature of one gram of water 1 C. The Centigrade heat unit measures the heat necessary to raise one pound of water 1 G in temperature. 24. Specific Heat. Eef erence was made in Art. 20 to the different heat capacities of different substances, e.g. water and linseed oil. If we mix a stated quantity of water at a fixed temperature successively with equal weights of various materials, all initially at the same temperature, the final temperatures of the mixtures will all differ, indicating that a unit * There are certain grounds for preferring that definition which makes the B. t. u. the yj^ part of the amount of heat required to raise the temperature of one pound of water at atmospheric pressure from the freezing point to the boiling point, THE HEAT UNIT. SPECIFIC HEAT 13 rise of temperature of unit weight of these various materials represents a different expenditure of heat in each case. The property by virtue of winch materials differ in this respect is that of specific heat, which may be defined as the quantity of heat necessary to raise the temperature of unit weight of a body through one degree. The specific heat of water at standard temperature (Art. 22) is, meas- ured in B. t. u., 1.0 ; generally speaking, its value is slightly variable, as is that of all substances. Rankine's definition of specific heat is illustrative : " the specific heat of any substance is the ratio of the weight of water at or near 39.1 F. [62-6r3 F.] which has its temperature altered one degree by the transfer of a given quantity of heat, to the weight of the other substance under consideration, which has its temperature altered one degree by the transfer of an equal quantity of heat." 25. Mixtures of Different Bodies. If the weights of a group of mixed bodies be X, Y f Z, etc., their specific heats #, ?/, z, etc., their ini- tial temperatures t, u, v, etc., and the final temperature of the mixture be m, then we have the following as a general equation of thermal equi- librium, in which any quantity may be solved for as an unknown: ni + zZv-m =0. This illustrates the usual method of ascertaining the specific heat of any body. When all the specific heats are known, the loss of heat to sur- rounding bodies may be ascertained by introducing the additional term, + Jf2, on the left-hand side of this equation. The solution will usually give a negative value for R, indicating that surrounding bodies have absorbed rather than contributed heat. The value of R will of course be expressed in heat units. 26. Specific Heat of Water. The specific heat of water, according to Rowland's experiments, decreases as the temperature is increased from 39.1 to 80 P., at which latter temperature it reaches a minimum value, afterward increasing (Art. 359, footnote). The variation in its value is very small. The approximate specific heat, 1.0, is high as com- pared with that of almost all other substances. 27. Problems Involving Specific Heat. The quantity of heat re- quired to produce a given change of temperature in a body is equal to the weight of the body, multiplied by the range of temperature and by the specific heat. Or, symbolically, using the notation of Art. 25, 14 APPLIED THERMODYNAMICS If the body is cooled, then m, the final temperature, is less than t, and the sign of H is - ; if the body is warmed, the sign of II is -f , indicating a reception of heat. 28. Consequences of the Mechanical Theory. The Mechanical Equivalent of Heat. Even before Joule's formulation (Art. 2), Eumford's ex- periments had sufficed for a comparison of certain effects of heat with an expenditure of mechanical energy. The power exerted by the Bavarian horses used to drive his machinery is uncertain ; but Alexander has computed the approximate relation to have been 847 foot-pounds = 1 B.t.u. (1), while another writer fixes the ratio at 1034, and Joule cal- culated the value obtained to have been 849. Carnot's work, although based throughout on the caloric theory, shows evident doubts as to its validity. This writer suggested (1824) a repetition of Ruinford's experiments, with provision for accurately measuring the force employed. Using a method later employed by Mayer (Art. 29) he calculated that 0.611 units of motive power" were equivalent to "550 units of heat"; a relation which Tyndall computes as representing 370 kilogram-meters per calorie, or 676 foot- pounds per B. t. u. Montgolfier and Seguin (1839) may possibly have anticipated Mayer's analysis. 29. Mayer's Calculation. This obscure German physician published in 1842 (2) his calculation of the mechanical equivalent of heat, based on the difference in the specific heats of air at* constant pressure and constant volume, giving the ratio 771.4= foot-pounds per B. t. u. (Art. 72). This was a substantially correct result, though given little consideration at the time. Mayer had previously made rough calculations of equivalence, one being based on the rise of temperature occurring in the " beaters " of a paper mill. 30. Joule's Determination. Joule, in 1843, presented the first of his exhaustive papers on the subject. The usual form of apparatus employed has been shown in Fig. 1. In the appendix to his paper Joule gave 770 as the best value deducible from his experiments. In 1849 (3) he presented the figure for many years afterward accepted as final, viz. 772. In 1878 an entirely new set of experiments led to the value 772.55, which Joule regarded as probably slightly too low. Experiments in 1857 had given the values 745, 753, and 766. Most of the tests were made with water at about 60 F. This, with the value of g at Manchester, where the experiments were made, in- volves slight corrections to reduce the results to standard conditions (4), 31. Other Investigators. Of independent, though uncertain, merit, were the results deduced by the Danish engineer, Colding, in 1843. His value of the equivalent is given by Tyndall as 038 (5). Helmholtz (1847) treated the matter of equivalence from a speculative standpoint. Assuming that "perpetual motion " is impossible, he contended that there must be a definite relation between heat energy and mechanical energy. As early as 1845, Holtzmann (6) had apparently MECHANICAL EQUIVALENT OF HEAT 15 independently calculated the equivalence by Mayer's method. By 1847 the reality of the numerical relation had been so thoroughly established that little more was heard of the caloric theory. Clausius, following Mayer, in 1850 obtained wide circulation for the value 758 (7) . 32. Hirn's Investigation. Joule had employed mechanical agencies in the heating of water. Him, in 1865 (8), described an experiment by which he trans- formed into heat the work expended in producing the impact of solid bodies. Two blocks, one of iron, the other of wood, faced with iron in contact with a lead cylinder, were suspended side by side as pendulums. The iron block was allowed to stnke against the wood block and the rise in temperature of water contained in the lead cylinder was noted and compared with the computed energy of impact. The value obtained for the equivalent was 775. Far more conclusive, though less accurate, results were obtained by Him by noting that the heat in the exhaust steam from an engine cylinder -was less than that which was present in the entering steam. It was shown by Clausius that the heat which had ' disappeared was always roughly proportional to the work done by the engine, the average ratio of foot-pounds to heat units being 753 to 1. This was virtually a reversal of Joule's experiment, illustrating as it did the conversion of heat into work. It is the most striking proof we have of the equivalence of work and heat. 33. Recent Practice. In 1876 a committee of the British Association for the Advancement of Science reviewed critically the work of Joule, and as a mean value, derived from his best 60 experiments, recommended the use of the figure 774.1, which was computed to be correct within ? fo. In 1879, Rowland, having conducted exact experiments on the specific heat of water, carefully redetermined the value of the equivalent by driving a paddle wheel about a vertical axis at fixed speed, in a vessel of water prevented from turning by counterbalance weights. The torque exerted by the paddle was measured. This permitted of a calculation of the energy expended, which was compared with the rise in temperature of the water, Rowland's value was 778, with water at its maximum density. This was regarded as possibly slightly low (9). Since the date of Rowland's work, the subject has been, investigated by Griffiths (10), who makes the value somewhat greater than 778, and by Reynolds and Moorby (11), who report the ratio 778 as the mean obtained for a range of temperature from 32 to 212 F. This they regard as possibly 1 or 2 foot-pounds too low. 34. Summary. The establishing of a definite mechanical equivalent of heat may be regarded as the foundation stone of thermodynamics. Accord- ing to Merz (12), the anticipation of such an equivalent is due to Poncelet and Carnot ; Bumf ord's name might be added. " The first philosophical generalizations were given by Mohr and Mayer j the first mathematical 16 APPLIED THERMODYNAMICS treatment by Helmholtz; the first satisfactory experimental verification by Joule." The constr action of the modern science on this foundation has been the work chiefly of Kankine, Clausius, and Kelvin. 35. First Law of Thermodynamics. Heat and mechanical energy are mutually convertible in the ratio of 778 foot-pounds to the British thermal unit. This is a restricted statement of the general principle of the conservation of energy, a principle which is itself probably not susceptible to proof. We have four distinct proofs of the first law : (a) Joule's and Rowland's experiments on the production of heat by mechanical work. (J) Hirn's observations on the production of work by the ex- penditure of heat. (V) The computations of Mayer and others, from general data. (J) The fact that the law enables us to predict thermal proper- ties of substances which experiments confirm. 36. WormelPs Theorem. There cannot be two values of the mechanical equivalent of heat. Consider two machines, A and B, in the first of which work is transformed into heat, and in the second of which heat is transformed into work. Let J be the mechanical equivalent of heat foi A, W the amount of work which it consumes in pi educing the heat $; then W = JQ or Q = W /. Let this heat Q be used to drive the machine B, in which the mechanical equivalent of heat is, say K. Then the work done by B is V = KQ = KW - J. Let this work be now expended in driving .4. It will produce heab R, such that JR = V or R = F -T- /.* If this heat R be used in -B, work will be done equal to KR ; but KR = KV-J = (Y W. Similarly, after n complete periods of operation, all parts of the machines occupy- ing the same positions as at the beginning, the work ultimately done by B will be If K is less than 7, this expression will decrease as n increases; i.e. the system will tend continually to a stale of rest, contiary to the first law of motion. If K be greater than J, then as n increases the work constantly increases, involving the assumed fallacy of perpetual motion. Hence K and / must be equal (13). 37. Significance of the Mechanical Equivalent. A very little heat is seen to be equivalent to a great deal of work. The heat used in raising the temperature of *The demonstration assumes that the value of the mechanical equivalent is con- stant for a given machine. FIRST LAW OP THERMODYNAMICS 17 one pound of water 100 represents energy sufficient to lift one ton of water nearly 39 feet. The heat employed to boil one pound of water initially at 32 F. would suffice to lift one ton 443 feet. The heat evolved in the combustion of one pound of hydrogen (62,000 B. t. u.) would lift one ton nearly five miles. (1) Treatise cm Thermodynamics, London, 1892. (2) Wohler and Liebig's Annalen der Pharmncie : Bemerkungen iiber die Krafte der unbelebten Natur, May, 1842. (3) Phil. Trans., 1850. (4) Joule's Scientific Papers, Physical Society of London, 1884. (5) Probably quoted by Tyndall from a later article by Colding, in which this figure is given. Colding's original paper does not seem to be accessible. (6) Ueber die Wdrme und Elasticitdt der Gase und Dampfe^ Mannheim, 1846. (7) Poggendorff, Annalen, 1860. (8) Theorie Mecanique, etc , Paris, 1865 (9) Proc. Amer. Acad. Arts and Sciences, New Series, VII, 1878-79. (10) Phil. Trans. Boy. Soc , 1893. (11) Phil. Trans , 1897. (12) History of European Thought, II, 137. (13) K. Wormell: Thermodynamics, 1886. SYNOPSIS OF CHAPTEH H Heat and temperature : heat quantity vs. heat intensity. Principles : (a) heat movement proportional to weight of substance ; (&) temperature range does not accurately measure heat movement ; (c) loss during cooling equals gain during warming, for idntical ranges. The British thermal unit: other units of heat quantity. Specific heat : mixtures of bodies ; quantity of heat to produce a given change of tem- perature ; specific beat of water. The mechanical equivalent of heat : early approximations. First law of thermody- namics : proofs \ oaly one value possible ; examples of the motive power of heat. PROBLEMS 1. How many Centigrade heat units are equivalent to one calorie? (Ana., 2.2046.) 2. Find the number of gram-calories in one B. t. u. (Ans., 252.) 3. A mixture is made of 5 Ib. of water at 200, 3 Ib. of linseed oil at 110, and 22 Ib. of iron at 220 (all Fahrenheit temperatures), the respective specific heats being 1.0, 0.3, and 0.12. Find the final temperature, if no loss occurs by radiation. (Ans., 196.7 F.) 4. If* the final temperature of the mixture in Problem 3 is 189 F., find the num- ber of heat units lost by radiation. (Ans., 65.7 B. t. u.) 5. Under what conditions, with the weights, temperatures and specific heats of Problem 3, might the final temperature exceed that computed? 6. How much heat is given out by 7J Ib. of linseed oil in cooling from 400 F. to 32 F.? (Ans., 828 B. t. u.) 7. In a heat engine test, each pound of steam leaves the engine containing 125.2 B.t.u, less heat than when it entered the cy Under. The engine develops 155 horse- power, and consumes 3160 Ib. of steam per hour. Compute the value of the mechani- cal equivalent of heat. (Ans., 775.7.) 18 APPLIED THERMODYNAMICS 8. A pound of good coal will evolve 14,000 B, t. u. Assuming a train resistance of 11 Ib. per ton of train load, how far should one ton (2000 Ib.) of coal burned in the locomotive without loss, propel a tiam weighing 2000 tons? If the locomotive weighs 125 tons, how high would one pound of coal lift it if fully utilized? (Ans., a, 187.2 miles, 6, 43.5ft.) 9. Find the number of kilogram-meters equivalent to one calorie, (1 meter = 39.37 in., 1 kilogram = 2.2046 Ib.) (Ans., 426.8.) 10. Transform the following formula (P being the pressure in kilograms per square meter, V the volume in cubic meters per kilogram, T the Centigrade temperature plus 273), to English units, letting the pressure be in pounds per square inch, the volume in cubic feet per pound, and the temperature that on the Fahrenheit scale plus 459.4, and eliminating coefficients in places where they do not appear in the original equation : P7=47.1 !T-P(14-0.000002 P) I" 0.031 (~\ 3 -0.0052~| . L \ / I !., PF=0.5962 T-P(1+0.0014P) 11. There are mixed 5^ Ib. of water at 204, 3 J Ib. of linseed oil at 105 and 21 Ib. of a third substance at 221, The final temperature is 195 and the radiation loss is known to be 8.8 B. t. u. What is the specific heat of the third substance? CHAPTER III LAWS OF GASES: ABSOLUTE TEMPERATURE: THE PERFECT GAS 38. Boyle's (or Marietta's) Law. The simplest thermodynamie relations are those exemplified by the so-called permanent gases. Boyle (Oxford, 1662) and Mariotte (1676-1679) separately enun- ciated the principle that at constant temperature the volumes of gases are inversely proportional to their pressures. In other words, the product of the specific volume and the pressure of a gas at a given temperature is a constant. For air, which at 32 F. has a volume of 12.387 cubic feet per pound when at normal atmospheric pressure, the value of the constant is, for this temperature, 144 x 14.7 x 12.387 = 26,221. Symbolically, if c denotes the constant for any given tempera- ture, pv = P t r or, pv = c. "Figure 2 represents Boyle's law graphically, the ordinates being pres- sures per square foot, and the abscissas, volumes in cubic feet per pound. The curves are a series of equilateral hyperbolas,* plotted from the second of the equations just given, with various values of c. 39. Deviations from Boyle's Law. This experimentally determined principle was at first thought to apply rigorously to all true gases. It is now known to be not strictly correct for any of them, although very nearly so for air, hydrogen, nitrogen, oxygen, and some others. All gases may be liquefied, and all liquids may be gasified. When far from the point of liquefaction, gases conform with Boyle's law. When brought near the liquefying point by the combined influences of high pressure and low temperature, they depart widely from it. The four gases just mentioned ordinarily occur at far higher temperatures than those at which they will liquefy. Steam, carbon dioxide, ammonia vapor, and some other well-known gaseous substances which may easily be liquefied do not confirm the law even approximately. Conformity with Boyle's law may be regarded as a measure of the "perfectness" of a gas, or of its approximation to the truly gaseous condition. * Kef erred to their common asymptotes as axes of P and V. 19 20 APPLIED THERMODYNAMICS 8000 10 20 30 40 50 60 FIG. 2. Arts. 38, 91. Boyle's Law. 40. Dalton's Law, Avogadro's Principle. Dalton has been credited (though erroneously) with the announcement of the law now known as that of Gay-Lussac or Charles (Art. 41). What is properly known as Dalton's law may be thus stated . A mixture of gases having no chemical action on one another exerts a pres- sure which is the sum of the pressures which would be exerted by the component gases separately if each in turn occupied the containing vessel alone at the given temperature. The ratio of volumes, at standard temperature and pressure, in which two gases combine chemically is always a simple rational fraction (J, J, J, etc.). Taken in conjunction with the molecular theory of chemical combination, this law leads to the principle of Avogadro that all gases contain the same number of molecules per unit of volume, at the same temperature and pressure. Dalton's law has important thermodynamic relations (see Arts. 52 6, 382 6). 41. Law of Gay-Lussac or of Charles (1). Davy had announced that the coefficient of expansion of air was independent of the pressure. Gay-Lus- sac verified this by the apparatus shown in Pig. 3. He employed a glass tube with a large reservoir A 9 containing tlie air ; which, had been previously LAWS OF GASES 21 dried. An index of mercury mn separated the air from the external atmos- phere, while permitting it to expand. The vessel B was first filled with melting ice. Upon applying heat, equal in- tervals of temperature shown on the ther- mometer were found to correspond with equal displacements of the index mn. When a pressure was applied on the atmospheric side of the index, the proportionate expansion of the air was shown to be still constant for equal intervals of temperature, and to be equal to that observed under atmospheric pressure. Precisely the same results were obtained with FIG. 3 Arts. 41, 48. Verifica- other gases. The expansion of dry air was lono ar es w< found to be 0.00375, or -^ of the volume at the freezing point, for each degree C. of rise of temperature. The law thus established may be expressed : For all gases, and at any pressure, maintained constant, equal increments of volume accompany equal increments of temperature. 42. Increase of Pressure at Constant Volume. A second statement of this law is that all gases, when maintained at constant volume, undergo equal increases of pressure with equal increases of temperature. This is shown experimen- tally by the apparatus of Fig. 4. The glass bulb A contains the gas. It communicates with the open tube manometer Mm, which measures the pressure P is a tube containing mercury, in which an iron rod is submerged to a sufficient depth to keep the level of the mercury in m at the marked point a, thus maintaining a constant volume of gas. 43. Regnault's Experiments. The constant 0.00375 obtained by Gay- Lussac was pointed out by Rudberg to be probably slightly inaccurate. Begnault, by employing four distinct methods, one of which was sub- stantially that just described, determined accurately the coefficient of increase of pressure, and finally the coefficient of expansion at constant pressure, which for dry air was found to be 0.003665, or -j^ per degree 0., of the volume at the freezing point. -^ 1 -^ FIG. 4. Arts. 42, 48. Coefficient of Pressure. 22 APPLIED THERMODYNAMICS 44. Graphical Representation. In Fig. 5, let db represent the volume of a pound of gas at 32 F. Let temperatures and volumes be represented, respectively, by ordinates and /* abscissas. According to Charles' Law, if the / pressure be constant, the volumes and tempera- _ v tares \vill increase proportionately ; the volume ab increasing 3^ for each degree C. that the temperature is increased, and vice versa. The straight line cbe then represents the successive relations of volume and temperature as the gas FIG 5 Arts. 44, M. is heated or cooled from the temperature at b. Charles' Law. ^ t t ] ie p O i n fc ^ where this line meets the a? 7 axis, the volume of the gas will be zero, and its temperature will be 273 C., or 491.4 F., lelow tlie freezing point. 45. Absolute Zero. This temperature of 459.4 F. suggests the absolute zero of thermodynamics. All gases would liquefy or even solidify before reaching it. The lowest temperature as yet attained is about 450 F. below zero. The absolute zero thus experi- mentally conceived (a more strictly alxsolute scale is discussed later, Art. 156) furnishes a convenient starting point for the measure- ment of temperature, which will be employed, unless otherwise speci- fied, in our remaining discussion. Absolute temperatures a) c those in which the zero point is the absolute zero. Their 'numerical values are to be taken, for the present, at 459.4 greater than those of the cor- responding Fahrenheit temperature. 46. Symbolical Representation. The coefficients determined by Gay-Lussac, Charles, and Regnaulfc were those for expansion from an initial volume of 32 F. If we take the volume at this temperature as unity, then letting T represent the absolute temperature, we have, for the volume at any temperature, V= r^. 40 1.4. Similarly, for the variation in pressure at constant volume, the initial pressure being unity, P = T^- 491.4. If we let a de-note tho value 1 *- 401.4, the first expression becomes V - aT, and the second, P = aT. Denoting temperatures on the Fahrenheit scale Ly t, we obtain, for an initial volume v at 32" and any other volume F corresponding to the temperature , produced without change of pressure, 7=v[l + a(*-32)]. Similarly, for variations in pressure at constant volume, P,ppH"a(i-82);|. LAWS OF GASES : ABSOLUTE TEMPERATURE 23 The value of a is experimentally determined to be very nearly the same for pres- sure changes as lor volume changes , the difference m the case of air being less than \ of one per cent. The temperature interval between the melting ot ice and the boiling of water being 180, the expansion of volume of a gtus between those 180 x 1 limits is = 0.365, whence Rankine's equation, originally derived from the 491.4 experiments of Regnault and Rudberg, T= 1.365, po in which P, V refer to the higher temperature, and p> v to the lower. 47. Deviations from Charles 1 Law. The laws thus enunciated are now known not to hold rigidly for any actual gases. For hydrogen, nitrogen, oxygen, air, caibon monoxide, methane, nitric oxide, and a few others, the disagreement is ordinarily very slight. For carbon dioxide, steam, and ammonia, it is quite pro- nounced. The leason for this is that stated in Art. 30. The first four gases named have expansive coefficients, not only almost unvarying, but almost exactly identical. They maybe legarded as our most nearly perfect gases. For air, for example, Regnaulfc found over a range of temperature of ISO F., and a range of pressure of from 109.72 mm. to 499i\0<) mm,, an extreme vai iation in the coefficients of only 1 G7 per cent. For caibon dioxide, on the other hand, with the same range of lempeiatures and a de- creased pressure range of from 78o.47 mm. to 4759.03 mm., the variation was 4.72 per cent of the lower value (2). \ 48. The Air Thermometer. The law of Charles sug- gests a form of thermometer far more accurate than the ordinary mercurial instrument. If we allow air to expand with- out change in pressure, or to increase its pressure without change in volume, then we have by measurement of the volume or of the pressure respectively a direct indication of absolute tem- perature. The apparatus used by Gay-Lussac (Fig, 3), or, equally, that shown in Fig. 4, is in fact an air ther- mometer, requiring only the establishment of a scale to fit it for practical use. A. simple modern form of air thermometer is shown in Fig. 6. The FIG. 6. Art, 48. Air Ther- mometer, FIG. 7. Art. 48. Preston Air Thermometer. APPLIED THERMODYNAMICS bulb A contains dry air, and communicates through a tube of fine bore with the short arm of the manometer BB> by means of which the pressure is measured. The level of the mercury is kept constant at a by means of the movable reservoir E and flexible tube m. The Preston air ther- mometer is shown in Fig. 7. The air is kept at constant volume (at the mark a) by admitting mercury from the bottle A through the cock B. In the Hoadley air ther- mometer, Fig. 8, no attempt is made to keep the volume of air constant; expansion into the small tube below the bulb increasing the volume so slightly that the error is com- puted not to exceed 5 in a range of 600 (o). 49. Remarks on Air Thermometers. Following Renault, the instrument is usually constructed to measure pressures at constant volume, using either nitrogen, hydrogen, or air as a medium, thily one "fixed point" need be marked, that of Iho temperature of melting ice. Having marked at 32 the atmos- pheric pressure registered at this temperature, the degrees aie spaced so that one of them denotes an augmentation of pressure of 14.7 - 491-4 = 0.0290 Ib. per square inch. It is usually more convenient, however, to determine the two fixed points as usual and subdivide the intervening distance into 180 equal degrees. The air thermometer readings differ to some extent from those of the most accurate mercurial instruments, principally because of the fact that mercury expands much less than any gas, and the modifying effect of the expansion of the glass container is there- fore greater in its case. The air thermometer is itself nob a E perfectly accurate instrument, since air doos not ewtetly follow Charles' law (Art. 47). The instrument is used for standardizing mercury thermometers, for direct measurements of temperatures belov the melting point of glass (000-800 F.) as in Regnault's experiments on vapors; or, "by using porcelain "bulbs, for measur- ing much higher temperatures. 50. The Perfect Gas, If actual gases conformed pre- cisely to the laws of Boyle and Charles, many of their thermal properties might be computed directly. The slightness of the deviations which actually occur sug- gests the notion of a perfect gas, which would exactly and invariably follow the laws, Fia.8. Art 48. Hoadley Air Ther- Any deductions which might be made from these sym- -- riii A n-p /wiirqA he rigorously true only THE PERFECT GAS 25 for a perfect gas, which does not exist in nature. TJie current thermo- dynamic method is, however, to investigate the properties of such a gas, modi- fying the results obtained so as to make them applicable to actual gases, rather than to undertake to express symbolically or graphically as a basis for computation the erratic behavior of those actual gases. The error involved in assuming air, hydrogen, and other "permanent 3 ' gases to be perfect is in all cases too small to be of importance in engineering applications. Zeuner (4) has developed an " equation of condition " or "characteristic equation" for air which holds even for those extreme con- ditions of temperature and pressure which are here eliminated. 51. Properties of the Perfect Gas. The simplest definition is that the perfect gas is one which exactly follows the laws of Boyle and Charles. (Rankine's definition (5) makes conformity to Daltoa's law the criterion of perfectness.) Symbolically, the perfect gas con- forms to the law, readily deduced from Art. 50, in which R is a constant and T the absolute temperature. Consid- ering air as perfect, its value for Jt may be obtained from experi 7 mental data at atmospheric pressure and freezing temperature-: R = PV+ ^=(14.7 x 144 x 12.387)-*- 491.4 = 53.36 foot-pounds. For other gases treated as perfect, fche value of R may be readily calculated when any corresponding specific' volumes, pressures, and temperatures are known. Under the pressure and temperature just assumed, the specific volume of hydrogen is 178.83 ; of nitrogen, 12.75; of oxygen, 11.20. A useful form of the perfect gas equation may be derived from that just given by noting that _PF"-*- 2 7 = -B, a constant : ]PV pv IF** t ' 52. Significance of ff. At the standard pressure and temperature specified in Art. 51, the values of E for various gases are obviously proportional to their specific volumes or inversely proportional to their densities. This leads to the form of the characteristic equation some- * At the temperature < 1? let the pressure and volume be p 1? tv If the gas were to expand at constant temperature, it would conform to Boyle's law, Pi&i=*c, or <^ = Ci. Let the pressure be raised to any condition p z while the volume remains t? a , &i the temperature now becoming t^. Theu by Charles' law, ^=-7, jpa^Pi T> P&* Pi &i *i =P&i=Pii 4 2 =Ci<2, where ^ is a constant to which we give the symbol 5. 26 APPLIED THERMODYNAMICS times given, PV= rT -+ M, in which Mia the molecular weight and r a constant having the same value for all sensibly perfect gases. TABLE : PROPERTIES OF THE COMMON GASES APPROXIMATE ATOMIC WEIGHT MOLECULAR WEIGHT SPECIFIC HEAT AT CONSTANT PRESSURE SYMBOL Hvdroeen 1 2 3.4 H Nitrogen Oxygen Carbon dioxide 14 16 28 32 44 2438 0.217 215 N C0 2 Alcohol 46 4534 C 2 H fl O Carbon monoxide Ether 28 74 2438 4797 CO (C 2 H 6 ) 2 Ammonia Sulphur dioxide . . . Chlo'rofQTrn. . T . . 17 64 119 5 15 1567 NHa S0 2 CHC1 3 Methane 16 5929 CEU Olefiant gas 14 4040 CH 2 Air 2375 SteRTY) . . ,.,.,.. 18 5 H 2 O 52a. Principles of Balloons. A body is in vertical equilibrium in a fluid medium when its weight is equal to that of the fluid which it dis- places. In a balloon ; the weight supported is made up of (a) the car, envelope and accessories, and (6) the gas in the inflated envelope. The equation of equilibrium is W=w+V(d-d') 9 where TT = weight in Ibs., item (a), above; w = weight of air displaced by the car, framework, etc., in Ibs.; "F = volume of inflated envelope, cu. ft.; d = density of surrounding air, Ibs. per cu. ft.; d r = density of gas in envelope, Ibs. per cu. ft. The term w is ordinarily negligible. The pressure of the gas in the envelope is only a small fraction of a pound above that of the atmos- phere. When gas is vented from the balloon, the latter is prevented from collapsing by pumping air into one of the compartments (ballonets), so that the effect of venting is, practically speaking, to decrease the size of the envelope. If the balloon is not in vertical equilibrium, then W w 7(d d') is the net downward force, or negatively the upward pull on an anchor rope which holds the balloon down, A considerable variation in the THE PERFECT GAS 27 conditions of equilibrium arises from variations in the value of d. Atmospheric pressure varies with the altitude about ^as follows: Altitude Normal Atmospheric in Miles. Pressure, Lbs per Sq. In. 14.7 i 14.02 \ 13.33 | 12.66 1 12.02 li 11.42 1J 10.88 2 9.80 52b. Mixtures of Gases. By Dalton's law (Art. 40), if wi, w z , w* be the weights of the constituents of a mixture at the state V (volume of entire mixture, Tiot its specific volume), T, P; and if the R values for these constituents be Ri, R*, Rz, then If W be the weight of the mixture =t0i+wfc+v\ then the equivalent R value for the mixture is PV r> .. ~ W Then, for example, P! _VRiTwi If vi t v Z) v 3 denote the actual volumes of several gases at the conditions P, T' 3 and Wi, 102, w>3, their weights, then Pvi=wiRiT } Pv 2 ^w z RzTj Pvz= V = U PV-WRT t- PV-W&l, V ~WRTP f From expressions like the last we may deal with computations relating to mixed gases where the composition is given by volume. The equivalent molecular weight of the mixture is, of course, (Art. 52). K Dalton's law, like the other gas laws, does not exactly hold with any actual gas: but for ordinary engineering calculations with gases or even with superheated vapors the error is negligible. 53. Molecular Condition. The perfect gas is one in which the molecules move with perfect freedom, the distances between them being so great in comparison with their diameters that no mutually attractive forces are exerted. No per- formance of disgregation work accompanies changes of pressure or temperature. 28 APPLIED THERMODYNAMICS Hirschfeld (6), in fact, defines the perfect gas as a substance existing in such a physical state that its constituent particles exert no interattraction. The coefficient of expansion, according to Charles 7 law, would be the exact reciprocal of the abso- lute tempeiature of melting 1 ice, for all pressures and temperatures. Zeuner has shown (7) that as necessary consequences of the theory of perfect gases it can be proved that the product of the molecular weight and specific volume, at the same pressure and temperature, is constant for all gases; whence he derives Avogadro's principle (Art. 40). Rankine (8) has tabulated the physical properties of the "perfect gas." 54. Kinetic Theory of Gases. Beginning with Bernoulli! in 1738, various investigators have attempted to explain the phenomena of gases on the basis of the kinetic theory, which is now closely allied with the mechanical theory of heat. According to the former theory, the molecules of any gas are of equal mass and like each other. Those of different gases differ in proportions or structure. The intervals between the molecules are relatively very great. Their tendency is to move with uniform velocity in straight lines. Upon contact, the direction of mo- tion undergoes a change. In any homogeneous gas or mixture of gases, the moan energy due to molecular motion is the same at all parts. The pressure of the gas per unit of superficial area is proportional to the number of molecules in a unit of volume and to the average energy with which they strike this area. It is there- fore proportional to the density of the gas and to the average of the squares of the molecular velocities. Temperature is proportional to the average kinetic energy of the molecules. The more nearly perfect the gas, the more infrequently do the molecules collide with one another. When a containing vessel is heated, the mole- cules rebound with increased velocity, and the temperature of the gas rises; when the vessel is cooled, the molecular velocity and the temperature are decreased. " When a gas is compressed under a piston in a cylinder, the particles of the gas rebound from the inwardly moving piston with unchanged velocity relative to the piston, but with increased actual velocity, and the temperature of the gas con- sequently rises. When a gas is expanded under a receding piston, the particles of the gas rebound with diminished actual velocity, and the temperature falls" (9). Recent investigations iu molecular physics have led to a new terminology but in effect serve to verify and explain the kinetic theory. 55. Application of the Kinetic Theory. Let w denote the actual molecular velocity. Resolve this into components #, y, and z, at right angles to one another. Then w a =; a 2 -f y 2 + z*. Since the molecules move at random in all directions, x = y = 2, and to 2 = 3 x 2 . Consider a single molecule, moving in an x direction back and forth between two limiting surfaces distant from each other d, the x component of the velocity of this particle being a. The molecule will make (a 2 rf) oscillations per second. At each impact the velocity changes from + a to a, or by 2 a, and the momentum by 2 am, if m represents the mass of the molecule. The average rate of loss of momentum per single impact is 2 am X (a * 2 d) =ma z +d; and this is the average force exerted per second on each of the limiting surfaces. The total force exerted by all the molecules on these surfaces is then equal to F = j-N = ~JV =* [ N ' m w ^ c ^ ^ ^ tne totia ^ number of molecules THE PERFECT GAS 29 in the vessel. Let q be the area of the limiting surface. Then the force per unit of aurfaw-p-F+g-^ + a-^.whencepB-^-g-w*, in which , is the volume of the gas = gd and W is its weight in Ibs. (10). See Art. 127 a. 56. Applications to Perfect Gases. Assuming that the absolute temperature is proportional to the average kinetic energy per molecule (Art. 54), this kinetic energy being ^ miv 2 , then letting the mass be unity and denoting by R a constant relation, we have pv = RT. The kinetic theory is perfectly consistent with Dai- ton's law (Art. 40). It leads also to Avogadro's principle. Let two gases be pres- ent. For the first gas, p - nmw 2 - 3, and for the second, P = N3IW 2 - 3. If t T, mw 2 = MW 2 , and if p = P, then n = N. If M denote the mass of the gas, M = mN 9 and pv = Mw 2 3, or w 2 = 3jow M 9 from which the mean velocity of the molecules may be calculated for any given temperature. For gases not perfect, the kinetic theory must take into account, (a) the effect of occasional collision of the molecules, and (b) the effect of mutual attractions and repulsions. The effect of collisions is to reduce the average distance moved between impacts and to increase the frequency of impact and consequently the pressure. The result is much as if the volume of the containing vessel were smaller by a constant amoant, ft, than it really is. For w, we may therefore wiite v b. The value of b depends upon the amount and nature of the gas.* The effect of mutual attractions is to slow down the molecules as they approach the walls. This makes the pressure less than it otherwise would be by an amount which can be shown to be inversely proportional to the square of the volume of the gas. For p, we therefore write p 4- (a i> 2 ), in which a depends similarly upon the quantity and nature of the gas. We have then the equation of Van der Waals, (1) Cf. Verdet, Legons de Chemie et de Physique, Paris, 1862. (2) ReL des Exp., I, 111, 112. (3) Trans. A. S. M. E., VI, 282. (4) Technical Thermodynamics (Klein tr.), II, 313. (5) "A perfect gas is a substance in such a condition that the total pressure exerted by any number of portions of it, against the sides of a vessel in which they are inclosed, is the sum of the pressures which each such portion would exert if enclosed in the vessel separately at the same temperature.' 1 The Steam Engine, 14th ed., p. 220. (6) Engineering Thermodynamics, 1007. (7) Op. cit., I, 104-107. (8) Op. eft., 593-595. (9) Nichols and Franklin, The Elements of Physics, I, 199-200. (10) Ibid., 199 ; Wormell, Thermodynamics, 167-161. (11) Over de Continuiteit van den Gas en Vloeistoestand, Leinden, 1873, 76 ; tr. by Roth, Leipsic, 1887. SYNOPSIS OF CHAPTER 3H Boyle's law, pw = PF: deviations. Dalton's law, Avogadro's principle. Law of Gay-Lussac or of Charles: increase of volume at constant pressure; increase of pressure at constant volume; values of the coefficient from 32 F.j deviations with actual gases. * Strictly, it depends upon the space between the molecules ; but Richardslsuggests (Science, XXXIV, N, S., 878), that it may vary with the pressure and the temperature. 30 APPLIED THERMODYNAMICS The absolute zero. 459.4 F , or 491 4 F. below the freezing point. Air thermometers Preston's ; Hoadley's ; calibration } gases used. The perfect gas, =-=- , definitions; properties, values of R ; absence of inter- t j. molecular action, the kinetic theory; development of the law PF^T there- from ; conformity with Avogadro's principle , molecular velocity. Table ; the common gases j Constants for gas mixtures . R~ l ^ * ' ' Balloons: weight -weight of fluid displaced. The Van der Waals equation for imperfect gases : PROBLEMS 1. Find the volume of one pound of air at a pi ensure of 100 Ib. per square inch, the temperature being 32 F., using Boyle's law only. (Ans,, 1.821 cu. ft.) 2. From Charles' law, find the volume of one pound of air at atmospheric pres- sure and 72 F. (Ans , 13.4 cu. ft.) 3. Find the pressure exerted by one pound of air having a volume of 10 cubic feet at 32 F. (Ans,, 18.2 Ib. per sq. in.) 4. One pound of air is cooled from atmospheric pressure at constant volume from 32 F. to 290 F. How nearly perfect is the vacuum pioduced? (Ans., 65.5%.) 5. Air at 50 Ib. per square inch pressure at the freezing point is heated at con- stant volume until the temperature becomes 2900 F. Find its pressure after heating. (Ans , 341,8 Ib. per sq. in.) 6. Five pounds of air occupy 50 cubic feet at a temperature of F. Find the pressure. (Ans., 17.03 Ib. per sq. in.) 7. Find values of R for hydrogen, nitrogen, oxygen. (Ans., for hydrogen, 770.3 ; for nitrogen, 54.9 ; for oxygen, 48.2.) 8. Find the volume of three pounds of hydrogen at 15 Ib. pressure per square inch and 75 F. (Ans , 571.8 cu. ft.) 9. Find the temperature of 2 ounces of hydrogen contained in a 1-gallon flask and exerting a pressure of 10,000 Ib. per square inch. (Ans., 1536 F.) 10. Compute the value of r (Art. 52). (Ans , 1538 to 1544.) 11. Find the mean molecular velocity of l Ib. of air (considered as a perfect gas) at atmospheric pressure and 70 F. (Ans., 1652 ft. per sec.) 12. How large a flask will contain 1 Ib. of nitrogen at 3200 Ib. pressure per square inch and 70 F. ? (Ans., 0.0631 cu. ft.) 13. A receiver holds 10 Ib. of oxygen at 20 C. and under 200 Ib. pressure per square inch. What weight of air will it hold at 100 F. and atmospheric pressure ? 14. For an oxy-hydrogen light, there are to be stored 25 Ib. of hydrogen and 200 Ib. of oxygen. The pressures m the two tanks must not exceed 500 Ib. per square inch at 110 F Fhul their volumes. 15. A receiver containing air at normal atmospheric pressure is exhausted until the pressure is 0.1 inch of mercury, the temperature remaining constant. "What per PROBLEMS 31 cent of the weight of air has been removed ? (14.697 Ib. per sq. in. =29. 92 ins. mercury.) 16. At sea level and normal atmospheric pressure, a 60,000 cu. ft, hydrogen balloon is filled at 14.75 Ib. pressure. The temperature of the hydrogen is 70 F.; that of the external air is 60 F. The envelope, car, machinery, ballast, and occu- pants weigh 3500 Ib. Ignoring the term w, Art 52a, what is the upward pull on the anchor rope ? 17. How much ballast must be discharged from the balloon in Prob. 16 in order that when liberated it may rise to a level of vertical equilibrium at an altitude of 2 miles ? 18. In Problem 17, there are vented from the balloon, while it is at the 2-mile altitude, 10 per cent of its gas contents. If the ballonet which has been vented is kept constantly filled with air at a pi ensure just equal to that of the external atmos- phere, to what approximate elevation will the balloon descend ? What is the net amount of force available for accelerating downward at the moment when descent begins ? 19. In Problem 17, while at the 2-mile level, the temperature of the hydrogen becomes 60 anil that of the surrounding air 0, without change in either internal or external pressure. What net amount of ascending or descending force will be caused by these changes ? How might tins be overcome ? 20. In a mixture of 5 Ib. of air with 1C Ib. of steam, at a pressure of 50 Ib. per square inch at 70 IT., what is the value of R for the mixture ? What is its equiva- lent molecular weight ? The difference of k and I * The partial pressure due to air only ? 21 A mixed gas weighing 4 Ib. contains, by volume, 35 per cent of CO, 16 per cent of II and 3 poi cent of CH 4 , the balance being N. The pressure ib 50 Ib. per square inch and the temperature 100 F. Find the value of E for the mixture, the partial pressure due to each constituent, and the percentage composition by weight. 22. Compute (and discuss) values of R and y for gases listed in the table, page 26. (See Arts. 69, 70.) CHAPTER IV THERMAL CAPACITIES : SPECIFIC HEATS OF GASES : JOULE'S LAW 57. Thermal Capacity. The definition of specific heat given in Art. 24 is, from a thermodynainic standpoint, inadequate. Heat jtroducea other effects than change of temperature. A definite movement of heat cun l>o estimated only \vlion all of these effects are defined. For example, the quantity of heat necessary to raise the temperature of air one degree in a constant volume air thermometer is much less than that used in raising the temperature ono degree in the constant pressure typo. The specific heat may be .satisfactorily defined only by referring to the condition of the substance during the changu of tein)>e,raturo. Ordinary specific heals assume constancy of jwYMWwrp, that oC tho atmosphere, whilo the volume increases with the temperature in a ratio "which is determined by the coeffi- cient of expansion of the material. A specific heat determined in this way as are those of solids and liquids generally I'M the specific heat at constant pressure. Whenever the term te ^ecffia heat" 'fit 'tittcd without qualification, this yur- tictdar specific Jieat in intended. Heat may be absorbed during changes of either pressure, volume, or temperature, while porno other of these proper- ties of the substance is kept constant. For a specific change of property, the amount of heat absorbed represents a specific thermal capacity. 58. Expressions for Thermal Capacities. If 7/ represents heat absorbed, c a constant specific heat, and (T ) a range of temperature, then, by definition, H=c(Tf) and c? = //-*- (T f). If c be variable, then H= \ cdT and c =* fl H -*- tl'F. If in place of c we wish to denote the specific heat at constant pressure (k), or that nt constant volume (f), we may apply subscripts to the differential coefficients ; thus, and I ' the subscripts denoting the property which remains constant during the change in temperature. We have also the thermal capacities, ' \W>/r' The first of these denotes the amount of heat necessary to increase the specific volume of the substance by unit volume, while the temperature remains constant; SPECIFIC HEATS OF GASES 33 this is known as the latent heat of expansion. It exemplifies absorption of heat without change of temperature. "No names have been assigned for the other thermal capacities, but it is not difficult to describe their significance. 59. Values of Specific Heats. It was announced by Dulong and Petit that the specific heats of substances are inversely as their chemical equivalents. This was shown later by the experiments of Regnault and others to be approximately, though not absolutely, correct. Considering metals in the solid state, the product of the specific heat by the atomic weight ranges at ordinary temperatures from 6.1 to 6.5. This nearly constant product is called the atomic heat. Determination of the specific heat of a solid metal, therefore, permits of the approximate computa- tion of its atomic weight. Certain n on -metallic substances, including chlorine, bromine, iodine, selenium, tellurium, and arsenic, have the same atomic heat as the metals. The molecular heats of compound bodies are equal to the sums of the atomic heats of their elements ; thus, for example, for common salt, the specific heat 0.219, multiplied by the molecular weight, 58.5, gives 12.8 as the molecular heat ; which, divided by 2, gives 6.4 as the average atomic heat of sodium and chlorine; and as the atomic heat of sodium is known to be 6.4, that of chlorine must also be 6.4 (1). 60. Volumetric Specific Heat. Since the specific volumes of gases are in- versely as their molecular weights, it follows from Art. 59 that the quotient of the specific heat by the specific volume is practically constant for ordinary gases. In other words, the specific heats of equal volumes are equal. The specific heats of these gases are directly proportional to their specific volumes and inversely pro- portional to their densities, approximately. Hydrogen must obviously possess the highest specific heat of any of the gases. 61. Mean, "Real," and "Apparent" Specific Heats. Since all specific heats are variable, the values given in tables are mean values ascertained over a definite range of temperature. The mean specific heat, adopting the notation of Art. 58, is c H-^(T f); while the true specific heat, or specific heat " at a point," is the limiting value c = dH-s- dT Rankine discusses a distinction between the real and apparent specific heats ; meaning by the former, the rate of heat absorption necessary to effect changes of temperature alone, without the performance of any disgregation or external work and by the latter, the observed rate of heat absorption, effecting the same change of temperature, but simultaneously causing other effects as well. For example, in heating water at constant pressure from 62 to 63 F., the apparent specific heat is 1.0 (definition, Art. 22). To compute the real specific heat, we must know the external work done by reason of expansion against the constant pressure, and the disgregation work which has readjusted the molecules. Deducting from 1.0 the heat equivalent to these two amounts of work, we have the real specific heat, that which is used solely for making the substance hotter. Specific heats determined by experiment are always apparent; the real specific heats are known only by computation (Art. 64). 34 APPLIED THERMODYNAMICS 62. Specific Heats of Gases. Two thermal capacities of especial importance are used in calculations relating to gases. The first is the specific heat at constant pressure, k, which is the amount of heat necessary to raise the temperature one degree while the pressure is kept constant; the other, the specific heat at constant volume, 1, or the amount of heat necessary to raise the temperature one deyree while the volume is kept constant. 63. Regnault's Law. As a result of his experiments on a large number of gases over rather limited ranges of temperature, Regnault announced that the specific heat of any gas at constant pressure is constant. This is now known not to be rigorously true of even our most nearly perfect gases. It is not even approxi- mately true of those gases when far from the condition of perfectness, a'.e. at low temperatures or high pressures. At very liigli temperatures, also, it is well known that specific heats rapidly inci ease. This pailicular variation is perhaps due to an approach toward that change of state described as dmocwtion. When near any change of state, combustion, fusion, evaporation, dissociation, every sub- stance manifests erratic thermal properties. The specific heat of carbon dioxide is a conspicuous illustration. Recent determinations by Holborn and Ilenning (2) of the mean specific heats between and x C. give, for nitrogen, k = 0.255 + 0.000019 x\ and for carbon dioxide, Jt = 0.201 + 0.0000742 j;- 0.000000018^: while for steam, heated from 110 to x C., *= 0.4000 -0.0000108 x+ 0.000000044 a* The specific heats of solids also vary. The specific heats of substances in general increase with the temperature. Kegnault's law would hold, however, for a perfect gas; in this, the specific heat would be constant under all conditions of tempera- ture, For our "permanent" gases, the specific heat is practically constant at ordinary temperatures. The table in Art. 52 shows that in general the specific heats at constant pressure vary inversely as the molecular weights. Carbon dioxide, sulphur dioxide, ammonia, and steam (which are highly imperfect gases) vary most widely from this law. 64. The Two Specific Heats. When a gas is heated at constant pressure, its volume increases against that pressure, and external work is done 111 consequence. The external work may be computed by multiplying the pressure by the change in volume. When heated at constant volume, no external work is done ; no movement is made against an external resist- ance. If the gas be perfect, then, under this condition, no disgregation work is done ; arid the specific heat at constant volume is a true specific heat, according to Kankine's distinction (Art. 61). The specific heat at constant pressure is, however, the one commonly determined by experi- ment. The numerical values of the two specific heats must, in a perfect gas, differ by the heat equivalent to the external work done during heating at constant pressure. Under certain conditions, as with, water at its SPECIFIC HEATS OF GASES 35 maximum density, no external work is done when heating at constant pressure ; and at this state the two specific heats are equal, if we ignore possible differences in the disgregation work. 65. Difference of Specific Heats. Let a pound of air .at 32 F. and atmospheric pressure be raised 1 F. in temperature, at constant pressure. It will expand 12.387-7-491.4 = 0.02521 cu. ft., against a resistance of 14.7 x 144 = 2116.8 lb. per square foot. The external work which it performs is consequently 2116.8 x 0.02521 = 53.36 foot- pounds. A general expression for this external work is W=P V+ T\ and as from Art. 51 the quotient P V~- T is a constant and equal to R, then IP" is a constant for each particular gas, and equivalent in value to that of R for such gas. The value of TFfor air, expressed in heat units, is 53.36-7-778 = 0.0686. If the specific heat of air at constant pressure, as experimentally determined, be taken at 0.2375, then the specific heat at constant volume is 0.2375 0.0686 = 0.1689, air being regarded as a perfect gas. 66. Derivation of Law of Perfect Gas. Let a gas expand at constant pres- sure P from the condition of absolute zero to any other condition F, T. The total external work which it will have done in consequence of this expansion is PV. The work done per degree of temperature is PF T. But, by Charles' law, this is constant, whence we have PV=RT. The symbol R of Art. 51 thus represents the external work of expansion during each degree of temperature increase (3). 67. General Case. The difference of the specific heats, while constant for any gas, is different for different gases, because their values of R differ. But since values of R are proportional to the specific volumes of gases (Art. 52), the differ- ence of the volumetric specific heats is constant for all gases. Thus, let , I be the two specific heats, per pound, of air. Then k - I = r. Let d be the density of the air; then, d(kT) is the difference of the volumetric specific heats. For any other gas, we have similarly, K L = R and D(K L) ; but, from Art. 52 R:r -d:D, or R - rd - D. Hence, K- L = rd - D = (k - l)(d - Z>), or D(K L) = d(k Z). The difference of the volumetric specific heats is for all gases equal approximately to 0.0055 B. t. u. (Compare Art. 60.) 68. Computation of External Work. The value of JK given in Art. 52 and Art. 65 is variously stated by the writers on the subject, on account of the slight uncertainty which exists regarding the exact values of some of the con- stants used in computing it. The differences are too small to be of consequence in engineering work. 69. Ratio of Specific Heats. The numerical ratio between the two specific heats of a sensibly perfect gas, denoted by the symbol y, is a constant of prime importance in thermodynamics. 36 APPLIED THERMODYNAMICS For air, its value is 0.2375 -^0.1689 =1.4 +. Various writers, using other funda- mental data, give slightly different values (4). The best direct experiments (to be described later) agree with that here given within a narrow margin. For hydrogen, Lummer and Pringsheim (5) have obtained the value 1 408; and for oxygen, 1 396. For carbon dioxide, a much leas perfect gas than any of these, these observers make the value of y, 1.2961; while Dulong obtained 1.338. The latter obtained for carbon monoxide 1.428. The mean value for the "permanent" gases is close to that for air, viz., The value of y is about the same for all common gases, and is practically inde- pendent of the temperature or the pressure. From Arts. 59, 60, 65, we have, letting m denote chemical equivalents and V specific volumes, m ~ where a and b are constants having the same value 'for all gases. 70. Relations of R and y. A direct series of relations exists between the two specific heats, their ratio, and their difference. If we denote the specific heats by Jc and ?, then in proper units, k l-R l-k-R i-v- -* .y. A-Z-.B. i-k t. t -y k _ E y fFor air, this gives ' 237 ^ ^ = 1.402.) 9^7^ ^"J-t5v / 778 ~k = Tcy yTJ. fcyJc = yR. k = R ^-r c/ t/ / ^ j_ 71. Rankme's Prediction of the Specific Heat of Air. The specific heat of air was approximately determined by Joule in 1852. Earlier determinations were unreliable. Eankine, in 1830, by the use of the relations just cited, closely ap- proximated the result obtained experimentally by Reguault three years later. Using the values y = 1.4, R = 53.15, Rankine obtained y Regnault's result was 2375. = R -- = (53.15 - 772) x (1.4 - 0.4) = 0.239. y 1 SPECIFIC HEATS OF GASES 37 72. Mayer's Computation of the Mechanical Equivalent of Heat Reference was made in Art. 29 to the computation of this constant prior to the date of Joule's conclusive experiments. The method is substantially as follows : a cylinder and piston having an area of one square foot, the former containing one cubic foot, are assumed to hold air at 32 F., which is subjected to heat. The piston is balanced, so that the pressure on the air is that of the atmosphere, or 14.7 Ib. per square inch ; the total pressure on the piston being, then, 144 x 14.7 = 2116.8 Ib. While under this pressure, the air is heated until its temperature has increased 491.4. The initial volume of the air was by assumption one cubic foot, whence its weight was 1 -4- 12.387 = 0.0811 Ib. The heat imparted was therefore 0.0811 x 0.2375 x 491.4 = 9.465 B. t. u. The external work was that due to doubling the volume of the air, or 1 x 14.7 x 144 = 2116.8 foot-pounds. The piston is now fixed rigidly in its original position, so that the volume cannot change, and no external work can be done. The heat required to produce an elevation of temperature of 491.4 is then 0.0811 x 0.1689 x 491.4 = (3.731 B. t, u. The difference of heat corresponding to the external work done is 2.734 B. t. u., whence the mechanical equivalent of heat is 2116.8 -5- 2.734 = 774.2 foot-pounds. 73. Joule's Experiment. One of the crucial experiments of the science was conducted by Joule about 1844, after having been previously attempted by Gay- Lussac. Two copper receivers, A and B, Fig. 9, were connected by a tube and stopcock, and placed in a water bath. Air was compressed in A to a pressure of 22 atmospheres, while a vacuum was maintained in . When the stopcock was opened, the pressure in each re- ceiver became 11 atmospheres, and the temperature of the air and of FIG. 9. Arts. 73, so. -Joule's Experiment. the water bath remained practically unchanged. This was an instance of expansion without the perform- ance of external work; for there was no resisting pressure against the augmentation of volume of the air. 38 APPLIED THERMODYNAMICS 74. Joule's and Kelvin's Porous Plug Experiment. Minute observations showed that a slight change of temperature occurred in the water bath. Joule and Kelvin, in 1852, by their celebrated "porous plug" experiments, ascertained the exact amount of this change for various gases. In all of the permanent gases the change was very small ; in some cases the tem- perature increased, while in others it decreased ; and the inference is jus- tified that in a perfect gas there would be no change of temperature (Art. 156). 75. Joule's Law. The experiments led to the principle that when a perfect gas expands without doing external work, and without receiving or discharging heat, the temperature remains unchanged and no disgregation work is done, A clear appreciation of this law is of fundamental importance. Four thermal phenomena might have occurred in Joule's experiment : a movement of heat, the performance of external work, a change in temperature, or work of disgregation. From Art. 12, these four effects are related to one another in such manner that their summation is zero; (-9"= T+I+ W). By means of the water bath, which throughout the experiment had the same temperature as the air, the movement of heat to or from the air was prevented. By expanding into a vacuum, the performance of external work was prevented. The two remaining items must then sum up to zero, i.e. the temperature change and the disgregation work. But the temperature did not change ; consequently the amount of disgre- gation wort must have been zero. 76. Consequences of Joule's Law. In the experiment described, the pres- sure and volume changed without changing the internal energy. !N"o dis- gregation work was done, and the temperature remained unchanged. Considering pressure, volume, and temperature as three cardinal thermal properties, internal energy is then independent of the pressure or volume and depends on the temperature only, in any perfect gas. It is thus itself a cardinal property, in this case, a function of the temperature. "A change of pressure and volume of a perfect gas not associated with change of temperature does not alter the internal energy. In any change of tem- perature, the change of internal energy is independent of the relation of pressure to volume during the operation ; it depends only on the amount by which the temperature has been changed" (6). The gas tends to cool in expanding, but this effect is "exactly compensated by the heat which JOULE'S LAW 39 is disengaged through the friction in the connecting tube and the im- pacts which destroy the velocities communicated to the particles of gas while it is expanding" (7) TJiere is ^racfr'raZ/;/ no disgregation work in heating a sensibly perfect gas; all of the interned energy is evidenced by temperature alone. When such a gas passes from one state to another in a variety of ways, the external work done varies; but if from the total movement of heat the equivalent of the external work be deducted, then the remainder is always the same, no matter in what way the change of condition has been produced. Instead of H = T -f 1 4- T7, we may write #= T+W. 77. Application to Difference of Specific Heats. The heat absorbed dur- ing a change in temperature at constant pressure being H=Jc(T), and the external work during such a change being W= P(Vv) = R(T ), the gain of internal energy must be H- W=(k-R)(T-t}. The heat absorbed during the same change of temperature at constant volume is H=l(T ). Since in this case no external work is done, the whole of the heat must have been applied to increasing the internal energy. But, according to Joule's law, the change of internal energy is shown by the temperature change alone. In whatever way the temperature is changed from T to f, the gain of internal energy is the same. Consequently, -t) = l(T-t) and fc- J? = Z, a result already suggested in Art. 65. 78. Discussion of Results. The greater value of the specific heat at constant pressure is due solely to the performance of external work dur- ing the change in temperature. The specific heat at constant volume is a real specific heat, in the case of a perfect gas ; no external work is done, and the internal energy is increased only by reason of an elevation of tem- perature. There is no disgregation work. All of the heat goes to make the substance hot. So long as no external work is done, it is not neces- sary to keep the gas at constant volume in order to confirm the lower value for the specific heat; no more heat is required to raise the tempera- ture a given amount when the gas is allowed to expand than when the volume is maintained constant. For any gas in which the specific heat at constant volume is constant, Joule's law is inductively established ; for no external work is done, and temperature alone measures the heat absorp- tion at any point on the thermometric scale. If a gas is allowed to expand, doing external work at constant temperature, then, since no change of inter* 40 APPLIED THERMODYNAMICS nal energy occurs, it is obvious from Art. 12 that the external work is equal to the heat absorbed. Briefly, the important deduction from Joule's experi- ment is that item (6), Art. 12, may be ignored when dealing with sensibly perfect gases. 79. Confirmatory Experiment. By a subsequent experiment, Joule showed that when, a gas expands so as to perform external work, heat dis- appears to an extent proportional to the work done. Figure 10 illustrates the apparatus. A receiver A, containing gas compressed to two atmos- pheres, was placed in the calorimeter B and connected with the gas holder Of placed over a water tank. The gas passed from A to G through the coil D } depressed the water in the gas holder, and divided itself be- tween the two vessels, the pressure falling to that of one atmosphere. The work done was computed from the augmentation of volume shown Fl ^ 10 " Art> 7 a 9 ' ~~ J J oul A e ' s T . . , . .,* . Experiment, Second Ap- by driving down the water in G against atnios- pa ratus. pheric pressure; and the heat lost was ascertained from the fall of temperature of the water. If the temperature of the air were caused to remain constant throughout the experiment, then the work done at G would be precisely equivalent to the heat given up by the water. If the temperature of the air were caused to remain constantly the same as that of the water, then H= = T+ 1+ W, (T+ 1)= - W, or internal energy would be given up by the air, precisely equivalent in amount to the work done in (7. 80. Application of the Kinetic Theory. In the porous plug experiment referred to in Art. 74, it was found that certain gases were slightly cooled as a result of the expansion, and others slightly warmed. The molecules of gas are very much closer to one another in A than in B, at the beginning of the experiment. If the mole- cules are mutually attractive, the following action takes place : as they emerge from A, they are attracted by the remaining particles in that vessel, and their velocity decreases. As they enter B, they encounter attractions theie, which tend to in- crease their velocity; but as the second set of attractions is feebler, the total effect is a loss of velocity and a cooling of the gas. In another ga>s, in which the molecules repel one another, the velocity during passage would be on the whole augmented, and the temperature increased. A perfect gas would undergo neither increase nor decrease of temperature, for there would be no attractions or repulsions between the molecules. (1) A critical review of this theory has "been presented by Mills The Specific Heats of the Elements, Science, Aug. 24, 1908, p. 221. (2) The Engineer, January 13, 1908. (3) Throughout this study, no attention will be paid to the ratio 778 as affecting the numerical value of constants in formulas involving both heat and work SYNOPSIS OF CHAPTER IV PROBLEMS 41 53.36 quantities. R may by either 63.36 or -ZZT-- The student should discern whether 778 heat units or foot-pounds are intended. (4) Zeuner, Technical Thermodynamics, Klein tr., I, 121. (5) Ibid., loc. tit. (6) Ewing: The Steam Engine, 1906. (7) Wormell, Thermodynamics. SYNOPSIS OF CHAPTER IV Specific thermal capacities; at constant pressure, at constant volume; other capacities. Atomic heat = specific heat X atomic weight; molecular heat. The volumetric specific heats of common gases are approximately equal. * 77" (JTT Mean specific heat = ; true specific heat = -; real and apparent specific heats. T t aT EegnauWs law : u the specific heat is constant for perfect gases." Difference of the two specific heats E = 53.36 ; significance of R. The difference of the volumetric specific heats equals 0.0055 B. t. u. for all gases. Ratio of the specific heats : y = 1.402 for air ; relations between A', Z, ?/, J?. Rankine's prediction of the value of k: Mayer 1 s computation of the mechanical equiva- lent of heat. Joule* s Law : no disgregation work occurs in a perfect gas. If the temperature does not change, the external work equals the heat absorbed. If no heat is received, internal energy disappears to an extent equivalent to the external work done. The condition of intermolecular force determines whether a rise or a fall of temperature occurs in the porous plug experiment. PROBLEMS 1. The atomic weights of iron, lead, and zinc being respectively 56, 206.4, 65 ; and the specific heats being, for cast iron, 0.1298 ; for wrought iron, 0.1138 ; for lead, 0.0314 ; and for zinc, 0.0956, check the theory of Art. 69 and comment on the results. (Ans., atomic heats are: lead, 6.481; zinc, 6.214; wrought iron, 6.373; cast iron, 7.259.) 2. [Find the volumetric specific heats at constant pressure of air, hydrogen, and nitrogen, and compare with Art, 60. ( k = 3.4 for H and 0.2438 for N.) (Ans., air 0.01917; hydrogen 0.01901; nitrogen 0.01912.) 3. The heat expended in warming 1 ib. of water from 32 F. to 160 F. being 127.86 B, t. u., find the mean specific heat over this range. (Ans., 0.9989.) 4. The weight of a cubic foot of water being 59.83 Ib. at 212 F. and 62.422 Ib. at 32 ff F., find the amount of heat expended in performing external work when ont> pound of water is heated between these temperatures at atmospheric pressure. (Ans., 0,00189 B.t.u,) 5. (a) Find the specific heat at constant volume of hydrogen and nitrogen. (Ans., 2.41; 0.1732.) (6) Find the value of y for these two gases. (Ans*, 1.4108; 1.4080.) 6. Check the value 0.0055 B. t. u. given in Art. 67 for hydrogen and nitrogen. (Ans., 0.00554; 0.00554.) 42 APPLIED THERMODYNAMICS 7. Compute the elevation in temperature, in Art. 72, that would, for an expansion of 100 per cent, under the assumed conditions, and with the given values of k and Z, give exactly 778 as the value of the mechanical equivalent of heat. What law of gaseous expansion would be invalidated if this elevation of temperature occurred ? (Ans , 489.05 F ) 8. In the experiment of Art. 79, the volume of air in C mci eased by one cubic foot against normal atmospheric pressure. The weight of water in B was 20 Ib The tem- perature of the air remained constant throughout the experiment. Ignoring radiation losses, compute the fall of temperature of the water. {Ans., 0.13604 F.) CHAPTER V GRAPHICAL REPRESENTATIONS: PRESSURE- VOLUME PATHS OF PERFECT GASES 81. Thermodynamic Coordinates. The condition of a body being fully defined by its pressure, volume, and temperature, its state may be repre- sented on a geometrical diagram in winch these properties are used as coordinates. This graphical method of analysis, developed by Clapeyron, is now in universal use. The necessity for three coordinates presupposes the use of analytical geometry of three dimensions, and representations may then be shown perspectively as related to one of the eight corners of a cube; but the projections on any of the three adjacent cube faces are commonly used ; and since any two of three properties fix the third when the characteristic equation is known, a protective representation is suffi- cient. Since internal energy is a cardinal property (Arts. 10, 76), this also may be employed as one of the coordinates of a diagram if desired. 82. Illustration. In Fig. 11 we have one corner of a cube constituting an origin of' coordinates at O. The temperature of a substance is to be represented by the distance upward from 0; its pressure, by the distance to the right ; and its volume, by the dis- tance to the left. The lines forming the cube edges are correspond- ingly marked OT, OP, 0V* Consider the condition of the body to be represented by the point A., within the cube. Its temperature is then represented by the distance AB, parallel to TO, the point B being in the plane VOP. The distance AD, parallel to PO, from A to.the plane TO F", indicates the pressure; and by drawing AQ paral- lel to VO, being the intersection of this line witli the plane TOP, we may represent the volume. The state of the substance is thus fully shown. Any of the three projections, Figs. 12-14, would equally fix its condition, providing the relation between P, V, and T is known. In each of these projections, two of the properties of the substance are shown ; in the three projections, each property appear^ 43 44 APPLIED THERMODYNAMICS twice; and the corresponding lines AB, AC, and AD are always equal in length. 1 1 FIG. 11 Perspe gram. A x-f x^ o ) Art. 82. FIG ictive Dia- V t C ...A. D . V \ ! o <-, " B P C u B .12. Art. 82. FIG. 13. Art. 82. FIG. 14. Art 82 TP Diagram. VP Diagram. TV Diagram. 83. Thermal Lines. In Tig. 15, let a substance, originally at A, pass at constant pressure and temperature to the state JB ; thence at constant temperature and volume to the state 0\ and thence at constant pressure 'D B,C, r FIG. 15. Art. 83. Perspective Ther- mal Line. FIG. 16. Art. 83. TP Path. FIG. 17. Art. 83. VP Path. FIG. 18. Art. 83. TV Path. and volume to D. Its changes are represented by the broken line ABCD, which is shown in its various projections in Figs. 16-18. The thermal line of the coordinate diagrams, Figs. 11 and 15, is the locus of a series of successive states of the substance. A path is the projection of a thermal line on one of the coordinate planes (Figs. 12-14, 16-18). The path of a substance is sometimes called its process curve, and its thermal line, a thermogram. The following thermal lines are more or less commonly studied : (a) Isothermal, in which the temperature is constant; its plane is perpendicular to the O^axis. (5) Isometric, in which the volume is constant ; having its plane per- pendicular to the OF' axis. (c) Isopiestic, in which the pressure is constant; its plane being per- pendicular to the OP axis. (d) Isodynamic, that along which no change of internal energy occurs. GRAPHICAL REPRESENTATIONS 45 (e) Adiabatic, that along which no heat is transferred between the substance and surrounding bodies; the thermal line of an. insulated body, performing or consuming work. 84. Thermodynamic Surface. Since the equation of a gas in- cludes three variables, its geometrical representation is a surface; and the first three, at least, of the above paths, must be projections of the intersection of a plane with such surface. Figure 19, from Pea- FIQ. 19. Arts, 84, 103. Thermodynamic Surface for a Perfect Gas. body (1), admirably illustrates the equation of a perfect gas, RT. The surface pmnv is the characteristic surface for a perfect gas. Every section of this surface parallel to the PV plane is an equilat- eral hyperbola. Every projection of such section on the PV plane is also an equilateral hyperbola, the coordinates of which express the law of Boyle, PF"=(7. Every section parallel with the TV plane gives straight lines pm, a?, etc., and every section parallel with the TP plane gives straight lines vn, xy, etc. The equations of these 212 32 46 APPLIED THERMODYNAMICS lines are expressions of the two forms of the law of Charles, their appearance being comparable with that in Fig. 5. 85. Path of Water at Constant Pressure. Some such diagram as that of Fig. 20 would represent the behavior of water in its solid, liquid, and vaporous forms when heated at constant pressure. The coordinates are temperature and volume. At A } the substance is ice, at a temperature below the freezing point. As the ice is heated from A to B, it undergoes a slight expansion, like other solids. At B, the melting point is reached, and as ice contracts in melting, there is a decrease in volume at constant temperature. At C, the sub- stance is all water; it contracts until it reaches the FIG 20 Art 85 Water ^ ^ ' . , . . _ _ , T . at Constant Pressure. temperature of maximum density, 39.1 F., at D, then expands until it boils at E 9 when the great increase in volume of steam over water is shown by the line EF. If the steam after formation conformed to Charles' law, the path would con- tinue upward and to the right from F, as a straight line. 86. The Diagram of Energy. Of the three coordinate planes, the PV is most commonly used. This gives a diagram corresponding with that produced by the steam engine indicator (Art. 484). It is sometimes called Watt's diagram. Its importance arises principally from the fact that it represents directly the external work done during the movement of the substance along any path. Consider a vertical cylinder filled with fluid, at the upper end of which is placed a weighted piston. Let the piston be caused to rise by the expansion, of the fluid. The force exerted is then equivalent to the weight of the piston, or total pressure on the fluid ; the distance moved is the movement of the piston, which is equal to the aug- mentation in volume of the fluid. Since work equals force multiplied by distance moved, the external work done is equal to the total uniform pressure multiplied by the increase of volume. 87. Theorem. On a PV diagram, the external work done along any path is represented by the area included be- p tween that path and the perpendiculars from its extremities to the horizontal axis. Consider first a path of constant pressure, a5, Fig. 21. From Art. 86, the external work is equivalent to the pressure multiplied by the in- FlG 21< Art 87 -~ f 1 r ^ z IJF yv 7 External Work at crease of volume, or- to ca x ab = cabd. General constant Pressure CYCLES case : let the path be arbitrary, ab, Fig. 22. Divide the area aide into an infinite number of vertical strips, amnc, mopn, oqrp, etc., each of which may be regarded as a rectangle, such that ac = mn, win = op, etc. The external work done along am, mo, oq, etc., is then repre- sented by the areas amnc, mopn, oqrp, etc., and the total external work along the path ab is repre- sented by the sum of these areas, or by aide. c L p r - FIG 22 Arts 87,t>8 Corollary L Along a path of constant volume External Work, no external work is done. y at ' Corollary II. If the path be reversed, i.e. from right to left, as along ba, the volume is diminished, and negative work is done ; work is expended on the substance in compressing it, instead of being per- formed by it. 88. Significance of Path. It is obvious, from Fig. 22, that the amount of external work done depends not only on the initial and final states a and b, but also on the nature of the path between those states. According to Joule's principle (Art. 75) the change of internal energy (T+ 1, Art. 12) between two states of a perfect gas is dependent upon the initial and final temperatures only and is independent of the path. The -external work done, however, depends upon the path. The total expenditure ofJieatj which, includes both effects, can only be known when the path is given. The internal energy of a perfect gas (and, as will presently be shown, Art. 109, of any substance) is a cardinal property; external work and heat transferred are not. They cannot be used as elements of a coordinate diagram. 89. Cycle. A series of paths forming a closed finite figure con- stitutes a cycle. In a cycle, the substance is brought back to its initial conditions of pressure, volume, and temperature. Theorem. In a cycle, the net external work done is represented on the PV diagram by the en- closed area. Let abed, Fig. 23, be any cycle. Along abc, the work done is, from Art. 87, represented by the area abcef. Along cda, the negative work done is similarly repre- J FIG. 23. Art. 89. External Work in Closed Cycle. 48 APPLIED THERMODYNAMICS sented by the area adcef. The net positive work done is equivalent to the difference of these two areas, or to abed. If the volume units are in cubic feet, and the pressure units ^Q pounds per square foot, then the measured area abed gives the work in foot-pounds. This principle underlies the calculation of the horse power of an engine from its indicator diagram. If the cycle be worked in a negative direction, e.g. as cbad, Fig. 23, then the net work will be negative ; i.e. work will have been expended upon the substance, adding heat to it, as in an air compressor. 90. Theorem. la a perfect gas cycle, the expenditure of heat is equivalent to the external work done. Since the substance has been brought back to its initial tempera- ture, and since the internal energy depends solely upon the tempera- ture, the only 'heat effect- is the external work. In the equation #= 2 7 + J-h W, F+I= 0, whence H= W, the expenditure of heat being equivalent to its sole effect. If the work is measured in foot-pounds, the heat expended is calcu- lated by dividing by 778. (See Note 3, page 37.) Conversely, in a reversed cycle, the expenditure of external work is equivalent to the gain of heat. 91. Isothermal Expansion. The isothermal path is one of much importance in establishing fundamental principles. By definition (Art. 83) it is that path along which the temperature of the fluid is constant. For gases, therefore, from the characteristic equation, if T be made constant, the isothermal equation is p v = RT - 0. Taking R at 53.36 and 2* at 491.4 (32 F.), (7-53.36X491.4 = 26,221; whence we plot on Fig. 2 the isothermal curve al> for this tempera- ture; an equilateral hyperbola, asymptotic to the axes of P and V. An infinite number of isothermals might be plotted, depending upon the temperature assigned, as cd, ef, gh, etc. The equation of the isothermal may le regarded as a special form of the exponential equation PV n = 0^ in whieh n = 1. ISOTHERMAL EXPANSION 92. Graphical Method. For rapidly plotting curves of the form PV = C, the construction shown in Fig. 24 is useful. Knowing the three corresponding prop- erties of the gas at any given state enables us to fix one point on the curve ; thus the volume x 12.387 and the pressure 2116.8 give us the point C on the isothermal for 491.4 absolute. Through C draw CM parallel to 0V. From draw lines OD, ON, OM to meet CM. Draw CB parallel to OP. From tha points 1, 5, 6, where OD, ON, OM intersect CB, draw lines 1 2, 5 7, 6 8 parallel to 0V. From D, N, M, draw lines perpendicular to 0V. The points of intersection 2, 7, 8 are points on the required curve. Proof : draw EC, .F6, parallel to OV, and 8 A parallel to OP. In the similar tri- angles 0GB, OMA, we have 6 B : MA \\OB\OA, or 8 A : CB : : EC : FQ, whence SA xF8=CBxEC,or P 8 F 8 = P c Vc- 93. Alternative Method. In Fig. 25 let 6 be a known point on the curve. Draw aD through & and lay off DA = ab. Then A is another point on the curve. Additional points may be found by either of the constructions indicated: e.g. by -drawing dh and laying off hf=db, or by drawing BK and laying off Kf= BA. These methods are prac- tically applied in the examination of the expansion lines of steam engine indicator diagrams. FIG. 24. Ait. 92, 93. Construction of Equilateral Hyperbola. 94. Theorem: Along an isothermal path for a per- fect gas, the external work done is equivalent to the heat absorbed (Art. 78). ~KT~ i a """""d v The internal energy FIG. 25. Art. 93. Second Method for Plotting is Unchanged, as indi- Hyperbolas. cate d by Joule's law (Art. 75) ; hence the expenditure of heat is solely for the performance of external work. H=T+I+ W, l>ut 2^=0, T+I=Q, and H= W. Conversely, we have Mayer's principle, that " the work done in compressing a portion of gas at constant temperature from one volume to another is dynamically equivalent to the heat emitted hy the gas during the compression" (2). 95. Work done during Isothermal Expansion. To obtain the ex- ternal work done under any portion of the isothermal curve, Fig. 24, we must use the integral form, 50 APPLIED THERMODYNAMICS in which v, "Fare the initial and final volumes. But, from the equa- tion of the curve, pv = P V, P = pv -f- V, and when p and v are given, XV fiy JT- Y p V V V JL The heat absorbed is equal to this value divided by 778. 96. Perfect Gas Isodynamic (Art. 87). Since in a perfect gas the internal energy is fixed by the temperature alone, the internal energy along an isothermal is constant, and the isodynamic and isothermal paths coincide. 97. Expansion in General. We may for the present limit the consideration of possible paths to those in which increases of volume are accompanied by more or less marked decreases in pressure ; the latter ranging, say, from zero to infinity in rate. If the volume in- CO.ST.NT PRESSURE n ~ o , creases without any fall in pressure, the path is one of constant pressure ; if the volume increases only when the fall of pressure is infinite, the path is one of con- stant volume. The paths under considera- tion will usually fall between these two, FIG. 26. Art. 97. -Expansive like 5, aw, ad, etc., Fig. 2<3. The general Paths - law for all of these paths is PV n s\> con- stant, in which the slope is determined by the value of the exponent n (Art. 91). Foi"M=0, the path is one of constant pressure, ae, Fig. 2G. For 7i= infinity, the path is one of constant volume.* The "steepness" of the path increases with the value of n. (Note that the exponent n applies to V only, not to the whole expression.) 98. Work done by Expansion. For this general case, the external work area, adopting the notation of Art. 95, is, 5 8 But since pv n = PF", P = pu n F'*; whence, when p and v are given, _ I n\ J n I n1 J. -L * F n =l, where w=0. If rt oc, we may write PccF=pa > y, or F=i;. THE ADIABATIC 51 When F= infinity, P = 0, and the work is indeterminate by this expression; but we may write W = -^~ (l - } = -PL. [~l _ f-HV" 1 ], in which, for V= in- nI \ pv J n 1 L \VI J finity, W pv (n 1), a finite quantity, The work undev an exponential curve (when n>l) is thus finite and commensurable, no matter how far the expansion be continued. 99. Relations of Properties. For a perfect gas, in which - = H, we have PVt= pvT. If expansion proceeds according to the law P V n = pu n t we obtain, dividing the first of these equations by the second, Z^ V n v n This result permits of the computation of the change in temperature following a given expansion. We may similarly derive a relation between temperature and pressure. Since pv n = PV n , v(p*) n V(P) n . Dividing the expression pv T = PVt by this, we have L ^1 /pN n _//-Z>N ? whence - = [ - By interpretation of these formulas of relation, we observe that for values of n exceeding unity, during expansion (i.e. increase of volume), the pressure and temperature decrease, while external work is done. The gain or loss of heat we cannot yet determine. On the other hand, during compression, the volume decreases, the pressure and temperature increase, and work is spent upon the gas. In the work expression of Art. 98, if p, v } t are always understood to denote the initial conditions, and P, V, T 9 the final conditions, then the work quantity for a compression is negative. 100. Adiabatic Process. This term (Art. 83) is applied to any process conducted without the reception or rejection of heat from or to surrounding bodies by the substance under consideration. It is by far the most important mode of expansion which we shall have to consider. The substance expands without giving heat to, or taking heat from, other bodies. It may Iqse heat, by doing work; or, in com- pression, work may be expended on the substance so as to cause it to gain heat : but there is no transfer of heat between it and surrounding bodies. If air could be worked in a perfectly non-conducting cylinder, we should have a practical instance of adiabatic expansion. In practice we sometimes approach the adiabatic path closely, by causing expansion to take place with great rapidity, so that there is no time 52 APPLIED THERMODYNAMICS for the transfer of heat. The expansions and compressions of the air which occur in sound waves are adiabatic, on account of their rapidity (Art. 105). In the fundamental equation H= T+ 1+ TF, the adi- abatic process makes JI= 0, whence W= (7+ J) ; or, the external work done is equivalent to the loss of internal energy, at the expense of which energy the work is performed. 101. Adiabatic Equation. Let unit quantity of gas expand adiabatically to an infinitesimal extent, iucreasing its volume by dv, and decreasing its pressure and temperature by dp and dt. As has just been shown, TF (^4- 1), the expression in the parenthesis denoting the change in internal energy during expansion. The heat necessary to produce this change would be Idt, I being the specific heat at constant volume. The ex- ternal work done is W=pdo\ consequently, pdv = Idt. Prom the equation of the gas, pv = Rt, t =^ 9 whence, dt = -=(pdv 4- vdp). Using this value for dt, M H pdv = -- (pdv + vdp). IL But It is equal to the difference of the specific heats, or to & Z; so that pdv = - (jpdv + vdp), K t ypdv pdv = pdv vdp, = -- E 9 giving by integration, v p ylog e v + log e p = constant, or pv y = constant, y being the ratio of the specific heats at constant pressure and con- stant volume (Art, 69.) 102. Second Derivation. A simpler, though less satisfactory, mode of derivation of the adiabatic equation is adopted by some writers. Assum- ing that the adiabatic is a special case of expansion according to the law PV n , the external work done, according to Art. 98, is E(t - T) ADIABATIC EXPANSION 53 During a change of temperature from t to T, the change in internal energy is l(t T) } or from Art. 70, since I = R -t-(y 1), it is Jffi - T) y-i But in adiabatic expansion, f/te external icork done is equivalent to the change in internal energy ; consequently n y 1 rc = 2/, and the adiabatic equation ispu v = PFX For air, the adiabatic is then represented by the expression ^(V) 1 ' 402 = a constant. 103. Graphical Presentation. Since along an adiabatic the external work is done at the expense of the internal energy, the temperature must fall during expansion. In the diagram of Fig. 19, this is shown by com- paring the line ab, an isothermal, with ae, an adiabatic. The relation of p to v, in adiabatic expansion, is such as to cause the temperature to fall. The projections of these two paths on the pv plane show that as expansion proceeds from a, the pressure falls more rapidly along the adiabatic than along the isothermal, a result which might have been anticipated from comparison of the equations of the two paths. If an isothermal and an adiabatic be drawn through the same point, the latter will be the "steeper" of the two curves. Any number of adiabatics may be constructed on the pv diagram, depending upon the value assigned to the constant (ptf) ; but since this value is determined, for any particular perfect gas, by contemporaneous values of p and v, only one adiabatic can be drawn for a given gas through a given point. 104. Relations of Properties. By the methods of Art. 98 and Art. 99, \ve find, for adiabatic changes, During expansion, the pressure and temperature decrease, external work is done at the expense of the internal energy, and there is no reception or rejection of heat. 105. Direct Calculation of ^the Value of y. The velocity of a wave in an astic medium is, according to a fundamental proposition in dynamics, equal to the square root of the coefficient of elasticity divided by the mass density:* that is, w * See, for example, Appendix A to Vol. HI of Nichols and Franklin's Elements of Physics. 54 APPLIED THERMODYNAMICS V being in feet per second and w in Ibs. per cubic foot. When a volume of gas of cross-section =n and length I is subjected to the specific pressure increment dp, producing the extension (negative compression) dl, dp e- dl+V The volume of this gas is In -v: so that -y = and e = -f-* The pulsations which constitute a sound wave are very rapid, hence adiabatic, so that pvv = constant, and ypvi - l dv = v * For 32 F. and p = 14. 697X144, w?=0.081. Taking ^ at 32.19 and V at the experimental value of 1089, 1089X1089X0081 2/ "32.19X14.697Xl44 105 a. Velocity with Extreme Pressure Changes. The preceding computation applies to the propagation of a pressure wave of very small intensity from a local- ized starting point. Where the pressure rises considerably say from JP to P, the volume meanwhile decreasing from v to Fo, then Now F (velocity) = V^~ and e = - ^ y ^ for finite changes. If v is the vol- ume of W Ib. of gas (not the specific volume), pv = Jft PF, PF = ^, and we have for the velocity, F = ADIABATIC EXPANSION For t = 530, p = 100, P = 400, this becomes 55 F = 32.2 x 53.36 x 530 x 800 x 144 =v^ (39,200,000,000 9060 = 2078 ft. per second. This would he the velocity of the explosion in the cylinder of an internal com- hustion engine if the pressure were generated at all points simultaneously. As a matter of fact, the combustion is local and the velocity and pressure rise are much less than those thus computed (Art. 319). 106. Representation of Heat Absorbed. Theorem: The heat ab- sorbed on any path is represented on the PV diagram by the area en- closed between that path and the two adiabatics through its extremities, indefinitely prolonged to the right. Let the path be ab, Fig. 27. Draw the adiabatics an, IN. These may be conceived to meet at an infinite dis- J> tance to the right, forming with the path the closed cycle abNn. In such closed cycle, the total expenditure of heat is, from Art. V N 90, represented by the enclosed area ; but _ v since no heat is absorbed or emitted along FIG. 27 Arts. 106, 109. Rep- the adiabatics, all of the heat changes in the resentation of Heat Ab- cycle must ] lave occurred along the path ab, sorbed. J D and this change of heat is represented by the area abNn. If the path be taken in the reverse direction, i.e. from b to a, the area abNn measures the heat emitted. 107. Representations of Thermal Capacities. Let ab, cd, Pig. 28, be two isothermals, differing by one degree. Then efnN represents the specific heat at constant volume, egmN the specific heat at constant pressure, eN, fn } and gm being adiabatics. The latter is apparently the greater, as it should be. Similarly, if ab denotes unit increase of volume, the area abMN represents the latent heat of expansion. The other thermal capacities men- tioned in Art. 58 may be similarly represented. ** FIG. 28. Art. 107, Thermal Capacities. 56 APPLIED THERMODYNAMICS 108. Isodiabatics. An infinite number of expansion paths is possible through the same point, if the n values arc different. An infinite number of curves may be dra\vn, having the name n value, if they do not at any of their points intersect. Through a given point and with a given value of n, only one curve can be drawn. When two or more curves appear on the same diagram, each having the same exponent (n ^87- ' (i) FIG. 29. Art. 108. Isodiabatics. (c) value), such curves are called isodiabatics. In most problems relating to heat motors, curves appear in isodiabatic pairs. Much labor may be saved in computation by carefully noting the following relations: 1. In Fig. 29 (a), let the isodiabatics pv Ml =const. be intersected by lines of constant pressure at a, b, c and 4 d. Then 711-1 ni-l rn m l b (Art. 99). n\ ; 2. In Kg. 29 (6), let the same isodiabatics be intersected by lines of constant volume, determining points a, b, c and d. Then "T c) (Art. 99). '14; ISODIABATICS 57 3. In Fig. 29 (c), the same isodiabatics are intersected by isothermals at a, b, c and d. Now (Art. 99). &T nr =Y W * c J = p^ and -p- a = -p-- (I) In this case, it is easy to show also, tnat V a Va .__, V a V (ID d K c but in this case (I) is not equal to (II) : the volume ratio is not equal to the pressure ratio. Note also that in each of the three cases the equality of ratios exists between properties other than that made constant along the intersecting lines; thus, in (a), the pressure is constant, and the volume and temperature ratio is constant. 109. Joule's Law. From the theorem of Art. 106, Rankine has illustrated in a very simple manner the principle of Joule, that the change of internal energy along any path of any substance depends upon the initial and final states alone, and not upon the nature of the path. In Fig, 27, draw the vertical lines ax, by. The total heat absorbed along ab = nabNj the external work done = xaby. The difference = nabN xaby = nzbN xazy, is the change in internal energy; H = T + I + W, whence H-W*=(T+r)} and the extent of these areas is unaffected by any change in the path ab f so long as the points a and b remain fixed. 58 APPLIED THERMODYNAMICS 110. Value of y. A method of computing the value of y for air has been given in Art. 105. The apparatus shown in Pig. 30 has been used by several observers to obtain direct values for various gases. The vessel was filled with gas at P, F, and T, T being the temperature of the atmos- phere, and P a pressure somewhat in excess of that r tQh-* of the atmosphere. I>y opening the stopcock, a sudden expansion took place, the pressure falling to that of the atmosphere, and the temperature falling to a point considerably below that of the atmosphere. Let the state of the gas after this adiabatic expansion be p, v, t. Then, since y = 7j?' i T -r FIG. 30. Art, 110, -De- 2 log;?- log P, sormes' Apparatus. log F log V After this operation, the stopcock is closed, and the gas remaining in the vessel is allowed to return to its initial condition of temperature, T. During this operation, the volume remains constant; so that the final state is pa % T\ whence p z v = PF, or log F log v = logjp a log P. Sub- stituting this value of log F log v in the expression for y, we have J io g y? 2 ~iogp' so that the value of y may be computed from tlie pressure changes alone. Clement and Desormes obtained in this manner for air, y = 1.3524 ; G-ay- Lussac and Wilter found ?/ = 1 .3745. The experiments of Hirn, Weisbach, Masson, Cazin, and Kohlrausch were conducted in the same manner. The .method is not sufficiently exact. 11L Expansions in General. In adiabatic expansion, the external work done and the change in internal energy are equally represented by the expression P v ~~ 9 derived as in Art. 98. For expansion from p, v to infinite volume, this becomes ' _.- The external work done during any " * rt-rr expansion according to the law pv n = PF n from pv to PF, is ' The stock of internal energy at p, v, is -- = It ; at P, F, it is -- = IT. V 1 y 1 The total heat expended during expansion is equal to the algebraic sum of the external work done and the internal energy gained. Then, * The final condition being that of the atmosphere, all of the gas, both within and without the vessel, is at the condition p, -y, t. The change in quantity (weight) of gas in the vessel during the expansion does not, therefore, invalidate the equation. POLYTROPIC PATHS 59 = Z( y)[ <y ~? )> i n which Z is the initial, and T the final temperature. \n ly This gives a measure of the net heat absorbed or emitted during any ex- pansion or compression according to the law po n = constant. When n exceeds y, the sign of II is minus ; heat is emitted ; when n is less than y but greater than 1.0, heat is absorbed : the temperature falling in both cases. When ^=y, the path is adiabtitic, and heat is neither absorbed nor emitted. 112. Specific Heat. Since for any change of temperature involving a heat absorption H 3 the mean specific heat is * = T^? we derive from the last equation of Art. Ill the expression, ,!=, 711 giving the specific heat along any path pv n = PV n . Since the values of n are the same for isodidbatics, the specific heats along such paths are equal (Art. 108). 113. Ratio of Internal Energy Change to External Work. For any given value of n, this ratio has the constant value n-1 y-i' 114. Polytropic Paths. A name is needed for that class of paths following the general law pv PP 1 , a constant. Since for any gas y and I are constant, and since for any particular one of these paths n is constant, the final formula of Art. Ill reduces to In other words, the rate of heat absorption or emission is directly pro- portional to the temperature change; the specific heat is constant. Such paths are called polytropic. A large proportion of the paths exempli- fied in engineering problems may be treated as polytropics. The polytropic curve is the characteristic expansive path for constant weight of fluid. 60 APPLIED THERMODYNAMICS 115. Relations of n and 5. We have discussed such paths in which the value of n ranges from 1.0 to infinity. Figure 31 will make the concep- tion ruore general. Let a represent the initial condition of the gas. If FIG. 31. Art. 115. Poly tropic Paths. it expands along the isothermal a& 5 n = 1, and s s the specific heat, is infi- nite ; no addition of heat whatever can change the temperature. If it expands at constant pressure, along ae, n = Q, and the specific heat is finite and equal to ly = k. If the path is ag, at constant volume, n is infinite and the specific heat is positive, finite, and equal to ?. Along the isother- mal of (compression), the value of n is 1, and s is again infinite. Along the adiabatic ah, n = 1.402 and s = 0. Along ai, n = and $ = k. Along ad, n is infinite and s = L Most of these relations are directly derived from Art. 112, or may in some cases be even more readily apprehended by drawing the adiabatics, en, gN", fm, iM, dp, bP, and noting the signs of the areas representing heats absorbed or emitted with changes in temperature. Tor any path lying between ah and af or between etc and a&, the specific heat is negative, i.e. the addition of heat cannot keep the temperature from fall- ing: nor its abstraction from rising. 116. Relations of Curves : Graphical Representation of n. Any number of curves may be drawn, following the law pv n = C, as the value of C is changed. RELATIONS OF n AND 61 In Fig. 32, let a&, ctf, e/be curves thus drawn. Their general equation is pv n = C, whence = or civ v If M TV is the angle made by the tangent to one of the curves with the axis 0V, and MOV the angle formed by the radius vector RM with the axis 0V, then, since dp dv is the tan- gent of MTV, andjp v is the tangent of MOV, FIG. 32. Art. 116. Determination of Exponent. - tan MTV = n tan MO V. If the radius vector be produced as ItMNQ the relations of the angles made be- tween the OF axis and the successive tangents MT, NS, QU, are to the angle MOV as just given; hence the various tangents are parallel (4) . Since tan MTV = Mg ^ gT and tan3/OF = Mg -r Of/, the preceding equation gives whence n = Og -=- gT. (The algebraic signs of 0(j and ^T, measured from g, are different.) In order to determine the value of n from a given curve, we need therefore only draw a tangent MT and a radius vector J/0, whence by drop- ping the perpendicular Mg the relation Og gT is established. If we lay oif from the distance OA as a unit of length, drawing A C parallel to the tangent, and CB through C, parallel to the c FIG. 33. Art 116. Negative Exponent. radius vector, then by similar triangles OgigTiiOBiOA and Og -r gT = 05= n. Figure 33 illustrates the generality of this method by showing its application to a curve in which the value of n is negative. 117. Plotting of Curves: Brauer's Method. The following is a simple method for the plotting of exponential curves, in- cluding the adiabatic, which is ordinarily a tedious process. Let the point Af, Fig. 34, be given as one point on the re- quired curve. Draw a line OA making an angle VOA with the axis OF, and a line OB making an angle POB with the axis FIG, 34, Art. 117. Brauer's Method. 62 APPLIED THERMODYNAMICS OP. Draw the vertical line MS and the horizontal line MT. Also draw the line TU making an angle of 45 with OP, and the line SJR making an angle of 45 with MS. Draw the vertical line EN through 11, and the horizontal line UN through U. The coordinates of the point of intersection, JV, ot these lines, are OR and RN. Let the coordinates of Jl/, TM (= OQ), and MQ be designated by v, p ; and those of N, OR, and RN (= 0L), by F, P. Then tan J r 0. 1 = QS - OQ, = Q,R^TM = (V-v)-v', and tan 7>0JB = UL - = 7X- NR = (p-P) -P; whence 7= v (tan FO4 -f 1) and jt? = P (tan PO5 H- 1). If the law of the curve through M and N is to be^y n = PF n , we obtain P(tanP05 + !>= 7>{i'(fcan F6L4 + 1)3% whence (tan POE 4- 1) - (tan FCU + !)" If now, in the first place, we make the angles POB, VGA such as to fulfill this condition, then the point N and others similarly determined will be points on a curve following the law pv n = PF n . 118. Tabular Method. The equation pv n = P V n may be written p = P( j or logjt? log P = n log (F i). Tf we express P as a definite initial pressure for all P V n curves, then for a specific value of n and for definite ratios F v we may tabulate successive values of log p and of p. Such tables for various values of n are commonly used. In employing them, the final pressure ia found in terms of the initial pressure for various ratios of final to initial volume. 119. Representation of Internal Energy. In Fig. 35, let An represent an adiabatiu. Daring expansion from A to a, the external work done is Aabc, which, from the law of the adiabatic, is equal to the expenditure of internal energy. If expansion is continued indefinitely, the adiabatio An gradually approaches the axis OF, the area below it continually representing expenditure of internal energy, until with infinite expansion An and OF coincide. The internal energy is then ex- 35. Art. no. Repro- h^usted. The total internal energy of a substance sentation of Internal may therefore be represented by the area between Ener gy- the adiabatic through its state, indefinitely prolonged to the right, and the horizontal axis. Representing this quantity by JB ; then from Art. Ill, where v is the initial volume, p the initial pressure, and y the adiabatio exponent. This is a finite and commensurable quantity. 120. Representation by Isodynamic Lines. A defect of the preceding representation is that the areas cannot be included on a finite diagram. GRAPHICAL REPRESENTATIONS 63 In Fig. 36, consider the path. AB. Let BG be an adiabatic and AC an- isodynamic. It is required to find the change of internal energy between A and B. The external work done daring adi- abatic expansion from B to G is equal to BCcb ; and this is equal to the change of internal en- ergy between B and 0. But the internal energy is the same at G as at A, because AC is an isodynamic. Consequently, the change of in- ternal energy between A and B is represented by the area BCcb; or, generally, by the area included between the adiabatic through the final state, extended to its intersection witli the iso- dynamic through the initial state, and the hori- zontal axis. FIG. 36. Arts. 120, 121. In- ternal Energy, Second Dia- gram. 181. FIG. 37 Art 121. External Work and Internal Energy. Source of External Work, If in Fig. 36 the path is such as to increase the temperature of the substance, or even to keep its temperature from decreasing as much as it would along an adiabatio, then heat must be absorber! . Thus, comparing the paths ad and ac, Fig. 37, aN and cm being adiabatics, the external work done along ad is adef, no heat is absorbed, and the internal energy decreases by adef. Along ac, the external work done is acef, of which arfe/was done at the ex- pense of the internal energy, and acd by reason of the heat absorbed. The total heat absorbed was Ncicm, of which acd was expended in doing external work, while Ndcm went to increase the stock of internal energy. 122. Application to Isothermal Expansion. If the path is isothermal, Fig. 38, line A B, then if BN t An are adiabatics, we have, W + X = external work done, X 4- Y = heat absorbed = W + X, W -f Z = internal energy at A, Y 4- Z = internal energy at B, W = work done at the expense of the in- ternal energy present at A, X = work done by reason of the absorption of heat along AB, Z = residual internal energy of that originally present at A , Y = additional internal energy imparted by the heat absorbed; and since in a perfect gas isothermals are isodynamics, we note that FIG. 38, Art. 122. Heat and Work in Isothermal Expansion, 64 APPLIED THERMODYNAMICS 123. Finite Area representing Heat Expenditure. In Fig. 39, let ab be any path, In and aN adiabatics, and nc an isodynamic. The exteinal work done along ab is abtle't while the increase of internal energy is befit. The total heat absorbed is then represented by the combined areas abcfe. If the path ab is iso- thermal, tins construction leads to the known result that there is no gain of internal energy, and that th? total heat absorbed equals the external work. If the path be one of those de- scribed in Art. 115 as of negative specific heat, \ye may represent il as ag, Fig. 40. Let Igm be an adiabatic. The external "S. FIG. 39. Art 123 Represen- tation of Heat Absorbed. m -V FIG. 40. Art. 13.1. Nega- tive Specific Heat, work done is ac/dc. The change of internal energy, from Art. 120, is bydf, if ab is an isodynamic; and this being a negative area, we note that internal en- ergy has been expended, although heat has been ab- sorbed. Consequently, the temperature has fallen. It seems absurd to conceive of a substance as receiving heat while falling in tem- perature. The explanation is that it is cooling, "by doing external work, faster than the supply of heat can warm it. Thus, H- T+ /+ W', but //< W\ con- sequently, (T 4- 7) is negative. 123 a. Ordnance. Some such equation as that given in Art. 105 a may apply to the explosion of the charge in a gun. Ordinary gunpowder, unlike various de- tonating compounds now used, is scarcely a true explosive. It is merely a rapidly burning mixture. A probable expression for the reaction with a common type of powder is 4 KN0 3 + C 4 + S = K 2 C0 3 + K 3 S0 4 + N 4 + 2 CO 2 + CO. It will be noted that a largo proportion of the products of combustion arc solids; probably, in usual practice, from 55 to 70 per cent. As first formed, these may be in the liquid or gaseous state, in which case they contribute large quantities of heat to the expanding and cooling charge as they liquefy and solidify. When the charge is first fired, if the projectile stands still, the temperature and pressure will rise proportionately, and the rise of the former will be the quotient of the heat evolved by the mean specific heat of the productw of combustion. Fortu- nately for designers, the projectile moves at an early stage of the combustion, so that the rise of pressure and temperature is not instantaneous, and the shock is more or less gradual. After the attainment of maximum pressure, the gases expand, driving the projectile forward. Work is done in accelerating the latter, but the process is not adiabatic because of the contribution of heat by the ultimately solid combustion products. The temperature does fall, however, so that the expansion is one between the isothermal and the adiabatic. The ideal in design is to obtain the highest possible muzzle velocity, but this should be accomplished without excessive maximum pressures. The more nearly ORDNANCE 65 the condition of constant pressure can be approximated during the travel of the projectile from breech to muzzle, the better. Both velocities and pressures during this traverse have been studied experimentally; the former by the chronoscope, the latter by the crusher gage. The suddenness of pressure increase may be retarded by increasing the density of the powder, and is considerably affected by its fineness and by the shape and uniformity of the grains. Suppose 1 + s Ib. of charge to contain 5 Ib. of permanently solid matter of spe- cific heat = c, and that the specific heat of the gaseous products of combustion, during their combustion, is I. Let the initial temperature be F. Then the temperature attained by combustion is T I + cs where H is the heat evolved in combustion. During any part of the subsequent expansion, H= T + I + W=E + W, dH = Idt -f pdv. The only heat contributed is that by the solid residue, and is equal to dH- - scdt = Idt +pdv, so that - (sc + dt = pdv = Rt~, and between the limits T and t, where V is the initial and v the final volume of the charge. Now since p - = --, i-Mo * T = = f J-Y* st< The external work done during expansion is W = (pdv = - ldt -scdt = (sc + (T - t) If we wish to include the effect due to the fact that a portion, say r, of the original volume of charge forms non-gaseous products, we may write or V, F(l r) ? and for v, v rF, and the complete equation becomes Suppose r = 4000 F., s = 0.6, c = 0.1, I = 0.18, fc = 0.25, 7=0.02, r = 0.6, v = 0.20; then 66 APPLIED THERMODYNAMICS 007 W= 0.24 x 4000 x 778 { 1 - (j^) "} = 1,000 ft-lb. If w be the weigh b of projectile, F the velocity imparted thereto, and / the " factor of effect " to care for practical deductions from the computed value of W, then = -V* and which for our conditions, with w - 5, /= 0.90, gives = IG4.4 x 491,000 x 0.90 _ * 5 The maximum work possible would be obtained in a gun of ample length, the products of combustion expanding down to their initial temperature, and would be, for our conditions, W= 778 H = 778 T(l + cs) = 778 x 4000 x 0.24 = 814,080 ft.-lb. The equation of the expansion curve is pv n = const., where n has the value + sg ; or, for our conditions " =13. nearly. / H- sc 0.24 Viewing the matter in another way : the heat contributed by the solid residues is that absorbed by the gases ; or ! = -*!, where $1 is the specific heat of the gases during expansion. Then s, = I n ~~ V and n = + Jtc , as before. 1 n - 1 1 + sc The external work done during expansion is from which the equation already given may be derived. MODIFICATIONS IN IRREVERSIBLE PROCESSES 124. Constrained and Free Expansion. In Art. 86 it was assumed that the path of the substance was one involving changes of volume against a resistance. Such changes constitute constrained expansion. In this pre- liminary analysis, they are assumed to take place slowly, so that no mechanical work is done by reason of the velocity with which they are effected. When a substance expands against no resistance, as in Joule's experiment, or against a comparatively slight resistance, we have what is known as free expansion, and the external work is wholly or partly due to velocity changes. IRREVERSIBLE PROCESSES 67 125, Reversibility. All of the polytropic curves which have thus far been discussed exemplify constrained expansion. The external and in- ternal pressures at any state, as in Art. 86, differ to an infinitesimal extent only ; the quantities are therefore in finite terms equal, and the processes may be worked at icill in either direction. A polytropic path having a finite exponent is in general, then, reversible, a characteristic of fundamental importance. During the adiabatic process which occurred in Joule's experiment, the externally resisting pressure was zero while the internal pressure of the gas was finite. The process could not be reversed, for it would be impossible for the gas to flow against a pressure greater than its own. The generation of heat by friction, the absorption of heat by one body from another, etc., are more familiar instances of irreversible process. Since these actions take place to a greater or less extent in all actual thermal phenomena, it is impossible for any actual process to be perfectly reversible. "A process affecting two substances is reversible only when the conditions existing at the commencement of the process may be directly restored without compensating changes in other bodies." 126. Irreversible Expansion. In Fig. 41, let the substance expand unconstrainedly, as in Joule's experiment, from a to &, this expansion being produced by the sudden decrease in ex- p ternal pressure when the stopcock is opened. Along the path ab, there is a violent movement of the particles of gas ; the kinetic energy thus evolved is transformed into pressure at the end of the expansion, causing a rise of pressure to c. The gain or loss of internal energy depends solely upon the states a, c; the external work done does FIG. 41. Art, uo. irre- not depend on the irreversible path ab, for with versible Path, a zero resisting pressure no external work is done. The theorem of Art. 86 is true only for reversible operations. 127. Irreversible Adiabatic Process, Careful consideration should be given to unconstrained adiabatic processes like those exemplified in Joule's experiment. In that instance, the temperature of the gas was kept up by the transformation back to heat of the velocity energy of the rapidly moving particles, through the medium of friction. We have here a special case of heat absorption. No heat was received from without ; the gas remained in a heat-insulated condition. While the process conforms to the adiabatic definition (Art. 83), it involves an action not contemplated when that definition was framed, viz , a reception of heat, not from, sur- 68 APPLIED THERMODYNAMICS rounding bodies, but from the mechanical action of the substance itself* The fundamental formula of Art. 12 thus becomes jy= r+ /+ w+ F, in which V may denote a mechanical effect due to the velocity of the particles of the substance. This subject will be encountered later in important applications (Arts. 175, 176, 426, 513). FUKTHEB APPLICATION OP THE KINETIC THEORY 127 a. The Two Specific Heats. The equation has been derived (Art, 55), *-?,*-"+ in which p = the specific pressure exerted by a gas on its bounding surfaces ; v = the aggregate volume (not the specific volume) of the gas, W = the weight of the gas, whence r = its specific volume, M its mass, w = the average velocity of all of the molecules of the gas. The kinetic theory asserts that the absolute temperature is proportional to the mean kinetic energy per molecule. In a gas without intermolecular attractions the application of heat at unchanged volume can only add to the kinetic energy of molecular vibration. In passing between the temperatures ^ and t 9 then, the ex- penditure of heat may be written M, 2N , A ^ 2 If the operation is performed at constant pressure instead of constant volume the expenditure of heat will be greater, by the amount of heat consumed in per- forming external work, jo(u 2 1^). From Charles' law, The external work is then and the total heat expended is H k = A + B = (u>** - w^). (C) If we divide C by JL, we obtain M 10 FURTHER APPLICATION OF THE KINETIC THEORY 69 which would be the ratio of the specific heats for a perfect monatomic gas. In such a gas, the molecules are relatively far apart, and move in straight lines. In a polyatomic gas (in which each molecule consists of more than one atom), there are interattractions and repulsions among the atoms which make up the molecule. Clausius has shown that the ratio of the intramolecular to the " straight line" or translational energy is constant for a given gas. If we call this ratio m, then for the polyatomic gas H k = 3/l +m 3 + 3m If m = 0, this becomes J, as for the monatomic gas. The equation gives also, m = 5 ""J/g- For oxygen, with y = 1.4, m = 5 ~ ^ = = 0.667. 127 &. Some Applications. Writing the first equation given in the i'brm. we have foi 1 Ib. of air at standard conditions w a = V3 x 53.36 x 492 x 8J.2 ^ 1593 ft. per second, the velocity of the air molecule. Noting also that w = (/) \/tJ under standard con- ditions, we obtain for hydrogen w h = 1593-Ji^ii = 6270 ft. per second. * lli.OOY These are mean velocities. Some of the molecules are moving more rapidly, some more slowly. The molecular velocities of course increase with the temperature and are higher for the lighter gases. A mixture of gases inclosed in a vessel containing an orifice, or in a porous container, will lose its lighter constituents first ; because, since their molecular velocities are higher, their molecules will have briefer periods of oscillation from side to side of the containing vessel and will more frequently strike the pores or orifices and escape. This principle explains the com- mercial separation of mixed gases by the pi ocess of osmosis. In any actual (polyatomic) gas, the molecules move in paths of constantly changing direction, and consequently do not travel far. The diffusion or perfect mixture of two or more gases brought together is therefore not an instantaneous process. High temperatures expedite it, and it is relatively more rapid with the lighter gases. We may assume that intramolecular energy is related to a rotation of atoms about some common center of attraction. The intramolecular energy has been shown to be proportional to the temperature. A temperature may be reached at which the total energy of an atomic system may be so greatly increased that the 70 APPLIED THERMODYNAMICS system itself will be broken up, atoms flying off perhaps to form new bonds, new molecules, new substances. This breaking up of molecules is called dissociation. In forming new atomic bonds, heat may be generated ; and when this generation of heat occurs with sufficient rapidity, the process becomes self-sustaining ; i.e. the temperature will be kept up to the dissociation point without any supply of heat from extraneous sources. If, as in many cases, the generation of heat is less rapid than this, dissociation of the atoms will cease after the external source of heat has been lemoved. According to a theorem in analytical mechanics,* there is an initial velocity, easily computed, at which any body projected directly upward will escape from the sphei e of gravitational attraction and never descend. For earth conditions, this velocity is, irrespective of the weight of the body, 6.95 miles per second .-= 36,650 ft. per second, ignoring atmospheric resistance. Now there is little doubt that some of the molecules of the lighter gases move at speeds exceeding this ; so that it is quite possible that these lighter gases may be gradually escaping from our planet. On a small asteroid, where the gravitational atti action was less, much lower velocities would suffice to liberate the molecules, and on some of these bodies there could be no atmosphere, because the velocity at which liberation occurs is less than the normal velocities of the nitrogen and oxygen molecules. (1) Thermodynamics, 1907, p. 18. (2) Alexander, Treatise on Thermodynamics, 1893, p 105. (3) Wormell, Thermodynamics, 123, Alexander, Thermodynamics, 103; Rankine, The Steam Engine, 249, 321; Wood, Thermodynamics, 71-77, 437. (4) Zeuner, Technical Thermodynamics, Klein tr., I, 156. (5) Ripper, Steam Engine Theory and Practice, 1895, 17. SYNOPSIS OF CHAPTER V Pressure, volume and temperature as therwodynanuc coordinates. Thermal line, the locus of a series of successive states , path, a projected thermal line. Paths : isothermal, constant temperature ; wodynamic, constant internal energy ; adiabatic, no transfer of heat to or from surrounding bodies. The geometrical representation of the characteristic equation is a surface. The PV diagram: subtended areas represent external work; a cycle is an enclosed figure ; its area represents external work ; it represents also the net expenditure of heat. The isothermal : pv n = c, in which n = 1, an equilateral hyperbola ; the external work done is equivalent to the heat absorbed, = pv log e : with a perfect gas, it coin- cides with the isodynamic. v Paths in general: pv" = c ; external work =^^T ; 1= (V~ n ; J= (\ n n\ The adiabatio ; the external work done is equivalent to the expenditure of internal energy ; pvv=c ; y = 1.402 ; computation from the velocity of sound in air ; wave velocities with extreme pressure changes. The heat absorbed along any path is represented by the area between that path and the two projected adiabatics ; representation of k and L * See, for example, Bowser's Analytic Mechanics, 1908, p. 301. SYNOPSIS 71 Isodiabatics : n^ = n^ ; equal specific heats ; equality of property ratios. Rankine's derivation of Joule's law : the change of internal energy between two states is independent of the path. Apparatus for determining the value of y from pressure changes alone. Along any path pv n = c, the heat absorbed is l(t !T)(^~ ? M ; the mean specific heat is i n ~y. Such paths are called polytropics. Values of n and s for various paths. n l Graphical method for determining the value of n ; Brauer^s method for plotting poly- tropics ; the tabular method. Graphical representations of internal energy ; representations of the sources of external work and of the effects of heat ; finite area representing heat expenditure. Poly tropic expansion in ordnance. Irreversible processes: constrained and fiee expansion ; reversibility ; no actual proc- ess is reversible , example of irreversible process ; subtended areas do not repre- sent external work , in acliabatic action, heat may be received from the mechani- cal behavior of the substance itself; H=T + I+ W+V; further applications of the kinetic theory. Use of Hyperbolic Functions : Tyler's Method. Given x m = a, let x m = e'. Then m logg x = s and x m = e mloSeX . Adopting the general forms & = cosh t + sinh t, e -t cogn i _ s i n h ^ we have x m = cosh (m log x) + sinh (m log fl or), where m log e x is positive ; x m _. cogh ( m i O g p x } _ giuh ( m i O g e a;^ where m log a x is negative. If now we have a table of the sums and differences of the hyperbolic functions, and a table of hyperbolic logarithms, we may practically without computation ob- tain the value of x m . Thus, take the expression Here x = 0.1281, m = 0.29, m log a x = - 0.596, (cosh sinh) m log e x = 0.552. The limits of value of x may be fixed, as in the preceding article, as and 1.0. For x = 0, m log e x = oc, and the method would require too extended a table of hyperbolic functions. But if we use the general form in which x > 1.0 and usually <10.0, m loge ^ will rarely exceed 10.0, and the method is practicable. For a fuller discussion, with tables, see paper by Tyler in the Polytechnic En- gineer, 1912. o . 30103 written 30103 1 30103 written 1 30103 2 30103 written 2 30103 -1 0.30103 wiitton 1 30103 72 APPLIED THERMODYNAMICS NOTES ON LOGARITHMS Definitions; log x or com log s-n, where 10 n =z. --m where e m =x, e = 27183+. = (2.3026) log x, Characteristic and Mantissa, the log consists of a characteristic, integral and either positive or negative; and a mantissa, a positive fraction or decimal Dividing or multiplying a number by 10 or any multiple thereof changes the characteristic of the log, but not its mantissa. Thus, Characteristic Mantissa log 2 log 20 log 200 log 0.2 = and equivalent to 69897 log 02 - -2 30103 written 2 30103 and equivalent to 1,69897 Operations with logarithms. log (aX&) =log a+log b. Remember also: i log (a-j-6) =log a log b. zfr^tyx. 1 log(a) n -nlogo. x n =~. Negative sign: the signs of negative characteristics must be carefully con- sidered. Thus, to find the value of 0.02~ 37 : log 0,02=2 30103 = -2.0+0 30103. -0.37 log 0,02= -0,37(-2 0+0,30103) =0,74-0.1114 = 0.6286. -log When the final logarithm comes out negative, it must be converted into loga- rithmic form (negative characteristic and positive mantissa) by adding and sub- tracting 1. Thus -0.6286 = 1,3714 -log 0.2352. For example, to find the value of 0.02 37 : log 0.02 = 2.30103= -2.0+0 30103 0.37 log 0.02 -0.37( -2.0+0 30103) = -0.74+0.1114= -0.6286-1.3714 -log (0.2352 =0.02' 37 ). PROBLEMS 73 PEOBLEMS 1. On a perfect gas diagram, the coordinates of which are internal energy and volume, construct an isodynamic, an isothermal, and an isometric path through E (internal energy) =2, F=2. 2. Plot accurately the following: on the TV diagram,* an adiabatic through T=270, F=10; an isothermal through T=300, F=20; on the TPf diagram, an adiabatic through T=230, P = 5; an isothermal through T=190, P = 30. On the JSV diagram,} show the shape of an adiabatic path through 22 = 240, F= 10. 3. Show the isometric path of a perfect gas on the PT plane ; the isopiestic, on the FT plane. 4. Sketch the TV path of wax from to 290 F., assuming the melting point to be 90, the boiling point 290, that wax expands m melting, and that its maximum density as liquid is at the melting point. 5. A cycle is bounded by two isopiestic paths through P = 110, P = 100 (pounds per square foot), and by two isometric paths through F= 20, F=10 (cubic feet). Find the heat expended by the working substance. (Ans., 0.1285 B. t. u.) 6. Air expands isothermally at 32 F. from atmospheric pressure to a pressure of 6 Ib. absolute per square inch. Find its specific volume after expansion. (Ans., 36.42 cu. ft.) 7. Given an isothermal curve and the 0V axis; find graphically the OP axis. 8. Prove the correctness of the construction described in Art. 93. 9. Find the heat absorbed during the expansion described in Problem 6. (Ans., 36.31 B.t.u.) 10. Find the specific heat for the path PF 1 - 2 = c, for air and for hydrogen. (An$., air, -0.1706; hydrogen, -2.54.) 11. Along the path PF 1 * 2 = c, find the external work done in expanding from P=1000, F=10, to F=100. Find also the heat absorbed, and the loss of internal energy, if the substance is one pound of air. Units are pounds per square foot and cubic feet. (Ans., W= 18,450 f t.-lb. ; J3T= 11,796 B. t. u. ; Jfy Jfc - 11.8 B. t. u.) 12. A perfect gas is expanded from #=400, t?=2, = 1200, to P = 60, F=220* Find the final temperature. (Ans., 19,800 aba.) 13. Along the path PF 1 - 2 = c, a gas is expanded to ten tjsnes its initial volume of 10 cubic feet per pound. The initial pressure being 1000, and the value of It 53.36, find the final pressure and temperature. (See Problem 11.) (Ans., p = 63.1 Ib. per sq. ft., t = 118.25 a"bs.) 14. Through what range of temperature will air "be heated if compressed to 10 atmospheres from normal atmospheric pressure and 70 F., following the law pi>i- 3 =c ? What will be the rise in temperature if the law is pW=c ? If it is # c ? (Ans., Cf, 371.3 ; 6, 495 ; c, 0). 5 Absolute pressures are pressures measured above a perfect vacuum. The abso- lute pressure of one standard atmosphere is 14.697 Ib. per square inch, 74 APPLIED THERMODYNAMICS 15. Find the heat imparted to one pound of this air in compressing it as described according to the lawjpw 1 3 = c, and the change of internal energy. (Ana , ZT 2 -A= -21. 6 B.t.u. , ^^ = 63.1 B.t.u.) 16. In Problem 14, after compression along the path pu 13 = c, the air is cooled at constant volume to 70 F., and then expanded along the isodiabatic path to its initial volume. Find the pressure and temperature at thu end of this expansion. (Ans., p =8.64 Ib. per sq. in., t =311 abs.) 17. The isodiabatics ob, cd, are intersected by lines of constant volume ac, &d. v *& -*-0 j * Q, *- 6 Prove = 1 and r sr- 18. In a room at normal atmospheric pressure and constant temperature, a cylinder contains air at a pressure of 1200 Ib. per square inch. The stopcock on the cylinder is suddenly opened. After the piessurc in the cylinder has fallen to that of the atmosphere, the cock is closed, and the cylinder left undisturbed for 24 hours. Compute the pressure in the cylinder at the end of this time. (Ans , 51.94 Ib. per sq, in.) 19. Find graphically the value of n for the polytropic curve ob, Fig. 41. 20. Plot by Brauer's method a curve jp / u 1 -S = 2G,200. Use a scale of 1 inch per 4 units of volume and per 80 units pressure. Begin the curve with p= 1000. 21. Supply the necessary figures in the following blank spaces, for ft = 1.8, and apply the results to check the curve obtained in Problem 20. Begin with u = 6.12, # = 1000. =2.0, 2.25, 2.50, 3.0, 4.0, 5.0, 6.0, 7.0, S.O P- P- P= 22. The velocity of sound in air being taken at 1140 ft. per second at 70 F. and normal atmospheric pressure, compute the value of y for air. (Ans., 1.4293.) 23. Compute the latent heat of expansion (Art. 58) of air from atmospheric pressure and at 32 F. (Ans., 2.615 B. t. u.) 24. Find the amount of heat converted into work in a cycle 1234, in which P 1 = P 4 = 100, 7i = 5, 1?; = 1, Pj = 30 (all in Ib. per sq. ft.), and the equations of the paths are as follows: for 41, PF = c; for 12, PF^cj'for 32, P7=c; for 43, PY 1B = c. The working substance is one pound of air. Find the temperatures at the points 1, 2, 3, 4. (Ans.,2&= 1.386 B.t.u.; ^^9.37; ^ = 1.097; T 2 = 1.097; r 4 =1.874.) 25. Find the exponent of the polytropic path, for air, along which the specific heat is k. Also that along which it is L Represent these paths, and the amounts of heat absorbed, graphically, comparing with those along which the specific heats are k and Z, and show how the diagram illustrates the meaning of negative specific heat. (Ans., f or 3 = k, n = 1. 167 ; f or s = Z, n = 1.201.) 26. A gas, while undergoing compression, has expended upon it 38,900 ft. Ib. of work, meanwhile, it loses to the atmosphere 20 B. t. XL of heat. What change occurs in its internal energy? PROBLEMS 75 27. One pound of air under a pressure of 150 Ib. per sq. in. occupies 4 cu. ft. What is its temperature? How does its internal energy compare with that at atmos- pheric pressure and 32 F, 9 28. Three cubic feet of air expand from 300 to 150 Ib. pressure per square inch, at constant temperature. Find the values of B", E and W. 29. How much work must be done to compress 1000 cu. ft. of normal air to a pres- sure of ten atmospheres, at constant temperature ? How much heat must be removed during the compression? 30. Air is compressed in a water-jacketed cylinder from 1 to 10 atmospheres; its specific volume being reduced from 13 to 2,7 cu. ft. How much work is consumed per cubic foot of the original air? 31. Let p = 200, u = 3, P=100, 7=5. Find the value of n in the expression jtt>"=P7. 32. Draw to scale the PT and TV representations of the cycle described in Prob. 24. 33. A pipe line for air shows pressures of 200 and 150 Ib. per square inch and tem- peratures of 160 and 100 F., at the inlet and outlet ends, respectively. What is the loss of internal energy of the air during transmission? If the pipe line is of uniform size, compare the velocities at its two ends. 34. If air is compressed so that #i)i-35=c, find the aonount of heat lost to the cyl- inder walls of the compressor, the temperature of the air rising 150 F. during com- pression. CHAPTER VI THE CARNOT CYCLE 128. Heat Engines. In a heat engine, work is obtained from heat energy through the medium of a gas or vapor. Of the total heat received by such fluid, a portion is lost by conduction from the walls of containing vessels, a portion is discharged to the atmosphere after the required work has been done, and a third portion disap- pears, having been converted into external mechanical work. By the first law of thermodynamics, this third portion is equivalent to the work done ; it is the only Jieat actually used. The efficiency of a heat engine is the ratio of the net heat utilized to the total quantity of heat supplied to the engine, or, of external work done to gross heat -^5 in which fi denotes the quantity of heat W absorbed; to _Z rejected by tlie engine, if radiation effects be ignored. 129. Cyclic Action. In every heat engine, the working fluid passes through a series of successive states of pressure, volume, and temperature ; and, in order that operation may be continuous, it is necessary either that the fluid work in a closed cycle which may be repeated indefinitely, or that a fresh supply of fluid be admitted to the engine to compensate for such quantity as is periodically discharged. It is convenient to regard the latter more usual ar- rangement as equivalent 'to the former, and in the first instance to study the action of a constant body of fluid, conceived to work continuously in a closed cycle. 130. Forms of Cycle. The sev- eral paths described in. Art. 83, and others less commonly considered, sug- gest various possible forms of cycle, some of which are illustrated in Fig. FIG. 42. Art. 130, Problem 2. Possible Cycles. 42. Many of these have been given names (1). The isodidbatic cycle, bounded by two isothermals and any two isodiabatica (Art. 108), may also be mentioned. 76 THE CARNOT CYCLE 77 131. Development of the Carnot Cycle. Carnot, in 1824, by describing and analyzing the action of the perfect elementary heat engine, effected one of the most important achievements of modern physical science (2) Carnot, it is true, worked with insufficient data. Being ignoiant of the fiist law of theimodynamics, and holding to the caloric theoiy, he asserted that no heat was lost during the cyclic process; but, though to this extent founded on error, his main conclusions were correct. Before his death, in 1832, Carnot was led to a more just conception of the true nature of heat; while, left as it was, his work has been the starting point for nearly all subsequent -investigations. The Cainot engine is the limit and standard for all heat engines. Clapeyron placed the arguments of Carnot in analytical and graphical form ; Clausius expressed them in terms of the mechanical theory of heat ; James Thomp- son, Rankiue, and Clerk Maxwell corrected Carnot's assumptions, redescribed the cyclic process, and redetennined the results ; and Kelvin (3) expressed them iu their final and satisfactory modern form. 132. Operation of Carnot's Cycle. Adopting Kelvin's method, the operation on the Carnot engine may be described by reference to Fig. 48. A working piston moves in the cylinder c, the walls of which are non-conduct- ing, while the head is a perfect conductor. The piston itself is FIG. 43. Arts. 132, 138. Operation of the Carnot Cycle. -, , , a non-conductor and moves without friction. The body s is an infinite source of heat (the furnace, in an actual power plant) maintained constantly at the temperature T, 110 matter how much heat is abstracted from it. At r is an infinite condenser, capable of receiving any quantity of heat whatever without undergoing any elevation of temperature above its initial temperature t. The plate f is assumed to be a per- fect non-conductor. The fluid in the cylinder is assumed to be initially at the temperature T of the source. The cylinder is placed on s. Heat is received, but the tempera- ture does not change, since both cylinder and source are at the same temperature. External work is done, as a result of the recep- tion of heat ; the piston rises. When this operation has continued for some time, the cylinder is instantaneously transferred to the non- conducting plate f. The piston is now allowed to rise from the expan- sion produced by a decrease of the internal energy of the fluid. It continues to rise until the temperature of the fluid has fallen to , 78 APPLIED THERMODYNAMICS that of the condenser, when the cylinder is instantaneously trans- ferred to r. If eat is now yiven up by the fluid to the condenser, and the piston falls ; but no change of temperature takes place. When this action is completed (the point for completion will be determined later), the cylinder is again placed on /, and the piston allowed to fall further, increasing the internal energy and temperature of the gas by compressing it. This compression is continued until the temperature of the fluid is T and the piston is again in its initial position, when the cylinder is once more placed upon s and the opera- tion may be repeated. No actual engine could be built or operated under these assumed conditions. 133. Graphical Representation. The first operation described in the preceding is expansion at constant temperature. The path of the fluid Is then an isothermal. The second operation is expansion without transfer of heat, external work being done at the expense of the internal energy; the path is consequently adiabatie. Dur- FIG. 44. Arts. 133-136, 1,38, 142. ing the third operation, we have isothermal The Carnot c y cle compression; and during the fourth, adiabatie compression. The Carnot cycle may then be represented by abed. Fig. 44. 134. Termination of Third Operation. In order that the adiabatie compression da may bring the fluid back to its initial conditions of pressure, volume, and tem- perature, the isothermal compression cd must be terminated at a suitable point d. From Art. 99, i-v lor the adiabatie da, T I V \i- ~ = ( .iJ? ) t \\ &/ T / T 7 " \ i~v and - = ( ~5 J for the adiabatie Ic \ 4-g-fc-& that is, the ratio of volumes during isothermal expansion in the first stage must be equal to the ratio of volumes during isothermal compression in the third stage, if the final adiabatie compression is to complete the cycle. (Compare Art. 108.) 135. Efficiency of Carnot Cycle. The only transfers of heat dur- ing this cycle occur along ab and cd. The heat absorbed along ab is THE CARNOT CYCLE 7 f= RT\og e & Similarly, along cd, the heat rejected r a 'a is Rt log e - The net amount of heat transformed into work is the y d difference of these two quantities ; whence the efficiency, defined in Art. 128 as the ratio of the net amount of heat utilized to the total amount of heat absorbed, is since -^ = -^, from Art. 134. e ' a 136. Second Derivation. The external work done under the two adiabaties , da is y-i y-i Deducting the negative work from the positive, the net adiabatic work is but PO.VO, = PT>VI>, from the law of the isothermal al\ similarly, P^V* = P e V c , and consequently this net work is equal to zero; and if we express efficiency by the ratio of work done to gross heat absorbed, we need consider only the work areas under the isothermal curves ab and cd, which are given by the numerator in the expression of Art. 135. The efficiency of the Carnot engine is therefore expressed by the ratio of the difference of the temperatures of source and condenser to the absolute temperature of the source. 137. Garnet's Conclusion. The computations described apply to any sub- stance in uniform thermal condition ; hence the conclusion, now universally accepted, that the motive power of heat is independent of the agents employed tc develop it ; it is determined solely by the temperatures of the bodies between which the cyclic transfers of heat occur. 138. Reversal of Cycle. The paths which constitute the Carnot cycle, Fig. 44, are polytropic and reversible (Art. 125); the cycle itself is rever- sible. Let the cylinder in Fig. 43 be first placed upon r, and the piston allowed to rise. Isothermal expansion occurs. The cylinder is trans- ferred to /and the piston caused to fall, producing adiabatic compression, The cylinder is then placed on s, the piston still falling, resulting In iso- thermal compression ; and finally onf, the piston being allowed to -rise, s as to produce adiabatic expansion. Heat has now been taken from the 80 APPLIED THERMODYNAMICS condenser and rejected to the source. The cycle followed is dcbad, Fig. 44. Work has been expended upon the fluid ; the heat delivered to the source s is made up of the heat taken Jrom the condenser r, plus the heat equivalent of the work done upon the fluid. The apparatus, instead of being a heat engine, is now a sort of heat pump, ti an sf erring heat from a cold body to one warmer than itself, by reason of the expenditure of external work. Every operation of the cycle has been reversed. The same quantity of heat originally taken from s has now been given up to it ; the quantity of heat originally imparted to r is now taken from it; and the amount of external work originally done by the fluid has now b^en expended upon it. The efficiency, based on our present definition, may exceed unity ; it is the quotient of lieat imparted to the source by work expended. Tho cylinder c must in this case be initially at the temperature t of the con- denser r. 139. Criterion of Reversibility. Of all engines working between the same limits of temperature, that which is reversible is the engine of maximum efficiency. If not, let A be a more efficient engine, and let the power which this engine develops be applied to the driving of a heat pump (Art. 138), (which is a reversible engine), and let this heat pump be used for restor- ing heat to a source s for operating engine A. Assuming that there is no friction, then engine A is to perform just a sufficient amount of work to drive the heat pump. In generating this power, engine A will consume a certain amount of heat from the source, depending u^on its efficiency. If this efficiency is greater than that of the heat pump, the latter will di$- charye more heat than the former receives (see explanation of efficiency, Art. 138) ; or will continually restore more heat to the source than engine A removes from it. This is a result contrary to all experience. It is impossible to conceive of any self-acting machine which shall continually produce heat (or any other form of energy) without a corresponding con- sumption of energy from some other source. 140. Hydraulic Analogy The absurdity may be illustrated, as by Heck (4-), by imagining a water motor to be used in driving a pump, the pump being em- ployed to deliver the water back to the upper level which supplies the motor. Obviously, the motor would be doing its best if it consumed no more water than bhe pump returned to the leservoir; no better performance can be imagined, and with actual motors and pumps this performance would never even be equaled. Assuming the pump to be equally efficient as a motor or as a pump (i.e. reversible), the motor cannot possibly be more efficient. 141. Clausius' Proof. The validity of this demonstration depends upon the 3orrectness of the assumption that perpetual motion is impossible. Since the iui- THE CARNOT CYCLE 81 possibility of perpetual motion cannot be directly demonstrated, Cflausius estab- lished the criterion of reversibility by showing that the existence of a more effi- cient engine A involved the continuous transference of heat from a cold body to one warmer than itself, without the aid of external agency : an action which is axio- matically impossible. 142. The Perfect Elementary Heat Engine. It follows from the analysis of Art. 135 that all engines working in the Carnot cycle are equally efficient ; and from Art. 139 that the Carnot engine is one of that class of engines of highest effi- ciency. The Carnot cycle is therefore described as that of the perfect elementary heat engine. It remains to be shown that among reversible engines working be- tween equal temperature limits, that of Carnot is of maximum efficiency. Con- sider the Carnot cycle abed, Fig. 44. The external work done is abed, and the efficiency, abed + nabN. For any other reversible path than &, like ae or fb, touching the same line of maximum temperature, the work area abed and the heat absorption area nabN are reduced by equal amounts. The ratio expiessing effi- ciency is then reduced by equal amounts in numeiator and denominator, and since the value of this ratio is always fractional, its value is thus always reduced. For any other reversible path than cd, like ch or gd, touching the same line of mini- mum temperature, the work area is reduced without any reduction in the gross heat area nabN. Consequently the Carnot engine is that of maximum efficiency among all conceivable engines worked between the same limits of temperature. A practical cycle of equal efficiency will, however, be considered (Art. 257). 143. Deductions. The efficiency of an actual engine can therefore never reach. 100 per cent, since this, even with the Carnot engine, would require t in. Art. 135 to be equal to absolute zero. High efficiency is con- ditioned upon a wide range of working temperatures ; and since the mini- mum temperature cannot be maintained below that of surrounding bodies, high efficiency involves practically the highest possible temperature of heat absorption. Actual heat engines do not work in the Carnot cycle; but their efficiency nevertheless depends, though less directly, on the tem- perature range. With many working substances, high temperatures are necessarily associated with high specific pressures, imposing serious con- structive difficulties. The limit of engine efficiency is thus fixed by the possibilities of mechanical construction. Further, an ordinary steam boiler furnace may develop a maximum temperature, during combustion, of 3000 F. If the lowest available OQQQ temperature surrounding is F., the potential efficiency is =0.87. But in getting the heat from the hot gases to the steam the temperature usually falls to about 350 F. Although 70 or 80 per cent of the energy originally in the fuel may be present in the steam, the availability of this energy for doing work in an engine has now been 82 APPLIED THERMODYNAMICS Off /"i decreased to ^ ft 4fi =0.43, or about one-half. (A boiler is of course not a heat engine.) (1) Alexander, Treatise on Thermodynamics, 1893, 38-40. (2) Garnet's Reflec- tions is available in Thurston's translation or in Magie's Second Law of TJiermody- namics. An estimate of his part in tlie development of physical science is given by Tait, Thermodynamics, 18(18, 44. (3) Trans. Roy. Soc. Edinburgh, March, 1851 ; Phil. Mag., IV, 1852 ; Math, and Phys. Papers, I, 174. (4) The Steam Engine, I, 50. SYNOPSIS OF CHAPTER VI Heat engines efficiency = heat utilized - heat absorbed = "I * = Cyclic action . closed cycle , forms of cycle. Carnot cycle: historical development; cylinder, source, insulating plate, condenser graphical representation; termination of third operation, when - = J-5; ^jl- rp j. YC rk ciency -=-^ Carnot's conclusion : efficiency is independent of the working substance. Reversal of cycle: the reversible engine is that of maximum efficiency; hydraulic Carnot cycle not surpassed in efficiency by any reversible or irreversible cycle. Limitations of efficiency in actual heat engines. PROBLEMS 1. Show how to express the efficiency of any heat-engine cycle as the quotient of two areas on the PV diagram. 2. Draw and explain six forms of cycle not shown in Fig. 42. 3. In a Carnot cycle, using air, the initial state is P= 1000, F= 100. The pres- sure after isothermal expansion ia 500, the temperature of the condenser 200 F. Find the pressure at the termination of the " third operation," the external work done along each of the four paths, and the heat absorbed along each of the four paths. Units axe cubic feet per pound and pounds per square foot. Ans. p 3 =13.1; TF 12 = 69,237ft. lb.; ^=88,943. t.u. ; TT 23 = 161,200 ft. lb.; Bi 3 =0; W S t~ 24,368ft. lb.; #34 = 31,32 B. t. u.; W* =1 61,200 ft lb.; J7 4l =0. 4 A non-reversible heat engine takes 1 B. t. u. per minute from a source and is used to drive a heat pump having an efficiency (quotient of work by heat imparted to source) of 0.70. What would be the rate of increase of heat contents of the source if the efficiency of the heat engine were 0.80? (Ans., O.U3 B. t. u. per min<) 5. Ordinary non-condensing steam engines use steam at 325 F. and discharge it to the atmosphere at 215 F. What is their maximum possible efficiency? (An$., 0,14,) PROBLEMS 83 6. Find the limiting efficiency of a gas engine in which a maximum temperature of 3000 F. is attained, the gases being exhausted at 1000 F. (Ans^ 0.578.) 7. An engine consumes 225 B. t. u. per indicated horse-power (33,000 foot-pounds) per minute. If its temperature limits are 430 F and 105 F., how closely does its efficiency approach the "best possible efficiency? (Ans., 51.59 per cent.) 8. How many B. t. u. per indicated horse power per hour would be required by a heat engine haying an efficiency of 15 per cent? 9. A power plant uses 2 Ib. of coal (14,000 B. t.u. per Ib.) per kilowatt-hour. (1 kw. = 1.34 h,p.) What is its efficiency from fuel to switchboard? 10. A steam engine working between 350 F. and 100 F. uses 15 Ib. of steam con- taining 1050 B. t. u. per Ib,, per indicated horse power per hour. "What proportion of the heat supplied was utilized by the engine? How does this proportion, compare with the highest that might have been attained? 11. Determine as to the credibility of the following claims for an oil engine: Temperature limits, 3000 F. and 1000 F. Fuel contains 19,000 B. t u. per Ib. Engine consumes 0.35 Ib. per kw.-hr. Loss between cylinder and switchboard, 20 per cent. 12. If the engine in Problem 3 is double-acting, and makes 100 r.p.m., what is its horse power? CHAPTER VII THE SECOND LAW OF THERMODYNAMICS 144. Statement of Second Law. The expression for efficiency of the Cariiot cycle, given in Art. 135, is a statement of the second law of thermodynamics. The law is variously expressed ; but, in general, it is an axiom from which is established the criterion of reversibility (Art. 139). With Clausius, the axiom was, (a) " Heat cannot of itself pass from a colder to a hotter body; " while the equivalent axiom of Kelvin was, (6) " It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the tempera- ture of ike coldest of surrounding objects" With Carnot, the axiom was that perpetual motion is impossible; while Ran- kine's statement of the second law (Art. 151) is an analytical restatement of the efficiency of the Carnot cycle. 145. Comparison of Laws. The law of relation of gaseous properties (Art. 10) and the second law of thermodynamics aie justified by their results, while thejirst law of thermodynamics is an expression of experimental fact. The second law is a " definite and independent statement of an axiom resulting from the choice of one of the two propositions of a dilemma" (1). For example, in Carnot's form, we must admit either the possibility of perpetual motion or the criterion of reversi- bility ; and we choose to admit the latter. The second ]aw is not a proposition to be proved, but an. "axiom commanding universal assent when its terms are understood." 146. Preferred Statements. The simplest and most satisfactory statement of the second law may be derived directly from inspection of the formula for effi- ciency, (T - t) ^ T (Art. 135). The most general statement, (c) rt The availability of heat for doing work depends upon its temperature" leads at once to the axiomatic forms of Kelvin and Clausius j while the most specific of all the statements directly underlies the presentation of Rankine : (c?) " If all of the heat be absorbed at one temperature, and rejected at another lower temperature, the heat transformed to 84 THE SECOND LAW OF THERMODYNAMICS 85 external work is to the total heat absorbed in the same ratio as that of the difference between the temperatures of absorption and rejec- tion to the absolute temperature of absorption ;" or, H- h = T- t H T ' in which H represents heat absorbed ; and 7i, heat rejected. 147. Other Statements. Forms (a), (ft), (c), and (d) are those usually given the second law. In modified forms, it has been variously expressed as follows (e) "All reversible engines working between the same uniform tem- peratures have the same efficiency." (/) " The efficiency of a reversible engine is independent of the nature of the working substance." (g) " It is impossible, by the unaided action of natural processes, to transform any part of the heat of a body into mechanical work, except by allowing the heat to pass from that body into another at lower temperature. " Qi) "If the engine be such that, when it is worked backward, the physical and mechanical agencies in every part of its motions are reversed, it produces as much mechanical effect as can be produced by any therm o- dynamic engine, with the same source and condenser, from a given quan- tity of heat." 148. Harmonization of Statements. It has been asserted that the state- ments of the second law by different writers involve ideas so diverse as, apparently, not to cover a common principle. A moment's consideration of Art. 144 will explain this. The second law, in the forms given in (a), (&), (c), ({/), is an axiom, from ichich the criterion of reversibility is estab- lished. In (r?), (e) (/), it is a simple statement of the efficiency of the Car- not cycle, with which the axiom is associated ; while in (7i), it is the criterion of reversibility itself. Confusion may be avoided by treating the algebraic expression of (VZ), Art. 146, as a sufficient statement of the second law, from which all necessary applications may be derived. 149. Consequences of the Second Law. Some of these were touched upon in Art. 143. The first law teaches that heat and work are mutually convertible, the second law shows how much of either may be converted into the other under stated conditions. Ordinary condensing steam engines work between tempera- tures of about 350 F. and 100 F. The maximum possible efficiency of such engines is therefore 350 - 100 350 + 459.4 = 0.31. 86 APPLIED THERMODYNAMICS The efficiencies of actual steam, engines range from 2J to 25 per cent, with an average probably not exceeding 7 to 10 per cent. A steam engine seems therefore a most inefficient machine ; but it must be remembered that, of the total heat supplied to it, a large prupoition is (by the second law) unavailable for use, and must be refected when its temperature falls to that of surrounding bodies. We can- not expect a water wheel located in the mountains to utilize all of the head of the water supply, measured down to &ea level. The available head is limited by the elevation of the lowest of surrounding levels. The performance of a heat engine should be judged by its approach to the efficiency of the Carnot cycle, rather than by its absolute efficiency. Heat must be regarded as a " low unorganized " form of energy, which pro- duces useful work only by undergoing a fall of temperature. All other forms of energy tend to completely transform themselves into heat. As the universe slowly settles to thermal equilibrium, the performance of work by heat becomes impossible and all energy becomes permanently degenerated to its most unavailable form.* 150. Temperature Fall and Work Done. Consider the Carnot cycle, abed, Fig. 45, the total heat absorbed being nabNaxKl the efficiency abcd-^-nabN Draw the isothermals , ij, successively differing by equal temperature intervals ; and let the tem- peratures of these isothermals be T 19 T 2 , T s Then the work done in cycle abfe is nabN x (T T^) * T >, that in cycle abhg is nabNx(TT 2 )---T; that in cycle abji is nabNx(T T$-*~T. As (T-T 3 ) = 3(T-2 r7 1 ) and (T-T 3 ) = 2(!T-2 7 1 )> abji = 3(abfe) and abhg = 2(a&/e); whence abfe = efhy = glvjL FIG. 45. Arts. 150, 153, 154, i.w. Second In otlier wor ^ s th e external work Law of Tuermodynamu-s. avai i able f rom a definite temperature fan is the same at all' parts of tlie thermometric scale. The waterfall analogy of Art. 149 may again be instructively utilized. 151. Rankine's Statement of the Second Law. " If the total actual heat of a uniformly hot substance be conceived to be divided into any number of equal parts, the effects of those parts in causing work to be performed are equal. If we re- member that by "total actual heat" Rankine means the heat corresponding to ab- solute temperature, his terse statement becomes a form of that just derived, dependent solely upon the computed efficiency of the Carnot cycle. 152, Absolute Temperature. It is convenient to review the steps by which the proposition of Art. 150 has been established. We have derived a conception of absolute temperature from the law of Charles, and have found that the effi- ciency of the Carnot cycle bears a certain relation to definite absolute temperatures. * *' Each time we alter our investment in energy, we have thus to pay a commis- sion, and the tribute thus exerted can never be wholly recovered by us and must be regarded, not as destroyed, but as thrown on the waste-heap of the Universe." Griffiths, KELVIN'S ABSOLUTE SCALE 87 Our scale of absolute temperatures, practically applied, is not entirely satisfactory ; for the absolute zero of the air thermometer, 459.-4 F., is not a true absolute zero, because air is not a perfect gas. The logical scale of absolute temperature would be that in "which temperatures were denned by reference to the work done by a reversible heat engine- Having this scale, we should be in a position to com- pute the coefficient of expansion of a perfect gas. 153. Kelvin's Scale of Absolute Temperature. Kelvin, in 1848, was led by a perusal of Carnot's memoir to propose such, a scale. His first defini- tion, based on the caloric theory, resulted only in directing general atten- tion to Carnot's great work ; his second definition is now generally adopted. Its form is complex, but the conception involved is simply that of Art. 150: " The absolute temperatures of two bodies are proportional to the quanti- ties of heat respectively taken in and given out in localities at one temperature and at the other, respectively, by a material system subjected to a complete cycle of perfectly reversible thermodynamic operations, and not allowed to part with or take in heat at any other temperature." Briefly, " The absolute values of two temperatures are to each other in the propor- tion of the quantities of heat taken in and rejected in a perfect thermodynamic engine, working with a source and condenser at the higher and the lower of the temperatures respectively." Symbolically, This relation may be obtained directly by a simple algebraic trans- formation of the equation for the second law, given in Art. 146, (d). 154. Graphical Representation of Kelvin's Scale. He turning to Fig. 45, but ignoring the previous significance of the construction, let ab be an iso- thermal and an, bN adiabatics. Draw isothermals ef, gh, ij, such that the areas abfe, efhg, ghji are equal. Then if we designate the temperatures along ab, ef, gh, ij by T, T 19 T 2 , T s , the temperature intervals T T l9 TI T 2J T 2 T 3 are equal. If we take ab as 212 F., and cd as 32 F., then by dividing the intervening area into 180 equal parts, we shall have a true Fahrenheit absolute scale. Continuing the equal divisions down below cd, we should reach a point at which the last remaining area be- tween the indefinitely extended adiabatics was just equal to the one next preceding, provided that the temperature 32F. could be expressed in an even number of absolute degrees. 155. Carnot's Function. Carnot did not find the definite formula for effi- ciency of his engine, given in Art. 135, although he expressed it as a function of the temperature range (T t). We may state the efficiency as 88 APPLIED THERMODYNAMICS z being a factor having the same value for all gases. Taking the general expres- sion for efficiency, f- ^ (Art. 128), and making H= h + d7i, we have H ^' "^ ^ ~~\ f ?h' ~ h + (111 ~~ A + rlh Tor e = z(T f)> we ^a-J write e zdt or s = -f, giving tn * = 7 -^ 7 - - <ft, equivalent to -^L But = (Art. 153) ; whence -^ = -^t and = -, and t = -^ = -. t h t h t h (Ih z Then z = - and e = - = - -~- in finite terniS; as already found. The factor z is known as Camofs function* It is the reciprocal of the absolute temperature* 156. Determination of the Absolute Zero. The porous plug experiments con- ducted by Joule and Kelvin (Art. 74) consisted in forcing various gases slowly through an orifice. The fact has already been mentioned that when this action was conducted without the performance of external work, a barely noticeable change in temperature was observed ; this being with some gases an increase, and with others a decrease. When a reMbting pressure was applied at the outlet oC the orifice, so as to cause the performance of some external work during tho flow of gas, a fall of temperature was observed ; and tin's fall wan different for dijicrcnt #<7,se,s*. The "porous plug" was a wad of silk fibers placed in the orifice for the purpose of reconverting all energy of velocity back to heat. Assume a slight hill of tem- perature to occur iu passing the plug, the velocity energy being reconverted to heat at the decreased temperature, giving the equivalent paths w/, rfc, Fig. 45. Then expend a sufficient measured quantity of work to bring the substance back to its original condition a, along cba. By the second law, , and -- = - nefN nabN - abfe' T^ nal)N-abfe' T T = T ( rcafrJV _ j \ __ rn (life 1 L \nal>N - altfe 1 _ __ \nal>N - altfe 1 x nabN - altfe ' If (T T^) as determined by the experiment = a, and nabN be put equal to unity, rp _ aCl alfe) A - abft ' In which abfe is the work expended in bringing the gas back to its original tem- perature. This, in outline, was the Joule and Kelvin method for establishing a location for the true absolute zero the complete theory is too extensive for pres- entation here (2). The absolute temperature of inciting ice is on this scale 491.58 F. or 273.1 C. The agreement with the hydrogen or the air thermometer is close. The correction for the former is generally less than yj^ 0., and that for THE SECOND LAW OF THERMODYNAMICS 89 the latter less than -j^ C. Wood has computed (3) that the true absolute zero must necessarily be slightly lower than that of the air thermometer. According to Alexander, (4) the difference of the two scales is constant for all temperatures. The Kelvin absolute scale establishes a logical defini- tion of temperature as a physical unit. Actual gas thermometer tempera- tures may be reduced to the Kelvin scale as a final standard. In the further discussion^ the temperature 459.6 J?. will be regarded as the absolute zero. (5) (1) Peabody, Thermodynamics, 1907, 27. (2) Phil Trans., CXVTV, 349. (3), Thermodynamics, 1905, 116. (4) Treatise on Thermodynamics, 1892, 91. (5) See the papers, On the Establishment of the Thermodynamic Scale of Temperature by Means of the Constant Pressure Thermometer, by Buckingham; and On the Standard Scale of Temperature in the Interval to 100 C., by Waidner and Dickinson; Bulletin of the Bureau of Standards, 3, 2; 3, 4. Also the paper by Buckingham, On the Definition of the Ideal Gas, op. cit., 6, 3. SYNOPSIS OF CHAPTER VII Statements of the second law an axiom establishing the criterion of reversibility ; jg h __ T t Of h _ _ a statement of the efficiency of the Carnot cycle ; the cri- H ~~ T H~~ T terion of reversibility itself. The second law limits the possible efficiency of a heat engine. The fall of temperature determines the amount oE external work done. Temperature ratios defined as equal to ratios of heats absorbed and emitted. The Carnot function for cyclic efficiency is the reciprocal of the absolute temperature. The absolute zero, based on the second law, is at 459,6 F. PROBLEMS 1. Illustrate graphically the first and the second laws of thermodynamics. Frame a new statement of the latter. 2. An engine works in a Carnot cycle between 400 F. and 280 F., developing 120 h.p. If the heat rejected by this engine is received at the temperature of rejection by a second Carnot engine, which works down to 220 F., find the horse power of the second engine. (Ans., 60). 3. Find the coefficient of expansion at constant pressure of a perfect gas. What is the percentage difference between this coefficient and that for air ? (Ans. t 0.0020342 ; percentage difference, 0.03931.) 4. A Carnot engine receives from the source 1000 B. t. u., and discharges to the condenser 500 B. t. u. If the temperature of the source is 600 F., what is the tem- perature of the condenser ? (.4ns., 70.2 F.) 5. A Carnot engine receives from the source 190 B.t. u. at 1440.4 F., and dis- charges to the condenser 90 B.t.u. at 440.4 F. Find the location of the absolute zero. (Ans., -459.6 F.) 6. In the porous plug experiment, the initial temperature of the gas being that of 90 APPLIED THERMODYNAMICS melting ice, and the fall of tempeiature being T J ff of the range from melting to boiling of water at atmospheric pressure, the work expended in restoring the initial tempera- ture was 1.5S foot-pounds. Find the absolute temperature at 32 F. (Ans., 492.39.) 7. The temperature range in a Camot cycle being 400 F., and the work done being equivalent to 40 pei cent of the amount of heat rejected, find the values of T and t. REVIEW PROBLEMS, CHAPTERS I- VII 1. State the precise meaning, or the application, o the t olio wing expressions : k 778 I (-} = - H = T+I+W r y E 53.36 PV = RT R -459.6 F. \P/v t n-1 I P V T pv logg y- ( J *= pi) n =c 42.42 pijy c 2545 pv c s = Z r n 1 2. A heat engine receives its fluid at 350 F. and discharges it at 110 F. It con- sumes 200 B. t. u. per Ihp. per minute. Find its efficiency as compared with that of the corresponding Carnot cycle. (Ans., 0.712.) 3. Given a cycle a&c, in which P a =P 6 = 100 Ib. per sq. in., V a - 1, ^rr= 6 (cu. ft.), YO, PfiVj, 1 8 =P c V c 1 ' B ,P a V a P c Y ct find the pressure, volume, and temperature at c if the substance is 1 Ib. of air. 4. Find the pressure of 100 Ib. of air contained in a 100 cu.-ft. tank at 82 F. (Ans., 28,900 Ib. per sq. ft.) 5. A heat engine receives 1175.2 B. t. u. in each pound of steam and rejects 1048.4 B. t. u. It uses 3110 Ib. of steam per hour and develops 142 lip. Estimate the value of the mechanical equivalent of heat. (Ans., 712.96.) 6. One pound of air at 32F . is compressed from 14.7 to 2000 Ib. per square inch, without change of temperature. Find the percentage change of volume. (Ana., 99.3%.) 7. Prove that the efficiency of the Carnot cycle is ^-. 8. Air is heated at constant pressure from 32 F. to 500 F. Find the percentage change in its volume. (Ans., 95.2 % increase.) 9. Prove that the change of internal energy in passing from a to 6 is independent of the path ab. "P V ~P "\7" 10. Given the formula for change of internal energy, & & -, prove that 11. Given It for air=53.36, V= 12,387; and given F= 178.8, fc=3.4 for hydro- gen : find the value of y for hydrogen. (Ans., 1.412.) 12. Explain isothermal, adiabatic, isodynamic, isodiabatic. 13. Find the mean specific heat along the pathpvi-8 =c for air (2=0.1689). (Ans., 0.084.) PROBLEMS 91 14. A steam engine discharging its exhaust at 212 F receives steam containing 1100 B, t. u. per pound at 500 F. What is the minimum weight of steam it may use per Ihp.-hr. ? (Arts , 7.71 Ib.) 15. A cycle is bounded by polytropic paths 12, 23, 13. We have given P i =P 2 100,000 Ib. per sq. ft. V 2 = V z =40 cubic feet per pound. T 1= :3000 F. PiFx-P.F,. Find the amount of heat converted to work in the cycle, if the working substance is air. (4ns., 4175 B.t.uJ CHAPTER VIII ENTROPY 157. Adiabatie Cycles. Let abdc, T?ig. 46, be a Carnot cycle, an and bJ$ the projected adiabatics. Draw intervening adiabatics em, g^f } etc., so located that the areas naem, megM', M<jl)N, are equal. Then since the effi- ciency of each of the cycles aefc, eyhf, gbdJi, is (T t) -=- T, tJie work areas represented by these cycles are all equal. To measure these areas by mechani- cal means would lead to approximate results only. 158. Rectangular Diagram. If the adiabatics and isothermals were straight lines, simple arithmetic would suffice for the measure- ment of the work areas of Fig. 46. We have seen that in the Carnot cycle, bounded by isothermals and adiabatics, = (Art. 158). Applying this for- mula to Rankine's theorem (Art. 106), we have the quotient of an area and a length as a constant. If the area h is a part of .fiT, then there must be some constant property, which, -when, multi- plied by the temperatures T or , will FIG. 40. Arts. 157, 158, 15<>, 100. Adiabatie Cycles.* 710 050 600 650 1191 G give the areas H or h. Let us conceive of a diagram in which only one coor- dinate will at present be named. That coordinate is to be absolute temperature. Instead of specifying the other coordi- nate, let it be assumed that subtended areas on this diagram are to denote quantities of heat absorbed or emitted, just as such areas on the JPV diagram represent external work done. As an example of such a diagram, consider Fig, 47. Let the substance be one * The adiabatics are distorted for clearness. In reality they are asymptotic. Many of the diagrams throughout the "book are similarly u out of drawing" for the same reason. 92 FIG. 4:7* Arts. 158, 163, ]71. En- tropy Diagram. ENTROPY 93 pound of water, initially at a temperature of 32 F., or 491.6 abso- lute, represented by the height #5, the horizontal location of the state b being taken at random. Now assume the water to be heated to 212 F., or 671.6 absolute, the specific heat being taken as con- stant and equal to unity. The heat gained is 180 B. t. u. The final temperature of the water fixes the vertical location of the new state point cZ, i.e. the length of the line cd. Its horizontal lo- cation is fixed by the consideration that the area subtended between the path bd and the axis which we have marked ON shall be 180 B. t. u. The horizontal distance ac may be computed from the properties of the trapezoicl abdc to be equal to the area abdc divided by [(a& -f- cd) + 2] or to 180 -f- [(491.6 + 671.6) -f- 2] = 0.31. The point d is thus located (Art. 163). 159. Application to a Carnot Cycle. Ordinates being absolute temperatures, and areas subtended being quantities of heat absorbed or emitted, we may conclude that an isothermal must be a straight horizontal line ; its temperature is constant, and a finite amount of heat is transferred. If the path is from left to right, heat is to be conceived as absorbed; if from right to left, heat is rejected. Along a (re- versible) adiabatic, no movement of heat occurs. The only line on this diagram T which does not subtend a finite area is a straight vertical line. Adiabatics are 1 consequently vertical straight lines. (But see Art. 176.) The temperature must N constantly change along an adiabatic. FIG. 48. Arts. 169, 160, 161, ics, The lengths of all isothermals drawn be- 106. Adiabatic Cycles, Entropy tween fc h e game two adiabatipq a pnnal UWCCll UL1C &ClJ.llt5 u \V \J LLLJ.CtUc1i LlUo GiL\3 dJ LiCuJ.. Diagram. The Carnot cycle on this new diagram may then be represented as a rectangle enclosed by vertical and hori- zontal lines ; and in Fig. 48 we have a new illustration of the cycles shown in Fig. 46, all of the lines corresponding. 160. Physical Significance. The new diagram is to be conceived as so related to the P V diagram of Fig. 46 that while an imaginary \, M 94 APPLIED THERMODYNAMICS pencil is describing any stated path on the latter, a corresponding pencil is tracing its path on the former. In the PV diagram, the subtended areas constantly represent external work done by or on the substance; in the new diagram they represent quantities of heat ab- sorbed or rejected. (Note, however, Art. 176.) The area of the closed cycle in the first case represents the net quantity of work done; iu the second, it represents the net amount of heat lost^ and conse- quently, also, the net work done. But subtended areas under a single path on the PV diagram do not represent heat quantities, nor in the new diagram do they represent work quantities. The validity of the diagram is conditioned upon the absoluteness of the properties chosen as coordinates. We have seen that temperature is a cardinal property, irrespective of the previous history of the substance ; and it will be shown that this is true also of the horizontal coordinate, so that we may legitimately employ a diagram in which these two properties are the coordinates. 161. Polytropic Paths. For any path in which the specific heat is zero, the transfer of heat is zero, and the path on this diagram is consequently vertical, an adiabatic. For specific heat equal to infinity, the temperature cannot change, and the path is horizontal, an iso- thermal. For any positive value of the specific heat, heat area and temperature will be gained or lost simultaneously; the path will be similar to ai or #/, Fig. 48. If the specific heat is negative, the tem- perature will increase with rejection of heat, or de- crease with its absorption, as along the paths ak, al, Fig, 48. These results may be compared with those of Art. 115. Figure 49 shows on the new diagram the paths corresponding with those of Fig. 31. It may be noted that, in general, though not invariably, increases of FIG. 49. Arts. 101, 1 05. Polytropic Paths on Entropy Diagram. ENTROPY 95 volume are associated with increases of the horizontal coordinate of the new diagram. 162. Justification of the Diagram. In the PV diagram of Fig. 50, consider the cycle ABCD. Let the heat absorbed along a portion of this cycle be repre- sented by the infinitesimal strips nabN, NbcM, Mcdm, formed by the indefinitely projected adiabatics. In any one of these strips, as nabN, we have, in finite terms, nabN _ T Qr negN t' nabN _ neqN T t Considering the whole series of strips from A to C, we have nabN __ ^ neqN v or, using the symbol H for heat trans- ferred, FIG. 50. Art. 162. Entropy a Cardinal Property. S ^7T = > in which T expresses temperature generally. Let the substance complete the cycle ABCD A] we then have, the paths leing reversible, JP P A dH_ I riff I rlH__~ ^r~ \* "F + P "F-' C/^i *Jo while for the cycle ADCDA, whence, The integral f thus has the same value whether the path is A DC or ABC, or, indeed, any reversible path between A and C; its value is independent of the path of the substance. Now this integral will be shown immediately to be the most general expression for the horizontal coordinate of the diagram under discussion. Since it denotes a cardinal property, like pressure or temperature, it is permissible to use a diagram in which the coordinates are T and f-m- 96 APPLIED THERMODYNAMICS 163. Analytical Expression. Along any path of constant tem- perature, as al> Fig. 48, the horizontal distance nN may be computed from the expression, nN=H+ T, in which S represents the quan- tity of heat absorbed, and T the temperature of the isothermal. If the temperature varies, the horizontal component of the path during the absorption of dH units of heat is dn = dff-i- T. For any path along which the specific heat is constant, and equals 7c, ?cdT= dn = , and = k. = k log, . If the specific heat is variable, say Jc a + IF, then The line Id of Fig. 47 is then a logarithmic curve, not a straight line ; and the method of finding it adopted in Art. 158 is strictly accurate only for an infinitesimal change of temperature. Writing the expression just derived in the form n = &log e (jF-*- 1) and remem- bering that T= PV-r- 72, while t = pv -*- 72, we have n = k log e (P V+- pv) . The expression Jclog e (T-r-) is the one most commonly used for calculating values of the hori- zontal coordinate for polytropic paths. The expression dn = dH-t- T is general for all re- versible paths and is regarded by Ranldne as the fundamental equation of thermodynamics. 164. Computation of Specific Heat. If at any FlG . 51 . Art . 16 L_ Graphi _ point on a reversible path a tangent be drawn, the cal Determination of length of the subtangent on the JV-axis represents the Specific Heat, value of the specific heat at that point. In Fig. 51, draw the tangent nm to the curve AB at the point nand construct the infinitesimal "~ triangle dtdn. From similar triangles, mr : nr : : dn : dt, or mr = Tdn - dt = dH - dt = s (Art. 112). 165. Comparison of Specific Heats. If a gas is heated at constant pressure from a, Pig. 52, it will gain heat and temperature, following some such path as ab. If heated at constant volume, through an equal range of temperature, a less FIG. 52. Art. 165. Com- parison of Specific Heats. ENTROPY 97 quantity of heat will be gained ; i.e. the subtended area aefd will be less than the area abed. In general, the less the specific heat, the more nearly vertical will be the path. (Compare Fig. 49.) When & == 0, the path is vertical ; when 7c = oo, the path is horizontal. 166. Properties of the Carnot Cycle. In Fig. 48, it is easy to see that since efficiency is equal to net expenditure of heat divided by gross ex- penditure, the ratio of the areas abdc and abNn expresses the efficiency, and that this ratio is equal to (T ?) H- T. The cycle abdc is obviously the most efficient of all that can be inscribed between the limiting iso- thermals and adiabatics. 167. Other Deductions. The net enclosed area on the TN diagram represents the net movement of heat. That this area is always equivalent to the corresponding enclosed area on the PV diagram is a statement of the first law of thermodynamics. Two statements of the second law have just been derived (Art. 166). The theorem of Art. 106, relating to the representation of heat absorbed by the area between the adiabatics, is accepted upon inspection of the TN diagram. That of Art. 150, from which the Kelvin absolute scale of temperature was deduced, is equally obvious. 168. Entropy. The horizontal or N coordinate on the diagram now presented was called by Clausius the entropy of the body. It may be mathematically defined as the ratio n = ^- - Any physical / J- definition or conception should be framed by each reader for himself. Wood calls entropy " that property of the substance which remains constant throughout the changes represented by a [reversible] adia- batic line." It is for this reason that reversible adiabatics are called isentropics, and that we have used the letters H, JT in denoting adiabatics. 169. General Formulas. It must be thoroughly understood that the validity of the entropy diagram is dependent upon the fact that entropy is a cardinal prop- erty, like pressure, volume, and temperature. For this reason it is desirable to become familiar with compu- tations of change of entropy irrespective of the path pursued, Otherwise, the method of Art, 163 is usually FIG. 54. Arts. 169, 329a. _ more convenient. ange o n opy. Consider the states a and b } Fig. 54. Let the substance pass at constant pressure to c and thence at constant volume 98 APPLIED THERMODYNAMICS T T to &. The entropy increases by 7c log e -^ -{-I log a * (Art. 163), 7c and I -LO, J- c in this instance denoting the respective special values of the specific heats. An equivalent expression arises from Charles 5 law : n = k log e Z* + i log. = k log e |*+ 1 log, J 6 , (A) r -*c r a * a in which last the final and initial states only are included. We may also write, Z = i io go *V+ ot* T7" = Z log a ^ + (ft - log, |>, Arts. 51, 65 : (B) -'a ' a and further, The entropy is completely detei mined by the adiabatic through the state point. T In the expression n^=.k^ log e , the value of n L apparently depends upon that of k^ which is of course related to the path ; along another path, the gain or loss of T entropy might be n 2 = & 3 log, > a different value ; but although the temperatures would be the same at the beginning and end of both processes, the pressures or volumes would differ. The states would consequently be different, and the values of n should therefore differ also. A graphical method for the transfer of perfect gas paths from the PFto the TN plane has been developed by Berry (1). 169a. Mixtures of Liquids. When m Ib. of water are heated from 32 to t absolute, the specific heat being taken at unity, the gain, of entropy is Let m Ib. at t be mixed with n Ib. at 1, the resulting temperature of (m+n) Ib. being (from Art. 25), without radiation effects, yy This, if heated from 32 to Z ; would have acquired the entropy (m+ri) log* ^~2, and the change in aggregate entropy due to the mixture is i t , i t\ f , \ i / nti + mt \ m log, m +n log e m - (+) log, m(m+n} ) The mixing of substances at different temperatures always in- creases the aggregate entropy. Thus, let a body of entropy n, at the temperature t, discharge a small amount, H, of its heat to an adjacent body of entropy N and temperature T. The aggregate entropy before the transfer is n + N; after the transfer it is TT rj and since t>T f <TF and the loss of entropy is less than the gain: t j. or 170, Other Names for n. Rankine called n the thermodynamic func- tion. It has been called the " heat factor." Zeuner describes it as " heat weight." It has also been called the " mass " of heat. The letters T, N, which we have used in marking the coordinates, were formerly replaced by the Greek letters theta and phi, indicating abso- lute temperatures and entropies; whence the name, theta-phi diagram. The TN diagram is now commonly called the temperature-entropy diagram, or, more briefly, the entropy diagram. 171. Entropy Units. Thus far, entropy has been considered as a horizontal distance on the diagram, without reference to any particular zero point. Its units are B. t. u. per degree of absolute temperature. Strictly speaking, entropy is merely a ratio, and has no dimensional units. Changes of entropy are alone of physical significance. The choice of a zero point may be made at random ; there is no logical zero of entropy. A convenient starting point is to take the adiabatic of the substance through the state P =2116.8, T=32 F., as the OT axis, the entropy of this adiabatic being assumed to be zero, as in ordinary tables. 100 APPLIED THERMODYNAMICS Thus, in Fig. 47 (Art. 158), the OT axis should be shifted to pass through the point b, which was located at random horizontally. 172. Hydraulic Analogy. The analogy of Art. 140 may be extended to illus- trate the conception of entropy. Suppose a certain weight of water W to be maintained at a height H above sea level; and that in passing through a motor its level is reduced to h. The initial potential energy of the water may be regarded as WH-, the final residual energy as Wh, the energy expended as W(H A). Let this operation be compared with that of a Carnot cycle, taking in heat at T and discharging it at t. Eegarding heat as the product of N and T, then the heat energies corresponding to the water energies just described are NT, Nt, and N(T t) ; N being analagous to W, the weight of the water. 173. Adiabatic Equation. Consider the states 1 and 2, on an adiabatic path, Fig. 55. The change of entropy along the constant volume path 13 D rp is I log e 3 ; that along the constant pressure path 32 T is Jc log fl -^ The difference of entropy between * i 1 and 2, irrespective of the path, is . V\ FIG. 55. Art. 173. ^ or a reversible adiabatic process, this is equal to Adiabatic Equation. zero; whence e or y lo & Fi + lo & A = y log, F! + log.P,, L from which we readily derive P^Vf = P a F^ #ie equation of the adiabatic. 174. Use of the Entropy Diagram. The intelligent use of the entropy diagram is of fundamental importance in simplifying thermodynamic con- ceptions. The mathematical processes formerly adhered to in presenting the subject have been largely abandoned in recent text-books, largely on account of the simplicity of illustration made possible by employing the TN coordinates. Belpaire was probably the first to appreciate their usefulness. Gibbs, at about the same date, 1873, presented the method in this country and first employed as coordinates the three properties volume, entropy, and internal energy. Linde, Schroeter, Hermann, Zeuner, and Gray used TN diagrams prior to 1890. Cotterill, Ayrton and Perry, Dwelshauvers Dery and Ewing have employed them to a con- siderable extent. Detailed treatments of this thermodynamic method have been given by Boulvin, Reeve, Berry, and Golding (2). Some precautions necessary in its practical application are suggested in Arts. 45i-458. IRREVERSIBLE PROCESSES 101 FIG. 56. Art. 175. Irreversible Cycle. IRREVERSIBLE PROCESSES 175. Modification of the Entropy Conception. It is of importance to distinguish between reversible and irreversible processes in relation to entropy changes. The significance of the term reversible, as ap- plied to a path, was discussed in Art. 125. A process is reversible only when it consists of a series of successive states of thermal equilib- rium. A series of paths constitute a reversible process only when they foim a closed cycle, each path of which is itself reversible. The Carnot cycle is a perfect example of a reversible process. As an example of an irreversible cycle, let the substance, after isothermal expansion, as in the Carnot cycle, be transferred directly to the condenser. Heat will be abstracted, and the pressure may be reduced at constant vol- ume, as along be, Fig. 56. Then allow it to compress isothermally, as in the Carnot cycle, and finally to be transferred to the source, where the temperature and pressure increase at constant volume, as along da. This cycle cannot be operated in the reverse order, for the pressure and temperature cannot be reduced from a to d while the substance is in communication with the source, nor increased from c to b while it is in communication with the condenser. 176. Irreversibility in the Porous Plug Experiment. We have seen that in this instance of unresisted expansion, the fundamental formula of Art. 12 becomes H= T + I + W + V (Art. 127). Knowing H = 0, W = 0, we may write (T + I) = V, or velocity is attained at the expense of the internal energy. The velocity evidences kinetic energy ; mechanical work is made possible ; and we might expect an exhibition of % such work and a fall of internal energy, and consequently of temperature. But we find no such utilization of the kinetic energy of the rapidly flowing jet; on the contrary, the gas is gradually brought to rest and the velocity derived from, an expenditure of internal energy is reconverted to internal energy, The process was adiabatic, for no transfer of heat occurred ; it was at the same time isothermal, for no change of temperature occurred ; and while both adiabatic and isothermal, no external work was done, so that the PV diagram is invalid. Further : the adiabatic path here considered was not isen tropic, like an ordinary adiabatic. The area under the path on the TN diagram no longer represents heat absorbed from surrounding bodies. Neither does dn = , for H is zero, while n is finite. The expression for increase of entropy, C f , along a reversible path, does not hold for irreversible operations. In irreversible operations, the expression C ( r ceases to represent a cardinal property. "We have the following propositions : 102 APPLIED THERMODYNAMICS (a) In a reversible operation, the sum of the entropies of the participating substances is unchanged. During a reversible change, the temperatures of the heat-absorbing and heat-emitting bodies must differ to an infinitesimal extent only; they are in finite terms equal. The heat lost by the one body is equal to the heat gained by the other, so that the expression f '" denotes both the loss of entropy by the one substance and the gain by the othei , the total stock of enti opy remaining constant. (1} During meuersible operations, the aggregate entropy increases. Consider two engines working in the Carnut cycle, the first taking the quantity of heat H : from the souice, and dischaiging the quantity H to the condenser; the second, acting as a heat pump (Art. 130), taking the quantity II j from the condenser and restoring H^ to the source. Then if the work produced by the engine is expended in driving the pump, without loss by friction, HI-HI = UJ-HI. If the engine is irreversible, H^ > ///, or IT l - If/ > 0, whence, H 2 - H 2 ' > 0. Tf we denote by a a positive finite value, H l = HJ + a and H 2 = H 2 ' +a. But ^L = , or y ~ ^ - 0, and consequently <H$ J'i J i ^2 PL -a H*-a n , H } H, (1 1 ~ --- =Q an<i = " Since T, > T* 1 - < 0, or > , or, generally, < 0. The value of C ( UL j s thus, for irreversible cyclic operations, negative. Now let a substance work irreversibly from A to JS, thence revemlly from B to A. We may write. (irruv ) Ciev ) (irrev ) (lev) r B dii But the cJiange of entropy of the substance in passing from A toBi$N B -N' A = I -, JA * (IE being the amount of heat absorbed along any reversible path, while the change of entropy of the source which supplies the substance with heat (reversibly) is jyy Njf = C 7-, the negative sign denoting that heat has been abstracted, Jj. V We have then, from equation (A), i.e. the sum of the entropies of the participating substances increases when the process is irreversible. (c) The loss of work due to irrerentibtlity is ptopoitional to the increase of entropy. Consider a partially completed cj cle : one which might be made complete were all of the paths reversible. Let the heat absorbed be Q, at the temperature !T, in- creasing the entropy of the substance by -,; and let its entropy be further increased IRREVERSIBILITY 103 by N f N during the process The total increase of entropy is then n = A T ' A 7 + y,, whence Q = T(n - N' + A 7 ) T \ he work done may be written as // H ' 4- Q, in which H and H' are the initial and final heat contents respectively. Calling this W, we have W = // - H 1 4- T(n - JV' + A 7 ). In a reversible cycle J = n , whence W R = H H' + 2"(JV" - A"') and ^ - W = Tn. (A careful distinction should be made at this point between the expression j TT and the term entropy. The former is merely an expression for the latter under specific conditions In the typical irreversible process furnished by the porous plug experiment, the entropy increased; and this is the case generally with such processes, in which dn > Internal transfers of heat may augment the entropy even of a heat-insulated body, if it be not in uniform thermal condition. Perhaps the most general statement possible for the second law of thermody- namics is that all actual processes tend to increase the entropy ; as we have seen, this keeps possible efficiencies below those of the perfect reveisihle engine. The prod- uct of the increase of entropy by the temperature is a measure of the waste of energy (3).) Most operations in power machinery may without serious error be analyzed as if reversible ; unrestricted expansions must always be excepted. The entropy diagram to this extent ceases to have " an automatic meaning." (1) Tlie Temperature-Entropy Diagram, 1008. (2) See Berry, op. cit. (3) The works of Preston, Swinburne, and Planck may be consulted by those interested in this aspect of the subject. See also the paper by M'Cadie, in the Journal of Electricity, Power and Gas, June 10, 1911, p. 505. SYNOPSIS OF CHAPTER VIII It is impracticable to measure PFheat areas "between the adiabatics. The rectangular diagram : ordinates = temperature; areas = heat transfers. Application to a Carnot cycle : a rectangle. 1?he validity oj the diagram is conditioned upon the absoluteness of the horizontal coordinate. The slope of a path of constant specific heat depends upon foe value of the specific heat. The expression C has a definite value for any reversible change of condition, regardless of the path pursued to effect the change. fj FT T T* dn = , or n = Tc log e for constant specific heat = k, or n = a log e -+ &( T for T t ' t variable specific heat = a + & T. The value of the specific heat along a poly tropic is represented "by the length of the sub- tangent. Illustrations : comparison of k and I ; efficiency of Camot cycle ; the first law j the second law ; heat area between adiabatics ; Kelvin's absolute scale. 101 APPLIED THERMODYNAMICS Entropy units are B. t. u.per degree absolute. The adiabatic for zero entropy is at . = nog c ^^ The mixing of substances at different temperatures increases the aggregate entropy. Hydraulic analogy ; physical significance of entropy ; use of the diagram. Derivation of the adiabatic equation. Irreversible Processes A reversible cycle is composed of reversible paths ; example of an irreversible cycle. Joule's experiment as an example of irreversible operation. Heat generated by mechanical friction of particles ; the path both isothermal and adia- batic, but not isen tropic. S- T+I + W+ For irreversible processes, d?i is not equal to ~- 3 the subtended area does not repre- sent a transfer of heat ; non-isentropic adiabatics. In reversible operations, the aggregate entropy of the participating substances is unchanged. During irreversible operations, the aggregate entropy increases, and J <0. The loss of work due to increase of entropy is nT\ du>d. T PROBLEMS 1. Plot to scale the TJVpath of one pound of air heated (a) at constant pressure from 100 F. to 200 F., then (Z>) at constant volume to 300 F. The logarithmic curves may be treated as two straight lines. 2. Construct the entropy diagram for a Carnot cycle for one pound of air in -which T= 400 F., t = 100 F., and the volume in the first stage increases from 1 to 4 cubic feet. Do not use the formulas in Art 169. 3. Plot on the TJV diagram paths along which the specific heats are respectively 0, oo,* 3.4, 0.23, 0.17, -0.3, -10.4, between T = 459.0 and T= 910.2, treating the logarithmic curves as straight lines. 4. The variable specific heat being 020-0.0004 T- 0.000002 T 2 (T being in Fahrenheit degrees), plot the TF path from 100 F. to 140 F. m four steps, using mean values for the specific heat in each step. Find by integration tlie change of entropy during the whole operation. 5. What is the specific heat at T=40 (absolute) for a path the equation of which on the TN diagram is TN= c = 1200 ? (Ans., 32.) 6. Confirm Art. 134 by computation from the TN diagram. 7. Plot the path along which 1 unit of entropy is gained per 100 absolute, What is the mean specific heat along this path from to 400 absolute? Begin at 0. 8. What is the entropy measured above the arbitrary zero per pound of air at normal atmospheric pressure in a room at 70 F.? (Ans. t 0.01766.) PROBLEMS 105 9. Find the arbitrary entropy of a pound of air in the cylinder of a compressor at 2000 Ib pressure per square inch and 142 F. (Ans., 0.301.) 10. Find the entropy of a sphere of hydrogen 10 miles in diameter at atmospheric pressure and 175 F. (Ans., 289,900,000,000.) 11. The specific heat being 0.24 -f- 0.0002 T, find the increase in entropy between 459.6 and 919.2 degrees, all absolute. What is the mean specific heat over this range ? (Ans., increase of entropy 0.25809 ; mean specific heat, 0.378.) 12. In a Carnot cycle between 500 and 100, 200,000 ft. Ib. of work are done. Find the amount of heat supplied and the variation in entropy during the cycle. 13. A Carnot engine works between 500 and 200 and between the entropies 1.2 and 1.45. Find the ft. Ib. of work done per cycle. 14. To evaporate a pound of water at 212 F. and atmospheric pressure, 970 4 B. t. u. are required If the specific volume of the water is 0.016 and that of the steam 26.8, find the changes in internal energy and entropy during vaporization. 15 Five pounds of air in a steel tank are cooled from 300 F. to 150 F. Find the amount of heat emitted and the change in entropy. (I for air =0.1689.) 16. Compare the internal energy and the entropy per pound of air when (a) 50 cu. ft. at 90 F. are under a pressure of 100 Ib. per sq. in., and (&) 5 cu. ft. at 100 F. are subjected to a pressure of 1200 Ib. per sq. in. 17. Air expands from p=100, u = 4 to P=40, F=8 (Ib. per sq in. and cu. ft. per Ib.). Find the change in entropy, (a) by Eq (A) Art. 169, (&) by the equation n 2 - Hi =s log e j-, where s=l n 18 A mixture is made of 2 Ib. of water at 100, 4 Ib. at 160, and 6 Ib. at 90 (all Fahr.). Find the aggregate entropies before and after mixture. CHAPTER IX COMPRESSED AIR (1) 177. Compressed Air Engines. Engines are sometimes used in which the working substance is cold air at high pressure. The pressure is previously pro- duced by a separate device ; the air is then transmitted to the engine, the latter being occasionally in the form of an ordinary steam engine. This type of motor is often used in mines, on locomotives, or elsewheie where excessive losses by con- densation would follow the use of steam. For small powers, a simple form of rotary engine is sometimes employed, on account of its convenience, and in spite of its low efficiency. The absence of heat, leakage, danger, noise, and odor makes these motors popular in those cities where the public distribution of compressed air from central stations is practiced (la). The exhausted air aids in ventilating the rooms in which it is used. 178. Other Uses of Compressed Air. Aside from the driving of engines, high- pressure air is used for a variety of purposes in mines, quarries, and manufac- turing plants, for operating hoists, forging and bending machines, punches, etc,, for cleaning buildings, for operating "steam" hammers, and for pumping water by the ingenious "air lift" system. In many works, the amount of power trans- mitted by this medium exceeds that conveyed by belt and shaft or electric wire. The air is usually compressed by steam power, and it is obvious that a loss must occur in the transtormation. This loss may be offset by the convenience and ease of transmitting air as compared with steam ; the economical generation, distribu- tion, and utilization of this form of power have become matters of first importance. The first applications were made during the building of the Mont Cenis tun- nel through the Alps, about 18GO (2). Air was there employed for operating locomotives and rock drills, following Colladon's mathematical computation of the small loss of pressure during comparatively long transmissions, A general introduction in mining operations followed. Two-stage compressors with inter- coolers were in use in this country as early as 1881. Among the projects sub- mitted to the international commission for the utilization of the power of Niagara, there were three in which distribution by compressed air was contemplated. Wide- spread industrial applications of this medium have accompanied the perfecting of the small modern interchangeable "pneumatic tools." 179. Air Machines in General. In the type of machinery under consideration, a considerable elevation of pressure is attained. Centrifugal fans or paddle-wheel blowers, commonly employed in ventilating plants, move large yolumes of air at very slight pressures, usually a fraction of a pound, and the thermodynamio 106 THE AIR ENGINE 107 relations are unimportant. Rotary blowers are used for moderate pressures, up to 20 lb., but they are generally wasteful of power and are principally employed to furnish blast for foundry cupolas, forges, etc. The machine used for coin- pressing air for power purposes is ordinarily a piston compressor, mechanically quite similar to a reciprocating steam engine. These compressors are sometimes employed for comparatively low pressures also, as " blowing engines."' rr 3 \g \^ -c. THE AIR ENGINE 180. Air Engine Cycle. In Fig. 57, ABOD represents an ideal- ized air engine cycle. AB shows the admission of air to the cylin- der. Since the latter is smull as compared with the transmitting pipe line, the specific volume and pres- P sure of the fluid, and consequently its temperature as well, remain un- A changed. BO represents expansion after the supply from the mains is cut off. If the temperature at B is that F of the external atmosphere, and ex- pansion proceeds slowly, so that any fall of temperature along BC is offset u by the transmission of heat from the outside air through the cylinder walls, this line is an isothermal. If, however, expansion is rapid, so that no transfer of heat occurs, BO will be an adidbatic. In practice, the expansion line is a polytropic, lying usually between the adiabatic and the isothermal. CD represents the expulsion of the air from the cyl- inder at the completion of the working stroke. At _Z), the inlet valve opens and the pressure rises to that at A. The volumes shown on this diagram are not specific volumes, but volumes of air in the cylinder. Subtended areas, nevertheless,*represent external work. 181. Modified Cycle. The additional work area LMC obtained by ex- pansion beyond some limiting volume, say that along onf, is small. A slight gain in efficiency is thus made at the cost of a much larger cylin- der. In practice, the cycle is usually terminated prior to complete expan- sion, and has the form ABLMD, the line LM representing the fall of pressure which occurs when the exhaust valve opens. FIG. 57. Arts. 180-183, 189, 222, 223, 226, Prob. 6. Air Engine Cycles. 108 APPLIED THERMODYNAMICS 182. Work Done. Letting p denote the pressure along AB, P the pressure at the end of the expansion, q the "back pressure" along MD (slightly above that of the atmosphere), and letting <Q denote the volume at B, and Fthat at the end of expansion, both volumes being measured from OA as a line of zero volumes, then, for isothermal expansion, the work done is V T7 e -qV\ and for expansion such that pv n = PV n , it is (In these and other equations in the present chapter, the air will be regarded as free from moisture, a sufficiently accurate method of procedure for ordinary design. For air constants with moisture effects considered, see Art. 3S2&, etc.) 183. Maximum Work. Under the most favorable conditions, expan- sion would be isothermal and "complete"; i.e. continued down to the back-pressure line CD. Then, q = P~pv+ F, and the work would be pv log e (F-4- v). For complete adiabatic expansion, the work would be y -PF= 0* -PF) 184. Entropy Diagram. This cannot be obtained by direct transfer from the PV diagram, because we are dealing with a varying quantity of air. The method of deriving an illustrative entropy diagram is explained in Art. 218. 185. Fall of Temperature. If air is received by an engine at P, F, and expanded to p, t, then from Art, 104, if P+p= 10, and T 529 absolute, with adiabatic expansion, t = 187 F. This fall of temperature during adiabatic expansion is a serious matter. Low final temperatures are fatal to successful working if the slightest trace of moisture is present in the air, on account of the formation of ice in the exhaust valves and passages. This difficulty is counteracted in various ways: by circulating warm air about the exhaust passages; by specially designed exhaust ports 5 by a reduced range of pressures; by avoidance of adiabatic expansion (Art. 219) ; and by thoroughly drying the air. The jacketing of the cylinder with hot air has been proposed. Unwin mentions (3) the use of a spray of water, injected into the air while passing through a preheater (Art. 186). This reaches the engine as steam and condenses during expansion, giving up its latent heat of PREHEATERS 109 vaporization and thus raising the temperature. In the experiments on the use of compressed air for street railway traction in ]N~ew York, stored hot water was employed to preheat the air. The only commercially suc- cessful method of avoiding inconveniently low temperatures after expan- sion is by raising the temperature of the inlet air. 186. Preheaters. In the Paris installation (4), small heaters were placed at the various engines. These were double cylindrical boxes of cast iron, with an intervening space through which the air passed in a circuitous manner. The inner space contained a coke fire, from which the products of combustion passed over the top and down the outside of the outer shell. For a 10-hp. engine, the extreme dimensions of the heater were 21 in. in diameter and 33 in. in height. 187. Economy of Preheaters. The heat used to produce elevation of temperature is not wasted. The volume of the air is increased, and the weight consumed in the engine is correspondingly decreased. Kennedy esti- mated in one case that the reduction in air con- sumption due to the in- crease of volume should have been, theoretically, 0.30; actually, it was 0.25. The mechanical efficiency (Art. 214) of the engine is improved by the use of preheated air. In one instance, Ken- nedy computed a saving of 225 cu. ft. of "free" air (i.e. air at at- mospheric pressure and tem- perature) to have been ef- fected at an expenditure of 0.4 Ib. of coke. Unwin found that all of the air used by a 72-hp. engine could be heated to 300 F. by 15 Ib. of coke per hour. Figure 58 represents a modern form of preheater. FIG. 58. Art. 187. Band Air Pieheater. 110 APPLIED THERMODYNAMICS 188. Volume of Cylinder. If n be the number of single strokes per minute of a double-acting engine, V the cylinder volume (maximum vol- ume of fluid), W the number of pounds of air used per minute, v the specific volume of the air at its lowest pressure p and its temperature t, N the horse power of the engine, and U the work done in foot-pounds per pound of air, then, ignoring clearance (the space between the piston and the cylinder head at the end of the stroke), the volume swept t through by the piston per minute = Wv=nV = WR- f whence P T , WRt , . TTrrr 00 nAr ^ SSOOON , SSQQQNRt 7= - ; and since TFE/=33,00(W, W = , and V = - ^ - np U ' nup 189. Compressive Cycle. For quiet running, as well as for other reasons, to be discussed later, it is desirable to arrange the valve movements so that some air is gradually compressed into the clear- ance space during the latter part of the return stroke, as along JSa, Fig. 57. This is accomplished by causing the exhaust valve to close at jE, the inlet valve opening at a. The work expended in this com- pression is partially recovered during the subsequent forward stroke, the air in the clearance space acting as an elastic cushion. 190. Actual Design. A single-acting 10-hp. air engine at 100 r. p. m., working between 114.7 and 14.7 lb. absolute pressure, with an " appar- ent " (Art. 450) volume ratio during expansion of 5 : 1 and clearance equal to 5 per cent of the piston displacement, begins to compress when the return stroke of the piston is -^ completed. .The expansion and compres- sion curves are PV 13 c. Assuming that the actual engine will give 90 per cent of the work theoretically computed, find the size of cylinder (diameter = stroke) and the free air consumption per Ihp.-hr. In Fig. 59, draw the lines ab and cd representing the pressure limits. "We are to construct the ideal PV diagram, making its enclosed length represent, to any convenient scale, the displacement of the piston per stroke. The extreme length of the diagram from the oP axis will be 5 per cent greater, on account of clear- ance. The limiting volume lines ef and gh are thus sketched in ; EC is plotted, making -^ = 5 ; the point E is taken so that =^? = 0.9, and EF drawn. Then ABCDEF is the ideal diagram. We have, putting Di = D t P A = P = H4.7. V c = V D = 1.05 D. =0.15 D. DESIGN OF AIR ENGINE 111 = 61.31. Work per stroke =jABi + iBCm - EDmk -jFEk D ,r T. x . P*V*-PoVG r> (V V\ PrVr-P*V* = PA( I B t A) -\ ~[ -^s( \ D VE) w _ ]_ = 144[(114.7 y, 0.20 -D) + ^ L ' x "*** ' ~^ - (14 7 x 0.9 D) - f 81 -" X - 5 ^ C 1 " X - 1S J) J = 5803.2 D foot-pounds. The actual engine will then give 0.9 x 5803.2 D = 5222.88 D foot-pounds per stroke or 5222.88 D x 100 foot-pounds per minute, which is to be made equal to 10 hp., or b 114.7 17.75 FIG. 59. Art. 190. Design of Air Engine. to 10 x 33,000 foot-pounds. Then 522,288 Z> = 330,000 and D = 0.63 cu. ft. Since the diameter of the engine equals its stroke, we write 0.7854 rf 2 x d 0,63 x 1728, where d is the diameter in inches; whence d = 11.15 in. To estimate the air consumption : at the point .B, the whole volume of air is 0.25 D. Part of this is clearance air, used repeatedly, and not chargeable to the engine. The clearance air at E had the vulueie V s and the pressure P E . If its 112 APPLIED THERMODYNAMICS behavior conforms to the law PF LS = c, then at the pressure of 1147 Ib. (point G) we would have _i The volume of fresh air brought into the cylinder per stroke is then 0.25 D - 0.0309 D = 0.2191 D or, per hour, 0.2191 x 0.63 x 100 x 60 = 828 cu. ft. Reduced to free air (Art. 187), this would be 828 x ^jy = 6450 cu. ft., or C45 cu, ft. per Ihp.-hr. (Compare Art. 192.) l 191. Effect of Early Compression. If compression were to begin at a suffi- ciently early point, so that the pressure were raised to that in the supply pipe before the admission valve opened, the fresh air would find the clearance space already completely filled, and a less quantity of such fresh air, by 0.05 D, instead of 0.0309 D, would be required. 192 Actual Performances of Air Engines. Kennedy (5) found a con- sumption of 890 cu. ft. of free air per Ihp.-hr., in a small horizontal steam engine. Under the conditions of Art. 183, the theoretical maximum work which this quantity of air could perform is 1.27 hp. The cylinder effi- ciency (Art. 215) of the engine was therefore 1.0-r- 1.27 = 0.79. With small rotary engines, without expansion, tests of the Paris compressed air system showed free air consumption rates of from 1946 to 2330 cu. ft. By working these motors expansively, the rates were brought within the range from 848 to 1286 cu. ft. A good reciprocating engine with, pre- heated air realized a rate of 477 cu. ft., corresponding to 36,4 lb. ? per brake horse power per hour. The cylinder efficiencies in these examples varied from 0.368 to 0.876, and the mechanical efficiencies (Art. 214) from 0.85 to 0.92. THE AIR COMPRESSOR 193. Action of Piston Compressor. Figure 60 represents the parts concerned in. the cycle of an air compressor. Air is drawn from the atmosphere through the spring check valve a, Ming the space Q in the cylinder. This inflow of air continues until the piston has reached its extreme right-hand position. On the return stroke, the valve a being closed, compres- sion proceeds until the pressure is slightly greater than that in the receiver D. The balanced outlet FIG co. Art. 103 valve 5 then opens, and air passes from Q to D Piston Compressor. J __ , , -\\r\- ,, at practically constant pressure. vV hen the pis- THE AIR COMPRESSOR 113 ton reaches the end of its stroke, there will still remain the clear- ance volume of air in the cylinder. This expands during the early part of the next stroke to the right, but as soon as the pressure of this air falls slightly below that of the atmosphere, the valve a again opens. 194. Cycle. An actual diagram is given, as ADCB) Fig. 61. Slight fluctuations in pressure occur, on account of fluttering through the valves, during discharge along AD and during suction along CB; the mean discharge FIG. ci. Art. 194. Cycle pressure must of course be slightly greater ir om P ressor - than the receiver pressure, and the mean suction pressure slightly less than atmospheric pressure. Eliminating these irregularities and the effect of clearance, the ideal diagram is adcb. 195. Form of Compression Curve. The remarks in Art. 180 as to the conditions of isothermal or adiabatic expansion apply equally to the compression curve BA. Close approximation to the isothermal path is the ideal of compressor per- formance. Let A, Fig. 62, be the point at which compression begins, arid let DE represent the maximum pressure to be attained. Let the cycle be completed through the states F, #. Then the work expended, if com- pression is isothermal, is v ACFG; if adiabatic, the FIG. 62. Arts. 195, 197, 2^18. -Forms of Compression work expe nded is -45^G. The same amount of air has been compressed, and to the same pressure, in either case; the area AEG represents, therefore, needlessly expended work. Furthermore, dur- ing transmission to the point at which the air is to be applied, in the great majority of cases, the air will have been cooled down practically to the temperature of the atmosphere ; so that even if compressed adia- batically, with rise of temperature, to B, it will nevertheless be at the state C when ready for expansion in the consumer's engine. If it there il APPLIED THERMODYNAMICS again expand adiabatically (along GH} instead of isothermally (along CA) 9 a definite amount of available power will have been lost, repre- sented by the area CI1A. t During compression, we aim to have the work area small ; during expansion the object is that it be large j the adiabatic path prevents the attainment of either of these ideals. The loss of power by adiabatic compression is so great that various methods are employed to produce an approximately isothermal path. As a general rule, the path is consequently intermediate between the iso- thermal and the adiabatic, a polytropic, pv n = 0. The relations derived in Arts. 183 and 185 for adiabatic expansion apply equally to this path, excepting that for y we must write n, the value of n being somewhere between 1.0 and 1.402, The effect of water in the cylinder, whether in- troduced as vapor with the air, or purposely injected, is to somewhat reduce the value of n, to increase the interchange of heat with the walls, and to cause the line FG, rig. 62, to be straight and vertical, rather than an adiabatic expansion, thus slightly increasing the capacity of the com- pressor, as shown in Art. 222. 196. Temperature Rise. The rise of temperature due to compression may be computed as in Art. 185. Un'der ordinary conditions, the air leaves the com- pressor at a tempeiature higher than that of boiling water. Without cooling devices, it may leave at such a temperature as to make the pipes red hot. It is easy to compute the (not very extreme) conditions under -which the rise in tern perature would be* sufficient to melt the cast-iron compressor cylinder. 197. Computation of Loss. The uselessly expended work during adiabatic (and similarly, during any other than isothermal) compression may be directly computed from the difference of the work areas CAKI and CBAKI, Fig, 62. The work under the isothermal is (jo, u, referring to the point C, and P, V, to the point -4), pv log e (V v) = pv log c (p P) ; while if Q is the volume at B, the work under ABC is = PV* and Q = F(-) V ; so that the percentage of loss corresponding to any ratio of initial and final pres- sures and any terminal (or initial) volume may be at once computed. 198. Basis of Methods for Improvement. Any value of n exceeding 1.0 for the path of compression is due to the generation of heat as the pressure rises, faster than the walls of the cylinder can transmit it to the atmosphere. The high temperatures thus produced introduce serious difficulties in lubrication. Economi- cal compression is a matter of air cooling; while, on the consumer's part, economy depends upon air heating. COMPEESSION CURVE 115 199. Air Cooling. In certain applications, where a strong draft is available, the movement of the atmosphere may be utilized to cool the compressor cylinder walls and thus to chill the working air during compression. While this method of cooling is quite inadequate, it has the advantage of simplicity and is largely employed on the air " pumps " which operate the brakes of railway trains. 200. Injection of Water. This was the method of cooling originally em- ployed at Mont Cenis by Colladon. Figure 63 shows the actual indicator card (Art. 484) from one of the older Colladon compressors. EP> CD is the coi responding ideal card with isothermal compression. The cooling by stream injection was evi- dently not very effective. Figure 61 rep- resents another diagram from a compressor in which this method of cooling was em- ployed ; oh representing the isothermal an*. ac the adiabatic. The exponent of the actual curve ad was 1.36; the gain over adiabatic compression was very slight. B/ introducing ths FIG. 03. FIG. Art. 200. Cooling by Jet Injection. Art. 20Q. Card from Colladon Compressor, water in a very fine spray, a somewhat lower value of the exponent was obtained in the compressors used by Colladon on the St. Gothard tunnel. Ganse and Post (6) have re- duced the value of n to 1.2G by an atomized spray. Figme 65 shows one of their diagrams, ab oeing the isothermal and ac the adiabatic. In all cases, spray injection is better than solid stream in- jection. The value n = 1.3(5, above given, was obtained when a solid jet of half-inch diameter was used. It is estimated that errors of the indicator may introduce an uncer- tainty amounting to 0.02 in the value of n. Piston leakage would cause an apparently low value. The comparative efficiency of spray injection is sho\vn from the fairly uniform temperature of dis- charged air, which can be maintained even with a varying speed of the compressor. In the Gause and Post experiments, with inlet air at 81 F., the temperature of dis- charge was 95 F. Spray injection has the objection that it fills the air with vapor, and it has been found that the orifices must be so small that they soon clog and become inoperative. The use of either a spray or FIG. 65. Art. 200. Cooling by Atomized Spray. a solid jet causes cutting of the cylinder and piston by the gritty substances carried in the water. In American practice the injection of water has been abandoned. 116 APPLIED THERMODYNAMICS 201. Water Jackets. These reduce the value of n to a very slight ex- tent only, but are generally employed "because of their favorable influence on cylinder lubrication. Gause and Post found that with inlet air at 81 F , and jackets on the barrels of the cylinders only (not on the heads), the temperature of the discharged air was 320 F. Cooling occurred dur- ing expulsion rather than during com- pression. The cooling effect depends largely upon the heat transmissive power of the cylinder walls, and the value of n consequently increases at cards With FIG. Art. 201.-CooliB by Jackets. are given in Fig. 66 ; ab being the isothermal and ac the adiabatic. more thorough cooling, jacketed heads, etc., a lower value of n may be obtained ; but this value is seldom or never below 1.3. Figure 67 shows a card given by Unwinfrom a Cockerill com- pressor, D O indicating the ideal isothermal curve. At the higher pressures, air is appar- ently more readily cooled; its own heat-conducting power is increased. D 1 FIG. 67. Art. 201. Cockerill Compressor with Jacket Cooling. 202. Heat Abstracted. In Fig. 68, let AB and AC be the adiabatic and the actual paths, An and CN adiabatics ; the heat to be abstracted is then equivalent to NO An = IAOL + nAIE - NCLK >v-PF , PV -. nAIEi = . 2/-1 FIG. 68. Arts. 202, 203. -Heat Ab- This is the heat to be abstracted per stracted by Cooling Agent. volume Fat pressure P > compressed to MULTI-STAGE COMPRESSION 117 p, expressed in foot-pounds. For isothermal compression; as along AD, IACL=pv log e (F-s-fl), and the total heat to be abstracted is measured by this area. If the path is adiabatic, AB, n = y, and the expression for heat abstraction becomes zero.* 203. Elimination of v. It is convenient to express the total area NCAn in tei ins of p, Pj and V only. The area FATT pv- pv - i PV W(i= ,-I--,^rT(^ Also, y-1 0-1 whence MM = 1 [() V - 1 ] + -^ - ^-(^. 204. Water Required. Let the heat to be abstracted, as above com- puted, be H 9 in heat units. Then if S and s are the final and initial temperatures of cooling water, and Q the weight of water circulated, we have C=H-r-(S s), the specific heat of water being taken as 1.0. In practice, the range of temperature of the cooling water may be from 40 to 70 F. 205. Multi-stage Compression. The effective method of securing a low value of n is by multi-stage operation^ the principle of which is illustrated in Fig. 69. Let A be the state at the beginning of compres- sion, and let it be assumed that the path is practically adiabatic, in spite of jacket cooling, as AB. Let AC be an isothermal. In multi-stage compression, the air follows the path AB up to a moderate pressure, as at ), and is then discharged and cooled Art. 205. Multi-stage Com- pression. at constant pressure in an external *' G- vessel, until its temperature is as nearly as possible that at which it was admitted to the cylinder. The path representing this cooling is DE. The air now passes to * More simply, as suggested by Chevalier, the specific heat along AC is s = 1 1^1^. (Art. 112) ; the heat to be abstracted is then, per Ib. of air n l ^B^!h working airatid cushion air must be cooled. 118 APPLIED THERMODYNAMICS a second cylinder, is adiabatically compressed along HF, ejected and cooled along ]?G/-, and finally compressed in still another cylinder along G-H. The diagram illus- trates compression in three " stages " ; but two or four stages are sometimes used. The work saved over that of single stage adiabatic compression is shown by the irregular shaded area HGrFUDB, equivalent to a re- duction in the value of n, under good conditions, from 1.402 to FIG. 70. Azts 205, 206. Two-stage Com- pressor Indicator Diagram. about 1 .25. Figure 70 shows the diagram from a two-stage 2000 hp. compressor, in which solid water jets were used in the cylinders. The cooling water was at a lower temperature than the inlet air, causing the point h to fall inside the isothermal curve AB. The compression curves in each cyl- inder give w = 1.36. Figure 71 is the diagram for a two-stage Biedler compressor with spray in- jection, AB being an isothermal and A an adiabatic. JFio 71. Arts. 205, 214. Two-stage Kledler Compressor Diagram. 206. Interceding. Some work is always wasted on account of the friction of the air passing through the intercooling device. In early compressors, this loss often more than outweighed the gain due to compounding. The area ghij, Fig. 70, indicates the work wasted from this cause. In this particular instance, the loss is exceptionally small. Besides this, the additional air friction through two or more sets of valves and ports, and the extra mechanical friction due to a multi- plication of cylinders and reciprocating parts must be considered. Multi-stage compression does not pay unless the intercooling is thoroughly effective. It seldom pays when the pressure attained is low. Incidental advantages in multi-stage operation arise from reduced mechanical stresses (Art. 462), higher volumetric efficiency (Art. 226), better lubrication, and the removal of moisture by precipita- tion during the intercooling. 207. Types of Intercoolers. The " external vessel " of Art. 205 is called the iatercooler. It consists usually of a riveted or cast-iron cylindrical shell, with cast- INTEKCOOLESTG 119 iron heads. Inside are straight tubes of brass or wrought iron, running between steel tube sheets. The back tube sheet is often attached to a stiff cast-iron inter- FIG. 72. Art. 207. Allis-Chalmers Horizontal Intercooler nal head, so that the tubes, sheet, and head are free to move as the tubes expand (Fig. 72). The air entering the shell sur- rounds the tubes and is compelled by baffles to cross the tube space on its way to the out- let. Any moisture precipitated is drained off at the pipe a. The water is guided to the tubes by internally projecting ribs on the heads, which cause it to circulate from end to end of the intercooler, several times. If of ample volume, as it should be, the intercooler serves as a receiver or storage tank. The one illustrated is mounted in a horizontal position. A vertical type is shown in Pig. 73. The funnel provides a method of ascertaining at all times whether water is flowing. i 208. Aftercoolers. In most manufacturing plants, the pres- ence of moisture in the air is ob- jectionable, on account of the difficulty of lubrication of air tools, and because of the rapid de- struction of the rubber hose used for connecting these tools with the pipe line. To remove the moisture (and some of the oil) p^. Ta . Art. 207.- Ingersoll-Seigeant Vertical present after the last stage of com- Intercooler, 120 APPLIED THERMODYNAMICS pression, and by cooling the air to decrease the necessary size of transmitting pipe, aftercoolers are employed. They are similar in design and appearance to mter- coolers. The cooling of the air deci eases its capacity for holding water vapor, and the latter is accordingly precipitated where it may be removed before the air has reached its point of utilization. An incidental advantage arising from the use of an aftercooler is the decreased expansive stress on the pipe line following the introduction of air at a more nearly noimal temperature. 209. Power Consumed. From Art. 98, the work under any curve pv n =PV n is, adopting the notation of Art. 202, pv J } The work along an adiabatic is expressed by the last formula if we make n = y = 1.402. The work of expelling the air from the cylinder after com- pression is pv. The work of drawing the air into the cylinder, neglecting AP\~"~ clearance, is PV=pv( } - The net work expended in the cycle is the algebraic sum of these three quantities, which we may write, It is usually more convenient to eliminate v } the volume after compres- sion. This gives the work expression, If pressures are in pounds per square inch, the foot-pounds of work per minute will be obtained by multiplying this expression by the number of working strokes per minute and by 144; and the theoretical horse power necessary for compression may be found by dividing this product by 33,000. If we make F=l, P=14.7, we obtain the power necessary to compress one cubic foot of free air. If the air is to be used to drive a motor, it will then, in most cases have cooled to its initial temperature (A.rt. 195), so that its volume after compression and cooling will be PV^-p. The work expended per cubic foot of this compressed and cooled air is then obtained by multiplying the work per cubic foot of free air by ^- - 210. Work of Compression. In some text-books, the work area under the compression curve is specifically referred to as the work of compression. This ig not the total work area of the cycle. RECEIVER PRESSURE 121 211. Range of Stages in Multi-stage Compression. Let the lowest pres- sure be g, the highest p, and the pressure during interceding P. Also let intercooling be complete, so that the air is reduced to its initial tempera- ture, so that the volume V after intercooling is ^, in which r is the volume at the beginning of compression in the first cylinder. Adopting the second of the work expressions just found, and writing z for n ~~ , we have n Work in first stage = 21 j /TV _ 1 j . Work in second stage = T( (&}' - 1 } = 2T ( (. Y_ 1 } . * \\PJ J * \\PJ J Total Differentiating with respect to P, we obtain w dP q\q Por a minimum value of W, the result desired in proportioning the pres- sure ranges, this expression is put equal to zero, giving P 2 =pq, or P = Vpq, or = f An extension of the analysis serves to establish a division of pressures for four-stage machines. From the pressure ranges given, it may easily be shown that in the ideal cycle the condition of rmnimmn work is that the amounts of work done in each of the cylinders be equal. The number of stages increases as the range of pressures increases; in ordinary prac- tice, the two-stage compressor is employed, with final pressures of about 100 Ib. per square inch above the pressure of the atmosphere. In low- pressure blowing engines,the loss due to a high exponent for the compres- sion curve is relatively less and these machines are frequently single stage. For three-stage machines, working between the pressures pi (low) and p 2 (high), with receiver pressures of PI (low) and P 2 (high), the conditions of minimum work are P2 ^PIP2 2 &&& Pi~^p2pi 2 , the amounts of work done in the three cylinders will be equal, and the cylinder volumes will be inversely as the suction pressures. 122 APPLIED THERMODYNAMICS ENGINE AND COMPRESSOR RELATIONS 212. Losses in Compressed Air Systems. Starting with mechanical power delivered to the compressor, we have the following losses (a) friction of the compressor mechanism, affecting the mechanical efficiency ; (b) thermodynamic loss, chiefly from failure to realize isothermal com- pression, but also from friction and leakage of air, clearance, etc., indicated by the cylinder efficiency; (c) transmissive losses in pipe lines ; (c?) thermodynamic losses at the consumer's engine, like those of (&) ; (e) friction losses at the consumer's engine, like those of (a). 213. Compressive Efficiency. While not an efficiency in the true sense of the term, the i elation of -work geueiated during expansion iii the engine to that ex- pended during compression in the compressor is sometimes called the compressive efficiency. It is the quotient of the areas FCTIG and FBA (9, Fig. 62. From the expression in Art. 209 for work under a polytropic plus work of discharge along BF or of admission along PC, we note that, the values o P andp being identical for the two paths, AB and CH< in question, the total work under either of these paths is a direct function of the volume V at the lower pressure P. In this case, providing the value of n be the same for both paths, the two work areas have the ratio V x, where Fis the volume at J, and x that at H. It follows that all the ratios of volumes LN - LIT, OQ - OP, etc , are equal, and equal to the ratio of areas. The compressive efficiency, then, = = T - t, where t is the temperature at A (or that at C% and I* that at II. For isothermal paths, T= t, and the com- pressive efficiency fs unity. In various testa, the compressive efficiency has ranged from 0.488 to 898. It depends, of course, on the value of n, increasing as n decreases. 214. Mechanical Efficiency. For the compressor, this is the quotient of work expended in the cylinder by work consumed at the flywheel; for the engine, it is the quotient of work delivered at the fly wheel by work done in the cylinder. Friction losses in the mechanism measure the mechanical inefficiency of the compressor or engine. With no friction, all of the power delivered would be ex- pended in compression, and all of the elastic force of the air would be available for doing work, and the mechanical efficiency would be 1.0. In practice, since compressors are usually directly driven from steam engines, with piston rods in common, it is impossible to distinguish between the mechanical efficiency of the compressor and that of the steam engine. The combined efficiency, in one of the best recorded tests, is given as 0.92. For the compressor whose card is shown in Fig. 71, the combined efficiency was 0.87. Kennedy reports an average figure of 0,845 (7). Uuwin states that the usual value is fiom 0.85 to 0.87 (8). These efficiencies are of course determined by comparing the areas of the steam and air indicator cards. 215. Cylinder Efficiency. The true efficiency, thermodynamically speaking, is indicated by the ratio of areas of the actual and ideal PV diagrams. For the PLANT EFFICIENCY 123 compressor, the cylinder efficiency is the ratw of the work done in the ideal cycle, without clearance, drawing in air at atmospheric pre sure, compressing it isothermally and discharging it at the constant receiver pressure, to the work done in the actual cycle of the same maximum volume It measures item (6) (Art. 212). It is not the "com- pressive efficiency " of Art. 213 For the engine, it is the ratw of the work done in the actual cycle to the work of an ideal cycle without clearance, with isothermal expan- sion tp the same maximum volume from the sameinitial volume, and with constant pressures during reception and discharge , the former leing that of the pipe line and the latter that of the atmosphere. Its value may range from TO to 0.00 in good machines, in gen- eral increasing as the value of n decreases. An additional influence is fluid fric- tion, causing, in the compressor, a fall of pressure through the suction stroke and a rise of pressure during the expulsion stroke ; a id in the engine, a fall of pressure during' admission and excessive Lack pressme during exhaust. All of these condi- tions alter the area of tlie PV cycle. In well-designed machines, these losses should be small. A large capacity loss in the cylinder is still to be considered. 216. Discussion of Efficiencies. Considering the various items of loss sug- gested in Art. 212, we find as average values under good conditions, (V) mechanical efficiency, 0.85 to 0.90; say 0.85; (5) cylinder efficiency of compressor, 0.70 to 0.90; say 0.80; (<?) transmission losses, as yet undetermined ; (d) cylinder efficiency of air engine, 0.70 to 90.0; say 0.70; (e) mechanical efficiency of engine, 0.80 to 0.90; say 0.80. The combined efficiency from steam cylinder to work performed at the con- sumer's engine, assuming no loss by transmission, would then be, as an average, 0.85 x 0.80 x 0.70 x 0.80 = 0.3808. For the Paris transmission system, Kennedy found the over-all efficiency (includ- ing pipe line losses, 4 per cent) to be 26 with cold air or 0.384 with preheated air, allowing for the fuel consumption in the preheaters (9). 217. Maximum Efficiency. In the processes described, the ideal efficiency in each case is unity. We are here deahng not with thermodynamic transformations between heat and mechanical energy, but only with transformations from one form of mechanical energy to another, in part influenced by heat agencies. No strictly thermodynamic transformation can have an efficiency of unity, ou account of the limitation of the second law. 218. Entropy Diagram. Figure 62 may serve to represent the com- bined ideal PV diagrams of the compressor (GABF) and engine (FGHGT). The quotient - is the compres&ive efficiency. The area representing net expenditure of work, that is, waste, is CBAH, bounded ideally by two 124 APPLIED THERMODYNAMICS adiabatics or in practice by two polytropics (not ordinarily isodiabatics) and two paths of constant pressure. This area is now to be illustrated on the TN coordinates. For convenience, we reproduce the essential features of Fig. 62 in Fig. 74. In Fig. 75, lay off the isothermal T, and choose the point A at random. Now if either T B or T H be given, we may complete the diagram. Assume that the former is given ; then plot the correspond- ing isothermal in Fig. 75. Draw AB, an adiabatic, BO and AS as lines of constant pressure FIG. Art. 218. Engine and Compressor Diagrams (n = k log e - J, the point O falling on the isothermal F. Then draw OB, an adiabatic, de T T termining the point S\ or, from Art. 213, noting that ^ = 4, we may find the point H di- rectly. If the paths AB and OH are not adia- batics, we may compute the value of the specific heat from that of n and plot these paths on Fig. 75 as logarithmic curves ; but if the values of n are different for the two paths, it no longer holds +1 f %B _ Zj. Tl FlG 75f Arts ' 218 ' 219 221. Compressed Air System, tnat J.ne area Entropy Diagram. OBAH in Fig. 75 now represents the net work expenditure in heat units. 219. Comments. As the exponents of the paths AB and OH decrease, these paths swerve into new positions, as AE, CD, decreasing the area representing work expenditure. Finally, with n = 1 9 isothermal paths, the area of the diagram becomes zero ; a straight line, OA. Theoretically, ENTROPY DIAGRAMS 125 with water colder than the air, it might be possible to reduce the tempera- ture of the air during compression, giving such a cycle as AICDA, or even, with isothermal expansion m the engine, AICA; in either case, the net work expenditure might be nega- tive; the cooling water accomplish- p ing the result desired. | ( j c E B 220. Actual Conditions. Under the more usual condition that the temperature of the air at admission to the engine is somewhat higher than that at which it is received by the compressor, we obtain Figs. 76, 77. T, T c and either T B or T H must now be given. The cycle in which the temperature is reduced during compression now appears FIG. 76. Art 220. Usual Combination of as AICDA or AIJA. 220. Usual Combination Diagrams. FIG. 77. Ait. 220. Combined Entropy Diagrams. 221. Multi-stage Operation. Let the ideal pv path be DECBA, Fig. 78. The "triangle" ABC of Fig. 75 is then replaced by the area DECBA, Fig. 79, bounded by lines of constant pressure and adiabatics. The area p F FIG. 78. Art. 221. Three-stage Com- pression and Expansion. FIG. 79. Art. 221. Entropy Diagram, Multi-stage Compression. 126 APPLIED THERMODYNAMICS saved is BFEC, which approaches zero as the pressure along CE } Pig. 78, approaches that along AB or at I), and becomes a maximum at an inter- mediate position, already determined in Art. 211. With inadequate inter cooling, the area representing work saved would be yFEx. Figures 80 and 81 represent the ideal pv and nt diagrams respectively for compressor and engine, each three-stage, with perfect intercooling and aftercooling and preheating and with no drop of pres- sure in transmission. BbA and AliB would be the diagrams with single-stage acliabatic compression and expansion. 6 Fm NO Art 22] Three-stage Compression and Expansion. FIG 81. Art 221. Thiee-stage Compression and Expansion. COMPRESSOK CAPACITY 222. Effect of Clearance on Capacity. Lei A BCD, Fig. 57, be the ideal pv dia- gram of a compressor without cleaiance. If there is clearance, the diagram will be aBCE; the air left in the cylinder at a will expand, nearly adiabatically, along , so that its volume at the intake pressure will be somewhat like DE. The total volume of fresh air taken into the cylinder cannot be DC as if there were no clearance, but is only EC. The ratio EC (Vc-V a ) is called the volumetric efficiency. It is the ratio of free air drawn in to piston displacement. 223. Volumetric Efficiency. This term is sometimes incorrectly applied to the factor 1 c, in which c is the clearance, expressed as a fraction of the cylinder volume. This is illogical, because this fraction measures the ratio of clearance air at final pressure, to inlet air at atmospheric pressure (Aa DC, Fig. 57) ; while the reduction of compressor capacity is determined by the volume of clearance air at atmospheric pressure. Jf the clearance is 3 per cent, the volumetric efficiency is much lew than 97 per cent. 224. Friction and Compressor Capacity, If the intake ports or pipes are small, an excessive suction will he necessary to draw in the charge, and the cylinder will VOLUMETRIC EFFICIENCY 127 !G be filled with air at less than atmospheric pressure. Its equivalent volume at atmospheric pressure "will then be less than that of the cylinder. This is shown in Fig. 82. The line of atmospheric pressure is DP, the capacity is reduced by FG, and the volumetric efficiency is DP HG. The capacity may be seriously affected from this cause, in the case of a badly designed machine. 225. Volumetric Efficiency ; Other Factors. Where jackets or water jets are used, the air is often somewhat heated during the intake stroke, increas- ing its volume, and thus, as in Art. 224, lowering the volumetric efficiency. The effect is more notice- able with jacket cooling, FIG 82 Art. 224. -Effect of Suction Friction. with which the cylinder walls often remain con- stantly at a temperature above that of boiling water. Tests have shown a loss of capacity of 5 per cent, due to changing from spray injection to jacketing. A high altitude for the compressor results in its being supplied with rarefied air, and this decreases the volumetric efficiency as based 011 air under standard pressure. At^an elevation of 10,000 ft. the capacity falls off 30 per cent. (See table, Art. 52a.) This is sometimes a matter of importance in mining applications also. Volumetric efficiency, in good designs, is principally a matter of low clearance. The clearance of a cylinder is practically constant, regardless of its length; so that its percentage is less in the case of the longer stroke compressors. Such compressors are com- paratively expensive. When water is injected into the cylinder, as is often the case in European practice, the clearance space may be practically filled with water at the end of the discharge stroke. Water does not appreciably expand as the pressure is lowered; so that in these cases the volumetric efficiency may be determined by the expression 1 c of Art. 223, being much greater than in cases where water injection is not practiced. 226. Volumetric Efficiency in Multi-stage Compression. Since the effect of multi-stage compression is to reduce the pressure range, the expansion of the air caught in the clearance space is less, and the dis- tance DE, Fig. 57, is reduced. This makes the volumetric efnciencjr, EC+ (V c V a ), greater than in single-stage cylinders. If FGH repre- sent the line of intermediate pressure, the ratio JE *- (V c 7 a ) is the gain in volumetric efficiency. 227. Refrigeration of Entering Air. Many of the advantages following multi- stage operation and intereooling have been otherwise successfully realized by the plan of cooling the air drawn into the compressor. This of course increases the density of the air at atmospheric pressure, and greatly increases the volumetric efficiency. Incidentally, much of the moisture is precipitated. At the Isabella furnace of the Carnegie Steel Company, at Etna, Pennsylvania, a plant of this 128 APPLIED THERMODYNAMICS kind has been installed. An ordinary ammonia refrigerating machine cools the air from 80 to 28 F. This should decrease the specific volume in the ratio (450 + 28) (459. G -f- 80) = 0.90. The free air capacity should consequently be increased in about this ratio (10). 228. Typical Values. Excluding the effect of clearance, a loss in ca- pacity of from 6 to 22 per cent has been found by Uiiwin (11) to be due to air friction losses and to heating of the entering air. Heilemann (12) finds volumetric efficiencies from 0.73 to 0,919. The volumetric efficiency could be precisely determined only by measuring the air drawn in and discharged. 229. Volumetric and Thermodynamic Efficiencies. The" volumetric effi- ciency is a measure of the capacity only. It is not an efficiency in the sense of a ratio of " effect " to " cause." In Pig. 83 the solid lines show an actual compressor diagram, the dotted lines, EGHB, the corresponding perfect diagram, with clearance and isothermal compression. In the actual case we have the wasted work areas, HLJQ, due to friction in discharge ports ; GQKD 9 due to non-isothermal compression; DFMC, due to friction during the suction of the air. At BHC, there is an area representing, apparently, a saving in work expenditure, due to the expansion of the clearance air; this saving in work has been accomplished, however, with a decreased capacity in the pro- portion BC-t-BE, a proportion which is greater than that of BHC to the total work area. Further, expansion of the clearance air is made possible as a result of its previous compression along 1PDK\ and the energy given up by expansion can never quite equal that expended in compression. The effect of excessive FIG. 8'X Art. 229. Volumetric and , . ,. , . ... , . ,, Thenuodynamic Efficiencies. friction during suction, reducing the capacity in the ratio DE -r- J3E, is usually more marked on the capacity than on the work. Both suction friction and clearance decrease the cylinder efficiency as well as the volumetric efficiency, but the former cannot be expressed in terms of the latter. In fact, a low volumetric efficiency may decrease the work expenditure absolutely, though not relatively. An instance of this is found in the case of a compressor working at high altitude. Friction during dis- charge decreases the cylinder efficiency (note the wasted work area HLJQ), but is practically without effect on the capacity. COMPRESSOR DESIGN 129 COMPRESSOR DESIGN 230. Capacity. The necessary size of cylinder is calculated much as in Art. 190. Let p, v, t, be the pressure, volume, and temperature of dis- charged air (v meaning the volume of air handled per minute), and P, F, T, those of the inlet air. Then, since jPF-s- T ' =f>v -s- t, the volume drawn into the compressor per minute is V=pvT-t- Pt } provided that the air is dry at both intake and delivery. If n is the number of revolutions per minute, and the compressor is double-acting, then, neglecting clearance, the piston displacement per stroke is V-*- 2 u = This computation of capacity takes no account of volumetric losses. In some cases, a rough approximation is made, as described, and by slightly varying the speed of the compressor its capacity is made equal to that required. Allowance for clearance may readily be made. Let the suction pressure be P 9 the final pressure p, the clearance volume at the final pressure of the piston displacement. Then, if expansion in the m clearance space follows the law pv n = PV n , the volume of clearance air at atmospheric pressure is of the piston displacement For the displacement above given, we there- fore write, zTi+i /ivm 2n '[_ m \mJ\Pj J This may be increased 5 to 10 per cent, to allow for air friction, air heating, etc. 231. Design of Compressor. The following data must be assumed : (a) capacity, or piston displacement, (Z>) maximum pressure, (c) initial pressure and temperature, (d) temperature of cooling water, (e) gas to be compressed, if other than air. Let the compressor deliver 300 cu. ft. of compressed air, measured at 70 F., per minute, against 100 Ib. gauge pressure, drawing its supply at 14.7 Ib. and 70 F., the clearance being 2 per cent. Then, ideally, the free air per minute will be 300 x (114.7 -r- 14.7) = 2341 cu. ft., or allowing 5 per cent for losses due to air friction and heating during the suction, 2341 -r- 0.95 = 2464 cu. ft. To allow for clearance, we may use the ex- pression in Art. 230, making the displacement, with adiabatic expansion of the clearance air, 130 APPLIED THERMODYNAMICS 2464+- [1-0.02 + 0.02] = 2640 cu. ft. Assuming for a compressor of this capacity a speed of 80 r. p. m., the necessary piston displacement for a double-acting compressor is then 2640 -r- (2 x 80) = 16.5 cu. ft. per stroke, or for a stroke of 3 ft., the piston area would be 792 sq. in. (13). The power expended for any assumed compressive path may be calculated as in Art. 190, and if the mechanical efficiency be assumed, the power necessary to drive the compressor at once follows. The assumption of clearance as 2 per cent must be justified in the details of the design. The elevation in temperature of the air may be calculated as in Art. 185, and the necessary amount of cooling water as in Art. 203, the exponents of the curves being assumed. 232. Two-stage Compressor. From Art. 211 we may establish an inter- mediate pressure stage. This leads to a new correction for clearance, and to a smaller loss of capacity due to air heating. Using these new values, we calculate the size of the first-stage cyliuder. For the second stage, the maximum volume may be calculated on the basis that intereoolitig is com- plete, whence the cylinder volumes are inversely proportional to the suc- tion pressures. The clearance correction will be found to be the same as in the low-pressure cylinder. The capacity, temperature rise, water con- sumption, power consumption, etc., are computed as before. A considera- ble saving in power follows the change to two stages. 233. Problem. Find the cylinder dimensions and power consumption of a double-acting single-stage air compressor to deliver 4000 cu. ft. of free air per minute at 100 Ib. gauge pres- sure at 80 r. p. m., the intake air being at 13.7 Ib. absolute pressure, the piston speed 640 ft. per minute, clearance 4 per cent, and the clearance- expansion and compression curves following the law FIG. 84 Art. 233. Design of Compressor. Lay off the distance Gff t Fig. 84, to repiesent the (un- known) displacement of the piston, which we will call D. Since the clearance is 4 per cent, lay off GZ = 0.04 D 9 determining as a coordinate axis. Draw the lines TU, VW, YX 7 representing the absolute pres- sures indicated. The compression curve 1 CE may now be drawn through C, and the clearance expansion curve DI through D. The ideal indicator diagram is CEDL We have, COMPRESSOR DESIGN 131 (P\0-7 P) V a = ( ^+= \ 1.04 D = 0.2158 D. 4 0.04 D = 0.1829 D, 1.04 D = 0.9872 ZX But j4-B = FB FA = 0. 8043 D is the volume of free air drawn into tlie cylinder : AB f7.E?"= 0.8043 is the volumetric efficiency:* to compress 4000 cu. ft. of free air per minute the piston displacement must then be 4000 0.8043 = 407^ cu. ft. per minute. Since the compiessor is double-acting, the necessary cylinder area is the quotient of displacement by piston speed or 4973 640, giving 7.77 sq. ft., or (neglecting the loss of area due to the piston rod), the cylinder diameter is 37.60 in. From the conditions of the problem, the strolce is 640 (2 x SO) = 4 ft. For the power consumption, we have W = GDEF + FECH - JICH - GDIJ 035 035 = 144*[(114.7x0.1758 )+ -(13.7 x O.S473)- ^ J 0.35 = 144 Z>[20.16 + 30.01 - 11.61 - 5.59] = 144 -D x 32,97. This is the work for a piston displacement = D cubic feet. If we take D at 4973 per mmute, the horse power consumed in compression is 144 x 32.97 x 4073 3300U ' = 715. 48.7 234. Design of a Two- stage Machine. With condi- tions as in the preceding, con- sider a two-stage compressor with complete interceding and a uniform friction of one pound between the stages. Here the combined diagrams appear as in Fig. 85. For economy of ' power, the intermediate pres- FIG. 85. Art. 23. Design of Two-stage Compressor. *This is not quite correct, because the air at J5 is not "free" air, i.e., air at atmospheric temperature. There is a slight rise in temperature between C and B, If T R is the atmospheric temperature, and b = - a = -, the volumetric efficiency is TR l-= -=~\ . If there is no cooling during discharge (along ED\ T A =T&, and \ & -A-/ the volumetric efficiency becomes -^(ba). 132 APPLIED THERMODYNAMICS sure is V114.7 x 13.7 = 39.64, whence the first-stage discharge pressure and the second-stage suction pressure, corrected for friction, are respectively 40.14 and 39.14 Ib. For thejirst stage, Fig. 85, P P = P Q = 40.14, P A = P = 14.7, P q = P M = 13.7, V H = 1-04 D, V F = 0.04 D. / p \ V4 / 1 ^ 7 \ 0.74 or V G = / V s = (^j 1 04 D = 0.4701 D* ' 74 0.04 D = 0.08864 tf. V P = (j'0 04 /> =0.08412 />, 74 / 73 \ n.74 / 1 o fr V 74 P, JV- = PjrIV-" or 7, = (JJ*) F* = (i|| J 1.04 D =0.987 D, The volumetric efficiency is jiJ3 - D = (V - FJ --D = 0.987 - 0.08412 = 0.90288. The piston displacement per minute is 4000 0.003 = 44SO. The piston diameter is V(4430 - 040) x 144 - 0.7854 = 35.6 in. for a stroke of 640 - (2 x 80) = 4 ft. The power consumptive for this first stage is, W = ^ 1 w 1 = [40.14(0.4701 - 0.04) + f*M*x <U701)-(18.7 x 1.M) , I- O.oo - 13.7(1.04 - 0.0886) - C 40 ' 14 x - M) " 3 ( 5 13 - 7 X - 0886 >]l44 D = 2348.64 D f oot-pounda or 10,404,475 foot-pounds per minute, equivalent to 315.3 horse power. SECOND STAGE Complete interceding means that at the beginning of compression in the sec- ond stage the temperature of the air will be as in the first stage, 70 F. The p volume at this point will then be V z = i-fV n = ~ 1.04 D =. 0.364 D. We thus Jr z oy.lJ. locate the point Z^ Fig. 85, and complete the diagram ZCE1, making V B = 0,04 (Fs-Fj?) =0.0141), Pc = <P# = 114.7, Pj=aP^=39.14, and compute as follows: y, = '0.3645= 0.3574 D. = 0.1642 D. or Fj-= ' K, = T' 0.014 2)= 0.0305 fl. \/ jl viU.!*/ or F/ = ^' 74 VB = (r^ ^ 0.014 D= 0.0311 D. *Note that ^ very nearly; so that = --?^-^; an approximation Pv. PH VH VQ v z YI which makes only one logarithmic computation necessary. COMPRESSOR DESIGN 133 The piston displacement is Vz VE = 035 D; the volumetric efficiency is the quo- tient of (Vx Vj) =0 3269 D by this displacement, or 0.934. For a stroke of 4 ft-, the cylinder diameter is \/[(0.35 D = 1550) -^640] X 144 -i- 0.7854=21 .05 in. The power consumption for this stage is qi47X0.1642)-f39.14X0.364> 0.35 - (39.14X0 3329; ,^4-7X0 OW-gQ 14XO.OB11)] =816 5 horse power. The total horse power for the two-stage compressor is then 631 8 and (within the limit of the error of computation) the work is equally divided between the stages. 235. Comparisons. We note, then, that in two-stage compression, the saving i-r-t e ftQO in power is ^ "^ = 12 of the power expended in single-stage compression; that the low-pressure cylinder of the two-stage machine is somewhat smaller than the cylinder of the single-stage compressor; and that, in the two-stage machine, the cylinder areas are (approximately) inversely proportional to the suction pressures. The amount of cooling water required will be found to be several times that neces- sary in the single-stage compressor. 236. Power Plant Applications. On account of the ease of solution of air in water, the boiler feed and injection waters in a power plant always carry a con- siderable quantity of air with them. The vacuum pump employed in connection with a condenser is intended to remove this air as well as the water. It is esti- mated that the waters ordinarily contain about 20 tunes their volume of air at atmospheric pressure. The pump must be of size sufficient to handle this air when expanded to the pressure in the condenser. Its cycle is precisely that of any ah* compressor, the suction stroke being at condenser pressure and the discharge stroke at atmospheric pressure. The water present acts to reduce the value of the exponent n, thus permitting of fair economy. 237. Dry Vacuum Pumps. In some modern forms of high vacuum apparatus, the air and water are removed from the condenser by separate pumps. The amount of air to be handled cannot be computed from the pressure and tempera- ture directly, because of the water vapor with which it is saturated. From Dai- ton's law, and by noting the temperature and pressure in the condenser, the pressure of the air, separately considered, may be computed. Then the volume of air, cal- culated as in Art. 236, must be reduced to the condenser temperature and pressure, and the pump made suitable for handling this volume (14) . COMMERCIAL TYPES OF COMPRESSING MACHINERY 238. Classification of Compressors. Air compressors are classified according to the number of stages, the type of frame, the kind of valves, the method of driving, etc. Steam-driven compressors are usually mounted as one unit with the steam cylinders and a single common fly wheel. ^Regulation is usually effected by varying the speed. The ordinary centrifugal governor on the steam cylinder im- poses a maximum speed limit; the shaft governor is controlled by the air pressure, which automatically changes the point of cut-off on the steam cylinder. Power- driven compressors may be operated by electric motor, belt, water wheel, or in APPLIED THERMODYNAMICS TYPES OF COMPRESSOR 135 other ways. They are usually regulated by means of an " unloading valve," which either keeps the suction valve closed during one or more strokes or allows the air to discharge into the atmosphere whenever the pipe lines aie fully supplied. In air lift practice, a constant speed is sometimes desire d, irrespective of the load. In the "variable volume" type of machine, the delivery of the compiessor is varied by closing the suction valve before the completion of the suction stroke. The air in the cylinder then expands below atmospheric pressure. 239. Standard Forms. The ordinary small compressor is a single-stage machine, with poppet air valves on the sides of the cylinder. The frame is of the " fork " pattern, with bored guides, or of the " duplex " type, with two single-stage cylinders. These machines maybe either steam or belt driven. The "straight line" compressors differ from the duplex in having all of the cylindeis in one straight line, regardless of their number. For high-grade service, in large units, the standard form is the cross-compound two-stage machine, the low-pressure steam and air cylinders being located tandem beside the high -pressure cylinders, and the air cylinders being outboard, as in Fig. 86. Ordinary standard machines of this class are built in capacities ranging up to 6000 cu. ft. of free air per minute. The other machines are usually con- structed only in smaller sizes, ranging down to as small as 100 cu. ft. per minute. Some progress has been made in the development of rotary compressors for direct driving by steam turbines. The efficiency is fully as high as that of an ordinary reciprocat- ing compressor, and the mechanical losses are much less. A paper by Rice (Jour. A. S. M. E. xxxiii, 3) describes a 6-stage turbo - machine at 1650 r p. m., direct- connected to a 4- stage steam turbine. With the low dis- charge p r e s- sure (15 Ib. gauge), num- erous stages and intercool- FIG. 87. Art. 240. Sommeiller Hydraulic Piston Compressor, ers, compression is practically isothermal. 240. Hydraulic Piston Compressors: Sommeiller's. In Fig. 87, as the piston F moves to the right, air is drawn through C to G, together with cooling water from B. On the return stroke, the air is compressed and discharged through D and A. Indicator diagrams are given in Fig. 88. 136 APPLIED THERMODYNAMICS The value of n is exceptionally low, and clearance expansion almost elimi- nated. This \vas the first commercial piston compressor, and it is still used to a PIG 88. Art, 240. Variable Discharge Pressure Indicator Diagrams, Sommeiller Compressor. limited extent in Europe, the large volume of water present giving effective! cool- ing. It cannot be operated at high speeds, on account of the inertia of the water. The Leavitt hydraulic piston compressor at the Calumet and Hecla copper mines, Michigan, has double-acting cylinders GO by 12 m., and runs at 25 i evolu- tions per minute, a compaiatively high speed. The value of n from the card shown in Fig. 89 is 1.23. ~~ 241. Taylor Hydraulic Compressor. "Water is conducted through a vei tical shaft at the necessary head (2 3 ft. per pound pressure) to a separating cham- FIG. 89. Art. 248 Cards from Leavitt Compressor. FIG. 90. Art. 241. Taylor Hydraulic ComDresssor. TYPES OF COMPRESSOR 137 her. The shaft is lined with a riveted or a cast-iron cylinder, and at its top is a dome, located so that the water flows downward around the inner circumference of the cylinder. The dome is so made that the water alternately contracts and expands during its passage, producing a partial vacuum, by means of which air is drawn in through numerous small pipes. The air is compressed at the tempera- ture of the water while descending the shaft. The separating chamber is so large as to permit of separation of the air under an inverted bell, from which it is led by a pipe. The efficiency, as compared with that theoretically possible in isothermal compression, is 60 to 70, some air being always carried away in solution. The initial cost is high, and the system can be installed only where a head of water is available. Figure 90 illustrates the device (15). The head of water must be at least equal to that corresponding to the pressure of air. The "cycle" of this type of compressor may be regarded as made up of two constant pressure paths and an isothermal, there being no clearance and no "valve friction." 242. Details of Construction. The standard form of cylinder for large machines is a two-piece casting, the working barrel being separate from the jacket, so that the former may be a good wearing metal and may be quite readily removable. Access to the jacket space is provided through bolt holes. On the smaller compressors, the poppet type of valve is frequently used for both inlet and discharge (Fig. 91). It is usually considered best to place these valves FIG. 9L Art. 242. Compressor Cylinder with Poppet Valves. (Clayton AJr Compressor Works.) in the head, thus decreasing the clearance. They are satisfactory valves for auto- matically controlling the point of discharge, excepting that they are occasionally 138 APPLIED THERMODYNAMICS noisy and uncertain in closing, and if the springs are made stiff for tightness, a con- siderable amount of power may be consumed in opening the valves. Poppet valves work poorly at very low pressures, and are not generally used for conti oiling the intake of air. Some form of mechanically opei a ted valve is preferably employed, such as the semi-rocking type of Fig. 92, located at the bottom of the cyhnder, which has poppet valves for the discharge at the top. For large units, Corliss inlet valves are usually employed, these being rocking cylindrical valves running crosswise. As in steam engines, they are so diiven from an eccentric and wrist plate as to give rapid opening and closing of the port, with a com- paratively slow interven- ing movement. They are not suitable for use as discharge valves in single- stage compressors, or in the high-pressure cylin- ders of multi-stage com- pressors, as they become fully open too late in the stroke to give a suffi- ciently free discharge. In Fig. 93 Corliss valves SUCTJON FIG. 92. Art 242 Compressor Cylinder with Rocking Inlet Valves. (Clayton Air Comprobsor Woiks ) are used for both inlet and discharge. The auxiliary poppet shown is used as a safety valve. FIG. 93. Art. 242. Compressor Cylinder with Corliss Yalves. (AUis-Ohalmers Oo.) COMPRESSED AIR TRANSMISSION 139 A gear sometimes used consists of Corliss inlet valves and mechanically operated discharge valves, which latter, though expensive, are free from the disadvantages sometimes experienced with poppet valves The closing only of these valves is mechanically controlled. Their opening is automatic, A common rule for proportioning valves and passages is that the average velocity of the air must not exceed 6000 ft. per minute. COMPRESSED Am TRANSMISSION 243. Transmissive Losses. The air falls in temperature and pressure in the pipe line. The fall in temperature leads to a decrease in volume, which is farther reduced by the condensation of water vapor; the fall in pressure tends to increase the volume. Early experiments at Mont Cenis led to the empirical formula F = 0.00000936 (n z l d), for a loss of pressure F in a pipe d inches in diameter, I ft. long, in which the velocity is n feet per second (1C). In the Paris distributing system, the main pipe was 300 mm. in diameter, and about f in. thick, of plain end cast iron lengths connected with rubber gaskets. It was laid partly under streets and sidewalks, and partly in sewers, involving the use of many bends. There were numerous drainage boxes, valves, etc., causing resistance to the flow ; yet the loss of pressure ranged only from 3.7 to 5.1 lb., the average loss at 3 miles distance being about 4.4 lb., these figures of course including leakage. The percentage of air lost by leakage was ascertained to vary from 0.38 to 1.05, including air consumed by some small motors which were unintentionally kept running while the measurements were made. This loss would of course be proportionately much greater when, tlie load was light. 244. TTnwin's Formula. Unwinds formula for terminal pressure after long transmission is commonly employed in calculations for pipe lines (17). It is. in which p = terminal pressure in pounds per square inch, P = initial pressure in pounds per square inch, f au experimental coefficient, u = velocity of air in feet per second, L = length of pipe in feet, d = diameter of (circular) pipe in feet, T = absolute temperature of the air, F. A simple method of determining/is to measure the fall of pressure under known conditions of P, , T, , and d 9 and apply the above formula. Unwin has in this way rationalized the results of Riedler's experiments on the Paris distributing system, obtaining values ranging from 0.00181 to 00449, with a mean value /= 0.00290. For pipes over one foot in diameter, he recommends the value 0.003 ; for 6-inch pipe,/= 0.00435; for 8-inch pipe,/ = 0.004. Biedler and Gutermuth found it possible to obtain pipe lengths as great as 10 miles in their experiments at Paris. Previous experiments had been made, on 140 APPLIED THERMODYNAMICS a smaller scale, by Stookalper. For cast-iron pipe, a harmonization of these experiments gives /= 0.0027(1 -f 0.3 e?), d being the diameter of the pipe in feet. The values of f for ordinary wrought pipe are probably not widely different. In any well-designed plant, the pressure loss may be kept very low. 245. Storage of Compressed Air. Air is sometimes stored at very high pres- sures for the operation of locomotives, street cars, buoys, etc. An important con- sequence of the principle illustrated in Joule's porous plug experiment (Art. 74) here comes into play. It was remarked in Art. 74 that a slight fall of temperatuie occurred during the reduction of pressure. This was expressed by Joule by the formula in which F was the fall of temperatuie in degrees Centigrade for a pressure drop of 100 inches of mercury when T was the initial absolute temperature (Centigrade) of the air. For air at 70 F., this fall is only l F., but when stored air at high pressure is expanded through a reducing valve for use in a motor, the pressure change is frequently so great that a considerable reduction of tempera- ture occurs. The efficiency of the process is very low ; Peabody cites an instance (IS) in. which with a reservoir of 7o cu. ft. capacity, carrying 450 Ib. pressure, with motors operating at 50 Ib. pressure and compression in three stages, the maximum computed plant efficiency is only 0.29. An element of danger arises in compressed air storage plants from the possibility of explosion of minute traces of oil at the high temperatures produced by compression. 246. Liquefaction of Air ; Linde Process (19). The fall of temperature accom- panying a reduction of pressure has been utilized by Linde and others in the manufacture of liquid air. Air is compressed to about 2000 Ib. pressure in a three-stage machine, and then delivered to a cooler. This consists of a double tube about 400 ft. long, arranged in a coil. The air from the compressor passes through the inner tube to a small orifice at its farther end, where it expands into a reservoir, the temperature falling, and returns through the outer tube of the cooler back to the compressor. At each passage, a fall of temperature of about 37J C. occurs. The effect is cumulative, and the air soon reaches a temperature at which the pressure will cause it to liquefy (Art. 610). 247. Refrigeration by Compressed Air. This subject will be more particularly considered in a later chapter. The fall of temperature accompanying expansion in the motor cylinder, with the difficulties which it occasions, have been men- tioned in Art. 185. Early in the Paris development, this drop of temperature was utilized for refrigeration. The exhaust air was carried through flues to wine cellars, where it served for the cooling of their contents, the production of ice, etc. In some 1 cases, the refrigerative effect alone is sought, the performance of wort during the expansion being incidental. (1) As text books on the commercial aspects of this subject Peele's Compressed Air Plant (John Wiley & Sons) and Wightman/s Compressed Air (American School of COMPRESSED AIR 141 Correspondence, 1909), may be consulted, (la) Riedler, Neue Erfahrungen uber die Kraftversorgung von Pans dmch Druckluft, Berlin, 1891. (2) Pernolet (L'Air Compnme) is the standard reference on this work. (3) Experiments upon Trans- mission, etc. (IdeU ed ), 1903, 98. (4) Unwin, op. at , 18 et seq. (5) Unwin, op. cit., 32 (6) Graduating Thesis, Stevens Institute of Technology, 1891. (7) Umvin, op. ait , 48. (8) Op cit , 109. (9) Unwin, op at., 48, 49; some of the final figures are deduced from Kennedy's data. (10) Power., February 23, 1909, p 382. (11) Development and Transmission of Power, 182 (12) Engineering News, March 19, 1908, 325. (13) Peabody, Thermodynamics, 1907, 378. (14) Ibid., 375. (15) Hiscox, Compressed Air, 1903, 273. (16) Wood, Thermodynamics, 1905, 306. (17) Transmission by Compressed Air, etc , 68; modified as by Peabody. (18) Thermodynamics, 1907, 393, 394 (19) Zeuner, Technical Thermodynamics (Klein); II, 303-313: Trans. A. S. M. E. t XXI, 156. SYNOPSIS OF CHAPTER IX The use of compressed cold air for power engines aud pneumatic tools dates from I860. The Air Engine The ideal air engine cycle is bounded by two constant pressure lines, one constant volume line, and a polytropic. In practice, a constant volume drop also occurs after expansion. Work formulas : -rr -,. _ PIT TT / , \ pv + pv log, -!-- gF; pv +^ ^ - qV- t pv log e - ; O-PF) -2- ) . v n i v \y ly Preheaters prevent excessive drop of temperature during expansion ; the heat em- ployed is not wasted. Cylinder volume = 33,000 NItt %n Up, ignoring clearance. To ensure quiet running, the exhaust valve is closed early, the clearance air acting as a cushion. This modifies the cycle. Early closing of the exhaust valve also reduces the air consumption. Actual figures for free air consumption range from 400 to 2400 cu. ft. per Uip-hr. Vie Compressor The cycle differs from that of the engine in having a sharp "toe 17 and a complete clear- ance expansion curve. Economy depends largely on the shape of the compression curve. Close approximation to the isothermal, rather than the adiabatic, should be attained, as during expan- sion in the engine. This is attempted by air cooling, jet and spray injection of water, and jacketing. Water required^ C= Multi-stage operation improves tfo compression curve most notably and is in other respects beneficial. Intercooling leads to friction losses but is essential to economy; must be thorough. 142 APPLIED THERMODYNAMICS Work, neglecting clearance (single cylinder), = T ^ r =-]; The area under the compression curve is called the ioork of compression. Minimum work, in two-stage compression, ih obtained when P 2 = qp. Engine and Compressor Relations Compressive efficiency : ratio of engine work to compressor work ; 0.5 to 0.9. Mechanical efficiency : ratio of work in cylinder and work at shaft , 80 to 0.90. Cylinder efficiency ratio of ideal diagram area and actual diagram area ; 0.70 to 0.90 Plant efficiency . ratio of work delivered by air engine to work expended at compressor shaft; 0.25 to 045 , tlieoietical maximum, 1.00. The combined ideal entropy diagram is bounded by tan constant pressure curves and two pulytropics. The economy of thorough mtercooling with multi-stage operation is shown , as is the importance of a low exponent for the polytropics. With very cold water, the net power consumption might be negative. Compressor Capacity Volumetric efficiency =ratio of free air drawn in to piston displacement; it is decreased by excessive clearance, suction friction, heating during suction, and installation at high altitudes. Long stroke compressors have proportionately less clearance. Water may be used to Jill the clearance space: multi-stage operation makes clearance less detrimental; refrigeration of the entering air increases the volumet- ric efficiency. Its value ranges ordinarily from 70 to 0.02. Suction friction and clearance also decrease the cylinder efficiency, as does discharge friction. Compressor Design Theoretical.pzstoft displacement per stroke ~ or including clearance, to be increased 5 to 10 per cent in practice. In a, multi-stage compressor with perfect interceding, the cylinder volumes are inversely as the suction pressures. The power consumed in compression may be calculated for any assumed compressive path. A typical problem shows a saving of 12 per cent by two-stage compression, The " vacuum pump" used with a condenser is an air compressor. Commercial Types of Compressing Machinery Classification is by number of stages, type of frame or valves, or method of driving. Governing is accomplished by changing the speed, the suction, or the discharge pressure. Commercial types include the single, duplex, straight line and cross-compound two-stage forms, the last having capacities up to 6000 cu, ft. per minute. Some progress has been made with turbo- compressors. Hydraulic piston compressors give high efficiency at low speeds. The Taylor hydraulic compressor gives efficiencies up to 0.60 or 0.70. PROBLEMS 143 Cylinder barrels and jackets are separate castings. Access to water space must be provided. Poppet, mechanical inlet, Corliss, and mechanical discharge valves are used. Compressed Air Transmission Loss in pressure = 0.00000936 n-l-rd In Paris, the total loss in 3 miles, including leakage, was 4.4 Ib. ; the percentage of leak- age was 0.3S to 1,05, including air unintentionally supplied to consumers. Unwin'*sformula; p = P\ l_-j^L_ 2 . Mean value of /= 0. 0029 /= 0.0027(1+ 0.3d). (9*79 ITV > --I Stored high pressure air may be used for driving motors, but the efficiency is low. The fall of temperature induced by throttling may be used cumulatively to liquefy air, The fall of temperature accompanying expansion m the engine may be employed f or refrigeration. PROBLEMS 1. An air engine works between pressures of 180 Ib. and 15 Ib. per square inch, absolute. Find the work done per cycle with adiabatic expansion fioni v = 1 to F 4, ignoring clearance. By what percentage would the work be increased if the expansion curve were PF 1 3 =c ? (Ans., 44,800 ft. Ib, 4.3 %.) 2. The expansion curve is PF 1 3 = c, the pressure ratio during expansion 7 : 1, the initial temperature 100 F. Find the temperature after expansion. To what tempera- ture must the entering air be heated if the final temperature is to be kept above 32 F. ? (Ans., -103 F., 310 F.) 3. Find the cylinder dimensions for a double-acting 100 hp. air engine with clear- ance 4 per cent, the exhaust pleasure being 15 Ib. absolute, the engine making 200 r. p. m., the expansion and compression curves being PF 135 c, and the air being received at 160 Ib. absolute pressure. Compression is carried to the maximum pres- sure, and the piston speed is 400 ft. per minute. A 10-lb. drop of pressure occurs at the end of expansion. (Allow a 10 per cent margin over the theoretical piston dis- placement.) (Ans., 13.85 ins. by 12.0 ins.) 4. Estimate the free air consumption per Ihp.-hr. in the engine of Problem 3. (Ans., 612cu.ft.) 5. A hydrogen compressor receives its supply at 70 F. and atmospheric pressure, and discharges it at 100 Ib. gauge pressure. Find the temperature of discharge, if the compression curve is PF 1 32 = c. (Ans., 412 F.) 6. In Problem 5, what is the percentage of power wasted as compared with iso- thermal compression, the cycles being like CBAD, Fig. 57 ? 7. In Problem 3, the initial temperature of the expanding air being 100 F., find what quantity of heat must have been added during expansion to make the path PF 1 36 c rather than an adiabatic. Assuming this to be added by a water jacket, the water cooling through a range of 70, find the weight of water circulated per minute. 8. Find the receiver pressures for minimum work in two and four-stage compres- sion of atmospheric air to gauge pressures of 100, 125, 150, and 200 Ib. 9. What is the minimum work expenditure in the cycle compressing free air at 70 F. to 100 Ib. gauge pressure, per pound of air, along a path PF 1 - 35 = c, clearance being ignored ? (Ans., 76,600 ft. Ib.) 10. Find the cylinder efficiency in Problem 3, the pressure in the pipe line being 165 Ib. absolute. (Ans., 62.5%.) 11. Sketch the entropy diagram for a four-stage compressor and two-stage air 144: APPLIED THERMODYNAMICS engine, in which n is 1.3 for the compressor and 1.4 for the engine, the air is inade- quately mtercooled, perfectly af tercooled, and inadequately preheated between the engine cylinders. Compaie with the entropy diagram for adiabatic paths and perfect interceding and such preheating as to keep the temperature of the exhaust above 32 F, 12. Find the cylinder dimensions and theoretical power consumption of a single- acting smgle-stage air compressor to deliver SOOO cu. ft. of free air per minute at ISO Ib. absolute pressure at GO r. p. m , the intake air being at 13 Ib. absolute press- ure, the piston speed 640 ft. pel minute, clearance 3 per cent, and the expansion and compression curves following the law PV 1 31 = c. (Ans , 80 by 64 in.) 13. "With conditions as in Problem 12, find the cylinder dimensions and power consumption if compression is in two stages, intercooling is perfect, and 2 Ib. of f ric- tiun loss occurs between the stages. (Ans., 74 by 38 by 64 in.) 14. The cooling water rising from *6S F. to 89 F. in temperature, in Art. 233, find the water consumption in gallons per minute. 15. Find the water consumption for jackets and intercoolmg in Art. 234 t the range of temperature of the water being from 47 to 68 F. 16. Find the cylinder volume of a pump to maintain 26" vacuum when pumping 100 Ib. of air per hour, the initial temperature of the air being 110 F , compression and expansion curves PT ri28 c, clearance 6 per cent., and the pump having two double-acting cylinders., The speed is 60 r. p. m. Pipe friction may be ignored. 17. Compare the liiedler and Gutermuth formula for / (Art. 244) with Unwin's values. What apparent contradiction is noticeable m the variation of / with d ? 18 In a compressed air locomotive, the air is stored at 2000 Ib. pressure and de- livered to the motor at 100 Ib. Find the temperature of the air delivered to the motor if that of the air in the reservoir is 80 F., assuming that the value of F (Art. 245J is directly proportional to the pressure drop. 19. \Vith isothermal curves and no friction, transmission loss, or clearance, what would be the combined efficiency from compiessor to motor of an air storage system m. which the storage pressure was 450 Ib. and the motor pressure 50 Ib.? The tem- perature of the air is 80 F. at the motor reducing valve. (Assume that the f ormula in Art. 245 holds, and that the temperature drop is a direct function of the pressure drop.) 20. Find by the Mont Gems formula, the loss of pressure in a 12-m. pipe 2 miles long 111 which the air velocity is 32 ft. per second. Compare with Unwin's formula, using the Eiedlcr and Gutermuth value for/, assuming P = 80, 2 T =70 F. 21. Find the free air consumption per Jhp,-hr. if the action of the engine in Art. 190 is modified as suggested in Ait. 191. 22. Find under what initial pressure condition, in Art. 183, an output of 1.27 Ihp. may theoretically be obtained from 890 cu. ft. of free air per hour, the exhaust pressure being that of the atmosphere, and the expansive path being (a) isothermal, (b) adiabatic. (/i?is., (a), 56 Ib. absolute ) 23. A compressor having a capacity of 500 cu. ft. of free air per minute (p= 14.7, t = 70) is requiied to fill a 700 cu. ft. tank at a pressure of 2500 Ib. per square inch. How long will this require, if the temperature in the tank is 140 at the end of the operation, and the discharge pressure is constant? 24. In Problem 10, what is the theoretical minimum amount of power that might be consumed, with no clearance and no abstraction of heat during compression? How does this compare vvith the power consumption in the actual case? 25. A Taylor hydraulic compressor (Art. 241), with water at 40 Q , compresses air to 50 Ib. gauge pressure. If the efficiency is 0.65 of that possible in isothermal compres- sion, rind the horse power consumed in compressing 4000 cu. ft. of free air per minute, CHAPTER X HOT-AIR ENGINES 248. General Considerations. From a technical standpoint, the class of air engines includes all heat motors using any permanent gas as a working substance. For convenience, those engines in which the fuel is ignited inside the cylinder are separately discussed, as internal combustion or gas engines (Chapter XI). The air engine proper is an external combustion engine, although in some types the products of combustion do actually enter the cylinder; a point of mechanical disadvantage, since the corro- sive and gritty gases produce rapid wear and leakage. The air engine employs, usually, a constant mass of working substance, i.e., the same body of air is alternately heated and cooled, none being discharged from the cylinder and no fresh supply being brought in; though this is not always the case. Such an engine is called a " closed " engine. Any fuel may be employed; the engines require little attention; there is no danger of explosion. Modern improvements on the original Stirling and Ericsson forms of air engine, while reducing the objections to those types, and giving excellent results in fuel economy, are, nevertheless, limited in their application to small capacities, as for domestic pumping service. The recent development of the gas engine (Chapter XI) has further served to minimize the importance of the hot-air cycle. In air, or any perfect gas, the temperature may be varied independ- ently of the pressure ; consequently, the limitation referred to in Art. 143 as applicable to steam engines does not necessarily apply to air engines, which may work at much higher initial temperatures than any steam en- gine, their potential efficiency being consequently much greater. When a specific cycle is prescribed, however, as we shall immediately find, pres- sure limits may become of importance. 249. Capacity. One objection to the air engine arises from the ex- tremely slow transmission of heat through metal surfaces to dry gases. This is partially overcome in various ways, but the still serious objection is the small capacity for a given size. If the Carnot cycle be plotted for one pound of air, as in Fig. 94, the enclosed work area is seen to be very small, even for a considerable range of pressures. The isothermals and adiabatics very nearly coincide. For a given output, therefore, the air en- gine must be excessively large at anything like reasonable maximum pres- sures. In. the Ericsson engine (Art. 269), for example, although the cycle was one giving a larger work area than that of Carnot, four cylinders were required, each having a diameter of 14 ft. and a stroke of 6 ft.; it 145 115 APPLIED THERMODYNAMICS was estimated that a steam engine of equal power would have required only a single cylinder, 4 ft in diameter and of 10-ft. stroke, running at 17 revolutions per minute and using 4 Ib. of coal per horse power per hour. The air engine ran at 9 r p. m v and its great bulk and cost, noisiness and rapid deterioration, overbore the advantage of a much lower fuel con- sumption, 1.S7 Ib. of coal per hp.hr. At the present time, with increased - g - 3~ 4 5 o 7 8 10 FIG 94. Arts. 249, 250. Carnot Cycle for Air. steam pressures and piston speeds, the equivalent steam engine would be still smaller. 250. Carnot Cycle Air Engine. The efficiency of the cycle shown in Fig 94 has already been computed as (T t) -*- T (Art. 135). The work done per cycle is, from Art. 135, -0 log, 2- -t) log. . K4 Another expression for the work, since POLYTROPIC CYCLE 147 But from Art. 104 ? *=(y-\ whence Pa=J>t j and Tr -- B ( z '-oi*. This can have a positive value only when - 1 ( }*-* exceeds unity ; which s ~P f T*\ y is possible only when =-i exceeds ( \ y ~ L . Now since P l and P 3 are the *s \t J limiting pressures in the cycle, and since for air y -f- (y 1) = 3.486, the minimum necessary ratio of pressures increases as the 3.486 power of the ratio of temperatures.* This alone makes the cycle impracticable. In Eig. 94, the pressure range is from 14.7 to 349.7 Ib. per square inch, although tlie temperature range is only 100. Besides the two objections thus pointed out large size for its capacity and extreme pressure range for its efficiency the Carnot engine would be distinguished by a high ratio of maximum to average pressure; a condition which would make friction losses excessive. 251. Polytropic Cycle. In Fig. 05, let T, t be two isothermals, el and dft\vo like polytropic curves, following the law pv n =. c, and ed arid bf two other like polytropic curves, following the law pu m = c. Then ebfd is a polytropic cycle. Let T, t, P b , P e n-l be given. Then T e = T\ " . In the en- tropy diagram, Fig. 96, locate the isothermals T, t, T e . Choose the point e at i andom. From Art. Ill, the specific heat along a path pv n c is and FIG. 96. Arts. 251, 256. Poly- tropic Cycle. FIG. 95. Arts. 251, 256, Prob. 4a. Polytropic Cycle. from Art. 163, the increase of entropy when the specific heat is s, in passing from e to &, is T N = s log,s . This permits of plotting the curve *It has been shown that ^= (-^ ) *~ . But P 3 <P^ if a finite work area is to v ~P fT\ vT be obtained; hence ^< ( ] . The efficiency of the Carnot cycle may of course be written as 1 "-ia i/-i 148 APPLIED THERMODYNAMICS el in successive short steps, in Fig, 96. Along ed, similarly, s 1 = / - ^) and rn \m I/ N^ = s, log e --^ between d and e. We complete the diagram by di awing bf and 2d df, establishing the point of intersection which determines the temperature at /. We find T f \ T b : : T d : T e . The efficiency is equal to ne of N , or to [nebx - ydfN - nedy~\ - [nebx , - f - T r j) the negative sign of the specific heat s x being disregarded. 252. Lorenz Cycle. In Fig. 97 let ^ and bh be adiabatics, and let the curves gb and <Zfe be polytropics, but unlike, the former having the exponent n, and the latter the exponent q. This constitutes the cycle of Lorenz. We find the tempera* FIG. 97. Arts. 252, 256, Prob, 5. Lorenz Cycle. FIG. 98. Arts. 252, 256. Lorenz Cycle, Entropy Diagram. tare at g as in Art. 251, and in the manner just described plot the curves gb and dh on the entropy diagram, Fig. 98, P g , P b , T b , T d , n and q being given, dg and bh of course appear as vertical straight lines. The efficiency is 253. Reitiinger Cycle. This appears as aug, Figs. 99 and 100. It is bounded by two isothermals and two like polytropics (isodiabatics). The Carnot is a special example of this type of cycle. To plot the entropy diagram, Fig, 100, we assume the ratio of pressures or of volumes along ai or cj. Let V a and 7^ be given. Then the gain of entropy from a to i is (p a V a lo&]r) +T. The curves ic and aj are JOULE AIR ENGINE 149 plotted for the given value of the exponent n. This is sometimes called the isodia- batic cycle. Its efficiency is f TJ J_ JT 77" T-T \ /" 77" _L 7" \ \ n <u ~r -"ic -fljc -H-aj) (/"ai r -Miejj which may be expanded as in Arts. 251, 252. p FIG. 99. Arts. 233, 256. Reit- linger Cycle. FIG. 100. Arts 253, 250, 257, 258 f 259. Reitlmger Cycle, Entropy Diagram. 254. Joule Engine. An air engine proposed by Ericsson as early as 1833, and revived by Joule and Kelvin in 1851, is shown in Fig. 101. A chamber contains air kept at a low temperature t by means of circulating water. Another chamber A contains hot air in a state of compression, the heat being supplied at a constant temperature T by means of an ex- ternal furnace (not shown). M is a pump cylinder by means of which air Fia. 101. Arts. 254, 255, 275. Joule Air Engine. may be delivered from C to A, and ^T is an engine cylinder in which air from A may be expanded so as to perform work. The chambers A and are so large in proportion to M and N that the pressure of the air in these chambers remains practically constant. 150 APPLIED THERMODYNAMICS The pump M takes air from (7, compresses it adiabatically, until its pressure equals that in A, then, the valve v being opened, delivers it to A at constant pressure. The cycle is fdoe, Fig. 102. In this special modification of the polytropic cycle of Art. 251, fd represents the drawing in of the air at con- stant pressure, do its adiabatic compression, and oe its discharge to A. Negative work is done, equal to the area fdoe. Concur- rently with this operation, hot FIG. 102. Arts. 254, 255, 256. Joule Cycle air has been flowing from A to through the valve u, then expand- ing adiabatically while u is closed ; finally, when the pressure has fallen to that in C, being discharged to the latter chamber, the cycle being ebqf, Fig. 102. Positive work has been done, and the net positive work per- formed by the whole apparatus is ebqf fdoe = obqd. 255. Efficiency of Joule Engine. We will limit our attention to the net cycle obqd. The heat absorbed along the constant pressure line ob is Hj J ='k(r T }. The heat rejected along qd is H qd = k(T q t). But fp rp rp _ t t from Art. 251, 2 = , whence, -=^- -=- = -=, , and the efficiency is t JL JL JLo t q -_^ _ T-T, T ~ T This is obviously less than the efficiency of the Carnot cycle between T and t. The entropy diagram may be readily drawn as in Tig. 103. The atmos- phere may of course take the place of the cold chamber C, a fresh supply being drawn in by the pump at each stroke, and the engine cylinder likewise discharging its contents to the atmosphere. The ratio fd -*- fq, FIG. 103. Arts. 255, 256, Joule Cycle, Entropy Diagram. in Pig. 102, shows the necessary ratio of volumes of pump cylinder and engine cylinder. The need of a large pump cylinder would be a serious drawback in practice ; it would make the engine bulky and expensive, and REGENERATOR 151 would lead to an excessive amount of mechanical friction. The Joule engine has never been constructed. 256. Comparisons. The cycles just described have been grouped in a single illustration in Fig. 104. Here we have, between the temperature limits T and , the Oarnot cycle, abed ; the polytropic cycle, debfi the Lorenz cycle, dglh ; that of Reit- linger^ aicj ; and that of Joule, obqd. These illus- trations are lettered to correspond with Figs. 95-100, 102, 103. A graphical demonstration that the. Carnot cycle is the one of maximum efficiency suggests itself. We now consider the most successful attempt yet made to evolve a cycle having a potential effi- ciency equal to that of Carnot. 257. Regenerators. By reference to Fig. 100, it may be noted that the heat areas under aj and ic are equal. The heat absorbed along the one path is precisely equal to that rejected along the other. This fact does not prevent the efficiency from being less than that of the Carnot cycle, for efficiency is the quotient of work done by the gross heat absorption. If, however, the heat under ic were absorbed not from the working substance, and that under ja were rejected FIG. 104. Arts. 256, 266. Hot-air Cycles. 152 APPLIED THERMODYNAMICS not to the condenser ; but if some intermediate body existed having a storage capacity for heat, such that the heat rejected to it along ja could be afterward taken up from it along ic, then we might ignore this quantity of heat as affecting the expression for efficiency, and the cycle would be as efficient as that of Carnot. The intermediate body suggested is called a regenerator. 258. Action of Regenerators. Invented by Robert Stirling about 1816, and improved by James Stirling, Ericsson, and Siemens, the present foim of regener- ator may be regarded as a long pipe, the walls of which have so large a capacity for heat that the temperature at any point remains practically constant. Through this pipe the air flows in one diiection when working along iY, Fig. 100, and in the other direction while working along ja. The air encounters a gradually changing temperature as it traverses the pipe. Let hot exhaust air, at i, Fig. 100, be delivered at one end of the regenerator. Its temperature begins to fall, and continues falling, so that when it 'leaves the regenerator its temperature is that at c, usually the temperature of the atmosphere. The temperatuie at the inlet end of the regenerator is then T, that at its outlet t. During the admission of fresh air, along;//, it passes through the regenerator in the opposite direction, gradually increasing in temperature from t to T 9 without appreciably affecting the temperature of the regenerator. Assuming the capacity of the regenerator to be unlimited, and that there are no losses by conduction of heat to the atmosphere or along the material of the regenerator itself, the process is strictly reversible. We may cause either the volume or the pressure to be either fixed or variable according to some definite law, during the regenerative move- ment. Usually, either the pressure 01 the volume is kept constant. As actually constructed, the regenerator consists of a mass of thin perforated metal sheets, so arranged as not to obstruct the flow of air. Some waste of heat always accompanies the regenerative process; in the steamer Ericsson, it was 10 per cent of the total heat passing through. Siemens appears to have reduced the loss to 5 per cent. 259. Influence on Efficiency. Any cycle bounded by a pair of isothermals and a pair of like polytropics (Reitlinger cycle), if worked with a regenerator, lias an efficiency ideally equal to that of the Carnot cycle. To be sure, the heated air is not all taken in at T, nor all rejected at t; but the heat absorbed from the source is all at I 7 , and that rejected to the condenser is all at t. The regenerative operations are mutually compensating changes which do not affect the general principle of efficiency under such conditions. The heat paid for is only that under the line ai, Fig. 100. The regenerator thus makes the efficiency of the Carnot cycle obtainable by actual heat engines. THE STIRLING ENGINE 153 As will appear, the cycles in which a regenerator is commonly employed are not otherwise particularly efficient. Their chief advantage is in the large work area obtained, which means increased capacity of an engine of given dimensions. For highest efficiency, the regenerator must be added. 260. The Stirling Engine. This important type of regenerative air engine was covered by patents dated 1827 and 1840, by Robert and James Stirling. Its action is illustrated in Fig. 105. G is the engine cylinder, containing the piston H, and receiving heated air through the pipe F from the vessel A A in which the air is alternately heated and cooled. The vessel A A is made \vith hollow walls, the inner lining being marked aa. The hemispherical lower portion of the lining is perforated ; while from A A up to CC the hollow space constitutes the regener- ator, being filled with strips of metal or glass. The plunger E fits loosely in the machined inner shell aa. This plunger is hollow and filled with some non-conducting material. The spaces DD contain the condenser, consisting of a coil of small copper pipe, through which water is circulated by a sepa- rate pump. An air pump discharges into the pipe F the necessary quantity of fresh air to compensate for any leakage, and this is utilized in some cases to maintain a pressure which is at all stages con- siderably above that of the atmosphere. The furnace is built about the ABA of the heating vessel. FIG. 105. Arts. 260, 361, 262, 263, 264. Stirling Engine. bottom 261. Action of the Engine. Let the plunger E and the piston H be in their lowest positions, the air above E being cold. The plunger E is raised, causing air to flow from X downward through the regenerator to the space 6, while H remains motionless. The air takes up heat from the regenerator, increasing its temperature, say to T, while the volume remains constant. After the plunger has come to rest, the piston H is caused to rise by the expansion produced by the absorption of heat from the furnace at constant temperature, the air reaching H by passing around the loose-fitting plunger E, which remains stationary. H now pauses in its "up" position, while E is lowered, forcing air through the regenerator from the lower space & to the upper space X, this air decreasing in temperature at con- stant volume. While E remains in its "down" position, H descends, forcing the air to the condenser D, the volume decreasing, but the temperature remaining con- stant at t. The cycle is thus completed. The working air has undergone four changes : (a) increase of pressure and temperature at constant volume, (&) expansion at constant tempera- ture, (c) a fall of pressure and temperature at constant volume, and (d) compression at constant temperature. 154 APPLIED THERMODYNAMICS 262. Remarks. With action as described, the piston II and the plunger E (sometimes called the " displacer pistou '') do not move at the same time , one is always nearly stationary, at or near the end of its stroke, while the other moves. In practice, uniform rotative speed is secured by modifying these conditions, so that the actual cycle merely approximates that described. The vessel A A is sometimes referred to as the "leceiver." It is obvious that a certain residual quantity of air is at all times contained in the spaces between the piston H and the plunger E. This does not pass through the regenerator, nor is it at any time subjected to the heat of the furnace. It serves merely as a medium for transmit- ting pressure from the "working air" to 77; and in contradistinction to that working substance, it is called " cushion air.*' Being at all times in communica- tion with the condenser, its temperature is constantly close to tlie minimum attained in the cycle. This is an important point in facilitating lubrication. 263. Forms of the Stirling Engine. In some types, a separate pipe is carried from the lower part of the receiver to the working cylinder G, Fig. 105. This removes the necessity for a loose-fitting plunger; in double-acting engines, each end of the cylinder is connected with the hot (lower) side of the one plunger and with the cold (upper) side of the other. In other forms, the regenerator has been a separate vessel ; in still others, the displacer plunger itself became the regen- erator, being perforated at the top and bottom and filled with wiie gauze. The Laubereau-Schwartzkopfl: engine (1) is identical in principle with the Stirling, excepting that the regenerator is omitted. The maintenance of high minimum pressure, as described in Art. 260, and the low ratio of maximum to average pressure, while not necessarily affecting the theo- retical efficiency, greatly increase the capacity, and (since friction losses are practi- cally constant) the mechanical efficiency as well. P \ \ FIG. 106. Arts. 204, 205, 267. Stirling Cycle. 264. Pressure-Volume Diagram. The cycle of operations described in Art. 261 is that of Fig. 106, ABCD* Considering the cushion air, the THE STIRLING ENGINE 155 actual diagram which would be obtained by measuring the pressures and volumes is quite different. Assume, for example, that the total volume of cushion air at maximum pressure (when E is at the top of its stroke and H is just beginning to move) is represented by the distance NE. Then if AT" be laid off equal to NE, the total volume of air present is NL Draw an isothermal EFHG, representing the path of the cushion air 3 sep- arately considered, while the temperature remains constant. Add its vol- umes, PF, ZH } QG, to those of working air, by laying off BK= PF, DM=ZH, CL=QG 9 at various points along the stroke. Then the cycle IKLM is that actually experienced by the total air, assuming the cushion air to remain at constant temperature throughout (Art. 262). The actual indicator diagrams obtained in tests are roughly similar to the cycle IKLM, Fig. 10G ; but the corners are rounded, and other distortions may appear on account of non-conformity with the ideal paths, sluggish valve action, errors of the indicating instrument, and various other causes. 265. Efficiency. The heat absorbed from the source along AB, Fig. 106, is e-^- That rejected to the condenser along CD is P^Folog^-^' The work done is the difference of these two quantities, YD and the efficiency is T-t T ' that of the Carnot cycle. Losses through the regenerator and by imper- fection of cycle reduce this in prac- tice. 266. Entropy Diagram. This is given in Fig. 108. T and t are the limiting isothermals, DA and BO the constant volume curves, along each of which the increase of en- tropy is n s= llQ%,(T-*rt\ I being the specific heat at constant volume. The gain of entropy along the iso- thermals is obtained as in Art. 253. Ignoring the heat areas EDAF and GCBH, the efficiency is ABCD + FABH, that of the Carnot cycle. The Stirling cycle appears in the PV diagram of Fig. 104 as dkbl. FIG. 108. Art. 266. Stirling Cycle, Entropy Diagram. 156 APPLIED THERMODYNAMICS 267. Importance of the Regenerator. Without the regenerator, the non- reversible Stirling cycle would have an efficiency of (P.- P,)^ log. ? *'vl This is readily computed to be far below that of the corresponding Carnot cycle. The advantage of the regenerative cycle lies in the utilization of the heat rejected along J3<7, Fig. 106, thus cancelling that item in the analysis of the cycle. Another way of utilizing this heat is to be described ; but while practical difficulties, probably insurmountable, limit progress in the application of the air engine on a commercial scale, the regenerator, upon which has been founded our modern metallurgical in- dustries as well, has offered the first possible method for the realization of the ideal efficiency of Carnot (2). 268. Trials. As early as 1847, a 50-hp. Stirling engine, tested at the Dun- dee Foundries, was shown to operate at a thermal efficiency of 30 per cent, esti- mated to be equivalent, considering the rather low furnace efficiency, to a coal con- sumption of 1.7 Ib. per hp.-hr. This latter result is not often surpassed by the aver- age steam engines of the present day. The friction losses in the mechanism were only 11 per cent (3). A test quoted by Peabody (4) gives a coal rate of 1.66 Ib., but with a friction loss much greater, about 30 per cent. There is no question as to the high efficiency of the regenerative air engine. 269. Ericsson's Hot-air Engine. In 1833, Ericsson constructed an unsuccess- ful hot-air engine in London. About 1855, he built the steamer Ericsson, of 2200 tons, driven by four immense hot-air engines. After the abandonment of this experiment, the same designer in 1875 introduced a third type of engine, and more recently still, a small pumping engine, which has been extensively applied. The principle of the engine of 1855 is illustrated in Fig. 109. B is the receiver, A the displacer, H the furnace. The displacer A fits loosely in B excepting near its upper portion, where tight contact is insured by means of packing rings. The lower portion of A is hollow, and filled with a non-conductor. The holes aa admit air to the upper surface of A. D is the compressing pump, with piston (7, which is connected FIG. 109. Arts. 2(>9, 270, 275. Ericsson Engine, with A by the rods dd. E is a pis- ton rod through which the de- veloped power is externally applied. Air enters the space above C through the check valve c, and is compressed during the up stroke into the magazine F ERICSSON ENGINE 157 through the second check valve e. G is the regenerator, made up of M'ire gauze. The control valves, worked from the engine mechanism, are at b and f. \Vhen b is opened, air passes from F through G to B, raising A. Closing of b at part completion of the stroke causes the air to work expansively foi the remainder of the stroke. During the return stroke of A, air passes through G, /, and g to the atmosphere. 270. Graphical Illustration. The PV diagram is given in Fig. 110. EBCF is the network diagram, ABCD being the diagram of the engine cylinder, AEFD that of the pump cylinder. Beginning with A in its lowest position, the state point in Fig. 110 is, for the engine (lower side of -.4), at A, and for the pump (upper side of C), at F. During about half the up stroke, the path in the engine is AB, air passing to B from the re- generator through s, and being kept at constant pressure by the heat from the furnace. During the second half of this stroke, the supply of air from the regenerator ceases, and the pressure falls rapidly as expansion occurs, but the heat im- parted from the furnace keeps the temperature practically constant, giving the isothermal path BC. Meanwhile, the pump, receiving air at the pressure of the atmosphere, has been fiist compressing it isothermally, or as nearly so as the limited amount of cooling surface will permit, along FE, and then discharging it through e at constant pressure, along EA, to the receiver F. On the down stroke, the engine steadily expels the air, now expanded down to atmospheric pressure, along the constant pressure line CD, while the pump simi- larly draws in air from the atmosphere at constant pressure along DF. At the end of this stroke, the air in F } at the state A^ is admitted to the engine. The ratio of pump volume to engine volume is FD DC, or FIG. 110. Arts. 270, 272, 273. Ericsson Cycle. T FIG. 111. Art. 271. Ericsson Cycle, Entropy Diagram. 271. Efficiency. The Ericsson cycle be- longs to the same class as that of Stirling, being bounded by two isothermals and two like polytropics ; but the polytropics are in this case constant pressure lines instead of constant volume lines. The net entropy diagram EBUF, Fig. Ill, is similar to that of the Stirling engine, but the isodiabatics swerve more to the right, since Jc exceeds l> while the efficiency (if a regenerator is employed) is the same as that rp of the Stirling engine, - 272. Tests, As computed by Rankine from Norton's tests, the effi- ciency of the steamer Ericsson's engines was 26.3 per cent; the efficiency of the furnace was, however, only 40 per cent. The average effecti v e pres- 158 APPLIED THERMODYNAMICS sure (EBCF-r- XC, Fig. 110) was only 2.12 Ib. The friction losses were enormous. A small engine of this type tested by the writer gave a con- sumption of 15.64 cu. ft. of gas (652 B. t. u. per cubic foot) per Ihp.-hr. ; equivalent to 170 B. t. u. per Ihp.-minute; and since 1 horse power = 33,000 foot-pounds =33,000 -*- 778 = 42.45 B. t. u. per minute, the thermodynamic efficiency of the engine was 42.45 -f- 170 = 0.25. 273. Actual Designs. In order that the lines FC and EB, Fig. 110, may be horizontal, the engine should be triple or quadruple, as in the steamer Ericsson, in which each of the four cylinders had its own compressing pump, but all were con- nected with the same receiver, and with a single crank shaft at intervals of a quarter of a revolution. Specimen indicator diagrams are given in Figs. 107, 112. FIG 107. Art. 273. Indicator FIG. 112. Art. 273. Indicator Card from Ericsson Engine. Diagram, Ericsson Engine. 274. Testing Hot-air Engines. It is difficult to directly and accurately meas- ure the limiting temperatures in an air engine test, so that a comparison of the actually attained with the computed ideal efficiencies cannot ordinarily be made. Actual tests involve the measurement of the fuel supplied, determination of its heating value, and of the indicated and eifective horse power of the engine (Art. 487). These data permit of computation of the thermal and mechanical efficiencies, the latter being of much importance. In small units, it is sometimes as low as 0.50. 275. The Air Engine as a Heat Motor. In nearly every large application, the hot-air engine has been abandoned on account of the rapid burning out of the heating surfaces due to their necessarily high temperature. Napier and Rankine (5) proposed an " air heater," designed to increase the transmissive efficiency of the heating surface. Modern forms of the Stirling or Ericsson engines, in small units, are comparatively free from this ground of objection. Their design permits of such amounts of heat-transmitting surface as to give grounds for expecting a much less rapid destruction of these parts. It has been suggested that exceLSsive bulk may be overcome by using higher pressures. (Zeuner remarks (6) that the bulk is not excessive when compared with that of a steam- engine with its auxiliary boiler and furnace). Rankine has suggested the introduction of a second com- pressed air receiver, in Fig. 109, from which the supply of air would be drawn through GJ and to which air would be discharged through/. This would make the engine a "closed" engine, in which the minimum pressure could be kept fairly high ; a small air pump would be required to compensate for leakage. A " con- denser " would be needed to supplement the action of the regenerator by more HOT-AIR ENGINES 159 thoroughly cooling the discharged air, else the introduction of " back pressure " would reduce the working range of temperatures. The loss of the air by leakage, and consequent waste of power, would of course increase with increasing pressures. Instead of applying heat externally, as proposed by Joule, in the engine shown in Fig. 101, there is no reason why the combustion of the fuel might not proceed within the hot chamber itself, the necessary air for combustion being supplied by the pump. The difficulties arising from the slow transmission of heat would thus be avoided. An early example of such an engine applied in actual practice was Cayley's (7), later revived by AVenham (8) and Buckett (9). In such engines, the working fluid, upon the completion of its cycle, is discharged to the atmos- phere. The lower limit of pressure is therefore somewhat high, and for efficiency the necessary wide range of temperatures involves a high initial pressure in the cylinder. The internal combustion air engine even in these crude forms may be regarded as the forerunner of the modern gas engine. (1) Zeuner, Technical Thermodynamics (Klein), 1907, I, 340. (2) The theoreti- cal basis of regenerator design appears to have been treated solely by Zeuner, op. cit,, I, 314-323. (3) Rankme, The Steam Engine, 1897, 368. (4) Thermodynamics of the Steam Engine, 1907, 302. (5) The Steam Engine, 1897, 370. (0) Op. eft., I, 381. (7) Nicholson's Art Journal, 1807; Min. Proc. Inst. C. E., IX. (8) Proc. Inxt. Mech. Eng., 1873. (9) Inst. Civ. Eng^ Heat Lectures, 1883-1884; Min. Proc. Inst. C. E., 1845,1854. SYNOPSIS OF CHAPTER X The hot-air engine proper is an external combustion motor of the open or closed type. The temperature of a permanent gas may be varied independently of the pressure ; this makes the possible efficiency higher than that attainable in vapor engines. 3-486 ; the Carnot cycle leads to either excessive pressures or an enormous ) cylinder. The poly tropic cycle is bounded by two pairs of isodiabatics. The Lorenz cycle is bounded by a pair of adiabatics and a pair of unlike polytropics. The Eeitlinger (isodiabatic) cycle is bounded by a pair of isothermals and a pair of isodiabatics. The Joule engine works in a cycle bounded by two constant pressure lines and two adiabatics ; its efficiency is ~" . The regenerator is a "fly wheel for heat." Any cycle bounded by a pair of iso- thermals and a pair of like polytropics, if worked with a regenerator, has an ideal efficiency equal to that of the Carnot cycle ; the heat rejected along one poly tropic is absorbed by the regenerator, which in turn emits it along the other polytropic, the operation being subject to slight losses in practice. The Stirling cycle, bounded by a pair of isothermals and a pair of constant volume curves : correction of the ideal PV diagram for cushion air : comparison with indi- cator card ; the entropy diagram ; efficiency formulas with and without the regen- erator ; coal consumption, 1,7 Ib. per hp.-hr. The Ericsson cycle, bounded by a pair of isothermals and a pair of constant pressure curves : efficiency from fuel to power, g$ per cent. 160 APPLIED THERMODYNAMICS By designing as "closed" engines, the minimum pressure may "be raised and the capacity of the cylinder increased. The air engine is unsatisfactory in large sizes on account of the rapid "burning out of the heating surfaces and the small capacity for a given "bulk. PROBLEMS (NOTE. Considerable accuracy in computation will he found necessary in solving Prob- lems 4 a and 5). 1. How much greater is the ideal efficiency of an air engine working "between tem- perature limits of 2900 F. and 600 F. than that of the steam engine described in Prob- lem 5, Chapter YI ? 2 Plot to scale (1 inch = 2 cu. ft. = 40 Ih. per square inch) the P 7" Carnot cycle for r=GOO, = 500 (both absolute) the lowest pressure being 14.7 Ib. per square inch, the substance being one pound of air, and the volume ratio during isothermal expansion being 12 C. 3. In Problem 2, if the upper isothermal be made 700 absolute, what will be the maximum pressure ? 4 a. Plot the entropy diagram, and find the efficiency, of a polytropic cycle for air between 000 F. and 500 F , in which m = 1.3, n = - 1.3, the pressure at d (Fig. 95) is 18 Ib. per square inch, and the pressure at e (Tig 95) is 22 Ib. per square inch. 4 6. In Art. 251, prove that 7> T b : : T d : T e , and also that P d P e : : P f : P io 5. Plot the entropy diagram, and find the efficiency, of a Lorenz cycle for air between 600 F. and 500 F., in which n = ~ 1.3, q = 0.4, the highest pressure being 80 Ib. per square inch and the temperature at g, Fig. 97, being 550 F. 6. Plot the entropy diagram, and find the efficiency, of a Reitlinger cycle between 000 F. and 500 F., when n = 1.3, the maximum pressure is 80 Ib. per square inch, the ratio of volumes during isothermal expansion 12, and the working substance one pound of air. rji rp 7. Show that in the Joule engine the efficiency is ^, Art. 255. 8. Plot the entropy diagram, and find the efficiency, of a Joule air engine working between C00 F. and 200 F., the maximum pressure being 100 Ib. per square inch, the ratio of volumes during adiabatic expansion 2, and the weight of substance 2 Ib. 9. Plot PFand NT diagrams for one pound of air worked between 3000 F. and 400 F. : (a) in the Carnot cycle, (&) in the Ericsson cycle, (c) in the Stirling cycle, the extreme pressure range being from 50 to 2000 Ib. per square inch. 10. Find the efficiencies of the various cycles in Problem 9, without regenerators. 11. Compare the efficiencies in Problems 4 a, 5, and 6, with that of the correspond- ing Carnot cycle. 12. Au air engine cylinder working in the Stirling cycle between 1000 F. and 2000 F., with a regenerator, has a volume of 1 cu ft. The ratio of expansion is 3. By what percentages will the capacity and efficiency be affected if the lower limit of pressure is raised from. 14.7 to 85 Ib. per square inch ? 18. In the preceding problem, one eighth of the cylinder contents is cushion air, at 1000 F, Plot the ideal indicator diagram for the lower of the two pressure limits, cor- rected for cushion air. HOT-AIR ENGINES 161 14. In Art. 268, assuming that the coal used in the Dundee foundries contained 14,000 B. t. u. per pound, what was the probable furnace efficiency? In the Peahody test, if the furnace efficiency was 80 per cent, and the coal contained 14,000 B. t. u., what was the thermal efficiency of the engine ? 15. What was the efficiency of the plant in the steamer Ericsson ? 16. Sketch the TJVand PF diagrams, within the same temperature and entropy limits, of all of the cycles discussed in this chapter, with the exception of that of Joule. "Why cannot the Joule and Ericsson cycles be drawn between the same limits ? Show graphically that in no case does the efficiency equal that of the Carnot cycle. 17. Compare the cycle areas in Problem 9. 18. In Problem 2, what is the minimum possible range of pressures compatible with a finite work area ? Illustrate graphically. 19. Derive a definite formula for the efficiency of the Eeitlinger cycle, Art. 253. 20. Derive an expression for the efficiency of the Ericsson cycle without a regenerator. CHAPTER XI GAS POWER, THE GAS PRODUCER 276, History. The bibliography (1) of internal combustion engines is exten- sive, although their commercial development is of recent date. Coal gas was dis- tilled as early as 1691 , the waste gases from blast furnaces were first used for heating in 1809. The first English patent for a gas engine approaching modern form was granted iu 1794. The advantage of compression was suggested as early as 1801 , but was not made the subject of patent until 1838 in England and 1861 in France. Lenoir, in 1800, built the first practical gas engine, which developed a thermal efficiency of 0.04. The now familiar polyti opic " Otto " cycle was pro- posed by Beau de Rochas at about this date. The same inventor called attention, to the necessity of high compression pressures in 1862 ; a principle applied in practice by Otto in 1874. Meanwhile, in 1S70, the first oil engine had been built. The four-cycle compressive Otto "silent" engine was brought out in 1876, show- ing a thermal efficiency of 0.15, a result better than that then obtained in the best steam power plants. If the isothermal, isometric, isopiestic, and adiabatic paths alone are considered, there are possible at least twenty-six different gas engine cycles (2). Only four of these have had extended development; of these four, only two have survived. The Lenoir (3) and Hugon (4) non-compressive engines are now represented only by the Bischoff (5) . The Barsariti " free piston " engine, although copied by Grilles and by Otto and Langen (1866) (6), is wholly obsolete. The variable vol- ume engine of Atkinson. (7) was commercially unsuccessful. Up to 1885, illuminating gas was commonly employed, only small engines were constructed, and the high cost of the gas prevented them from being com- mercially economical. Nevertheless, six forms were exhibited in 1887. The Priestman oil engine was built in 1888. ' With the advent of the Dowson process, in 1878, with its possibilities of cheap gas, advancement became rapid. By 1897, a 400-hp. four-cylinder engine was in use on gas made from anthracite coal. At the present time, double-acting engines of 5400 hp. have been placed in operation ; still larger units have been designed, and a few applications of gas power have been made even, in marine service. Natural gas is now transmitted to a distance of 200 miles, tinder 300 Ib. pres- sure. Illuminating gas has been pumped 52 miles. Martin (8) has computed that coal gas might be transmitted from the British coalfields to London at a delivered cost of 15 cents per 1000 cu. ft. His plan calls for a 25-inch pipe line, at 500 Ib. initial pressure and 250 Ib. terminal pressure, carrying 40,000,000,000 cu. ft. of 162 GAS POWER 163 gas per year. The estimated 46,000 hp. required for compression "would be derived from the waste heat of the gas leaving the retorts. Producer gas is even more applicable to heating operations than for power production. It is meeting with extended use in ceramic kilns and for ore roast- ing, and occasionally even for firing steam boilers. 277, The Gas Engine Method. The expression for ideal efficiency, (T t) -r- T, increases as T increases. In a steam plant, although boiler fur- nace temperatures of 2500 F. or higher are common, the steam passes to the engine, ordinarily, at not over 350 IT. This temperature expressed in absolute degrees limits steam, engine efficiency. To increase the value of T 9 either very high, pressure or superheat is necessary, and the practicable amount of increase is limited by considerations of mechanical fitness to withstand the imposed pressures or temperatures. In the internal com- bustion engine, the working substance reaches a temperature approximat- ing 3000 F. in the cylinder. The gas engine has therefore the same ad- vantage as the hot air engine, a wide range of temperature. Its working substance is, in fact, for the most part heated air. The fuel, which may be gaseous, liquid, or even solid, is injected with a proper amount of air, and combustion occurs within the cylinder. The disadvantage of the ordi- nary hot air engine has been shown to arise from the difficulty of trans- mitting heat from the furnace to the working substance. In this respect, the gas engine has the same advantage as the steam engine, large capa- city for its bulk, for there is*no transmission of heat; the cylinder is the furnace, and the products of combustion constitute the working sub- stance. A high temperature of working substance is thus possible, with large work areas on the pv diagram, and a rapid rate of heat propagation. In the gas engine, then, certain chemical changes which constitute the pro- cess described as combustion, must be considered ; although such changes are in gen- eral not to be included in the phenomena of engineering thermodynamics, 278. Fuels, (See Arts. 561, 561 a.) The common fuels are gases or oils. In* so;ne sections, natural gas is available. This is high in heating value, consisting mainly of methane, CH 4 . Carbureted water gas, used for illumination, is nearly as high in heating value, consisting of approximately equal volumes of hydrogen, carbon monoxide, and methane, with some methylene and traces of other substances. .Uncarbureted (blue) water gas is almost wholly carbon monoxide and hydrogen. Its heating value is less than half that of the carbureted gas. Both water gas and coal gas are uneconomical for power production; in the processes of manufacture, large quantities of coal are left behind as coke. Coal gas, consisting principally of hydrogen and methane, is slightly lower in heating value than carbureted water gas. It is made by distilling soft coal in retorts, about two thirds of the weight of coal becoming coke. Coke oven gas is practically the same product; the main output in its case being coke, while in the former it is gas. Producer gas (" Dowson " gas, " Mond " gas, etc.) is formed by the 164 APPLIED THERMODYNAMICS partial combustion of coal, crude oil, peat or other material, in air. It is essentially carbon monoxide? diluted with large quantities of nitro- gen and consequently low in heating value. Its exact composition varies according to the fuel from which it is made, the quantity of air supplied, etc. When soft coal is used, or when much steam is fed to the producer, large proportions of hydrogen are present. It is of no value as an illuminant. Blast furnace gas is producer gas obtained as a by-product on a large scale in metallurgical operations. It contains less hydrogen than ordinary producer gases, since steam is not employed in its manufacture, and is generally quite variable in its composition on account of the exigencies of furnace operation. Acetylene, C 3 H 2 , is made by combining calcium carbide and water. It has an extremely high heating and illuminating value. All hydiocarbonaceous substances maybe gasified by heating in closed vessels; gases have in this way been produced from peat, sawdust, tan bark, wood, garbage, animal fats, etc. 279. Oil Gases. Many liquid hydrocarbons may be vaporized by appropriate methods, under conditions which make them available for gas engine use. Some of these liquids must be vaporized by artificial heat and then immediately used, or they will again liquefy as their temperatures fall. The vaporizer or gt carburetor " is therefore located at the engine, where it atomizes each charge of fuel as required. Gasoline is most commonly used ; its vapor has a high heating value. Kerosene, and, more recently, alcohol, have been employed. By mixing gasoline and air in suitable proportions, a saturated or " carbureted " air is produced. This acts as a true gas, and must be mixed ^ ith more air bo permit of combustion. A gas formed in the proportion of 1000 cu. ft. of air to 2 gallons of liquid gasoline, for example, does not liquefy. A thiid form of oil gas is produced by heating certain hydrocarbons without air; the "cracking" process produces, first, less dense liquids, and, finally, gaseous bodies, which do not condense. The process must be carried on in a closed retort, and arrangements must be made for the removal of residual tar and coke. 280. Liquid Fuels. These have advantages over solid or gaseous fuels, aris- ing from the usually large heating value per unit of bulk, and from ease of trans- portation. All animal and vegetable oils and fats may be reduced to liquid fuels; those oils most commonly employed, however, are petroleum products. Crude petroleum maybe used; it is more customary to transform this to "fuel oil" by removing the moisture, sulphur, and sediment; and some of these "fuel oils*' are used in gas engines. Of petroleum distillates, the gaaolires are most commonly utilized in this country. They include an 80 liquid, too dangerous for commer- cial purposes; the 74 "benzine," and the 69 naphtha. "Distillate," an impure kerosene, from which the gasoline has not been removed, is occasionally used. Both grain alcohol (C 2 H 6 0) and wood alcohol (CH 4 0) have been used in gas en- gines (9). Various distillates from brown and hard coal tars have been employed in Germany. Their suitability for power purposes varies with different types of engines. The benzol derived from coal gas tar has been successfully used ; the brown coal series, C n H 2n , C n H 2n+2) C n H 2n _ 2 , contains many useful members (10). THE GAS PRODUCER 165 281. The Gas Producer. This essential auxiliary of the modern gas engine is made in a large number of types, one of which is shown in Fig. 113. This is a brick-lined cylindrical shell, set over a water-sealed pit P, on which the ash bed rests. Air is forced in by means of the steam jet blower A, being distributed by means of the conical hood B, from which FIG. 113. Art, 281. The Amsler Gas Producer. it passes up to the red-hot coal bed above. Here carbon dioxide is formed and the steam decomposes into hydrogen and oxygen. Above this " com- bustion zone" extends a layer of coal less highly heated. The carbon dioxide, passing upward, is decomposed to carbon monoxide and oxygen. The hot mixed gases now pass through the freshly fired coal at the top of the producer, causing the volatile hydrocarbons to distill off, the entire product passing out at C. The coal is fed in through the sealed hopper D. 166 APPLIED THERMODYNAMICS At E are openings for the bars used to agitate the fire. At F are peep- holes. An automatic feeding device is sometimes used at D. The air may be forced in by a blower, or sucked through by an exhauster, or by the engine piston itself, displacing the steam jet blower A. The fuel may be supported on a solid grate, or on the bottom of a producer without the water seal; grates may be either stationary or mechanically operated. Mechanical agitation may be employed instead of the poker bars inserted through E, Sometimes water gas, for illumination, and producer gas, for power, are made in the same plant. Two producers are then employed, the air blast being applied to one, while steam is decomposed in the other. Provision must be made for purifying the gas, by deflectors, wet and dry scrubbers, filters, coolers, etc. For the removal of tar, which would be seriously objectionable in engines, mechanical separation and washing are useful, but the complete destruction of this substance involves the passing of the gas through a highly heated chamber; this may be a portion of the producer itself, as in " under-feed," " inverted combustion/ 7 or " down-draft " types : causing the trans- formation of the tar to fixed gases. On account of the difficulty of tar removal, anthracite coal or coke or semi-bituminous, non-caking coal must generally be used in power plants. The air supplied to the producer is sometimes preheated by the sensible heat of the waste gases, in a " recuperator." The " regenerative " prin- ciple heating the air and gas delivered to the engine by means of the heat of the exhaust gases is inapplicable, for leasons which will appear. 282. The Producer Plant. The ordinary producer operates under a slight piessure; in the suction type, now common in small plants, the engine piston draws air through the producer in accordance with the load requirements. Pres- sure producers have been used on extremely low grade fuels : Jahn, in Germany, has, it is reported, gasified mine waste containing only 20 per cent of coal. Suc- tion producers, requiring much less care and attention, are usually employed only on the better grades of fuel. Most producers require a steam blast; the steam must be supplied by a boiler or " vaporizer," which in many instances is built as a part of the producer, the superheated steam being generated by the sensible heat carried away in the gas. Automatic operation is effected in various ways: in the Amsler system, by changing the proportion of hydrogen in the gas, involving control of the steam supply ; in the Pintsch process, by varying the draft at the producer by means of an inverted bell, under the control of a spring, from beneath which the engine draws its supply; and in the Wile apparatus, by varying 1 the drafb by means of valves operated from the holder. Figure 114 shows a complete producer plant, with separate vaporizer, economizer (recuperator), and holder for storing the gas and equalizing the pressure. 283. By-product Recovery. Coal contains from 0.5 to 3 per cent of nitrogen, about 15 per cent of which passes off in the gas as ammonia. The successful development of the Mond process has demonstrated the possibility of recovering this in the form of ammonium sulphate, a valuable fertilizing agent. THE GAS PRODUCER 167 168 APPLIED THERMODYNAMICS 284. Action in the Producer. Coal is gasified on the producer grate. In suction producers, the rate of gasification may be anywhere between 8 and 50 Ib. per sq ft. of grate per hour. Anthracite pro- ducers are in this country sold at a rating of 10 to 15 Ib. Ideally, the coal is carbon, and leaves the producer as carbon monoxide, 4450 B. t. u. per pound of carbon having been expended 111 gasification. Then only 10,050 B. t. u. per pound of carbon are present in the gas, and the efficiency cannot exceed 10,050 -*- 14,500 = 0.694. The 4450 B. t. u. con- sumed m gasification are evidenced only in the temperature of the gas. With actual conditions, the presence of carbon dioxide or of free oxygen is an evidence of improper operation, further decreasing the efficiency. By introducing steam, however, decomposition occurs in the producer, the tem- perature of the gas is reduced, and available hydrogen is carried to the engine ; and this action is essential to producer efficiency for power pur- poses, since a high temperature of inlet gas is a detriment rather than a benefit in engine operation. The ideal efficiency of the producer may thus be brought up to something over 80 per cent; a limit arising when the proportion of steam introduced is such as to reduce the temperature of the gas below about 1800 F., when the rate of decomposition greatly decreases. The proportion of steam to air, by weight, is then about 6 per cent, the heating value of the gas is increased, the percentage of nitrogen decreased, and nearly 20 per cent of the total oxygen delivered to the producer has been supplied by decomposed steam. A similar result may be attained by introducing exhausted gas from the engine to the producer. The carbon dioxide in this gas decomposes to monoxide, which is carried to the engine for further use. This method is practiced in the Mond system, and has had other applications. To such extent as the coal is hydrocarbonaceous, however, the ideal efficiency, irrespective of the use of either steam or waste gas, is 100 per cent. Figure 115 shows graphically the results com- puted as following the use of either steam or waste gases with pure car- bon as the fuel. The maximum ideal efficiency is about 3 per cent greater when steam is used, if the temperature limit is fixed at 1800 F., but the waste gases give a more uniform (though less rich) gas. The higher ini- tial temperature of the waste gases puts their use practically on a parity with that of steam. Either system tends to prevent clinkering. The maximum of producer efficiency, for power gas purposes, is ideally from 5 to 10 per cent less than that of the steam boiler. High percentages of hydrogen resulting from the excessive use of steam may render the gas too explosive for safe use in an engine (10 a) (25). 285. Example of Computation. Let 20 per cent of the oxygen necessary for gasifying pure carbon be supplied by steam. Each pound of fuel requires 1J- Ib. of oxygen for conversion to carbon monoxide. Of this amount, 0.20 x !$= 0.2666 Ib. will then be supplied by steam ; and the balance, 1.0667 Ib., will be derived from PRODUCER EFFICIENCY 169 the air, bringing in with it Jxi 0667=3 57 Ib. of nitrogen. The oxygen derived from steam will also carry with it 4X02666=0.0333 Ib. of hydrogen. The pro- duced gas will contain, per pound of carbon, 2 33 Ib. carbon monoxide, 3 57 Ib. nitrogen, 0.0333 Ib. hydrogen. Waste GassupphedjPercentageof Fuel gasified by Weight _g 109 202 256 382 J I I I 34 5 6 7 8 9 10 It tZ 13 M- 15 16 17 Percentage of Steam by Weight.- FIG. 115. Art. 284. Reactions in the Producer. The heat evolved in burning to monoxide is 4450 B. t. u. per pound. A por- tion of this, however, has been put back into the "gas, the temperature having been lowered by the decomposition of the steam. Under the conditions existing in the 170 APPLIED THERMODYNAMICS producer, the heat of decomposition is about 62,000 B t u per pound of hydrogen. The net amount of heat evolved is then 4450 - (0,0333 X 62,000) = 2383 B. t. u., and the efficiency is ' ~" = 0.84. The rise in temperatme is computed as li,t)UU follows : to heat the gas 1 F. there are required SPECIFIC HEAT For carbon monoxide, 2.33 X 0.2479 = 0.378 B. t. u. For nitrogen, 3.57 X 0.2438 = 0.800 15. t u. For hydrogen, 0.0333 x 3.4 = 113 B. t. u. a total of 1.500 B. t. u. The 2383 B. t. u. evolved will then cause an elevation of temperature of . 2 3?3 = 1527 F. 1.560 With pure air only, used for gasifying pure carbon, the gas would consist of 2J Ib. of carbon monoxide and 4.45 Ib. of nitrogen ; the percentages being 34.5 and 65.5. For an actual coal, the ideal gas composition may be calculated on the assumptions that the hydrogen and hydrocarbons pass oif unchanged, and that the carbon requires 1J times its own weight of oxygen, part of which is contained in the fuel, and part derived from steam or from the atmosphere, carrying with it hydrogen or nitrogen. Multiplying the weight of each constituent gas in a pound by its calorific value, we have the heating value of the gas. As a mean of 54 analyses, Fernald finds (11) the following percentages ly volume : Carbon monoxide (CO) ............ 19.2 Carbon dioxide (COj) ............. 9.5 Hydrogen (H) ............... 12.4 Marsh gas and ethylene (CH 4 , C 2 H 4 ) ....... 3.1 Nitrogen (N) ................ 55.8 100.0 285 a. Practical Study of Producer Reactions. This subject has presented unexpected complications. Tests made by Allcut at the University of Birming- ham (Power, July 18, 1911, page 90) call attention to three characteristic processes : C + H 3 = CO + H a , (A) C -f 2 H 2 = C0 2 + 2 H 2 , (5) CO + H 3 O = C0 3 + H 2 . (C) Of these, (-4) takes place at temperatures above 1832, is endothermic, and results m the absorption of 4300 B. t, u. per pound of carbon. The corresponding figure for reaction (), also endothermic, which occurs at temperatures below 1112, is 2820 B. t. u. The former of the two is the reaction desired, and is facilitated by high temperatures. The operation (C) is chemically reversible j taking place as stated at temperatures above 932, but gradually reversing to the opposite (and preferred) transformation when the temperature reaches 1832. The tests show that increasing proportions of C0 2 may be associated with increasing proportions of steam introduced. The maximum decomposition reached was 0.535 Ib. of steam per pound of anthracite pea coal, at 1832 F. The maxi- THE GAS ENGINE 171 FIG. 116. Art. 287. Single-acting Gas Engine, Four Cycle. (Prom " The Gas Engine, 1 ' by Cecil P. Poole, with the permission of the Hill Publishing Company ) FIG. 117. Art. 288. Piston Movements, Otto Cycle. (From "The Gas Engine," by Cecil P. Poole, with the permission of the Hill Publishing Company.} 172 APPLIED THERMODYNAMICS mum heat value iu the gas was obtained when 0.72 Ib. of steam was introduced (only 0.52 Ib. of which was decomposed) per pound of coal. If we take the ratio of air to coal by weight at 9 Ib., the ratio of steam decomposed to air supplied at highest heat value and heat efficiency is 0.52 -r- 9.0 = 0.058 ; approximately 6 per cent, as in Art. 284. An interesting study of the principles involved may be found in Bulletins of the University of Illinois ; vi, 16, by J. K. Clement, On the Rate of Formation of Carbon Monoride in Gas Producers, and is., 2i, by Garland and Kratz, Tests of a Suction Gas Producer. 286. Figure of Merit. A direct and accurate determination of efficiency is generally impossible, on account of the difficulties in gas measurement (12). For comparison of results obtained from the same coals, the figure of merit is sometimes used. This is the quotient of the heating value per pound of the gas by the weight of carbon in a pound of gas : it is the heating value of the gas per pound of carbon contained. In the ideal case, for pure carbon, its value would be 10,050 B. t. u. For a hydrocarbonaceous coal, it may have a greater value. GAS ENGINE CYCLES 287. Four-cycle Engine. A gas engine of one of the most commonly used types is shown in Fig. 116. This represents a single-acting engine; i.e. the gas is in contact with one side of the piston only, the other end being open. Large en- gines of this type are frequently made double-acting, the gas being then con- tained on both sides of a piston moving in an entirely closed cylinder, exhaust occurring on one side while some other phase of the cycle is described on the other side. 288. The Otto Cycle. Figure 117 illustrates the piston move- ments corresponding to the ideal pv diagram of Fig, 118. The cycle includes five distinctly marked paths. During the out stroke of the piston from position A to position jB, Fig. 117, gas is sucked in by its movement, giving the line 5, Fig. 118. During the next in- ward stroke, B to 9 the gas is com- pressed, the valves being closed, along the line Ic. The cycle is not yet completed : two more strokes are necessary. At the beginning Fio.ll*. Arts.SSS.m-TheOttoCrcle. being at c, Fig. 118, the gas is ignited and practically instantaneous combustion ogqurs at constant volume, giving the line <?(7, An out THE TWO-CYCLE ENGINE 173 stroke is produced, and as the valves' remain closed, the gas expands, doing work along Cd, while the piston moves from Q to -D, Fig. 117. At d) the exhaust valve opens, and during the fourth stroke the piston moves in from D to J?, expelling the gas from the cylinder along de, Fig. 118. This completes the cycle. The inlet valve has been open from a to 5, the exhaust valve from d to e. During the remainder of the stroke, the cylinder was closed. Of the four strokes, only one was a " working " stroke, in which a useful effort was made upon the piston. In a double-acting engine of this type, there would be two working strokes in every four. FIG. 110. Arts. 1289-201, -TO, 3#J. Two-cycle Gas Engine. (From "The Gas Engine," t>y Cecil P Poole, with the permission of the Hill Publishing Company ) 289. Two-stroke Cycle. Another largely used type of engine is shown in Fig. 119. . The same five paths compose the cycle ; but the events are now crowded into two strokes. The exhaust opening is at E ; no valve is necessary. The inlet valve is at A, and ports are provided at C, and /. The gas is often delivered to the engine by a separate pump, at a pressure several pounds above that of the atmosphere ; in this engine, the otherwise idle side of a single-acting piston becomes itself a pump, as will appear. Starting in the position shown, let the piston move to the left. It draws a supply of combustible gas through A, B and the ports into the chamber D. On the outward return stroke, the valve A closes, and the gas in D is compressed. Compression continues until the edge of the piston passes the port I, when this high pressure gas rushes into the space F : at 174 APPLIED THERMODYNAMICS practically constant pressure. The piston now repeats its first stroke. Following the mass of gas which we have been considering, we find that it undergoes compression, beginning as soon as the piston closes the ports E and /, and continuing to the end of the stroke, when the piston is in its extreme left-hand position. Ignition there takes place, and the next out stroke is a working stroke, during which the heated gas expands. Toward the end of this stroke, the exhaust port E is uncovered, and the gas passes out, and continues to pass out until early on the next backward stroke this port is again covered. 390. Discussion of the Cycle. We have here a two-stroke cycle ; for two of the four events requiring a perceptible time interval are always taking place simultaneously. On the first stroke to the left, while gas is entering D, it is for a brief interval of time also flowing from 7 to F, from F through E, and afterward being compressed in F. On the next stroke to the right, while gas is compressed in Z>, ignition and expansion occur in F] arid toward the end of the stroke, the exhaust of the burned gases through E and the admission of a fresh supply through J, both begin.. The inlet port I and the exhaust port E are both open at once during part of the operation. To prevent, as far as possible, the fresh gas from, escaping directly to the exhaust, the baffle G is fixed on the piston. It is only by skillful proportioning of port areas, piston speed, and pressure in D that large loss from this cause is avoided. * The burned gases in the cylinder, it is sometimes claimed, form a barrier between the fresh enter- ing gas and the exhaust port. 291. PV Diagram. This is shown for the working side (space F) in Fig. 120 and for the pumping side (space D) in Fig. 121. The exhaust port is uncovered at tf, Fig. 120, and the pres- sure rapidly falls. At a, the inlet port opens, the fresh supply of gas holding up the pres- sure. From a out to the end of the diagram, and back to 6, both ports are open. At & the inlet port closes, and at c the exhaust port, when compres- sion begins. The pump diagram of FIG. 120. Art 291. Two-stroke Fig. 121 COrre- y cle - spends with the negative loop deal of Fig. 118. Aside from FIG. 13L Art. 291. Two-stroke the slight difference at dabc, Fig. 120, the Cycle Pump Diagram. * Two cycle gas engines should never be governed by varying the quantity of mixture drawn in (Art. 348) because of the disturbing effect which such variations would have on these factors. THE OTTO CYCLE 175 diagrams for the two-cycle and four-cycle engines are precisely the same; and in actual indicator cards, the difference is yery slight. 292. Ideal Diagram. The perfect PV diagram for either engine would be that of Fig. 122, ebfd, in which expansion and com- pression are adiabatic, combustion instan- taneous, and exhaust and suction unre- FIG. 122. Arts. 292, 293, 29i, Stricte(1 I so that the area of the negative 295, 314, 329, 1329a, 329Z>, loop dg becomes zero, and eb and fd are 331. Prob. 15. Idealized . * -, -,** Gas Engine Diagram. lmes of constant volume. From inspection of the diagram we find 293. Work Done. The work area under Jfis under ed is TT _ p I?" ' ' - - ; that ; the net work of the cycle is This may be written in terms of two pressures and two volumes only, for P e V e = PtVjfVJ-v and P f V f = P d F 6 "F^, giving a- P* W V^ - P Vf VJ-* 294. 4. Relations of Curves. Expressing ^ = ^Y and ^ = (" Y, and I/ v^i/ -fd \y*j remembering that F 5 = V* 7,= V d , we have ^ = ^ and ^ = ?f . This / *, d ** *d permits of rapidly plotting one of the curves when the other is given. We also find ^- and = - * *d -* -Li 176 APPLIED THERMODYNAMICS 295. Efficiency. In Fig. 122, heat is absorbed along 06, equal to l(T b Tg); this is derived from the combustion of the gas. Heat is rejected along fd, =l(T f T a ). Using the difference of the two quantities as an expression for the work done, we obtain for the efficiency T t - T e - T f The efficiency thus depends solely upon the extent of compression TF ) > while ^ ~r-=the clearance of the engine, V d / V d~ V e 'and since & 1 LOO .80 .60 .40 ,20 i \ \ X ^x, ^ ^. ""- JO .20 .30 .40 .50 .60 .70 .80 .90 LOO Clearance PIG. 122a. Art. 295. Relation between Efficiency and Clearance in the Ideal Cycle. the efficiency may be expressed in terms of the clearance only. (See Fig. 122a.) 295 a. The Sargent Cycle. Let the engine draw in its charge at atmos- pheric pressure, along ad, Fig. 122 c. The inlet valve closes at d and the charge expands somewhat, along dc. It is then compressed along cde, ignited along e& ; and expanded along bg. The exhaust valve opens at g, the pressure falls to that of the atmosphere along gh, and the cylinder contents are expelled along ha. The work area is debfgh^ there is 110 negative loop work area dhc. The entropy diagram shows the cycle to THE SARGENT AND THE FRITH CYCLES 177 be more efficient than the Otto cycle debf between the same temperature limits ; the superior Otto cycle ebgc has wider temperature limits. The gain by the Sargent cycle is analogous to that in a steam engine by an increased ratio of expansion (Art. 411), and involves a reduction in capac- ity in proportion to the size of cylinder. The efficiency is debgh __ mebn mdhgn mebn mebn _ T h T d ~~ y T b -T< T,-T e 295 o. The Frith Regenerative Cycle (Jowr. A. S. M. ., XXXII, 7). In Fig. 122 d, abed is an ordinary Otto cycle. Suppose that during expansion some of the fluid passes through a regenerator, giving up heat, following some such path as ae. Then let the regenerator in turn impart this heat to the working substance during or just before combustion, as along di in the entropy diagram. If the regenerator were perfect, and the transfers as described could occur, the heat absorbed from external sources would be jiah and the work would be daec. The quotient of the latter by the former, if the path through the regenerator were ac (limiting case), would be unity. But this would involve a contravention of the second law, since heat would have to pass from the regenerator (at c) to a sub- stance hotter than itself (at d). If, however, we make the temperature range T d T e very small, a large proportion of the heat transferred to the regenerator may again be absorbed along da, and as the output of the engine approaches zero, its efficiency approaches 100 per cent. If, as in Fig. 122 5, the expansion curve strikes the point c, we may assume that of all the heat (fcaty delivered to the regenerator, only that portion (Ikah), the temperature of which exceeds T& can be redelivered to the fluid along da. The efficiency is then dac _ fdah fcah fdah Ikah fdah Ikah n 2/ - T d ) -s(T a - T e ) _ n- 1 T a T d - - -r (7 T a - T d ) where s = I ~ is the specific heat along the path akc, the equation of which is pv n '= const. Since ' P a Va n = PcV c n t while PdVd v =P G Vc v , PC , PC 178 APPLIED THERMODYNAMICS FIG. 1226 Art. 2955. T FIG. 122e. Art T / FIG. 122d. Art. 295b. Let P c = 14.7, P d = 147, P tf = 294. Then n-y 0.561 n - 1 0.963 log 0.10 = 0.582. 1.963, -JY -IV ATKINSON ENGINE 179 Now if T c = 300 P. = 760 abs,, T d = 1470 abs., and if T a = 3000 abs., the efficiency becomes 1530 - (0.582 x 2240) = 230 = 1530 - (0.582 x 1530) 640 " * while that of the Otto cycle is T d - T c _ 1470 - 760 = Q 4g T d 1470 For a discussion of limiting values, see the author's paper in Polytechnic Engineer^ 1914. 296. Carnot Cycle and Otto Cycle; the Atkinson Engine. Let nbcd, Fig. 123, represent a Carnot cycle drawn to pv coordinates, and bfde, the corresponding Otto cycle between the same temperature limits, T and t. For the Carnot cycle, the efficiency is (T t) -=- T 7 ; for the Otto, it is, as has been shown, (T e T d ) H- T e . It is one of the disad- vantages of the Otto cycle, as shown in Art. 294, that the range of temperatures during expansion is the same as that dur- ing compression. In the ingenious Atkin- son engine (13), the fluid was contained in the space between two pistons, which space was varied during the phases of the cycle. This permitted of expansion independent of compression ; in the ideal case, expansion continued down, to the temperature of the atmosphere, giving such a diagram as ebcd, Fig. 123. The entropy diagrams for the Carnot, Otto, and Atkinson cycles are correspondingly lettered in Fig. 124. For the Atkinson cycle, in the ideal case, we have iii Fig. 124 the elementary strip vicxy, which may stand for dH, and the isothermal dc at the temperature t. Let the variable temperature along eb be T x , having for its limits T b and T^ Then, for the area ebcd, we have FIG. 124. Arts 296, 297, 305, 307, 3296. Efficiencies of Gas Engine Cycles. The efficiency is obtained by dividing by I (T b T e ) and is equal to f /TT ^ v i J- / FIG. 123. Art. 2% Carnot, Otto, and Atkinson Cj eles. C^dff fr b dT x ( = I -Tjr^* V = ' L ~m~ ( 2 * " +J TC ^ x *J T e -L s 297. Application to a Special Case. Let T e = 1060, T whence, from Art, 294, T/ = 1688. We then have the following ideal efficiencies: 180 APPLIED THERMODYNAMICS Carnot, Atkinson, 1 Otto, T-t_ 3440 -520 T ~ 3440 .. n . 620,. = 0.85. 3440 A _ = 0.74. T e -t_ 1060 - 520 T e ~~ 1060 = 0.51. The Atkinson engine can scarcely be regarded as a practicable type ; the Otto cycle is that upon which most gas engine efficiencies must be based; and they depend solely on the ratio of temperatures or pressures during compression. 298. Lenoir Cycle. This is shown in Fig. 125. The fluid is drawn into the cylinder along Ad and exploded along df. Expansion then occurs, giving the path/, when the exhaust valve opens, the pressure INSTANT VOLUME FIG. 125. Arts 298, 301, 302. Lenoir Cycle. FIG. 12(5. Art 298. Entropy Diagram, Lenoir Cycle. falls, g7ij until it reaches that of the atmosphere, and the gases are finally expelled on the return stroke, liA. It is a two-cycle engine. The net entropy diagram appears in Fig. 126. The efficiency is Heat absorbed - heat rejected _ ?(2> - !T d ) - l(T 9 - T h ) - k(T h - T d ) Heat absorbed "~ ( 7/ T d ) 299. Brayton Cycle. This is shown in Fig. 127. A separate pump is employed. The substance is drawn in along Ad, compressed along dn, and forced into a reservoir along n. The engine begins to take a charge from the reservoir at -B, which is slowly fed in and ignited as it enters, so that combustion proceeds at the same rate as the piston movement, giving the constant pressure line 1. Expan- sion then occurs along lg, the exhaust valve opens at g, and the charge is expelled along Ji A. The net cycle is dnbgh^ the net ideal entropy diagram is as in Fig. 128. This is also a two-cycle BRAYTON CYCLE 181 FIG. 127. Arts, 2<>9, P>02. Bray ton Cycle. FIG. 128 Art. 2VI9 Bray ton Cycle, Entropy Diagram. engine. The " constant pressure " cycle which it uses was suggested in 1865 by Wilcox. In 1873, when first introduced in the United States, it developed an efficiency of 2.7 Ib. of (petroleum) oil per brake hp.-hr. The efficiency is (Fig. 127) If expansion is complete, the cycle becoming dnli, Pigs. 127, 128, then T g = T h = T t} and the efficiency is /in rrj fTi fTi __ fji HH r r* ^r* r r* ' a result identical with that in Art. 295 ; the efficiency (with complete ex- pansion) depends solely upon the extent of compression. 300. Comparisons with the Otto Cycle. It is proposed to compare the capacities and efficiencies of engines working in the Otto,* Brayton, and Lenoir cycles; the engines being of the same size, and working "between the same limits of temperature. For convenience, pure air will be regarded as the working substance. In each case let the stroke be 2 ft., the piston area 1 sq. ft., the external atmosphere at 17 C., the maximum temperature attained, 1537" 1 C. In the Lenoir engine, let ignition occur at half stroke; in the Brayton, let compression begin at half stroke and con- tinue until the pressure is the same as the maximum pressure attained in the Lenoir cycle, and let expansion also begin at half stroke. These are to be compared with an Otto engine, in which the pump compresses 1 cu. ft. of free air to -iO Ib. net pressure. This quantity of free air, 1 cu. ft., is then supplied to each of the three engines. 301. Lenoir Engine. The expenditure of heat (in work units) along df, Fig. 125, is Jl(T - 0> in which T = 1537, t - 17, J is the mechanical equivalent of a Centigrade heat unit, and / is the specific heat of 1 cu. ft. of free air, * The " Otto cycle " in this discussion is a modified form (as suggested by Clerk) in which the strokes are of unequal length. 182 APPLIED THERMODYNAMICS heated at constant volume 1 (J. Now J ' 778 x 3 = HOO.i, and *// ]d a;>p mately 0.1689 x 0.075 x 1400.4 = 17.72. The expendituie of heat is then 17.72(1537 - 17) = 26,900 ft.-lb. The pressure at /is UJ 1587 + 273 == Q1A lb< absolufce . 17 + 27*3 and the pressure at g is 91.4 (i)v = 34.25 Ib. absolute. The work done under fg is then = 8190 ft.-lb. The negative woik under fid is 14 7 x 144 x 1 = 2107 ft.-lb., and the net work is 8190 - 2107 = 6083 ft.-lb. The efficiency is then 6083 - 26,900 = 226. 302. Brayton Engine. We first find (Fig. 127) v--! Tn = T d (^\ v = (273 + 17) (~~}^ = 489 absolute or 216 C. Proceeding in the same way as with the Lenoir engine, we find the heat expendi- ture to be Jk(Ti - T n ) = 2375 x 0.075 x 1400.4(1537 - 216) = 33,000 ft.-lb. The pressure at n is by assumption equal to p f in the case of the Lenoir engine; the pressure at g in the Brayton type then equals that at g in the Leuoir. The work under Ig is the same as that under fg in Fig. 125. The work under nb is found by first ascertaining the volume at n. This is UTj^LO =0.272. 9.14/ The work under nb is then 91.4 x 144 x (1 - 0.272) = 9650 ft.-lb., and the gross work is 9650 + 8190 = 17,840 ft.-lb. Deducting the negative work under hd, 2107 ft.-lb., and that under dn, i44 *_-- x = 3650 ft.-lb., the net work area is 12,083 ft.-lb., and the efficiency, 12,083 - 33,000 = 0.366'. 303. Clerk's Otto Engine. In Fig. 129, a separate pump takes in a charge along AB, and compresses it along BC, afterward forcing it into a receiver along CD at 40 Ib. gauge pressure. Gas flows from the receiver into the engine along DC, is ex- ploded along CEj expands to F, and is expelled along GA. The net cycle is BCEFQ. The volume at C is ~ y = 0.393 cu. ft. FIG. 129. Arts. 303, 305, Clerk's Otto Cycle. CLERK'S GAS ENGINE 183 The temperature at C is 0.393) (278 + 17) 14.7 x 1 The pressure at E is then (1537 + 273)54.7 = 2311 Io3 + 273 The pressure at F is 231 (^f^V = 23.64 Ib. absolute. The work under EF is _ 27:3 = 133 o a that under 5(9 is 2107 ft.-lb., and that under EC is ltf / (54.7 x 0.393) -(14. 7 x V 1.402 - 1.0 The net work is 15,600 - 2107 - 2430 = 11,063 ft.-lb. The heat expenditure in this case is Jl(T E - T c ) = 17.72 x (1537 - 153) = 24,500 ft.-lb., and the efficiency is 11,063 - 24,500 = 0.453; considerably greater than that of either the Lenoir or the Brayton engine (14). If we express the cyclic area as 100, then that of the Lenoir engine is 52 and that of the Brayton engine is 104. (See Art. 295a.) 304. Trial Results. These comparisons correspond with the consumption of gas found in actual practice with the three types of engine. The three efficiencies are 0.226, 0.366, and 0.453. Taking 4 cu. ft. of free gas as ideally capable of giv- ing one horse power per hour, the gas consumption per hp.-lir. in the three cases would be respectively 4 - 0.226 = 17.7, 4 - 0.3C6 = 10.9, and 4 - 0.453 = 8.84 cu. ft. Actual tests gave for the Lenoir and Hugon engines 90 cu. ft. ; for the Brayton, 50 ; and for the modified Otto, 21. The possibility of a great increase in economy by the use of an engine of a form somewhat similar to that of the Brayton will be discussed later. 305. Complete Pressure Cycle. The cycle of Art. 303 merits detailed exami- nation. In Fig. 129, the heat absorbed is l(T E - 7 j that rejected is the efficiency is The entropy diagram may be drawn as ebmnd, Fig. 124, showing this cycle to be more efficient than the equal-leugth-stroke Otto cycle, but less efficient than the Atkinson. With complete expansion down to the lower pressure limit, the cycle becomes BCEFH, Fig. 129, or ebo<U Fig. 121; the strokes are still of unequal length, and the efficiency is (Fig. 129) 184 APPLIED THERMODYNAMICS If the strokes be made of equal length, with incomplete expansion, T G =T i the cycle becomes the ordinary Otto, and the efficiency is 1 T F -Tn = Tc-Tn r r r n / 7 T " 1 E - 1 c J-C 306. Oil Engines : The Diesel Cycle. Oil engines may operate in either the two-stroke or the four-stroke cycle, usually the latter; and combus- tion may occur at constant volume (Otto), constant pressure (Brayton), or constant temperature (Diesel). Diesel, in 1893 (15), first proposed what has proved to be from a thermal standpoint the most economical heat engine. It is a four-cycle engine, approaching more closely than the Otto to the Carnot cycle, and theoretically applicable to solid, liquid, or gaseous fuels, although actually used only with oil. The first engine, tested by Schroter in 1897, gave indicated thermal efficiencies ranging from 0.34 to 0.39 (16). The ideal- ized cycle is shown in Fig. 130. The opera- tions are adiabatic compression, isothermal ~ v expansion, adiabatic expansion, and dis- FIG. 130. Arts. 306, 307. Diesel _/ ' , . 1 ^ ' . . Cycle, charge at constant volume. Pure air is com- pressed to a high pressure and temperature, and a spray of oil is then gradually injected by means of external air pressure. The temperature of the cylinder is so high as at once to ignite the oil, the supply of which is so adjusted as to produce combustion practically at constant temperature. Adiabatic expansion occurs after the supply of fuel is discontinued. A considerable excess of air is used. The pressure along the combustion line is from 30 to 40 atmospheres, that at which the oil is delivered is 50 atmospheres, and the temperature at the end of compression approaches 1000 P. The engine is started by compressed air; two or more cylinders are used. There is no uncertainty as to the time of ignition; it begins immediately upon the entrance of the oil into the cylinder. To avoid pre-ignition in the supply tank, the high-pressure air used to inject the oil must be cooled. The cylinder is water-jacketed. Figure 131 shows a three- cylinder engine of this type; Fig. 132, its actual indicator diagram, reversed. The Diesel engine has recently attracted renewed interest, especially in small units: although it has boen built in sizes up to 2000 hp. It has been applied in marine service, and has successfully utilized by- product tar oil. THE DIESEL ENGINE FIG. 131. Art. 306. Diesel Engine. (American Diesel Engine Company.) FIG. 1J2. Art. OOC. Indicator Diagram, Diesel Engine. (16 X 24 In. engine, 100 r.p.m. Spring 400.) 186 APPLIED THERMODYNAMICS 307. Efficiency. The heat absorbed along J, Fig. 130, is The heat rejected along/c? is l(T f T^. We may write the efficiency as i i But 2>= r- rancl Z^; whence y For the heat rejected tilong/d we may therefore write *rfY r 'Y~ 1 il -i,/ 1 -- 1 i , y LVT' a / J and for the efficiency, This increases as T a increases and as -~~ decreases. The last conclu- Ka sion is of prime importance, indicating that the efficiency should in- crease at light loads. This may be apprehended from the entropy diagram, abfd, Fig. 124. As the width of the cycle decreases (If moving toward ad), the efficiency increases, 307 &. Diesel Cycle with Pressure Constant. In common present practice, the engine is supplied with fuel at such a rate that the pressure, rather than the temperature, is kept constant during combustion. This gives a much greater work area, in a cylinder of given size, than is possible with isothermal combustion. The cycle is in this case as shown in Fig. 132 a, combining features of those of Otto and Brayton. The entropy diagram shows that the efficiency exceeds that of the Otto cycle ebfd between the THE DIESEL ENGINE 187 same limits ; but it is less than that of the Diesel cycle with isothermal combustion. The definite expression for efficiency is r,) T f -T d mabn Inspection of the diagram shows that the efficiency decreases as the load increases. (For a description of the Junkers engine, see the papers by Junge, in Power, Oct. 22, 29, Nov. 5, 1912 ) P T -N m FIG. 132a. Art. 307&. Constant-pressure Diesel Cycle. 3070. Entropy Diagram, Diesel Engine. In constructing the entropy diagram from an actual Diesel indicator card a difficulty arises similar to one met with in steam engine cards; the quantity of substance m the cylinder is not constant (Art. 454 ) . This has been discussed by Eddy (17), Frith (18), and Reeve (19). The illustrative dia- gram, constructed as in Art. 347, is sugges- tive. Figure 133 shows such a diagram for an engine tested by Denton (20). The initially hot cylinder causes a rapid ab- sorption of heat from the walls during the early part of compression along db. Later, along be, heat is transferred in the opposite direction. Combustion occurs along cd, the temperature and quantity of heat increas- ing rapidly. During expansion, along de, the temperature falls with increasing rapidity, the path becoming practically adiabatic during release, along ef. The TV diagram of Fig; 133 indicates that no further rise of temperature would accompany increased compression; the actual path at y has already become practically isothermal. 308. Comparison of Cycles. Figure 134 shows all of the cycles that have been discussed, on a single pair of diagrams. The lettering cor- responds with that in Pigs. 122-128, 130. The cycles are, FIG. 133. Art. ^07. Diesel Engine Diagrams. 188 APPLIED THERMODYNAMICS Garnet, abed, Lenoir, d/o0^o>#Mb Diesel, ddbf, Otto, ebfd, Brayton, diibgli, dnU, Atkinson, ebcd, Complete pressure, debgh, debi. FIG. 134. Art. 308, Probs. 7, U5. Comparison of Gas Engine Cycles. 3080. The Humphrey Internal Combustion Pump. In Kg. 134a, C is a chamber supplied with water through the check valves V from the storage tank .ET, -and connected by the discharge pipe D with the delivery tank F. Suppose the lower part of C, with the pipe D and the tank F, to be filled with water, and a combustible charge of gas to be present in the upper part of C, the valves I and E being closed. The gas charge is exploded, and expansion forces the water down in C and up in F. The movement does not stop when the pressure of gas in C falls to that equivalent to the difference in head between F and C ; on the contrary, the kinetic energy of the moving water carries it past the normal level in F, and the gases in C fall below that pressure due to head. This causes the opening of E and F, an inflow of water from ET to C, and an escape of burnt gas from C through E. The water rises in C. Meanwhile, a partial return flow from F aids to fill C, the kinetic energy of the moving water having been exhausted, and the stream having come to rest with an abnor- mally high level in F. Water continues to enter C until (1) the valves V are closed, (2) the level of E is reached, when that valve closes by the impact of water; and (3) the small amount of burnt gas now trapped in the space Ci is compressed to a pressure higher than that correspond- ing with the difference of heads between F and Ci. As soon as the returning flow of water has this time been brought to rest, the excess pressure in C\ starts it again in the opposite direction from Ci toward F. When the pressure in Ci has by this means fallen to about that of the atmosphere, a fresh charge is drawn in through J. Frictional losses prevent the water, this time, from rising as high in F as on its first outflow; but nevertheless it does rise sufficiently high to acquire THE HUMPHREY INTERNAL COMBUSTION PUMP 189 a static head, which produces the final return flow which finally com- presses the fresh charge. The water here takes the place of a piston (as in the hydraulic piston compressor, Art. 240). The only moving parts are the valves. Gas ft FIG. 134a. Art. 308a. Humphrey Pump. The action is unaccompanied by any great rise of temperature of the metal, since nearly all parts are periodically swept by cold water. The pump as described works on the four-cycle principle, the operations being (Fig. 1346): a. Ignition (a&) and expansion b. Expulsion of charge (cd, de), suction of water, com- pression of residual charge (ef) ; c. Intake (feg, gh) ; d. Compression (ha). Disregarding the two loops ehg, dcm, the cycle is bounded by two polytropics, one line of constant volume and one of constant pressure. Between the temperature limits T b and T h it gives more work than the Otto cycle habj, and if the curves be and ah were adiabatic would necessarily have a higher efficiency than the Otto cycle. The actual paths are not adiabatic: during expansion (as well as during ignition) some of the heat must be given up to the water; while the heat generated by compression is similarly (in part) transferred to the water along ha. With the adiabatic assumption adopted for the purpose of classification, the cycle is that described in Art. 305 and shown in Fig. 134 as debi. The strokes are of unequal length. (See Power, Dec. 1, 1914.) The gases are so cool toward the end of expansion that a fresh FIG. 1346. Art. 308a. Cycle of Humphrey Pump. 190 APPLIED THERMODYNAMICS charge may be safely introduced at that point, by outside compression on the two-cycle principle (Art. 289). The pump may be adapted for high heads by the addition of the hydraulic intensifies It has been built in sizes up to 40,000,000 gal. per twenty-four hours, and has developed a thermal efficiency (to water) under test of about 22 per cent. (See American Machinist, Jan. 5, 1911.) PRACTICAL MODIFICATIONS OF THE OTTO CYCLE. 309. Importance of Proper Mixture. The working substance used in gas engines is a mixture of gas, oil vapor or oil, and air. Such mixtures will not ignite if too weak or too strong Even when so proportioned as to permit of ignition, any variation from the correct ratio has a detrimental effect; if too little air is present, the gas will not burn completely, the exhaust will be dart colored and odorous, and unburned gas may explode in the exhaust pipe when it meets more air. If too much air is admitted, the products of combustion will be unnecessarily diluted and the rise of temperature daring ignition will be decreased, causing a loss of work area on the PV diagram. Figure 185 shows the effect on rise of temperature and pressure of varying the proportions of air and gas, assuming the variations to remain within, the limits of ~~ possible ignition. Fail Lire to ignite may occur Bto. 133. Art 309.- Effect T ^^ of the ^ of eMM(| rf ^ ag Mixture otrengLn. A ,.-,,., n 7 , well as when the air supply is deficient. Rapidity of flame propagation is essential fur efficttnry, and this is only possible with a proper mixture. The gas may in some ca*es bum so slowly as to leave the cyl- inder partially unconsumed In an engine of the t\pe shown hi Fig. 119, this may result in a spread of flame through /, B, and C back to D, with dangerous consequences. 310. Methods of Mixing. The constituents of the mixture must be intimately mingled in a finely divided state, and the governing of the engine should prefei - ably be accomplished by a method which keeps the proportions at those of highest efficiency. Variations of pressure in gas supply mains mav interpose serious dif- ficulty in this respect. Fluctuations in the lights which may be supplied from the same mains are also excessive as the engine load changes. Both difficulties are sometimes obviated in. small units by the use of a rubber supply receiver. Varia- tions in the speed of the engine often change the proportions of the mixture. "When the air is drawn from out of doors, as with automobile engines, variations in the temperature oE the air affect the mixture composition. In simple types of engine, the relative openings of the automatic gas and air inlet valves are fixed when the engine is installed, and are not changed unless the quality or pressure of the gas changes, when a new adjustment is made by the aid of the indicator or by observation of the exhaust. Mechanically operated mixing valves, usually of the "butterfly" type, are used on high-speed engines; these are positive in their ALLOWABLE COMPRESSION 191 action. The use of separate pumps for supplying air and gas permits of proportion- ing in the ratio of the pump displacements, the volume delivered being constant, regardless of the pressure or temperature. Many adjustable mixing valves and carbureters are made, in which the mixture strength may be regulated at will. These are necessary where irregularities of pressure or temperature occur, but require close attention for economical results. In the usual type of carbureter or vaporizer, used with gasoline, a constant level of liquid is maintained either by an overflow pipe or by a float. The suction of the engine piston draws air through a nozzle, and the fuel is drawn into and vaporized by the rapidly moving air current. Kerosene cannot be vaporized without heating it: the kerosene carbureter may be jacketed by the engine exhaust, or the liquid may be itself spurted directly into the cylinder at the proper moment, air only being present in the cylinder during compression. The presence of burned gas in the clearance space of the cylinder affects the mixture, retarding the flame propagation. The effect of the mixture strength on allowable compression pressures remains to be considered. 311. Actual Gas Engine Diagram. A typical indicator diagram from a good Otto cycle engine is shown in Fig. 136. The various lines differ somewhat from those established in Art. 28S. These differences we now discuss. Figure 137 shows the portion bcde of the diagram in Fig. 133 to an enlarged vertical scale, thus representing the action more clearly. The line/0 is that of atmospheric pressure, omitted in Fig. 136. TVe will begin our study of the actual cycle with the compression line. FIG. 136. Arts. 311, 342, 345. FIG. 137. Arts. 311, 3^H, 328. - Eii- Otto Engine Indicator Diagram. larged Portion of Indicator Diagram. 312. Limitations of Compression. It has been shown that a high degree of compression is theoretically essential to economy. In practice, com- pression must be limited to pressures (and corresponding temperatures) at which the gases will not ignite of themselves ; else combustion will occur before the piston reaches the end of the stroke, and a backward impulse will be given. Gases differ widely as to the temperatures at which they will ignite; hydrogen, for example, inflames so readily that Lucke (21) estimates that the allowable final pressure must be reduced one atmosphere for each 5 per cent of hydrogen present in a mixed gas. The following are the average final gauge compression pressures recommended by Lucke (22) : for gasoline, in automobile engines, 45 to 95 lb., in ordinary engines, 60 to 85 Ib. ; for kerosene, SO to 85 lb.; for natural gas, 75 to ISO lb. ; for coal gas or carbureted water gas, 192 APPLIED THERMODYNAMICS 60 to 100 II. ; for producer gas, 100 to 160 Ib. ; and for blast furnace ffas, 120 to 190 Ib. The range of compression depends also upon the pressure existing in the cylinder at the beginning of compression ; for two-cycle engines, this varies from 18 to 21 Ib., and for four-cycle engines, from 12 to 14 Ib., both absolute. The pre-compression temperature also limits the allowable range below the point of self -ignition. This temperature is not that of the entering gases, but it is that of the cylinder contents at the moment when compression begins ; it is determined by the amount of heat given to the incoming gases by the hot cylin- der walls, and this depends largely upon the thoroughness of the water jacketing and the speed of the engine. This accounts for the rather wide ranges of allow- able compression pressures above given. Usual pre-compression temperatures are from 140 to 300 F. " Scavenging" the cylinder \uth cold air, the injection of water, or the circulation of water in tubes in the clearance space, may reduce this. Usual practice is to thoroughly jacket all exposed sm faces, including pistons and valve faces, and to avoid pockets where exhaust gases may collect. The primary object of jacketing, however, is to keep the cylinder cool, both for mechanical reasons (e g., for lubrication) and to avoid uncontrollable explosions at the moment when the gas reaches the cylinder. 313. Practical Advantages of Compression. Compression pressures have steadily increased since 1881, and engine efficiencies have increased correspond- ingly, although the latter gain has been in part due to other causes. Improved methods of ignition have permitted of this increased compression. Besides the therm odynarnic advantage already discussed, compression increases the engine capacity. In a non-compressive engine, no considerable range of expansion could be secured without allowing the final pressure to fall too low to give a large work area; in the compressive engine, wide expansion limits may be obtained along with a fairly high terminal pressure. Compression reduces the exposed cylinder surface in proportion to the weight of gas present at maximum temperature, and so decreases the loss of heat to the walls. The decreased proportion of clearance space following the use of compression also reduces the proportion of spent gases to be mixed with the incoming charge. 314. Pressure Rise during Combustion. In Art. 292, the pressure P b after combustion was assumed. "VVhile, for reasons which will appear, any computation of the. rise of pressure by ordinary methods is unreliable, the method should be described. Let H denote the amount of heat liberated by combustion, per pound of fuel. Then, Fig. 122, H= l(T b - IT.), T b - T e = and T b = + TV But = S * + i. Tim ft -!>. = . But * whence 11 e P. IT. COMPUTED MAXIMUM TEMPERATURE 193 Then 315. Computed Maximum Temperature. Dealing now with the constant volume ignition line of the ideal diagram, let the gas be one pound of pure carbon monoxide, mixed with just the amount of air necessary for com- bustion (2.48 lb.), the temperature at the end of compression being 1000 absolute, and the pressure 200 lb. absolute. Since the heating value of 1 lb. of CO is 4315 B. t. u., while the specific heat at constant volume of C0 2 is 0.1692, that of N being 0.1727, we have rise in temperature = - 4315 - = 7265F. (1.57 x 0.1692}+ (1.91 x 0.1727) The temperature after complete ignition is then 8265 absolute. The pressure is 200 x -^ = 1653 lb. If the volume increases during igni- 1000 tion, the pressure decreases. Suppose the volume to be doubled, the rise of temperature being, nevertheless, as computed : then the maximum pres- sure attained is 826.5 lb. Compression Ratio ( \* "Fijr.122 ) FIG. 137a. Art. 316. Rise of Pressure in Practice. 316. Actual Maxima. No such temperature as 8265 absolute is attained. In actual practice, the temperature after ignition is usually 194 APPLIED THERMODYNAMICS about 3500 absolute, and the pressure under 400 Ib. The rise of either is less than half of the rise theoretically computed, for the actual air supply, with the actual gas delivered. The discrepancy is least for oil fuels and (mixtures being of proper strength) is greatest for fuels of high heat value. It is difficult to measure the maximum temperature, on account of its extremely brief duration. It is more usual to ni3asure the pressure and compute the temperature. This is best dons by a graphical method, as with the indicator. Fig. 137a gives the results of a tabulation by Poole of pressure rises obtained in usual practice. 317. Explanation of Discrepancy. There are several reasons for the disagree- ment between computed and observed results. Charles' law does not hold rigidly at high temperatures; the specific heats of gases are known to increase with the temperature (Meyer found in one case the theoretical maximum temperature to be reduced from 4250 E. to 3330 F. by taking account of the increases iu specific heats as determined by Mallard and Le Chatelier); combustion is actually not instantaneous throughout the mass of gas and some increase of volume always occurs ; and the temperature is lowered by the cooling effect of the cylinder walls. Still another reason for the discrepancy is suggested in Art. 318. 318. Dissociation. Just as a certain maximum temperature must be attained to permit of combustion, so a certain maximum temperature must not be exceeded if combustion is to continue. If this latter temperature is exceeded, a suppression of combustion ensues. Mallard and Le Chatelier found this "dissociation " effect to begin at about 3200 F. with carbon monoxide and at about 4500 F. with steam. Deville, however, found dissociative effects with steam at 1800 F., and with car- bon dioxide at still lower temperatures. The effect of dissociation is to produce, at each temperature within the critical range for the gas in question, a stable ratio of combined to elementary gases, e.g. of steam to oxygen and hydrogen, which cannot widely vary. No exact relation between specific temperatures and such stable ratio has yet been determined. It has been found, however, that the maximum temperature actually attained by the combustion of hydrogen in oxygen is from 3500 to 3800 C-, although the theoretical temperature is about 9000 C. At constant pressure (the preceding figures refer to combustion at constant vol- ume), the actual and theoretical figures are 2500 and 6000 C. respectively. For hydrogen burning in a,ir, the figures are 1830 to 2000, and 3800 C. Dissociation here steps in to limit the complete utilization of the heat in the fuel. In gas en- gine practice, the temperatures are so low that dissociation, cannot account for all of the discrepancy between observed and computed values ; but it probably playa a part. (See Art. 1276.) 319. Rate of Flame Propagation. This has been mentioned as a factor influ- encing the maximum temperature and pressure attained. The speed at which flame travels in an inflammable mixture, if at rest, seldom exceeds 65 ft. per sec- ond. If under pressure or agitation, pulsations may be produced, giving rise to "explosion waves," in which the velocity is increased and excessive variations in pressure occur, as combustion is more or less localized (23). Clerk (24), experi- RATE OP FLAME PROPAGATION 195 meriting on mixtures of coal gas with air, found maximum pressure to be obtained in minimum time \\hen the proportion of air to gas by volume was 5 or 6 to 1 : for pure hydrogen and air, the best mixture was 5 to 2. The Massachusetts Insti- tute of Technology experiments, made with carbureted water gas, showed the best mixture to be 5 to 1 ; with 86 gasoline, the quickest inflammation was obtained ^lien 0.0217 parts of gasoline were mixed with 1 part of air; with 76 gasoline, when 0.0203 to 0278 parts were used.* Grover found the best mixture for coal gas to be 7 to 1 ; for acetylene, 7 or 8 to 1, acetylene giving higher pressures than coal gas. Vt'ith coal gas, the weakest i^nitible mixture was 15 to 1, the theoreti- cally perfect mixture being 5.7 to 1. The limit of weakness with acetylene was 18 to 1. Both Grover and Lucke (2G) have investigated the effect of the presence of "neutrals" (carbon dioxide and nitrogen, derived either fiom the air, the incom- ing gases, or from residual burnt gas) on the rapidity of piopagation. The re- tJ> 5 5.5 PARTS AIR PER ONE PART GAS 6.6 FIG. l.TS. Art 319 Effect of Presence of Neutrals. (From Button's " The Gas Engine, 11 by permisbion of Joku Wiley L Sons, Publishers ) suits of Lucke's study of water gas are shown in Fig. 13$. The ordinates show the maximum pressures obtained -with various propoitions of air and gas. These are highest, for all percentages of neutral, at a ratio of air to gas of 5 to 1 ; but they decrease as the proportion of nentnil increases. The experiments indicate that the speed of flame travel varies widely with the nature of the mixture and tlie conditions of pressure to which it is subjected. If the mixture is too weak or too strong, it will not Inflame at alL (See Art. 105a.) 320. Piston Speed. The actual shape of the ideally vertical ignition line will depend largely upon the speed of flame propagation as compared with the speed of the piston. Figure 139, after Lucke, illustrates this. The three diagrams were taken from the same engine under exactly the same conditions, excepting that the speeds in the three cases were 150, 500, and 750 r. p. m. Similar effects may be obtained by varying the mixture (and consequently the flame speed) while keep- ing the piston speed constant. High compression causes quick ignition. Throt- * The theoretical ratio of air to C 6 H 14 is 47 to 1. 196 APPLIED THERMODYNAMICS tlinrg of the incoming charge increases the percentage of neutral from the burnt gases and retards ignition. 150 r. p. m. 500 i. p. m 750 r. p. m. FIG. 139. Art. 320. Ignition Line as affected by Piston Speed. (From Lucke's "Gas Engine Design.") 321, Point of Ignition. The spreading of flame is at first slow. Ignition is, therefore, made to occur prior to the end of the stroke, giving a practically verti- cal line at the end, where inflammation is well under way. Figure 140, from Poole (27), shows the effects of change in the point of ignition. In (a) and (b), ignition was so early as to produce a negative loop on the diagram. This was cor- rected in (c), but (d) represents a still better diagram. In () and (/), ignition was so late that the comparatively high piston speed kept the pressure down, and the work area was small. It is evident that too early a point of ignition causes a backward impulse on the piston, tending to stop the engine. Even though the inertia of the fly wheel carries the piston past its " dead point," a large amount of power is wasted. The same loss of power follows accidental pre-ignition, whether due to excessive compression, contact with hot burnt gases, leakage past piston rings, or other causes. Failure to ignite causes loss of capacity and irregularity IGNITION 197 IGNITION 25% EARLY IGNITION 20 ft EARLY IGNITION 10 ^ LAT FIG. 140. ArL. C.I. 7i-.c of Ii^...L_. (From Poole'a " The Gas Engine," by permission of the Hill Publishing Company.) 366- -fife- 355- 350- 345- 340- 2.-500 335- I a 117666 320- ^K-^ 10,500 w 10 11 12 13 14 15 16 17 18 Ignition Advance, Per Cent FIG. 140a. Art. 321. Mixture Strength and Ignition Point. 198 APPLIED THERMODYNAMICS Oi apeed, but theoretically at least does not affect economy. For reasons already suggested, light loads (where governing is effected by throttling the supply) and weak mixtures call for early ^qnltwn Fig. 140a, based on tests of a natural gas engine reported by Poole, shows the effect of a simultaneous varying of mixture strength and ignition point. The splitting of each curve at its left-hand end is due to the use of two mixture strengths at 10 per cent ignition advance. 322. Methods of Ignition. An early method for igniting the gas was to use an external flame enclosed in a rotating chamber which at proper intervals opened communication between the flame and the gas. This arrangement was applicable to slow speeds only, and some gas always escaped. In early Otto engines, the external flame with a sliding valve was used at speeds as high as 100 r. p. m. (28). The insertion periodically of a heated plate, once practiced, was too uncertain. The use of an internal flame, as in the Brayton engine, was limited in its applica- tion and introduced an element of danger. Self-ignition by the catalytic action of compressed gas upon spongy platinum was not sufficiently positive and reliable. The use of an incandescent wire, electrically heated and mechanically brought into contact with the gas, was a forerunner of modern electrical methods. The "hot tube "method is still in frequent use, particularly in England. This in- volves the use- of an externally heated refractory tube, which is exposed to the gas either intermittently by means of a timing valve, or continuously, ignition being then controlled by adjusting the position of the external flame. In the Hornsby- Akroyd and Diesel engines, ignition is self-induced by compression alone; but external heating is necessary to start these engines. What is called "automatic ignition" is illustrated in Fig 151. Here the external vaporizer is constantly hot, because unjacketed. The liquid fuel is sprayed into the vaporizer chamber. Pure air only is taken in by the engine during its suction stroke. Compression of this air into the vaporizer during the stroke next succeeding brings about proper conditions for self-ignition. 323. Electrical Methods. The two modern electrical methods are the (t make and break " and " jump spark." In the former, an electric , current, generated from batteries or a small dynamo, is passed through two separable contacts located in the cylinder and connected in series with a spark coil. At the proper instant, the contacts are separated and a spark passes between them. In the jump spark system, an induction coil is used and the igniter points are stationary and from 0.03 to 0.05 in. apart. A series of sparks is thrown between them when the primary circuit is closed, just before the end of the compression stroke. Occasionally there are used more than one set of igniter points. 324. Clearance Space. The combustion chamber formed in the clearance space must be of proper size to produce the desired final pressure. A common ratio to piston displacement is 30 per cent. Hutton has shown (29) that the limits for best results may range easily from 8.7 to 56 per cent (Arts. 295, 332). IGNITION AND EXPANSION 199 FIG. 141. Art. 323. After Burning. '325. Expansion Curve. Slow inflammation has been shown to result in u decreased maximum pressure after ignition. Inflammation occurring during expan- sion as the result of slow spreading of the flame is callod "after burning. 1 ' Ideally, the expansion curve should be adiabatio; actually it falls m many cases above the air adiabatic, py 1402 = constant, although it is known that during expansion from 80 to 40 per cent of the total heat in the gas is being earned away ly the jacket water. Figure 141 repre- sents an extreme case; after-burning has made the expansion line almost horizontal, and some uuburnt gas is being discharged to the exhaust. Those who hold to the dissociation theory would explain this line on the ground that the gases dissociated during combustion are gi adually combining as the temperature falls ; but actually, the temperature is not falling, and the effect which we call after binning is most pronounced with weak mix- tures and at such low temperatures as do not permit of any considerable amount of dissociation. Practically, dissociation has the same effect as an increasing specific heat at high temperature. It affects the ignition line to some extent; but the shape of the expansion line is to a far greater de- gree determined by the slow inflammation of the gases. The eifect of the transfer of heat between the fluid and the cylinder walls is dis- cussed in Art. 347. The actual exponent of the expansion curve varies from 1.25 in large engines to 1.38 in good small engines, occasionally, however, rising as high as 1.55. The compression curve has usually a somewhat higher exponent. The adiabatic exponent for a FIG. 142. Art. 325. -Explosion Waves. mixture of hydrocarbon gases is lower than that for air or a perfect gas; and in many cases the actual adiabatic, plotted for the gases used, would be above the determined expansion line, as should normally be expected, in spite of after burning. The presence of explosion waves (Art, 319) may modify the shape of the expansion curve, as in Fig. 142. The equivalent curve may be plotted as a mean through the oscillations. Care must be taken not to confuse these vibrations with those due to the inertia of the indicating instrument. 326 The Exhaust Line. This is shown to ati enlarged vertical scale as 6c, Fig. 137. "Low q>ring" dia- grams of this form are extremely u^e- F I0 . 143. Art. '^-Delayed Exhaust Valve f^ ^ ^^ ^ ^ ^ ^ " lost motion " becomes present in the valve-actuating gear, an 4 the tendency of this is to vary the instant of opening or closing the inlet or the exhaust valve. The effect of delayed opening of the latter is shown in Fig. 143; that of an inadequate exhaust passage, in Fig. 144. An early opening^of the exhaust valve may cause loss also, as in Fig. 145. There 200 APPLIED THERMODYNAMICS FIG. 144. Ait SCO Thi-ottlod Exhaust Passages. is always a loss of this kind, more or less pronounced: the expansion ratio is never quite equal to the compression ratio The exhaust valve begins to open when the expansion stroke is only from 80 to 93 per cent completed In multiple cylinder engines having common exhaust and suction mains, early exhaust from one cylinder may produce a rise of pressure during the latter part of the exhaust stroke of another. Obstructions to suction and discharge movements of gas are com- monly classed together as " fluid friction. " This may in small engines amount to as much as 30 per cent of the power developed. In good engines of large or moderate size, it should not exceed 6 per cent. It increases, pro- portionately, at light loads; and possibly absolutely as well if governing is effected by throttling the charge FIQ. lj Art. 326 Exhaust Valve Opening too Early, 327. Scavenging. To avoid the presence of burnt gases in the clear- ance space, and their subsequent mingling with the fresh, charge, " scav- enging," or sweeping out these gases from the cylinder, is sometimes prac- ticed. This may be accomplished by means of a separate air pump, or by adding two idle strokes to the four strokes of the Otto cycle. In the Crossley engines, the air admission valve was opened before the gas valve, and before the termination of the exhaust stroke. By using a long ex- haust pipe, the gases were discharged in a rather violent puff, which pro- duced a partial vacuum in the cylinder. This in turn caused a rush of air into the clearance space, which swept out the burnt gases by the time the piston had reached the end of its stroke. Scavenging decreases the danger of missing ignitions with weak gas, tends to prevent pre-ignition, and appears to have reduced the consumption of fuel. 328. The Suction Stroke. This also is shown in Fig. 137, line cd. The effect of late opening of the valve is shown in Fig. 146 ; that oi an obstructed passage or of throttling the supply, in Fig'. 147. If the opening is too eaily, exhaust gases will enter the supply pipe. If closure is too early, the gas will expand during the re- mainder of the suction stroke, but the net work lost is negligible; if too late, some gas will be discharged back to the supply pipe during the beginning of the compression stroke, FIG. 146. Art. 328. Delayed Opening of Suction Valve. DIAGRAM FACTOR 201 ACTUAL FIG 147. Art. 328. Throttled Suction. as in Fig. 148. Excessive obstruc- tion in the suction passages de- creases the capacity of the engine, in a way already suggested in the study of air compressors (Art. 224). 329. Diagram Factor. The discussion of Art. 309 to Art. 328 serves to show why the work area of any actual dia- gram must always be less than that of the ideal diagram for the same cylinder, as given in Fig. 122. The ratio of the two is called the diagram factor. The area of the ideal card would constantly increase as compression increased ; that of the actual card soon reaches a limit in this respect; and, consequently, in general, the diagram factor decreases as compression increases. Variations in excellence of design are also responsible for variations of diagram factor. FIG. 148. Art. 328. Late Closing ot Suction Valve. -Gasolene Vapor -Kerosene Spray -Natural and Dlmnmating Gases -Mond Producer Gas ^Jp"^ - Suction Anthracite Producer Gag 10 75 85 100 115 ISO 145 160 Absolute Pressure at the End of Compression, Lbs.per Sa.In, FIG. 148a. Art. 329. Maximum Mean Effective Pressures Realized in Practice. 202 APPLIED THERMODYNAMICS In the best recorded tests, its value has ranged from 0.38 to 0.59; in ordinary practice, the values given by Lucke (30) are as follows: for kerosene, if previously vaporized and compressed, 30 to 0.40, if injected on a hot tube, 20; for gasoline, 0.25 to 50; for producer gas, 0.40 to 0.56; for coal gas, 0.45; for carbureted water gas, 0.45; for blast furnace gas, 0.30 to 0.48; for natural gas, 0.40 to 0.52. These figures are for four- cycle engines. For two-cycle engines, usual values are about 20 per cent less. Figure 149 shows on the PV and entropy planes an actual indica- tor diagram with the corresponding ideal cycle. Some of the highest mean effective pressures obtained in practice with various fuels, tabulated by Poole, have been charted in Fig. 148a, ACTUAL DIAGRAM IDEAL DIAGRAM FIG 149. Art. 329. Actual and Ideal Gas Engine Diagrams. MODIFIED ANALYSIS 329 a. Specific Heats Variable. Suppose k = c 4- bt, I a + bt, M=Jcl = c a. For a differential adiabatic expansion Idt = pdv, Also, from pv = adt . -,-,, ndv f- oat = H . pdv + vdp = Edt ??-|- = ; whence v p t dv (1) p MODIFIED ANALYSIS 203 (a 4- H) log e v + <i log fi p -\-bt= constant, c log e u + a log c /> + it = constant, - log fl v + log, 2? + = constant. a a r M 2iv eft = constant, where e is the Napierian logarithmic base. Between given limits, the approximate value of n may be obtained as follows: from Equation (1), log.g + (a + JB) log, & = & ft - *,) (2) If we assume an equation in the form p^vf =^ 2 v 2 n to be possible, then log, "=-= ^ 71 log 9?2 ^l Substituting in Equation (2), ^i - (3) a al S-J The external work done during the expansion is J/ 6 Idt = I (CL -J- oi) dt'=' ct \t% ti) ^2 ~ ^i)^ or x 2 n-1 "' where n has the value given in Equation (3). We may find a simple expression for n by combining these equations : 204 APPLIED THERMODYNAMICS The efficiency of the Otto cycle debf, Fig. 122, may now be written C\a+bt)dt in which - (t, - t d + t e - %) = log, f?*?*} = log, (&A a relation obtained by dividing the equation of the path bf by that of the path ed. Following the method of Art. 169, the gain of entropy between the states a and b is, for example, a log. k + &(*. - * a ) + o log. 4 + 6ft - 0, If we apply an equation in this general form to each of the constant volume paths eb, df, Fig. 122, we find a log & + 6&-0 = ^ log C e C log/^ = -fe- V^// a as already obtained. o W 329 b. Application of the Equations. The expression pv a e*= con- stant is exceedingly cumbersome in application excepting as t is employed independently. If t is to be assumed, however, we may write log a p + log e v H = log, constant , a a -Gog. -R + log. t log.p) 4- = log, constant, a a ^log. p +-(log. JJ + log. t) + - = log. constant. a & a MODIFIED ANALYSIS 205 Consider one pound of air at the absolute pressure of 100 Ib. per square inch and a volume of 1 cu. ft. Let = 0.23327+0.00002652, Z=0.1620 + 0000265*. We find __ tl ~~R ptle^=100X 144X 1X2.7183 **- 15030. a-c 0.1620 -0.23327 c 0.23327 ,.. . _ , A _ = - 01620 - = -- 4il ; a~oi62 L44; lo &*-3-7- Let < 2 =200. Then ^ 2 = 0.0327, Iog e f 2 = 5.3, -(log. B+logA)- 13.32, -T i oge p s =log e 15030-13.32-0.0327= 9.61- 13.35= -3.74, Iogp 3 = 8.48, log p 2 = 3.685, p 2 =4845 Ib. per square foot =33.63 Ib. per square inch. Also Rk_ 53.36X200 . 01 V% - - 7n* ,- - ^ &.]. p 2 4845 (0 233^7\ ' 1g ^ ) we should have had U.lO^ / pi \tj \.70J ' log p 2 = 2+(3.27x-0.131) = 1.571, p 2 =37.23 Ib.. per square inch, and v 2 = - = ' Qx/1 . .= 1.99. Proceeding in this way, we plot the two PZ &* & X curves as required. The y curve is the steeper of the two, and for expansion to a given lower temperature reaches a point of considerably less volume. By Equation (3) ; for the upper of the two curves, between P! = 100 ; *i=270, ri=l, and p 2 , AA 0.0000265X170 ,3 log 11.14 206 APPLIED THERMODYNAMICS the curve being somewhat less steep than the y curve. This value of n (1.43) will be found to fit the whole expansion with reasonable accuracy. Also, by Equation (4), _, , 5336-^778 _, n i H /o nooo f >fi ^ "" 9 0163 + f a fairly close check value. If we take p at 50 lb. p?r s uare inch, and ti at 135 absolute, instead of the conditions given, we have, iwe a " = 50 x 144 x 1 X 2.7183 (>2L '= 7360. If we let f s = 100, ^ = 0.01635, Iog e i 2 = 4.6 ; - (log. 22 + log. f a ) = 12.3, ^^log.p^log, 7360-12.3 -0.01635 = -3.42, logj>,=7.75, log^ 2 =3.37, 0.0000205 x 35 0.162X2.3 log 2.2 53.36 --778 0.182 + x 283 . . =l-42. The value of n is thus about the same for this curve as for that formerly considered, and (approximately), in Fig. 122, hjL te U' If this relation were exact, the efficiency of an Otto cycle would be expressed by the same formula as that which holds when the specific heats are constant. In Fig. 124 ; the efficiency of the strip cycle qvwp is = l t a , and if -2. = - = j2 = -^ etc., the efficiency of the whole cycle *d * q t p V 1 ^ = 1 tf^-t^t.-t, *. *6 *. t, For a path of constant volume, in Equation (5), = 1, = -$i, and ^a Pa, t a the gain of entropy is GAS ENGINE DESIGN 207 In the case under consideration, t b 270, t a = 135, a = 0.162, b = 0.0000265, so that Equation (6) gives for the path eb, 0.162 x 2.3 log -^4 + (0.0000265 x 13,)) = 01123 + 0.0036 = 0.1158. If m Fig. 122 the temperature at d is 100, we may write 0.1158 = 0.162 x 2.3 log -^- + 0.0000265 ($,- 100) 100 = 0.372 log t, - 0.744 + 0.0000265 t f - 0.00265, log *, + 0.0000712^ = 2.32, from which t f is, nearly, log- 1 2.32, and ^ = 200, about. In expanding from 270 to 200, the volume increased from 1.0 to 2.21 ; in expand- ing from 135 to 100, it increased from 1 to 2 28. We have computed the change of entropy from p = 50, v = 1, t = 135, to p = 100, r = 1.0, t = 270, as 0.1158. This must equal the change from p = -\\\ 3 - = 1(5.85, = 100, = 2.28, to ^ = 33.6, <y = 2.28, * = ? Now for ^ = 33.6, = 2.21, it was found that t = 200, Adiabatic expansion from this point to the greater volume 2.28 means that t f must be slightly less than 200; but a very slight change in temperature produces a large change in volume since the isothermals and the adiabatics nearly coincide. GAS ENGINE DESIGN 330. Capacity. The work done per stroke may readily be computed for the ideal cycle, as in Art 293. This may be multiplied by the diagram factor to determine the probable performance of an actual engine. To develop a given power, the number of cycles per minute must be established. Ordinary piston speeds are from 450 to 1000 ft. per minute, usually lying between 550 and 800 ft., the larger engines having the higher speeds. The stroke ranges from 1.0 to 2.0 times the diameter, the ratio increasing, generally, with the size of the engine. A gas engine has no overload capacity, strictly speaking, since all of the factors entering into the determination of its capacity are intimately related to its effi- ciency. It can be given a margin of capacity by making it larger than the computations indicate as necessary, but this or any other method involves a con- siderable sacrifice of the economy at normal load. 331. Mean Effective Pressure. Since in an engine of given size the extreme volume range of the cycle is fired, the mean net ordinate of the work area measures the capacity. The quotient of the cycle area by the volume range gives what is called the mean effective pressure (m. e. p.). In Fig. 122, it is ebfd -(V d - 7). We y-l then write m. e. p. = W - ( V 4 - 7 e ); but from Art. 295, W = Q[I - (fr) * ] 5 208 APPLIED THERMODYNAMICS being the gross quantity of heat absorbed iu the cycle. Then, in proper units, without allowance for diagram factor, 332. Illustrative Problem To determine the cylinder dimensions of a four-cycle^ two-cylinder, double-acting engine of 500 Tip.) using producer gas (assumed to contain CO, 394; N, 60; If, 06; parh in 100 by weight) (Art. 285), at 150 r. p. m. and a piston speed of 825ft. per minute. We assume (Fig. 150), P L = 12, P 2 = 144.7, I\ = 200 F., and diagram factor = 0.48 (Arts 312, 329). V /P\y /1447\- 718 Since P l 7^ = P 2 IV, L = ( y~ 1 = f ' I = 5.9. Let the piston displace- I'z vPi/ \ 12 / ment V l - 7 3 = D. Then 7 9 = 0.2045 D and V l = 1.2045 D. The clearance is =0.2045 (Art. 324)*. Also T* = ^ = 659 ' 6 ^ ^^ = 1357 absolute. The heat evolved per pound of the mixed gas (taking the calorific value of hydrogen burned to steam as 53,400) is (0.394 x 4315) + (0.006 x 53,400) = 2021 B. t. u. The products of com- p bustion consist of $ x 0.394 = 3, 0.619 Ib. of C0 2 (specific heat = 0.1G92), 0.006 x 9 = 0.054 Ib. of H 2 (steam, specific heat 0.37), and H (0-619 - 0.394) = 0.751 Ib. of N" accompanying the oxygen introduced to burn the CO, with (0.054- 0.006)H =0.1007 Ib. of N" accompanying the oxygen in- troduced to burn the H; and 0,60 Ib. of K originally in the gas, making a total of 1.5117 Ib. of N (specific lieat 0. 1727) . To raise the temperature of these constituents 1 F. at constant FIG. 150. Arts. 332-335. Design of Gas Engine. volume requires (0.619 x 0.1692) + (0.054x0.37) + (1.5117 x 0.1727) = 0.3849 B. t, u. Adding the heat required for the clearance gases always present, this may be taken as 0.3849 X 1.2045 = 464 B. t u. The rise in temperature T 3 - T 2 is then 2021 -f 0.464 = 4370, and T* = 4370 + 1357 = 5727 absolute. Then P 3 144.7 : ,5727 1357 = 613, and > P* - ir 6 13 _ KO q ] F;" 12 144.7" 509 - * While the use of a " blanket " diagram factor as in this illustration may be justi- fied, in any actual design the clearance at least must be ascertained from the actual exponent of the compression curve. The design as a whole, moreover, would better be based on special assumptions as in Problem 15, (i), page 227. GAS ENGINE DESIGN 209 The work per cycle is y- i = 144 x 0.48 D [ (613 x0 ' 2045 ) " ( 5Q -9 x 1.2045) -(144.7 x 0.2045) + (12 x 1.2045)1 L 0.402 J = 8410 D foot pounds. In a two-cylinder, four-cycle, double-acting engine, all of the strokes are work- ing strokes ; the foot-pounds of work per stroke necessary to develop 500 hp. are - -^- = 55,000. The necessary piston displacement per stroke, D, is 55,000 + 8410 = 6.52 cu. ft, The stroke is 825 -s- (2 X 150) = 2.75 ft. or 38 in. The piston area is then 6.52 + 2.75 = 2.37 sq. ft. or 342 sq. in. The area of the water- cooled tail rod may be about 33 sq. in., so that the cylinder area should be 342 + 33 = 375 sq. in. and its diameter consequently 21.8 in. 333. Modified Design. In an actual design for the assumed conditions, over- load capacity was secured by assuming a load of 600 hp. to be carried with 20 per cent excess air in the mixture. (At theoretical air supply, the power developed should then somewhat exceed 600 hp.) The air supply per pound of gas is now [(0.394 x Jf) + (0.006 x 8)] VJfx 1.2 = 1.422 Ib. Of this amount, 0.23 x 1.422 = 0.327 Ik is oxygen. The products of combustion are f f x 0.394 = 0.619 Ib. C0 21 0.006 x 9 = 0.054 Ib. H 2 O, (1.422 - 0.327) + 0.60 = 1.693 Ib. N, and 0.327 - (if x 0.394) - (8 x 0.006) =0.054 Ib. of excess oxygen ; a total of 2.422 Ib. The rise in temperature r 3 - T 2 is _ 2021*1.2045 (0.619 X 0.1692)'+ (0 054 X 37) + (1,693 X 0.1727) + (0.054 X 0.1551) Then !T 3 3950 + 1357 - 5307 absolute, " * P - P . Pz - P ' 4 ~ l Pi " 1447 and the work per cycle is i AA v n AC n ["(569 X O.C045) -(47.2JX 1 2045) -f 144.7 X 0.2045y-H(12 X 1.2045)1 144XU.4S^ 04Q2 J =7630 D fooir-pounds. 600 X 33000 The piston displacement per stroke is 2 v 150 x 7630 ~ 8 ' 6 ^ CU " **"' tbe ^tinder area is (8.65 -i- 2.75)144 + 33 = 486 sq. in., and its diameter $4.9 in. The cylinders were actually made 23J by 33 in., the gas composition being independently assumed. 334. Estimate of Efficiency. To determine the probable efficiency of the engine under consideration : each pound of working substance is supplied with 1.422 Ib. of air. Multiplying the weights of the constituents by their respective specific volumes, we obtain as the volume of mixture per pound of gas, 31.33 cu. ft. at 14.7 Ib. pressure and 32 F., as follows : 210 APPLIED THERMODYNAMICS CO, 394 x 12 75 = 5.01 H, 006 x 178 t>3 = 1.07 N. 0.600 x 12.75 = 7.65 Air, 1.422 x 12.387 =1760 31.33 At the state 1, Fig. 150, 7^ = 659 6, P l = 12, whence v = PI P,.r, l= 659.6 x 147 x 31.83 51 1 P,T 12 x 491.6 The piston displaces 8.65 X 300 = 2595 cu, ft. of this mixture per minute. The heat taken in per minute is then 2021 X (2595 -s- 51.2) = 102,400 B. t. u. The work done fiOO V ^l^ftOfl per minute is - ^ - = 25,500 B t. u. The efficiency is then 25,500 -f- 102,400 = 0.249. An actual test of the engine gave 0.282, with a load somewhat under 1 ^ t\7 fi f\Q fi 600 hp. The Otto cycle efficiency is - 1357 = 0.516.* 335. Automobile Engine. To ascertain, the probable capacity and economy of a four-cylinder^ four-cycle, single-acting gasoline engine with cylinders 4- by 5 in. 3 at J.500 r. p. m. In Fig. 150, assume P 3 = 12, P 2 = 84.7, 7\ = 70 F., diagram factor, 0.375 (Arts. 312, 329). Assume the heating value of gasoline at 19,000 B. t. u , and its composition as C Q H^: its vapor density as 3.05 (air = 1.). Let the theoretically necessary quantity ot air be supplied. The engine will give two cycles per revolution. Its active piston displacement is then ' 7854 x W* x 5 x 3000 = 145.5 cu. ft. per minute, which may be repre- 1728 seated as V^ - F 2 , Fig. 150. We now find s = -" = 0,2495: F 2 = 0.2495 F I; 0.7505 7i = 145. 5; 7 1 = V 1 V84.7/ Clearance = = 0.384 (Art. 324); r a = ,- = 936 absolute. 145.0 1 x iy^ To burn one pound of gasoline there are required 3.53 Ib. of oxygen, or 15.3 Ib. of air. For one cubic foot of gasoline, we must supply 3.05 x 15.3 = 46.6 cu. ft. of air. The 145.5 cu. ft. of mixture displaced per minute must then consist of * The actual efficiency will always be less than the product of the Otto cycle efficiency by the diagram factor. Thus, let the actual cycle be described as 1234, Fig. 160, and let the corresponding ideal cycle be 123'4 ; . The efficiencies are, respectively, 1234 The quotient 1234 -f- 123'4' = the diagram factor Then write 1( ? T3} * diagram factor x Jf= -JSL^ CURRENT GAS ENGINE FORMS 211 145.5 ^- 47.6 = 3.06 cu. ft. of gasoline and 142.44 cu. ft. of air, at 70 F. and 12 Ib. pressure. The specific volume of air at this state is 52<Q ' 6 x 14 '' 7 x 12 - 3S7 _ 16.33 491.6 x 12 cu. ft. ; that of gasolene is 16.38 -*- 3.03 = 3.37 cu. ft. The weight of gasoline used per minute is then 3.06 - 5.37 = 0.571 II. The heat used per minute is 0.571 x 19,000 = 10,840 B. t. u. The combustion reaction may be written 86 + 304 = 264 + 126 W= 3.06 lb.C0 2 per lb.C 6 H 14 V/ = 1-35 Ib. II 2 per Ib. C 6 H 14 11 x W = 11-82 Ib. N per Ih. C fl H 14 16.23 = 1. + 15.3, approximately. The heat required to raise the temperature of the products of combustion 1 F. is [(3.06 X 0.1692) + (1.35 X 0.37) + (11.82 X 0.1727)] 0571 = 1.746 B. t. u. per minute. Adding for clearance gas, this becomes 1,746 X 1.334 = 2.327 B. t. u. The rise in temperature T* - T is then 10,840 -=- 2 327 = 4660, T* - 4660 -|- 936 = 5596 absolute, P 3 = 84.7^- - 508, P 4 = 12 - 72 0, and the wr* per m^n- ute K 0.375X144[^ 508X48 ' 5 1 200 000 foot-bounds. This is equivalent to '^ * = 1540 B. t u. per minute or to 77o 1 20' A OCO 33 000 = 86 * l TSe ' power ' Tli e e ff lcienc y is 1540 * 10,840 = 0.142. In an auto- mobile running at 50 miles per hour, this would correspond to 50 -s- (0.571 X 60) = 1.46 miles run per pound of gasohne. In practice, the air supply is usually incor- rect, and the power and economy less than those computed. It is obvious that with a given fuel, the diagram factor and other data of assumption are virtually fixed. An approximation of the power of the engine may then be made, based on the piston displacement only. This justifies in some measure the various rules proposed for rating automobile engines (30 a). One d*n of these rules is, brake hp. == -, where n is the number of four-cycle cylinders of 2.5 diameter d inches, running at a piston speed of 1000 ft. per minute, CURRENT GAS ENGINE FOKVIS 336. Otto Cycle Oil Engines. This class includes, among many others, the Mietz and Weiss, two-cycle, and the Daimler, Priebtman, and Hornsby-Akroyd, four-cycle. In the last-named, shown in Fig. 151, kerosene cil is injected by a small pump into the vaporizer. Air is drawn into the cylinder during the suction stroke, and compressed into the vaporizer on the compression stroke, where the simultaneous presence of a critical mixture and a high temperature produces the explosion. External heat must be applied for starting. The point of ignition is determined by the amount of compression; and this may be varied by adjusting 212 APPLIED THERMODYNAMICS the length of the connecting rod on the valve gear. The engine is governed by partially throttling the charge of oil, thus weakening the mixture and the force of FIG. 151. Arts 322, 336, -Kerosene Engine with Vaporizer. (From " The Gaa Engine," by Cecil P. Poole, with the permission of the Hill Publishing Company ) the explosion. The oil consumption may be reduced to less than 1 Ib. per brake hp, per hour. In the Priestman engine, an earlier type, air under pressure sprayed the oil into a vaporizer kept hot by the exhaust gases. The method of governing was to reduce the quantity of chaige without changing its proportions. A hand pump and external heat for the vaporizer were necessary in starting. An indicated thermal efficiency of 0.1 Go has been obtained. The Daimler (German) engine uses hot-tube ignition without a timing valve, the hot tube serving as a vaporizer. Extraordinarily high speeds are attained. 337. Modern Gas Engines : the Otto. The present-day small Otto engine is ordi- narily single-cylinder and single-acting, governing on the "hit or miss" principle (Art. 343). It is used with all kinds of gas and with gasoline. Ignition is elec- trical, the cylinder water jacketed, the jackets cast separately from the cylinder. The Foos engine, a simple, compact form, often made portable, is similar in princi- ple, lu the Crossley-Otto, a leading British type, hot-tube ignition is used, and the large units have two horizontal opposed single-acting cylinders. In the Andrews form, tandem cylinders are used, the two pistons being connected by external side rods. TYPES OF GAS ENGINE 213 338. The Westinghouse Engine. This has recently been developed in very large units. Figure 132 shows the \\oiking side of a two-cylinder, tandem, double acting engine, representing the mlt valves on top of the cylinders. FIG. 152. Arts. 338, 330. Westinghouse Gas Engine. Two-cylinder Tandem, Four-cycle. Smaller engines are often built vertical, with one, two, or three single-acting cylinders. All of these engines are four-cycle, with electric ignition, governing by varying the quantity and proportions of the admitted mixture. Sections of the cylinder of the Riverside horizontal, tandem, double-acting engine are shown in Fig. 15;?. It has an extremely massive frame. The Allis-Chalmers engine is built in laige units along similar general lines. Thirty-six of the latter engines of 4000 hp. capacity each on blast furnace gas are now (1009) being constructed. They weigh, each, about 1,500,000 lb., and run at83J r. p. m. The cylinders are 44 by 54 in. Nearly all are to be direct-connected to electric generators. 339. Two-cycle Engines. In these* the explosions are twice as frequent as with the four-cycle engine, and cooling is consequently more difficult. With an equal number of cylinders, single- or double-acting, the two-cycle engine of course gives better regulation. The first important two-cycle engine was introduced by Clerk in 18SO. The principle was the same as that of the engine shown in Fig. 11>. The Oechelhaueser engine has two single-acting pistons in one cylinder, which are connected with cranks at ISO , so that they alternately approach toward and lecede from each other. The engine frame is excessively long. Changes in the quantity of fuel supplied control the speed. The Eoerting engine, a double-acting horizontal form, has two pumps, one for air and one for gas. A ' scavenging " charge of air is admitted just prior to the entrance of the gas, sweeping out the burnt gases and acting as a cushion between the incoming charge and the exhaust ports. The engine is built in large units, with electrical ignition and compressed air starting gear. The speed is conti oiled by changing the mixture propoitions. 340. Special Engines. For motor bicycles, a single air-cooled cylinder is often used, with gasoline fuel. Occasionally, tv\o cyhndeis are employed. The engine 214 APPLIED THERMODYNAMICS TYPES OF GAS ENGINE 215 is four-cycle and runs at high speed. Starting is effected by foot power, which can be employed whenever desired. Ignition is electiical and adjustable. The speed is controlled by throttling. Extended surface air-cooled cylinders have also been used on automobiles, a fan being employed to circulate the air, but the limit of size appears to be about 7 hp. per cylinder. Most automobiles have water- cooled cylinders, usually four in number, four-c\cle, single-acting, running at about 1000 to 1200 r. p. m., normally. Governing is by throttling and by chang- ing the point of ignition. The cylinders are usually vertical, the jacket water being circulated by a centrifugal pump, and being used repeatedly. Both hot-tube and electiical methods of ignition have been employed, but the former is now almost wholly obsolete. The number of cylinders varies from one to six ; occa- sionally they are arranged horizontally, duplex, or opposed. Two-cycle engines have been introduced. The fuel in this country is usually gasoline. For launch engines, the two-cycle piinciple is popular, the crank case forming the pump chamber, and governing being accomplished by throttling. Kerosene or gasoline are the fuels. 341. Alcohol Engines. These are used on automobiles in France. A special carburetor is employed. The cylinder and piston an angement is sometimes that of the Oechelhaueser engine (Art. 330). The speed is controlled by varying the point of ignition. In launch applications, the alcohol is condensed, on account of its high cost, and in some cases is not burned, but serves merely as a working fluid in a " steam " cylinder, being alternately vaporized by an externally applied gaso- line flame and condensed in a surface condenser. The low value of the latent heat of vaporization (Art. 360) of alcohol permits of * getting up steam " more rapidly than is possible in an ordinary steam engine. 342. Basis of Efficiency. The performances of gas engines may be compared by the cubic feet of gas, or pounds of liquid fuel, or pounds of coal gasified in the producer, per horse power hour ; but since none of these figures affords any really definite basis, on account of variations in heating value, it is usual to express the results of trials in heat units consumed per horse power per minute. Since one horse power equals 33,000 - 778 = '2A'2 B. t. u. per minute, this constant divided by the heat unit consumption gives the indicated thermal efficiency. In making tests, the over-all efficiency of a producer plant may be ascertained by weighing the coal. When liquid fuel is used, the engine efficiency can readily be determined separately. To do this with gas involves the measurement of the gas, al\\ ays a matter of some difficulty with any but small engines. The measurement of power by the indicator is also inaccurate, possibly to as great an extent as o per cent, which may be reduced to 2 per cent, according to Hopkinson, by employing mirror indicatoi s. This error has resulted in the custom of expressing performance in heat units consumed per brake horse power per hour or per kw.-hr., where the engines are directly connected to generators. There is some question as to the proper method of considering the negative loop, bcde, of Fig. 136. By some, its area is deducted from the gross work area, and the difference used in computing the indicated horse power. By others, the gross work area of Fig. 136 is alone considered, and the "fluid friction " losses 216 APPLIED THERMODYNAMICS producing the negative loop aie then clashed with engine friction as reducing the "mechanical efficiency." Yaiious codes for testing gas engines aie in use (31). 343. Typical Figures Small oil or gasoline engines may easily show 10 per cent brake efficiency. Alcohol engines of small size consume less than 2 pt. per brake hp.-hr. at full load (->2). A well-adjusted Otto engine has given an indicated thennal efficiency of 010 \\itli gasoline and 023 with kerosene (33). Ordinary producer gas engines of average size under test conditions have repeatedly shown indicated thermal efficiencies of 2.5 to 2,9 per cent. A Cockerill engine gave 30 per cent. Hubert (3-1) tested at Seramg an engine shov\ ing neaily 32 per cent indicated thermal efficiency. Mathob (3o) reports a test of an Ehihardt and Lehiner double- acting, fom-cycle 000 hp engine at Ilemitz which reached nearly 38 per cent. A blast furnace gas engine gave at full load 25.4 per cent. Expressed in pounds of coal, one plant with a low load factor gave a kilowatt-hour per 1 8 Ib. In another case, 1.59 Ib. was reached, and in another, 2 97 Ib of wood per kw.-hr. It is common to hear of guarantees of 1 Ib of coal per brake hp.-hr., or of 11,000 B. t. u. in gas. A recent test of a Crossley engine is reported to have shown the result 1.13 Ib. ot coal per kw.-hr. Under ordinary running conditions, 1.5 to 2.0 Ib ^ith varying load may easily be realized. These latter figures are of course for coal burned iu the producer. They repiesent the joint efficiency of the engine and the pioducer. The best results have been obtained m Germany. For the engine alone, Schroter is reported to have obtained on a Guldner engine an indicated thermal efficiency of 0.<t27 at full load with illuminating gas (30). The efficiency cannot exceed that of the ideal Otto cycle. In one test of an Otto cycle engine an indicated thermal efficiency of 0.37 was obtained, while the ideal Otto efficiency was only 0.41. The engine was thus within 10 per cent of perfection for its cycle * The Diesel engine has given from 0.32 to 0.412 indicated thermal efficiency. Its cycle, as has been shown, peimits of higher efficiency than that of Otto. Plant Efficiency. Frames ba\e been given on coal consumption. Over- all efficiencies from fuel to indicated u oik have ranged from 0.14 upward. At the Maschinenfabrik Wiuterthur, a consumption of 0.7 Ib. of coal (13,850 B. t. u.) per brake hp.-hr. at full load has been reported (37). This is closely paralleled by the 285 indicated plant efficiency on the Guldner engine mentioned in Art. 343 when opeiated with a suction producer on anthiacite coal. At the Royal Foundry, Wurtemburg (38), 0.78 Ib. ol anthracite weie burned per 1 hp.-hr., and at the Imperial Post Office, Hamburg, O.U3 Ib. of coke. In the best engines, variations of efficiency with reasonable changes of load below the normal have been greatly reduced, largely by impi oved methods of governing. 345. Mechanical Efficiency. The ratio of work at the brake to net indicated work ranges about the same for gas as for steam engines having the same arrange- ment of cylinders. When mechanical efficiency is understood in this sense, its * At the present tune, any reported efficiency much above 30 per cent should be regarded as needing authoritative confirmation, GAS ENGINE TRIALS 217 value is nearly constant for a given engine at all loads, decreasing to a slight extent only as the load is reduced In the other sense, suggested in Art. 342, i.e. the mechanical efficiency being the ratio of work at the brake to gross indicated work (no deduction being made for the negative loop area of Fig 136), its value falls off sharply as the load decreases, on account of the increased proportion of "fluid friction." Lucke gives the following as average values for the mechanical efficiency in the latter sense: ENGINE MECHANIC \L AT FULL EFFICIENCY LOVD F<nn>~ryalf 7V/ n-cyvlc Lar^e, 500 Ihp. and over, ..... SI to SO 63 to 0.70 Medium, 25 to 500 Ihp , ..... O.VO to O.S1 U.D i to 06 Small, 4 to 25 Ihp., 0.74 to 0.80 0.00 to 70 The friction losses for a single-acting engine are of course relative!} 71 greater than those for an ordinary double-acting steam engine. 346. Heat Balance. The principal losses in the gas engine are due to the cooling action of the jacket water (a necessary evil in present prac- tice) and to the heat carried away in the exhaust. The arithmetical means of nine trials collated by the writer give the foil owing percentages representing the disposition of the total heat supplied: to the jacket, 40.52; to the exhaust, 33.15; work, 21.87; unaccounted for, 6.23. Hutton (40) tabulates a large number of trials, from which similar arithmetical averages are derived as follows: to the jacket, 37.96; to the exhaust, 29.84; work, 22.24; unaccounted for, 8.6. In general, the larger engines show a greater proportion of heat converted to work, an increased loss to the exhaust and a decreased loss to the jacket. In working up a "heat balance," the loss to the exhaust is measured by a calorimeter, which cools the gases below 100 F. The heat charged to the engine should therefore be based on the " high" heat value of the fuel (Arts. 561, 561a). The * k work " item in the above heat balance is indicated work, not brake work. Unlike the jacket water heat (Art. 352), the heat carried off in the exhaust gases is at fairly high temperature. There would be a decided gain if this heat could be even partly utilized. Suppose the engine to have consumed, per hp., 10,000 B. t. u. per hour, of which 30 per cent, or 3000 B. t. u., passes off at the exhaust. At 80 per cent efficiency of utilization, 2400 B. t. u. could then be recovered. In form- ing steam at 100 Ib. absolute pressure from feed water at 212 F., 1006.8 B. t. u. are needed per pound of steam. Each horse power of the gas engine would then give as a waste gas by-product 2400 1006.3 = 2.39 Ib. of steam. Or if the steam plant had an efficiency of 10 per cent, 240 B. t. u. could be obtained in work from the steam engine for each horse power of the gas engine. This is 240 - 2545 = Q\ per cent of the work given by the gas engine. A much higher gain would be possible if the steam generated by the exhaust gases were used for heating rather than for power production. 218 APPLIED THERMODYNAMICS 347. Entropy Diagram. When the PV diagram is given, points may be trans- Vb Pb f erred to the entropy plane by the formula n 6 -7? a = fc log e - + / log e (Art. y a * o 169). The state a may be taken at 32 F. and atmospheric pressure; then the entropy at any other state b depends simply upon V d and P&. To find V a , we must know the equation of the gas. According to Richmond, (41) the mean value of k may be taken at 0.246 on the compression curve and at 0.26 on the ex- pansion curve, while the mean values of I corresponding are 0.17G and 0.189. The values of R are then 778(0.240 - 0.176) = 54.46 and 778(0.260 - 0.189) = 55.24. The characteristic equations are, then, PV = 54.46 T along the compression curve; and PV = 55.24 T along the expansion curve. The formula gives changes of en- tropy per pound of substance. The indicator diagram does not ordinal ily depict the behavior of one pound; but if the weight of substance used per cycle be known, the volumes taken from the PV diagram may be converted to specific volumes for substitution in the formula. It is sometimes desirable to study the TFielations throughout the cycle. In Fig. 154, let ABCD be the PV diagram. Let EF be any line of constant volume intersecting this diagram at G, H. By Charles' law, T : T B : : P G - P H . The Pqr T FIG. 154. Art. 347. Gas Engme TV Diagram. ordinates JG, JH may therefore serve to represent temperatures as well as pres- sures, to some scale as yet undetermined. If the ordinate JG represent tempera- turf, then the line OG is a line of constant pressure. Let the pressure along this line on a TV diagram be the same as that along IG on a PV diagram. Then (again by Charles' law) the line OH is a line of constant press ui e on the TV plane, corresponding to the line KH on the PV plane. Similarly, OL corresponds to MJT and OQ to RB. Pioject the points 5, T, R, B, where MN and RB intersect the PV diagram, until they intersect OL, OQ. Then points Z7, Q, W, X are GOVERNING 219 points on the corresponding TV diagram. The scale of T is determined from the characteristic equation; the value of R may be taken at a mean between the two given. A tiansfer may now be made to the NT plane by the aid of the equation n^-n a -l log e |f + (k - Z)log. -^ (Art. 169), in which T a = 491.6, * ra .54.46 x 491.6 2116.8 = 12.64. Figure 155, from Reeve (42), is from a similar four-cycle engine. The enor- mous area BA CD represents heat lost to the water jacket. The inner dead center of the engine is at x ; thereafter, for a short period, heat is evidently abstracted from the fluid, being afterward restored, just as in the case of a steam engine (Art. 431), because during expansion the temperatm e of the gases falls below that of the cylinder walls. This agrees with the usual notion that most of the heat is discharged to the jacket early in the expansion stroke. It would probably be a fair assumption to consider this loss to occur during ^gn^t^on t as far as its effect on the diagram is concerned. Reeve gives several instances in which the expansive path resembles xBzD; other investi- gators find a constant loss of heat during expan- sion. Figure 156 gives the PV and NT dia- grams for a Hornsby-Akroyd engine; the expan- sion line be here actually rises above the iso- thermal, indicative of excessive after burning. FIG. 155. Art. 347. Gas Engine 348. Methods of Governing. The Entropy Diagram, power exerted by an Otto cycle engine may be varied in accordance with the external load by various methods; in order that efficiency may be maintained, the governing should not lower the ratio of pressures during compression. To ensure -N FIG. 156. Art. 347. Diagrams for Hornsby-Akroyd Engine. 220 APPLIED THERMODYNAMICS this, variation of the clearance, by mechanical means or water pockets and outside compression have been proposed, but no practicably efficient means have yet been developed. The speed of an engine may be changed by varying the point of ignition, a most wasteful method, because the reduction in power thus effected is unaccompanied by any change whatever in fuel consumption. Equally wasteful is the use of excessively small ports for inlet or exhaust, causing an increased nega- tive loop area and a consequent reduction in power when the speed tends to increase. In engines where the combustion is gradual, as in the Brayton or Diesel, the point of cut-off of the charge may be changed, giving the same sort of control as in a steam engine, Three methods of governing Otto cycle engines are in more or less common use. In the "Jiitor-miss" plan, the engine omits drawing in its charge as the external load decreases. One or more idle strokes ensue. No loss of economy results (at least from a theoretical standpoint), but the speed of the engine is apt to vary on account of the increased irregularity of the already occasional impulses. Governing by changing the proportions of the mixture (the total amount being kept constant) should apparently not affect the compression; actually, however, the compression must be fixed at a sufficiently low point to avoid danger of pre-ignition to the strongest probable mixture, and thus at other proportions the de- gree of compression will be less than that of highest efficiency. A change in the quantity of the mix- ture, without change in its propor- tions, by throttling the suction or by entirely closing the inlet valve Art. 348 -Effect of lOirottlin/ tOWard the end f the suction stroke, results in a decided change of compression pressure, the superimposed cards being similar to those shown in Pig. 157. In theory, at least, the range of compression pressures would not be affected; but the variation in proportion of clearance gas present requires injurious limitations of final compression pressure, just as when governing is effected by variations in mixture strength. Besides, the rapidity of flame propagation is strongly influenced by variations in the pressure at the end of compression. 349. Defects of Gas Engine Governing. The hit-or-miss system may be regarded as entirely inapplicable to large engines. The other practicable methods sacrifice the efficiency. Further than this, the DETAILS 221 governing influence is exerted during; the suction stroke, one full revolu- tion (in four-cycle engines) previous to the working stroke, which should be made equal in effort to the external load. If the load changes during the intervening revolution, the control will be inadequate. Gas engines tend therefore to irregularity in speed and low efficiency under variable or light loads. The first disadvantage is overcome by increasing the number of cylinders, the weight of the fly wheel, etc., all of which entails additional cost. The second disadvantage has not yet been overcome. Tn most large power plant engines, both the quantity and strength of the mixture are varied by the governor. 350. Construction Details. The irregular impulses characteristic of the gas engine and the high initial pressures attained require excessively heavy and strong frames. For anything like good regulation, the fly wheels must also be exceptionally heavy. For small engines, the bed casting is usually a single heavy piece. The type of frame usually employed on large engines is illustrated in Fig. 152. It is in contact with the foundation for its entire length, and in many cases is tied together by rods at the top extending from cylinder to cylinder. Each working end of the cylinder of a four-cycle engine must have two valves, one for admission and one for exhaust. In many cases, three valves are used, the air and gas being admitted separately. The valves are poppet, of the plain disk or mushroom type, with beveled seats; in large engines, they are sometimes of the double-beat type, shown in Fig. 153. Sliding valves cannot be employed at the high temperkture of the gas cylinder.* Exhaust opening must always be under positive control; the inlet valves may be automatic if the speed is low, but are generally mechanically operated on large engines. Alljshould be finally seated by spring action, so as to avoid shocks. In horizontal four-cycle engines, a earn shaft is driven from an eccentric at half the speed of the engine. Cams or eccen- trics on this shaft operate each of the controlling valves by means of adjustable oscillating levers, a supplementary spring being empolyed to accelerate the closing of the valves. In order that air or gas may pass at constant speed through the ports, the cam curve must be carefully proportioned with reference to the varia- tion in conditions in the cylinder (43). Hutton (44) advises proportioning of ports such that the mean velocity may not exceed 60 ft. per second for automatic inlet valves, 90 ft. for mechanically operated valves, and 75 ft. for exhaust valves, on small engines. 351. Starting Gear. No gas engine is self-starting. Small engines are often started by turning the fly wheel by hand, or by the aid of a bar or gearing. An auxiliary hand air pump may also be employed to begin the movement. A small electnc motor is sometimes used to drive a gear-faced fly wheel with which the motor pinion meshes. In all cases, the engine starts against its friction load only, and it is usual to provide a method for keeping the exhaust valve open during part of the compression stroke so as to decrease the resistance. In multiple-cylinder engines, as in automobiles, the ignition is checked just prior to stopping. A com- pressed but unexploded charge will then often be available for restarting. In the * The sleeve valve, analogous to the piston valve commonly used on locomotives^ has been successfully developed for automobile work 222 APPLIED THERMODYNAMICS Clerk engine, a supply of unexploded mixture was taken during compression from the cylinder to a strong storage tank, from which it could be subsequently drawn Gasoline railway motor cars are often started by means of a smokeless powder cartridge exploded in the cylinder Modern lar^e enpines are started by com- pressed air, furnished by a direct-driven or independent pump, and stored in small tanks. Kecent automobile practice has developed two new starting methods: (a) By acetylene generated from calcium carbide and watei under pressure, and (6) by an electric motor, operated from a storage battery which is charged while the engine is running The same batteiy lights the car. 352. Jackets. The use of water-spray injection during expansion has been abandoned, and air cooling is practicable only in small sizes (say, for diameters less than 5-inch). The cylinder, piston, piston rod, and valves must usually be thoroughly water-jacketed * Positive circulation must be provided, and the water cannot be used over again unless artificially cooled. At a heat, consumption of 200 B t u. per minute per Ihp v with a 40 per cent loss to the jacket, the theoretical consumption of water heated from 80 to 160 F is exactly 1 Ib. per Ihp per minute. This is greater than the water consumption of a non-condensing steam plant, but much less than that of a condensing plant The discharge water fiom large engines is usually kept below 130 F. In smaller units, it may leave the jackets at as high a temperature as 160 F. The usual nss of temperature of water while passing through the jackets is from 50 to 10 j F. The circulation may be produced either by gravity or by pumping. 353. Possibilities of Gas Power. The gas engine, at a comparatively early stage in its development, has surpassed the best steam engines in thermal effi- ciency. Mechanically, it is less perfect than the latter ; and commercially it is regarded as handicapped by the greater lehabihtv, moie geneial field of applica- tion, and much lower cost (excepting, possibly, in the Idrgest sizes j) of the steam engine. The use of producer gas f 01 power eliminates the coal .smoke nuisance , the stand-by losses of producers are low ; and gas may be stored, in small quanti- ties at least The small gas engine is quite economical and may be kept so. The small steam engine is usually wasteful. The Otto cycle engine regulates badly, a disadvantage which can be overcome at excessive cost; it is not self -starting ; the cylinder must be cooled. Kveu if the mechanical necessity for jacketing could be overcome, the same loss would be experienced, the heat being then earned off in the exhaust. The ratio of expansion is too low, cau&ing excessive waste of heat at the exhaust, which, however, it may prove possible to reclaim. The heat in the jacket water is large in quantity and losv in temperature, so that the proV lem of utilization is confronted with the second law of thermodynamics. Methods of reversing have not yet been worked out, and no important marine applications of gas power have been made, although small producer plants have been installed for ferryboat service with clutch reversal, and compressed and * The piston need not be cooled in single-acting four-cycle engines. f Piston speeds of large gas engines may exceed those of steam engines. Unless special care is exercised in the design of ports, the efficiency will fall off rapidly with increasing speed. Gas engines have been built in units up to 8000 hp :-2000 hp. from each of the four twin-tandem double-acting cylinders. GAS POWER 223 stored gas has been used for driving river steamers in France, England, and Germany. The proposed combinations of steam and gas plants, the gas plant to take the uniform load and the steam units to care for fluctuations, really beg the whole question of comparative desirability. The bad k - characteristic " curve low effi- ciency at light loads and absence of bona fide overload capacity "will always bar the gas engine from some services, even where the storage battery is used as an auxiliary. Many manufactui ing plants nuist have steam in any case for process work. In such, it will be difficult for the gas engine to gain a foothold. For the utilization of blast furnace waste, even aside from any question of commercial power distribution, the gas engine has become of prime economic importance. [A topical list of research problems in gas power engineering, the solution of which is to be desired, is contained m the Report of the Gas Power Research Com- mittee of the American Society of Mechanical Engineers (1910).] [See the Resume of Producer Gas Investigations, by Fernald and Smith, Bulletin No. 13 of the United States Bureau of Mines, 1911.] (1) Button, The Gas Engine, 190S, 545; Clerk, Theory of the Gas Engine, 1903, 75. (2) Hutton, The Gas Engine, 1908, 158. (3) Clerk, The Gas Engine, 1890, 119-121. (4) Ibid., 129. (5) Ibid, 133. (6) Ibid., 137. (7) Ibid, 198. (8) Engineering News, October 4, 1906, 357. (9) Lucke and Woodward, Tests of Alcohol Fuel, 1907. (10) Junge, Power, December, 1907. (10 a) For a fuller exposi- tion of the limits of producer efficiency with either steam or waste gas as a diluent, see the author's paper, Trans. Am. Inst. Chem. Engrs T Vol. II. (11) Trans. A. S. M. E., XXVIII, 6, 1052. (12) A test efficiency of 657 was obtained by Parker, Holmes, and Campbell. United States Geological Survey, Professional Paper No. 48. (13) Ewing, The Steam Engine, 1906, 418. (14) Clerk, The Theory of the Gas Engine, 1903. (15) Theorie und Construction eines rationdlen Warmemotors. (16) Zeuner, Technical Thermodynamics (Klein), 1907, I, 439, (17) Trans. A. S. M. E., XXI, 275. (18) Ibid., 286. (19) Op. ciL, XXIV, 171. (20) Op. cit., XXI, 276. (21) Gas Engine Design, 1897, 33. (22) Op. at., p 34 et seq. (23) See Lucke, Trans. A.S.M. E., XXX, 4, 418. (24) The Gas Engine, 1890, p 95 et seq. (25) A. L. Westcott, Some Gas Engine Calculations based on Fuel ami Exhaust Gases; Power, April 13, 1909, p. 693. (26) Hutton, The Gas Engine, 1908, pp. 507, 522. (27) The Gas Engine, 1908. (28) Clerk, op. cit , p. 216. (29) Op. *., p. 291. (30) Op. cit., p. 38. The corresponding usual mean effective pressures are given on p. 36. (30 a) See the author's papers, Commercial Ratings for Internal Combustion Engines, in Machinery, April, 1910, and Design Constants for Small Gasolene Engines, with Special Reference to the Automobile, Journal A. S. M. E., September, 1911. (31) Zeits. Ver. Deutsch. Ing., November 24, 1906; Power, February, 1907. (32) The Electrical World, December 7, 1907, p. 1132. (33) Trans. A. S. M. E., XXIV, 1065. (34) Bui. Soc. de V Industrie Mineral, Ser. Ill, XIV, 1461. (35) Trans. A. S. M. E., XXVIII, 6, 1041. (36) Quoted by Mathot, supra. (37) Also from Mathot. (38) Mathot, supra. (40) Op. cit., pp. 342-343. (41) Trans. A. S. M. E., XIX, 491. (42) Ibid., XXIV, 171. (43) Lucke, Gas Engine Design, 1905. (44) Op. tit., 483. 224 APPLIED THERMODYNAMICS SYNOPSIS OF CHAPTER XI The Producer The importance of the gas engine is largely due to the producer process for making cheap gas. In the gas engine, combustion occurs in the cylinder -, and the highest temperature attained by the substance determines the cyclic efficiency. Fuels are natural gas, carbureted and uncarburetcd water gas, coal gas, coke oven gas, producer gas, blast furnace gas ; gasoline, kerosene, fuel oil, distillate, alcohol, coal tars. The gas producer is a lined cylindrical shell in which the fixed carbon is converted into carbon monoxide, while the hydrocarbons are distilled off, the necessary heat being supplied by the fixed carbon burning to CO. The maximum theoretical efficiency of the producer making power gas is less than that, of the steam boiler. Either steam or exhaust gas from the engine* must be intro- duced to attain maximum efficiency. The reactions are complicated and partly reversible. The mean composition of producer gas, by volume, is CO, 10.2 ; C0 2l 05, H, 12.4 ; CH 4 , C 2 H 4 , 3.1; N, 55.8. The "figure of merit ^ is the heating value of the gas per pound of carbon contained. Gas En (tine The Otto cycle is bounded by two udialH.it ics and two lines of constant volume; the engine may operate in either thefour-s'rftkc eyrie or the two-stroke cycle. In the two-stroke cycle, the inlet and exhaust ports are loth open at once. In the Otto cycle, 5> = t and ^ = If J ' P e P d T e T d Efficiency - Tf ~ T * = 1 - f ?*\ IT = 7& " T '= 1 - / r\ V; it depends solely on the extent of compression. The Sargent and Frith cycles. Efficiency of Atkinson engine (isothermal rejection of heat) = l " log e ~J 10 J. e J. e higher thaii that of the Otto cycle. Lenoir cycle: constant pressure rejection of heat, efficiency = 1 - ^ - y h ~" - Tj- T d T f T^" Brayton cycle : combustion at constant pressure; efficiency = I fr g ~~ ,, -- I?"" ~ > 2/( A J-n) Tb T n T T 1 or, with complete expansion, " "" <l TH A special comparison shows the Clerk Otto engine to give a much higlier efficiency than the Brayton or Lenoir engine, but that the Brayton engine gives slightly the largest work area. The Clerk Otto (complete pressure) cycle gives an efficiency of 1 'II 'II rri g rn JL e JLo J. e intermediate between that of the ordinary Otto and the Atkinson. SYNOPSIS 225 The Diesel cycle: isothermal combustion; efficiency = 1 -- L \ a/ - =!; increases as ratio of expansion decreases. . yRT a log e Va The Diesel cycle . constant pressure combustion. The Humphrey internal combustion pump. Modifications in Practice The PV diagram of an actual Otto cycle engine is influenced by (a) proportions of the nurture, \tkich must not be too weak or too strong, and must be controllable , (&) maximum allowable temperature after compression to avoid pre-ignition ; the range of compression, \\hich determines the efficiency, depends upon this as well as upon the pre-coinpiesaion pressure and temperature; (c) the rise of pressure and temperature during combustion; always less than those theoretically computed, on account of (1) divergences from Charles' law, (2) the variable specific heats of gases, (0) slow combustion, (4) disso- ciation ; (<Z) the shape of the expansion curve, usually above the adiabatic, on account of after burning, in spite of loss of heat to the cylinder wall; (e) the forms of the suction and exhaust lines, which may be affected by badly proportioned ports aud passages and by improper valve action. Dissociation prevents the combustion reaction of more than a certain proportion of the elementary gases at each temperature within the critical limits. The point of ignition must somewhat precede the end of the stroke, particularly with weak mixtures. Methods of ignition are by hot tube, jump spark, and make and break. Cylinder clearance ranges from 8.7 to 56 per cent. It is determined by the compression pressure range. Scavenging is the expulsion of the burnt gases in the clearance space prior to the suction stroke. The diagram factor is the ratio of the area of the indicator diagram to that of the ideal cycle. Analysis with specific heats variable. 4'-( Mean effective pressure - , r Gas Engine Design In designing an engine for a given power, the gas composition, rotative speed and piston speed are assumed. The probable efficiency may be estimated in advance. Overload capacity must be secured by assum- ing a higher capacity than that normally needed ; the engine will do no more work than that for which it is designed. 226 APPLIED THERMODYNAMICS Current Forms Otto cycle oil engines include the Mietz and TVeiss, two-cycle, and the Daimler, Priest- man, and Hornsby-Akroyd, four-cycle. Modem forms of the Otto got* engine include the Otto, Foos, Crossley-Otto, and Andrews. The TTestinghouse, Riverside, and Allis-Chalmers engines are built in the largest sizes. Two-cycle gas engines include the Oechelhaueser and Koertmg. Special engines are "built for motor bicycles, automobiles, and launches, and for burn- ing alcohol. The basis of efficiency is the heat unit consumption per horse power per minute The mechanical efficiency may be computed from either gross or net indicated work. Recorded efficiencies of gas engines range up to 42. 7 per cent; plant efficiencies to 0.7 Ib. coal per brake hp.-hr. The mechanical efficiency increases with the size of the engine, and is greater with the four-stroke cycle. About 38 per cent of the heat supplied is carried oS by the jacket water, and about S3 per cent by the exhaust (jases^ in ordinary practice. The entropy diagram may be constructed by transfer from the PFor TV diagrams. Governing is effected (a) by the hit-or-miss method; economical, but unsatisfactory for speed regulation, V) by throttling, 1 both witehil. (c) by changing mixture proportions, J In all cases, the governing effort is exerted too early in the cycle. Gas engines must have heavy frames and fly wheels; exhaust valves (and inlet valves at high speed) must be mechanically operated by carefully designed cams; pro- vision must be made for starting; cylinders and other exposed parts are jacketed. About 1 Ib. of jacket water is required per Ibp. -minute. Gas engine advantages: high thermal efficiency; elimination of coal smoke nuisance ; stand-by losses are low ; gas may be stored ; economical in small units ; desirable for utilizing blast furnace gas. Disadvantages : mechanically still evolving ; of unproven reliability ; less general field of application ,- generally higher first cost ; poor regulation ; not self-starting ; cylinder must be cooled ; low ratio of expansion ; non-reversible ; no overload capacity ; no available by-product heat for process work in manufacturing plants, PROBLEMS 1. Compute the volume of air ideally necessary for the complete combustion of 1 cu. ft. of gasoline vapor, C fa Hii. 2* Find the maximum theoretical efficiency, using pure air only, of a power gas producer fed with a fuel consisting of 70 per cent of fixed carbon and 30 per cent of volatile hydrocarbons. 3. In Problem 2, what is the theoretical efficiency if 20 per cent of the oxygen necessary for gasifying the fixed carbon is furnished by steam ? 4. In Problem 3, if the hydrocarbons (assumed to pass off unchanged) are half pure hydrogen and half marsh gas, compute the producer gas composition by volume, PROBLEMS 227 using specific volumes as follows, nitrogen, 12.75; hydrogen, 17R.83; carbon mon- oxide, 12.75; marsh gas, 22.3. 5. A producer gasifying pure carbon is supphed with the theoretically necessary amount of oxygen from the atmospheie and from the gas engine exhaust. The latter consists of 28.4 per cent of CO., and 71.6 per cent of X, by -weight, and is admitted to the extent of 1 Ib. per pound of pure carbon gasified. Find the rise in temperature, the composition of the produced gas, and the efficiency of the process. The heat of decomposition of CO., to CO may be taken at 10,050 B. t. u. per pound of carbon. 6. rind the figures of merit in Piobleins 4 and 5. (Take the heating value of H at 53,400, of CH 4 , at 22,500.) 7. In Fig. 134, let ^ = 4, P d = 30 (Ibs. per sq. m ), P a = P ff =P d +W, T 6 = 3000, * e T d = 1000 (absolute). Find the efficiency and area of each of the ten cycles, for 1 Ib. of air, without using efficiency formulas. 8 In Problem 7, show graphically by the XT diagram that the Carnot cycle is the most efficient. 9 What is the maximum theoretical efficiency of an Otto four-cycle engine in which the fuel used is producer gas ? (See Art. 312.) 10. What maximum temperature should theoretically be attained in an Otto en- gine using gasoline, with a temperature after compression of 780 F. ? (The heat liber- ated by the gasoline, available for inci easing the temperature, may be taken at 19,000 B. t. u per pound.) 11. Find the mean effective pressure and the work done in an Otto cycle between volume limits of 0.5 and 2.0 cu. ft. and pressure limits of 14.7 and 200 Ib. per square inch absolute. 12. An Otto engine is supplied with pure CO, with pure air in just the theoretical amount for perfect combustion. Assume that the dissociation effect is indicated by the formula * (1.00 a) (COOO 7") = 300, in which a is the proportion of gas that will combine at the temperature T F. If the temperature after compression is 800 F., what is the maximum temperature attained during combustion, and what proportion of the gas will burn during expansion and exhaust, if the combustion line is one of con- stant volume ? The value of I for CO is 0.1758. 13. An Otto engine has a stroke of 24 in., a connecting rod 00 in. long, and a pis- ton speed of 400 ft. per minute. The clearance is 20 per cent of the piston displace- ment, and the volume of the gas, on account of the speed of the piston as compared with that of the flame, is doubled during ignition. Plot its path on the PV diagram and plot the modified path when the piston speed is increased to 800 ft, per minute, assuming the flame to travel at uniform speed and the pressure to increase directly as the spread of the flame. The pressure range during ignition is from 100 to 200 Ib. 14. The engine in Problem 11 is four-cycle, two-cylinder, double-acting, and makes 100 r. p. m. with a diagram factor of 0.40. Find its capacity. 15. Starting at P d = 14.7, F</ = 43.45, T<j = 32 F. (Fig. 122), plot (a) the ideal Otto cycle for 1 Ib. of CO with the necessary air, and (b) the probable actual cycle * This is assumed merely for illustrative purposes. It has no foundation and is irrational at limiting values. 228 APPLIED THERMODYNAMICS modified as described in Arts 309-328, and find the diagram factor. Clearance is 25 per cent of the piston displacement in both cases. 16. Find the cylinder dimensions in Art. 332 if the gas composition be as given in Art. 285. (Take the average heating value of C II 4 and C^ at 22,500 B t. u. per pound, and assume that the gas contains the same amount of each of these constituents ) 17. Find the clearance, cylinder dimensions, and probable efficiency in Art. 332 if the engine is two-cycle. 18. Find the size of cylinders of a four-cylinder, four-cycle, single-acting gasoline engine to develop 30 blip, at 1200 r. p. in , the cylinder diameter being equal to the stroke. Estimate its thermal efficiency, the theoretically necessary quantity of air being supplied. 19. An automobile consumes 1 gal. of gasoline per 9 miles run at 50 miles per hour, the horse power developed being 23. Find the heat unit consumption per Ihp. per minute and the thermal efficiency , assuming gasoline to weigh 7 Ib. per gallon SO. A two-cycle enyine gives an indicator diagram in which the positive work area is 1000 ft.-lb., the negative work area 00 ft -Ib. The work at the brake is 700 ft,-lb. Give two values for the mechanical efficiency 21. The engine in Probtem 17 dischaiges 30 per cent of the heat it receives to the jacket. Find the water consumption in pounds per minute, if its initial temperature is 72" F. 22. In Art 344, what was the producer efficiency in the case of the Guldner en- \ 0.20 0.40 0.00 u.SO 100 FIG. 158. Prob. 23. Indicator Diagram for Transfer. PROBLEMS 229 gine, assuming its mechanical efficiency to have been 0.85? If the coal contained 13,800 B. t. u. per pound, what was the cual consumption per brake hp -lir. ? 23 Given tbe indicator diagram of Fi. 158, plot accurately the TV diagram, the engine using 0.0402 Ib. of substance per cycle. Draw the compressive path on the NT diagram by both of the methods of Art. 347. 24. The engine in Problem 17 governs by throttling its charge. To what percent- age of the piston displacement should the clearance be decreased in ordei that the pres- sure after compression may be unchanued when the pre-compression pressure drops to 10 Ib. absolute ? What would be the object (if huch a change in clearance ? 25. In the Diesel engine, Problem 7, by what, percentages will the efficiency and capacity be affected, theoretically, if the supply of fuel, is cut off 30 per cent earlier in T r T* the stroke ? (i.e , cut-off occurs when the volume is u ~~ * + F, Fig. 134.) 2 26. Under the conditions of Art. 835, develop a relation between piston displace- ment in cubic inches per minute, and Ihp., lor four cylinder four-cycle single acting gasolene engines Also find the relation between cylinder volume and Ihp. if endues run at 1500 r. p. m., and the relation between cylinder diameter and Ihp. if bore = stroke, at 1500 r. p. in. 27. In an Otto engine, the range of pressures during compression is from 13 to 130 Ib,, the compression curve pa 1 -* = /. Find the percentage of clearance. 28 The clearance space of a 7 by 12 in. Otto engine is iound to hold Ib. of water at 70 F. Find the ideal efficiency of the engine. (See Art. 295.) 29. An engine uses 220 cu. ft. of gas, containing 800 B. t. u. per cubic foot, in 39 minutes, while developing 12.8 hp. Find its thermal efficiency. 30. In the formula, brake hp. = - (Art. 335), if the mechanical efficiency is 80, what mean effective pressure is assumed in the cylinder ? 31. A six-cylinder four-cycle engine, single-acting, with cylinders 10 by 24 in., develops 500 hp. at 200 r. p. m. What is the mean effective pressure ? 32. An engine uses 1.62 Ib. of gasolene (210! K> B. t. u. per pound) per Blip -hr. What is its efficiency from fuel to shaft ? If it is a 2-cycle engine with a pressure of 00 Ib. gage at the end of compression, estimate the ideal efficiency. 33. Derive an expression for the mean effective pressure in Ait. 293. CHAPTER XII THEORY OF VAPORS 354. Boiling of Water. If we apply heat to a vessel of water open to the atmosphere, an increase of temperature and a slight increase of volume may be observed. The increase of temperature is a gain of internal energy; the slight increase of volume against the constant resisting pressure of the atmosphere represents the performance of external work, the amount of which may be readily computed. After this operation has continued for some time, a temperature of 212 F. is attained, and steam begins to form. The water now gradually disappears. The steam occupies a much larger space than the water from which it was formed ; a considerable amount of external work is done in thus augmenting the volume against atmospheric pressure ; and the common temperature of the steam and the water remains con- stant at 212 F. during evaporation. 355. Evaporation under Pressure. The same operation may be performed in a closed vessel, in which a pressure either greater or less than that of the atmosphere may be maintained. The water will now boil at some other temperature than 212 F. ; at a lower temperature, if the pressure is less than atmospheric^ and at a higher temperature^ if greater. The latter is the condition in an ordinary steam boiler. If the water be heated until it is all boiled into steam, it will then be possible to indefinitely increase the temperature of the steam, a result not possible as long as any liquid is present. The temperature at which boiling occurs may range from 32 F. for a pressure of 0.089 Ib. per square inch, absolute, to 428 F. for a pressure of 336 Ib. ; but for each pressure there is a fixed temperature of ebullition.* * A striking illustration is in the case of air, which has a boiling point of 314 B 1 . at atmospheric pressure. As we see "liquid air," it is always boiling. If we attempted to confine it, the pressure which it would exert would "be that corresponding with the room temperature, several thousand pounds per square inch. Hydrogen has an atmospheric boiling point of 423 2T. 230 SATURATED AND SUPERHEATED VAPOR 231 356. Saturated Vapor. Any vapor in contact with its liquid and in thermal equilibrium (i.e. 7 not constrained to receive or reject heat) is called a saturated vapor. It is at the minimum temperature (that of the liquid) which is possible at the existing pressure. Its density is consequently the maximum possible at that pressure. Should it be deprived of heat, it cannot fall in temperature until after it has been first completely liquefied. If its pressure is fixed, its temperature and density are also fixed. Saturated vapor is then briefly definable as vapor at the minimum temperature or maximum density possible under the imposed pressure. 357. Superheated Vapor. A saturated vapor subjected to ad- ditional heat at constant pressure, if in the presence of its liquid, cannot rise in temperature ; the only result is that more of the liquid is evaporated. When all of the liquid has been evaporated, or if the vapor is conducted to a separate vessel where it may be heated while not in contact with the liquid, its temperature may be made to rise, and it becomes a superheated vapor. It may be now regarded as an imperfect gas; as its temperature increases, it constantly becomes more nearly perfect. Its temperature is always greater, and its density less, than those properties of saturated vapor at the same pressure ; either temperature or density may, however, be varied at will, excluding this limit, the pressure remaining constant. At constant pressure, the temperature of steam separated from water increases as heat is supplied. The characteristic equation, PV = R T, of a perfect gas is inapplicable to steam. (See Art. 390.) The relation of pressuie, volume, and temperature is given by various empirical formulas, including those of Joule (1), Rankine (^), Him (3), Racknel (4), Clausius (5), Zeuner (6), and Knoblauch Linde and Jakob (7). These are in some cases applicable to either saturated or superheated steam. SATURATED STEAM 358. Thermodynamics of Vapors. The remainder of this text is chiefly concerned with the phenomena of vapors and their application in vapor engines and refrigerating machines. The behavior of vapors during heat changes is more complex than that of perfect gases. The temperature of boiling is different for different vapors, even at the same pressure ; but the following laws hold for all other vapors as well as for that of water ; 232 APPLIED THERMODYNAMICS (1) The temperatures of the liquid and of the vapor in contact with it are the same ; (2) The temperature of a specific saturated vapor at a specified pres- - sure is always the same ; (3) The temperature and the density of a vapor remain constant during its formation from liquid at constant pressure ; (4) Increase of pressure increases the temperature and the density of the vapor ; * (5) Decrease of pressure lowers the temperature and the density ; (6) The temperature can beincreased and the density can be decreased at will, at constant pressure, when the vapor is not in contact with its liquid ; (7) If the pressure upon a saturated vapor be increased without allow- ing its temperature to rise, the vapor must condense ; it cannot exist at the increased pressure as vapor (Art. 356). If the pressure is lowered while the temperature remains constant, the vapor becomes superheated. 359. Effects of Heat in the Formation of Steam. Starting with a pound of water at 32 F., as a convenient reference point, the heat expended during the formation of saturated steam at any temperature and pressure is utilized in the following ways : (1) h units in the elevation of the temperature of the water. If the specific heat of water be unity, and t be the boiling point, h = t 32 ; actually, h always slightly exceeds this, but the excess is ordinarily small. -f-J * Since mercury boils, at atmospheric pressure, at 675 F., common thermometers cannot be used for measuring temperatures higher than this ; but by filling the space in the thermometric tube above the mercury with gas at high pressure, the boiling point of _the mercury may be so elevated as to permit of its use for measuring flue gas temperatures exceeding 800 F. t According to Barnes 1 experiments (8), the specific heat of water decreases from 1.0094 at 32 F. to 91)735 at 100 3 P., and then steadily increases to 1.0476 at 428 F. J In precise physical experimentation, it is necessary to distinguish between the value of h measured above 32 F. and (ttinrispheric pressure, and that measured above 32 F. and the corresponding pressure nf the saturated vapor. This distinction is of no consequence in ordinary engineering work. FORMATION OF STEAM 233 (2) P^ ' v ) units in the expansion of the water (external work), p ( To being the pressure per square foot and v and T^the initial and final specific volumes of the water respectively. This quantity is included in item , as above defined ; it is so small as to be usually negligible, and the total heat required to bring the water up to the boiling point is regarded as an internal energy change. (3) e = ^^ } - units to perform the external work of increasing 7 ( 8 the volume at the boiling point from that of the water to that of the steam, HHbeing the specific volume of the steam. (4) r units to perform the disgregation work of this change of state (Art. 15) ; items (3) and (4) being often classed together as L. The total heat expended per pound is then The values of these quantities vary widely with different vapors, even when at the same temperature and pressure; in general, as the pressure increases, h increases and L decreases. Watt was led to believe (erroneously) that the sum of h and L for steam was a constant; a result once described as expressing ^ Watt's Law." This sum is now known to slowly increase with increase of pressure. 360. Properties of Saturated Steam. It has been found experimentally that as p, the pressure, increases, t } 7i, e, and H increase, while r and L decrease. These various quantities are tabulated in what is known as a steam table.* * Begnaalt's experiments were the foundation of the steam tables of Rankine (9), Zeuner (10), and Porter (11). The last named have been regarded as extremely accu- rate, and were adopted as standard for use in reporting trials of steam boilers and pumping engines by the American Society of Mechanical Engineers. They do not give all of the thermal properties, however, and have therefore been unsatiKtactory lor some purposes. The tables of Dwelshauevers-Dery (12) were based on Zeuner's ; Duel's tables, originally published in Weisbach's Jtfeefaz/ucs (13), on Rankine's, Peabody's tables are computed directly from Regnaulfs work ( 14). The principal differences in these tables were due to Rome uncertainty as to the specific volume of steam (15). The precise work of Holborn and Henning (16) on the pressure-tempera- ture relation and the adaptation by Davis (17) of recent experiments on the specific heat of superheated steam to the determination of the total heat of saturated steam (Art. 388) have suggested the possibility of steam tables of greater accuracy. The most recent and satisfactory of these is that of Marks and Davis (1R), values from which are adopted in the remainder of the present text. (See pp. 247, 248.) 234 APPLIED THERMODYNAMICS Our original knowledge of these values was derived from the com- prehensive experiments of Regnault, whose empirical formula for the total heat of saturated steam was ff = 1081.94 + 0.305*. The recent investigations of Davis (17) show, however, that a more accurate ex- pression is ff = 1150.3 + 0.3745(*-212)-0.00055(-212)2 (Art. 388). (The total heat at 212 F, is represented by the value 1150 3.) Barnes' and other determinations of the specific heat of water permit of the com- putation of h; and L =H h. The value of e may be directly calculated if the volume W is known, and r=Le. The value of r has a straight line relation, approximately, with the temperature. This may be expressed by the formula r = 1061.3- 0.79 t F. The method of deriv- ing the steam volume, always tabulated with these other thermal properties, will be considered later. When saturated steam is con- densed, all of the heat quantities mentioned are emitted in the reverse order, so to speak. Regnault's experiments were in fact made, not by measuring the heat absorbed during evaporation, but that emitted during condensation. Items h and r are both internal energy effects; they are sometimes grouped together and indicated by the symbol E] whence H=E + e. The change of a liquid to its vapor furnishes the best possible example of what is meant by disgregation work. If there is any difficulty in conceiving what such work is, one has but to com- pare the numerical values of L and r for a given pressure. What becomes of the difference between L and e? The quantity L is often called the latent heat, or, more correctly, the latent heat of evapora- tion. The " heat in the water " referred to in the steam tables is h\ the " heat in the steam " is #", also called the total heat. 361. Factor of Evaporation. In order to compare the total expen- ditures of heat for producing saturated steam under unlike condi- tions, we must know the temperature T, other than 32 F. (Art. 359), at which the water is received, and the pressure p at which steam is formed ; for as T increases, h decreases ; and as p increases, S increases. This is of much importance in comparing the results of steam boiler trials. At 14.7 Ib. (atmospheric) pressure, for ex- ample, with water initially at the boiling point, 212 F., A = and H~ L*= 970. 4 (from the table, p. 247). These are the conditions adopted as standard, and with which actual evaporative performances PRESSURE-TEMPERATURE 235 are compared. Evaporation under these conditions is described as being From (a feed water temperature of) and at (a pressure correspond- ing to the temperature of) 212 F. Thus, for p = 200, we find L = 843.2 and h = 354.9 ; and if the tem- perature of the water is initially 190 F., corresponding to the heat contents of 157,9 B. t. u., H= L + (354.9 - 157.9) = 843.2 + 197 = 1040.2. The ratio of the total heat actually utilized for evaporation to that necessary " from and at 212 F/' is called the factor of evaporation. In this instance, it has the value 1040.2 -r- 970.4 = 1.07. Generally, if L> h refer to the assigned pressure, and A is the heat correspond- ing to the assigned temperature of the feed water, then the factor of evaporation is F = \L+ (h -A )]-*- 970.4. 362. Pressure-temperature Relation. Regnault gave, as the result of his ex- haustive experiments, thirteen temperatui es corresponding to known pressures at saturation. These range from - 32 C. to 220 C. He expressed the relation by four formulas (Art. 19); and no less than fifty formulas have since been. devised, representing more or less accurately the same experiments. The deter- minations made by Holborn and Kenning (16) agree closely with those of Reg- nault; as do those by Wiebe (19) and Thiesen and Scheel (20) at temperatures below the atmospheric boiling point. The steam table shows that, beginning at 32 F. ; the pressure rises with the temperature, at first slowly and afterward much more rapidly. The fact that slight increases of temperature accompany large increases of pressure in the working part of the range seems fatal to the development of the engine using saturated steam, the high temperature of heat absorption shown by Caraot to be essential to efficiency being unattainable without the use of pressures mechanically objection- able. A recent formula for the relation between pressure and temperature is (Power> March 8, 1910) 6 - in which t is the Fahrenheit temperature and p the pressure in pounds per square inch. This has an accuracy within 1 or so for usual ranges. Marks gives (Jour. A.S.M. E., XXXIII, 5) the equation, log p- 10.515354 -- -0.00405096 T+ 0.00000 1392964 T 2 , T being absolute and p in pounds per square inch. This has an established accuracy within i of 1 per cent for the whole range of possible temperatures. 236 APPLIED THERMODYNAMICS 363. Pressure and Volume. Fairbairn and Tate ascertained experimentally in 1860 the relation between pressure and volume at a few points; some experi- ments were made by Hira; and BatteUi has reported results which have been examined by Tumhrz (21) who gives BT where p is in pounds per square inch, c = 0.256, 5 = 0.5962 and T is in degrees absolute. More recent experiments by Knoblauch, Linde, and Klebe (1905) (22) give the formula j-0 5962 r-p(l+0.0014 p) ( 150 ' 3 ff' 00 -0.0833] , in which p is in pounds per square inch, *> in cubic feet per pound, and T in degrees absolute. Goodenough's modified form of this equation is more convenient: in which =0.5963, log w = 13.67938, n=5, c=0.088, a=00006. A simple empirical formula is that of Rankine, pptt = constant, or that of Zeuner, ppri.owe constant. These forms of expression must not be confused with the PV n = c equation for various polytropic paths. An indirect method of determin- ing the volume of saturated steam is to observe the value of some thermal piop- erty, like the latent heat, per pound and per cubic foot, at the same pressure. The incompleteness of experimental determinations, with, the diffi- culty in all cases of ensuring experimental accuracy, have led to the use of analytical metliods (Art. 368) for computing the specific volume. The values obtained agree closely with those of Knoblauch, Linde, and Klebe. 364. Wet Steam. Even when saturated steam is separated from the mass of water from which it has been produced, it nearly always contains traces of water in suspension. The presence of this water produces what is described as wet steam, the wetness being an indi- cation of incomplete evaporation. Superheated steam, of course, cannot be wet. Wet steam is still saturated steam (Art. 356) ; the temperature and density of the steam are not affected by the pres- ence of water. The suspended water must be at the same temperature as the steam; it therefore contains, per pound, adopting the symbols of Art. 359, h units of heat. In the total mixture of steam and water, then, the proportion of steam being x, we write for L, xL ; for r, xr ; for e, xe i for j, xr + h ; while, h remaining unchanged, J3T= Ji + xL. FORMATION OF STEAM 237 o FNJ K>l) Arts. :*M, 3f>6, 371). Paths of Steam Formation. The factor of evaporation (Art. 361), wetness considered, must be correspondingly reduced ; it is F= [sL + (Ji - 7/ )] -H 070.4. Tlie specific volume of wet steam is TF, r = V+x( TF F)=^^+ T", where #= TF T. For dry steam, .r= 1, and TF; r = V+ ( W V) = TF- The error involved in assuming W n = ./ TFis usually inconsiderable, since the value of T r is comparatively small. 365. Limits of Existence of Saturated Steam. In Fig. 160, let ordinates represent temperatures, and abscissas, volumes. Then db is a line representing possible condi- tions of water as to these two proper- ties, which may be readily plotted if the specific volumes at various tem- peratures are known; aud cd is a similar line for steam, plotted from the values of TFand t in the steam table. The lines db and cd show a tendency to meet (Art. 370). The curve cd is called the curve of saturation, or of con- stant steam weight; it represents all possible conditions of constant weight of steam, remaining 1 saturated. It is not a path, although the line db is (Art. 3G3). States along db are those of liquid; the area lade includes all wet saturated states ; along rfc, the steam is dry and saturated; to the right of dc^ areas include superheated states. 366. Path during Evaporation. Starting at 32, the path of the substance during heating and evaporation at constant pressure would be any of a series of lines aef, old, etc. The curve db is sometimes called the locus of boiling points. If superheating at constant pres- sure occur after evaporation, then (assuming Charles' law to hold) the paths will continue as fg* ij, straight lines converging at 0. For a saturated vapor, wet or dry, the isothermal can only be a straight line of constant pressure, 367. Entropy Diagram. Figure 161 reproduces Fig. 160 on the entropy plane. . The line ab represents the heating of the water at constant pressure. Since the specific heat is slightly variable, the 238 APPLIED THERMODYNAMICS increase of entropy must be computed for small differences of tem- perature. The more complete steam tables give the entropy at various boiling points, measured above 32. Let evaporation occur when the g M FIG. 161. Arts. 367, 3. M73, 370, 370, 386, 426 The Steam Dome. temperature is T b . The increase of entropy from the point b (since the temperature is constant during- the formation of steam at constant pressure) is simply L -s- (2^ + 459. G), which is laid off as be. Other points being similarly obtained, the saturation curve cd is drawn. The paths from liquid at 82 to dry saturated steam are ale, a VN, aUS, etc. The factor of evaporation may be readily illustrated. Let the area eUSf represent L^ the heat necessary to evaporate one pound from and at 212 P. The area gjbcJi represents the heat necessary to evaporate one pound at a pressure b from a feed-water temperature j. The factor of evaporation is gjbch-** eUSf. For wet steam at the pressure b, it is, for example, gjbik -5- eUSf. 368. Specific Volumes* Analytical Method. This was developed by Clapeyron in 1834, In Fig. 102, let abed represent a Carnot cycle in which steam is the working substance and the range of temperatures is dT. Let the substance be liquid along da and dry saturated vapor along be. VOLUME OF VAPOR 239 The heat area alfe is L\ the work area abed is (L -+ T)dT. In Fig. 163, let abed represent the corresponding work area on the pv diagram. Since the range of temperatures is only dT, the range of pressures may be FIGS. 162 and 1(>3 Arts. otiS, 400, ^Ou. nj \ oiuuieb taken as c/P; whence the area abed in Fig. 1C3 is dP( W F), where W is the volume along be, and Fthat along ad. This area must by the first law of thermodynamics equal (778 L -=- T)dT\ whence 78 L d'. Thus, if we know the specific volume of the liquid, and the latent heat of vaporization, at a given temperature, we have only to determine the dT differential coefficient in order to compute the specific volume of the vapor. The value of this coefficient may be approximately estimated from the steam table; or may be accurately ascertained when any correct formula for relation between P and T is given. The advantage of this indirect method for ascertaining specific volumes arises from the accuracy of experimental determinations of T, L, and P. 369. Entropy Lines. In Fig. 161, let ab be the water line, cd the saturation curve ; then since the horizontal distance between these lines at any absolute temperature T is equal to i-s-2 7 , we deduce that, for steam only partially dry, the gain of heat in passing from the water line toward cd being xL instead of i, the gain of entropy is xL -*- T instead of L -+ T. If on be and ad we lay off bi and al = x be and x ad, respectively, we have two points on the constant dryness curve -e7, along which the proportion of dryness is x. Additional points will fully determine the curve. The additional curves zn, pq, etc., are similarly plotted for various values of 2:, all of the horizontal intercepts between ab and cd being divided in the same proportions by any one of these curves. 210 APPLIED THERMODYNAMICS 370. Constant Heat Curves. Let the total heat at o be H. To find the state at the temperature be, at which the total heat may also equal IT, we remember that for wet steam H= li -I- xL, whence x = (-ff h) -*- L = bj> -f- be. Additional points thus determined for this and other assigned values of H give the constant total heat curves op, mr, etc. The total heat of saturated vapor is not, however, a cardinal property (Art. 10). The state points on this diagram determine the heat contents only on the assumption that heat has been absorbed at constant pressure ; along such paths as abc, aUS, aVN, etc. 371. Negative Specific Heat. If steam passes from o to r, Tig. 161, heat is absorbed (area sort) while the temperature decreases. Since the satu- ration curve slopes constantly downward toward the light, the specific heat of steam kept saturated is therefore negative. The specific heat of a vapor can be positive only when the saturation curve slopes downward to the left, like CM, as in the case, for example, of the vapor of ether (Fig. 315). The conclusion that the specific heat of saturated steam is negative was reached independently by Kankine aud Clausins in 1850. It was experi- mentally verified by Him in 1862 aud by Cazin in 1866 (24). The physical significance is simply that when the temperature of dry saturated steam is increased adiabatically, it becomes superheated ; heat must be abstracted to keep it saturated. On^the other hand, when dry saturated steam expands, the temperature falling; it tends to condense, and lieat must be supplied to keep it dry. If steam at c, Fig. 161, having been formed at constant pressure, works along the saturation curve to 2?, its heat contents are not the same as if it had been formed along aVN, but are greater, beiug greater also than the " heat contents " at c. 372. Liquefaction during Expansion. If saturated steam expand adia- batically from c, Fig. 161, it will at v have become 10 per cent wet. If its temperature increase adiabatically from y, it will at c have become dry. If the adiabatic path then continue, the steam will become superheated. Generally speaking, liquefaction accompanies expansion and drying or superheating occurs during compression. If the steam is very wet to begin with, say at the state #, compression may, however, cause liquefaction, and expansion may lead to drying. Water expanding adiabatieally (path bz) becomes partially vaporized. Vapors may be divided into two classes, depending upon whether they liquefy or dry during adiabatic expansion under ordinary conditions of initial dryness. At usual stages of dryness and temperature, steam liquefies during expansion, while ether becomes dryer, or superheated. INTERNAL ENERGY OF VAPOR 241 373. Inversion. Figure 161 shows that when x is about 0.5 the constant dry- ness lines change their direction of curvature, so that it is possible for a single adiabatic like DE to twice cut the same dryness curve ; x may therefore have the same value at the beginning and end of expansion, as at D and E. Further, it may be possible to draw an adiabatic which is tangent to the dryness curve at A. Adiabatic expansion below A tends to liquefy the steam ; above A, it tends to dry it. During expansion along the dryness curve below A, the specific heat is nega- tice; above .4, it is positive. By finding other points like A, as F 9 (7, on similar constant dryness curves, a hue BA may be drawn, which is called the zero line or line of inversion. During expansion along the dryness lines, the specific heat becomes zero at their intersection with AB, where they become tangent to the adiabatics. If the line AB be projected so as to meet the extended saturation curve dc, the point of intersection is the tempeiature of imersion. There is no temperature of inversion for dry steam (Art. 379), the saturation curve reaching an upper limit before attaining a vertical direction. 374. Internal Energy. In Fig. 164, let 2 he the state point of a wet vapor. Lay off 2 4 vertically, equal to (T-L)(L- r). Then 1 2 4 3 (3 4 being drawn hoiizontally and 1 3 vertically) is equal to This quantity is equal to the external work of vaporization = xe, which is accordingly repre- sented by the area 1243. The irregular area 651347 then represents the addition of internal energy, 6518 having been ex- pended in heating the water, and 8 3 4 7=xr being the disgregation work of vaporization. FIG. 161. Art 374. Internal Energy and External Work. 375. External Work. Let Jl/jV, Fig. 165, be any path in the saturated region. The heat absorbed is mMNn. Construct J/cfa, Nfed, as in Art. 374. The inter- nal energy has increased from Oabcm to Odefn, the amount of increase being adefnmcb. This is greater than the amount of heat absorbed, by dei^fcba iNf, which difference consequently measures the external work done upon the substance. Along some such curve as XY 9 it will be found that external work has been done by the substance. o FIG. 165. Art. 375. In- ternal Energy of Steam. 376. The Entropy Diagram as a Steam Table. In Fig. 161, let the state point be H. We have T= HI, from -which P may be found. HJ is made equal to (T L)(L r), whence Oa VKJI E and VH.TK xe. Also x = VH -s- FTV", the entropy measured from the water line is VH 9 the momentary specific heat of the water along the dif- ferential path jL is g}LH-^Tj\ xL = PVHI, xr - KJIP, A = OaVP, and H = Oa VHI. The specific volume is still to be considered. 242 APPLIED THERMODYNAMICS FIG 166. Art. 377 Constant Tolume Lines. 377. Constant Volume Lines. In Fig. 166, let JA be the water line, JBGf the saturation curve, and let vertical distances below ON represent specific volumes. Let xs equal the volume of boiling water, sensibly constant, and of comparatively small numerical value, giving the line ss. From any point B on the saturation curve, draw BD vertically, making QD represent by its length the specific volume at B. Draw BA horizontally, and AH vertically, and connect the points J^andD. Then ED shows the relation of volume of vapor and entropy of vapor, along AB, the t\vo increasing in arithmetical ratio. Find the similar lines of relation KL and HJFioT the temperature lines JTand YGr. Draw the constant volume line TD, and find the points on the entropy plane w, v, JB, corresponding to t, u, D. The line of constant volume wB may then be drawn, with similar lines for other specific volumes, qz, etc. The plotting of such lines on the entropy plane permits of the use of this diagram for obtaining specific volumes (see Fig. 175). 378. Transfer of Vapor States. In Fig. 167, we have a single represen- tation of the four coordinate planes pt, tn, m\ and pv. Let ss be the line of water volumes, db and ef the satura- tion curve, Od the pressure-tempera- ture curve (Art. 362), and Op the water line. To transfer points a, 5 on the saturation curve from the pv to the tn plane, we have only to draw a (7, Cfe, bd, and df. To transfer points like i, Z, representing wet states, we first find the vn lines qh and rg as in Art. 377, and then project (7, jk> Im, and mn (25). FIG. 167 Art. 378 Transfer of Vapor States. CRITICAL TEMPERATURE 243 Consider any point t on the pv plane. By drawing tu and uv we find the vertical location of this point in the tn plane. Draw w A and #2?, making zB equal to the specific volume of vapor at x (equal to EF on the pv plane). Draw AS and project t to c. Projecting this last point upward, we have D as the required point on the entropy plane. 379. Critical Temperature. The water curve and the curve of saturation in Figs. 160 and 161 show a tendency to meet at their upper extremities. Assuming that they meet, what are the physical conditions at the critical temperature existing at the point of intersection ? It is evident that here L = 0, T = 0, and e = 0. The substance would pass immediately from the liquid to the superheated condition ; there would be no intermediate state of saturation. "No external work would be done during evaporation, and, conversely, no expenditure of external work could cause liquefaction. A vapor cannot be liquefied, when above its critical temperature, by any pressure whatsoever. The density of the liquid is here the same as that of the vapor : the two states cannot be distinguished. The pressure re- quired to liquefy a vapor increases as the critical temperature is approached (moving upward) (Arts. 358, 360) ; that necessary at the critical temperature is called the critical pressure. It is the vapor pressure corresponding to the temperature at that point. The volume at the intersection of the saturation curve and the liquid line is called the critical volume. The " specific heat of the liquid 5 ' at the critical temperature is infinity. The critical temperature of carbon dioxide is 88.5 F. This substance is sometimes used as the working fluid in refrigerating machines, particularly on shipboard. It cannot be used in the tropics, however, since the available supplies of cooling water have there a temperature of more than 88-5 F., making it im- possible to liquefy the vapor. The carbon dioxide contained in the microscopic cells of certain minerals, particularly the topaz, has been found to be in the critical condition, a line of demarcation being evident, when cooling was produced, and disappearing with violent frothing when the temperature again rose. Here the substance is under critical pressure; it necessarily condenses with lowering of temperature, but cannot remain condensed at temperatures above 88.5 F. Ave- narius has conducted experiments on a large scale with ether, carbon disulphide, chloride of carbon, and acetone, noting a peculiar coloration at the critical point (26). For steam, Regnault's formula for H (Art 360), if we accept the approximation h = / - 3*2, would give L = H - h = 1118.94 - 0.695 1, which becomes zero when t = 1603 F. Davis* formula (Art 360) (likewise not intended to apply to temper- atures above about 400 F.) makes L - when t - 1709 F. The critical tempera- ture for steam has been experimentally ascertained to be actually much lower, the best value being about 689 F. (27). Many of the important vapors have been studied in **"' direction by Andrews. 214 APPLIED THERMODYNAMICS 380. Physical States. We may now distinguish between the gaseous conditions, including the states of saturated vapor, superheated van) or, and true gas. A saturated vapor, which may be either dry or icet, is a gaseous substance at its maximum, density for the given temperature or pressure ; and below the critical temperature. A superheated vapor is a gaseous sub- stance at other than maximum density whose temperature is either less than, or does not greatly exceed, the enticed temperature At higher tempera- tures, the substance becomes a true gas. All imperfect gases may be regarded as superheated vapors. Air, one of the most nearly perfect gases, shows some deviations from Boyle's law at pressures not exceeding 2500 Ib. per square inch. Other substances show far more marked deviations. In Fig. 168, QP is an equilateral hyperbola. The isothermals for air at vaiious temperatures centi- grade are shown above. The lower curves are isothermals for carbon di- oxide, as determined by Andrews (28). They depart widely from the perfect gas isothermal, PQ. The dotted lines show the liquid curve and the satura- tion curve, running together at , at the critical temperature. There is an evi- dent increase in the irregularity of the curves as they approach the ei itical tem- perature (from above) and pass below it. The cuive for 21.5 C. is paiticu- larly interesting. From I to c it is a liquid curve, the volume remaining practically constant at constant temperature in spite of enormous changes of "pres- sure. From b to d it is a nearly straight horizontal line, like that of any vapor between the liquid and the dry saturated states; T\hile fiom d to e it approaches the perfect gas form, the equilateral hyperbola. All of the isothermals change their direction abruptly whenever they ap- T proach either of the limit curves ctf or ag. 381. Other Paths of Steam Formation. The discussion has been limited to the formation of steam at constant pressure, the method of practice. Steam might con- ceivably be formed along any arbitrary path, as for instance in a closed vessel at constant volume, the pressure steadily in- creasing. Since the change of internal energy of a substance depends upon its initial and final states only, and not on the intervening path, a change of path affects the external work only. " For formation at constant volume, the total heat equals E> no external work being done. Jf in Fig. 169 water at c could be com- so- S 85 ' Soft 2*' I 7fi $70- W- 60 55- 50- FIG. 168. Art 380. Critical Temperature. FIG. 109. Art. 381. Evaporation at Constant Volume. VAPOR ISODYNAMIC 245 pletely evaporated along en at constant volume, the area acnd would represent the addition of internal energy and the total heat received. If the process be at con- t>tatit pressure, along cbn, the area acbnd lepresents the total heat received and the area cbn represents the external work done. 382. Vapor Isodynamic. A saturated vapor contains heat above 32 F. equal to li -f r -f e ; or, at some other state, to \ -f- r L -f e r If the two states are isody- narnic (Art. 83), h + r = 7^ -f r 1? a condition which is impossible if at both states the steam be dry. If the steam be wet at both states, h + xr = 7^ 4- a^. Let y>, p r v be given ; and let it be required to find v r the notation being as in Art. 304. "We have x 1 = - xr ~~ \ all of these quantities being known or readily ascertain- able. Then i = ^ + ^ (W, - V^x^ + l\ ^V. + Z^h + zr- h,). r i If x = 1.0, the steam being diy at one state, x l = * "^ r "" ' and Substitution of numerical values then shows that if p exceed pi, v is less than vi; i.e. the curve slopes upward to the left on the pv diagram and x is less than x r The curve is less " steep" than the satuiatiou curve. Steam cannot be worked isodynaimcally and remain dry; each isodynamic curve meets the saturation curve at a single point. 382ft. Sublimation. It has been pointed out that a vapor cannot exist at a temperature below that which "corresponds" to its pressure. It is likewise true that a substance cannot exist in the liquid form at a temperature above that which " corresponds ' ' to its pressure. When a substance is melted in air, it usually becomes a liquid; and if a further addition of heat occurs it will at some higher temperature become a vapor. If, however, the saturation pressure at the melting temperature exceeds the pressure of the atmosphere, then at atmospheric pressure the saturation temperature is less than the melting temperature, and the substance cannot become a liquid, because we should then have a liquid at a higher temperature than that which corresponds to its pressure. Sublimation (Art. 17), the direct passage from the solid to vaporous condition, occurs because the atmospheric boiling point is below the atmospheric melting point. Water at 32 has a saturation pressure of 0.08&6 Ib. per square inch. If the moisture in the air has a lower partial pressure than this, ice cannot be melted, but will sublime, because water as a liquid cannot exist at 32 at a less pressure than 0.0886. THERMODYNAMICS OF GAS AND VAPOR MIXTURES 3825. Gas Mixture. (See Art. 52 b.) When two gases, weighing ?i and w Ib. respectively, together occupy the same space at the conditions p, v, /, we may write the characteristic equations, using subscripts to represent the different gases, conforming to Dalton's law, 246 APPLIED THERMODYNAMICS WEIGHTS OF AIR, VAPOR OF WATER, AND SATURATED MIXTURES OF AIR AND VAPOR AT DIFFERENT TEMPERATURES, UNDER THE ORDINARY ATMOSPHERIC PRESSURE OF 29 921 INCHES OF MERCURY. Temperature Fahrenheit MIXTURES OP AIR SATURATED WITH VAPOR Elaatio Force of the Air in the Mixture of Air and Vapor in ins of Mercury Weight of Cubic Foot of the Mixture of Air and Vapor Weight of the Air in Pounds Weight of the Vapor in Pounds 29 877 .0863 000079 12 29 849 .0840 000130 22 29 803 .0821 000202 32 29 740 .0802 000304 42 29 654 .0784 .000440 52 29 533 .0766 000627 62 29 365 0747 .000881 72 29 136 .0727 .001221 82 28 829 .0706 .001667 92 28 420 ,0684 .002250 102 27 885 .0659 .002997 112 27 190 .0631 003946 122 26 300 .0599 005142 132 25 169 .0564 .006639 142 23.756 .0524 .008475 152 21 991 .0477 .010716 162 19 822 .0423 .013415 172 17.163 .0360 .016682 182 13 961 0288 020536 192 10 093 .0205 .025142 202 5 471 .0109 030545 212 000 0000 036820 These yield as the equation of the mixture, where -R (R\w\-{-Ry.w^^-(wi-\-w^^ For pure dry air, containing by weight 0.77 nitrogen to 0.23 oxygen, the value of R should then be (48.2X0,23) +(54.9X0 77)=53 2. 382c. Air and Steam. We are apt to think of the minimum boiling point of water (except in a vacuum) as 212 F. But water will boil at temperatures as low as 32 F. under a definite low partial pressure for each temperature. Thus at 40 F., if an adequate amount of moisture is exposed to the normal atmosphere it will THERMODYMAMICS OF GAS AND VAPOR MIXTURES 247 be vaporized until the mixture of air and steam contains the latter at a partial pressure of 0.1217 Ib. per square inch, the partial pressure of the air then being only 14.697-0.1217 = 145753 Ib. per square inch. Such air is saturated. If there is a scant supply of moisture, the partial pressure of vapor will be less than that corresponding with its temperature, and such vapor as is evaporated will be super- heated The weight of moisture in a cubic foot of saturated air is the tabular density of the vapor at its temperature. What is commonly called the absolute humidity of air may be expressed either in terms of the weight of vapor per cubic foot of mixture or of the partial vapor pressure. The weight of gas or superheated vapor in any assigned space at any stated temperature is directly proportional to the partial pressure thereof. The relative humidity of moist air may therefore be expressed either as or as , where w and W PI W are respectively the weights of water vapor in a cubic foot of moist air, unsatu- rated and saturated, and p*., P* are the corresponding partial pressures. The value of R in the characteristic equation is obtained, for moist air at a relative humidity below 1.0, by the method of the first paragraph, using for the water vapor Rz =85.8. If the air temperature is 92 F., and a wick-covered ("wet bulb") thermometer reads 82, the partial pressure of the vapor is that corresponding with saturation at 82, that is, 0.539 Ib. per square inch; for the air about the wet-bulb thermometer is saturated, evaporation from the moist wick causing the cooling. Saturated air at 92 would have a partial vapor pressure of 0.741 Ib. per square inch. The air in 539 question has therefore a relative humidity of o~74i~^'^" ^ e va ^ ue f -^ * or ^^ air is not 53.2, but a subordinate relation being (14.697-0.539)144 - 53.2X552 - - - 069 ' If the respective specific heats are fci and kz t then the specific heat of the mix- ture is which for our conditions, with fa =0.2375, fe =0.4805, gives A; =0.248. 382^. Thennodynamic Equations. When dealing with mixtures of wet vapors, or of wet vapors and air, the ordinary equations for expansion do not in general apply. This is the more unfortunate in that any general analysis of the subject must include consideration of expansion paths which will partially liquefy one or more of the constituents of even a wholly superheated mixture. The internal energies of the constituents and their entropies are dependent upon and may be computed from their thermal conditions alone, however; mixing 248 APPLIED THERMODYNAMICS does not affect the energy, and adiabatic expansion does not affect the entropy; so that it is by no means impracticable to study the phenomena accompanying (a) the operation of mixing and (&) the expansion or compression of the mixture. 3820. Wet Vapor and Gas. As a simple case, consider a mixture of wet steam and air. the condition of a super-saturated atmosphere. Let such a mixture be at the state p, v, t' } the steam state being w*, p*, x, and that of the air wi, pi. Then P=pi+Pz, and v = u> 2 X2V2= - -, where v 2 is the specific volume of the dry steam. PL The internal energy of the mixture is E where I is the specific heat of air at constant volume and A 2 and r 2 are tabular thermal properties at the pressure pa- The entropy of the mixture is l Iog fl +$ -0 where k is the specific heat of air at constant pressure, v is the volume of w\ Ib. of air under standard conditions and n^ and n$ are the entropies of steam at the pressure #. In an isothermal change of such mixture, EI remains constant and (the dryness of the steam changing to xs) E 2 increases by w&sfa #2). The air conforms to its usual characteristic equation , piViRt lf In reaching the expanded volume zfy the external work done by the air is then The steam remaining wet expands at constant pressure, and does the external work s v) } so that the whole amount of external work done is W = piv log fl +#2(03 - v) . The heat absorbed may be expressed as the sum of the external work done and the internal energy gained; or as H=piv log, ^H-p2(Ps v) +Wzr 2 (xs 3%) =piv log* ^+w 2 I z (or 3 a^), where k is the latent heat of vaporization corresponding with the pressure p. Alternatively, the heat 'absorbed is equal to the product of the temperature by the increase of entropy; or H=t I Wi(k-l) log* j as before; ^e-r being the entropy of vaporization at the pressure pa- Let it be noted also that vs=w<tx*vs=~l, so that =, Pz r denoting the partial pressure of air in the mixture after expansion. The mixing of air with saturated steam produces a total pressure which is higher than the saturation pressure of steam at the given temperature. Such a mixture THERMODYNAMICS OF GAS AND VAPOR MIXTURES 249 may therefore be regarded as the reverse of superheated vapor, in which latter the pressure is less than that corresponding with the temperature. In adiabatic expansion, let the final condition be x 3 , U, p z . The entropy remaining constant, wik loge y + wi (k I) log e , -+- Ws(n w ' -\-x 3 ne n w xne) 0, where n w is entropy of liquid and primes refer to final conditions. The paHial pressure of the vapor is tabular for Z 3 . If v$ is the specific volume of steam for . then where p 3 " and p z f are the partial pressures of air and steam, respectively. The external work is written as the loss of internal energy, or, as 382 /. High Pressure Steam and Air. The pressure attained by mixing cannot exceed the initial pressure of the more compressed constituent. Assume 1 Ib. of steam, 0.85 dry, at an absolute pressure of 200 Ib., to be mixed with 2 Ib. of air at 220 Ib. pressure and 400 F. The respective volumes are ,=0.85X2 29 = 1.945; *-& and the volume of mixture will be, under the usual condition of practice, 1.945+2.9=4.845. The internal energy before (and after) mixing is (2X0.1689X860)+354.9+(0.85+759.5)=1288 B. t. u. This we put equal to (2X0. 1689X0+^+^^; 3a= = - ; and (assuming values of t) we find by trial and error, (1285) ft <=314(+460), fo=2S4, 7- 2 pi = 118.2, p -200.5. Mixing has caused an increase in dryness of steam, a considerable reduction of tem- perature, and a final pressure between the two original pressures. The entropy of the mixture is now 2 | (o.!689X2.3 log ^ + ^0.0686X2.3 log |g +0.456+(0.908X1.1617) =1.438 250 APPLIED THERMODYNAMICS Let isothermal expansion increase the dryness to 0.95. The volume then becomes 0.95X5.33 = 5.08 =z> 8 . The external work done is =i= j (l44XHS 2X4.845X2.3 log f|f) +144X82.3(508-4.845) [ =8.45 B. t. u. 778 \ The internal energy increases by 042 X759.5 = 31 9 B. t. u , and the heat absorbed should then be 31.9+8.45=40.35 B. t. u. The entropy in the expanded condition is j (0.1 2 0.1689X2.3 log + o 0686X2.3 log +0456+(0.95X1-1617)=1.49, and the check value for heat absorbed is (460 +314) X (1.49-1. 438) - 40.3 B. t. u. The partial pressures after expansion are Air, 393' =PI-= 118.2 = us; and steam, 82.3, as before. In the usual expression for external work, pv-py pv ~py+w W ~ n-l > n ~ W the equivalent value of n is 1441 (200 5X4.845)-(195.3X5.08)} + (8.45X778) 845X778 Consider next the adidbatic expansion from the same initial condition to a temperature * 3 = 50(+460); when v 3 ' = 1702, p 3 '=0178, n*' -0.0361, rc e '=2.0865, v z = 1702%. Then 1.438 =2 j ^0.1689 X2.3 log ||) + ^0.0686 X2.3 log ^-^) j- +0.0361 +2.0865^, and a* =0.47, v 3 = 802. The internal energy in the expanded condition is 18.08 +(0.47X1007.3) +2(0.1689X510) =665 B. t. u., and the external work done is 1288 665=623 B. t u. The steam expanding alone from its original condition would have had a final dryness of 0.65, and would have afforded external work amounting to 354.9+(0.85X759.5)-18 08 -(0.65X1007.3)= 323 B. t. u. The air expanding alone to 50 , according to the law piv i y =p 1 'v ] .' v would have given .1^220X2 9-0085X80^ . THERMODYNAMICS OF GAS AND VAPOR MIXTURES 251 The total work obtainable without mixture, down to the temperature 3 =50 would then have been 262+323 = 585 B. t. u. The equivalent value of n for the expansion of the mixture is 144) (200.5 X4.845) - (0 263 X802)| + (665 X778) 665X778 Since y for steam initially 0.91 dry is 1.126, and y for air is 1.402, the value of n might perhaps have been expected to be about (2X1.402) +1.126 3 * 382^. Superheated Steam and Air. If the steam is superheated, its initial volume is (from the Tumlirz equation, Art. 363), where B=* 0.5962, c= 0.256. The internal energy of superheated steam may be written as that at saturation (hz+xzr z ) plus that of superheating, where k s is the specific heat of the superheated steam, y, = 1.298, and t s is the satu- ration temperature for the partial pressure pt. The entropy of the steam is Its behavior during expansion may be investigated by the relations previously given. 382/j. Mixture of Two Vapors. Let two wet vapors at the respective conditions w*j Pa, 2, s, ht, k, r z , and w 2 , p2, ts, 12* hj, *2r r 2, be so mixed that the volume of the aggregate is v =02+v 4 . The internal energy of the mixture is the numerical value of which may be computed for the conditions existing prior to mixing. After mixing, the temperature t being attained, the internal energy is the same as before, and the drynesses are ** where v ' is the tabular volume at the temperature t. The known internal energy may then be written as a function of tabular properties at the temperature t, and the 252 APPLIED THERMODYNAMICS value of t found by trial and error The equation for adiabatic expansion entirely in the saturated field to the state fe is w'2(Hw5+2rc e )+W2fn w +X2He) = w*(n v ,' +X2 f n & f ) +W2(n w '+x 2 'ne'), primes denoting final conditions. Thus, let 1 Ib of steam at 107 Ib. pressure, 90 dry, be mixed with 2 Ib. of carbon tetrachloride at the same pressure, 95 dry. The tables give t 2 = 320, h 2 -61 2, r 2 = 58.47, v =0.415, n w = 0.1003, n e = 0855; .=333, ^=303 4, r 2 =802 5, r =4 155, 7^ = 0,4807, n e = l 1158. Then r 2 = 090X4.155*3 75, v 2 =0 95X2X0.415 = 789, z>= 3 75+0.789=4.539. The internal energy is 303.4+(090XS02.5)+2{61.2 + (0.95X58.47)} =1258 B. t. u. Since * 2 '>1 for values of t between 320 and 333 the carbon tetrachloride is superheated after mixture occurs. We must then express the energy as ~t 2 ') j =1258, k in which k =0.056, # = 1.3, - = 0.043, and t 2 ' is the saturation temperature corre- sponding with the partial pressure of the carbon tetrachloride. Assuming that this vapor when superheated conforms with the usual characteristic equation for gases, and putting 5 = 100, P2'=jg^f|||= 0,0307 2. Assuming values of t, the trial and error method gives a resulting mixture temperature close to 319, at which p 2 '=239 I t 2 '=200, and 7-814.0+2(34 59+72.64+0.043X119) -1255(1258) B.t.u. 4.980 The entropy computed as before mixing Is 0.481+(0.9Xl-1152)+2(0.1003+0.95X0.0855) after mixing, it is 0.4627+ (HJ2-L1492) +2 (o. 1846 +0.056X2.3 log gg) =1.89. Mixing has again lowered the temperature. Let adiabatic expansion proceed until the temperature is 212. The tetrachloride will stall be superheated, and 0.3118+1.4447jr 2 r +2 20 X 672 For every assumed value of t 2 ', the whole volume of mixture is 7 r v', say: 144p2 Then 2'=% where 0/=26.79, the volume of saturated steam at 212. At THERMODYNAMICS OF GAS AXD VAPOR MIXTURES 253 t 2 ' = 106^ p 2 ' =4.37,0' =21^4, z 2 '=|^ =0.798, n w '+ n</ =0.1865; and the en- tropy is 0.3118+1.15+2(01865 +0.0094) = 1.89. The internal energy is now 180 0+(0.798X897.6)+2(14.92+81.76+O.Q43+106) =1098 B. t. u., and the external work done during expansion is 12581098 = 160 B. t. u. If the two vapors had expanded from their original condition to 212 separately, the external work done would have been, very nearly, 126 B. t. u. 382 1. Technical Application of Mixtures in Heat Engines- The preceding illustration shows that the expanded mixture, although at 212 F., has a pressure 4.37 Ib. per sq. in. greater than that of the atmosphere. A mixture at an absolute pressure of 1 Ib. (about the lowest commercially attainable) might similarly exist at a temperature considerably lower than the 102 F. which is characteristic of steam alone. A lowering of the temperature of heat-rejection is thus the feature which makes the use of a fluid mixture of practical interest. This is the more important, since from a power-producing standpoint the most fruitful part of the cyclic temperature range is the lower part. The operation of mixing itself reduces the initial temperature, but it in no way impairs the stock of internal energy of the constituents If one of the constituents is at the lower temperature of the cycle a superheated vapor, it cannet be condensed at that temperature: but since cooling water con- ditions permit of normal condensing temperature around 65, the use of a mixture, even one of air and steam, may permit the attainment of that temperature without the necessity for an impracticably high vacuum. The total heat of saturated steam increases less than J B. t. u. per degree of temperature; that of superheated steam increases from 0.5 to 0.6 B. t.u. It follows that at the same temperature superheated steam "contains" more heat than saturated steam. The internal energy of saturated steam increases about 0.2 B. t. u. per degree of temperature; that of superheated steam, about 0.4 to 0.45 B. t. u. The total internal energy at a given temperature is thus also greater with super- heated than with saturated steam. The less the internal energy at the end of the expansion, the greater is the amount of external work performed during expansion for given initial conditions. The analyses show that in general the effect of mixing air or vapor with steam is to decrease the dryness of the steam after expansion, and thus to decrease its final stock of internal energy and to increase the external work performed. Saturated steam expands (i e., increases in volume) more rapidly than air, as its temperature is lowered. Similarly, for a given rate of increase in volume, the temperature of air falls more rapidly than that of steam. TVhen the two fluids are mixed, a condition of uniform temperature must prevail. This necessitates a transfer of heat from the steam to the air, decreasing the entropy of the former and increasing that of the latter. The decrease in entropy of the steam is responsible for its decreased dryness at the end of expansion. 254 APPLIED THERMODYNAMICS SUPERHEATED STEAM 383. Properties : Specific Heat. In comparatively recent years, superheated steam has become of engineering importance in application to reciprocating en- gines and turbines and in locomotive practice. Since superheated steam exists at a temperature exceeding that of saturation, it is important to know the specific heat for the range of superheating. The first determination was by Regnault (1S62), who obtained as mean values k = 0.4805, I = 0.346, y = 1.39. Fenner found I to be variable, ranging from 0.341 to 0.351. Hirn, at a later date, concluded that its value must vary with the temperature. Weyrauch (29), -who devoted himself to this subject from 1876 to 1904, finally concluded that the value of k increased both with the pressure and with the amount of superheating (range of temperature above saturation), basing this con- clusion on his own observations as collated with those of Regnault, Hirn, Zeuner, Mallard and Le Chatelier, Sarrau and Teille, and Langen. Rankine presented a demonstration (now admitted to be fallacious) that the total heat of superheated steam was independent of the pressure. At very high temperatures, the values obtained by Mallard and Le Chatelier in 1883 have been generally accepted by metallurgists, but they do not apply at temperatures attained in power engineer- ing. A list by Dodge (30) of nineteen experimental studies on the subject shows a fairly close agreement with Regnault's value for k at atmospheric pressure and approximately 212 F. Most experimenters have agreed that the value increases with the pressure, but the law of variation with the temperature has been in doubt. Holborn's results (31) as expressed by Kutzbach (32) would, if the em- pirical formula held, make k increase with the temperature up to a certain limit, and then decrease, apparently to zero. 384. Knoblauch and Jakob Experiments. These determinations (33) have attracted much attention. They were made by electrically super- heating the steam and measuring the input of electrical energy, which was afterward computed in terms of its heat equivalent. These experi- menters found that k increased with the pressure, and (in general) decreased with the temperature up to a certain point, afterward increas- ing (a result the reverse in this respect of that reported by Holborn). Figure 170 shows the results graphically. Greene (34) has used these in plotting the lines of entropy of superheat, as described in Art. 398. The Knoblauch and Jakob values are more widely used than any others experimentally obtained. They are closely confirmed by the equation derived by Goodenough (Principles of Thermodynamics, 1911) from fundamental analysis : SUPERHEATED STEAM 255 where k is the true or instantaneous value of the specific heat at the constant pressure p (Ibs. per sq, in.) and at the temperature T abso- FiG. 170. 340 280 320 360 400 440 480 520 SCO 600 G40 680 720 TEMPERATURE- DECREES FAHRENHEIT Arts. 384, 421. Specific Heat of Superheated Steam. Knoblauch and Jakob Results. lute, and log (7 = 14.42408. Values given by this equation should correspond with those of the curves, Fig. 170. The values in Fig. 171 are for mean specific heat at the pressure p from saturation to the temperature T, for which Goodenough's equation is 256 APPLIED THERMODYNAMICS Amp(n+l) (l+^p hyT-n- To being the saturation temperature, a=0.367, 6=00001. log m = 13.67938, n = o 3 A =-7YT> 1S {Am(n-\-l) } =11.566. 385. Thomas' Experiments. In these, the electrical method of heating and a careful system of radiation corrections were employed (35). The conclusion reached was that 7c increases with increase of pressure and decreases with increase of temperature. The variations are greatest near the saturation curve. The values given included pressures from 7 to 500 Ib. FIG. 171. Aits. 385, 388, $, 417, Prob 42. Specific Heat of Superheated Steam. Thomas' Experiments. per square inch absolute, and superheating ranging up to 270 F. The entropy lines and total heat lines are charted in Thomas' report. Within rather narrow limits, the agreement is close between these and the Knob- lauch and Jakob experiments. The reasons for disagreement outside these limits have been scrutinized by Heck (36), who has presented a table of the properties of superheated steam, based on. these and other data. The steam tables of Marks and Davis (see footnote, p. 202) contain a complete set of values for superheated states. Figure 171 shows the Thomas results graphically. 386. Total Heat. As superheated steam is almost invariably formed at constant pressure, the path of formation resembles dbcW, Fig. 161, ab SPECIFIC HEAT 257 being the water line and cd the saturation curve. Its total heat is then H c -i-k(T t), where T, t refer to the temperatures at W and c. If we take Begnault's value for H c , 1081.94 + 0.305* (Art. SCO), then, using 7c = 0.4805, we find the total heat of superheated steam to be 108] .94 0.1755 1 -f- 0.4805 T. A purely empirical formula, m which P is the pres- sure in pounds per square foot, is ff= OASOo(T 10. 37 jP ^; -f 857.2. For accurate calculations, the total heat must be obtained by using correct mean values for & during successive short intervals of temperature between t and T. 387. Variations of k. Dodge (37) has pointed out a satisfactory method for computing the law of variation of the specific heat. Steam is passed through a small orifice so as to produce a constant reduction in a constant pressure. It is superheated on both sides the orifice ; but, the heat coii- tents remaining constant during the throttling operation, the temperature changes. Let the initial pressure be ^>, the final pressure j^ Let one observation give for an initial temperature t, a final temperature t x ; and let a second observation give for an initial temperature T, a final tempera- ture 2i. Let the corresponding total heat contents be 7d, 7^, H, JI^ Then h H= Je p (t T) and 7^ H : = fc, (^ T 2 ). But k = 7^ H= H^ whence TP f m h H= hi H^ and -*- = ^ -^ - If ive know the mean value of k for any K D t jt *\ given range of tem2}erature, we may then ascertain the mean value for a series of ranges at various pressures. 388. Davis' Computation of H. The customary method of deter- mining k has been by measuring the amount of heat necessarily added to saturated steam in order to produce an observed increase of tem- perature. Unfortunately, the value of H for saturated steam has not been known with satisfactory accuracy ; it is therefore inade- quate to measure the total heat in superheated steam for comparison with that in saturated steam at the same pressure. Davis has sho\N ri (17) that since slight errors in the yalue of H lead to large errors in that of &, the reverse computation using known values of k to determine H must be extremely accurate ; so far so, that while additional determinations of the specific heat are in themselves to be desired, such determinations cannot be expected to seriously modify values of ^BTas now computed. The basis of the computation is, as in Art. 387, the expansion of superheated steam through a non-conducting nozzle, with reduction 258 APPLIED THERMODYNAMICS of temperature. Assume, for example, that steam at 38 Ib. pres- sure and 300 F. expands to atmospheric pressure, the temperature becoming 286 F. The total heat before throttling we may call H c = H b -+ kyT c 2&), i n which H b is the total heat of saturated steam at 38 Ib. pressure, T e = 300 F., and T b is the temperature of saturated steam at 38 Ib. pressure, or 264.2 F. After throttling, similarly, H d = 2Zi + * a (Ztf 2^), in which H e is the total heat of saturated steam at atmospheric pressure, T & is its temperature (212 FO, and T d is 286 F. Now JZ d = H e , and H e = 1150.4 ; while from Fig. 171 we find * x = 0.57 and * a = 0.52 ; whence S b = - 0.57(800 - 264.2) + 1150.4 + 0.52(286 - 212) = 1168.47. The formula given by Davis as a result of the study of various throttling experiments may be found in Art. 360. The total heat of saturated steam at some one pressure (e.g. atmospheric) must be known. A simple formula (that of Smith), which expresses the Davis results with an accuracy of 1 per cent, between 70 and 500, was given in Power, February 8, 1910. t being the Fahrenheit temperature. 389. Factor of Evaporation. The computation of factors of evapora- tion must often include the effect of superheat. The total heat of super- heated steam which we may call H t may be obtained by one of the methods described in Art. 386. If ?IQ is the heat in the water as sup- plied, the heat expended is H t Ji^ and the factor of evaporation is (H 9 -o)-*- 970.4. 390. Characteristic Equation. Zeuner derives as a working formula, agreeing with Hirn's experiments on specific volume (38), PF= 0.64901 T- 22.5819 P 03 *, in which P is in pounds per square inch, V in cubic feet per pound, and T in degrees absolute Fahrenheit. This applies closely to saturated as well as to superheated steam, if dry. Using the same notation, Tumlirz gives (39) from Battelli's experiments, PV= 0.594 T- 0.00178 P. The formulas of Knoblauch, Linde and Jakob, and of Goodenough, both given in Art. 363, may also be applied to superheated steam, if not too PATHS OF VAPORS 259 highly superheated. At very high temperatures , steam behaves like a perfect gas, following closely the law PV=RT. Since the values of R for gases are inversely proportional to their densities, we find R for steam to be 85.8. 391. Adiabatic Equation. Using the value just obtained for 72, and Regnault's constant value 0.4805 for k, we find y 1.208. The equation of the adiabatic would then be ^?i 1298 = c. This, like the characteristic equation, does not hold for wide state ranges; a more satisfactory equation remains to be developed (Art. 397). The exponential form of expression gives merely an approximation to the actual curve. PATHS OF VAPORS 392. Vapor Adiabatics. It is obvious from Art. 372 that during adiabatic expansion of a saturated vapor, the condition of dryness must change. We now compute the equa- tion of the adiabatic for any vapor. In Fig. 172, consider expansion from J to c. Draw the isothermals T, t. We have FIG. 172. Art. 392. Equa- ing the variable temperature along da. But tion of Vapor Adiabatic. ^ = ^ ^ . f the specific heat of the liquid be constant and equal to <?, ^=6 j log t! I- ^, the desired equation. t t JL If the vapor be only X dry at J, then 393. Applications. This equation may of course be used to derive the results shown graphically in Art. 373. For example, for steam initially dry, we may make X = 1, and it will be always found that x e is less than 1. To show that water expanding adiabatically partially vaporizes, we mate X 0. To determine the condition under which the dryness may be the same after expansion as before it, we make x = X. 394. Approximate Formulas. Rankine found that the adiabatic might be represented approximately by the expression, PP"^ = constant; which holds fairly well for limited ranges of pressure when the initial dryness is 1.0, but which gives a curve lying decidedly outside the true adiabatic for any con- siderable pressure change. The error is reduced as the dryness decreases, down to a certain limit. Zeuner found that an exponential equation might be written in 260 APPLIED THERMODYNAMICS the form P V n = constant, if the value of were made to depend upon the initial dryness. He represented this by n = 1.035 + 0.100 X, for values of X ranging from 0.70 to 1.00, and found it to lead to sufficiently accu- rate results for all usual expansions. For a compression from an initial dry ness r, n = 1.034 + 0.11 x. "Where the steam is initially dry, n = 1.135 for expansion and 1,144 for compression. There is seldom any good reason for the use of exponential formulas for steam adiabatics. The relation between the true adiabatic and that described by the exponential equation is shown by the curves of Fig. 173, after o & 10 FIG. 173. Arts. 394, 395. Adiabatic and Saturation Corves. Heck (40). In each of these five sets of curves, the solid line represents the adiabatic, while the short-dotted lines are plotted from Zeuner's equation, and the long-dotted lines represent the constant dryness curves. In I and II, the two adiabatics apparently exactly coincide, the values of x being 1.00 and 0.7o. In IH, IV, and V, there is an increasing divergence, for x = 0.50, 0.25 and 0. Case V is for the liquid, to which no such formula as those discussed could be expected to apply. 395. Adiabatics and Constant Dryness Curves. The constant dryness curves I and II in Fig. 173 fall above the adiabatic, indicating that heat is absoj-bed during expansion along the constant dryness line. Since the temperature falls during expansion, the specific heat along these constant diyness curves, within the limits shown, must necessarily be negative, a result otherwise derived in Art. 373, The points of tangency of these curves with the corresponding adiabatics give the points of inversion, at which the specific heat changes sign. STEAM ADIABATICS 261 396. External Work. The work during adiabatic expansion from PVto pV) assuming pv n = PF", is represented by the formula PV-pv 71-1 ' More accurately, remembering that the work done equals the loss of internal energy, we find its value to be H h -f- XR xr, in which H and h denote the initial and final heats of the liquid, 397. Superheated Adiabatic. Three cases are suggested hi Fig. 174, paths //, jk t de, the initially superheated vapor being either dry, ^wet, or superheated at the / J ij / w & f 1^ / I / \ k N FIG. 174. Art. 397. Steam Adiabatics. end of expansion. If k be the mean value of the specific heat of superheated steam for the range of temperatures in each case, then for>, c log. + ^ 2 T for jk,c log, + 398. Entropy Lines for Superheat. Many problems in superheated steam are conveniently solved by the use of a carefully plotted entropy diagram, as shown in Fig. 175.* The plotting of the curves within the saturated limits has already been explained. At the upper right-hand corner of the diagram there appear constant pressure lines and constant total heat curves. The former may be plotted when we know the mean specific heat fc at a stated pressure between the temperatures T and t : the T entropy gained being Tc log e --. The lines of total heat are determined * This diagram, is based on saturated steam tables embodying Regnault's results, and on Thomas' values for k ; it does not agree with the tables given on pages 247, 248. The same remark applies to Figs. 159 and 177. 262 APPLIED THERMODYNAMICS 0.5 00 017 0>9 1.0 11 12 13 14 1.5 16 17 1.8 1 Fio. 175. Arts. 377, 398, 401, 411, 417, 516, Problems. Temperature-entropy for Steam. ENTROPY LINES FOR SUPERHEAT 263 by the following method: For saturated steam at 103.38 Ib. pressure, #=1182.6, T= 330 F. As an approximation, the total heat of 1200 B. t. u. will require (1200 - 1182. 6 j-s- 0.4805 = 36.1 F. of superheating. For this amount of superheating at 100 Ib. pressure, the mean specific heat is, according to Thomas (Fig. 171), 0.604; whence the rise in tem- perature is 17.4 -r- 0.604 = 28.7 F. For this range (second approxima- tion), the mean sp3cific heat is 0.612, whence the actual rise of temperature is 17.4 -4- 0.612 = 28.4 F. No further approximation is necessary ; the amount of superheating at 1200 B. t. u. total heat may be taken as 28 F., which is laid off yertically from the point where the satu- ration curve crosses the line of 330 F., giving one point on the 1200 B. t. TL total heat curve. A few examples in the application of the chart suggest themselves. Assume steam to be formed at 103.38 Ib. pres- sure ; required the necessary amount of superheat to be im- parted such that the steam shall be just dry after adiabatic expansion to atmos- pheric pressure. Let rs, Fig. 176, be the line of atmospheric pressure. Draw st vertically, intersecting di\ then t is the required initial condition. Along the adiabatic ts, the heat contents decrease from 1300 B. t u. to 1150.4 B. t. u., a loss of 149.6 B. t. u. To find the condition of a mixture of unequal weights of water and super- heated steam after the establishment of thermal equilibrium, the whole operation being conducted at constant pressure : let the water, amounting to 10 Ib., be at r, Fig. 176. Its heat contents are 1800 B, t. u. Let one pound of steam be at t, having the heat contents 1300 B. t. u. The heat gained by the water must equal that lost by the steam ; the final heat con- tents will then be 3100 B. t. u., or 282 B. t u. per pound, and the state FIG. 17G. Arts. 398, 399, 401. Entropy Diagram, Superheated Steam. 1460 1440 1480 1420 141 1400 1380 1380 1370 1360 1350 1340 1330 1320 131 1COO 1200 1280 127 1260 x25 124 121 1200 1190 1180 i fc a " g * i i * l ". EtfTHOPV THE MOLLIER HEAT CHART 265 be /, where the temperature is 312 F. ; the steam "will have been completely liquefied. We may find, from the chart, the total heat in steam (wet, dry, or superheated) at any temperature, the quality and heat contents after adiabatic expansion from any initial to any final state, and the specific volume of saturated steani at any temperature and dryness. 399. The Mollier Heat Chart. This is a variant on the temperature entropy diagram, in a form rather more convenient for some purposes. It has been developed by Thomas (41) to cover his experiments in the superheated region, as m Fig. 177. In this diagram, the vertical coordi- nate is entropy ; and the horizontal, total heat. The constant heat lines are thus vertical, while adiabatics are horizontal. The saturation curve is inclined upward to the right, and is concave toward the left. Lines of constant pressure are nearly continuous through the saturated and super- heated regions. The quality lines follow the curvature of the saturation line. The temperature lines in the superheated region are almost vertical. It should be remembered that the " total heat" thus used as a coordinate is nevertheless not a cardinal property. The " total heat '' at t, Fig. 176, for exam pie, is that quantity of heat which would have been imparted had water at 32 F. been converted into superheated steam at constant pressure. It will be noted that within the portion of saturated field which is shown, the total heat at a given pressure is directly proportional to the total entropy. This would be exactly true if the water line in Fig. 175 PRESSURE. POUNDS PER SQUARE INCH 1550 80 100 & UO MO 180 SOO *2D iltt 2<KI 2W 300 SSO MO SCO 3*0 400 4204401004 SATURATED TEAM TEMPERATURE DEGREES P. FIG. 185, Art, 399, PruWw*, Total Heavpressure Diagram, 266 APPLIED THERMODYNAMICS were a straight line and if at the same time the specific heat of water could be constant. An empirical equation might be written in the form where n s , H and P are the total entropy, total heat and pressure of a wet vapor. The so-called total heat-pressure diagram (Fig 185) is a diagram in which the coordinates are total heat above 32 F. and saturation temperature; it usually includes curves of (a) constant volume, (b) constant dryness, and (c) in the superheated field, constant temperature. Vertical lines show the loss or gain of heat corresponding to stated changes of volume or quality at constant pressure. Horizontal lines show the change in pressure, volume, and quality of steam resulting from throttling (Art. 387). This diagram is a useful supplement to that of Mollier. Heck has developed a pressure-temperature diagram for both saturated and superheated fields, on which curves of constant entropy and constant total heat (throttling curves) are drawn. By transfer from these, there is derived a new diagram of total heat on pressure, on which are shown the isothermals of superheat. A study of the shape of these isothermals illustrates the variations in the specific heat of superheated steam. VAPORS IN GENERAL 400. Analytical Method: Mathematical Thermodynamics. An expression for the volume of any saturated vapor was derived in Art. 368: Where the specific volume is known by experiment, this equation may be used for computing the latent heat. A general method of deriving this and certain related expressions is now to be described. Let a mixture of x Ib. of dry vapor with (1 - x) Ib. of liquid receive heat, dQ. Then dQ = kxdT + c (1 - x)dT + Ldx, in which k is the "specific heat" of the continually dry vapor, L the latent heat of evaporation, and c the specific heat of the liquid. If P,V are the pressure and volume, and E the internal energy, in foot-pounds, of the mixtuie, then dQ = PdV + dE = IxdT + c (1 - x) dT 4- Ldx, whence / 78 dE = 778 [kx + c (1 - a;)] dT + 778 Ldx - PdV. Now V = (/) T, x] whence d V = f dT + 1? dx, whence bjT Sx dE = 778 [for + c (1 - or)] dT + 778 Ldx - P-* dT-P ^dx = J778 [for + C (l _ *)] _ p|ZJ dT + ( 77SL -P^\ dx. Moreover, E = (/) T, x, whence VAPORS IN GENERAL 267 giving (all properties excepting V and x being functions of T only). The volume, V, may be written xu -f r, where u is the volume of the liquid and X T r w the increase of volume during vaporization. This gives 8 J r = wSx or = u. ox Also, since F= (/) T, or, |1|- = JJ5L, and equation (A) becomes Now if the heat is absorbed along any reversible path, = dN, or dN _ kzdT + cQ - x*)dT + Ldx = kx + c(l - s) -/' 6V +-*-. ( which may be combined with (B), giving 778 = u = F - 0, as in Art 369. (D) 401. Computation of Properties. Equation (D), as thus derived, or as obtained in Art. 369, may be used to compute either the latent heat or the rdume of any vapor when the other of these properties and the relation of temperature and pres- sure is known. The specific heat of the saturated vapor may be obtained from (C) ; the temperature of inversion is reached when the specific heat changes sign. For steam, if L - 1113.94 - 0.695 1 (Art. 379), where t is in degrees P., or 1113.94 - 0.695(2 T - 459.6) where T is the absolute temperature: ~ T = - 0.695. Also c = 1 ; whence, from equation (C), k = 0.305 - , which equals zero when T= 1433 absolute.* At 212 *\k= 0.303 - = - 1.135. This may be roughly Oil.Q * This would be the temperature of inversion of dry steam if the formula for L held : but L becomes zero at 689 F. (Art. 379), and the saturation carve 'for steam slopes downward toward the right throughout its entire extent. For the dry vapors of chloroform and ben- zine, there exist known temperatures of inversion. 268 APPLIED THERMODYNAMICS checked fiom Fig. 175. In Fig. 176, consider the path ,s^ from 212 F, to 157 F., and fiom n = 1 735 to n = 1.835 (Fig. 175). The average height of the area ctibe representing the heat absorbed is 459.6 + 212 * ln/ = 644.1 ; whence, the area is fiU 1(1 835 - 1.735) = 04.41 B. t. u., and the mean specific heat between s and b is 61.11 - (212 - 157) = 1.176. The properties of the volatile vapors used in refriger- ation are to some extent known only by computations of this sort. When once the pressure-temperature i elation and the characteristic equation are ascertained by experiment, the other propeities follow. 402. Engineering Vapors'. The properties of the vapors of steam, carbon dioxide, ammonia, sulphur dioxide, ether, alcohol, acetone, carbon disulphide, carbon tetrachlonde, and chloroform have all been more or less thoroughly studied. The firnt five are of considerable importance. For ether, alcohol, chloroform, carbon disul- phide, carbon tetrachloride, and acetone. Zeuner has tabulated the pressure, tempera- tui e. volume, total heat, latent heat, heat of the liquid, and internal and external woik of vaporization, in both French and English units (42), on the basis of Regnault's experiments. The properties of these substances as given in Peabody's "Steam Tables" (1890) are reproduced from Zeuner, excepting that the values - 273.7 and 426.7 aie used instead of - 273.0 and 424.0 for the location of the absolute zero centigrade and the centigiade mechanical equivalent of heat, respectively. Peabody's tables for these vapors are in Fiench units only. Wood has derived expressions for the properties of these six vapois, but has not tabulated their values (40). Rankine (44) has tabulated the pressure, latent heat, and density of ether, per cubic foot, in English units, fiom Regnault's data. Forcrzr&n/i dioxide, the experimental results of Andrews, Cailletet and Hautefeuille, Cailletet and Mathias (45), and, finally, Ainagat (46), have been collated by Mollier, whose table (47) of the properties of this vapor has been reproduced and extended, in French and English units, by Zeuner (48). The vapor tables appended to Chapter XVIII, it will be noted, are based on those of Zeuner. The entropy diagrams for am- monia, ether, and carbon dioxide, Figs. 314-316, have the same foundation The present writer (in Vapor* for Heat Engines, D. Van No-strand Co., 1911) has computed the entropies and prepared temperature-entropy diagrams for alcohol, acetone, chloroform, carbon chloride and carbon disulphide. 403. Ammonia. Anhydrous ammonia, largely used in refrigerating machines, was first studied by Regnault, who obtained the relation Q IA-O = S.40<9 t in which p is in pounds per square foot and t is the absolute temperature. A " characteristic equation " between p, v, and t was derived by Ledoux (49) and employed by Zeuner to permit of the computation of V> L, e, r and the specific heat of the liquid (the last having recently been deter- mined experimentally (50)). The results thus derived were tabulated by Zeuner (51) for temperatures below 32 P. ; 'while for higher temperatures he uses the experimental values of Dietrici (52). Peabody's table (53), also derived from Ledonx, uses his values for temperatures exceeding 32 F. 5 Zeuner regards Ledgux's values in this region as unreliable. VAPORS IN GENERAL 269 Peabody's table is in French units ; Zeuner's is in both French arid Eng- lish units. The latent heat of evaporation has been experimentally de- termined by Regnault (54) and Ton Strombeck Coo). The specific volume of the vapor at 26.4 F. and atmospheric pressure is 17.51 cu. ft. ; that of the liquid is 0.025; whence from equation (D), Art. 400, = 778 " ^ dT = 433.2 (17 51 _ oog) /2196 x 2.3026 X 14 7 X 144\ 778 ^ " V 433.2x433.2 / dP the value of being obtained by differentiating Regnault's equation, above given. From a study of Regnault's experiments, Wood has derived the characteristic equation, PF == oi _ 16920 T ~~ which is the basis of his table of the properties of ammonia vapor (56). Wood's table agrees quite closely with Zeuner's, as to the relation between pressure and temperature ; but his value of L is much less variable. For temperatures below C., the specific volumes given by Wood are rather less than those by Zeuner; for higher temperatures, the volumes vary less. Zeuner's table must be regarded as probably more reliable. The specific heat (0.508) and the density (0.597, when air = l.Q) of the super- heated vapor have been determined by experiment. 404. Sulphur Dioxide. The specific heat of the superheated vapor is given by Regnau.lt as 0.15438 (57). The. specific volume, as compared with that of air, is 2.23 (58). The specific volume of the liquid is 0.0007 (oO) ; its specific heat is approximately 0.4. A characteristic equation for the saturated vapor has been derived from Regnault's experiments : P F = 26.4 !T - 184 P - 22 ; in which Pis in pounds per square foot> Tin cubic feet per pound, and T in abso- lute degrees. The relation between pressure and temperature has been studied by Reguault, Sajotschewski, Blumcke, and Miller. Regnault's observations were made between - 40 and 149 F. ; Miller's, between 68 and 211 F. ; a table repre- senting the combined results has been given by Miller (00). lu the usual form of the general equation, log p = a bd* ce *, the values given by Peabody for pleasures in pounds per square inch are (61) a = 3.9527847, log b = 0.4792425, log d = 1.9984994, log c = 1J659562, logc = 1.99293890, n = 18.4 -f Fahrenheit temi>erature. The specific volumes, determined by the characteristic equation and the pressure-temperature formula, permit of the computation of the latent heat from equation (D), Art, 400. An empirical formula 270 APPLIED THERMODYNAMICS for this property is L = 176 0.27( - 32), in which t is the Fahrenheit tempera- ture. The experimental icsults of Cailletet and Mathias, and of Mathias alone (62) , have led to the tables of Zeuner (63). Peabody, following Ledoux's analysis, has also tabulated the properties in French units. Wood (61) has independently com- puted the properties in both French and English units. Comparing Wood's, Zeu- ner's, and Peabody's tables, Zeunei's values for L and V are both less than those of Peabody. At F., he makes L less than does Wood, departing even more widely than the latter from. Jacobus' experimental results (65) ; at 30 F., his value of L is greater than Wood's, and at 104 F., it is again less. The tabulated values of the specific volumes differ correspondingly. Zeuner's table may be regarded as sus- tained by the experiments of Cailletet and Mathias, but the lack of concordance with the experimental results of Jacobus remains to be explained 405. Steam at Low Temperatures. Ordinary tables do not give the properties of water vapor for temperatures lower than those corresponding to the absolute pressures reached in steam engineering. Zeuner has, however, tabulated them for temperatures down to -4 F. (66). 40 5. Vapors for Heat Engines. Engines have been built using, instead of steam, the vapors of alcohol, gasolene, ammonia, ether, sulphur dioxide and carbon dioxide, with good results as to thermal efficiency, if not with commercial success. In a simple condensing engine, with a rather low expansive ratio, a considerable saving may be effected with some of these vapors, as compared with steam; and the cost of the fluid is not a vital matter, since it may be used over and over again. Strangely enough, in the case of none of the vapors is a very low discharge temperature practically desirable, under usual simple condensing engine conditions. This statement applies even T t to steam. The Carnot criterion - -, does not exactly apply, sinca it refers to potential efficiency only: but the use of a substitute vapor might perhaps be justified on one of the two grounds, (a) an increased upper temperature without excessive pressures or (6) a decreased lower temperature at a reasonable vacuum, say of 1 Ib. absolute. To meet both requirements the vapor would have to give a pt curve crossing that of steam. It is probable that carbon tetrachloride is uch a vapor, bearing such a relation to steam as alcohol does to it. ITo great gain is possible in respect to the lower temperature limit, since this limit is in any case established by the cooling water. The criterion given in Art. 630 measures the relative efficiencies of fluids working in the Clausius cycle. On this basis steam surpasses all other common vapors in potential thermal efficiency. The lower " heat content " per pound of the more volatile and heavy vapors leads to a greatly reduced nozzle velocity with adiabatic flow, and this suggests the possibility of developing a turbine expanding in one operation without excessive peripheral speeds (see Chapter XIV). STEAM PLANT CYCLE 271 The greater density of the volatile sapors also leads to the con- clusion that the output from a cylinder of given size might in the cases of some of them be about twice what it is from a steam cylinder. On the whole, the use of a special vapor seems to be more promising, technically and commercially, than the binary vapor principle (Art. 4S3). For a fuller discussion of this subject, reference may be made to the work referred to in Art. 402. STEAM CYCLES 406. The Carnot Cycle for Steam. This is shown in Figs. 163, 179. The efficiency of the cycle abed may be rend from the entropy T-t diagram as T The external work done per pound of steam T t is L - ; or if the steam at I T-t is wet, it is xL T If the 1 I FIG. 179. Art 40t> Carnot Cycle for Steam. fluid at the beginning of the cycle (point a) is wet steam instead of water, the dryness being x^ then the work per pound of steam is L(x # ) m * . i In the cycle first discussed, in order that the final adiabatic compression may bring the substance back to its initially dry state at a, such compression must begin at d, where the dryness is md -s- mn. The Carnot cycle is impracticable with steam; the substance at d is mostly liquid, and cannot be raised in temperature by compression. What is actually done is to allow condensation along cd to be com- pleted, and then to warm the liquid or its equivalent along ma by trans- mission of heat from an external source. This, of course, lowers the efficiency. 407. The Steam Power Plant. The cycle is then not completed in the cylinder of the engine. In Fig. 180, let the substance at d be o FIG. 180. Arts. 407, 408, 410 T 412, 413. The Steam Power Plant. 272 APPLIED THERMODYNAMICS cold water, either that resulting from the action of the condenser on the fluid which luis passed through the engine, or an external supply. This water is now delivered by the feed pump to the boiler, iu which its temperature und pressure become those along al. The work done by the feed pump per pound of fluid is that of raising unit weight of the liquid against a head equivalent to the pressure; or, what is the same thing, the product of the specific volume of the water by the range in pressure, in pounds per square foot. From a to b the substance is in the boiler, being changed from water to steam. Along fit-, it is expanding in the cylinder; along ed it is being liquefied in the condenser or being discharged to the atmos- phere. In the former case, the resulting liquid reaches the feed pump at <Z. In the latter, a fresh supply of liquid is taken in at d, but this may be thermally equivalent to the liquid resulting from atmospheric exhaust along cd. (See footnote, Art. 502.) The four organs, feed pump, boiler, cylinder, and condenser, are those essential in a steam power plant. The cycle rep- resents the changes undergone by the fluid in its passage through them. 408. Clausius Cycle. The cycle of Fig. ISO, worked without adialatic fiompresxion, is known as that of Chutius. Its entropy diagram is shown as dele in Fig. 181, that of the corresponding Carnot cycle being dhbc. The Carnot efficiency is obviously greater than that of the Clausius cycle. For wet steam the corresponding cycles are deM and dhkl. FIG. 181. Arts 40&-41.1. Rteain Cydes. 409. Efficiency. dele In Fig. 181, cycle dele, the efficiency is _ ft, Ji a + L b xjj f But x c = % if the specific heat of the RANKIXE CYCLE 273 liquid be unity. Then letting 7, L refer to the state J, and t, I to the state <?, the efficiency is T-t+ L which is determined s0ZeZ# by tJie temperature limits Tand t. For steam initially wet, the efficiency is T- 410. Work Area. In Figs. 180, 181, -we have W= W ab + W bc - W cd - W da ignoring the small amount of work done by the feed pump in forcing the liquid into the boiler. But p b (v b a ) = e b and j^Oy i\i) J'Sj (Art. 359), whence W=h e + L b -h-x t Lsi a result identical with the numerator of the first expression in Art. 409. 411. Rankine Cycle. The cycle delgq, Fig. 181, af>gq<J, Fig. 180. is known as that of Rankine (67). It differs from that of Clansuis merely in that expansion is incomplete, the "toe"" gey, Fig. ISO, being cut off by the limiting cylinder volume line gq. This is the ideal cycle nearest which actual steam engines work. The line yy in Fig. 181 is plotted as a line of constant volume (Art. 877). The efficiency is obviously less than that of the Clausius cycle ; it is elgqd __ W ab +W^~ W qd (Fig. 180) " - O] + (** + ?b - K - The values of h^ X T r t , x q , depend upon the limiting volume v g = v r and may be most readily ascertained by inspecting Fig, 175. The computation of these properties resolves itself into the problem : given 274 APPLIED THERMODYNAMICS the initial state, to find the temperature after adidbatic expansion to a given volume. We have v g - v r = x g (v s - fl r ), n g = w 6 , -^ 9 n s n r n s ?? r L s + T r whence in which v ff , T e , LI, are given, v r =0.017, and v sj L s are functions of T T , the value of which is to be ascertained. The greater the ratio of expansion, ^-s-r*, Fig. 181, with given cyclic limits, the greater is the efficiency. 412. NoB-expansive Cycle. This appears as debt, Fig. 181 ; and'a&ed, Fig. 180. No expansion occurs; work is done only as steam is evaporated or condensed. The efficiency is (Fig. 181) del* = W* - W ed (Fig. 180) = p b (v b - r a ) - p t (r - v d ) t h e -h d +L b h t - h d + 5 This is the least efficient of the cycles considered. 413. Pambonr Cycle. The cycle debf. Fig. 181, represents the operation of a plant in which the steam remains dry throughout expansion. It is called the Pamhtur cycle. Expansion may be incomplete, giving such a diagram as debuq* Let abed in Fig. 180 represent debfiu. Fig. 181. The efficiency is external work done _ _ external work done _ gross heat absorbed ~" heat rejected + external work done _ TFqft + TtV - W ed _ in which the saturation curve If may be represented by the formula pv& = con- stant (Art. 363). A second method for computing the efficiency is as follows: & T L the area debf= \ ~=dT, in which T and t are the temperatures along eb and df jt y respectively, and L =(J)T= 1433 - 0.695 T (Art. 379). This gives debf= 1433 log.- - 0.695(T - *)* and the efficiency is 1433 log fl - - 0.695( T - 1) debf __ debf _ -debf+idfv imiogf I_ M g 6(T _ t)+L/ SUPERHEATED CYCLES 275 The two computations will not precisely agree, because the exponent $ does not exactly represent the saturation curve, nor does the formula for L in terms of T hold rigorously. Of the whole amount of heat supplied, the portion Kbfv was added during expattswi, as by a steam jacket (Art. 439). To ascertain this amount, we have heat added by jacket = whole heat supplied heat present at beginning of expansion = 1433 log,^- O.G95 (!T- /) + Zy- h, + U d - L L The efficiency is apparently less than that of the Clausius cycle (Pig. 181). In practice, however, steam jacketing increases the efficiency of engines, for reasons which will appear (Art. 439). 414. Cycles with Superheat. As in Art. 397, three cases are pos- sible. Figure 182 shows the Clausius cycles debzw, debgf, debzAf, in which the steam is respectively wet, dry, and superheated at the end of expansion. To appreciate the gain in efficiency due to super- heat, compare the first of these cycles, not with the dry steam Clausius cycle dele, but with the superior Oarnot cycle dhbe. If the path of superheating were b C, the efficiency would be unchanged; the actual path is Jj?, and the work area bxO is gained at 100 per cent efficiency. The cycle dhbxw is thus more efficient than the Car- not cycle dhbc, and the cycle debxw is more efficient than the Clausius cycle debc. It is not more efficient than a Carnot cycle through its own temperature limits, The cycle debyf shows a further gain in efficiency, the work area added at 100 per cent effectiveness being byE. The cycle debzAf shows a still greater addition of this desirable work area, but a loss of area AfB now appears. Maximum efficiency appears to be secured with such a cycle as the second of those considered, in which the steam is about dry at the end of expansion. The Carnot formula FIG 182. Art. 414. Cycles with Superheat. 276 APPLIED THERMODYNAMICS suggests the desirability of a high upper temperature, and superheating leads to this ; "but when superheating is carried so far as to appreciably raise the temperature of heat emission, as in the cycle debzAf, the efficiency begins to fall. 415. Efficiencies. The work areas of the three cycles discussed may be thus expressed : in which Jc v Jc# k# k# refer to the mean specific heats over the re- spective pressure and temperature ranges. The efficiencies are obtained by dividing these expressions by the gross amounts of heat absorbed. The equations given in Art. 397 permit of computation of such quantities as are not assumed. 416. Itemized External Work. The pressure and temperature at the beginning of expansion being given, the volume may be computed and the external work during the reception of heat expressed in terms of P and F. The temperature or pressure at the end of expansion being given, the volume may be computed and the negative external work during the rejection of heat calculated in similar terms. The whole work of the cycle, less the algebraic sum of these two work quantities (the feed pump work being ignored), equals the work under the adiabatic, which may be approximately cheeked from the formula py-pv^ ^ suitable value being used for n (Art. 394). A second n 1 approximation may be made by taking the adiabatic work as equivalent to the decrease in internal energy, which at any superheated state has the value h + r + - (T f), T being the actual temperature, and A, r, t referring to the condition of saturated steam at the stated pressure. The most simple method of obtaining the total work of the cycle is to COMPARISONS 277 read from Fig. 177 the " total heat " values at the beginning and end of expansion. (See the author's " Vapors for Heat Engines/' D. Van Nostrand Co., 1912.) 417. Comparison of Cycles. In Fig. 183, we have the following cycles: \ tpawq FIG. 183. Arts. 417, 441, 442. Seventeen Steam Cycles. Clausius, Rankine, Non-expansive, with dry steam, dele (the corresponding Carnot cycle being dhbe) ; with wet steam, dekl ; with dry steam, debgq ; with wet steam, dekJq; with dry steam, debt ; with wet steam, dekK- 9 Pambour, complete expansion, debf; incomplete expansion, debuqi Superheated to a;, complete expansion, debxw ; incomplete expansion, debxLuq\ no expansion, debxNp; Superheated toy, complete expansion, debyfi incomplete expansion, debyMuqi BO expansion, debyRs; Superheated to z, complete expansion, debzAfi incomplete expansion, debzTuq ; no expansion, debt Vw. 278 APPLIED THERMODYNAMICS The lines tl, pNx, sRy, icTz, quT, are lines of constant volume, Superheating without expansion would be unwise on either technical or practical grounds ; superheating with incomplete expansion is the condition of "universal practice in reciprocating engines. The seventeen cycles are drawn to PJ 7 " coordinates in Fig. 184. x y z Iff J FIG 184. Arts 417, 420, 424, 517 Seventeen Steam Cycles. ILLUSTRATIVE PROBLEM To compare the efficiencies, and the cyclic areas as related to the maximum volume at- tained: let the maximum pressure be 110 lh.,the minimum pressure 2 Ib , and consider the Clausiua cycle (a) with steam initially dry, () with steam initially 90 per cent dry ; the Rankine with initially dry steam and a maximum volume of 13 cu. ft , the same Kankine with steam initially 90 per cent dry; the non-expansive with steam dry and 00 per cent dry ; the Pambour (a) with complete expansion and (6) with a maximum volume of 13 cu- ft. ; and the nine types of superheated cycle, the steam being; (a) 06 per cent dry, (ft) dry, (c) 40 F. superheated, at the end of complete expansion ; and expansion being (a) complete, (/>) limited to a maximum volume of 13 cu. ft., (c) eliminated. L Cla usius cycle. The gross heat absorbed is h lta - 7< a -f 140 = 324 - 6 - 9^-0 + 86"-6 = 1098. S. The</rytt&MJattheend of expansion is dc -*- df, Fig. 183, ~(n e n d + n ab ') -n d/ = (0.5072 - 0.174!) 4- 1.0075) - 1.74;U = O.SOJ* The teat rejected along cd is x<.L f = 0.80S X 1021 = 8194. 1098.2" The uork done is 1008.2 - 819.4 = 273.8 B. t. u. The efficiency is ^ = 0354. The efficiency of the corresponding Carnot cycle is TW-T* 353.1 -,120.15 = 0.88, n. !T 14() 353.1+ 459.0 ' Clawdwt cycle with tret steam. The gross teat absorbed is h l4D -h Si + x*L J40 =324.6 - 94.0 + (0.00 x 8C7.G) = 1015.44* The dryrwif at the end of expansion is dl -5- df (n - nj -f n&) -*- n^ = (0.5072 - 0.174D + 0.90 x 1.0b73) *- 1.7431 = 0.741. COMPARISONS 279 The heat rejected along Iff is XiL f 0.741 x 1021 = 756. The work done is 1013.44: - 73fj = 359.44 B> * * The efficiency is (It is in all cases somewhat less than that of the initially dry steam cycle.) til. Rankine cycle* dry steam. The grouts Iwtt absorbed, as in T, is 10QS.2. The work along rte, Fig. 181, is 14 1 x liJS x 0.017 = ;A75. ~> foot-pounds (Art. 407); along eb is 144 x 140 x (Fi 0.017) = 64,300 foot-pounds ; " is A c + r 6 Ji z ay^ = 103.76 B. t. u. (Prom Fig. 175, f,=247 P., whence ,=947.4, F a = 11.52, *,= 0.8950 = [0.5072-2.3 (log T n - log 491.6) + 1.0075] TV 1433 - 0.093 T ff For !T ff = 247 F. = 700.6 absolute, this equation gives x ff = 0.905 ; a suffi- cient check, considering that Fi. 173 in based on a different set of values than those used in the steam talle. Then It 2 = 213., I* r g = 871.6. The work along qd is P d ( F f - T d ) = 144 x 2 x (13 - 0.017)= 3740 foot- pounds. The whole work of the cycle is 64anft ~ : ^ 8 ' 5 " 374 + 100.76 = . ^7$ The efficiency is IV- ^an^tfne cyr/e, ?e </eai. The ^ro.w Aca^ afoorbed is as in IT, 1015 J4* The negatire work along </<? and ^ is, as iu III, 338.5 -f 3740 = 4078.5 foot- pounds* The work along ek Is 14i x 140 X 0.90(T" 6 - 0.017)= 57J70 foot-powids. The work along kJ is A -f Xtf* A x jrj-r r = 99.8 B. t, u. (From Fig. 175, t x = 242 F., whence A x = 210.3, r r = 875.3, V r = 15.78, IS -0.017 35.78- 0.017 ^ The taAo/c M7orJt of the cycle is 5787 "I 4 078>5 + 99.8 = ^5.1 B. t. The efficiency is V. Non-*xpQn*ive cycle, dry steam. The gross heat absorbed, as in I, is The wrb <dong d*> s in III, is 33B J foot-pounds; along eb? as in IFI, iw $4,300 foot-poundt ; along td is^(F fc - T*) = 144 x 2 x (3.21& - 0.017)= 9$2 foot-pound*. The wA^ tcorjfe efttie cycle is - 338.5 - 922 = 63,039.5 foot-pounds = 81.0$ B. is 280 APPLIED THERMODYNAMICS VI. Non-expansive cycle t wet steam. The gross heat absorbed, as in II, is 1015.44+ The work along de, ek, as in IV, is - 338.5 + 57,870 = 57,531.5 foot-pounds* The work along Kd is J>(r*- 0.017)= 144 x 2 x 0.90 x (3.219 - 0.17)= 829.8 foot-pounds. The whole work of the cycle is 57,531.5 - 829.8 = 56,701.7 foot-pounds = 73 B. t. u. The efficiency is - = 0.072*. VII. Pambour cycle, complete expansion. The heat rejected is L f 102LO. The work along de, eb, as in in, is - 338.5 -f 64300 = 63,961.5 foot-pounds. The work along bfis = ^ 800 foot _ pounds . The work along fd is P d ( r, - Vd) = 2 x 144 (173.5 - 0.017) = 49,900 foot- pound*. The whole work of the cycle is 63,961.5 -1- 236,800 - 49,900 = 250J61.5 foot- pounds. (Otherwise 1433 log, - 0.695 (2^- /)= 312 B. t. u. = 42, 000 foot-pounds (Art. 413).) ' Using a mean of the two values for the whole work, the gross Jieat absorbed is ?iMp + 1021 = 1340 B. t. u. and the efficiency is ^ 2464 ^ = M8. The heat supplied by the jacket is 1340 - 1098.2 = S46.S B. t. u. VIII. Pambour cycle, incomplete expansion (debuq). In this case, we cannot directly find the heat refected, nor can we obtain the work area by inte- gration.* From Fig. 175 (or from the steam table), we find T u =253.8 F., P M = 31.84. The heat area under bu is then, very nearly, T + r (n u - 712 ' 6 + 812 ' 7 (1.6953 - 1.5747) = 9S B. t. u. 2 2i The whole heat absorbed is then 1098.2 -f 92 = 1190 S B. t. u. The work along de, eb y as in VII, is 6^96 1.5 foot-pounds. The work along bu is 144 x 16[(140 x 3.219) - (31.84 x 13)] = 85,800 foot- pounds. The work along qd, as in III, is 37 40 foot-pounds. The whole work of the cycle is 63,961.5 + 85,800 - 3740 = 146,021,5 foot-pounds = 188.2 B. t. u. The efficiency is = 0.1585. * A satisfactory solution may be had by obtaining the area of the cycle in two parts, a horizontal line being drawn through u to de. The upper part may then be treated as a com- plete-expansion Fambour cycle and the lower as a non-expansive cycle. The gross heat absorbed IB equal to the work of the upper cycle plus the latent heat of vaporization at the division temperature plus the difference of the heats of liquid at the division temperature and the lowest temperature. A somewhat similar treatment leads to a general solution for any Rankine cycle : in which, if the temperature at the end of expansion be given, the use of charts becomes unnecessary. COMPARISONS 281 IX. Superheated cycle, steam 0.96 dry at the end of expansion ; complete expansion; cycle debxw. We have n v ,=n d +x 1D n^ / = 0.1749 + (0.96 x 1.7431) = 1.8449. The state x(n x = n tt ) may now be found either from Fig. 175 or from the superheated steam table. Using the last, we find 7*, = 081.1 F., .7*= 1481.8, V x = 5.96. The whole heat absorbed, measured above T d , is then 1481.8 - 94.0 = 1387.8. The heat rejected is x v L f = 0.96 x 1021 = 981. The external work done is 1387.8 981 = 4063, and the efficiency is SB (The efficiency of the Carnot cycle within the same temperature limits is 931. 1 - 126.15 ^p^v 931.1 + 459.6 " *' X. !T&e same superheated cycle, with incomplete expansion. The whole heat absorbed, as before, is 1387.8, The work done along de, eb, as in HI, is 63,961.5 foot-pounds. The work done along bx is P b (V, - T 5 ) = 144 x 140(5.96 - 3.219)= 55,000 foot-pounds. The w?0r& cfone atony a; is x 5J>51.1 x 13 = 81 j 00 foot-pounds. (V L = 13, P*F s i* = p z ?yj, p, = 140^y ** = 51.1 ; a procedure which is, however, only approximately correct (Art. 391).) The work along gd, as in III, is 3740 foot-pounds. The whole work of the cycle is 63,961.5 + 55,000 + 81,500 - 3740 = 196,721.5 footpounds = 2SS.5 B. t. u. The efficiency is XI. T?ie *ame superheated cycle, worked non-ezpansively. The (7r(w fta/ alwrbed is ... The j<?rA: a/on^ <?, eb 9 bx, as in X, is 118,961.3 foot-pounds. The worfc along pd is 2 x 144 X (5.96 - 0.017)= 1716 foot-pounds. The whole work of the cycle is 118,961.5 - 1716 = 111 ',246.5 footpounds = 150.6 B. t. u. The efficiency is ^| = OJ086. XIL Superheated cycle, steam dry at Ihe end of expansion, complete expansion ; cycle debyf. We have n f = n/= 1.018. This makes the temperature at y above the range of our table. Figure 171 shows, however, that at high tempera- tures the variations in the mean value of k are less marked. We may perhaps then extrapolate values in the superheated steam table, giving r r = 1120.1 F., H 9 - 1578.5, T r r = 6.81. The whole heat absorbed, above T* is then 157&5 - 94.0 = U79J. The heat refected is L/st 1Q&1. 282 APPLIED THERMODYNAMICS The external work done is 1479.5 - 1021 = 458.6 S. t. u., and the efficiency XIIL Superheated cycle as above^ but with incomplete expansion. The gross heat absorbed is 1470.5. The work done along de, eb, as in III, is 63,961. o font-pounds f The work done ahmy ly is 144 x 140 X (6.81 - 3 219) = 72,200 foot-pounds. (6 81\ 1>23 ^ Hi J =60.3poun d.% approximately, s The we* done along yX is lf I 140 * ^T.f ' 3 X 13) ) = *V< >* v o.ijyo / pounds, alf*o approximately. The ipori ^/ir a/o/zy ^/, as in III, is 3740 foot-pounds. The 7r&0/e ttorl' of the cycle is 63,961.5 + 72,200 + 81,100 - 3740 = 213,521.5 foot-pounds = 875 B. t. u. The efficiency is "j , - ^?.^5r. XTV, Superheated cycle as above, but without expansion. The #ros$ Aeaf absorbed The warX: a?on^ /^, eb y by, as in XIII, is 136,161.5 foot-pounds. The zconfc atofl' *rf is 2 x 144 x (6.81 - 0.017) = 1952 foot-pounds. The tota/ wor/fc' is 136,161-5 - 1952 = 134,209.5 foot-pounds = 172.7 B. t. u. The efficiency is 2j=jL. = 0.117. - XV. Superheated cyde^ steam superheated 4&* F* at the end of expansion; expan- sion complete ; cycle debzAf. TV T e have n A = n x = 1.9486. A rather doubtful extrapolation now makes T s = 1202.1 F., #, = 1613.4, V* = 7.18. The irhule heat absorbed is 1613.4 - 94.0 = 1519.4- The heat re- jected is H A = 1133.2. The total work is 1519.4 - 1133.2 = 386.2 B. t. u., SS6 ^ and the efficiency is ' '" = 0355. lolU.4 XVL The same superheated cycle, with incomplete expansion. The pressure at T is 140 (-TQ-) 65.8 pounds. The work along zT (approximately) is 144 ((140 x 7.18) -(65.8 x!3)\ = 7Sj900 foot _ pmnds . TheoZ, work is \ o.2yo / 63,961.5 + [144 x 140 x (7.18 - 3.219)] + 73,900 - 3740 = 213,921.5 foot- pounds = 875.$ B. t. u., and the efficiency is 1519.4 XYII. The same superheated cycle without expansion. The total work is 63,961.5 + [144 x 140 x (7.18 - 3.219)] - [2 x 144 x (7.18 - 0.017)] =141,701.5 foot- pounds = 1833 B. t. u. T aud the efficiency is 0.1803. 418. Discussion of Results. The saturated steam cycles rank in order of efficiency as follows: Carnot, 0.28; Clausius, with, dry steam, COMPARISONS 283 0.254; with wet steam, 0.254 (a greater percentage of initial wetness would have perceptibly reduced the efficiency); Pambour, with com- plete expansion, 0.238 ; with incomplete expansion, 0.1585 ; Rankine, with dry steam, 0.1704 ; with wet steam, 0.1667; non-expansive, with dry steam 0.074; with wet steam, 0.0722. The economical impor- tance of using initially dry steam and as much expansion as possible is evident. The Pambour type of cycle has nothing to commend it, the average temperature at which heat is received being lowered. The Rankine cycle is necessarily one of low efficiency at low expan- sion, the non-expansive cycle showing the maximum waste. Comparing the superheated cycles, we have the following efficiencies : CYCLE COMPLETE EXPANSION INCOMPLETE EXPANSION No EXPANSION debxw 0.293 0.183 0.1086 debyf debzAf 0.31 0.255 0.187 0.182 0.117 0.1203 The approximations used in solution* will not invalidate the conclusions (a) that superheating gives highest efficiency when it is carried to such an extent that the steam is about dry at the end of complete expansion; (J) that incomplete expansion seriously re- duces the efficiency ; (V) "that in a non-expansive cycle the effi- ciency increases indefinitely with the amount of superheating. As a general conclusion* the economical development of the steam en- gine seems to be most easily possible by the use of a superheated cycle of the finally-dry-steam type, with as much expansion as pos- sible. We shall discuss in Chapter XIII what practical modifica- tions, if any, must be applied to this conclusion. The limiting volumes of the various cycles are F c for the Garnet, I, = 139.3. V w for IX = 166.5. V l for H = 128.2. V x for XI = 5.96. F* for V = 3.219. V k for VI = 2.9. F^for VII, XII =: 173, 5. * See footnote, Problem 53, page 296. A for XV = 186.1. ; for XVII = 7.18. 284 APPLIED THERMODYNAMICS The capacity of an engine of given dimensions is proportional to cyclic area ^ w hich. quotient has the following values* : maximum volume Car/not, temperature range x entropy range 31 = 226.95(1.5747 - 0.1749)= 317.5 : quotient = ^|i^ = 2.29, L 278.8-5-130.3 = 2.0 II. 259.44-5-128.2 = 2. III. 187.29-13 = 14.4. IV. 169.1 + 13 =13.0. V. 81.05-* 8.219 = 25.1. VL 73.0-5-2.9=25.1. VII. E18-f- 173.5 =1.84. VIII. 188.2-13 = 14.5. IX, 400.8^-166.5 = 2.445. 139.3 X. 253.5 -i- 13 = 19.45. XL 150.6 -f- 5.96:= 25.3. XII. 458.5 -r- 173.5 = 2.65. XIII. 275-13 = 21.1. XIV. 172,7 -6.81 =25.4. XV. 386,2-186.1 = 2.075. XVI. 275.3-13 = 21.1. XVII. 182,2-5-7.18 = 25.5. Here we find a variation much greater than is the case with the efficiencies ; but the values may be considered in three groups, the first including the five non-expansive cycles, giving maximum capacity (and minimum efficiency); the second including the six cycles with incomplete expansion, in which the capacity varies from 13 to 21.1 and the efficiency from 0.1585 to 0.187; and the third including six cycles of maximum efficiency hut of minimum capacity, ranging from 1,84 to 2.65. In this group, fortunately, the cycle of maximum efficiency (XII) is also that of maximum capacity. * The assumption of a constant limiting volume line Tuq, Pig. 183, is scarcely fair to the superheated steam cycles. In practice, either the ratio of expansion or the amount of constant volume pressure-drop at the end of expansion is assumed. As the firKt increases and the second decreases, the economy increases and the capacity figure decreases. The following table suggests that with either an equal pressure drop or an equal expansion ratio the efficiencies of the superheated cycles would compare still more favorably with that of the Rankine : CYCLES WITH INCOMPLETE EXPANSION ClCLB RATIO OP EXPAXSIOX PEESSETEE DROP Rankine ^i- r* = 13 - 3.219 = 4.04 P ff - P 9 - 26.3 Superheat I VL- r, = 18 -* 5.06 =2.185 PL P q = 49.1 Superheat II VM^- V, = IS ^- 0.81 = 1.91 Pjf P 9 = 68.3 Superheat III TV- F^=13-7.18 =1.815 P T - P f = 63.3 THE STEAM TABLES 285 Practically, high efficiency means fuel saving and high capacity means economy in the first cost of the engine. The general incom- patibility of the two affords a fundamental commercial problem in steam engine design, it being the function of the engineer to estab- lish a compromise. 419. The Ideal Steam Engine. No engine using saturated steam can develop an efficiency greater than that of the Clausius cycle, the attainable temperature limits m present practice being between 100 and 400 Q F., or, for non-condensing engines, between 212 F, and 400 F. The steam engine is inherently a wasteful machine ; the wastes of practice, not thus far considered in dealing with the ideal cycle, are treated with in the succeeding chapter, THE STEAM TABLES 420. Saturated Steam. The table on pages 247, 248 is abridged from Marks' and Davis' Tables and Diagrams (18). In computing these, the absolute zero was taken at 459.64 F. ; the values of h and n w were obtained from the expei i- ments of Barnes and Dietrici (68) on the specific heat of water; the mechanical equivalent of heat was taken at 777.52 ; the pressure-temperature relation as found by Holborn and Henning (Art. 360); the thermal unit is the "mean B. t, u."(se footnote, Art. 23) ; the value of H is as in Art. 388 ; and the specific volumes were computed as in Art. 368. The symbols have the following significance : P = pressure in pounds per square inch, absolute ; T temperature Fahrenheit ; V = volume of one pound, cubic feet ; h = heat in the liquid above 32 P., B. t. u. ; H= total heat above 32 F., B. t. u.; L = heat of vaporization = ZT A, B. t. ti. ; r = disgregation work of vaporization = L e (Art. 359), B. t. u.; n^ = entropy of the liquid at the boiling point, above 32 F* ; n, = entropy of vaporization = ; n, = total entropy of the dry vapor = n -f n+ 421. Superheated Steam. The computations of Art. 417 may suggest the amount of labor involved in solving problems involving superheated steam. This is' largely due to the fact that the specific heat of superheated steam is variable. Figure 177, representing Thomas' experiments, may be employed for calculations which do not include volumes; and volumes may be in some cases dealt with by the Linde formula (Art. 3fl#). The most convenient procedure is to use a table, such as that of Heck (71) T or of Marks and Davis, in the work already referred to. On the following page is an extract from the latter table. The values of naed are the result of a harmonization of the determinations of Knoblauch and Jakob (Art 384) and Holborn and Henning (&9) and other data (70). They differ somewhat from &OSB given in, Fig. 170. The total heat values are obtained by 286 APPLIED THERMODYNAMICS adding the values of k(T-t) over successive short intervals of temperature to the total heat at saturation ; the entropy is computed iu a corresponding manner. The specific volumes are from the Linde formula. PROPERTIES OF SUPERHEATED STEAM hirmiiBAT, *F 40 90 200 300 400 500 600 Absolute Pre&Rnre- Lbfl. per Square Inch 1 ' * = 141.7 V = 357.8 ' # = 1122.6 11)1.7 387.9 1145.3 301.7 4:03-7 1195.6 401.7 513.4 1241.5 501.7 573.1 1287.6 601.7 632.7 1334.1 701.7 692.4 1381.0 n = 2.0060 2.0434 2.1145 2.1701 2.2218 2.2679 2.4100 ' t = 166 1 216.1 326.1 426,1 526.1 626.1 726.1 F= 186.1 201.2 234.2 264.1 293.9 323.8 353.6 9 1 IT =1133.2 1156.1 1206-4 1252.4 1298.6 1345.2 1392.2 n = 1.9486 1.0836 2.0529 2.1071 2.1586 2.2044 2.2459 (t = 280.1 330.1 440.1 540.1 640.1 740.1 840.1 35 #=1179.6 18.61 1203.4 21.32 1255.6 23.77 1302.8 26.20 1350.1 28.61 1397.5 31.01 1445.4 n = 1.7402 1.7712 1.8330 1.8827 1.9277 1.9688 2.0078 ' t = 367.8 417.8 527.8 627.8 727.8 827.8 927.8 100 # = 1208 4 5.07 1234.6 5.80 1289.4 6,44 1337.8 7.07 1385.9 7.69 1434.1 8.31 1482.5 n = 1.6294 1.6600 1.7188 1.7656 1.8079 1.8468 1.8829 ' t = 393.1 443.1 553.1 653.1 733.1 853.1 953.1 140 F=3.44 ' # = 1213.8 3.70 1242.8 4,24 1298.2 4.71 1346.9 5.16 1395,4 5.61 1443.8 6.06 1492.4 n = 1.6031 1.6338 1.6916 1.7376 1.7792 1.8177 1.8533 ' 1 = 398 5 448.5 558.5 658.5 758.5 858.5 958.5 150 1 #=1217.3 3.46 1244.4 3.97 1300.0 4.41 1348.8 4.84 1397.4 5.25 1445.9 5.67 1494.6 n = 1.5978 1.6286 1.6862 1.7320 1.7735 1.8118 1,8474 t = temperature Fahrenheit ; V = specific volume ; H =s total heat above 32 P. ; n = entropy above 32 F. (Condensed from Steam Tables and Diagram, by Marks and Davis, with the per- mission of the publishers, Messrs. Longmans, Green, & Co.) THEORY OF VAPORS 287 PKOPERTIES OF DRY SATURATED STEAM (Condensed from Steam Tables and Dirrr/ntmn, by Marks and Duvis, with the permit sion of the publishers, Messrs Longmans, Green, & Co.) JP T r h L H r "a " n t 1 101.83 333.0 698 1034 1104.4 072.9 1327 1.8427 1.0754 2 126.15 173.5 94.0 1U21.0 1115.0 030.7 0.1740 1.74,31 1.0180 a 141.52 118.5 109.4 1012.3 1121.0 040.4 02008 1.0840 1.8848 4 153.01 90.5 120.9 1003.7 11"J<J.3 038.0 !>108 1.641(5 1.8614 5 162.28 73.33 130.1 1000.3 1130.5 932.4 0.2:J48 1.6084 1.8432 6 170.00 61 80 137.9 905.8 1133.7 027.0 0.2471 1.5814 1,8285 7 170.85 53.50 144.7 991.8 1 ];](>. 5 02.4 0.2570 1.6582 1.8161 8 182.86 47.27 150.8 988.2 1130.0 018 2 0.2073 1.5380 1.8053 9 188.27 42.30 156.2 985,0 1141.1 OH.4 2750 1.&WW 1.7958 10 193.22 38.38 161.1 982.0 1143.1 010.9 2832 1.6042 1.7874 11 197,75 35.10 165.7 970.2 1144.0 007.8 0.2002 1.4805 1.7797 12 201.96 32.36 160.9 9706 1140 5 904.8 0.2007 1.4700 1.7727 18 205.87 30.03 173.8 074.2 1148.0 90:!. 0.3025 1.4080 1.7064 14 209.55 28.02 177.5 971.0 1140.4 81W.3 0.3081 1.4523 1.7604 15 213.0 26.27 181.0 909.7 1150.7 81*0.8 0.3133 1.4410 1.7549 16 216.3 24.79 184.4 007.0 1152.0 804.4 3183 1.4311 1.7494 17 219.4 23.38 187.5 905.6 1163.1 802.1 3220 1.4215 1.7444 18 222.4 22.16 100.5 903.7 1154.2 880.9 0.3273 1.4127 1.7400 19 225.2 21.07 193.4 961.8 1155.2 887.8 3316 1.4045 1.7360 20 228.0 20.08 190.1 960.0 1156.2 885.8 0.3355 1.3005 1.7320 21 230.6 19.18 198.8 958.3 1157.1 883.0 0.3303 1.3887 1.7280 22 233.1 18.37 201.3 950.7 1158.0 882.0 0.3430 1.3811 1.7241 23 235.5 17.62 203.8 0551 115H.8 880.2 0,3405 1.3730 1.7204 24 237.8 16.93 206.1 953.5 1100.0 878.5 O.:)409 1.0070 1.7169 25 240.1 16.30 208.4 952.0 1100.4 870.8 0.8532 1.3004 1.7136 26 242.2 15.72 210.6 950.6 1161.2 876.1 0.3504 1.3642 1.7106 27 244.4 15.18 212.7 949.2 1101.9 873.6 0.3504 1.848$ 1.7077 28 246.4 14.67 214.8 JH7.8 1162.0 872,0 0.3023 1.3425 1.7048 29 248,4 14,19 216.8 946.4 1KS3.2 870.5 o.30:>2 1.8807 1.7019 80 250.3 13.74 218.8 945.1 1103.9 809.0 0.3080 1.8311 1.6991 81 252.2 13.32 220.7 943.8 1164.5 867.0 0.3707 1.3357 1.6KH 82 254.1 12.93 2^22.6 942.5 11(15.1 800.2 0.978$ 1.8315 1.6938 88 255.8 12.57 224.4 941,3 1105.7 804.8 0.3759 1.315-3 1.6V14 84 257.6 12.22 226.2 940.1 netu 803.4 0.3784 1.3107 1.6891 85 259.3 11.89 227.9 938.9 1106,8 8C2.1 0.3808 1.30BO 1.0868 86 261.0 11,58 229.6 937.7 1167.3 860.8 0.&&2 1.3014 1.U846 87 262.6 11.29 28tS 930.6 1167.8 869.5 0.3865 1.2960 1JW34 88 264.2 11.01 232.9 955.5 1108.4 858.3 0.3877 1.2&25 1.0803 89 265.8 10.74 234.5 934.4 1108.9 867.1 0.3SiJ 1.288i 1.6781 40 267.3 10.49 236.1 m$ IMHU &66.0 0.3920 1.2841 1.6761 41 268.7 10.26 237.6 932.2 116&.8 864.7 0.3941 1.2800 1 6741 42 270.2 10,02 *89.i &U.2 1170.8 $5#.6 0.3962 1.2769 1,6721 48 271.7 9.80 240.5 930.2 1170.7 852.4 0.3982 , 1.2720 1.6702 44 273.1 9,69 242.0 929.3 1171/2 51.3 0.4002 1.2081 1.0683 45 274.5 9.39 243.4 028.2 1171.6 860.3 0.4021 1.2644 1.6665 46 275.8 9.20 244.8 927.2 1178.0 649. 2 0.4040 1.2607 1.6647 47 277.2 9.02 246.1 9U H75U 848.1 0.4059 12671 1.6630 48 278.6 8.S4 247.6 926.3 ira.s 847.1 0.4077 1.25S6 1.6613 49 S79.8 8.07 248.8 924.4 117.^ &4<U 0.40% 1.2502 1.6697 50 281.0 8.61 fctfU 02S-6 1175U &45.0 0,411$ 1.2468 1.65&1 i 288 APPLIED THERMODYNAMICS PROPEKTIES OF DRY SATURATED STEAM (Condensed from Steam Tables and Diagrams, by Marks and Davis, with the permis- sion of the publishers, Me&srs. Longmans, Green, & Co.) p T r i h L n r n u e i n s 51 282.3 8.35 i 231 4 ! 922.0 11740 8440 0.4130 1 2435 i 1 6505 52 283.5 8.20 2526 921.7 1174.3 843.1 0.4147 1 2402 1 0549 58 284.7 8.05 253.0 920.8 1174.7 8421 0.4164 1.2370 16534 54 285.9 7.01 5*55.1 911>.0 11750 841.1 0.4180 1.2330 1 6519 55 287.1 7,78 256.3 9U.0 1175.4 840.2 0.4196 1.2309 16505 56 288.2 7.65 257.5 0182 1175.7 839.3 0.4212 1.2278 1.6400 57 289.4 7.02 258.7 917.4 1170.0 838.3 0.4227 1.2248 1 6475 5S 200.5 7.40 250.8 910.5 1176.4 837.4 0.4242 1.2218 16460 59 291.6 7.28 261.0 013.7 1176.7 836.5 4257 1.2189 16446 60 21*2.7 7.17 262 1 9140 1177.0 8356 0.4272 1.2100 1.6432 61 293.8 7.06 203.2 914.1 1177.3 834.8 0.4287 1.2132 1.6419 62 204.1) 6.05 2(54 3 913.3 1177.6 833.0 0.4302 1.2104 1.6406 63 295.9 6.85 205.4 912.5 1177,9 833.1 0.4316 1.2077 1.6393 64 297.0 0.75 2<XU 911.8 1178 2 832.2 0.4330 1.2050 16380 65 2080 6.65 207.5 911.0 1178.5 831.4 0.4344 1.2034 16368 66 290.0 656 268.5 910.2 11788 830.5 0.4368 1.2007 16355 67 300.0 6.47 200.6 000.5 1170.0 829.7 0.4371 1.1972 16343 68 301.0 638 270.6 008.7 1170.3 828.9 0.4385 1.1946 1.6331 69 302.0 6.29 271.0 908.0 11796 828.1 0.4398 1.1921 1.6319 70 302.9 6.20 j 272.6 907.2 1179.8 827.3 0.4411 1.1896 1.6307 71 303.9 6.12 2736 906.5 1180.1 82.5 0.4424 1.1872 1 6296 72 304.8 6.04 274.5 905.8 11804 825.8 04437 1.1848 16285 78 305.8 5,90 275.5 9051 1180.6 825.0 0.4440 1.1825 1.6274 74 30.7 5.89 270.5 904.4 1180.0 824.2 0.44G2 1.1801 16263 75 307.6 581 277.4 903.7 1181.1 823.5 0.4474 1.1778 1.6262 80 i 312.0 5.47 282.0 900.3 1182.3 819.8 0.4535 1.1665 1.6200 85 310.3 516 286. 3 897.1 1183.4 816.3 0.4590 1.1561 1 6151 90 320.3 4.89 290.5 893.9 1184.4 818.0 0.4644 1.1461 1.6105 95 324.1 4.65 294.5 890.9 1185.4 809.7 0.4604 1.1367 16061 100 327.8 4.429 298.3 888.0 1186.3 800.6 0.4743 1.1277 1.6020 105 331.4 4.230 302.0 885.2 1187.2 803.6 04780 1.1191 1.5980 110 334.8 4.047 305.5 8825 1188.0 800.7 04834 1.1108 1.5942 115 338.1 3.880 309.0 879.8 1188.8 797.9 04877 1.1030 15907 120 3413 3.720 312.3 877.2 1189.6 795.2 0.4919 1.0964 1.5873 125 344.4 3.583 315.5 874.7 1190.3 792.6 0.4959 1.0880 1.5839 180 347.4 3.452 318.6 872.3 i 1191.0 790.0 0.4998 1.0809 15807 140 353.1 3.219 324.6 8076 1192.2 785.0 0.5072 10675 15747 150 358.5 3.012 330.2 863.2 1193.4 7804 0.5142 1.0550 15692 160 303.C 2.834 336.6 858.8 1194.5 775.8 0.5208 1.0431 1.5639 170 368.5 2.675 340.7 854.7 1195.4 771.5 0.5269 1.0321 15590 180 373.1 2.533 34-5.6 850.8 1196.4 767.4 0.5328 10215 15543 190 377.6 2,406 350.4 846.9 1197.3 763.4 0.5384 1.0114 16498 200 381.9 2.290 354.9 843.2 1198.1 769.5 0.6437 10019 1.5456 210 386.0 2,187 3592 839.6 1198.8 766.8 0.5488 0.9928 1.6416 220 389.9 2.091 363.4 836.2 1109.6 752.3 0.5538 0.9841 15379 280 393.8 2.004 367.5 832.8 1200,2 748.8 0.6686 0.9758 1.5344 240 397.4 1.924 371.4 829.5 1200.9 7454 0.5633 0.9676 15309 250 401.1 1.850 375.2 82J.3 1201.5 742.0 05676 0.9600 1.6276 THEORY OF VAPORS 289 (1) PhiL Trans., 1851, CXLIV, 360. (2) Phil. T/V/TW., 1854, 330 ; 1862, 579. (3) Theorie Mecanique de la Chaleur, 2d ed., I, 195. (4) Wood, Th*rmoi~?ynrunf t 1905, 390. (5) Wiedemann, Ann. Her Phys. und Chem., 1880, Vol. IX. ff5) Technical Thermodynamics (Klein), 1907, II, 215. (7) Mitteilungcn Wter ForirtuntwirMteu. etc., 21 ; 33. (8) Peabody, Steam Tables, 1908, 9 ; Marks and Davis, Tables awl Diagrams, 1909, 88; Phil. Trans., 199 A (1902), 149-2(33. (0) The AV^wi Buying 1897, 001. (10) Op. rft., II, App. XXX. (11) The EicharOs Strim, Etujhie Indica- tor, by Charles T. Porter. (12) Trans. A. S. .If. E., XL (13) Dubols ed , II. 11, 1H84. (14) Peabody, op. ctt. (15) Trans A S. M. E., XII, 590. (10) Ann for Ffty*rt\ t, 26,1908,833. (17) Trans. A. S. M. E.. XXX, 1419-1432. (18) Tables and &WJMM* of The Thermal Properties of Saturated and Superheated Steam, Itttf. (1'J) Zrttx. fur Instrumentenkunde, XIII, 329, (20) Wissenschnftliche Ahhandlungpii, III, 71. (21) Sitzungsberichte K. A. W. in Wien, Math.-natur *Kla$se, CVII, II, Oct, 1809. (22) Loc. tit., note (7), (24) Comptes Rendus, LXII, 56; Bull, de la Soc. Industr. de Mulhouse, CXXXIII, 129. (25) Boulvin's method: see Berry, The Tempera- ture Entropy Diagram, 1906, 34. (26) Zeuner, op. cd., II, 207-208 (27) Nichols and Franklin, Elements of Physics, I, 194. (28) Phil. Trans., 1869, II, 575. (29) Zeits. Ver. Deutsch Ing., 1904, 24. (30) Trans. A. S. M. E., XXVIII, 8, 1264. (81) Ann. der Phys., Leipzig, 1905, IV, XVIII, 739. (32) Zeits. Ver. Deutsch. Ing., Oct. 19, 1907. (33) MitteiL uber Forschungsarb., XXXVI, 109. (34) Tnttt*. A. fl. M. E., XXVIII, 10, KJ95. (35) Trans. A. S. M. E., XXIX, 0, 033. (30) Hid., XXX, 5, 533. (37) Ibid., XXX, 9, 1227. (38) Op. cit., II, 239. (39) Pea- body, Op. cit., 111. (40) The Steam Engine, 1905, flg. (41) Trantt. A. S. M. E., XXIX, 6. (42) Op. tit., II, Apps. XXXIV, XXXV, XL, XLIV, XLU, XXXVIII. (43) Op. cit., 407 et. seq. (44) Qp. c#.,600. (45) Cvmpte* Rrwlu*, Oil, 1886, 1202. (40) Ibid., CXIV, 1892, 1093 ; CXIII, 1891. (47) Zetts.JVr die gesamte Katie-Industrie, 1895, 66-85. (48) Op. cit., II, App. L. (49) JfcirAjn** a froid, Paris, 1878. (60) Elleau and Ennis, Jour. Frank. Inst., Mar., Apr., 1K98 ; Dietrici, Zeits. Kalte-Ind., 1904. (51) Op. cit., II, App. XL VI. (52) Zrit*.fur die gesamte Kalte-Industrie, 1904. The heavy line across the table on pa^e 422 indicates a break in continuity between the two sources of data. The same break is resj>onsible for the notable irregularity in the saturation and constant dry ness carves on the ammonia entropy diagram, Fig. 316. (53) Tables of the Properties of Saturated Steam and other Vapors, 1890. (54) See Jacobus, Trans. A. S. M. E., XH. (55) Jour. Fmnl\ /**., Dec., 1890. (56) Op. cit., 466. (57) Mem. de rinstttiit de France, XXI, XXVI. (58) Landolt and Bdrnstein, Physitetbache-chemische Tabfllen; Gmeliii ; Peabody, Thermodynamics, 118. (59) Andreeff, Ann. Chem. Phartn., 1859. (tiO) Trans. A. $. M. E., XXV, 176. (61) Tables, etc., 1890. (02) Comptes Sendw, CXIX, 1894, 404-407. (63) Op. tit-, App. XLVIII. (04) Op. cit., 48. (to) Trans. A. 8. M. E. t XEL (66) Op. cit., IE, App. XXXII. (07) Trans. A. S. 3f. E.. XXI, 3, 406. (08) WieA. Annallen,, (4), XVI, 1905, 5S)3-620. (09) WM. AnnaUen, (4), XVIII, 1905, 73&~756; (4), XXIII, 1907, 809-845. (70) Marks and Davis, op. cit., 5. (71) Trans. A. S. M. E., May, 1908. SYNOPSIS OP CHAPTER XII The temperature remains constant during evaporation j that of the liquid is the same as that of the vapor; increase of pressure raises the toiling point, and wre wr<i; it also increases tM density. There is a definite boiling point for each pr**ure. Saturated vapor is vapor at minimum temperature and maximum density for the given pressure. Superheated vapor is *n imperfect gas, produced by adding heat to a dry saturated vapor. 290 APPLIED THERMODYNAMICS Saturated Steam FC W V") The principal effects of heat are, h = t 32, e = s ^ p ^ ( to Asp increases, i, h, e and H increase, and r and X decrease. &= Mm + 0.3745(2 -212) -0.00065 ($- o/ evaporation, = X + ~ ^' The pressure increases more rapidly than the temperature. Characteristic equation for steam, JH? = o!T j)(l + Z>p) rjg Saturated steam may he dry or wef . 3Tor -wet steam, and the /actor of evaporation is t 7 - Tne volume is TF=F+a:(TFo- 7). The zoa^r Zine shows the volume of water at various temperatures j the saturation curve shows the relation between volume and temperature of saturated steam. Approxi- mately, pv$ = constant. The isothermal is a line of constant pressure. The path during evaporation is (a) along the water line (&) across to the saturation curve at constant pressure and temperature. If superheating occurs, the path pro- ceeds at constant pressure and increasing temperature to the right of the satura- tion curve. T On the entropy diagram, the equation of the water line is n = clog, . The distance between the water line and the saturation curve is JV r =^- Constant dry ness curves divide this distance in equal proportions. Lines of constant total heat may be drawn. The specific heat of steam kept dry is negative. The dryness changes during adiabatic expansion. The temperature of inversion is that temperature at Trtiich the specific heat of dry steam is zero. The change of internal energy and the external work along any path of saturated steam may be represented on the entropy diagram. W= F| . Constant volume lines may be plotted on the entropy diagram, permitting of the trans- fer of any point or path from the PFto the T2? plane. The temperature after expansion at oontant entropy to a limiting volume can best be obtained from the entropy diagram, The critical temperature is that temperature at which the latent heat becomes zero (68SP F.}. Saturated vapor (dry or wet), superheated vapor, gas ; physical states in relation to the critical temperature ; shape of isothermals. The i&odynamic path for saturated steam touches the saturation curve at one point only. SYNOPSIS 291 Sublimation occurs if the saturation pressure at the melting temperature exceed* that of the surrounding medium. Gax and Tapor ^fixtures Value of E for gas mixtures : mixture of air and steam ; absolute and relative humidities ; wet and diy bulb thermometers ; in mixtures, mixing does not affect the internal energy and adiabatic expansion, ih without influence on the aggregate entropy. Mixture and expansion of (a) wet vapor and jras, (6; hi^h-pressure steam and air, (c) superheated steam and air, (d\ two vapors. Equivalent values of n. In the heat engine, mixtures may lower the temperature of heat rejection* Superheated Steam The specific heat has been in doubt. Its value increases with the pressure, and varies with the temperature. j5T=jff, + %,(:r-o. r = ^ rL r- -Hi-J3i=-*C^-afi) + ^(2l-r.). Kpi T t Factor of evaporation = Saa + *?' f) ~ h PF= 0.64901 T- 22.5819 J- y iU.4 PF= 0.694 T - 0.00178 P. J? = 85.8. y = 1.298. Paths of Vapors Adiabatic equation : = doge + - Approximately, PF n =constank Values of n. t t T External work along an adiabatic = h Continuously superheated adiabattc, e Adtabatfc crossing the saturation curve : m Method of drawing constant pressure lines on the entropy diagram : n = Aplog. -- Method of drawing lines of constant total heat. Use of the entropy diagram for graphically solving problems: dryness after expansion j work done during expansion ; mixing ; heat contents. The Mollier coordinates, total heat and entropy. The total heat^pressvre diagrams. Vapors in General &--* *--* ' When the pressure-temperature relation and the characteristic equation are given, -we may compute L for various temperatures, and the specific heat of the vapor. *=0.608, 292 APPLIED THERMODYNAMICS vapor density =0.597 (air = l), specific volume of lic][uid= 0.025, its specific heat = 1.02. Sulphur dioxide: =0.15438, vapor density = 2. 23, specific volume of liquid = 0.0007, its specific heat = 0.4. PY = 26.4T-1S4P 23 . Pressure-tem- perature relation. L = 176- 0.27(2-32). Engine capacity and economy is influenced by the vapor employed. Steam Cycles Efficiency = work done -r gross heat absorbed. The Carnot cycle is impracticable , the steam power plant operates in the Clausius cycle. Efficiency of Glauslus cycle - J. Sankine cycle (incomplete expansion) determination of efficiency, with steam initially wet or dry. tfbn-opanszoe cycle: efficiency = (fr-lX 3 **- - 017 ). 1483 log. 0.695(r-0 Pambour cycle : steam dry during expansion ; efficiency = - computation of heat supplied by jacket. Superheated cycle : efficiency is increased if the final dryness is properly adjusted and the ratio of expansion is not too low. Numerical comparison of seventeen cycles for efficiency and capacity : steam should be initially dry. The ratio of expansion should be large for efficiency and small for capacity. The Steam Tables Computation is from p (or ) to t (or j>), H, h, L, 3?, F, e, r, n u , ^ n,. at The superheated tables give /*, F, H, f, for various superheats at various pressures ; all values depending on H^t, w, and kp. PROBLEMS NOTE. Problems not marked T are to be solved without the use of the steam table. In all cases where possible, computed results should be checked step by step with those read from the three charts, Figs. 175, 177, 185. Tl. The weight per cubic foot of water at 32 P. being 62.42, and at 250.3 F , 58.84, compute in heat units the external work done in heating one pound of water at pressure from 32 to 250.3. (The pressure is that of saturated steam at a tempeiature of 250,3.) (J.ns., 0.0055 B. t. u.) T la. 10 Ib. of water at 212 are mixed with 20 Ib. at 170.06. What is the total heat per pound, above 32 F., of the resulting mixture? 2. Forp^lOO, =327.8, FW.429, compute h (approximately), fl", X, e, r, in the order given. Why do not the results agree with those in the table? ., ^=295.8, J5T=1186.3, ^890.5, e = 81.7, r=SOS.8.) PROBLEMS T 2a. Water at 90 F. is fed to a boiler in which the pressure is 105 Ib. per sq. in. absolute. How much heat must be supplied to evaporate one pound ? T 3. Find the factor of evaporation for dry steam at 95 Ib. pressure, the feed- water temperature being 153 F. (Ans^ 1.097.) 273*^ 396945 T 4. Given the formula, log p = c ^ ^j-, T being the absolute tempera- ture and p the pressure per square foot, find the value of ~ f or p = 100 Ib, per square inch, t = 327.8 F. Check roughly by observing nearest differences in the steam table. T 5. What increase in steam pressure accompanies an increase in temperature from 353.1 F. to 393.8 F? Compare the percentages of increase of absolute pressure and absolute temperature. T 6. Find the values of the constants in the KanMne and Zeuner equations (Art. 363), at 100 Ib. pressure. T 7. From Art. 363, find the volume of dry steam at 240.1 F. hi four ways. Compare with the value given in the steam table and explain the disagreement. 8. At 100 Ib. pressure, the latent heat per pound is 888.0 j per cubic foot, it is 200.3. Find the specific volume. (Ans., 4.433.) 9. For the conditions given in Problem 2, W being the volume of dry steam, find the five required thermal properties of steam 95 per cent dry. Find its volume. T 9a. How much heat is consumed in evaporating 20 Ib. of water at 90 F. into steam 96 per cent dry at 100 Ib. absolute pressure per sq. in. ? T 96. What is the volume occupied by the mixture produced in Problem 9a ? T 9c. Five pounds of a mixture of steam and water at 200 Ib. pressure have a volume of 3 cu. ft. How much heat must be added to increase the volume to 6 cu. ft. at the same pressure ? T 9d. A boiler contains 2000 Ib. of water and 130 Ib. of dry steam, at 100 Ib. presssure. What is the temperature ? What are the cubic contents of the boiler ? T 9e. Water amounting to 100 Ib. per min. is to be heated from 65 to 200 by passing through a coil surrounded by steam 90 per cent dry, kept at 100 Ib. pressure. What is the TninimiTm weight of steam required per hour ? T 9f. Water amounting to 100 Ib. per min. is to be heated from 55 to 200 by blowing into it a jet of steam at 100 Ib. pressure, 90 per cent dry. What is the minimum weight of steam required per hour f T10. State the condition of steam (wet, dry, or superheated) when (a)p=100, <=327.8; (&)p=95, 0=4.0; (c) jp= 80, 2=360. II. Determine the path on the entropy diagram for heating from 200 to 240 F. a fluid the specific heat of which is LOOfoft, in which t is the Fahrenheit temperature and a =0.0044. T 12. Find the increases in entropy during evaporation to dry steam at the f o 1 - lowing temperatures : 228% 261, 386 F. T 13. Compute from Art. 368 the specific volume of dry steam at 327.8 F. What is its volume if 4 per cent wet 1 (See Problem 4.) Tl3a. Steam at 100 Ibs. pressure 2 per cent wet, is blown into a tank having a capacity of 175 cu. ft. The weight of steam condensed in the tank, after the flow is discontinued, is 60 Ib. What weight of steam was condensed during admission ? 294 APPLIED THERMODYNAMICS T 14. Find the entropy, measured from 32 F., of steam at 327.8 F., 65 per cent dry, (a) by direct computation, (5; from the steam table. Explain any discrepancy, T15. Dry steam at 100 IK pressure is compressed without change of internal energy until its pressure is 200 IK rind its dryness after compression. T 16. Find the diyness of steam at 300 F. if the total heat is 800 B. t. u. T ita. One pound of steam at 200 Ib. pleasure occupies 1 cu. ft. "What per cent of moisture is present in the steitm ? T 17. Pind the entropy of steam at 130 Ib. pressure -when the total heat is 840 B. t. u. T 18. One pound of steam at 327.8 E., having a total heat of 800 B. t. u., expands adiabatically to 1 Ib. pressure, rind its diyness, entiupy, and total heat after expan- sion. What weight of steam wab condensed during expansion ? 18 a. Three pounds of water at 760 absolute expand adiabatically to 660 absolute. What weight of steam is pretext at the end of expansion ? (Use Pig. 175.) 19. Transfer a wet steam adiabatic from the TJUfto the PV plane, by the graphi- cal method. 20. Transfer a constant dryness line in the same manner. 21. Sketch on the T^anrl PV planes the saturation curve and the water line in the region of the critical temperature. T22, At what stage of dryness, at 300 F., is the internal energy of steam equal to that of dry steam at 228 F. ? T23. At what specific volume, at 300 F., is the internal energy of steam equal to that of dry steam at 228 F? T 23 a. A boiler contains 4000 Ib. of water and 400 Ib. of steam, at 200 Ib. absolute pressure. If the boiler should explode, its contents cooling to 60 F. and completely liquefying, in 1 sec., how much energy would be liberated ? What horse power would be developed during the second following the explosion ? 724. Compute from the Thomas experiments the total heat in steam at 100 Ib. pressure and 440 F. T 25. Find the factor of evaporation for steam at 100 Ib. pressure and 500 F. from feed water at 153 F. T26. In Problem 18, find the volume after expansion, and compare with the vol- ume that would have been obtained by the use of Zeuner's exponent (Art. 394). Which result is to be preferred? T 27. Using the Knoblauch and Jakob values for the specific heat, and determin- ing the initial properties in at least five steps, compute the initial entropy and total heat and the condition of steam after adiabatic expansion from P=100, T=7QQ F. to p = 13. Find its volume from the formula in Art. 390. Compare with the volume given by the equation PV 1 aWw^oi^w. (Assume that the superheated table shows the steam to be superheated about 55 F. at the end of expansion.) T27a. Steam at 100 Ib. pressure, 95 per cent dry, passes through a superheater in which its temperature increases to 450 F. Find the heat added per Ib. and the increase of volume, T2S. Compute the dryness of steam after adiabatic expansion from P=140 r T 753.1 F, t to t = 153 F. -Find the change in volume during expansion. ^29. Find the external work done in Problems 27 and 28, along the expansive paths. PROBLEMS 295 T29a. Three pounds of steam, initially dry, expand adiabatically from 100 Ibs. to 1 Ib. pressure. Find the initial and final volumes and the external work done. T 30. At what temperature is the total heat in steam at 100 Ib. pressure 1200 B. t. u. ? 31. Find the efficiency of the Carnot cycle between 341.3 F. and 101 83 F. T 32. Find the efficiency of the Clausius cycle, using initially dry steam between the same temperature limits. T 33. In Problem 32, find the efficiency if the steam is initially 60 per cent dry. T 34. In Problem 32, find the efficiency if expansion terminates when the volume is 12 cu. ft. (Rankine cycle). T 35. In Problem 32, find the efficiency if there is no expansion. T36. Find the efficiency of the Pambour cycle between the temperature limits given in Problem 31. How much heat is supplied by the jacket ? T 37. Find the efficiency of this Pambour cycle if expansion terminates when the volume is 12 cu. ft. T 38. Steam initially at 140 Ib. pressure and 443.1 F. is worked (a) in the Clau- sius cycle, (5) in the Rankine cycle, with the same ratio of expansion as in Problem 37. Find the efficiency in each case, the lower temperature being 101.83 F. Find the efficiency of the Rankine cycle in which the maximum volume is 5 cu. ft. (See foot- nqte, Case VIII, Art. 417.) T 39. At what per cent of dryness is the volume of steam at 100 Ib. pressure 3 cu. ft. ? 7*40. Steam at 100 Ib. pressure is superheated so that adiabatic expansion to 261 F. will make it just dry. Find its condition if adiabatic expansion is then carried on to 213 F. Find the external work done during the whole expansion, T 41. Steam passes adiabatically through an orifice, the pressure falling from 140 to 100 Ib. When the inlet temperature of the steam is 500 F. 7 its outlet temperature is 494 F. ; and when the inlet temperature is 000 F., the outlet temperature is 505 F. The mean value of the specific heat at 140 Ib. pressure between 600 F. and 600 F. is 0.498. Find the mean value at 100 Ib. pressure between 505* F. and 404 F. How does this value agree with that found by Knoblauch and Jacob ? T 42. Find from Problem 41 and Fig. 171 the total beat in saturated steam at 140 Ib. pressure, in two ways, that at 100 Ib. pressure being 1186 3. T 43. Plot on a total heat-pressure diagram the saturation curve, the constant dryness curve for x = 0.8$, the constant temperature curve for T= 500 F^ and a constant volume curve for V = 13, passing through both the wet and the superheated regions. Use a vertical pressure scale of 1 in. = 20 Ib., and a horizontal heat scale of 1 in. = 20 B. t. n, 44- Compute the temperature of inversion of ammonia, given the equation, L = 666.6 - 0,613 T F M the specific heat of the liquid being 1.0, What is the result if L = 656.5 - 0.01S r- 0.00021& f* (Art 401) t 45. Compute the pressure of the saturated vapor of sulphur dioxide at 60 F (ArL 404). (Compare Table, page 424,) T 48*. Compare the capacities of the cycles in Problems 81-37, as in Art. 418. 47. Sketch the water line, the saturation curve, an adiabatic lor saturated, steam, and a constant dryness line on the PT plane. 296 APPLIED THERMODYNAMICS 7 T 48. A 10-gal. vessel contains 0.1 Ib. of water and 0.7 Ib. of dry steam. What is the pressure ? T 49. A cylinder contains 0.25 Ib. of wet steam at 58 Ib. pressure, the volume of the cylinder being 1.3 cu. ft. What is the quality of the steam ? T 50. What is the internal energy of the substance in the cylinder in Problem 49 ? T&I. Steam at 140 Ib. pressure, superheated 400 F., expands adiabatically until its pressure is 5 Ib. Find its final quality and the ratio of expansion. T 52. The same steam expands adiabatically until its dryness is 98. Find its pressure. T 53. * The same steam expands adiabatically until its specific volume is 50. Find its pressure and quality. T 54. Steam at 200 Ib. pressure, 94 per cent dry, is throttled as in Art. 387. At what pressure must the throttle valve be set to discharge dry saturated steam ? T 55. Steam is throttled from 200 Ib. pressure to 15 Ib. pressure, its temperature becoming 235.5 F. What was its initial quality ? (Use Fig. 175.) 56. Represent on the entropy diagram the factor of evaporation of superheated steam. 57. Check by accurate computations all the values given in the saturated steam table for t = 180 F., using 459.64 F. for the absolute zero, 14.696 Ib. per square inch for the standard atmosphere, 777.52 for the mechanical equivalent of heat, and 0.017 as the specific volume of water. Use Thiesen's formula for the pressure : (t 4- 459.6) log ~L- = 5.409 (- 212)- 8.71 x 10-w[(689- O 4 - 477*]; t being the Fahrenheit temperature and p the pressure in pounds per square inch. Use the Knoblauch, Linde and Klebe formula for the volume and the Davis formula for the total heat. Compute the entropy and beat of the liquid in eight steps, using the following values for the specific heat of the liquid : at 40, 1.0045; at 120, 0.9974 ; at 60, 0.9991; at 140, 0.9987 ; at 80, 0. 997 ; at 160, 1.0002 ; at 100, 0.99675 ; at 180, 1.002e. Explain the reasons for any discrepancies. * This is typical of a class of problems the solution of which is difficult or impos- sible without plotting the properties on charts like those of Figs. 175, 177, 185. Prob- lem 53 may be solved by a careful inspection of the total heat-pressure and Mollier diagrams, with reasonable accuracy. The approximate analytical solution will be found an interesting exercise. We have no direct formula for relation between V and T, although one may be derived by combining the equations of Bankine or Zeuner (Art. 363) with that in Problem 4. The following expression is reasonably accurate between 200 and 400 F., where a is in cu. ft. per Ib. and t is the Fahrenheit temperature : (0.005 1 +0.505) 8 0**=477. For temperatures between 200 and 260 F., an approximate equation is PROBLEMS 297 T58. Check the properties given m the superheated steam table for P^ 25 with 200 of superheat, UMIU; Knoblauch values for the specific heat, in at least three steps, and using the Knoblauch, Lmde and Klebe formula for the volume. Explain any discrepancies. 59. Represent on the entropy diagram the temperature of inversion of a dry vapor. 60. Sketch the Molher Diagram (Art, 399) from T=0 to JBT=r400, n = to 7i = 0.5. CHAPTER THE STEAM ENGINE PBACTICAL MODIFICATIONS OF THE RANKINE 422. The Steam Engine. Figure 186 shows the working parts. The piston P moves in the cylinder A, communicating its motion through the piston rod R, crosshead (7, and connecting rod M to the disk crank D on the shaft S, and thus to the belt wheel W. The guides on which the crosshead moves are indicated by 6r, -H", the frame which supports the working parts by J. Journal bearings at B and support the shaft. The function of the mechanism is to transform the to-and-fro rectilinear motion of the piston to a rotatory movement at the crank. Without entering into details at this point, it may be noted that the valve V, which alternately admits of the passage of steam through either of the ports JT, Y", is actuated by a valve rod I traveling from a rocker J", which derives its motion from the eccentric rod N and the eccentric E. In the end view, L is the opening for the admission of steam to the steam chest JI", Q is a sim- ilar opening for the exit of the steam (shown also in the plan), and the valve. 423. The Cycle. With the piston in the position shown, and moving to the left, steam is passing from the steam chest through Y into the cylinder, while another mass of steam, which has expended its energy, is passing from the other side of the piston through the port JTand the opening Q to the atmosphere or the condenser. When the piston shall have reached its extreme left-hand position, the valve will have moved to the right, the port Y will have been cut off from communication with 2> and the steam on the right of the piston will be passing through Yto Q. At the same time the port X will be cut off from Q and placed in communication with -E The piston then makes a stroke to the right, while the valve moves to the left. The engine shown is thus 298 THE STEAM ENGINE 299 300 APPLIED THERMODYNAMICS If the valve moved instantaneously from one position to the other precisely at the end of the stroke, the PV diagram representing the changes in the fluid on either side of the piston would resemble efcd, Fig. 184. Along eb, the steam \vould be passing from the steam chest to the cylinder, the pressure being practically constant because of the comparatively enormous storage space in the boiler, while the piston moved outward, doing work. At 5, the supply of steam would cease, while communication would be immediately opened with the atmosphere or the condenser, causing the fall of pressure along It. The piston would then make its return stroke, the steam passing out of the cylinder at practically constant pressure along id, and at d the position of the valve would again be changed, closing the exhaust and opening the supply and giving the instan- taneous rise of pressure indicated by de. 424. Expansion. This has been shown to be an inefficient cycle (Art. 41 7j, and it would be impossible, for mechanical reasons, to more than approximate it in practice. The inlet port is nearly always closed prior to the end of the stroke, producing such a diagram as debgq, Fig. 184, in _B which the supply of steam to the cylin- der is less than the whole volume of the piston displacement, and the work -area under bg is obtained without the supply of _ v heat, but solely in FIG. 187. Arts. 424, 42o, 427, 430, 431, 436, 441, 445, 446, consequence of the 448, 449, 450, 451,452, 454. Indicator Diagram and . ,. r RanJdnc Cycle. expansive action of the steam. Appar- ently, then, the actual steam engine cycle is that of RanMne * (Art. 411) . But if we apply an indicator (Art. 484) to the cylinder, an instru- * It need scarcely be said that the association of the steam engine indicator dia- gram and its varying quantity of steam with the ideal Bankine cycle is open to objection (Art. 454). Yet there are advantages on the ground of simplicity in this method of approaching the subject. WIREDRAWING 301 ment for graphically recording the changes of pressure and volume during the stroke of the piston, we obtain some such diagram as abodes, Fig. 187, which may be instructively compared with the cor- responding Rankine cycle, ABGDE. The remaining study of the steam engine deals principally with the reasons for the differences between these two cycles. 425. Wiredrawing. The first difference to be considered is that along the lines 6, AB. An important reason for the difference in volumes at ft and B will be discussed (Art. 430) ; we may at present note that the pressures at a and b aie less than those at A and B, and that the pressure at b is less than that at a. This is due to the frictional resistance of steam pipes, valves, and ports', which caufes the steam to enter the cylinder at a pressure somewhat less than that in the boiler ; and produces a further drop of pressure while the steam enters. The action of the steam in thus expanding with considerable velocity through constricted pas- sages is described as "wiredrawing." The average pressure along ab will not exceed 0.9 of the boiler pressure; It may be much less than this. A loss of \voik area ensues. The greater part of the loss of pressure occurs in the ports and pas- sages of the cylinder and steam chest. The friction of a suitably designed &team pipe is small. The pressure-drop due to wiredrawing or "throttling," as it is sometimes called, is greatly aggravated when the steam is initially wet; Clark found that it might be even tripled. Wet steam may be produced as a result of priming or frothing in the boiler, or of condensation in the steam pipes. Its evil effect in this as in other respects is to be prevented by the use of a steam separator near the engine; this automatically separates the steam and entrained moisture, and the water is then trapped away. 426. Thermodynamics of Throttling. Wiredrawing is a non~rever$- ible process, in that expansion proceeds, not against a sensibly equivalent external pressure, but against a lower and comparatively non-resistant pressure. If the operation be conducted with sufficient rapidity, and if the resisting pressure be negligible, the external work done should be zero, and the initial heat contents should be equal to the final heat contents; i.e., the steam expands adiabatically (though not isentropic- ally) along a line of constant total heat like nir, Fig. 161. The steam is thus dried by throttling; but since the temperature has been reduced, the heat has lost availability. Figure 188 represents the case in which the steam remains superheated throughout the throttling process. A is the initial state, DA aixd EC Enee of constant pressure, AB an adiabatic, A.F a line of constant total heat, and C the final state. The areas SHJDAG and SHECK, and, consequently, the areas JDABEH and GBCK, are equal; the temperature at C is less than that at A. (See the superheated steam tables : at p~140 ; H = 1298.2 when -553.1 F.; 302 APPLIED THERMODYNAMICS at p-100, H = 1298.2 when t is about 548 F.) The effect of wire- drawing is generally to lower the temperature, while leaving the total quantity of heat unchanged. FIG. 188. Art. 426. Throttling of Superheated Steam. inn FIG. 189. Arts. 426, 445, 453. Converted Indicator Dia- gram and Rankine Cycle. 427. Regulation by Throttling. On some of the cheaper types of steam engine, the speed is controlled by varying the extent of opening of the admis- sion pipe, thus producing a wiredrawing effect throughout the stroke. It is obvious that such a method of regulation cannot be other than wasteful; a better method is, as in good practice, to vary the point of cut-off, &, Fig. 187. (See Art. 507.) 428. Expansion Curve. The widest divergence between the theo- retical and actual diagrams appears along the expansion lines 6c, BC, Fig. 1ST. In neither shape nor position do the two lines coincide. Early progress in the development of the steam engine resulted in the separation of the three elements, boiler, cylinder, and condenser. In spite of 'this separation, the cylinder remains, to a certain extent, a condenser as well as a boiler, alternately condensing and evaporating large proportions of the steam supplied, and producing erratic effects not only along the expansion line, but at other portions of the diagram as well. 429. Importance of Cylinder Condensation. The theoretical analysis of the Ran- kine cycle (Art. 411) gives efficiencies considerably greater than those actually attained in practice. The principal reason for this was pointed out by Clark's experiments on locomotives in 1855 (1); and still more comprehensively by Isherwood, in his classic series of engine trials made on a vessel of the United States Navy (2). The further studies of Loring and Emery and of Ledoux (3), and, most of all, those conducted under the direction of Him (4), served to point out the vital importance of the question of heat transfers within the cylinder. Recent accurate measure- ments of the fluctuations in temperature of the cylinder walls by Hall, Callendar and Nicholson (5) and at the Massachusetts Institute of Technology (6) have furnished quantitative data. CYLINDER CONDENSATION 303 430. Initial Condensation. When hot steam enters the cylinder at or near the beginning of the stroke, it meets the relatively cold surface of the piston and cylinder head, and partial liquefaction immediately occurs. By the time the point of cut-off is reached the steam may contain from 25 to 70 per cent of water. The actual weight of steam supplied by the boiler is, therefore, not determined by the volume at b, Fig. 1ST; it is practically from 33 to 233 per cent greater than the amount thus determined. If ABCDE, Fig. 1ST, represents the ideal cycle, then b will be found at a point where V b =from 0.30 V B to 0.75 V B (Art. 436). Behavior during Expansion. The admission valve closes at 6, and- the steam is permitted to expand. Condensation may continue for a time, the chilling wall surface increasing ; but as expansion pro- ceeds the pressure of the steam falls until its temperature becomes less than that of the cylinder walls, when an opposite transfer of heat begins. The walls now give up heat to the steam, drying it, i.e., evaporating a portion of the commingled water. The behavior is complicated, how- ever, by the liquefaction which necessarily accompanies expansion, even if adiabatic (Art. 372). The reevaporation of the water during expansion is effected by a withdrawal of heat from the walls; these are consequently cooled, resulting in the resumption of proper conditions for a repetition of the whole destructive process during the next succeed- ing stroke. Reevaporation is an absorption of heat by the fluid. For maximum efficiency, all heat should be absorbed at maximum tempera- ture, as in the Camot cycle. The later in the stroke that reevaporation occurs, the lower is the temperature of reabsorption of this heat, and the greater is the loss of efficiency. Data on Condensation. Even if the cylinder walls were per- fectly insulated from the atmosphere, these internal transfers would take place. The Callendar and Nicholson experiments showed that the temperature of the ianer surface of the cylinder walls followed the fluctuations of steam temperature, but that the former changes were much less extreme and lagged behind in point of time. Clayton has demonstrated (7) that the expansion curve may be represented (in non-condensing ttnjacketed cylinders) by the equation * constant, n*0.&c- 0.465, where x is the proportion, of dryness at cut-off: the value of n being independent of the initial pressure or ratio of expansion. The initial 304 APPLIED THERMODYNAMICS wetness is thus the important factor in determining the rate of reevapora- tion during expansion. With steam very dry at cut-off (due to jacket- ing or superheat) heat may be lost throughout expansion. In ordinary cases, the condensation which may occur after cut-off, during the early part of expansion, can continue for a very brief period only: the prob- ability is that in most instances such apparent condensation has been in reality nothing but leakage (Art. 452), and that condensation prac- tically ends at cut-off. 432. Continuity of Action. When unity of weight of steam condenses, it gives up the latent heat L] when afterward reevaporated, it reabsorbs the latent heat Li; meanwhile, it has cooled, losing the heat h hi. The net result is an increase of heat in the walls of L-Li+h-h^H-Hi, and the walls would continually become hotter, were it not for the fact that heat is being lost by radiation to the external atmosphere and that more water is reevaporated than was initially condensed; so much more, in fact, that the dryness at the end of expansion zs usually greater than it would have been, had expansion been adiabatic, from the same condition of initial The outer portion of the cylinder walls remains at practically uniform tem- perature, steadily and irreversibly losing heat to the atmosphere. The inner portion has been experimentally shown to fluctuate in temperature in accordance with the changes of temperature of the steam in contact with it. The depth of this " peri- odic " portion is small, and decreases as the time of contact during the cycle decreases, e.g., in high speed engines* 433. Influences Affecting Condensation. Four main factors are related to the phenomena of cylinder condensation: they are (a) the temperature range y (6) the size of the engine, (c) its speed and (most important), (d) the ratio of volumes during expansion. Of extreme importance, as affecting condensation during expansion, is the condi- tion of the steam at the beginning of expansion. The greater the range of pressures (and temperatures) in the engine, the more marked are the alternations in temperature of the walls, and the greater is the dif- ference in temperature between steam and walls at the moment when steam is admitted to the cylinder. A wide range of working temperatures, although practi- cally as well as theoretically desirable, has thus the disadvantage of lending itself to excessive losses. 434. Speed. At infinite speed, there would be no time for the transfer of heat, however great the difference of temperature. Willans has shown the percentage of water present at cut-off to decrease from 20.2 to 5.0 as the speed increased from 122 to 401 r. p. m., the steam consumption per Ihp-hr. concurrently decreasing from 27.0 to 24.2 Ib. (8). In another test by Willans, the speed ranged from 131 to 405 r. p. m., the moisture at cut-off from 29.7 to 11.7, and the steam consumption from 23.7 to 20 3; and in stifl another, the three sets of figures were 116 to 401, 20.9 to 8.9, and 20.0 to 17.3. In all cases, for the type of engine under <5onsideca- EXPANSION AND CYLINDER CONDENSATION 305 tion, increase of speed decreased the proportion of moisture and increased the economy: but it should not be inferred from this that high speeds arc necessarily or generally associated with highest efficiency. 435. Size. The volume of a cylinder is -sD*L+4 and its exposed wall surface is (3cZ)L)-h(xD 2 -^2), if D denotes the diameter and L the exposed length. Tie volume increases more rapidly than the wall surface, as the diameter is increased for a constant length. Since the lengths of cylinders never exceed a certain hn.it, it may be said, generally, that small engines show greater amounts of condensation, and lower efficiencies, than large engines. 436. Ratio of Expansion. This may be defined as Fd-*-7, Fig. 187 (Art. 450). The greater the ratio of expansion, the greater is the initial condensation. This would be true even if expansion were adiabatic; with early cut-off, moreover, the time during which the metal is exposed to high temperature steam is reduced, and its mean temperature is consequently less. Its activity as an agent for cooling the steam during expansion is thus increased. Again, the volume of steam during admission is more reduced by early cut-off than is the exposed cooling; surface, since the latter includes the two constant quantities, the surfaces of the piston and of the cy Under head (clearance ignored Art. 450). The following Bhows the results of several experiments: OBSERVERS BATIO or PEE CENT. OF WATEB I-TEA.W CuNMrMrrmv. EXPANSION AT Cl T-OFF Poi"KI*s PEC laV-IIB L&ie High Low High Zo/r ! ///i/1 Loring and Emery Willans (9) 4.2 4.0 16.8 8.0 8.9 25.0 21.2 ! 5.1 20.7 ' 2JU Barrus (10) gives the following as average results from a large number of of Corliss engines at normal speed : CUT-OFT, PBB CEMTT. OF STBOKB PBBOEWTAGE or GOXWBXSATCOX CtTT*>rr T Fm* CENT. OP STUCK* Pl-nCETTAl/E >F ( COMPSSPITIOH ' 2.5 62 25.0 24 5.0 54 SO.O 20 10.0 44 40.0 16 : 15.0 36 45.0 15 20.0 28 j In these three sets of experiments, it was found that the propor- tion of water steadily decreased as the ratio of expansion decreased. The steam consumption, however, decreased to a certain mfriininm figure, and then increased (a feature not shown by the tabulation) see Fig. 189a. The beneficial effect of a decrease in condensation 306 APPLIED THERMODYNAMICS was here, as in general practice, offset at a certain stage by the thenno- dynamic loss due to relatively incomplete expansion, discussed in Art. 418. The proper balancing of these two factors, to secure best efficiency, is the problem of the engine designer. It must be solved by recourse to theory, experiment, and the study of standard practice. In American stationary engines, the ratio of expansion in simple cylinders is usually from 4 to 5. RATIO OF EXPANSION FIG. I89a. Art 436. Effect of Ratio of Expansion on Initial Conden- sation and Efficiency. 437. Quantitative Effect. Empirical formulas for cylinder condensation have been presented by Marks and Heck, among others. Marks (11) gives a curve of condensation, showing the proportion of steam condensed for various ratios of expansion, all other factors being eliminated. A more satisfactory relation is established by Heck (12), whose formula for non-jacketed engines is 0.27 in which M is the proportion of steam condensed at cut-off, N is the speed of the engine (r. p. m.)> is the quotient of the exposed surface of the cylinder in square feet by its volume in cubic feet 12 /2Z) -- +4 ) where D and L are in inches, p is the D\L TABLE: VALUES FOR T Pa Const. Po Const. PO Const. Po Const. 170 45 262 115 348 185 409 1 175 50 269} 120 353 190 413 2 179 55 277 125 358 195 416} 3 183 60 284 130 362} 200 420 4 186 65 291 135 367 210 427 6 191 70 297} 140 371} 220 434 8 196 75 304 145 376 230 441 10 200 80 310 150 380} 240 447} 15 210 85 316 155 385 250 454 20 220 90 321} 160 389 260 460} 25 229 95 327 165 393 270 467 30 238 100 332} 170 397 280 473 35 246 105 338 175 401 290 479 40 254 110 343 180 405 300 485 (T in the formula is equal to the difference in constants corresponding with the highest and lowest absolute pressures in the cylinder.) STEAM JACKETS 307 absolute pressure per square inch at cut-off, e. is the reciprocal of the ratio of expan- sion, and T is a function of the pressure range in the cylinder, which may be obtained from the table on p. 306. Heck estimates that the steam consumption of an engine may be computed from its indicator diagram (Art. 500) within 10 per cent by the application of this formula. If the steam as delivered from the boiler is wet, some modification is necessary. 438. Reduction of Condensation. Aside from careful attention to the factors already mentioned, the principal methods of minimizing cylinder condensation are by (a; the use of steam-jackets, (b) super- heating the steam, and (r) the employment of multiple expansion. 439. The Steam Jacket. Transfers of heat between steam and cylinder walls would be eliminated if the walls could be kept at the momentary temperature of the steam. Initial condensation is elimi- nated if the walls are kept at the temperature of steam during admis- sion : it is mitigated if the walls are kept from being cooled by the low-pressure steam during the latter part of expansion and exhaust. The steam jacket, invented by Watt, is a hollow casing enclosing the cylinder walls, within which steam is kept at high pressure. Jackets have often been mechanically imperfect, and particular difficulty has been experienced in keeping them drained of the condensed water. In a few cases, the steam has passed through the jacket on its way to the cylinder; a bad arrangement, as the cylinder steam was thus made wet. It is usual practice, with simple engines, and at the high-pressure cylindeis of compounds, to admit steam to the jacket at full boiler pressure; and in some cases the pressure and temperature in the jacket have exceeded those in the cylinder. Hot-air jackets have been used, in which flue gas from the boiler, or highly heated air, was passed about the body of the cylinder. 440. Arguments for and against Jackets. The exposed heated surface of the cylinder is increased and its mean temperature is raised; the amount of heat lost to the atmosphere is thus increased. The jacket is at one serious disadvantage : its heat must be transmitted through the entire thickness of the walls; while the internal teat transfers are effected by direct contact between the steam and the inner " skin " of the walls. Unjacketed cylinder walls act like heat sponges. The function of the jacket is preventive, rather than remedial, opposing the formation of moisture early in the stroke, liquefaction being transferred from the cylinder to the jacket, where its influence is less harmful. The walls are kept hot at all times, instead of being periodically heated and cooled 308 APPLIED THERMODYNAMICS by the action of the cylinder steam. The steam in the jacket does not expand; its temperature is at all times the maximum temperature attained in the cycle. The mean temperature of the walls is thus raised. i 441. Results of Jacketing. In the ideal case, the action of the jacket may be regarded as shown by the difference of the areas dekl and debf, Fig. 183 The total heat supplied, without the jacket, is Ideb2, but cylinder condensation makes the steam wet at cut-off, giving the work area dekl only. The additional heat 2&/3, supplied by the jacket, gives the additional work area kbfl, manifestly at high efficiency. In this country, jackets have been generally employed on well-known engines of high efficiency, particularly on slow speed pumping engines; but their use is not common with standard designs. Slow speed and extreme expansion, which suggest jackets, lead to excessive bulk and first cost of the engine. With normal speeds and expansive ratios, the engine is cheaper and the necessity for the jacket is less. The use of the jacket is to be determined from considerations of capital charge, cost of fuel and load factor, as well as of thermodynamic efficiency. These commercial factors account for the far more general use of the jacket in Europe than in the United States. From 7 to 12 per cent of the whole amount of steam supplied to the engine may be condensed in the jacket. The power of the engine is almost invariably increased by a greater percentage than that of increase of steam consumption. The cylinder saves more than the jacket spends, although in some cases the amount of steam saved has been small. The range of net saving may be from 2 or 3 up to 15 per cent. The increased power of the engine is represented by the ^ , ^j ^j^ i i t difference between the areas abodes and aXYdes, 4 Vis 7 6 J M aJ w il f Fig. 187. The latter area approaches much more 3 P^T^mrr.: dosely the ideaj fflrea ABCDEf Jacketing pays best when the conditions are such as to naturally induce excessive initial condensation. The diagram of Fig. 190, after Donkin (14), shows the variation in value of a steam jacket at varying ratios of expansion in the same engine run at constant speed and initial pressure. With the jacket, the best ratio of expansion was about 10, giving 25 Ib. of steam per hp.-hr: without the jacket, the lowest steam consump- tion (of 39 Ib. per hp.-hr) was reached at an expansion ratio of 4. 442. Use of Superheated Steam. The thermodynamic advantage of superheating, though small, is not to be ignored, some heat being taken in at a temperature higher than the mean temperature of heat absorp- tion; the practical advantages are more important. Adequate super- heat fills the " heat sponge " formed by the walls, without letting the steam become wet in consequence. If superheating is slight, the steam, during admission, may be brought down to the saturated condition, and may even become wet at cut-off, following such a path as debxbkl, Fig. 183. With a greater amount of superheat, the steam may remain Vis POINT OF CUT-OFF FIG. 190. Art. 441. Effect of Jackets at Various Ex- pansion Ratios. SUPERHEAT 309 dry or even superheated at cut-off, giving the paths debzijf, deblzA. The minimum amount of superheat ordinarily necessary to give dry- ness at cut-off seems to be about 100 F.; it may he much greater. Ripper finds (15) that about 7.5 F. of superheat are necessary for each 1 per cent of wetness at cut-off to be expected when working with saturated steam. We thus obtain Fig. 191, in which the increased work areas acbd, cefb, eghf are obtained by superheating along jk, kl f Im, each path representing 7o of superheat. Taking the pressure along aj as 120 lb. ; and that along hb as 1 lb., the absolute temperatures are 800 S)~ .UK! 561.43, respectively, and since the latent heat at 120 lb. is 87T.1 1 U. t. u., the work gained by each of the areas in question is aceg ms 2iS 000 Slik dbJK 800.9 ' * If we take the specific heat of superheated steam, roughly, at 0.48, the heat used in secur- ing this additional work area is 0.48 x 75 = 36 P> t. u. The efficiency of superheating is then 1^.1-5-36 = 0.73, while that of the non-super- FIG. 101. Art. 442. Snper- heated cycle as a whole, even if operated at Car- heat for overcoming Initial nnt efficiency, cannot exceed 239. 47 -=-800.9= 0.30. Condensation. Great care should be taken to avoid loss of heat in pipes between the super- heater and the cylinder; without thorough insulation the fall of tem- perature here may be so great as to considerably increase the amount of superheating necessary to secure the desired result in the cylinder. 443. Experimental Results with Super- heat The AJaace teats of 1892 showed, with from 60 to 80 of superheat, mi average net saving of 12 per cent, baaed on fuel, even when the coal consumed in the separately fired superheaters was considered; and when the superheaters were fired by waste heat from the boilers, the average saving was 20 per cent. WiUflns found a considerable saving by superheat, even when cutn>ff was at half stroke, a ratio of expansion certainly not unduly favorable to superheating. As with jackets, the advantage of superheat is greatest in engines of low speeds and high expansive ratios. Striking results have been obtained by the use of high superheats, ranging from 200 to 300 F. above the temperature of saturation. The mechanical design, of the engine must then be considerably modified. Vaughan INDICATED HORSE POWER FIG. 193. Art. 443, Prob. 7. Steam Economy in Relation to Superheat. 310 APPLIED THERMODYNAMICS (16) has reported remarkably large savings due to superheating in locomotive practice. Figure 193 shows the decreased steam consumption due to various degrees of superheat in a small high speed engine. 444, Actual Expansion Curve. In Fig. 187, bY represents the curve of constant dryness, bC the adiabatic. The actual expansion curve in an un jacketed cylinder using saturated steam will then be some such line as be, the entropy increasing in the ratio xz+xy and the fraction of dryness in the ratio xz+xw. Expressed exponentially, the value of n for such expansion curve depends on the initial dryness (Art. 4316); it is usually between 0.8 and 1.2, and averages about 1,00, when the equation of the curve is PV=pv. This should not be confused with the perfect gas isothermal; that the equation has the same form is accidental. The curve PV =pv is an equilateral hyperbola, commonly called the hyperbolic line. The actual expansion path be will then appear on the entropy dia- gram, Fig, 189, as be, bc f , usually more like the former. The point b (cut-off) specifies a lower pressure and temperature than does B in the ideal diagram, and lies to the left of B on account of initial condensa- tion. If expansion is then along bc 3 the walls are giving up, to the steam, heat represented by the area mbcn. This is much less than the area mbBAf, which represents roughly the loss of heat to the walls by initial condensation. 445. Work done during Expansion: Engine Capacity. From Art. y 95, this is, for a hyperbolic curve, BC, Fig. 187, P B V B log, ^ Assume no clearance, and admission and exhaust to occur without change of pressure; the cycle is then precisely that represented by ABODE, excepting that the expansive path is hyperbolic. Then the work done during admission is P B V B ] the negative work during exhaust is Pj)V c ; and the net work of the cycle is The mean effective pressure or average ordinate of the work area ia obtained by dividing this by V c , giving -p a MEAN EFFECTIVE PRESSURE 311 Y or, letting =- =r, it is Pg(l + log.r) Letting m stand for this mean effective pressure, in pounds per square inch, A for the piston area in square inches, L for the length of the stroke in feet, and N for the revolutions per minute, the total average pressure on the piston (ignoring the rod) is mA pounds, the distance through which it is exerted per minute is in a double-acting engine 2 LN feet, and the work per minute is 2 mALN foot-pounds, or 2 mALX -4- 33,000 horse power. This is for an ideal diagram, which is always larger than the actual diagram abcdes; the ratio of the latter to the former gives the diagram factor, by which the computed value of m must be multiplied to give actual results. Diagram factors for various types of engine, as given by Seaton, are as follows: Expansion engine, with special valve gear, or with a separate cut-off valve, cylinder jacketed . . . 0.90; Expansion engine having large ports and good ordinary valves, cylinders jacketed . . , 0.86 to 0.88; Expansion engines with ordinary valves and gear as in general practice, and unjacketed . . . 0.77 to 0.81; Compound engines, with expansion valve on high pressure cylinder, cylinders jacketed, with large ports, etc. . . . 0.86 to 0.88; (see Art. 466), Compound engines with ordinary slide valves, cylinders jacketed, good ports, etc. . . . 0.77 to 0.81; Compound engines with early cut-off in both cylinders, without jackets or separate expansion valves . . . 0.67 to 0.77; Fast-running engines of the type and design usually fitted in warships . . . 0.57 to 0,77. The extreme range of values of the diagram factor is probably between 0.50 and 0.90. Regulation by throttling gives values 0.10 to 0.25 lower than regulation by cut-off control. Jackets raise the value by 0.05 to 0.15. Extremely early cut-off in simple unjacketed engines (less than 1) or high speed (above 225 r. p. m.) may decrease it by 0.025 to 0.125. Features of valve and port design may cause a variation of 0.025 to 0175. Piston speeds of large engines at around 100 r. p. m. now range from 720 ft. per minute upward. The power output of an engine of given size is almost directly proportional to the piston speed. Rotative speeds (r. p. m.) depend largely on the type of valve gear, and are limited by the strength of the flywheel. Releasing gear engines do not ordinarily run at over 100 r. p. m. (Art. 507): nor do four-valve engines often exceed 240 r. p. m. The smaller engines are apt to have the higher rotative speeds and the larger ratios of cylinder diameter to stroke. Long strokes favor small clearances, with many types of valve* Engines of high rotative speed will generally have short strokes. Speeds of stationary reciprocating engines seldom exceed 325 r. p. m. 312 APPLIED THERMODYNAMICS 446, Capacity from Clayton's Formula. If the expansion curve can be repre- sented by the equation pv n = const., in which n^l, the mean effective pressure (clearance ignored) is, with the notation of Art. 445, nPs ~ Ps -- lilt f Z~T X if f TT . r(n 1) r n (n I) The best present basis for design is to find n as suggested in Arts. 4316, 437, to assume a moderate amount of hyperbolic compression (see Art 451) and to allow for clearance. This is in fact the only suitable method for use where there is high superheat: in which case n> LO. Thus, let the pressure limits be 120 and 16 Ib. absolute, the apparent ratio of expansion 4, clearance 4 per cent, compression to 32 Ib. absolute, n = 1.15. The approximate equation above gives 1.15X120 120 _ , m 060 16 ~4i-i* X 0.15 More exactly, calling the clearance volume 0.04, the length of the diagram is 1.0, the volume at cut-off is 0.29, and the maximum volume attained is 1.04. The mean effective pressure is f- 6Xl -^-16 (1.04-0.08) - (30 X0.04 log 2) = 54.5 Ib. per square inch, (0 29\ 1 - 15 r-^-j J - 27.6 Ib. and the volume at the beginning of compression being 0.04X11=0.08. Any diagram factor employed with this method will vary only slightly from 1.0, depending principally upon the type of valve and gear. Unfortunately, we do not as yet possess an adequate amount of information as to values of n in condensing and jacketed engines 447. Capacity K$. Economy. If we ignore the influence of con- densation, the Clausius cycle (Art. 409), objectionable as it is with regard to capacity (Art. 418), would be the cycle of maximum effi- ciency ; practically, when we contemplate the excessive condensation that would accompany anything like complete expansion, the cycle of Rankine is superior. This statement does not apply to the steam tur- bine (Chapter XIV). The steam engine may be given an enormous range of capacity by varying the ratio of expansion ; but when this falls above or below the proper limits, economy is seriously sacrificed. In purchasing engines, the ratio of expansion at normal load should be set fairly high, else the overload capacity will be reduced. In marine service, economy of fuel is of especial importance, in order to save storage space. Here expansive ratios may therefore range CLEARANCE AXD COMPRESSION 313 higher than is common in stationary practice, where economy in first cost is a relatively more important factor. 448. The Exhaust Line : Back Pressure. Considering now the line de of Fig. 187, \\e find a noticeable U)hS of \\ork area as compared with that in the ideal catse. (Line J)E represents the pressure existing outside the cylinder.) This is due to several causes. The f notional resistance of the ports and exhaust pipes (greatly increased by the prepuce of water) produces a wiredrawing effect, mak- ing the pressure in the cylinder higher than that of the atmosphere or of the con- denser. The presence of air in the exhaust passages of a condensing engine may elevate the pressure above that corresponding to the temperature of the steam, and fto cause undesirable resistance to the backward movement of the piston. This air may be present as the re>ult of leakage, under poor operating conditions; more or less air is always bi ought in the cycle with the boiler feed and condenser water. The effect of these causes is to increase the pressure during release, even in good engines, from 1 to 3.0 Ib. above that ideally obtainable. Hee'vaporation may be incomplete at the end of expansion; it then proceeds during exhaust, sometimes, in flagrant cases, being still incomplete at the end of exhaust. The moisture then present greatly increases initial condensation. The evaporation of any water during the exhaust stroke seriously cools the cylinder walls. In general good practice the steam is about dry during exhaust; or at least during the latter portion of the exhaust. 449. Effect of Altitude. The possible capacity of a non-condensing engine is obviously increased at low barometric pressures, on account of the lowering of the line DE, Fig. 187. "With condensing engines, the absolute pressure attained along DE depends upon the proportion of cooling water supplied and the effectiveness of the condensing apparatus. It is practically independent of the barometric pres- sure, excepting at very high vacua; consequently, the capacity of the engine is unchanged by variations in the latter. A slightly decreased amount of power, however, will suffice to drive the air pump which delivers the products of conden- sation against any lessened atmospheric pressure. 450. Clearance. The line e*a does not at any point come in contact with the ideal line EA, Fig. 187. In all engines, there ia necessarily a small space left tatween the piston and the inside of the cylinder heat! at the end of the stroke. This space, with the port spaces back to the contact surfaces of the inlet valves, is filled with steam throughout the cycle. The distance t* in the diagram represents the volume of these " clearance " spaces. In Fi;. 195, the apparent ratio of ex- pansion is ^ . If the zero volume line OP be found, the real ratio of expansion, ab FD clearance volume included, IB , The proportion of clearance (always ex- Ab Aa pressed in terms of the piston displacement) is . The clearance in actual engines 314 APPLIED THERMODYNAMICS varies from 2 to 10 percent of the piston displacement, the necessary amount depending largely on the type of valve gear. In such an engine as that of Fig. 186, it is necessarily large, on account of the long ports. In these flat slide valve engines it averages 5 to 10 per cent* with rotary (Corliss) valves, 3 to 8 per cent; with single piston valves, 8 to 15 per cent. These figures are for valves placed on the side (bar- rel) of the cylinder. When valves are placed on heads, the clearance may be reduced 2 to 6 per cent. In the unidirectional flow (Stumpf) engine (Art. 507), it is only about 2 per cent. It is proportionately greater in -v small engines than in those of large size. FIG. 195. Arts. 450, 451. -Real and Ap- Tt ma y be accurately estimated by placing parent Expansion. the piston at the end of the stroke and fill- ing the clearance spaces with a weighed or measured amount of water. All waste spaces, back to the contact surfaces of the valves, count as clearance. 451. Compression. A large amount of steam is employed to fill the clearance space at the beginning of each stroke. This can be avoided by closing the exhaust valve prior to the end of the stroke, as at e, Fig. 187, the piston then compressing the clearance steam along es, so that the pressure is raised nearly or quite to that of the entering steam. This compression serves to prevent any sudden reversal of thrust at the end of the exhaust stroke. If compression is so complete as to raise the pressure of the clearance steam to that carried in the supply pipe, no loss of steam will be experienced in filling clearance spaces. The work expended in compression, eahg t Fig. 195, will be largely recovered during the next forward stroke by the expan- sion of the clearance steam: the clearance will thus have had httle effect on the efficiency; the loss of capacity efa will be just balanced by the saving of steam, for the amount of steam necessary to fill the clearance space would have expanded along ae, if no other steam had been present. Complete compression would, however, raise the temperature of the com- pressed steam so much above that of the cylinder walls that serious condensation would occur. This might be counteracted by jacketing, but in practice it is cus- tomary to terminate compression at some pressure lower than that of the entei ing steam. A certain amount of unresisted expansion then takes place during the entrance of the steam, giving a wiredrawn admission line. If the pressure at s, Fig. 187, is fixed, it is, of course, easy to determine the point e at which the exhaust valve must close. Considered as a method of warming the cylinder walls so as to prevent initial condensation, compression is " theoretically less desirable than jacketing, for in the former case the heat of the steam, once transformed to work, with accompanying heavy losses, is again transformed into heat, while in the latter, heat is directly applied." For mechanical reasons, some compression is usually considered necessary. It makes the engine smooth-running and probably iecreases condensation if properly limited. Compression must not be regarded as bringing about any nearer approach to the Carnot cycle. It is applied to a very 3mall portion only of the working substance, the major portion of which is jxternally heated during its passage through the steam plant. VALVE ACTION: LEAKAGE 315 452. Valve Action: Leakage. We have now considered most of the differences between the actual and ideal diagrams of Fig. 187. The rounding of the corners at b, and along cdu, is due to sluggish valve action; valves must be opened slightly before the full effect of their opening as realized, and they cannot close instantaneously. The round corner at e is due to the slow closing of the exhaust valve. The inclined line sa shows the admission of steam, the shaded work area being lost by the slow movement of the valve. In most cases, admission is made to occur slightly prior to the end of the stroke, in order to avoid this very effect. If admission is too early, a n3gative lost work loop, mno, may be formed. Important aberrations in the diagram, and modifications of the phenomena of cylinder condensation; may result from leakage past valves or pistons. In an engine like that of Fig. 186, steam may escape directly from the steam chest to the exhaust port. Valves are more apt to leak than pistons, A valve may be tight when stationary, but leak when moving; it may be tight when cold and leak when hot. Unbalanced slide valves, poppet and Corliss valves tend to wear tight; piston valves and balanced slide valves become leaky with wear. Leakage is increased when the steam is wet. Jacketing the cylinder decreases leakage. The steam valve may allow steam to enter the cylinder after the point of cut-off has been passed. Fortunately, as the difference in pressure between steam chest and cylinder increases, the overlap of the valve also increases. Leakage past the exhaust valve is particularly apt to occur just after admission, because then (unless there is considerable compression) the exhaust valve has only just closed. The indicator diagram cannot be depended on to detect leakage, excepting as the curves are transferred to logarithmic coordinates (7). Such steam valve leakage as has just been described produces the same apparent effect as reevaporation occurring shortly after cut-off. Leakage from the cylinder to the exhaust, occurring during this period, produces the effect which was formerly regarded as due to cylinder condensation immediately following cut-off. In engines known to have tight exhaust valves, this latter effect is not found. * An engine may be blocked and examined for leakage (Trans. A.S.M. E., XXIV, 719) but it is difficult to ascertain the actual amount under running conditions. In one test of a small engine, leakage was found to be 300 Ib. per hour. Tests have shown that with sin pie flat slide or piston valves the steam consumption ir creases about 15 per cent in from 1 to 5 years, on account of leakage alone, A large number of tests made on all types of engine gave steam consumptions averaging 5 per cent higher where leakage was apparent than where valves and pistons were known to be tight. THE STEAM ENGUTE CYCLE ON THE ENTROPY DIAGRAM 453. Cylinder Feed and Cushion Steam. Fig. 189 has been left incomplete, for reasons which are now to be considered. It is convenient to regard the working fluid in the cylinder as made up of two masses, the " cushion steam/' which alone nils the compression space at the end of each stroke, and is constantly present, and the " cylinder feed," which enters at the beginning of each stroke, and leaves before the completion of the next succeeding stroke. In testing steam engines by weighing the discharged and condensed steam, the cylinder feed is alone measured ; it alone is chargeable as heat consumption ; but for an accurate conception of the cyclical relations in the cylinder, the influence of the cushion steam must be con- sidered. ' 316 APPLIED THERMODYNAMICS In Fig. 196, let abcde be the PV diagram of the mixtme of cushion steam and cylinder feed, and let gh he the expansion line of the cushion steam if it alone were present. The total volume rq at any point q of the combined paths is made up of the cushion bte<im volume co and the cylinder feed volume, obviously equal to og. If we wish to obtain a diagram shoeing the behavior of the cylinder feed alone, -we must then deduct from the volumes around alc<U the correspond- ing volumes of cushion steam. The point p is then derived by making rp = vq vo, and the point t by making rt = ru rs. Proceeding thus, we obtain the diagram nzjklm, representing the behavior of the cylinder feed. Along nz the diagram coincides with the OP axis, indicating that at this stage the cylinder contains cushion steam only. o FIG. 196. Arts. 453, 457 Elimination of Cushion Steam. 454, The Indicator Diagram. Our study of the ideal cycles in Chapter XII has dealt with representations on a single diagram of changes occurring in a given mass of steam at the boiler, cylinder, and condenser, the locality of changes of condition being ignored. The energy diagram abcdes of Fig. 187 does not represent the behavior of a definite quantity of steam working in a closed cycle. The pressure and volume changes of a varying quantity of fluid are depicted. During expansion, along he, the quantity remains constant; during compression along es, the quantity is likewise constant, but diiferent. Along sab the quantity increases ; while along cde it decreases. The quality or dryness of the steam along es or be may loe readily determined by comparing the actual volume with the volume of the same weight of dry steam ; but no accurate information as to quality can be obtained along the admission and release lines sab and cde. The areas under these lines represent work quantities, however, and it is desirable that we draw an entropy diagram which shall represent the corresponding heat expenditures. Such a diagram will not give the thermal history of any definite amount of steam ; it is a mere projection of the PV diagram on diiferent coordinates. It tacitly assumes the indicator diagram to represent a reversible cycle, whereas in fact the operation of the steam engine is neither cyclic nor reversible. 455. Boulvin's Method. In Fig. 197, let abode be any actual indicator diagram, YZ the pressure temperature curve of saturated steam, and QR the curve of satu- ration, plotted for the total quantity of FIG. 197 Art 455. Transfer from PV steam in the cylinder during expansion. to JVT Diagram (Boulvin's Method). The water line OS and the saturation curve CONVERTED DIAGRAMS 317 MT are now drawn for 1 Ib. of steam, to any convenient scale, on the entropy plane. To transfer any point, like B, to the entropy diagram, we draw BD, DK, EH, KT, BA, AT, HT, BG, and GF as in Art. 378. Then F is the required point on the temperature entropy diagram. By transferring other points m the same way, we obtain the area NVFU. The expansion line thus traced correctly represents the actual hLstory of a definite quantity of fluid; other parts of the diagram are imaginary. It is not safe to make deductions as to the condition of the substance from the NT diagram, excepting along the expansion curve. For example, the diagram apparently indicates that the dryness is decreasing along the exhaust line SU; although we have seen (Art. 448) that at this stage the dryness is usually increasing (17). 456. Application in Practice. In order to thus plot the entropy diagram, it is necessary to have an average indicator card from the engine, and to know the quantity of steam in the cylinder. This last is determined by weighing the dis- charged condensed steam during a definite number of strokes and adding the quantity of clearance steam, assuming this to be just dry at the beginning of com- pression, an assumption fairly well substantiated by experiment. 45705. Reeve's Method. By a procedure similar to that described in Art. 453, an indicator diagram is derived from that originally given, representing the behav- ior of the cylinder feed alone, on the assumption that the clearance steam works adiabatically through the point e, Fig. 196. This often gives an entropy diagram in which the compression path passes to the left of the water line, on account of the fact that the actual path of the cushion steam is not adiabatic, but is occasion- ally less " steep." The Reeve diagram accurately depicts the process between the points of cut- off and release and those of compression and admission with reference to the cylinder feed only. 4575. Preferred Method. The most satisfactory method is to make no attempt to represent action between the points of admission and cut-off and of Release and compression. During these two portions of the cycle we know neither the weight nor the dryness of steam present at any point. The method of Art. 155 should be used for the expansion curve alone. For compression, a new curve corresponding with RQj Fig. 197, should be drawn, representing the pv relation for the weight of clearance steam alone. Points along the compression curve may then be transferred to the upper right-hand quadrant by the same process as that described in Art. 455. The TN diagram then shows the expansion and compression curves, both correctly located with reference to the water line OS and the dry steam curve TM } for the respective weights of steam; and the heat transfers and dryness changes during the operations of expansion and compression are perfectly illustrated. 458. Specimen Diagrams. Figure 199 shows the gain by high initial pressure and reduced back pressure. The augmented work areas befc, cfho, are gained at high efficiency; adji and adlh cost nothing. The operation of an engine at back pressure, 318 APPLIED THERMODYNAMICS to permit of using the exhaust steam for heating purposes, results in such losses as adji, adlk. Similar gains and losses may be shown for non-expansive cycles. Figure 200 shows four interesting diagrams plotted from actual indicator cards from a small FIG. 199. Art. 458. Initial Pressure and Back Pressure. FIG. 200. Art. 458. Effects of Jacket- ing and Superheating. engine operated at constant speed, initial pressure, load, and ratio of expansion (18). Diagrams A and C were obtained with saturated steam, B and D with super- heated steam. In A and B the cylinder was un jacketed; in C and D it was jacketed. The beneficial influence of the jackets is clearly shown, but not the expenditure of heat in the jacket. The steam consumption in the four cases was 45.6, 28.4, 27 25 and 20.9 Ib. per Ihp-hr., respectively. MULTIPLE EXPANSION 459. Desirability of Complete Expansion. It is proposed to show that a large ratio of expansion is from every standpoint desirable, excepting as it is offset by increased cylinder condensation ; and to suggest multiple expansion as a method for attaining high efficiency by making such large ratio practically possible. From Art. 446, it is obvious that the maximum work obtainable from a cylinder is a function solely of the initial pressure, the back pressure, and the ratio of expan- sion. In a non-conducting cylinder, maximum efficiency would be realized when the ratio of expansion became a maximum between the pressure limits. Without expansion, increase of initial pressure very slightly, if at all, increases the efficiency. Thus, in Fig. 201, the cyclic work areas abed, aefg, ahij, would all be equal if the line XY followed the law po == PV. As the actual law (locus of points representing steam dry at cut-off) is approximately, \ d the wort areas increase slightly as the pressure in- creases; but the necessary heat absorption also increases, and there is no net gain. The thermody- FrQ ^ Art 459 _ Non _ namic advantage of high initial pressure is realized only ' expaiig i ve Cycles. wksn the ratio of expansion is large. By condensing the steam as it flows from the engine, its pressure may be re- duced from that of the atmosphere to an absolute pressure possibly 13 Ib. lower. The cyclic work area is thus increased ; and since the reduction of pressure is ac- MULTIPLE EXPANSION 319 companied by a reduction in temperature, the potential efficiency is increased. Figure 202 shows, however, that the percentage gain in efficiency is smaU with no expansion, increasing as the expansion ratio increases. Wide ratios of expansion are from all of these standpoints essential to efficiency. We have found, however, that wide ratios of expansion are associated with such excessive losses from condensation that a compromise is necessary, and that in practice the best efficiency is secuied with a rather limited ratio. The practical attain- ment of large expansive ratios without correspond- ing losses by condensation is possible by multiple expansion. By allowing the steam to pass suc- cessively through two or more cylinders, a total expansion of 15 to 33 may be secured, with condensa- tion losses such as are due to much lower ratios. FIG. 202. Art. 439. Gain due to Vacuum. 460. Condensation Losses in Compound Cylinders. The range of pres- sures, and consequently of temperatures, in any one cylinder, is reduced by compounding. It may appear that the sum of the losses in the two cylinders would be equal to the loss in a single simple cylinder. Three considerations may serve to show why this is not the case : (a) Steam ree'vaporated during the exhaust stroke is rendered avail- able for doing work in the succeeding cylinder, whereas in a simple engine it merely causes a resistance to the piston; (&) Initial condensation is decreased because of the decreased fluctua- tion in wall temperature; (c) The range of temperature in each cylinder is half what it is in the simple cylinder, but the whole wall surface is not doubled. 461. Classification. Engines are called simple, compound, triple, or quadruple, according to the number of successive expansive stages, ranging from one to four. A multiple-expansion engine may have any number of cylinders ; a triple expan- sion engine may, for example, have five cylinders, a single high-pressure cylinder discharging its steam to two succeeding cylinders, and these to two more. In a multiple-expansion engine, the first is called high-pressure cylinder and the last the low-pressure cylinder. The second cylinder in a triple engine is called the intermediate; in a quadruple engine, the second and third are called the first intermediate and the second intermediate cylinders, respectively. Compound en- gines having the two cylinders placed end to end are described as tandem ; those having the cylinders side by side are cross-compound. This last is the type most commonly used in high-grade stationary plants in this country. The engines may be either horizontal or vertical ; the latter is the form generally used for triples or quadruples, and in marine service. Sometimes some of the cylinders are horizon- tal and others vertical, giving what, in the two-expansion type, has been called the angle compound. Compounding may be effected (as usually) by using cylinders of various diameters and equal strokes i or of equal diameters and varying strokes, 320 APPLIED THERMODYNAMICS or of like dimensions but unequal speeds (the cylinders driving independent shafts), or by a combination of these methods 462. Incidental Advantages. Aside from the decreased loss through cylinder condensation, multiple-expansion engines have the following points of superiority . (1) The steam consumed m filling clearance spaces is less, because the high- pressure cylinder is smaller thau the cylinder of the equivalent simple engine; (2) Compression in the high-pressure cylinder may be carried to as high a pressure as is desnable without beginning it so early as to greatly i educe the woik area; (3) The low-pressure cylinder need be built to withstand a fraction only of the boiler pressure ; the other cylinders, which carry higher pressures, are com- paratively small; (4) In most common types, the use of two or more cylinders permits of using a greater number of less powerful impulses on the piston than is possible with a single cylinder, thus making the rotative speed more unifonn; (5) For the same reason, the mechanical stresses on the crank pin, shaft, etc., are lessened by compounding. These advantages, -with that of superior economy of steam, have led to the general use of multiple expansion in spite of the higher initial cost which it en- tails, whei ever steam pressures exceed 100 Ib. 463. Woolf Engine. This was a form of compound engine originated by Horn- blower, an unsuccessful competitor of Watt, and revived by Woolf in 1800, after the expiration of Watt's principal patent. Steam passed dhectly from the high to the A - low-pressure cylinder, entering the latter while being exhausted from the former. This necessitated having the pistons either in phase or a half revolution apart, and there was no improvement over any other double-acting engine with regard to uni- D formity of impulse on the piston. Figure 203 represents the ideal indicator diagrams. 1^.303. Arts. 463, 466. Woolf Engine. A BCD is the action in the high-pressure cylinder, the fall of pressure along CD being due to the increase in volume of the steam, now passing into the low-pressure cylinder and forcing its piston out- ward. EFGH shows the action in the low-pres- sure cylinder; steam is entering continuously throughout the stroke along EF. By laying off MP - LK, etc., we obtain the diagram TABRS, representing the changes undergone by the steam during its entire action. This last area is ob- viously equal to the sum of the areas ABCD and EFGH. Figure 204, from Ewing (19) shows a pair of actual diagrams from a Woolf engine, the length of the diagrams representing FIG. 204 Art. 463, Prob. 31. Dia- grams from Woolf Engine. RECEIVER ENGINE 321 the stroke of the pistons and not actual steam volumes. The low-pressure dia- gram has been reversed for convenience Some expansion in the low-pressure cylinder occurs after the closing of the high -pressure exhaust valve at a. Some loss of pressure by wiredrawmg in the passages between the two cylinders is clearly indicated. 464# . Receiver Engine. In this more modern form the steam passes from the high-pressure cylinder to a closed chamber called the receiver, and thence to the low-pressure cylinder. The receiver is usually an independent vessel connected by pipes with the cylinders; in some cases, the intervening steam pipe alone is of sufficient capacity to constitute a receiver. Receiver engines may have the pistons coin- cident in phase, as in tandem engines, or opposite, as in opposed beam engines, or the cranks may be at an angle of 90, as in the ordinary cross-compound. In all cases the receiver engine has the characteristic advantage over the Woolf type that the low-pressure cylinder need not receive steam during the whole of the working stroke, but may have a definite point of cut-off, and work in an expansive cycle. The dis- tribution of work between the two cylinders, as will be shown, may be adjusted by varying the point of cut-off on the low-pressure cylinder (Art. 467). Receiver volumes vary from to 1 times the high-pressure cylinder volume. 4645. Reheating. A considerable gain in economy is attained by drying or superheating the steam during its passage through the WITH REHEATERS \ AND JACKETS WITHOUT REHEATERS AND JACKETS HEHEATEfie AND JACKETS .WITHOUT ~ 1EATERS JACKETS FIGS. 215 and 216. Art. 4646. Effect of Reheaters and Jackets (25). receiver, by means of pipe coils supplied with high-pressure steam from the boiler, and drained by a trap. The argument in favor of reheating is the same as that for the use of superheat in any cylinder (Art. 442). It is not surprising, therefore, that the use of reheaters is only profit- able when a considerable amount of intermediate drying is effected. Reheating was formerly unpopular, probably because of the difficulty 322 APPLIED THERMODYNAMICS of securing a sufficient amount of superheat with the limited amount of coil surface -when saturated steam was used in the receiver coils. With superheated steam, this difficulty is obviated. Reheating increases the capacity as well as the economy of the cylinders. 465. Drop. The fall of pressure occurring at the end of expansion (cdj Fig. 196) is termed drop. Its thermodynamic disadvantage and practical justification have been pointed out in Arts. 418, 447. In a compound engine, some special considerations apply. If there is no drop at high-pressure release, the diagram showing the whole expan- sion is substantially the same as that for a simple cylinder. With drop, the diagram is modified, the ratio of expansion in the high-pressure cylinder is decreased, and the ideal output is less. The orthodox view is that there should be no drop in the high- pressure cylinder (21). The cylinders of a compound engine work with less fluctuation of temperature than that of a simple engine, and may therefore be permitted to use higher ratios of expansion (i.e., less drop) than does the latter. In the design method to follow, dimen- sions will be determined as for no drop. Changes of load from normal may introduce varying amounts of drop in operation. 466. Combination of Actual Diagrams: Diagram Factor. Figure 210 shows the high- and low-pressure diagrams from a pTrm.11 compound*engine. These are again H.P. FIG. 210. Art. 466. Compound Engine Diagrams. FIG. 211. Art. 466. Compound Engine Diagrams Combined. shown in Fig. 211, in which the lengths of the diagrams are proportioned as are the cylinder volumes, the pressure scales are made equal, and the proper amounts of setting off for clearance (distances a and 5) are regarded. The cylinder feed per single stroke was 0.0498 lb., the cushion steam in the high-pressure cylinder 0.0074 lb., and that in the other cylinder 0,0022 lb. No single saturation curve is possible; the Lne *s is drawn for 0.0572 lb. of steam, and SS for 0.0520 lb. As in Art. 453, we may obtain equivalent diagrams with the cushion steam eliminated. COMBINED DIAGRAMS 323 In Fig. 212, the single curve SS then represents saturation for 0.0498 Ib. of steam. The areas of the diagrams are unaltered, and correctly measure the work done; they may be transferred to the entropy plane as in Art. 455. The moisture present at any point during expansion is still represented by the dis- tance cd, corresponding with the distance similarly marked in Fig. 211. The ratio of the area of the combined actual diagrams to that of the Ran- kine cycle through the same extreme limits of pressure and with the same ratio of expansion is the diagram factor, the value of which may range up to 95, being higher than in simple engines (Art. 459). 467. Combined Diagrams. Figure 205 shows Fro. 212. Art. 466. Combined the ideal diagrams from a tandem receiver engine. Diagrams for Cylinder Feed. Along CD, as along CD in Fig. 203, expansion into the low-pressure cylinder is taking place The corresponding line on the low- pressure diagram is EF. At F the supply of steam is cut off from the low-pressure cylinder, after which hyperbolic expansion occurs along FS. Meanwhile, the FIG. 205. Arts. 467, 475. Elimination of Drop, Tandem Receiver Engine. FIG. 214. Art. 468. Effect of Low- pressure Cut-off. exhaust from the high-pressure cylinder is discharged to the receiver; and since a constant quantity of steam must now be contained in the decreasing apace between the piston and the cylinder and receiver walls, some compression occurs, giving the line DE. The pressure of the receiver steam remains equal to that at E after the high-pressure exhaust valve closes (at E) and while the high-pressure cylinder continues the cycle along EABC. If the pressure at C exceeds that at E, then there will be some dropr As drawn, the diagram shows none. If cut-off in the low-pressure cylinder occurred later in the stroke, the line DE would be lowered, P c would exceed P s , and drop would be shown. An incidental advantage of the receiver engine is here evident. The intro- duction of cut-off in the low-pressure cylinder raises the lower limit of tempera- 324 APPLIED THERMODYNAMICS FIG. 206. Art. 468. Effect of Changing Low- pressure Cut-off ture in the high-pressure cylinder from D in Fig. 203 to D in Fig. 205. This reduced range of temperature decreases cylinder condensation 468. Governing Compound Engines. Fig. 214 shows that delayed cut-off on the high-pressure cylinder greatly increases the output of the low-pressure cylinder while (the receiver pressure being raised) scarcely affecting its own output. In Fig. 206, is shown the result of varying low-pressure cut-off in a tandem receiver engine with drop, the low-pressure clearance being exaggerated for clearness. The high-pressure diagram is fabcde, the low-pressure is ghjkl, p f =p d = p a and p e =p. Low-pressure cut-off occurs at h (point e in the high-pressure diagram). If this event occur earlier, the corresponding point on the high-pressure diagram is made (say) n, and compression then raises the receiver pressure to o instead of /. The result is that the drop decreases to cp instead of cd (p p =p )- The admission pressure of the low-pressure cylinder thus becomes Pxpp^po instead of p ffj -and the gain qmg due to such increased pressure more than offsets the loss mhj due to the fact that low-pressure cut-off now occurs at p m = p n . The same results will be found with cross-compound engines. The total output of the engine is very little affected by changes in low-pressure cut-off: but (contrary to the result in simple cylinders) the output of the low-pressure cylinder varies directly as its ratio of expansion. With delayed cut-off, the low-pressure cylinder performs a decreased proportion of the total work. When the load changes in a compound engine which has a fixed point of low-pressure cut-off , equality of work distribution becomes impossible. The output of the engine should be varied by varying the point of high-pressure cut-off. Equal distribution of the work should then be accomplished by variation of low-pressure cut-off. The two points of cut-off will be changed in the same direction as the load changes. At other than normal load, there will then be some drop. The aim in design will be, after fixing upon a suitable receiver pressure, to select a normal corresponding point of low-pressure cut-off at which, with the given receiver volume and cylinder ratio, drop will be eliminated. (Arts. 475-478), DESIGN OP COMPOUND ENGINES 325 DESIGN OF COMPOUND ENGINES 469. Preliminary Diagram. We first consider the action as repre- sented in Fig. 205, which shows the combined ideal diagrams without clearance or compression, and with hyperbolic expansion. Losses or gains between the cylinders are ignored. The following notation is adopted : P= initial absolute pressure, Ib. per sq. in., along a&; FIG. 205. Arts. 469, 470, 473 -Pre- p 0=t receiver absolute pressure, Ib. liminary Compound Engine Diagram. per sq. in., along dc; p=back pressure, absolute, Ib. per sq. in,, along gf; Pmh= mean effective pressure, Ib. per sq. in., high-pressure cylinder; effective pressure, Ib. per sq. in., low-pressure cylinder; =# A = ratio of expansion, high-pressure cylinder; =Ri = ratio of expansion, low-pressure cylinder; v c ~=R = whole ratio of expansion; z> 6 r ' -1 =C = ratio of cylinder volumes, or " cylinder ratio." v c The following relations are useful: R=R h Ri=CR h ; C=R t j p^^-Tr log c ^; ^O=-D~; ) p. 470. Bases for Design. The values of P, p and R being given, whatever fixes the pressure or volume at c determines the proportions of the engine. We may assume either * (a) the receiver pressure, p ; (b) the cylinder ratio, (7=; ^c (c) equal division of the temperature ranges; that is, T b -T c = T c -T f , or r. * Some designers of marine engines aim at equalization of maximum pressures on the cranks. This requires careful consideration of clearance and compression. 326 APPLIED THERMODYNAMICS and PO is the pressure corresponding with the temperature T c - or, (d) equal division of the work; that is, abcd=dcefg, attained when Any one of these four assumptions may be made, but not more than one. Having made one, the pressures and volumes at &, c, e and / are all fixed. 471. Diagrams with Clearance. We now employ Fig. 213, in which clearance is allowed for. The expansion curve is still assumed to be a continuous hyperbola, and inter- cylinder losses are ignored. (These last need not be important.) If dn is the high-pressure clearance ^hd-r-dc, Fig 213), the apparent ratio of expansion in the high-pressure cylinder is *'-!- FIG. 213. Arts. 471-473. Design Similarly, the apparent ratio of expansion Diagram i Compound Engine. j^ the low-pressure cylinder is Ri' = where dj=--T ls the low-pressure clearance. Engines are usually designed by specifying the whole apparent ratio of expansion, (dD-\-gf)-s-ab. In terms of the real ratios, this is The mean effective pressures in the cylinders are now P P and Ri=C only when d* = dj. The mean effective pressures reached in practice will differ from these by some small amount, the ratio of probable actual to com- puted pressure being described as the diagram factor. Generally speak- DESIGN OF COMPOUND ENGINES 327 ing, the diagram factor to be used for the cylinder of a multiple expan- sion engine of n expansion stages and R ratio of expansion is the same as that for a simple engine of expansion ratio R s when 472. Size and Horse Power. In general, diagram factors, piston speeds and strokes are the same for all the cylinders of the engine. Then following Art. 446, _2fLN E-P* DO* f\r\ \Pmh-"- ft -\-pml A. i) . OOUUU where /= diagram factor and A^ and A l are the areas of high and low- pressure cylinders respectively, in sq. in. Letting C denote the cylinder ratio, , 2fLNA l in which ^ describes what is called the "high-pressure mean-effective pressure referred to the low-pressure cylinder." 473. Division of Work: Equivalent Simple Engine. The work will be divided between the cylinders in the same ratio as the two areas abed, Dcefg, Fig. 213; or in the ratio, When the assumption of equal output is made (Art. 469), the mean effective pressures must be inversely as the cylinder areas. The power of the compound engine is very nearly the same as that which would be obtained from a simple cylinder of the same size as the low-pressure cylinder of the compound, with a ratio of expansion equal to the whole ratio of expansion of the compound. This would bo exactly true if the diagram factor were the same for the simple as for the compound and if the no-clearance diagram, Fig. 213, were used for finding p m . An approximate expression for the area of the low- pressure cylinder of a compound is then hn = 2/LJVA,(P(l+logefl) , ( R l 328 APPLIED THERMODYNAMICS 474. Cylinder Ratio. Ratio of Expansion. Non-condensing com- pound engines usually have a cylinder ratio C =3 to 4. With condensing engines, the ratio is 1 or 5, increasing with the boiler pressure. In triple engines, the ratios are from 1 : 2.0 : 2.0 up to 1 : 2.5 : 2.5 in sta- tionary practice. With quadruple expansion the ratios are succes- sively from 2.0 to 2.5 : 1. Tests by Rockwood (22) of a triple engine in which the intermediate cylinder was cut out r permitting of running the high- and low-pressure cylinders as a compound with the high cylinder ratio of 5.7 to 1, give the surprising result that with the same initial pressure and expansive ratio, the compound was more economical than the triple. This was a small engine, with large drop. The pointing out of the fact that the conditions were unduly favorable to the compound as compared with the triple did not explain the excellent economy of the former as com- pared with all engines of its class. Somewhat later, exceptionally good results were obtained by Barrus (23) with a compound engine having the extraordinary cylinder ratio of 7.2 : 1.0. Thurston, mean- while, experimented in the same manner as Rockwood, determining, in addition, the economy of the high-pressure and intermediate cylinders when run together as a compound. There were thus two compounds of ratios 3.1 : 1 and 7.13 : 1 and a triple of ratio 1 : 3.1 : 2 3, available for test. The results showed the 7.1 compound to be much better than the 3.1, but less economical than the triple (24) . As the ratio of expan- sion decreased, the economy of the intermediate compound closely approached that of the triple; and at a very low ratio it would probably have equaled it. It is a question whether the high economy of these " intermediate compounds " has not been due primarily to the high ratio of expansion which accompanied the high cylinder ratio. The best performances have been reached by compounds and triples alike at ratios of expansion not far from 30. Ordinary compound engines probably have the high-pressure cylinders too large for best economy. This is due to the aim toward overload capacity. As in a simple engine, the less the total ratio of expansion, the greater is the output: but in a compound, the lowest ratio of expansion cannot be less than the cylinder ratio. Values of R for multiple expansion engines range normally from 12 to 36, usually increasing with the number of expansive stages. Superheat, adequate reheating or jacketing justify the higher values The use of Compound (two-stage) engines is common practice every- where. For stationary service, since the development of the turbine, the triple, even, is an almost extinct type. The extra mechanical losses necessitated by the triple arrangement often offset the slightly DESIGN OF COMPOUND ENGINES 329 greater efficiency. The gain by the compound over the simple is so great (where condensing operation is possible) that excepting under peculiarly adverse conditions of fuel cost or load factor the compound must be regarded as the standard form of the reciprocating steam engine using saturated steam. 475, Determination of Low-pressure Cut-off. Tandem Compound. In Fig. 205, let ABCD be a portion of the indicator diagram of the high-pressure cylinder of a tandem receiver engine, release occurring at C. At this point, the whole volume of steam consists of that m the receiver plus that in the high-pressure cylinder. Let the receiver volume be represented by the distance CX. Then the hyperbolic curve XY may represent the expansion of the steam between the states C and D, and by deducting the constant volumes CX, LR, MZ, etc., we obtain the curve CCr, representing the expansion of the steam in the two cylinders. For no drop, the pressure at the end of compression into the receiver must be equal to that at C. We thus find the point E y and draw EF, the admission line of the low-pressure cylinder, such that ac+ad = ae, etc ; the abscissa of cC being to that of Ed in the same ratio as the respective cylinder volumes. By plotting ED we find the point D at its intersection with CD. A horizontal projection from D to EF gives F. The point F is then the required point of cut-off in the low-pressure cylinder. The diagram EFSHI maj 7 be completed, the curve FS being hyperbolic. 476. Analytical Method. Let the volume of high-pressure cylinder be taken as unity, that of the receiver as R, that of the low-pressure cylinder as L. Let x be the fraction of its stroke completed by the low-pressure piston at cut-off, and let p be the pressure at release from the high-pressure cylinder, equal to the receiver pressure at the moment of admission to the low-pressure cylinder. The volume of steam at this moment is 1+-K; at low-pressure cut-off, it is 1 -\-R-i-xL x If expansion follows the law pv = PV t and P be the pressure in the low-pressure cylinder at cut-off, -x), or P^ The remaining quantity of steam in the high-pressure cylinder and receiver has the volume 1 re+fl, which, at the end of the stroke, will have been reduced to R. If the pressure at the end of the stroke is to be p, then ~ or Combining the two values of P, we find R+l 477. Cross-compound: Cranks at Right Angles. In Fig. 208, let dbC be a portion of the high-pressure diagram, release occurring at C. Communication is now opened with the receiver. Let the receiver volume be laid off as Cd, and let de be a hyperbolic curve. Then the curve C/, the volume of which at any pressure is Cd less than that of de, represents the path in the high-pressure cylinder. This continues until admission to the low-pressure cylinder occurs at g. The whole volume of steam is now made up oiL that in the two cylinders and the receiver; the volumes in the cylinders alone are measurable out to /C. In Fig. 209, lay off hi = 1C and jk so that jk+hi is equal to the ratio of volumes of low- and high-pressure 330 APPLIED T cylinder. At the point C of the cycle, the high-pressure crank is at i, the low-pres- sure "crank 90 ahead or behind it. When the high-pressure crank has moved from ^ to m, the volume of steam in that cylinder is represented by the distance hn, the low-pressure crank is at o and the volume of steam in the low-pressure cy under is represented by pk. Lay off qr, in Fig. 208, distant from the zero volume line al by an amount equal to hn+pk. Draw the horizontal line is. Lay off tu=hn and tv=n,s^pk. Then u is a point on the high-pressure exhaust line and v is a point on the low-pressure admission line. Similarly, we find corresponding crank posi- tions w and x, and steam volumes hy and zk t and lay off AB = hy+zk, Ac = hy, AD=cB=zk, determining the points c and Z? t The high-pressure exhaust line FIG. 208, Arts. 477-479. Elimination of Drop, Cross-compound Engine. guc is continued to some distance below I. For no drop, this line must terminate at some point such that compression of steam in the high-pressure cylinder and receiver will make I the final state. At I the high-pressure cylinder steam volume is zero; all the steam is in the receiver. Let IE represent the receiver volume and EF a hyperbolic curve. Draw IG so that at any pressure its volumes are equal to those along EF, minus the constant volume IE. Then T, where IG intersects guc, is the state of the high-pressure cycle at which cut-off occurs in the low-pressurq cylinder. By drawing a horizontal line through H to intersect vD, we find the point of cut-off J on the low-pressure diagram. If we regard the initial state as that when admission occurs to the low-pressure cylinder, then at low-pressure cut-off the JTC* high-pressure cylinder will have completed the -^- proportion of a full stroke. to Modifications of this construction permit of determining the point of cut-off for no drop in triple or quadruple engines with any phase relation of the cranks. 478. Cross-compound Engine: Analytical Method. In this case, the fraction of the stroke completed at low-pressure cut-off is different for the two cylinders. Let X be the proportion of the high-pressure stroke occurring between admission and cut-off in the low-pressure cylinder. Proceeding as before, the volume of the steam at low-pressure admission is 0.5 +R, and that at low-pressure cut-off is 0.5 X+R -}-xL. The volume of steam in the high-pressure cylinder and the receiver at the end of the high-pressure exhaust stroke is R; the volume just after low-pressure cut-off occurs is 0.5 X+R. The volume at the beginning of exhaust from the high- pressure cylinder is l+R. In Fig. 208, let the pressure at C and I be p; let that at ffbeP. Then CROSS-COMPOUND ENGINE 331 XI + ) = ^(0.5 + R-) 01 P= Let the pressure at H be Q : then P(0.5 + R) = Q(0.5 - X + tf + . 0.5 + Jrc 0.5 - X + ,tt + sL* f (0.5 - X + j whence, -y=0.5 + ^-: (A) In Pig. 209, we have the crank circles corresponding to the discussed movements. If Ow and Ox are at right angles, then for a high-pressure pis- ton displacement Oy, we have the corresponding low-pres- sure displacement kz. If these displacements be taken as at low-pressure cut-off, then ~~ A* "jk We may also draw OwP^ PQ, and write X = -~z In the mm Arts. 47r i 478.-Cr^k Circles and Piston ^^ O p^ Q ^ QQ = xzjk . X, xz 2 + ~Oz'^ 5?, and (jk - X) z + (*- x - jk\ = ( 4-J J whence X = Var ar 2 . Substituting this value iu Equation (A), we find jR (ar 1) = 0.5 Var z 2 as the condition of no drop. 479. Practical Modifications. The combined diagrams obtained from actual engines conform only approximately to those of Figs. 205 and 208. The receiver spaces are. usually so large, in proportion to the volume of the high-pressure cylinder, that the fluctuations of pressure along the release lines are scarcely notice- able. The fall of pressure during admission to the low-pressure cylinder is, how- ever, nearly always evident. Marked irregularities arise from the angularity of the connecting rod and from the clearance spaces. The graphical constructions may easily be modified to take these into account. In assuming crank positions and piston displacements to correspond, we have tacitly assumed the rod to be of infinite length; in practice, it seldom exceeds five or six times the length of the crank. We have assumed all expansive paths to be hyperbolic; an assumption not strictly justified for the conditions considered. 482. Superheat and Jackets. Since multiple expansion itsalf decreases cylinder condensation, these refinements cannot be expected 332 APPLIED THERMODYNAMICS to lead to such large economies as in simple engines. Adequately superheated steam has, however, given excellent results, eliminating cylinder condensation so perfectly as to permit of wide ranges of expan- sion without loss of economy and thus making the efficiency of the engine, within reasonable limits, almost independent of its load. ^ The best test records have been obtained from jacketed engines, A simple engine with highly superheated steam (see Chapter XV) will be nearly as economical as a compound with saturated steam. 483. Binary Vapor Engine. This was originated -by Du Tremblayin 1850 (26). The exhaust steam from a cylinder passed through a vessel containing coils filled with ether. The steam being- at a temperature of almost 250 F., vrhile the atino^ihmc boiling point of ether is 94 F., the latter was rapidly vaporized at a considerable pressure, and was then used for performing work in a second cylinder. Assuming the initial temperatuie of the steam to have been 320 F., and the final temperature of the ether 100 F,, the ideal efficiency should thus be increased from 320 - 250 = 09 to 320 - 100 _. 320 + 460 ~~ ' 320 + 460 a gain of over 200 per cent. The advantage of the binary vapor principle arises from the low boiling point of the binary fluid. This permits of a lower tempera- ture of heat emission than is possible with ^ater. Binary engines must be run condensing. Since condensing water is generally not available at temperatures below f>0' or TO J F., the fluid should be one which may be condensed at these tem- peratures. Etliw satisfies this requirement, and gives, at its initial temperature of, >ay, SoO^ F., a woiking pressure not far fiom 151) Ib. On account of its high boiling point* however, its pressure is less than that of the atmosphere at 70 F. ? and an air pump w m- cessary to discharge the condensed vapor from the condenser just us is the case with condensing steam engines. Sulphur dioxide has a much lower boiling point, and may be used without an air pump: but its pressure at 250 would be excessive, and the best results are secured by allowing the steam cylinder to run condensing at a final temperature as low as pos- sible ; at 104 3 F., the pressure of sulphur dioxide is only OO.o Hi. The best steam engines have about this lower temperature limit; the maximum gain due to the use of a binary fluid cannot exceed that corresponding to a reduc- tion of this temperature to about 60 or 70 F., the usual temperature of the available supply of cooling- water. The steam-ether engines of the vessel Brestt operated at 43.2 Ib. boiler pressure and 7.G Ib. back pressure of ether. The cylinders were of equal size, and the mean effective pressures were 11,6' and 7. 1 Ib. respectively. The Fio.217. Art.483, Prob. coal colwlllll ption was brought down to 2.44 Ib. per ra-Biuaiy Vapor Eu- Ihl) _ br> . ft ^ favorab le result than that obtainable from glne> good .steam engines of that time. Several attempts have THE INDICATOR 333 been made to revive the binary vapor engine on a small scale, the most important recent experiments are those of Josse (27), on a 200-hp. engine using steam at 160 Ib. pressure and 200 of superheat, including four cylinders. The first three cylinders constitute an ordinary triple-condensing steam engine, a vacuum of 20 to 25 in of mercury being maintained in the low-pressure cylinder by the circula- tion of sulphur dioxide in the coils of a surface condenser. The dioxide then enters the fourth cylinder at from 120 to 180 Ib .pressure and leaves it at about 35 Ib. pressure. The best result obtained gave a consumption of 167 B t u, per Ihp. per minute, a result scarcely if ever equaled by a high-grade steam engine (Art 550). The ideal entropy cycle for this engine is shown m Fig. 217, the three steam cylinders being treated as one. The steam diagram is abode, and the heat delivered to the sulphur dioxide vaporizer is aerm This heated the binary liquid along M and vaporized it along ?/, giving the work area hifg. The different liquid lines and saturation curves of the two vapors should be noted The binary vapor principle has been suggested as applicable to gas engines, in which the temperature of the exhaust may exceed 1000 F. ENGINE TESTS* 484. The Indicator. Two special instruments are of prime importance in measming the perfoimance of an engine. The first of these is the indicator, one of the secret inventions of Watt (28), which shows the action of the steam in the cylinder. Some conception of the influence of this device on progress in economical engine operation may be formed from the typically bad and good dia- grams of Fig. 218. The indicator furnishes a method for computing the mean effective pres- sure and the horse power of any cylinder. Figure 219 shows one of the many common forms. Steam is admitted from the engine cylin- der through 6 to the lower side of the movable piston 8. The fluctuations of pressure in the FIG. 218. Arts. 484, 486. Good and Bad Indicator Diagrams. cylinder cause this piston to rise or fall to an extent determined by the stiffness of the accurately calibrated spring above it. The piston movements are trans- mitted through, the rod 10 and the parallel motion linkage shown to the pencil 23, where a perfectly vertical movement is produced, in definite proportion to the movement of the piston 8. By means of a cord passing over the sheaves 37, 27, a to-and-fro movement is communicated from the crosshead of the engine to the drum, 24. The movements of the drum, under control of the spring, 31 J are made just proportional to those of the piston; so that the coordinates of the diagram traced by the pencil on the paper are pressures and piston movements. 485. Special Types. Various modifications are made for special applications. For gas engines, smaller pistons are used on account of the high pressures; springs of various stiffnesses and pistons of various areas are employed to permit of accu- rately studying the action at different parts of the cycle, safety stops being pro- vided in connection with the lighter springs. The Mathot instrument, for example, gives a continuous record of the ignition lines only of a series of suc- * See Trans, 4, 8* M, E., XXIV 7 713; Jour, 4.& M. J&, XXXIV, 11, 334 APPLIED THERMODYNAMICS cessiye gas engine diagrams. "Outside-spiing" indicators are a recent type, in which the spring is kept away from the hot steam. The Ripper mean-pressure indicator (29) is a device which shows continuously the mean effective pressure in the cylinder. Instruments are often provided with pneumatic or electrical operating mechanisms, permitting one observer to take exactly simultaneous dia- grams from two or more cylinders. Indicators for ammonia compressors must luive ail internal parts of steel; special forms are also constructed for heavy hy- FIG. 219. Art. 484. Crosby Steam Engine Indicator. draulic and ordnance pressure measurements. For very high speeds, in \\hich the inertia of the moving parts would distort the diagram, optical indicators are used. These comprise a small mirror which is moved about one axis by the pressure and about another by the piston movement. The path of the beam of light is pre- served by photographing it. Indicator practice constitutes an art in itself; for the detailed study of the subject, with the influence of drum reducing motions, methods of calibration, adjustment, piping, etc., reference should be made to such works as those of Carpenter (30) or Low (31). In general, the height of the dia- gram is made of a convenient dimension by varying the spring to suit the maxi- mum pressure; and accuracy depends upon a just proportion between (a) the movements of the drum and the engine piston and (6) the movement of the indi- cator piston and the fluctuations in steam pressure. INDICATOR DIAGRAMS 335 486. Measurement of Mean Effective Pressure. This may be accomplished by averaging a laige number of equidistant ordinates across the diagram, or, mechanically, by the use of the planimeter (32). In usual practice, the indicator is either piped, with intervening valves, to both ends of the cylinder, in which case a pair of diagrams is obtained, as in Fig. 218, one cycle after the other, representing the action on each side of the piston ; or two diagrams are obtained by separate indicators. In order that the diagrams may be complete, the lines ab, representing the boiler pressuie, cd, of atmospheric pressure, and efof vacuum in the condenser, should be drawn, together with the line of zero volume ea^ determined by measur- ing the clearance, and the hyperbolic curve (/, constructed as in Art. 92. The saturation curve gh for the amount of steam actually in the cylinder is sometimes added. As drawn in Fig 218, the position of the saturation curve indicates that the steam is dry at cut-off scarcely the usual condition of things. 487. Deductions. By taking a "full-load" card, and then one with the ex- ternal load wholly removed, the engine overcoming its own frictional resistance only, we at once find the me- chanical efficiency, the ratio of power exerted at tie shaft to power developed in the cylin- der; it is the quotient of the difference of the two diagrams by the former. By measur- ing the pressure and the vol- ume of the steam at release, and deducting the steam pres- ent during compression, we may in a rough way com- \ pute the steam consumption per Ihp.-hr., on the assumption that the steam is at this point dry; and, as in Art. 500, by properly estimating the per- centage of wetness, we may closely approximate the actual steam consumption. Some of the applications of the indicator are suggested by the diagrams of Fig. 220. In a, we have admission oc- curing too early; in b, too late. Excessively early cut-off is shown in c ; late cut-off, with excessive terminal drop, in d. Figure e indicates too early release ; the dotted curve would give a larger wort area; in f, release is late. The bad effect of early compression is indicated in g ; late com- pression gives a card like that of h, usually causing noisiness. Figure * shows excea- FIG. 320. Art. 487. Indicator Diagiams and Valve Adjustment. 336 APPLIED THERMODYNAMICS sire throttling during admission; / indicates excessive resistance during exhaust \\hich may be due to thiotthng or to a poor vacuum. The effect of a small supply pipe is shown in k, in which the upper line repiesents a diagram taken with the indicator connected to the steam chest. The abrupt rise of pressure along LC is due to the cutting off of the flow of steam from the steam chest to the cylinder. Figure I shows the fomi of card taken when the drum is made to derive its mo- tion from the eccentric instead of the croabhead. This is often done in order to study more accurately the conditions near the end of the stroke when the piston moves veiy slowly, while the eccentric moves more rapidly. Figure m is the coi- responding ordinary diagram, and the two diagrams are correspondingly letteied. Figure is an excellent card from an air compressor ; o shows a card from an air pump with excessive poit friction, particularly on the suction side. Figure j> shows what is called a stroke card, the dotted line representing net pressures on the piston, obtained by subtracting the back pressure as at cib from the initial pressure uc, i.e. by making tic = alt. Figure q shows the effect of varying the point of cut-off; r, that of throttling the supply. Negative loops like that of g must be deducted from the remainder of the diagram in estimating the work done. 488- Measurement of Steam Quality. The second special instrument used in engine testing is the steam calorimeter, so called because it determines the percent- age of dryness of steam by a series of heat measurements. Carpenter (33) classi- fies steam calorimeters as follows : (a) Condensing Calorimeters ' Barrel or tank Continuous Jet - Surface .Kent External Barrus Barrus Continuous Hoadley () Superheating J Separator ^ ' \ Chemical 489. Barrel or Tank Calorimeter. The steam Is discharged directly into an insulated tank containing cold water. Let W, w be the weights of steam and water respectively, t, ti the initial and final temperatures of the water, correspond- ing to the heat quantities h, hi ; and let the steam pressure be P 0j corresponding to the latent heat L Q and heat of liquid ho, the percentage of dryness being zo- The heat lost by the steam is equal to the heat gained by the water ; or, the steam and water attaining the same final temperature, W(x*Lo + ho - Ai) = (*! - A), whence * = M" + ^)-^-W%o . The value of IT is determined by weighing the water before and after the mix- ture. The radiation corrections are large, and any slight error in the value of W CALORIMETERS 337 greatly changes the result; this foim of calorimeter is therefore seldom used, its average error even under the best conditions ranging from 2 to -t per cent. Some improvement is possible by causing condensation to become continuous and tak- ing the weights and temperatures at frequent intervals, as in the " Injector " or " Jet Continuous " caloi imeter. 490. Surface-condensing Calorimeter. The steam is in this case condensed in a coil ; it does not mingle with the water. Let the final temperatui e of the steam be fe, its heat contents //a ; then - h) and x = More accurate measurement of W is possible with this arrangement. In the Hoadley form (34) a propeller wheel was used to agitate the u ater about the coils; in the Kent instrument, arrangement was made for removing the coil to peimit of more accurately determining W. In that of Barrus, the flow was continuous and a series of observations could be made at short intervals. 491. Superheating Calorimeters. The Peabody throttling calorimeter is shown in Fig. 221 ; steam entering at b through a partially closed valve expands to a lower steady pressure in A and then flows into the atmos- phere. Let L Q , 7i , x be the condition at b, and assume the steam to be superheated at A, its temperature being T, t being the temperature corresponding to the pressure p, and the cor- responding total heat at saturation H. Then, the total heat at I equals the total heat at A, or (%L + AO)= #+ Tt(T- f), where 7c is the mean specific heat of superheated steam at the pressure p between Tand ; whence If we assume the pressure in A to be that of the atmos- phere, 27" =1150.4, and superheating is possible only when x L Q + h exceeds 1150.4. For each initial pressure, then, there is a corresponding minimum value of x^ beyond which measurements are impossible; tlms, for 200 lb., FIG. 221 Art 401. L Q = 843.2, ft = 354.9, and a*, (minimum) is 0.94. Aside from this limitation, the throttling calorimeter is exceed- ingly accurate if the proper calibrations, corrections, and methods of sampling are adopted. In the Barrus throttling calorimeter, the valve at b is replaced by a diaphragm through which a fine hole is drilled, and the range of C values is increased by mechanically separating some of the moisture. The same advantage is realized in the Barrus superheating calorimeter by initially and externally heating the sample of steam. The Superheat- ing Calorimeter. 338 APPLIED THERMODYNAMICS amount of heat thus used is applied in such a way that it may be ac- curately measured. Let it be called, say, Q per pound. Then -f)-h Q - Q 492. Separating Calorimeters. The water and steam are mechanically sepa- rated and separately -weighed. In Fig. 222, steam enters, through 6, the jacketed chamber shown. The water is intercepted by the cup 14, the steam reversing its direction of flow at this point and entering the jacket space 7, 4, whence it is discharged through the small orifice 8. The water ac- cumulates in 3, its quantity being indicated by the gauge glass 10. The quantity of steam flowing is de- termined by calibration for each reading of the gauge at 9. The instrument is said to be fairly accurate un- less the percentage of moisture is very small. The steam may be, of course, run off, condensed, and actually weighed. 493- Chemical Calorimeter. This depends for its action on the fact that water will dissolve certain salts (e.g. sodium chloride) which are insoluble in. dry steam. 494- Electric Calorimeter. The Thomas superheat- ing and throttling instrument (35) consists of a small soapstoue cylinder in which are embedded coils of German silver wire, constituting an electric heater. 3 . is inserfced in a brass ea8e thr U S h which fl WS a current of steam. The electrical energy correspond- to heat-augmentation to any superheated condition being known, say, as . t.u. per pound (1 B, t. u. per minute = 17.59 watts), we have, as in Art. 491, or Z n + 7/ + E = JT+ k(T- ), whence ar = H + k ( T ~ ')-* " B . ing E B 495. Engine Trials: Heat Measurement. "We may ascertain the heat supplied in the steam engine cycle either by direct measurement, or by adding the heat equivalent of the external work done to the measured amount of heat rejected. In the former case the amount of water fed to the boiler must be determined, by weighing, measuring, or (in approximate work) by the use of a water meter. The heat absorbed per pound of steam is ascer- tained from its temperature, quality, and pressure, and the temperature of the water fed to the boiler. In the latter case, the steam leaving the engine is condensed and, in small engines, weighed; or in larger engines, determined by metering or by passing it over a weir. This latter of the two methods of testing has the advantage with small engines of greater ENGINE TEST 339 accuracy and of giving accurate results in a test of shorter duration. Where the engine is designed to operate non-condensing, the steam may be con- densed for the purposes of the test hy passing it over coils exposed to the atmosphere, so that no vacuum is produced by the condensation. If jackets are used, the condensed steam from them must be trapped off and weighed. This water would ordinarily boil away when discharged at atmospheric pressure, so that provision must be made for first cooling it. 496. Heat Balance. By measuring loth the heat supplied and that rejected, as well as the work done, it is possible to draw up a debit and credit account show- ing the use made of the heat and the unaccounted for losses. These last are due to the discharge of water vapor by the air pump, to radiation, and to leakage. The weight of steam condensed may easily be four or five per cent less than that of the water fed to the boiler. Let 71, h, be the heat contents of the steam and the heat in the boiler feed water respectively; the heat absorbed per pound is then H h. Let Q be the heat contents of the exhausted steam (measured above the feed water temperature) and W the heat equivalent of the work done per pound. Then for a perfect heat balance, H h = Q -f W. In practice, W is directly computed from the indicator diagrams ; H and Q must be corrected for the quality of steam as determined by the calorimeter or otherwise. The heat charged to the engine is measured from the ideal feed- water temperature corresponding with the pressure of the atmosphere or condenser to the condition of steam at the throttle: that is, it is (in general symbols), R =Q(H- A ), B. t. u. per Ihp. hr., where Q represents the steam consumption in Ib. per Ihp. hr. Then 2545 +R is the thermal efficiency =E. Let H be the total heat above 32 after adiabatic expansion in the Clausius cycle: then the ideal efficiency is and the " efficiency ratio " or relative efficiency is E 2545 The efficiency ratio referred to the Carnot cycle is correspondingly 2545 T ^ c 'Q(H-h )(T-ty where T and t are, respectively, the absolute temperatures at the throttle and corresponding with atmospheric or condenser pressure. In working up a heat balance, it is convenient to measure all heat 340 APPLIED THERMODYNAMICS quantities above 32. The gross heat charged to the engine is then HQ, less any transmission losses between boiler and engine. If the engine runs" condensing, and Qi Ib. of condenser water circulated rise from *i to t 2 F., the heat rejected to the circulating water is -^i) B. t. u. There are also rejected, in the condensed steam, ~. t. u., where h 3 is the heat of liquid corresponding with the tem- perature t 3 of the condensed steam. (Note that t 3 = fe in jet condensing engines.) Some of the heat thus rejected may, however, be returned to the boiler, and should then be credited, the amount of credit being the sum of the weights returned each multiplied by the respective heat of liquid. Any steam condensed in the jackets is charged to the engine, but the heat rejected from the jackets (usually returned to the boiler) is then credited as Q 2 h where Qj is the weight of steam con- densed and h the heat of liquid corresponding with its pressure (usually the throttle pressure). 497. Checks; Codes. Where engines are used to drive electrical generators the measurement of the electrical energy gives a close check on the computation of indicated horse power. Let G= generator output in kilowatts, E& = generator efficiency ,E m = mechanical efficiency of the umt,#=Ihp. of engine- then 1.34G = HEffEm. In locomotive trials a similar check is obtained by comparison of the drawbar pull and speed (36). In turbines, the indicator cannot be employed, measurement of the mechanical power exerted at the shaft is effected by the use of the friction brake. Standard codes for the testing of pumping engines (37), and of steam engines generally (38;, have been developed by the American Society of Mechanical Engineers. 3TiG. 224. Arts. 498, 499, 500. Indicator Cards from Compound Engine. 498. Example of an Engine Test* Figure 22 i, from Hall (39), gives the indicator diagrams from a 30 and 56 by 72-in. compound engine at 58 r. p. m. The piston rods were 4J and 5J in. diameter. The boiler * Values from steam tables, used in this article, do not precisely agree with those given on pp. 287, 288. ENGINE TEST 341 pressure was 124.0 Ib. gauge: the pressure in the steam pipe near the engine, 122.0 Ib. The temperature of jacket discharge was 338 F. The conditions during the calorimetric test of the inlet steam were P = 122.08 Ib. gauge, T = 302.1 F. (Art. 491), pressure in calorimeter body (Fig. 221), 11.36 Ib. (gauge). The net weight of boiler feed water in 12 hours was 231,861.7 Ib. ; the weight of water drained from the jackets, 15,369.7 Ib. Prom the cards, the mean effective pressures were 44.26 and 13.295 Ib. respectively; and as the average net piston areas were 697.53 and 2452.19 square inches respectively, the total piston pressures were 44.26 X 697.53=30872.7 and 13.295 x 2452.19=32601.9 Ib. respectively. These were applied through a distance of if X 2 x 58 = 696 feet per minute; whence the indicated horse power was (30872.7+32601.9) X 696 = 33000 From Art. 491> A>+^o = &+& (Tfy or in this case, 866.5 x + 322.47 = 1155.84 + 0.48* (302.1-242.3) whence X Q = 0.995. The weight of cylinder feed was 231,861.7 15,369.7 = 216,492.0 Ib. At its pressure of 136.7 Ib. absolute, =866.5, ft = 322.4. Tor the ascertained dryness, the total heat per pound, above 32, is 322.4 + (0.995x866.5) =1184.5 B. t u. The heat left in the steam at discharge from the condenser (at 114 F.) was 82 B. t. u. ; the net heat absorbed per pound of cylinder feed was then 1184.5 82.0 = 1102.5; for the total weight of cylinder feed it was 1102.5 x 216,492 = 238,682,430 B. t. u. The total heat in one pound qf jacket steam was also 1184.5 B. t. tu This was discharged at 338 F. (7i = 308.8), whence the heat utilized in the jackets was 1184.5 308.8 = 875.7 B. t. u. (The heat discharged from both jackets and cylinders was transferred to the boiler feed water, the former at 338, the latter at 114 F.) The supply of heat to the jackets was then 875.7 x 15,369.7 =i 13,459,246.29 B. t. u: the total to cylinders and jackets was this quan- tity plus 238,682,430 B. t. u., or 252,141,676.29 B. t. u. Dividing this by 60 x 12 we have 350,196.77 B. t. u. supplied per minute. 499. Statement of Results. We have the following : (a) Pounds of steam per Ihp.-hr. = 231,861.7 -s- 12 -=- 1338.62 = 14.43. (This is the most common measure of efficiency, but is wholly unsatisfactory when superheated steam is used.) * Value taken for the specific heat of superheated steam. 342 APPLIED THERMODYNAMICS (6) Pounds of dry steam per Ihp.-hr. = 14.43 x 995 * = 14.36. (c) Heat consumed per Ihp. per minute = 350,196.77 -*- 1338.62 = 261.61 B. t. u. ($) Thermal efficiency = ^f^-*- 261.61 = 0.1621. (e) Work per pound of steam=?^%^^ X 0.1621 = 176 B. t. u. l. I CO Camotefficiency^- =0.293. (gf) Clausius efficiency (Art. 409), with dry steam, 810.82 ^1=0.265. 351.22-114+866.5 (&) Ratio of (<*)-*-&) = 0.1621 -4- 0.265 = 0.61. 500. Steam Consumption from Diagram. The inaccuracy of such estimates will be shown. In the high-pressure cards, Pig. 224, the clearance space at each end of the cylinder was 0.932 cu. ft. The piston displacement per stroke on the side opposite the rod was 706.86 x 72 - 1728 = 29.453 cu. ft.; the cylinder volume on this side was 29.453 + 0.932 = 30.385 cu. ft. The length of the coriespond- ing card (a) is 3.79 in. ; the clearance line Ic is then drawn distant from the admission line 3.79 x -^?i = 0.117 in. 29.453 At rf, on the release line, the volume of steam is 30.385 cu. ft., and its pressure is 31.2 Ib. absolute. From the steam table, the weight of a cubic foot of steam at this pressure is 0.076362 Ib.; whence the weight of steam present, assumed dry, is 0.076362 x 30.385 = 2.3203 Ib. At a point just after the beginning of compres- sion, point e, the volume of steam expressed as a fraction of the stroke plus the clearance equivalent is 0.517 *- 3.907 = 0.1321, 3.907 being the length bg iu inches. The actual volume of steam at e is then 0.1321 x 30.385 = 4.038 cu. ft., and its pressure is 28.3 Ib. absolute, at which the specific weight is 0.069683 Ib. The weight present at e is then 4.038 x 0.069683 = 0.2SO Ib. The net weight of steam used per stroke is 2.3203 - 0.280 = 2.0403 Ib., or, per hour, 2.0403 x 58 x 60 = 7090 Ib., for this end of the cylinder only. For the other end, the weight, similarly obtained, is 7050 Ib. ; the total weight is then 14,140 Ib. The horse power developed being 1339, the cylinder feed per Ihp -hr. from high-pressure diagrairs is 10.6 Ib., or 26 per cent less than that which the test shows. The same process may be applied to the low-pressure diagrams. It is best to take the points d and e just before the beginning of release and after the beginning of compression respec- * The factor 0.995 does not precisely measure the ratio of energy in the actual steam to that in the corresponding weight of dry steam, but the correction is usually made in this way. MEASUREMENT OF REJECTED HEAT 343 tively. The method is widely approximate, but may give results of some value in the absence of a standard trial (Arts 448, 4iO). 501. General Expression. In Fig. 224a, let -7=^, =D. Let the cylinder L L area be A sq in., the stroke S ft , the clearance d = m(Ld)=mAS: and let the speed be n r, p. m. The horse power of the double- acting engine is ZpnASn n 33,000 ' for p m Ibs. mean effective pressure per square inch. The weight of steam used per stroke, in pounds, is w = BAS(1 +m) DASQ +m) 501. Steam am. M/1+ \ (JL-JL\ FlG 224a ' Ali SOI. St 144 \ / \xv XVo) ' Consumption from Diagr where v and V are the specific volumes of dry steam and x and X are the dryncss of the actual steam, at d and e respectively Making X-x = l.Q, we find (from the indicator diagrams alone) the weight of steam consumed per Ihp. hour to be in pounds, 13,750(1 +m)/B D\ HP. " p m U vj" In applying this to compound engines, p m must be taken as the total equivalent mean effective pressure "referred to" to the cylinder of area A (Art. 472). For the conditions of Art. 500, p*=44.26+ f|^X13.295J =90.36, and the steam rate is /1.0317\ / 30.385 4.038 \ , n41 , *' U \ 90.36 / \30.3S5X13.24 30.385 X14.53/ U< * 1Dt 502. Measurement of Rejected Heat A common example is in tests in which the steam is condensed by a jet condenser (Art. 584). In a test cited by Ewing (40), the heat absorbed per revolution measured above the temperature of the boiler feed was 1551 B. t u. ; that converted into work was 225 B. t. n. The exhaust steam was mingled with the condensing water, a combined weight of 51.108 Ib. being found per revolution. The temperature of the entering water was 50 F., that of the discharged mix- ture was 73.4 F. ? and the cylinder feed amounted to 1.208 Ib. per revolu- tion. The temperature of the boiler feed water was 59 F. We may compute the injection water as 51.108 1.208 = 49.9 Ib. and the heat absorbed by it as approximately 49.9(73.4 50) = 1167 B. t. u. The 1.208 Ib. of feed were discharged at 73.4, whereas the boiler feed was at 59 ; a heat rejection of 73.4 - 59 = 14.4 occurred, or 14.4 x 1.208 = 17.4 344 APPLIED THERMODYNAJV1ICS B. t. u. The total heat rejection was then 1167 + 17.4 = 1184.4 B. t. u., to which we must add 47 B. t. u. from the jackets, giving a total of 1231.4 B. t. u. Adding this to the work done, we have 1231.4 + 225 = 1466.4 B. t. u. accounted for of the total 1551 B. t. u. supplied; the discrepancy is over 6 per cent. "When surface condensers are used, the temperatures of discharge of the condensed steam and the condenser water are different and the weight of water is ascertained directly. In other respects the computation would be as given.* 503. Statements of Efficiency. Engines are sometimes rated on the basis of fuel consumption. The duty is the number of foot-pounds of work done in the cylinder per 100 pounds of coal burned (sometimes and preferably the number of foot-pounds of work per 1 } 000,000 B. t. u consumed at coal. The efficiency of the plant is the quotient of the heat converted into work per pound of coal, by the heat units contained in the pound of coal. In the test in Art. 498, the coal consumption per Ihp.-hr. was 2068.84-J-1338,62 = 1.54 Ib. In some cases, all state- ments are baaed on the brake horse power instead of the indicated horse power. The ratio of the two is of course the mechanical efficiency. It may be noted that the engine is charged with steam, not at boiler pressure, but at the pressure in the steam pipe. The difference between the two pressures and qualities represents a loss which may be considered as dependent upon the transmissive efficiency. The plant efficiency is obviously the product of the efficiencies of boiler (Art. 574), transmission, and engine. 504. Measurement of Heat Transfers: Hirn's Analysis. In the refined methods of studying steam engine performance developed by Hirn (41), and expounded by Dwelshauvers-Dery (42), the heat absorbed and that rejected are both measured. Dur- ing any path of the cycle, the heat inter- change between fluid and walls is computed from the change in internal energy, the heat externally supplied or discharged, and the external work done. The internal energy of steam is, in general symbols, h+xr. The heat received being Q, _ __ ___ ___ and the heat lost by radiation Q', we have . 225. Art. 504. Hirn's Analysis, the general form where the path is, for example, from 1 to 2, and the weight of steam increases from wi to tr*. Applying such equations to the cycle, Fig. 225, made up of the four * It is most logical to charge the engine with the heat measured above the tem- perature of heat rejection. This, in Tig. 182, for example, makes the efficiency d&bc dsoc rather than ~-~ *m the ordinate FJT representing the feed-water temperature, HIRN'S ANALYSIS 345 operations 01, 12, 23, 30, we have, M Q denoting the weight of clearance steam and M that of cylinder feed, per stroke, in pounds. Ei = (M +M) (^ Let Q a , Q&, Q c , Qd, represent amounts of heat transferred to the walls along the paths a, b, c, d. Consider the path a. Let the heat supplied by the incoming steam be Q. Then Along the path 5, -Q&-TPH-(#--Ei); along d, - Along c , heat is carried away by the discharged steam and by the cooling water. Let G denote the weight of cooling water per stroke, k 5 and hi its final and initial heat contents, and h the heat contents of the discharged steam. The heat rejected by the fluid per stroke is then G(h & -h 4 )-\-Mk 6 . Then Q c -G(h tt -h^Mh^ = -W C +(E S -E 2 ), and Q c ^-G(k s -h t )-Mh 9 +W c -(St-S^ Values for the h and r quantities are obtained from the steam table for the pres- sures shown by the indicator diagram. The diagram gives also the work quantities along each of the four " paths." The conditions of the test give Q 3 O t h s , h& t h*, and M. The remaining unknown quantities are M Q and the drynesses. MQ is found by assuming #3 = ! (see Art 500). Then the dryness at any of the remaining points 0, 1, 2, may be found by writing v x = , WVo where v is the volume shown by the indicator diagram, v is the specific volume of dry steam and w is the weight of steam present, at the point in question. The quantity w will be equal to M or (M +Mo) as the case may be. 505. Graphical Representation. In Fig. 226, from the base line xy, we may lay off the areas oefs representing heat lost during admission, smba showing heat gained during expansion, mhcr showing heat gained during release, and oakr showing heat lost during compression. If there were no radiation losses from the walls to the atmosphere, the areas above the line xy would just equal those below it. Any excess in upper areas represents radiation losses. Ignoring these losses, Him found by comparing the work done with the value of Q Mh* G(h & A 4 ) an approximate value for the mechanical equivalent of heat (Art. 32). Analytically, if Q T denote the loss by radiation, its value is the algebraic sum of Q a , Q&, Q c , Q&. If the heat Q 3 be supplied by a steam jacket, then Q r = Qj + sQo, 6 . c, d- The heat transfer during release, Q c , regarded by Him as in a special sense a measure of wastefulness of the walls, may be expressed as Q T Qj~- S^ fl , b . d In a non-condensing engine, Q r can be determined only by direct experiment. 505a. Testing of Regulation, The " regulation " of a steam engine refers to its variations in speed. In most applications uniformity of rotation is important. This is particularly the case when engines drive electric generators, and the momen- FIG. 226. Art. 505. Heat Transfers. 346 APPLIED THERMODYNAMICS tary or periodic variations in speed must be kept small regardless of fluctuations in initial pressure, back pressure, load or ratio of expansion. This is accomplished by using a sensitive governor and a suitably heavy fly-wheel. Regulation cannot be studied by unaided observation with a revolution counter or by an ordinary recording instrument. An accurate indicating tachometer or some special optical device must be employed (Trans. A. S. M. E., XXIV, 742). TYPES OF STEAM ENGINE 506.; Special Engines. "We need not consider the commercially unimportant class of engines usmg vapors other than steam, those experimental engines built for educational institutions which belong to no special type (43), engines of novel and limited application like those employed on motor cars (44), nor the " fireless " or stored hot-water steam engines occasionally employed for locomotion (45). 507. Classification of Engines. Commercially important types may be con- densing or non-condensing. They are classified as right-hand or left-hand, accord- ing as the flywheel is on the right or left side of the center line of the cylinder, as viewed from the back cylinder head. They may be simple or multiple-expan- FIG. 23$. Art- 607- An^le-Compouud Engine. (American Ball Engine Company.) TYPES OF ENGINE 347 sion, with all the successive stages and cylinder arrangements made possible in the latter case. They may be single-acting or double-acting ; the latter is the far more usual arrangement. They may be rotative or non-rotative. The direct-acting pumping engine is an example of the latter type; the work done consists in a rectilinear impulse at the water cylinders. In the duplex engine, simple cylinders are used side by side. The terms horizontal, vertical, and inclined refer to the posi- tions of the center lines of the cylinders. The horizontal engine, as in Figs. 186 and 229, is mostly used in land practice ; the vertical engine is most common at FlG. 229. Art. 607. Automatic Engine. (American Ban Engine Company.) sea. Cross-compound vertical engines are often direct-connected to electric gen- erators. Vertical engines have occasionally been built with the cylinder below the shaft. This type, with the inclined engine, is now rarely used. Inclined engines have been built with oscillating cylinders, the use of a crosshead and connecting rod being avoided by mounting the cylinder on trunnions, through which the steam was admitted and exhausted. Figure 228 shows a section of 348 APPLIED THERMODYNAMICS the interesting angle-compound, in which a horizontal high-pressure cylinder exhausts into a vertical lou -pressure cylinder. A different type of engine, but with a similar structural arrangement, has been used in some of the largest power stations. Engines are locomotive, stationary, or marine. The last belong in a class by themselves, and will not be illustrated hei e ; their capacity ranges up to that of our laigest stationary power plants. Stationary engines are further classed as pumping engines, mill engines, power plant engines, etc. They may be further grouped accoiding to the method of absorbing the power, as belted, direct-con- nected, rope driven, etc An engine directly driving an air compressor is shown in Pig. 86. <k Rolling mill engines'* undergo enormous variations in load, and must have a correspondingly massive (tangye) frame. Power plant engines gen- erally mast be subjected to heavy load variations; their frames are accordingly usually either tangye or semi-tangye. Mill engines operate at steadier loads, and have frequently been built with light girder frames. Modern high steam pressures have, however, led to the general discontinuance of this frame in favor of the semi-tangye. A slow-speed engine may run at any speed up to 125 r. p.m. From 125 to 200 r.p.m. may be re- garded as medium speed. Speeds above 200 r.p.m. are regarded as high. Certain types of engine are adapted only for certain speed ranges ; the ordinary slide-valve engme, shown in Fig. ISO, may be oper- ated at almost any speed. For ]arge units, speeds range usually from 80 to 100 r.p.m. The higher- speed engines are considered mechanically less re- liable, and their valves do not lend themselves to quite as economical a distribution of steam. An important / /\ 8 class of medium-speed engines has, however, been in- troduced, in which the independent valve action of the Coiliss type has been retained, and the promptness of cut-ofE only attainable by a releasing gear has been approximated. In some cheap high-speed engines governing is effected simply but uneconomically by throttling the steam supply. Such engines may have shallow continuous frames or the sub-base, as in Fig. 220, which represents the large class of automatic high-speed engines in which regulation is effected by automatically varying the point of cut-off. Figure 230 shows three sets of indicator diagrams from a com- pound engine of this type, running non-condensing at various loads. Some of the irregulations of these diagrams are without doubt due to indicator inertia; but they should be care- fully compared with those showing the steam distribution with a slow-speed TYPES OF ENGINE 349 releasing gear, in Fig. 218. All of the so-called " automatic " engines run at medium or high rotative speeds. The throttling engine is used only in special or unimportant applications. The automatic type is employed where the comparatively high speed is admissible, in units of moderate size. Better distribution is afforded by the four -valve engine, in BacKCqlinderHtoJ. 3och Cylinder Heo Back Cyl Head Steam pipe feom Flanq* ^Throttle Valve Planished 5te) Laqqinq Heat insulating Filling iss STeomVolve Chamber 1 ^Cyl.ndtrH.od itCtjImdtr Hcod 5tud >^pnflod Gland Studs Piston Rod G'and Corliss tihomtVafvi Eihaustthe^T , ^ ^Erhoust Openmq -trhaust Pipe FiG. 231. Art. 607. Corliss Engine Details- (Murray Iron Works Company ) 350 APPLIED THERMODYNAMICS which the four events of the stroke may be independently adjusted, and this type is often tised at moderately high speeds. Sharpness of cut-off is usually obtainable only with a releasing geai, in which the mpchaiiihiu operating the valves is discon- nected, and the steam valve is au- tomatically and instantaneously closed. This feature distinguishes the Corliss type, most commonly used, in high-grade mill and power plant service. AVith the releasing -gear, usual speeds seldom exceed 100 r. p.m. The valve in a Cor- liss engine is cylindrical, and ex- tends across the cylinder. Some details of the mechanism are shown in Fig. 231. In very large engines, the releasing principle is sometimes retained, but "with poppet or other forms of valve. Figure 232 shows the parts of a typical Corliss engine with semi' tangye frame. 507a. The Stumpf Engine. Re- markable reductions in cylinder loss have been effected by the unidirec- tional-flow piston-exhaust engine of Stumpf, shown in Fig. 23 la. The piston itself acts as an exhaust valve by uncovering slots in the barrel of the cylinder at i 9 o strike* The jacketed heads form steam chests for the poppet admission valves. The piston is about half as long- as the cylinder. The ad- vantages of the engine are, very slight piston leakage, no special exhaust valve, ample exhaust ports, low clearance (1J to 2 per cent) and reduced cylinder condensation. This last is due to the continuous flow of steam from ends to center of the cylinder, which keeps the cooled and expanded steam from sweeping over the heads. (The steam in an engine cylinder is by no means in a condition of thermal equilibrium.) The condensation is so small that very large ratios of expansion arc employed, and the simple engine with either saturated or superheated steam seems to give an efficiency about equal to that attained by a triple expan- sion engine of the ordinary type. Compression is necessarily excessive: so much po that when the engine is used non-condensing a special piston valve, working ID THE STUMPF ENGINE 351 the piston, is used to prolong the exhaust period during part of the return stroke. Some of the advantages are thereby sacrificed : this modification is not necessary on condensing engines. The device has been applied to locomotives on the Prussian state railways (Engi- neering Magazine, March, 1912). The cylinders are of excessive lengths: a special valve gear, highly economical in power consumption, has been developed. The early compression (no supplementary exhaust valve being used) requires large clearance: but it is claimed that with a concave-ended hollow piston the wall surface of the clearance space (which influences the loss) is from 40 to 60 per cent less than that in an ordinary locomotive cylinder. Any initial condensation is automatically PEG. 231a. Art. 507a. The Stumpf Engine. discharged through holes hi the Cylinder wall, so that it ceases to be a factor in producing further condensation. 508.' The Steam Power Plant Figure 233, from Heck (4=6), is introduced at this point to give a conception of the various elements composing, with the engine, the complete steam plant. Fuel is burned on the grate 1; the gases from the fire follow the path denoted by the arrows, and pass the damper 4 to the chimney 5. Water enters, from the pump IV, the boiler through 29, and is evaporated, the steam passing through 8 to the engine. The exhaust steam from the engine goes through 18 to the condenser III, to which water is brought through 21. Steam to drive the condenser pump comes from 26. Its exhaust, with that of the feed pump 31, passes to the condenser through 27. The condensed steam and warmed water pass out through 23, and should, if possible, be used as a source of supply for the boiler feed. The free exhaust pipe 19 is used in case of breakdown at the condenser. 352 APPLIED THERMODYNAMICS 509. The Locomotive, This is an entire power plant, made poi table. Fig me 234 shows a typical modern form. The engine consists of t^o horizontal double acting cyl- inders coupled to the ends of the same axle at light an- gles. These are located tin- der the front end of the boiler, which is of the type described in Art. 563. A pair of heavy frames sup- ports the boiler, the load be- ing earned on the axles by means of an , intervening " spring rigging." The stack is necessarily short, so that artificial draft is provided by means of an expanding noz- zle in the "smoke bos," through which the exhaust steam passes; live steam may be used when necessary to .supplement this. The engines are non-condensing, but superheating and heat- ing of feed water, particu- larly the former, are being introduced extensively. The water is carried in an aux- iliary tender, excepting in light locomotives, in. v\ inch a *' saddle " tank may be built over the boilei . The ability of a locomo- tive to start a load depends upon the force which it can exert at the rim of the diiv- ing wheel. If d is the cylin- der diameter in inches, L the stroke in feet, and p the maximum mean etfective pressure of the steam per square inch, the work done per revolution by two equal cylinders is vd*Lp. Assume THE LOCOMOTIVE 353 this work to be trans- mitted to the point of contact between wheel and rail without loss, and that the diameter of the wheel is D feet, then the tractive power, the force exerted at the rim of the wheel, The value of p, with such valve gears as are employed on locomo- tives, may be taken at 80 to 85 per cent of the boiler pressure. The actual tractive power, and the 'pull on the drawbar, are reduced by the friction of the mechanism ; the latter from 5 to 15 per cent. Under ordinary con- ditions of rail, the wheels will slip when the tractive power ex- ceeds 0.22 to 0.25 the total weight carried by the driving wheels. This fraction of the total weight is called the adhesion, and it is useless to make the tractive power greater. In locomotives of cer- tain types, a " traction increaser " is sometimes used. This is a device for shifting some of the weight of the machine from trailer wheels to driving wheels. The weight on the drivers and the adhesion are thereby increased. The engineman, upon ap- 354 APPLIED THERMODYNAMICS preaching a heavy grade, may utilize a higher boiler pressure or a later cut-off than would otherwise be useful. 510. Compounding. Mallet compounded the two cylinders as early as 1876. The steam pipe between the cylindeis wound through the smoke box, thus becom- ing a reheating receiver. Mallet also proposed the use of a pair of tandem compound cylinders on each side. The Baldwin type of compound has two cylinders on each side, the high pressure being above the low pressure. Webb has used two ordinary outside cylinders as high-pressure elements, with a very large low-pressure cylinder placed under the boiler between the wheels. In the Cole compound, two outside low-pressure cylinders receive steam from trwo high-pressure inside cylinders. The former are connected to crank pins, as in ordinary practice, the latter drive a forward driving axle, involving the use of a crank axle. The four crank efforts differ in phase by 90. This causes a veiy regular rotative impulse, whence the name balanced compound. Inside cylinders, with crank axles, are almost exclusively used, even with simple engines, in Europe: two-cylinder compounds with both cylinders inside have been employed. The use of the crank axle has been complicated in some locomotives with a splitting of the connecting rod from the inside cylinders to cause it to clear the forward axle. G-reater simplicity follows the standard method of coupling the inside cylinders to the forward axle. 511. Locomotive Economy. The aim in locomotive design is not the greatest economy of steam, but the installation of the greatest possible power-producing capacity in a definitely limited space. Notwithstanding this, locomotives have shown very fair efficiencies. This is largely due to the small excess air supply arising from the high rate of fuel consumption per square foot of grate (Art. 564). The locomotive's normal load is what -would be considered, in stationary practice, an extreme overload. Its mechanical efficiency is therefore high. For the most complete data on locomotive trials, the Pennsylvania Railroad Report (47) should be consulted. The American Society of Mechanical Engineers has published a code (48) j Reeve has worked out the heat interchange in a specimen test by Hirn's analysis (49). (See Art. 554.) (1) D. K. Clark, Railway Machinery. (2) Isherwood, Experimental Researches in Steam Engineering ', 1863. (3) De la condensation de la vapeur, etc., Ann. des mines, 1877. (4) Bull, de la Soc Indust. de Mulhouse, 1855, et seq. (5) Proc. Inst. Civ. Eng., CXXXI. (6) Peabody, Thermodynamics, 1907, 233. (8) Min. Proc. Inst. C. E., March, 1888; April, 1893 (9) Op. cit. (10) Engine Tests, G. H. Barrus. (11) The Steam Engine, 1892, p. 190. (12) The Steam Engine, 1905, 109, 119, 120. (13) Proc Inst. Mech. Eng., 1889, 1892, 1895. (14) Ripper, Steam Engine Theory and Practice, 1905, p. 167. (15) Ripper, op. tit., p. 149. (16) Trans. A.8.M. E. f XXVIII, 10. (17) For a discussion of the interpretation of the Boulvin diagram, see Berry, The Temperature-Entropy Diagram, 1905. (18) Proc.Inst Mech. Eng., January, 1895, p. 132. (19) The Steam Engine, 1906. (21) Trans. A. S. M. E., XV. (22) Ibid., XIII, 647. (23) Ibid , XIX, 189. (24) Ibid., loc. cit. (25) Ibid., XXV, 482, 483, 490, 492. (26) Manuel du Conducteur des Machines Binaires, Lyons, 1850-1851. (27) Peabody, Thermodynamics, 1907, 283. (28) Thurston, Engine and Boiler Trials, p. 130. (29) Ripper, Steam Engine Theory and Practice, 1905, p. 412. (30) Experimental Engineering, 1907. (31) The Steam Engine Indica- SYNOPSIS 355 tor, 1898. Reference should also be made to Miller's and Hall's chapters of Prac- tical Instructions for using the Steam Engine Indicatory published by the Crosby Steam Gage and Valve Company, 1905. (32) Low, op. at., pp. 103-107; Carpen- ter, op. Git , pp. 41-55, 531, 780. (33) Op. tit., p. 391. (34) Trans. A. S. M. E., VI, 716. (35) Ibid , XXV. (36) Ibid , 1892, also XXV, 827. (37) Ibid., XI. (38) Ibid , XXIV, 713. (39) Op. ciL, 144. (40) The Steam Engine, p. 212. (41) Bull. delaSoc.Ind deMulhouse, 1873. (42) Expose Succinct, etc.; Revue Unwerselle des Mines, 1880. (43) Carpenter, Experimental Engineering, 1907, 657; Peabody, Thermodynamics, 1907, 225. (44) Trans. A S. M. E, XXVIII, 2, 225. (45) Zeuner, Technical Thermodijnamics (Klein), II, 449 (46) The Steam Engine, 1905, I, 2, 3. (47) Locomotive Tests and Exhibits at the Louisiana Purchase Expositionj 1906. (48) Trans. A. S. M. E., 1892. (49) Ibtd., XXVIII, 10, 1658. SYNOPSIS OP CHAPTER Practical Modifications of the Rankine Cycle With valves moving instantaneously at the ends of the stroke, the engine would operate in the non-expansive cycle. The introduction of cut-off makes the cycle that of Rankine, modified as follows : (1) Port friction reduces the pressure during admission. This causes a loss of availa- bility of the heat Regulation by throttling is wasteful. (2) The expansion curve differs in shape and position from that in the ideal cycle. Expansion is not adiabatic. The steam at the point of cut-off contains from 25 to 70 per cent of water on account of initial condensation. Further condensation may occur very early in the expansion stroke, followed by reevaporation later on, after the pressure has become sufficiently lowered. The exponent of the expansion curve is a function of the initial dryness. The inner surfaces only of the walls fluctuate m temperature. Condensation is influenced by (a) the temperature range : wide limits, theoretically desirable, introduce some practical losses ; (6) the size of the engine : the exposed surface is proportionately greater in small engines , (c) its speed : high speed gives less time for heat transfers ; (d) the ratio of expansion : wide ratios increase condensation and decrease efficiency, particularly because of increased initial condensation. Initial wetness facilitates the formation of further moisture. In good design, the ratio should be fixed to obtain reasonably complete expansion without *^7 Is T * excessive condensation, say at 4 or 5 to 1. M= -^i-vl . Values of T. Steam jackets provide steam insulation at constant temperature ; they oppose initial condensation in the cylinder and are used principally with slow speeds and high ratio of expansion. Some saving is always shown. Superheat, used under similar conditions, increases the mean temperature of heat absorption. Each 75 of superheat may increase the dryness at cut-off by 10 per cent. The actual expan- sion curve averages PV=pv. M.E.P.=-Pj> with the RanMne 356 APPLIED THERMODYNAMICS form of cycle. H . P . , 2 X dlagm itrtorXiiwUjr Diagnim faotor =0 .5 to 33000 0,9, With polytropic expansion, M.E.P.= ^ p D - JJ (3) The exhaust line shows back pressure due to friction of ports, the presence of air, and reevaporation. High altitudes increase the capacity of non-condensing engines. (4) Clearance varies from 2 to 15 per cent. "Real" and "apparent" ratios of expansion. (5) Compression "brings the piston to rest quietly ; though theoretically less desirable than jacketing, it may reduce initial condensation. (6) Valve action is not instantaneous, and the corners of the diagram are always somewhat rounded. Leakage is an important cause of waste. The Steam Engine Cycle on the Entropy Diagram. Cushion steam, present throughout the cycle, is not included in measurements of steam used. Its volumes may be deducted, giving a diagram representing the behavior of the cylinder feed alone. The indicator diagram shows actions neither cyclic nor reversible : it depicts a varying mass of steam. The Boulvin diagram gives the NT history correctly along the expansion curve only. The Reeve diagram eliminates the cushion steam J; it correctly depicts both expan- sion and compression curves, as referred to the cylinder feed. The preferred diagram plots the expansion and compression curves separately. Diagrams may show (a) loss by condensation, (&) gains by increased pressure and decreased back pressure, (c) gains by superheating and jacketing. Multiple Expansion Increased initial pressure and decreased back pressure pay best with wide expansive ratios. Such ratios are possible, with multiple expansion, without excessive condensation. Condensation is less serious because of (a) the use made of reevaporated steam, (6) the decrease in initial condensation, and (c) the small size of the high- pressure cylinder. Several numbers and arrangements of cylinders are possible with expansion in two, three, or four stages. Incidental advantages : less steam lost in clearance space ; compression begins later ; the large cylinder is subjected to low pressure only j more uniform speed and moderate stresses. The Woolf engine had no receiver ; the low-pressure cylinder received steam through- out the stroke as discharged by the high-pressure cylinder. The former, there- fore, worked without expansion. The piston phases coincided or differed by 180. In the receiver engine, the pistons may have any phase relation and the low-pressure cylinder works expansively. Early cut-oS in the low-pressure cylinder increases its proportion of the load, and is practically without effect on the total work of the engine. SYNOPSIS 357 The approximate point of low-pressure cut-off to eliminate drop may "be graphically or analytically determined for tandem and cross-compound engines. In combining diagrams, twi saturation curves are necessary, unles3 the cushion stcnm be deducted. The diagram factor has an approximate value the same as that in a simple engine hav- ing Wn expansions, in which n is the number of expansions in the compound engine and c its number of expansive stages. Cylinder ratios are 3 or 4 to 1 if non-condensing, 4 or 6 to 1 if condensing, iu com- pounds ; triples have ratios from 1 : 2.0 : 2.0 to 1 : 2.5 : 2.5. A large high-pressure cylinder gives high overload capacity. The engine may be designed so as to equalize work areas, or by assuming the cylinder ratio. " Equivalent simple cylinder." Values of E. Governing should be by varying the point of cut-off in both cylinders. Drop in any but the last cylinder is usually considered undesirable. Exceptionally high efficiency is shown by compounds having cylinder ratios of 7 to 1. The high-pressure cylinder in ordinary compounds is too large for highest efficiency. The binary vapor engine employs the waste heat of the exhaust to evaporate a fluid having a lower boiling point than can be attained with steam. Additional work may then be evolved down to a rejection temperature of 60 or 70 F. The best result achieved is 167 B. t. u. per Ihp.-minute. Engine Tests The indicator measures pressures and volumes in the cylinder and thus shows the ''cycle." Its diagram gives the m. e. p. and points out errors in valve adjustment or control. ^, . , , , A ftiCio + TF) wh Who Calorimeters : the barrel type : XQ = - - ^j- - J - , . whi 4- Wh z wh surface condensing : XQ = i-Z_i __ superheating : XQ = -"--) . limits of capac j ty . H+lctT-^-Jio-Q JBarrus : XQ = - - =-^ - - ; JL4) separating : direct weighing of the steam and water; chemical : insolubility of salts in dry steam ; electrical : 1 B. t. u. per minute = 17.59 watts. Engine trials : we may measure either the heat absorbed or the heat rejected + the work done. By measuring both, we obtain a heat balance. Results usually stated : Ib. dry or actual steam per Ihp.-hr.; B. t. u. per Ihp.-minute ; thermal efficiency ; work per Ib. tteam ; Carnot efficiency ; Clausius efficiency ; efficiency ratios. By assuming the steam dry at compression and cut-off or release, and knowing the clearance, we may roughly estimate steam consumption from the indicator diagram. ft.-lb. of work per 100 Ib. coal (or per 1,000,000 B. t.u.) Plant efficiency 358 APPLIED THERMODYNAMICS ffirn's analysis: E X =2M (h x +x r x Y, H X =E X +W X ; heat transfer to and from walls may be computed from the supply of heat, the change in internal energy, and the -work done. The excess of losses over gains represents radiation. Testing of regulation (speed control). Types of Steam Engine Standard engines : non-condensing or condensing, light-hand or left-hand, simple or multiple expansion ; single-acting or double-acting ; rotative or non-rotative , duplex or single ; horizontal, vertical, or inclined , locomotive, stationary (pump- ing, mill, power plant), or marine , "belted, direct-connected, or rope-driven ; air compressors ; girder, tangye, or semi-tangye frames ; slow, medium, or high speed ; throttling, automatic, four-valve, or releasing gear. The Stumpf uniflow engine. The power plant: feedpump, boiler, engine, condenser. The locomotive: tractive power =^rrS adhesion =0.22 to 0.25Xweight on drivers; two-cylinder and four-cylinder compounds , the balanced compound \ high econ- omy of locomotive engines. PROBLEMS 1. Show from Art. 426 that the loss by a throttling process is equal to the prod- uct of the increase of entropy by the absolute temperature at the end of the process. 2. Ignoring ladiation, how fast are the walls gaining heat because of transfers during expansion in an engine running at 100 r. p. m,, in which J pound of steam is condensed per revolution at a mean pressure of 100 lb., and 0.30 pound is reevaporated at a mean pressure of 42 lb. (Ans., 3637 B. t. u. per minute). 3 a. Plot curves representing the lesults of the tests given in Art. 434. 3 6. Represent Toy a curve the results of the Barms tests, Art. 436. 4. All other factors being the same, how much less initial condensation, at \ cut- off, should be found in an engine 30J"X48" than in one 7"x7"? (Art. 437). 5. Sketch a curve showing the variation hi engine efficiency with ratio of expan- sion. 6. Find the percentage of initial condensation at J cut-off in a non-condensing engine using dry steam, running at 100 r. p. m. with a pressure at cut-ofE of 120 lb. t the engine being 30|"X48" (Art. 437). 7. In Fig. 193, assuming the initial pressure to have been 100 lb., the feed-water temperature 90 I\, find the approximate thermal efficiencies with the various amounts of superheat at a load of 15 hp. 8. In an ideal Clausius cycle with initially dry steam between p = 140 and p = 2 (Art. 417), by what percentage would the efficiency be increased if the initial pressure were made 160 lb. ? By what percentage would it be decreased if the lower pressure were made 6 lb. ? 9. Find the mean effective pressure in the ideal cycle with hyperbolic expansion and no clearance between pressure limits of 120 and 2 lb., with a ratio of expansion of 4. (Ans., 69.6 lb.) 10. Find the probable indicated horse power of a double-acting engine with the best type of valve gear, jackets, etc., operating as in Problem 9, at 100 r. p. m., the cylinder being 30J"X4S". (Ignore the piston rod.) (Ans., 1107 hp.) PROBLEMS 359 11. In Problem 9, what percentage of power is lost if the lower pressure is raised to 3J Ib. ? 12. By what percentage would the capacity of an engine be increased at an altitude of 10,000 ft. as compared with sea level, at 120 Ib. initial gauge pressure and a back pressure 1 Ib, greater than that of the atmosphere, the ratio of expansion being 4 ? (Atmospheric pressure decreases | Ib. per 1000 ft. of height.) 3. An engine has an apparent ratio of expansion of 4, and a clearance amounting to 0.05 of the piston displacement, TVhat is its real ratio of expansion ? (Aiis., 3.5.) 14. In the dry steam ClausiiiR cycle of Problem 8, by what percentage are the ca- pacity and efficiency affected if expansion is hyperbolic instead of adiabatic ? Discuss the results. 15. In an engine having a clearance volume of 1.0 and a back pressure of 2 Ib., the pressure at the end of compression is 40 Ib. If the compression curve is PF 1 - 03 =c, what is the volume at the beginning of compression ? (Ans., 18.28.) 16. An engine works between 120 and 2 Ib. pressure, the piston displacement being 20 cu. ft., clearance 5 per cent, and apparent ratio of expansion 4. The expan- sion curve is PV 1 02 = c, the compression curve PV 1 3 = c, and the final compression pressure is 40 Ib. Plot the PV diagram with actual volumes of the cushion steam eliminated. 17. In Problem 16, 1.825 Ib. of steam are present per cycle. Plot the entropy dia- gram from the indicator card by Boulvin's method. 18. In Problems 16 and 17, compute and plot the entropy diagram by Keeve's method, assuming the steam dry at the beginning of compression. (See Art. 457.) Discuss any differences between this diagram and that obtained in Problem 17. 19. In a non-expansive cycle, find the theoretical changes in capacity and economy by raising the initial pressure from 100 to 120 Ib., the back pressure being 2 Ib. (Ans., 1.2 per cent gain in capacity : 8.5 per cent increase in efficiency.) 20. A non-expansive engine with limiting volumes of 1 and 6 cu. ft. and an initial pressure of 120 Ib., without compression, has its back pressure decreased from 4 to 2 Ib. Find the changes in capacity and efficiency. The same steam is now allowed to expand hyperbolically to a volume of 21 cu. ft. Find the effects following the reduction of back pressure in this case. The steam is in each case dry at the point of cut-off. (Ans., (a) 1.7 per cent increase in capacity and efficiency; (&) 3.2 per cent increase in capacity and efficiency. 21. rind the cylinder dimensions of an automatic engine to develop 30 horse power at 300 r. p. m., non-condensing, at J cut-off, the initial pressure being 100 Ib. and the piston speed 300 ft. per minute. The engine is double-acting. 22. Sketch a possible cylinder arrangement for a quadruple-expansion engine with seven cylinders, three of which are vertical and four horizontal, showing the receivers and pipe connections. 23. Using the ideal combined diagram for a compound engine with a constant receiver pressure, clearance being ignored, what must that receiver pressure be to divide the diagram area equally, the pressure limits being 120 and 2 and the ratio of expansion 16 ? 24. Consider a simple engine 30J"X48" and a compound engine 15 J" and 30J"X48", all cylinders having 5 per cent of clearance and no compression. What 9je the amounts of steam theoretically wasted in filling clearance spaces in the simple 360 APPLIED THERMODYNAMICS engine and in the high-pressure cylinder of the compound, the pressures being as in Problem 23 ? 25. Take the same engines. The simple engine has a real ratio of expansion of 4; the compound is as in Problems 23 and 24. Compression is to be carried to 40 Ib. in the simple engine and to 60 Ib. in the compound in order to prevent waste of steam. By what percentages are the work areas reduced in the two engines under consideration ? 26. A cross-compound double-acting engine operates between pressure limits of 120 and 2 Ib. at 100 r. p. m. and 800 ft. piston speed, developing 1000 hp. Find the sizes of the cylinders under the following assumptions, there being no drop . (a) dia- gram factor 0.85, 20 expansioas, receiver pressure 24 Ib. ; (&) diagram factor O.S5, 20 expansions, work equally divided ; (c) diagram factor 0.85, ^0 expansions, cylinder ratio 5:1; (d) diagram factor 0.83, 32 expansions, work equally divided. Find the power developed by each cylinder in (a) and (c). Find the size of the cylinder of the equivalent simple engine having a diagram factor of 0.80 with 20 expansions. Draw up a tabular statement of the five designs and discuss their comparative merits. 27. lit Problem 26, Case (a), the receiver volume being equal to that of the low- pressure cylinder, find graphically and analytically the point of cut-off on the low- pressure cylinder. 28. Trace the combined diagram for one end of the cylinder from the first set of cards in Fig. 230, assuming the clearance in each cylinder to have been 15 per cent of the piston displacement, the cylinder ratio 3 to 1, and the pressure scales of both cards to be the same. 29. Show on the entropy diagram the effect of reheating. 30. In Art. 483, what was the Carnot efficiency of the Josse engine ? Assuming it to have been used in combination with a gas engine, the maximum temperature in the latter being 3000 F., by what approximate amount might the Carnot efficiency of the former have been increased ? (The temperature of saturated sulphur dioxide at 35 Ib. pressure is 52 F.) 31. An indicator diagram has an area of 82,192.5 foot-pounds. What is the mean effective pressure if the engine is 30"X48" ? What is the horse power of this engine if it runs double-acting at 100 r. p. m. ? (Ans^ 28. 1 Ib. ; 498 hp.) 32. Given points l r 2 on a hyperbolic curve, such that V* 7i = 15, P J =120, jP 2 = 34.3, find the OP-axis. 33. An engine develops 500 hp. at full load, and 62 hp. when merely rotating its wheel without external load. What is its mechanical efficiency * (Ans., 0.876.) 34. Steam at 100 Ib. pressure is mixed with water at 100. The weight of water increases from 10 to 11 Ib., and its temperature rises to 197J. What was the per- centage of dryness of the steam ? ( Atis., 95 per cent.) 35. The same steam is condensed in and discharged from a coil, its temperature becoming 210, and 10 Ib. of surrounding water rise in temperature from 100 to 204 J. Find the quality of the steam. What would have been an easier way of determining the quality ? 36. What is the maximum percentage of wetness that can be measured in a throt- tling calorimeter m steam at 100 Ib. pressure, if the discharge pressure is 30 Ib. ? (Ans., 2.5 per cent.) 37. Steam at 100 Ib. pressure has added to it from an external source 30 B. t. u. PROBLEMS 361 per pound. It is throttled to 30 Ib. pressure, its temperature becoming 270.3. What was its diyness ? (Ans , 0.955.) 38. Under the pressure and temperature conditions of Problem 37, the -added heat is from an electric current ot 5 amperes provided for one minute, the Toltage f ailing from 220 to 110. What was the amount of heat added and the percentage of dryness of the steam ? (See Art. 494.) (Ans., 95.4 per cent.) 39. An engine consumes 10,000 Ib. of dry steam per hour, the moisture having been completely eliminated by a receiver separator which at the end of one hour is found to contain 285 Ib. of water. What was the dryness of the steam entering the separator ? (Ans., 97.2 per cent.) A double-acting engine at 100 r. p. m. and a piston speed of 800 feet per minute gives an indicator diagram in which the pressure limits are 120 and 2 Ib., the volume limits 1 and 21 cu. ft. The apparent ratio of expansion is 4. The expansion curve follows the lawPF 1 - 02 ^ c. Compression is to 40 Ib., according to the law PV 1 03 =c. Disregard rounded corners. The boiler pressure is 130 Ib., the steam leaving the boiler is dry, the steam at the throttle being 95 per cent dry and at 120 Ib. pressure. The boiler evaporates 26,500 Ib. of steam per hour ; 2000 Ib. of steam are supplied to the jackets at 120 Ib. pressure. The engine runs jet-condensing, the inlet water weighing 530,000 Ib. per hour at 43.85 F., the outlet weighing 554,000 Ib. at 90 P. The coal burned is 2700 Ib. per hour, its average heat value being 14,000 B. t. u. Compute as follows : 40. The mean effective pressure and indicated horse power. (NOTE. The work quantities under the curves must be computed with much accuracy.) (Ans., 68.57 Ib.; 1196.8 hp.) 41. The cylinder dimensions of the engine. (Ans., 30.24 by 48 in.) 42. The heat supplied at the throttle per pound of cylinder and jacket steam, and the B. t. u. consumed per Ihp. per minute ; the engine being charged with heat above the temperatures of the respective discharges. 43. The dry steam consumption per Jhp.-hr., thermal efficiency, and work per pound of dry steam. 44. The Carnot efficiency, the Clausius efficiency, and the efficiency ratio, taking the limiting conditions as at the throttle and the condenser outlet, 45. The cylinder feed steam consumption computed as in Art. 500 ; the consump- tion thus computed but assuming x = 0.80 at release, z= 1.00 at compression. Com- pare with Problem 43. 46. The percentage of steam lost by leakage (all leakage occurring between the boiler and the engine); the transmissive efficiency ; the unaccounted-for losses. 47. The duty, the efficiency of the plant, and the boiler efficiency. 48. The heat transfers and the loss of heat by radiation, as in Art. 504, assuming x 1.00 at compression. Compare the latter with the unaccounted-for heat obtained In Problem 46. 49. The value of the mechanical equivalent of heat which might be computed from the experiment, (Jns., 720.) 362 APPLIED THERMODYNAMICS 50. Explain the meaning of the figure 2068.84 in Art. 503. 51. Revise Fig. 233, showing the arrangement of machinery and piping if a sur- face condenser is used. 52. A locomotive weighing 2uO,000 Ib. carries, normally, 60 per cent of its weight on its drivers. The cylinders are 19"X26", the wheels 66" in diameter. What is the maximum boiler pressure that can be profitably utilized ? If the engine has a traction increaser that may put 12,000 Ib. additional weight on the drivers, what maximum boiler pressure may then be utilized ? 53. Represent Fig. 217 on the PV diagram. 54. Find the steam consumption in Ib. per Ihp.-hr. of an ideal engine working in the Clausius cycle between absolute pressures of 150 Ib. and 2 Ib., the steam contain- ing 2 per cent of moisture at the throttle. What is the thermal efficiency ? 55. What horse power will be given by the engine in Problem 10 if the ratio of expansion is made (a) 5, (b) 3 ? 56. If an engine use dry steam at 150 Ib. absolute pressure, what change in efficiency occurs when the back pressure is reduced from 2 to | Ib. absolute, if the ratio of expansion is 30 ? If the ratio of expansion is 100 ? CHAPTER XIV THE STEAM TURBINE 512- The Turbine Principle. Figure 235 shows the method of using steam in a typical impulse turbine. The expanding nozzles discharge a jet of steam at high velocity and low pressure against the blades or buckets, the im- pulse of the steam causing ro- tation. We have here, not expansion of high pressure steam against a piston, as in the ordi- nary engine, but utilization of the kinetic energy of a rapidly flowing stream to produce move- ment. One of the assumptions of Art. 11 can now no longer hold. All of the expansion oc- curs in the nozzle ; the expansion j i -A 4.u / , j Fro- SSS- Arts. 512, 524, 536. De Laval Turbine produces velocity, the velocity does me ' el ^ Nozzles> work. The lower the pressure at which the steam leaves the nozzle, the greater is the velocity attained. It will presently be shown that to fully utilize the energy of velocity, the buckets must themselves move at a speed proportionate to that of the steam. This involves ex- tremely high rotative speeds. The steps in the design of an impulse turbine are (a) determination of the velocity produced by expansion, (6) computation of the nozzle dimensions necessary to give the desired expansion, and (c) the propor- tioning of the buckets. 513. Expansive Path. There is a gradual fall of pressure -while the steam passes through the nozzle. With a given initial pressure, the pres- sure and temperature at any stated point along the nozzle should never change. There is, therefore, no tendency toward a transfer of heat be- tween steam and walls. Further, the extreme rapidity of the movement gives no time for such transfer ; so that the process in the nozzle is truly adiabatic, although friction renders it non-isentropic. The first problem of turbine design is then to determine the changes of velocity, volume, temperature or dryness, and pressure, during such adiabatic expansion, for a vapor initially wet, dry, or superheated ; the method may be accu- 363 364 APPLIED THERMODYNAMICS rate, approximate (exponential), or graphical. The results obtained are to include the effect of nozzle friction. 514. The Turbine Cycle. Taking expansion in the turbine as adiabatic and as carried down to the condenser pressure, the cycle is that of Clansius, and is theoretically more efficient than that of any ordinary steam engine working through the same range. The turbine is free from losses due to interchange of heat tcitJi the icalls. The practical losses are four: (a) Friction in the nozzles, causing a fall of temperature -without the performance of work ; (&) Incomplete utilization of the kinetic energy by reason of the assumed blade angles and residual velocity of the emerging jet (Art. 528); (c) Friction along the buckets, increasing as some power of the stream speed ; (rZ) Mechanical friction of journals and gearing, and friction between steam and rotor as a whole. 515. Heat Loss and Velocity. In Fig. 236, let a fluid flow adiabatically from the vessel a through the frictionless orifice b. Let the internal en- ergy of the substance be e in a and E in 6; the velocities v and V\ the pressures p and P\ and the specific volumes w and W. If the velocities could be ignored, as in previous computations, the volume of each pound of fluid in a would decrease by w in passing out at the constant pressure^; and the volume of each pound of FIG aril Art 515. Flow fl u jfl i n i W0 uld increase by W at the constant ioug n ee pressure P. The net external work done would be PW2M, the net loss of internal energy e E, and these two quan- tities would be equal. With appreciable velocity effects, we must also consider the kinetic energies in a and b ; these are and Zf; 2ff 2& and we now lave H=T+I+W+V, (T+I)+W+V=0, 2g " '- ' 2g' or = DW HEAT DROP AND VELOCITY 365 Let X, Z7, -H~, JR, and x, u, Ji, r, be the dry ness, increase of vol- ume during vaporization, heat of liquid, and internal latent heat, at P W'diidpw respectively ; let * be the specific volume of water ; then for expansion of a vapor from pw to P JF within the saturated region, - in which q, Q represent total heats of wet vapor above 32 degrees. If expansion proceeds from the superheated to the saturated ret/ion, y-1 in which n = u 4- s is the volume of saturated steam at the pressure p, w is the volume of superheated steam, and p(w n) is the internal energy measured above saturation.* This also re- duces to q Q -f s(p P), where q is the total heat in the super- heated steam, and the same form of expression will be found to apply to expansion wholly in the superheated region. The gain in kinetic energy of a jet due to adiabatic expansion to a lower pressure is thus equivalent to the decrease in the total heat of the steam plus the work which would be required to force the liquid back F^- 337 - Art -' against the same pressure head. In Fig. 237, let al, AB, CD, represent the three paths. Then the losses of heat are represented by the areas dale, deABc, deCDfc* * For any gas treated as perfect, the gain of internal energy from t to T is tf J. ~~ VJ Q JL ~~ tj ~" ~~~ ~~ ^ * ~ Uj ^ , !/ y ~~ * y "~* ^ or in this case, since internal energy is gained at constant pressure, - Adiabatic Heat 366 APPLIED THERMODYNAMICS The term s(p-P) being ordinarily negligible, these areas also rep- resent the kinetic energy acquired, which may be written V 2 r v 2 In the turbine nozzle, the initial velocity may also, without serious error, be regarded as negligible; whence =7-0 or 7= V50103.2(g -Q) =223.84Vg :l Q feet per second. 20 516. Computation of Heat Drop. The value of q Q may be determined for an adiabatic path between stated limits from the entropy diagram, Fig. 175, or from the Mollier diagram, Fig. 177. Thus, from the last named, steam at 100 Ib. absolute pressure and at 500 F. contains 1273 B. t. u. per pound; steam 85 per cent dry at 3 Ib. absolute pressure contains 973 B. t. u. Steam at 150 Ib. absolute pressure and 600 F. con- tains 1317 B. t. u. If it expand adiabatically to 2.5 Ib. absolute pressure, its condition becomes 88 per cent dry, its heat contents 1000 B. t. u., and the velocity produced is 223.84 V317 = 3980 ft. per second. 517. Vacuum and Superheat. The entropy diagram indicates the nota- ble gain due to high vacua and superheat. Comparing dry steam expanded from 150 Ib. to 4 Ib. absolute pressure with the same steam superheated to 600 and expanded to 2.5 Ibs. absolute pressure, we find q Q in the former case to be 248 B. t. u., and in the latter, 317 B. t. u. The corre- sponding values of V are 3330 and 3980 ft. per second. The turbine is peculiarly adapted to realize the advantages of wide ratios of expansion. These do not lead to an abnormally large cylinder, as in ordinary engines; the "toe" of the Clausius diagram, Fig. 184, is gained by allowing the steam to leave the nozzle at the condenser pressure. Superheat, also, is not utilized merely in overcoming cylinder condensation 5 it increases the available " fall " of heat, practically without diminution. 518. Effect of Friction. If the steam emerging from the nozzle were brought back to rest in a closed chamber, the Mnetic energy would be reconverted into heat, as in a wiredrawing process, and the expanded steam would become super- heated. Watkinson has, in fact, suggested this (1) as a method of supei heating steam, the water being mechanically removed at the end of expansion, before re- conversion to heat began. In the nozzle, in piactice, the friction of the steam against the walls does partially convert the velocity energy back to heat, and the heat drop and velocity are both less than in the ideal case. The efficiencies of nozzles vary according to the design from 0.90 to 0.97. The corresponding variation in ratio of actual to ideal velocity is 0.95 to 0.99. EFFECT OF NOZZLE FRICTION 367 In Fig. 238, for adiabatic expansion from j>, v, q, to P, V, Q, the velocity imparted is 223 84 V?^- P During expansion from p, v, g, to P^ Vi, Qi, the velocity imparted is 223.84 V^ft- Since Fi exceeds F, the steam is more nearly dry at Fi; i.e. Q l exceeds Q. The loss of energy due to the path pvq P^ViQi as compared with puy PVQ, is FIG. 238. Art. 518 Abiabatic Expansion with and without Friction, in which X 2 is the difference of the squares of the velocities at Q and ft. This gives X 2 = 50103.2 (Q l - Q). In Fig. 239, let NA be the adiabatic path, NX the modified path due to fric- tion. NZ represents a curve of constant total heat ; along this, no work would be done, but the heat would steadily lose its availability. As NX recedes from NA toward NZ, the work done during expan- sion decreases. Along NA, all of the heat lost (area FHNA) is transformed into work: along NZ, no heat is lost and no . . , ,, -r>-nrT*-m j work 1S done ^ the areas BFHNQ and BFZD being equal. Along NX, the heat transformed into work is BFHNC - BFXE = FHNA CAXE, less than that during adiabatic expansion by the amount of work converted back to heat. Considering expansion from _ZVto Z 9 IE !o FIG. 239 Art. 518. Expansive Path as Modified by Friction. F= 223.84 Vq=~& = 0, since q = ft. Nozzle friction decreases the heat drop, the final velocity attained, and the external work done. 519. Allowance for Friction Loss. For the present, we will assume nozzle friction to reduce the heat drop by 10 per cent. In Fig. 240, which is an enlarged view of a portion of Fig. 177, let AB represent adiabatic (isentropic) expansion from the condition A to the state B. Lay off ' 368 APPLIED THERMODYNAMICS and draw the line of constant heat CD. H Then D is the equivalent final state at the same pressure as that existing at B, and AC represents the heat drop corrected for friction. Similarly by laying off FIG. 240. Arts 519, 524, 3;i3, rJl\ 3;U The Steam Path of the Tuibine and drawing GE to inter- sect the 35-lb. pressure line, we find the point E on the path AD of the steam through the nozzle. We may use the new heat drop thus obtained in de- termining "T; or generally, N if m is the friction loss, and If m = 0.10, F = = 223.84 212.42 vq - 520. Analytical Relations. The influence of friction in determining the final condition of the steam may be examined analytically. For example, let the initial condition be wet or dry ; then friction will not ordinarily cause superheating, so that the steam will remain saturated throughout expansion. Without friction, the final dryness X Q would he given by the equation (Art. 392), Friction causes a return to the steam of the quantity of heat m(q Q). This in- creases the filial dryness by - W-^-S/, making it ' >o If the initial condition is superheated to t g , and the final condition saturated, adiabatic expansion -would give f, _ # n / and friction would make the final condition T flog, 3,+ ' + * log, f j j + m(q - NOZZLE PROPORTIONS 369 If the steam is superheated throughout expansion, we have for the final tem- perature T st without friction, log. 3, + \ = in which the value of k Q must be obtained by successive approximations. 521. Rate of Flow. For a flow of G pounds per second at the velocity F, when the specific volume is W, the necessary cross-sectional area of nozzle is F = . The values of W and V may be read or inferred from the heat chart or the formulas just given. In Fig. 241 (2), let ab represent frictionless adiabatic expansion on the TN plane, a'b' the same process on the PV plane. By finding q a and values of Q at various points along ab, we may obtain a series of successive values of V. The correspond- ing values of W being read from a chart or computed, we plot the curve MN, representing the re- lation of specific volume and velocity throughout the expan- FIG 241 Art. 521. Graphical Determination of sion. Draw yy' parallel to W, Nozzle Area, making Oy = G, to some con- venient scale. Draw any line OD from to MN, intersecting yy f at k. From similar triangles, yk : yO : : On : nD, or yk = - F. To find the prewnre at any specified point on the nozzle, lay off yk = F> draw OkD, Dn, and project z to the PT plane. The minimum value of F is reached when OD is tangent to j\TN. It becomes infinite when V = 0. The conclusion that the crobs-sectionul area of the nozzle reaches a minimum at a certain stage in the expansion will be presently verified. 522. Maximum Flow (2a). For a perfect gas, y-l'~ y-r k If the initial velocity be negligible, we have, as the equation of flow (Art. 515), 9W PW y-l y-l y-l and since (pw-PW); Edt 370 APPLIED THERMODYNAMICS Then. From Art. 521, Taking the value of V at we obtain G= This reaches a maximum, for air, when P p 0.5274 (3). The velocity is then equal to that of sound. For dry steam, on the assumption that y = 1.135, and that the above relations apply, the ratio for maximum flow is 0.577. Using the value just given for the ratio P p, with y = 1.402, the equation for G simplifies to the equation of flow of a permanent gas, which has been closely confirmed by experiment. With steam, the ratio of the specific heats is more variable, and the ratio of pressures has not been as well confirmed experimentally. Close approxi- mations have been made. Claike (4), for example, shows maximum flow with saturated steam to occur at an average ratio of 0.56. The pressure of maximum flow determines the minimum or throat diameter of the nozzle, which is independ- ent of the discharge pressure. The emerging velocity may be greater than that in the throat if the steam is allowed to further expand after passing the throat. The nozzle should in all cases continue beyond the throat, either straight or ex- panding, if the kinetic energy is all to be utilized in the direction of flow. In all cases, the steam velocity theoretically attained at the throat of the nozzle will be 1450 ft per second. 523. Experiments. Many experiments have been made on the flow of fluids through *nozzles and orifices. Those of Jones and Rathbone (5), Rosenhain (6), Gutermuth (7), Napier (8), Rateau (9), Hall (10), Wilson (11), Kunhardt (12), Buchner (13), Kneass (14), Lewicki (15), Durley (16), and chiefly, perhaps, those of Stodola (17), should be studied. There is room for further advance in our knowledge of the friction losses in nozzles of various proportions. There are sev- eral methods of experimentation : the steam, after passing the orifice, may be con- densed and weighed; the pressure at various points in the nozzle may be measured by side orifices or by a searching tube ; or the reaction or the impulse of the steam at its escape may be measured. The velocity cannot be measured directly. TYPES OF TURBINE 371 A greater rate of flow is obtainable through an orifice in a thin plate (Fig. 242) than through an expanding nozzle (Fig. 243). For pressures under 80 lb., with discharge into the atmosphere, the plain oiifice is more efficient in producing velocity. For wider pressure ranges, a divergent nozzle is necessary to avoid deferred expansion occurring after emergence. Expansion should not, however, be carried to a pres- sure lower than that of dischaige. The rate of flow, but not the emeiging velocity, depends upon the shape of the inlet; a slightly rounded edge (Fig. 243) gives the greatest rate ; a greater amount ^ IG 342. Art. of rounding may be less desirable. The experimentally observed 523. Diverg- critical pressure ratio ( , Art. 522 J ranges with various fluids mg n Ce * from, 0.50 to 0.85. Maximum flow occurs at the lower ratios with rather sharp corners at the entrance, and at the higher ratios when a long divergence occurs beyond the throat, as in Fig. 243. The "most efficient" nozzle will have different proportions for different pressure ranges. The pressure is, in general, greater at all points along the nozzle than theory would indicate, on account of 243. Arts. 523, friction ; the excess is at first slight, but increases more and 525 Expanding more rapidly during the passage. Most experiments have necessarily been made on very small orifices, discharging to the atmosphere. The fiiction losses in larger orifices are probably less. The experimental method should include at least two of the measurements above mentioned, these checking each other. The theory of the action in the nozzle has been presented by Heck (18). Zeuner (19) has discussed the flow of gases to and from the atmosphere (20), both under adiabatic and actual conditions, and the efflux of gases in general through orifices and long pipes. 524. Types of Turbine. The single stage impulse turbine of Fig. 235 is that of De Laval. Its action is illustrated in Fig. 244. The pressure falls in the nozzle, and remains PRESSURES constant in the buckets. The Curtis and ......... Rateau turbines use 'a series of wheels, with ex- panding nozzles between the va- FIG. 244. Art. 524. De Laval Turbine. FIG. 245. rious series (Figs. 245, 246). The steam is only partially ex- Art. 524. Curtis panded in each nozzle, until it reaches the last one. Such turbines are of the multi- stage impulse type. During passage through the blades, the ve- locity decreases, while the pressure remains unchanged. In the 372 APPLIED THERMODYNAMICS pressure turbine of Parsons, there are no expanding nozzles ; the steam passes successively through the stationary guide vanes OS g, _ FBESSUSES^ and movable wheel buckets, TFJ w. Fig. 247. ^ A gradual fall of pressure occurs, the buck- "T*SI!! OF ets being at all times full of steam. In impulse turbines, the buckets need not be full of steam, and the pressure drop occurs FIG. 2-46 Art. 524. Rateau i n the nozzle only. Turbme " A lower rotative speed results from the ** use of several pressure stages with expanding nozzles total heat drop of 317 B. t. u., in Art. 516, be divided into three stages by three sets of nozzles. The exit velocity from G| each nozzle, corrected for friction, is ^ = 2180 ft. per sec- Let the Arts. 524, 533. Parsons Turbine. then 212. ond, instead of 3980 ft. per second; lay- ing off in Fig. 240 the three equal heat drops, we find that the nozzles expand between 150 and 50, 50 and 13, and 13 and 2.5 Ib. respectively. The rotative speeds of the wheels (proportional to the 'emerging velocities), Art. 52S ; are thus reduced. 525. Nozzle Proportions ; Volumes. The specific volume W of the. steam at any point along the path AD, Fig. 240, having been obtained from inspection of the entropy chart, or from the equation of condition, and the velocity V at the same point having been computed from the WQ- heat drop, the cross-sectional area of the nozzle, in square feet, is F= - (Art. 521). Finding values of jPfor various points along the expansive path, we- may plot the nozzle as in Fig. 243, making the horizontal inter- vals, abj be, cd, etc., such that the angle between the diverging sides is about 10, following standard practice.* It has been shown that I 1 reaches a minimum value when tlie pressure is about 0.57 of the initial pres- sure, and then increases as the pressure falls further. If the lowest pressure exceeds 0.57 of the initial pressure, the nozzle converges toward the outlet. Otherwise, the nozzle converges and afterwards expands, as in Fig. 243. Let, in such ease, o be the minimum diameter, the outlet diameter, L the length between these diameters; then for an angle of 10 between the sides, ~ = L tan 5, or L = 5.715(0 o). 2i 2 * A variable taper may be used to give constant acceleration of the steam VELOCITY DIAGRAMS 373 526. Work Done. The work done in the ideal cycle per pound of steam is 778(2 Q) foot-pounds. Since 1 horse power = 1,980,000 foot-pounds per hour, the steam consumption per hp.-hr. is theoreti- cally 1,980,000 -r- 778(2 - <?) = 2545 -s- (y - <?). If H is the effi- ciency ratio of the turbine, from steam to buckets, and e the efficiency from steam to shaft, then the actual steam consumption per indicated horse power is 2545 -5- E(q Q), and per brake horse power is 2545 -f- e(q Q*) pounds. The modifying influences of nozzle and bucket friction in determining ]3 are still to be considered. 527. Relative Velocities. In Fig. 248, let a jet of steam strike the bucket A at the velocity t;, the bucket itself moving at the speed u. The velocity of the steam rela- tive to the bucket is then repre- sented in magnitude and direction by V. The angles a and e made with the plane of rotation of the bucket wheel are called the absolute entering and relative entering angles respectively. Analytically, sin e = v rr rp, . J , FIG. 248. Art. 527. Velocity Diagram, sin a -5- v. 1 he stream traverses the surface of the bucket, leaving it with the relative velocity a/, which for convenience is drawn as x from the point 0. Without bucket friction, x = V. The angle / is the relative angle of exit. Laying off w, from 2, we find Y as the absolute exit ve- locity, with g as the absolute angle of exit. Then, if x = V, To include the effect of nozzle and bucket friction, we proceed as in Fig. 249, decreasing v to VI m of its original value (Art. 519), and making x less than F'by from 5 to 20 per cent, as in ordinary practice. As before, sin e = v sin a-*~V\ but for a bucket friction of 10 per cent, sin^ = 0.9 F"sin/-f- Y. FIG. 240, Arts. 527, 532, 534. Velocity Corrected for Friction. 374 APPLIED THERMODYNAMICS FIG. 250 Arts 528,529. Rotative and Thrust Components. 528. Bucket Angles and Work Done. In. Fig. 250, the absolute velocities v and Y may be resolved into components ab and db in the direction of rotation, and ac and de at right angles to this direction. The former compo- nents are those which move the wheel ; the lat- ter produce an end thrust on the shaft. Now ab 4- M (Id being negative) is the change in velocity of the fluid in the direction of rotation ; it is the acceleration; the force exerted per pound is then (ab + M)-*-ff= s (ab 4- Brf) -4- 32.2 = (y cos a 4- Fcos y) -f- 32. 2. This force is exerted through the distance u feet per second ; the work done per pound of steam is then. u(v cos a 4- T"cos5r)-7- 32.2 foot-pounds. This, from Art. 526, equals 778 U (2 <?) whence J= (z; cos a 4- rcos^)-*- 25051.6(2 <?) The efficiency is thus directly related to the bucket angles. To avoid splashing, the entrance angle of the bucket is usually made equal to the relative entering angle of the jet, as in Fig. 251. (These formulas hold only when the sides of the buckets are enclosed to prevent the lateral spreading of the stream.) In actual turbines, Id (Fig. 250) is often not negative, on account of the extreme reversal of direction that would be necessary. With positive values of Id, the maximum work is obtained as its value ap- proaches zero, and ultimately it is uv cos #~-32.2. o Since the kinetic energy of the jet is , the efficiency 2 - cos a. FIG 251. Art 528 Velocities and Bucket Angles. 5? from steam to buckets then becomes In designing, we may either select an exit bucket angle which shall make Id equal to zero (the relative exit velocity being tangential to the surface of the bucket), or we may choose such an angle that the end thrust components de and ca^ Fig. 250, shall bal- VELOCITY EFFICIENCY 375 ance. In marine service, some end thrust is advantageous ; in stationary work, an effort is made to eliminate it. This would be accomplished by making the entrance and exit bucket angles equal, for a zero retardation by friction. With friction considered, the angle of exit 1C, in Fig. 251, must be greater than the entering an- gle e. In any case, where end thrust is to be eliminated, the rota- tive component of the absolute exit velocity must be so adjusted as to have a detrimental effect on the economy. 529. Effect of Stream Direction on Efficiency. Let the stream strike the bucket in the direction of rotation, so that the angle a = 0, Fig. 250, the relative exit velocity being perpendicular to the plane of the wheel. The work done is _v kinetic energy is The efficiency, 2 u u becoines a maximum at 0.50 when u = - With a ciip-shaped vane, as _j in the Pelton wheel, Pig. 252, complete reversal of the jet occurs 5 the absolute exit velocity, ignoring friction, is v-2u. The change in FIG. 252 Arts. 529, 536. -Pel- velocity is v + v 2 u = 2(v ?*), and the work ton Bucket< is 2u(v u) -f- g, whence the efficiency, 7" ? becomes a maximum at 100 per cent when u = ^- Complete reversal in turbine buckets is im- practicable. ^ 530. Single-Stage Impulse Turbine. The absolute velocity of steam enter- ing the buckets is computed from the heat drop and nozzle friction losses. In a u turbine of this type, the speed of the v ! buckets can scarcely be made equal to half that of the steam; a more usual proportion is 0.3. The velocity u thus seldom exceeds 1400 ft. per second. Fixing the bucket speed and the absolute entering angje of the steam (usually 20) we determine graphically the entering angle of the bucket. The bucket may now be de- signed with equal angles, which would eliminate end thrust if there were no FIG. 253. Art. 530. Bucket Outline. friction, or, allowance being made for 376 APPLIED THERMODYNAMICS friction, either end thrust or the rotative component of the absolute exit velocity may be eliminated. The normals to the tangents at the edges of the buckets being I drawn, as ec, Fig. 253, the radius r is made equal to about 0.965 ec. The thickness t may be made equal to 0.2 times the width kl. The bucket as thus drawn is to a scale as yet undetermined; the widths kl vary in practice from 0.2 to 1.0 inch. (For a study of steam trajectories and the relation there- of to bucket design, see Roe, Steam Tur- bines, 1911.) It should be noted that the back, rather than the front, of the bucket is made tan- gent to the relative velocity V. The work per pound of steam being computed from the velocity diagram, and the steam con- sumption estimated for the assumed out- put, we are now in a position to design the nozzle. 531. Multi-stage Impulse Turbine. If the number of pres- sure stages is few, as in the Curtis type, the heat drop may be di- vided equally between the stages. In the Bateau type, with a large number of FIG. 254. Art. 531. Curtis Turbine. (General Electric Company ) Stages, a proportion- ately greater heat drop occurs in the low-pressure stages. The corresponding intermediate pressures are determined from the heat diagram, and the various stages are then designed as DESIGN OF MULTI-STAGE TURBINE 377 separate single-stage impulse turbines, all having the same rotative speed. The entrance angles of the fixed intermediate blades in the Curtis turbine are equal to those of the absolute exit velocities of the steam. Their exit angles may be adjusted as desired; they may be equal to the entrance angles if the latter are not too acute. The greater the number of pressure stages, the lower is the economical limit of circumferential speed; and if the number of revolutions is fixed, the smaller will be the wheel. Figure 254 shows a form of Curtis turbine, with five pressure stages, each containing two rows of moving buckets. The electric generator is at the top. 532. Problem. Preliminary Calculations for a Multi-stage Impulse Tnrline. To design a 1000 (brake) hp. impulse turbine with three pleasure stages, having two moving wheels in each pressure stage. Initial pressuie, 130 Ib. absolute; temperature, 600 F. ; final pressure, 2 Ib. absolute; entering stream angles, 20; peripheral velocity, 500 ft. per second ; 1200 revolutions per minute. By reproducing as in Fig. 240 a portion of the Molher heat chart, we obtain the expansive pat,h AB, and the heat drop is 1316.6 - 987.5 = 329.1 B. t. u. Divid- ing this into three equal parts, the heat drop per stage becomes 329.1 3 = 109.7 B. t. u. This is without correction for friction, and we may expect a somewhat unequal division to appear as friction is considered. To include friction in deter- mining the change of condition during flow through the nozzle, we lay off, in Fig. 240, AH = 109.7, HG = -, and project GE, finding/* = 50, t = 380, at the out- lets of the first set of nozzles. The velocity attained (with 10 per cent loss of available heat by friction) is v = 212.42 V109.7 = 2225 ft. per second t u *n f FIG. 255. Art. 532. Multi-stage Velocity Diagram. We now lay off the velocity diagram, Fig. 249, making a =20, ^ = 500, v=2225. The exit velocity x may be variously drawn; we will assume it so that 378 APPLIED THERMODYNAMICS the relative angles e and/ are equal, and, allowing 10 per cent for bucket friction, will make x 0.9 F. For the second wheel, the angle a' is again 20, while v', on account of friction along the stationar} T or guide blades, is 0.9 Y. After locating F', if the angles e 1 and/ 7 were made equal, there would in some cases be a back- ward impulse upon the wheel, tending to stop it, at the emergence of the jet along T. On the other hand, if the angle/' weie made too acute, the stream would be unable to get away from the moving buckets. With the particular angles and velocities chosen, some backward impulse is inevitable. AVe will limit it by mak- ing/' = 30. The rotative components of the absolute velocities may be computed as follows, the values being checked as noted from the complete graphical solution of Fig. 255 : ab = v cos 20 = 2225 x 0.93969 = 2090.81. (2080) cd = cz - rfz = 0.9 Fcos/- u- 0.9 Fcose - u = 0.9(2090.81 - 500)- 500 = 931.73. (925) ef= eg cos20 = 0.9 c#*cos20 = 0.9 x 1158 x 0.93969 = 979. (975) U = km - Im = 500 - x 1 cos 30 = 500 - 0.9 V cos 30 = 500 - (0.9 x 596.2 f x 0.80603)= 36, The work per pound of steam is then ("* + "* +/"*') = 30fiG x 50 = 61500 v O < w / O**'i foot-pounds, in the first stage. This is equivalent to 61,500 778 = 79.2 B. t. u. The heat drop assumed foi this stage was 109.7 B. t. u. The heat not converted Into work exists as lesidual velocity or has been expended in overcoming nozzle and bucket friction and thus indirectly in superheating the steam. It amounts to 109 7 - 79 2 = 30.5 B, t. u. Returning to the construction of Fig. 240, we lay off in Fig. 256 aw, = 79.2 B. t. u. and project no to r?, finding the condition of the steam after passing the first stage buckets. Bucket friction has moved the state point from m to o, at which latter point Q = 12:37.2, p = 50, t = 414. This is the condition of the steam which is to enter the second set of nozzles. These nozzles are to expand the steam down to that pressure at which the ideal (adiabatic) heat drop from the initial condition is 2 x 109.7 = 219.4 B. t. u. Lay off ae = 219.4, and find the line eg of 12 Ib. absolute pressure. Drawing the adiabatic op to intersect eg, we find the heat drop for the second stage, without friction, to be 1237.2 1120 = 117.2 B. t. u., giving a velocity of 21 2.42 Vl 17.2 = 2299.66 ft. per second. * To find cgr, we Lave cb = Fcos e = 2090.81 - 500 = 1590.81, bj = v sin a = 2225 x S4202 = 760.99, F= ^cb 2 + ty* = Vi5iio.81* + 700.W* = 1705, T: = 9 F= 0.9 x 1765 = 1688.5, ch = 3 sin/ = 1588 5 sin e = 1688.5^ = 1588.5 700 - 99 = 685, _ _ F 1765 eg = V'ch 2 + hf = Veg^+MTfl 2 = 1158. t To find F', we have gf= v' sin 20 = Tsin 20 = 0.9 x 1158 x 0.34202 = 355, ft/= tf- u = 979 - 500 = 470, F/ = rftf* + gf* =^ / 479 2 + 35? = 596.2. STEAM PATH, MULTI-STAGE TURBINE 379 The complete velocity diagram must now be drawn for the second stage, fol- lowing the method of Fig. 255. This gives for the rotative components, ab - 2160.97, cd = 994.87, ef= 1032.59, LI = 8.06. (There is no backward impulse from kl in this case.) The work per pound of steam is 500(2160.97+994.87+1032.59+8.06) = or 83.76 B. t, u. Of the available heat drop, 117.2 B. t. u., 33.44 have been ex- pended in friction, etc. Laying off, in Fig. 256, pq = 33.44, and projecting qr to meet pr, we have r as the state point for steam entering the third set of nozzles. Here p = 12, *i=223, <?'i=115344. In expanding to the final condenser pressure, the ideal path is rs, terminating at 2 Ib. absolute, and giving an uncorrected heat drop of Q r -& = 1153.44-1039 = 114.44 B. t. u. The velocity attained is 212.42 VlU.44 =2271.83 feet per second. A third velocity diagram shows the work per pound of steam for this stage to be 63,823 foot-pounds, or 82.04 B.t.u. We are not at present con- cerned with determining the condition of the steam at its exit from the third stage, The whole work obtained from a pound of steam passing through the three stages is then 79 .2 +83. 76 +82.04= 245.0 B. t. u. (20a). The horse power required is 1000 at the brake or say 10000.8 = FIG. 256. Art. 532. Steam Path, Multi- stage Turbine. 1250 hp. at the buckets. This is equivalent to 1250 X 1980000 778 ' 3,181,250 B. t. u. per hour. The pounds of steam necessary per hour are 3,181,2504-245.0=12,974. This is equivalent to 12.97 Ib. per brake hp,-hr., a result sufficiently well confirmed by the test results given in Chapter XV. GW Proceeding now to the nozzle design, we adopt the formula F= from Art. 521. It will be sufficiently accurate to compute cross-sectional areas at throats and outlets only. The path of the steam, in Fig. 256, is as follows: through the first set of nozzles, along am; through the corresponding buckets, along mo; thence alternately through nozzles and buckets along ou, ur, n>, vt. The points u, v 3 etc., are found as in Fig. 240. It is not necessary to plot accurately the whole of the paths am, ou, rv] but the condition of the steam must be determined, for each nozzle, at that point at which the pressure is 0.57 the initial pressure (Art. 522). The three initial pressures are 150, 50, and 12; the corresponding throat pressures are 85.5, 28.5. and 6.84. Drawing these lines of pressure, we lay off, for example, , project xy to wy, and thus determine the state y at the throats of the 380 APPLIED THERMODYNAMICS first set ot nozzles. The corresponding states are similarly determined for the other nozzles. We thus find, at y, p = 85.5, t = 474, at m, p = 50, t = 380, q = 1260.5 ; q = 1217.87 ; at A, p = 28.5, t = 313, at u, p = 12, x = 0.989, q = 1192 ; q = 1131.72 ; at B, p = 6.84, ar = 0.9835, at u, ^ = 2, a; = 0,932, = 1118; 5=1050.44. We now tabulate the corresponding velocities and specific volumes, as below. The former are obtained by taking V = 223.84 V^ - q 2 ; the latter are computed from the Tumlirz formula, W = 0.5963 - 0.256. Thus, at the throat of the first nozzle, V = 223.84 V1316.8 - 12(50.5 = 1683 ; while W = 0.5963 4GO + 474 _ 0.256 = 6.26. 80. 5 In the wet region, the Tumlirz formula is used to obtain the volume of dry steam at the stated pressure and the tabular corresponding temperature ; this is applied to the wet vapor : W w = 0.017 + x( W - 0.017) . The tabulation f ollows. At y, V = 1683, W = 6.26 ; at m, V = 2225, W = 9.724 : at A V= 1507, W = 15.92 ; at ti, F= 2299, TT= 32.24; at B, V = 1330, TF = 53.92 ; at v, 7 = 2271, W = 162.62. The value of G 9 the weight of steam flowing per second, is 12,974- 3600 = 3.604 Ib. For reasonable proportions, we will assume the number of nozzles to be 16 in the first stage, 42 in the second, and 180 in the third. The values of G per nozzle for the successive stages are then 3.604 16 = 0.22525, 3.604 - 42 = 0.08581 and 3.604 -^ 180 = 0.02002. We find values of F as follows : Aty, at m, 0.22525 x 6.26 1683 0.22525 x 9.724 2225 0.08581 x 15.92 = 0.000839; at u, = 0.000989; at 3, = 0.000903; at u, 0.08581 x 32.24 2299 0.02002 x 53.92 1330 0.02002x162.62 = 0.001205 ; = 0.000809; = 0.00144. ' 1507 ' ' 2271 Completing the computation as to the last set of nozzles only, the throat area is 0.000809 sq. ft, that at the outlet being 0.00144 sq. ft. These corre- spond to diameters of 0.385 and 0.515 in. The taper may be uniform from throat to outlet, the sides mak- ing an angle of 10. This requires a length from throat to outlet of (0.515 - 0.385) -- 2 tan 5 = 0.742 in. The length from inlet to throat may be one fourth this, or 0.186 in., the FIG. 267. Axt.532.-mrd Stage Nozzle. ^f * i*^ l^* ^ oT^' The nozzle is shown in Fig. 257. The diameter of the bucket wheels at mid-height is obtained from the rotative speed and peripheral velocity. If d be the diameter, 3.1416 d x 1200 = 60 x 500, or d = 7.98 feet. PRESSURE TURBINE 381 The forms of bucket are derived from the velocity diagrams. For the first stage, we proceed as in Art. 530, using the relative angles e and /given in Fig. 255 for determining the angles of the backs of the moving blades, and the absolute angles for determining those of the stationary blades. 533. Utilization of Pressure Energy. Besides the energy of impulse against the wheel, unaccompanied by changes in pressure, the steam may expand while traversing the buckets, producing work by reaction. This involves incomplete expansion in the nozzle, and makes the velocities of the discharged jets much less than in a pure impulse turbine. Lower rotative speeds are therefore practicable. Loss of efficiency is avoided by carrying the ultimate expansion down to the condenser pressure. In the pure pressure turbine of Parsons, there are no expanding nozzles ; all of the expansion occurs in the buckets (Art. 524). (See Fig. 247.) Here the whole useful effort is produced by the reaction of the expanding steam as it emerges from the working blades to the guide blades. No velocity is given up during the passage of the steam ; the velocity is, in fact, increasing, hence the name reaction turbine. The impulse turbine, on the contrary, performs work solely because of the force with which the swiftly moving jet strikes the vane. It is sometimes called the velocity turbine. Turbines are further classified as horizontal or vertical, according to the position of the shaft, and as radial flow or axial flow, according to the location of the successive rows of buckets. Most pressure turbines are of the axial flow type. 534. Design of Pressure Turbine. The number of stages is now large. The heat drop in any stage is so small that the entering velocity is no longer negligible. The velocity of the steam will increase continually throughout the machine, being augmented by expansion more rapidly than it is decreased by friction. If the effective velocity at entrance to a row of moving blades is Fi, increasing to F a by reason of expansion occurring in the blades, the energy of reaction, available for 7 2 2_7j2 performing work, is - . The effective velocity entering the stationary blades being Fa, and increasing to V by expansion therein, energy is produced equal to 7 4 z_y 3 2 - - , which is given up to the following set of moving blades, in the shape of an impulse. Each moving blade thus receives an impulse at its entrance end and a reaction at its outlet end. By making the forms and angles of fixed and moving blades the same, the work done by impulse equals the work done by reaction, or In Fig. 259, lay off the horizontal distance F0 } representing the aggregate axial length of four drums composing a pressure turbine. The peripheral speeds of drums vary from 100 to 350 ft. per sec., increasing as the pressure decreases and 382 APPLIED THERMODYNAMICS as the size of the machine increases, and being generally less in marine than in stationary service The successive drum diameters and peripheral speeds frequently have the ratio A/2 : 1 (21) Assume, in this case, that the peripheral speed of the first drum is 130 it per sec., and that A of the last drum 350 ft per sec. The * usual plan is to increase the successive i drum speeds at constant ratio. This makes the speeds of the blades on the intermediate drums 181 and 251 ft per sec , respectively. The steam velocity will be usually between If and 3 times the blade ve- locity. it will increase more rapidly as Art. 534, Piob. 17. Design of the f ower pressures are reached The Pressure Turbine. yalue of thifl ratlo should vary between about the same limits for each drum. The curve EA is sketched to represent steam velocities assumed: the ordinate FE may be 130X2 = 260 ft. per second, and the ordinate OA say 973 ft. per sec. The shape of this cuive is approximately hyperbolic. It is now desirable to lay off on the axis FO distances representing approximately the lengths of the various drums. An empirical formula which facilitates this is C Fro. 259. where %=number of rows of blades when the blade speed is u ft. per sec., C = a constant, =1,500,000 for marine turbines, =2,600,000 for turbo-generators. When (as in our case) u is different for different drums, we have ni being the number of stages on a drum of blade velocity wi> developing the s pro- portion of the total power. The power developed by the successive drums increases toward the exhaust end : let the division in this case be }, , 1, f , of the total respect- ively. Then for (7 = 2,600,000, 2,600,000 1 ~ X 6"~ 2b} 2,600,000 1 - ~ X ' 2,600,000 2,600,000 3 350~ X 8 The total number of stages is then approximately 60. The distances FC, CD, DB, BO, are then laid off, equal respectively to !, i, $ and & of FO, At any point like G, then, the steam velocity is ZG and the blade velocity is that for the drum in question: for G, for example, it is 181 ft. per sec. Knowing the steam velocity and peripheral velocity for any state like <?, we construct a velocity diagram as in Fig. 249, choosing appropriate angles of entrance and exit. In ordinary practice, the expansion in the buckets is sufficient, not- PRESSURE TURBINE 383 withstanding friction, to make the relative exit and absolute entrance angles and velocities about equal. (This equalizes the amounts of work done by impact and by reaction.) In such case, we have the simple graphical construction of Fig. 260. Since abbc, db=*be, and ad=ec, we ob- tain . u(ah+he) ad(hc + hd) W0rk ' Drop the perpendicular bh, and with h as a center describe the arc aj. Draw dg per- pendicular to ac. Then dg 2 = adXdc = ad(dh+hc), and foot-pounds, or B. t. u. FIG. 260 Art. 634, Prob. 18. Velocity Diagram, Pressure Turbine. This result represents the heat converted into work at a stage located vertically in line with the point G, Fig. 259, Let this heat be laid off to some convenient scale, as GH. Similar determinations for other states give the heat drop curve IJKELMNOP. The average ordinate of this curve is the average heat drop or work done per stage. If we divide the total heat drop obtained by the average drop per stage, we have the number of stages, the nearest whole number being taken.* Suppose the machine to be required to drive a 2000 kw. generator (2400 kw. overload capacity) at 175 to. initial absolute pressure and 50 of superheat, the condenser pressure being 1 Ib absolute, the r. p. m. 3600, the generator efficiency 0.94 and the losses as follows: steam friction, 25; leakage, 06, windage and bearings, 0.16; residual velocity in exhaust, 0.03. The theoretical heat drop is 1227890=337 B. t. u. The drop corrected for steam friction is 337X0.75 =253 B. t. u. The average ordinate of the heat drop curve in Fig. 259 being 4.16 B. t. u., the corrected number of stages is 253 =61 (nearest whole number) instead of 60. The curve of heat drops may now 4.16 be corrected for the necessary revised numbers of stages in the various drums: thus, 253 the whole heat drop being 253 B. t. u., that in the first drum must be =42 2 6 B.\ u. The average heat drop per stage for the first drum being (average ordinate 42.2 of U) 1.56 B, t. u., the number of stages on that drum is ~ = 27 (instead of 26). r -- 1.56 For the other drums, proceeding in the same way, the numbers of stages work out as before, 16, 10 and 8. The aggregate of losses exclusive of steam friction is 0.25. The heat available for producing power is then 253X0.75 = 190 B. t. u. per Ib. of steam. With the given generator efficiency, the weight of steam required per kw.-hr. is 2545 X 1.34 190X0.94 5 = 19.0. * Dividing the total heat drop at a state in a vertical line through C by the average drop per stage from F to C, we have the number of stages on the first drum. 384 APPLIED THERMODYNAMICS At normal rate, the weight of steam used at the overload condition is 19.0X2400 3600 12.67 Ib. per sec. 535. Specimen Case. To determine the general characteristics of a pressure turbine operating between pressures of 100 and 3 5 Ib., with an initial superheat of 300 F., the heat drop being reduced 25 per cent by friction. There are to he 3 drums, and the heat drop is to be equally divided between the drums. The per- ipheral speeds of the successive drums are 160, 240, 320 ft. per second. The rela- tive entrance and absolute exit velocities and angles are equal; the absolute entrance angle is 20. The turbine makes 3000 r. p. m. and develops 2500 kw. with losses between buckets and generator output of 65 per cent. ^ri_ i A ^ 3s& Y.,0 I x 1 ^0 \J$ 1 V- if* 1290 i \ ~J~ ~*2Q .r$ I \ >i * "I" c 1270 T 2% 3-3**" X j[ ft \ 1 XJ T < t < J % 1246 "*" i rX ^ jj y u ' (* 9 I \t o ' *j *0 ^s ^ 122Q l i r~ ~3 h ^ ^ ^ \J ^ 1210 3l '> ' s. s^ i s T zd V <* T \ flu ^ V so S V V 9 dp Sy - J X ~Zfj 1g 1 T *D ^2 -4- F iJ Tfr T . 260 a. Art. -535. Expansion Path, Pressure Turbine. PRESSURE TURBINE 385 In Fig. 260 a, the expansive path is plotted on a portion of the total heat- entropy diagram. The total heat drop is shown to be 1342 1130 = 212 B. t. u., and the heat drop per drum is 212 - 3 = 70| B. t. u. In Fig. 260 b, lay off to any scale the equal distances ab, be, cd, and the vertical distances ae, bg, ci, rep- resenting the drum speeds. Lay ofE also ak, bm, co, equal respectively to 1 x (ae, bg, ci), and al, bn, cp, equal respectively ale FIG. 260 b. Art. 535 Elements of Pressure Turbine. of entrance absolute velocities is now assumed, so as to lie wholly within the area llsntpuvowmx. Figure 260 c shows the essential parts of the velocity diagram for the stages on the first drum. Here ab represents aq in Fig. 260 b> ad represents ae, the angle bad is 20, and (-^-} 2 = f^ZV =3.12 B. t. u. is the heat drop \lo8.o/ \158.o/ for the first stage in the turbine. Making ac represent by and drawing dc, ch, af, = 3.70 B. t. u. as the heat drop for the last stage on we find ( r-z TVT \lo8.3/ Vlob.o/ the first drum. For intermediate stages between these two, we find, IlHTIAt AnSOTUTE VELOCITY OEDINATB PROM d HEAT DROP, B. T.U ab = 350 de - 279.7 3.12 356J 282.8 3.20 362J 285.9 3.26 368f 289.0 3.34 375 202.1 3.40 381J 295.3 3.48 387J 298.4 3.56 393 i 301.5 3.63 ac = 400 df= 304.7 3.70 386 APPLIED THERMODYNAMICS In Pig. 260 5, we now divide the distance ab into 8 equal parts and lay off to any convenient vertical scale the heat drops just found, obtaining the heat drop curve zA. The average ordinate of this curve is 3.41 and the number of stages on the first drum is 70 j 3.41 = 21 (nearest whole number). The number of stages FIQ. 260 c. Art. 535, Velocity Diagram, Pressure Turbine. on the other drums is found in the same way, the peripheral velocity ad, Fig. 260 c, being different for the different drums. The diameter d of the first drum is given by the expression 3000^ = 60X160 or d = The weight of steam flowing per second is 2500 x 1.34 x 2545 0.65x212x3600 _, 71 1/ * 11D ' PRESSURE TURBINE 387 In the first stage of the first drum, the condition of the steam at entrance to the guide blades is (Fig 260 a) # = 1342, p = 100; at exit from the moving blades, it is H = 1338 59, p = 98. From the total heat-pressure diagram, or by computa- tion, the corresponding specific volumes are 6.5 and 6.6. The volumes of steam flowing are then 6.5 X 17.1 = 111 and 6.6 X 17.1 = 113 cu ft per second. The absolute steam velocities are (Fig 260 6) 350 and 356i ft. per second. The axial components of these velocities (entrance angle 20) are 034202X350 = 120, and 0.34202 X356 = 122 The drum periphery is 1 02 X3 1416 = 3 2 f t. If the blade thicknesses occupy J this periphery and the width for steam passage between the buckets is constant, the width for passage of steam is f X3 2 =2 133 ft., and the necessary height of fixed buckets is = 434 ft. or 5 2 in. at the beginning of the stage'and 2. loo X 1^0 2133X122 = 0.434 ft. or 5 2 in. at the end. The fixed blade angles are determined by the velocities be and ab, Fig. 260, those of the moving blades by bd and be. There is no serious error involved in taking the velocitv and specific volume as constant throughout a blade. The height of the movmg buckets should of course not be less than that of the guide blades; this may be accomplished by increasing the thick- ness of the former The blade heights should be at least 3 per cent of the drum diameter, if excessive leakage over tips is to be avoided. The clearance over tips varies from 0.008 to 0.01 inch per foot of drum diameter. Blade widths vary from | to 1J in , with center-to-center spacing from If to 4 ins. The method of laying out the blades is suggested in Fig 260 d. Let ab be the absolute steam velocity at entrance to a row of moving blades, cb the blade velocity. Then the relative velocity ac determines the enter- ing angles at c and e The movmg blade is made with a long straight tapering tail, in which expan- sion occurs after the steam passes the point r. Let hjj parallel with the center line of the expanding portion of the blade (fa), represent the velocity attained at the outlet of this blade, and let jk again represent the blade velocity. Then hk represents the absolute velocity of exit and determines the entering angles of the following fixed blades, on and ml being parallel with hk. Finally, since the steam must emerge from the fixed blades with a velocity parallel with ab, we draw pq parallel with ab, determining the direction of the expanding posi- FJQ. 260 d. Art 535 Blading tion (beyond s) of the fixed blade. The angles abc of Pressure Turbine, and kjk are made equal, and range between 20 and 30. It should be noted that the velocities indicated by the curve qr, Fig. 260 6, are those of the steam at exit from the fixed blades and entrance to the moving blades. The diagram of Fig. 260 gives the absolute velocity of the steam entering the next set of fixed blades. COMMERCIAL FORMS OF TURBINE. 536. De Laval; Stumpf. Figure 235 illustrates the principle of the De Laval machine, the working parts of which are shown in Fig. 261. Entering through divergent nozzles, the steam strikes the buckets around the periphery of the wheel b. The shaft c transmits power through the helical pinions a, a, which drive the gears e, e> e t e, on the working shafts /, /. The wheel is housed with the iron cas- 388 APPLIED THERMODYNAMICS ing g. This is a horizontal single-stage impulse turbine with a single wheel. Its rotative speed is consequently high; in small units, it reaches 30,000 r. p. m. It is b.iilt principally in small sizes, from 5 to 300 h.p. The nozzles make angles of 20 with the plane of the wheel; the buckets are symmetrical, and their angles DE LAVAL TURBINE 389 range from 32 to 36, increasing with the size of the unit. For these proportions, the most efficient values of u would be about 950 and 2100 for absolute steam veloci- ties of 2000 and 4400 feet per second, respectively; in practice, these speeds are not attained, u ranging from 500 to 1400 feet per second, according to the size. The high rotative speeds require the use of gearing for most applications. The helical gears used are quiet, and being cut right- and left-hand respectively they practically eliminate end thrust on the shaft. The speed is usually reduced in the proportion of 1 to 10. The high rotative speeds also prevent satisfactory balanc- ing, and the shaft is, therefore, made flexible ; for a 5-hp. turbine, it is only J inch in diameter. The bearings h, /are also arranged so as to permit of Rome movement. The pressure of steam in the wheel case is that of the atmosphere or condenser, all expansion occurring in the nozzle. A centrifugal governor controls the speed by throttling the steam supply and by opening communication between the wheel case and atmosphere when necessary. The nozzles of the De Laval turbine are located as in Fig. 235. Those of the Stumpf, another turbine of this class, are tangential, while the buckets are of the Pelton form (Fig. 252), and are milled in the periphery of the wheel. A very large wheel is employed, the rotative speeds being thus reduced. In a late form of the Stumpf machine, a second stage is added. The reversals of direction are so extreme that the fluid friction must be excessive. 537. Curtis Turbine. This is a multi-stage impulse turbine, the principle of operation having been shown in Fig. 245. In most cases, it is vertical ; for marine applications, it is necessarily made horizontal. Figure 262 illustiates the stationary and moving blades and nozzles. Steam enters through the nozzle A, strikes a row of mov- ing vanes at a, passes from them through stationary vanes B to another row of moving vanes at e, then passes through a second set of expanding nozzles at li to the next pressure stage. This particu- lar machine has four pressure stages with two sets of moving buckets in each stage. The direc- tion of flow is axial. The number of pressure stages may range from two to seven. From two to four velocity stages (rows of moving buckets) are used in each pressure stage. In the two-stage machine, the second stage is disconnected when the turbine runs non-con- densing, the exhaust from the first stage being discharged to the at- mosphere. Governing is effected FIG. 262. Art. 637. Curtis Turbine. 390 APPLIED THERMODYNAMICS by automatically varying the number of nozzles in use for admitting steam to the first stage. A step bearing carries the whole weight of the machine, and must be supplied with lubricant under heavy piessure ; an hydiauhc accumulator system is commonly employed. 538. Rateau Turbine. This is a hoiizontal, axial flow, multi-stage impulse turbine. The number of pressure stages is very laige from twenty-five upward. There is one velocity stage in each pressure stage. Very low speeds are, theiefore, possible. Figure 203 shows the general airangement ; the tranveise partitions e, e form cells, in which i evolve the wheels/, /, the nozzles are merely slots in the partitions. The blades aie pressed out of sheet steel and riveted to the wheel. The wheels themselves are of thin pressed steel. FIG. 2G3 Art. 538. -Rateau Turbine. 539. Westinghouse-Parsons Turbine. This is of the axial 'flow pressure type, and horizontal. The steam expands through a large number of successive fixed and moving blades. In Fig. 204, the steam enters at A and passes along the vari- ous blades toward the left; the movable Buckets are mounted on the three drums, and the fixed buckets project inward from the casings. The diameters of the drums increase by steps ; the iuci easing volume of the steam within any section is accommodated by varying the bucket heights. The balance pistons P, P, P are used to counteract end thrust. The speed is fairly high, and special provision must be made for it in the design of the bearings. Governing is effected by inter- mittently opening the valve T r ; this valve is wide open whenever open at all. The length of this machine is sometimes too great for convenience. To over- come this, the " double-flow " turbine receives steam near its center, through expanding nozzles which supply a simple Pelton impulse wheel. This utilizes a large proportion of the energy, and the steam then flows in both directions axially, through a series of fixed and moving expanding buckets. Besides reduc- ing the length, this arrangement practically eliminates end thrust and the neces^ sity for balance pistons. APPLICATIONS OF TURBINES 391 392 APPLIED THERMODYNAMICS 540. Applications of Turbines. Turbo-locomotives have been experimented with in Germany ; the direct connection of the steam turbine to high-pressnre rotary air compressors has been accomplished. In stationary work, the diiect driving of genei ators by turbines is common, and the high rotative speeds of the latter have cheapened the former. At high speeds, difficulties may be experi- enced with commutation; so that the turbine is most successful with aJteinating- current machines. When driving pumps, turbines permit of exceptionally high lifts with good efficiencies for the centrifugal type, and low first costs. For low- pressure, high-speed blowers, the turbine is an ideal motor. (See Art. 239.) The outlook for a gas turbine is not promising, any gas cycle involving combustion at constant pressure being both practically and thermodynamically inefficient. The objections to the turbine in marine application have arisen from the high speed and the difficulty of reversing. A separate reversing wheel may be em- ployed, and graduation of speed is generally attained by installing tuibines 111 pairs. A small reciprocating engine is sometimes employ ed for maneuvering at or near docks. Since turbines are not well adapted to low rotative speeds, they are not recommended for vessels rated under 15 or 16 knots. The advantages ot turbo-operation, in decreased vibration, greater simplicity, smaller and more deeply immersed propellers, lower center of gravity of engine-room machinery, decreased size, lower first cost, and greater unit capacity without excessive size, have led to extended marine application. The most conspicuous examples are in the Cunard liners Lusitania and Mauretama. The former has two high-pi essure and two low- pressure main turbines, and two astern turbines, all of the Parsons type (22). The drum diameters are respectively 96, 140, and 104 in. An output of 70,000 hp. is attained at full speed. 541. The Exhaust-steam Turbine. From the heat chart, Fig. 177, it is obvious that sfceam expanding adiabatically f rom 150 Ib. absolute pressure and 600 F. to 1.0 Ib. absolute pressure transforms into work 365 B. t. u. It has been shown that in the ordinary reciprocating engine such complete expansion is unde- sirable, on account of condensation losses. The final pressure is rarely below 7 Ib. absolute, at which the heat converted into work in the above illustration is only 252 B. t. u. The turbine is particularly fitted to utilize the remaining 113 B. t. u. of available heat. The use of low-pressure turbines to receive the exhaust steam from reciprocating engines, has, therefore, been suggested. Some progrebs has been made in applying this principle in plants where the engine load is intermit- tent and condensation of the exhaust would scarcely pay. With steel mill en- gines, steam hammers, and similar equipment, the introduction, of a low-pressure turbine is decidedly profitable. The variations in supply of steam to the tuibine are offset by the use of a regenerator or accumulator, a cast-iron, water-sprayed chamber having a large storage capacity, constituting a " fly wheel for heat," and by admitting live steam to the turbine through a Deducing valve. When a sur- plus of steam i caches the accumulator, the pressure rises; as soon as this falls, some of the watei is evaporated. The maximum pressure is kept low to avoid back pressure at the engines. A steam consumption by the turbine as low as 35 Ib. per brake hp.-hr. has been claimed, with 15 Ib. initial absolute pressure and a final vacuum of 26 in. Other good results have been shown in various trials (23). (See Art. 552.) Wait (24) has described a plant at South Chicago, 111., in EXHAUST STEAM TURBINES 393 which a 42 by 60 inch double cylinder, reversible rolling-mil I engine exhausts to an accumulator at a pressure 2 or 3 Ib. above that of the atmosphere. This delivers steam at about atmospheric pressure to a 500 kw. Rateau turbine operated with a 28-m vacuum. The steam consumption of the turbine was about 35 Ib. per electrical hp -hr , delivered at the switchboard. The S S. Turbinia, in 1897, was fitted with low-pressure turbines receiving the exhaust from reciprocating engines and operating between 9 Ib. and 1 Ib. absolute. One third of the total power of the vessel was developed by the turbines, although the initial pressure was 160 Ib. 542. Commercial Considerations. The best turbines, in spite of their thermo- dynaimcally superior cycle, have not yet equalled in thermal efficiency the best reciprocating engines, both operating at full load. (This refers to work at the cylinder. The heat consumption referred to work at the shaft has probably been brought as low, with the turbine, as with any form of reciprocating engine ) The combination of reciprocating engine and turbine (Art. 552) has probably given the lowest con- sumption ever reported for a vapor engine. The average turbine is more economical than the average engine; and since the mechanical and fluid friction losses are disproportionately large, it seems reasonable to expect improved efficiencies as experimental knowledge accumulates. The turbine is cheaper than the engine; it weighs less, has no fly wheel, requires less space and very much less foundation. It can be built in larger units than a reciprocating cylinder. Power house buildings are cheapened by its use; the cost of attendance and of sundry operating supplies is reduced. It probably depre- ciates less rapiflly than the engine. The wide range of expansion makes a high vacuum desirable; this leads to excessive cost of condensing apparatus. Similarly, superheat is so thoroughly beneficial in reducing steam friction losses that a con- siderable investment in superheaters is necessary* The choice as between the turbine and the engine must be determined with reference to all of the conditions, technical and commercial, including that of load factor. Turbine economy cannot be measured by the indicator, but must be determined at the brake or switchboard, and should be expressed on the heat unit basis (B t u. consumed per unit of output per minute). For results of trials of steam turbines, see Chapter XV. (1) Tram. Inst. Engrs and Shipbuilders in Scotland, XLVI, V. (2) Berry, The Temperature-Entropy Diagram, 1905. (2 a) For the general theory of fluid flow, see Cardullo, Practical Thermodynamics, 1911, Arts, 55-60; Goodenough, Principles of Thermodynamics, 1911, Arts. 148-150, 153; for empirical formulas, see Goodenough, op cit. } Art. 154. (3) To show this, put the expression in . v the brace equal to m, and make -p 0; then ( - | , which may be solved for any given value of y. (4) Thesis, Polytechnic Institute of Brooklyn, 1905. (5) Thomas, Steam Turbines, 1906, 89. (6) Proc. Inst. Civ. Eng,, CXL, 199. (7) Zetts. Ver. Deutsch. Ing , Jan. 16, 1904. (8) Rankine, The Steam Engine, 1897, 344. (9) Experimental Researches on the Flow of Steam, Brydon tr.; Thomas, op. tit., 106. (10) Thomas, op. tit., 123. (11) Engineering, XIII (1872). (12) Trans. A.S.M. E., XI, 187. (13) MUM. uber Forschungsarb., XVIII, 47. (14) Practice and Theory of the Injector, 1894, (15) Peabody, Thermodynamics, 1907, 443. 394 APPLIED THERMODYNAMICS (16) Trans. A. S. M. E., XXVII, 081. (17) Stodola, Steam Turbines. (18) The Steam Engine, 1905, I, 170. (19) Technical Thermodynamics, Klem tr., 1907: I, 225; II, 153. (20) Trans. A. S. M. E , XXVII, 081. (20 a) For a method for equalizing the three quantities of work, see Caidullo's paper, " Energy and Pressure Drop in Compound Steam Turbines," Jour. A. S M. E., XXXIII, 2. (21) See H. F. Schmidt, in The Engineer (Chicago), Dec. 16, 1907; Trans. Inst Engrs. and Shipbuilders in Scotland, XLXIX. (22) Power, November, 1907, 770. (23) Trans. A. S. M. E., XXV, 817; Ibid, XXXII, 3, 315. (24) Proc. A. I. E. E., 1907. SYNOPSIS OF CHAPTER XIV The turbine utilizes the velocity energy of a jet or stream of steam. Expansion in a nozzle is adiabatic, but not isentropic , the losses in a turbine are due to residual velocity, friction of steam through nozzles and buckets and mechanical friction. JS + PW+ =e+pw + ^-,oT^ = q-Q, approximately ; 2(7 ly <*g whence V = 223 .84 \ 'q - Q. The complete expansion secured in the turbine warrants the use of exceptionally high vacuum. Nozzle friction decreases the heat converted into work and the velocity attained; F= 212.42 V^Q. The heat expended in overcoming friction reappears in drying or superheating the steam. F- # , which reaches a minimum at a definite value of - Tor steam, this value V P is about 0.57. If the discharge pressure is less than 57 p, the nozzle converges to a "throat" and afterward diverges. The multi-stage impulse turbine uses lower rotative speeds than the single stage. The diverging sides of the nozzle form an angle of 10 ; the converging portion may be one fourth as long. Steam consumption per Ihp.-hr. = 2545 ->- JE(q - ). The rotative components of the absolute velocities determine the work ; the relative velocities determine the (moving) bucket angles. Bucket friction may decrease relative velocities by 10 per cent during passage. Work = (0 cos a Ycosg*) -. ff Efficiency = E = Work 778(7 $) . Bucket angles may be adjusted to equalize end thrust, to secure maximum work, or may be made equal For a right-angled stream change, maximum efficiency is 0.50 ; with complete reversal, it is 1.00. TVith practicable buckets, it is always less than 1.0. The backs of moving buckets are made tangent to the relative stream velocities. The angles of fixed blades are determined by the absolute velocities. In the pure pressure turbine, expansion occurs in the "buckets. No nozzles are used. Turbines may be horizontal or vertical, radial or axial flow, impulse or pressure type. In designing a pressure turbine, - = 0.33 to 0.67. The heat drop at any stage may equal f -O 2 5 Fig. 200, The number of stages is the quotient of the whole heatj PROBLEMS 395 drop, corrected for friction; by the mean value of this quantity. Friction through buckets may be from 20 to 30 per cent. The accumulated heat diop to any stage is ascertained and the condition of the steam found as in Pig. 240 Typical design, Arts. 534, 535. Commercial forms include the De Laval, single-stage impulse : Stumpf , single- or two-stage impulse, with Pelton buckets. Curtis, multi-stage impulse, usually vertical, axial flow. Bateau, multi-stage impulse, axial flow, horizontal, many stages. Westinghouse-Parsons, pressure type, axial flow, horizontal ; sometimes of the " double flow " form. Marine applications involve some difficulty, but have been satisfactory at high speeds. The turbine may utilize economically the heat rejected by a reciprocating engine. A regenerator is sometimes employed. The best recorded thermal economy has been attained by the reciprocating engine ; but commercially the turbine has many points of superiority. PROBLEMS 1. Show on the 7W diagram the ideal cycle for a turbine operating between pressure limits of 140 Ib. and 2 lb., with an initial temperature of 600 F. and adiabatic (isen tropic) expansion. What is the efficiency of this cycle ? (Ana., efficiency is 0.24 ) 2. In Problem 1, what is the loss of heat contents and the velocity ideally attained ? 3. In Problem 1, how will the efficiency and velocity be affected if the initial pressure is 150 lb.? If the initial temperature is 600 F.? If the final pressure is 1 lb.? 4. Solve Problems 1, 2, and 3, making allowance for friction as in Art. 519. 5. Compute analytically, in Problem 3, first case, the condition of the steam after expansion, as in Art 520, assuming the heat drop to have been decreased 10 per cent by friction. (Ans , dry ness =0.877.) 6 An ideal reciprocating engine receives steam at 150 lb. pressure and 550 F., and expands it adiabatically to 7 lb. pressure. By what percentage would the efficiency be increased if the steam were afterward expanded adiabatically in a turbine to 1.5 lb. pressure. (Ans. 9 47 per cent.) 7. Steam at 100 lb. pressure, 92 per cent dry, expands to 16 lb. pressure. The heat drop is reduced 10 per cent by friction. Compute the final condition and the velocity attained. (Ans^ dryness= 0.846 ; velocity = 2375 ft. per sec.) 8. In Problem 5, find the throat and outlet diameters of a nozzle to discharge 1000 lb. of steam per hour, and sketch the nozzle. (Ans. t throat diameter =0.416 in.) p 9. Check the value = 0.5274 for maximum flow in Art. 522. P 396 APPLIED THERMODYNAMICS 10. Check the equation of flow of a permanent gas, in Art. 522. 11. If the efficiency in Problem 5, from steam to shaft, is 0.60, find the steam consumption per brake hp -hr, and the thermal efficiency. 12. In Problem 5, let the peripheral speed be it =480, the angle a =20, and find the work done per pound of steam in a single-stage impulse turbine (a) with end thrust eliminated, (&) with equal relative angles. Allow a 10 per cent reduction of relative velocity for bucket friction. 13 In Problem 12, Case (&), what is the efficiency from steam to work at the buckets ? (Item J7, Art. 526.) Find the ideal steam consumption per Ihp.-hr. 14. Sketch the bucket in Problem 12, Case (6), as in Art. 530. 15. Compute the wheel diameters and design the first-stage nozzles and buckets for a two-stage impulse turbine, with two moving wheels in each stage, as m Art. 532, operating under the conditions of Problem 5, the capacity to be 1500 kw., the enter- ing stream angles 20, the peripheral speed 600 ft. per second, the speed 1500 r. p. m., the heat drop reduced 0.10 by nozzle friction. Arrange the bucket angles to give the highest practicable efficiency,* the stream velocities to be reduced 10 per cent by bucket f notion. State the heat unit consumption per kw.-minute. 16. In Problem 5, plot by stages of about 10 B*t.u. the N'T expansion path in a pressure turbine in which the heat drop is decreased 0,25 by bucket friction. 17. In Problem 16, the drums have peripheral speeds of 150, 250, 350. Construct a reasonable curve of steam velocities, as in Fig. 259, the velocity of the steam enter- ing the fiist stage being 400 ft. per second, and the outputs of the three drums as t, J, }. 18. In Problem 17, let the absolute entrance angles be 20 7 and let the velocity diagram be as in Pig. 260. Find the work done in each of six stages along each drum. Find the average heat drop per stage, and the number of stages in each drum, the total heat drop per drum having been obtained from Problem 16. 19. The speed of the turbine in Problem IS is 400 r.p.m. Find the diameter of each drum. 20. In Problems 16-19, the blades are spaced 2" centers. The turbine develops 1500 kw. Find the heights of the moving blades for one expansive state, assuming losses between buckets and generator of 45 per cent. Design the moving bucket. 21. Sketch the arrangement of a turbine in which the steam first strikes a Pelton impulse wheel and then divides ; one portion traveling through a three-drum pressure rotor axially, the other through a two-pressure stage velocity rotor with three rows of moving buckets in each pressure stage, also axially, the shaft of the velocity turbine being vertical. 22. Compare as to effect on thermal efficiency the methods of governing the De Laval, Curtis, and Westinghouse-Parsons turbines* 23. Detemtine whether the result given in Art. 541, reported for the S.S. is credible. * The angle / must not be less than 24 in any case. CHAPTER XV RESULTS OF TRIALS OF STEAM ENGINES AND STEAM TURBINES 543. Sources. The most reliable original sources of information as to con- temporaneous steam economy are the Transactions or Proceedings of the various national mechanical engineering societies (1). The reports of the Committee of the Institution of Mechanical Engineers on Marine Engine Trials aie of special interest (2). The Alsatian experiments on superheating have already been le- f erred to (Art. 443). The works of Barrus (3) and of Thomas (4) present a maso of results obtained on reciprocating engines and turbines respectively. The investigations of Isherwood are still studied (5). The Code of the American Society of Mechanical Engineers (Trans. A. S. M. E. t XXIV) should be examined. 543 a. Steam Engine Evolution. The Cornish simple pumping engines (9) which developed from those of the original Watt type had by 1840 shown dry steam rates between 16 and 24 Ib. per Ihp.-hr. They ran condensing, with about 30 Ib. initial pressure, and ratios of expansion between 3 and 1, and were unjacketed. Excessive wiredrawing and the single-acting balanced exhaust (which produced almost the temperature conditions of a compound engine) led to a virtual absence of cylinder condensation. The advantage of a large ratio of expansion was understood, and was supposed to be without definite limit until Isherwood (1860) demonstrated that expansion might be too long continued, and that increased condensation might arise from excessive ratios. Early compound engines, without any increase in expansion over the ratios common in simple engines, failed to produce any improvement, steam rates attained being around 19 IK As higher boiler pressures (150 Ib ) became possible, the ratio of expansion of 14, then adopted for compounds, promptly reduced steam rates to 15 Ib. These have been gradually brought dovn to 12 Ib. in good practice. The 5400 hp. Westinghouse compound of the New York Edison Co., with a 5.8 : 1 cylinder ratio, 185 Ib. steam pressure and 29 expansion^, reached the rate of 11.93 Ib. Triple engines, using still higher ratios of expansion, soon attained steam rates around 12 \ Ib. The best record for a triple with saturated steam seems to be 11 05 Ib., reached by the Hackensacfc, N. J., pumping engine, with 188 Ib. throttle pressure and 33 expansions. Quadruple engines, and engines with superheat, have shown still better results: see Arts. 549c, 549d, 550. 544. Limits and Measures of Efficiency. Art. 496 gives expressions for the Clausius (EJ and relative (E R ) efficiencies corresponding with 397 398 APPLIED THERMODYNAMICS given steam rates and pressure and temperature " conditions. The efficiency of the turbine cannot exceed E r That of the reciprocating engine has for a still lower limit the Rankine efficiency, which is with saturated steam, where pi upper pressure, Ib. per sq. in., absolute; P2=tenninal pressure, Ib. per sq. in., absolute (end of expansion) ; p 3 Blower pressure, Ib. per sq. in., absolute (atmosphere or condenser) ; Xi =initial dryness (beginning of expansion); 0:2= terminal dryness (end of expansion); vi =specinc volume at pressure pi m , 7; 2 =specific volume at pressure p%; hi =heat of liquid at pressure pi; 7z, 2 =heat of liquid at pressure p 2 ; 7z, 3 =heat of liquid at pressure pz] TI ^internal heat of vaporization at pressure p\] r 2 =mtemal heat- of vaporization at pressure p%] LI =latent heat of vaporization at pressure pi. With the regenerative cycle (Art. 550) the Carnot efficiency is the limit. With superheated steam, the Rankine cycle efficiency is 144 where ,H"=total heat in the superheated steam, B. t. u.; fi"2=total heat above 32 at the end of adiabatic expansion; ^2=specific volume of the actual steam at the end of adiabatic expansion; 2>2= pressure of steam at the end of adiabatic expansion, Ib, per sq. in. ; Pa=lowest pressure, Ib. per sq. in.; h 3 heat of liquid at the pressure p 3 . The efficiency ratio E R is almost always between 0.4 and 0,8; in important practice, between 0.5 and 0.7. Attention is called * The backpressure p 9 of best efficiency is not necessarily the lowest attainable. RESULTS OF TRIALS 399 to the table, Art. 551. Average values of the efficiency ratio seem to be: Condensing. Non-condensing. Simple 0.4 0.6 Compound 0.5 . 65 Triple 0.6 0.8 (Art. 5490). With saturated steam, it is from 0.15 to 2 higher in non-condensing than in condensing engines, and increases by 0.05 to 0.1 as the number of expansive stages increases from 1 to 2 or from 2 to 3. With high superheat, E R seems to be between 0.6 and 0.7 for both condensing and non-condensing engines having either one or two expansive stages. The figures given for saturated steam are increased 0.03 to 0.05 by jacketing. The steam rate (Ib. dry steam per hp.-hr.) is scarcely a precise measure of performance, and is of very little significance when superheat is used. Results should preferably be expressed in terms of the thermal efficiency or B. t. u. consumed per Ihp.-min. (See Art. 551.) 545. Variables Affecting Performance. Some of these can be weighed from thermodynamic considerations alone: but in all cases it is well to confirm computed anticipations from tests The essentially thermodynamic factors are: (a) Initial pressure (Art. 549 e)\ (6) Dryness or superheat (Arts. 549/, 549 ff); (c) Backpressure (Art. 549 /z); (d) Ratio of expansion (Art 549 ). The factors influencing relative efficiency, to be considered primarily from experi- mental evidence, are (e) Wire-drawing (Art. 549.7); (/) Cylinder condensation including : Leakage (Art. 549 fc); (h) Compression 1 (i) Clearance / (Art ' (1) Jacketing (Art. 549m); (2) Superheating (Art. 5490); (3) Multiple expansion and reheating (Arts. 549m, 549 n); (4) Speed, Size (Art. 549 o); (5) Ratio of expansion (Art. 549 i) ; 0") valve action (Arts. 546-548 &, 549 o, 551). SUMMARY OF TESTS 546. Saturated Steam: Simple Non-condensing Engines, without Jackets. Steam Kate, S& ^ Initial Lb Gage Ratio of oa Type of Valve. Pres- R p m Size, Hp Expan- |ss sure, Lb sion. Avge. Timits III as Single, automatic, high compression 70-100 100-300 20-100 34 324 30-38 0.141 0.55 Double automatic 75-80 50-150 4 30 134 63 Four-valve, non-releas- ing 100 below 225 above 50 3-4J 29 26-32 0.15 58 Four-valve, releasing. 100 below 100 above 75 3-4i 26 24-28 15 0.65 400 APPLIED THERMODYNAMICS 547. Saturated Steam, Simple Condensing Engines, without Jackets: Improved Valve Gear. Valves Initial Gage Pressure, Lb. R p m Size, Hp Ratio of Expan- sion Steam Rate, Lb (Approximate) Average Range *1 E R Non-releasing Releasing. . 90-110 90-100 below 225 below 100 over 60 over 100 3|-5 3J-5 24 21* 22-26 19J-23i 268 266 35 40 548#. Saturated Steam, Compound Non-condensing Engines, without Jackets. Valve. Initial Gage Pressure, Lb R p m Size, Hp Stpam Rate, Lb. (Approximate) */ E R Single, automatic 110-165 120 130 250-300 265 100 50-250 165 350-450 23 6 23 2 21 9 20 9 167 162 166 0.166 63 67 70 72 Double, automatic. . , . Four-valve, releasing , , Willans 5485. Saturated Steam, Compound Condensing, without Jackets, Normal Cylinder Ratio, Initial (Approximate) Valve. Gage Pressure, Lb. R p. m Size, Hp Steam Rate, Lb , t ER Single, automatic 110-130 200-300 100-500 19 1 275 43 Double, automatic. . . . 120 160-170 100-300 16 3 275 50 Four-valve, releasing. . . . 100-150 underlOO abovelOO 14 6 278 56 (15). . Saturated Steam, Triple Expansion, without Jackets (12), (13), (14), Back Pressure. Initial Gage Pressure, Lb Steam Rate, Lb. (Approximate) E; E* Non-condensing (8) 1! Average. 12i ^ Range. 11J-45 0.169 0.295 80 61 Condensing 12^-200 RESULTS OF TEIALS 401 5495. Jacketed Engines, High Grade, Saturated Steam, Compounds Usually with Reheaters. Steam Us ite, Lb. Type. Jacketed Same Type of Engine, Unjacketed Small, non-condensing simple, 5 exp., 75 Ib. gage pressure . ... 25 26-32J Simple condensing, 120-150 Ib. pressure. . . Woolf compound, condensing, 16 exp., 12 r p. m., 120 Ib. pressure 17-20 13.6 19|-26 rinmpoiinfi non-nondensing 19 20 9-23 6 Compound condensing, ordinary cylinder ratio * (Saving due to jackets, 1J to +10 9 per cent: per cent of total steam consumed in jackets, about 5.0.) Compound condensing, high cylinder ratio, 150- 175-lb. pressure, about 30 expansions, 8 to 14 per cent of total steam used in jackets and reheaters 13.5 11.9 14.6-19.1 Triple condensing, 85 to 175 Ib. pressure, 25 to 33 expansions 11 05-11 75 11 75-15 * One engine gave, with jackets, 13.85; without jackets, 14 99. 5490. Superheated Steam, Reciprocating Engines. Type. Steam Rate, Lb B t u. per Hp -mm Approxi- mate Clausms Efficiency Relative Efficiency. Simple non-condensing, no jackets, slight superheat 23 Simple non-condensing, no jackets, 620 F . Simple condensing, 800 hp., 4 exp., 65 Ib. pressure, 450 F 15.3 16 319 317 0.182 259 66 52 Simple condensing, 620 F 11 6 234 27 67 doTTipoupd non-condensing, locomotive. . . . 17 6 Compound condensing, 500 F 12 9 253 291 57 Compound condensing, 620 F 10.6 224 0.3 '0 63 Compound condensing, 45 hp., 600-lb. pres- sure, 800 F. (19) 10 8 246 375 46 Triple condensing, 500 10 9 221 0.299 0.64 Triple condensing, 620 9 6 200 309 69 402 APPLIED THERMODYNAMICS 549J. Comparative Tests, Saturated and Superheated Steam. Type. Steam Rate, Lb B. t. u. per Hp -mm. Saturated Super- heated Saturated Super- heated Compound condensing, 150-lb pressure, 9 56 8 99 213-246 247 225 199-223 205 192 Compound condensing, 140-lb. pressure, superheated 400 (18) Compound condensing, 130-lb. pressure, fliirbprhpntpd 307 13 84 11 98 (126 r. p m. f 250 hp , 32 exp )(11) 5490 Initial Pressure. Increased pressures have been so associated with development in other respects that it is difficult to show by experimental evidence just what gain in economy has been due to increased pressure alone. Art. 546 gives usual steam rates from 24-28 Ib. for simple non-condensing engines of the best type, in this country, with initial pressures around 100 Ib, In Germany, where pressures range from 150 to 180 Ib., the corresponding rates are between 19 and 23 Ib. per Ihp -hr. 549/1 Initial Dryness. The efficiency of the Clausius or Rankine cycle is greater as the initial dryness approaches 1.0 (Art 417). No considerable amount of moisture is ever brought to the engine in practice, and tests fail to show any influence on dry steam consumption resulting from variations in the small proportion of entrained water. . Superheat. There is no thermodynamic gain when superheating is less than 100, because the steam is then brought to the dry condition by the time cut-off is reached Tables 549 c and 549 d show that heat rates for compound engines with low superheat are around 250 B. t. u , and for triples about 220 B. t. u., while with high superheat the compound or the triple may reach about 200 B. t. u. With high superheat, exceeding 200 F,, some gain due to temperature is realized in addition to the elimination of cylinder condensation. To properly weigh the effect of high superheat, all steam rates given for saturated steam should be reduced to the heat unit equivalent. This is done in the table shown at the top of page 403 Adequate superheating thus causes a large gain in simple engines, either condens- ing or non-condensing In either case, the simple engine using superheated steam is as economical as the ordinary compound engine using saturated steam, so that superheat may be regarded as a substitute for compounding. The best compounds and triples with superheat are (though in a less degree) superior to the same types of engine using saturated steam. RESULTS OF TRIALS 403 Type. Steam Rate, Lb Initial Absolute Preraure, Lb. Feed Tempera- ture, B t u per Ihp -mm. B. t u per Ihp -mm , Super- heated Per Cent Gain by Super- heating. Simple non-condensing, best 26 100 200 434 319 26 Simple condensing, best Compound non-condens- inff .... 21J 22 110 120 150 200 383 380 234 332 (loco- 39 13 Compound condensing, ordinary Compound condensing, high cylinder ratio (see Art. 5496) 15 12-13 150 175 150 150 268 213-247 motive) 224-253 192-223 5-17 0-22 Triple condensing, aver- age 12| 175 150 224 205-221 1-9 5497i Back Pressure. This is best investigated by considering the difference in performance of condensing and non-condensing engines. Arts. 546-549 c give: Steam Rate, Lt s per Ihp.-hr. Per Cent Type. Non-condens- ing Condensing. Saving Due to Condensing. f Simple non-releasing. Saturated Steam, 1 Simple, releasing. . . not jacketed 1 Compound (average) L Triple 29 26 22 2 18 5 24 21.5 16 7 12 5 17 17 17 32 _, , ( Simple 25 18 5 26 Saturated steam, 1 ^ , , , . . , , \ Compound (usual jacketed. \ ^ \ 19 (average) 13 5 29 Superheated / Simple, average 19.15 13 8 28 steam. 1 Compound, average . . . 17.6 11.4 35 The arithmetical averages give about the results to be expected: (1) Condensing saves 24 per cent in simple engines, 27 per cent in compounds, 32 per cent in triples; (2) Condensing is relatively more profitable when jackets or superheat are used. 549t. Ratio of Expansion. This has been discussed in Art. 436. Since engines are usually governed (i. e., adapted to the external load) by varying the ratio of expansion, a study of the variation in efficiency with output is virtually a study of the effect of a changing ratio of expansion. (The question of mechanical efficiency (Arts. 554^-558) somewhat complicates the matter.) Figure 266 gives the results 404 APPLIED THERMODYNAMICS of such an investigation. The shape of the economy curve is of great importance. A flat curve means fairly good economy over a wide range of probable loads. The 43 40 ^ 3S \ 36 V V f ai f\ I M 32 \ s ^v^ -I ^ TT7 ^ ^ ^ A a^ sj - . _r * Z. 28 -< 25 -1- s ^ & 24 ^ S" 00 ^ *"*! .. ?. 1 20 i c u - 1 )- ' 16 16 7 7 14 5k 1?, "^ f )_ o- 0- 50 00 70 80 90 100 110 120 130 LOAD PER CENT OF RATING FIG. 266a. Art. 649i. Efficiency at Various Ratios of Expansion. flatness varies with different types of engine. A few typical curves are given in Fig, 266a. Curve I is from a single-acting Westinghouse compound engine, run- ning non-condensing Curve II is from the same engine, condensing (Trans A S. M . E , XIII, 537). Curve III is for the 5400-hp. Westinghouse compound condensing engine mentioned in Art. 543a. Curve IV is for a small four-valve simple non-condensing engine : curve V for a single-valve high-speed simple non-condensing engine If we regard the usual ratio of ex- pansion in a compound as 16, in a simple engine, 4, and in a triple or high-ratio compound as 30, with cor- responding steam rates of_15, 26, and 12J (condensing engines), we obtain the curves of Fig. 266 6, showing a steady gain of efficiency as the total ratio of expansion increases, provid- ing two stages of expansion are used when the ratio exceeds some value 8 10 a . a . Jt s between : * T* W- K W" that RATIO OF EXPANSION no considerable further gain can be FIG. 2666. Art. 549i. Efficiency and Ratio of expected by increasing either the Expansion. ratio of expansion or the number of RESULTS OF TRIALS 405 549;'. Wire-drawing. None of the tests above quoted applies to throttling engines. Cut-off regulation is now almost universal. Moderate throttling may be desirable at high ratios of expansion (Art. 426) A large part of the 8 per cent differ- ence in steam consumption between the single-valve and double-valve engines of Art. 546 (15 per cent in Art. 548 6) is due to the partial throttling action of the single valve at cut-off. The difference between the performances of four-valve engines with and without releasing gear is very largely due to the comparative absence of wire-drawing in the former. This difference is 10 per cent in Art. 546 Leakage. The average steam rate ascertained on engines which had run from 1 to 5 years without refitting of valves or pistons (7) was 39.3 Ib. This was for simple single-valve non-condensing machines, for which the figure given in Art 546 is 32 J. Some of the difference was due to the fact that the engines tested ran at light loads (| to f normal: see Art. 549 i and Fig. 266) but a part must*also have been due to leakage resulting from wear In 65 tests reported by Barrus, the average steam rate of engines known to have leaking valves or pistons was 4 8 per cent higher than that of those which were known to be tight. Leakage is less in compound than in simple engines. (See Art. 452.) 549 1. Compression, Clearance. The theory of compression has been discussed (Art. 451). High-speed engines have more compression than those running at low speed. The compression in compound engines is less than that in simple engines. There is an amount of compression (usually small) at which for a given engine and given conditions the efficiency will be a maximum. No general results can be given, The maximum desirable compression occurs at a moderate cut-off: at other points of cut-off, compression should be less Within any range that could reasonably be prescribed, the amount of compression does not seriously influence efficiency Clearance is a necessary evil, and the waste which it causes is only partially offset by compression. Designers aim to make the amount of clearance (which depends upon the type and location of the valves) as small as possible. The pro- portion of clearance in steam engines of various types is given in Art. 450. The differences between the steam rates of single valve and Corliss valve engines, shown in Arts. 546 to 548 &, already mentioned as partly due to wire-drawing, are also in part attributable to differences in clearance. 549m. Jackets. The saving due to jackets may range from nothing (or a slight loss) up to 20 per cent or more. Art. 549 6 shows minimum savings of 6 to 9 per cent and maximum of 19 to 23 per cent, for one, two or three expansive; stages. Yet there are undoubtedly cases where jackets have not paid, and they are not usually applied (excepting on pumping engines) in American stationary practice to-day. The best records made by compounds and triples have been in jacketed engines. This is with saturated steam. With superheat, jackets are not warranted. The proportion of steam used in jackets (of course charged to the engine) ranges usually between 0.03 and 0.08, increasing with the number of expansive stages. Jacketing pays best at slow speeds and hiejh ratios of expansion. Reheaters for compound engines can scarcely be discussed separately from jackets. It is difficult to get an adequate amount of transmitting surface without making the receiver very large. The objection to the reheater is the same as that to the jacket increased attention is necessary in operation and maintenance. There is an irreversible drop of temperature inherent in the operation of the reheater. 406 APPLIED THERMODYNAMICS 549 n. Multiple Expansion. The tables already given furnish the following: UNJACKETED ENGINES -Steam Rate, Lb. per Ihp -hr - Condensing Non-condensing. No. of expansion stages .1 2 3 123 Type Single-valve .... 19 1 32J 23 6 . . Double-valve . . 16 3 . . 30 23 2 Four-valve, non-releasing. 24 . . 29 . Four-valve, releasing . 21* 14 6 12 5 26 21 9 18 5 Superheat, good valve 11 6-16 10 6-12 9 9 6-10 9 15 3-23 17 6 The non-condensing engine \vith a cheap type of valve is 23 to 27 per cent more economical in the compound form than when simple. (The non-condensing compound is on other grounds than economy an unsatisfactory type of engine, see American Machinist, Nov. 19, 1891 ) In four-valve releasing engines, non-condensing, the compound saves 16 per cent over the simple and the triple saves 16 per cent over the compound. The same engines, condensing, give a saving of 32 per cent by com- pounding and an additional saving of 14 per cent by triple expansion With super- heat, non-condensing, the compound is from 15 per cent worse to 23 per cent better than the simple engine Condensing, the compound saves 15 per cent over the simple and the triple saves 13 per cent over the compound. High Ratio Compounds have been discussed in Art. 473 The tests in Art. 548 b include only compound engines of normal cylinder ratio. The following results have been attained with wider ratios : Lbs per Ihp hr. 150-lb pressure, 26 exp , ratio 7:1 12 45 (jacketed) 150-lb pressure, 120 r. p. m , 33 exp 12 1 (head jacketed) 130-lb. pressure. 126 r. p. m., 32 exp 11 98 (jacketed) These figures are practically equal to those reached by triple engines. They are due to (a) high ratios of expansion, (6) jacketing, and (c) the high cylinder ratio 5499. Speed and Size: Efficiency in Practice. None of the tests shows a steam rate below 16.3 Ib. at speeds above 140 r. p. m. Low rotary speed is essential to the highest thermal efficiency. Between very wide limits say 100 or 200 to 2500 hp. the size of an engine only slightly influences its steam rate. Very small units are wasteful (some direct-acting steam pumps have been shown to use as much as 300 Ib. of steam per Ihp -hr)(6) and very large engines are usually built with such refinement of design as to yield maximum efficiencies. All figures given are from published tests. It is generally the case that poor performances are not published. The tabulated steam rates will not be reached in ordinary operation: first, because the load cannot be kept at the point of maximum efficiency (Art. 549 1} nor can it be kept steady and second, because under other than test conditions engines will leak Probably few bidders would guarantee results, even at steady load, within 10 per cent of those quoted. In estimating the probable steam rate of an engine in operation, this 10 per cent should first be added, correction should then be made for actual load conditions, based on such a curve as that of Fig. 266, and an additional allowance of 5 per cent or upward should then be made Tor leakage. RESULTS OF TRIALS 407 13 \ 11 \ N X 13 V ' . - . a <y . - "o~~ ! |i 1 \ ! ? ( ! i | i INDICATED HORSE POWLR FIG. 266. Arts. 549i, 556. Test of Rice and Sargeut Engine (10). 550. Quadruple Engines: Regenerative Cycle, ances on record with saturated steam have been made in quadruple-expansion engines. The Nordberg pumping engine at Wildwood (16), although ot only 6.000,000 gal. capacity (712 horse power) and jacketed on barrels of cylinders only, gave a heat consumption of 186 B. t. u. with 200 Ib. initial pressure and only a fair vacuum. The high efficiency \\ as obtained by drawing off live steam from each of the receivers and trans- tcrrmg its high-temperature heat direct of the boiler feed water by means of coil heaters. Heat was thus absorbed more nearly at the high temperature Some of the best perform- \. . 267. Ait. 550. Nordberg Engine Diagrams 2B8_78 ^THROTTLE limit, and a closer approach made to the Carnot cycle than in the ordinary en- gine. Thus, in Fig. 267, BCDS represents the Clausius cycle. The heat areas lil HE) gKJh, NMLg represent the withdrawal of steam from the various receivers, these amounts of heat being applied to heating the water along Bd, de, ef. The heat imparted from without is tben only cfCDE. The work area DHIJKLMRS hag been lost, but the much greater heat area ABfc has been saved, so that the effi- ciency is increased. The cycle is regenerative 5 if the number of steps were infinite, the expansive path would be DF, parallel to BO, and the cycle would be equally efficient with that of Carnot. The actual efficiency was 68 per cent of that of the Carnot cycle. The steam rate was not low, being increased by the system of drawing off steam for the heaters from 11.4 to 12.26; but the leal efficiency was, at the time, unsurpassed. A later test of a Nord- berg engine of similar construction, used to drive au air com- pressor, is reported by Hood (17). Here the combined diagrams were as in Fig. 268. Steam was received at 257 Ib. pressure, the vacuum being rather poor. At normal capacity, 1000 hp. ; the mechanical effi- ciency was 90.35 per cent, and the heat consumption ^ t , 169 29 B. t. u 13.85 RECEIVER M.24-CONDENSER FIG. ?6$. Art. 550- Hood Compressor Diagrams. 408 APPLIED THERMODYNAMICS 551. Summary of Best Results, Reciprocating Stationary Engines. Lbs Steam n Q Cy B. t.u. per perlhp.-hr. ^ 4Q6) Ihp.-mm. Saturated steam, simple, non-condensing, single valve, without j ackets. . . . 32 051-0 55 548 Saturated steam, simple, non-condensing, double valve, without jackets . . 30 0.63 502 Saturated steam, simple, non-condensing, four valve, releasing, without jackets . . 26 . 65 434 Saturated steam, simple, non-condensing, with jackets .............. 25 68 418 Saturated steam, compound non-condensing, without jackets ...... . .22 0.63-0.72 353 Saturated steam, simple condensing, four valve releasing, without jackets ...... 21 0-40 383 Saturated steam, compound non-condensing, with jackets, four valve ....... 19 0.71-0.82 305 Saturated steam, compound condens- ing, normal ratio, single valve, no jackets .................... 19 0.43 359 Saturated steam, simple condensing, with jackets .................. 18J 0.45 330 Superheated, compound non-condensing (locomotive) ........ ....... 17J 58-0 72 332 Superheated (620 F.) steam, simple, non- condensing ....... ... 15 0,66-084 319 Saturated steam, compound condensing, four valve, no jackets .............. 14J 56 274 Saturated steam, compound condensing, normal ratio, four valve, with jackets. . 13J- . 50-0 . 60 255 Saturated steam, triple condensing, no jackets ...................... 12J 0.61 234 Saturated steam, high ratio compound con- densing, jacketed .................. 12 0.63 226 Superheated (620 F.) steam, simple, con- densing ........ . ................. HJ 0. 67 234 Saturated steam, triple condensing, with jackets .......................... llf 0.66 205 Superheated (620 F.) steam, compound con- densing ....................... 10J- 0.63 224 Superheated (620 F.) steam, triple con- densing ....................... 9} 0.69 200 Saturated steam, quadruple, condensing ..... * 169 * Efficiency is 77 per cent, that of the Carnot cycle between the same extreme temperature limits. TURBINES 409 552. Turbines. With pressures of from 78.8 to 140 lb.,* and vacuum from 24.3 to 26 4 in , steam rates per brake horse power of 18.0 to 23 2 have been obtained with saturated steam on De Laval turbines. Dean and Main (20) found correspond- ing ratea of 15.17 to 16 54 with saturated steam at 200 lb. pressure, and 13.94 to 15.62 with this steam superheated 91. Parsons turbines, with saturated steam, have given rates per brake horse power from 14 1 to 18 2, with superheated steam, from 12 6 to 14 9. This was at 120 lb pressure. A 7500-kw. unit tested by Sparrow (21) with 177.5 lb. initial pressure, 95.74 of superheat, and 27 in. of vacuum, gave 15 15 lb of steam per kw.-hr. Bell reports for the Lusitama (22) a coal consumption of 1.43 lb. per horse power hour delivered at the shaft. Denton quotes (23) 10.28 lb. per brake horse power on a 4000 hp. unit, with 190 of superheat (214 B t. u. per minute); and 13.08 on a 1500- hp. unit using saturated steam. A 400-kw unit gave 11 2 lb. with 180 of super- heat. A 1250-kw. turbine gave 13.5 lb. with saturated steam, 12.8 with 100 of superheat, 13.25 with 77 of superheat (24). (All per brake hp.-hr.) A Rateau machine, with slight superheat, gave rates from 15.2 to 19.0 lb. per brake horse power. Curtis turbines have shown 14.8 to 18.5 lb. per kw.-hr., as the superheat decreased from 230 to zero, and of 17.8 to 22.3 lb. as the back pressure increased from 08 to 28 lb. absolute. Kruesi has claimed (25) for a 5000-kw Curtis unit, with 125 of superheat, a steam-rate of 14 lb. per kw.-hr.; and for a 2000-kw. unit, under similar conditions, 16.4 lb. A 2600-kw. Brown-Bo veri turbo-alternator at Frankfort consumed 11.1 lb. of steam per electrical horse-power-hour with steam at 173 lb. gauge pressure, super- heated 196 and at 27.75 ins. vacuum. The 7500-kw. ALLis cross-compound engines of the Interborough Rapid Transit Co., New York, with 190 lb. gauge pressure and 25 ins vacuum (saturated steam) used 17.82 lb. steam per kw.-hr. When exhaust turbines were attached (Art. 541) the steam rate for the whole engine became between 13 and 14 lb. per kw.-hr., or (at 28 ins. vacuum) the B. t. u. consumed per kw> min., ranged from 245 to 264; say, approximately from 156 to 168 B. t. u. per Ihp.-min , which was better than any result ever reached by a reciprocating engine or a turbine alone Heat unit consumptions below 280 B. t. u. per kw.-min. (190 per Ihp.-min.) have been obtained in many turbine tests. 553. Locomotive Tests. The surprisingly low steam rate of 16.60 lb. has been obtained at 200 lb. pressure, with superheat up to 192. This is equivalent bo a rate of 17.8 lb. with saturated steam. The tests at the Louisiana Purchase Exposition (20) showed an average steam, rate of 20.23 lb. for all classes of engines tested, or of 21.97 for simple engines and 18.55 for compounds, "with steam pres- sures ranging from 200 to 225 lb. These results compare most favorably with any obtained from high-speed non-condensing stationary engines. The mechanical 3/ficiency of the locomotive, in spite of its large number of journals, is high ; in bhe tests referred to, under good conditions, it averaged 88.3 per cent for consoli- iation engines and 89.1 per cent for the Atlantic type. The reason for these high efficiencies arises from the heavy overload carried in the cylinder in ordinary ser- vice. The maximum equivalent evaporation per square foot of heating surface varied from 8 55 to 16.34 lb. at full load, against a usual rate not exceeding 4.0 lb. n stationary boilers ; the boiler efficiency consequently was low, the equivalent evaporation per pound of dry coal (14,000 B. t. u.) falling from 11.73 as a maxi- num to 6.73 as a minimum, between the extreme ranges of load. Notwithstand- * Pressures in this chapter, unless otherwise stated, are gauge pressures 410 APPLIED THERMODYNAMICS ing this, a coal consumption of 2.27 Ib. per Ihp.-hr. has been reached. These trials were, of course, laboratory tests; road tests, reported by Hitchcock (27), show less favorable results ; but the locomotive is nevertheless a highly economical engine, considering the conditions under which it runs* 554. Engine Friction. Excepting in the case of turbines, the figures given refer usually to indicated horse power, or horse power developed by the steam in the cylinder. The effective horse power, eseited by the shaft, or brake horse power, is always less than this, by an amount depending upon the friction of the engine. The ratio of the latter to the former gives the mechanical efficiency, which may range from 85 to 0.90 in good piactice with rotative engines of moderate size, and up to 0.965 in excep tional cases. (See Art. 497.) The brake horse power is usually determined by measuring the pull exerted on a friction brake applied to the belt wheel. When an engine drives a generator, the power indicated in the cylinder may be compared with that developed by the generator, and an over-all efficiency of mechanism thus obtained. The difficulties involved have led to the general custom, in turbine practice, of reporting steam rates per kw-hr. Thurston has employed the method of driving the engine as a machine from some external motor, and measuring the power required by a transmission dynamometer. In direct-driven pumps, air compressors and re- frigerating machines, the combined mechanical efficiency is found by comparing the indicator diagrams of the steam and pump cylinders. These efficiencies are high, on account of the decrease in number of bearings, crank pins, and crosshead pins. Art 555. Engine Friction. r-700- _100. 555. Variation in Friction. Theoretically, at^ 10 - 269 - least, the friction includes two parts: the initial friction, that of the stuffing boxes, which remains practically constant ; and the Ijad friction, of guides, pins, and bearings, which varies with the initial pressure and expansive ratio. By plotting concurrent values of the brake horse power and friction horse power, we thus obtain such a diagram as that of Tig. 269, in which the height ab represents the constant initial fric- tion, and the variable ordinate xy the load friction, incieasing in arith- metical proportion with the load. It has been found, however, that in practice the total friction is more affected by accidental variations in lubrication, etc., thau by changes in load, and that it may be regarded as practically constant,_for a given en- gine, at all loads. ^^ 2|0 40 BRAKE HORSE POWCT FIG. 270. Art. 555. Willans Line for Varying Initial Pressure. MECHANICAL EFFICIENCY 411 The total steam consumption of an engine at any load may then be regarded as made up of two parts : a constant amount, necessary to overcome friction ; and a variable amount, necessary to do external work, and varying with the amount of that work. Willans found that this latter part varied in exact arithmeti- cal proportion with the load, 1.2200 / A 1800 1600 / *s / A y 1200 ^ y / -8004 / 7 -100. 10 20 30 40 50 00 70 8 90 100 110 1*20 ELECTRICAL HORSE POWtR with the when the output of the engine was varied by changing the initial pressure; a condition repre- sented by the Willans line of Fig. 270 (28). The steam rate was thus the same for all loads, excepting as modified by fric- tion. (Theoretically, this should not hold, since lowering of the initial pressure lowers the efficiency.) When the load is changed by varying the ratio of expansion, the corrected steam rate tends to decrease with increasing ratios, and to increase on account of increased condensation; there is, however, some gain up to a certain limit, and the Willans line must, therefore, be concave up- ward. Figure 271 shows the practically straight line obtained from a series of tests of a Parsons turbine. If the line for an ordinary engine were perfectly straight, with varying ratios of expansion, the indication would be that the gain by more complete expansion was exactly offset by the increase in cylinder con- densation. A jacketed engine, in which the influence of condensation is largely eliminated, should show a maximum curvature of the Willans line. FIG. 271. Art. 556, Prob. 10. Willans Line for a Parsons Turbine. 559. Variation in Mechanical Efficiency. With a constant friction loss, the mechanical efficiency must increase as the load increases, hence the desirability of running engines at full capacity. This is strikingly illustrated in the locomotive (Art. 554). Engines operating at serious variations in load, as in street railway power plants, may be quite wasteful on account of the low mean mechanical efficiency. The curve in Fig. 266 gives data for the " Total " curve of Fig. 271a, which is plotted on the assumption that the horse power consumed in overcoming friction is 100, and the corresponding total weight of steam 1000 Ib. per hour. Thus, at 700 Ihp., the steam rate from Fig. 266 is 12.1 Ib., and the steam consumed per hour is 8470 Ib. The corresponding ordinate of the second curve in Fig. 271a is then (8470 - 1000) + (700 - 100) =7470 *600 = 12.45, where the abscissa is 600. 412 APPLIED THERMODYNAMICS 13,000 12,000 3 11,000 = 10,000 0,000 8,000 7,000 6,000 5,000 100 200 300 400 500 600 700 800 900 BRAKE HORSE POWER >FiG. 271a. Art. 556. Effect of Mechanical Efficiency. 557. Limit of Expansion. Aside from cylinder condensation, engine friction imposes a limit to the desirable range of expansion Thus, in Fig 272, the line ab may be drawn such that the constant pressure Oa represents that necessary to overcome the friction of the engine. If expansion is carried below ab, say to c, the force exerted by the steam along be will be less than the resisting force of the engine, and will be without value. The maximum desirable expansion, irrespective of cylinder condensation, is reached at 6. FIG. 272 Art 557. Engine Friction and Limit of Expansion. 558. Distribution of Friction. Experi- menting m the manner described in Art, 555, Thurston ascertained the distribution of the friction load by successively removing various parts of the engine mechanism. Extended tests of this nature, made by Carpenter and Preston (29) on a horizontal engine indicate that from 35 to 47 per cent of the whole friction load is imposed by the shaft bearings, from 22 to 49 per cent by the piston, piston rod, pins, and slides (the greater part of this arising from the piston and rod), and the remaining load by the valve mechanism. (1) Trans. A. 8 M. E , Proc. Inst, Jf. E , Zeits. Ter Deutsch. Ing., etc. (See The Engineering Diciest, November, 1908, p. 542.) (2) Proc. Inst. Mech. Eng., from 1889. (3) Engine Tests, by Geo. H. Barrus. (4) Steam Turbines, 1900, 208-207. (5) Be- searches in Experimental Steam Engineering. (6) Peabody, Tliermoaynamics, 1907, 244 , White, Jour. Am. Sue. Ifav. Engrs., X. (7) Trans, A. S. M. E. t XXX, 6, 811. PROBLEMS 413 (8) Ewing, The Steam Engine, 1006, 177. (9) Denton, The Stevens Institute Indi- cator, January, 1905. (10) Trans. A. JS. M. E., XXIV, 1274. (11) Denton, op. cit. (12) Ewing, op cit., 180. (13) Trans. A S. M. E., XXI, 1018. (14) Ibid., XXI, 327. (15) J&M , XXI, 793. (10) J6 M f,XXI,181. (17) Hid., XXVIII, 2, 221. (18) Ibid., XXV, 2G4. (19) Ibid , XXVIII, 2, 226. (20) Thomas, Steam Turbines, 1906, 212. (21) Power, November, 1907, p. 772. (22) Proc. List. Nav. Archls., Apnl 9, 1908. (23) Op cit. (24) Trans, A. S. M E , XXV, 745 et seq. (25) Power, December, 1907. (20) Locomotive Tests and Exhibits, published by the Pennsylvania Railroad. (27) Ttans. A S. M. E., XXVI, 054. (28) Mm. Proc. Inst. G. E., CXIV, 1893. (29) Peabody, op. cit., p. 29G. SYNOPSIS OF CHAPTER XV Sources of information : development of steam engine economy. Limit of efficiency (Rankme cycle) , with the regenerative engine, the Carnot cycle; with the turbine, the Clausius cycle. Efficiency vs. steam zate. Variables affecting performance : Efficiency vanes directly with initial pressure ; is independent of initial dryness ; is increased by high superheat (superheat is a substitute for compounding)^ varies inversely as the back pressure, and is greater in condensing than in non-condensing engines ; reaches a maximum at a moderate ratio of expansion and decreases for ratios above or below this ; varies directly with the number and independence of valves ; may be seriously reduced by leakage or high compression ; is usually somewhat increased by jacketing; increases with the number of expansive stages, though more and more slowly ; is low in very small engines or at very high rotative speeds ; in ordinary practice is below published records. Typical figures for reciprocating engines and turbines, with saturated and super- heated steam, simple vs. compound, condensing vs. non-condensing, with and without jackets, triple and quadruple regenerative. PROBLEMS (See footnote, Art. 552.) 1. Find the heat unit consumption of an engine using 30 Ib. of dry steam per Ihp.-hr. at 100 Ib. gauge pressure, discharging this steam at atmospheric pressure. How much of the heat (ignoring radiation losses) in each pound of steam is rejected ? What is the quality of the steam at release ? (Ans., a, 504.4 B. t. u. per minute ; 6, 1088.8 B. t.u. ; c, 93.6 per cent.) 2. What is the mechanical efficiency of an engine developing 300 Ihp., if 30 hp. is employed in overcoming friction ? (Ans., 90 per cent.) 3. Why is it unprofitable to run multiple expansion engines non-condensing ? 414 APPLIED THERMODYNAMICS 4. Find the heat unit consumptions corresponding to the figures in Art. 552 for De Laval turbines, assuming the vacuum to have been 27 in. * (Aiis., a, 295 , 6, 286 B t. u. per minute.) 5. Find the heat unit consumption for the 7500-kw. unit in Art. 552. (AM., 296.3 B.t.u.) 6. Estimate the probable limit of boiler efficiency of the plant on the S.S. Lusttama, if the coal contained 14,200 B. t. u. per Ib. {Ana., if engine thermal efficiency were 0.20, mechanical efficiency 0.90, the boiler efficiency must have been at least 0,69 ) 7. Determine from data given in Art. 553 whether a coal consumption of 2.27 Ib. per. Ihp.-hr. is credible for a locomotive. 8. Using values given for the coal consumption and mechanical efficiency, with how little coal (14.000 B. t. u. per pound), might a locomotive travel 100 miles at a speed of 50 miles per hour, while exerting a pull at the drawbar of 22,0001b. ? Make comparisons with Problem 8, Chapter n, and determine the possible efficiency from coal to drawbar. 9. An engine consumes 220 B t. u. per Ihp.-min., 360 B. t. u. per kw.-min. of generator output. The generator efficiency is 0.93. What is the mechanical efficiency of the direct-connected unit ? (Ans., 88 per cent.) 10. From Fig. 271, plot a curve showing the variation in steam consumption per kw.-hr. as the load changes. 11. An engine works between 150 and 2 Ib. absolute pressure, the mechanical efficiency being 0.75. What is the desirable ratio of (hyperbolic) expansion, friction losses alone being considered, and clearance being ignored ?_ (Ans., 12.25.) 12. If the mechanical efficiency of a rotative engine is 0.85, what should be its mechanical efficiency when directly driving an air compressor, based on the minimum values of Art. 558 ? (Ans^ 0.94.) 13. In the jacket of an engine there are condensed 310 Ib. of steam per hour, the steam being initially 4 per cent wet. The jacket supply is at 150 Ib. absolute pressure, and the jacket walls radiate to the atmosphere 52 B t. u. per minute. How much heat, per hour, is supplied by the jackets to the steam in the cylinder ? ' 14. A plant consumes 1.2 Ib. of coal (14,000 B. t. u. per Ib.) per brake hp.-hr. What is the thermal efficiency ? * Vacua are measured downward from atmospheric pressure. One atmosphere 14.690 Ib per square inch= 29.921 inches of (mercury) vacuum. If p = absolute pressure, pounds per square inch, -0= vacuum hi inches of mercury, -as ~- > CHAPTER XVI THE STEAM POWER PLASTT 560. Fuels. The complex details of steam plant management arise largely from differences in the physical and chemical constitution of fuels. "Hard" coal, * for example, is compact and hard, while soft coal is friable ; the latter readily breaks up into small particles, while the f orfner maintains its initial form unless subjected to great intensity of draft. Hard coal, therefore, requires more draft, and even then burns much less rapidly than soft coal ; and its low rate of combustion leads to important modifications in boiler design and operation in cases where it is to be used. Soft coal contains large quantities of volatile hydrocarbons ; these distill from the coal at low temperature, but will not remain ignited unless the temperature is kept high and an ample quantity of air is supplied. The smaller sizes of anthracite coal are now the cheapest of fuels, in propor- tion to their heating value, along the northern Atlantic seaboard ; but the supply is limited and the cost increasing. In large city plants, where fixed charges are high, soft coal is often commercially cheaper on account of its higher normal rate of combustion, and the consequently reduced amount of boiler surface necessary. The sacrifice of fuel economy in order to secure commercial economy with! low load factors is strikingly exemplified in the "double grate" boilers of the Philadelphia Rapid Transit Company and the Interborough Rapid Transit Company of New York (1). 561. Heat Value. The heat value or heat of combustion of a fuel is determined by completely burning it in a calorimeter, and noting the rise in temperature of the calorimeter water. The result stated is the number of heat units evolved per pound with products of combustion cooled down to 32 F. Fuel oil has a heat value upward of 18,000 B. t. u. per pound, its price is too high, in most sections of the country, for it to compete with coal. Wood is in some sections available at low cost; its heat value ranges from 6500 to 8500 B. t. u. The heat values of com- mercial coals range from 8800 to 15,000 B. t. u. Specially designed furnaces are usually necessary to burn wood economically. * A coal may be called famZ, or anthracite, when from 89 to 100 per cent of its combustitle is fixed (non-volatile, uncombined) carbon. If this percentage is between 83 and 89, the coal is semi^bituminou^ ; if less than S3, it is bituminous, or soft. 415 416 APPLIED THERMODYNAMICS TABLE COMBUSTION DATA FOB VARIOUS FUELS Symbol Equivalent Reaction f B t u per Lb Hydrogen . ... H H 2 +0 = H 2 O 62, lOOt Carbon C C+O-CO 4,450 Carbon . C C+0 2 = C0 3 14,500 Carbon monoxide. CO CO+0-C0 2 4,385 Acetylene C 2 H 2 C 2 H 2 +0 6 =2C0 2 +H 2 21,4001 Methane CH 4 CH 4 +04 = C0 2 -|-2H 2 23.842J Ethylene C 2 H 4 C 2 H4-f-O 6 =2C0 2 +2H 2 O 21,250t Sulphur . ... S S-K) 2 =S0 2 4,100 Gasolene* . . CeHu C 6 Hu+0 19 =6C0 2 +7H 2 1 9,000 1 * Gasolene IB a variable mixture of hydrocarbons, CeHu being a probable approximate formula t The number of atoms m the molecule is disregarded j These figures represent the 4< high values " When hydrogen, or a fuel containing hydrogen, is burned, the maximum heat is evolved if the products of combustion are cooled below the tem- perature at which they condense, so that the latent heat of vaporization is emitted The *' low neat value " would be (970 4 XHJ) B t u less than the high value when w is the weight in pounds of steam formed during the combustion, if the final temperatures of the products of combustion were the same in both " high " and " low " determinations When the products of combustion are permanent gases there is no distinction of heat values Computed Heat Values. When a fuel contains hydrogen and carbon only, its heat value may be computed from those of the constituents. If oxygen also is present, the heat of combustion is that of the substances uncombined with oxygen. Thus in the case of cellulose, C 6 Hi O & , the hydrogen is all combined with oxygen and unavailable as a fuel. The carbon constitutes the yVu = 0.444 part of the substance, by weight, and the computed heat value of a pound of cellulose is therefore 0.444X14,500 = 6430 B. t. u. The heat of combustion of a compound may, however, differ from that of the combustibles which it contains, because a compound must be decomposed before it can be burned, and this decomposition may be either exothermic (heat emitting) or endothenric (heat absorbing). In the case of acetylene, C 2 H 2 , for example, if the heat evolved in decomposition is 3200 B. t. u., the " high " heat of combustion is computed as follows: C =f|X14,500 =13,400 E=AX62,100 - 4,790 Heat of decomposition = 3,200 Heat value =21,390 With an endothennic compound the heat of combustion will of course be less than that calculated from the combustibles present Suppose 0.4 cu. ft. of gas to be burned in a calorimeter, raising the temperature of 10 Ib. of water 25 F. The heat absorbed by the water is 10X25 =250 B. t. u , and the heat value of the gas is 250-r0.4=#25 B. t. u. per cu. ft. If the tempera- ture of the gas at the beginning of the operation were 40 F., and its pressure 30.5 ins. of mercury, then from the relation PV^pv 30.57 29.920 T t' 40+460 32+460' EFFICIENCY OF COMBUSTION 417 -we find that a cubic foot of gas under the assigned conditions would become 1 001 cu. ft. of gas under standard conditions (32 F. and 29 92 in. barometer) The heat value per cubic foot under standard conditions would then be 625 -T- 1 001 = 624 4 B. t. u. These are the " high " heat values. Suppose, during the combustion, & Ib. of water to be condensed from the gas, at 100 F. Taking the latent heat at 970.4 and the heat evolved m cooling from 212 to 100 at 112 B. t. u., the heat con- tributed during condensation and cooling would be 05(970 4-f-112) =54 12 B t. u., and the " low " heat value of the gas under the actual conditions of the experiment would be 625-54.12 =570.88 B t. u The tabulated " heat value " of a fuel is usually the amount of heat liberated by 1 Ib. thereof when it and the air for combustion are supplied at 32 F and atmos- pheric pressure, and when the products of combustion are completely coolerl down to these standard conditions. In most applications, the constituents are supplied at a temperature above 32 F., and the products of combustion are not cooled down to 32 F. Two corrections are then necessary: an addition, to cover the heat absorbed in raising the supplied fuel and air from 32 to their actual temperatures, and a deduction, equivalent to the amount of heat which would be liberated by the products of combustion in cooling from their actual condition to 32. 562. Boiler Room Engineering. While the limit of progress in steam engine economy has apparently been almost realized, large opportunities for improvement are offered in boiler operation. This is usually committed to cheap labor, with insufficient supervision. Proper boiler operation can often cheapen power to a greater extent than can the substitution of a good engine for a poor one. New designs and new test records aie not necessary. Efficiencies already reported equal any that can be expected; but the attainment of these efficiencies in ordi- nary operation is essential to the continuance in use of steam as a power produc- ing medium. 563. Combustion. One pound of pure carbon burned in air uses 2.67 Ib. of oxygen, forming a gas consisting of 3.67 Ib. of carbon dioxide and 8.90 Ib. of nitrogen. If insufficient air is supplied, the amount of carbon dioxide decreases, some carbon mon- oxide being formed. If the air supply is 50 per cent, deficient, no carbon dioxide can (theoretically at least) be formed. With, air in excess, additional free , . AJR SUPPLY. PERCENTAGE OP AM'T THEOR._NECeSSARY FOR COMBUSTION oxygen and mtro- FJQ 373 Arts. 563, 564. Air Supply and Combustion. 418 APPLIED THERMODYNAMICS gen will be found in the products of combustion. Figure 273 illus- trates the percentage composition by volume of the gases formed by combustion of pure carbon in varying amounts of air. The propor- tion of carbon dioxide reaches a maximum when the air supply is just right. 564. Temperature Rise. In burning to carbon dioxide, each pound of carbon evolves 14, 500 B. t u. In burning the carbon monoxide, only 4450 B. t. u. are evolved per pound. Let W be the weight of gas formed per pound of carbon, ./Tits mean* specific heat, Tt the elevation of tempera- ture produced ; then for combustion to carbon dioxide, T t = and 4450 . for combustion to carbon monoxide, T t = . The rise of tempera- ture is much less in the latter case. As air is supplied in excess, W increases while the other quantities on the right-hand sides of these equa- tions remain constant, so that the temperature rise similarly decreases. The temperature elevations are plotted in Fig. 273. The maximum rise of temperature occurs when the air supply is just the theoretical amount. 565. Practical Modifications. These curves ti;uly represent "the phenomena of combustion only when the reactions are perfect. In practice, the fuel and air are somewhat imperfectly mixed, so that 1 we commonly find free oxygen and carbon monoxide along with carbon dioxide. The presence of even a very small amount of carbon monoxide appreciably reduces the evolution of heat. The best results are obtained by supplying some excess of air; instead of the theoretical 11.57 lb., about 16 lb. may be supplied, in good practice. In poorly operated plants, the air supply may easily run up to 50 or even 100 lb., the percentage of carbon dioxide, of course, steadily decreasing. Gases * JSTis quite variable for wide temperature ranges. (See Art. 63.) In general, it may be written as a& or as adb&e 2 where a, & and c are constants and t the temperature range from some experimentally set state. For accurate work, then = \Kdt= \adt r*6fttt Jti Jti Jti the last term disappearing when 2T may be written as a function; of the first power only of the temperature. STEAM BOILERS 419 containing 10 per cent of dioxide by volume are usually considered to represent fair operation. 566. Distribution of Heat. Of the heat supplied to the boiler by the fuel, a part is employed in making steam, a small amount of fuel is lost through the grate bars, some heat is transferred to the external atmosphere, and some is carried away by the heated gases leaving the boiler. This last is the important item of loss. Its amount depends upon the weight of gases, their specific heat and temperature. The last factor we aim to fix in the design of the boiler to suit the specific rate of combustion; the specific heat we cannot control; but the weight of gats is determined solely by the supply of air, and is subject to operating control. Efficient operation involves the minimum possible air supply in excess of the theoretical requirement; it is evidenced by the percentage of carbon dioxide in the discharged gases. If the air supply be too much decreased, however, combustion may be incomplete, forming carbon monoxide, and another serious loss will be experienced, due to the potential h j?at carried off by the gas. 567. Air Supply and Draft. The draft necessary is determined by the physical nature of the fuel; the air supply, by its chemical composition. The two are not equivalent; soft coal, for example, requires little draft, but ample air supply. The two should be subject to separate regulation. Low grade anthracite requires ample draft, but the air supply should be closely economized. If forced draft, by steam jet, blower, or exhauster, is employed, the necessary head or pressure should be provided without the delivery of an excessive quantity or volume of air. Drafts required vary from about 0.1 in. of water for free-burning soft coal to 1.0 in. or more for fine anthracite. A chimney is seldom designed for less than 0.5 in,, nor forced blast apparatus for less than 0.8 in. 568. Types of Boiler. Boilers are broadly grouped as fire-tube or water-tube, internally or externally fired. A type of externally fired water- tube boiler has been shown in Fig. 233. In this, the Babcock and Wilcox design, the path of the gases is as described in Art. 508. The feed water enters the drum 6 at 29, flows downward through the back water legs at a, and then upward to the right along the tubes, the high tem- perature zone at 1 compelling the water above it in tubes to rise. Figure 274 shows the horizontal tubular boiler, probably most generally used in this country. The fire grate is at S. The gases pass over the bridge wall 0, under the shell of the boiler, up the back end F", and to the right through tubes run- ning from end to end of the cylindrical shell. The tubes terminate at C, and 420 APPLIED THERMODYNAMICS the gases pass up and away Feed water enters the front head, is carried in the pipe about two thirds of the distance to the back end, and then falls, a compensating I upward current being generated over the grate. This is an eternally fired fire-tube boiler. Figure 275 shows the well-known locomotive boiler, which is internally fired. The coldest part of this boiler is at the end farthest from the grate, on the exposed sides. The feed is consequently admitted here. Figure 276 shows a boiler com- monly used in marine service. The grate is placed in an internal furnace ; the gases pass upward in the back end, and return through the tubes. The feed pipe is located as in horizontal tubular boilers. STEAM BOILERS 421 569. Discussion. The internally fired boiler requires no brick furnace setting, and is compact. The water-tube boiler is rather safer than the fire- tube, and requires less space. It can be more readily used with high steam pressures. The im- portant points to observe in boiler types are the paths of the gases and of the water. The gases should, for economy, im- pinge upon and thoroughly circulate about all parts of the heating surface; the circulation of the water for safety and large capacity should be posi- tive and rapid, and the cold feed water should be introduced at such a point as to assist this circula- tion. There is no such thing as a "most economical type" of boiler. Any type may be economical if the proportions are right. The grade of fuel used and the draft attain- able determine the neces- sary area of grate for a given fuel consumption. The heating surface must be sufficient to absorb the heat liberated by the fuel. The higher the rate of combustion (pounds of fuel burned per square foot of grate per hour), the greater the relative amount of heating surface necessary. 422 APPLIED THERMODYNAMICS LONGJTUDINAL SECTION FIG. 276, Ait. 5fJ8 Marine Boiler. (The Bigelow Company.) Rates of combustion, range from 12 Ib, with, low grade hard coal and natural draft up to 30 or 40 Ib. with soft coal ; * the corresponding ratios of heating surface to grate surface may vary from 30 up to CO or 70. The best economy has usually been associated with high ratios. The rate of evaporation is the number of pounds of water evaporated per square foot of heating surface per hour; it ranges from 3.0 upward, de- pending upon the activity of circulation of water and gases, f An effective heating surface usually leads to a low flue-gas temperature and relatively small loss to the stack. Small tubes increase the efficiency of the heat- ing surface but may be objectionable with certain fuels. Tubes seldom exceed 20 ft. in length. In water-tube boilers, the arrangement of tubes is important. If the bank of tubes is comparatively wide and shallow, the gases may pass off without giving up the proper proportion of their heat. If the bank is made too high and narrow, the grate area may be * Much, higher rates are attained in locomotive practice ; and in torpedo boats, with intense draft, as much as 200 Ib. of coal may be burned per square foot of grate per hour. f Former ideas regarding economical rates of evaporation and boiler capacity are being seriously modified. Bone has found in " surface combustion " with gas fired boilers an efficiency of 0.94 to be possible with an evaporation rate of 21,6 Ib. Power, Nov. 21, 1911, Jan. 16, 1912.) STEAM BOILER ECONOMY 423 too much restricted. The gases must not be allowed to reach the flue too quickly. 570. Boiler Capacity. A boiler evaporating 3450 Ib. of water per hour from and at 212 F. performs 970.4X778X3450 =2,600,000,000 foot-pounds of work, or 1300 horse power. No engine can develop this amount of power from 3450 Ib. of steam per hour; the power developed by the engine is very much less than that by the boiler which supplies it. Hence the custom or rating boilers arbitrarily. By defini- tion of the American Society of Mechanical Engineers, a boiler horse power means the evaporation of 31J Ib. of water per hour from and at 212 F. This rating was based on the assumption (true in 1S7G, when the original definition was established) that an ordinary good engine required about this amount of steam per horse power hour. This evaporation involves the liberation of aboujfc 33,000 B. t. u. per hour. Under forced conditions, however, a boiler may often transmit as many as 25 B. t. u. per' square foot of surface per hour per degree of tem- perature difference on the two sides of its surface. 571. Limit of Efficiency. The gases cannot leave the boiler at a lower temperature than that of the steam iu the boiler. Let t be the initial temperature of the fuel and air, x the temperature of the steam, and T the temperature attained by combustion ; then if W be the weight of gas and K its specific heat, assumed constant, the total heat generated is WK(T ), the maximum that can be utilized is WK(T a;), and the limit ol efficiency is T-x T-t In practice, we have as usual limiting values T= 4850, #= 350, =60; whence the efficiency is 0.94 a value never reached in practice. 572. Boiler Trials. A standard code for conducting boiler trials has been published by the American Society of Mechanical Engineers (2). A boiler, like any mechanical device, should be judged by the ratio of the work which it does to the energy it uses. This involves measuring the fuel supplied, determining its heating value, measuring the water evaporated, and the pressure and superheat, or wetness, of the steam. The result is usually expressed in pounds of dry steam evaporated per pound of coal from and at 212 F., briefly called the equivalent evaporation. Let the factor of evaporation be F. If W pounds of water are fed to the boiler per pound of coal burned, the equivalent evaporation is FW. 424 APPLIED THERMODYNAMICS If C be the heating value per pound of fuel, the efficiency is 970 FW + C. Many excessively high values for efficiency have been reported in con- sequence of not correcting for wetness of the steam; the proportion of wetness may range up to 4 per cent in overloaded boilers. The highest well-confirmed figures give boiler efficiencies of about S3 per cent. The average efficiency, considering all plants, probably ranges from 0.40 to 0.60, 573. Checks on Operation. A careful boiler trial is rather expensive, ""and must often interefere with the operation of the plant. The best indication of cur- rent efficiency obtainable is that afforded by analysis of the flue gases It has been shown that maximum efficiency is attained when the percentage of carbon dioxide reaches a maximum Automatic instruments are in use for continuously determining and recording the proportion of this constituent present in flue gases. 575. Chimney Draft. In most cases, the high temperature of the flue gases leaving the boiler is utilized to produce a natural upward draft for the maintenance of combustion. At equal temperatures, the chimney gas would be heavier than the external air in the ratio (n+l] I -s-n, where n is the number of pounds of air supplied per pound of fuel. If T, t denote the respective absolute temperatures of air and T /n I 1\ gas, then, the density of the outside air being 1, that of the chimney gas is ( - ) At 60 F., the volume of a pound of air is 13 cu. ft. The weight of gas in a chimney of cross-sectional area A and height H is then The " pressure head," or draft, due to the difference in weight inside and outside is, per unit area, This is in pounds per square foot, if appropriate units are used ; drafts are, how- ever, usually stated in " inches of water," one of which is equal to 5.2 Ib. per square foot. The force of draft therefore depends directly on the height of the chimney; and since n -f 1 is substantially equal to n, maximum draft is obtained when T t is a minimum, or (since T is fixed) when t is a maximum; in the actual case, however, the quantity of gas passing would be seriously reduced if the value of t were too high, and best results (3), so far as draft is concerned, are obtained when *.r::25-12. To determine the area of chimney: the velocity of the gases is, in feet per second, v = V2~h = 8.03 V^ = 8.03 V 7 h being the head corresponding to the net pressure p and density d of the gases in the chimney. Also 4 T ( n+1 \ 13\ n )' CHIMNEY DESIGN 425 Then if C Ib. of coal are to be burned per hour, the weight of gases per second is ,, , -3600 ' their V lume 1S and the area of the chimney, in square feet, is A slight increase may be made to allow for decrease of velocity at the sides. The results of this computation will be in line with those of ordinary " chimney tables," if side friction be ignored and the air supply be taken at about 75 Ib. jper pound of fuel. 576. Mechanical Draft. In lieu of a chimney, steam-jet blowers or fans may be employed. These usually cost less initially, and more in maintenance. The ordinary steam-jet blower is wasteful, but the draft is independent of weather con- ditions, and may be greatly augmented in case of overload. The velocity of the air moved by a/<w is ,_ _ v = v2 gh, where 7i is the head due to the velocity, equal to the pressure divided by the density. Then If a be the area over which the discharge pressure p is maintained, the work necessary is w = pav = We may note, then, that the velocity of the air and the amount delivered vary as the peripheral speed of the wheel, its pressure as the square, and the power consumed as the cube, of that speed. Low peripheral speeds are therefore economical in power. They are usually fixed by the pressure required, the fan width being then made suitable to deliver the required volume. 577. Forms of Fan Draft. The air may be blown into a closed fire room or ash pit or the flue gases may be sucked out by an induced draft fan. In the last case, the high temperature of tho gases reduces the capacity of the fan by about one half; i.e., only one half the weight of gas will be discharged that would be delivered at 60 F. Since the density is inversely proportional to the absolute temperature, the required pressure can then be maintained only at a considerable increase in peripheral speed; which is not, however, accompanied by a concordant increase in power consumption Induced draft requires the handling of a greater weight, as well as of a greater volume of gas, than forced draft; the necessary pres- sure is somewhat greater, on account of the fnctional resistance of the flues and passages; high temperatures lead to mechanical difficulties with the fans. The difficulty of regulating forced draft has nevertheless led to a considerable applica- tion of the induced system. 578. Furnaces for Soft Coal; Stokers. Mechanical stokers are often used when soft coal is employed as fuel Besides saving some labor, in large plants at least, they give more perfect combustion of hydrocarbons, with reduced smoke produc- 426 APPLIED THERMODYNAMICS tion. Figure 277 shows, incidentally, a modern form of the old " Dutch oven " principle for soft coal firing. The flames are kept hot, because they do not strika the relatively cold boiler surface until combustion is complete. Fuel is fed alter- nately to the two sides of the grate, so that the smoking gases from one side meet the hot flame from the other at the hot baffling " wing walls " a, &. The principle FIG. 277. Arts. 578, 579. Sectional Elevation of Foster Superheater combined with Boiler and Kent Wing Wall Furnace. (Power Specialty Company ) FIG. 278. Arts. 578,579. Babcoek and Wilcox Boiler with Chain Grate Stoker and, Superheater. SUPERHEATERS 427 involved in the attempt to abate smoke is that of all mechanical stokers, which may be grouped into three general types. In the chain grate, coal is carried forward continuously on a moving chain, the ashes being dropped at the back end. The gases from the fresh fuel pass over the hotter coke fire on the back portion of the grate. (See Fig. 278.) The second type comprises the vndined grate stokers. The high combustion chamber above the lower end of the grate is a decided advan- tage with many types of boilers. The smoke is distilled off at the " coking plate." The underfeed stoker feeds the coal by means of a worm to the under side of the fire, and the smoke passes through the incandescent fuel. All stokers have the advantage of making firing continuous, avoiding the chilling effect of an open fire door, Airing soft coal furnaces not associated with stokers, one of the best known is the Hawley down draft. In this, there are two grates, coal being fired on the upper, through which the draft is downward. Partially consumed particles of coal (coke) fall through the bars to the lower gate, where they maintain a steady high temperature zone through which the smoking gases from the upper grate must pass on their way to the flue, 579. Superheaters ; Types. Superheating was proposjd at an early date, and given a decided impetus by Hirn. After 1870, as higher steam pressures were introduced, superheating was partially abandoned. Lately, it has been reintro- FlG. 279. Art. B79. Cole Superheater. (American Locomotive Company ) duced, and the use of superheat is now standard practice in France and Germany, while being quite widely approved in this country. Superheaters may be sepa- rately fired, steam from a boiler being passed through an entirely separate machine, or, as is more common, steam may be carried away from the water to some space 428 APPLIED THERMODYNAMICS provided for it within the boiler setting or flue, and there heated by. the partially spent gases. When it is merely desired to dry the steam, the "superheater" may be located in the flue, using waste heat only. When any considerable increase of temperature is desired, the superheater should be placed in a zone of the furnace where the temperature is not less than 1000 F. With a difference m mean temperature between gases and steam of 400 F , from 4 to 5 B t. u may be transmitted per degree of mean temperature difference per square foot of surface per hour (4) . According to Bell, if 8 ~ amount of superheat, deer. F , T = temperature of flue gases reaching superheater, ^tem- perature of saturated steam, x sq. ft. of superheater surface per boiler horsepower; 108f FIG. 280. The location of the Babcock and Wilcox superheater is shown in Fig. 277; a similar arrangement, in which the chain grate stoker is incidentally represented, is shown in Fig 278. In locomotive service (in which superheat has produced unexpectedly large savings) Field tubes may be em- ployed, as in Fig 279, the steam emerging Art. 579. -Superheater Element. frQm ^ bmlfff ftt ^ an d passing through (Power Specialty Company.) thc header b to the small tubes c, c, c, in the fire tubes d,d,d(5). A typical superheater tube or " element " is shown in Fig 280. This is made double, the steam passing through the annular space. Increased heating surface is afforded by the cast iron rings a, a. In some single-tube elements, the heating surface is augmented by internal longitudinal ribs. The tubes should be located so that the wettest steam will meet the hottest gases. 580. Feed-water Heaters. In Fig. 233, the condensed water is returned directly from the hot weU 24, by way of the feed pump IV, to the boiler. This water is seldom higher in temperature than 125 F. A considerable saving may be effected by using exhaust steam to further heat the water before it is delivered to the boiler. The device for accomplislung this is called the feed-water heater. With a condens- ing engine, as shown, the water supply may be drawn from the hot well and the necessary exhaust steam supplied by the auxiliary exhausts 27 and 31; I Ib. of steam at atmospheric pressure should heat about 9.7 Ib. of water through 100. Accurately, W(xL+ h-qJ=w(Q-q), in which W is the weight of steam condensed, x is its dryness, L its latent heat, h its heat of liquid, and w is the weight of feed water, the initial and final heat contents of which are respectively q and (?. The heat contents of the steam after condensation are q Q . Then With non-condensing engines, the exhaust steam from the engines themselves is used to heat the cold incoming water. FEED WATER HEATERS 429 581. Types. Feed-water heaters may be either " open," the steam and water mixing, or " closed," the heat being transmitted through the surface of straight or curved tubes, through which the water circulates. Figure 281 shows a closed heater; steam enters at A and emerges at JS, wator enters at C, passes through the tubes and out at D. The openings E, E are for drawing off condensed steam. An open heater is shown in Fig. 282. Water enters through the automatically controlled valve a, steam enters at 6. The water drips over the trays, becoming finely divided and effectively heated by the steam. At c there is provided storage space for the mixture, and at d is a bed of coke or other absorbent material, through which the water filters upward, passing out at e. The open heater usually makes the water rather hotter, and lends itself more readily to the re- claiming of hot drips from the steam pipes, returns from heating systems, etc., than a heater of the closed typo. Live steam is sometimes used for feed- water heating ; the greater effective- ness of the boiler-heating surface claimed to arise from introducing the water at high temperature has been disputed (6) ; but the high temperatures possible with live steam are of decided value in removing dissolved solids, and the waste of steam may be only slight. Closed heaters are, of course, used for this service, as also with the isodiabatic multiple expansion cycle described in Art. 550, Removal of some of the suspended and dissolved FIG. 281. Art. 581. Wheeler Feed Water Heater. FIG. 282. Art. 581. Open Feed Heater. (Harrison Safety Boiler "Works.) solids is also possible in ordinary open-exhaust steam heaters. Various forms of feed- water filters are used, with or without heaters. 582. The Economizer. This is a feed-water heater in which the heating medium is the waste gas discharged from the boiler furnace. It may increase the feed temperature to 300 F. or more, whereas no ordinary exhaust steam heater can produce a tem- perature higher than 212 F. The gam by heating feed water is about 1 B. t. u. per pound of water for each degree heated, or since average steam contains 1000 B. t. u. net, it is about 1 per cent for each 10 that the temperature is raised; precisely, the gain is (# /0-rQ, in which Q is the total heat of the steam gained from the temperature of feed to the state at evaporation and h and H the total heats in the water before and after heating. If T, t be the temperatures 430 APPLIED THERMODYNAMICS of flue gases and steam, respectively, W the weight, and K the mean specific heat of the gases (say about 0.24), then the maximum saving that can be effected by a peifect economizer is WK(T t). Good operation decreases W and T and thus makes the possible sav- ing small. A typical economizer installation is shown in Fig. 283; arrangement is always made for by-passing the gases, as shown, to permit of inspecting and cleaning. The device consists of vertical cast- iron tubes with connecting headers at the ends, the tubes being some- times staggered so that the gases will impinge against them. The external surface of the tubes is kept clean by scrapers, operated from a small steam engine. The tubes obstruct the draft, and some form of mechanical draft is em- ployed in conjunction with econo- mizers. From 3J to 5 sq. ft. of economizer surface are ordinarily used per boiler horse power. The rate of heat transmission (B. t. u. per square foot per degree of mean temperature difference per hour) is usually around 2.0. 583. Miscellaneous Devices. A steam separator is usually placed on the steam pipe near the engine. This catches and more or less thoroughly removes any condensed steam, which might otherwise cause damage to the cylinder. Steam meters are being introduced for approximately indicating the amount of steam flowing through a pipe. Some of them record their indications on a chart. Feed-water measuring tanks are sometimes in- stalled, where periodical boiler trials are a part of the regular routine. The steam loop is a de- vice for returning condensed steam direct to the boiler. The drips are piped up to a convenient height, and the down pipe then forms a radiating coil, in which a considerable amount of condensation occurs. The weight of this column of water in the down pipe offsets CONDENSERS 431 a corresponding difference in pressure, and permits the return of drips to the boiler even when their pressure is less than the boiler pressure. The ordinary steam trap merely removes condensed water without permitting the discharge of un condensed steam. Oil separators are sometimes used on exhaust pipes to keep back any traces of cylinder oil. 534. Condensers. The theoretical gain by running condensing is shown by the Carnot formula (2 1 t) + T. The gain m practice may be indicated on the PV diagram, as in Fig. 284 The shaded area represents work gamed due to condensation; it may amount to 10 or 12 Ib. of mean effective pressure, which means about a 25 per cent gain, in the case of an ordinary simple engine.* This gain is principally the result of the intro- duction of cooling water, which usually costs merely the power to pump it; in most cases, some additional powor is needed to drive an air pump as well. In the surface condenser the steam and the water do not come into contact, so that impure water may be used, jp I(J> 2 84. Art. 584. Sav- as at sea, even when the condensed steam is returned to ing Due to Condensation, the boilers, f Such a condenser needs both air and circulating pumps. The former ordinarily carries away air, vapor and condensed steam. In some cases, the discharge of condensed steam is separately cared for and the dry vacuum pump (which should always be piped to the condenser at a point as far as possible from the steam inlet thereto) handles only air and vapor. The amount of condensing surface should be computed from Orrok's formula (Jour. A. S. M. E., XXXII, 11): \vhere Z7 = B. t. u. transmitted per sq. ft. of surface per hour per degree of mean temperature difference between steam and water; C = a cleanliness coefficient (tubes), between 1.0 and 0.5; r ratio of partial pressure of steam to the total absolute pressure in the condenser, depending on the amount of air present, and varying from 1.0 to 0; m = Si coefficient depending upon the material of the tubes; 1.0 for copper, 0.63 for Shelby steel, 0.98 for admiralty, etc., ranging down to 0.17 for a tube vulcanized on both sides, all of these figures being for new metal. Corrosion or pitting may reduce the value of m 50 per cent; V velocity of water in tubes, ft. per sec., usually between 3 and 12; Tin -mean temperature difference between steam and water, deg. F. For T m = 18.3 (corresponding with 28" vacuum, 70 temperature of inlet water , 90 temperature of outlet water), this becomes 435Cr 8 mVy. The former approximate expression of Whitham (T^was * In the case of the turbine, good vacuum is so important a matter that extreme refinement of condenser design has now "become essential. f There is always an element of danger involved in returning condensed steam from reciprocating engines to the "boilers, on account of the cylinder oil which it contains. 432 APPLIED THERMODYNAMICS where S was the condenser surface in sq. ft , W the weight of steam condensed, Ib. per hour, L the latent heat at the temperature T of the steam, and t the mean temperature of circulating water between inlet and outlet. With the same nota- tion, Orrok's formula gives ~_WL WL U(T-ty nearly. With C=0.8, r = OS, m = 0.50, TV, = 18.2, T = 16, V becomes approximately ISO, as in the Whitham formula Let u, U be the initial and final temperatures of the water; then the weight w of water required per hour is WL-7-(U u). The weight of water is often permitted to be about 40 times the weight of steam, a considerable excess being desirable. The outlet temperature of the water in ordinary surface condensers will be from 5 to 15 below that of the steam. The direction of flow of the water should be opposite to that of the steam. The jet condenser is shown in Fig. 285. The steam and water mix in a chamber above the air pump cylindei, and this cylinder is utilized to draw in the water, if the lift is not excessive. Here U = T; the supply of water necessary is less than in surface condensers. With ample water supply, the surface condenser gives the better vacuum. The boilers may be fed from the hot well, as in Fig 233 (which shows a jet condenser installation), only when the condensing water is pure. The siphon condenser is shown in Fig 286 Condensation occurs in the nozzle, a, and the fall of water through b produces the vacuum. To preserve this, the lower end of the discharge pipe must be sealed as shown. The vacuum would draw water up the pipe 6 and permit it to flow over into the engine, if it were not that the length cd is made 34 ft. or more, thus giving a height to which the atmospheric pressure cannot force the water. Excellent results have been obtained with these con- densers without vacuum pumps. In some cases, however, a " dry" vacuum pump is used to remove air and vapor from above the nozzle. The device is then called a barometric condenser. The vacuum will lift the inlet water about 20 ft., so that, unless the suction head is greater than this, no water sup- FIG. 285. Art. 584. Horizontal Independent Jet ply pump is required after the Condenser, condenser is started. Either the jet or the siphon (or barometric) condenser requires a larger air pump than a surface condenser. Experience has shown that there will be present 1 cu. ft. of free air (Art 187) per 10 to 50 cu. ft. of water entering the air pump of a surface condenser or per 30 to 150 cu. ft. of water entering a jet or barometric condenser. The surface condenser air pump handles the condensed steam only; the other condensers add the circulating water (which mav be 20 to 40 times this) to the steam. The volume of air at the low absolute pressure prevailing in the condenser is large, and the necessity for reducing the partial pressure due to air has led to the employ- ment of pumps still larger than the influence of air volume, alone would warrant. CONDENSERS 433 (For a discussion of air pump design and the importance of clearance in connection with high vacuums, see Caidullo, Practical Thermodynamics , 1911, p. 210.) 585. Evaporative Condensers; Cooling Towers. Steam has occasionally been condensed by allowing it to pass through coils over which fine streams, of v ater trickled. The evaporation of the water (which may be hastened by a fan) cools the coils and condenses the steam, which is drawn off by an air pump. With ordinary condensers and a limited water supply cooling towers are sometimes used. These may be identical in construction with the evaporative condensers, excepting that warm water enters the coils instead of steam, to be cooled and used over again, or they may consist of open wood mats over which the water falls as in the open type of feed-water heater. Evaporation of a portion of the water in question (which need not bo a, large proportion of the whole) and warming of the air then cools the remainder of the water, the cooling being facili- tated by placing the mats in a cylindrical tower through which FlG m Art. C8A.Bulkley Iniectoi Condenser. there is a rapid upward current of air, naturally or artificially produced (8). The cooling pond (8a) is equivalent to a tower. 586. Boiler Feed Pump. This maybe either steani-driven or power-driven (as may also be the condenser pumps). Steam-driven pumps should be of the duplex type, with plungers packed from the outside, and with individually acces- sible valves. If they are to pump hot water, special materials must be used for exposed parts. The power pump has usually three single-acting water cylinders. There is much discussion at the present time as to the comparative economy of steam- and power-driven auxiliaries. The steam engine portion of an ordinary small pump is extremely inefficient, while power-driven pumps can be operated, at little loss, from the main engines. The general use of exhaust steam from aux- iliaries for feed-water heating ceases to be an argument in their favor when econo- mizers are used ; and in large plants the difference in cost of attendance in favor of motor-driven, auxiliaries is a serious item. 587. .The Injector. The pump is the standard device for feeding stationary boilers; the injector, invented by Giffard about 1858, is used chiefly as an auxil- iary, although still in general application as the prime feeder on locomotives. It 434 APPLIED THERMODYNAMICS consists essentially of a steam nozzle, a combining chamber, and a delivery tube. In Fig 287, steam enters at A and expands through B, the amount of expansion being regulated by the valve C. The water enters at D, and condenses the steam in Ef. We have here a rapid adiabatic expansion, as in the turbine; the ve'ocity of the water is augmented by the impact of the steam, and is in turn con- veited into pressure at F. In starting the injector, the water is allowed to flow away through G ; as soon as the velocity is sufficient, this overflow closes. An in- jector of this form will lift the water from a reasonably low suction level ; when the water flows to the device by gravity, the valve C may be omitted. FIG. 287 Art 587. Injector. A self-starting injector is one in which the adjustment of the overflow at G is automatic. The ejector is a similar device by which the lifting of water from a we,!! or pit against a moderate delivery head (or none) is accomplished. The siphon condenser (Art. 584) involves an application of the injector principle. The double injector is a series of successive injectors, one discharging into another. 588. Theory. Tet x, L, h be the state of the steam, .fftheheat in the water, and v its velocity; Q the heat in the discharged water at its veloc- ity V. The heat in one pound of steam is xL + h; the heat in one pound of water supplied is H f and its kinetic energy v 2 -5- 2 g j the heat in one pound of discharge is Q, and its kinetic energy F 2 -s- 2 0. Let each pound of steam draw in y pounds of water ; then THE INJECTOR 435 v 2 V 2 The values of and may ordinarily be neglected, and ~~ Q-H ' In another form, y(Q J/)= xL + h Q, or the heat gained by the water equals that lost by the steam. This, while not rigidly correct, on account of the changes in kinetic energy, is still so nearly true that the thermal efficiency of the injector may be regarded as 100 per cent ; from this stand- point, it is merely a live-steam feed-water heater. 589. Application. The formula given shows at once the relation between steam state, water temperature, and quantity of water per pound of steam. As the water becomes initially hotter, less steam is required ; but injectors do not handle hot water well. Exhaust steam may be used in an injector : the pressure of discharge is determined by the velocity induced, and not at all by the initial pressure of the steam ; a large steam nozzle is required, and the exhaust injector will not ordinarily lift its own water supply. 590. Efficiency. Let S be the head against which discharge is made ; then the work done per pound of steam is (! + ?/)$ foot-pounds ; the efficiency is /S(l + #)-*- (xL + h Q), ordinarily less than one per cent. This is of small consequence, as practically all of the heat not changed to work is returned to the boiler. Let W be the velocity of the steam issuing from the nozzle; then, since the momentum of a system of elastic bodies remains con- stant during impact, W + yv = (1 + y) V. The value of W may be expressed in terms of the heat quantities by combining this equation with that in Art. 588. The other velocities are so related to each other as to give orifices of reasonable size. The practical proportioning of injectors has been treated by Kneass (9). (1) Finlay, Proc. A. I. E. E., 1907. (2) Trans A. S. M. E., XXI, 34. (3) Ran- kme, The Steam Engine, 1897, 289. (4) Longridge, Proc. Inst. M. E., 1896, 175. (5) Trans. A. S. M. E., XXVIII, 10, 1606. (6) Bilbrough, Power, May 12, 1908, p. 729. (7) Trans. A. S. M. E., IX, 431. (S) Bibbins, Trans. A. S. M. E. t XXXI, 11; Spangler, Apphed Thermodynamics, 1910, p. 152. (8a) Cardullo, Practical Thermodynamics, 1911, p. 264. (9) Practice and Theory of the Injector. SYNOPSIS OF CHAPTER XVI Hard coal requires high, draft ; soft coal, a high rate of air supply. In spite of its higher cost, commercial factors sometimes make soft coal the cheaper fuel. Heating values: fuel oil, 18,000; wood, 6500-8500; coals, 8800-15,000; B.t.u. per Ib. Method of computing heat value. The proportion of carbon dioxide in the flue gases reaches a maximum when the air supply is just right ; this is also the condition of maximum temperature and theoretical efficiency. 436 APPLIED THERMODYNAMICS Advance in steam power economy is a matter of regulation of air supply j economy may be indicated by automatic records of carbon dioxide. Types of boiler : water-tube, horizontal tubular, locomotive, marine ; conditions of efficiency. Attention should be given to the circulation of the gases and water. A boilT fcp.34J Ib, of water per hour from and at 212 P., approximately 33,000 B. t. u. per hour. Limit of efficiency = % . say ^94. . never reached in practice. T t Boiler efficiency = 5 usually 0.40 to 0.00 , may be 0.83. Furnace efficiency = "**** . Heating surface efficiency = ^at in steam . heat in fuel neat in gases en*.** *** = jr[i-(ll) j -is Fan draft : w= ^/2gh, p = , W= - a<bS 3 slow speeds advantageous. 2 Q 2 g In induced draft, the fan is between the furnace and the chimney ; in forced draft, it delivers air to the ash pit. Mechanical stokers (inclined grate, chain grate, underfeed), used with soft coal, aim to give space for the hydrocarbonaceous flame without permitting it to impinge on cold surfaces. Superheaters may be located in the flue, or, if much superheating is required, may be separately fired. About 5 B. t. u. may be the transmission rate. Feed-water heaters may be open or closed: w = TC^ "~ g) ; for open heaters, q = Q. Q~q The economizer uses the waste heat of the flue gases : saving per pound of fuel = WK(T t). From 3| to 5 sq.. ft. of surface per boiler hp. Condensers may be surface, jet, evaporative, or siphon, w = WL+('O' u), S = W. L-r- U(T-t); U = 630 r \ . The siphon condenser may operate with- * OT* out a vacuum pump. The use of steam-driven, auxiliaries affords exhaust steam for feed-water heating. The injector converts heat energy into velocity: y= ^ \ efficiency = PROBU3MS 1. One pound of pure carbon is burned in 16 Ib. of air. Assuming reactions to be perfect, find the percentage composition of the flue gases and the rise in tempera- ture, the specific heats being, C0 2 , 0,215 ; N, 0.245 ; 0, 0.217. 2. A boiler evaporates 3000 Ib. of water per hour from a feed-water temperature of 200 3T. to dry steam at 160 Ib. pressure. What is its horse power? 3. In Problem 1, what proportion of the whole heat in the fuel is carried away PROBLEMS 437 m the flue gases, if their temperature is 600 F., assuming the specific heats of the gases to be constant ? The initial temperature of the fuel and air supplied is F. 4. The boiler in Problem 2 burns 350 Ib. of coal (14,000 B. t.u. per pound) per hour. What is its efficiency ? 5. In Problem 1, if the gas temperature is 600 F., the air temperature 60 F., compare the densities of the gases and the external air. What must be the height of a chimney to produce, under these conditions, a draft of 1 in. of water ? Find the diameter of the (round) chimney to burn 5000 Ib. of coal per hour. (Assume a 75 Ib. air supply in finding the diameter.) 6. Two fans are offered for providing draft in a power plant, 15,000 cu. ft. of air being required at 1J oz. pressure per minute. The first fan has a wheel 30 in. in diameter, exerts 1 oz. pressure at 740 r. p. m., delivers 405 cu. ft. per minute, and consumes 0.10 hp., both per inch of wheel width and at the given speed. The second fan has a 54-inch wheel, runs at 410 r. p. m. when exerting 1 oz. pressure, and delivers 726 cu. ft. per minute with 0.29 hp., both per inch of wheel width and at the given speed. Compare the widths, speeds, peripheral speeds, and power consump- tions of the two fans under the required conditions. 7. Dry steam at 350 F. its superheated to 450 F. at 135 Ib. pressure. The flue gases cool from 900 F. to 700 F. Find the amount of superheating surface to pro- vide for 3000 Ib. of steam per hour, and the weight of gas passing the superheater. If 180 Ib. of coal are burned per hour, what is the air supply per pound of coal ? 8. Water is to be raised from 60 F. to 200 F. in a feed-water heater, the weight of water being 10,000 Ib. per hour. Heat is supplied by steam at atmospheric pres- sure, 0.95 dry. Find the weight of steam condensed (a) in an' open heater, (Z>) in a closed heater. Find the surface necessary m the latter (Art. 584). 9. In Problem 3, what would be the percentage of saving due to an economizer which reduced the gas temperature to 400 F. ? 10. An engine discharges 10,000 Ib. per hour of steam at 2 Ib. absolute pressure, 0.95 dry. Water is available at 00 F. Find the amount of water supplied for a jet condenser. Find the amount "of surface, and the water supply, for a surface con- denser in which the outlet temperature of the water is 85 F. If the surface con- denser is operated with a cooling tower, what weight of water will theoretically be evaporated in the tower, assuming the entire cooling to be due to such evaporation. (N. B. A large part of tho cooling is in practice effected by the air.) 11. Find the dimensions of the cylinders of a triplex single-acting feed pump to deliver 100,000 Ib. of water per hour at 60 F. at a piston speed of 100 ft. per minute and 30 r. p. m. 12. Dry steam at 100 It), pressure supplies an injector which receives 3000 Ib. of water per hour, the inlet temperature of the water being 60 F. Find the weight of steam used, if tho discharge temperature is 165 F. 13. In Problem 12, the boiler presBure is 100 Ib. What is the efficiency of the injector, considered as a pump ? 14. In Problem 12, the velocity of the entering water is 12 ft. per second, that of the discharge is 114 ft. per second. Find the velocity of the steam leaving the discharge nozzle. 15. What is the relation of altitude to chimney draft ? (See Problem 12, Chapter 438 APPLIED THERMODYNAMICS 16. Circulating water pumped from a surface condenser to a cooling tower loses 4J per cent of its weight by evaporation and is cooled to 88 I\ If the loss is made up by city water at 55, fed continuously, what is the temperature of the water entering the condenser ? 17. Steam at 100 Ib. absolute pressure and 500 F. is used in an open feed-water heater to warm water from 60 to 210. How much water will be heated by 1 Ib. of steam ? 18. Steam at 150 Ib. absolute pressure, 2 per cent wet, passes through a super- heater which raises its temperature to 500 F. How much heat was added to each Ib. of steam ? 19. 20,000 Ib, of steam at 150 Ib. absolute pressure, 2 per cent wet, are super- heated 200 in a separately-fired superheater of 0.70 efficiency. What weight of coal, containing 14,000 B, t. u, per Ib,, will be required ? CHAPTER XVII FIG. 288., Art 501, Still. DISTILLATION FUSION LIQUEFACTION OF GASES VACUUM DISTILLATION 591. The Still. Figure 288 represents an ordinary still, as used for purifying liquids or for the recovery of solids in solution by concentration. Externally applied heat evaporates the liquid in A, which is condensed at g B. All of the heat ab- | sorbed in A is given up at B to the cooling water; the only wastes, in theory, arise from radiation. Con- ceive the valve c to be closed, and the space from the liquid level d to this valve to be filled with satu- rated vapor, no air being present in any part of the apparatus. Then when the value c is opened, a vacuum will gradually be formed throughout the system, and evaporation will proceed at lower and lower temperatures. Since the total heat of saturated vapor decreases with decrease of pressure, evaporation will thus be facilitated. In practice, however, the apparatus cannot be kept free from air ; and, notwithstanding the opera- tion of the condenser, the vacuum would soon be lost, the pressure increas- ing above that of the atmosphere. This condition is avoided by the use of a vacuum pump, which may be applied at e, removing air only; or, in usual practice, at/, removing the condensed liquid as well. Evaporation now proceeds continuously at low pressure and temperature. The possi- bility of utilizing low-temperature heat now leads to marked economy. Solutions are usually assumed to contain about 5 per cent of their volume of free air. The condenser, if of the jet type, should be designed to handle about 150 times the water volume of actual air; if of the surface type (which must be used when the distilled product is to be recovered), about 100 times the water volume. 592. Application. Vacuum distillation is employed on an important scale in sugar refineries, soda process paper-pulp mills, glue works, glucose factories, for the preparation of pure water, and in the manufacture of gelatine, malt extract, 439 APPLIED THERMODYNAMICS MULTIPLE-EFFECT EVAPORATION 441 wood extracts, caustic soda, alum, tannin, garbage products, glycerine, sugar of milk, pepsin, and licorice. In most cases, the multiple-effect apparatus is employed (Art. 594). 593. Newhall Evaporator. This is shown in Fig. 289. Steam is used to supply heat ; it enters at A, and passes through the chambers A 1 , A 2 9 to the tubes B, B. After passing through the tubes, it collects in the chambers C 2 , C l , from which it is drawn off by the trap D. The liquid to be distilled surrounds the tubes. The vapor forms in E, passes around the baffle plate F and out at G. The concentrated liquid is drawn off from the bottom of the machine. 594. Multiple-effect Evaporation. Conceive a second apparatus to be set alongside that just described ; but instead of supplying FIG. 290. Art. 595. Triple Effect Evaporator. steam at A, let the vapor emerging from 6- of the first stage be piped to A in the second, and let the liquid drawn off from the hot- 442 APPLIED THERMODYNAMICS torn of the first be led into the second ; then further evaporation may proceed without the expenditure of additional heat, the liquid being partially evaporated and the vapor partially condensed by the inter- change of heat in the second stage, the pressure in the shell {outside the tubes) being less than that in the first stage. The construction will be more clearly understood by reference to Fig. 290 (la). 595. Yaryan Apparatus. Here the heat is applied outside the tubes, the liquid to be distilled being inside. The liquid is forced by a pump through a small orifice at the end of the tube, breaking into a fine spray during its pas- sage. The fine sub- division and rapid movement of the liquid facilitate the transfer of heat. The baffle plates E, E, Tig. 291, serve to separate the liquid and no. 291, Art. 595. Yaryan Evaporator, its vapor, the former settling in the chamber b, the latter passing out at c. Figure 290 shows a "triple-effect" or three-stage evaporator; steam (preferably exhaust steam) enters the shell of the first stage. The liquor to be evaporated enters the tubes of this stage, becomes partly vaporized, and the separated vapor and liquid pass off as just described. From the outlet c, Fig. 291, the vapors pass through an ordinary separator, which removes any ad- ditional entrained liquid, discharging it back to &, and then proceed to the shell of the second stage. Meanwhile the liquid from the chamber b of the first stage has been pumped, through a hydrostatic tube which permits of a difference in pressure in two successive sets of tubes, into the tubes of the second stage. As many as six successive stages may be used; * the vapors from the last being drawn off by a condenser and vacuum pump. The liquid from the chamber b of the last stage is at maximum density. 596. Condition of Operation. The vapor condensed in the various shells is ordinarily water, which in concentrating operations may be * The number of effects that can be used is limited by the difference in tempera- ture of steam supphed and final condensate discharged. MULTIPLE-EFFECT EVAPORATION 443 drawn off and wasted, or, if the temperature is sufficiently high, employed in the power plant. The condenser is in communication with the last tubes, and, through them, with all of the shells and tubes excepting the first shell; but between the various stages we have the heads of liquid in the chambers b, which permit of carrying different pressures in the different stages. A gradually decreasing pressure and temperature are employed, from first to last stage; it is this which permits of the further boiling of a liquid already partly evaporated in a former effect. The pressure in the tubes of any stage is always less than that in the surrounding shell; the pressure in the shell of any stage is equal to that in the tubes of the previous stage. 597. Theory. Let TFbe the weight of dry steam supplied; the heat which it gives up is WL. Let w be the weight of liquid enter- ing the first stage, H its heat, and h and I the heat of the liquid and latent heat corresponding to the pressure in the first-stage tubes. If x pounds of this liquid are evaporated in the first stage, the heat supplied is xl + w(li .fl), theoretically equal to WL m > whence x= \WL-wQi- ny\ -s- 1. Then x pounds of vapor enter the shell of the second stage, giving up the heat xl. The weight of liquid entering the tubes of the second stage is w x. Let the latent heat and heat of liquid at the pressure in the tubes of this stage be m and i: then the heat ab- sorbed, if y pounds be evaporated, is ym + (w oi)(i A), the last term being negative, since i is less than h. Tlien y = [xl (w #)0* /O] *" m " Consider now a third stage. The heat supplied may be taken at ym ; the heat utilized at zM+ (w x y)(J (z being the weight of liquid evaporated, AT its latent heat, and I the corresponding heat of the liquid), whence z = \_yrn (w x y)(I 01 "*" ^ The analysis may be extended to any number of stages. 598. Rate of Evaporation. Ordinarily, the evaporated liquid is an aqueous solution; the total evaporation per pound of steam supplied increases with the number of stages, being practically limited by the additional constructive expense 444 APPLIED THERMODYNAMICS and radiation loss. For a triple-effect evaporator, the total evaporation per W pounds of steam supplied is a? + y + a. Let W = 1, and let the steam be siipplied at atmospheric pressure, the vacuum at the condenser being 0.1 Ib. absolute, and the successive shell pressures 14 7, 8.1, 1.5. The pressures in the tubes are then 8.1, 1.5, and 1 : whence L= 970.4, /= 987.9, h = 151.3, m = 1027.8, *=81.9, 7=6.98, M = 1048.1. Let H be 100, the liquid being supplied at 132 F. A definite re- lation must exist between w and W, in order that the supply of vapor to the last effect, y t may be sufficient to produce evaporation, yet not so great as to burden the apparatus; this is to be detei mined by the degree of conceiitiation desired in any particular case, whence x + y + z = (/) w, in which (/) represents the proportion of liquid to be evaporated. Let (/) = 1.0, as is practically the case in the distillation of water; then w = x + y+z. We now have, x =0.982 - 0.0521 ?, y=O.S8 + 0.0211 w, a = 0.726 + 0.094 w, x + y + z = w = 2.588 4- 0.063 w, whence w = 2.76. This is equivalent to about 27.6 Ib. of water evaporated per pound of coal burned under the best conditions. By increasing the number of effects, evaporation rates up to 37 Ib. have been attained in the triple-effect machine. A sextuple-effect apparatus has given an evaporation of 45 Ib. of water per Ib. of combustible in the coal. 599. Efficiency. The heat expended in evaporation is in this case xl+ym+zM =3080 B. t. u. The heat supplied by the steam was WL = 970.4 B. t. u. The efficiency is, therefore, apparently 3 18, a result exceeding unity. A large amount of additional heat has, however, been furnished by the substance itself, which is delivered, not as a vapor, but as a liquid, at the condenser. 600. Water Supply. The condenser being supplied per pound of steam supplied to the first stage with v pounds of water, its heat increasing from n to N, the heat interchange is zM=v(tf-~n), whence, v=zM+ (#71), the liquid being discharged at the boiling point corresponding to the pressure in the condenser. In this case, for JT-n = 25, v = 40.2 Ib., or the water supply is 40.2 -*- 2,76 = 14.5 Ib. per pound of liquid evaporated. Some ex- cess is allowed in practice : the greater the number of effects, the less, gen- erally speaking, is the quantity of water required. 601. The Goss Evaporator. This is shown in Pig. 292. Steam enters the first stage F from the boiler G-, say at 194 Ib. pressure and 379 F. The liquid to be evaporated (water) here enters the last stage A, say at 62 F. 5 the boiling of the liquid in each successive stage from F to A produces steam which passes to the interior tube of the next succeeding stage, along with the water resulting: from condensation in the interior tube of the previous stage. The condensed steam from the first stage, is, how- ever, returned to the boiler, which thus operates like a house-heating boiler, with closed circulation. Let 1 Ib. of liquid be evaporated in F\ its pressure and temperature are so adjusted that, in this case, the whole temperature range between that of the steam (379 F.) and that of the liquid finally dis- charged from A (213 F.) is equally divided between the stages. The THE GOSS EVAPORATOR 445 446 APPLIED THERMODYNAMICS amount of vapor produced in any stage may then be computed from the heat supplied for the assigned temperature and corresponding pressure. Finally, in A, no evaporation occurs, the incoming liquid being merely heated; and it is found that the weights of discharged liquid and incoming liquid are equal, amounting each to 4.011 lb. The steam supplied by the boiler may be computed ; in F } we condense steam at 379 !F., at which its latent heat per pound is 845.8. It is assumed that 3 per cent of the heat supplied in each effect is lost by evaporation; the available heat in each pound of steam supplied is then 0.97 x 845.8 = 820.426. This heat is expended in evaporating 1 Ib. of water at 312.6 to dry steam at 345.8% requiring 1187.44 - 282.26 = 905.18 B. t. u., for which = 11 lb - of steam ai ' e 8 JO. 43 required. The whole evaporation for the six-effect apparatus is = 3.646 lb. per pound of steam. For the second effect, E, the heat supplied is LW 8 = 870.66, gross, or 0.97 x 870.66 = 844.54, net. The heat utilized is 1.873(282.22 -248.7) +(0.873 x 895.18) =844.54. In D, the heat supplied is 0.97 [(0.873 x 3126 ) + 1(316,98 - 282 22)] -= 790.8 ; that utilized is 2.633(248.7 -215.3) + (076x918.42) = 790.8. The heat interchange is perfect ; it should be noted that the liquid to be evaporated and the heat- ing medium are moving in opposite direction-) This involves the use of a greater amount of heating surface, but leads, '-o higher efficiency, than the customary arrangement. An estimated ecoi omy of 60 lb. of water per pound of coal is possible with seven stages (1). The Petleton evaporator, instead of reducing the pressure over the liquid to permit of easier vaporization, mechanically compressed the vapor previously removed and thus enabled it to further vaporize the remaining liquid. Steam was used to start the apparatus. The vapor generated was compressed by a separate pump to a higher pressure and temperature and was then passed back through a coil in contact with the residual liquid. Here it gave up its heat and was condensed and trapped off. Enough additional vapor was thus produced to maintain operation without the further supply of steam. With an efficient pump, the fuel consump- tion may be less than half that ordinarily reached in triple effect machines. FUSION 602. Change of Volume during Change of State. The foimuia, T dP was derived in Art. 368. The specific volume of a vapor below the criti- cal temperature exceeds that of the liquid from which it is produced; FUSION 447 dT consequently V v has in all cases a positive value, and hence must UiJL be positive; i.e. increase of pressure causes an increase in temperature. It is universally true that the boiling points of substances are increased by increase of pressure, and vice versa, at points below the critical tempers ture. If for any vapor we know a series of corresponding values of V> L, T, and v, we may at once find the rate of variation of temperature with pressure. 603. Fusion. The same expression holds for the change of state de- scribed as fusion ; the Carnot cycle, Pigs. 162, 1C3, may represent melting along ab, adiabatic expansion of the liquid along be, solidification along crt, and adiabatic compression of the solid to its melting point along da. In this case, V does not always exceed v ; it does for the majority of sub- stances, like wax, spermaceti, sulphur, stearine, and paraffin, which con- tract in freezing ; and for these, we may expect to find the melting point raised by the application of pressure. This has, in fact, been found to be the case in the experiments of Bunsen and Hopkins (2). On the other hand, those few substances, like ice, cast iron, and bismuth, which expand in freezing, should have their melting points lowered by pressure \ a result experimentally obtained, for ice, by Kelvin (3) and Moussou (4). The melting point of ice is lowered about 0.0135 F, for each atmosphere of pressure. The expansion of ice in freezing is of practical consequence. A familiar illustration is afforded by the bursting of water pipes in winter. 604. Comments. As the result of a number of experiments with non-metallic substances, Person (5) found the following empirical formula to hold : in which L is the latent heat of fusion, C, c are the specific heats in the liquid and solid states respectively, and T the Fahrenheit temperature of fusion. Another general formula is given for metals. A body may be reduced from the solid to the liquid state by solution. This operation is equivalent to that of fusion, but may occur over a wider range of temperatures, and is accompanied by the ab- sorption of a different quantity of heat. The applications of the fundamental formulas of thermodynamics to the phenomena of solution have been shown by Kirchofi (6). The temperature of fusion is that highest temperature at which the substance can exist in the solid state, under normal pressure. The latent heat of fusion of ice has a phenomenally high value. 448 APPLIED THERMODYNAMICS LIQUEFACTION OF GASES 605. Graphical Representation. In Fig. 293, let a represent the state of a superheated vapor. It may be reduced to saturation, and liquefied, either at constant pressure, along acd> the temperature being reduced, or at constant temperature along ale, the pressure being in- creased. After reaching the state of satura- tion, any diminution of volume at constant temperature, or any de- crease in temperature at FIG 293 Art. 606 -Lique- constant volume, must faction of Superheated , . . , , . Vapor. produce partial lique- faction. Constant tem- perature liquefaction is not applicable to gases having low critical temperatures. Thus, in Fig, 294,