# Full text of "Hydraulics"

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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/ HYDRAULICS BY F. C. LEA, B.Sa (London Engineering), 8SNI0B WmrWOBTH 8CH0LAB; ASSOC. B. COL. 80.; A. M.INST. 0. £. ; TELVORD PBIZBMAM; LBCTUBEB IK APPLIED MECHANICS AND ENOINEEBINO DESIGN m THE CITT AND GUILDS OF LONDON CBNTBAL TECHNICAL COLLEGE. LONDON EDWARD ARNOLD 41 & 43, MADDOX STREET, BOND STREET, W. 1908 [All Rights rexerrad] THE NEW YORK ASTOR, LffNOX AMD TIL^EN FOuS'OATlONa. J XJSTHEN the author undertook fioma time ago to writ^ thi IT worlE, it was under the iiopressionj which irapression wa shared by many teachers, that a book was required by Engineering etndents dealing with the subject of Hydraulics in a wider sense th^D that covered by existing tejtt books. In addition tho author ' ' ' ^ thoagh several excellent text books were in existence, ,r amount of experimental re&earch carried out during the last 10 or lt5 years, very little of which has been done in this eotantry, on the subject of the flow of water, had not received the atteBtiun it deserved. The great developments in turbines and ceotrifagal pumps also mt^'ited some notice* An attempt has been made to embody the results of the latest r€>9earcha^ in the book, and to give sufficient details to indicate the methods used in obtaining these resuhs, especially in those cables where such information and the references thereto, are hkely to prove of value to those desirous of carrying out ex- perimeTits on the flow of water. Perfiapa in no branch of Applied Science is it more difficult to co-ordinate results and express them by general formulae than in Hydraulics. Practical Engineers engaged in the design of WEter channels frequently complain of the large differences they obtain in the calculated dimensions of such channels by using thp ff»rmnlae put forward by different authorities. Before any can be used with assurance it is necessary to have son Mi-.i...4ge of the data used hi determining the empirical con^ wmniM in the formula. For this reason a little attention Jms been giv^n to the historical development of the formulae for determining the flow in pipes and channels, and some particulars of the data from which the constants were determined are given. In thia respect the loganthmic analysis of experimental data, especially in Chapter YI, tngetlier with the plottings of Fig. 114 and the references to experiments, will it is hoped be of assistance to IV PREFACE cngineerB in enabling them to choose the coefficients saitable to given circujiiatances, and it is further hoped that the methods of analysis given will be educational and useful to studentSj and h^jlpful ill the interpretation of experiments. The chapter on the flow of w^ater in pipes is an*anged so that a. student who reads as far as section 93 should be able to solve a large number of problems on flow of water in pipes, without further readings At the end of the chapter the formulae derived in the chapter are Bummarisedj and various kinds of practical problems solved, and arithmetical examples worked out. In the chapter on flow in channels the student who reads to section 119, and then sections 124 and 129 should be able to foHow the problems at the end of the chapter, and to work the examples. Chapter YIII enables the student who is desirous of studying the elementary theory of the impact of water on vanes, and of turbines, to do so apart from the details of turbines, and the more practical problems that arise in connection with their design. The principles of construction of the various types of turbines are illuBti-ated in Chapter IX by diagrams of the simpler and older types, as well as by drawings of the more complicated modern turbines. The dravdngs have been made to scale, and in particular cases sufficient dimensions are given to enable the student acquainted with the principles of machine design to design a turbine. The author believes the analysis given of the form of the vanes for mixed flow turbines and also for parallel flow turbines is new. The subject of centrifugal pumps is treated somewhat fully, because of the complaint the author has often heard of the difficulty engineers and students have in determining what the performance of a centrifugal pump is likely to be under varying conditions* The method of analysis of the losses at entrance and exit as given in the text, the author believes, is due to Professor Unwin, and he willingly acknowledges his obligation to him, Tlie general formula given in article 237 is believed to be new, and the examples given of its application in sections 235, etc,, shuw that by such an equation, w^hich may be called the character- istic equation for the pump, the performance of the pump under varying conditions can be approximately determined* The effects of inertia forces in plunger pumps and the effect of air vessels in diminishing these forces are only imperfectly treated, as no attempt is made to deal with the variations of pressure in the air vessel Sufficient attention is however given to the snbject to emphasise the importance of it, and it is probably tr^ted as fully as is desirable, considered from a practical PREFACE V engineeriiig standpoint. The analysis of section 260, although too refined for practical purposes, is of vahitj to the student in ihat, neglecting losses which cannot very well be determined, it enables htm to realise how the energy given as velocity head to the vrat*;r both in the cylinder and in the suction pipe is recovered before the end of the stroke is reached. The examples given of ** Hydraulic Machines " have been chosen as types* and no attempt himB been made to introduce veiy special kinds of machines* Tlie antbor has had a wide experience of this cla^ of machinery^ and lie think« the examples illustrate sufficiently the principles and prBCtice of the design of such machines. The last two chapters have been introduced in the hope that ibrnf will be of assistance to Umversity studentii^ and to candidates fofT the Institution of Civil Engineers examinations* if r Fronde's experiments, on the frictional resistance of boards \mg through water, are considered in Chapter XII simply in I ir relationship to the resistance of ships^ and no attempt has n made, as is frequently done, to use them to determine so- ciiilJed taw*s of fluid friction for water flowing in pipes and cbanneb. The author harfUy dares to hope that in the large amount of arithmetical work involved in the exercises given, mistakes will liMt have crept in^ and he ^vill be grateful if those discovering miMtake<t will kindly point them out. The author wishes to express his sincereat thanks to his friend, Mr W. A. Taylor, Wh,Sc., A.K.CS., for his kindness in reading proofs, and for many valuable suggestions, and also to Mr W, Hewsunj B.8c., who has kindly read through some of the proofs. Ty the following firms the author is ander great obligation for the ready way in which they acceded to Ins request for information; Messrs Kscher, Wj^ss and Co, of Zurich for drawings of torbtnes and for loan of block of turbine filteri Messrs Piccai*d, IHctet and Co. of Geneva for di^wings of turbinas* Messrs Worthington and Co* for drawings of centrifugal pctm^j^^ and fur loan of block. Meagre fielding and Piatt of Gloucester for drawings of BceixmvlBtoT. Messrs Tangye of Birmingham for drawings of pumps. &li*^rs Glen Held and Kennedy of Kilmarnock for drawijigs of tneter and for loan of blocks. Messrs G, W. Kent of London for description and loan of blockjt of Vt*niuri meter recording gear* 1 VI PREFACE Messrs W. and L. E. Gurley of Troy, N.Y., U.S.A. for loan of block of current meter. Messrs Holden and Brooke of Manchester for drawing of Leinert meter. Messrs W. H. Bailey and Co. of Manchester for drawing of hydraulic ram. Messrs Armstrong, Whitworth and Co. for drawings of crane valves. Messrs Davy of Sheffield for loan of block of forging press. F. C. LEA. Gbntral Tbchnical GOLLSaE, November, 1907. CONTENTS. CHAPTER I. FLUIDS AT REST. Introdnction. Fluids and their properties. Compressible and incom- pressible fluids. Density and specific gravity. Hydrostatics. Intensity of pressure. The pressure at a point in a fluid is the same in aU directions. The pressure on any horizontal plane in a fluid must be constant. Fluids at rest with free surface horizontal. Pressure measured in feet of water. Pressure head. Piezometer tubes. The barometer. The differential gauge. Transmission of fluid pressure. Total or whole pressure. Centre of pressure. Diagram of pressure on a plane area. Examples . Page 1 CHAPTER II. FLOATING BODIES. Conditions of equilibrium. Principle of Archimedes. Centre of buoyancy. Condition of stability of equilibrium. Small displacements. Metacentre. Stability of rectangular pontoon. Stability of floating vessel containing water. Stability of floating body wholly immersed in water. Floating docks. Stability of floating dock. Examples . . Page 21 CHAPTER III. FLUIDS IN MOTION. i^teady motion. Stream line motion. Definitions relating to flow of ^*ter. Energy per pound of water passing any section in a stream line. ^niottilli's theorem. Venturi meter. Steering of canal boats. Extension ^ Bernouilli^s theorem. Examples Page 37 Vlll CONTENTS CHAPTER IV. FLOW OF WATER THROUGH OEIFICES AND OVER WEIRS. Velocity of clischargc? frgni aii oriUce. Coefficient of contraction for sharp -edged orifice. Coefficient of Telocity for ehaj^- edged orifice. Bazin'st experiments on a sharp -edged orifice. Distribution of Telocity in tlie plane of the orifice. Pressure in the plajie of tlie orifice. Coefficient of discharge Efi^ect of fjiipprcBBed con traction on the coefficient of discharge. The form oi the jet from sharp-edged orifices. Large orifices. Drowned orifices. Partially drowned orifice. Velocity of approadx. Coefficient of resistance, Sndden enlargement of a current of water. Sudden contraction of a current of water. Loss of head due to sharp* edged entrance into a pipe or mouthpiece. MouUipieceB. Borda*8 mouthpiece. Conical mouthpieces and nozzleu. Flow througli orifices and mouth pieces under constant pressure. Time of emptying a tank or reservoir. Notches and weirs. Bectangulai* sharp -edged weir. Derivation of the weir formula from that of a large orifice* Thomson'^ principle of similarity* Discliarge tlirongh a trianglar notch by the principle of similarity. Diaeharge through a rectangular weir by the principle of similarity. Rectangular w^eir with end contractions, Bazin's formula for the discharge of a wx*ir. Bazin*s and tlio CorneO experiments on weirs. Velocity of approach. Influence ei the height of the weir sill above the beil of tlie stream on the contractioii, Disdiarge of a weir when ttie air is not fi*eely admitted beneath the nappe. Form of the nappe. Depressed nappe. Adhering nappes. Drovnied or wetted nappes. Instability of the form of the nappe. Drowned weirs with sharp crests* Yei'tical weirs of small thickness. Depressed and wetted nappes for flat-crested weirs. Drowned nappes for flat-crested weirs. Wide flat-created weirs. Flow over dams. Form of weir for accurate gau Boussinesq'a theory of the discharge over a weir* Determining by proximation the discliarge of a weir, when the velocity of approach 1?" unknown. Time required to lower the water in a reservoir a given dbtanoe by means of a weir. Examples ....... Page 60 CHAPTER V. FLOW THROUGH PIPEK. Resistances to the motion of a fluid in a pii>e. Loss of head by friction* Head lost at the entrance to tlie pipe. Hydrauhc gradient and virtual slope* Determination of the loss of head due to friction. Reynold'a apparatus. Equation of flow in a pipe of uniform diameter and determi- nation of the head lost due to friction. Hydrauhc mean dex>th. Empiriod CONTENTS IX LtUae for lo^ at head due to friction, Formola of Darcy, Variatioa ol C m the formula v — Cs^mi with service, Gangtdllet and Kutter*s foo^itilA. BeynoUrsi experiroents and the logarithmic formula. Critical f«locit>% Critical yelocity by the method of colour baade. Law of frictional resistance for velocities above tlie critical velocity* The de* terminAlidii ol the values of C given in Table XII. Variation of k, in the lulft i = It*, with Hm diameter, Oriiieifiin of experiments. Piezometer fittings. Effect of temperature on the velocity of flow, Lo«s of head due to bands and elbows. Variations of the velocity at the cross section of a liadrical pipe. Head necessary to give the meaa velocity v^ to the in Uie pipe. Practical problcniB. Velocity of flow in pipeB, Trans- snteioil of power along pipes by hydraulic pressure. The hmiting diameter «l cwai iitm pipes. Pressures on pii>e bends. Pressure on a plate in a pipe filled With flowing water. Pressure on a cylinder. Examples , Page 112 ^Bfosmi ^■fittJl! CHAPTER VI. FLOW IN OPEN CHANNELS. Vamly of the forma of channels. Steady motion in uniform channeli* ^ for the flow when the motion is uniform in a channel of uniform land slope. Formula of Chezy, Fonnulae of Prony and Eytelwein. i1a €d Darcy and Bajdn, GangniUet and K utter *8 formula. Bazin's bi. Variations of the coefficient C. Logarithmic formula for flow In Approximate formula for the flow in earth channels. Diwtribu- L of velocity in tiie cross section of open ohannek, Fomi of the curve I velocitiets on a vertical section. The dopes of channels and the velocities rallown] in them. Sections of a(|uo<liictft and sewerH. Siphons forming pan of a^edncte. The best form of channel. Depth of flow in a circular diasne] for maximum velocity and maximum chscharge. Curves of velocity ttud discharge for a channel. Apphcations of the formulae. Problems. Dpleci ,...-, Page 178 CHAPTER VIL QAtrOiNG THE FLOW OF WATER [ the flow of water by weighing. Meters. Measuring the flow of an orifice. JleaJHiiring the flow in oi^en channels. Surface Doable floats. Bod floats). The current meter, Pitot tube. Cali^ bvmtiim of Pitot tubes. Gauging by a weir. The hook gauge. Oaujring |h# (low in pipes ; Venturi meter. Deacon 'u waste -water meter, Kennedy's Oanging the flow of streams by chemical means. Examples Page 2H CONTENTS CHAPTER YllL IMPACT OF WATER OK VAlflTS. Deflnitioii of ?ector. Sum of two vectors. Resultant of two Telocitiee, Difference of two sectors. Impulse of water on vanes* Relative velocity* Befinition of relative velocity as a vector. To find the presstire on a moving vane, and the rate of doing work. Impact of water on a vane when the directions of motion of the vane and jet are not paralleL Conditions which the vanes of hydra viltc machines should fiatisfy* De&nition of angular m omen torn. Change of angnl&r moment nm. Two important principles. Work tlone on a series of vanea filled to a wheel expressed in terms of the velocities of whirl of the water entering and leaving the wheel. Carved vanes. Pel ton whecL Force tending to move a vessel from which water is issuing through an orifice. The propulsion of ships by water jets. Examples . , « . > . Pa^fe 261 CHAPTEE IX- WATER WHEELS AND TURBINES. Overehot water wheeb, Brea4^t wheel Sagehien wheels. Impulse wheelB. Poncelet wheel. Turbines. Reaction turbines. Outward flow turbines. Losses of head due to frictional and other resistances in outward flow turbines. Some actual outward flow turbines. Inward flow turbiueo. Some actual inwarrl flow turbines. The best peripheral velocity for inward and outward flow ttirbines, Exi^erimental determination of the beat peripheral vekjcity for inward and outward flow turbines. Value of e Yv to be used in the formula — =^H. The ratio of the velocity of wliirl Y to the velocity of the inlet periphery t\ The velocity with which water leaves a turbine. Bernouilli's equations for inward and outward flow turbinei neglecting friction. Bemouilli*s eixuations for the inward and outward flow turbinea including friction. Turbine to develope a given horse -power. Parallel or axial flow turbinea. Regulation of the flow to parallei flow turbines, Bernouilli's equations for axial flow turbii Mixed flow turbinea. Cone turbine. Effect of changing the direction the guide blade > when altering the flow of inward flow and mixed flow turbines. Effect of diminishing the flow thi^ough turbinea on the velocity of exit. Kegulatiou of the flow by means ^ cylindrical gates. The Bwain ptte. The form of the wheel vanes between the inlet and outlet of turbines* The limiting head for a single stage reaction tnrbine. Series or multiple stage reaction turbines. Impulse turbines. The form of tlie vanes for impulse turbines, neglecting ifriction. Triangles of velocity for an axial flow impulse turbine considering friction. Impulse turbine lor Mgh head, Pelton wheel. Oil pressure governor or regulator. Water pressure regulators for impulse turbines. Hammer blow in a long turbine supply pipe. Examples *,,.,,.< Page 283 J CONTENTS CHAPTER X, PUMPS. Cesitrifiig&l &nd tor&me pimips. Starting centrifugal or torbme pnmps. I of the v^nes of centrifugal pumps. Work done on the water by the ^ifbeel* lUMo ol velocity of whirl to peripheral velocity. The kinetic energy of ihm water &t exit from the wbeeL Gross lift of a centrifugal punip^ £fficsietiides of a centrifugEl pump. Experimental deteimlnatioti of the dficieocy of a centrifiigal pump. Desi^ of pump to give a discharge Q, Tlie centiiftigal head impressed on the water by the wheel. Head -velocity citiiFe ef & centrifugal pump at zero discbarge. Variation of the discbarge of & oenfcrifngal pomp with tlio head when the speed ia kept constant* BemomUi's equations applied to centrifugal pumps. Ltoeses in centrifugal Variation of the bead with diHcbarge and with the speed of a I pump. The effect of the variation of the centrifugal head and the low hf friction on the discharge of a pump. The c^ect of the diminu- ol ibe oentiifugal head and tlie increase of the friction head as the ft, (m the velocity. Di^targe curve at constant head» Special ftnmiigeniOBta for converting tbe velocity bead — , with which the water laftves ttie wheel., into pressure bead. Tm-bino pumps. Losses in tbe flpltml aiam^ of centrifugal pumps* General equation for a oentrifngal |»r^| i- The T jpii t.ing height to which a single wheel centrifugal pump caa ] lie laed to raiae water. The suctioa of a centrifugal pump. Series or mulsi-ti&age ttirfaine pumps^ Advanta^^es of ceotrifngal pumps. Pump «Urfwiig into a long pipe line. Parallel flow turbine pump. Inward flow tBrMoe pnmp* Reciprocating pumps- Coeflicient of discharge of the {mmp. Slip^ Diagram of work done by tbe pump* The accelerations ol the pomp plunger and tbe water in tlie suction pipe. The effect of ' oioeelonition of tbe plonger on tbe pressure in Uie cylinder during the sncttoii steoike. Accelerating forces in the delivery pipe. Variation of IHiiiiiirr in tbe eylinder due to friction. Air vessel on the suction pipe. Air TCBBcl on the delivery pipe. Separation daring the suction stroke. K^sliTe slip, Heimration in tbe delivery pipe, Diagram of work done i ooiyddering tbe variable qnantity of water in the cylinder. Head lost at tbe ww^&OMk talve. Variation of tbe pressure in bydrauhc motors due to iaciriia incces. Worked examples. High pressure plunger pump, Tangye Oeplex fiiisi|i. Tbe hydraulic ram. Lifting water by comprised air. Kxaifitiipa Page 392 CHAPTER XI, HYDRAULIC MACHI!?ES. Jomta tmd packings used in hydraulic work. The acciimulator. Dif- feffpiiftl oocamnk.tor. Air accumulator. Intenaifiers. Steam intensiiiera., HydeMiUo forging preis. Hydraulic cranes. Double power crane Bjdrsnlie crane valveSi Hydraubc press. Hydraulic riveter. Brother* hood Odd Bigg hydraulic engines. Examples . , . . Page 4B^ Xll CONTENTS CHAPTER XII. RESISTANCE TO THE MOTION OF BODIES IN WATER. Froude*8 experiments on the resistance of thin boards. Stream line theory of the resistance offered to motion of bodies in water. Determination of the resistance of a ship from that of the model. Examples . Page 507 CHAPTER XIII. STREAM LINE MOTION. Hele Shaw's experiments. Curved stream line motion. Scouring of river banks at bends ...•••... Page 517 Answers to Examples • . Page 521 Index Page 525 HYDEAULICS. CHAPTER I. FLUIDS AT REST. 1. IntrodnctioiL The science of Hydraulics in its limited sense as originally understood^ had for its object the consideration of the laws regulating the flow of water in channels, but it has come to have a wider significance, and it now embraces, in addition, the study of the principles involved in the pumping of water and other fluids and their application to the working of different kinds of machines. The practice of conveying water along artificially constructed channels for irrigation and domestic purposes dates back into it antiquity. The Egyptians constructed transit canals for purposes, as early as 3000 B.C., and works for the better of the waters of the Nile were carried out at an even According to Josephus, the gardens of Solomon |utiful by fountains and other water works. Tlie some of which were constructed more than among the " wonders of the world," and \ Athens is partially supplied with water by bt constructed probably some centuries before ydraulics, however, may be said to have oi ce at the end of the seventeenth century wh osophers was drawn to the problems involv of the fountains, which came into considerable idscape gardens, and which, according t great beauty and refreshment." The fou' Torricelli and Marriott from the exper' from flie theoretical, side. TThe experip of Marriott to determine the discharge -^ ^ ™* fees in the sides of tanks and through shr ... , ° / cubic inches. L. B. The Aqueducts of Rome. FroDtinus, translajf 85*9 cubic feet. 1—2 2 HYDRAUUGS mark the first attempts to determine the laws regulating the flow of water, and Torricelli's famous theorem may be said to be the foundation of modem Hydraulics. But, as shown in the chapter on the flow of water in pipes, it was not until a century later that any serious attempt was made to give expression to the laws regulating the flow in long pipes and channels, and firacti- cally the whole of the knowledge we now possess has been acquired during the last century. Simple machines for the utilisation of the power of natural streams have been made for many centuries, examples of which are to be found in an interest- ing work Hydrostatika and Hydrauliks written in English by Stephen Swetzer in 1729, but it has been reserved to the workers of the nineteenth century to develop(» all kinds of hydraulic machinery, and to discover the principles involved in their correct design. Poncelet's enunciation of the correct principles which should regulate the design of the "floats" or buckets of wafer wheels, and Foumeyron's application of the triangle of velocities to the design of turbines, marked a distinct advance, but it must be admitted that the enormous development of this class of macliinery, and the very high standard of efficiency obtained, is the outcome, not of theoretical deductions, but of experience, and the careful, scientific interpretation of the results of experiments. 2. Fluids and their properties. The name fluid is given, in general, to a body which offers very small resistance to deformation, and which takes the shape of the body with which it is in contact. If a solid body rests upon a horizontal plane, a force is required to move the body over the plane, or to overcome the friction between the body and the plane. If the plane is very smooth the force may be veiy small, and if we conceive the plane to be perfectly smooth the smallest imaginable force would move the body. If in a fluid, a horizontal plane be imagined separating the '^lid into two parts, the force necessary to cause the upper ' slide over the lower will be very small indeed, and however small, applied to the fluid above the plane ' to it, Avill cause motion, or in other words will cause of the fluid. a very thin plate be immersed in the fluid in any te can be made to separate the fluid into two '^tion to the plate of an infinitesimal force, !t fluid this force would be zero. FLUIM AT REST 3 luide found in nature are not perfect and are cosity- but when they are at rest the conditions equilibritiin can be obtained, with sufficient accuracy, on aasumption that they are perfect fluids, and that therefore ^tangential stresses can exist along any plane in a fluid. Its branch of the study of fluids is called Hydrostatics; when laws of movement of fluids are consideredj as in Hydraulics, 5)8 tangential, or firictional forces have to be taken into ideration. 1 Compressible and inooDELpresslble fluidB. There are two kinds of fluids, gases and liquids, or those which are easily compressed, and those which are compressed with * "" * V, The amount by which the volumes of the latter are ftir a very large variation in the pressure is so small that practical problems this variation is entirely neglected, and bey are therefore considered as incompressible fluids. In this volume only incorapreasible fluids are considered, and . Attention is eonflned, almost entirely, to the one fluid, water. ^H 4* Density and specific gravity. The deusity uf any substance is the weight of unit volume at he standard temperature and pressure. f specific gravity of any substance at any temperature and Bnpe id the ratio of the weight of unit volume to the weight tmit volume of pure water at the standard temperature and ttxre. The variation of the volume of liquid fluids, with the pressure, as eluted above, is negligible, and the variation due to changes of tempfn^lnrc, such as are ordinarily met with, is so small, that in practical problems it is unnecessary to take it into account. In tht? case of water, the presence of salta in solution is of ,ter importance in determining the density than variation temperature, as will be seen by comparing the densities of sea Vator and pure water given in the foUowing table. ■f t€ TABLE I. Useful data. One cnbio foot of water at 391' F. weighs 62-425 lbs. 60* F. „ 62-86 „ One oablo foot of average sea water at 60° F. weighs 64 Ibe* 0De gallon of pure water at 60* F, weighs 10 lbs. One gallon of ptue water bas a volome of 277*25 cubic mches* One kai ol pure water at 60" F. has a volume of 85*9 cubic feet. 1—2 HTDRA.ULICB Table of dendtiea of pure water. Temperatnn ^rees Fahxenheii Demdty 82 -09967 89*1 1-000000 M) OiW978 60 0-99906 80 0-99664 104 0il9288 From the above it will be seen that in practical {irobleiiis it will be KufHciently near to take the weight of one cubic foot of fresh water as 62*4 lbs., one gallon as 10 ponnds, 6124 gallons in a cubic foot, and one cubic foot of sea water as 64 pounds. 5. Hydrostatics. A knowledge of the principles of hydrostatics is very helpful in approaching the subject of hydraulics, and in the wider sense in which the latter word is now used it may be said to include the former. It is, therefore, advisable to consider the laws of fluids at rest. There are two cases to consider. First, fluids at rest under the action of gravity, and second, those cases in which the fluids are at rest, or are moving very slowly, and are contained in closed vessels in which pressures of any magnitude act upon the fluid, as, for instance, in hydraulic lifts and presses. 6. Intensity of pressure. The intensity of pressure at any point in a fluid is the pressure exerted upon unit area, if the pressure on the unit area is uniform and is exerted at the same rate as at the point. Consider any little element of area a, about a point in the fluid, and upon which the pressure is uniform. If P is the total pressure on a, the Intensity of Pressure p, is then P or when P and a are indefinitely diminished, ^ap ^ da' 7. The pressure at any point in a fluid is the same in all directions. It has been stated above that when a fluid is at rest its resist- ance to lateral deformation is practically zero and that on any plane in the fluid tangential stresses cannot exist. From this experimental fact it follows that the pressure at any point in the fluid is the same in all directions. FLUIDS AT BEST Consider a smaU wedge ABC, Pig, 1^ floating immersed in a II aid at rest. Sine© there caimot be a tangential reag on any of the planes AB, BC, or AC, tf prussurad on them must be normaL Lei p, pi and ps be the intenaitiea of tires on these planes respectively, H© weight of the wedge will be very fftnal] and may he neglected* An thtt wedge m in equihbriiim under the forces acting on three faces, the resolved components of the force acting on ;C in the directions of p and pi mnst balance the forces acting AB and BC respectively* Therefore p, . AC cos - p , AB, d psACsin^^PaBC, But AB = ACcose, And BC^ACsin^* Thert*fore p = pj = p, , 8. The pressure on any horizontal plane in a fluid must eonfilant. Confiider a small cylinder of a fluid joining any two points A nd B OQ the same horissontal plane in the fluid. Since there can be no tangential forces acting on the cylinder purallel to the axiB^ the cylinder must be in equilibrium under the pressures on the ends A and B of the cylinder, and since these re of equal area, the pressure must be the same at each end of he cylinder* 9. Fluids at rest, with the free surface horizontal, 'Hie pres.sure per unit area at any depth h below the free of a finid due to the weight of the fluid is equal to the |b.t of a column of fluid of height k and of unit sectional area. Let the pressure per unit area acting on the surface of the Sold be p lbs. If the fluid is in a closed vessel, the pressure p may^ bave any assigned value, but if the free surface is exposed to tin aCiQO&phere, p will be the atmospheric pressure. klf a small open tube AB, of length h^ and cross sectional area a, placed in the fluid, the weight per unit volume of which is Ib&, with its axis vertical, and its uppor end A coincident with 9 0tif£ftC0 of the fluid, the weight of fluid in the cylinder uiUBt be ip.apA Ibfls. The, pressure acting on the end A of the colunm is pa Ib^. a HTDRAUUCS Since there cannot be any force acting on the colnmn parallel to the eddes of the tube, the force of imiA lbs. + pa lbs. most be kept in equilibrium by the pressure of the external fluid acting on the fluid in the cylinder at the end B. The pressure per unit area at B, therefore, = ^ = (wh + p) lbs. The pressure per unit area, therefore, due to the weight of the fluid only is tch lbs. In the case of water, w may be taken as 62*40 lbs. per cubic foot and the pressure per sq. foot at a depth of h feet is, therefore, 62*40% lbs., and per sq. inch '433b lbs. It should be noted that the pressure is independent of the form of the vessel, and simply depends upon the vertical depth of the point considered below the surface of the fluid. This can be illustrated by the different vessels shown in Fig. 2. If these were all connected together by means of a pipe, the fluid when at rest would stand at the same level in all of them, and on any horizontal plane AB the pressure would be the same. -T-UJ- .^/ Pressure an the Pixuie A^^w-h lbs persq Foot, Fig. 2. If now the various vessels were sealed from each other by closing suitable valves, and the pipe taken away without disturbing the level CD in any case, the intensity of pressure on AB would remain unaltered, and would be, in all cases, equal to wh. Example, In a condenser containing air and water, the pressure of the air is 2 lbs. per sq. inch absolute. Find the pressure per sq. foot at a point 8 feet below the free surface of the water. I> = 2xl44 + dx62*4 =475*2 lbs. per sq. foot. FLUIDS AT REST 10. PreiBiires meaBiired in feet of water. Fressure head. It is oonvenient in hydrostatics and hydraulics to express the inreesore at any point in a fluid in feet of the fluid instead of pounds per sq. foot or sq. inch. It follows from the previous section that if the pressure per sq. foot is p lbs. the equivalent pressure in feet of water, or the pressure head, is A = ^ ft. and for any other fluid w having a specific gravity p, the pressure per sq. foot for a head h of the fluid is p = w.p.hy or A = — . top IL Piesometer tubes. The pressure in a pipe or other vessel can conveniently be measured by fixing a tube in the pipe and noting the height to which the water rises in the tube. Such a tube is called a pressure, or piezometer, tube. The tube need not be made straight but may be bent into any form and carried, within reasonable limits, any distance horizon- tally. The vertical rise h of the water will be always where p is the pressure per sq. foot in the pipe. If instead of water, a liquid of specific gravity p is used the height h to which the liquid will rise in the tube is w .p Example. A tube having one end open to the atmosphere is fitted into a pipe containing water at a pressure of 10 lbs. per sq. inch above the atmosphere. Find the height to which the water will rise in the tube. The ^*ater will rise to such a height that the pressure at the end of the tabc in the pipe due to the column of water will be 10 lbs. per sq. inch. Therefore ^ 10x144 ^„^^ ^ , h= =23-08 feet. 12. The barometer. The method of determining the atmospheric pressure by means of the barometer can now be understood. If a tube about 3 feet long closed at one end be completely filled with mercury, Fig. 3, and then turned into a vertical position with its open end in a vessel containing mercury, the liquid in the tube falls until the length h of the column is about 30 inches above the surface of the mercury in the vessel. Fig. 3. HYDRAULICS Since the pressure p on the top of the mercury is now zero, iYm-^ pressure per unit area acting on the section of the tube, level witX^ the surface of the mercury in the vessel, must be equal to ih^ weight of a column of mercury of height h. The specific gravity of the mercury is 13*596 at the standard temperature and pressure, and therefore the atmospheric pressure per sq. inch, pa, is, 80" X 13-596 X 62-4 -,.^„ . , Pa= ^2 X 144 ~ ' • P®^ ^* "^ Expressed in feet of water, 14*7 X 141 A = -i> I— = 33-92 feet. 62 4 This is so near to 34 feet that for the standard atmospheric pressure this value will be taken throughout this book. A similar tube can be conveniently used for measoring low pressures, lighter liquids being used when a more sensitive gauge is required. 13. The differential gauge. A more convenient arrangement for measuring pressures^ and one of considerable utility in many hydraulic exx>eriment8, iB known as the differential gauge. Let ABCD, Fig. 4, be a simple U tube containing in the lower part some fluid of known density. If the two limbs of the tube are open to the atmosphere the two surfaces of the fluid will bo in the same horizontal plane. If, however, into the limbs of the tube a lighter fluid, which does not mix with the lower fluid, be poured until it rises to C in one tube and to D in the other, the two surfaces of the lower fluid ^vill now be at different levels. Let B and E be the common surfaces of the two fluids, h being their difference of level, and hi and th the heights of the free surfaces of the lighter fluid above E and B respectively. Let p be the pressure of the atmosphere per unit area, and d and di the densities of the lower and upper fluids respectively. Then, since upon the horizontal plane AB the fluid pressure must be constant, p + rfjfes = p + djii + dh, d\ (Jh — hi) = dh. Fig. 4. FLUIDS AT REST 9 li now^ instead of the two limbs of the U tube being open to »tmcMipha?e, they are ooimected by tubes to closed vessels in ich the pressures are pi and p^ pounds per sq* foot respectively^ d ^1 and fh are the vertical lengths of the colnnms of fluid above ind B respectively^ then Pt^di^kt =Pi + di^hi + d.hf p^- Pi^d,k-di{fh-hi). kik application of snch a tube to determine the difference of fmuTe at two pointa in a pipe containing flowing water is shown I % 88, page 116. Fluidk generally used in such U ttihee. In hydraulic experiments f Bpper part of the tube is filled with water, and therefore the lid b the lower part must have a greater density than water. fWthe difference of pressure is fairly large, mercury is generally tli8$peci£c gravity of which ia 13'596. When the difference ' pemife is small, the height k is difficult to measure iv^th m that| if this form of gauge is to be used, it is desirable i»pli«e the mercury by a lighter liquid. Carbon bisulphide s Ijeen used but its action is sluggish and the meniscus between and the water is not always well defined, jth)-bi*imne ^v^ good results^ its prin- '1*1 fault being that the falling meniscus ^ DOt very quickly assume a definite ape. Hf inverted air gauge, A mure sen- ^*^ gauge can be made by inverting a ^ and enclosing in the upper part **rt&iii quantity of air as in the tube Be, Fig. 5. I^ the pressure at D in the limb DF Pi poimds per square foot, equivalent ;*liead hi of the fluid in the lower part ^ gauge, and at A in the limb AK let * Iffmure be p%^ equivalent to a head h%. W A bt* the difference of level of G and C* Tfien if CGH contains air^ and the weight of the air be [beted, being very small, the pressure at C muBt equal the at G I and since in a fluid the pressure on any horizontal constant the pressure at C is equal to the pressure at D, the pressure at A equal to the pressure at B. Again the at G is equal to the pressure at K. Th^tiefore ht-h^ht^ Pf-pt-pw.h, H ».|^ iBlCJ ^ F u Fig. &. v.i uy using, in tli« upper part of tlie tube, an oil liglitei than water instead of air, as sliowii in Fig. 6. Let pi and pj be the pressures in the two limbs of the tube on a given horizontal plane AB, hi and }h being the equivalent heads of water. The oil in the bent tube will then take up some such position as shown, the plane AD being supposed to coincide with the lower surface C. Then^ since upon any horizontal plane in a homogeneous fluid the pressure must be constant, the pres- ■es at G and H are equal and also >se at D and C. Let pi be the specific gravity of ) water, and p of the oil. m Pihi-ph = Pi(hi'-h), i irefore fc (pi-p) =Pi (^-fei) (Pi - p) )stituting for hi and h^ the values FLUIDS AT EEST thi^t either kerosene, gasoline, or sperm oil gave excellent results, h\x% flperm oil waa too sluggish in its action for rapid work. Kei?«fieiia gave the be^t results. nkmperaiure coeffictent of the inverhd oil gauge, Dnhke the inverted far gaog^ the oil gauge has a considerable temperature oc^^rfficietit, aa will be seen from the table of specific gravities at viLriotis t^'Uipera tares of water and the kerosene and gasoline used br >Villiams, HubbeU and Fenkell In this Cable the specific gravity of water is taken as unity Wat^ T 40 ! 60 1-00092 100 EefQseiie I'OOOO *9041 40 60 100 ■7955 *7879 '7725 Gft«oliii6 40 60 80 *72147 71587 '70547 jf^^ caltbrafwn of ike hiverted oil gauge. Messrs Williams, i^ >-il>beU and Fenkell have adopted an ingenious method of <^^lihniting the oil gauge. This will readily be understood on '**f»PBiioe to Fig. 6- *The difference of level of E and F clearly gives the difference ^ liand acting on the plane AD in feet of water, and this from <^^^tion (1) equals M?l:ip) , Pi "VFatar is put into AE and FD so that the surfaces E and F *''^*^ <m the same levels the common surfaces of the oil and the *^^ter abo being on the 'same level, this level being aiero for the ^^^ Water is then run out of FD until the surface F is f^^djy I inch below E and a reading for h taken. The surface F ^ ^^in lowered 1 inch and a reading of h taken* This process * csontinued tmtfl F is lowered as far as convenient, and then irt^ wati*r in EA is drawn out in a similar manner. When E ^^itrl p are again level the oil in the gauge should read zero. 14. Tranamission of fluid pressure. If an external pressure be applied at any point in a fluid, it ia ^^nsmifcted equally in all direc- ^u^B through the whole mass. T^UJ is proved experimentally h means of a simple apparatus ^ch aa shown in Fig. 7, I^ lire P is exerted upon >^' , on Q of a sq- inches Fig. 7- 12 HYDRAULICS j P area, the pressure per unit area p = — , and the piston at B on the same level as Q, which has an area A, can be made to lift a load W P equal to A — ; or the pressure per sq. inch at R is equal to the pressure at Q. The piston at R is assumed to be on the same level as Q so as to eliminate the consideration of the small differences of pressure due to the weight of the fluid. If a pressure gauge is fitted on the connecting pipe at any point, and p is so large that the pressure due to the weight of the fluid may be neglected, it will be found that the intensity of pressure is p. This result could have been anticipated from that of section 8. Upon this simple principle depends the fact that enormous forces can be exerted by means of hydraulic pressure. If the piston at Q is of small area, while that at R is large, then, since the pressure per sq. inch is constant throughout the fluid, W__A P "a' or a very large force W can be overcome by the application of a small force P. A very large mechanical advantage is thus obtained. It should be clearly understood that the rate of doing work at W, neglecting any losses, is equal to that at P, the distance moved through by W being to that moved through by P in the ratio of P to W, or in the ratio of a to A. Example, A pomp ram has a stroke of 3 inches and a diameter of 1 inch. The pump supplies water to a lift which has a ram of 5 inches diameter. The force driving the pomp ram is 1500 lbs. Neglecting all losses dne to friction etc., determine the weight lifted, the work done in raising it 5 feet, and the nnmber of strokes made by the pomp while raising the weight. Area of the pump ram ='7854 sq. inch. Area of the lift ram = 19*6 sq. inches. Therefore W = 1?1|J^= 37,500 lbs. Work done = 37,600 x 5 = 187,500 ft. lbs. Let N equal the number of strokes of the pump ram. Then N x A x 1500 lbs. = 187,600 ft. lbs. N= 600 strokes. or ^ or whole pressure. From (2) it leessure acting on a surface is the sum of all the different from thacting on the surface. K the surface is plane all differences of presiLel, and the whole pressure is the sum of these •Frou PLUIJDS AT REST 13 Ijet any sarfece, which need not be a planej be immerBed a finid. Let A be the area of the wetted aiirface, and k the head at the centre of gravity of the area. If the area is inLHiersed in a fluid the presisure on the surface of which m zero, tiie free snrface of the fluid will be at a height h above the centre of gravity of the area* In the case of the area being immeraed in a flnidf the surface of which is exposed to a pressure p, and below vliick the depth of the centre of gravity of the area is ih^ then w If the area exposed to the fluid pressure is one face of a body^ the opposite faee of which m tsxposed to the atmospheric pT*e&aure, in the case of the side of a tank containing water, or the dam of Fig, 14, or a valve closing the end of a pipe as Fig. 8| the pressure due to the itmosphere is the same on the two ■fec^ts and therefore may be neglected. Let IP be the weight of a cubic foot of the fluid. Then, the whole preesure on the area is If the surface is in a horizontal plane the theorem is obviously true, fliDce the intensity of pressure is eon- slant and equals w , K In general, imagine the surface. Fig- % divided into a large number of small areas a> ai, o^ ... * Let « be the depth below the free surface FS^ of any element of area a ; the pressure on this element = tc . a; , a. Tlie whole pressure P ^ Sw, ac * a. But tr is constant, and the sum of the moments of the elements [of the area about any axis equals the moment of the whole area* fal>ont the same axis, therefore 5iE , fi = A . ftj juid P = W7 . A . A. 16* Centre of pressure. The centre of pressurt? of any plane snrfo^re acted upon by a fluid is the point of action of the resultant pressure acting upon the surface. J^^ of ihs centre of presswre. Let DBC, Fig. 9, be any plane surface exposed to fluid pressure. * Em %eit-book8 on Me^haoiei. Fig. 8. B( ffi C Fig. 9. 14 HYDRAULICS Let A be the area, and h the pressure head at the centre of gravity of the surface, or if FS is the free surface of the fluid, h is the depth below FS of the centre of gravity. Then, the whole pressure F = w.A.h. Let X be the depth of the centre of pressure. Imagine the surface, as before, divided into a number of small areas a, ai, Os, ... etc. The pressure on any element a = w .a.x, and P = ^wax. Taking moments about FS, P.X= (wax* + waia>i*+ ...) = Siooa;', or X = wAh Ah ' When the area is in a vertical plane, which intersects the surface of the water in FS, Soa?* is the "second moment" of the area about the axis FS, or what is sometimes called the moment of inertia of the area about this axis. Therefore, the depth of the centre of pressure of a vertical area below the free surface of the fluid moment of inertia of the area about an axis in its own plane _ ' and in the free surface ~" area x the depth of the centre of gravity ' or, if I is the moment of inertia, Areao A.h ^^ t of Inertia about any axis. Calling I© the Moment Thereforebout an axis through the centre of gravity, and I the _ , - nertia about any axis parallel to the axis through the Work done .. , . ,. . i. jl. -.l » ^, , xty and at a distance h from it. Let N equal ti*^ t t a 7 • Then I = Io^AA«. *ea is a rectangle breadth 6 and depth d. P=ir.6.d./i, or ' or 6^ From(2)itieess. "12+^^'*' different from thacti. ^+wfc« differences of presdel, X=- Proc. hdh FLUIDS AT REST 16 If the free sorfiuie of the water is level with the upper edge of the rectangle, ,= |, mod X = |.^ 2' is a drole of radios B. X= ^+xR.*. TSFT If the top of the circle is jast in the free sorfaoe or A=B, X = 1R. TABLE n. Table of Moments of Inertia of areas. Form of area Moment of inertia abont an axle AB through the C. of G. of the section Rectangle JlJ A^' Triangle k'''' Circle 1©° 64 Semicircle rj^ About the axis AB 8 Paarabola i > 16 HYDRAUUCS 17. Diagram of pressure on a plane area. If a diagram be drawn showing the intensity of pressure on a plane area at any depth, the whole pressure is equal to the volume of the solid thus formed, and the centre of pressure of the area is found by drawing a line through the centre of gravity of this solid perpendicular to the area. For a rectangular area ABCD, having the side AB in the surface of the water, the diagram of pressure is AEFCB, Fig. 10. The volume of AEFCB is the whole pressure and equals ^bcPw, b being the width and d the depth of the area. Since the rectangle is of constant width, the diagram of pressure may be represented by the triangle BCF, Fig. 11, the resultant pressure acting through its centre of gravity, and therefore at id from the surface. ^''' ""-s. 12 and 13. h g f^ ow cucL. Fig. 12. For a vertical circle the diagram of pressure is as shown in The intensity of pressure ah on any strip at a o» oovii }i^ ig ^h^^ The whole pressure is the volume of the truncated ^Sk'er efkh and the centre of pressure is found by drawing a 'T)endicular to the circle, through the centre of gravity ^^^<uncated cylinder. Work done Let N eqaal-i Then or ' Oi From (2) it lee^o) different from thaci differences of presde. ♦ IVoc Pig. 18. ^, FLUIDS AT REST It Another, and fretiuently a very convenient method of dpfcer- tniniiig^ the depth of the centre of pressure, when the whole of the «x*ea is itt some dietanco below the surface of the water, is to eoiiisider the pressure on the area as made up of a uniform pressure 1 over the whole surface, and ii pressure of variable intensity. Take again, as an example^ the vertical circle the diagrams of pressure for which are shown in Figs. 12 and 13. At any depth h the intensity of pressure on the strip ad is wh„ - wHa ^ y^fh ' The pressure on any strip ad is^ therefor*?, made up of a 4' ' presssure per unit area wtiA and a variable pressure wJh ; ii : ^vhole pressure is equal to tlie volume of the cylinder efghj tSg. 12, together with the circular wedge fkg. The wedge fkg is equal to the whole pressure on a vertical circle, the tangent k> which is in the free surface of the water and equals *r * A . ^ , and the centre of gravity of this wedge will he at the same vertical distance from the centre of the circle as the centre of pressure when the circle touches the surface* The whole pressure P umy be supposed therefore to be the resultant of two pttndlel forces Pi and Pa acting through the centres of gravity of the cylinder efght and of the circular wedge fkg respectively, the ma^tudes of P^ and Pj being the volumes of the cylinder and the wedge respectively* Tu find the centre of pressure on the circle AB it is only iieoeeeary to find the resultant of two parallel forces Pi^A.whj, and P^^w.Al^jt <jt irhicli P, act** at the centra c, and P, at a point c-t which is at a distance from A of | r. £iai^. A tsJMonry dam, Fig* 1-1, jM«i hetirtit of 80 fe*t tmm the founda- Ml aa4 tbe »&ter tmi^ is ineliJaed At *^^K^tm 10 the vcTiical ; find the whole f^*o»* m the fkce tlue to the water per •*^»iiilii ijf the dftm^ «nd the oehtr^ of y*^*Oft, «h«i3 th* Wfttcr Harf*o« t«* levpl ^*4ie tii|) of lh« imm, Thu atmo- *9^ pnmam m^ be »etlecicd< ^^ vJlole pnsmme will be the force J^J^ to flff •rtura tii«} iUtti, Biace Uie "•**w»Ul oQiufMJiii^iii *yf I 111 prefisute « A» 4«f to ^ wiU be IJ«wit* of .c LiieBfittre on ^ ^*^ " !.■ the prsMure •&*!»« a ijuimiii. jiiiii the miaufiiiy U proportional to tUa ^eplli, RUtAereSfttLajit tAtitst on tlie- /jasi^^ IIBoftd arfs oJt tJie poirU^ E, Fig, 14. i-n. ^■ii 18 HYDRAULICS the diagram of pressure on the face AB will be the triangle ABC, BC being equal to wd and perpendicular to AB. The centre of pressure is at the centre of gravity of the pressure diagram and is, therefore, at | the height of the dam from the base. The whole pressure acts perpendicular to AB, and is equal to the area ABC = i iccP X sec 10° per foot width = i . 62-4 X 6400 X 1-054 =r 20540 lbs. Combining P with W, the weight of the dam, the resultant thrust R on the base and its point of intersection £ with the base is determined. Example. A vertical flap valve closes the end of a pipe 2 feet diameter ; the pressure at the centre of the pipe is equal to a head of 8 feet of water. To determine the whole pressure on the valve and the centre of pressure. The atmospherie pressure may be neglected. The whole pressure P = irirR^ . 8' = 62-4.x. 8 = 1670 lbs. Depth of the centre of pressure. The moment of inertia about the free surface, which is 8 feet above the centre of the valve, is 82 Therefore 1= — +irR*''. X = ^**^=8'(^" IT . O That is, } inch below the centre of the valve. The diagram of pressure is a truncated cylinder efkh. Figs. 12 and 13, ef and hk being the intensities of pressure at the top and bottom of the valve respectively. Example. The end of a pontoon which floats in sea water is as shown in Fig. 15. The level WL of the water is also shown. Find the whole pressure on the end of the pontoon and the centre of pressure. Fig. 15. The whole pressure on BE = 64 lbs. X lO' X 4-5' x 2-25' = 6480 lbs. The depth of the centre of pressure of BE is 1.4-6=8'. The whole pressure on each of the rectangles above the quadrants =ii>. 6 = 320 lbs., and the depth of the centre of pressure is | feet. Tlie two quadrants, since they are symmetrically placed about the vertieal centre line, may be taken together to form a semicircle. Let d be the distance below the centre of the semicircle of any element of area, the distance of the element below the surface being h^ . FLUIDS AT REST 19 i the mteBsitT- of preisare Jil deplh h^ wlM>le prcsAiire on the lemicirde Le P = I Hp lemlciyeto «1i«d the diameter \& la ^be eurfnee of tb^ waler. ^TW diateiioe of the t^niu of gmvitj of & s^toitiirde from the oentre of tho 2 2' -*- thi whole presstire Ik '^^ =S01B* + 42 66R>^ 1256 4- 666 lbs. I Tb# ^etAb of the oentni of pteaBUre c>f tb@ semieir^^Ie when t!i« sorfiaoe of the P=wfB*+ virB> 4B li ml the oenire at the cir^ ptefiBU X^= 8 ,R 75^ s 16 -M7'> ^^■■^^MiM^the whrrk pressure ort th« 9emi«!ircle is the tium of two Forces, ^^^HHHH|HH6 Ibi^.^ acU at th^ ceutre of gr&vity, ar At & diKtance of 3 06' from ■iPBiCBBff 0( r>66 Ibi^. acta ut u diati^tice of 3^47' from AD. T^tfi taklti^ momenta al»out AD thi^ product of the preBSiire on the whole area isiCc* the (t«.'nth of lii& ceiitr«* of pressure is equal to the moiuetittt of i^ll th« forced OB the afv», a boat AI), The depth of this centre of priMiiF& in, therefore, 6%m Ibi. X ^' + 320 Ibi. X 2 X 1' + 1266 lbs. x 306 + S66 Ibw, x 3^47' ^a*9a feet. 64ti€ + ti40'f 1355 + 666 EXAMPLES. l\ A f«AMfiBl*^ ^&o^ 1^ ^^^ l<^^g* ^ ^^^ wide, a^d 5 leet deep is with ni^ F'ind ih« iota) prassure on an end and side of the tank. Finil the total presj^ure and the centre of preaaure, on a vertical e, eLrctdiir in fortn, 2 feet in diameter, the centre of which is 4 feet Ibe sorface of tlje water. [M. S. T. Cambridge, 1901,] A oiaaonry dam vertical on the water side supports water of I feet- depth. Find the pressure per square foot at depths of 20 feet and . iitiBi the Miirface ; also the total pressni^ on 1 foot length of Uie dam, 0) A dodc gate is hinged horiEontallj at the hottoni and Bttpported in Im ^viiicai pcMttion hj horiacontal chains at the top. i of ^te 45 feet, width 80 ft. Depth of water at ope siJe of the [ And 20 feet on the other side. Find the tension in the diains. cr weigbfl ©4 poitada per cubic foot, II ui<erctir>' ia ISJ times as heavy as water, find the height of a com^sponduig to a pressure of 100 lbs. per square inch, , sirai^t pipe 6 inches diameter has a right-angled bend connected ^ the end of ilie bend being closed by a flange. \ oaDtaifis water at a pressure of 700 !bs, per sq. inch. Determine I In ttie bults at both ends of tlie elbow, 2—2 ^^ 20 HYDRAULICS (7) The end of a dock caisson is as shown in Fig. 16 and the water level is AB. Determine the whole pressure and the centre of pressure. Ut — 44) -H Fig. 16. (8) An U tube contains oil having a specific gravity of 1*1 in the lower part of the tube. Above the oil in one limb is one foot of water, and above the other 2 feet. Find the difference of level of the oil in the two limbs. (9) A pressure gauge, for use in a stokehold, is made of a glass U tabs with enlarged ends, one of which is exposed to the pressure in the stokehold and the other connected to the outside air. The gauge is filled with water on one side, and oil having a specific gravity of 0*95 on the other — ^the surface of separation being in the tube below the enlarged ends. If the area of the enlarged end is fifty times that of the tube, how many inches of water pressure in the stokehold correspond to a displacement of one inch in the surface of separation ? [Lond. Un. 1906.] (10) An inverted oil gauge has its upper U filled with oil having a specific gravity of 0*7955 and the lower part of the gauge is filled with water. The two limbs are then connected to two different points on a pipe in which there is flowing water. Find the difference of the pressure at the two points in the pipe when the difference of level of the oil surfaces in the limbs of the U is 15 inches. (11) An opening in a reservoir dam is closed by a plate 8 feet sqnare^ which is hinged at the upper horizontal edge ; the plate is inclined at an angle of 60"* to the horizontal, and its top edge is 12 feet below the suifaoe of the water. If this plate is opened by means of a chain attached to the centre of the lower edge, find the necessary pull in the chain if its line of action makes an angle of 45° with the plate. The weight of the plate ia 400 pounds. [Lond. Un. 1905.] (12) The width of a lock is 20 feet and it is closed by two gates at each end, each gate being 12' long. If the gates are closed and the water stands 16' above the bottom on one side and 4' on the other side, find the magnitude and position of the resultant pressure on each gate, and the pressure between the gates. Show also that the reaction at the hinges is equal to the pressure between the gates. One cubic foot of water=62-5 lbs. [Lond. Un. 1905.] CHAPTER 11. FLOATING BODIES. IB. ConditionB of equilibxium. Wlieii a body floats in a fluid the surface of the body in contact -^th the fluid is subject to hydrostatic pressures, the intensity of pressure on any element of the surface depend- ing upon its depth below the surface. The resultant of the vertical components of these hydrostatic forces is called the buoyancy, and its magnitude must be exactly equal to the weight ol the body, for if not the body will either rise or sink. Again the horizontal components of these hydrostatic forces must be in equilibrium amongst themselves, otherwise the body will have a lateral movement. The position of equilibrium for a floating body is obtained when (a) the buoyancy is exactly equal to the weight of the body, and (b) the vertical forces — the weight and the buoyancy — act in the same vertical line, or in other words, in such a way as to produce no couple tending to make the body rotate. Let G, Fig. 17, be the centre of gravity of a floating ship and BK, which does not pass through G, the line of action of the resultant of the vertical buoyancy forces. Since the buoyancy Fig. 17. Fig. 18. must equal the weight of the ship, there are two parallel forces each equal to W acting through G and along BK respectively, and these form a couple of magnitude Wx, which tends to bring the ship into the position shown in Fig. 18, that is, so that BK 22 HYDRAULICS passes through G. The above condition (6) can therefore only be realised, when the resultant of the buoyancy forces passes through the centre of gravity of the body. K, however, the body is displaced from this position of equilibrium, as for example a ship at sea would be when made to roll by wave motions, there will generally be a couple, as in Fig. 17, acting upon the body, which should in all cases tend to restore the body to its position of equilibrium. Consequently the floating body will oscillate about its equilibrium position and it is then said to be in stable eqm- librium. On the other hand, if when the body is given a small displacement from the position of equilibrium, the vertical force* act in such a way as to cause a couple tending to increase the displacement, the equilibrium is said to be unstable. The problems connected with floating bodies acted upon by forces due to gravity and the hydrostatic pressures 011I7, resolve themselves therefore into two, (a) To find the position of equilibrium of the body. (b) To find whether the equilibrium is stable. 19. Principle of ArchimedeB. When a body floats freely in a fluid the weight of the body i» equal to the weight of the fluid displaced. Since the weight of the body is equal to the resultant of the vertical hydrostatic pressures, or to the buoyancy, this principle will be proved, if the weight of the water displaced is shown to ^ equal to the buoyancy. Let ABC, Fig. 19, be a body floating in equilibrium, AC bei^^ n the surface of the fluid. thv 16 L (11 which « angle of i of the wal centre of th. action make(» 400 pounds. [ (12) Thewi. end, each gate be. sideand^d'ontheoa^^*^' element ab of the surface, of area a and pressoie on each gattPf the element being inclined at any angle 6 to the reaction at the hinen, if to is the weight of unit volume of the cubic foot of water«62 sure on the area a is wha, and the vertical sure is seen to be wha cos 0, Fig. 19. FLOATIJfO BODIES 23 IniRgine now a vertical cylinder standing on this area^ the top ^ which is in the snrface AC. le horizontal sectional area of this cylinder m a cos^, the is ha cos and the weight of the water filling this volume fshaccmB^ and is, therefore, equal to the buoyancy on the stmilar cylinders be imagined on all the little elements I area which make up the whole immersed surface, the total iome of these cylinders is the volume of the water displaced, L ihe total buoyancy is, therefore, the weight of this displaced li the body is wholly immersed as in iy is supposed to he mmle up of small ffrticttl cylinders intersecting the surface of j\\e hAj in the elements of area ah and ab\ ch are inclined to the horizontal at angles ^ and having areas a and aj resi^ctively, w vertical component of the pressure on ah 111] be itha cos and on a'^' wnll be whiaj cos ^. « C(M§ must equal aj cos ^, each being ^^^^ ^*'' I to the horizontal section of the small cylinder. The whole is therefore ^wha cos - Swh^at cos <^j fiBafam equal to the weight of the water displaced* h this case if the fluid be assumed to be of conBtant density ^i th weight of the body as equal to the weight of the Huid r^ tie same volume, the body will float at any depth. The F«ligtt*^ increase in the weight of the body would cause it to («iik njitil it reached the bottom of the vessel containing the Huid, ^litfe a very small diminution of its weight or increase in ita ^*^iiiii(? would cause it to rise immediately to the surface. It wcruld cle?fcrly be practically impossible to maintain such a body I *n t'qiiilibrium, by endeavouring to adjust the weight of the body, |%piimping out* or letting in» water, as has been attempted in a |ni type of submarine boat. In recent submarines the lowering '^liringof the boat are controlled by vertical screw propellers, 20. Centre of buoyancy. ^irict* the bunyancy on any element of area is the weight of Tierrical cylinder of the fluid above this area, and that the \ ancy is the sum of the weights of all these cylinders, it ■llowH, that the resultant of the buoyancy forces rami 1 through the eentre of gravity of the water displaced, and this Bt is, therefore, called the Centre of Buoyancy. 24 HYDRAUUCS 21. Condition of stability of equilibrium. Let AND, Fig. 21, be the section made by a vertical plane containing G the centre of gravity and B the centre of buoytincy of a floating vessel, AD being the surface of the fluid when the centre of gravity and centre of buoyancy are in the same vertical line. Fig. 21. Fig. 92. Let the vessel be heeled over about a horizonal axis, PE being J now the fluid surface, and let Bi be the new centre of buoyancy, J the above vertical sectional plane being taken to contain G, B, and Bi. Draw BiM, the vertical through Bi, intersecting the Kne GB in M. Then, if M is above G the couple W . x will tend to restore the ship to its original position of equilibrium, but if M i&^ below G, as in Fig. 22, the couple will tend to cause a furthe^^ displacement, and the ship will either topple over, or will heel ova^-^ into a new position of equilibrium. In designing ships it is necessary that, for even large displac^^. ments such as may be caused by the rolling of the vessel, tkr^^ point M shall be above G. To determine M, it is necessary "fc^ determine G and the centres of buoyancy for the two positiox:ii of the floating body. This in many cases is a long and somewH^ tedious operation. 22. Small displacements. Metacentre. When the angular displacement is small the point M is call^^ the Metacentre, and the distance of M from G can be calculated. Assume the angular displacement in Fig. 21 to be small an(^ equal to 0. Then, since theVolume displacement is constant the volume of the wedge CDE muk equal CAF, or in Fig. 23, CiCaDE must equal CiCAF. 25 Let Gi and Gj be the centres of gravity of the wedges C1C2AF and CiCsDE reflectively. df D Fig. 23. The heeling of the ship has the effect of moving a mass of witer equal to either of these wedges from Gi to G2, and this movement causes the centre of gravity of the whole water di^Iaced to move from B to Bj. Let Z be the horizontal distance between Gi and Gj, when FE is horizontal, and S the perpendicular distance from B to BiM. Let V be the total volume displacement, v the volume of the ^wredge and tc the weight of unit volume of the fluid. Then tr.v.Z = 'M7. V. S = w .Y . BM . sin 0, Or, ance ^ is small, ='m?.V.BM.^ (1). The restoring couple is tr.V.HG = ir.V.GM.^ = ii?.V.(BM-BG)^ =^w.v,Z-w.Y.BG,e (2). Bat If . v . Z = twice the sum of the moments about the axis CiCijOf all the elements such as acdh which make up the wedge CCiDE. Taking ah as ar, hf is x6, and if ac is SZ, the volume of the element is ia^^.aZ. The centre of gravity of the element is at fa; from C1C2. 3 Therefi :ore tr . r . Z = 2w 7?dl p .(3). But, -^ is the Second Moment or Moment of Inertia of the element of area aceb about C2C1, and 2 / .5 - is, therefore, the .'0 'J Moment of Inertia I of the water-plane area AC1DC2 about C1C2- Therefore w ,v ,Z = w ,1.0 (4). 26 HYDRAULICS The restoring couple is then wI0-w.Y.BG.e. If this 18 positive, the equilibrium is stable, but if negative it is unstable. Again since from (1) wv.Z = w.Y.BU.e, therefore w.Y.BM.0 = wI0y and BM = ;5 (5), If BM is greater than BG the equilibrium is stable, if less than BG it is unstable, and the body will heel over until a new position of equilibrium is reached. If BG is equal to BM the equilibrium is said to be neutral. The distance GM is called the Metacentric Height, and varies in various classes of ships from a small negative value to a positive value of 4 or 5 feet. When the metacentric height is negative the ship heels until it finds a position of stable equilibrium. This heeling can be corrected by ballasting. ExampU, A ship has a displacement of 15,400 toDS, and a dranght of 27*5 feet. The height of the centre of buoyancy from the bottom of the keel is 15 feet. The moment of inertia of the horizontal section of the ship at the water line is 9,400,000 feet« units. Determine the position of the centre of gravity that the metacentric height shall not be less than 4 feet in sea water. M00^000x64 16,400x2240 = 171 feet. Height of metacentre from the bottom of the keel is, therefore, 32*1 feet. As long as the centre of gravity is not higher than 0*6 feet above the anrfiace of the water, the metacentric height is more than 4 feet. 23. Stability of a rectangular pontoon. Let RFJS, Fig. 24, be the section of the pontoon and G its centre of gravity. Let VE be the surface of the water when the sides of the pontoon are vertical, and AL the surface of the water when the pontoon is given an angle of heel 0, Then, since the weight of water displaced equals the weight of the pontoon, the area AFJL is equal to the area VFJE. Let B be the centre of buoyancy for the vertical position, B being the centre of area of VFJE, and Bi the centre of buoyancy for the new position, Bi* being the centre of area of AFJL. Then the line joining BGr must be perpendicular to the surface VE and * In the Fig., B, is not the centre of area of AFJL, as, for the sake of cleamesi^ it is farther removed from B than it actually should be. FLOATING BODIES 27 s the direction in which the buoyancy force acts when the sides ){ the pontoon are vertical, and BiM perpendicular to AL is the direction in iwrliich the buoyancy force acts when the pontoon is heeled over ttrough the angle 0. M is the metacentre. Fig. 24. The forces acting on the pontoon in its new position are, W the weight of the pontoon acting vertically through G and an equal and parallel buoyancy force W through Bj . There is, therefore, a couple, W • HG, tending to restore the pontoon to its vertical position. If the line BiH were to the right of the vertical through G, or in other words the point M was below G, the pontoon would be in unstable equilibrium. The new centre of buoyancy Bi can be found in several ways. The following is probably the simplest. The figure AFJL is formed by moving the triangle, or really the wedge-shaped piece CEL to CVA, and therefore it may be imagined that a volume of water equal to the volume of this wedge is moved from G2 to Gi . This will cause the centre of buoyancy to move parallel to GiGj to a new position Bi, such that BBi X weight of pontoon = G1G2 x weight of water in CEL. Let b be half the breadth of the pontoon, I the length, D the depth of displacement for the upright position, d the length LE, or AV, and w the weight of a cubic foot of water. Then, the weight of the pontoon W = 2b.B.l.w and the weight of the wedge CLE = -^ ^ I .w. 28 H7DRAUUCS Therefore BBi.26.D = - 2 and BBi = -tjz GriGg. Besolving BBi and GiGj, which are parallel to each other, along and perpendicular to BM respectively, d r. ^ d /2^A bd 6»tan^ B.Q = ^G.K=^(|26) = 3D 3D ' A TiT>-Tin G^ _hd d _ iT _ ^'tan'tf and ^*^-^»'^G.K"3D26~6D" 6D ' To find the distance of the point Mfrom G and the value of the restoring couple. Since BiM is perpendicular to AL and BM to YE, the angle BMBi equals 6. Therefore QM = B,Q cot« = ^ cottf = ^. Let z be the distance of the centre of gravity G from C. Then QG = QC -z = BC-BQ -a _D 5'tan*g 2 6D '• Therefore rnur rnur nr ^* D^fc'tan'tf^ And since HG = GM sin 0, the righting couple, w Tin w • af^' D . 5'tan *g . \ The distance of the metacentre from the point B, is QM + QB = B,Q cot« + ^^^ 3D "^ 6D • When is small, the term containing tan*tf is negligible, and This result can be obtained from formula (4) given in section 22. I for the rectangle is j\l (25)» = ilb\ and V = 2bl>l. Therefore BM = ^. If BG is known, the metacentric height can now be found. FLOATING BODIES 29 Example, A pontoon hM a displaoemeDt of 200 tons. Its length is 50 feet. The centre of gravity is 1 foot above the centre of area of the oross section. Find tbe fareadth and depth of the pontoon so that for an angular displacement of 10 degrees the metaoentre shall not be less than 8 feet from the centre of gravity, and the free- board shall not be less than 2 feet. BeCerring to Fig. 24, G is the centre of gravity of the pontoon and O is the eeotre of the eroes section BJ. Then, GO =1 foot, Fo=2 feet, GM=3 feet. Ijet D be the depth of displacement. Then D X 26 X 62-4 x 50 lbs. =200 tons x 2240 lbs. Therefore D6=71-5 (1). The height of the centre of bnoyancy B above the bottom of pontoon is BT=JD. Since the free-board is to be 2 feet, OT=4(D-|-2). Then B0 = 1' and BG=2'. Therefore BM=5'. Bnt BM = QM + BQ __^ 6»tan«g ■"8D"*" 6D •' (2). Moltiplying namerator and denominator by 6, and substitating from equation (1) 6» &»tan«^ ^, -=6, from which therefore and 214-6 ^ 429 6»(2 + (-176)«)=5x429, 6=101 ft.. D = 71 ft., The breadth B= 20-2 ft.) ,, depth 3 = 20-2 ft.) = 7-lft.| Ans, 24. Stability of a floating vessel containing water. If a vessel contains water with a free surface, as for instance the compartments of a floating dock, such as is described on page 31, the surface of the water in these compartments will remain horizontal as the vessel heels over, and the centre of gravity of the water in any compartment will change its position in such a way as to increase the angular displacement of the vessel. In considering the stability of such vessels, therefore, the taming moments due to the jj ^^^'""""' Q water in the vessel must be taken into account. -^ As a simple case consider E the rectangular vessel. Fig. 2o, H ^ \ jj^ which, when its axis is vertical, floats with the plane AB in the ^^8- ^5. ■€^ K D 30 HYDRAULICS surface of the fluid, DE being the surface of the fluid in the vessel. When the vessel is heeled through an angle ^, the surface of fluid in the vessel is KH. The effect has been, therefore, to move the wedge of fluid OEH to ODK, and the turning couple due to this movement is tu . t? .Z, V being the volume of either wedge and Z the distance between the centre of gravity of the wedges. If 2b is the width of the vessel and I its length, v is-^l tan 6, Z is |6 tan ^, and the turning couple is w ft' I tan* 0. If is small wvZ is equal to wI0y I being the moment of inertia of the water surface KH about an axis through 0, as shown in section 22. For the same width and length of water surface in the compartment, the turning couple is the same wherever the compartment is situated, for the centre of gravity of the wedge OHE, Fig. 26, is moved by the same amount in all cases. If, therefore, there are free fluid surfaces in the floating vessel, for any small angle of heel ^, the tippling-moment due to these surfaces is 'S.wIOy I being in all cases the moment of inertia of the fluid surface about its own axis of oscillation, or the axis through the centre of gravity of the surface. E H 0-4 pH D Fig. 26. Fig. 27. 25. Stability of a floating body wholly immersed. It has already been shown that a floating body wholly im- mersed in a fluid, as far as vertical motions are concerned, can only with great difficulty be maintained in equilibrium. If further the body is made to roll through a small angle, the equilibrium will be unstable unless the centre of gravity of the body is below the centre of buoyancy. This will be seen at once on reference to Fig. 27. . Since the body is wholly immersed the centre of buoyancy cannot change its position on the body itself, as however it rolls the centre of buoyancy must be the centre of gravity of the displaced water, and this is not altered in form by FLOATING BODIES 31 any movement. of the body. K, therefore, G is above B and the body be given a small angular displacement to the right say, G will move to the right relative to B and the couple will not restore the body to its position of equilibrium. On the other hand, if G is below B, the couple will act so as to brin^ the body to its position of equilibrium. 26. Floatiiig docks. Figs. 28 and 29 show a diagrammatic outline of the pontoons forming a floating dock, and in the section is shown the outline of a ship on the dock. Fig. 29. To dock a ship, the dock is sunk to a sufficient depth by admitting water into compartments formed in the pontoons, and the ship is brought into position over the centre of the dock. Water is then pumped from the pontoon cliarubers, and the dock in consequence rises until the ship just rests on the keel blocks of the dock. As more water is pumped from the pontoons the dock rises with the ship, which may thus be lifted clear of the water. Let Gi be the centre of gravity of the ship, Ga of tlie dock and its water ballast and G the centre of gra\4ty of the dock and the ship. The position of the centre of gi-avity of the dock >W11 vary 32 HTDRAULICS relative to the bottom of the dock, as water is pumped from the pontoons. As the dock is raised care must be taken that the metacentre is above G or the dock will " list." Suppose the ship and dock are rising and that WL is the water line. Let Ba be the centre of buoyancy of the dock and Bi of the portion of the ship still below the water line. Then if Vi and V2 are the volume displacements below the water line of the ship and dock respectively, the centre of buoyancy B of the whole water displaced divides BsBj, so that BBi_Va BBa"Vx- The centre of gravity G of the dock and the ship divides GiGt in the inverse ratios of their weights. As the dock rises the centre of gravity G of the dock and the ship must be on the vertical through B, and water must be pumped from the pontoons so as to fulfil this condition and as nearly as possible to keep the deck of the dock horizontal. The centre of gravity Gi of the ship is fixed, while the centre of buoyancy of the ship Bi changes its position as the ship is raised. The centre of buoyancy Ba of the dock will also be changing, but as the submerged part of the dock is symmetrical about its centre lines, Ba will only move vertically. As stated above, B must always lie on the line joining Bi and Ba, and as G is to be vertically above B, the centre of gravity Ga and the weight of the pontoon must be altered by taking water from the various compartments in such a way as to fulfil this condition. Quantity of water to he 'pumped from the pontoons in raising the doch. Let V be the volume displacement of the dock in its lowest position, Vo the volume displacement in its highest position. To raise the dock mthout a ship in it the volume of the water to be pumped from the pontoons is V - Vo . If, when the dock is in its highest position, a weight W is put on to the dock, the dock will sink, and a further volume of water W . . — cubic feet will be required to be taken from the pontoons to raise the dock again to its highest position. To raise the dock, therefore, and the ship, a total quantity of water w cubic feet will have to be taken from the pontoons. FLOATING BODIES 33 Ifft^vlf, K lUmtmg dock ms fibotrtt di mentioned m Fir. 2B is made np of s not! 540 feet long x OG fc«l wide x 14-75 feet deep, two aide pontooDt gx 13 ft»et wide y 4^8 feet de«p, the bottom of these poutoonB being ^ "<.*vf- Ihc bcittoin of the dock^ aud two side chambers on the top of thm I poatooti 447 feel long by 8 feet deep imd 2 feet wide at the top and 8 fe«l ftt | . itoiD. The kf^l blocks may be taken as 4 feet deep. I h« dock is Ui lift » sbip of 15,400 ttiris dij^plncetneul and 27' 6" dratlght. l)ttj^nriiine the amouDt of water that mu^t be pumped from the dock, to raJBe ifi tm that the deck of the lowest pontoon is m the water surface. »:ji the xbip jnst t^kes to the keel bloeks on the dock^ the bottom of the atw * im 27*5' ^ 14 '75' + 4' = 46 '25 feet below the water line, Tht voloiik^ displaovment of the dock is then I 14-76 * MO ^ 96 ^' 2 ^ 44-25 X X3 X aSO + 447 X 8 X 5' = 1,3K7,6(H) cabic feet, I Tb« foltime of dock diaplacemeat when ih© deck is just awash m I 540 X 96 M 14-75 + 2 k 3B0 x 13' >^ ( l4*7o - 2) = H90,OC)O mhm feel. I The TolTim« djsplacemeat of the ahip ii I 15.400 >c 2240 e.rt^^ t- , , ^7 = 640, 000 ctibio feet , ASid tlik ttfOftlt Ihe weight of the ship in cubic feet. of tb« ^1,000 citbic feel di»placemeni wheu the ship is olear of the water^ &S1 /Wll eotiic feel J!i therefore reiiTtired to support Ihe dock aloDe. 'I riiply lei tninm the dock through 31*5 feet the amount of wat^r to l^e pumped ia Teretkcs i>f the diaplacemenlH^ and is, tlierefon*, 347i*K]0 cubic feet. "'^ lh# ehip with the dock an additiooal ^40,000 cubits feet must be ':7aD the pontoous* :%) q^tAolitjt Iherefore, to be taken from the pontoons from the time the kkem to iht ked blocks lo when the pontoon deck is in the HurCaoe of the ^ 887.600 cubic feet ^ 25,380 tons, , 27. Stability of the floatiBg dock. As some (jI the compartments of tlie dock are partially filled w^iiierT it in necesHaiy, in considering the stability, to take • 'f the tipplirig-moments eauaed by the movement of the ^ r.ce of the water in these compartments. kL Ii G i» the centre of gra^nty of the dock and ship on the " ih^^c^ B the centre of bnoyancyj I the moment of inertia of the n of the ship and dock by the water-plane abont the axis of ] ti, and Ii, Is etc. the moments of inertia of the water in the compartments about their axes of oscillation^ the n^ moment when the dock receives a amaU angle of ifW- ti? (Yi + Va) BO^- t£vi9 (I, + 1, +...). I llir moment of inertia of the water-plane section YarieAl eoiu»iclerahly ub the dock ib raised, and the stability varies When tht; i*hip is immersed in the water, I is equal to the jnamenT of inertia of the horizontal section of the ship at the ter fiarfAC«», together with the moment of inertia of the ■ ' -ection of the side pontoons, about the axis of t- a- li^Mlktflki 34 HTDRAUUCS When the tops of the keel blocks are just above the SDrEaea: of the water, the water-plane is only that of the side pontoon^, and I has its minimum value. If the dock is L-shaped as Fig. 30, which is a very convenient form for some purposes, the stability when the tops of the keel blocks come to the surface simply depends upon the moment of inertia of the area AB about an axis through the centre of AB. This critical point can, however, be eliminated by ii fitting an air box, shown dotted, on the p. ^ outer end of the bottom pontoon, the top of which is slightly higher than the top of the keel blockB* Example. To find the height of the metacentre above the centre of baoymoflj if " the dock of Fig. 28 when {a\ the ship just takes to the keel blocks, {h) the keel is just clear of the water, \c) the pontoon deck is just above the water. Take the moment of inertia of the horizontal section of the ship at tht water line as 9,400,000 ft> units, and assume that the ship is symmetricallj placed on the dock, and that the dock deck is horizontal. The horizontal distanee between the centres of the side tanks is 111 ft. (a) Total moment of inertia of the horizontal section is 9, 400,000 4- 2 (380 x 1 3' x 66 S' 4- tV x 380 x 13») = 9,400,000 + 30,430,000 + 139,000. The volume of displacement =640,000 + 1,237,600 cubic feet. The height of the metacentre above the centre of buoyancy is therefore 39,968,000 «... . ^^=1,932,000=^^^^^*- (h) When the keel is just clear of the water the moment of inertia is 30,569.000. The volume displacement is 540 X 96 X 14-75 + 380 x 2 x 13 x (14-75 + 4-2) = 930,000 cubic feet. Therefore BM = 32*8 feet. (c) When the pontoon deck is just above the surface of the water, I = 30,569,000 + iV x 5 40' X 96» y.. =70,269,000. W "^he volume displacement is 890,000 cubic feet. — CUL . W '^fow BM= 79-8 feet. raise the (X^^ of ^^^ centre of buoyancy above the bottom of the dock can be rn -^ -^p ' finding the centre' of buoyancy of each of the parts of the dock, and lo raise » ^ jjj jjj^ water, and then taking moments about any axis, water To find the height h of the centre of buoyancy of the dock and ^ ship just comes on the keel blocks, oyancy for the ship is at 15 feet above the bottom of the keel, vancy of the bottom pontoon is at 7*375' from the bottom. I.- r i. ^^^\. ^ »» side pontoons „ 24-125' ,, cubic feet Will have ,, „ chambers ., i7-94' ,; IXOATUIQ BODIES 35 \ tnometits &boat the bottom of the dock h (540,000 + 437,000 + 765,0004 35,760) = 540.000 X 33*75 + 765,000 x 7'375 + 437,00O X *i4'i25 i- 3S,760 X 17 "95, A -19 -7 feet, {^) ihe metiio^iitre is^ tberefi^re, 40*3' above the bottom of the dock. If «si]iT« of griL^i^ of Ibe dcHsk &ad sUip u kuowa the metacentrio height EXAMPLES. A «titp ^bcn fully loaded has a total burden of 10,000 tons. Find i displacemeDt in aea water. The aides d a ship are vertical near the water lino and the area of sittl section at the water line ia 22,000 aq. feet. The total weight r sJiip IB 10,000 tons when it leaves the river dock, tbe diffei^iice in draught in the dock and at sea after the weight F ship ii&a been rednoed by consiimption of ooaU etc., by 1500 totis, > 8 be Ibe difference in draught, I c K 22.000= the difference in volume displacement 10,000 ic 2340 _ 6500x 2240 " 62-48 64 =61S0 cnbic feet. ^■i*2'rSf6et sS 34 inches. Tlie moment of inertia of the section at the water line of a boat foot* tmita; the weight of the boat is 11 '5 tons. I the height of the metacentre above tlie centre of buoyancy. {%} k ship has a total displacement of 15.000 tons and a draught of 1 the ship is lifted by a floating dock so that the depth of the bottom \ ked is 16*5 feet, tlie centre of buoyancy is 10 feet from the bottom of ^kael aad tlie displacement is 9000 tons. f nwiiDeait of inertia of Uie water-plane is 7t600,000 foot* units. Iiorueoiital section ol the dock* at tlie plane 16*5 feet above the of ttw keel^ consists of two rectangles 380 feet k 11 feet, the distance i td the «enlre lines of the rectangles b^ing 114 feot- The TobtBie displacement of tlie dock at this level is 1,244,000 cubic feet. Tbeciotra of buoyancy for the dock alone is 24*75 feet below the surface f water. line ia] The centre of buoyancy for the whole ship and the dock. Tlw height of tlie metacentre above the centre of buoyancy. h fV A rectangulai' pontoon 60 feet long is to have a displacement of % fre^^ board of not leas than 3 feet, and the metacentre in not to than S feet above the centre of gravity when the angle of heel The centre of gravity coincides with the centre ol tigure. tbe vddUi and depth ol the i>ontooii. 3—2 36 HYDRAULICS (6) A rectangular pontoon 24 feet wide, 50 feet long and 14 feet deep» has a displacement of 180 tons. A vertical diaphragm divides the pontoon longitudinally into two compartments eadi 12 feet wide and 50 feet long. In the lower part of each of these compartments there is water ballast, the 8iir&<» of the water being free to move. Determine the position of the centre of gravity of the pontoon that it may be stable for small displacements. (7) Define "metacentric height** and show how to obtain it graphicallj or otherwise. A ship of 16,000 tons displacement is 600 feet long, 60 leel beam, and 26 feet draught. A coefficient of ^ may be taken in the moment of inertia term instead of ^ to allow for the water-line section not beii^ a rectangle. The depth of the centre of buoyancy from the water line » 10 feet. Find the height of the metacentre above the water line and determine the position of the centre of gravity to give a metacentric hei^ of 18 inches. [Lond. Un. 1906.] (8) The total weight of a fully loaded ship is 5000 tons, the water line encloses an area of 9000 square feet, and the sides of the ship are vertical at the water line. The ship was loaded in fresh water. Find the changB in the depth of immersion after the ship has been sufficiently long at sea to bum 500 tons of coal. Weight of 1 cubic foot of fresh water 62} lbs. Weight of 1 cubic foot of salt water 64 lbs. CHAPTER III. FLUIDS IN MOTION. 28. Steady motion. The xnotioii of a fluid is said to be steady or permanent, when the particles -which succeed each other at any point whatever have the same density and velocity, and are subjected to the same pressure. In practice it is probably very seldom that such a condition of flow is absolutely realised, as even in the case of the water flowing steadily along a pipe or channel, except at very low velocities, the velocities of succeeding particles of water which arrive at any point in the channel, are, as will be shown later, not the same either in magnitude or direction. For practical purposes, however, it is convenient to assume that if the rate at which a fluid is passing through any finite area is constant, then at all points in the area the motion is steady. For example, if a section of a stream be taken at right angles to the direction of flow of the stream, and the mean rate at which water flows through this section is constant, it is convenient to assume that at any point in the section, the velocity always remains constant both in magnitude and direction, although the velocity at different points may not be the same. Mean velocity. The mean velocity through the section, or the mean velocity of the stream, is equal to the quantity of flow per unit time divided by the area of the section. 29. Stream line motion. The particles of a fluid are generally regarded as flowing along definite paths, or, in other words, the fluid may be supposed to flow in thread-like filaments, and when the motion is steady these filaments may be supposed to be fixed in position. In a pipe or channel of constant section, the filaments are generally supposed to be parallel to the sides of the channel. 38 HYDRAULICS An DaUuiLlLe^ Fig. 81. 30. Definitions relating to flow of water. Pressure head. The pressure head at a point in a fluid at rest has been defined as the vertical distance of the point from the free surface of the fluid, and is equal to — , where p is the pressure per sq. foot and w is weight per cubic foot of the fluid. Similarly, the pressure head at any point in a moving fluid at which the pressure is p lbs. per sq. foot, is - feet, w and if a vertical tube, called a piezometer tube. Fig. 31, be inserted in the fluid, it will rise in the tube to a height h, which equals the pressure head above the atmo- spheric pressure. K p is the pressure per sq. foot, above the atmospheric pressure, h= —, but if p is the absolute pressure per — sq. foot, and px the atmospheric pressure, W W Velocity head. If through a small area around the point B, the velocity of the fluid is v feet per second, the velocity head is rt- , g being the acceleration due to gravity in feet iter second per second. Position head. If the point B is at a height z feet above any convenient datum level, the position head of the fluid at B above the given datum is said to be z feet. 31. Energy per pound of water passing any section in a stream line. The total amount of work that can be obtained from every pound of water passing the point B, Fig. 31, assuming it can fall to the datum level and that no energy is lost, is 2 + ^' + 2 ft. lbs. w 2g Proof. Work available due to pressure head. That the work which can be done by the pressure head per pound is ~ foot pounds can be sho'WTi as follows. Imagine a piston fitting into the end of a small tube of cross sectional area a, in which the pressure is h feet of wator as in I FLUIDS IN MOTION 39 g. 32, And let a small quantity ^Q cubic feet of water enter the ibe and move the piston through a small dis- knce 6as. Then dQ^a.dx. The iwork done on the piston as it enters Nrill he w.h.a.dx = w .hdQA But the ^weight of 5Q cubic feet iaw.dQ pounds, ^'8. 82. and the ^wrork done per pound is, therefore, A, or — foot pounds. A. pressure head h is therefore equivalent to h foot pounds of energy ijer ix>und of water. Work available due to velocity. When a body falls through a height h feet, the work done on the body by gravity is h foot pounds i)er pound. It is shown in books on mechanics that if the body is allowed to fall freely, that is without resistance, the velocity the body acquires in feet per second is V = J2ghy And since no resistance is offered to the motion, the whole of the work done on the body has been utilised in giving kinetic energy to it, and therefore the kinetic energy per pound is ^ ~ • In the case of the fluid moving with velocity v, an amount of energy equal to ^y foot pounds per pound is therefore available before the velocity is destroyed. Work available due to position. If a weight of one pound falls through the height z the work done on it by gravity will be z foot pounds, and, therefore, if the fluid is at a height z feet above any datum, as for example, water at a given height above the sea level, the available energy on allowing the fluid to fall to the datum level is z foot pounds per pound. 32. Bemouilli's theorem. In a steady moving stream of an incompressible fluid in which the particles of fluid are moving in stream lines, and there is no loss by friction or other causes p v^ w 2g is constant for all sections of the stream. This is a most important theorem and should be carefully studied by the reader. 40 HYDRAULICS It has been shown in the last paragraph that this expression represents the total amount of energy per pound of water flowing through any section of a stream, and since, between any two points in the stream no energy is lost, by the principle of the conservation of energy it can at once be inferred that this expression must be constant for all sections of a steady flowing stream. A more general proof is as follows. Let DE, Fig. 33, be the path of a particle of the fluid. Fig. 33. Imagine a small tube to be surrounding DE, and let the flow in this be steady, and let the sectional area of the tube be so small that the velocity through any section normal to DE is uniform. Then the amount of fluid that flows in at D through the area AB equals the amount that flows out at E through the area CF. Let Pj) and Vd, and p^ and ve be the pressures and velocities at D and E respectively, and A and a the corresponding areas of the tube. Let z be the height of D above some datimi and Zi the height of E. Then, if a quantity of fluid ABAiBi equal to 3Q enters *at D, and a similar quantity CFCiFi leaves at E, in a time dtj the velocity at D is Vj,= AuV and the velocity at E is Ve = aot' The kinetic energy of the quantity of fluid uQ entering at D FLUIDS IN MOTION 41 and tJiat of tlie liquid leaving at E Since tlie flow in the tabe is steady, the kinetic energy of the portion ABGF does not alter, and therefore the increase of the kinetic energy of the quantity dQ The work done by gravity is the same as if ABBiAi fell to CFFiCi and therefore equals The total pressure on the area AB is po • A, and the work done at X> in time ot = PdAi?d9^=Pd9Q, and the work done by the pressure at E in time t But the gain of kinetic energy must equal the work done, and therefore ^^ . (ve'- V) = wdq (z - 2,) + Pd ^Q-PeC'Q. From which 2g 2g w w^ or ^«%2?^^^ = V+e5 + ;, = constant. 2g w 2g w From this theorem it is seen that, if at points in a steady moving stream, a vertical ordinate equal to the velocity head plus the pressure head is erected, the upper extremities of these ordinates will be in the same horizontal plane, at a height H equal to — + s~ "•" ^ above the datum level. ^ ttj 2g Mr Froude* has given some very beautiful experimental illus- trations of this theorem. In Fig. 34 water is taken from a tank or reservoir in which the water is maintained at a constant level by an inflowing stream, through a pipe of variable diameter fitted >vith tubes at various points. Since the pipe is short it may be supposed to be frictionless. If the end of the pipe is closed the water will rise in all the tubes to the same level as the water in the reservoir, but if the end C is opened, water will flow through the pipe and the water surfaces in the tubes will be found to be at different levels. * British Assoc. Report 1875. 42 HYDRAULICS The quantity of water flowing per second through the pipe can be measured, and the velocities at A, B, and C can be found by dividing this quantity by the cross-sectional areas of the pipe at these points. Fig. 34. If to the head of water in the tubes at A and B the ordinates S 2 1^ and g^ be added respectively, the upper extremities of these ordinates will be practically on the same level and nearly level with the surface of the water in the reservoir, the small difference being due to frictional and other losses of energy. At C the pressure is equal to the atmospheric pressure, and neglecting friction in the pipe, the whole of the work done by gravity on any water leaving the pipe while it falls from the surface of the water in the reservoir through the height H, which is H ft. lbs. per pound, is utilised in giving velocity of motion to the water, and, as will be seen later, in setting up internal motions. Neglecting these resistances. Due to the neglected losses, the actual velocity measured will be less than vq as calculated from this ^nation. If at any point D in the pipe, the sectional area is less thaai the area at C, the velocity will be greater than Vc, and the pressure will be less than the atmospheric pressure. If V is the velocity at any section of the pipe, which is supposed to be horizontal, the absolute pressure head at that section is -" -" "- ^2g w^'2g 2g' w w Pa being the atmospheric pressure at the surface of the water in the reservoir. At D the velocity Vd is greater than Vo and therefore p© is less FLUIDS IN MOTION 43 than p.. If coloured water be put into the vessel B, it will rise in the tube DE to a height w w If the area at the section is so small, that p becomes negative, the fluid will be in tension, and discontinuity of flow will take place. If the fluid is water which has been exposed to the atmosphere and which consequently contains gases in solution, these gases will escape from the water if the pressure becomes less than the tension of the dissolved gases, and there will be discontinuity even before the pressure becomes zero. Figs. 35 and 36 show two of Froude's illustrations of the theorem. Fig. 86. Fig. 36. At the section B, Fig. 36, the pressure head is h^ and the velocity head is = /l-/lB = H. If a is the section of the pipe at A, and a^ at B, since there is continuity of flow. and If now a is made so that 2g K the pressure head /ia becomes equal to the atmospheric pressure, and the pii)e can be divided at A, as shown in the figure. Professor Osborne Reynolds devised an interesting experiment, to show that when the velocity is high, the pressure is small. He allowed water to flow through a tube | inch diameter under a high pressure, the tube being diminished at one section to 01)5 inch diameter. £4 W 2ff = 2? w 2ff w 2*7' H = Pi" 2ff If 'Ua is equal to Vi, pa is theoretically equal to pi, but there is vays in practice a slight loss of head in the meter, the difference - Pa being equal to this loss of head. • Tramactiom Am,S.C,E,, 1887. "1 HYDRAULICS At this diminished section, the velocity was very high and the assure fell so low that the water boiled and made a hissing jse. 33. Venturi meter. An application of Bemouilli's theorem is found in the Venturi ter, as invented by Mr Clemens Herschel*. The meter takes name from an Italian philosopher who in the last decade of the h. century made experiments upon the flow of water through lical pipes. In its usual form the Venturi meter consists of two mcated conical pipes connected together by a short cylindrical ►e called the throat, as shown in Figs. 37 and 38. The meter is erted horizontally in a line of piping, the diameter of the large is of the frustra being equal to that of the pipe. Piezometer tubes or other pressure gauges are connected to ) throat and to one or both of the large ends of the cones. Let a be the area of the throat. | ^ Let ai be the area of the pipe or the large end of the cone A. Let Oa be the area of the pipe or the large end of the cone C. Let p be the pressure head at the throat. Let pi be the pressure head at the up-stream gauge A. Let Pa be the pressure head at the down-stream gauge C. Let H and Hi be the differences of pressure head at the throafe i large ends A and C of the cone respectively, or w w^ . ^ d H. = ^-2. w w Let Q be the flow through the meter in cubic feet per sec. Let V be the velocity through the throat. Let Vi be the velocity at the up-stream large end of cone A. Let Va be the velocity at the down-stream large end of cone 0. Then, assuming Bemouilli's theorem, and neglecting friction, PiOTDS IN MOTION 46 HYDRAULICS The velocity t? is — , and i?i is — Oi Therefore Q^ (^, - ^,) = 2flf . H, and Q = aai Joa - d^ V2^TH. Due to friction, and eddy motions that may be set up in the meter, the discharge is slightly less than this theoretical value, or 4 being a coefficient which has to be determined by experiment. For a meter having a diameter of 25*5 inches at the throat and 54 inches at the large end of the cone, Herschel found the following values for fc, given in Table III, so that the coefficient varies but little for a large variation of H. TABLE m. Herschel Coker Hfeet h in cu. ft. k 1 2 6 12 18 28 •995 •992 •985 •9785 -977 •970 -0418 -0319 •0254 •0185 •0096 •0084 •9494 •9587 •9572 •9920 1-2021 1-8588 Professor Coker*, from careful experiments on an exceedingly well designed small Venturi meter. Fig. 38, the area of the throat of which was '014411 sq. feet, found that for small flows the coefficient was very variable as shown in Table III. These results show, as pointed out by Professor Coker from an analysis of his own and Herschel's experiments on meters of various sizes, that in large Venturi meters, the discharge is very approximately proportional to the square root of the head, but for small meters it only follows this law for high heads, and for low heads they require special calibration. Example. A Venturi meter having a diameter at the throat of 86 inohes is inserted in a 9 foot diameter pipe. The pressure head at the throat gauge is 20 feet of water and at the pipe gauge is 26 feet. * Canadian Society of Civil Engineers, 1902. FLUIDS IN MOTION 47 ?xiid the diaeharge, and the Telocity of flow through the throat. The area of the pipe is 63*5 sq. feet. throat 7-06 The differenee in prewore bead at the two gauges is 6 feet. -^ , ^ 63-6 X 7-06 , Therefore Q= =====: ^2x32*2x6 = 137 c. ft. per second. The Tdoeity of flow in the pipe is 2*15 ft. per sec. „ „ through the throat is 19*4 ft. per sec. 34. Steering of canal boats. An interesting application of Bemouilli*8 theorem is to show the effect of speed and position on the steering of a canal boat. l\Tien a boat is moved at a high velocity along a narrow and shallow canal, the boat tends to leave behind it a hollow which is filled by the water rushing past the boat as shown in Figs. 39 and 40, while immediately in front of the boat the impact of the bow on the still water causes an increase in the pressure and the water is " piled up '' or is at a higher level than the still water, and what is called a bow wave is formed. Fig. 39. Fig. 41. A Fij?. 40. Let it be assumed that the water moves past the boat in stream lines. If vertical sections are taken at E and F, and the points E and F are on the same horizontal line, by Bemouilli's theorem w 2g w 2g' At E the water is practically at rest, and therefore Vs is zero, and w w 2g' The surface at E will therefore be higher than at F. 48 HTDRAUUCS Wlien the boat is at tlie centre of the canal the stream lines on both sides of the boat will have the same velocity, but if the boat is nearer to one bank than the other, as shown in the figures, the velocity vr of the stream lines between the boat and the neaarer bank, Fig. 41, will be higher than the velocity Vr on the other side; But for each side of the boat w w 2g w 2g ' And since vr is greater than vv, the pressure head pf is greater than pn or in other words the surface of the water at the right side D of the boat will be higher than on the left side R The greater pressure on the right side D tends to push the boat towards the left bank A, and at high speeds considerably increases the diflSculty of steering. This difficulty is diminished if the canal is made sufficiently deep, so that flow can readily take place underneath the boat. 35. Extension of Bemooilli's theorem. In deducing this theorem it has been assumed that the fluid is a perfect fluid moving with steady motion and that there are no losses of energy, by friction of the surfaces with which the fluid may be in contact, or by the relative motion of consecutive ele- ments of the fluid, or due to internal motions of the fluid. In actual cases the value of p v^ w 2g diminishes as the motion proceeds. If hf is the loss of head, or loss of energy per pound of fluid, between any two given points A and B in the stream, then more generally £^ + |i.%,, = 2B^|LV,3.;,, (1). w 2g "^ w 2g Tht EXAMPLES. . ^ "^ The diameter of the throat of a Venturi meter is | inch, and of various S.^ which it is connected IJ inches. The discharge through the approxima-) minutes was found to be 814 gaUons. small metenerence in pressure head at the two gauges was 49 feet, heads they ree coefficient of discharge. Example. A V^i meter has a diameter of 4 ft. in the large part and inserted in a 9 foot oat. With water flowing through it, the pressure head is is 26 feer^^^^"^ ^^^^ P*^ ^^^ ^"^ ^' ** ^® ^i^oai. Find the velocity in the 1 discharge through the meter. Coefficient of Cat. FLUIDS IN MOTION 49 A pipe AB, 100 ft loiig^ has an inclinatton ot 1 in 5. The haad dae iliie pre^xne at A is 45 tt^ the velocity Is 3 ft. per aecondf ajid the section kf the pipe is 3 sq, ft. Find the head due ta the preasure at B, where the is 1| sq. ft. Take A as the lower end of the pipe. (4^ The section pipe d a pump h laid at an inclinatioii of 1 in 5, and is pnnaped ihrongb it at 6 ft. per fiecond. Suppose the air in the >ter is disengaged if the pressure ialls to more than 10 lbs. below ^eric pceesare. Then deduce the greatest practicable length of ptpa Friction neglected. Wsler is delivered to an Inward-flow turbine under a head of 100 feet IX j. The pressure just outside the wheel ia 25 lbs. per jliy gan^e. Fihe velocity with which the water approaches the wheel* Friction (0^ A sh£»rt conical pipe varying in diameter from 4^ 6'' at the large end feel at tiie small end forms part of ei horizontal water main. Tlie ! head at the large end is found to be 1 00 feet, and at the small end [ the disohaige through the pipe. Coefficient of discharge nnity^ Three cubic l^t of water per second flow along a pipe which as it in diameter from 6 inches to 12 inches. In 50 feet the pipe ieei. Pne to rarions causes there is a leas of head of 4 feet* (aj the lo6S ol energy in fcxjt ponndfi par minute^ and in horse- id the difference in pressure head at the two points 50 feet apart. eqnmtioii 1, section 35*) |9i A boriso&tal pipe in whidi the sections vary gradually has sections 1 10 9go«r« feet* I square foot, and 10 square feet at sections A, B, and G* haad at A i% 100 feet, and the yelocity S feet per second., head and velocity at B. tliftt in another case the difference of the pressure heads at A , B It* 2 f«?et. Find the velocity at A, A Ventnri meter in a water main consists of a pipe converging to fcod enlarging again gradually. The section of main is 9 sq. ft. of UuToat 1 sq. ft. The difference of presstire in the main and ibroAi is 12 feet of water* Find the discharge of the main per hour* If If tbe inlet area of a Venturi meter is n times the throat area, and Ijp mm the ^lelodty and preB«!.ure at the throat, and the inlet pressure thmt Up Asd mp are observed, t^ can be found. CHAPTER IV. FLOW OF WATER THROUGH ORIFICES AND OVER WEIRS. 36. Flow of fluids through orifices. The general theory of the discharge of fluids through orifices, as for example the flow of steam and air, presents considerable difficulties, and is somewhat outside the scope of this treatise. Attention is, therefore, confined to the problem of detennining the quantity of water which flows through a given orifice in a given time, and some of the phenomena connected therewith. In what follows, it is assumed that the density of the fluid k constant, the effect of small changes of temperature and pressure j in altering the density being thus neglected. ! Consider a vessel. Fig. 42, filled with water, the free surEace of which is maintained at a constant level ; in the lower part of tbe vessel there is an orifice AB. Fig. 42. Let it be assumed that although water flows into the vessel ^ as to maintain a constant head, the vessel is so large that at soi^^ surface CD, the velocity of flow is zero. Imagine the water ir^ the vessel to be divided into a number ^ stream lines, and consider any stream line EF. Let the velocities at £3 and F be Ve and t^p, the pressure heads h^ and h^ and the positio^ heads above some datum, z^ and 9ft j respectively. FLOW THBOUGH ORIFICES 61 Then, applying Bemouilli's theorem to the stream line EF, If tf is zero, then 1 t^E* T V9 217 = fcp~fcB + 2ji— 2£, But from the figure it is seen that is equal to A, and therefore Ve =fc, or 2g Since Ae is the pressure head at E, the water would rise in a tabe having its end open at E, a height /^e> and h may thus be called — ^following Thomson — the fall of "free level for the point B." , At some section GK near to the orifice the stream lines are all practically normal to the section, and the pressure head will be equal to the atmospheric pressure ; and if the orifice is small the fall of free level for all the stream lines is H, the distance of the centre of the section GK below the free surface of the water. If the orifice is circular and sharp-edged, as in Figs. 44 and 45, the section GK is at a distance, from the plane of the orifice, about equal to its radius. For vertical orifices, and small horizontal orifices, H may be taken as equal to the distance of the centre of the orifice below the free surface. The theoretical velocity of flow through the small section GK is, therefore, the same for all the stream lines, and equal to the velocity which a body will acquire, in falling, in a vacuum, through a height, equal to the depth of the centre of the orifice below the free surface of the water in the vessel. The above is Thomson's proof of Torricelli's theorem, which was discovered experimentally, by him, about the middle of the 17th century. The theorem is proved experimentally as follows. If the aperture is turned upwards, as in Fig. 43, it is found that the water rises nearly to the level of the water in the vessel, and it is inferred, that if the resistance of the air and of the orifice could be eliminated, the jet would rise exactly to the level of the surface of the water in the vessel. 1! 1^ ' Fig. 43. 4—2 52 HTDBAUU08 Other experimentB deflcribed on pages 54— S6, also bIiow thal^ with carefully constracted orifices, the mean velocity throogh the orifice differs from ^2gB, by a very small quantify. ; 37. Coeffloient of contraotion for aharp-edged orifice. If an orifice is cut In the flat side, or in the bottom of a veeBe^. and has a sharp edge, as shown in Figs. 41 and 45, the stream Knfli; set up in the water approach the orifice in all directions, as shovi in the figure, and the directions of flow of the particles of wato^ [ except very near the centre, are not normal to the plane of Am ^ orifice, but they converge, producing a contraction of the jet \ Fig. 44. Fig. 46. At a small distance from the orifice the stream lines become practically parallel, but the cross sectional area of the jet is considerably less than the area of the orifice. If w is the area of the jet at this section and a the area of the orifice the ratio - is called the coefficient of contraction and may a be denoted by c. Weisbach states, that for a circular orifice, the jet has a minimum area at a distance from the orifice slightly less than the radius of the orifice, and defines the coefficient of contraction as this area divided by the area of the orifice. For a circular orifice he gives to c the value 0*64. Recent careful measurements of the sections of jets from horizontal and vertical sharp-edged circular and rectangular orifices, by Bazin, the results of some of which are shown in Table IV, show, however, that the section of the jet diminishes continuously and in fact hss no minimum value. Whether a minimum occurs for square orifice^ is doubtful. The diminution in section for a greater distance than tha-* given by Weisbach is to be expected, for, as the jet moves aw»y from the orifice the centre of the jet falls, and the theoretical velocity becomes J2g (R + y),y being the vertical distance betweeO- the centre of the orifice and the centre of the jet. FLOW THROUGH ORIFICES 5a M a Muall distance away from the orifice, however, the stream iiueaitpe prscdcally parallel, and very h'ttb error is introduced in tile coeieieut of contraction by measuring the stream near the Poncelet and Lesbros in 1828 fotmdj for an orifice '20 m. square, I a minimom section of the jet at a distance of *3 uu from the orifice ianti Bt lim section c was '.56^3. M. Bazin^ in discussing these IiwdI^ re-marks tlmt at distances greater than 0*3 m, the section j becoinf» very difficult to measure^ and althongh the vein appears jto expand, the sides become hollow, and it is uncertain whether [tlieur^ is really diminiBhed, Compkii* eontractimi. The maxinnim contraction of the jet aitef place when the orifice is sharp edged and is well removed Mm Ike Hides and bottom of the vesseL In this case the contrac- on i» said to be complete* Experiments show, that for complete »&tfaeti(>n the dista^nce from the orifice to the sides or bottom of p reesel should not be lea® than one and a half to twice the least ijter of the orifice. fipMe or supprmmd contractimi. An example of incom- tmtraction is shown in Fig. 46, the lower edge of the Jar orifice being made level with the bottom of the vessel. time effect is produced by pla^ring a horizontal plate in VB»el level with tlie bottom of the orifice. The stream ii rlie lower part of the orifice are normal to its plane 'the contraction at the lower edge is consequently suppressedp Fig, a. Similarly, if the width of a rectangular orifice is made equal ^?6selj or the orifice ahcd m pro\'ided with side walls J he side or lateral contraction is suppressed. In any ttippressed contraction the discharge is increased, but, as later, the discharge coeificient umy vary more than n traction is complete. To suppress the contraction ly» the orifice must be made of such a form that the liiie& biM^otne parallel at the orifice and normal to its plane. 54 HYDRAULICS Experimental deiermination of c. The section of the stream from a circular orifice can be obtained with considerable accu- racy by the apparatus shoT^Ti in Fig, 40, which consists of a ring having four radial set screws of fine pitch, Tlie screws are adjusted until the points thereof touch the jet. M. Bazin has recently lised an octagonal frame v^ith twenty- four set screws^ all radiating to a common centre, to deter- mine the form of the section of jets from various kinds of orifices. The screws were adjusted until they just touched the jet. The frame was then placed upcm a sheet of paper and the positions of the ends of the screwi marked upon the paper. The forms of the 8^:stions could theo be obtained, and the areaa measured with considerable accuracy. Some of the results obtained are shown in Table IV and also in the section on the form of the liquid vein, 38. Coefflcient of velocity for sliarp-edged orifice. The theoretifai velocity through the contracted section is, m shown in section 36, equal to V2^H, but the actual velocity Vi is slightly less than this due to friction at the orifice. The ratio — = 4 is called the coeflicient of velocity. ^ Eayperimental determination of k, Tliere are two methods adopted for determining k experimentally. First method. The velocity is determined by measuring the discharge in a given time under a given head, and the cross sectional area w of the jet* as explained in the last paragraph, is also obtained. Then, if Vi is the actual velocity, and Q the discharge per second, and Jc^-^ Second method. An orifice, Fig, 50, is formed in the aide of a vessel and water alloweil to fiow from it. The water after leaving the orifice flows in a parabolic curve. Above the orific e is fixed a horizontal scale on which is a slider carrying a vertical scale, i to the bottom of which is clamped a bent piece of wire, w ^tli a sharp 1 i FLOW THROUGH ORIFICES 55 point The vertical scale can be adjusted so that the point touches the upper or lower surface of the jet, and the horizontal and vertical distances of any point in the axis of the jet from the centre of the orifice can thus be obtained. Fig. 50. -Assume the orifice is vertical, and let Vi be the horizontal velocity of flow. At a time t seconds after a particle has passed the orifice, the distance it has moved horizontally is X = Vit (1). The vertical distance is y = hgt' (2). Therefore y = \g —% and -. = V|- The theoretical velocity of flow is Therefore h = -J= = ^ ?-— . j2gK 2'JyB. h is better to take two values of x and y so as to make allowance for the plane of the orifice not being exactly perpen- dicular. B the orifice has its plane inclined at an angle to the ^^cal, the horizontal component of the velocity is Vi cos and ^e verrical component Vi sin 9. At a time t seconds after a particle has passed the orifice, the horizontal movement from the orifice is, X = Vi COS Ot (1), and the vertical movement is, y = v,smet + yt^ (2). After a time ii seconds ah = ViCos^^ (3), yi = ViamOti + igt{' (4). 56 HYDRAULICS Substituting the value of t from (1) in (2) and U from (3) in (4), y=''*^^*^2i#^ ®' and, y, = «,tan« + 2^,^ (6). From (5), 2V^.^W1 Substituting for Vi' in (6), i^^e^yi^^^^ (8). Having calculated tan 6^ sec can be found from mathematics^ tables, and from (7) Vi can be calculated. Then ^ sl2gR' 39. Bazin's experiments on a sharp-edged orifice. In Table IV are given values of A; as obtained by Bazin from experiments on vertical and horizontal sharp-edged orifices, for various values of the head. The section of the jet at various distances from the orifice waa carefully measured by the apparatus described above, and the actual discharge per second was determined by noting the time taken to fill a vessel of known capacity. The mean velocity through any section was then Q being the discharge per second and A the area of the section. The fall of free level for the various sections was different, and allowance is made for this in calculating the coefficient h in the fourth column. Let y be the vertical distance of the centre of any section below the centre of the orifice ; then the fall of free level for that section is H + y and the theoretical velocity is V2sr(H + y). The coefficients given in column 3 were determined by dividing the actual mean velocity through different sections of the jet by J2gT3.y the theoretical velocity at the centre of the orifice. Those in column 4 were found by dividing the actual mean velocity through the section by \/2g (H + y), the theoretical velocity at any section of the jet. The coefficient of column 3 increases as the section is taken further from the jet, and in nearly all cases is greater than unity. \ FLOW THBOUGH ORIFICES 57 TABLE IV. Sharp-^dged Orifices Contraction Complete. Table showing the ratio of the area of the jet to the area of the orifice at definite distances from the orifice, and the ratio of the mean velocity in the section to >/2grH and to J2g.(R + y)f H being the head at the centre of the orifice and y the vertical distance of the centre of the section of the jet from the centre of the orifice. Vertical circular orifice 0*20 m. ('656 feet) diameter, H = '990 m. (3-248 feet). Coefficient of discharge m, by actual measurement of the flow is m = -5977*. Mean Velocity >/2</(H + y) = k I^tanoe of the Beetion ^ni the plane of the orifice in metres (K)8 013 017 0-235 0-335 0-516 Area of Jet Area of Orifice =c •6079 •6971 •5951 •6904 •5830 •5690 Mean Velocity 0-983 1001 1-004 1-012 10^5 1-050 •998 •999 1003 1^007 1^010 Horizontal circular orifice 0*20 m. ('656 feet) diameter, fl = -975m. (3-198 feet). 0-075 0-093 0-110 0128 0-145 0163 m = 0*6035. 0-6003 0-5939 0-5824 0-5734 0-5658 0-5597 1-005 1016 1036 1053 1-067 1-078 0-968 0-971 0-982 0-990 0-996 0-998 Vertical orifice '20 m. (-656 feet) square, H = '953 m. (3126 feet). m = 0'6066. 0-151 0-6052 1-002 •997 0175 0-6029 1-006 1-000 0-210 0-5970 1-016 1-007 0-248 0-5930 1-023 1-010 0-302 0-5798 1046 1-027 0-350 0-5788 1-049 1024 The real value of the coefficient for the various sections is lowever that given in column 4. For the horizontal orifice, for every section, it is less than inity, but for the vertical orifice it is greater than unity. Bazin's results confirm those of Lesbros and Poncelet, who in See section 42. 58 HTDRAULICS 1828 found that the actual velocity through the contracted section of the jet, even when account was taken of the centre of the section of the jet being below the centre of the orifice, waa ■^ greater than the theoretical value. This result appears at first to contradict the principle of the conservation of energy, and Bemouilli's theorem. 1 It should however be noted that the vertical dimensions of the ! orifice are not small compared with the head, and the explanation of the apparent anomaly is no doubt principally to be found in the fact that the initial velocities in the different horiuzontal filaments of the jet are different. Theoretically the velocity in the lower -ps^rt of the jet is greater than J2g (R + y), and in the upper part less than J2g (H + y). Suppose for instance a section of a jet, the centre of which is 1 metre below the free surface, and assume that all the filaments have a velocity corresponding to the depth below the free surface, and normal to the section. This is equivalent to assuming that the pressure in the section of the jet is constant, which is probably not true. Let the jet be issuing from a square orifice of '2 m. ("656 feet) side, and assume the coefficient of contraction is '6, and for simplicity that the section of the jet is square. Then the side of the jet is '1549 metres. The theoretical velocity at the centre is \/2gr, and the discharge assuming this velocity for the whole section is '6 X '04 X J2g = '024 J2g cubic metres. The actual discharge, on the above assumption, through any horizontal filament of thickness dh, and depth fe, is oQ = 0-1549xdAxN/2^, and the total discharge is /-l-OTTS Q = 0-1549^2^ h^dh ^ ^ y-9225 se^.. = '0241 n/2^. 'e theoretical discharge, taking account of the varying heads The co?^®' r004 times the discharge calculated on the assumption jn^ , -head is increased this diiierence dimmisnes, and when ^ J ®. CTeater than 5 times the depth of the orifice, is very Those m cc velocity througi^ ^jg^^ agrees very approximately with that given velocity at any se. gquare orifice, where the value of k is given bs The coefficient . further from the jet, rum THROUGH ORIFICES 59 Uris partly then, ejq^lams the anomalous values of k^ but it UPO< be Ifxiked upon as a complete explanation* B^ < tual jet are not. exactly those assumed, H)» fry normal to the plane of the section is EtttUly much more complicated than here Essumed. iAm Baein further points out, it m probable that, in jets like |pn^m the square urifiee, which, as will be seen later when the jbof the jet i« considered, are sabject to considerable deformation, ^diverfent^-e of m^me of the iilaments gives rise to pressures less BQ that of the atmosphere. ^k litteinpted to demonstrate thist experimen tally , and Hti> ', Fig, 150, registered pressures less than that of the Imosphen*; but he doubts the reliability of the results, and Mte out the extreme difficulty of satisfactorily determining the ^pix* in the jet, ^^imt Uie inequality of the velocity of the filaments is the tmmrf CHUt^e, receives support from the fact that for the ^vnoQlal orifice, discharging downwards, the coefficient k is luili difhtly less than unity. In this case, in any horizontal lolimj below tlie orifice, the head is the same for all the stream 1166, fti>d the velocity of the filaments is practically constant. np: '*^" ^ of velocity is never less than '96, bo that the loss ^P'' rnal friction of the liquid is very small, 40. Distribtitioii of velocity in the plane of the orifice. Btttn has examined the distribution of the velocity in the ioois sections of the jet by means of a fine Pi tot tube (see ^ 215). In the plane of the orifice a minimum velocity teow, which for vertical orifices is just aliove the centre, but at a |||4kiaiice from the orifice the minimum velocity is at the top F()r orifices having complete contraction Bazin found the toimaiii velocity to be '62 to '64 n/2^H, and for the rectaTigular ttifioft, frith lateml contraction suppressed, 0'69 v^2^H. Ab the ilii^tance from the plane of the orifice increases, the riucities in the transverse section of the jets from horizontal rtfices, rapidly become uniform throughout the transverse section* For vertical orifices, the velocities below the centre of the jet 1*^ greater than those in the upper part. 41. Presstire in the plane of the orifice. M* Lagerjelm stated in 1826 that if a vertical tnbe open at was placed with its lower end near the centre, and not uiy below the plane of the inner edge of a horizontal I I \ 60 HYDRAULICS orifice made in the bottom of a large reservoir, the water rose in the tube to a height equal to that of the water in the reservoir, that is the pressure at the centre of the orifice is equal to the head over the orifice even when flow is taking place. M. Bazin has recently repeated this exx)eriment and founds that the water in the tube did not rise to the level of the water in the reservoir. If Lagerjelm's statement were correct it would follow that the velocity at the centre of the orifice must be zero, which again does not agree with the results of Bazin's experiments quoted above. 42. Coefficient of discharge. The discharge per second from an orifice, is clearly the area of the jet at the contracted section GK multiplied by the mean velocity through this section, and is therefore, Q=c.fc.aN^H. Or, calling m the coefficient of discharge, This coeflScient m is equal to the product c . A;. It is the only coefficient required in practical problems and fortunately it can be more easily determined than the other two coefficients c and h. Experimental determination of the coefficient of discharge. The most satisfactory method of determining the coefficient of discharge of orifices is to measure the volume, or the weight of water, discharged under a given head in a known time. The coefficients quoted in the Tables from M. Bazin*, were determined by finding accurately the time required to fill a vessel of known capacity. The coefficient of discharge m, has been determined with a great degree of accuracy for sharp-edged orifices, by Poncelet and Lesbrost, WeisbachJ, Bazin and others §. In Table IV Bazin's values for m are given. The values as given in Tables V and VI may be taken as representative of the best experiments. For vertical, circular and square orifices, and for a head of about 3 feet above the centre of the orifice, Mr Hamilton Smith| junr. II, deduces the values of m given in Table VI. ♦ Annales dfs Pont$ et ChausBies^ October, 1888. t Flow through Vertical Orijices, X Mechanics of Engineering. § Experiments upon the Contraction of the Liquid Vein. Bazin translated bj Trautwine. II Tfw. Flow of Water through Orijices and over Weirs and through open CondmUi ' and Pipes, Hamilton Smith, junr., 1886. FLOW THBOUOH ORIFICES 61 TABLE V. Si|MriiiieDtar Partioulara of orifice Coefficient of discharge m Baxm Pbooelet and Leabitn Bam ' n n " 1 Vertioal aquaie orifice side of square 0-6662 ft. » »» t« 9, Vertical Rectangular orifice -666 ft. high x 2-624 ft wide with side contraction suppressed Vertical circular orifice 0*6662 ft. diameter Horizontal 0-8281 0-606 0-606 0-627 0-698 0-6086 0-6068 TABLE VI. Cvrcvlar orifices. DiuKterof infti 0O197 I 0-627 0O296 0^17 0-089 0-611 0O492 0-606 0-0984 0-608 I 0-164 0-600 0-328 0-699 0-6662 0-698 0-9848 0-597 Square orifices. IT Side^of^uare ^^^ I ^y^^^ \ ^^^ 0-681 0-612 0-607 0-197 ! 0-6906 I 0-9843 0-605 I 0-604 I 0-603 I ! TABLE VIL ' Table showing coefficients of discharge for square and rect- angular orifices as determined by Poncelet and Lesbros. 1 HeftJ of water ' Width of orifice -6502 feet Width (»f orifice 1 -908 feet i ilwre the top , of the orifice : Depth of orifice in feel • in feet 0328 -0656 •0984 •1640 •607 •3287 6562 •0656 6562 •0328 701 •660 •630 •0656 694 -659 -634 •615 -596 572 •643 -1312 683 •658 •640 •623 •603 582 •642 ] 595 •26-24 670 -656 •638 •629 •610 589 •640 1 601 •3937 663 •653 •636 •630 •612 593 •638 1 603 •6562 655 •648 •633 •630 •615 598 •635 1 605 1-640 1 642 •638 •630 •627 •617 604 •630 1 607 3-281 632 -633 -628 •626 •615 605 •626 1 605 4-921 616 -619 -620 •620 •611 602 •623 1 602 6-562 611 -612 -612 •613 •607 601 •620 , 602 9-^*48 609 •610 -608 •606 •603 601 •615 1 601 62 HTDRAUUCT The heads for which Bazin determined the coefficientB Tables IV and V varied only from 2"6 to 3"3 feet, but, as wilJ seen from Table YII, deduced from reaults given by Poiicelet Lesbros* in their cla^cal work, when the variation of head is small, the coefficienta for rectangular and square orifices vary considerably with the head. 43. Effect of suppressed contraction on the coeffideni of discharge* Sharp^ged orifice. When &ome part of the contraction of & transverse section of a jet issuing from an orifice is suppressed^ the cross sectional area of the jet can only be obtained witk diflSculty. The coefficient of ditwharge can, however^ be easily obtained, as before, by determining the discharge in a given time. Ttfr^ most complete and accurate experimenta on the effect of contrac- tion are those of Lesbros, some of the results of which are quoted in Table VIII. The coefficient is most constant for square or rectangular orificee when the lateral contraction is suppressed. The- reason being, that whatever the head, the %'ariation in the section of the jet i^ confined to the top and bottom of the orifice, the width of the stream remaining constant, and therefore in a greater part of the transverse section the stream lines are normal to the plane of the orifice* According to Bid one, if iP is the fraction of the periphery of a sharp-edged orifice upon which the contraction is suppressed, and m the coefficient of discharge when the contraction is complete, then the coefficient for incomplete contraction is, Wi = m (1 + "IStc), for rectangular orifices, and for circular orifices, Bidone's formulae give result* agreeing fairly well with Lesbros' experiments. His formulae are, howeverj unsatisfactory when x approaches unity, as in that case mi should be nearly unity. If the form of the formula is preserved, and m taken ae '606^ for mi to be unity it would require to have the valne, i jrii - m (1 + '65ir), ^ For accurate measurements, either orifices with perfect a traction or, if possible, rectangular or square orifices with lateral contraction completely suppressed, should be used. It ^ * Experiences hydrauliquee tur Us lois de Vicoulement de Veau h ^eSf etc., 1882. Ponoelet and Lesbros. FLOW THK017OH ORIFICES 6S Qf be neceeeaiy ki caljbmte the orifice for variotis heads, W as shown above the coefficient for the latter kind is more likely b] be constant. TABLE Vni, Table showing the effect of soppressing the contraction on the efiidetit of discharge. Lesbroa *, Square vertical orifica 0"65d feet square. 1 R«v4 ,j walCT Side con- Contrfttitiot] Contraction Ppet Sh&rp^^dged tr&otjon Buppres^d al HnppFeased at the hmet tmd Bide edgen t ■ ■ -mfiee Bappreseed the lower edge 1 mm 0572 0-599 0-1640 i>585 0*6S1 0^608 09281 0-592 0'631 0-615 0*6562 0-596 0*632 0-621 0*708 , imo 0-6(^ 0^631 0-623 0-680 W81 0-606 (>-628 0-624 0676 4^931 0-602 0*627 0*624 0'672 fl^62 0-601 0^626 0-619 0-668 ^^§& 0^1 0-624 0*614 0^665 ' Fig* 51, Section of Jet from cireokr orifice. W, Tlie fonn of tlie jet from sharp-edged orifices. From a circular orifice the Jet emerges like a cylindrical rod iirf retaimt a form nearly cylindrical for some distance from the hg. 51 3how8 three sections of a jet from a vertical circular [(►nfice u varying distances from the e, m given by M, Bazin, The flow from square orifices m Dpanied by an int-eresting and |c«riou« phenomenon called the in- of the jet. At a vitry small distance from llhH tjrifice the section becomes as Y^ifmi in Fi|f* 52, The aides of the t ant concave and the comers* are ve sections. The lies octagonal as in \ and afterwards takes the form of a square with concave [and rounded comers, the diagonals of the square being cuhir to the sides of the orifice, Fig. 54» DOO Figs. 62 — 54. Section of jet froni aqtiare orifice* * Kjp^fimefiU h§drmUiqueM tur Ut loi$ d^ ricmdent^nt de Vtau^ ^?':riare orifice ^« tlxe coe .j^T- GO- ^"^ c^^^ 64 HTDRAUUCS 45. Large orifioea Table VII shows very clearly that if the depth of a vertical orifice is not small compared with the head, the coefficient of discharge varies very considerably with the head, and in the discussion of the coeflScient of velocity fc, it has already been shown that the distribution of velocity in jets issuing from such orifices is not uniform. As the jet moves through a large orifice the stream lines are not normal to its plane, but at some section of the stream very near to the orifice they are practically normal. If now it is assumed that the pressure is constant and equal to the atmospheric pressure and that the shape of this section is known, the discharge through it can be calculated. Rectangular orifice. Let efgh, Fig. 55, be the section by a vertical plane EF of the stream issuing from a vertical rectangular orifice. Let the crest E of the stream be at a depth h below the free surface of the water in the vessel and the under edge F at a depth h^. Fig. 66. At any depth h, since the pressure is assumed constant in the section, the fall of free level is h, and the velocity of flow through the strip of width dh is therefore, kJ2gh, and the discharge is kbJ2ghdh. If & be assumed constant for all the filaments the total discharge in cubic feet per second is Q = kJ2^j\hidh = f J2gkb (hi*- ho*). Here at once a difficulty is met with. The dimensions ^, hi and b cannot easily be determined, and experiment shows tJial they vary with the head of water over the orifice, and that they cannot therefore be written as fractions of Ho, Hi, and B. FLOW THBOUQH ORIFICES 65 By replacing %«, &i and b by Ho, Hi and B an empirical formnla of the same form is obtained wbicli, by introducing a coefficient c, can be made to agree with experiments. Then Q = 5c^^.B(Hlt-Ho*), or replacing |c by n, Q = nV^.B(Hi»-Ho») (1). The coefficient n varies with the head Ho, and for any orifice the simpler formnla Q=m.a.^J2gR (2), a being the area of the orifice and H the head at the centre, can be used with equal confidence, for if n is known for the particular orifice for yarious values of Ho, m will also be known. From Table YII probable values of m for any large sharp- edged rectangular orifices can be interpolated. Rectangular slvices. If the lower edge of a sluice opening is some distance above the bottom of the channel the discharge through it will be {vractically the same as through a sharp-edged orifice, but if it is flush with the bottom of the channel, the contraction at this edge is suppressed and the coefficient of discharge will be slightly greater as shown in Table VIII. 46. Drowned orifices. "WTien an orifice is submerged as in Fig. 56 and the water in the up-stream tank or reservoir is moving so slowly that its velocity may be neglected, the head causing velocity of flow through any filament is equal to the difference of the up- and down-stream levels. Let H be the difference of level of the water on the two sides of the orifice. Fig. 56. L. H. 66 HYDRAULICS ■h^^hf Consider any stream line FE which passes through the orifice at B. The pressure head at E is equal to Aj, the depth of E below the down-stream level. If then at F the velocity is zero, 29[ or Ve = ^2g (h - A,) or taking a coefficient of velocity k VE = kJ2g.B., which, since H is constant, is the same for all filaments of the orifice. If the coefficient of contraction is c the whole discharge through the orifice is then Q = ckas/2gR = m.a, J2gK. 47. Partially drowned orifice. If the orifice is partially drowned, as in Fig. 57, the discharge may be considered in two parts. Through the upper part AC the discharge, using (2) section 45, is and through the lower part BC Q2=mi.ai.N/2sr.Hi. 48. Velocity of approach. BS^= I Fig. 57. It is of interest to consider the efEect of the water approaching an orifice having what is called a velocity of approach, which will be equal to the velocity of the water in the stream above the orifice. In Fig. 56 let the water at F approaching the drowned orifice have a velocity Vi. Bemouilh's equation for the stream line drawn is then 2g + h,: J.^'"' ■'^^2^' and Ve=>/2^(r^^) , which is again constant for all filaments of the orifice. Then Q = m.a.N^.(H+gy. SUDDEN ENLABOEMEirr OF A STREAM 67 49. Effecit of velocity of approach on the discharge throng a large reetangnlar orifice. If the water approaching the large orifice, Fig. 55, has a velocity of approach t?i, Bemonilli's equation for the stream hne passing throug^h the strip at depth hy will be w 2g w 2gr' Pm being the atmospheric pressure, or putting in a coefficient of velocity, The discharge through the orifice is now, 50. CoefELcient of resistance. In connection with the flow through orifices, and hydraulic plant generally, the term " coefficient of resistance " is frequently nsed. Two meanings have been attached to the term. Some- times it is defined as the ratio of the head lost in a hydraulic system to the effective head, and sometimes as the ratio of the head lost to the total head available. According to the latter method, if H is the total head available and h/ the head lost, the coefficient of resistance is Cr jj. 51. Sudden enlargement of a current of water. It seems reasonable to proceed from the consideration of flow throagh orifices to that of the flow through mouthpieces, but before doing so it is desirable that the effect of a sudden enlargement of a stream should be considered. Suppose for simplicity that a pipe as in Fig. 58 is suddenly enlarged, and that there is a continuous sinuous flow along the pipe. (See section 284.) I At the enlargement of the pipe, the J stream suddenly enlarges, and, as shown L in the figure, in the comers of the large pipe it may be assumed that eddy motions p. gg are ^t up which cause a loss of energy. 5—2 68 HYDRA ULICS Ck>nsider two sections oa and dd at each a distance from bh that the flow is steady. Then, the total head at dd equab the total head at oa minaii the loss of head between oa and dd^ orH hm the loss of head doe to shock, then Va Vj Pd Vd 1 w 2g w 2g Let A« and A^ be the area at oa and dd respectively* Since the flow past oa a ' ^ ^^^^ ^^ Then, assuming that each velocity Vo, and r^ at dd^ tlie m' which passes aa in unit time i^ < of the water that passes dd is t nt of fluid at aa has iin of the quantity of waterl w - Afli'rt'i and the tDOmentmii 9 ' the momentum of a mass of M ide moving with a velo V feet per second being Mt* pounds feet. The change of tnonientum is therefore, The forces acting on the water between tm and dd to produce this change of moment um, are p<tAa acting on aa, p^A^ acting on dd^ and, if p is the mean pressure per unit area on the annular riu^ hhy an additional force p{Ad- A^)^ There is considerable doubt as to what is the magnitude of the pressure p, but it is generally assiuwed that it is equal to pa, for the following reason. The water in the enlarged portion of the pipe may be looked upon as divided into two parts, the one part having a motion of translation, while the other part, which is in contact with the annular ring, is practically at rest* (See section 284 J K this assumption is correct, then it is to be expected that the pressure throughout this atill water will be practically eqtial at all points and in all directions, and must be equal to the pressure ia the stream at the section 65, or the pressure p is equal to p«. Therefore PcjA<| - Pa (Ad - A«) -paAa = 'M?-— -(V«-Vd), KkVa 9 from which (pd - po) A4 = to (t?o — t?d) ; SUDDEN ENLARGEMENT OF A STREAM 69 and aince A«t7« = A^rVd, 9 Adding ^ to both sides of the equation and separating 2Sr ~ into two parts. vr or & the loss of head dae to shock is eqaal to According to St Venant this quantity should be increased by 1 Vd^ wi amount equal to 5 nZ > ^^^ this correction is so small that as & nile it can be neglected. 52. Sudden contraotion of a current of water. Suppose a pipe partially closed by means of a diaphragm as in Fig. 59. Afi the stream approaches the diaphragm ""■which is supposed to be sharp-edged — it contracts in a similar way to the stream passing through an orifice on the side of a vessel, so that the minimum cross sec- tional area of the flow will be less than the Fig. 59. area of the orifice. The loss of head due to this contraction, or due to passing through the orifice is small, as seen in section 39, but due to the sodden enlargement of the stream to fill the pipe again, there is a considerable loss of head. Let A be the area of the pipe and a of the orifice, and let c be the coefficient of contraction at the orifice. Then the area of the stream at the contracted section is ca, and, therefore, the loss of head due to shock 2g 70 HTDRAULICS If the pipe simply diminishes in diameter as in Fig. 58, the section of the stream enlarges from the contracted area oa to fill j the pipe of area a, therefore the loss of head in this case is "-'iil-^)' «• Or making St Venant correction ^-m-')'^i} «)• Valtte of thecoefficient c. The mean valneof cfor aaharp-edged circular orifice is, as seen in Table IV, about 0*6, and this maj be taken as the coefiicient of contraction in this formula. Substituting this value in equation (1) the loss of head k found to be -^— , and in equation (2), -g— ,v being the velodUy iB J 0*5o^ the small pipe. It may be taken therefore as i^a'' ^^^'*'^^] experiments are required before a correct value can be assigned. 53. Loss of head due to sharp-edged entrance Into a pipi^ or mouthpiece. When water enters a pipe or mouthpiece from a vessel throogk a sharp-edged entrance, as in Fig. 61, there is first a contraction, and then an enlargement, as in the second case considered in section 52. The loss of head may be, therefore, taken as approximately -5-- and this agrees with the experimental value of — ^ given by Weisbach. This value is probably too high for small pipes and too low far large pipes*. 54. Mouthpieces. If an orifice is provided with a short pipe or mouthpiece, througfc which the liquid can flow, the discharge may be very differem.'^ from that of a sharp-edged orifice, the difference depending npo* the length and form of the mouthpiece. If the orifice is cylindric^i^ as shown in Fig. 60, being sharp at the inner edge, and so shor^ that the stream after converging at the inner edge clears th-^ outer edge, it behaves as a sharp-edged orifice. Short external cylindrical mouthpieces. If the mouthpiece 1J| cylindrical as ABFE, Fig. 61, having a sharp edge at AB anC3 a length of from one and a half to twice its diameter, the je^*^ * See M. Bazin, Exp€rience» nouvelUs sur la distribution des vite$se» dam^ le$ tuyaux. FLOW THROUGH MOUTHPIECES 71 contracts to CD, and then expands .to fill the pipe, so that at EF it discharges foil bore, and the coefficient of contraction is then onitF. Experiment shows, that the coefficient of discharge is — B Fig. 60. Fig. 61. from 080 to 0"85, the coefiicient diminishing with the diameter of the tube. The coefficient of contraction being unity, the coefficients of velocity and discharge are equal. Good mean ^iies, according to Weisbach, are 0*815 for cylindrical tubes, w»d 0*819 for tubes of prismatic form. These coefficients agree with those determined on the assump- ^on that the only head lost in the mouthpiece is that due to sndden enlargement, and is 0-5i;" ^ being the velocity of discharge at EF. Applying Bernouilli's theorem to the sections CD and EF, and taking into account the loss of head of -p— , and pa as the atmo- spheric pressure, w 2g w 2g 2g w^ or -7^^ — = H. Therefore 2g v^ = '66 X 2gR md i? = -812N/2^H. The area of the jet at EF is a, and therefore, the discharge er second is a.v = '8V2as/2gR. Or m, the coefficient of discharge, is 0*812. The pressure head at the section CD. Taking the area at CD s 0-606 the area at EF, rcD = l'66t7. 72 HYDRAULICS Therefore S^ = S^ . ^' - 2|2l^ Pa _ lf^\ or the pressure at C is less than the atmospheric pressure. K a pipe be attached to the mouthpiece, as in Fig. 61, and tt^ lower end dipped in water, the water should rise to a height of abo^c- — s — feet above the water in the vessel. 55. Borda's mouthpiece. A short cylindrical mouthpiece projecting into the vessel, as ^B Fig. 62, is called a Borda's mouthpiece, arid is of interest, as tfcr^ coefficient of discharge upon certain assumptions can be readiK- calculated. Let the mouthpiece be so short that the jet issuing at EF falls clear of GH. The orifice projecting into the liquid has the effect of keeping the liquid in contact with the face AD practically at rest, and at all points on it except the area EF the hydrostatic pressure will, therefore, simply " depend upon the depth below the free ^^' surface AB. Imagine the mouthpiece produced to meet tli0 face EC in the area IK. Then the hydrostatic pressure on AID, neglecting EF, will be equal to the hydrostatic pressure on BC, neglecting IK. Again, BC is far enough away from EF to assume that the pressure upon it follows the hydrostatic law. The hydrostatic pressure on IK, therefore, is the force whicl gives momentum to the water escaping through the orifice, ove^ comes the pressure on EF, and the resistance of the mouthpiece. Let H be the depth of the centre of the orifice below the free surface and p the atmospheric pressure. Neglecting frictional resistances, the velocity of flow v, through the orifice, is j2glEL. Let a be the area of the orifice and cu the area of the transverse section of the jet. The discharge per second will be it; . w J2gK lbs. The hydrostatic pressure on IK is pa + wdK lbs. The hydrostatic pressure on EF is pa lbs. The momentum given to the issuing water per second, is M = -.o,.2^H. Therefore pa + — o> 2gK = pa + wdR, and (I) = ia. FLOW THROUGH MOUTHPIECES 73 The coefficient of contraction is then, in this case, equal to one half. Experiments by Borda and others, show that this result is justified, the experimental coefiicient being slightly greater than \. 56. Conical monthpieoes and nozzles. These are either convergent as in Fig. 63, or divergent as in Fig. 64. j=^ Fig. 63. Fig. 64. Calling the diameter of the mouthpiece the diameter at the ontlet, a divergent tube gives a less, and a convergent tabe a greater discharge than a cylindrical tube of the ®^ diameter. Experiments show that the maximum discharge for a convergent tube is obtained when the angle of the cone is from 12 to 13^ degrees, and it is then 0*94 . a . J2gh, K instead of making the convergent mouthpiece conical, te sides are curved as in Fig. 65, so that it follows as ear as possible the natural form of the stream lines, the ^efficient of discharge may, with high heads, approxi- ate very nearly to unity. Weisbach*, using the method described on page 55 determine the velocity of flow, obtained, for this >nthpiece, the following values of k. Since the mouth- p. gg ?ce discharges full the coefficients of velocity k and icharge m are practically equal. Head in feet 0-66 1-64 11-48 55-8 888 k and m •959 •967 •975 •994 •994 According to Freeman t, the fire-hose nozzle shown in Fig. 66 3 a coefficient of velocity of *977. • Mechanics of Engineering. t TramaetionM Am, Soc. C.E., Vol. xxi. 74 HYDRAULICS If the mouthpiece is first made convergent^ and then divergent, Fig. 66. as in Fig. 67, the divergence being sufficiently gradual for the stream lines to remain in contact with the tube, the coefficient of contraction is unity and there is but a small loss of head. The velocity of efflux from EP is then nearly equal to >/2gfH and the discharge is m . a . N/2grH, a being the area of EP, dnd the coefficient m approximates to unity. It would appear, that the discharge could be increased indefinitely by length- ening the divergent part of the tube and thus increasing a, but as the length increases, the velocity decreases due to the friction of the sides of the tube, and farther, as the discharge increases, the velocity through the contracted section CD increases, and the pressure head at CD consequently falls. Calhng Pa the atmospheric pressure, pi the pressure at CSD> and Vi the velocity at CD, then w and 2g w w w 2g If s^ is greater than H + — , 2^ becomes negative. As pointed out, however, in connection with Proude's apparatoBi page 43, if continuity is to be maintained, the pressure cannot be negative, and in reaUty, if water is the fluid, it cannot be lew than 7 the atmospheric pressure, due to the separation of the air from the water. The velocity Vi cannot, therefore, be incr indefinitely. FLOW THROUGH MOUTHPIECES 75 Assmnin^ th© pressure can J list become zero, a,nd taking the eric preamre as equivalent to a head of 34 tL of water, the mm possible Yelocitjr, is Ur=^%/2ff CH + 34ft.) id the tnaiciTnnm ratio of the area of EF to CD is ^A^ S4ft H Praclically^ tlie maximtim value of vi may be taken as and the maximnin ratio of EF to CD as v/-^. The maxim HID discharge is y- ^ Tlie ratio gWen of EF to CD may be taken as the masdnaum beiween the area of a pipe and the throat of a Venturi meter [lo be used in the pipe. 57. Flow tliroiigh orifices and mouthpieces under constant freasure. The head of water causing flow through an orifice may be produced \yf a pomp or other mechanical means, and the discharge iB^ tdce place int-o a vessel, such as the condenser of a steam , in which the pressure is less than that of the atmosphere. example, suppose water m be discharged from a cylinder I a vemel B, Fig, 6H^ through It orifice or moathpiece by means U iMtoEi loaded with P lbs., and fft^preettre per sq. foot in B Lpt the area of the piston be A ijuare feet. Let h be the height of the water in the cylinder above *fc« ORiti« of the orifice and fh of ^ water in the vesael B. The tlworetical effective head forcing water through the orifice may be writlm Fig, 68. Aw w 76 HTDRAITLICS If P is large K and h will generally bo negligible. At the orifice the pressure head la K-^^t a^id therefore for w any stream line through the orifice, it there is no friction^ 2g w Aw The actual velocity will be less than u, due to frictioG, and Kl] IB a coefiicient of velocity, the velocity is then and the discharge i^i Q-m.a*J2gH. In practical examples the cylinder and the vessel will geneiaDy be connected by a short pipe, for which the coefficient of velocity will depend upon the length. If it is only a few feet long the principal loss of head will be at the entrance to the pipe, and the coefficient of discharge will probably vary between 0*65 and 0'85, The effect of lengthening the pipe mil be understood aft'er tbd chapter on flow through pipes has been read. Example. Water is di^ehai-t^ed from a pamp inta & condenaer in ^hioh Hift pressure is 3 lbs. per aq. inch Ibroui^li a f^ort pipe S^ inchea diameter. The pressure in the puiup i& 2Q iba. per kij. tneh. Find the dischargi^ into the condenser, takinR the coefficieut of discharge 0'7S* The effective head is ^^20x144 3jcU4 as k^ 62-4 section v^. =39 a feet. falls. f Q=-?i>v '7M54 ^^- V . /- ' i** ' jjij4:^4i y. 3li*2 cubic feet per sec i Calling J =1'84 cubic ft. per sec. and Vi the Vi e of emptying a tank or reservoir. reservoir to have a sharp-edged horizontal orifice It is required to find the time taken to empty and -UjS vf the horizontal section of the reservoir at any If ^ is greater t^ ^^^^ ^^ ^ gq f^^^ ^^^ ^1^^ ^^^ ^j ^^ As pointed out,ho>^ let the ratio ~ be sufficiently large that the page 43, if continuity . ? , , -. negative, and in reaUt? the reservoir may be neglected, than i the atmospheric J* *he water is at any height h above the from the water. The veV Ao^s through the orifice m any time dt indefinitely. FLOW THROUGH MOUTHPIECES 77 The amoimt dh by which the surface of water in the reservoir falls in tKe time dt is j_ maJ2ghd t A. ;.. Aoh or or = - - . — -. . ma v2gA* The time for the water to fall from a height H to Hi is ^^ f^ Adh_ ^ i_ r« Adh J ^tma ^gh a J2g J h, mh^ ' If A is constant, and m is assumed constant, the time required for the surface to fall from a height H to Hi above the orifice is _ . Adh ma ^2g • ^Vc-^H-n/Hi), ma J2g and the time to empty the vessel is ^ ^ 2 A n/H mu \/2g* or is equal to twice the time required for the same volume of wwter to leave the vessel under a constant head H. Time of emptying a lock with vertical drowned sluice. Let the 'water in the lock when the sluice is closed be at a height H, Fig^. 56, above the down-stream level. Then the time required is that necessary to reduce the level in the lock by an amount H. "When the flow is taking place, let x be the height of the water Ij^ the lock at any instant above the down-stream water. \jet A be the sectional area of the lock, at the level of the ^vrater in the lock, a the area of the sluice, and m its coefficient of discharge. The discharge through the sluice in time dt is cQ = m . a >/2gx . ot. If da? is the distance the surface falls in the lock in time cty then Adx = m^ J2gxdt, Aox or ^^ =" " /?r- 1 • m^ y/2gx^ \ To redace the level by an amount H, f ^ Adx ma *J2gxi H .J > : ni: of -: -.■■t±;:t-n: cf "-:- iTri A is r • i :•:&.; ^f 40 a To iischai^e 10 eft FLOW THROUGH ORIFICES AND MOUTHPIECES 79 (5) A jet 3 in. diameter at the orifice rises vertically 50 ft. Find its at 25 ft. above the orifice. (6^) An orifice 1 sq. ft. in area discharges 18 c. ft. per second under a head of 9 ft. Assuming coefficient of velocity =0*98, find coefficient of ooatraction. {7) The pressure in the pump cylinder of a fire-engine is 14,400 lbs. per sq. ft.; assuming the resistance of the valves, hose, and nozzle is such that the coefficient of resistance is 0*5, find the velocity of discharge, and the height to which the jet will rise. <8) The pressure in the hose of a fire-engine is 100 lbs. per sq. inch; the jet rises to a height of 150 ft. Find tlie coefficient of velocity. <9) A horizontal jet issues under a head of 9 ft. At 6 ft. from the orifice it lias faUen vertically 15 ins. Find the coefficient of velocity. i 10) Required the coefficient of resistance corresponding to a coefficient of velocity =0-97. Ill) A fluid of one quarter the density of water is discharged from a Tessel in which the pressure is 50 lbs. per sq. in. (absolute) into the fttmosphere where the pressure is 15 lbs. per sq. in. Find the velocity of d&Mharge. (12) Find the diameter of a circular orifice to discharge 2000 c. ft. per hour, under a head of 6 ft. Coefficient of discharge 0*60. (18 1 A cylindrical cistern contains water 16 ft. deep, and is 1 sq. ft. in cross section. On opening an orifice of 1 sq. in. in the bottom, the water lervel fell 7 ft. in one minute. Find the cocfficicut of discliarge. (14) A miner*s inch is defined to be the discharge through an orifice in a vertical plane of 1 sq. in. area, under an average head of 6J ins. Find the supply of water per hour in gallons. Coefficient of discharge 0*62. (15 1 A veJisel fitted with a piston of 12 sq. ft. area discharges water imder a head of 10 ft. What weight placed on the piston would double the jmte of discharge? (16) An orifice 2 inches square discharges under a bead of ICX) feet 1*888 cubic feet per second. Taking the coefficient of velocity at 0*97, find the coefficient of contraction. (17) Find the discharge per minute from a circular orifice 1 inch diaiiieter, under a constant pressure of 84 lbs. per sq. inch, taking 0*60 as the coefficient of discharge. (18) The plunger of a fire-engine pump of one quarter of a sq. ft. in ana is driven by a force of 9542 lbs. and the jet is observed to rise to a hdigjit of 150 feet. Find the coefficient of resistance of the apparatus. (19) An orifice 8 feet wide and 2 feet deep has 12 feet head of water ■bofe Hb centre on the up-stream side, and the backwater on the other ie is at the level of the centre of the orifice. Find the discharge if fli.»i«CH$2. 80 HYDRA tTLlCS (20) Tan e. ft, of watat per second flow thfQu^ a pipe of 1 aq. ft. Bxe^ which suddenly enlarges to 4 sq. ft. area. Taking the pressure at 100 lbs; par sq. ft. in the mmaUar part of the pipe, find (1) the head lost in aho (*2j the preaaure in the larger part, (3) the work expended in forcing water through the enlargement. (21) A pipe of 3" diameter ia suddenly enlarged to 5'^ diameter* A \ tuba containing marcnry is connected to two points, one on eaeh side of enlargement, at points where the flow ia steady. Find the differenee : level in the two Linabs of the U when water flows at the rate of 2 c, ft» second from tlie small to the large section and i^ice versd. The sp gravity of mercury is IB'6, Lond» Un, (22) A pipe is suddenly enlarged from 2| indies in diameter to inches in diameter. Water flows tlirough these two pipes from the i to the larger, and the discharge from the end of the higger pipe is twd gaDons per second. Find:— (a) The loss of head, and gain of pressure head, at the ment. (b) The ratio of head lost to velocity bead In small pipe. (28 J The head and tail water of a vertical -sided lock differ in levdl 12 ft The area of the lock bai^in is TOO sq. ft. Find the time of emp^flQf I the lock, through a sluice of 5 sq. ft. area, with a coefficient 0"6, Tli»1 sluice discharges helow tail water level. (24 1 A tank X200 sq. ft. in area discharges through an orifice 1 a^ilt J in area. Calculate the time retj^aired to lower the level in the tank frodi j 60 ft. to 26 ft. ahove the orifice. Coefficient of discharge 0"6. (26) A vertical-sided lock is 65 ft. long and 18 ft, wide. Lift li5 H ] Find the area of a sluice helow tail water to empty tlie lock in 5 minute!. | Coefficient 0'6* (26) A reservoir has a bottom width of 100 feet and a length of 3S0 J feet. The sides of the reservoir are vertical. The reservoir ia connected to a second reservoir of the same dimensioM I by means of a pipe 2 feet diameter. The surface of the water in tlie &til| reservoir is 17 feet above tliat in the other. The pii>6 is Iwlow the su of the water in both reservoirs. Find the time taken for the water in 1 two reservoirs to become level* Coefficient of discharge 0*8. 59* Notches and Weirs. Wben the sides of an orifice are prod need J so that they extend be- yond the free surface of the water, as in Figs. 69 and 70, it is called a notch, Notches are generally made tri- angular or rectangular as shown in the fignres and are largely used for gauging the flow of water* Fig. 09* Triangukt KoKfh, FLOW OYER WEIRS 81 For example, if the flow of a small stream is required, a dam is ooQstrticted scroti t^e stream and the wate^r allowed to pasg tliTotigli a noteli cut in a board or metal plate * Fig. 70, B£eta.»gulftr Notch. They can conveniently be used for measuring the compensation [irater to be supplied from collecting reservoirSj and also to gauge f fiupplj of water to water wheels and turbines. The tenn weir is a name givon to a structure used to dam np I a strnun and over which the water flows. The conditiomi of flow are practically the same as through ■ a rectangtilar notchj and hence such notches are generally caUedl ifreiri, and in wlmt folio wft ihe latter term only is used. The top of the weir corresponds to the horisfiontal edge of the notch and is called the sill of the weir. r ijf water flowing over a weir or through a notch is g* V 1 lied the vein, sheetj or nappe* The shape of the nappe depends upon the form of the sill and sid«6 of the weir, tlie height of the sill above the bottom of the oi>€ti^«ra channel J the width of the up-stream channel, and the c^iDitrocd/m of the channel into which the nappe falls. The effect of the form of the Bill and of the down-stream, lehaimiel will be considered later, butj for the present, attentioi J will lie confined to weirs with sharp edges, and to those iu which [the air has free access under the nappe so that it detaches itself ply from the weir as shown iu Fig, 70, 60, Rectangular sharp-edged weir. If the crest and sides of the weir are made sharp-edged^ aa ^wn in Fig. 70, and the weir i^ narrower than the approaching ",and the sill some distance above the bed of the stream, at the sill and at the ddes, contraction similar to that at » abarp^ged orifice, Tbr imrface of the water m it approaches the weir falls, taking J eitmid form, so that the thickness K, Fig. 70, of the vein over^ weir, id teea than H, the height, above the sill, of the water at t.a. 6 &ni&iULIC8 some distance from the weir. The height H, whicjh is ealledl|fl head over the weir, should be carefully measured at such a distaoM from itj that the water surface has not commenced to cumaj Fteley and Steams state, that this distance should be equal tol 2^ times the height of the weir above the bed of the stream^ 1 For the present, let it be assumed that at the point where H if J measured the water is at rest. In actual cases the water will I always have some velocity, and the effect of thi^ velocity will haFel to be considered later, H may be called the still water head oyer I the weir, and in aU the fonnulae following it has this meaning, I Side cofUraclwi, According to Fteley and Stearns the amount I by which the stream is contracted when the weir is sharp-edg^ I is from 0*06 to 0'12H at each side, and Francis obtained a mean of I O'lH. A wide weir may be divided into several baj^B by parti- 1 tions, and thert* may then be mt>re than two contractions, at each 1 of which the effective width of the weir will be diminished, if 1 Francis' value be taken, by 0*1 H. I If L is the total width of a rectangLilar weir and N the mimber I of contractions, the effective ^^'idth l^ Fig. 70, is then, I (L-O'lN). I When L is very long the lateral contraction may be neglect^* I Suppresdo7i of the cmitractton. The side contraction can be I completrely suppressed by mating the approaching channel with I vertical sides and of the same width as the weir, as was done fof I the orifice shown in Fig. 47. The width of the stream ia then I equal t-o the \\^dth of the sill. I 61. Derivation of the v^eir formula f^om that of a liidtJ orifice. I If in the formula for large orifices, p, 64, /i& is made equal te| zero and for the effective \\idth of the stream the length I ill substituted for b^ and k is unity, the formula becomes I Q = t^2gJ.h^ ....a), j If instead of hi the head H, Fig, 70, is substituted^ ondl a coefficient H introduced, I The actual width I is retained instead of L, to make allowBan for the end contraction which as explained above is equal to 0*13 for each contraction, I If the width of the approaching channel is made equal to thi width of the weir I is equal to L, J With N contractions I = (L - OIN), I and Q-|C^.(L-01N)H*. I If C is given & mean value of 0*625, I Q = 333CL-0aN)H* ..,-,,..(2). I FLOW OVER WEXBS 83 well-known formula deduced by Prancb* from^ of €jcperiments on sharp-edged w^eirs, itila, as an empirical one, is approximately cMjrrect and'j Me \*alne^ for the discharge, Ki^thcid of obtaining it from that for large orifices is, Lipen to very serious objection, as the velocity at F on EP, Fig* 70, is clearly not equal to zero, neither is the flaw at the surface perpendicnlar t^ the section EF, [»re«i&ttre on EF, as will be underst*:jod later (section 83) (Tkely to be constant, Ilhc* directions and the velocities of the stream lines are irorn those through a section taken near a sharp-edged [seen by comparing tho thickness of the jet in the two | li the Ci:»effictent of discharge, %e tthiirp-edged oritic^ ^vith mde contractions suppressed, of the thicknees of the jet Uj the depth of the orifice is not afferent fr*itn tlie coefficient of discharge, being about 625, llicknetos EF of the nappe of the weir is very nearly 0'78Hj lie coefficient of discharge is practically 0'625, and the [is therefore V2A times the coefficient of discharge, pars therefore, that although the assumptions made in the t!ow through an orifice may be justifiable, pro\nding [above the top of the orifice is not very small ^ yet when riies xero, the assumptions are not approximately true. Igles which the stream lines make with the plane of EF I very different from 90 degrees, so that it would appear, ?rror principally arises fa^om the asaamption that the bronghout the section is uniform. f«ir special cases has carefully measured the fall of the i* and the tliickness EF, and if the assumptions of constant •e and stream lines perpendicular to EF are madei the •ge through EF can Vie calculated. r, example, the height of the point E above the sill of the ae of Baziu'S experiments was 0'n2H and the tliickness HBU. The fall of the point F is, therefore, 0*1 08H. constant pressure in the section, the discharge per foot He weir is, then, Mmm ^f V2^,Hi {C888}*^C108)^l • Lowall, BudrnMe Mi^erimtm, New York, 186S. 6—2 84 HYDRAULICS The actual discharge per foot width, by experiment, was g = 0-433 n/2^.H*, so that the calculation gives the discharge 1*228 greater than the actual, which is approximately the ratio of the thickness EF to the thickness of the stream from a sharp-edged orifice having a depth H. The assumption of constant pressure is, therefore, quite erroneous. 62. Thomson's principle of similarity. " When a frictionless liquid flows out of similar and similarly placed orifices in similar vessels in which the same kind of liquid is at similar heights, the stream lines in the different flows are similar in form, the velocities at similar points are proportional to the square roots of the linear dimensions, and since the areas of the stream lines are proportional to the squares of the linear dimensions, the discharges are proportional to the linear dimensions raised to the power of 4*-" Let A and B, Figs. 71 and 72, he exactly similar vessels with similar orifices, and let all the dimensions of A be n times those of B. Let c and Ci be similarly situated areas on similar stream lines. Fig. 71. Fig. 72. Then, since the dimensions of A are n times those of B, the fall of free level at c is n times that at Ci. Let v be the velocity at c and Vi at Ci. Then, since it has been shown (page 36) that the velocity in any stream line is proportional to the square root of the &I1 of free level, .'. V : Vi :: vn : 1. Again the area at c is n^ times the area at Ci and, therefore, the discharge through c ^ .- . the discharge through Ci ~ ^ s/n^n , which proves the principle. * British Association Reports 1858 and 1876. FLOW OVER WEIRS S5 63. Discharge throngh a triangular notch by the principle of similarity. Lies AI>C\ Figs, 73 and 74, b© a triangular notch. ^gr rr^ ^=^y^ — 4 '*-'• yV^ — D Fig. 74. the depth of the flow through the notch at one time be H and at another n , H. Stippfj^^ the area of the stream in the two cases to be divided into the same number of horissontal elements, such as ab and aibi , Then clearly the thickness of ab will be n times the thickness of Oi^w Let di&i be at a distance m from the apex B, and ah at a distance m^; then the width of ab is clearly n times the width The area of ab will therefore ben* times the area of ^i&i , Again, the head above ab is n tiniea the head above ai6i and therefore the velocity thraugh ah will be \/n times the velocity through a^bj and the discharge through ab will be n' times that through ai&i. More generally Thomson expresses this as follows : ** If tm'o triangular notches, similar in form, have water flowing through them at different depths, but with similar passages of apfiroach, the cross sections of the jets at the notches may be limitarly divided into the same number of elements of area, and the area^ of corresponding elements will be proportional to the aquaree of the Hnea! dimensions of the cross sections, or pro- portionai to the squares of the heads.*^ Ab the depth h of each element can be expressed as a fraction €f the head H, the velocities through these elements are propor- tional to the stjuare root of the head, and, therefore, the discharge ift propiirtional to H*. ~ Therefore Q a«o H^ being a coefficient which has to be determined by experiment. Prom experiments with a sharp-edged notch having an angle the rertex of 90 degrees, he found C to be practically constant 'jM baads and equal to 2*635. Tlien Q-2 635.H* .,,,..(3), Mmta 86 HTDRAULICS 64. Flow tbrongh a triangular notch. The flow through a triangalar notch is frequently given as Q=Aw>/27.BHt. in which B is the top width of the notch and n an experimental coefficient. It is deduced as follows: Let ADC, Fig. 74, be the triangular notch, H being the still water bead over the apex, and B the width at a height H above the apex. At any depth h the width b of the strip a, 6, is r^ — - • If the velocity through this strip is assumed to be v=kj2gh^ the width of the stream through a^fii , — — , and the thickness dh, the discharge tbrongh it is The section of the jet just outside the orifice is really less than the ares EFD. The width of the stream through any strip a,&j is less than a^bj, the surface is lovff than EF, and the apex of the jet is some distance above B. The diminution of the width of a^b^ has been allowed for by the coefficient c, vA the diminution of depth might approximately be allowed for by integrating between ^=0 and ^=:H, and introducing a third coefficient c^. Then Q = kecy^ /' ^B(H-fe) ^ — ^^ =TVcCifcN/2^-B-H*, Replacing ee-Jt by n Q=T^.nV27.BHt (4). Calling the angle ADC, 6, B=2Htan|, and Q=^^tn^/2^.tan|.H*. When 6 is 90 degrees, B is equal to 2H, and Taking a mean value for n of 0*617 Q= 2-636. H*, which agrees with Thomson's formula for a right-angled notch. The result is the same as obtained by the method of similarity, but the method of reasoning is open to very serious objection, as at no section of the jet are all th0 stream lines normal to the section, and A; cannot therefore be constant. The assumption that the velocity through any strip is proportional to Jh is also open to objection, as the pressure throughout the section can hardly be uniform. 65. Discharge through a rectangoiar weir by the principle of similarity. The discharge through a rectangular weir can also be obtained by the principle of similarity. FLOW OVSB WEIRS 87 Consider two rectangular weirs each of length L, Figs. 75 and 76, and let the head over the sill be H in the one case and Hi, or nH, in the other. Assume the approaching channel to be of such a form that it does not materially alter the flow in either case. K- L ^ ^ Fig. 75. B Fig. 76. To simplify the problem let the weirs be fitted with sides projecting up stream so that there is no side contraction. Then, if each of the weirs be divided into any number of equal XiartB the flow through each of these parts in any one of the weirs will be the same. Supxx)se the first weir to be divided into N equal parts. K N H then, the second weir is divided into J^ equal parts, the parts ±±1 in the second weir will be exactly similar to those of the first. By the principle of similarity, the discharge through each of the parts in the first weir will be to the discharge in the second H* as — ; , and the total discharge through the first weir is to the Hi* discharge through the second as N.H^ H^_ 1 N.H .Hi^ Hi^ n^' Hi Instead of two separate weirs the two cases may refer to the same weir, and the discharge for any head H is, therefore, pro- pjrtional to H^ ; and since the flow is proportional to L Q = C.L.H^ in which C is a coefficient which should be constant. 66. Rectangular weir with end contractions. If the width of the channel as it approaches the weir is greater than the width of the weir, contraction takes place at each side, and the effectual width of the stream or nappe is diminished ; the amount by which the stream is contracted is practically inde- pendent of the width and is a constant fraction of H, as explained above, or is equal to &H, k being about 0*1 88 HTDRAULTCS I Let the total width of each weir be now divided into three I parts, the width of ea4?h end part being equal to » . fc , H, The I width of the end parts of the transTerae section of the stream will ] each be i7i — l)k . H, and the width of central part L- 2«tH. I The flow through the central part of the weir will be equal to I Qi = C(L-2tim)Hi 1 Now, whatever the head on the weir, the end pieces of the I stream, since the width is (n — 1) JcK and fe is a conBtsjit, will he I similar figures, and, therefore, the flow through them can be I expressed as I Qa = 2C,(n-l)fcHHl I The total flow is, therefore, I Q = C (L - 2rifcH) H^ + 2C» (n - 1) fcHH^- I If now Ci is assumed equal to C I Q = C(L-2^H)Hl 1 If instead of two there are N contractions, due to the weir] being divided into several bays by posts or partitions, the formula ] becomes I Q = C(L-NOa.H)Hi I This is Francis* formula, and by Thomson's theory it is thual shown to be rational* I 67- Bazin's* formula for the discharge of a weir, ] The discharge through a weir with no side contraction may be I written _ J or Q = mL%/2^.H, ^J the coefficient m being equal to -7=. . ^H v2^ ^1 Taking Francis* value for C as 3'33, m is then 0'415. From experiments on sharp-crested weirs with no side traction Bazin deduced for mt the value m ^ 0'405 ^ ■□■ . In Table IX, and Fig. 77, are shoviTi Bazin'ti values for m foi different heads, and also those obtained by Rafter at Cornell upon a weir similar to that used by Bazin^ the maximum head in thi Cornell experiments being much greater than that in Baxin': experiments. In Fig* 77 are also shown several values of m, lu calculated by the anthorj from Francis' experimental data* t " Eiperimeots oa flow over Weira^" Am.8.C.E^ Td. xx^n. FLOW OVEK WEIRS TABLE IX. 89 Valueti of the coeflScient m in the f onmda Q = mL ^2g H^* W^ir* aharp-crestedj 6 '56 feet wide with free oyerfall and lateraX Dotitrmctioti suppressed, H being the atill water head over the weir, >r ihe nuiafliirted head A* corrected for irelocity of approach. Bazin, bileet a-164 0^328 0*656 0'984 1*312 1-64 1968 fit 0448 0483 0421 0-417 0414 0412 0-409 m -tHOS^"-^. Rafter. ^^H He&d in f«et m C ^^^^ 0^1 0-4286 3437 ^^^H 0*5 0-4230 8-392 ^^^H 1*0 0*4174 3348 ^^^^H l'& 0-4136 3-317 ^^^H M 0-4106 8-293 ^^^H S^ 0'4094 3283 ^^^^1 8^ 0*4094 s-m^ ^^^H S'5 0*4099 8-288 ^^^H 4-0 0*4112 8^298 ^^^H 4'fi 04125 8-808 ^^^f m 0-4188 8-815 ^^^^ §'& 04135 8-816 H e-0 0-4136 3317 W. Bazta's and the Cornell experiments on weirs* Bazin's experinient« were made on a weirt 6'56 feet long faATiiig the approaching channel the same width as the weir, so lluit the lateral contractions were suppressed, and the discharge was meftsiired by noting the time taken to fill a concrete trench of Iehovth capacity. The bead over the weir was measured by means of the hook lESOge, page 249. Side chambers wei^e constructed and connected to the channel by means of circular pipes 0*1 nu diameter. The water in the chambers was very steady, and its level cotaUd therefore be accurately ganged. The gauges were placed h metree from the weir. Tht^ maximum head over the weir in Bazin% experiments was however only 2 feet. Tlic experiments for higher heads at Cornell University were tuaile on a weir of practically the same width as fiassin's, 6'53 feet, fher conditioDii being made as nearly the aame as possible; Liiaximum head on tlie weir was 6 feet. * Sed page 90. i Mnnalm 4e$ Font* tl Vkau9s^f*, p. 445, Yq], ii. Ig91. J 00 HYDRAULICS The results of these experiments, Fig, 77, show that the cciefficient m diminishes and then increases, having a minimmn value when H is between 2'5 feet and 3 feet. "I 3 3 ^ JfiKzO' oitffxajmt atryes ^r Sharp -^d^mt H'oLrs A FnMfuxa* " (Deduced by ths ajuUh^r} Fig, 77. It is doubtful, however, although the experiments were made with great care and skill, whether at high heads the deduced coefficients are absolutely reliable* To measure the iiead over the weir a 1 inch galvanised pipe with holes Jinch diameter and opening downwards^ 6 inchea apart, was laid acrosus the channel To this pipe were connected \ inch pipes pfi-ssing through the weir to a convenient point beloir the weir where they couki be connected to the gauges by rubber tubing. The gauges were glas^ tubes J inch diameter mounted on a frame, the height of the water being read on a seal* graduated to zmm* spaces. 69, Velocity of approach. It should be clearly understood that in the formula given, it^ has been assumed, in giving values to the coefficient m that H is the height, above the sill of the weir, of the still water surface. A FLOW OVER WEIRS 91 In actual cases the water where the head is measured will have ome velocity, and due to this, the discharge over the weir will be acreased. If Q is the actual discharge over a weir, and A is the area of he up-stream channel approaching the weir, the mean velocity in he channel is t? = ? . A There have been a number of methods suggested to take into iccouit this velocity of approach, the best perhaps being that idopted by Hamilton Smith, and Bazin, This consists in considering the equivalent still water head H, Dfver the weir, as equal to a being a coefficient determined by experiment, and h the measured head* The discharge is then <i.mj2ih(h*^' (5), <3...L(<..f)y2,(A.g). CLV Expanding (5), and remembering that ^-7 is generally a small quantity, The velocity v depends upon the discharge Q to be determined and is equal to ^ • Therefore Q = ^LA>/2^ ( 1 + | 2£0 ^^^• From five sets of experiments, the height of the weir above the bottom of the channel being different for each set, Bazin found the mean value of a to be 1*66. This form of the formula, however, is not convenient for use, since the unknown Q appears upon both sides of the equation. If, however, the discharge Q is expressed as Q = nL ^2gh, h, the coefficient n for any weir can be found by measuring Q and h. It will clearly be different from the coefficient m, since for m to be used h has to be corrected. From his experimental results Bazin calculated n for various heads, some of which are shown in Table X. ..,u). :iC .: .:t' :o the f-r: above FLOW OVER WEIRS 93 coefficient to determine Q, h most first be corrected, or Q calculated from formula 9. Baher in determining the values of m from the Cornell ex- periments, increased the observed head ^ by o;- only, instead of by 1-66^. 2g Fteley and Steams*, from their researches on the flow over weirs, f onnd the correction necessary for velocity of approach to be from 1-45 to 1-5^. Hamilton Smith t adopts for weirs with end contractions sappressed the valaes 1-33 to 1-40^, and for a weir with two end contractions, 11 to 1-251^. TABLE X. Coefficients n and m as calculated by Bazin from the formulae Q= nL'J2gh^ and Q = mLN/^H^ h being the head actually measured and H the head corrected for velocity of approach. Head h in feet Height of sill p in feet Coefficient n Coefficient m 0-164 0-656 6*560 0-458 0-448 0-448 0-984 0-656 6-560 0-500 0-421 0-417 1-640 0-656 6-560 0-500 0-421 0-4118 An example is now taken illustrating the method of deducing ilie coefficients n and m from the result of an experiment, and the difference between them for a special case. EaoM^U. In one of Bazin*8 experiments the width of the weir and the ■ffMiMiiliiiHj, channel were hoth 6-56 feet. The depth of the channel approaching mm wmr mminrnd at a point 2 metres up stream from the weir was 7-544 feet and hflad menBiired over the weir, which may be denoted by A, was 0-984 feet. The ~ diaehazge was 21*8 cubic ft. per second. ♦ TranMoetiom Am,S,C.E,, Vol. xn. t Hydrauliet, 94 HTBRAULICS The Telooitj at the sectiou whsre h was me&Bnrfld» and which may be called Ihe I Tilooitj of approach was, therefore, Q _ 2ia ^ ~ 7'54i X B 56' ' 7 544 jc eW = 0^44 feet jier aeccmd. If now the formala for discharge be written and n ia qaleuLated from this formula by Bubstftuimg the known valaea of] Q, L and h n = 0*421. Correeting h for veiooitj of approacb. 2ff Then from whtoh = •9888, 918 'emj2g.^dms = 0-415. It will sieem froin Table X that when the h«ip;hi p of the dll of the weir abore the stream bed is amall compared with tba head, Ihe dilFerenoe majr be much ' larger than for thia eiample. When the head is 1-64 feet and larger than p, the coeffiolent n ia eighteen] per oent. greater than m. In auoh eases failnTe to eoireet the eoefficient wUJ letil to considerable inaccDracy, 70. Influence of the height of the weir slU above the bed of the stream on the contraction. The nearer the sill is to the bottom of the streamj the less the contraction at the sill, and if the depth is small compared with H, ] the diminution on the contraction may conBiderably affect the] flow. When the sill was Vlb feet above the bottom of a channel, I of the same width as the weir, Bazin found the ratio ^ (Fig. 85) to be 0^097, and %vhen it was S70 feet, to be 0'112. For greater ] heights than these the mean value of ^ was OlS. 71. Discharge of a weir when the air is not fireelj ' admitted beneath the nappe. Form of the nappe. Francis in the Lowell experiments, found that, by making the width of the channel below the weir equal to the width of the weir, and thus preventing free access of air to the underside of the nappe J the discharge was increased. Bazin*, in the experiments | already referred to, has investigated very fully the effect upon the discharge and upon the form of the nappe, of restricting the free passage of the air below the nappe. He finds, that when the flow is sufficient to prevent the air getting under the nappe, it niay assume one of three distinct forms, and that the discharge for * Annalev d^t FmU ti Chau4s€ea, 1891 and 1896. L=> ite^ FLOW OVXB WEIBS 95 I them may be 28 per cent, greater tlian when the air is admitted, or the nappe is ''free.'' Which of these three the nappe assomes and the amount by which the discharge ater than for the ''free nappe/' depends largely upon the over the weir, and also upon the height of the weir above ater in the down-stream channel. 18 phenomenon is, however, very complex, the form of the ' for any head depending to a very large extent upon ler the head has been decreasing, or increasing, and for a head may possibly have any one of the three forms, so that jscharge is very uncertain. M. Bazin distinguishes the forms ^pe as follows : ) Free nappe. Air under nappe at atmospheric pressure, 70 and 78. ) Depressed nappe enclosing a limited volume of air at a ire less than that of the atmosphere. Fig. 79. ) Adhering nappe. No air enclosed and the nappe adher- the down-stream &ce of the weir. Fig. 80. The nappe in this oay take any one of several forms. Tcpafauam£l\ ! Fig. 79. Fig. 80. Drowned or wetted nappe, Fig. 81. No air enclosed but ppe encloses a mass of turbulent water which does not move le nappe, and which is said to wet the nappe. Fig. 81. 96 ETBEA0LICS 72* Depressed nappe. The air below the nappe being at less than the atmofipherie" pressure the excess pressure on the top of the nappe causes it to^ be depressed* There is also a rise of water in the down-st channel under the nappe. The discharge is slightly greater than for a free nappe. On i weir 2*46 feet above the bottom of the up-stream channel ^ thej nappe was depressed for heads below 0'77 feet, and at this head the coefficient of discharge was 1*08 wii, mi being the absolute coefficient for the free nappe, 73* Adhering nappes. As the head for tliis weir approached 0*77 feet the air rapidly expelledj and the nappe became vertical as in Fig, 80, ita surface having a corrugated appearance* The coefficient of di charge changed from r08 Wi to r28TOi- This large change ia| the coefficient of discharge caused the head over the weir to fall to 0'69 feet, but the nappe still adhered to the weir, 74. Browned or wetted nappes. As the head was further increased, and approached 0'97 fe the nappe came away from tlie weir face, assuming the drowned| form, and the coefficient suddenly fell to 119 mi. As the he was further increased the coefficient diminished^ becoming V12\ when the head was above IS feet. The drowned nappes are more stable than the other two, bu whereas for the depressed and adliering nappes the discharge not affected by the depth of wat^r in the down-stream channd, the height of the water may influence the flow of the dro^ nappe. If when the drowned nappe falls into the down streanl the rise of the water takes place at a distance from the foot of thai nappe^ Fig, 81 j the height of the down-stream water does not affe the flow. On the other hand if the rise encloses the foot of tia nappe. Fig. 82, the discharge is affected. Let K^ be the differenc Fig. as. FLOW OVER WEIRS 97 erf level of the siJI of the weir and the water below tho weir. The coefficieiit of discharge in the first ease iB independent of h^ bat is dependent upon p the height of the sill above the head of the up- stream channel, and is m^ = m, ^0878 + 0128 |Y.. ..(11). Bazin funnd that the drowned nappe could not be formed if h is lees than 0*4 p and, therefore, r cannot be greater than 2'5, Snbatitnting for f»i its value ■ £rom (10) page 92 w», = 0"470 + 0TO75^' .,....„... (12). In the second case the coefficient depends upon ^, and ia, m^mi(im + 0'my(^-om)^ „,(13), for iprhich, with a safficient degree of approximation, may be nbptitated the simpler formula, m, = mj(r05+M5^) ...,....(14). The limiting value of ?% is 1*2 m^ for if hi becomes greater ths^ h the nappe is no longer drowned. Further, the rise can only enclose the foot of the nappe when h^ i» lesa than Q p- k). Ay h^ passes this value the rise is pushed duwn arream away from the foot of the nappe and the coefficient chAHgr^e to that of the preceding case. fB* Instability of the form of the nappe, ' ul at wliich the form of nappe changes depends upon ^rh*- i*' head is increasing or diminishing, and the depressed ajid adiiering nappes are very nnstable^ an accidental admission 'jr other mterference causing rapid change in their form, r, the adhering nappe is only formed under special circum- , and as the air is expelled the depressed nappe generally ,1.1 --^. r^ directly to the drowned form, tf, therefore, the air is not freely admitted below the nappe iven head is very uncertain and the discharge L with any great degree of assurance. With the weir 2 46 feet above the bed of the channel and 6 56 ' '* i obtained for the same head of 0*656 feet, the four ^ ; : . , the coefficients of discharge being as follows ; 98 fiTDEAULlCS Free nappe, 0*433 Depre^ed nappe, 0*460 Drowned nappe, level of water down stream O'^l feet below the crest of the weir, 0"4OT Nai>pe aflhering to down-stream ince, 0'554 The dii?charge for tliis weir w^hile the head was kept oonstant, thus vari<.Hi 26 per cent. 76. Drowned weirs with sharp crests*. When the surface of the water n stream is higher than t^ sill of the weir, as in Fig. 83, the w s said to be drowned- T r : ^^ — Fig, 83. * Bazin gives a formula for deducing the coefficients for such a weir from those for the sharp-edged weirs with a free nappe, w^bicli in its simplest form is, ITIe --bH'^'^^/'n^] "^'. ^ Aa being t!ie height of the down-stream water above the sill of the weir, k the head actually measured abov^e the weir, p the height of the sill above the up-stream channel, and mi the coefficient ((10), p, 02) for a sharp-edged weir. This oxpre^isioti gives the same value within 1 or 2 per cent, as the formulae (13) and (14). Example. The heiid ovex a mnr is I foot, and tbe height of the sill abova np-stream channel Ia fj feet. The leugth is ti Seet &iid the eurface of the in the down -stream channel is 6 inches alM>vi) the Rill, Find the diachajge. From formnla (10)i pa^e f^2, the eoeffici«nt fitj for a b harp-edged weir with tim nappe is * AttempU have been made to esprega the diiieharge over a drowned wdx equivalent to thnt throagh a drowned orifice of an area equal to LA^, under a hi h-ha* together with a disobarge over a weir of length L when the head ia h^hm. I^e dUscharge is then n^2ghh^{h-h^)i + m^lj{h-h^)^, n and m being coefficients. ^m FLOW O^EB WEIRS 99 ^^^K» m,== '4215 [1^(1 +-031} 0-761] ^^^^^ ^3440, I Thai Q = *a44>^V2^.1* I =22i)S eubio ft* }ser second. I 77. Vertical weirs of small thickness. I likhtt^ail of makiiig the sill of a weir sharp-edged, it niaj^ I bave a flat sill of thickness c. This will frequently be the ease in nbetict^ the weir being constructed of timbers of uniform width placed mvi* upon the other. The conditions of flow for these weirs I My be v^ry dilferent from those of a sharp-edged weir, I I'he Ttappes of such weirs present two distinct forms, according I as the WHter is in contact with th€+ crent of tlie weir, or becomes I tWtat'liwl at the up-stream edge and leaps over the crest without I *<itjcliiTig the down-stream edge. In the second case the discharge I » tile hHtne as if the weir were sharp-edged. When the head k I 'jrt'er the weir is more than 2c this condition is realised, and may I *»^tftiii when h pa?+ses |c. Between these two value** the nappe is I t«»coTuHtian of unstable equilibrium; when k is less than fc the I oapjx' adheres to the sill, and the coefficient of discharge is I m^ = mi (070 + 0*185^), V *3^ oxtemal perturbation such as tlie entrance of air or the F ^'^'^^-r 1,1 a floating body causing the detachment. \ li^ppL* adheres between ic and 2c the coefBcient m^ varies I ^lik Mnii to r07iwi, bat if it is free the coefficient w^^mi. I " n«D B = Jc, m^ is 79mi . If therefore the coefficients for a I *wtfp-etiin?d weir are used it is clear the error may be con- HfnabliC HP^ formula for iWis gives approximately correct results when *fc^ width of the sill m great, from 3 to 7 feet for example. If the up-{^tream edge of tlie weir is rounded the discharge is iDCi^.Hed. Ilic di&charge* for a weir having a crest 6'bG feet ^di\ when the up-stream edge was rountk?d to a radius of 4 inches, •^ mcreaeed by 14 per cent., and that of a weir 2'624 feet wide ky 12 per cent. The roofiding of the comers^ due to wear, of timber weirs of dmienmonsi to a radius o£ 1 inch or less, will, therefore, IpcI the flow eonsiderahly. 78. Depressed and wetted nappes for flat-crested weirs. The nappes of weirs having tiat sills may be depre^ed, and ijf become drowned as for sharp-edged weirs, • dnmla du Pmtt et Ch&u»»iti, Val u. 18^. 7—2 416V02 100 HYDRAULICS The coefficient of discharge for the depressed nappes, whether the nappe leaps over the crest or adheres to it, is practically the same as for the free nappes, being slightly less for low heads and becomes greater as the head increases. In this respect they differ from the sharp-crested weirs, the coefficients for which are always greater for the depressed nappes than for the free nappes. 79. Drowned nappes for flat-crested weirs. As long as the nappe adheres to the sill the coefficient m may be taken the same as when the nappe is free, or When the nappe is free from the sill and becomes drowned, the same formula 77io = mi(o-878+0-128|), as for sharp-crested weirs with drowned nappes, may be used. For a given limiting value of the head h these two formulae give the same value of m©. When the head is less than this limiting value, the former formula should be used. It gives values of m slightly too small, but the error is never more than 3 to 4 per cent When the head is greater than the limiting value, the second formula should be used. The error in this case may be as great as 8 per cent. 80. Wide flat-crested weirs. When the sill is very wide the surface of the water fali^ towards the weir, but the stream lines, as they pass over the weimr^ are practically parallel to the top of the weir. Let H be the height of the still water surface, and h the depbii of the water over the weir, Fig. 84. T L' Fig. 84. Then, assuming that the pressure throughout the section of tb^ nappe is atmospheric, the velocity of any stream line is v=J2g{B."h), and if L is the length of the weir, the discharge is Q = ^Lhs/W^) (16). FLOW OVER WBIBS 101 For the flow to be pennanent (see page 106) Q must be a naximom for a given value of h, or -^ must equal zero. Therefore From which 2(H-fe)-fe = 0, ind h = f H. Substituting for h in (16) = 0-385L n/^ . H = 308L VH . H. The actual discharge will be a little less than this due to friction on the sill, etc. Bazin found for a flat-crested weir 6*56 feet wide the coefficient m was 0-373, or C = 2-991. Lesbros' exx>eriment8 on weirs sufficiently wide to approximate to the conditions assumed, gave '35 for the value of the co- efficient w. In Table XI the coefficient C for such weirs varies from 2'66 to 310. 81. Flow over dams. Weirs of various forms, M. Bazin has experimentally investi- ?^^rf the flow over weirs having (a) sharp crests and (6) flat crests, the up- and down-stream faces, instead of both being vertical, (1) vertical on the down-stream face and inclined on the "P-stream face, (2) vertical on the up-stream face and inclined on the down- ^ream face, (3) inclined on both the up- and down-stream faces, ^^d (c) weirs of special sections. The coefficients vary very considerably from those for sharp- c^ted vertical weirs, and also for the various kinds of weirs. ^^fficients are given in Table XI for a few cases, to show the ^^essity of the care to be exercised in choosing the coefficient for ^^y weir, and the errors that may ensue by careless evaluation of the coefficient of discharge. For a full account of these experiments and the coefficients ^otained, the reader is referred to Bazin's* original papers, or to Rafter's t paper, in which also will be found the results of experi- • Afmaie* de$ Fonts et Chau*s4ett, 1898. t Traruactions of the Am.S.C.E., Vol. XLiv., 1900. 102 HYDRAULICS TABLE XI. ^^ Values of the coefficient C in the formula Q = CL . fc% for weirs of the sections shown, for various values of the observed head A. Bazva. Section of Head iu feet weir 0-3 0-6 10 1-3 20 30 40 6-0 6-0 V31S 2*66 2-66 2*90 8*10 1 1 -i 8-61 8*80 401 8*91 y^ ^ s 402 416 418 416 < \s 8-46 8-67 8-49 8*86 8*80 *< ^^^^C 8*46 8*59 8*68 gc:^^ 808 808 819 8*22 FLOW OVER WEIBS 103 TABLE XI (continued). Baain. Section of Head in feet weir 0-8 0-6 10 1-3 20 30 40 60 60 — ^ '€S 810 8-27 8-05 8-78 8-90 8-78 i^-Y^ T^ 2-76 8-62 Section of weir — -J3^ Rafter. Head in feet 0-3 0-6 10 1-3 20 30 40 50 60 8-85 814 2-95 8-68 8-42 816 8-88 8-52 8-27 8-77 8-61 8-45 8-68 8-66 8-56 8-70 8-66 8-71 8-64 8-7] 8-6£ 8-61 8-66 8-6^ 104 ByDRACTLlCS inents made at Cornell University on the discharge of weirs, sitaiJar to those used by Bazin and for heads higher than lie used, and also weirs of sections approximating more closely to tho&e oi existing masonry dams, nsed as weirs* From Bazin's and Rafter's experimentSj curves of discharge for varying heads for some of these actual weirs have been dra\vn up, 82. Form of weir for accurate gauging. The uncertainty attaching itself to the correction to be applied to the measured head for velocity of approach, and the difficulty of making proper allowance for the imperfect contraction at the sides and at the sill, when the sill is near the bed of tlie channel and is not sharp-edged, and the inetabihty of the nappe and uncertainty of the form for any given head when the admission of air below the nappe is imperfect, make it desirable that as far as possible, when accurate gaugings are reqniredj the %veir should comply with the following four conditions, aa laid down by Bas&in, (1) The sill of the weir must ba made as high as possible above the bed of tlie stream, (2) Unless the weir is long compared with the head, the lateral contraction should be suppressed by making the channel approaching the weir with vertical sides and of the same width as the weir. (3) The sill of the weir must be made sharp-crested. (4) Free access of air to the sides and under the imppe of the weir must be ensured. 83. Boussinesq's* theory of the discharge over a weir. As stated above, if air is freely admitted below the nappe of a weir there is a contraction of the stream at the sharp edge of the sillj and also due to the falling curved surface. If the top of the sill is well removed from the bottom of thd channel, the amount by which the arched under side of the nappe is raised above the sill of the weir is assumed by Boussinesq— And this assumption has been verified by Bazin's expeinments—to be some fraction of the head H on the weir. Let CDj Fig, 85, be the section of the vein at wliich the maximum nBe of the bottom of the vein occurs above the sill, and let e be the height of D above S. Let it be assumed that through the section CD the stream lines are moving in curved paths normal to the section^ and that they have a common centre of cur^^ature 0. * Comptf* Eendm, XSSI ftjad 1889. FLOW OYER WEIRS 105 Let H be the height of the snr&ce of the water up stream ove the silL Ijet R be the radius of the stream line at any bt E in CD at a height x above S, and Bi and Bs the radii of rvatore at D and C respectively. Let V, Vi and Vj be the locities at E, D, and C respectively. Fig. 85. Consider the equilibrium of any element of fluid at the point E, the thickness of which is 8R and the horizontal area is a. If w is the weight of unit volume, the weight of the element is w . a3R. Since the element is moving in a circle of radius R the centri- V8R fngal force acting on the element is wa — ^~ lbs. gix The force acting on the element due to gravity is iraSR lbs. Let p be the pressure per unit area on the lower face of the element and p + Sp on the upper face. Then, equating the upward and downward forces, / ^ \ ST> . 'M?aV'SR (p + op) a + tcaoK = pa+ -^ — . gti From which ^dR=-l^^ ^^^- Assuming now that Bemouilli's theorem is applicable to the ream line at EF, w 2g Differentisitingy and remembering H is constant, , dp YdV w = 0, ldp_ ^ YdY ■w dx g.dx IM ffTTntjfcrxics fa xR- vrv tSt. ' iW* ■ -V,iK = 0- trakos of die the pressme is = *-ig H--U "^n = T 3i, JBBt. 3, i^m die ignre is (B, + * — «), therefore, Kt --'^^-''^ZJZ, (2). '-^^^^'(s;^.)'^ = ^ix 3-* Rt I Bi^jr-e n.-K =-2? ^-^ B.Ioe^=g-^ (3). ^'^ 1 -at TOW .»v«r -ii? WOT is permanent, the thickness K <rf :t» :»|f|K» auBC iiUit&gf :t&ik£. iu due for the given head H the '*Sr ^HiuatttflBt ^w iuwi*Ter cui only take place if each 'flSff ^'v^'xvtt i>F Jft^ tai? maximnm velocity possible to :at« iliuneiLGs will be accelerated; and ^•i- - .;^v«t iteiciMciQi^ -n* tan:icii>tst^ «• k therefore a minimnm, or vc -*r .^r*««> ^^^ "C ^ ^iK xbscfurae is a maximnm. That is, when ^ '::feto<^<ktK^ A ^litt J* wmtMi J6 a fimction of fc,, the valae of ^ ^NiUv.u "iiat^ 4 ^ -^^^^ni ^iin^ can be determined by differ- A. Therefore, A« = (H - e) (1 - »«), Ri = w(l + n)(H~e). Sabstitatm£^ this value of Bi in the expression for Q, Q=V2^.(H~6)*(n + n»)logi FLOW OVER WEIRS 107 n' lAich, smce Q is a Tnaximnin when ^ = 0, and A is a function % is a maxiiniiin when --p = 0. Bifferentiatincr &i^d equating to zero^ (l+2n; D solution of which gives (l+2n)log--(l + n) = 0, TV n = 0-4685, and tlierefore, Q = 05216 ■J2g(a- e)* = 0-5216>/^(l-g)*H* = 0-5216 (l-g)*-s^.H* • Ae coefficient m being equal to ^ 0-5216 (l - g)*. M. Bazin has found by actual measurement, that the mean i^^ iralne for ^^ when the height of the weir is at considerable < disfcanoe from the bottom of the channel, is 0*13. Then, (l-g)* = 0-812, and m = 0'423. It will be seen on reference to Fig. 77, that this value is very near to the mean value of m as given by Francis and Bazin, and the Cornell experiments. Giving to g the value 32*2, Q = 3'39 H* per foot length of the weir. If the length of the weir is L feet and there are no end con- tractions the total discharge is Q = 3-39L.Ht, and if there are N contractions Q = 3-39(L-N01H)Hi 108 HYDRAULICS The coefficient 3*39 agrees remarkably well with the mean value of C obtained from experiment. The value of a theory must be measured by the closeness of the results of experience with those given by the theory, and in this respect Boussinesq's theory is the most satisfactory, as it not only, in common with the other theories, shows that the flow is proportional to H*, but also determines the value of the constant C. 84. Solving for Q, by approximation, when the veloeitj of approach is nnknown. A simple method of determining the discharge over a wdr when the velocity of approach is unknown, is, by approximation, as follows. Let A be the cross-sectional area of the channel. First find an approximation to Q, without correcting for velocity of approach, from the formula Q = mLh J2gh, The approximate velocity of approach is, then. and H is approximately «=!. A nearer approximation to Q can then be obtained by sub- stituting H for fe, and if necessary a second value for v can be found and a still nearer approximation to H. In practical problems this is, however, hardly necessary. Example, A weir without end contractions has a length of 16 feet The beal as measured on the weir is 2 feet and the depth of the channel of approaeh belov the sill of the weir is 10 feet. Find the discharge. „»=0-405 + ^5^=.4099. Therefore C=3«28. Approximately, Q=3-28 2^.16 =r 148 cubic feet per second. The velocity v = — — -^ = '77 ft. per sec. , and 1^=. 0147 feet. A second approximation to Q is, therefore, Q = 3-28 (20147)^16 = 150 cubic feet per second. A third value for Q can be obtained, but the approximation is snffioieiiily Mtf for all practical purposes. In thin case the error in neglecting the velocity of approaoh altogether, ii probably less than the error involved in taking m as 0*4099. PLOW OVER WEIRS 109 85. Time required to lower the water in a reservoir a given distance hy means of a weir. A reservoir has a weir of length L feet made in one of its sides, and kavin^ its siU H feet below the original level of the water in tihe reservoir. It is reqnired to find the time necessary for the water to fall to m level H« feet above the sill of the weir. It is assumed that the area of the reservoir is so large that the velocity of the water as it approaches the weir may be neglected. When the snr&ce of the water is at any height h above the sill the flow in a time dt ia Let A be the area of the water surface at this level and dh the distance the surface falls in time dt. Then, CUfidt = Adh, The time required for the surface to fall (H-Ho) feet is, therefore, ^^1 (^ Adh The coefficient C may be supposed constant and equal to 3'34. If then A is constant ^2A/_1 1_\ CLWilo n/H/ To lower the level to the sill of the weir, Ho must be made equal to and t is then infinite. That is, on the assumptions made, the surface of the water never could be reduced to the level of the sill of the weir. The time taken is not actually infinite as the water in the reservoir is not really at rest, but has a small velocity in the direction of the ^Bveir, which causes the time of emptying to be less than that ^ven by the above formula. But although the actual time is not infinite, it is nevertheless very great. 9.x When Ho is iH, t = WTien EU is ^H, t = CLn/H' 6A clVh' So that it takes three times as long for the water to fall from \n to iVH as from H to iH. 110 HYDRAULICS Example 1. A reservoir has an area of 60,000 sq. yards. A weir 10 feet long has its sill 2 feet below the surface. Find the time required to redaoe the level of the water 1' 11". H,=A'. H = 2'. Therefore t=^-^^^^ (Sid -0109), o'o4 . lU 2.640.000 ^ 3-34.10 '^ = 89,000 sees. = 24-7 hours. So that, neglecting velocity of approach, there will be only one inoh of the weir after 24 hours. Example 2. To find in the last example the dischaige from the reeerfoir in 15 hours. 2. A Therefore 54 000=p^^-^ ( -l=- - -^) . C.LV^Ho V2/ From which jEQ = 0'i21, Ho=0176 feet. The discharge is, therefore, (2-0-176) 540,000 cubic feet = 984,960 cubic feet. EXAMPLES. (1) A vreir is 100 feet long and the head is 9 inches. Find the disdiaigs in c. ft. per minute. C = 3'34. (2) The discharge through a sharp-edged rectangular weir is 500 gallons per minute, and the still vrater head is 2^ inches. Find the effective length of the vreir. m = -43. (3) A vreir is 15 feet long and the head over the crest is 15 incheii Find the discharge. If the velocity of approach to this weir were 5 feel per second, what would be the discharge ? (4) Deduce an expression for the discharge through a right-ao^fid triangular notch. If the head over apex of notch is 12 ins., find the discharge in c. ft. per sec. (5) A rectangular weir is to discharge 10,000,000 gallons per di^ (1 gallon =10 lbs.), with a normal head of 15 ins. Find the length at tilt ! weir. Choose a coefficient, stating for what kind of weir it is applioaUai or take the coefficient C as 8*38. (6) What is the advantage in gauging, of using a weir without end contractions? (7) Deduce Francis' formula by means of the Thomson principle d similarity. Apply the formula to calculate the discharge over a weir 10 feet widr under a head of 1*2 feet, assuming one end contraction, and neglecting thi effect of the velocity of approach. I FLOW OVER WEIRS 111 W (S> A ratnfAtl of fy inch per hoar i» discharged from a catohment area ai 5 aqua^re mtlea. Find tbe still water head when this Tolutue flowB ot^^er a weir mill free oTerfftll 30 feet m longib^ (x»nstract^ in m% bays, each ^ iee4 iride, tulriiyg 0*415 as Basin's coefficdent. A dteSrici of 6500 acxes (1 acreB43,560 aq. ft.) drains into a large I neserroir. The ma^iuum late at which rain falls in the dii^trict ia jl ins^ is 24 hoars. Whan r&in talk after tbe reservoir is full^ the water x^aii^es lo be discharged over a weir or bye-wanh which has its crest at tbe ordinary top- water level of the reservoir. Find tVie length of such a m^mtMT fof tlie abore reaervoir^ under the condition that the water in the seMSrroir aball never rin^ more than 18 ins. above it^ top- water level Tlie top of tlie weif may be Bopposed flat and abotit 18 inches wide (10> Compare rectangular and V notches in regard to aocnraey and ^amTenieDae when there is considerable variation in tl)e flow. In ft rectangnl&r notch 50^' wide the still water surface level is 15'' above iba BilL If the same quantity of water Bowed over a right-angled V notch, what ^vroiiltl be the height of the still water surface above the apex ? Lf the channels are narrow how would you correct for velocity of jkpproAch tti each case? Lton. Ua. 1906. ll 1 \ Tbe heaviest daily record of rainfall for a catchment area was if!Hi.iid to be 4*2^ mlUion gallonB. Assuming two-thirds of the rain to reach ^^iotage retnervoir and to ftass over the waste weir, find the length of •mi erf the wa&te weir* so that the water shall never riae more than two f[0^ above the fiOL (12) A weir is 300 yards long. What Ib the disebarge when the liead hk 4 feoi t Take Basdn's coeMcieut m="40SH , — , (iSf Sa|ipoae tbe water approaches the weir m the last question in a g b att Pf*' f$* 6'' ^e#f> and nOO yarda wide. Find by approximation the dis- cbaqfe. t*kiitg into acoouut tlie velocity of approach. (14^ The area of the water surface of a re^rvoir is 20,000 square ymt^m- Ffa>d Uie ttme rei|u]ru(l for tbe surface to fall one foot, when the wmier diadiuiges over a sbarp^edged weir 5 feet long and the original head crrrr tbe weir is 2 feet, 1 15> Find, from the following data, the horse-power available is a given wsterCaU:- Available height of fall 120 feet. A tTH-tangitlar notch above the fall, 10 feet long, is used to measure %km qoantity of water, and the mean head over the notch la found to be tDicbefet when tlie velocity of approach at the point where ttie head i m 100 feet per miniita Lon. Un. 1905. CHAPTER V. FLOW THROUGH PIPES. 86. ResiBtances to the motion of a fluid in a pipe. When a fluid is made to flow through a pipe, certain resistances are set up which oppose the motion, and energy is consequently dissipated. Energy is lost, by friction, due to the relative motion of the water and the pipe, by sudden enlargements or contractions of the pipe, by sudden changes of direction, as at bends, and by obstacles, such as valves which interfere with the free flow of the fluid. It will be necessary to consider these causes of the loss of energy in detail. L088 of head. Before proceeding to do so, however, the student should be reminded that instead of loss of energy it is convenient to speak of the loss of head. It has been shown on page 39 that the work that can be obtained from a pound of water, at a height z above datum, moving with a velocity v feet per second, and at a pressure head * is — + s- + « foot pounds. w^ w 2g ^ If now water flows along a pipe and, due to any cause, K foot pounds of work are lost per pound, the available head is clearly diminished by an amount h. In Fig. 86 water is supposed to be flowing from a tank throngli a pipe of uniform diameter and of considerable length, the end B being open to the atmosphere. Fig. 86. Lobs of head by friction in a pipe. FLOW THROUGH PIPES 113 Let *^ l>e the head due to the atmospheric pressure. Then if there were no resistances and assuming stream line flow, Bemonilli's equation for the point B is w 2g w* from which st = Zp-Zb = H, or rB= V2grH. The whole head H above the point B has therefore been utilised to give the kinetic energy to the water leaving the pipe at B. Experiment would show, however, that the mean velocity of the water would have some value v less than Vb, and the kinetic energy would be oZ« Ahead /, = ___ = H-2^ ha3 therefore been lost in the pipe. By carefully measuring H, the diameter of the pipe d, and the discharge Q in a given time, the loss of head h can be determined. For "-7^' and therefore A = H ^*--; The head h clearly includes all causes of loss of head, which, in this case, are loss at the entrance of the pipe and loss by friction. 87. Loss of head by friction. Suppose tubes 1, 2, 3 are fitted into the pipe AB, Fig. 86, at equal distance apart, and with their lower ends flush with the inside of the pipe, and the direction of the tube perpendicular to the direction of flow. If flow is prevented by closing the end B of the pipe, the water would rise in all the tubes to the level of the water in the reservoir. Further, if the flow is regulated at B by a valve so that the mean velocity through the pipe is v feet per second, a permanent- regime being established, and the pipe is entirely full, the mean velocity at all points along the pipe will be the same ; and there- fore, if between the tank and the point B there were no resistances offered to the motion* and it be assumed that all the particles L.H. 8 114 HYDRAULICS have a velocity equal to the mean velocity, the water would again rise in all the tubes to the same height, but now lower than the surface of the water in the tank by an amount equal to sr- It is found by experiment, however, that the water does nofc i rise to the same height in the three tubes, but is lower in 2 thaa j in 1 and in 3 than in 2 as shown in the figure. As the fluid movai \ along the pipe there is, therefore, a loss of head. The difference of level fh of the water in the tubes 1 and 2 iB called the head lost by friction in the length of pipe 12. In any . length I of the pipe the loss of head is h. This head is not wholly lost simply by the relative movement of the water and the surface of the pipe, as if the water wotb a solid body sliding along the pipe, but is really the sum of tiia losses of energy, by friction along the surface, and due to relatifft motions in the mass of water. It vrill be shown later that, as the water flows along the pipe^ ; there is relative motion between consecutive filaments in the jripe^ and that, when the velocity is above a certain amount, the waler has a sinuous motion along the pipe. Some portion of this headk is therefore lost, by the relative motion of the filaments of water, and by the eddy motions which take place in the mass of fliB water. When the pipe is uniform the loss of head is proportional to the length of the pipe, and the line CB, drawn through the tops of the columns of water in the tubes and called the hydraulic gradient, is a straight line. It should be noted that along CB the pressure is equal to that of the atmosphere. 88. Head lost at the entrance to the pipe. For a point B just inside the pipe, Bernouilli's equation is — + j^ + head lost at entrance to the pipe = Aa + ~ , 2? being the absolute pressure head at E. The head lost at entrance has been shown on page 70 to be about -P5 — , and therefore, 2g w w 2g That is, the point C on the hydraulic gradient vertically above l'5v' E, is -^ — below the surface FD. ' 2g FLOW THKOUQH PIPES 115 If the pipe is bell-moathed, there will be no head lost at entrance, id the point C is a distance equal to ^ below the surface. 89. Hydraulic gradient and virtual slope. The line CB joining the tops of the columns of water in the nhe, is called the hydraulic gradient, and the angle i which it Doakes with the horizontal is called the slope of the hydraulic gradient, or the virtual slope. The angle i is generally small, and mi may be taken therefore equal to i, so that j = ^« In what follows the virtual slope -j is denoted by i. More generally the hydraulic gradient may be defined as the line, the vertical distance between which and the centre of the pipe gives the pressure head at that point in the pipe. This line will only be a straight line between any two points of the pipe, wben the head is lost uniformly along the pipe. If the pressure head is measured above the atmospheric fffessure, the hydraulic gradient in Fig. 87 is AD, but if above lero, AiDi is the hydraulic gradient, the vertical distance between LD and AiDi being equal to ressuru per sq. inch. w -, Pa being the atmospheric Fig. 87. Pipe rising above the Hydraulic Gradient. If the pipe rises above the hydraulic gradient AD, as in Fig. 87, c* pressure in the pipe at C will be less than that of the atmosphere r a head equal to CE. If the pipe is perfectly air-tight it will :t a> a siphon and the discharge for a given length of pipe will jt }je altered. But if a tube open to the atmosphere be fitted at 8—2 116 HYDRAULICS the highest point, the pressure at C is equal to the atmospheric pressure, and the hydraulic gradient will be now AC, and the flow will be diminished, as the available head to overcome the resSst* ances between B and C, and to give velocity to the water, will only be CF, and the part of the pipe CD will not be kept full. In practice, although the pipe is closed to the atmosphere, yeA air will tend to accumulate and spoil the siphon action. As long as the point C is below the level of the water in tihs reservoir, water will flow along the pipe, but any accumulation ol air at C tends to diminish the flow. In an ordinary pipe line it it desirable, therefore, that no point in the pipe should be allowed to rise above the hydraulic gradient. 90. Determination of the loss of head due to frietton. Reynolds' apparatus. Fig. 88 shows the apparatus as used by Professor Reynolds* for determining the loss of head by friction in a pipe. Fig. 88. BeyuoldB' apparatus for determining loss of head by friction in a pipe^ A horizontal pipe AB, 16 feet long, was connected to the water main, a suitable regulating device being inserted between tlie main and the pipe. At two points 5 feet apart near the end B, and thus at a distance sufiiciently removed from the point at which the water entered the pipe, that any initial eddy motions might be destroyed and i steady regime established, two holes of about 1 mm. diameter wero pierced into the pipe for the purpose of gauging the pressoie, it these points of the pipe. Short tubes were soldered to the pipe, so that the hofc communicated with these tubes, and these were connected 1 * PhiU TratiB. 1883, or Vol. n. Scienti/ic Papen^ Bejmolds. rnSianil FLOW THROUGH PIPES 111 ibber pipes ki the limbs of a siphon gauge G^ made of glass] and which contained mercuiy or bisulphide of carbon. Sealetl wi?re fixed behiiKl tbt* tubes ^y that the height of the columns in each limb of the gauge could be n?ad. For i'ery small differences of level a cathetometer was used*, '""' < made to flow through the pipe, the diffeiience in ' columns in the two limbs of the siphon measured presfure at the two points A and B of tlie pipeJ ouM iiiM- fiM.i'Mied the lo&s of head due to friction. ■ If * is the specific gravity of the liquid, and H the difference in ht'tght of the columns, the loss of head due to friction in feet oil waitir is A^H (^-1). I The quantity of water fl(»wing in a time t was obtained b|il actual measttTemeEit in a graduated flask . Calling r th© mean velocity in the pipe in feet per eecondi Q the dischafg^ in cubic feet per secondi and d the diameter of thai {ripe m f00l, J The lo» of head at different velocities was carefully measured^ tlia law connecting head lost in a given length of pipe, with T«lodty» determined* Hie resolU obtained by Reynolds, and others^ using tins of expt^rimeniing, will be refen-ed to later. Bqmation of flow in a pipe of uniform diameter detenaination of the head lost due to friction. Lei ci he the length of a ginall element of pipe of uniform r. Fig. m A Fig. 81). I,, rea of the transverse section be uj, P the length of lu . jutact of the water and the surface on this section, or wetu-tl perimeter, n the inclination of the pipe, p the pressm-e ' unit area on AB, iind p-dp the pressure on CD* • p. 268, Vol. I, ScieniifU Papif*^ Bejnoldf. 118 HYDRAULICS Let V be the mean velocity of the fluid, Q the flow in cubic feet per second, and w the weight of one cubic foot of the fluid. The work done by gravity as the fluid flows from AB to CD = Qw .dz-io,v .w .dz. The work done on ABCD by the pressure acting upon the area AB = p.io.v it. lbs. per sec. The work done by the pressure acting upon CD against the flow = (p — dp) . ci> . 17 f t. lbs. per sec. The frictional force opposing the motion is proportional to the area of the wetted surface and is equal to F . P . oZ, where P is some coefficient which must be determined by experiment and is the frictional force per unit area. The work done by friction per aecX is, therefore, F . P . 5Z . v. The velocity being constant, the velocity head is the same ttt both sections, and therefore, applying the principle of the con- servation of energy, p.tt},v + <o,v.w.dz= (p- dp) CO . i; + F . P . 3Z . «. Therefore w . w . 92? = -82) . w + F . P . 3Z, , dp F.P.dZ or dz = — - + . Integrating this equation between the limits of z and Zi, p and Pi being the corresponding pressures, and I the length of the pipe^ z- -Zi w w F.PZ W ci» w + z =£1 w + Zl + FP I W CD* Therefore, FPZ The quantity — is equal to A of equation (1), page 52, and it the loss of head due to friction. The head lost by friction it therefore proportional to the area of the wetted surface of the pipe PZ, and inversely proportional to the cross sectional area of the pipe and to the density of the fluid. 92. Hydraulic mean depth. The quantity p is called the hydraulic radius, or the hydranlio mean depth. If then this quantity is denoted by m, the head h lost hf friction, is w .m FLOW THBOtJOH PIPES 119 Thio quantity F, which hns been oalled above the friction per imit mr^A, is found by experiments to vaiy with the density^ ^imoom^^ and velocity c»f the flaid, and v%nth the diameter and lOO^hneas cif the internal mirface of the pipe. In Hydraulics, the fluid considered is water, and any variationB in dtrnnity or viscosity, doe to changes of temperature^ are generally nie-Gfliiribl^, F^ therefore, may be taken as proportional to the «^ * <r to the weight w per cubic foot, to the roughness of the ; , . -. i as some function, /{v) of the mean velocity, and f{d) of rh** «liaaEM»ter of the pipe, Then, j^^pfM/m m in which expresion ^ may be called the coefficient of friction. li will be seen later^ that the mean velocity v is different from the Ttdathe Telocity u of the water and the surface of the pipe, arMi it probably would be better to express F as a function of «, ' 1 itftelf probably varies? with the roughness of the pipe and] ^\__ -:uer circumstances, and cannot directly be deteruiined, it, mnplifictd matters to express F, and thus A, as a function of i?. 93. Empirical formulae for loss of head due to friction, Tha difficulty of con^ctly determining the exact value of /(t) f{d)^ has led to the use of empirical formulae, which have poporcd of great practical service, to express the head h m terms of the trelodty and the dimenMons of the pipe. Tbe fiiniplest formula assumes that the friction simply varies as tlif* nqaare of the velocity, and is independent of the diameter of ape, or f{v) fid) = at^. Then, I or writiiig gi for a, m i-^*^ '^"C^ •(1), .(2), frcmi which is deduced the well-known Chezy formula, v = C ^/m.j^ or 1? - C ^Jmt. Another form in which formnla (1) ia often found is 120 HYDRA ITLICS or since wi = 7 for a circular pipe full of water, t 4,f.vH ^~2g.d .(3), in which for a of (1) is substituted L 29 • The quantity 2g was introduced by Weisbach so that h is expressed in terms of the velocity head. Adopting either of these forms, the values of the coefficients C and / are determined from experiments on various classes of pipes. It should be noticed that C = a/ -? . Values of these constants are shown in Tables XTT to XTV for different kinds and diameters of pipes and different velocities. TABLE Xn. Values of C in the formula v = G Jrni for new and old cast-iron pipes. New oast- iron pipes Old oaafc. iron pipes Velocities in ft. per second 1 3 6 10 1 3 6 10 Diameter of pipe 8" 95 98 100 102 68 68 71 78 6" 96 101 104 106 69 74 77 79 9" 98 105 109 112 78 78 80 84 12" 100 108 112 117 77 82 85 88 16" 102 110 117 122 81 86 89 91 18" 105 112 119 125 86 91 94 97 24" 111 120 126 131 92 98 101 104 30" 118 126 131 186 98 108 106 109 86" 124 181 136 140 108 108 111 114 42" 180 186 140 144 106 111 114 117 48" 185 141 145 148- 106 112 115 118 60" 142 147 150 162 For method of determining the values of C given in the tables, see page 102. On reference to these tables, it will be seen, that C and / are by no means constant, but vary very considerably for different kinds of pipes, and for different values of the velocity in any given pipe. FLOW THROUGH PIPES 121 The fact that C varies with the velocity, and the diameter of the pipe, suggests that the coefficient C is itself some function of the velocity of flow, and of the diameter of the pipe, and that /i/(r) f{d) does not, therefore, equal at;'. TABLE Xni. Values of / in the formula , _4/t;»J 1 New east-iron pipes Old cast-iron pipes VeloeitiMm ft.p«rieeoDd 1 3 6 10 1 8 6 10 DiAiiLofpipe 8" •0071 -0067 •0064 •0062 •0152 •0189 •0128 •0122 6" •007 •0068 •006 •0057 •0135 •0117 •0108 •0103 9^ •0067 •0058 •0055 •0051 •0122 •0105 •010 •0092 IT -0064 •0056 •0051 •0048 •0108 •0096 •0089 •0084 15" -0062 •0058 •0048 •0048 •0099 •0087 •0081 •0078 1 18" •0058 •0051 •0045 •0041 •0087 •0078 •0073 •0069 1 24" •0058 •0045 •0040 •0037 •0076 •0067 •0063 •0060 1 W •0046 •0040 •0087 •0035 •0067 •0061 •0067 •0066 86" •0042 •0087 •0085 •0038 •0061 •0056 •0062 •0060 1 42" -0068 •0085 •0038 •0081 •0058 •0052 •005 •0048 1 48" -0086 •0082 •0081 •0029 •0067 •0061 •0049 •0046 60" -0032 •0030 •0029 •0028 TABLE XIV. Valaes of C in the formula v-C -Jmi for steel riveted pipes. Velocities in ft per second 1 8 5 10 Diameter of pipe 8" 81 86 89 92 11" 92 102 107 115 111" 15^ 93 99 102 105 109 112 114 117 88" 113 113 113 118 42" 102 106 108 111 48" 105 105 105 105 72"* 110 110 111 111 72" 93 101 105 110 108" 114 109 106 104 See pages 124 and 137. 122 HYDRAULICS 94. Formula of Daroy. In 1857 Darcy* published an account of a series of experiments on flow of water in pipes, previous to the publication of which, it had been assumed by most writers that the friction and consequently the constant C was independent of the nature of the wetted surface of the pipe (see page 232). He, however, showed by experimenis upon pipes of various diameters and of different materiah^ including wrought iron, sheet iron covered with bitumen, lead, glass, and new and old cast-iron, that the condition of the internal surface was Of considerable importance and that the resistance was by no means independent of it. He also investigated the influence of the diameter of the pipe upon the resistance. The results of his experiments he expressed by assuming the coefficient a in the formula T al 2 m Q was of the form a = o + - r ' r being the radius of the pipe. For new cast-iron, and wrought-iron pipes of the same roughness, Darcy's values of a and P when transferred to English units are, a = 0-000077, /J = 0-000003235. For old cast-iron pipes Darcy proposed to double these values. Substituting the diameter d for the radius r, and doubling ft for new pipes, ;.=(o-oooo77.«:«o^)^ = 0-00000647 (1-2^1) ^^ or "^^^^V i2dTl^^ (*) ^^^s/mri-^' (5)- Substituting for m its value 2 > ^^^ multiplying and dividing by 29, \-'M'^mW^ w- For old cast-iron pipes, ;.=o-ooooi294(i-^)^ \ a / m -«'"('^is)S-s o- 2g * Reeherehea ExpirimentaUt. ^^^^1 FLOW THROUGH PIPES 123 To!^ v = 27S^^^^^i (8) I -^^\^l^l^^ t»>' ■ As the atudant cannot possibly retain, witliout unnecessary Biboiff, Tallies of / sod C for different diameters it is convenient ■to Temember the rim pie fonnSj I for Dew pipes, and ■ for old pipes. H According to Darcy, therefore, the coefficient C in the Chezy ■ fontiala varies only with the diameter and roughness of the pipe. I The %*mlae8 of C afi calcolated from his experimental results, for ■ t^m? of the pipes, were practically constant for all velocitieBj and ■Jiotubty for those pipes which had a comparatively rough internal ^■picef hut for smooth pipes, the value of C varied from 10 to ^Hper ct*nL for the siame pipe as the velocity changed. The Experiments of other workers show the same results, V aption that ^f{v)f(d) = av^ in which a is made to Tii^ : h the diameter and roughness, or in other words, the aoiunptian that h is proportional to ij" is therefore not in general stilted by experiments. As stated above, the formulae given must be taken as ^npiriea], and though by the introduction of suitable they can be made to agree with any particular experi- even set of experiroents, yet none of them probably rprt!!Si^s truly the laws of fluid friction. The formula of Chezy by its simplicity has found favour, and I is likely, that for aome time to come^ it i^411 continue to be used, in the form t? = C vmi, or in its modified f onu ^ 2gd ' In making calculations, values of C orf, which most nearly suit ly given case, can be taken from the tables, 00. Yariatioii of C in the formula v = C -/mi with aervioe. It should be clearly borne in mind, however, that the dis- diaf^ging capacity of a pipe may be considerably diminished after m few yeanf* service. Dwtjy'B re«u]t« show that the loss of head in an old pipe may doable that in a new one, or since the velocity v is taken as ^hm doi 124 HYDRAULICS proportional to the square root of A, the discharge of the old pif for the Bame head will be -^ time^ that of the new pipe, or aboafj 30 per cent. less. An experiment by Sherman* on a 3d-inch cast-iron main sh( that after one year's gervice the discharge was dmiinishei 23 iiev cent., but a second year's service did not make any furthi?r altc^ration. Experiments by Kuichlingt on a 36-inch cast-iron main showed that the discharge during four years diminished 36 per cent*| while experiments by Fitzgerald t on a cast-iron main, coated with tar, which had been in use for 16 years, showed that cleaning inci^eased the discharge by nearly 40 per cent, Fitzgerald also found tlat the discharge of the Sudbury aqueduct diminished 10 per ceni. in one year due to accumulation of slime. The experiments of Marx, Wing, and Hoskins| on a 72-inc)i stt'el main, when new, and after two years' service, showed that ther# had been a change in the condition of the internal surface of tfiB pipe, and that the disscharge had diminished by 10 per cent, at lo^ velocities and about 5 per cent, at the higher velocities. If, therefore, in calculations for pipes, values of C or /are used for new pipes, it will in most cases be advisable to make the pipe of such a size that it will discharge under the given bead at least from 10 to 30 per cent, more than the calculated value. 97* Ganguiliet and Kiitter*s formula. Granguillet and Katter endeavoured to determine a form foi the coefficient C in the Chezy formula v = G ^mi, applicable to all forms of channels, and in which C is made a function of the virtual slope i, and also of the diameter of the pipe. They gave C the value, 1-811 0-00281 41-6 C = ^ 1,(41-0^2:29^) « This formula is very cumbersome to use^ and the value of th coefficient of roughness n for different cases is uncertain. Tabk have however been prepai*ed which considerably facilitate the of the formula. * Tram. Am^SX.E. Vol xjurv, p, 85. t Tram. Am.S.CE, Vol. eliv, p. 56. t Tram. Am.S.C.E. ToL xw¥, p* 87. g Sea Table No. KIV, FLOW THROirOH PIPES Iti Folties of H in Ganguilht ami Kutler*s formula. Wood pip€*s ^ '01, may be as high as "015. Cast-iron and sttnA pipes = *011, „ „ 02* Glaj^ eartheuwar© = *013* &8* Keynolds' experiments and the logarithmic formtila. The formulae for \om of head due to friction previously givea iiave all teen founded upon a probable law of %*ariation of h "Wiik V, bat oo rational basis for the assumptions has been adduced. It has been ^tjited in Bection 93, that on the assumption that h vam with 17^, the ci:^etticient C in the formula -Cy/ h ilitolf a ftinction of the \'elocity. Tilt? experiments and deductions of Reynolds, and of lati^r workers, throw considerable light upon this subject, and show that i i» proportional to t'**, where n is an index which for very small ^locities*— as previously shown by Poiseuille by experiments on CBfiillary tubes — is equal to unity, and for higher velocities may We jk variable vahte, which in many cases approximates to 2. } '^ experLments marked a decifled advaneej in showing ^1" - -ily that the roughness of the wetted surface has an fiffisct upon the loss due to friction^ so Heynolds' work marked her step in showing that the index w depends upon the state intt'mal surface, being generally greater the rougher the I student will be better able to follow Reynolds, by a brief | tion of one of his experiments, n Table XV are shown the results of an experiment made | f Reynolds with apparatus as illustrated in Fig. 88. In calumiis 1 and 5 are shown the experimental values = J, and « peKpjectively, The curves. Fig. 90, were obtained by plotting v as abscissae kd t 88 ordinates. For velocities up to 1*347 feet per second, the points lie very close A straight line and i m simply proijortional to the velocity, or i = hv „, (IIX bein^ a coefficient for this particular pipe. Above 2 feet per second j the points he very near to a continuous^ I the equation to which is i-itu** ,.........(12). • Phil Tmns, 1863, 126 HYDRAULICS Taking logarithms, log i = log h + nlog t;. The curve, Fig. 90 a, was determined by plotting log i as ordinate and logv as abscissae. Reynolds calls the lines of this figure .the logarithmic homologues. Calling logi, y, and log v, a?, the equation has the form y = k-i- 7WJ, which is an equation to a straight line, the inclination of which to the axis of x is ^ = tan""^7i, or n = tan 0, Further, when » = 0, y = fc, so that the value of Jc can reardily be found as the ordinate of the line when x or logv = 0, that is, when v = 1. Up to a velocity of 1*37 feet per second, the points lie near to a line inclined at 45 degrees to the axis of v, and therefore, n is unity, or as stated above, i = kv. The ordinate when v is 6qual to unity is 0*038, so that for the first part of the curve k = '038, and i = '0381?. FLOW THROUGH PIPES 127 Above the velocity of 2 feet i)er second the points lie about ft second strai^Ht line, the inclination of which to the axis of t; is » = tan-U-70. Therefore log i = 1*70 log v + k. The ordiixate -when v equals 1 is 0*042, so that fc = 0-042, and t = 0-042t;i^. Fig. 90 a. Logarithmic plottings of t and v to determine the index n in the formula for pipes, t = A;i;*^. In the table are given values of i as determined experimentally and as calculated from the equation i = k, v^. The quantities in the two columns agree within 3 per cent. 128 HTDRAUUCS Lead Pipe. TABLE XV. Experiment on Resistance in Pipes. Diameter 0*242". Water from Mancliester Main. Slope "1 k n Velocity ft per second Experimental value Calculated from •0086 •0092 •038 -289 •0172 • •0172 •038 •451 •0258 •0261 •038 •690 •0346 •0347 •038 •914 •0480 •0421 •038 1109 •0516 •0512 -038 1-849 •0602 • •> 1482 •0682 ... 1573 •0861 ... 1-671 •1033 ... 1775 •1206 ... 1-857 •1378 •1362 •042 1-70 1-987 •1714 •1610 •042 1-70 2-208 •3014 •2944 •042 170 8141 •4306 •4207 •042 1^70 8-98 •8185 •8017 •042 1-70 5-66 1021 1033 •042 170 6-57 1-438 1-476 •042 170 811 2-455 2^404 •042 170 10-79 8^274 3-206 •042 1^70 12-79 3^878 3-899 •042 1-70 14-29 Note. To make the columns shorter, only part of Beynolds' results are ghcs. 99. Critical velocity. ^t appears, from Reynolds' experiment, that up to a certun - which is called the Critical Velocity, the loss of head li which is a)nal to v, but above this velocity there is a definite change the axis ow connecting i and t;. 'cperiments upon pipes of different diameters and the Qj, variable temperatures, Reynolds found that the critical hich was taken as the point of intersection of the two Further, ^^^^ag found as the _ '0388? when t? = 1. ^* " D ' Up to a velooeing a line inclined at p_. 1 ,^qx unity, or as stated a 1 +0-0336T + •0000221P ^^^^' The ordinate wheature in degrees centigrade and D the diameter first part of the curve. ^ ¥upw through pipes 129 Critical velocity by the method of colour bands. exisftnice of the rrirical velocity has been beautiftill}' by Reynolds, by tho metliod of colour baTid^^ and his - r^ slIs*j explain why there is a sudden change in the law ^ I and r. Water was drawn through tubes (Figs. 91 and 92), ont of ^rigp gla?^ tank in which the tnbcg were immersedj and in th ihi* water had been allowe<l to come to rest, arrangements made %B shown in tlie figure so that a streak or streaks of liy coloured water entered the tubes with the clear water," ^ 3 Fig. 91. ^ Fig. 92, ' Tfsults were a^ follows :— " (I) When the velocities were sufficiently low, the streak <iliiur extended in a beautiful straight line through the tube*' **{2> A« the velocity was increa^^d by small stages, at r pc»int in the rube, always at a considerable distance from the jimjjrt-«ha|)ed intake, the colour band would all at once mix up the Huri'MnTnlitig water, and fill the rest of the tube with »if coltHired water" (Fig. 92), ij* Httdden change takes place at the cntical velocity. such a change takes place is also shown by the appai-atus Bjstrated in Fig. ^S'f when the critical velocity is reached there is liiknt disturbance of the meruury in the U tube. iJij therefore, a definite and sudden change in the con- of fl€>w. For vehjcities below the critical velocity, the flow to the tubes, or is ** Stream Line" flow, but after the Jocity ha.s been passed, the motion parallel to the tube is Qinpaniecl liy eddy motions, which cause a definite change to plut-e iu the law of resistance. Banie^i and Coker* have determined the critical velocity by l^<iting the sudden change of temperature of the water when its ttduii changes. They have also found that the critical velocity, determined by noting the velocity at which stream-line flowr PrxK4t4imji> *>/ iht Hmjat Socuqf, Vol. lxMV. 1904; Phil, TranMctioni, Tliat There 130 HYDRAULICS breaks up into eddies, is a much more variable quantity tlian that determined from the points of interBection of the two \mm as in Fig* 90* In the former case the critical velocity depends upon the oondition of the water in the tank, and when it k perfectly at rest the stream lines may be maintained at much higher velocities than those given by the formula of Reynolds. If the water is not perfectly at rest, the results obtained by both methods agree with the formula. Barnes and Cokor have called the critical velocity obtained by the method of colour bands the upper limit, and that obtained by the interiiiection of the logarithmic homologues the lower critical velocity* The first gives the veloc^ity at which water flowing from rest in stream-line motion break.s up into eddy motion, while the second gives the velocity at which water that is initially dii^turbed persists in flowing with eddy motions throughout a long pipe, or in other words the velocity ib too high to allow stream line^ to be formed. That the motion of the water in large conduits is in a similar condition of motion is shown by the experiment of Mr G* H, Benzenberg* on the discharge through a sew-er 12 feet in diameter* 2534 ft,, long. In order to measure the velocity of water in the sewer, red eosine dissolved in water was suddenly injected into the sewer, and the time for the coloured water to reach the outlet half a mile away was noted. The colour was readily perceived and it was found that it was never distributed over a length of more than Q feet. As >vill be seen by reference to section 130, the velocities of translation of the particles on any cross section at any instant are very difFerentj and if the motion were streatn line the colour must have been spread gut over a much greater length. 101. Law of factional resistance for velocities above critical velocity* As seen from Reynolds' formula, the critical velocity ea for very small pipes is so very low that it is only nece&sary S^ practical hydraulics to consider the law of frictional resistance for velocities above the critical velocity* For any particular pipe, and it remains to determine k and n» From the plottings of the results of his own and Dmr * Tramaeiiom Am.SM,E, 18t3| and alto Fn>eeeMttfft Am,S.€.E.^ VoL m.rm* FLOW THROUGH PIPES 131 experiments, Beynolds found that the law of resistance " for all pipes and all velocities" could be expressed as -t - Transposing, , /BD \« B'D'.u'.P' t = and 4= AP-.D* B"P-" .(14). .(15), A D"-* D is diameter of pipe, A and B are constants, and P is obtained from formula (13). Taking' the temperature in degrees centigrade and the metre as nnit length, A = 67,700,000, B = 396, 1 P = or in w t = y=. 1 + -00361 + -000221T" B'.t)'.F-» y.v* 67,700,000 D»-»" D»-» B»P^" .(16), ^^^ ' 67,700,000- Values of y tchen the temperature is 10' C. n 7 1-76 1-86 1-96 2-00 0-000266 0000888 0-000687 0-000704 \ The values for A and B, as given by Reynolds, are, however, only applicable to clean pipes, and later experiments show that alt^oagli it is doubtful whether p = 3 - w, as given by Reynolds, is correct. Value of n. For smooth pipes n appears to be nearly I'lo. Reynolds found the mean value of n for lead pipes was 1*723. Saph and Schoder*, in an elaborate series of experiments carried out at Cornell University, have determined for smooth * TranBoeticnt of the American Society of Civil Engineers, May, 1903. See czerdte 31, psge 172. 9—2 132 HYDRAULICS brass pipes a mean value for n of 1*75. Coker and Clements found that n for a brass pipe *3779 inches diameter was 1*731. In column 5 of Table XVI are given values of n, some taken from Saph and Schoder's paper, and others as determined by the author by logarithmic plotting of a large number of experiments. It will be seen that n varies very considerably for pipes of different materials, and depends upon the condition of the surface of a given material, as is seen very clearly from Nos. 3 and 4. The value for n in No. 3 is 1*72, while for No. 4, which is the same pipe after two years' service, the value of n is 1*93. The internal surface had no doubt become coated with a deposit of some kind. Even very small differences in the condition of the surfafCe, such as cannot be seen by the unaided eye, make a considerable difference in the value of 7i, as is seen by reference to the xdines for galvanised pipes, as given by Saph and Schoder. For large pipes of riveted steel, riveted wrought iron, and cast iron, the value of n approximates to 2. The method, of plotting the logarithms of i and v determined by experiment, allows of experimental errors being corrected without difficulty and with considerable assurance. 102. The determination of the values of C given in Table XH. The method of logarithmic plotting has been employed for determining the values of C given in Table XII. If values of C are calculated by the substitution of the experimental values of v and i in the formula s/mi many of the results are apparently inconsistent with each other due to experimental errors. The values of C in the table were, therefore, determined as follows. Since i = kv^ and in the Chezy formula v = C J mi, . mC or 1 = — 3-, therefore —pa = fry* and 2 log C = 2 log v - (log m + log A; + n log t?) (17). The index n and the coefficient k were determined for a number of cast-iron pipes. FLOW THROUGH PIPES Valoes of C for i.'tJodties fnim 1 to 10 were calculated. Carvi w<-T\^ tlieTi plf»tt^^ for diffei^nt velocities, having C as oMinai atid dianieters as abscisisae, and the values given in the table we; Ai?<iaced from the curves. The vBlneB of C so interpolated diifer very conaiderably, fOiue cases, from the es(>t*riinental values* The diffieulties &Umding the accurate deterii*ination of i and v are very great, and tbe %*alties of C, for any given pipe, as calculated by substi titing in the Chezy formula the losses of head in friction and th vt^liiciries as determined in the experiments, were frequent!; inamsiMeiit wHth each father. As, for example, in the pipe of 3*22 ins. diameter given i Tible XTI which was one of Darcy's pipes, the variation of C a calculated from h and r given by Darcy m from 78'8 to 100. Chi plotting log ft and logr and correcting the readings a tiw they all lie on one line and recalculating C the variation wa ^ ' r > be only from ^5*9 to 101. ■tar correotioni^ liave been made in other cases. Thv author thinks this procedure is justified by the fact that mauy of the best experiments do not show any such inconsistencies. An attempt to draw up an interpolated table for riveted pipes mt-i not 8atisfjictor>% The author has therefore in Table XI Y givtm the values of C as calculated by formula (17), for %^arioua Vidoctties, and the iliameters of the pipes actually experimentec ap»kn. If curves are plotted from the values of C given ii Tntde XIV, It will be seen that, except for low velocities, th< -es are not continuous, and, until further experimental evidenc< i^ i^irthcoming for riveted pipes, the engineer must be content with ch*X)sing values of C, which most nearly coincide, as far be can judge, with th© case he is considering. 103. Vwiation of k, in the formula i = kv", with the diameter. It has beeii shown in section 98 how the value of fc, for iveo pipe, can be obtiiined by the logarithmic plotting of t and t?; In Ttihle XVI, are given values of A\ as determined by the authcirt by plotting the results of different experiments, Saph - hoder found that for smooth hard-drawn In-ass pipes us siases n varied between 173 and 1*77, the mean value being 1'75. ■ By plotting logff tm abscissae and log ft as ordinates, as in' Fig* I'ti, for these bniss pi|X'S the points He nearly in a straight line which hoA aJi inelination & with the axis of d, such that tan^--P25 134 HYDRAULICS and the equation to the line is, therefore, \ogh = logy-p\ogd, where p = 1*26, and log y = log Jc when d = 1. From the figure y = 0-000296 per foot length of pipe. •05 -03. (m EqucUwntu liive Zo€f. 'k^Log m - 1-2SLog d ' dz 03 Of -06 -08 ho W^ '3 Log di Fig. 93. Logarithmio plottings of fc and d, to determine the index p in the fonnul . 7 . r* On the same figure are plotted logd and log A;, as deduce from experiments on lead and glass pipes by various workers. I will be seen that all the points lie very close to the same line. For smooth pipes, therefore, and for velocities above tb critical velocity, the loss of head due to friction is given by % = _yv ~ d^ the mean value for y being 0*000296, for n, 1*75, and for p l'2o. From which, v = 104i""d'^', or log V = 2-017 + 0-572 log i + 0*715 log d. FLOW THROUGH PIPES 135 The value of p in this formula agrees with that given by Reynolds in his formnla Pnrfessor Unwin* in 1886, by an examination of experiments on cast-iron pipes, deduced the formula, for smooth cast-iron pipes, •0004t?^* * = cf and for rough pipes, % = ^^ • M. Flamantt in 1892 examined carefully the exi)eriment8 available on flow in pipes and proposed the formula, for all classes of pipes, and suggested for y the following values : Lead pipes ^ Glass „ [ •000236 to -00028, Wrought-iron (smooth) J Cast-iron new '000336, „ „ in service •000417. If the student plots from Table XVI, log d as ordinates, and log" A- as abscissae, it will be found, that the points all lie bqtween two straight lines the equations to which are log k = log -00069 - 1-25 log d, and log & = log '00028 - 1-25 log d. Further, the points for any class of pipes not only He between these two lines, but also lie about some line nearly parallel to these lines. So that p is not very different from r25. From the table, n is seen to vary from 1*70 to 208. A general formula is thus obtained, , -00028 to •00069t;^'^^^^-^/ The variations in y, n, and p are, however, too great to admit of the formula being useful for practical purposes. For new cast-iron pipes, , -000296 to -0004181?^^ '«^*^Z fe = ^ • If the pipes are lined with bitumen the smaller values of y and T< may be taken. * Industries, 1SS6. t AimaUs des Pouts et Chauss^et, 1892, Vol. ii. 136 HYDRAULICS For new, steel, riveted pipes, ^_' 0004to'00054i;^"^'«'Z ^- d}^ Fig. 94 shows tlie result of plotting logfe and logd for all the pipes in Table XVI having a value of n between 1*92 and 1*94. They are seen to lie very close to a line having a slope of 1^, and the ordinate of which, when d is 1 foot, is '000364. , •000364r^"*'Z Therefore d}'' or r = 59i-««d"' very approximately expresses the law of resistance for particular pipes of wood, new cast iron, cleaned cast iron, and galvanised iron. Fig. 94. Taking a pipe 1 foot diameter and the velocity as 3 feet per second, the value of i obtained by this formula agrees with that from Darcy's formula for clear cast-iron pipes within 1 per cent. Use of the logarithmic formula for 'practical calculations. A very serious difficulty arises in the use of the logarithmic formula, as to what value to give to n for any given case, and consequently it has for practical purposes very little advantage over the older and simpler formula of Chezy. TABLE XVI. 1 Diameter (in int.) Velocity in Value of n ^ /alue of k enter 1 Kind of pipe fk per seo. from to in formula 1 n formula le ^ Wood 44 8-46 — 4-416 1-78 0001254 «» 54 2-28 — 4-68 1-75 000088 skissV w 72-5 1 ~ 4 172 000061 M 72-6 1 — 6-6 1-98 000048 witcham Riveted 8 1^88 00245 oith Wrought 11 1-81 000516 iron or steel 11} 1^90 000470 ff 16 194 000270 ding »» 88 •506— 1-254 2^0 000099 ^ »» 42 210 - 499 198 00011 »» 48 2 - 6 (?) 2-0 000090 ^ipg) »» 72 1—4 199 000055 99 72 1 — 66 1^85 000077 2bel »9 108 1 — 45 2-08 000036 cy Cast iron 8*22 •289— 1071 1-97 00156 new 6-89 •48 — 158 1-97 00079 tf 7-44 •678—1617 1^956 00062 ff 12 1779 000823 iZHfl }f 16-25 1^858 000214 pe »» 16-5 2-48 — 8-09 1^80 000267 «f 19-68 1-88 — 87 1^84 00022 nan »i 36 4 — 7 2* 000062 -ns »» 48 1-248- 8-28 1^92 0000567 FenkeU 30 2 00003 ZJ Cast iron 1-4136 •167— 2077 1-99 0098 old and 31296 •403- 3-747 1-94 0085 tuberculated 9-575 1-007—12-58 1-98 0009 aan ^1 20 2-71 — 5-11 ?i 36 11 — 4-5 2 000105 raid 48 1.176— 3-533 2-04 0(KX)83 ♦> 48 1-135— 3-412 2-00 00(X)85 :y Cast-iron 1-4328 -371— 3-69 1-85 (K)41 old pipes 3-1536 -633— 5-0 1-97 (K)lHo cleaned 11-68 -8 —10-368 20 (KK);J75 raid yj 48 3-67 — 5-6 202 0(K)082 " 48 •895— 7-245 1-94 ()(KX)59 :y Sheet- iron 1055 -098— 8-225 1-76 (K)74 ,^ 3-24 •328-12-78 1-81 00154 ,*,' 7-72 -591-19-72 1-78 (X)059 11-2 1-296— 10-52 1-81 00039 Gas -48 -113— 3-92 1-83 0278 1-55 •205— 8-521 1-86 1-91 00418 0072 Sclioder Galvanised -364 1-96 0352 •494 1-91 0181 ,, -623 1-86 0182 •824 1-80 1 0095 ,^ 1-048 1-93 •(K)82 Hard -drawn 15 pipes 1-75 •00025 to brass up to 1-84 -0(K)85 .Ids / Lead 1-782 •55 1 1-61 1 1-761 \ -0\^e> 1^783 \ •0^^^ 138 HTDBAUUCS TABLE XVn. Showing reasonable values of y, and n, for pipes of TarioDf kinds, in the formula, Takes for j 1 1 i^ « y i Clean caBt-iron pipes Old cast iron pipes Riveted pipes Galvanieod pipes Sheet-iron pipes cover- ed with bitumen Clean wood pipes BrasR and lead pipes -00029 to -000418 '00O47 to '00069 '00040 to *00054 -00035 to -00045 '00030 to *00088 '00056 to '00063 1*80 to 1*97 1'94 to 2*04 1-93 to 2'08 1*80 to 1-96 l'76tol'81 ; 1-72 to 1-75 ' ■00086 -00060 •00050 '00040 -00034 ■O0060 *00080 2 t vm m When further experiments have been performed on pipes, rf which the state of the internal surfaces is accurately known, vA special care taken to ensure that all the loss of head in a git® length of pipe is due to friction only, more definiteness maybe given to the values of y, n, and p. Until such evidence is forthcoming the simple Chezy formnl* may be used with almost as much confidence as the mow complicated logarithmic formula, the values of C or/ being takea from Tables XII — XIV. Or the formula h = fcr* may be used, values of k and n being taken from Table XVI, which most nearly fits the case for which the calculations are to be made. I 104. Criticism of experiments. The difficulty of differentiating the loss of head due to friction . from other sources of loss, such as loss due to changes in direction, change in the diameter of the pipe and other causes, as well as the possibilities of error in experiments on long pipes of large diameter, makes many experiments that have been performed of very little value, and considerably increases the difficulty of arriving at correct formulae. The author has found in many cases, when log i and log d were plott<3d, from the records of experiments, that, although the results seemed consistent amongst themselves, yet compared with other e-Yperiments, they seemed of little value. \ for one ol Couplot^s* e3:periments on a lead and ipe being &b low as 1*56, while the results of an iSunpecint cm a cust-iroii pipe gave n as 2*5. In the fe were a number uf bends in the pipe* experiments for loss of head due to friction, it is I the pipe should be of uniform diameter and aa Uible between the points at which the pressure head Further, special care Rhould be taken to ensure the^ I air, and that a perfectly steady flow m established Fhere the pressure is taken* Eometer fittings, h-eme iuiixirtance that the ktmections shall be made JBference in the pressures any two pcjints shall be fiction^ and friction only, points, Hiiptes that there shall ^P^B to interfere with the Se w*ater, and it h, there- bntial that all burrs sliall hom the inside of the pipe, bients un small pipits in JT the best results are no bd by cutting the pipe irough at the connection t'ig. 95, which illustrates I connection use<l by Dv p|- gg (u experiments cited on lie two ends of the pipe ai*e not more than u^nr ^B the method adopted by Marxj Wing and Hoskins riments on a 72^inch wooden pipe to ensure a correct B pressure. I? X was connected to the top of the pipe only while jted at four points as shown. (erences were observed in the readings of the t%vo ji they thought were due to some accidental circum- big the gauge X only, as no change was obsoiTcd of Y when the points of communication to Y were of the cocki*, utici, HamUton Smith. Jnnr. iinfft fif the Ttntitufe of Vint Euffineertf 1855, ^Ita 140 HYDRAULICS 106. Effect of temperature on the velocity of flow. Poiseuille found that by raising the temperature of the water from 50** C. to 100' C. the discharge of capillary tubes was doubled. Fig. 96. Piezometer connections to a wooden pipe. Reynolds* showed that for pipes of larger diameter, the effect of changes of the temperature was very marked for velocities below the critical velocity, but for velocities above the critical velocity the eifect is comparatively small. The reason for this is seen, at once, from an examination of Reynolds'* formula. Above the critical velocity n does not differ very much from 2, so that P"* is a small quantity compared >vith its value when n is 1. Saph and Schodert, for velocities above the critical velocity, found that, as the temperature rises, the loss of head due to friction decreases, but only in a small degree. For brass pipes of small diameter, the correction at 60° F. was about 4 per cent, per • Scientific Papers, Vol. ii. + See also Barnes and Coker, Proceeding* of the Royal Society ^ Vol. utx. 1904 ; Coker and Clements, Transactions of the Royal Society, Vol. cci. Proceeding* Am.S.C.E. Yol xxix. FLOW THROUOH PIPES 141 degrees F. With galvanised pipes the correction appears to le from 1 per cent, to 5 per cent, per 10 degrees F. Since the head lost increases, as the temperature falls, the lischarge for any given head diminishes with the temperature, mt for practical purposes the correction is generally negligible. 107. Ij088 of head due to bends and elbows. The loss of head due to bends and elbows in a long pipe is generally so small compared with the loss of head due to friction Q the straight part of the pipe, that it can be neglected, and onsequently the experimental determination of this quantity has lot received much attention. Weisbach*, from experiments on a pipe 1^ inches diameter, rith bends of various radii, expressed the loss of head as *-(«^*T)g. being the radius of the pipe, B the radius of the bend on the pntre line of the pipe and v the velocity of the water in feet per ?<rond. If the formula be written in the form 7 at?* le table shows the values of a for different values of ^ . r R •1 -167 •2 -250 •6 -626 St Venantt has given as the loss of head h^ at a bend, Ab = -001524 y^^=Ol| 4 yi nearly. being* the length of the bend measured on the centre line of the ?nd and d the diameter of the pipe. ANTien the bend is a right angle RVR 2VR' When d = 1, •5, % d = 1-57, Ml, •702 iB = ■ -'"g' •111 29' < . • Ueehatue$ of Engineering, i Compte* Rendtii, 1862. 143 HTTDRiUXICS Kt^ent experiments by Williams, Hubbell and Fenkell^onc iron inpes asphalted, hy Saph and &boder on brajss pipe.**, oth* r* by Alexander t on wooden pipes, show that the loss of ] in lK>nds, as in a straight pipe, can be expressed as n Wing a variable for different kinds of pipes, while Ar- y'Hr y being a constant coeffieii lipe. For the cast-ircin pipes or and Penkell, y^ n, m, and f ♦ have approximately the {ollowm^ les. Dia&Mter of pipe ir ir mr <€0« 1-TB i-m i-m ^^^len r is 3 feet per second and p" is i, the bend being a righl angle^ tlie lo^ of head as calculated by thia formula for difi'j i2-ineh pipe is \^ , and for ta© dU-mch pipe -^- - . For the brass pipes of Saph and Schoder, 2 inches diametO'^ Alexander found, and for vanitshecl wood pipes when ^ is less than 0'2, and wlieii u is betn^een 0"2 and O'o, 11 He further found for \*amished wood pipes that^ a bend of radias equal to 5 times the radius of the pipe gives the mini mum loss + ^^-'rf} '»*^'i f^->* ^*^ !-<»si?^tance is equal to a straight pipe 3"3S times the length of the bend. Messrs Williams, Hubbell and Fenkell also state at the end their elaborate paper, that a bend having a radius equal to : • Proe. Amer. Soc, Civil EmgiHeen, Y6L xxm. t Proc, Intt. Civil Emgineen, Vol. clix. rLOW THROUGH TlfES 143 loffers hm remstanoe to the flow of water than those of aditm. It should not be overlooked, howeverj that although nf liead in a bend of radius equal to 2A diameterB of the ess than for any other, it does not follow that the loss of JT unit length of the pipe measured along its centre line cniiiit&um value for bends of this radiuB. Variations of the velocity at the cross section of a rieal pipe* (eriinents show that when water flows through conduits of rm, the velocities are not the same at all points of any turn aeetion, but decrease from the centre towards the terence. f first experimentsi to determine the law of the variation of [ocity in cylindrical pipes were those of Darcy, the pipes If in diameter frfjm 7"8 inches to 19 incites. A complete tf of the exi>eriraents is to be found in his Recherchm tnmiiales dan^ les iuyauit^ • Telocity was me^isured by means of a Pitot tube at five on a vertical diamfetjer, and suits plotted as iihown in r. ling V the velocity at the of a pipe of radins R^ u the Y at the circumference, tv san velocity, y the vt^wity distance r from the centre, losB of head per unit [the pipe, Darcy deduced the formulae \ the unit is the metre the value of k is \\% and 20'4 when ft IB the English foot. 'jBT earperiments commenced by Darcy and continued by on the distribution of velocity in a semicircular channel, rface of the water being maintained at the horizon tal g" and in which it was assumed tho conditions were similar ^Ki a cylindrical pipe, showed that the velocity near the ^OT the pipe diminished much more rapidly than indicated iormula of Darcy, 144 HYDRAUUCS Bazin substituted therefore a new formula, Y-v = 38y/Ri(^J (1), or since t7» « C Jmi = —j^ JTU -C'^ii)' ■■■ <^>- It was open to question, however, whether the conditions of flow in a semicircular pipe are similar to those in a pipe discharging full bore, and Bazin consequently carried out at Dijon*, experi- ments on the distribution of velocity in a cement pipe, 2'73 feel diameter, the discharge through which was measured by means of a weir, and the velocities at different points in the transverse section by means of a Pitot tubet. From these experiments Bazin concluded that both formulae (1) and (2) were incorrect and deduced the three formulae V-..38^RS((0-(iy*(g} (3), V-.^VBi{38(^)\4 9(g(l- rigy (4), V-r = N/Rt53-5{l-yi--95(j)'} (5), the constants in these formulae being obtained from Bazin's by changing the unit from 1 metre to the English foot. Equation (5) is the equation to an ellipse to which the sides of the pipes are not tangents but are nearly so, and this formula gives values of v near to the surface of the pipe, which agree much more nearly with the experimental values, than those given by any of the other formulae. Experiments of WilliamSy Hvhhell and FenkellX* An elaborate series of experiments by these three workers have been carried out to determine the distribution of velocity in pipes of various diameters, Pitot tubes being used to determine the velocities. The pipes at Detroit were of cast iron and had diameters of 12, 16, 30 and 42 inches respectively. The Pitot tubes § were calibrated by preliminary experimentB on the flow through brass tubes 2 inches diameter, the total * ' * Memoire de TAcad^mie des Soiences de Paris, Becueil des SavantsEtraxigdiv," Vol. XXXII. 1897. Proc, Am,S.C,E. Vol. xxvii. p. 1042. t See page 241. X ** Experiments at Detroit, Mich., on the effect of carvatare on the flow of water in pipes,*' Proc. Am.S.C.E. Vol. xxvu. p. 318. § See page 246. FLOW THROUGH PIPES 145 discliarge being determined by weighing, and the mean velocity thas determined. From the results of their experiments they came to the conclusion that the curve of velocities should be an ellix>8e to winch the sides of the pipe are tangents, and that the velocity at the centre of the pipe V is TlOvm, t?» being the mean velocity. These results are consistent with those of Bazin. His experi- V mental value for — for the cement pipe was 1*1675, and if the constant "95, in formula (5), be made equal to 1, the velocity curve be<x>me8 an ellipse to which the walls of the pipe are tangents. V The ratio — can be determined from any of Bazin's formulae. Substituting —^ for >/Ri in (1), (3), (4) or (5), the value of V at radius r can be expressed by any one of them as '-^-#--/(S)- Then, since the flow past any section in unit time is VmyfR\ and that the flow is also equal to 2'7rrdr . v, f therefore v^^W = 2t J^|v - 5^"/(^)jrdr (6). (t\ ^lA*** -g^j , its value -^ from equation (1), and integrating. ;; = ^"-c- (7), and by substitution oifi^j from equation (4), l-'*v <«. V so that the ratio — is not very different when deduced from the simple formula (2) or the more complicated formula (4). Wlien C has the values C = 80, 100, 120, from (8) ~ = 1-287, 123, ri9. The value of C, in the 30-inch pipe referred to above, varied between 109"6 and 123*4 for different lengths of the pipe, and i^ H. 10 /: 146 HYDRAUUCS the mean value was 116, so that there is a remarkable agreement between the results of Bazin, and Williams, Hubbell and Fenkell. The velocity at the 8v/rface of a pipe. Assuming that the velocity curve is an ellipse to which the sides of the pipe are tangents, as in Fig. 98, and that V= ri9t?«, the velocity at the surface of the pipe can readily be determined. Let u = the velocity at the surface of the pipe and v the velocity at any radius r. Let the equation to the ellipse be in which a? = v - u, and b = y — u. Then, if the semi-ellipse be revolved about its horizontal axis, the volume swept out by it will be |^rR''6, and the volume of discharge per second will be r^ 7rR'»t7^ = 27rrdr . i; = ttR' . u + §7rR«6, J itt + f X ri9r«, and u = '621v„,. u Using Bazin's elliptical formula, the values of — for C = 80, are u -=•552, -642, Vm 100, 120, 702. The velocities, as above determined, give the velocity of translation in a direction parallel to the pipe, but as shown by Reynolds' experiments the particles of water may have a much more complicated motion than here assumed. 109. Head necesisaky to give the mean velocity Vm to the water in the pipe. It is generally assumed that the head necessary to give a mean V ^ velocity Vm to the water flowing in a pipe is ^, which would be correct if all the particles of water had a common velocity Vm. If, however, the form of the velocity curve is known, and on the assumption that the water is moving in stream lines with definite velocities parallel to the axis of the pipe, the actual head can be determined by calculating the mean kinetic energy per lb. of water flowing in the pipe, and this is slightly greater than ^ . FLOW THBOUOH PIPES 147 As bef ore, let v be the velocity at radius r. The kinetic energy of the quantity of water which flows past any aectioii per second w . 2vrdr . t; . ^r- , 2g' ir being tlie ^vreight of 1 c. ft of water. The kinetic energy per lb., therefore, f^w.2irrdrv* «io 2g ^^^ 1 w . 2xr dn? 2g}oV V2r. ^/r\l» , (9). The simplest value for / (^ is that of Bazin's formula (1) above. from which ■(-^-F) and /G D=^i" Substituting these values and integrating, the kinetic energy per 1^- is |— , and when C is 80, 100, a is 112, 1-076. On the assumption that the velocity curve is an ellipse to which the ijvalls of the pipe are tangents the integration is easy, and the value of a is 1*047. Using the other formulae of Bazin the calculations are tedious and the values obtained differ but slightly from those given. The head necessary to give a mean velocity Vm to the water in the pipe may therefore be taken to be o" > t;he value of a being ftbout 1'12. This value* agrees with the value of 1*12 for a, obtained by M. Boussinesq, and with that of M. J. Delemer who finds for a the value 11346. IIO. Practical problems. Before proceeding to show how the formulae relating to the loss of head in pipes may be used for the solution of various problems, it will be convenient to tabulate them. * Flamant's Hydrauliqw. 10—2 148 hydraulics Notation. fc = los8 of head due to friction in a length Z of a straight pipe. . i = the virtual slope = y . t? = the mean velocity of flow in the pipe, d = the diameter. m = the hydraulic mean depth A i*ftfli A fl Fcyrmuh,!. h = ^=^. This may be written y = 7^5— , or 1; = C J mi. The values of C for cast-iron and steel pipes are shown in Tables XII and XIV. Formula 2. h = rf-^ , f , . . 1 ^ in this formula being equal to 7^ of formula (1). Values of /are shown in Table XIII. Either of these formulae can conveniently be used for calculating fc, t?, or d when /, and Z, and any two of three quantities fe, v, and d, are known. Formula 3. As values of C and / cannot be remembered for variable velocities and diameters, the formulae of Darcy are convenient as giving results, in many cases, with sufficient accuracy. For smooth clean cast-iron pipes A = -005(l^|^^, or ^=m^j^M For rough and dirty pipes or « = i39y_|_V5f =27«yi2^^- FLOW THROUGH PIPES 149 II d is the unknown, Darcy^s formulae can only be used to solve for d by approximation. The coefficient ( 1 + t^ j is first neglected and an approximate value of d determined. The coefficient can then be obtained from this approximate value of d with a greater degree of accuracy, and a new value of d can then be found, and so on. (See examples.) Formula 4. Known as the logarithmic formula. h . y . r* Values of y, n, and p are given on page 138. By taking logarithms log h = log y + n log v + log I — p log d, from which h can be found if i, v, and d are known. If A, Z, and d are known, by writing the formula as n log V = log h - log I - log y + p log d, T can be found. If A, /, and v are known, d can be obtained from p log d = log y + n log v + log I - log h. This formula is a little more cumbersome to use than either (1) or (2) but it has the advantage that y is constant for all velocities. Formula 5. The head necessary to give a mean velocity v to ri2i7' the iJvater flowing along the pipe is about — ^ — , but it is generally convenient and sufficiently accurate to take this head as ^ , as was done in Fig. 87. Unless the pipe is short this quantity is negligible compared with the friction head. Formula 6.^ The loss of head at the sharp-edged entrance to a pij>e is about -^ and is generally negligible. Formula 7. The loss of head due to a sudden enlargement in a pipe where the velocity changes from Vi to rj is ^ . ^9 Formula 8. The loss of head at bends and elbows is a very variable quantity. It can be expressed as equal to -y- in which 9 a varies from a very small quantity to unity. ProbUm 1. The difference in level of the water in two reservoirs is h feet, FUr. 99, And thej are connected by means of a straight pipe of length I and diameter d ; to find the discharge through the pipe. 150 HYDRAULICS Let Q be the number of eabic feet discharged per second. The head h is utilised in giving velocity to the water and in overcoming resistance at the entrance to the pipe and the frictional resistances. Fig. 99. Pipe connecting two reservoirs. Let V be the mean velocity of the water. The head necessary to give the water this mean velocity may be taken as —= — , and to overcome the resistance at the entrances Then •6t;« ^'- 2g "^ 2g'^2g,d' Or using in the expression for friction, the coefficient 0, ;i =-0174v3+ -0078179+ = •025r2 + 4fa« C^d I . If - is greater than 300 the head lost dae to friction is generally great compared with the other quantities, and these may be neglected. iflv^ _ 4Zt7« ~C^,d* C /dh Then h = - and "2 As the velocity is not known, the coefficient C cannot be obtained from the table, but an approximate value can be assumed, or Darcy*s value C = 394 y 12d + l for clean pipes, C=278 a/toT-i *^ *^® P^P® " ^*y» and can be talcen. An approximation to v — which in many cases will be sufficiently near or will be as near probably as the coefficient can be known — ^is thus obtained. From the table a value of C for this velocity can be taken and a nearer approximation to V determined. Then Q=^dM;. The velocity can be deduced directly from the logarithmic formula A=^^, provided y and n are known for the pipe. FLOW THROUGH PIPES 151 The bydxmnlic gradient is EF. At any point C distant x from A the pressore head - is eqaal to the distance rtween the centre of the pipe and the hydranlio gradient. The pressure head Lftt inside the end A of the pipe is Aa kz— « ^^^ ^^ ^^® ^^^ ^ ^^^ pressure head last be eqaal to Ab. The hesd lost due to friction is h, which, neglecting the oiall qnantity — ^ — , is eqoal to the difference of level of the water in the two inks. Example 1. A pipe 3 inehes diameter 200 ft. long connects two tanks, the ifferenoe of lerel of the water in which is 10 feet, and the pressure is atmospheric. find the discharge assuming the pipe dirty. Using Darey'8 coefficient 17 = 278 ^y^ ^/SA^A"=69•5^/,i^ =3'8dft. per sec. For a pipe 3 inches diameter, and this velocity, C from the table is about 69, so iiat the approximation is sufficiently near. ^ ^ ,. -00064^1^/ Taking h= -^,^ , r=3-88 ft. per sec, . oooei?*! pves v=d'85 ft. per sec. Example 2. A pipA 18 inches diameter brings water from a reservoir 100 feet &bove datum. The total length of the pipe is 15,000 feet and the last 5000 feet ^re at the datum level. For tbis 5000 feet the water is drawn off by service pipes at me Quifonn rate of 20 cubic feet per minute, per 500 feet length. Find the pressure &t the end of the pipe. The total quantity of flow per minute is ^ 6000x20 ^^ u .r . = =200 cubic feet per minute. Area of the pipe is 1*767 sq. feet. The velocity in the first 10,000 feet iw, therefore, 200 The head lost due to friction in this length, is 4./^0^.2;888a 2i/.l-5 In the last 5000 feet of the pipe the velocity varies uniformly. At a distance 1*888^ X feet from the end of the pipe the velocity is ^^ . In a length dx the head lost due to friction is 4./. l'888».j'dg 2^.1-5.5000» ' and the total loss by friction is .^/ll^??. /"^ .^_4/. (l-888)« 6000 ^~2^.1'5.6000»jo 2^.1-5 ' 3 * The total head lost due to friction in the whole pipe is, therefore, H=;r-^. 1-8882 (10,000 + A«yui). 152 HYDRAULICS Taking / as *0082, H = 14*3 feet. Neglecting the velocity head and the loss of head at entrance, the pressure head at the end of the pipe is (100 - H) feet =85-7 feet Problem 2. Diameter of pipe to give a given discharge. Beqoired the diameter of a pipe of length I feet which will discharge Q onbie feet per second between the two reservoirs of the last problem. Let V be the mean velocity and d the diameter of the pipe. Then v=^-^ (1), and ft=-025t>»+^. Therefore, ^/W A- / .nos_i Squaring and transposing, ^g 0-0406. (yd If Hs long compared with d, h .(2). A- vs. (8). Since v and d are unknown G is unknown, and a value for C must be pro- visionally assumed. Assume G is 100 for a new pipe and 80 for an old pipe, and solve equation (3) for d. From (1) find v, and from the tables find the value of G corresponding to the values of d and v thus determined. If G differs much from the assumed value, reoaloultfte d and v using this aeoond value of C, and from the tables find a third value for G. This will generally be found to be sufficiently near to the second value to make it unnecessary to oaloolate d and v a third time. The approximation, assuming the values of G in the tables are correct, can be taken to any degree of accuracy, but as the values of G are uncertain it will not as a rule be necessary to calculate more than two values of d. Logarithmic formula. If the formula h = ^^^ be used, d can be found direct, from jp log d=n log V +log7+log £ -log ^. Example 3. Find the diameter of a steel riveted pipe, which will discharge 14 cubic feet per second, the loss of head by friction being 2 feet per mile. It is assumed that the pipe has become dirty and that provisionally G = 110. From equation (3) ^ 2-55.14 /6280 ^=^[Io-V^-' or ^^ log <i= log 16-63, therefore d = 3-08 feet. For a thirty-eight inch pipe Euichling found G to be 113. The assumption that G is 110 is nearly correct and the diameter may be taken as 37 inches. Using the logarithmic formula ^ •00045ri-»Z FLOW THROUGH PIPES 153 od sabetitating for v the yalne --£- Tom whieh 5-15 log d == log -000i5 - 1-95 log 0-7854 + 1*95 log 14 + log 2640, ind <i=d-07 feet. Short pipe. If the pipe Ib short so that the velooity head and the head lost at Atruftee are not negligible oompared with the loss due to friction, the equation •0406Q«d _ 6-5/Q* ^ h ^ C*h ' rhen a Talne is given to C, can be soWed graphically by plotting two carves LZld yi= _ -0406<y .d + 6-5«Q« h ' Ch ' The point of intersection of the two corves will give the HaxaetesT d. It is however easier to solve by approximation in the Dlloving manner. Neglect the term in d and soNe as for a long pipe. Chooee a new value for C corresponding to this ap- roximate diameter, and the velocity corresponding to it, nd then plot three points on the cnrve y=d^t choosing mines of d which are nearly eqnal to the caloolated value f d, and two points of the straight line •0406QSd yi= r^ + 6-6/Q* Fig. 100. •6 The enrve y=(^ between the three points can easily e drawn, as in Fig. 100, and where the straight line cuts be cnr-re, gives the required diameter. KxamepU 4. One hundred and twenty cubic feet of water are to be taken «T minate from a tank through a cast-iron pipe 100 feet long, having a square- diged entrance. The total head is 10 feet. Find the diameter of the pipe. Neglecting the term in d and assuming G to be 100, ^=6M?4. = -026. nd Therefore 100.100. (i=: 4819 feet. 2 j(-4819)« 10" = 10-9 ft. per sec. From Table XIl, the value of C is seen to be about 106 for these values of and r. A second value for d" is ^,^6-5.100.4 .10 = •0233, 106* rom which d= -476'. The schedule shows the values of d' and y for values of d not very different rom the calculated value, and taking C as i06. d -4 -5 -6 d» -01024 08125 0776 yi -0297 0329 The line and curve plotted in Fig. 100, from this schedule, intersect at|> for which d= -498 feet. 154 HYDRAULICS It is seen therefore that taking 106 as the Talae of C, oeglectiDg the term in i, makes an error of -022' or -264". This problem shows that when the ratio -z is about 200, and the virtual slope k even as great as ^, for all practical purposes, the friction head only need be oon- sidered. For smaller values of the ratio -r the quantity *025t^ may become in- portant, but to what extent will depend upon the slope of the hydraulic gnuiieol. The logarithmic formula may be used for short pipes but it is a little man cumbersome. Using the logarithmic formula to express the loss of head for short pipes witt square-edged entrance, •025Q» 7.Q».t or d*»-»^i« - •0406Q2d«*-«^= V^t'^ • When suitable values are given to y and n, this can be solved by plotting tin two curves and j,^=.0406(yd*»-aT5+l:^^ ^ the intersection of the two curves giving the required value of d. Problem 3. To find what the discharge between the reservoirs of problem (1) would be, if for a given distance l^ the pipe , of diameter d is divided into two branches I i • laid side by side having diameters d, and dj, k— J,^ — >W L — H Fig. 101. I ^ ii _ t ! Assume all the head is lost in friction. A^ ^ fi ^ cL \ C. Let Qi be the discharge in cubic feet. ^ *; "^ /^ * Then, since both the branches BC and BD , j ^ V t are connected at B and to the same reservoir, j ^^ ^ | D the head lost in friction must be the same in | j BC as in BD, and if there were any number I* ^ *i of branches connected at B the head lost in Yia, 101 them all would be the same. The case is analogous to that of a conductor joining two points between iHikh a definite difference of potential is maintained, the conductor being divided between the points into several circuits in parallel. The total head lost between the reservoirs is, therefore, the head lost in AB together with the head lost in any one of the branches. Let 17 be the velocity in AB, v^ in BC and Vj in BD. Then vd^^v^d^^ + v^^ (1), and the difference of level between the reservoirs 4?3r2 4/,V C*d "^Ci^di ' I the same as i iqual to Cg C*d ^ Ci*di <^'- And since the head lost in BC is the same as in BD, therefore, \ C,»d, C,«d, (')• If proTirionally Ok be taken as equal to C,, FLOW THROUGH PIPES 155 ThaeCore, d,*+ '-'■A .(4). Fitni (2), V ean be found by sabstitatiiig for v^ Arom (4), and thus Q can It dBiarmiiied. If AB, BC, mnd CD are of the same diameter and 2^ is eqaal to 2,, then ProfrinR 4. Pipes eonneeting three reservoin. As in Fig. 102, let three pipes AB, EC, and BD, connect three reservoirs A, C, D, the level of the water in each •f which remains constant. Lei V,, V,, and v, be the velocities in AB, BG, and BD respectively, Q,, Qs, md Q, Uie qoantities flowing along these pipes in cubic feet per sec., Z] , l^, and ^ |h0 Vngtha of the pipes, and d^ , d, and d, their diameters. Fig. 105^. Jjgt x^, r,, and £, be the heights of the surfaces of the water in the reservoirs, lad X, the height of the junction B above some datum. Liet h^ be the pressure head at B. /^ ffgyiTTM* all losses, other than those due to friction in the pipes, to be negligible. Xhe head lost due to friction for the pipe AB is tud for tlie pipe BC, .(1), .(2), ihe upper or lower signs being taken, according as to whether the flow is from, or Uywards, the reservoir G. For the pipe BD the head lost is (3). Cs'di"^'"*" ""''' Sinee the flow from A and G must equal the flow into D, or else the flow from A most equal the quantity entering G and D, therefore, or Vi'±r2^'»=rA' (4). There are foor equations, from which four unknowns may be found, if it is farther known which sign to take in equations (2) and (4). There are two cases to eooaider. 156 HYDRAULICS Ccue (a). Given the levels of the surfaces of the water in the reflerfoin aai of the junction B, and the lengths and diameters of the pipes, to find the quaali^ flowing along each of the pipes. To solve this problem, it is first necessary to obtain by trial, whether water floH to, or from, the reservoir C. First assume there is no flow along the pipe BC, that is, the pressure bead \ al B is equal Xo z^-Zq. Q Then from (1), substituting for v^ its value -^^f <J.=t/^ » from which an approximate value for Qj can be found. By solving (3) in the Mi way, an approximate value for Q,, is. Q, =T\/'^^ (^ If Q, is found to be equal to Qj, the problem is solved ; but if Q. is greater te; Qi , the assumed value for \ is too large, and if less, h^ is too smaU, for a diBur~ tion in the pressure head at B will clearly diminish Q, and increase Qj, and ^ also cause flow to take place from the reservoir C along GB. Increasbig te pressure head at B will decrease Q|, increase Q,, and cause flow from B to C. This preliminary trial will settle the question of sign in equations (2) snd (A and the four equations may be solved for the four unknowns, v^, v^, v^ and \. A is better, however, to proceed by "trial and error." The first trial shows whether it is necessary to increase or diminish h^ and : values are, therefore, given to h^ until the calculated values of v, , v^ and r| sati^^ equation (4). Case (&). Given Q^, Q^, Q,, and the levels of the surfaces of the water i& the reservoirs and of the junction B, to find the diameters of the pipes. In this case, equation (4) must be satisfied by the given data, and, therelbn^ only three equations are given from which to calculate the four unknowns 6y d], d^ and Iiq. For a definite solution a fourth equation must consequently n fonnd, from some other condition. The further condition that may be taken ii that the cost of the pipe lines shall be a minimum. The cost of pipes is very nearly proportional to the product of the length tai diameter, and if, therefore, Udi + l^+l^^ is made a minimum, the cost dL tbi pipes will be as small as possible. Differentiating, with respect to h^ , the condition for a minimnm is, that ^dh^^^dK^^dh, ^ Substituting in (1), (2) and (3) the values for 17^, r, and v,, V =_2l, ■■^' -dJ 4^ Q, -d^ differentiating and substituting in (7) 1 FXOW THBOUOH PIPES 157 Pottiiic the ^mlaes of Qi, Qa, and Q, in (1), (2), (8), and (8), there are four mitaoM as before for four anlmown quantities. U wUI be beUer however to solve by approximation. Qiie some arbitrary valoe to say d,, and oalcolate Aq from equation (2). Then calenlate cL and dm by putting h^ in (1) and (8), and substitute in t|Mli0ll(8). If this equation ie satisfied the problem is solved, but if not, assume a seoond ^ihw for dL and try again, and so on until such values of d,, dL, d. are obtained «al (8) is satisfied. In this, as in simpler ^stems, the pressure at any point in the pipes ought not to hJl below the atmoepherie pressure. flow through a pipe of constant diameter when the flow i$ diminishing at a ms\f$ru rate. Let I be the length of the pipe and d its diameter. Let h be the total loss of head in the pipe, the whole loss being assumed to be ^frietion. Let Q be the number of oobio feet per second that enters the pipe at a section A, mi. Q| the number of cnbie feet that passes the section B, I feet from A, the eaanti^ Q-Qi being taken from the pipe, by branches, at a uniform rate of Q-Q, ■ . * eoUe feet per foot. I Then, if the pipe is assumed to be continued on, it is seen from Fig. 103, that M the rate of discharge per foot length of tlie I is kepi oonstant, the whole of Q will be 1 in a length of pipe. The diaeharge past any seetion, x feet from CwiUbe ! lO.^ H t-^--^B M- j^. . The vekxity at the seetion is Fig. 103. 4( Q-Q,)x Aseoming that in an element of length dx the loss of head due to friction is nd eabetituting for v^^ its value -d« L^d» 4 le lose of head due to friction in the length I is *dx [^ / 4Q \»x»< JL-i'^ULd'y d' _ 7 / 4Q Y If Qj is zero, / is equal to L, and » {L«*^-'-(L-0*+*} di« _7_/4Q\»J_ n + l\rd^J di-»' The result is simplified by taking for dh the value 4v^dx dh= od asBTiming C constant. Then C«d 3T«C«d» |M, r FLOW THROUGH PIPES 159 IjbA Y be the ^eloci^ of the water in the pipe. Then, nnoe there is oontinaitT of flow, v the velooity .with which the water V.D" BMB the noszle ie ^ . The bead lost by friotion in the pipe is 2g.D~ 2gD^ ' r» i Tbm kinetie energy of the jet per lb. of flow as it leaves the nozzle is ~ . »«*" ^^=25 V^ V ) W' fan vfaieh by transposing and taking the square root, / 2gT>^h \4 ^-[D^hm) ^'>- The weight of water which flows per second szjtP.v.w where 10 = the weight of .(3). % oobie foot of water. T h erefore, the kinetie energy of the jet, is t Thia is a maximnm when j^=0> neiefore irir / 2gWi \* ~ d (I>»+4/W*)* (2^/iD»)* -ll^^ (2pftD»)* (16/W») (D» + 4/W*)*=0...(4), vlikh D» + 4/W*=12/W*, D»=8/W\ t- ^=^/87! (^)- If the nozzle id not circular but has an area a, then since in the circular nozzle pf tiie same area -d'=a, ! ^y anbstituting the value of D' from (5) in (1) it is at once seen that, for iMBmam kinetic energy, the head lost in friction is [ f " •'■ I UnHem 7. Taking the same data as in problem 6, to find the area of the ^%tm^ thai the momentum of the issuing jet is a maximum. I Dm momentum of the quantity of water Q which flows per second, as it leaves lMi«Uif !_iE lbs. feet. The momentum M is, therefore, 9 9 4 lor v* from equation (1), problem 6, 160 HYDRAULICS and Differentiating, and equating to zero, D»-4/W*=0, D» 4/r If the nozzle has an area a. d= D» = 5/^», and .=•392^: Ti' Sabstitating for D^ in eqoation (1) it is seen that when the momentom ii a maximum half the head h is lost in friction. Problem 6 has an important application, in determining the ratio of the mm of the supply pipe to the orifice supplying water to a Pelton Wheel, while problem 7 gives the ratio, in order that the pressure exerted by the jet on a fixed plaa* perpendicular to the jet should be a maximum. Problem 8. Lom of head due to friction in a pipe, the diameter of wkieh vmUi uniformly. Let the pipe be of length I and its diameter vary anifonnly from 4^ to d,. Suppose the sides of the pipe produced until they meet in P, Fig. 104. 8. = ^.«. «=^ Then r = ^ and S= ..av 8 + i do do-di The diameter of the pipe at any distance x from the small end ia ~~ s • The loss of head in a small element of length dx is ^^ , v being the veloeity when the diameter is d. Fig. 104. If Q is the flow in cubic ft per second t; = -^ = i Q The total loss of head % in a length I is ^ 64Q». dx ir»C»d» 64.Q«8»dj? oir«C«di»(S+j;)» 16Q«.S» (s* (S+OV' Substituting the value of S from equation (1) the loss of head doe to can be determined. Problem 9. Pipe line comitting of a number ofpipei ojf different dtOMetefi. tk- practice only short conical pipes are used, as for mstance in the limbs of a VaBtn meter. | If it is desirable to diminish the diameter of a long pipe line, instead of ndq a pipe the diameter of which varies uniformly with the length, the line is mads ^ of a number of parallel pipes of different diameters and lengths. I FLOW THBOUQH PIPES 161 tiel L« l^* t^ .«« he tiie leogth* and d^, d^^d^.,, the diiuneters retpeotiv^elj, of Ibi w^timmi df ibid mpe, 1!be loial loss of B«a^ due lo bielioti^ if G b^ aaaiimdd oonatanti is (?\d,^ d,^ d, '-} Tlie iTIiitifrtfr 4 of the pip«, which, for ihe mme total length , would ghe the iMne diTJiTgn for the i&me lou of head due (o friotiou, can be found from the Tlic leDftb Js for the oX ft pp^T c^f 4^ostant diameter D, which wiU give the lame ae loe« of head by friction, is ■'"ii- + — ^ + — ^PrtMam 10. Pipe aetinff at a tiptore. It U iome times i:ieoesBAry to t^ke a pipe tiae otst aome obetraclion^ snch ae a hill, which neoe&Eitatea the pipe rising » ttol onl^ »tiOTe the hydraolic gradient as in Fig. 87, but even above the origlEial Ifffivl o(f tiM water in the reservoir from which the supply [fl derived » Ii«t It be BQippoaed* as in Fig. 105, that water ie to be delivered from the resefvoir 3 to lb» fieaervoir C through the pipe BAG, which at the poiat A rificfi h^ feet above Ihe iev^ af the lorlace of the water in the npper reeer voir. Fig. 105. X4»t atm iifftfenee is level of the s^rfaoes of the water in the xeservoiFB I«et *. b« the prea^ni^ bead equivBlent to the atmospbcrio prcaatire* To ttlan the flow in the pipe, it will he ncoeasary lo fill it by a pump or other \j^ tl be a^eumed that the flow ia allowed to take place and is regulated ao that ^ i» ixyfitiiioo^ and the velocity i? is aa large as p<»seible. aei^ecting the velocity heatl and reBistaoeee othef than that due to fVictionf (1). Xft asul il beilif the length and diameter of the pipe reepoctively^ Tkm h7draalie gradient is practleaUy the straight line DE. Tb^cMVtieally II AF i^ made greater than k^^ which is about 34 feet^ the pfefeure ■t A \mtif^mf'^ nr^^ative and the flow will oeaa^« Fcsrel>' nnot l»e made much greater than 25 feet. X«jl«'' '''»*Tn vettKit^ poa*ibU in the ruinfj limb AB, to that tfu preuure <^^§d mi A t^ini'ljiiit b^ UFO. Lei e^m ^ ^^ telomty. Lei the datum level be the snr&ce of the water in C^ £^ B. II 162 HYDRAUUC3 Then fiai Therefore h^-^hu-t^n — 20. d + H + ^. H = ftB + -»B + Ai* .-s/' 2g{h„-.h,).d 1/«AB .(»). If the pre&sare head is not to be lees thaa 10 f«et of w^ter^ If i^m la loBB Ihan i>i the dischafge of tha siphon will be determined lij i Umititig velociij, hhi] it will be nuce^arv to throttle tlie (jipe &t C by meiuis of I valve, Ao tkB to keop the limb AC full and to keep the *' aiphoa *' from bf^mg bfbk«D. 1 In designing Htidh a siphon it lA, therefore, ned&ss&iy to determine whether ihi flow through the pipe as a whole nnder a bead h^ is greater, or le^a Ibftn^ the flow in the rising limb under a head h^- hi. If AB is flhort, or A^ io imall that v^ is greater than t>, the head absorbed bf | friction in A0 will be 2pd ^ If the end Oof the pipe U open to the atmosphere instead of heing ootmeotad I a reserrotr, the total head available will be h^ tustead of h^. 111. Velocity of flow in pipes. The mean velocity of flow in pipes is generally about 3 f© per second, but in pipes supplying water to hydraulic machme and in short pipes, it may be a^^ high as 10 feet per second. If the velocity is high, the loss of head due to friction in lo pipes becomes excessive, and the risk of broken pipes and valve through att-empts to rapidly cheek the flow, by the sudden clodi of valves, or other causes, is considerahly increased. On the other hand, if the velocity is too small, unless the wnti is very free from suspended matter, sediment* tends to collt?ct i the lower parts of the pipe, and farther, at low velocities it probable that fresh water sponges and polyzoa will make thdd abode on the surface of the pipe, and thus diminish it% \ capacity. 112, Transmission of power along pipes by hydraoHa pressure. Power can be transmitted hydraulically through a constderabll distance, >vith very great efficiency, as at high pressures the centage loss due to friction is small* Let water be delivered into a pipe of diameter d feet under i head of H feet, or pressure of p lbs, per aq, foot, for which n equivalent head is H = - feet. * An interesting example of this is quoted on p. Vol. XUT. 82 Trnw, JflUS^ai FLOW THBOUOH PIPES 163 Let the velocity of flow be v feet per second, and the length of the pipe Li feet. The head lost due to friction is ^ 2g.d ^^^' and the energy per ix)und available at the end of the pipe is, therefore, w 2gd The efficiency is B.-h_. h H "^ H 2gdB.' The fraction of the giyen energy lost is h For a given pipe the eflSciency increases as the velocity diiniiiishes. If / and L are supposed to remain constant, the efficiency is constant if jj^ is constant, and since v is generally fixed from other conditions it may be supposed constant, and the efficiency then increases as the product dR increases. If W is the weight of water per second passing through the jape^ the work put into the pipe is W . H foot lbs. per second, the available work per second at the end of the pipe is W (H - fe), and the horse-iX)wer transmitted is XiX - 560 - 550 ^^ '"'• Since W = 62%^d^v, the horse-power 4 550 \^ 2gd J = -089»d'H(l-m). From (1) "*°- 2gd ' therefore, ..-1 VdmB. and the horse-power I = 0-357 ^^d*H*(l-m). 11—2 164 HYDRAULICS If p is the pressure per sq. incli TT_ pl44 ^^ 62-4 ' and the horse-power =1*24 a/ jj d*p* (1 - m). From this equation if m is given and L is known the diameter d to transmit a given horse-power can be found, and if c2 is known the longest length L that the loss shall not be greater than the given fraction m can be found. The cost of the pipe line before laying is proportional to its weight, and the cost of laying approximately proportional to its diameter. If t is the thickness of the pipe in inches the weight per foot length is 3T5irdt lbs., approximately. Assuming the thickness of the pipe to be proportional to the pressure, i.e. to the head H, t = Jcp=JcK, and the weight per foot may therefore be written w = kid . H. The initial cost of the pipe per foot will then be C=fefeidH = K.d.H, and since the cost of lajdng is approximately proportional to d, the total cost per foot is P = K.d.H + Kid. And since the horse-power transmitted is HP = -357 ^^ d*H» (1 -m), for a given horse-power and efficiency, the initial cost per horse- power including laying will be a minimum when 0-357 ,y^ d*H* (1 - m) Jk K.d.H + Kid is a maximum. In large works, docks, and goods yards, the hydraulic trans- mission of power to cranes, capstans, riveters and other machines is largely used. A common pressure at which water is supplied from the pumps is 700 to 750 lbs. per sq. inch, but for special purposes, it is sometimes as high as 3000 lbs. per sq. inch. These high preesnres are, however, frequently obtained by using an intensifier (Ch, XI) to raise the ordinary pressure of 700 lbs. to the pressure required. FLOW THROUGH PIPES 165 The demand for hydraulic power for the working of liftB, etc. 1 led to the laying down of a network of mains in several of the cities of Grieat Britain. In London a mean velocity of 4 feet second is allowed in the mains and the presaur^ is 750 Ihs* sq^. inch, In later installationSj pressures of 1100 lbs. per inch are used. 113. The limiting diameter of cast-iron pipes. The diameter d for a cast-iron pipe cannot be made very large [ if the presscre is high. If f> is the safe internal pressure per sq. inch, and s the safe i^3^ees per sq. inch of the metal^ and r^ and r^ the internal and external radii of the pipe^ p=- r,* + ri* For a pressnre p = 1000 lbs. per aq. inch, and a stress a of lbs. per sq. inch, Tt is 5' 65 inches when n is 4 inches, or the ipe req aires to be 1*65 inches thick. If, therefore, the internal diameter is greater than 8 inches, the jpe becomefi Teiy thick indeed. The largest cast-iron pipe used for this pressure is between and ST internal diameter. tTstng a nsanmam velocity of 5 feet per second, and a pipe inches diameter, the ma^mum horse-power, neglecting friction, it can be transmitted at 1000 lbs. per sq, inch by one pipe is „^ 4418x1000x5 ^*^^'^ 550^ -400, The following example shows that, if the pipe is 13,300 feet 15 per c^nt. of the power is lost and the maximum power can be transmitted with this length of pipe is, therefore, 320 torse-power. Steel mains are much more suitable for high pressures, as the rkin^ stress may be as high as 7 tons per sq, inch. The greater ity of the metal enables them to resist shock more readily cast-iron pipes and slightly higher velocities can be used, A pipe 15 inches diameter and | inch thick in which the is 1000 lbs. per sq. inch, and the velocity 5 ft, per second, to transmit 1600 horse-power, E^x^tmpU. Power !■ InutBmitted ulong ^ c&st^iron m&in 7^ mcliea diameter at A pwsmmrt of 1000 Iba. p«r ^, meli. The velooity of the wmUa u 5 feel per aeoond. Fioil ib» tnAiiniiim 4istMio© tbe p&w^r can be tranBmitted so tldat the effleienQy * Swing's Strength of Materktli, 4 166 therefore Then from which HTDRAULICS A = 0*l5 3ca3O0 4 X 0^0104 x_26^ 2^ X 0-626 • 345K&|-4x0^e25 S4fi' = L = 0-0104 X 100 :^ 13,900 f^t rest, the intensity of preasare 13^ £ t^ Fig, 106, Pig^m 114. Pre&aures on pipe her If a bent pipe contain a fluic being the same in all directiot the resultant force tending to mr the pipe in any direction will the pressure pernnitarea moltipl by the projected area of the on a plane perpendicular to direction. If one end of a right-angled elbow, as in Fig. 106, be bolted to a pipe full of wat-er at a pressure p pounds per sq. inch by gauge, and on the other end of the elboir is bolted a flat cover, the tension in the bolts at A will be tie same as in the bolts at B. The presi^ure on the cover B is clearly '7854pcP, d being the diameter of the pipe in inches. If the elbow be projected on to a vertical plane the projection of ACB is dmfc^ the projection of DEF is ahcfe. The resultant pressure on the elbow in the direction of the arrow is, therefore, p . ahcd = '7S54jxf*. If the cover H is removed, and water flows through the pipe with a velocity v feet per second, the horizontal momentum ol the water is destroyed and there is an additional force in the direction of the arrow equal to '78'54irrfV, When flow is taking place the vertical force tending to lift tbe elbow or to shear the bolts at A is If the elbow is less than a right angle, as in Fig. 108, the total tension in the bolts at A is T = p (daehgc - aefgc) + -7854ii;dVcos^, and since the area aehgcb is common to the two projected areas, T = '1854<P(p'-pcoQe'\-wv'co&e). PLOW THROUGH PIPES 167 [^Consider now a pipe bent m shown in Fig. 109, the limbs AA and the water being supposed at rest» direction AA ia FF being fmrallel, acting total f Mrce the i P^p {degkea - asfgcb ^ dcgKea - aef*ge¥\ dh dearly is equal to 0. V m^tead of the fluid being at rest it has a uniform ' ht* pressure must remain constantj and since there is no of velocity tljere is no change of momentum^ and the re- tant pK?asttre in the direction paraUel to AA is still zero. There is however a couple acting upon the bend tending to it in a clockwise direction. Let p and q be the centres of gravity of the two areas daekgc 1 a^fgch respectively, and m and « the centres of gravity of dWeiig'c and aef*gch\ Through these points there are parallel forces acting BM shown the arrows, and the couple iB M — K - m7i - P » pq. Tli^ ' • P*pg is also equal to the pressure on the semicircle Ic II'. i by the distance between the centres of gravity of k and efg^ and the couple P' . 7nn is equal to the pmssure on ad*c* iltiplic^d by the distance between the centres of gra-^nty of a'd't^ Ufg. Tlu*n the resiUtant couple is the pressure on the semicircle efg Inpbed by the distance between the centres of gravity of efg u'r9. of FF atid AA are on the same straight line the as the force J becomes aero, % can also be shown, by similar reasoningj that, as long as the it4?r« at F and A are equal, the velocities at these aections therefore equal, and the two ends A and F are in the same ight line, the force and the couple are both zero, whatever the of the pipe. If, therefore, as stated by Mr Fronde, " the J 168 HYDRAULICS two ends of a tortuous pipe are in the same straight line, there is no tendency for the pipe to move." 115. Pressure on a plate in a pipe filled with flowing water. The pressure on a plate in a pipe filled with fiowing water, with its plane perpendicular to the direction of fiow, on certain assump- tions, can be determined. Let PQ, Fig. 110, be a thin plate of area a and let the sectional area of the pipe be A. The stream as it passes the edge of the plate will be contracted, and the section of the stream on a plane gd will be c(A-a), c being some coeflScient of contraction. It has been shown on page 52 that for a sharp-edged orifice the coeflBcient Fig. no. of contraction is about 0*625, and when part of the orifice is fitted with sides so that the contraction is incomplete and the stream lines are in part directed perpendi- cular to the orifice, the coeflBcient of contraction is larger. If a coeflBcient in this case of 0*66 is assumed, it will probably be not far from the truth. Let Vi be the velocity through the section gd and V the mean velocity in the pipe. The loss of head due to sudden enlargement from gd to ef is 2g ' Let the pressures at the sections db, gd, ef be p, p, and p, pounds per square foot respectively. Bemouilli's equations for the three sections are then, w 2g w 2g ^^^, and £..|l^E,,^,(I^* (2). Adding (1) and (2) (V.-V)' The whole pressure on the plate in the direction of motion is then (V.-Y)' \w wJ F=(p-pi).a = w.a. 2 FLOW THBOUQH PIPES P ^ 4^a 5- nearly. P = 116, Pre&aitre om a cylinder, 1£ inBtBfid of a thiii plate a cylinder be placed in the pif Iwitli U« aads coincident with the axis of the pipe, Pig< 111, there fare two anlargements of the section of the water. As the stream passes the up-stream edge of the cylinder, it contracts to the section at ccl, and then enlarges to the section ef. It again eniargi?s at the down-stream end of the cylindc I ftom the section a/ to the section gh. «' |C .^ J9 Fig. 111- l^t ^i, 1%, v»^ v^ be the velocities at ai, cd, ef and lepectively, €4 and Vt being eqnah Betw^eti cd and ^/ there is a loss of head between e/and gh there is a loss of 2g • The BemouiUi's eqnations for the sections are w 2g w 2g'^' ' " gk re- w 2g w 2g w 2g ^ ' w 2g w 2g 2g ' Miing (2> and (3), «? 2g 2g ' (1), .(2), .(3). ii^Mik 170 HTDRAtTLlCS If the coefficient of oontractioti at cd is-e, the area at cd -c A Then v^= — > v' " v ^-nd i'>=^^^. Therefore atid the pressure on the cylinder is EXAMPLES, (1) A new caBt'iroB pipe is 2000 ft. loDg and 3 ins. diameter. Itisbo discharge 50 c. ft, of water per miniite. Find the lofis of bead in tndma and the virtual elope. (2) What is the head loet per mile in a pipe 2 ft. diameter* diflchitf|iiiC 6,000,000 gallons in 24 hours ? /= -007. (3) A pipe tig to r^upply 40,000 gallons in 24 hours. Head of witcf above point of discharge = 86 ft. Length of pipe=2J miles* Find iti diameter. Take C from Table XII. (4) A pipe is 12 ms. in diameter and 3 rnile^ in length. It oohmscH two FeaerroirB with a difference of level of 20 ft. Find the disehazga pflf minute in a ft. Use Darcy's coefficient for corroded pipes. {5) A water main has a Tirtual slope of 1 in 900 and di£chaf-ge636C*(L per second. Find the diameter of the main. Coefficient / is 0*007. (0) A pipe 12 inB. diameter is suddenly enlarged to IB ins.» and tlian to 24 ins. diameter. Each section of pipe is 100 foet long* Find the h&k d head in friction in each length, and the loss due to shock at eaeli ealMge- ment. The discharge is 10 c. ft. per second^ and the coefficient of fricti» /=^'O106, Draw, to scale, tlie hydraulic gradient of the pipe. (7) Find an oxpresaion for the relative discharge of a square^ aad * circular pipe of the same section and slope. (8) A pipe is 6 ins. diameter, and is laid for a quarter mile at a ^ops of 1 in 60: for another quarter mile at a slope of 1 in 100; and for ftUM quarter mile is level. The level of the water is 20 ft. above the inlet efli and 9 ft. above the outlet end. Find the diacharge (neglecting all except skin friction) and draw the hydrauMc gradient. Mark Uie in the pipe at each quarter mile. (9) A pipe 2000 ft. long discharges Q c. ft. per second. Find bybo* much the discharge would be increased if to the last 1000 ft* a second ppl of the same siase were laid alongside the first and the water allowed to equally well along either pipe. FLOW THBOUOH PIPES 171 .Ibei I) ►) A naacrffoir , the level of wkicl) is 50 ft, above datum, diBcharges neeood rmmvm 80 ft. ^bove datum, through a 12 in. pipe, 5000 ft. find the disc^hiurge. AIbd, taking the levelB of tlie pipe at the ; Rud at each successive 1000 ft., to be 40, 25, 12, 12, 10, 15, d»liiin, wriie down Uie pressure at each of tbeie poiats, and posltieii of the line of hydraulic gradient, 1% m reqained to draw off the water of a reservoir Uirough a lOed harixoatAllj. Diameter of pipe 6 ins. Length 40 ft* Ef- lead 20 ft Find the disch^^e per geoond. Given the data of Ex. 11 find the diBcharge^ taking into atccount oi h#ad if the pipe ifi not bell -mouthed at either end* A pipe 4 ins. diameter and 100 ft. long discharges ^ c. ft* per Find the head expended in giving velocity of entry^ in overcoming reatstasce, and in Mctdon, 14) BeqQired the diameter of a pipe having a fall of 10 ft. per mUe, Mm ol delivering water at a velocity of 3 ft per second when dirty. TaMsg the coefficient / as 0*01 (l + f^^)i ^^ ^ow much water be diaehatged through a 12^inch pipe a mile long, connecting two i with a differeoce ol level of 20 feet, Watetr flows through a 12 -inch pipe liaving a virtual slope of 8 feet I feel at a velocity of 8 feet per second. I tlie Action per sq. ft. of surface of pipe iu lbs, iihc Talne of / in the ordinary formula for flow in pipes. Find the relative discharge of a 6 -inch main witli a slope of 40Ov and a 4 Inch main with a slope of 1 in 50. A 6'inch main 7 mUes in length vrith a virtual slope of 1 in 100 hj 4 miles of Sinch main, and S miles of 4-iiich main. Wowld ► be altered, and, if so* by how much ? [ (i^) Find the velocity of flow in a water main 10 miles longi con- [ Iwo reaervoirs with a dificTence of level of 200 feet. Diameter of > ifldiea. Goef&cient /=U^009. ' (10) Ftod the discharge, if the pipe of the last question is replaced for I flxnt 5 BuLas by a pipe 20 inches diameter and the remainder by a pipe idkmeter. Ctl) Calculate the loss of head per mile in a 10- inch pipe (area of cross 1 0^54 fl(|. ftj when the discliarge is 2^ c, ft per second. A pipe canaiata 6f ^ a mile of 10 inch, and 4 a mile of 5 -inch pipe, I oomr^ya $| e* It per second. State from the answer to the previous k the losB of head in each section and sketch a hydraulic gradient* » head al the ontlet is 5 ft. What is the head lost in friction in a pipe 8 feet diameter 6,000,000 galioDS in 12 hours? ' {Uj A pipe 2000 feet long and 8 inches diameter is to discharge 65 o. ft. f ndnnte* What mufit be the head of water f ITi 1(19) ^^whargel J^^b 172 HTDRAULICS (25) A pipe 6 mcbes dlanaeter, 50 feet long, is connected to the bcitkrnti of a tank 50 feet long by 40 feet wide. The original head over the openl end of the pipe is 15 feet. Find the time of emptying the tank^ assmning] the entrance to the pipe in sharp -edged. If /i^the head ovet the exit of the pipe at any moment, t?* -St?" 4/t^5Q^ °2g^ %^2£rx05' from 'which, v In time dt^ the discharge is * 1*5 +400/' 144 l'5 + 400/ In time ct the Borface falls an amoimt dh. Therefore 0_m^^^^^^^^h 1-5 + 400/ ^4 Integratmg, ^_ 2QQ0 (1-5 + 400/) 2 ^j^^ 79QQ0(l'5 + 4OO/) ^^ 0-196^% V2g (26) The internal diameter of the tubea of a condenser is 5*654 Inc The tubes are 7 feet long and the n amber of tubes is 400* The number < gallons per minute flowing through the condenaer is 400. Find the lo» ot| head due to friction as the water flowB through the tubes* /^ 0*006. (27) Aasundng fluid friction to vary as the square of the velodty, J an expression for the work done in rotating a disc of diameter d at i angular velocity a in water. (26) What horse power can be convej'ed through a 6- in, main if ihs I working presBure of the water supplied from the hydraulic power station tt 700 lbs, per sq, in;? Assume that the velocity of the water Is limited l9 8 ft, per second, (29) Ten horae-power is to be transmitted by hydranlic presanie i distance of a mile. Find the diameter of pipe and pfdssm-e required for li efficiency of ^ when the velocity is 5 ft. per sec The frictional loss is given by equation 2g d (BO) Find the inclination necessary to produce a velocity of 4| leei p* I second in a steel water main 31 inches diameter, when roniiing full ifl4 digchargiug with free outlet, using the formula ■ ■0005 tJ^^ *" di-» ' (Bl) The following values of the slope i and the velocity v w«re determined from an experiment on flow in a pipe '1296 ft diam* i -00022 '00182 '00650 ^02889 '04348 -12815 '22409 V 206 *606 1*252 2*585 8'693 6310 8'521 di^ii FLOW THROOGH PIPES 173 Delermme k and n m the formola i—ki)'^. AIbo determine T&Itiee of O for this pipe for velocitieB of *5, 1, 3^ 5 and 1 feel per sec (S2^ The total length of the Coolgsrdie steel aqueduct is SOTf miles ftttd the di&nieter 30 inches. The discharge per daj Z£iay be 5f600|000 The water is lifted a total height of 1499 feet, (a) the head lo^t dtie to friction, {b} the total work done per minote In raising the water. A pipe 2 feet diameter and 500 feet long without bende furnishee wmlier to a tnrhine* The tnrbine works under a head of 25 feet and uses ID e. fL ol water per second. What |)ercentage of work of the fall is lost ^A MctioD in the pipe ? Find im Coiifficiaitt /-'007 {'""m)^ C84> Sight thousand gallons an hour ha^e to be disoharged through ^^ch of mx nozzles, and the jet haM to reach a height of 80 ft. If the water supply i& 1^ tuilee away, at what elevation above the ooj^lea wcpnld you place the required reseryoir, and what would you make the diameter of the supply main ? QtTB the dimensions of the refiervoir you would provide to keep a OODStant supply for six hours. Loud. Uu. 1903. (85 ^ The pipes laid to connect tlie Vymwy dam with Liverpool are 4S inebea diameter. How much water will snch a pipe snpply in gallons ^m diiBBi if the slope of the pipe is 4^ feet per mile 9 At 0Q« point on the line of pipes the gradient m 6| feet per mile« and the fipe diam^er ia reduced to 89 inches; is thJB a reasonable reduction in the ^BieBsioii of the croas section ? Loud. Un. 1905. "*'ater under a head of 60 feet is diachajged through a pipe I meter and 150 fe^t long, and then through a nozzle the area of - -tenth tlie area of the pipe. Neglecting all losaes except friction, locity witli which tbe water leaves the nozzle* ^ Two rectangular tanks each 50 feet long and 50 feet broad are ftmnected by a horiiEuntal pipe 4 inches diameter, 1000 feet long. The Wd over the centre of tlie pipe at one tank is 12 feet, and over the other i ke% when flow commences. Determine tlie time taken for the water in the two tanks to come to the ^am level. Aianme the coefficient € to be constant and equal to 90. (885 Two reaervoirs are cotinectcd by a pipe 1 mile long and 10'' Kuieter; Ih^e difference in the water surface levels being 25 ft. B^terminic the flow through the pipe in gallons per hour and find by modi tlia discharge would be increased if for the last 2000 ft. a second ol lOT diameter is laid alongside the first. Loud. Un. 1905. (99^ A pipe 18^ diameter leads from a reservoir, SflO ft. above the ; od ia continued for a lengtli of 5000 ft. at the datum, the length' iSjOOO ft For the Last 5000 ft. of its length water is drawn off by 174 HYDRAULICS fiendoe pipes at the rate of 10 c. it, per nun* per 500 fL tmiformly, FmiM the pressure at the end of the pipe. Loud. Un. 1906. I (40) B50 horse -power ia to be transmitted by hydraulic preasoxe m distance of 1^ miles. I Find the nmnber of 6 ins. diameter pipes and the preadoi^e required fDcl an efficiency of 92 per cent, /—Ol* Take t* as 3 ft, per see I (41) Find the loaa of head due to friction in a water maiD L feet Iohh whicJi receives Q cubic feet per second at the inlet end and dahTMl P cubic feet to branch mains for each foot of its length* I What is the form of the hydraulic gradient ? I (42) A reservoir A aupphes water to two other reservoirs B and CLl The difference of level between the surfaces of A and B is 75 feet, anil between A and C 97-5 feet, A common 8 -inch cast- iron main supplies foci the hrst 850 feet to a point T>* A 6'inch main of length 1400 feet m tbeal earned on in the same straight line to B^ ajid a 5 -inch main of lengtlil 630 feet goes to C* The entrance to the 8«iiich main is bell-tnouthed« and! losses at pipe exits to the reservoirs and at the junction may be neglaoleU Find tlie quantity discharged per minnte into the reservoirs B and GJ Take the coefficient of friction (/) as *01. Lond, 0n. 1907. 1 (43) Describe a method of finding tlie '* loss of head " in a pipe due tol the hydraulic resistances^ and sta,te how you would proceed to find IbJ loss as a function of the velocity. I (44) A pipe* I feet long and D feet in diameter, leads water from ■ tank to a nozzle who^e diameter is d, and whose centre is h feet bekM the level of water in the tank* The jet impinges on a fixed plaal surface. Assuming that the loss of head due to hydraulic resistanoe m given by M show that the preasure os the Boriace in a maxiiiiniii wheD I ^ m- 1 (45) Find the flow through a sewer consisting of a cast-iron ptpJ 12 inches diameter, and having a fall of 8 feet per mllet when dLschafgufl full bore, c = 100. 1 (4d) A pipe 9 inches diameter and one mile long alop^ for the Snfl half mile at 1 in 200 and for the aeoond half mite at 1 in 100. The pr€ii^ head at the higher end is found to be 40 feet of water and at the Iohm 20 feet. I Find the velocity and flow through the pipe. I Draw the hjrdrauhc gradient and find the pressure in feet at 500 ktm and lOOO feet from the higher end. I (47) A town of 250,000 inhabitants is to be suppUed with water. EmM the daily supply of 32 gallons per head is to be delivered in 8 hours* I The service reservoir is two miles from the town^ and a fall of 10 §&m per mile can be allowed in the pipe. M What must be the size of the pipe? C = 90, I PLOW THROtJOH PIPES 175 (4$) A w&iar pipe is lo be laid in a street 800 jardfl long with houseB Mbolli Biidea of ibe street of 24 feet frontage. The average number ol iiilml)itsiits ai aacb honse is 6, and the aTemge eoneumption of water far eacli peCBcm ia 80 gallon b in 8 brs. On the assumption that the pipe is laid In loar equal lengths of 200 yards and has a uniform dlope of j^, and that Hie wbo)e of tha water flows through the first lengthy three-fourtlis through Ibe Mooml* one half through the third and ona quarter through the fourth, lad Uwl Ibe Talne of G is 90 lor the whole pipe^ lind the diameters of the ter parte of the pipe. (49) A pipe 3 miles long has a nniform slope of 20 feet per mile, and is 19 inclMM diameter for the first mile, 30 inches for the second and 21 faicbes for the third. The pressure heads at the liigher and lower ends of fbe pipe are 100 feet and 40 feet respectively. Find the discharge through th«^ [Hprn and determine the pressure heads at the comman cement of the 3ij inchem diameter pipe, and abo of the 21 inches diameter pipe, (SOi The difference of lerel of two i^eserroirs ten miJes apart in 60 feet, A pq^ Is Tnquired to connect tliem and to convey 45,000 gaUona of water pm boor fr«>m the higher to the lower reservoir. Find the necefisary diameter of the pipe, and sketch the hydraulic gtafdienU aMsanusg/^OOl. Tlw middle part of the pii>e is 120 feet below the surface of the upper reaerrotr. Osiculate the pressure head in the pipe at a point midway betw^iSl tiie two reservoirs. f51) Sonne hydraulic machines are served with water under pressure hf a pipe 1000 feet long, the pressure at the machines being 600 lbs. per ai^oara indi. The horBe^power developed by the machine is 300 and ihe fricdoa iione-powes- in the pipes 120. Find the necessary diameter of the ptp&^ taking the loss of head in feet as 0*03 li^^ and *i3 lb. per square laeli aa ac^aiTiilefit lo 1 foot head. Also determine the pressure at which Aa w«t6r ia deliirered by the pump. Wliat ts the ma^mum horse ^power at which it would be possible to work the machines, the pump pressure remaining the same ? Lond. Un. 1906* (53} Discnsa Beynolds' work on the critical velocity and on a general law oi resistance, describing tJie experiments appai-atus, and showing the ooB&edioii with the experiments of Poisemlle and D'Arcy. Load. Un, (58) In a condenser, the water entars through a pipe (section A) at the ol the lower water head, passes through the lower neat of tubes, thzerii^ the uppernestof tubes into the upper water head (section B). I '9m teetional areas at sections A and B are 0-196 and 0^95 sq. ft. respec- #i«ly; the total sectional area ol flow of the tubes forming the lower nest Ji C^S14 81]* ft^ and of the npper nest 75 sq. ft., the number of tnbea being feoipecltrely 35a and 826. The length of all the tubes is 6 feet 2 incheSp IVIifiii the Tolume of the circulating water was 1^21 c, ft. per sec., the Ered dtfferance ol preasure head (by gauges) at A and B was 6 5 feet. Ihe total actual head necessary to overcome frictional resistancei and 176 BYDaATJLICa the coelScieiit of hydraulic reciBtanoe referred to A. If i^3^ oodEdest d Motion (4/) for the tubed is t&ken to be *015t find the coefficient of bj^ilttiiifi . reflifitanee for the tubes alone, and compare with the actual l niiiiTipcit ] Lend. Un. 1906. iCr = head lost dimded bp veL he^ at A.) (54) An open stream, which 19 discharging 20 c. ft. of water M^ BeGood ifl paeied under a road by bl siphon of smooth stoneware pipev |H eectioii of the siphon being cylindrical, and 2 feet in diameter. When toVi stream enters ttiis siphon, the siphon descends vertioally 12 feet, H then has a horizontal length of 100 feet« and again rises 12 feet^ If all Hm I bends are sharp right-angled bends, what is the total Iohb of head in ibi I tunnel due to the bends and to the friction ? G = 117. Loud. Un. 190?. I (50) It has been shown on page 159 that when the kinetic ene^y of jfl jet issuing from a nozzle on a long pipe line is a maximum^ H ^ 8/L' ^ Hence find the minimum diameter of a pipe that will supply a ¥^im I Wheel of 70 per cent, efficiency and 500 brake horse-power, the airaikUJ head being 600 feet and tlie length of pipe 3 miles. ^M (56) A fire engine supphes water at a pressure of 40 lbs, per sqo^H inch by gauge, and at a velocity of 6 feet per second into a pipe 3 indi^l diameter. The pipe is led a distance of 100 feet to a nozzle 25 feet aba^| the pump. If the coeJScient / (of friction) in the pipe be '01, and the ad^l lift of the jet is | of that due to the velocity of effiux, find the actnal tiei^| to which the jet will rise, and the diameter of the nozzle to eati^ ^| conditions of the problem. ^H (57) Obtain expressions (a) for the head lost by friction (e^kpre^ej^l feet of gas) in a main of given diameter, when the main is horizontaL i^| when the variations of pressure are not great enough to cause any importtiH change of volume, and (b) for the discharge in cubic feet per saoond. ^ Apply year resnlte to the following example:— I The main is 16 Inches diameter, the length of the main is 300 jrwfc j the density of the gas is 0'56 (that of air^ l),and the difference of presMiii] at the tw^o ends of the pipe is | inch of water j find:^ I (a) The head lost in feet of gas, " (h) The discharge of gas per hour in cubic feet. Weight of 1 cubic foot of air=0'08 lb,; weight of 1 cubic foot of waier J^ba.; coefficient / (of friction) for the gas against the waUs of the pipe J*^"^ Loud, Un. 1905. J on f t '^^ ^^^ ' substitute for w the weight of cubic foot of gas,) I Find thetee reservoirs A, B and C are connected by a pipe le«di4g Draw the a junction box P situated 450' above datum, and 1000 feet ^ of the pipes are respectively 10^000' ♦ 5000' and 6000" and ilm (47) A tow*^^ water sorface in A, B and C are SOff, 600' and 200' &bot9 ^j^^^^^^^magnitude and indicate the direction of mean Telocity itt ^ permilecanbeaD€^^=>10*>*^'*^*' ^^ pip^s being aU the same WliatmustbethP"'^^' FLOW THROUGH PIPES 177 (00) A pipe 8* fiT diameter bends through 45 degrees on a radius of 7 IS leek Deiennine the dieplaGing foroe in the direction of the radial line \ MinnHnc the aii|^ between the two limbs of the pipe, when the head of P wiler in the pipe is 260 feet ^ Show alao that, if a nniformly distributed pressure be applied in the / plftBe of the oentre lines of the pipe, normally to the pipe on its outer '^ ^w**^, and of intensity f _ 49M« ,^ par unit length, the bend is in equilibrium* B»Tadins of bend in feet d— diameter of pipe. h»head of water in the pipe. 12 CHAPTER VL FLOW IK OPEN CHANNELS, 117, Variety of the forms of channels. The study of the flow of water in open cliannelB is much more complicated than in the case of closed pipeSj because of the infinite variety of the forms of the channels and of tlie dii&reit degrees of ro ugliness of the wetted BurfaceSj varying^ as they do, from channels lined with smooth boarda or cementj to the irregnkr , beds of rivers and the rough, pebble or rock strewn, moiEidiift| stream. Attempts have been made to find formulae wliich are applicalJft I to any one of these very variable conditions, bat as in the case d I pipes, the logarithmic formulae vary with the roughness of ttel pipe, so in this case the formulae for smooth regular shaped chanBeU I cannot with any degree of assurance be applied to the CBlculahon I of the flow in the irregular natural streams. 118. Steady motion in imifomi channels. The experimental study of the distribution of velocitiefi water flowing in open channels reveals the fact that, as in tlil'l case of pipesj the particles of water at different points in a crowl section of the stream may have very difFerent velocities, and the] direction of flow is not always actually in the direction of the flow [ of the stream. The particles of water have a sinuous motion, and at any jkxuh it is probable that the condition of flow is continually changinfJ In a channel of uniform section and slope, and in which the totJ flow remains constant for an appreciable time, since the lyiii quantity of water passes each section, the mean velocity i? in direction of the stream is constant, and is the same for all sections, and is simply equal to the discharge divided by the ami of the cross section. This mean velocity is purely an artificial] quantity, and does not represe^nt, either in direction or magnitit<}«^| the velocity of the particles of water as they pass the section. FLOW IK OPEN CHANNELS 179 ?fits with current meters, to determine the distribution iii diannels, show, however, that &t any point in the tion, the corapOBent of velocity in a direction parallel to fefction of flow renmins practically constant. The considers- pf the motion m consequently simplified by assuming that iter moves in paralh*] fillets or stream lines, the velocities in •ne different, but the velocity in each sti-eam line remains This is the aesutnption that is made in investigating rational formulae for the velocity of flo^v in channels, [tt should not be overWked that the actual motion may be more complicated. Formula for the flow when the motion is uniform Otiftiuael of onifonn section and slope. I this sasumptionj the conditions of flow at similarly situated i C and D in any two cross sections AA and BB^ Figs. 112 [118^ of a channel of uniform slop© and section are exactly the I; the velocities are equal, and since C and D are at the same lice below the free surface the pressures are also equal For it CD, therefore, «? 2^ w 2g ' B, HiBce the same is true for any other filament, for the two sectii :»us. C i-""~" — '—^ mm fi».m. Fig, 113. k-H be the mean velocity of the stream, i the fall per foot J ^ the surface of the water, or the slope, al the length tn AA and BB, w the cross sectional area EF&H of the fttiit P the wetted perimetor, i.e, the length EF + FG- + GH, 1 *e the weight of a cubic foot of water. Let ^ s HI be called the hydraulic mean depth* I bet iff be the fall of the surface between AA and BB* 11 ds-i. cL Since 12—1 180 HTDBAUUCS If Q cubic feet per second fall from AA to BB, the wort dow upon it by gravity will be : Then, since ^ £ "^ 2^) is constant for the two sections, the work done by gravity be equal to the work done by the frictional and other resistaiios opposing the motion of the wa***** As remarked above, all the so that there is relative motion and since water is not a perfect done by gravity is utilised in ova relative motion. Energy is also eddy motions, which are neglect and some resistance is ako offer surface of the water. The principal cause of loss of the sides of the channel^ a;.» work done by gravity is utilised in overcoming this resistance Let F . v be the work done per unit area of the sides of tlie channel, v beiug the mean velocity of flow, F is often called tk frictional resistance per unit area, but this assumes that tlie relatire velocity of the water and the sides of the channel is equal to the mean velocity, which is not correct. The area of the surface of the channel between AA and BB isP.dZ. Then, wi^idl-FvPdl^ its have not the same velodtf^ bween consecutive likmeDti, id some portion of the work Euing the friction daetotliifi due to the cross currents or 1 assuming stream line flow, o the flow by the air on ih ever^ the frictional reskt&a* s assumed that the whole A therefore or « ,_ F . F w F is found by experiment to be a function of the velocity ani also of the hydraulic mean depth, and may be written J b being a numerical coefficient. Since for water to is constant - may be replaced by k and therefore, mi^k ./{v) f(m). ] The form of /(i?) /(?n) must be determined by experiment- 120. Formula of Chezy. The first attempts to determine the flow of water in channels FLOW IN OPEN CHANNELS 181 ith precision -wrere probably those of Chezy made on an earthen Jialy at Coax^alet in 1775, from which he concluded that /(t;) = t;»and/(m) = l, id therefore m% = kv' (1). Writing C for -^ v = C vmi, 'hich is known as the Chezy formula^ and has already been given 1 the chapter on pipes. lai. JPormulae of Prony and E3rtelwem. Prony adopted the same formula for channels and for pipes^ and Bsmned that F was a function of v and also of t?', and therefore, mi = av + W. By an examination of the experiments of Chezy and those of )a Buat* made in 1782 on wooden channels, 20 inches wide and 988 than 1 foot deep, and others on the Jard canal and the river layne, Prony gave to a and h the values a = -000044, 5 = -000094. This formula may be written mx ■(-:*')•■■ >r ^ = / J mi. s/~v + b The coefficient C of the Chezy formula is then, according to Prony, a function of the velocity v. If the first term containing v be neglected, the formula is the same as that of Chezy, or V = 103 Jmi. Eytelwein by a re-examination of the same experiments together with others on the flow in the rivers Rhine t and Weser J, gave values to a and b of a = -000024, 6 = -0001114. Xeglecting the term containing a, V = 95 y/mi, • Prifieipei d'hydraulique, t ExperimeDts by Funk, 180S-6. X Experimenis by Braoings, 1790-92 182 HTDRAULICB Afi in the ease of pipes, Prony and Eytelwein mcorrectlf assumed that the constants a and h were independent of the j nature of the bed of the channel. 122. Formula of Barcy and Bazin, After completing his classical experiments on flow in pipes I M, Darcy commenced a series of experiments upon open channels — afterwards completed by M. Bazin — to determine, how the ] frictional resistances varied with the material mth which the channels were lined and also with the form of the channel. Experimental channels of semicircular and rectang-ular secdon I were construct^ at Dijon, and lined mth different materiak. Experiments were also made upon the flow in small earthen channels (branches of the Burgoyne canal), earthen channels hned with stonesj and similar channels the beds of which were cove^^ with mud and aquatic herbs* The results of these experimental published in 1858 in the monumental work, Rechsi^he^ Hydrtnc* liquss^ very clearly demonstrated the inaccuracy of the assutop- tions of the old writers, that the frictional resistances were independent of the nature of the wetted surface. From the results of these experiments M. Bazin proposed for J the coefficient k^ section 120, the form used by Darcy for pipes^ -i'^i). a and bein^ coefficients both of which depend upon the nat of the lining of the channel. Thus, mi = {a-r -Av^ or V- 1 y a + mt- The coefficient C in the Chezy formula is thus made to with the hydraulic mean depth m^ as well as with the roughne of the surface* It is convenient to write the coefficient k as \ am/ Taking the unit m 1 foot, Bazin^s values for a and ft mi values of k are shown in Table XVIII. It will be seen that the influence of the second term inc very considerably with the roughness of the surface, 123. Ganguillet and Kutter, from an elimination of Baim*j FLOW IN OPEN CHANNEI^ 183 periments, together with some of their own, found that the efficient C in the Chezy formula could be written in the form I which a is a constant for all channels, and b is a coefficient of TABLE XVin. Showing the values of a, fiy and k in Bazin's formula for channels. 1 a P k Planed boards and smooth cement Bon^ boards, bricks and concrete Aiihl^r masonry ;Earth Gravel (Ganguillet and Katter) •0000467 •0000580 •0000780 •0000864 •0001219 •0000046 •0000188 •00006 •00085 •00070 •0000457 (l+^*^«) •000078 (l + ^^) •0000854(1+^-1) •0001219(1 + ^"^^) The results of experiments by Humphreys and Abbott upon the flow in the Mississippi* were, however, found to give results inconsistent with this formula and also that of Bazin. They then proposed to make the coefficient depend upon the slope of the channel as well as upon the hydraulic mean depth. From experiments which they conducted in Switzerland, upon the flow in rough channels of considerable slope, and from an examination of the experiments of Humphreys and Abbott on the flow in the Mississippi, in which the slope is very small, and a large number of experiments on channels of intermediate slopes, they gave to the coefficient C, the unit being 1 foot, the value 41-6 + 0-00281 1-811 C = n l.(41-6.-«2281) »' \ * / Vm ^0 which n is a coefficient of roughness of the channel and has the values given in Tables XIX and XIX A. . * Report on the Hydraulics of the Missunippl River, 1861 ; Flow of water in ^^ and canaU, Tnutwine mnd HeriDg, 1893. 184 HYDRAULICS TABLE XIX. Showing values of n m the formala of Gaugtiillet and Kuttjer. Chumd n Very smooth, cement and planed boards »„ -009 to -01 Smooth, boards, brickHi, concrete ^latoW Smooth, covered wiUi slime or tnbercula-ted • .,, -015 Roagh ashlar or rubble masonry ., «^« Ditto 019 Very firm gravel or pitched witli stone ,.. *., ... D2 Earth, in ordinary conditioii free fron^ m and weeds *.. D25 Earth, not free from atones and weed *. ., "030 Gravel in bad condition , ... H)65 to ^ Torrential streamH with rough stony » , *0S TABLE Ca. Showing values of n in the fa i of GangiiiUet and Kutter^ determined from recent experimc K Rectangular wooden flume, very mno ,», ..» „. .^, *0M Wood pipe 6 ft. diameter ... ... -OlSI Brick, washed with cementf basket sharped sewer^ 6'?c6'8*\ nearly new -Olto Brick, washed witli cement, basket shaped newer, 6'x6'B", one year old ,.. ,., '0148 Brick, washed with cement, basket shaped sewer, Wx&&\ font years old .,, •0153 Brick, washed with cement, circular sewer, 9 ft- diameter, nearly new „ „, ^116 Brick, washed with cement, circular sewer, 9 ft. diameter, four years old -Om Old Croton aqueduct, lined with brick *0i5 New Croton aqueduct* .-. , ... D12 Sudbury aqueduct ^ Dl Glasgow aqueduct, lined with cement ... D124 Steel pipe, wetted, clean, 1897 (mean) "0144 Steel pipe, 1899 (mean) DUB This formula has found favour with English, American and German engineers, but French writers favour the simpler formula of Bazin. It is a peculiarity of the formulaj that when m equals unity then C = - and is independent of the slope; and also when m^B n large, C increases as the slope decreases. It is also of iitiportance to notice that later experiments apou the Mississippi by a special commissionj and others on the flow of the Irrawaddi and various European rivers, are inconsistent with * Meport New York Aqueduct Committion^ ^ FLOW IK OPEN CHANNELS 185 early experiments of Humplireya and Abbott, to which G&Dg^illet and Katter attached very considerable import^ice in fraiiittig^ their formula, and the later experiments show, as described later^ that the experimental determination of the How in, and the dope of, large natural etreams is beset with anch great difBcidtieB, ■ ' ' any formula deduced for channels of uniform section and '*.* cannot with confidence be applied to natural streams, and trier v^rsS.^ The application of this formula to the calculation of uniform «5hannels gri%'^Sj however^ excellent results, and providing the value of n in known, it can be used with confidence* It is, however, very^ cumbersome, and does not appear to give results mare accurate than a new and simpler formula suggested I lecsently hy Bazin and which is given in the next section. 124- BI. Spin's later formula for the flow in chaxmela. M. Baxin had recently (AiirJiali^ des Fonts ei ChausseeB^ 1897, VbL IV- p, 20), made a careful examination of practically all the ble experinient'S upon channels, and has propa^ed for the icieut C in the Chezy formula a form originally proposed by Gan^uillet and Kufcter, which he writes 1 c=- «t + c= when the unit is one foot, 1+4= m 4: c = 157-5 m .(2), s/i in which a is ccmstant for all channels and ^9 is a coefficient of roo^hnass of the channel- Taking 1 metre as the unit a - '0115, and writing y for -, c = -2I- (1). tike ralne of y in (2) being I'Slly, in formula (1), The values of y as found by Bazin for various kinds of channels ar^ shown in Table XX, and in Table XXI are shown values of 186 HYDRAULICS C, to the nearest whole number, as deduced from Bazin's coefficients for values of m from "2 to 50. For the channels in the first four columns only a very feir experimental values for C have been obtained for values of m greater than 3, and none for m greater than 7*3. For the earth channels, experimental values for C are wanting for small valuei of m, so that the values as given in the table when m is greater than 7*3 for the first four columns, and those for the first ihiee columns for m less than 1, are obtained on the assumption, tlurff Bazin's formula is true for all values of m within the limits dt tiifr i table. TABLE XX. Values of y in the formula, C = 157.5 1 + -s/m Very smooth surfaces of cement and planed boards ... Smooth surfaces of boards, bricks, concrete Ashlar or rubble masonry Earthen channels, very regular or pitched with stones, tunnels and canals in rock Earthen channels in ordinary condition Earthem channels presenting an exceptional resistance, the wetted surface being covered with detritus, stones or weed, or very irregular rocky surface unit metre nnit fool •06 •16 •46 -106S •88 •86 1-80 1-54 2-86 1-7 817 125. Glazed earthenware pipes. Vellut* from experiments on the flow in earthenware pii>e8 has given to C the value in which or C = 41-7 + i n 1 + 75^' 71 = -0072, 181 C = , -54 ' This gives values of C, not very different from those given by Bazin's formula when y is 0*29. In Table XXI, column 2, glazed earthenware pipes have been included with the linings given by Bazin. ♦ Proe, I, C. E,, Vol. cli. p. 482. FX^OIV IM^ OPEX CHANNELS 187 TAJBLiE XXI. Values of C in the f ormiila v = C^/m^ calculated from Bazin's {ormiila, the unit of len^i^lx l>eiii^ 1 foot, 157-5 C = 1 + y/m ChannelR 126 Basin.*« xnetliod of determining a and JS. The meftiod used by Bazin to determine the values of a and /? ' of sufficieiit interest and importance tx) be considered in detail. 1 TTe first calculated values of -t= and ■^ vm >/: mi from experimental ^jita, and plotted these values as shown in Fig. - Jim abBciBme,and -^ 114, -^ vm as as ordinates. 188 HTDBADLICS As will be seen on reference to the figure, points Iiave bea plotted for four classee of channels^ and the points lie close to km straight lines passing through a common point P on the aja of y. -•i The equation to each of these lines is y = a + )8aj, 5 \\ I ■ s i \ B FI^OW m OPEN CHANNELS 189 a bein^ the intercept on the axis of ^, or the ordinate when -j^ is ieto, a^d /3, ^irluch is variable, is the inclination of any one of Hhcfwi linae to tKe ajds of oj ; for when -i= is ^ero, - — = a, and traBspoamg ihB ©qaation, whicli IB clearly the tangient of the angle of inclination of the line to the aiUB of ae. It should he notedj that since - — - = n j ^^^ ordinatea give acteikl eiqperimental values of ^ , or by inverting the Bcale, values of C. Two scales for ordinates are thua shown. In addition to the points shown on the diagram, Fig, 114, Bazin plotted the results of some hundreds of experiments for all londs of channels^ and found that the points lay abont a series of lines, all passing through the point P, Fig. 1 14, for which a is '00635, B and the values of - , i.e. y, are as shown in Table XX. c BsLsin therefore concluded, that for all channels ^* = '00636 + -^. V Sim I the iralae of p depending upon the roughness of the channel, For very smooth channels in cement and planed boards, Bazdn plotted a large number of points, not shown in Fig. 114, and the luie for which y = '109 passes very nearly through the centre of the »cirie occupied by these points. TTie line for which y is 0"29 coincides well with the mean of the plotted points for smooth channels, but for some of the points y msky he as high as 0'4* It is further of interest to noticej that where the surfaces and 0ectioiiH of the channels are as nearly as possible of the same character, as for instance in the Boston and New York aqueductS| the values of the coefficient C differ by about 6 per cent,, the differenix* being probably due to the pointing of the sides and ^rch of the New York aqueduct not being so carefully executed aa for the Boston atiueduct. By simply washing the walls of the latter with cement^ E*teley found that its discharge was increased 20 per cant. 190 HYDRAULICS y is also greater for rectangular-shaped channalB, or those which approximate to the rectangular form, than for those rf circular form^ as is seen by comparing the two channels in wood W and P, and also the circular and basket-shaped sewers, M* Bazin also found that y was slightly greater for a veiy smooth rectangular channel lined with cement than for one of semicircular section. In the figure the author has also plotted the results of some recent experimentSj which show clearly the effect of slime aid tuberculatiousj in increasing the resistance of very smooth channels. The %"alue of y for tlie basket-shaped sewer lined with brici, washed with cement, rising from % to "642 during 4 years* service. 12 7« Variations in the coefficient G. For channels lined ^dth rubble, or similar materials, some iif the experimental points give values of C differing very consideTjlily from those given by points on the line for which y is 0'B3, Fig. 1 but the values of C deduced from experiments on parti channels show similar discrepancies among themselves. On reference to Bazin^s original paper it will be seen that, channels in earthy there is a atilj greater variation between experimental values of C, and those given by the formula, but experimental results in these cases, for any given channeli even more inconsistent amongst themselves. An apparently more serious difficulty arises with respect Bazin's formula in that C cannot be greater than 157'5. maximum value of the hydraulic mean depth m recorded ii any series of experiments is 74"3, obtained by Humphreys Abbott from measurements of the Mississippi at Carrollton in l Taking y as 2*35 the maximum value for C would then be 121 Humphreys and Abbott deduced from their experiments valoeft of C as large as 254. If, therefore, the experiments are reliable the formula of Basin evidently gives inaccurate results for axoep* tional values of m. Tlie values of C obtained at Carrollton are, however, incon- sistent %vith those obtained by the same workers at Yicksbuig, and they are not confirmed by later experiments carried out at CatTollton by the Mississippi commission. Further the Telocitiil at Carrollton were obtained by double floats, and, according to Grordon*, the apparent velocities determined by such floats should be at least increased, when the depth of the water is large, by ten per cent, Bazin has applied this correction to the velocities obtained by * Gordon, Proeeedinff§ Intt Civil Eng,, 189S. FLOW IN OPEH CHAKNELS iphrejB and Abbott at Yickabm^ and also to those obtained be MiflstHsippi Commissioti at Catrollton, and shows, that the immB vala€> for C is theHj probably, only 122. %st the t-alties of C as deduced from the early experiments on Mwiesippi are unreliable, is more than probable, since the iUefi( slope, as measured, was only '00000*34, which is less than kch per mile* It is almost impossible to believe that such small ervnces of level could be measured with certainty, as the ^Il<?«t ripple would mean a very large percentage error, and ■^rther probable that the local variations in level would be HR than this measured difference for a mile length. Further, itoiug the slope is cx)rrectj it seems probable that the velocity gijch a fait won Id be less than some critical velocity similar that obtained in pipesp and that the velocity instead of being >portional t4> the square root of the slope t, is proportional i. That either the measured slope was unreliable, or that the ocity was less than the critical velocity, seema certain from the :, that experiments at other part8 of the Mississippi, upon the ftwaddi by Gordon, and upon the large rivers of Europe, in no gtva iralaas of C greater than 124. The experimental endence for these natural streams tends, er, clearly to show, that the formulae, which can with ifidtrnce be applied to the calculation of flow in channels of form, cannot with assurance be used to determine the of riirerH* The reason for tins is not far to seek, as lit ions obt-aining in a river bed are generally very far ed from those assumed, in obtaining the formula. The pttoti that the motion is uniform over a length sufficiently lo be able to measure with precision the fall of the surface, be ixLT from the truth in the case of riv^ers, as the irregu- ia the crosa section must cause a corresponding variation n velocities in those sections, deri%'ation of the formula, frictional resistances only n into account, whereas a considerable amount of the tie on the falling water by gravity is probably dissipated eddy motions, set up as the stream encounters obstructions in bed of the river, Tliese eddy motions must depend very li OB local circunuitance-s and will be much more serious in liar channels and those strewn with weeds, stones or other tioiis, than in the regTilar channels. Another and probably Otis difficulty is the assumption that the slope is uniform oui the whole length over which it is measured, whereas between two cross sections may vary considerably bank and bank. It is also doubtful whether locally 192 HYDRAULICS there is always eqailibriam between the resisting and accele forces. In those eases, therefore, in which the beds are rock covered Mrith weeds, or in which the stream has a very ir shape, the hypotheses of uniform motion, slope, and sectmi} i not even be approximately realised* 128. Logarithmic formula for the flow In channels. In the formulae discussed, it has been assumed that the fricti resistance of the channel varies as the square of the velocity, in order to make the formulae fit the qp^eriments, the coefficient has been made to vary with the velo As early as 1816, Du Buat* p increased at a less rate than the half a century later St Venant pr mi = W040' To determine the discharge of 1 Crimp has suggested the formula t^ = 124m* • and experiments show that for sewers that have been in use time it gives good results. The formula may be written 0*00006 1>* ited out, that the slope [uare of the velocity, >i»0d the formula ck-lined sew^ers, Mr i- m An examiTiatton of the results of experiments, by logarit plotting, shows that in any uniform channel the slope ^ ' m^ ' Ic being a numerical c inefficient which depends upon the rouglme«] of the surface of the channel, and n and p also vary with nature of tht.^ surface. Therefore, in the formula. fMfirn)^ .P't' From what follows it will be seen that n varies between IIS ] and 2'1, while p varies between 1 and I'S, Since m is constant, the formula t - — ^ may be written i - friT, m" b being equal to Therefore log i = log b + n log 17. ♦ Principcs d'Hydraulique, Vol. i. p. 29, 1816. FXOTV IN OPEN CHANNELS 193 Pig. 115 are shoiwii plotted the logarithms of i and v A from an eaqperiment by Bazia on the flow in a semi- r cem^nt-lixied pipe. The x)oint8 lie about a straight line, Qgent of the inclination of which to the axis of v is 1*96 le intercept on the axis of % through v = ly or log v = 0, is 08. Fig. 115. liOgarithmio plottings of t and v to determine the index n in the formola for channels, %=—=, For this experimental channel, therefore, i = -00008085t;^*«. In the same figure are shown the plottings of log i and log v for 18 siphon-aqueduct* of St Elvo lined with brick and for which I is 2*78 feet. In this case n is 2 and b is -000283. Therefore t=-000283i?^ If, therefore, values of v and i are determined for a channel, rhile m is kept constant, n can be found. • AnnaUi d£$ Fonts et Chatusiei, Vol. tv. 1897. U B. \z 194 HYDRAULICS To determine the ratio - . The formula, may be written in the form, or log m = log (-J + - log v. By determining experimentally m and r, while the slope i is kept constant, and plotting log m as ordinates and log v as abscissae, the plottings lie about a straight line, the tangent of the inclination of which to the axis of v is equal to -, and the P intercept on the axis of m is equal to JcV (;) In Fig. 116 are shown the logarithmic plottings of m and v for a number of channels, of varying degrees of roughness. The ratio - varies considerably, and for very regular channels increases with the roughness of the channel, being about 1"40 tat very smooth channels, lined with pure cement, planed wood or cement mixed with very fine sand, 1*54 for channels in unplaned wood, and r635 for channels lined with hard brick, smooth concrete, or brick washed with cement. For channels of greater roughness, - is very variable and appears to become nearly equal to or even less than its value for smooth channels. Only in one case does the ratio - become equal to 2, and the values of m and P V for that case are of very doubtful accuracy. As shown above, from experiments in which m is kept constant^ n can be determined, and since by keeping i constant ~ can be found, n and p can be deduced from two sets of experiments. Unfortunately, there are wanting experiments in which m is kept constant, so that, except for a very few cases, n cannot directly be determined. There is, however^ a considerable amount of experimental data for ciannels similarly lined, and of different slopes, but here FLOW IN OPBN CHANNELS 195 Fig. 116. Logarithmie plottings of m and v to determine the ratio - in the formula t= — - . TABLE XXII. Particulars of channels, plottings for which are shown in Fig. 116. 1. Semieirealar ehannel, very smooth, lined with wood 3. •> »t tf ft ft }» cement mixed with Tery fine sand 3. Bectangnlar channel, very smooth, lined with cement 4. ,. », „ „ „ „ wood, 1' 7" wide 5. f, tf smooth „ „ „ slope '00208 6- t» «f fi »» »» M ti '0043 #. »i »f »t ♦» »» »» It '004y 8. „ „ M „ M „ „ -00824 9. New Croton aqnednet, smooth, lined with bricks (Report New York Water Sapply) 10. Glasgow aqueduct, smooth, lined with concrete IProc. I, C. E. 1896) 11. Sodbary „ „ ,, ,, brick well pointed (Tr, Am. S.C.E. 1883) 12. Boston sewer, circular, smooth, lined with brick washed with cement {Tr. Am.S, C. E, 1901) 13. Hectangular channel, smooth, lined with brick 14. 9f tf •> tf tt ^o<xi 15. t, t, „ „ „ small pebbles 15^1. Bectangnlar slniee channel lined with hammered ashlar l«>w- tt tt tf ff »f ff If 16. tt ehannel lined with large pebbles 17. Torlofiia ionnel, rock, partly lined 18. Ordinary ehannel lined with stones covered with mud and weeds ... 19. tf 9» tt ff 50. BiTerWeeer 51. «. ff S3. «t tt 23. Eitrth dianneL Ores bois S4« Catoot canal S5. Bhrer Seiiifl n P 1-45 1-36 1-44 1-38 1-64 1-64 1-54 1-64 1-74 1-636 1-635 1-635 1-635 1-655 1-49 1-36 1-36 1-29 1-49 1-18 -94 1-615 1-65 2-1 1-49 1-5 XZ—'l 196 HYDRAULICS again, as will appear in the context, a difficulty is encountered, even with similarly lined channels, the roughness is in no t' cases exactly the same, and as shown by the plottings in Fig. 1. no two channels of any class give exactly the same yali for - , but for certain classes the ratio is fairly constant. Taking, for example, the wooden channels of the group (Nos to 8), the values of - are all nearly equal to 1*54. The plottings for these channels are again shown in Fig. 11 The intercepts on the axis of m vary from 0*043 to 0'14. Fig. 117. Logarithmio plottings to determine the ratio - for smooth ch&nne Let the intercepts on the axis of m be denoted by y, then JcV -(!). FLOW IN OPEN CHANNELS 197 and logy=-log*--logi. If fc and 'p are constant for these channels, and logt and logy are plotted as abscissae and ordinates, the plottings should lie about a straig^lit line, the tangent of the inclination of which to the axis of i is — • and when logy = 0, or y is unity, the abscissa %-lc. V we. the intercept on the axis of % is k. In Y\%. 118 are shown the plottings of log % and log y for these channels, froni which p=l'14 approximately, and % = '00023. Therefore, n is approximately 1*76, and taking - as 1*54 ._ '00023t?^^ ^ ' N ~ ' N. \ ^ - i \ 1 1 \ \ < ^ s s -S^-i s \ \ w 4 \ trr T \ « - J N *t-— _ .__ — '"" ~" -J " ^^ ' '" "" ~'^' '" :^ - ^*- — ~ _. ^- , ■'^—^—'—' ^ «IUi 1 , toLn.d}-%i i \ U- COOXt. S b^ ■oc Jff '^ br oc f -Qi n m IS ^i Log, iy Fi^. 118. Logmrithmio plottings to determine the value of p for smooth channels, in the formula t=— ;:• Since the ratio - is not exactly 1*54 for all these channels, the valaes of n and p cannot be exactly correct for the four channels, but, as will be seen on reference to Table XXIII, in which are shown yaloes of t? as observed and as calculated by the formula, the calculated and observed values of v agree very nearly. 198 HYDEAUtlCS TABLE XXm. Values of t?, for rectangular channels lined with wi determined experimentally, and as calculated from the formn Slope -O020e Slope -0019 Slope -00824 e ob- V ©ftJcu- V ob* t; ealea- V ob- r^ftleu- m m served lated m in served Jated m m served UUd metres metres metres metres metres meirea metres metres metres per B«o. per iee. per sec. per Bee. pel- sec. per »a«. 0*1881 0*962 0W2 0^042 1-825 1814 -0882 1-594 1-580 '1609 1'076 1-07 -1224 1-479 1-459 '1041 1-776 1-764 '1833 1162 1*165 -1882 1-612 1-58 -1197 1-902 liSSS 1076 1-259 1'228 -1585 1-711 1-690 -1313 2-053 2-051 '2146 1-324 1-290 -1668 1-818 1-782 -1420 2-186 2158 '2318 1-874 1*854 -1789 1-898 1-S5S ^548 2-268 2-276 ^2441 1-440 1402 a9l3 1-967 1-947 ■1649 2-S57 2-87T *2578 1-487 1*452 *2018 2-045 2014 *1744 2-447 2-460 ^2681 1-562 1-49 -2129 2-102 2-089 -1842 2-518 2*j^ ■2809 1'687 1-552 -2215 2-179 2143 -1919 2-612 2*618 As a further example, which also shows how n and p increase with the roughness of the channel, consider two channels built ia hammered ashlar^ for which the logarithmic plottings of m and are shown in Fig* 116j Nos. 15 a and 15 6^ and - is 1"36. The slopes of these channels are "101 and *037, By plotting log't and logi^, p is found to be r43 and k '000149. So thai for these two channels The calculated and observed velocities are shown in Table XXXI and agree remarkably well. Very smooth channels. The ratio - for the four very smooth p channels, shown in Fig. 116, varies between V36 and 1'45, tha average value being about 1'4. On plotting log^ and logt the points did not appear to lie about any particular line, so that conld not be determined, and indicates that k is different for tbo four channels. Trial values of n ^ 175 and p ^ 1*25 were teken, oe ._^k.v'^ and values of ft calculated for each channeL FUm IN OPEN CHAKKELS Velocities b& determined experimentally and as calculated for ' ^ ehaimels are shown in Tablo XXIII from which it will it fe vanes from '00006516 for the channel lined with eetnent, to "0001072 for tlie rectangular shaped section lined th cai^fuHy planed boards. It will be seen, that although the range of velocities is con- iderabie, there is a remarkable agreement between the calculated obfierved values of t% so that for very smooth channels the Ities of n and p taken, can be tised with considerable confidence, Channeb moderattly imwoik. Ite plottings of logm and logtj chatHiels Uned with brick, concrete, and brick washed with ^izLetit are shown in Fig. 116, Mos. 9 to 13. It will be seen that the value of - is not so constant as for the P ro classes prevjouBly considered, but the mean value is about "635, which is exactly the value of - for the Sudbury aqueduct, mi Tut the New Croton aqueduct - is as Mgh as r74, and, as shown ^m Fi^* 114j this aqueduct is a little rougher than the Sudbuiy. The variable values of - show that for any two of these P iel« either n, or p, or both, are different » On plotting log* r a* was done in Fig* 115, the points, as in the last casei lid not W said to lie about any particular straight line, and the la^ of p IB therefore uncertain* It was assumed to be 115, and n pre, teJdng - as 1*635, n is I'88. n Since no two channels have the same value for -, it is to be P ted that the coefficient k will not be constant. lo the Tables XXIV to XXXHI the values of v as observed as Gslciilated from the formula aW the value of k are given. It will be seen that k varies very considerably, but, for the large aquedncts which were built with care, it is fairly It. The effect of the sides of the channel becoming dirty with time^ i« revy well seen in the case of the circular and basket- sewers. In the one case the value of k^ during four years' varied from 00006124 to 00007998 and in the other from '•00006405 to 0001096- It is further of interest to note, that when 200 HYDRAULICS m and v are both unity and k is equal to ^000067, the value of i a the same as given by Bazin'a formula, when y is *29, and when k is ] '0001096, as in the case of the dirty basket-shaped sewer, th© value of y is '642^ which agrees with that shown for this sewer on Fig, 114 ' Channels in Tnasonry^ HamTnered ashlar and rfihble. AttentiaQ has already been called^ page 198, to the result-s given in Table XX XT for the two channels lined with hammered ashlar* The values of n and p for these two chaonels were determined directly from the logarithmic plottings, but the data is tnsuiBeietit to give definite values, in general, to n, p, and k. In addition to these two channels, the results for one id Bazin's channels lined with small pebbles, and for other channels lined with rabble masonry and large pebbles are given. The n ratio - is quoted at the head of the tables where possible. In the, other cases n and p were determined by trial. The value of n, for these rough channels, approximates to 2,] and appears to have a mean value of about r96, while p vari$lj from 1*36 to 1"5, Earthen channels, A very large number of experiments h*f been made on the fiow in canals and rivers, but as it is generally J impracticable to keep either i or m constant, the ratio - c&n only] be determined in a very few cases, and in these, as wiJl be seen! from the plottings in Fig, 116, the results are not satisfactory, and] n appear to be unreliable, as - varies between '94 and 2*18, P It probable that p is between 1 and 1*5 and n from 1*96 to 2'15< Logarithmic formulae for tmrimts classes of channels. Very smooth channels, lined with cement, or planed boards^ „17S 1- ("000065 to '00011)-^, m* Smooth channels, lined with brick well pointed, or concrete, * = '000065 to *00011^, Channels lined with ashlar masonry, or small pebbles, i = '00015^., m}* Channels lined with rubble masonry, large pebbles, rock^ and] exceptionally smooth earth channels free from deposits, i = *00023 ^,,^,;, . \ FLOW IN OPEN CHANNELS 201 Earth channels, h varies {rem '00033 to '00050 for channels in ordinary condition and from '00050 to *00085 for channels of exceptional resistance. 120. Approximate formula for the flow in earth nhamialii The author has by trial found n and p for a number of channels, and except for very rough channels, n is not very different from 2, and p is nearly 1*5. The approximate formula t? = C vm^i, may, therefore, be taken for earth channels, in which C is about 50 for channels in ordinary condition. In Table XXXTTT are shown values of t? as observed and calculated from this formula. The hydraulic mean depth varies from '958 to 14*1 and for all valaes between these external limits, the calculated velocities Wbgree with the observed, within 10 per cent., whereas the variation of C in the ordinary Chezy formula is from 40 to 103, and according to Bazin's formula, C would vary from about 60 to 115. With this formula velocities can be readily calculated with the ordinary slide rule. TABLE XXIV. Very smooth channels. Planed wood, rectangular, 1*575 wide. i = •0001072^, log* = 4-0300. mfeet V ft. per sec v observed ( ft. per »loalai •2872 8-65 8-57 •2811 4-00 408 •8044 4-20 4-26 •8468 4-67 4-68 •8717 4-94 4-94 •8980 511 512 •4124 5-26 5-80 •4811 5*49 5-47 202 HTDRAULICS TABLE XXIV (continued). Pure cement, semicircular. *=m-> •00006516^, log* = 5-8141. m V observed v oalonlated •508 8-72 8-66 •682 4-69 4-55 •760 4-87 4^87 •916 557 5-62 1-084 614 614 y fine sand, semicircular. ^I'n 1 = -0000759^, log* = 6-8802. t; ft. per seo. v ft per see. mfeet observed ( Bsloalated •879 2-87 2-74 •629 8-44 8*49 •686 8-87 8-98 •706 4-80 4-80 •787 4-51 4*59 •889 4-80 4-84 •900 4-94 610 •941 5-20 5-26 •988 5^88 5-48 1^006 5-48 5-58 1^02 555 5-58 1-04 5-66 TABLE XXV. 5-66 Boston circular sewer, 9 ft. diameter. Brick, washed with cement, t = 7xnnr (Horton). i = -00006124- m' log V = -6118 log m + -5319 log t + 2-2401. V ft. per sec. V ft. per seo. mfeet observed caloalated •928 2-21 2*84 1208 2*70 276 1^408 808 8-08 1-880 8*48 8^56 1*999 878 8-75 2809 4^18 410 FLOW IN OPEN CHANNELS 208 TABLE XXV (continued). The same sewer after 4 years' service. t = -00007998^, log t; = -6118 log m + -SSIO log i + 2-1795. m V ohsenred v oaloulated 1120 2*88 2*29 1-606 2*82 2-76 1-952 8-16 8-22 2-180 8-80 8-89 TABLE XXVI. New Croton aqueduct. Lined with concrete. i = -000073^, log V = -6118 log m + -5319 log i + 2'200. V ft. per 860. V ft. per seo. iifeet obsenred . ealcolated 1-000 1-87 1-87 1-260 1-59 1-67 1-499 1-79 1-76 1-748 1-96 1-98 2-001 211 210 2-260 2-27 2-26 2-600 2-41 2-40 2-749 2-52 2-65 2-998 2-66 2*68 8-261 2-78 2-82 8-508 2-89 2-96 8-750 8-00 8-08 8-888 8-02 8-12 TABLE XXVII. Sudbury aqueduct. Lined with well pointed brick. i = •00006427^., 081ogt? = •6118 log m + •5319 log 1 V ft. per 860. t; ft. per seo. iifeet obBenred oaloulated •4987 1-186 1-142 -6004 1-269 1-279 -8005 1-616 1-526 1-000 1-765 1-762 1-200 1-948 1-964 1-400 2149 2147 1-601 2-882 2-881 1-801 2-518 2-511 2-001 2-651 2-672 2-201 2-844 2-882 2-886 2-929 2-987 204 HYDRAULICS TABLE XXVm. Rectangular channel lined with brick (Baziii). i=-000107^. V ft. per 860. V ft. per seo. m feet observed calealated •1922 2-76 2-90 -2888 8-67 8*68 -8654 4-18 4-80 •4285 4-72 4-71 •4812 5^10 5-09 •540 5-84 5-46 •5828 5-68 577 •6197 6-01 5-94 -6682 615 6-22 -6968 6-47 689 -7888 6-60 6-62 •7788 6-72 688 Glasgow aqueduct. lined with concrete. i = *0000696^„ log V = -6118 log m + -5319 log i + 2-2118. V ft. per seo. t; ft. per eeo. m feet observed calculated 1-227 1-478 1-478 1^489 1-499 1-499 1-548 1-597 1-607 1-610 1-620 1-627 1-788 1-811 Charlestown basket-shaped sewer 6' x 6' 8". Brick, washed with cement, i = ^Tnnr (Horton), t = -00008405 ^5, log V = -6118 log m + -5319 log i + 2-1678. V ft. per sec. t; ft. per see. m feet observed calculated •688 1^99 2^05 •958 2-46 2-52 1187 2-82 2-87 1-589 8-44 8*86 1-87 1-89 2-07 2-11 2-106 2-11 2-214 218 2-18 214 215 214 218 2-22 2-21 2-28 228 2-28 2-22 2-24 2-24 2-24 2-25 2-27 2-26 2-88 2-47 2-40 3 XXTX, FLOW IN OPEN CHANNELS 205 TABLE XXIX (continued). rhe same se^irer after 4 years' service, i=-0001096- m' log V = -6118 log m + -5319 log i + 2* 1065. mfeet 1-842 1-506 1-645 V ft. per aeo. v ft. per see. obserred oalcnlated 2-66 2-86 804 2-68 2-88 8-04 TABLE XXX, Left aqnednct of the Solani canal, rectangular in section, lined ill rabble masonry (Cunningham), i = -00026^. i mfeet V ft. per sec. observed V ft. per seo. calculated -000225 •000206 •000222 •000207 •000189? 6-48 6-81 7-21 7-648 7-94 8-46 3-49 8-70 3-87 4-06 3-50 3-47 3-84 3-83 3-88 aeduct, i = -0002213 ^,. t m V observed V calculated •000195 -000225 -000205 -000198 -000198 -000190 3-42 5-86 6-76 7-43 7-77 7-96 2-43 3-61 3-73 3-87 3-93 406 2-26 3-58 3-76 3-89 4-04 4-06 "orlonia tunnel, partly in hammered ashlar, partly in solid i= 00104, •00022- m' m V observed V calculated 1-932 3-382 3-45 2172 3-625 8-73 2-552 4-232 4-16 2-696 4-324 4-32 8-251 5046 4-90 8-488 4-965 5-08 8-581 4-908 5-18 8-718 5-858 5-37 HYDRAUUCS TABLE XXXI. Channel lined with hammered ashlar, - = 1-36, m feet •824 •467 •580 •662 t = -000149 log * = 4-1740. m* {=•101 V ft. per seo. observed 12-80 16-18 18-68 21-09 t; ft. per sec. calculated 12-80 16-18 18-97 20-8 t = -087 m feet -424 -620 -746 -862 V ft. per see. observed 9-04 11-46 18-66 16-08 V ft. per MflL oal(»lattd 902 11-86 18-52 14-98 Channel lined with small pebbles, i = '0049 (w=l'96, p = l'32 will give equally good results). ^ = 1-49, ■ P i = -000152 ■ log ifc = 41913. m' V ft. per sec. V ft. per sec. nfeet observed calculated -250 216 2-84 -867 2-95 2-97 -450 8-40 8-47 •520 8-84 8-82 •588 414 4-16 -644 4-48 4-48 •700 4-64 4-66 -746 4-88 4^88 -786 612 5-05 -882 5-26 5-25 -871 5-48 648 -910 5-67 6-68 FU>W IN OPSN CHANNELS 207 TABLE XXXn. Cluumel lined with large pebbles (Bazin), t = -000229^, log i = 4-3605. V ft. per seo. v ft. per see. m feet observed oalcalated 1-79 1-84 2-48 2-44 2-90 2-90 8-27 818 8-56 8-45 8-86 8-67 4^08 8-91 4-28 4-88 4-48 4-58 4-60 4-69 4-78 4-84 4-90 6-00 TABLE XXXm. Velocities as obsery^and as calculated by the formula c^X' t;=Cvm* i. C = 50. f'-"" GaTiges Canal. i mfeet V ft. per sec. observed V ft. per sec. calcolated •000155 •000229 •000174 •000227 •000291 5-40 8-69 7-82 9-84 4-50 2-4 8-71 2-96 402 2-82 2-84 8-80 8-08 4-00 2d8 River Weser. t m V obserred t; calcolated •0006608 •0005608 •0002494 •0002494 8-98 18-86 141 10-6 6-29 7-90 6-69 4-75 60 8-18 6-70 4-78 Missouri. i m V observed V calculated •0001188 •0001782 •0001714 •0002180 10-7 12-8 15-4 17-7 8-6 4-88 6-03 619 8-28 4-87 4-80 6*26 208 HYDRAULICS Cavon/r Caned. i m V obsenred t; calculated •00029 •00029 •00088 •00088 782 5-16 6-63 4-74 870 8^10 840 8^04 8*80 2-92 8-14 2-91 Earth channel (branch ofBurgoyne canal). Some stones and a few herbs upon the swrfcLce. C = 48. 17 ft. per sec. v ft. per sec t m feet observed calculated •000957 -958 1^248 1^80 •000929 1-181 1-702 1-66 •000998 1-405 1-797 1^94 •000986 1588 1-958 2^06 •000792 •958 1-288 1-26 •000808 1-210 1-666 1-66 •000858 1-486 1-814 1-79 •000842 1-558 1^998 2-08 130. Distribution of the velocity in the oroas aectioii of open channels. The mean velocity of flow in channels and pipes of small ctosb sectional area can be determined by actually measuring the weight or the volume of the water discharged, as shown in Chapter Vll, and dividing the volume discharged per second by the ckw section of the pipe. For large channels this is impossible, and the mean velocity has to be determined by other means, usually by observing the velocity at a large number of points in the same transverse section by means of floats, current meters*, or Htot tubes t. If the bed of the stream is carefully sounded, the cross section can be plotted and divided into small areas, at tiie centres of which the velocities have been observed. K then, the observed velocity be assumed equal to the mean velocity over the smaD area, the discharge is found by adding the products of the areas and velocities. Or Q = Sa . t;. M. Bazint, with a thoroughness that has characterised lus exi)eriments in other branches of hydraulics, has investigated ths distribution of velocities in experimental channels and also in natural streams. In Figs. 119 and 120 respectively are shown the cross section of an open and closed rectangular channel with curves of eaas * See page 288. t See page 241. X Bazin, Recherches HydraiUique^ FLOW IN OPEN CHANNELS 209 velocity drawn on the section. Curves showing the distribution of velocities at different depths on vertical and horizontal sections are alao shown. ^aj^j^sar ofequat VeU^cify ion Rectcuiffular Chaitttd/, Fig. 119. VertLcaLS^cUjons. Eori^ciUal S fictions. 1 Fig. 120. L Xt will be seen that the maximum velocity does not occur in L^ free surface of the water, but on the central vertical section Sit some distance from the surface, and that the surface velocity \ may be very different from the mean velocity. As the maximum m velocity does not occur at the surface, it would appear t\iat \n I U H. \4 210 MTDRAULIC^ assuming the wetted perimeter to be only the wetted sarbeej the chaniii'l^ some error is iritroducefL That the air hsi aOll same influcfna; tm if the water wer*? in ccmtact with a sur similar to that of the sides of the channel, is very cle shown by comparing the carves of equal velocity for the ch rectangular channel as showii in Fig. 119 with thase of Fig. 1211 The air resistancej no doubt, accounts in some measure for tltl surface velocity not being the nmxinmm velocity, but that it dosij not wholly acc<mnt for it^ in shinm by the fact that^ whether to] wind is blowing up oi below the surface. J why the maximum veio the water is \em eonstraini movements of all k: utilised in giving mouio translation. Depth rm a7ty vsrtkal at velocity. Later is discussed, on the verticals of any crosa the maximum velocity i&«t3t I ggest^ as the principal reason [ ot ocrcur at the surface, e surface, and that ir op, and energy is thepel»| rater not in the direction dfl i$mheity u equal to ihi mmX 3^ tiie dktrtbntion of reloci^ , and it will be seen^ thst it \ is the mean veUx*ity on any vertical st^ctioti of the channt'I, tte] depth at which the velocity is equal to the mean velocity is about 0'6 of tht' total depth. This depth varies with the roughne«!i of the stream, and u deeper the greater the ratio of the depth to the width of the stream. It varies between '5 and *55 of the depth for rivers of small depth, ha\ing beds of fine sand, and from '5S to 'Q^ in large rivers from 1 to 3f feet deep and ha\ing stitffig bedst. As the banks of the stream are approached, the point at wliidi the mean veh^city occurs falls nearer still to the bed of the stream, but if it falls v^ery low there m generally a second point near tk surface at which the velocity is also equal to the mean velocity. When the river is covered with ice the maxim ima velocity of the current \^ at a depth of '3-5 to '45 of the total depth, and tl« mean velocity at two points at depths of '08 to '13 and '68 to'JV of the total depth J. ■ If, therefore, on various verticals of the cross section of a strfsS^ the velocity is determined, by means of a current meter, or I*ik>t tube, at a depth of about '6 of the total depth from the surface, the velocity obtained may be taken as the mean velocity upon ibe vertical. * Hydraulique. f I,f GSnie Civil, April, 1906, ''Analysis of a communication by Marphj to the Hydrological section of the Institute of Geology of the United States." "^ Cunningham, Experiments on the Ganges Canal. FLOW IN OPEN CHANNELS 211 The total discliarge can then be found, approximately, by ividing the cross section into a number of rectangles, such as M, Fig. 120a, and multiplying the area of the rectangle by the "velocity measured on the median line at 0*6 of its depth. Fig. 120 a. The flow of the Upper Nile has recently been determined in Akway. Gaptain Cunningham has given several formulae, for the mean idocity u upon a vertical section, of which two are here quoted. u=\(y-^^{) (1), tt = i(2t?j-t;j + 2yj) (2), V being the velocity at the surface, v^ the velocity at i of the depth, V| at one quarter of the depth, and so on. Form of the curve of velocities on a vertical 13L notion. M. Bazin* and Cunningham have both taken the curve of telocities upon a vertical section as a parabola, the maximum velocity being at some distance hm below the free surface of the water. Let V be the velocity measured at the centre of a current and as near the surface as possible. This point will really be at 1 inch iff more below the surface, but it is supposed to be at the surface. Let V be the velocity on the same vertical section at any depth A, and H the depth of the stream. Bazin found that, if the stream is wide compared to its depth, the relationship between v, V, h, and i the slope, is expressed by the formula, V-t ; »r i; = V-i(g)VHi (1), ; being a numerical coefficient, whicli has a nearly constant value »f 36'2 when the unit of length is one foot. ns. Recherches Hydrauliquef p. 228 ; Annales des Fonts et Chansn^es, 2nd Vol., \4— 'i tV tV ;,\ocw; • \- ...uve ^^V »:, v\vU«*V' , ,..t '>^'^;".„ and t^*' \w, " \\p ^^^•'l >rg>' "a\iuc^ \V ateY lAOt \n .•^'",f;; «--^:^s:^??t;>cs;--■:f> 3<' ivu" to tlieP^;", ..euv<:::t'^*"«^* to*?' opoi litits atvV'^^^faeptVv:- t\tet«t^^...^.otv ot t\^« ,Tio*! \cal» oi*«»T vevt^'^,. rf^ a>y ^^''*;e\octty vottv ,V.eBS^^!!lt:Ui>^ Cttii^ .tiit^»^*' 212 HYBRAULICS lb detetmtne ihe depth tyn aiiy vertical at wM^M the velocity i eqtml to the mean veI>ocity, Let u be the mean velocity on vertical section, and hu the depth at which the velocity is equal I the mean velocity. The discharge through a vertical Btrip of width* dl is rH uEdl^dlj v.dh. Therefore uR = j^ (y - ^s/Ri )dA, and u^y-^Jm -(2). Substitnting u and A, in (1) and equating to (2), and A« = '577H. This depth, at which the velocity is equal to the mean velocityJ is determined on the assumption that Jc is constant, which is true for sections very near to the centre of strea^ms which si«| wide compared with their depth. It will be seen from the curves of Fig. 120 that the deptli ai 1 which the maximum velocity occurs becomes greater as the sides J of the channel are approached, and the law of variation of \'elocitf J also becomes more complicated ♦ M, Baxin ako found that , depth at the centre of the stream, at winch the maximum vek occurSj depends upon the ratio of the width to the depth, j reason apparently being that, in a stream which is wide comf to its depth J the flow at the centre is but slightly affected byl resistance of the sides, but if the depth m large compared with 1 width, the effect of the sides is felt even at the centre of stream* The farther the vertical section considered is remO from the centre, the effect of the resistance of the aides* increased, and the distribution of velocity is influenced to greater degree. This effect of the sides, Bazin expressed making the coefficient k to vary with the depth h^ at wl the maximum velocity occurs. The coefficient is then, 36"2 Jfc- ('-fer Further, the equation to the parabola can be written in of v„^ the maximum velocity, instead of V. FLOW IN OPEN CHANNELS 213 Th», ,.,,. 88-2Vffift-t.)- (3, The mean velocity t*, upon the vertical section, is then, 1 f^ riJo Therefore '-"*(r:|)-(5-H-H.)-;:(^<'-w- When t7 = tA, fe = /i«, « 1 fcm /l«' 2huhm and therefore, 3 " H "^ H* W~ ' The depth ^ at which the velocity is a maximum is generally len than *2H, except very near the sides, and hu is, therefore, not -very different from '6H, as stated above. Ratio of maximum velocity to the mean velocity. From equation (4), ., - 36'2>/Hi7 1 K^K\ In a wide stream in which the depth of a cross section is fairly constant the hydraulic mean depth m does not differ very much rfrom H, and since the mean velocity of flow through the section is ( C ^mi and is approximately equal to u, therefore, v^ . 36-2 /I K , hj\ ^ Cfl-M^^^ H hV- Assuming A. to vary from to '2 and C to be 100, — varies from 1*12 to r09. The ratio of maximum velocity to mean "Vidocity is, therefore, probably not very different from Tl. 132. The slopes of channels and the velocities allowed in them. The discharge of a channel being the product of the area and ihe velocity, a given discharge can be obtained by making the rea small and the velocity great, or vice versa. And since the riocity is equal to Cvmi, a given velocity can be obtam^id \>y 214 HYDRAULICS varying' either vt or i. Since m will in general increase with the area> the area will be a niininuiuj when i is as lar^e as possible. ^Butf as the coat of a channel, including land, excavation and Icxmstructionj will^ in many cases, be almost proportional to ite tcroes eectiona! area, for the first cost to be small it is desirable rthat i should be large. It should be noted, however, that the discharge ia generally increased in a greater proportion, by an increase in A, than for the same proportional increase in L Assume, for instance, the channel to be semicirctilar* The area is proportional to ^, and the velocity t* to vd , i. Therefore Q oc cp Jdl If d is kept constant and i donfeled, the discharge i^ increased to %^2Q, but if d 18 doubled, i being kept constant^ the diacbargi? will be increased to 5"6Q, The maximum slope that can bt* given will in many cases be determined by the difference in level of til two points connected by the channel. When water is to be conveyed long distance-s, it is often necessary tci have several pumping stations en rout^^ as gutficient fall cannot be obtained to admit of the aqueduct or pipe line being laid in one continuous lengtb. The mean velocity in large aqueducts is about 3 feet per second, while the slopes vary from 1 in 2000 to 1 in 10,000, ITje slope may be as high as 1 in 1000^ but should not^ only in eioep" tional circumatancesj be less than 1 in 1 0,000, In Table XXXIV are given the slopes and the maximoiB velocities in them, of a number of brick and masonry liuai aqueducts and earthen channels, from which it will be seen thai the maximum velocities are between 2 and 5J feet per second, and the slopes vary from 1 in 2000 to 1 in 7700 for the brick aad masonry lined aciueducts, and from 1 in 300 to 1 in 20,000 for the- earth channels. The slopes of large natural streams are in iomt Leases even less than 1 in 100^000, If the velocity is too small P suspended matter is deposited and slimy growths adhere to the sidefi It is desirable that the smallest velocity in the channel aliall such, that the channel is " self -cleansing," and as £ar as poiiibi the growth of low forms of plant life prevented. In sewers, or ctiannels conveying unfiltered wTaters^ it especially desii-able that the velocity shall not be too stnall, ehouldj if possible, not be less than 2 ft, per second. ^ TABLE XXXIV. Showing the slopes of, and maximum velocities^ as deterauoe experimental lyj in some existing channels. FLOW IN OPEN CHANNELS 215 / Smooth aqueducts. Slope Maximum velocity New Croton aqueduct -0001326 3 ft. pel • second Sudbory aqoednct •000189 2-94 „ )) Glasgow aqueduct •000182 2-25 „ »» Paris Dhuis •000180 Avre, Istpart -0004 „ 2nd part -00038 Manchester Thirlmere -000815 Naples -00050 4^08 »> Boston Sewer -0005 844 „ >» »» >f •000388 Earth chamiels. 4^18 „ *» Slope Maximum velocity Lining ages canal •000806 4-16 ft. per second earth ±er „ -008 4-08 >» »» »» ith „ •00037 5-63 » »» gravel and some. stones FOOT „ •00083 342 )) fi amen „ •0070 874 » »♦ earth azilly cut •00085 1^70 t )9 1 earth, stony, \ few weeds •00048 1^70 » »» the bottom of the canal) •00005 3 » >» »» M TABLE XXXV. Showing for varying values of the hydraulic mean depth m, the limum slopes, which brick channels and glazed earthenware es should have, that the velocity may not be less than 2 ft. second. m feet slope •1 ] L in 93 •2 3 L „ 275 •3 J I „ 510 •4 ] L „ 775 •5 ] L „ 1058 •6 ] L „ 1880 •8 ^ L „ 2040 i-o ] L „ 2760 m feet slope 125 1 in 8700 15 1 „ 4700 175 1 „ 5710 20 1 „ 6675 25 1 „ 9000 30 1 „ 11200 40 1 „ 15850 lie slopes are calculated from the formula 157-5 ,--. 1 + n/w m he value of y is taken as 0*5 to allow for the channel becoming For the minimum slope for any other velocity v, multiply (2\* - j . For example, the minimum slope velocity of 3 feet per second when m is 1, is 1 in 1227. 216 HYDRAULICS Velocity offimv in, and $lope of earth channels* If th© relodtf is high in earth channels, the sides and bed of the ehaji&el sii | eroded, while on the other hand if it is too sroall, the c:i|>;u ify of the channel will be rapidly diminished by the depositi.u *►! ;<n4 and other suspend til nmfcter, and the growth of aijuittic plant*. Da Buat gives '*"> f(H>t per second as the minimum velocity that mad shall not be dei)o«ite<l, while Belgrand ttlluws a miminma of '8 foot per second. TABLE XXX\T Showing the velocities above ih^ according to Du Bttiit, ' and as quoted by Rankine, erofiioi hanuebof vanous matemli j takes place. Soft clay Om tL per sec^iid Fine sand 0^^ „ „ Coarse sand and gravel m large a^ p OW „ n Gravel 1 inch diameter 2*25 „ ^ Pebbles 1^ inchea diameteT 8*33 „ ,, Heavy sbmgle 4'00 „ «. Soft rock, brick, eartbeuware 4'50 „ |, Rock, various kinds 600 „ ,, and upwurii 133. Sections of aqueducts and sewers. The forms ^jf sections given to scjme aqueducts and sewers are shown in Figs. 121 to 131, In dt^signing snch aqueduct-s ^d sewers, con side nit ion has to be given to problems other than tlie comparatively simple one of determining the size and slope to be given to the channel to convey a certain quantity of water. The nature of the strata through which the aqueduct is to be cut, and whether the excavation can best be accomplished by tunnelling, or by cut and cover, and also^ whether the aqnediict is to be lined, or cut in eolid rock, must be considered. In many cases it is desinible that the aqueduct or sewer should have 9QcI a form that a man can conveniently walk along it, although its sectional area is not required to be exceptionally large. la such cases the section of the channel is made deep and narrow. For sewers, the oval section. Figs. 126 and 127, is largely adopted because of the facilities it gives in this respect, and it has ' the further advantage that, as the flow diminishes, the cross section also diniinishes, and the velocity remains nearly constant for all, except very small, discharges. ITiis is im]>ortant, as at small velocities sediment tends to collect at the bottom of the sewer. 134. Siphons forming part of aqueducts. It is frequently necessary for some part of an aqueduct to be constructed as a siphon, as when a valley has to be crossed or the FLOW IN OPEN CHANNELS 217 aciaedact taken nnder a stream or other obstruction, and the aqaednct must, therefore, be made capable of resisting con- siderable pressure. As an example the New Croton* aqueduct from Croton Lake to Jerome Park reservoir, which is 33*1 miles rrs Fig. 121. Fig. 122. <— 7.^— > I _- IT- Fig. 123. i<- is:s^ ->i Fig. 126. rr\ « Fig. 127. Fig. 128. Fig. 129. Fig. ISO. Fig. 131. 218 HYDRAULICS long, is made up of two pnrt^a. The first m a tnasonry conduil the section shown in Fig. 121, 23'9 miles long and haipHng a of '0001326, the second consists almo«t pntirely of a brick lio^ siphon 6*83 miles long, 12' 3' diameter, the maximum head in which is 126 feet, and the differenci* in level of the two ends is 6*19 feet. In such casen, however, the siphon is fretiuentljr made of steel, or cast-iron pij^es, as in the case of the new Edinburgh aqueduct (see Fig, 131) which j where it crosses the vaUeja, m made of cast-iron pipes S3 inches diameter. 135. The best form of cham The best form of channel, or nnel of least resistance, is that which, for a gii'-en slope and ?a, will give the majdmonL discharge. Since the mean velocity in a cl el of given slope is propor- A tional to p , and the discharge is A the best form of channel for a given area, is that for which P linimum. The form of the cliannel whicj i the minimura wetted meter for a given area is a semicir for which p if r is the nuiina, the hydraulic mean depth is ^. More convenient forms, for channels to be excavated in roct or eartli, are those of the rectangular or trapezoidal section, Fig. 133. For a given dis^charge, the best forms for these I channels, will be those for which both A and P are a minimmD; I that is, when the ilifTerentialw (^A and oV are respectively equal to ^ zero. Rectangular chanwL Lot L be the >\ndth and h the depttj Fig. 132, of a rectangular channel ; it is required to find the ratio ^ that the area A and the wetted perimeter P may both be a minimum, for a given discharge. A = L/i, therefore dA = h.dh+hdh = (1), P = L + 2/i, therefore dP = dh + 2Jh = (2> - Substituting the value of dL from (2) in (1), L = 2A. Therefore m = -rj- = ^ . 4/1 2 Since L = 2h, the sides and bottom of the channel touch a cir^^^ having h as radius and the centre of which is in the free surf^*^^ of the water. FLOW IN OPEN CHANNELS 219 Earth ch4innels of trapezoidal form. In Fig. 133 let I be tlie bottom width, h tbe depth, A. the cross sectional area FBCD, P tbe length of FBCD or the wetted perimeter, i tbe slope, and let tbe slopes of the sides be t horizontal to one vertical ; CG is then equal to th and tan CDG = t. th-H and Fig. 132. Fig. 133. Let Q be the discharge in cubic feet per second. Then K^hl-^th? P=Z + 2/iV^^"+~l h(l±th)_ "^'U^hjwvi .(3), .(4), .(5). For tbe channel to be of the best form ^P and dA both equal zero- From (3) K = hl-^th\ and tberefore dA = hdl + ldh + 2thdh = From (4) P = Z + 2hs/FTl ^^a dP = dl'i-2s/¥^ldh=^0 Substituting the value of dl from (7) in (6) Therefore, l=2hy/F^-2th . .(6). .(7). .(8). m = 4h^/¥Tl - 2ht h T L«et O be the centre of the water surface AD, then since from (8) I therefore, in Fig. 133, CD = EG = OD. ^SO STDBATUCS 1 Draw 'j¥ mii •'•£ geryniifnifar ta CD and BC respectirely. T^usi. 'i«£aaae -zsb ansfe «>FD s a TiaAt an^le, the mn^ks CD6 inii JOD M* OTiaL: ami mce OF = ODcc«FOD, auid DG = OE, J Ami I«5- = »:D ^:«CIfe- -iu»5t. », OE=OF; mod ance OEC and •jFC ir» T:jnir ing^i**. a orcie wim tj as centre will touch the sides if The 'hannt*!, a6 in die v:at«e of die rectangular channeL in a ebaimd of given fbim thai, a TBaiimi i m , (b; tiie diacharge majr and tracispiDfin^. For a grven slope and ron^faness of the channel v is, therefore, propc-rti'^cal to the h3fiiraiiUc mean depth and will be a mRYimnm wl^n H is a maxiniiim. That isj when the differential of p is zero, or PJA-ArfP = (1). For nummom discharge, Ar is a maximum, and therefore, .(p)'iB: f SL maxm[iQm. Differentiating and equating to zero, ^?^PdA-»A^P=0 (2). Affixing values to n and p this differential equation can be solved for special cases. It will generally be sufficiently accurate to assume w is 2 and p = 1, as in the Chezy formula, then n-^'p _ 3 n ""2' and the equation becomes 3PdA-AdP = (3). 137. Depth of flow in a circnlar channel of given radius and slope, when the velocity is a maxinoLum. Lot r bo tlio radius of the channel, and 2<^ the angle subtended by tho Hiirfaco of the water at the centre of the channel, Pig. 184, FLOW IN OPEN CHANNELS 221 Then the i^etted perimeter The area A = r»*-r»8in*co8* = r^(^-5^V ; and dA = r»<i^ -7^(508 2^(2^. Snbstitating these values of dP and dA in equation (3), ■ectaon 136, tan2<^ = 2<^. The solution in this case is obtained directly as follows, A r /^ sin 2<^N m P~2V 24 J' This will be a TnaTrimuTn when sin2<^ 18 n^;ative, and sin2<» 2* 18 a maximnni, or when d /sin2^\_^ d^\ 24 )^ ' Fig. 184. .'. 2*cos2<^-8in2<^ = 0, and tan2^ = 2^. The solution to this equation, for which 24 is less than 360^ is 2^ = 257'' 27'. Then A = 2-738r», P = 4-494r, m = -608r, and the depth of flow d = l-626r. 138. Depth of flow hi a ch'cular channel for maximum diflcharge. Substituting for dP and dA in equation (3), section 136, 6r*^<^ - 6r»«^ co8 2iM«^ - 2r»^«^ + r»sin 2«^d</» = 0, from which 4^-66 cos 2<^ + sin 2<^ = 0, and therefore <^ = 154'. Then A = 3044r», P = 5-30r, m = •573r, and the depth of flow d = l*899r. Similar solutions can be obtained for other forms of channels, and may be taken hj the student as useful mathematical exercises bat thejr |ure not of much practical utility. 222 HYDRA trues 139. Curves of velocity and (iiscliarge for a giif channel. The depth of How for maximum vrfocity, or discharge, aai U* determined very readily hy drawing curres of velocity atid dis- charge for different depths of flow in the channeL This metliud is useful and instructive, especially to those students who are not familiar with the differential calculus. As an exiumple, velocities and discharge, for different depths of flow, have l>een calculated for a ^--ge aqueductt the profUi^ of which is shown in Fig. 135, and the pe i of which is 000013^31 The velocities and discharges are m by the cnrves drawn in the figure. Fig. 135. Values of A and P for different depths of flow were first deter- mined and m calculated from them. Tlie velocities were calculated by the formula using values of C from column Z, Table XXI. It will be seen that the velocity does not vary very much for all depths of rt<jvv greater than 3 feet^ and that neither the velocity nor the discharge is a niaxinnini when the aqueduct is full; the reason being that, as in the circular channel, as the surface of the water approaches the top of the aqueduct the wetted perimeter \ increases muuh muic! i apidly than the area. The maximum velocity is obtained when m is a maximum and equal to 3*87, but the maximum discharge is given, when the depth of flow is greater than that which gives the greatest FLOW IX OPEN CHANNELS 223 Telocity. A. circle is shown on the figure which gives the same maxinmm discliarg^e. The student should draw similar curves for the egg-shaped aewer or otKer form of channel. 140. AjfpiiBmtiflM of the finrmiiU. Problewi 1. To find the flow in a channel of given section and slope. This is the simplest problem and can be solved by the application of either the logftnllmiie fonnuls or by Bazin's formula. The only difficoltj that presents itself, is to affix values to k, n, and p in the logsnthmie fonnohi or to 7 in Bazin's formula. (1) By the logdritkmie formula. Wnwt agn some value to k^ n, and p by comparing the lining of the channel mfA those given in Tables XXIV to XXXin. Let w be the cross sectional area of the water. Then since i = -^^ , log V = - log t + — log ffl - - log Ae, n n ti mnd Q=«.r, ^^ logQ = log« + -logi+^logm- -log*. n 11 n (3) By the Chezy formula, using BazirC$ coefficient. The coefficient for a given value of m must be first calculated from the formula c=. "^-^ or taken from Table XXI. Then 157-5 1 + ^^ Example'. Determine the flow in a circular culvert 9 ft. diameter, lined with sooth brick, the slope being 1 in 2000, and the channel half full. ^^ =i=2-25'. Wetted perimeter 4 (1) By the logarithmic formula t= 00073^ m 116 • Therefore, log ,=:^ log 0005+ J;^ log 2-25 - ^ log 00007. 17=4*55 ft. per sec., «='-2— =31-8 sq.ft., Q=145 cubic feet per sec. i2) By the Chery formula, unng Baxin's coefficient, ^2-25 r = 182 V2~25T^^= 4-43 ft. per sec. Q = 31-8 X 3-35 = 141 cubic ft. per sec. 224 HYDRAULICS Problem 2. To find the diameter of a eiroular ohaxm^ of giTen slope, for whidi the maximum discharge is Q cubic feet pear second. The hydraulic mean depUi m for maximum discharge is *678r (oeetion 138) tad A = 3044r». Then the velocity is t;=•767C^/^^ and Q=2•37Cr^^/^. therefore *'=Ri3V OT' and the diameter »— v^- The coefficient is unknown, but by assuming a value for it, an approzunatioa to D can be obtained ; a new value for can then be taken and a nearer appraxi- mation to D determined ; a third value for G will give a still nearer approximatiiii to D. Example. A circular aqueduct lined with concrete has a diameter of 6' 9" and a slope of 1 foot per mile. To find the diameter of two oast-iron siphon pipes 5 miles long, to be put m series with the aqueduct, and which shall have the same disd^arge ; the diffenoot of level between the two ends of the siphon being 12*6 feet. The value of m for the brick lined aqueduct of cirenlar section when tilt discharge is a maximum is *573r=-64 feet. The area A=:8'044rS=25 sq. ft. Taking C as 130 from Table XXI for the brick culvert and 110 for the cait-inn pipe from Table XII, then 64 5280 Therefore dl=__^__ ^_ 2-26' d=400 feet. Problem 3. Having given the bottom width {, the slope t, and the side slopes I of a trapezoidal earth channel, to calculate the discharge for a given depth. | First calculate m from equation (5), section 135. From Table XXI determine the corresponding value of C, or calculate C from Bazin's formula, then and A convenient formula to remember is the approximate formula for oidinHj earth channels t; = 50vm*i For values of m greater than 2, v as calculated from this formula ia yen equal to v obtained by using Bazin's formula. rrv r , . •CH)037t;«'» The formula i= ^^ — may also be used. FLOW IN OPEN CHANNELS 225 Examiple. An og6iDMXj euth duumel has a width 1= 10 feet, a depth, d=4 feet, adailope< = v^T- Side-alopes 1 to 1. To find Q A=46 aq. ft., P=21-212 ft., »=2*16ft, 157-6 C=- 1 + Ta -=60-6, From tbe formnla ^/?16 vs 1-625 ft per sec., Q=r74*7 oabio fk. per see. v=l*68 ft. per see., Q=75 cubic ft. per sec., From the logarithmio fbrmnla . •00087t?«-^ • W^^' vs 1-649 ft. per sec., Q=75*8 cubic feet per sec. Prtiblew^ 4. Haying given the flow in a canal, the slope, and the side slopes, to (hwl the dimensions of the profile and the mean velocity of flow, (a) When the canal is of the best form. (5) When the depth is given. In the first ease m=^ * ^'^^ ^^ equations (8) and (4) respectively, section 18i> T=l + 2h ,JF+T. A Therefore Sabetitoting ^ for si m= r=r UHl Bet Therefore uid 4hjt'»+l-2th 2>j€^ + l-t A«=M(2N/«*+i-t)2. 2 hH2y/wn-t)^ (1). c«.t(2^A>+r-t)« A ▼alne for C should be chosen, say C=70, and h calculated, from which a mean vmine for m=^aaihe obtained. A neeJ'er approximation to h can then be determined by choosing a new valae of C, fgffgo T»ble XXI corresponding to this approximate valae of m, and recalcalatiD^ h from equation (1). jr^ampU, An earthen channel to be kept in very good condition, having a slope ai 1 in 10,000, and side slopes 2 to 1, is required to discharge 100 cubic feet «MMad : to find the dimensions of the channel; take C=70. troni the ad» iloMt C» »e> an4 Uie M TV »Mui T^omKj c^ m whkh is nni tytr nf dJK Ke sol ja»i :^ r aoififlMnt C i* unknowfi, ainm k ^peads upon ih» ^ . ml «vm if 1 tmlm: fnvChv^ uemned the equAUaa cusot I ' * !< dcwftlkk, Uieretore, u> toltt bj ftppn)^iiuLtioQ> -ini. fisid from mlamn 4« T^e XXI, the con^sponjuf ate vhkli viU probfthlT not btttl^ Fv wmtm iDt ^ &oin the fbnnala TW ycv^tMi cift:^ hw fodtwl In m TW wiJ^M^M « loii^ I A Moe^ Tilac for L 1 Igr ■■hillT^hmt ^titdm fonnok (2), it ^ in ^paai!^ . if eo. ^ »FF^ , d as 4iiiMimil bj luing thi i vtiM <£ ' mmiij Avail )ir«eii«»l ptir poMp ' ««3r by ^ kifaritbi&k fom lik t«a-l^miI'$i^^EBttf^, mdllM-flOflg • FLOW TS OPEN CHANNELS 227 £xmmtpir. The depth of &a ardinary earth ch&nn&t Ib 4 feet, the uide slopes 1 ta I4 tbe Alope 1 ID €0€0 and the diiohaife is to be 7000 oublc feet per minute. Find the bottom width of the fhnnn^K Ammaaxe m "emi&t for m, iaj 2 feet. Frc»iii the lof^tilhinic formula tl log r=log 1 + 1*6 log TO -4-6682 , (3), V - 1*122 feet per aeo* 7000 '^^^ ^"1-022:00"^^ '^'l*^*^*' , 104 -IS „„,^ .. 1= — 3 ^32 feet. 4 SulvtitiiliiiK thi^ mine for I in eqtuitioo (2) 4x23 + 16 „ ,« 22 + 8^2 Bcwlcnlatinir p &om fonucLla (S^ Tlnflii A = 75 feet, 1=14-75 feet, id in =2*88 feet. Tbe ^iTst ralue of i ia, therefore, too large, and this second valati U too small Tbird irmloes were foiuijd to be 17 =£ 1*455, A=eo-2, f= 16*05. in = 2-9B5* X^i* ^vmlna cf 1 19 again too lafge. ▲ tatrtti ^coiatioD gave 17- 1*475, A = 79 2, i=15 8, afifirDiimatioii has been carried auffidentlj far^ and ereQ further tlian ii _ tor met chattnele the coeffiqient of roughness k eaiiuot b» trusted to mm Acenimcjr correvpondlng to the stDall differeDoe between tli^ third and four^i wm^lnes of ^ PfiM^tm 6. HftTiDg giTen the boltom width £, the slope t and the tilde alnpes of tjm p e« Qidal channel, to hud the depth d for a given diBcharge* Thlm problem is aolTsd exaetlj as the last, by first afisumlng a value for m, and J an approiima(« value for p from the formula v = C Vm7. bj BQbetitntion in ei^Qation (I) of the last i^oblem^ and solving the Tbm^ /q I I atmg this valae for d In equation (2), a new valne for m can be found » beoeev a Moond approiimation to fi, and bo on. Um«g t^ logadthjnio formula the procedure is exaotlj the same as for ^Pro^Uml*. HaTing a natural stieam BC, Fig. 13dri, of ^ven slope, it in required iluiafTiiIpe the point C, at whiqh a canal, of trapezoidal s^ciioti, whkh is to jf«r m ^finite qoantity of water to a given point A at a given level, shall be I to jptn ibe stream so that the cost of the canal is a minimum. Tbe aolnlion here given 10 praotioally the same as that given bj M. Flamant t fiim «3L0Qlktit treatise Hydmuli^ue, and tbKt^fore from which wfi= 228 Let I be the ilope of the ntrmtn, i of the msud, k Ihfi height ftboTt ioa» j of the Burfooe of the watef at A, and h^ of ib« water in the tttr^tu at B^ at «om€ diatanoe X4 from C. Let L be alfto the length and A the MCtioiial area of the canal, aod t«t it ba aaaamed that the ««ctLOD of the canal !» of the moat eoonomieo] form, or m = - . Th« nde elopes of Ihe canal wil) be tw^ aooording to the tiatai^ of the atrata through which the canal u eat, uid 1 8uppof»«<i to be known. Then the level of the water at C ia Therefore Let f be the bottom width of the ■ectio& la then dl^t^^ and i SobttitDtiug f m for d, 4 The ooefficient C in the formula v=C iJnU may be assmned constant. Then r«=C*iiM, and r*=C*m«i«. Q For V snbetitnting ^ , and for m' the above value, <y C^At< A*' and ^'•'= ^ <2 >/?+l - 1). Therefore The cost of the canal will be approximately proportional to the product of the length L and the eroes sectional area, or to the cubical content of the exctvitioii. Let £ik be the price per cubic yard including buying of land, excavation etc Let Ir be the total cost. Then £x=£k,h.A This will be a minimum when jt =0. Differentiating therefore, and equating to zero, |i*=tl.-i, and t = fL The most economical slope is therefore | of the slope of the natural streun. If instead of taking the channel of the best form the depth is find, tli^ slope tss^.I. FliOW IN OPEN CHANNELS 229 There hate heea l#o lusiiiDptions mftde m the calcuJAlioti, neitliei of whifih is ri^dljr trite^ tho &i^t beiog that the coefficieQt C ia cod slant, And the second th&X the price of the isftnaJ ia proportional to ita cross sectiontU aTCft. It win not &]vvjs be po««ible to adopt the elope tbuH founds as the mean ^reloctt^ mafit be maiDlained within thif limits Kiven on page 216, and It ta not adviaable llial the slope shonld be loss than 1 in 10,000. EXAMPLES. (1) The areft of flow in a sewer was found to be (h2d sq. feet; the peruneter 1*60 feet; the mcliaation 1 in 38' 7. The mean Telocity llo^r was S'12 ^t per second* Find the value of G in the formula (2) Tbe dramage area of a certain difitrict was 19'32 aerefli the whole being impermeahle to rain water. The maximum intensity of the was 0*300 ins^ per hour and the maximum rate of discharge regis - in the sewer was 96% of the total rainfall. Find the size of a circular glazed earthenware culvert having a slope of is 50 aoi table for carrying the storm water. (0) Draw a curve of moan velocities and a curve of discharge for an «i|g^Hili&ped brick iewer, using Bazin's coefficient. Sewer, 6 feet high by ^ ieei groatesi width; dope 1 in 1200. (4> The sewer of the previous question is required to join into a main Odt^ftll «ewer. To cheapen the junction with the main outfall it is thought ad^satile to make the last 100 feet of the sewer of a circular steel pipe feet diameter, the junction between tbe oval sewer and the pipe being Jy shaped so that tliere is no impediment to the flow* Find what fall the circular pipe should have no that its maximum ^ shall be equal to the maxlmmn discharge of the sewer. Having ihe slope* draw out a curve of velocity and discharge. i$) A oftnal in earth has a slope of 1 foot in 20,000, side slopes of faodflcmtal to 1 vertical > a depth of 22 feet, and a bottom width of find the volume of discharge. Baxin^ coefficient y^2'35. ^fi) Oire ihe diameter of a circular brick sewer to rnn half -full for a peipQlation of 80,000, the dinmal volume of sewage being 75 gallons per head, the period of maximum flow 6 hours^ and tlie available fall 1 in lOOO. Inst C. E. 1906, (7) A channel is to be cat with aide slopes of 1| to 1 ; depth of water, ak^iet 9 inches per mile: discharge, 6,000 cubic feet per minute, f approximation dimensions of clianneL 0f An area of irrigated land requires 2 cubic yards of water per hoar f acre. Find dimensions of a channel 3 feet deep and with a side slope ^1 lo L Fall. 1^ feet per mile. Area to be irrigated, 6000 acres, (Solve *F apfiroadmation^) y = 2*35* (9) A trapezoidal channel in earth of the most economical form has a i 0f 10 feet uad side slopes of 1 to 1. Find the discharge when the \ li IS inches per mile. y=2*35. S30 HYDRAULtCS (10) A river has tlie loUowiiig section :— top width, 800 feet ; depth water^ 20 feet ; mde slopes 1 to 1 ; fail, 1 foot per mile* Find the disdiAig^ iismg Bazin'B coefficient for earth channels* (U) A channel ib bo be ooiwtnicted for a, discharge of 2000 cubic feet per second ; the fall ii 1| feet per mile ; side ^opes, 1 to I ; bottom widlhr 10 times the depth. Find dimensions of diannel. Use the approximate' formula, v=50^ftrL (12) Find the dimenstoim of a trapezoidal earth channel, of the economical form, to convey 800 cubic feet per second, with a fall of per mile, and side slopes, 1| to h (Approxiniate formula,) (13) An irrigation channel, with side slopes of H to 1, receiver 600 cubic feet per second. Design a suitable channel of 3 feet depth lad determine its dimensions and slope* The mean velocity is not to exceed 2^ feet per second, y = 2 ■ 3 5 . (14) A eanal^ excavated in rock, has vertical sides, a bottom widtli d 160 feet, a depth of 22 feet, and the slope is 1 foot in 20^000 feet. Find tlm discharge, y = 1 '64, (15) A length of the canal referred to in question (14) is in earth. It has side slopes of 2 horizontal to 1 vertical ; its width at the water ]kt is 290 feet and its depth 22 feet. Fmd the slope this |x>rtion of the canal should have, taJdng y aa S"!^ (16) An aqueduct 95 iJ miles long is made op of a cud vert &0J miJn long and two steel pipes 8 feet diameter and 45 miles long laid side byst^ The gradient of the culvert is 20 inches to tlie mile, and of the pipes 2 1«e* to the mile. Find the dimensions of a rectangnlar culvert lined with ^ pointed bricks so that the deptli of flow shall be equal to the wid^ crf lb* culvert^ when the pipes are giving their maximum dischaxge. Take for the culvert the formula and for the pipes the formula . -000061 tJ^*88 i^ ■00050. t?" (17) The Ganges canal at Taoli waa found to ha^e a slope el and its hydraulic mean deptli m was 7^0 feet ; tlie velocity 8 by vertical floats was 2*80 feet per second; find the value of C io3"' value of y in Bazin's equation, (18) The following data were obtainad from an aqueduct lined fi^ brick carefully pointed : mmeties ID me Ires per S4». ■229 0-0001326 '336 ■S81 tt 484 •5S8 t* ■596 '680 It '691 '888 II *769 991 fi '848 1148 If '918 M70 n '922 PLOW IN OPEN CHANNELS 231. ^^= is oirdin&tes, — - ba absciesae; Imd T&liieaof a and ^ in Barb's 1^ and thos dedaee a valne of y for this aqueduct. Aa aqtiedoct l(fl\ miles long eonsbts of 13| miles of siphon, and niodsr of a masonry cnivert 6 feet 10| inches diameter with a gradient I iOPCl The niphone coniiit of two lines of caet-irQii pipes 43 inches Air tiaruig a slope of 1 in 500, Determine the maximum cUaaharge. An sqaednet ocmststs partly of the section shown in Fig, 131, iT» ft&d fMutly (ie, when crossing valleys) of 38 inches diameter cast- tbe minimnm dope of the siphons^ so that the aqueduct liacisafge 15,000,000 giillons per day^ aiid the slope of the masonry so that the water shall not be more than 4 feet 6 inches deep in ) Calealaie the qtiantity delivered by the water main in question (30), |Lper day of 34 hours, ^Hpiooiit, representing the water supply of a city, is discharged into Hn at tha rate of one -half the total daily volume in 6 hours, and m lebled by rainfalL Find the diameter of the circular brick outfall which wUi carry off the combined flow when running half full, the hie Ml being 1 in 1500, Use Bamn's eoefiicient for brick channels. \) Determine for a smooth eyhndrical cast-iron pipe the angle at the centre by the wetted perimeter, when tlie velocity of flow Determine the hydraulic mean depth of the pipe nnder Land, Un, 1905. % A 8*fliQh drain pipe is Laid at a slope of 1 in 150, and the valne of 17 {v^e^mi). Find a general expression for the angle subtended at mtre by the water hue. and the velocity of flow ; and indicate how Uie d e^uatioB^ may be solved when the discharge is given. Loud. Un. Ir Sifter^ Qit^e^unt ef tf%« hUiarical ihi^^hpm^nt of the pip^ and ehannel formulae. kui reniJirki^bte Uint, although tti^ pmctiise of conduGting water ulnng piped bapek for doniaitio and other purpnieg haa been cfLrried on for many ^Hhio aerioiii ittempt bo discover the la^'fi regulating th^ fiow fteems Wmmtk att^topted nntiJ Ih*^ eigblc^nth centurj. It soema diMcult to realise It giffmnlie «i?b ernes of wnti^r ihaii-ibiiiion of the ancstetit Gitiea cnuld hftve be^n ~ viibout E^ycb kouw ledge, but ci'rtain it i«, tb&t whatever infonnnticin they I. it WM lost diiriuf; ibe middle ages, ■i if pMoiiar interest to note tbe trouble taken by the Boman engineers in of their aqueducts. In order to keep the elope oonBtftnl they thnjugb biUs and a&rried their fujuedocts on mnf^iSeant arehea. The^ j a^nedtiet wai 38 milei loug and dad a coiii^tAnt dope of five feel per mile, mtiy they were unaware of ihe »tmple fact that it is not necessary for a pipe idaet aonneoting two r^aervoira to be laid perfectly straight t or sine they the water tt all parla of the aqoedticts to b^ at atmospheric pressure, Seliwet^er in his interesting treatise on bydroHtatiiiB and hydraulics in 1729 quotes experiments by Marriott Ehowitig that, a pipe 1400 yards i| loobaA diameter, only gave | of the diiM^harge which a bole 1| inches diameter idt of m tiitik wonld giv« under the saoie heudp and also eiplama that the of the Hquid in the pipes is diminished by friction, but he is entirely biws regulatinjj the flow of Huida through pipes. Even as late as 2^2 HYDRAULICS 1 17B6 Dq Bta^t* wrote, "We mm jet m absolute ignoranoe of the laws lo wblek ih«] moTement of wat^r is enbjected.*^ I Thei <»adii^»t recorded experimetils of any valu<} on long pipes lyne tboM oil Couplet, in whiish he measoced the flow through the pipes wm^ aupplicHl tbal Ibmoiia fountains of Versailles in 1732, In 1771 Abb^ Bossni made experimefite nal iow in pipes and ohannals, these bein^ followed by tba expedments of Do BuaU wbal eironeotiflly argued that the loss of head due |p f notion iu a pip« was indsiiendeall of the internal »urfa43e of the pipe, and gave a oomplieated formula for the Tdoei^l of ^ow when the head and the leo^th of the pipe were known. I In 1T75 M. Chezj from expenmeuts upon the flow in an open eatial, caine IdH the conclusion that the 6uiJ friction was propoTtioual to the velooitj squared, and' that the nlope of Ihe channel multiplied by the croB» sectional area of the stt«*in, wag equal to the prodoct of the length of the wetted surface measured on tbe enw aeetiont the relooitj iquared^ and aotue eotiatant* or iA^Fav* :,, .,.41}, I being the slope of the bed of the channel, A the croe^ sectional area of the atieam, P Ihe wetted perimeter^ and d a coefficient. From thi^ ie dedtieed the well-known Ghezj formula ^-^^ ^i-ojmu Pronyf , applying to the iow of water in pipes the results of the cl&aaical esp«ii* ments of Coulotoli on fluid frietion, from whieh Coulomb had dednoed the lav thst fluid frieiioB was proportionai to av-k-bty^t arriyed at the formula «i(= at? + ^« = [- + p\ r/^. This is elmilar to the Chesj formola* ( - + /9 j being equal io ^ , By an eiatnination of the experimentii of Couplet, Boseut, atid Du Bnat, Proii|| gave values to a and ^ whiah when trane formed into British units are, 0^-00001733, /3 = '00010614, For TclooitieSi above S feet per seeond, Pronj negleotod the term containing tbej first power of the velocity and deduced the formula Ue continued the mistake uf Du Bunt and assumed that the Criotion wai in< dependent of the eoodition of the internal ^orfaoe of the pipe and gave the foUowioir eiplanation : *' When the fluid flow? in a pipe or upon a wette^l surface a Slnx'' fluid adhere;; to the surface, and tltis film may be regarded an enoloiing the tail of fluid in raotiont-*' That such a film encloses the moving water reoeiyes soppenl from the eiperiments of ProfesHor Hele Shswf. The expei-iment!? were made npAl sueh K KmaD seals that it is dilBcult to say how far the results obtained arft indie**' tivti of the conditiona of flow in large pipes, and if the ilm exists it do^ not sera to m\ in the way argued by Prony. H The value of t in Prony's formula was equal tn y , H inaluding, not onlj tt» loaa of head doe to friction bat* as measured by Couplet, Bossut and Da I' it ako included the head neoensary to give velocity to the water and to overc. leflistance^ at the entrance to the pipe. Eytelwein and also Aubisson, both niade aUowaoces for these loisea^ hs ^ tracting from H a quantity x— , and then determined new yalnes for a and I ia tb« formnla * Zf XhVcowTf priliminaire de m* PHnciptt d'k^draulique, f See also Girard'a Movement dc9 fluids dans let tub^t eapittairew, 1817, ^ Traiti d*hydrauliqut, f Engineer, Aug, 1897 and Maj ISUB. FLOW m OPEN CHANNELS 233 Th gy iAv« lo a and b the rollowing TaJaes. Eytelwem (r - ^000023584, h= ^000085434. Aubiaaon' a == 'OOOO 1 8837, 6=1)00104392. By iMg)«ctui^ the term oonlaiiiiiig v to the first power, and transform ing the I Atibisflon'B formolA reduces to •'=«\^JT Hd 3S-6er ■ ToiiJDfc* tQ tbd Efuryc^cpQ^dta Bnl^nf^t^u, gave a complicated formula for ? when »cul ^ were known, bnt gave the fiimfilified formula, for veJocitiee unoh oa Are geiQ^imllj met with m practice, 8t Teoant made ft decided depftrtEue by making - prdportlonal to v'r imtead of I r* M to lite Ch«^ IbimiilA. Wbioi eipiMKjd IP EngHflh feei aa unitSi hia formula becomes v = 206(>?fi)T^^ I by an examination of the early experiments together with ten others bv avH and on* by M. GneyttArd gave to the ooeflScient a in the formula ft = — - rftlne , |b« be made it to vaiy with the Telocity* valiiia of a and ^ being ol^QOI U , Wram lins formnlA iablas were drawn up by Weiabach, and in England by * J, which were considerably need for calciilations relating to flow of in pipes. l^iflcy, as explained in Chapter \\ made the coefticient a to vary with the r, and Qag«n proposed to make it v&ry with both the velocity aod the M^ formula then beoame «■=&.-")'"• The fbrtnclae of Qangmllet and Etitter and of Biusin hare b#eii given in I V and VI. lAmpe &om axperimentB on the Dantzig maina and other pipes proposed ElonBUlft *-"ffi* < modifying St Tenant's formula «nd antioipattng the formulae of Beynoldif a^d Unwrnn, Id which, i ftfid p being variable coeMcienta, * Traiti d'htfdfaulique. CHAPTER VII. GAUGING THE FLOW OF WATER, 142. MeaBuidng the ^ow of water by weigMng. In the laboratory or workshop a flow of water can geiws be tueRsiired by collecting the water in tanks ^ and either direct weigh iiigi or by measuring the voltime from the \m capacity of the tank, the discharge in a given time cau determined, lliis is the most accurate method of measu water and should be adopt-ed where possible in ex per work. In pump trials or in measuring the supply of water to boilfl determining the quantity by direct weighing has the di^ct advantage that the results are not materially afTtctetl by changes of temperature* It is generally necessiirj" tu have W tanks, one of which is filling while the other is being weifbeJ and emptied. For facility in weighing the tanks should etJ on the tables of weighing machines. 143, Meters. Lirwri meter. An ingenious direct weighing meter gauging practically any kind of liquid* is constrnckHl . Figs. 136 and 137. It consists of two tanks A" and A*, each of which can on knife edges BB. The liquid is allowed to fall into a she wliich sl\^vels about the centre J, and fi'om which it falls either A* or A' according to the position of the shoot. The 1 have weights D at one end, which are so adjusted that wi ** -ta-in weight of water has run into a tank* it swings ovt*r i formu.-j^l^^ position, Fig. 136j and flow commences thrtvagt H "oe C. When the level of the liquid in tlie tank ^^H , j^ Dhc^^^^' ^^^ i^t^igbts D cause the tank to come back] ^^H t See also csitioUj but the siphon continues in action until I OAUOiya THE FLOW OF WATEB 235 lenly tilts over tlie shoot F^ and the liquid is discharged other tank. An indicator H registers th© nuTiiber of le tanks are tUledjatid as at each tippling a detinite weight is etnptied from the tank, the indicator can be marked >unds or in any other unit. Fig. im. Fig. VAl. Liner t direct weighing meier. Meaatirmg the flow by means of an orifice c<:»t?fficient of discharge of sharp-edged orifices can be ith cousiderablo precisioUj from the tables of Chapter IVj coefficient for any given orifice can be deterniined for hm he^ids by direct measurement of the How in a given time^ tdeicnbed above* Tlien, knowing the coefficient of discharge at ■tidii a curve of rate of discharge for the orifice, as in may be drawn, and the orifice can then b© uised to * a continuous flow of water. ! orilice should be made in the side or bottom of a tank. If I the side i>f the tank the lower edge sh<juld be at k^ast one and If to twice its depth above the bottom of the tank, and the of the orifice whether hori?.ontal or vertical should be at |0tie and a half to twice the width from the sides of the tank, ink i»hould be provided with baffle plates, or some other pnient, fur destroying the velocity of the incoming water t cnFuring quiet water in the neighbourhood of th© orifice. The pent of dischargu i^ otherwise indefinite. Th© head over the jnhould be observed at stated intervals. A head-time curve \ h^ftd aas ordinate^ and time as abscissae can then be plotted , 189. From the head-discharge curve of Fig. 138 the rate of discharge found for any head h^ and the curve of Fig. 139 plotted of this curve between any two ordinates AB and CD, I I I I I ] 236 HYDRAULICS which is the mean ordinate between AS and CD multiplied bytbej time tj gives the discharge from the orifice in tinae t. The head h can be measured by fixing a scale, hBrlng '• " coinciding ^vith the centre of the orifice, behind a tube on : of the tank. Fig, im. 145. Meaauring tlie flow in open channels. Large open channels : floats. The oldest and simplest me of determining approximately the discharge in an open cha by means of floats. A part of the channel as straight as possible is selected, i which i\m flow may be considered as uniform. The readings should be taken on a calm day as a down^st wind will accelei*ate the floats and an up-stream wind retard 1 Two cords are stretched across the channel, as near to J surface as possible, and perpendicular to the direction of flow, distance apart of the cords should be as great as possible coi: - with uniform fiow^ and should not be less than 150 feet, i boat, anchored at a point not less than 50 to 70 feet aboTe alreafflt j so that the float shall acquire before reaching the first Hneaj imiform velocity, the float is allowed to fall into the streiim audi GAUOIKG THE FLOW OF WATEH 237 m time carefuDy^ noted hy means of a clironometer at wliich it KM oa l>ot]i the first aod aecond line. If the velocity ig slow, the Issenrei* ma^y walk along the bank while the float is moving. from me cortl to the other, but if it is greater than 200 feet per minute M observers will generally be reqaired, one at each line, ^kA better method, and one which enables any deviation of the Hi fmni a path perpendicular to the lines to be determined, is, Sr two observers provided with box sextan tSj or theodoHtes, to be ioned at the points A and B, which are in the planes of the hneo. As the float passes the line AA at D, the observer L fign&Is^ and the observer at B measures the angle ABD if both ane ppivided with watches, each notes the time. en the float passes the line BB at E, the observer at B signals, the obsen^er at A measures the angle BAE, and both again note the time. The distance DE can then be arately determined by calculation of by a sc^le drawing, and \ mean velocity of the float obtained, by dividing by the time. I To ensure the mean velocities of the floats being nearly equal be mean velocity of the particles of water in contact with , their horizontal dimensions should be as small as possible, to reduce friction, and the portion of the float above the of the water should be very small to diminish the effect of ^.pointed out in section 130, the distribution of velocity in averse section is not by any means uniform and it is (T, therefore^ to obtain the mean velocity on a number of planes, by finding not only the surface velocity, but also JvekxHty at various depths on each vertical. 1 146. Surface floats, 5m^€e floats may consist of washers of cork, or wood, or small fioating bodies, weighted so as to Just project above surface- The surface velocity is, however, so likely to It?cted by wind, that it is better to obtain the velocity a distance below the surface* 147. I>ouble floats. To uiesi^ur*? the velocity at points below the aur&ce double are employed* They consist of two bodies connected by of a fine wire or cord, the upper one being made as small lible so as to reduce its resistance, 1% on the Irrawaddi, used two wooden floats connected \ fine fittkmg linej the lower float being a cylinder I foot longi • Proe.lmL V. E„ IS98. riib 238 HYDRAULICS and 6 iBches diameter, hollow undemeatli and loaded with gink it to any required depth } the upper floaty which swam od surface, waa of light wcKMi 1 inch tliiuk, and carried a small Am The surface velocity was obtained by sinking the lower I to a depth of 3J feet, the velocity at this depth being not i different from the surface velocity and the motion of the float i independent of the effect of the wind* Kg. 141. Gurby'f ourrent m©t«r< Subsurface velocities were measured by increasing the of the lower float by lengths of Si feet until the bof reached. OAnGINe THE FLOW OF WATER H Gordon has cM>mpared the results ob tamed hy floats with those ^Btftizied hf mestis of a cuiTeiit meter (see section 149). For HpaU depths and low velocities the results obtained by double ^■KtB are ^rly accurate^ but at high velocities and great depths, ^Ee Talocities obtained are too high. The error is from to 10 per cent. Double floats are soinetiinea made with two eimilar floats, of the same dimensioiiS| one of which is ballasted so as to float at any reiiuired depth and the other floats just below the surface. The wwiodtf^ of the float is then the mean of the surface velocity mad the velocity at the depth of the lower float. 148. Bod floats. The mean velocity, on any verticalj may be obtained ap- pro jdmately by means of a rod float, which consists of a long rod baring at the lower end a small hollow cylinder, which may be fillad with lead or other ballast so a^ to keep the rod nearly vertical. The rc»d is made sufficiently long, and the ballast adjusted, so thBt ita lower end is near to the bed of the stream, and its upper end project* slightly above the water. Its velocity is approximately tn^&n velocity in the vertical plane in wliich it floats. 149. The onrrent nieteT. The discharge of large channels or rivers can be obtained most |ccitiv'oni*?ntly and accurately by determining the velocity of flow number of pi>ints in a transverse section by means of a current The arrangement shown in Fig. 141 is a meter of the anemo- ineter type, A wheel is mounted on a vertical spindle and has fi%-e conical buckets. The spindle revolves in bearings, from ^rkicfa all water is excluded, and which are carefully made so tbat the fricrion shall remain constant. The upper end of the «tpixidle extends above its bearingj into an air-tight chamber, and im shaped to form an eccentric. A light spring presses against eecentric, and successively makes and breaks an electric r ■ - the wheel revolves. The number of revolutions of the r njcorded by an electric register, which can be arranged any convenient tlistance from the wheeL When the circuit is taade, an electro^magnet in the register moves a lever, at the end of which is a pawl carrying forward a mtchet wheel one tooth I for each revolution of the spindle. The fi'ame of the meter, which im made of bronze^ is pivoted to a hollow cyUnder which can be clamped in any desired position to a vertical rod. At the rightr 240 HTDBAtTttCS hand side is a rudder having four lig^ht metal wing^ which balances the wheel and its frame. Wlien the meter is being miei in deep waters it m suspended by means of a tine cable, and the lower end of the rod is fijced a lead weight. The el circuit wires are passed through the trunnion and so haw tendency to pull the meter out of the line of current, placed in a current the meter is free to move about the hoi axis, and also about a vertical axis, ao that it adjosta il the direction of the current. The meters are rated by experiment and the makers the following method. The meter should be attached to al a boati as shown in Fig. 1^, and immersed in still wmter less til an two feet deep* A thin rope should be attached boatj and pa^ed round a pulley in lint* %vith the course in the boat is to move. Two parallel lines about 200 feet should be staked on shore and at right angles to the course boat. The boat should be without a rudder, but in the huai the observer should be a boatman to keep the boat from Fig, 142, into the shore. The boat should then be liauled between i ranging lines at varying speeds, which during each passage i be as unifonn as possible. With each meter a reduction supplied from which the velocity of the stream in feet jm^^t b can be at once determined from the niunber of revolutions j per second of the wheel. The Haftkell meter has a wheel of the screw ] revolving upon a horizontal axis. Its mode of a. eimilar to the one described. Comparative tests of the discharges along a rectangoiarj as measured by these two meters and by a sharp-edged ' had been carefully calibrated, in no case differed by moit^ 5 per cent, and the agreement was generally much closer*. * Mnrphj on current Meter and Wdr lU^abargesv Pruceedinp i» Vol TZYiu p. 779. GAT7GINO THE FLOW OF WATEB 241 150. ntot tube. Another apparatus which can be used for determining the docity at a point in a flowing stream, even when the stream is of ull dimensions, as for example a small pipe, is called a Pitot die. In its simplest form, as originally proposed by Pitot in 1732, =*=, Jll E Fig. 143. Pitot tube. 3k ooDBisEtB of a glass tube, with a BmII orifice at one end which may be tomed to receive the impact of Ik stream as shown in Fig. 143. Be water in the tube rises to a iMght h above the free surface of Me water, the value of h depending i the velocity v at the orifice of Btnbe. If a second tube is placed I the first with an orifice parallel to the direction of flow, tte water will rise in this tube nearly to the level of the free ■Bzface, the &11 h being due to a slight diminution in pressure 9X the mouth of the tube, caused probably by the stream lines laving their directions changed at the mouth of the tube. A &rther depression of the free surface in the tube takes place> if the tube, as EF, is turned so that the orifice faces down stream. Theory of the Pitot tube. Let v be the velocity of the stream at the orifice of the tube in ft. per sec. and a the area of the orifice in sq. ft. The quantity of water striking the orifice per second is wav pound;s. to The momentum is therefore - . a . v' pounds feet. If the momentum of this water is entirely destroyed, the pressure on the orifice which, according to Newton's second law of motion is equal to the rate of change of momentum, is P = wav find the pressure per unit area is 9 The equivalent head h - V!^ = - According to this theory, the head of water in the tube, due to le impact, is therefore twice |- , the head due to the velocity v, and 242 the water shotild rise in the tube bo a height above th©( equal to h. Experimoiit allows that the actual height the water ] tube is more nearljr equal to the velocity head and the head b ig thum generally taken as c being a coefficient for any is fairly conKtantv Similarly for given tubes tube, which escperunent i and The coefRcienta are detei the velocities of which are body which moves through carefully meitj^uring h for diUereu^ velocities t 7 by placing the tubes in i I, or by attaching them \ iter with a known vclocit .i. Fig. 144. Darcy* was the first to use the Pitot tube as an instrumei precision. His improved apparatus as used in open channels sisted of two tubes placed side by side as in Fig. 144, the or in the tubes facing up-stream and down-stream respectively. Reeherchef Hydrauliques, etc., 1857. QAI7GIKO THE FLOW OF WATER 2m bobes were connected at the top^ a cock C^ being placed in the tnon tube to allow the tubes to be opened or closed to the At the lower end both tubes could be closed at the e time by uieans of cock C. Wlien the apparatus is put into in^ water, the cc»cks C and C* being open, the free surface is the tube B a height hi and is depressed in D an amount Tbfi cock C^ i&> then closed, and the apparatus can be taken i the wster and the difference in the leYel of the two columns, h-hi^ fhj with cotusiderable accuracy* deeiredi air can be aspirated from the tubes and the eolunms to rise to convenient levels for observation, without moving The difference of level will be the same, whatever in the upper part of the tubeB, i 145 shows one of the forms of Pitot tubes, as experimented by Pnifeasor Gardner Williams*, and used to determine ribution of velocities of the water flowing in circular pipes, arrangement shown in Fig. 146, ia a modified form of the tias used by Freeman t to determine the distribution of in a jet of water issuing from a fire hose under con- ile pressure. As shown in the sketch, the small orifice the impact of the stream and two small holes Q are drilled be T in a direction perpendicular to the Bow, The lower ^paratns OV, as shown in the sectional plan, is made so as to prevent the formation of eddies in the hood of the orifices. The pressure at the orifice is ed through the tube OS, and the pressure at Q through QR. To measure the difference of pressure, or head, two tubeS) OS and QR were connected to a differential SLttiilar to that described in section 13 and very small of head could thus be obtained with great accuracy, lobe shown iii Pig* 145 has a cigar-shaped bulb, the orifice O being at one end and communicating with the OS. There are four small openings in the side of the bulb, M any Tariations of pressure outside are equalised in the The pres^upes are transmitted through the tubes OS and > m differential gauge as in the case above, r I%» 147 is shown a special atuffing-bojc used by IVofessor mOBBtto allow the tube to be moved to the various positions in '•the I ipri of Fit0t tiibea api atetl by ProfeSBor WiHiflma, E. S< Oele tLnd of the Jm^\C,E,, VpL uli. 16^^ 1 I I 244 HYDRAULICS the cross section of a pipe, at which it was desired to detenmne the velocity of translation of the water*. Mr E. S. Colet has nsed the Pitot tube as a continaoas meteri the arrangement being shown in Fig. 148. The tabes were con- nected to a U tube containing a mixture of carbon tetrachloride and gasoline of specific gravity 1'25. The difference of level ct the two colunms was registered continuously by photography. ^s^szsi^^ssm \^A^WA^JJ?MJJ^JM'fJJ^J>>JJJi^M>}^>. ■^■-gry?g Fig. 147. Fig. 146. Fig. 148. The tubes shown in Figs. 149 — 150, were used by Bazin to determine the distribution of velocity in the interior of jets issoiiig * See page 144. t Proc, A.M.S.C.E., Vol. xztu. See also experiments by Murphy and TomuMi in B&me voiume. OAUGIKG THE FLOW OF WATER 245 {rem orifices, and in the interior of tlie nappes of weirs. Each tobe consisted of a copper plate 1*89 inches wide, by '1181 inch thick, sharpened on the upjier edge and having two brass tubes "0787 inch diameter, soldered along the other edge, and having orifices '059 inch diameter, 0'394 inch apart. The opening in tube A was arranged perpendicular to the stream, and in B on the face of the plate parallel to the stream. cai — I£ 2 Kg. 149. Fig. 160. 15L Calibration of Pilot tubes. Whatever the form of the Ktot tube, the head h can be expressed as h = cv' or 2g' = ksf2gh, Ic being called the coefficient of the tube. This coefficient h must be determined by exi)eriment under conditions as near as possible like those under which the tube ^vill be used to determine velocities. To calibrate the tubes used in the determination of the distri- bution of velocities in open channels, Darcy * and Bazin used three distinct methods. (a) The tube was placed in front of a boat which was drawn through still water at different velocities. The coefficient was 1"034. This was considered too large as the bow of the boat probably tilted a little, as it moved through the water, thus tilting the tube so that the orifice was not exactly vertical. (h) The tube was placed in a stream, the velocity of which was determined by floats. The coefficient was TOOG. (c) Readings were taken at different points in the cross section of a channel, the total flow Q through which was carefully measured by means of a weir. The water section was divided * J^eherehes Hydrauliquet, 246 HTDIUUUC3 [ two metikoAB of caUbit ^nbes throQgli still wfttlflr; a cmmmlerential i 8 inches de^p wa& buflt sntre line, which W8§ tbs hes. The tube to b^ rafe i a central shaft wkicli m and oeilitigrf and whicb observer. The gang^ ^as into areas, and about the cemm of mdt a raaiiitto' of tW waa taken. CaDtng m the araa of oa»of theaa Hctio&9, and ^ reading of the tube, the coefficient i = — 9= and was foimd to be *fl93- Darcy* and Ba^o also fotmd that by changing the positioti the orifice in the pressure tube the coefficients changed giderably. Williama, Hnbbell and Fenkel which gave very diffei^nt re^nlts The first methi^d was to mo'v known velocities. For this pc rectangolar in section, 9 inches i galvanised iron. The diameter t the path of the tube, wag 1 1 feet WB8 supported npon an arm atta^ free to revolve in bearings on i supported the gauge and a sear n. connected with the tabe by rubber hose. The arm carrying the tube was revolved by a man walking behind it, at as unifonaa rate as possible, the time of the revolution being taken by means of a watch reading to i of a second. The velocity was mail* tained as nearly constant as possible for at least a period of 5 minutes. The \'a!ue of A: as determined by this method was ■9'2( for the tube sho^^-n in Fig* 145, In the second method adopted by these workers, the tabe iras inserted into a brass pipe 2 inches in diameter, the discharge through which was obtained by weighing. Readings were tak^a at various positions on a diameter of the pipe, while the 6ow in th? pipe was kept constant. The values of J2gk^ which may be called the tube velocities, could then be calculated, and the mean value « Vfl, of them obtained. It was found that, in the caaes in which the I form of the tube was such that the volume occupied by it in the pipe was not sufficient to modify the fiow, the velocity was a maxim am at, or near, the centre of the pipe* Calling this maximum velocity V Vc, the ratio ^ for a given set of readings was found to be '8L J Previous experiments on a cast-iron pipe line at Detroit having shown that the ratio ^ was practically constant for all velocities, a similar condition was assumed to obtain in the case of the brass * Reeherchet Hydraulique$, ailFOmG THE FLOW OF WATER ii ipe. The fcube wae then fixed at the centre of the pipe, anfl lugs taken for varinaB rates of discharge, the mean velocity determined by weight, varying from ]- to 6 feet per second. For the valnes of h thna determined, it was found that /— sf2gh i practically cgnstanti This ratio was *729 for the tube shown Fig- 145. Then since for any reading h of the tuhe^ the velocity v is u ^_ U V. I acttml m^n velocity fe-. Bui Thi^rcffore &- ratio of U to V. 814' ^^' ratio of Ym to V* For the tube ahowTi in Fig. 146, some of the values of h 'by the two methods differed very considerably* i^'fm fff the values of k by the two methods. It will ihat the value of k as determined by moving the tube throug] iQ water differs very considerably from that obtained i l^rotining water. In the latter case the pressure was considerably higher than in the former, and it appears therefore, that k depends only upon the form of the tube but upon the pressure under ch it is workings It is, clearly, of considerable importance the value of k shall be determined for conditions similar those under which the tube is to be finally used. This aty of the value of the coefficient under varying con- of prensure, and the difficulty in any caae of accurately silling it, and the danger of its alteration by objects floating Lflw ftream, makes the use of the Pitot tube as a velocity somewhat uncertain, and it should be used with con* able care. In the handg of Darcy and Bazin it proved an client instrument in the measurement of small velocities in canakj but for the determination of velocities in closed ^Is in which the pressure is greater^ it does not seem so Hit. U2. Ganglxig by a weir. ( When a stream is so small that a barrier or dam can be easily ett?d acrc^s it, or when a large quantity of water is required ^ be gauged in the laboratory, the flow can be determined bjr ^*^*" of a notch or weir. it- 248 HYDRAtTHOS The chaimel as it approachBe the weir should be as far aa poBflible uniform in sectioTij aud it is desirable for accurate gauging*, that the sides of the channel be made verticalj and fhe width equal to the mdth of the weir. The sill should be sharp- edged, and perfectly horizontalj and as high as possible abore th& bed of the stream, and the dowii-sti^ani channel should be vender than the weir to ensure atmospheric pressure under the nappe* The difference in level of the sill and the surface of the water^ before it begins to slope towards the weir, should be ac- curately Tneasured. This is best done by a Boyden hook gauge. 153. Tlie hook gauge. A simple form of hook gauge as made by Gurley is shown in Fig> 15L In a rectangular groove foruied in a frame of wood, three or four feet long, slides another piece of wood S to which is attached a scale graduated in feet and hundredths, similar to a leTel staff* To the lower end of the scale is connected a hook Hj which has a sharp point. At the upper end of the scale is a screw T which passes through a lug, connected to a second sliding piece L. This sliding piece can be clamped to the frame in any position by means of a nut, not shown. The scale can then be moved J either up or down, by means of the milled nut. A vernier V is fixed to the frame by two small screw^s passing through slot holes, which allow for a slight adjustment of the zero. At some point a few feet up-stream from the weir*, the frame can be fixed t-o a post, or better still to the side of a box from which a pipe runs into the stream. The level of the water in the box will thus be the same as the level in the stream. The exact level of the crest of the weir must be obtained by means of a level and a line marked on the box at the same height as the crest. The slider L can be moved, so that the hook point is nearly coincident with the mark, and the final adjustment made by means of the screw T. The vernier can be adjusted so that its zero is coincident with the i^ero of the scale, and the slider again raised until the hook approaches the surface of By means of the screw, the hook is raised slowly^ until, * See Hootton 82. GA^UaiNQ THE FLOW OF WATER 249 J^ 15S. SMgJn'B Hook Gaage. 250 HYDRAULICS fche surface of the water, it causes a distortion of the light refleete from the surface. On moving the hook downwards again vm slightly, the exact surface will be indicated when the distortio disappears^ A more elaborate hook gauge, as used by Bazin for his expef mental work, is shown in Fig. 152. For rough gauging® a po«t can be driven into the bed of tl channel J a few feet above the weir, until the top of the post level with the sill of the weir. Tlie height of the water sutCm Fig. 154. Eecording App«ratms Kent Teaturi Meier. GAUGING THE FLOW OF WATER 251 the top of the post can then be measured by any convenient 154. Ga^gSjig the flow In plpea; Venturi meter* Sucb methods as already dei^ribed are inapplicable to the ?meiit of the flow in pipes, in which it is necessary that ah&II be no discontinuity in the flow, and special meters have i ingiy been devised* For large pipes, the Ventnri meter, Fig. 153^ is largely used in f and is coming into favonr in this country. The theory of the meter hae already been discussed (p, 4i), it waa shown that the discharge is proportional to the square * of the difference H of the head at the throat and the bend in ^pipCi or ^* being a coefficient. For measuring the pressure heads at the two ends of the conei W* G. Kent nses the arrangement shown in Fig. 154. Fig> 154. B«oording drum of the Kent TeDtmi Meter. • Seepage 4a i^^li 252 HYDBAULICS The two pressure tubes from the meter are connected to a U tube consisting of two iron cylinders containing mercury. Upon the surface of the mercury in each cylinder is a float made of iron and vulcanite; these floats rise or fall with the surfaces of the mercmy. Fig. 156. Integrating dram of the Kent Yentori Meter. When no water is passing through the meter, the mercury in tibl two cylinders stands at the same level. When flow takes platt the mercury in the left cylinder rises, and that in the xi^ cylinder is depressed until the difference of level of the sox&oh OAUGINO THE FLOW OF WATER 253 of the mercoiy is equal to — , « being the specific gravity of the mercury and H the difference of pressure head in the two cylinders. The two tabes are equal in diameter, so that the rise in the one is exactly equal to the fall in the other, and the move- ment of either rack is proportional to H. The discharge is proportional to VH, and arrangements are made in the recording apparatus to make the revolutions of the counter proportional to ^/H. To the floats, inside the cylinders, are connected racks, as shown in Fig. 154, gearing with small pinions. Outside the mercury cylinders are two other racks, to each of which vertical motion is given by a pinion fixed to the same spindle as the pinion gearing with the rack in the cylinder. The rack outside the left cylinder has connected to it a light pen carriage, the pen of which Fig. 157. Kent Vestori Meter. Development of Integrating drum. makes a continuous record on the diagram drum shown in Fig- 155. This drum is rotated at a uniform rate by clockwork, and on suitably prepared paper a curve showing the rate of discharge at any instant is thus recorded. The rack outside the right cylinder is connected to a carriage, the function of which is to regulate the rotations of the counter which records tlie total flow. Concentric with the diagram drum shown in Fig. 155, and within it, is a second drum, shown in Fig. 156, wliich also rotates at a uniform rate. Fig. 157 shows this internal drum developed. Hie snrface of the drum below the parabolic curve FEG is recessed. If the right-band carriage is touching the drum on ttie xece'g.^eA. HTDRAUUGS , the coonter gearmg is in sc^on^ but is pat out of a^tioB th* c&rmge tottches the cylinderr on the nuaad portm *boTe FB. Sn^pomB ihie mercury in the right cylindE^r to fill a beight i » o port io a»l to H, then the carriage will be m KXi-wm with the dram^ aa the dmm rofeales, along the line CD, bm i» wiU imly be in operBtion while the CArriage is ia along the kfngth CE, Since FG ts a parabolic carve ik frmctkm of the drnunlerence CE = m , ^fW^ vi being & constant; r for M*w «iiftn]st^«*>«n^«ti Tf cif the Hoats the ct juater k I action far a poricxl pToptff' ae top of the right cytind^frt im^ and in contact wiili i^ ferolntion and no flow i s in itB lowest position tht *inn» and flow is registW 111' recording apparati^cii « ism than 1000 feet fnM 'jp^ larger as the diHtaon of boulto^. Wk tfae carnage m at nised portifii for u Whm tl is al the b tianng the whole of a .«, bo placed ai any oonTOi- tlie meter, tbe conaecaog B tQcreaeed. 155. Deacon's waste-water meter. An ingenious and very diuple meter designed by Mr G. K Deacon prmcipally for detecting the leakage of water from pipes is a^ shc'Wn in Fig, lad. The body of the meter which is made of cast*iron, has fittefl | into it a hollow cone C made of brass. A disc D of the same diameter | 3fc^ the upper end of the cone is suspended in this cone bymeao^cl { a fine winv -— er a pulley not shown ; the other end of the >Wre carries a balance weight. GAUOtKG THK FLOW OF WATER 255 no water paaaes throu^li the meter the disc is drawn to {top of the cone, bat when water is drawn through, the disc is downwards to a position depending upon the quantity of pasmiig* A pencil is attached to the wire, and the motion imc can then be recorded upon a drum mad© to revolve by rork- The posirion of the pencil indicates the rate of flow through the meter at any instant. used as a waste-water meter, it is placed in a by-pass ^ from the main, as shown diagrammatically in Fig, 159, (^ s.v: Bi sy 3 s.\: D Fig, 16a \ulv»-ii A and B are closed and the valve C opened. The €v*ii!naniption in the pipe AD at those hours of the night the actual consumption is very amall, can thus be detei^ i^ and an estimate made as to the probable amount wasted. * If wm^te is taking place, a cai-efiil inspection of the district Hiliod by the main AD may then be made to detect where the Hte is occurring. ^i&B. Kennedy's meter. This is a positive meter in which the volume of water passing ragh the meter is measured by the displacement of a piston ing in the measuring cylinder. long hollow piston P^ Fig. 157, fits loosely in the cylinder hot m made water-tight bj* means of a cylindrical ring of iter which rijUs between the piston and the inside of the ider, the friction being thus reduced to a minimum. At each of the cylinder is a rubber ring, which makes a water-tight ii when the pisrton is forced to either end of the cylinder, so it the rtibber roller has only to make a joint while the piston is to move, water ©titers the meter at A, Fig. 161 i, and for the ition shown of the regulating cock, it Hows down the passage id under the piston* ^Flie piston rises, and as it does so the R turns the pinion 8, and thug the pinion p which is keyed he .same spindle as S* This spindle also carries loosely ighted !ever W, which is moved as the spindle revolves by r of two projecting fingers. As the piston continues to id, the weighted lever is moved by one of the fingers until its -^ -^ --— ^- 250 HYDRAULICS centre of gravity passes the vertical positioHi wlien it suddei falls on to a buffer j and in its motion moves the lever L^ whi tuma the cock. Fig. 161 fe, into a position at right angles to tl r~7 rn ^ R^ibb^SeaHiiff Fig. 160. GArOlKG THE rLOW OF WATER 257 lown. The water now passes from A throngh the paasage C, syUnder, and as the piston descends, Fig. 161 6, k jMilk HYBIUtrUCS Ao wmter that is below it pas(s<^ to the outlet B- The the pinkm H is now re^ersedj and the weight W lilted mgam x^eecbee the vertical podttion, wheti it faik, hxm% opek C into Ibo position shown in the fignre^ and another i Fig. 161 c, oke i« commenced. The rise illations of the pinion p bt^\ to the counter niecVmnisui through the pinions p, 161 a^ in each of which i^ a ratchet and pawl. The eofl thtta rotated in the same direction whichever way the p IBl. dauging the flow of streams by chemical meauL ilr Stromeyer* has very sacceesfully ganged the quantity water supplied to boilers, and also the flow of stretams by mixing with the stream doring a deliuite ime and at a uniform rate, a auwn quantity of a concentrated llation of some chemical, the 91100 of which in watar^ even in very small quantities, can be ]y detected by some sensitive &nt. Suppose for instance iraler ia flowing along a small Two stations at a known i apart are taken^ and the determined which it takes the water to traverse the dis- "HDce between them. At a stated *e, by means of a special ap- "ns— Mr Stromeyer uses the gLL '^ment shown in Fig. 162 Fig, 169. nc acid, say, of known run into the stream at a known rate» at the up. f Navai AnMifCti, 1896 ; Proctidingf inwL C,B*, Vol, CLI. ^ -^^i"^ K J GAUaiKG THE FLOW OF WATER 259 While the acid is being pat into the stream, a small k»ce op-stream from where the acid is introduced samples of rr are taken at definit€r ititervals* At the lower station tg is commenced, at a time, after the insertion of the the apper station is started, equal to that required by the W to trarerse the distance between the stations, and samples then taken, at the same intervalsj as at the upper station, quantity of acid in a known volume of the samples taken he tipper and lower station is then determined by analyaiB* I Ttjlnme V^ of the samples, let the difference in the amount of hfiric acid be equivalent to a volume tv of pure sulphuric k If in a time ^, a volume V of water j has flowed down the ^ and there has been mixed with this a volume v of pure ric acid, then, H the acid has mixed uniformly Anth the ^ the ratio of the quantity of water flowing down the stream BqBanlity of acid put into the stream, is the same as the ^fe the volnme of the sample tested to the difference of the me of the acid in the samples at the two stations, or Pf Stromeyer considers that the flow in the largest rivers can iterxnined by this method within one per cent, of its true value. d large streams special precautions have to be taken in the chemical solution into the water, to ensure a uniform and also special precautions must be adopted in taking [okliar important information upon this interesting method lorin^ the flow of water the reader is referred to the two uted. above. iratos for accurately gauging the flow of the solution in Fig, 162* The chemical solution is delivered into lindrieal tank by means of a pipe L On the surface of the 1 floats a cork which carries a siphon pipe SS, and a balance keep the cork horizontal. After the flow has been the head h above the orifice is clearly maintained irhaterer the level of the surface of the solution in the \1—1 260 HYDBAtJLlCS EXAMPLES, (1) Same obaerrfttions aro made by towing a coireot meter, witli ifa4 following resnlta:— Speed in ft per lec. 1 5 Find an eqii^tion far the m^ter^ (2) Deecribe two methods of gi in vertical and horizontal planes; , obtained. If the croBs section of a met is discharge may bo oaUmated by ob alone. (8) The following observations dl were made ill connection with a weir Head m feet ... ... 01 01 Discharge in cubic feet per sec per foot width ,., 017 Be?i. of meter per min, 80 560 bulge river, from observatioiii ^te the nature of the resultB explain how the apptosiiii«be n of the mid-anzface f^pi^ nd the oorresponding disdiai^H et wide. 8-0 ! 3-5 I 4i) ■1 19-82 25 18*08 17-03 21-54 126^4 1-2 1», Assuming the law connecting the hoad h with the discharge Q as Q = mh . A", find m and n. ^Plot log&ritlimH of Q and h,} (4) The following values of Q and h were obtained for a shacp-ed^ weir 6'58 feet longT without lateral contraction. Find the coefl&djent d discharge at veltious heads. lead A ... •1 j-4 -6 -8 1*0 1*5 2*0 1 25 30 3-5 4-0 €-5 W H 1 per foot- 1 tength ... •17 -87 j 1-56 237 sas ei 9-32 15-08 17-03 3154 26-4 31*62 37-09 43^81 (5) The following values of the head over a weir 10 feet long were obtained at 5 minutes Intervals, Head m feet 35 -36 37 37 38 80 40 *41 *42 -40 *8& "41 Taking tlie coefficient of discharge C as 8 36, find the discharge m one hour. (6) A Pitot tube was calibrated by moving it through still water in & tank, the tube being tixed to an arm wliidi was made to revolve at constant speed about a fixed centre. The following were the velocities of the tube and the heads measured in inches of water. Velocities ft. per sec. 1*432 Head in inches of water -448 1-788 •663 2-275 I 2-718 1-02 1 1-69 8-286 2-07 8-878 2-88 4-988 I 5*584 5-40 6-97 6143 8-51 Determine the coefficient of the tube. For examples on Venturi meters see Chapter U. CHAPTER VIIL IMPACT OF WATER ON VANES. 168. Definition of a vector. A right line AS, considered as haying not only length, bat also direction, and sense, is said to be a vector*. The initial point A is said to be the origin. It is important that the difference between sense and direction should be clearly recognised. Suppose for example, from any point A, a line AB of definite length is drawn in a northerly direction, then the direction of the line is either from south to north or north to soath, but the sense of the vector is definite, and is from A to B, that is from sooth to north. The vector AB is equal in magnitude to the vector BA, but they are of opposite sign or, AB = -BA. The sense of the vector is indicated by an arrow, as on AB, Fig. 163. Any quantity which has magnitude, direction, and sense, may be represented by a vector. D For example, a body is moving with a given velocity in a given direction, sense being now implied. Then a line AB drawn parallel to the direction of motion, and on some scale equal in * Sir W. Hamilton, QuaUmioru, 262 HYDRAULICS length to the velocity of the body is the velocity vector; the seise is from A to B< 159. ♦ Stun of two vectora. If a and ^, Fig. 163, are two vectors the euoi of these vecton is found, by drawing the vectors, so that the beginniiig of j8 ii il j the end of a, and joining the beginning of a to the end of ftj Thus y is the vector sum of a and fi. 160. ReanltaJEit of two When a body has velocities^ the resultant direction is the vector si may be stated in a way problems to be hereai moving with a given i velocity is inipressed op vector sum 6f the initial Teiocitias* )n it at any instant two 16 body in magnitude nni impressed velociti**^, TIlei iefiaitely applicable to the as follows. If a body ii en direction, and a ieoomd he resnltant Telocity is the velocities. Example* Bfippos^ ^ pAri with a velocitj V^, rolative U If the Tune is at rest, the |»i ^u*^ ^" at A with this velocity. If the ^ane iti timd« to move m the ^±44^iiiOEL EF with a velocity r, and l3ot particle haa still a Vf^locitj V^ r«lalive to the vane, and remams in contact with tbs vane until th« point A i» reach^, the vdocity of the water as it learea the ^n« it A, will be the ^^otor (lum 7 of a and ^, i.e^ of V,. and V, or h equal to i^, 161. DtfiTerencQ of two vectors. The difference of two vectors a and is f onnd by drawing bo^ vectors from a common origin A, and joining the end of ^ to thei end of a. Thus, CB, Fig, 165, is the difference of the two vectora a and A or y = a-ft and BC is equal to i^-of, or^-a^-7. Fig. 166. 162. Absolute yelooity. By the terms " absolute velocity " or " velocity " without the adjective, as used in this chapter, it should be clearly understood, is meant the velocity of the moving water relative to the earth, or to the fixed part of any machine in which the water is moving. Henrioi and Tamer, Veeton tmd Baton, IMPACT OF WATER ON VANES 263 To avoid repetition of the word absolute, the adjective is ■eqaently dropped and " velocity " only is used. 163. When a body is moving with a velocity U, Fig. 166, in ny direction, and has its velocity changed to U' in any other iirection, by an impressed force, the change in velocity, or the 'elocity that is impressed on the body, is the vector difference of he final and the initial velocities. It AB is U, and AC, U', the mpressed velocity is BC. By Newton's second law of motion, the resultant impressed Sorce is in the direction of the change of velocity, and if W is the wmght of the body in pounds and t is the time taken to change ihe velocity, the magnitude of the impressed force is W P = -T (change of velocity) lbs. gt This may be stated more generally as follows. The rate of change of momentum, in any direction, is equal to ihe impressed force in that direction, or P= — .;^lbs. g at In hydranlic machine problems, it is generally only necessary o consider the change of momentum of the mass of water that fccts open the machine per second. W in the above equation then lecomes the weight of water per second, and t being one second, W P = — (change of velocity). 164. Impulse of water on vanes. It follows that when water strikes a vane which is either noving or at rest, and has its velocity changed, either in magni- nde or direction, pressure is exerted on the vane. As an example, suppose in one second a mass of water, weighing iV lbs, and moving with a velocity U feet per second, strikes a ixed vane AD, and let it glide upon the vane at A, Fig. 167, and eave at D in a direction at right angles to its original direction >f motion. The velocity of the water is altered in direction but lot in magnitude, the original velocity being changed to a velocity kt right angles to it by the impressed force the vane exerts upon he water. The change of velocity in the direction AC is, therefore, W qual to U, and the change of momentum per second is — .U oot lbs. 264 HYDRAULICS Since W lbs. of water strike the vane per second, the pressi P, acting in the direction C A, required to hold the vane in positi is, therefore, W Pig. 167. Again, the vane has impressed upon the water a velocity U the direction DF which it originally did not possess. The pressure Pi in the direction DF is, therefore, W Pi = P = — .U. 9 The resultant reaction of the vane in magnitude and directi is, therefore, R the resultant of P and Pi. This resultant force could have been found at once by finding the resultant change in velocity. Set out clc^ Fig. 168, equal to the initial velocity in magnitude and direction, and ad equal to the final velocity. The change in velocity is the vector difference cd, or cd is the velocity that must be impressed on a particle of water to change its velocity from ac to od. ^«- ^^ The impressed velocity cd is V = VU* + TP, and the to) impressed force is W- W 9 9 n/2W 9 IMPACT OF WATEB ON VANES S0S It at once folIow*s, that if a jnt of water strikes a fixed plan€3 [perpendicularly^ with a velocitjr U, and glides along the plane, the [normal proas ure on the plane is^ ^ U. A ttrtuD of water 1 eq, fcN?t in seetion ft&d having a Telocity of ^inft par Mooad glMea on lo a fixed vane ia a dUreolion makiDg an angle of iifMft wUh a gifen direction AB. Tbm vaae tatna tbe jet ihroogh an angle ot §C degrees, Whid Iha Pfcwmm cm Ihe Tane io the direction parsiLlsl lo AB and the reittltant wmom tibe taaa. Fle» 167. A€ ia the oiiginal direction of the jet and DF the tnal direction. lae ctntply ehangea the directioti of the water* the ^nal velocity being oqnal triaagle it ar4, Fif?^ ISS^ ae and ad being equal, of vel<>£i^ In magnitude and direction is cd^ the vector difference of ; roaolTing cd parallel to, and perp^ndienlar to AB^ ce Is the ohangie of pafsOlel to aB. eallog off C9 and oalling it v^ » the for^e to be applied along BA to keep the •I rmt ia, Bat c4=j2,l0 C4 = cd oo» 1 5^ Tli0 premsum aomml to AB ii^ = 2fi4lbfl. ■ 9 Tbe rvsoltaat ia B= m,m4 , 1007^2.62 4 32*2 3SS 274 Ibe. IBS. RelaUye yeloclty. Betfore going on to the consideration of moving vanes it ^^Bfi^tit that the student should have clear ideaa as to what iS f/fmk' by relative i^elocity. I A rrain is said to have a velocity of dxty miles an honr when| iti . ' Tied in a Btraight line at a constant velocity for one ^^j-^ Ad travel sixty miles. What is meant is that the train ^ fir^ying «^ sixty miles an hour relative to the earth. ^^ t^wo trains run on parallel lines in the same direction, oi .ty And the other at forty miles an hour, they have a !• Other of 20 miles an hour. If they move ^^^^^^,_ [hey havo a relative velocity of 100 miles boor. If one of the trains T is travelling in the direction AB, ^ 169^ and th^ other T, in the direction AC, and it be supposed fnt the line© on which they are travelling cross each ottei at A^ Li n ^ n(^^ 266 HYDRAULICS and the trains are at any instant over each other at A, at the end of one minute the two trains will be at B and C respectively, at distances of one mile and two-thirds of a mile from A. Relatively to the train T moving along AB, the train Ti moving along AC has, therefore, a velocity equal to BC, in magnitude and direction, and relatively to the train Ti the train T has a velocity equal to CB. But AB and AC may be taken as the vectors of the two velocities, and BC is the vector difference % Fig. 169. of AC and AB, that is, the velocity of Ti relative to T is the vector difference of AC and AB. 166. Definition of relative velocity as a vector. If two bodies A and B are moving with given velocities v and t?i in given directions, the relative velocity of A to B is the vector difference of the velocities v and Vi. Thus when a stream of water strikes a moving vane the magnitude and direction of the relative velocity of the water and the vane is the vector difference of the velocity of the water and the edge of the vane where the water meets it. 167. To find the pressure on a moving vane, and the rate of doing work. A jet of water having a velocity U strikes a flat vane, tie plane of which is perpendicular to the direction of the jet, and which is moving in the same direction as the jet with a velocitv ft I Fig. 170. U^ Fig. 171. The relative velocity of the water and the vane is U— «, the vector difference of U and v. Fig. 170. If the water as it strikei the vane is supposed to glide along it as in Fig. 171, it will do IMPACT OF WATER ON VANES 267 80 with a velocity equal to (U— v), and as it moves with the vane it will still have a velocity v in the direction of motion of the vazie. Instead of the water gliding along the vane, the velocity U — V may be destroyed by eddy motions, but the water will still liave a velocity v in the direction of the vane. The change in nelocity in the direction of motion is, therefore, the relative relocity U-r, Fig. 170. For every pound of water striking the vane, the horizontal XJ — t? ihan^e in momentum is , and this equals the norm^/l pressure ^ on the vane, per pound of water striking the vane. The work done per second per pound is Pt? = . V foot lbs. 9 The original kinetic energy of the jet per pound of water . XP triking the vane is s— > and the efficiency of the vane is, therefore, "liicli is a maximum when v is ^U, and e = J. An application of icH vanes is illustrated in Fig. 185, page 292. Nozzle and single vane. Let the water striking a vane issue XMn a nozzle of area a, and suppose that there is only one vane. "Let the vane at a given instant be supposed at A, Fig. 172. At le end of one second the front of the jet, if perfectly free to love^ would have arrived at B and the vane at C. Of the water lat has issued from the jet, therefore, only the quantity BC will ave hit the vane. Fig. 172. The discbarge from the nozzle is W = 62-4.a.U, id the 'Weight that hits the vane per second is W.(U- t>) u The chftn^r^ of momentum per second is W(U-t))' g U ' HTDKAUUCa froiQ ilie posxkil ttitd hsm no ^ wmmm^ If them wre m €tiket, tlie wholB of tlie w^r mnti Urn work done is IP • ^ w»ter wheel, witb imdial bbdoi^l m P%- 19§^ cuiBoC IxftTe an effici^^cj^ of more than 50 per cent IG6- Impact of wftter on & T&ne when the dlxectioni € wrf^"^ of the Ymne and jet are not parallel. L«« r be tke ^ckiciix erf » j^ ol wuier and AB its directkitf J> ne fif. tm Ltfi «bt» f<48« A,<ii Ae na^ AC be morbg witli a Tielodtrn tbe i tti i iTT ^ velocity Vr of tbe water and the Taoe at A is DB tfe tnu^le DAB it is aeeti tbAt.^ tbe rector sunt of tht of the Tane and tb«^ r^latire f^kMriljr of the jet ai>d d4 «^W ii tqasl to tlie T^k^^it^ ol tbe jec; for clearl|^ IT is tbe t&^ aaB vi r at»l Tr. If tbe dbvetiiJii vi tbe tip ol tbe Tmne at A is made panUel td DB tbe water will gbde on to tbe T%ne in exaetlj^ the same vsf J IMPACT OF WATER ON VANIS 2G9 it were mt rest, and the water were moving in tha direction L This h the condition that no eDerg7 shall be lost by shock, "Wli^n the water leaves the vane, the relative velocity of the and the vBne must be parallel to the diroction of the ^mit to the vane at the point where it leaves, and it is equal to vector difference of the absolute velocity of the water, and Tane. Or the absolute velocity with which the water leaves mape is the vect-or sum of the velocity of the tip of the vane I the relative velocity of the water to the vane* l^t CO be the direction of the tangent to the vane at C. Let i be ri p the velocity of C in magnitude and direction, and let CF ihe absoliite velocity Ui with which the water leaves the vane. Draw EF pai-aUel to C(t to meet the direction CF in Fj then \ relative velocity of the water and the vane is EF, and the ocity with which the water leaves the vane is equal to CF. If Vi and the direction CG are given, and the direction in which I water leaves the vane is given^ the triangle CEF can be IWB, and CF determined. If on the other hand Vj is given, and the relative velocity Vr is in magnitude and direction, CF can be found by measuring along EF the known relative velocity iv and joining CF. If t?j and Ui are given, the direction of the tangent to the vane then, a« at inlet, the vector difference of Ui and t^i. It will be seen that when the water either strikes or leaves the ne, the relative velocity of the water and the vane m the vector erence of the velocity of the water and the vane, and the actual Oi3ty of the water as it leaves the vane is the vector sum of the ocity of the van© and the relative velocity of the water and rane. I I I ■ I I I I , The difectiott of tbe tip of the vivne at the out^r circumferrence of a witli T&fiai, m*ke« aa Angle of 165 degiee:^ with the direction of motioa _ kip of the TADe. 7li« Tclodty of the tip at the outer circtimf«rf tjc«; in 82 feet p&r F^cnzid^ «mier leaves the ^h&e\ m auch a direct ii>n and with such a Velocity that the ' ■enl i^ 13 fmx per second- wbmAute velocLty of tbe water in direetion and magnitude and the of the wat«r and the wheeL ^ m tbe tri*iigl« of velocities, set out A 15 uqual to B2 feet, and make tha^ ^ eQiiaJI to IS degneefi. BQ Ih then parallei to the tip of the vane. V irallel lo ABp and at a diittatjce from It eqn&l to 1^ fe«t &aA in C- ,\C t* tha vector Aum of AB and UC, and ia the absolate velocitj of the directiaa and magnttTide, Ihgoiiometrically AC^ - (82 - 13 oot isy + 133 = 33^5*HhlB* and AC = air? ft per seo. i 270 HYDRAULICS 169. Conditions wMch the vanes of bydraalio naadiini should satiflfy* In all properly deBigned hydraulic machines, sneh as tnrbti veater wheels, and centrifugal pumps, in which water flowing id a definite direction impinges on mo^^ng vanes, the relative ^ of the water and the vanes should be parallel to the din- the vaoes at the point of contact. If not, the water breaks into eddies aa it moves on to the vanes and energy is lost. Again, if in such machines the water is required to leave the Tanes with a given velocity in magnitude and direction, it is only neceaaary to make the tip of the vane parallel to the vect<jr difference of the given velocity with which the water is to leav^? the vane and the velocity of the tip of the vane, ExampU (1|, A jet of water, Fig. 174, movea in a direction AB making as as . • of 30 def^ees with the direct! od of motion AC of a vane moving in tlie atmosi/t The jet has a velocity of 30 ft. per second and the vaue of lo ft. per si^ood. Tt» umiI (ii} the direction ^f the vnoe at A eo tbal tlie water may cnttir wiChoQl aliook; {h} tk* direction of the taugeni to tiie vane where the water leaves it, so that the abtolulf velocity of th€ water when it teavei^ the vane is in a direction perpendiotiJAr to ^Cfi (r) the pressure on the vane and the work done par second per poand of ' striking the vane, Frictioii U neglected. Fig, 174. The relative Telocity T^ of Ihe water and the vatje %i A ie CB, and for no i the vane at A must be parallel to CB. Sinee there is no friction, the relative velocity V,, of the watar ajid ibe hm] cannot alter, and tharefore, the triangte of velocities at exit is ACD or FA*C*. The point D is fouodt by taking G as centre and CB as radiua uid sitwoff \ are ED to cut the known dlreotion AD in D. The total change of velocity of the jet i« the vector difference DB of the L^ltkl ] and final velocities, and the change of velocity in the direction of motion u E£^ 1 Calling this velocity Y^ the presaare exerted npon ihe vane in the direolkc <i motion In V — lbs, per lb. of water striking the vane^ Tlie work done p^ lb. is, therefore^ — ft. lbs, asd the efEdencj, sinc« Ibtl* ii | no loss by friction, or shock, h \v _ %Vv •'25 IMPACT OP WATER ON VANES 271 The change in the kinetic energy of the Jet i$ equal to the work done by the jet. The kinetio energy per lb. of the original jet is ^ and the final kinetic energy is El! The work done is, therefore, ^ — o^ ^^- ^^^* ^^^ ^^® efficiency is '2g It can at once be seen from the geometry of the figure that Vr __ Ua Ui« For AB«=AC«+CB«+2AC.CG, knd aince CD=CB and CD^=AC« + AD», lierefore. AB«-AD*=2AC(AC + CG) = 2i;V. Bat AB«-AD«=U«-Uj«, herefore, 2g g ' If the water instead of leaving the vane in a direction perpendicular to t\ leaves t with a velocity Uj having a component V^ parallel to v, the work done ou the -ane per pound of water is 9 If Ui be drawn on the figure it will be seen that the change of velocity in the V -V Lirection of motion is now (V - V^), the impressed force per pound is ^ , and / V — V \ be work done is, therefore, ( j ^'i '^' ^^^' P^^ pound. A!» before, the work done on the vane is the loss of kinetic energy of the jet, and .herefore, (V-V,)v,^U»-Ui« 9 'k The work done on the vane per pound of water for any given value of Uj , is, Jierefore, independent of the direction of U^ . Example (2). A series of vanes such as AB, Fig. 175, are fixed to a (turbine) whmel which revolves about a fixed centre C, with an angular velocity u. The radius of B is B and of A, r. Within the wheel are a number of guide na«rr(p"i through which water is directed with a velocity U, at a definite inchnatiou f with the tangent to the wheel. The air is supposed to have free accesn to the wbe«L To draw the triangles of velocity, at iulet and outlet, and to find the directions yi tlM tips of the vanes, so that the water moves on to the vanes without shock and Learee the wheel with a given velocity Uj. Friction neglected. Aa in the last example the velocity relative to the vane must remain constant, ^a%A therefore, V^ and v^ are equal, but v and v^ are unequal. The tangent AH to the vane at A makes an angle with the tangent AD to the irheel, so (hat CD makes an angle with AD. The triangle of velocities ACD at nlgt 18, therefore, as shown in the figure and does not need explanation. To draw the triangle of velocities at exit, set out BQ equal to t'j and perpen- lieolar to the radius BC, and with B and G as centres, describe circles with U, and . which is equal to V^— as radii respectively, intersecting in E. Then G£ is l^rmliel to the tangent to the vane at B. 272 HYDRAULICS If there is a loss of head, /t/, by friction, as the water moves over the vane tl Vf IB less than V^, if h/ is known, it oan be found from 29 2g "^^ (See Impulse turbines.) Work done on the wheel. Neglecting Motion etc. the work done per poond water passing through the wheel, since the pressure is constant, bong equal to ' atmospheric pressure, is the loss of kinetic energy of the water, and is 2g The work done on the wheel can also be found from the consideration of 1 change of the angular momentum of the water passing through the whed. Befi going on however to determine the work per pound by this method, tiie notati that has been used is summarised and sevoral important principles oonsideredL ^ - ^ ft. lbs. Notation used in connection with vaneSy tturbines and centrifug pwm/pa. Let U be the velocity with which the water approach the vane, Fig. 175, and v the velocity, perpendicular to the radii AC, of the edge A of the vane at which water enters the wheel. Let V be the component of U in the direction of t?, u the component of U perpendicular to t?, Vr the relative velocity of the water and vane at A, t?i the velocity, perpendicular to BC, of the edge B of the vw at which water leaves the wheel, ^ Ui the velocity with which the water leaves the wheel, Vi the component of Ui in the direction of t^i, IMPACT OF WATER OV VANES 273 Ui the component of Ui perpendicular to r,, or along BC, tv the relative velocity of the water and the vane at B. ITelociiies of whirl. The component velocities V and Vi are called the velocities of whirl at inlet and outlet respectively. This temi will frequently be used in the following chapters. 170. Definition of angular momentum. If a weight of W pounds is moving with a velocity U, Figs. 175 and 176, in a given direction, the perpendicular distance of which is S feet from a fixed centre C, the angular momentum of W is W — . U . S pounds feet. 171. Change of angular momentiun. If after a small time t the mass is moving with a velocity Ui in a direction, which is at a perpendicular distance Si from C, the W angular momentum is now — UiSi; the change of angular momentum in time t is -J (US -U, SO; and the rate of change of angular momentum is ^(US-UaSO. Fig. 176. Fig. 177. 172. Two important principles. (1) Work done by a coti/pUy or turning moment. When a body is turned through an angle a measured in radians, under the action of a constant turning moment, or couple, of T pounds feet, the -work done is Ta foot pounds. If the body is rotating with an angular velocity w radians per second, the rate of doing work is Tw foot pounds per second, and the horse-power is g^. I^ H. \^ 274 HYDRAULICS Suppose a body rotates about a fixed centre C, Fig. 177, and a force P lbs. acts on the* body, the perpendicular distance from C to the direction of P being S. The moment of P about C is T = P.S. If the body turns through an angle ca in one second, the distance moved through by the force P is « • S, and the work done by P in foot pounds is Pa>S=T«. And since one horse-power is equivalent to 33,000 foot pounds per minute or 550 foot pounds per second the horse-power is (2) The rate of change of angular momentv/m of a bodf rotating about a fixed centre is equal to the couple acting wpm the body. Suppose a weight of W pounds is moving at any instant with a velocity U, Fig. 176, the perpendicular distance of which from a fixed centre C is S, and that a couple is exerted upon W so as to change its velocity from U to Ui in magnitude and direction. The reader may be helped by assuming the velocity U is changed to Ui by a wheel such as that shown in Fig. 175. Suppose now at the point A the velocity Ui is destroyed in a time dt^ then a force will be exerted at the point A equal to P-W U g 'cV and the moment of this force about C is P . S. At the end of the time d^, let the weight W leave the wheel with a velocity Ui. During this time dt the velocity Ui might have been given to the moving body by a force p_WU, acting at the radius Si. The moment of Pi is Pi Si ; and therefore if the body has been acting on a wheel. Fig. 175, the reaction of the wheel thus exerting the couple upon the body, or a couple has been exerted upon it in any other way, the couple required to change the velocity of W from U to Ui is T = J^(US-U.S,) (1). Let the wheel of Fig. 175, or the couple which is acting npoA the body, have an angular velocity w. DCPAGT OF WATER ON VANES 275 In a time dt the angle moved through by the couple is cud^, and therefore the work done in time dt is T.a)3^ = — co(US-UiSO (2). Suppose now W is the weight of water in pounds per second iprhich strikes the vanes of a moving wheel of any form, and this -water has its velocity changed from U to Ui, then by making dt in either equation (1) or (2) equal to unity, the work done per second is Ta>= — co(US-U,S,), and the work done per second per pound of water entering the wheel is ^(US-UiSx). This result, as will be seen later (page 337), is entirely inde- pendent of the change of pressure as the water passes through the wheely or of the direction in which the water passes. 173. Work done on a aeries of vanes fixed to a wheel expreased in terms of the velocities of whirl of the water entering and leaving the wheel. Outvard flaw ta/rbine. If water enters a whgelat the inner circumference, as in Pig. 175, the flow is said to be outward. On reference to the figure it is seen that since r is perpendicular to V, and S to U, therefore r_U And for a similar reason R_Ui A^ain the angular velocity of the wheel " = r = R' therefore the work done per second is And the work done per pound of flow is g g Inward fl^ow twrhine. If the water enters at the outer cir- cunrference of a wheel with a velocity of whirl V, and leaves at the inner circumference with a velocity of whirl Vi, tYve veVoaVdfe^ 276 HYDRAULICS of the inlet and outlet tips of the vanes being v and ri respectively the work done on the wheel is still Vr ViTi V 9 The flow in this case is said to be inward. Parallel flour or axial fl^vc turbine. If vanes, such as those shown in Fig. 174, are fixed to a wheel, the flow is parallel to tiie axis of the wheel, and is said to be axial. For any given radius of the wheel, Vi is equal to Vy and the work done per pound is which agrees with the result already found on page 271. 174. Curved vanes. Pelton wheel. Let a series of cups, similar to Figs. 178 and 179, be moving with a velocity t?, and a stream with a greater velocity U in the same direction. The relative velocity is Vr=(U-r). Neglecting friction, the relative velocity Vr will remain con- stant, and the water will, therefore, leave the cup at the point B with a velocity, Vr, relative to the cup. If the tip of the cup at B, Fig. 178, makes an angle * with the direction of v, the absolute velocity with which the water leaves the cup will be the vector sum of v and Vr, jftnd is therefore Ui. The work done on the cups is then IP U^« 2g 2g IMPACT OF WATER ON VANES 277 per lb. of water, and the efficiency is For TJi, the value Ui = 'J{v - (U - v) cos e\' + (U - vy sin ^r-* can be substituted, and the efficiency thus determined in terms of r, U and 0. JPelton wheel (mps. If is zero, as in Fig. 178, and U-v is equal to t?, or XJ is twice v, Ui clearly becomes zero, and the water drops away from the cup, under the action of gravity, without possessing velocity in the direction of motion. The whole of the kinetic energy of the jet is thus absorbed and the theoretical efficiency of the cups is unity. TTie work done determined from consideration of the cltange of mofnenium. The component" of Ui, Fig. 178, in the direction of motion, is 17 — (U - V) COS d, and the change of momentum per pound of water striking the vanes is, therefore, U~t?-f (U - v)jDOS 9 The work done per lb. is t? {U - 1? + (U - 1?) coa^} 9 and the eflSciency is ^ 2v{U-v + (U-t7)cos 0} W When is 0, cosO is unity, and e=- u— > which is a maximum, and equal to unity, when v is -^^ . 175. Force tending to move a vessel firom which water is iflsnintf through an orifice. When water issues from a vertical orifice of area a sq. feet, in the side of a vessel at rest, in which the surface of the water is maintained at a height h feet above the centre of the OT\&ce^t\v^ 278 HYDRAULICS pressure on the orifice, or the force tending to move the vessel in the opposite direction to the movement of the water, is F=2w.a,h\hs., w being the weight of a cubic foot of water in pounds. The vessel being at rest, the velocity with which the water leaves the orifice, neglecting friction, is t7= J2ghy and the quantity discharged per second in cubic feet is The momentum given to the water per second is 9 = 2w ,a.h. But the momentum given to the water per second is equal to the impressed force, and therefore the force tending to move the vessel is F = 2w.a.h, or is equal to twice the pressure that would be exerted upon a plate covering the orifice. When a fireman holds the nozzle of a hose-pipe through which water is issuing with a velocity r, there is, therefore, a pressure on his hand equal to 2wav^ _ wav^ "W" 9 • If the vessel has a velocity V backwards, the velocity U of the water relative to the earth is and the pressure exerted upon the vessel is F=^--^'^lbs. 9 The work done per second is F . V = ^^ foot lbs., 9 or = — ^ foot lbs. 9 per lb. of flow from the nozzle. The efficiency is e = — ^—r — - _ 2V(t?-V) which is a maximum, when .i? = 2V and e = J. IMPACT OF WATER ON VANES 279 176. The proptdflion of Bhips by water jets. A method of proi)eUing ships by means of jets of water issuing from orifices at the back of the ship, has been used with some sxiccess, and is still employed to a very limited extent, for the propalsion of lifeboats. Wat^r is taken by pumjw carried by the ship from that surrounding the vessel, and is forced through the orifices. Let tr be the velocity of the water issuing from the orifice relative to the ship, and V the velocity of the ship. Then ^ is the hea.d h forcing water from the ship, and the available energy per pound of water leaving the ship is h foot pounds. The whole of this energy need not, however, be given to the water by the pumps. Ima^ne the ship to be moving through the water and having a pip^e with an open end at the front of the ship. The water in front of the ship being at rest, water will enter the pipe with a velocity V relative to the ship, and having a kinetic energy ^y per pound. K friction and other losses are neglected, the work that the pomps will have to do upon each pound of water to eject it at the back with a velocity v is, clearly, 2g 2g' As in the previous example, the velocity of the water issuing from the nozzles relative to the water behind the ship is v - Y, v-Y and the change of momentum per pound is, therefore, - . If a is the area of the nozzles the propelling force on the ship is and the work done is 9 9 The efficiency is the work done on the ship divided by the work done by the engines, which equals wav (5 — n " ) ^"*^' therefore, .2V(i;-V) e = 2V ■« + ¥' 280 HYDBAULICS which can be made ae near unity as is desired hy makiiig r and V approxiraate to equality. Bat for a given area a of the orifices^ and velocity t\ the nearer V approximates to V the less the propelling fore© F becomes, and the size of ship that can be driven at a g:iven velocity V for ihn given area a of the orifices diminishes. If ris2Y, e-|. EXAMPLES. (1) Ten cubic feet of water per second ore diiicharged ttom & ^tationtfj jet* llie eectional area of which is 1 square foot. The water irapiagea not* mally on a fiat surface, moring in the directiDii of the jet w^tb a velodiy of % feet per second. Find the pressure on the plane in lbB,« and the ^mak done on tlie plane in horse -iK>wer. (2) A jet of water deUyering 100 gallons per Becond with a Telocity d 20 feet per second iiQpiuge& perpendicularly on a wall. Find the pre^itre on the w^all* (8) A jet delivers 160 cubic feet of water per minute at a velocity of I feet per Heeond and strikes a plane perpendicularly. Find the preestiM tlie plane^(l) when it is at rest; {%} when it is moviug at 5 feet per second In tlie direction of the jet. En the latter case find the work c]oo« per second in driving the plane. (4) A fire*engine hose^ 8 inches bore, discharges water at a velocity of 100 feet per seccnd. Supposing the jet directed normally to tJie side d a binding, find the presaure. (6) Water iBsues horizontally from a fixed thin -edged orifice^ indbai sqnare, under a head of 25 feet. Tlie jet impinges normally on a pbii^ moving in tlie same direction at 10 feet per second. Find tlie pres^suns <A the plane in lbs., and the work dome in horse-power. Take the coefficicBl of discharge as *64 and the coefficient of velocity as '97. (6 1 A jet and a plane surface move in directions inclined at 30% witb velocities of 30 feet and 10 feet per second resjiectively. \Mm% i» tlw relative velocity of the jet and surface ? (7) Let AB and BC be two lines inclined at 80% A jet of water moffli in the direction AB, w^th a velocity of 25 feet per second, and a »eric» o( vanes move in the direction CB with a velocity of 15 feet per second. Fm^ the form of the vane so that the w^ater may come on to it tangentially, afiJ leave it in the direction ED, perpendicular to CB. Supposing that the jet is 1 foot wide and 1 inch thick before impinging find the effort of the jet on the vanes. (8) A enrved plate is mounted on a slide so that tlie plate i move along tlie shde. It receives a jet of water at an angle of 3i • normal to the direction of sliding, and the jet leaves tlie plate at m iagi» IMPACT OF WATEE ON VANES 281 "With the «eiiia noimial. Find the force which must be applied to be to the direction of slidiag to hold it at rest^ and also tii© nt^moL !e on the slide. Quantity of water flawing is 500 Iha. per minute ■vekxaty of B5 feet per second. A fixed Fane receives a jet of water at an angle of 120^ with a a AB- Find what angle the jet must be turned tlirough in order I pffcasore on the vane in the direction AB may be 40 Ibs.^ when the \ water m 45 Ibs^. per second at a velocity of 30 feet per second. Water under a head of 60 feet m discharged through a pipe 6 indies t and 150 feet long, and then tlirough a nois^le, the area of which nth the area of the pipe, all losses but the friciioti of the piper determine the preaaure ed plate placed in front of the nozzle. A jet of watier 4 tnehee diameter impinges on a fixed cone, the Adding with that of the jet, and the apex angle being BO degEeeSi llocity of 10 feet per second. Pind the pressure tending to move the tJie direction of its asds. A veeael containing water and having in one of it» vertical sides orMoe 1 inch diameter^ which at first is phigged up, in ided in such a way tliat any diiiplacing force can be accurately ^ed. On the removal of the plug, the horii^ontai force rec}uired to & venal in place, applied opposite to the orifice^ is 3*6 lbs. By the measimng tank the discharge m found to be 31 gallons (>er minute^ tol of the water in the vessel being maintained at a constant height M wihove the orifica Determine the coefficients of velc>cityi con^ X and di»cliaige. A train carrying a Eamsbottom's scoop for taking water into the 18 nmiiiiig at 24 miles an hour. What is the greatest height at Ibe 8000|k will deliver tlie water ? A locomotive going at 40 miles an hour scoops up water from a The tank is 8 feet above the mouth of the scoop* and tlie delivery M ftJi area of 50 square inches. If half the available head is wasted msce, find tlie velocity at wliich the water is delivered into the tank, number of tons Ufted in a trench 500 yards long. What, utider ions* is the increased resistance; and what is the minimum at which the tank can be filled ? Lond. Un. 1906. A stream deliyeiing 3000 gallons of water per minute with a <d 40 feet per second, by impinging on vanes is caused freely to Ihroogh an angle of 10'", the velc^city being dimmialied to M feet ond. Delejmine the pressure on tlie vanes due to impact. If the m moving in the direction of that pressure, find their velocity and the luefni hor»e-power* Lond. Un. 1906, Wsttif flows from a 2-inch pipe, without contraction, at 45 feet per imtne the maadmuni work done on a machine carrying moving I^AjoUowiDg emm and tlie respective efficiencieti;^ 282 HYDRAULICS (a) When the water impinges on a single flat plate at right angles and leaves tangentiaUy. (b) Similar to (a) but a large number of equidistant flat plates an interposed in the path of the jet. (c) When the water glides on and off a single semi-cylindrical cup. {(i) When a large number of cups are used as in a Pelton wheeL (17) In hydraulic mining, a jet 6 inches in diameter, discharged unte a hesid of 400 feet, is delivered horizontally against a vertical cliff ho$B Find the pressure on the face. What is the horse-power delivered by tin jet? (18) If the action on a Pelton wheel is equivalent to that ol a jet on a series of hemispherical cups, find the efficiency when the speed of the wheel is five-eighths of the speed of the jet. (19) If in the last question the jet velocity is 50 feet per second, and the jet area 0*15 square foot, find the horse-power of the wheeL (20) A ship has jet orifices 8 square feet in aggregate area, and dii- charges through the jets 100 cubic feet of water per second. The speed «( the ship is 15 feet per second. Find the propelling force of the jets, te efficiency of the propeller, and, neglecting friction, the horse-power of fli engines. CHAPTER IX. WATER WHEELS AND TURBINES. Water ^wKeels can be divided into two classes as follows. (a) Wheels upon which the water does work partly by mpolae but almost entirely by weight, the velocity of the water rfien it strikes the wheel being small. There are two types of bis class of wheel, Overshot Wheels, Figs. 180 and 181, and Ireast Wheels, Figs. 182 and 184. (6) Wheels on which the water acts by impulse as when he wheel utilises the kinetic energy of a stream, or if a head h is available the whole of the head is converted into velocity before lie water comes in contact with the wheel. In most impulse rheels the water is made to flow under the wheel and hence hey are called Undershot Wheels. It will be seen that in principle, there is no line of demarcation letween impulse water wheels and impulse turbines, the latter mly differing from the former in constructional detail. 177. Overshot water wheels. This type of wheel is not suitable for very low or very high leads as the diameter of the wheel cannot be made greater than he head, neither can it conveniently be made much less. Figs, 180 and 181 show two arrangements of the wheel, the mly difference in the two cases being that in Fig. 181, the top of he wheel is some distance below the surface of the water in the ip-0tream channel or penstock, so that the velocity v with which he water reaches the wheel is larger than in Fig. 180. Tliis has he advantage of allowing the periphery of the wheel to have a i^her velocity, and the size and weight of the wheel is conse- uently diminished. The buckets, which are generally of the form shown in the gures, or are curved similar to those of Fig. 182, are coll- ected to a rim M coupled to the central hub of the >N\\ee\ b^ 284 HYDRAULICS suitable spokes or framework. This class of wheel has considerably used for heads varying from 6 to 70 feet, but u becoming obsolete, being replaced by the modem turbine, y for the same head and power can be made much more com and can be run at a much greater number of revolutions pei time. Fig. 180. Overshot Water Wheel. Fig. 181. Overshot Water Wheel. The direction of the tangent to the blade at inlet for no si can be found by drawing the triangle of velocities as in Figs, and 181. The velocity of the periphery of the wheel is t? and velocity of the water U. The tip of the blade should be par to Vr. The mean velocity U, of the water, as it enters the wl WATER WHEELS 285 in Fig. 181, will be Vo •^kyj2gB,, v^ being the velocity of approach t|{ the water in the channel, H the fall of the free surface and k a ooeflScient of velocity. The water is generally brought to the wlieel along a wooden flume, and thus the velocity U and the •opply to the wheel can be maintained fairly constant by a simple dmce placed in the flume. The best velocity v for the periphery is, as shown below, equal to iU cos tf, but in practice the velocity v is frequently much greater than this. In order that XJ may be about 2v the water must enter the wheel at a depth not less than below the water in the penstock. When r = 4-5 feet, H = 0-63 feet, mnd when r = 8 feet, H = 1 foot. If the total fall to the level of the water in the tail race is h, the diameter of the wheel may, therefore, be between h and 9 Since IT is equal to >/2flrH, for given values of U and of /i, the larger the wheel is made the greater must be the angular distance from the top of the wheel at which the water enters. With the type of wheel and penstock shown in Fig. 181, the bead H is likely to vary and the velocity U will not, therefore, be constant. If, however, the wheel is designed for the required power at minimnm flow, when the head increases, and there is a greater quantity of water available, a loss in efficiency \vill not be important. The horse-^fxncer of the wheel. Let D be the diameter of the wheel in feet which in actual wheels is from 10 to 70 feet. Let N be the number of buckets, which in actual wheels is generally from 2| to 3D. Let Q be the volume of water in cubic feet of water supplied per second. Let •• be the angular velocity of the wheel in radians, and n the number of revolutions per sec. Let b be the width of the wheel. Let d, which equals rj — n, be the depth of the shroud, which en actual wheels is from 10" to 20". i 286 Wbmfii the form of tbe backete the capacity of each bucket j S Tfce BTiiiiber of bocketa whieh pa^ tbe str^m pet m:miii U « fnctbii i of each bucket: is filled mth water or Tlie Iractidii 1 % from | to |. If k » die &L flie level of die tail race ^i the efficiefic7 of ti l-power is . 50 ' and the widtli b for a given horse-power, HP, 18 llOOHP ^j^,^ HP ^<p*ci o/ reiUn/wi^of /ot^^. As the wteel revolves, the siir&« of the water in the buckets, due to centrifugal forces, takes up i parabolic form. It h shown on page 33-S that when a mass of water having tt inner radios r^ and outer radius ri revolves about a fijced cenlie ^^-ith angular velocity «, the pressure headj due to centrifogi forces, at any radius r, is ir" 2g To balance this prepare head the surface of the water in asj bucket, at the point Cj of radius r, must be raised above tb hori^>iitaI through A a distance This is the equation to a parabola, and the surface of the water, therefore, assumes the form of a parabolic curve. Let To be the radius at the centre of the surface of the watoin any cup and ^ the inclination of the radius r© to the horisontaL Then since n is nearly equal to ri, ^ = n nearly. WATER WHEELS 287 Then y=2^(n+r) (r-rO = ^r.(r-n) nearly. Therefore, y is approximately proportional to r— n, and the irface AB is approximately a straight line inclined at an angle , the tangent of which is tan B = cos ^. g Losses of energy in overshot wheels. V * . (a) The whole of the velocity head ^ is lost in eddies in the nekets. In addition, as the water falls in the bucket through the ertical distance EM, its velocity will be increased by gravity, nd the velocity thus given will be practically all lost by eddies. Again, if the direction of the tip of the bucket is not parallel to Tr the water will enter with shock, and a further head will be Dst. The total loss by eddies and shock may, therefore, be rritten '^^^^^ [)r Ai + *i2^, fc and Tci being coefficients and hi the vertical distance EM. (6) The water begins to leave the buckets before the level of the tail race is reached. This is increased by the centrifugal forces, as clearly, due to these forces, the water will leave the buckets earlier than it otherwise would do. If h^ is the mean Iieight above the tail level at which the water leaves the buckets, » h«id equal to fc» is lost. By fitting an apron GH in front of the wheel the water can be prevented from leaving the wheel until it i very near the tail race. (c) The water leaves the buckets with a velocity of whirl equal to the velocity of the periphery of the wheel and a further l^ead JT is lost. (d) If the level of the tail water rises above the bottom of Awheel there will be a further loss due to, (1) the head K equal to •Ke height of the water above the bottom of the wheel, (2) the ^pact of the tail water stream on the buckets, and (3) the ■C^ndency for the buckets to lift the water on the ascending side of ^ wheel. 288 HTDRAUUCS In times of flood there may be a considerable rise of Urn down-stream, and h^ may then be a large fraction of A. If on' the other hand the wheel is raised to such a height above the td water that the bottom of the wheel may be always clear^ Urn head km will be considerable during dry weather flow, and d» greatest possible amount of energy will not be obtained from tfa water, just when it is desirable that no energy shall be wasted. If h is the difF erence in level between the up and down-stresa surfaces, the maximum hydraulic efficiency possible is J-^'^t^^ ,. and the actual hydraulic efficiency will be ^^ h k, ki and h being coefficients. The efficiency as calculated from equation (1), for any giToi value of hm, is a maximum when 2g^2g is a minimum. From the triangles EKF and KDF, Fig. 180, (U cos e-vy^ (U sin ey = va Therefore, adding v^ to both sides of the equation, Vr" + i;' = U*cos''^-2Ut;cos^ + 2v» + U>sin*tf, which is a minimum for a given value of U, when 2Uv cos^-2f* is a maximum. Differentiating and equating to zero this, and therefore the efficiency, is seen to be a maximum, when U /J V = -^^ cos u. The actual efficiencies obtained from overshot wheels vary from 60 to 80 per cent. 178. Breast wheel. This type of wheel, like the overshot wheel, is becoming obsolete. Fig. 182 shows the form of the wheel, as designed by Fairbairn. The water is admitted to the wheel through a number of passages, which may be opened or closed by a sluice as shown in the figure. The directions of these passages may be made so thai the water enters the wheel without shock. The water is retained WATER WHEELS * backet) by the breast, until tb© bucket reaches the tail race, greater fraction of tlie head m therefore utilised than in verahot wheel. In order that the air may enter and leave i^keta freely, they are partly open at the inner vim. Since iter in the tail iBce runs in the direction of the motion of 9ttom of the wheel there is no serious objection to the tail level being 6 inches above the bottom of the wheel. The losses of head will be the same as for the overshot wheel S^t that k^ will be practically ^ero, and in addition, there will I08B by friction in the guide passages, by friction of the water t moves over the breast, and further loss due to leakage &n the breast and the wheel* ling to Rankine the velocity of the rim for ovei^hot and wbeeb, shoald Le from 4^ to 8 feet per second, and the j/dty XJ fihoaltl be about 2i^ depth o! the shroud which is equal to n-tt is from 1 to liet it be denoted by d. Let H be the total fall and let aed that the efficiency of the wheel is 65 per eeiit. Then, mmM mi 290 HYDRAUUGS the quantity of water required per second in cubic feet for a given horse-power N is ^_ N.550 ^"■62-4xHxO-65 _ 13'5N H • From ^ to f of the volume of each bucket, or from ^ to | of the total volume of the buckets on the loaded part of the wheel is filled with water. Let 6 be the breadth of the buckets. If now V is the velocity of the rim, and an arc AB, Fig. 183, is set off on the outer rim equal to v, and each bucket is half full, the quantity of water carried down per second is iABCD.6. Therefore «-»(^') vdb. Equating this value of Q to the above value, the width b is ^^ 27KD (ti + Ti) vdR ' D being the outer diameter of the wheel. Breast wheels are used for falls of from 5 to 15 feet and the diameter should be from 12 to 25 feet. The width may be as great as 10 feet. Example, A breaat wheel 20 feet diameter and 6 feet wide, working on a fifl of 14 feet and having a depth of shroud of 1' S", has its backets f falL The meta velocity of the backets is 5 feet per second. Find the horse-power of the wh«d» assaming the efficiency 70 per cent. xjxy K ^oK a ^ 62-4 X 0-70x14' HP = 5x 1-25x6x5 X =^77 o ooU = 26-1. The dimensions of this wheel should be compared with those calonlated for M inward flow turbine working under the same h^ and developing the aame hofM* power. See page 339. 179. Sagebien wheels. These wheels, Fig. 184, have straight buckets inclined to the radius at an angle of from 30 to 45 degrees. The velocity of the periphery of the wheel is very small, nerer exceeding 2i to 3 feet per second, so that the loss due to the witer leaving the wheel with this velocity and due to leakage betweoi the wheel and breast is small. WATER WHEELS 291 An efficiency of over 80 per cent, has been obtained wi^ wheels* The water enters the wheel in a horizontal direction wit velocity V equal to that in the penstock, and the triangle of reloeities is therefore ABC* If the bucket is made parallel to Vr the water entei-s without ck, while at the same time there is no lofts of bead due to icnon of guide passages, or to contraction as the water enters or iv€3 them ; moreover the direction of the stream has not to be aoged. Fig. 1S4. Sftgfbien WUeel. Tlie iucUned srraight bucket has one disadvantage ; when the [lower part of the whee! is drowaedj the buckets as they ascend are {more nearly perpendicular to the surface of the tail water than rheti the blades are radial, but as the peripheral speed is very flow the renistance due to this cause is not considerable, ISO. Impulse wheels. In OvL-rshot ami Breast wheels the work is done principally Iby the weight of the water. In the wheels now to be considered ie whole of the head available is converted into velocity before le wAter strikes the wheel, and the work is done on the wheel "rig the niomeutum of the mass of moving water, or in i -, by changing the kinetic energy o£ tlae w&^«nc. J 292 HYDRAULICS Undershot wheel with fiat blades. The simplest case is wlien a wheel with radial blades, similar to that shown in Fig*. 185, is put into a nmning stream. If 6 is the width of the wheel, d the depth of the stream under the wheel, and U the velocity in feet per second, the weight of water that will strike the wheel per second isb.d.w.TJ lbs,, and the energy available per second is b.d.WcT foot lbs. Let V be the mean velocity of the blades. The radios of the wheel being large the blades are similar to a series of flat blades moving parallel to the stream and the water leaves them with a velocity v in the direction of motion. As shown on page 268, the best theoretical value for the velocity v of such blades is iXJ and the maximum possible efficiency of the wheel is 0'5. Fig. 185. Impalse Wheel. By placing a gate across the channel and making the bed near the wheel circular as in Fig. 185, and the width of the wheel equal to that of the channel, the supply is more under control, and loss by leakage is reduced to a minimum. The conditions are now somewhat different to those assmned for the large number of flat vanes, and the maximum possiUe efficiency is determined as follows. Let Q be the number of cubic feet of water passing through the wheel per second. The mean velocity with which the wrtir leaves the penstock at ah is \5 = hJ2gh. Let the defyth of tht WATER WHEELS 293 stream at a& be ^. The velocity with which the water leaves the wheel at the section cd is Vy the velocity of the blades. K the width of the stream at c(2 is the same as at a& and the depth is A«, then, ^ X t? = ^ X XJ, or ^ = — . V Since TT is greater than v, h^ is greater than ty as shown in the figure. The hydrostatic pressure on the section cd is ^ho^bw and on the section a6 it is ifbw. The change in momentum per second is and this must be equal to the impressed forces acting on the mass of water flowing per second through ab or cd. These impressed forces are P the driving pressure on the wheel blades, and the difference between the hydrostatic pressures acting on cd and ab. If, therefore, the driving force acting on the wheel is P lbs., then, P + iWfcw; - ^^"fctr = Q^ (U - tj). Substituting for hoy — , the work done per second is Or, since Q = 6 . ^ . U, w-9=(n-.)-|«j(2-i). The efficiency is then, f (U-v) t /U v\ g 2\v U/ *= — — w ' 2g which is a maximum when 2v^ - 4u»U + gtU'' + gtv^ = 0. The best velocity, v, for the mean velocity of the blades, has been found in practice to be about 04U, the actual efficiency is from 30 to 35 per cent., and the diameters of the wheel are generally from 10 to 23 feet. Floating wheels. To adapt the wheel to the rising and lowering of the waters of a stream, the wheel may be mounted ou 294 HYDRAULICS a frame which may be raised or lowered as the stream rises, Gr thd axle carried upon pontoons so that the wheel rises automaticalljr r'th the stream. 181, Poncelet wheel. ^rhe efficiency of the straight hlade impulse wheels is small, due to the large amount of energj^ lost by shock, and to velocity with which the water leaves the w^heel in the direction motion, Tlie efficiency of the wheel is doubled, if the blades are of such a form, that the direction of the blade at enti-ance is parallel to ■the relative velocity of the water and the blade, as fiist suggestad l)y Poncelet, and the water is made to leave the wheel with do component in the direction of mution of the periphery of tha wheel. Fig, 186 shows a Poncelet wheel. I Fig. 186* Undershot WlieeL Suppose the water to approach the edge A of a blade mil velocity U making an angle ^ mth the tangent to the wheel at t Then if the direction of motion of the water is in the directifl AC the triangle of velocities for entrance is ABC* The relative velocity of the water and the wheel is V^ and if the blade is made sufficiently deep that the water does not overflow the upper edge and there is no loss by shock and by fiictiun, i particle of water will rise up the blade a vertical height WATER WHEELS 295 a begins to fall and arrives at the tip of the blade with the ty Vr relative to the blade in the inverse direction BE. he triangle of velocities for exit is, therefore, ABE, BE being 1 to BC. .'he velocity with which the water leaves the wheel is then AE = U,. It has been assomed that no energy is lost by friction or by »ck, and therefore the work done on the wheel is ad the theoretical hydraulic efficiency* is = l-§* (1). This will be a maximum when Ui is a minimum. Now since BE = BC, the perpendiculars EF and CD, on to AB and AB produced, from the points E and C respectively, are equaL And since AC and the angle are constant, CD is constant for all values of Vy and therefore FE is constant. But AE, that is Ui, is always greater than FE except when AE is perpendicular to AD. The velocity Ui will have its minimum value, therefore, when AE is equal to FE or Ui is perpendicular to v. The triangles of velocities are then as in Fig. 187, the point B bisects AD, and V = JU cos 6, For maximum efficiency, therefore, V = fU cos 0. * In what follows, the terms theoretical hydraulic efficiency and hydniulio •flSeiency will be freqaently osed. The niazimom work per lb. that can be utilised bj ftoy hydxmolio maohine supplied with water under a head H, and from which the water ezluuuts with a velocity u is H - ^t- . The ratio H Hi 11 the theoratieel hydraulic effidenoy. If there are other hydraulic losses in th m nifhiw^i eqairalent to a head h/ per lb. of flow, the hydraulic efficiency is — H — • The aetoal effioieiiey of the maohine is the ratio of the external work done per of water hj the machine to H. BTDftlDIiCS ' cmn »bo be found t>K oaoitdeiiiig Qm dtangf oT TW totfti cb»oge of Telocity jmprewed <m the water is CE^ aoJ ife cittiig^ in th^ directioii of tsMmx is fwm^ FD, 1%. im AIL^C A«d»aiBEi.ttloltoBC,FBiii L^^^^'h'^^ FD = 2(Uco6i-r). Kg. m. uid tke r 9-e) 4 DtteRaDtiMtmg wttli i ind eqoatiii^ to zero, or r = yU cos d. The velocity Ui with which the water leaves the wheel, is then perpendicular to r and is Ui=Usind. Substitnting for r its value JU cos B in (2), the maximum efficient is cos^ 0, The same result is obtained from equation (1), by substitutini for Ui, Usin^. The maximum efficiency is then ^ , IPsin'^ ,. E = 1 ™ — = cos'^. A common ^'alue for is 15 degrees, and the theoretia hydraulic efficiency is then 0"933. This increases as diminishes, and would become unity if could be made zero. If, however, 6 is zero, U and r are parallel and the tip of tl blade will be perpendicular to the radius of the wheel. This is clearly the limiting case, which practically is n realisable, Avithout modifying the construction of the wheel. Tl necessarj' modification is shown in the Pelton wheel described < page 377. The actual efficiency of Poncelet wheels is from 55 to 65 p nt. B'ATER WHEELS tirfthe bed. Water enters i\w wheel at all points between untl R, and for no shock the bed of the channel PQ should be it of such a fomi tlmt the diroction of the stream, where it, aliTs fhe wheel at any point A between E and Q, should make constant angle & ^vith the radius of the wheel at A, Willi as centre, draw a circle touching the line AS which, liriikef^ the given angle ^ with the radius AO. Take several point* on the circumference of the wheel between R and and draw tangents t4> the circle STV* If then a curve ' m drawn normal to thcfle several tangents, and the stream are parallel to PQ, the ivater entering any part of the fl between R and Qj will make a constant angle \rith the iiijss and if it enters without shock at A, it will do so at all rinti*. Tlje actual velocity of the water U, as it moves along the PQj will be less than V2gH, due to friction, etc. The efficient of velocity iv in most cases will probably be between I and 0*95, so that taking a mean value for fey of 0*92*1, U = 0-925 s/%H. The best value for tfte velocity v tahhig frictum mio ctcc&wni, \ determining the best velocity for the periphery of the wheel no Qowance has been made for the loss of energy due to friction in * wheel. If V, is the relative velocity of the water and wheel at entrance^ is Uy be expected that the velocity relative to the wheel at exit ! be le^ than Vr, due to friction and interference of the rising 3d filing particles of water. Tlie case is somewhat analogous to that of a stone thrown ertically up in the atmosphere with a velocity v. If there were Mnesistance to its motion, it would rise to a certain height, : A.= 2ff' , di^ii descend, and when it again reached the earth it would j a velocity equal to its initial velocity t*. Due to resistances,^ height to which it rises will be less than h^ j and the velocity ith which it reaches the gi^ound will be even less than that due foiling freely through this diminished height. Let the velocity relative to the wheel at exit be nVr, « being , fraction less than unity. Tlie triangle of velocities at exit will then be ABE, Fig, 188. j he change of velocity in the direction of motion is GH, which Itials BHH-GB = BH(l + n) = {1 +n)(U cos e-r). 298 HTDRAUUCS If the velocity at exit relative to the wheel is only nVn thai must have been lost by friction etc., a head equal to The work done on the wheel per lb. of water is, therefore, {(l+n)(Ucose-v)}v Yr\. „r, -g -^(l-"^- Fig. 188. Let (1 - TJ?) be denoted by /, then since V/ = BH" + CH» = (U cos «-«)' + U» sin'tf, the efficiency {(l + n)(Ucos«-«)}--^{(Ucostf-t>)» + U»8in»tf} e = a-^ . Differentiating with respect to v and equating to zero, 2 (1 + w) Ucos^-4 (1 + n) t; + 2U/cos 6 - 2t7/=0, from which _ {(l+n)+/}Ucos^ "" /+2(l + n) ^(2 + n~?i')Ucosg 3-n*4.2n If /is now supposed to be 0*5, i.e. the head lost by friction, et is — ^.^ — , n IS 71 and V = -5617 cos ^. If /is taken as 075, v = 0'6Ucos^. Dimemsiotis of Ponceht wheels. The diameter of the wheel should not be less than 10 feet when the bed is curved, and noi less than 15 feet for a straight bed, otherwise there will be con- siderable loss by shock at entrance, due to the variation of the angle which the stream lines make with the blades between B and Q, Fig. 186. The water will rise on the buckets to a heigte WATER WHEELS %m Bttrlj equal to ^ , and since the water first entera at a point R, be blade depth d must, therefore, be greater than this, or the f»t^* will overflow at the upper edge. The clearance between lie bed and the bottom of the wheel should not be less than f\ Ibe peripheral distance between the consecutive blades is taken torn S inches to 18 inches, ^ Morse^p&wer of Ponctlet wheels. If H is the height of the Ir&oe of water in the penstock above the bottom of the wheel, be velocity U will be about ^ 0*92 v^^, , tid V m&Y be taken aa 0*55 X 0*92 n/%H = 0'5 V%H. Let D be the diametei* of the wheel, and & the breadth, and let Ibe the depth of the orifice EP* Then the number of revolutions BT minute is I 0-5 V2ffH I IT, D The coeflicient of contraction c for the orifice may be from 0*6, I it is shaTp-edged, to 1 if it is carefully rounded^ and may be iketi as 0^8 if the orifice is formed by a flat-edged sluice. The quantity of water striking the wheel per second is, then^ Q = 0'92d6%/2^. 1 If the efficiency is taken as 60 per cent., the work done per bond is 0-6 ^ 624QH ft. lbs, I The horse-power N is then ^ ■ 5,50 • I 182« Turbines. Although the water wheel has been developed to a considemble kgre^ of perfection, efficiencies of over 80 per cent having been plained, it is being ahuost entirely superseded by the turbine. I The old water wheels were required to drive slow moving hichiiiery, and the great disadvantage attaching to them ot iring a small angular velocity was not felt. Such slow moving WMth are however entirely unsuited to the driving of modern lacKinery, and especially for the d^i^^Ilg of dynamos, and they 1^ further quite unsuited for the high heads which are now Billed for the generation of power. I Turbine wheels on the other hand can be made to run at either pr or veiy high speeds, and to work under any head vaTrjm^ ^1^ ^dM 800 BYBRAOLICS fn^jiii 1 foot to 2000 feet, and the speed can be regulated much greater precision. Due to the slow B|}eeds, the old water wheels ecmld not deirelop large power, the niaximmn being about 100 horse-power, wherea*! at Niagara Falls, turbines of 10,000 horse-pjower have recently | been installed. Types of Tm'hifM&. Turbines are generally divided into two cl&saes^ impulse,^ free deviation turbiues, and reaction or pressure turbines. In both kinds of turbines an attempt is made to shape vanes so that the water enters the wheel without shock ; tliat is the direction of the relative velocity of the water and the vi parallel to the tip of the vane^ and the direction of the leai edge of the vane is made so that the water leaves in a speciie^ direetion. In the first class, the whole of the available he-ad is com into velocity before the water strikes the turbine wheel, am pressure in the driving fluid as it moves over the vanes rei constant^ and equal to the atmospheric pressure. The wheel vanes, therefore, must be so formed that the air has free between the vanes, and the space between two consecutive vaneij must not be full of water. Work is done upon the vanes, or i other wordsj upon the turbine wheel to which they are fixedti! virtue of the change of momentum or kinetic energy of moving water, as in examples on pages 270 — 2* Suppose water supplied to a turbine, as in Fig. 258, under ic^ effective head H, which may be suppoiied equal to the total Ht^J miiius losses of head in the supply pipe and at the noEsle* Tin water issues from the nozzle with a velocity U = j2gKj and ihi available energy per pound is IP H = %^ Work is done on the wheel by the absorption of the who?e, c^r part, of this kinetic energy. If Uj is the velocity with which the water leaves the wb«r:u the energy lost by the water per pound is 2g 2g * and this is equal to the work done on the wheel together w'4 energy lost by friction etc* in the wheel. In the second class, only part of the available head ib verted into velocity before the water enters the wheel, and Blocity and preesore both vary as the water pasaefi through the rbeel- It is therelore essential, that the wlieel shall always be pe|»t full of water* Work is done upon the wlieel, ae will be seen B the sequence, partly by changing the kinetic energy the water when it enters the wheel, and partly by changing itB or potential energy, Suppose water is supplied to the turbine of Fig. 191, under [ effective hi?ad H ; the velocity U with which the water enters wheel, is only Borae fraction of J2gilj and the pressure head the inlet fco the wheel mil depend upon the magnitude of U And upon the position of the wheel relative to the head and tail ■vmter siirface^i* The turbine wheel always being full of water, jUiere is continuity of flow through the wheel^ and if the !iead faBpreeaed upon the water by centrifugal action is determint^d, as pa pAge 335, the equations of Bernouilli * can be used to determine Ed aajr gi^en case the difference of pressure head at the inlet and of the wheeL tlie preasure head at inlet is — and at outlet — , and the ity with which the water leavea the wheel is Ui, the v^ork on the wheel (see page 338) is ^ - — + ,ir- - TT- per pound of water, ir work is done on the wheel, partly by changing the velocity lead and partly by changing the presBure head. Such a turbine i called a reaction turbine, and the amount of reaction is measured ^ the ratio r p Pi riy, if p is made equal to p^ the limiting case is reached, f tho turbine becomes an impntsej or free-de\^ation turbine. Id be clearly understood that in a reaction turbine no ue on the wheel merely by hydrostatic pressure, in the III which work m done by the pressure on the piston of a engine or the ram of a hydraulic Hft. 13, Eeactioii turbiBes. tie oldest furm of turbine is the simple reaction, or Scotch ine, which in its simplest form is illustrated in Fig. 189. %*ertieal cnbe T has two horizontal tubes connected to it, the ends of which are bent round at right angles to the direction * See page SSC 302 HTDRAUUCS of length of the tube, or two holes O and Oi are drilled as in tiie figure. Water is supplied to the central tnbe at such a rate as to keep the level of the water in the tube constant, and at a height h above the horizontal tubes. Water escapes through the orifices O and Oi and the wheel rotates in a direction opposite to the direction of flow of the water from the orifices. Tur- bines of this class are frequently- used to act as sprinklers for distri- buting liquids, as for example for distributing sewage on to bacteria beds. A better practical form, known as the Whitelaw turbine, is shown in Fig. 190. K-R-s Fig. 189. Sootoh TuImim. 1 1 Fig. 190. Whitelaw Turbine. To understand the action of the turbine it is first necessary t< consider the effect of the whirling of the water in the arm upot TURBINES 303 4ie discharge from the wheel. Let v be the velocity of rotation ^ the orifices, and h the head of water above the orifices. Imagine the wheel to be held at rest and the orifices opened ; the head causing velocity of flow relative to the arm is iply A, and neglecting friction the water will leave the nozzle with a velocity Vo = 'J2gh. Now suppose the wheel is filled with water and made to rotate ttt an angular velocity f», the orifices being closed. There will aofw be a xnressure head at the orifice equal to h plus the head iminesBed on the water due to the whirling of each particle of water in the arm. Assume the arm to be a straight tube, Fig. 189, having a cross sectional area a. At any radius r take an element of thickness dr. The centrifugal force due to this element is a/= — - — . The pressure per unit area at the outer periphery is, therefore, 9 ~ 2g ' and the head impressed on the water is P __ Ctf V Let V be the velocity of the orifice, then t? = cor, and therefore w''2g' If now the wheel be assumed frictionless and the orifices are opened, and the wheel rotates with the angular velocity co, the head causing velocity of flow relative to the wheel is K=.h + ^ = h+^ (1). w 2g Let Vr be the velocity relative to the wheel >vith which the water leaves the orifice. -^-^ ^ = ^-^1 (2). The velocity relative to the ground, with which the water leaves the wheel, is Vr-t?, the vector sum of Vr and v. _1 f^ wau^rdr ^"ajo g 304 The water leaves the wheel, therefore, with a velocity relttnf | to the ground of M= Vr- », »nd the kinetic energy lost ia The theoretical hydraulic efficiency is theni Since from (2)| Tr becomes increases, the energy lost per | and the eflicieacy E, therefor The efficifmty of the reuctiou . As before, 6 nearly equal to t^ a& f diminishes b& v im a^a with !?• C whenfnetion is cofwidertd. ^9 -(3), kY/ Assuming the head lost by friction to be ^h^, the total head must be equal to 2ff -(4). The work done on the wheel, per pound, is now h- and the hydraulic efficiency is A- 2ff 2ff' Substituting for h from (4) and for ^j Y^-v^ 2v(Yr-v) Let then e = {l+k)Yr'-v'' Yr = nv, 2(71-1) (l+/L-)w»-r Differentiating and equating to zero, n2(l + A:)-2n(l4.fr) + l = 0. TURBINES 305 the efficiency is a maxiYnnni when T 1 + k' Fig. 191. Outward Flow Torbine. U. B. "l^ 806 HTDRAUUC8 184. Outward flow turbines. The outward ilow tarhine was invented in 1838 hy neyron. A cylindrical wheel W, Figs. 191, 192, and 201, har a number of suitably shaped vanes, is &xed to a vertical ami The water enters a cylindrical chamber at the centre of d»J turbine, and is directed to the wheel by auitable tiated blades G, and flows through the wheel in a radial dir outwards. Between the guide blades and the wheel is a ryhndiij cal sluice R which is used Uj ' >1 the flow of water th the wheel. *v f Fjg. 191 <t. This method of regnlating the flow is very imperfect, as wbm the gate partially closes the pas^sageSj tbere must be a suddii^ enlargement as the water enters the wheel, and a loss of h ensues, "llie efficiency at " part gate " is consequently reif much less than when the flow in unchecked. This difficaltf partly o%'ercome by dividing the wheel into sevei^l distinct compartments by horizontal diaphragmsj as shown in Pig. 19^ so that when working at part load, only the efficiency of oa* compartment is affected* The wheels of outward flow turbines may have their asse^ either horizuntal or vertical, and raay be put either above, * below, the tail water level. The '' atuclion hihty If plact.Mi above the tail water, ti* exhaust must take place down a '* suction pipe," as in Fig. 2!0l page 317, the end of which must be kept drowned, and the pip* air-tight, so that at the outlet of the wheel a pressure less tbii the atmospheric pressure raay be maintained* If A i is the he^ of the centre of the discharge periphery of the wheel al tail water level, and pa is the atmospheric pressure in pow square foot, the pressure head at the discharge circumferoiui? w TURBINES SOT I wiieel catmot be more than 34 feet above the level of the tail iter, or the preasore at the outlet of the wheel will be negativei \d practTcally, it cannot be greater than 25 feet- It is Ehowii later that the effective head, under which the rbine works, whether it h drowned, or placed in a suction tnbe, the total fall of the water to the level of the tail race. Fif, 19a, FoQTfieyron Out word Flow Turbine. ttm af the suction tube has the advantage of allowing the ' I be placed at some distance above the tail water J r.j bearings can be readily got atj and repairB caili r emmiy executed. mokiDg the suction tube to enlarge as it deseendBt the of exit can be dimimghed very gradually, aiii '\^ %mX. 308 HYDRAULICS value kept small. If the exhaust takes place direct froi wheel, as in Fig. 192, into the air, the mean head available head of water above the centre of the wheel. Triangles of velocities at inlet and outlet For the wai enter the wheel without shock, the relative velocity of the and the wheel at inlet must be parallel to the inner tips < vanes. The triangles of velocities at inlet and outlet are i in Figs. 193 and 194. Fig. 193. Let AC, Fig. 193, be the velocity XJ in direction and magr of the water as it flows out of the guide passages, and let A the velocity v of the receiving edge of the wheel. Then DC the relative velocity of the water and vane, and the recc edge of the vane must be parallel to DC. The radial comp GC, of AC, determines the quantity of water entering the ^ per unit area of the inlet circumference. Let this radial vel be denoted by u. Then if A is the peripheral area of the face of the wheel, the number of cubic feet Q per #3cond ent the wheel is Q = A.i^, or, if (2 is the diameter and b the depth of the wheel at inlet t is the thickness of the vanes, and n the number of vanes, Q = (vd -n.t) .b.u. Let D be the diameter, and Ai the area of the discharge phery of the wheel. The peripheral velocity Vi at the outlet circumference is v.D TtJRBINES 309 t Ui he tlie radial component of velocity of exit, then what- per the direction with which the water leayes the wheel the tciiAl component of velocity for a given discharge is coustant- The trmngle of velocity can now be drawn as follows : Set off BE equal to t^i. Fig. 194, and BX radial and equal Let it BOW be supposed that the direction EF of the tip of the Itne at discharge is knoiJ^Ti, Draw EF parallel to the tip of the ^ne at D, and tlirotigh K draw KF parallel to BE to meet EF F, Then BF is the velocity in direction and magnitude with which lie ir^ter leaves the wheel j relative to the gi'ound, or to the fixed of the turbine. Let this velocity be denoted by Uj. If, istead of the direction EF being given, the velocity Uj is given I direction and magnitude, the triangle of velocity at exit can be ri&im by setting out BE and BF equal to Vi and Ui respectively, ttd joining EF, Then the tip of the blade must be made parallel EF, Fof »uy given value of Ui the quantity of water flowing iroQgh the wheel is Q = AiUiCotj^=A,w,. Work drjtw tm the wheel neglecting friction, etc. The kinetic a©rgy of the wmter as it leaves the turbine wheel is ^ per pound, if the discharge is into the air or into the tail water this is of necessity lost. Neglecting friction and other losses, ' available energy per pound of water is then H-5i! foot lbs., . the theoretical hydraulic efficiency is E = H ' "ant for any given value of Ui, and independent of the .„., f Uj- This efficiency must not be confused with the 1 efficiency, which is much less than E» be smaller Uj , the greater the theoretical hydraulic efficiency, ^nnce for a given flow through the wheel, Uj will be least it 18 radial and equal to ttj, the greatest amount of work be obtained for the given flow, or the efficiency will be a imtuti, when the water leaves the wheel radially. M tToss ML 310 HTDRAULICS water leaves with a velocity Ui in any other direction, the efficiency will be the same, but the power of the wheel wfll be diminished. If the discharge takes place down a snction tnbe^ and there is no loss between the wheel and the outlet from the tube, the velocity head lost then depends upon the velocity TTi with which the water leaves the tube, and is independent of the velocity or direction with which the water leaves the wheel. The velocity of whirl at inlet and outlet The component of XJ, Fig. 193, in the direction of v is the velocity of whirl at inlel^ and the component of Ui, Fig. 194, in the direction of ih, is the velocity of whirl at exit. Let y and Yi be the velocities of whirl at inlet and oatlek respectively, then V = Ucos^ and Vi = Uisini8 = t^itani8. Work done on the wheel. It has already been shown, section 173, page 275, that when water enters a wheel, rotating about a fixed centre, with a velocity XJ, and leaves it with velodi^ Ui, the component Yi of which is in the same direction as Vi, the work done on the wheel is Yv Yii?i , per pound, and therefore, neglecting friction, y-ir-^-^ w- This is a general formula for all classes of turbines and shaoU be carefully considered by the student. Expressed trigonometrically, t;U cos S __ ViUitoxiP _ TT _ Hl fc%\ 9 9 -^ 2g <2). If F is to the left of BK, Yi is negative. Again, since the radial flow at inlet must equal the radial floir| at outlet, therefore AUsintf = AiXJiCos/9 ,,..,, ^...(g). When Ui is radial, Yi is zero, and th equals t^i tan a. '^^'^ T=°-|* • ^^ from which ?l£2i« = H-?^^ ® g 2g and from (3) ATTsin^^ AiVitana (f^ TURBINES 311 and If the tip of the Tana is radial at inlet, i.e. Vr is radial, V = r 2sr V* tan' a 9 9 ' = H-- 29 ..(7) .(8). In actual turbines P- is from '02H to '07H. Exaw^U. An oatwmrd flow turbine wheel. Fig. 195, has an internal diameter of 6'349 feet, and an external diameter of 6*25 feet, and it makes 250 revolutions per minnte. The wheel has 32 yanes, which may be taken as | inch thick at inlet and 1} inches thick at outlet The head is 141*5 feet above the centre of the wheel and the ezhaoat takes place into the atmosphere. The effective width of the wheel face mt inlet and outlet is 10 inches. The quantity of water supplied per second is 915 cubic feet. Neglecting all frictional losses, determine the angles of the tips of the vanes at inlet and oaUiet to that the water shall leave radially. The peripheral velocity at inlet is v=«- X 6-249 X W=69 ft- P«r 8«o-. ftodatootlet Vi=«'x6-25x V/=:82ft. „ „ Fig. 195. The i»dial Tcloeity of flow at inlet is 215 w X 5-249 x li - f I X J = 18*35 ft. per sec. The zadiAl Tekwty of flow at exit is ^ 215 *^"»x6*25xH-Hxf' = 16*5 ft. per sec. ^=4-23 ft. 312 Then and ^ = 14^5 = 137-27 ft, 137'«7>cS2-2 = 64 ft. p^ Mc» To dr&w the triangle of Tdocitks at mtel Bet out p and u at Hglit ai]ig]e& Then mnee V is 64^ and ia |b« tan^eotia] coEDpooeat of U> k\A u in tkniM compon<?tit of U, tb« ^ir^tioD and magnitude of U is determined. By joinmg B and C the relative velocity' \% is ohtained, and BC i« p&Ealkl lo Ik tip of thi' \'une. The triangle of v«lodtkfl at exit ii DBF, and the ti^ of the t-ane mmt be ptfiOK toEF. - ~-V^*»" -^ tf,-S^ ! B t/"'' "c" E Fig, m Fig. 197, -aitW — The angiefi i?» ^, and a can be caloulated; for tanS== ^-1^-0-2867, b'4 tan^= - — !^= -3-670 and and, therefore, tanB--^^0vt994, ^ = 105° ir, a=iri7'. It will be seen later how these aiiKles are modt^ed when friction ia oonsiderei Fig. 1^8 rHows the form the guide blades and ?anea of the wheel woold probably lake. 7*^ path of the water thrQugh thi wheel. The average l^adial vetoc^itj chroogfa the wheel may be taken aa 17*35 feet. The time taken for a i>article of water to get through the wheel i«t thereibrc^ R ^ r 0'5 ^ ^ The an^le turned through bj the wheel in thia time is 0^S9 radians. Set off the arc AB, Fig. 198, ec|ual to -39 radian, and divide it into four eqntl parts, and draw the radii fUffb, pc and Bit. Divide AD atiM into four equal parts, Aud draw cireles through A^^ A,, &nd A,' Suppose a pai-ticlo of ^ater to enk^r the vrhed &i A in contact wiib a vooe aM Buppoef it to remam id contact with the vau<* during itB poBsage through the vhts^ ' Then, a&snmiug the radial velocity is coo&tant, while the wheel turns throTJcb tbe ft the water will mote radially a dietanoe AA^ and a particle that came o^ ^ TURBINES 313 B TA&e ftt A will, therefore, be in oontaot with the Tane on the arc throngh A^ . i« Tane initially passing throngh A will be now in the position el, al being Qml to hJ and the partiele will therefore be at 1. When the particle arrives on e are throng A, the vane will pass throngh/, and the particle will consequently at S, 63 bdng eqoal to mn. The curve A4 drawn through Al 2 etc. gives the ^tli of the water relative to the fixed casing. Fig. 198. 185. Losses of head due to Motional and other resistances i outward flow turbines. The losses of head may be enumerated as follows : (a) Loss by friction at the sluice and in the penstock or ipply pipe. If Vo is the velocity, and ha the head lost by friction in 18 pipe, (b) As the water enters and moves through the guide assages there will be a loss due to friction and by sudden changes 1 the velocity of flow. This head may be expressed as being a coefficient. * See page 119. 814 HYDRAULICS (c) There is a loss of head at entrance due to shock as the direction of the vane at entrance cannot be determined with precision. This may be written V« 2a ' that is, it is made to depend upon Yr the relative velocity of the water, and the tip of the vane. (d) In the wheel there is a loss of head hd, due to fricticniy which depends upon the relative velocity of the water and the wheel. This relative velocity may be changing, and on any small element of surface of the wheel the head lost will diminish, as the relative velocity diminishes. It will be seen on reference to Figs. 193 and 194, that as the velocity of whirl Vi is diminished the relative velocity of flow tv at exit increases, but the relative velocity Vr at inlet passes through a minimum when V is equal to r, or the tip of the vane is radial If Vo is the relative velocity of the water and the vane at any radius, and b is the width of the vane, and dl an element of length, then, Jci being a third coefficient. If there is any sudden change of velocity as the water poonrw through the wheel there will be a further loss, and if the turbine has a suction tube there may be also a small loss as the water enters the tube from the wheel. The whole loss of head in the penstock and guide passages may be called H/ and the loss in the wheel h/. Then if U* is the Fig. 199. TUJEIBINES 315 city with which the water leaves the tufbme the effective is In well designed inward and outward flow turbines 2ff + A/+H/ ^-^' = eH ies from O'lOH to *22H and the hydraulic efficiency is, therefore^ [jm 90 to 78 per cent The efficiency of inward and outward flow tnrbines including Dhanical losses is fruin 75 to 88 per cent. Calling the hydraulic efficiency e, the general formula (1), tion 184, may now be written & 9 = 78to-9H Outward flow turbines were made by Boyden* about 1848 for rhich he claimed an efficiency of 88 per cent. The workmanship of the highest quality and great care was taken to reduce losses by friction and shock. The section of the crowns of the rhoel of the Boyden turbine is shown in Pig, 199, Outside of turbine wheel was fitted a "diffuser" through which, after Irving the wheel, the water moved radially with a continuously imlni^iung velocity, and finally entered the tail race with a !ity much less, than if it had done so direct from the wheeL lo«s by velocity head was thus diminished, and Boyden claimed that the diffuser increased the efficiency by 3 per cent. 186. Some actual outward flow turbines. l^'uhh' oufward flaw turhin&f. The genera! arrangement of an ontward flow turbine as installed at Ch^vres is shown in Fig. 200, bere are four wheels fixed to a vertical shaft, two of which ire the water from below, and two from above. The fall i.ries from 27 feet in dry weather to 14 feet in time of flood. Tlje uppt.*r wheels only work in time of flood, while at other imes the full power is developed by the lower wheels alone, the 'cylindrical sluices which surround the upi>er wheels being set in ^«uch a position as to cover completely the exit to the wheel. The water after leaving the wheels, diminishes gradually in locity, in the concrete passages leading to the tail race, and the of head due to the velocity with which the water enters the 4 Lamdl HffdrauUc E^eHmettttt J. B. Franoisi 1S56. i 316 tail race is consequently small. These passagee eemre _, ™^, purpose as Bo^den^s diifuser, and as tlie enlarging saction tabej in that they allow the velocity of exit to diminish gradiiaUj^. High Fig. 200. Double Outward Flow Turbine, (E so her W3*sa and Co.) Outward fioie furhine with horizontal tu^m Fig, 201 shows b section through the wheel, and the supply and exhaust pipes, of Rn outward flow turbine, having a horizontal axis and exhausting dowm a " suction pipe." The water after leaving the wheel enters a large chamber, and then passes down the exhaust pipe, the lower end of which is below the tail race. The supply of water to the wheel is regulated by a horizontal cylindrical gate S, between the guide blades G and the wheel. The gate is connected to the ring R, which slides on guides, outside the supply pipe P, and is under the control of the governor. The pressure of the water in the supply pipe is prevented from causing end thrust on the shaft by the partition T, and between T and the wheel the exhaust water has free access. Outward flow turbines at Niagara Falls. The first turbines installed at Niagara Falls for the generation of electric power, TURBINES 817 ^rere outward flow turbinee of the type shown in Figs* 202 and 203. TTier© are two wheels on the same vertical ahaft-^ the water being brought to the chamber between the wheels by a Tei-ticai penBtock 7' &* diameter. The water passes upwards to one wheel And downwards to the other. Fig, SOI. Ontw&rd Flow TnTbine with Hoetion Tube. Am shown in Fig. 202 the water preBBiire in the chamber is vented from acting on the lower wheel by the partition MN, fa allowed to act on the lower side of the upper wlieel, the {MrtitioD HK having holes in it to allow the watar free access iderneath the wheoL The weight of the vertical shaft; and of wheels, is thus balanced, by the water pressure itself. The lower wheel is fixed to a solid shaft, which passes through Iho centre of the upper wheel, and is connected to the hollow of the upper wheel as shown diagram matically in Fig. 202, i¥e this connection^ the vertical shaft is formed of a hollow thD 318 tube 38 inclieB diameter, except where it passes through tb bearii)gB, where it is solid, and 11 inches diameter. A thrust block is also provided to carry the tmbaiftnced I weight* The regulatiug sluice is external to the wheel. To maintaias I high efficiency at part gate, the wheel is divided into three sepaniti ] compartments as in Fourneyron's wheeL Fig. 20*2. Diagrammatic section of OQt\rard Flow Turbine it Niagtftft Fftlk A vertical section through the lower wheel is shown in Fig. 203, ami a part sectional plan of the wheel and guide blades in Fig. 1115. (Further particulars of these turbines and a description of the governor will be found in Cassier's Magazine, Vol. III., and in Turbines Actuelle) Buchetti, Paris 1901. 187. Inward flow turbines. In an inward flow turbine the water is directed to the wheel through guide passages external to the wheel, and after flowing radially finally leaves the wheel in a direction parallel to the axis. Like the outward flow turbine it may work drowned or with a suction tube. The water only acts upon the blades during the radial movement. 819 improvud by Francis •, in 1840, the wheel was of the form in Pig. 204 and was called by its inventor a "central rent I ted O 1 § M g 1^ 1^ I'.'fi m t ill runi uii a vertical shaft., resting cm a footstep, rted by a collar bearing placed above the gtaging S. • Ltwetl Hjfdmtilii E^peHmenUf F, B. Ffioeii, 1856. 320 HTURAULICS LboT0 tie wheel is a heavy castm^ C, supported by l>d| from the staging S, which acts as a guide for the cylindric ,,alaice F, and carries the bearing B for the shaft. There \ wmnm in the wheel shown, and 40 fixed guide blades, the f ormf aing made of iron one quarter of an inch thick and the lat three-eixteenthB of an inch. Fig. 204. Francis" Inward flow or Centrml v^ni TiubiB«. The triangles of velocities at inlet and outlet, Fig. ^'' drawn, exactly as for the outward flow turbine, the only d/> being that the velocities v, U, V, Vr and u refer to the owlet TURBINES 321 eriplieryy and th, Ui, Vi, Vr and lii to the inner periphery of the rheel. The work done on the wheel is YH_Yl5lft.ib8.perlb., 9 9 *~ ' iiid neglecting friction, 9 9 ^ For maximum efficiency, for a given flow through the wheel, Ji should be radial exactly as for the outward flow turbine. Fig. 205. The student should work the following example. , The oater diameter of an inward flow tortfine wheel is 7*70 feet, and the inner ^bmeler 6-8 feet, the wheel makes 55 revolntions per minute. The head is '^'8 feet, the Telocity at inlet is 25 feet per sec., and the radiid velocity may be — iiimJ constant and equal to 7*5 feet. Neglecting friction, draw the triangles of 2^oeities at inlet and outlet, and find the directions of the tips of the vanes at ^Jet and oatlet so that there may be no shock and the water may leave radially. Lass of head by friction. The losses of head by friction are •*«u3ar to those for an outward flow turbine (see page 313) and ^e general formula becomes 9 9 ■^^en the flow is radial at exit, 9 The value of e varying as before between 0*78 and 0*90. ^^uExaMj»£f (1). An inward flow turbine working under a head of 80 feet lias 2^ial blades at inlet, and discharges radially. The angle the tip of the guide rJJ^le makes with the tangent at the inlet is 30 degrees and the radial velocity is ^^^tant. The ratio of the radii at inlet and outlet is 1*75. Find the velocity of ^ inlet drenmference of the wheel. Neglect friction. L. H. ^\ ^-^^ = 6H. 322 HYDRAULICS Sinoe tlM diach&rge it radial, tli« Tvlodtj' ftt wt h Ui=:=Pjtan3(r V 1-75 taoSr. Then 7" 1T5«""^^* and sinoe the Uiid^ A^m radial «i inlet V ia eqnaX Ui i^, therefore from whieh /82 r an E^ r^ Fig. 206. Example (2). The outer diameter of the ^he«l of an inward flow tcrbiiii* 200 horse-power U 2*41^ fret, the inner diameter is l^^tfiS feet. The eDfeenT*iwhb of the wheel at inlets ILS feet. The head b 39'5 feet and 59 eobic feet of water per second ^iv supplied. The radial velocity with which the wat^r \^^^ the wheel may he taken us 10 feet per ftecond. Determine the theori^tical hydraulic eMciencj £ aod the aeitial effioien^'i^ the turbine, and de^-vign suitable vane>^. 300x550 ^i"a9-5x5yxti3-5" '"* Theoretical hydranUo efficiency S9'5" 10* m-b = 96% The radial velocity of flow at in let, 59 2 46xrxM5 = fi*7 feet per sec* TURBINES 323 ripbenJ yekmij r=2-46. XX W=88-6 feet loctty 0/ wkiH V. Aasamiiig a hydraulic effioienqj of B6%, from ^_ S9'5x32-ax'85 88-6 =:S8*0 feet per lee. gU e. Sinoe if=6-7 ft. per sec. and y=38*0 ft. per sec. tan ^=^=0-289, igle ^. Since V is ] ^=180 27'. \ than V, is greater than 90°. 0=162<>. le water to discharge radially with a velocity of 10 feet per sec. ^ 10x60 ^^^^ *"'= l-«68xxx800 =°"^' a =18° nearly, leoretical vanes are shown in Fig. 206. 9U (3). Find the valoes of ^ and a on the assumption that e is 0*80. nsorCa inward flow twrhvne. In 1851 Professor James n invented an inward flow turbine, the wheel of which rounded by a large chamber set eccentrically to the wheel, n in Figs. 207 to 210. reen the wheel and the chamber is a parallel passage, in ire four guide blades Gr, pivoted on fixed centres C and :jan be moved about the centres C by bell crank levers, 1 to the casing, and connected together by levers as shown 207. The water is distributed to the wheel by these guide and by turning the worm quadrant Q by means of the the supply of water to the wheel, and thus the power of bine, can be varied. The advantage of this method of ing the flow, is that there is no sudden enlargement from ide passages to the wheel, and the efficiency at part load much less than at full load. i, 209 and 210 show an enlarged section and part sectional )n of the turbine wheel, and one of the guide blades G. tails of the wheel and casing are made slightly different bose shown in Figs. 207 and 208 to illustrate alternative is. t sides or crowns of the wheel are tapered, so that the jral area of the wheel at the discharge is equal to the 3ral area at inlet. The radial velocities of flow at inlet itlet are, therefore, equal. 41—2 TURBINES 325 iriea of velocities for the inlet and outlet are shown in water leaving the wheel radially, ^of the water through the wheel, relative to the fixed "^ shown and was obtained by the method described •w turbines with adjustable guide blades, as made by !ral makers, have a much greater number of guide I'ig. 238, page 352). tion through wheel and casing of Thomson Inward Flow Turbine. »ine actual inward flow turbines. jnn of the Francis inward flow turbine as designed by /O., and having a horizontal shaft, is shown in Fig. 212. id is doable and is surrounded by a large chamber water flows through the guides G- to the wheel W. g tihe wheel, exhaust takes place down the two suction 38 allowing the turbine to be placed well above the hile ntilifliiig the full head. nhting dnioe F consists of a steel cylinder, which parallel to the axis between the wheel and 3S6 HYDEAULICS Fig. 209. Fig. 210* Detail of wbeel and galde blade of Thomson Inward Flow Turbine. Fig. 211, m-hl-Wi TURBINES 327 Hie wheel is divided into five separata compartments, so that iny time only one can be partially closedj and loss of head by ittTaction and sudden enlargement of the streanij only takes tee in this one compartment* TUBBINES 329 ita^e or mjh the ^■he tti Ha h Bessar Ripened or clewed as required by the steel cylindrical sluice CC ^VDUtidtng the distributor* ^"WThen one of the stages is only partially closed by the eluice, h loss of efficiency must take place, but the efficiency of this one ita^ only is diminisbedj the stages that are still open working their full efficiency. With this construction a high efficiency turbine is maintained for partial How, With normal flowSj head of about 62-5 feetj the three lower stages only are iry t-o give full power, and the efficiency is then a mnximum. In times of H<x>d there is a large volume of %vater S reliable, but the tail watar rises so that the head is only about \9 te0tf the two upper stages can then be brought int-o operation to accommodate a larger floWj and thus the same power may be obtained under a less head. The efficiency is less than when the Oiree stages only are working, but as there is plenty of wat-er availablCj the loss of efficiency is not serious* I The cylinder C is carried by four vertical spindles S, having ks R fixed to their upper ends. Gearing with these racks, are ions Pf Fig. 213, all of which are worked simultaneously by the ^lator, or by hand, A bevel wheel fixed Uj the vertical sliaft f9 with a second bevel wheel on a horizontal shaft, the velocity b being 3 to L ^89* The best peripheral velocity for inward and outward tow turbines. IjWheii the discharge is radial, the general formula, as showTi on b 315, is I — = eH = 0-78toO-90H L ^ K the blades are radial at inlet, for no shock, t^ should be equal r , and I ^^Yi^039 toO'45V2gH, f t^ -V-0-624to0 67v^2^, This is aametimes called the best velocity for t;, but it should be rly understood that it is only so when the blades are radial at .(1). W 190. Experimental determination of the best peripheral Telocity for inward and outward flow turbines. fw an outward flow turbine, working under a head of 14 feet, mdial at inlet* Francis* found that when v was 626 V2^, • Lowell, Hydmuth Exfifrlmentst ' 390 HYDRAULICS From the formula — - ■ 9 the efficiency was a maximum and equal to 79 "37 per cent. Tlie efficiency however was over 78 per cent, for all \^lues of tf between 0'545 -J^E aud *671 J2gR. If 3 per cent, be allowt'd for the mechanical losses the hydraulic efficiency may be taken ^B 82"4 per cent, VtJ - '824H, and taking V equal to i\ V ^ '64 %^yH, so that the result of the experiment agrees well with the formnla* For an inward flow turbine having vanes as shown in Fig, 205, I the ttjtal efficiency was over 79 per cent, for values of r between PO'624 V%B and 0'708 J2^, the greatest efficiency being W! per cent* when v was 0*708 v2gH and again when t? WM It will be seen from Fig. 205 that although the tip of the irwa^ at the convex side is nearly radial, the general direction of tha vane at inlet is inclined at an angle greater than 90 degree \f> the directioii of motion, and therefore for no shock Y ^should hi less than v, Wben V was 708 V2gH, V, Fig. 205^ was less than i\ Tbfi value of V was deduced from the following data» which is abo useful as being taken from a turbine of very high efficiency. Diameter of wheel 9'338 feet. Width between the crowns at inlet 0'999 foot. There were 40 vanes in the wheel and an equal number of fixed guides external to the wheel. The minimum width of each guide passage was 0"1467 foot ini the depth r0066 feet. The quantity of water supplied to the wheel per second *i* ni2'526 cubic feet, and the total fall of the water was 134 fe<*- Kthe radial velocity of flow u was^ therefore, 3*86 feet per secoii The velocity through the minimtmi section of the guide was 19 feet per second. When the efficiency was a maximum, t' was 20'8 feet per Then the radial velocity of flow at inlet to the wheel 3'S6 feet, and U being taken as 19 feet per second, the of velocities at inlet is ABC, Pig. 205, and V is 18'4 feet per K it is assumed that the water leaves the wheel radially, thfi^ L eH=— = 1185feet r ^ 11*85 The efficiency e should be j^rx ==88'5 per cent., which is 9| cent higher than the actual efficiency. TURBINES 331 The actual efficiency howeirer mcladea not only the fluid losses it also the mechanical losses, and these would probahly be from to 8 per c^nt,, and the actual work done by the turbine on the laft IB probably between 80 and S&5 per cent, of the work done r the water. j Vv I 19L Value of e to be used in the formula — - ^ eH* 1 i In general, it may be said that, in using the formula — = eH, e value of e to be used in any given case is doubtful, as even oQ^b the efficiency of the class of turbiues may be known, it is fficult to say exactly how much of the energy is lost mechanically id how moch hydrauUcally, A trial of a turbine without load, would be useless to deter- ine the mechanical efficiency, as the hydraulic losses in such a tal would be very much larger than when the turbine is working full load. By revolving the turbine without load by means of i electric motor, or through the meditnn of a dynamometer, the art to overcome friction of bearings and other mechanical losses uld be found. At all loads, from no load to full load, the iotional resistances of machines are fairly constant, and the schanical losses for a given class of turbineSj at the normal load r which the vane angles are calculated, could thus approximately \ obtained. If, however, in making calculations the difference tween the actual and the hydraulic efficiency be taken as, say, per cent., the error cannot be very great, as a variation of 5 per ntp in the value assumed for the hydraulic efficiency e, will only kke a difference of a few degrees in the calculated value of I angle <^. The beat value for 0, for inward flow turbines, is probably 0*80, 1 ejcperience shows tliat this value may be used with confidence, ffiwilpfr. Tftkiag thedjiU ikM u^^en in the exacnple of ae^tiou 184, and aasaming MMtWQT t^' ^^ tutbioe of 75 per eeat.^ the horfie-power k iKJK 215 y 62-4 >£Ul%5x 75x60 ■ ==2600 horse -power If the hjdr&iUie efficiencj ij anppoaed to be SO per centt^ the velocitj of d V sbonld be ^ eff.H ^ 08,32 -Ul-6 r " 60 I =53 feet per a«e* I riten tmm^ 18 35 -18 85 | Cbcti *~5a-6S~ 17 ' ^-182° 47', Kippoae the torbiise to be itill ^nersting 2600 hor»e*power, and to h&ye \j of 80 pet oenL, ^nd » hydrauito effioie^cy of 65 per eent. 332 HYDRAUUCS Then the qaantity of water reqaired per second, is ^ 216x0-76 o^ u- # * Q = — zr^ — = 200 cubic feet per sec. and the radial velocity of flow at inlet will be 18-36x200 215 = 17*1 ft, per sec. -, •86.32.141-6 __.-^ V = nn =66*4 ft. per sec. Then tan0=: 69 171 66-4-69 = 128°. 24'. -171 13-6 192. The ratio of the velocity of whirl V to the velodtj of the inlet periphery v. Experience shows that, consistent with Yv satisfying the gfenenl formula, the ratio ^y may vary between very wide limits without considerably altering the efficiency of the turbine. Table XXXVII shows actual values of the ratio • , taken from a number of existing turbines, and also corresponding valnai Fig. 214. TURBINES 333 =^ , V beinff calculated from — : ' 0'8H. The corresponding .tion in the angle <t>y Fig. 214, is from 20 to 150 degrees, or a given head, v may therefore vary within wide limits, h allows a very large variation in the angular velocity of the 1 to suit particular circumstances. TABLE XXXVII. howing the heads, and the velocity of the receiving circum- ice V for some existing inward and outward, and mixed flow nes. Hfeet vfeet per see. J29H Ratio V sl2gR H.P. V Ratio -7=- >/2pH V being calculated from— = -8H 9 .rdflaw: gara Falls* 146 70 96-8 0-72 5000 0-655 infelden 14-8 22 80-7 0-71 840 0-666 Theodor) and Co. j 28-4 89 42-6 0-91 0-44 60-4 82-2 62-8 0-52 0-77 et and Co. 188-7 611 76-8 0-47 800 0-85 »» 184-6 46-6 65-6 0-505 800 0-79 6-25 16-6 20 0-88 0-48 ji 80 25-75 44 0-58 700 0-69 ♦» 88-5 50-8 0-77 200 0-52 Lz and Co. 112 64-8 84-6 0-54 0-74 »9 226 64-7 120 0-64 682 0-58 ber and Co. 10-66 15-2 26 0-685 80 0-69 ^rdflow : gara Falls 141-6 69 95-2 0-725) 5000 0-55 ;et and Co. 180-6 69 91-6 0-750) 0-58 LZ and Co. 96-1 88-7 780 0-495 290 0-81 »» 228 65-6 1200 0-46 1200 0-87 ♦ Escher Wyss and Co. 'or example, if a turbine is required to drive alternators rt, the number of revolutions will probably be fixed by the nators, while, as shown later, the diameter of the wheel is tically fixed by the quantity of water, which it is required to through the wheel, consistent with the peripheral velocity of v^heel, not being greater than 100 feet per second, unless, as ie turbine described on page 373, special precautions are 1. This latter condition may necessitate the placing of two 3re wheels on one shaft. 334 BYDBAmJCa Suppose then, the number of reT-olutions of the wheel to given and d is fixed, then v has a definite value, and V muii 1 made to satisfy the equation s Fig. 214 is drawn to illustrate three cas€«s for which Yri] constant, Tht? angles of the vanes at outlet are the same fori three, but the guide angle ^ and the vane angle ^ at inlet considerably^. 193. The velocity witli In a well-deaigned turbine leaves tht* turbine should be as keeping the turbine whrel an dimensions. In actual turbines the 1 varies from 2 to 8 per cei suction pipe the water may t with a fairly high velocity a conical sv» ai^ to allow the actual water leaves a turbme. locity with which the ' as possible, condsteal i >ftTi-take within h ^due to this velocity I turbine is fitted with jd to learc the wheel iacharge pipe can be \ irge velocity to be as i as desired. It should however be noted that if the w^ater leavtf* the wheel with a high velocity it is more than probable that there ^^^ll be Jiome loss of head due to shock, as it is difficult to ensure that wattT ?4* di^chargtKi shall have its velocity changed gradiiallT. 194. BemoniUi's equations applied to inward and out- ward flow tnrbmes neglecting firiction. Ctntrifn^al head imprejtsed on the water by the wheei Tk theory of rhe reaction turbines is best considered from the point of N-iew of Bemouilli's equations ; but before proceeding to discuss them in detail, it is necessary to consider the " centrifugal head" impivssed un the water by the w^heeh This head has already been considered in connection with tk Scotch turbim*, page 303, Let r^ Kig. 216j be the internal radius of a wheel, and B tb external radius. At the internal circumference let the wheel be covered with i cylinder e S4> that there can be no flow through the wheel, and 1h it W suppixsed that the wheel is made to revolve at the angular vehxnty f^ which it has as a turbine, rhe wheel being full of wat€f and surrounded by water at rest, the pressure outside the wbed being sufficient to prevent the water being whirled out of the whet^l. Lei d be the depth of the wheel between the crowia Consider any element of a ring of radius n and thickness dr, and subtending a smaU angle 6* at the centre C, Fig. 210* TURBINES 335 The weight of the element is UDToO .dr,d, and the centrifugal force acting on the element is uTo^ ,dr,d, «Vo 9 lbs. Iiet p be the pressure per unit area on the inner force of the element and p-^dp on the outer. fjPToB ,dr .d . «Vo op = Then g.nO.d Fig. 215. Fig. 216. The increase in the pressure, due to centrifugal forces, between -r and R is, therefore, -r T'-*" I •'<"'"'■' i^nd For equilibrium, therefore, the pressure in the water surround- ing the wheel must be pe. If now the cylinder c be removed and water is allowed to flow tlnroiigh the wheel, either inwards or outwards, this centrifugal lieftd will always be impressed upon the water, whether the wheel u driven by the water as a turbine, or by some external agency, mnA acts as a pump. BernouilWa equations. The student on first reading these equations will do well to confine his attention to the inward flow terbine, Fig. 217, and then read them through again, confining his attention to the outward flow turbine, Fig. 191. 336 HYDRAULICS Let p be the pressure at A, the inlet to the wheel, or in the clearance between the wheel and the guides, pi the pressure at the outlet B, Fig. 217, and pa the atmospheric pressure, in pounds per square foot. Let H be the total head, and H« the statical head at the centre of the wheel. The triangles of velocities are as shown in Figs. 218 and 219. Then at A Pa TP w w 2g .(1). Between B and A the wheel impresses upon the water the centrifugal head 2g 2g' V being greater than Vi for an inward flow turbine and less for the outward flow. Fig. 217. Consider now the total head relative to the wheel at A and BL The velocity head at A is -^ and the pressure head is ^i>il||H at B the velocity and pressure heads are ^ and ^ respectivdy. If no head were impressed on the water as it flows thtongli the wheel, the pressure head plus the velocity head at A and B would be equal to each other. But between A and B thero i« impressed on the water the centrifugal head, and therefore, Pi Vr^ v^ ^^' = P I ^^* w 2^ 2g 2g w 2g .(2). TURBINES .337 This equation can be used to deduce the fundamental equation, ^-^ = fc (3). 9 9 From the triangles CDE and ADE, Fig. 218, Vr»=(V-t7)« + ti'andV» + tt« = U', ind from the triangle BFG, ¥ig 219, vr' = (vi - YiY + u,' and Va» + V = Vi\ Therefore by substitution in (2), i?"" 2^ 2g 2g^2g w^ 2g ^ 2g "'^^^' From which ti7 g 2g w 2g g ' ad g g w w 2g 2g Substituting f or *^ + ^5- from (1) w VV VlVl ^ ^ , Pa Pi Ul' 2sr .(5). (6). Pig. 218. Wheel in suction tube. K the centre of the wheel is ho feet the snr&ce of the tail water, and Uo is the velocity witli e water leaves the down-pipe, then ^^m th '- 2g ^ w 2g w Snbstitntingr for ^ + §^' in (6), w 2g 9 " 9 - - WW 2g = H-U'' 2ff- I. B. ^•1 338 HYDRAULICS IfVisO, —^R-^---h. 9 2flr The wheel can therefore take full advantag^e of the head H even though it is placed at some distance above the level of the tail water. Drowned wheel. If the level of the tail water is CD, Fig. 217, or the wheel is drowned, and hi is the depth of the centre of the wheel below the tail race level, W W and the work done on the wheel per pound of water is again vV Yiv, „ W , 9 9 ^9 vY IfViisO, — = h. From equation (5), vV _ ViYj ^ P _ Pi ^ U! _ W 9 9 w w 2g 2g ^ so that the work done on the wheel per pound is the difference between the pressure head plus the velocity head at entrance and the pressure head plus velocity head at exit. In an impulse turbine p and pi are equal, and the work doi« is then the change in the kinetic energy of the jet when it strikes and when it leaves the wheel. A special case arises when pi is equal to p. In this case a considerable clearance may be allowed between the wheel and the fixed guide without danger of leakage. Equation (2), for this case, becomes 2g 2g 2g 2g' and if at exit Vr is made equal to Vi, or the triangle BFG, Fig. 219, is isosceles, 2g 2g^ and the triangle of velocities at entrance is also isosceles. The pressure head at entrance is w 2g' and at exit is either — + fei, or — - ^. w ^ w TURBINES 339 lieref ore, since the pressures at entrance and exit are equal, <r else Ho + /iD = H. The water then enters the wheel with a velocity equal to that Lue to the total head H, and the turbine becomes a free-deviation NT impulse turbine. 195. Bemonilli's equations for the inward and outward low turbines including friction. If H/ is the loss of head in the penstock and guide passages, ^ the loss of head in the wheel, he the loss at exit from the wheel ttid in the suction pipe, and Ui the velocity of exhaust, £ + U!=H. + £e-H, (1), w^2g^2g-2g-w^2-g-^^ ^^^' w w w w rom which — = H-f -^^ + A/+H/+fee) (4). f the losses can be expressed as a fraction of H, or equal to KH, hen Vt? — = (l-K)H = eH ^ =0-78H to 0-90H*. 196. Turbine to develop a given horse-power. Let H be the total head in feet under which the turbine works. Let n be the number of revolutions of the wheel per minute. Let Q be the number of cubic feet of water per second required >y the turbine. Let E be the theoretical hydraulic efficiency. Let e be the hydraulic efficiency. Let Sm be the mechanical efficiency. Let fii be the actual efficiency including mechanical losses. Let Ui be the radial velocity with which the water leaves the irheel. Let D be the diameter of the wheel in feet at the inlet circum- erence and d the diameter at the outlet circumference. Let B be the width of the wheel in feet between the crowns it the inlet circumference, and b be the width between the crowns it the outlet circumference. Let N be the horse-power of the turbine. * See page 815. 340 HYDRAULICS The number of cubic feet per second required is ^ e,H. 62-4.60 ^^'• A reasonable value for ei is 75 per cent. The velocity Uo with which the water leaves the turbine, since is Uo=>/25f(l-E)Hft.per8ec (2). If it be assumed that this is equal to thy which would d necessity be the case when the turbine works drowned, or exhausts into the air, then, if Hs the peripheral thickness of the vanes at outlet and m the number of vanes, (ird-mOUo6=Q. If Uo is not equal to Ui, then (^d-mt)uib = Q (3). The number of vanes m and the thickness t are somewhat arbitrary, but in well-designed turbines t is made as small as As a first approximation mt may be taken as zero and (3) becomes 7rdbUi=Q (4). For an inward flow turbine the diameter d is fixed from consideration of the velocity with which the water leaves the wheel in an axial direction. If the water leaves at both sides of the wheel as in Pig. 208, and the diameter of the shaft is do, the axial velocity is t^= y ft. per sec. 2j(d'-d.') The diameter do can generally be given an arbitrary value, or for a first approximation to d it may be neglected, and u% may be taken as equal to i^. Then ^-^"^ <«• From (4) and (5) b and d can now be determined. A ratio for -v having been decided upon, D can be calcvlatedy and if the radial velocity at inlet is to be the same as at oatletk and to is the thickness of the vanes at inlet, (^D-mMB = S = (ird-m06 (6). til TURBINES 341 For rolled brass or wrought steel blades, to may be very small, and for blades cast with the wheel, by shaping them as in Fig. 227, to is practically zero. Then If now the number of revolutions is fixed by any special condition, such as having to drive an alternator direct, at some definite speed, the peripheral velocity is frDn -. ,^,^. t7 = -^ ft. per sec (/). Yv Then V"^^' and if e is given a value, say 80 per cent., V = '^ ft. per sec (8). V Since u, V, and v are known, the triangle of velocities at inlet can l>e drawn and the direction of flow and of the tip of vanes at inlet determined. Or B and <^, Fig. 214, can be calculated from tan^ = ^- (9) and ' tan<^ = ;^^ (10). Then U, the velocity of flow at inlet, is U = V sec 0, Trdn «, At exit fi = -^ ft. per sec, and taking Ux as radial and equal to Uy the triangle of velocities can be drawn, or a calculated from u tan a = - . If BU is the nead of water at the centre of the wheel and 11/ the head lost by friction in the supply pipe and guide passages, the pressure head at the inlet is Example- An inward flow turbine is required to develop 300 horse-power under A bead 60 feet, and to run at 250 revolutions per minute. To determine the leading dimensions of the turbine. Aaeuming ^i to be 75 per cent., Q_ 300x33,000 ^~ -76 x60 x~02Tx^60 = 58*7 cubic feet per sec. 342 HTDRAUUCS Assmmiig E is 95 per cent., or five per cent, of the head iq lost by Telo^ of exit and u^^u, =^=•05.60 2g and tt= 13*8 feet per sec. Then from (5), page 340, ^^-^W^-J^-^^ = 1-65 feet, say 20 inches to make allowance for shaft and to keep eyen dimenBion. Then from (4), * = fH "= '^ ^^* =9} inches say. Taking - as 1*8, D=3*0 feet, and a V =rir . 8 . W = 39*3 feet per sec., and B=5} inches say. Assuming « to be 80 per cent., „ -80x60x32 ^^^^ " "~ 39-3 ~*>''""*P«^"«- " * n 13*8 tan^= 39 ' and ^=19^80'. *«'*=^'.- - ^^ and 0=91'^ 15'. 13*8 X 1*8 **"*=-39*3-' and a = 32° 18'. The velocity U at inlet is U=>/39*0» + (13*8)« =41*3 ft. per sec. The absolute pressure head at the inlet to the wheel is n n 41 'V - = Ho + -^ — hf, the head lost by friction in the down pipe =Ho+34-26*6-/i/. The pressure head at the outlet of the wheel will depend upon the height of t wheel above or below the tail water. 197. Parallel or axial flow turbines. Fig. 220 shows a double compartment axial flow turbine, tl guide blades bein g placed above the wheel and the flow throuj the wheel being parallel to the axis. The circumferential secti of the vanes at any radius when turned into the plane of t paper is as shown in Fig. 221. A plan of the wheel is also shov The triangles of velocities at inlet and outlet for any radi are similar to those for inward and outward flow turbines, t velocities v and t^i, Figs. 222 and 223, being equal. TUUBINES 343 The general formula now becomes For maximum efficiency, for a given flow, the water should leave the wheel in a direction parallel to the axis, so that it has no momentum in the direction of v. Fig. 220. Doable Compartment Parallel Flow Turbine. Figs. 221, 222, 223. Then takii^ friction and other losses into account, 9 344 HrDRACILICS The ineloeit^ r will be proportional to the radius, mi th^t if 1 w^ter is to enter and lettTO the wheel withoat shock, the anfltsl^l ^ and m tous^ faiy with the radius. TW rariMion m the form of the r&ne with the imdios is skmr^] by an exaai{iki^ A Jcfowml wheel has an internal diameter of 5 feet mud extremal diameter of B"^. The depth of the wheel is 7 iiicli TTie h^ttd is 15 feet and the wheel makes 55 tevolationf minnre, The flow b 300 cubic feet per second. Ti> find the h^iriMes- power of th< — ^ id, and to design the wh Let Ti be the immn radins, a: ai the inner and oater ciirmnfer r - i'o feet attd r = 2a fi = ^^*> feet and rj = S ff^= 4l25 ket and r^ - & The mean axial Telocity is ind Ft the radii of the wh e^iectiTely. Then = 14"4 feet per sec,^ = 21*5 feet per sec,| =24^5 fe^ per sec. . 1 jv =8'la !i, per sec* *4-*^^ — ^:,_H%^;AML ~I^^M,j , Jt C F^. aa*. TriAa^J^s f»l Tdodlies ai mki md aailet %t tbree dif ei«nt imlit of m Pkimllel Fl^v Tarbine. Taking ^ as 0^80 at each radius, i4'4 ^ ^^ ®^'' 14 4 385 Vt - ^YjT^ ==17^ ft- per sec,, \ , = ^ , i = la f ft, per sec ImcJimatiim of ik^ ran^s at inlei. The triangles of velocities for the ihrt*^^ T*dii n rj, r- are siniwn in Fig. 224. For example, at nftditt^ r, ADC i^ the triangle of Yelocities at inlet and ABC tie TURBINES 345 ian^le of velocities at oatlet. The inclinations of the vanes at let are found from 8*15 tan <^ = 2&f:^UA ' ^^^ ^*^^^ * " ^^ ^ ' ^^"^^ 179 -^21-5 ^"^^ *! = 113^50', . 8'15 tan 4>i = 157^24^5 » ^^^ which «^ = 137** 6'. 2^he inclination of the guide blade at each of the three radii. tan^ = 267' om 'which ^ = 17', tan^i = j^ and ^,=24^30', tan^5 = ?~ and ^2 = 27'* 30'. lo7 The inclination of the vanes at exit, *''°" = 14-4 = 29'36', tan a, = 5'^ = 20° 48', Zl o tan 03= 1^^=18^ 22'. If now the lower tips of the guide blades and the upper tips [)f the wheel vanes are made radial as in the plan, Fig. 221, the inclination of the guide blade will have to vary from 17 to 27^ degrees or else there will be loss by shock. To get over this diflBculty the upper edge only of each guide blade may be made radial, the lower edge of the guide blade and the upper edge of each vane, instead of being radial, being made parallel to the npper edge of the guide. In Fig. 225 let r and R be the radii of the inner and outer crowns of the wheel and also of the guide hlades. Let MN be the plan of the upper edge of a guide blade and let DG be the plan of the lower edge, DG being parallel to HX. Then as the water runs along the guide at D, it will leave "the guide in a direction perpendicular to OD. At G it will leave in a direction HG perpendicular to OG. Now suppose the guide «M; the edge DG to have an inclination fi to the plane of the paper. H then a section of the guide is taken by a vertical plane XX XKrpendicnlar to DG, the elevation of the tip of the vane on this X^e will be AL, inclined at P to the horizontal line AB, awd AG »46 HYDRAULICS will be the intersection of the plane XX with the plane tangei to the tip of the vane. Now suppose DE and GH to be the projections on the plai of the paper of two lines lying on the tangent plane AC ao perpendicular to OD and OG respectively. Draw EF and HI perpendicular to DE and GH respectively, and make each < them equal to BC. Then the angle EDF is the inclination of tl stream line at D to the plane of the paper, and the angle HGE : the inclination of the stream line at G to the plane of the pape These should be equal to and 0^, -JC-Z/fl; btcut&l. of upper edg& ofiwrvt Fig. 225. Plan of guide blades and vanes of Parallel Flow Tarbines. Let y be the perpendicular distance between MN and DG Let the angles GOD and GrOH be denoted by <^ and a respectively Since EF, BC and HK are equal, ED tan^ = y tan)3 (1), and GH tan ^2 = y tan ^ (2). But and Therefore and Again, ^ = cos (a + «^), JL = cos a. GH tan B - cos (a + <^) tan fi. tan ^3 = cos a tan j3 V sina = R .(3), .(4). .(5). There are thus three equations from which a, ^ and fi can ta determined. Let X and y be the coordinates of the point D, O being tta intersection of the axes. TURBINES 347 len cos (a + ^) = •om (5) cosa bstitating for cos (a + ^) and cos a and the known values of »id tan ^s in the three equations (3 — 5), three equations are led with X, y^ and fi as the unknowns. Iving simultaneously X = ri4 feet, y = 2-23 feet, tan /3 = 0-67, «rhich i8 = 34\ Fip. 226. Fig. 227. Fig. 228. e length of the guide blade is thus found, and the constant at the edge DG so that the stream lines at D and G shall he correct inclination. now the upper edge of the vane is just below DG, and the : the vane at D and G are made as in Figs. 226 — 228, ^ and :U^ HTDKJLUIJCS ^ bein^ '^ '¥/ ami I-TT ^ respectively, the wmter wiD more on to the vTkEie wTtho'iit «hf xrk. The plane •:•£ the lower edge of the Tame may now be taken ii VG\ riz, 22^^ and che cErcnlar aectbos DIX, FQ, and GG' at th three nuifi. r, r, and Tj are th^i as in FigSL 226 — ^228. 198. BegoIationofthellowtopanaidlloirtiiiliiniaL To regulate the dow through a paralld flow tarbinesy FontuM placed Juices in the guide passagesy as in Fig^. 229, connected to a ring which could be raised or lowered by three Tertical lodi having nuts at the upper ends fixed to toothed pinions. What Fig. 32!^. Fontaine's Slniee*. Fig. 230. Adjustable guide blades for Parallel Flow Turbine. the sluices required adjustment, the nuts were revolved together by a central toothed wheel gearing with the toothed piniotf canying the nuts. Fontaine fixed the turbine wheel to a hoUoir shaft which was carried on a footstep above the turbine. In son* modem parallel flow turbines the guide blades are pivoted, as in Fig. 230, so that the flow can be regulated. The wheel may ^ made v^-ith the crowns opening outwards, in section, similar to the Girard turbine shown in Fig. 254, so that the axial vdodtf with which the water leaves the wheel may be small. The axial flow turbine is well adapted to low falls with variable head, and may be made in several compartments as in Fig. 220. In this example, only the inner ring is provided with gates. In dry weather flow the head is about 3 feet and the gates of the inner ring can be almost closed as the outer ring will give the full TURBINES 349 »r. During times of flood, and when there is plenty of water, head faJls to 2 feet, and the sluices of the inner ring are ed. A larger supply of water at less head can thus be red to pass through the wheel, and although, due to the shock le guide passages of the inner ring, the wheel is not so efficient, ibundance of water renders this unimportant. cample, A doable compartment Jonval turbine has an outer diameter of and an inner diameter of 6 feet. le radial width of the inner compartment is 1' 9" and of the outer compart- 1' 6". Allowing a velocity of flow of 8*25 ft. per second and supposing the Dum Call is V Q^, and the number of revolutions per minute 14, find the hone- : of the wheel when all the guide passages are open, and find what portion of iner compartment must be shut off so that the horse-power shall be the same > a head of 3 feet. Efficiency 70 per cent. >gleoting the thickness of the blades, the area of the outer compartments j (12-5'-9'5^ = 52*6 sq. feet. „ „ inner „ =^ (9'6»-6«) = 42-8 sq. feet. >tal area=95-4 sq. feet 16 weight of water passing through the wheel is W=95'4 X 62*4 X 3*25 lbs. per sec. = 19,800 lbs. per sec. he horse-power is 19,800 X 1-66 X 0-7 .^.^ °^= 660 = *^®- (saming the velocity of flow constant the area required when the head eet is 40-8x83,000 60x62-6x8x-7 =55-6 sq. feet, • outer wheel will nearly develop the horse-power required. 99. Bemouilli's equations for axial flow turbines. Tie Bemouilli's equations for an axial flow turbine can be ten down in exactly the same way as for the inward and ^rd flow turbines, page 335, except that for the axial flow ine there is no centrifugal head impressed on the water een inlet and outlet. Tien, P^l^^.P^^'f^h,, which, since v is equal to Vi, p V*-2Vt? + t?' u* pi r'~2ViV + V,^ t^» , p V Vi; t^'^p. V.'^t^' Y,v tore — +0:: — Z"^ oZ^ ~'^ lyZ'^ oZ — 1~ + %> tc 2g g 2g w 2g 2g g g g w 2g 2g w ^' 330 BrDRAin.lC3 If U« ie udal and equal to u, as in Fig. 228, 200. Mix^d flow tturbinas* By a taodtiicatioti of tht* sIuijh* uf the viineei of an jnwarl turbiiu*, tht* mixed flow turbine in obtained. In tie inwwt outward flow turbine the water only acti* upon the whfiel wl is raaving in a radial direction, but in tho mix<?d flow tarbi; viuies are so formed that the water aotfi u|)Oii tliem aiais flowing axially. modern Fig. 230, made with the (iirard tu w^th which tbt The axial fl head, and may \ In this example, dry weather flo>^^ ^ diagrammatic section through the whee inner rinff can b^^^^^^> ^^® ^^® ^^ which is vertical. Tlie w Fig. 231. Mixed Flow Turbine. k TU^KIS 351 tlie wheel in a horizontal direction and leaves it ¥@rtically^|] leaTes the di^hargiiig edge of the vanes in different ions. At the upper part B it leaves the vanes nearly lly, and at the lower part A, axially; The vanes are spoon- 1, BM shown in Fig. 2*32, and should be so formed, or in other fc, the inclination of the discharging edge gihould so vary, i^herever the water leaves the vanes it should do so with no ment in a direction perpendicular to the axis of the turbine, ith no velocity of whirl. The regulation of the supply to b] in the turbine of Fig, 231 is effected hy a cylindrical ] ' speed gate between tlie fixed guide blades and the wheel. Fig. 232. Wht^l i^f Mixed Flow Turbme. Sg- 2*i*^ shows a section through the wheel and casing of a le tnixrd flow turbine having adjustable guide blades to ate the How, Fig, 234 shows a half longitndinal section of rbine, and Fig. 23o an outside elevation of the guide blade idng gean The guide blades are surrounded by a lar^e 852 HYDRAULICS vortex chamber, and the outer tips of the guide blades are of variable shapes, Fig. 283, so as to diminish shock at the entrance to the guide passages. Each guide blade is really made in two parts, one of which is made to revolve about the centre C, while the outer tip is fixed. The moveable parts are made so that the flow can be varied from zero to its maximum value. It will be Fig. 233. Section through wheel and guide blades of Mixed Flow Turbine. noticed that the mechanism for moving the guide blades is entirely external to the turbine, and is consequently out of the water. A further special feature is that between the ring B and each of the guide blade cranks is interposed a spiral spring. In the event of a solid body becoming wedged between two of the guide blades, and thus locking one of them, the adjustment ol the other guide blades is not interfered with, as the spring con- nected to the locked blade by its elongation will allow the linf to rotate. As with the inward and outward flow turbine, the mixec flow turbine wheel may either work drowned, or exhaust into I "suction tube." ^ TURBINES 353 For a given flow, and width of wheel, the axial velocity vitli which the water finally flows away from the wheel being the tame for the two cases, the diameter of a mixed flow turbine can )e made less than an inward flow turbine. As shown on page 340, ;lie diameter of the inward flow turbine is in large measure flxed {. JL =L ZFegt Fig. 234. Half-loDgitudinal section of Mixed Flow Tarbine. \gy the diameter of the exhaust openings of the wheel. For the ■ame axial velocity, and the same total flow, whether the turbine is an inward or mixed flow turbine, the diameter d of the exhaust <^>eniiig8 must be about equal. The external diameter, therefore, of the latter wiQ be much smaller than for the former, aivd ticva L. H. 'i^ 354 HTDRAUUGS general dimensions of the turbine will be also diminished, a given head H, the velocity v of the inlet edge being the san the two cases, the mixed flow turbine can be run at a hi angular velocity, which is sometimes an advantage in dri dynamos. o a !2 'a TURBINES 355 J'Wm of ths varies. At the receiving edge, the direction of the is found in the same way as for an inward flow turbine* ABC, Fig, 236, is the triangle of velocities^ and BC is parallel the tip of the blade* This triangle has been drawn for the data he turbine shown in Figs. 23S— 235 ; v is 46*5 feet per secondj from 9 Y = 33*5 feet per second* The angle ^ is 189 degrees. Triangles of V^ctHjeG at raomrmff fdge^ Fig. 236, The best form for the vane at the discharge is somewhat icult to determine, as the exact direction of flow at any point the discharging edge of the vane is not easily found. The aditioii to be satisfied is that the water must leave the wheel ; any component in the direction of motion* The following constmction gives approximately the form of vane. Hake a section through the wheel as in Fig. 237. The outline the discharge edge FG-H is shown. This edge of the vane is til be on a radial plane, and the plan of it is, therefore, f Tsdins of the wheels and upon this radius the section is taken* It ifi now neoeasary to draw the form of the stream lines, as would be approximatelyj if tlie water entered the wheel ly and flowed out axially, the vanes being removed* Ihvide 04, Fig. 237, at the inlet, into any number of equal say four, and subdivide by the points a, i, /i, e. Tike any point A, not far from Cj as centre, and describe circle MMi touching the crowns of the wheel at M and M,. AM and AMi, ]>i^w a flat curve Mi Mi touching the hues AM and AM, in M Mt respectivety, and k% near as can be estimated, perpendicular j 856 HYDRAULICS to the probable stream lines through a, 6, d, e, which can sketched in approximately for a short distance from 04. Taking this curve MMi as approximately perpendicular to stream lines, two points/ and g near the centres of AM and A are taken. Fig. 287. Let the radius of the points g and / be r and n respectively- If any point Ci on MMi is now taken not far from A, tfe peripheral area of Mci is nearly 2irrMci, and the peripheral art* of MiCi is nearly 2w-riMiCi. On the assumption that the mean velocity through MiM ^ constant, the flow through Mci will be equal to that throng* MiCi, when, Mci.r = MiCi.ri. TURBINES 357 If, therefore, MM| ib divided at the point Cj ao that Mj_Ci _ r thB point rj will approximateljr be on the stream line through c. If now when the stream line cci is carefully drawn in^ it is perpendicular to MMi, the point c, cannot be much in error. A near€*r approximation to d can be found by taking new values for r and n, obtained by moving the points / and g bo that they more tiearly coincide with the centres of CjM and cjMi. If the two corves are not perpend icularj the cui've MM^ and the point Ci ajre not quite correct, and new vsiluea of r and Ti wiU have to be obtained by moving the points / and g. By approximation Cj can "be thus found ^vith considerable accuracy. By diawing other circles to touch the crown of the wheelSj the curv«i MrM«, M^Ma etc. normal to the stream lines, and the points 45i, Cs, etc. on the centre stream line, can be obtained. The curve 22, thereforej divides the stream lines into equal parf^ Proceeding in a similar manner, the curves 11 and 33 can be 'I'htaitied^ dividing the stream lines into four equal parts, and iii^-^ again subdivided by the curves oa, hh, rfd, and ee, which inttfrst»ct the outlet edge of the vane at the points F, G, H and e Tie*pectively. To dei ermine the direction of the tip of the varw at points mi the <^i^chirg{ng edge. At the pcdiUs F, Gr, H, the directions of the <to?am lines are known, and the velocities tip, t%, uu can be found, «mce the flows through 01, 12, etc, are equal, and therefore uvR^qt = Uii'RiTtui = tiuRiW't' Q 8tr- Draw a tangent FK to the stream line at F, This is the inter- *^*cti«ui, with the plane of the paper, of a plane perpendicular to ^ ^U' p*»|>er and tangent to the stream line at F. The piiint F in the plane of FK m moving perpendiculai* to the Wane of the paper with a velocity equal to *w,Rm, «»* being the ^n^kr velocity of the wheel, and H, the radius of the point F. If a circle be struck on this plane with K as centre, this circle ^ay be taken as an imaginary discharge circumference of an *^^ward Row rnrbinej the velocity v of which is mR^^ and the tip of ^^ = '* Made i& w have such an inclination, that the water shall '**>chiirgo radially* i.*;. along FK, with a velocity nr . Turning this 'ff'cle >nto the plane of the paper and drawing the triangle of ^>--l^K.nties FST, the inclination a^ of the tip of the blade at F in ^hm pliiTie FK is obtained, 358 HYDRAULICS At G the stream line is nearly vertical, but «R» can be set ont in the plane of the paper, as before, perpendicular to uq and the inclination oeq, on this plane, is found. At H, an is found in the same way, and the direction of the vane, in definite planes, at other points on its outlet edge* can be similarly found. Fig. 288. Fig. 239. Sections of the vane by planes OGby and OiHd. These are shown in Figs. 238 and 239, and are determined as follows. Imagine a vertical plane tangent to the tip of the vane at inlet. The angle this plane makes with the tangent to the whed at b is the angle <^, Fig. 236. Let BC of the same figure be the TURBINES 359 plan of a liomontal line lying in this plane, and BD the plan of tbe radius of the wheel at 6. The angle between these lines is y. Liet 0be the inclination of the plane Oiyh to the horizontal. From D, Fig, 236, set ont DE, inclined to BD at an angle ft aiul intersecting AB produced in E, and draw BF perpendicular to CB. Make BF equal i<.\ BE and join CF. Now set out a triangle BGDj ha^nng BG equal to CF, DiG eqtial to DE, and the angle BGDj a right angle* In the figure Di and D happen to coincide. The angle BGD is the angle yi, which the line of intersection of tte plane OG^, Fig, 237, witli the plane tangent to the inlet tip of tlie vane, makeB irith the radius O^. Ill Fig- 238 the inclination of the inlet tip of the blade is yi as To detamiine the angle a at the outlet edge, resolve tt©, Fig, *2S7\ along and perpendicular to OS, ti^ being the component along (Xt, Draw the triangle of velocities DEF, Fig. 238, Tlte tangent to the vane at D is parallel to FE, In the same way^ the section on the plane Hti, Fig. 237, may be letentiinetl ; the inclination at the inlet is y^. Fig, 239. MU*d Jl^tt hirhim worki^ig m open stream, A double turbine rnrking in open stream and discharging tlirough a suction tube shown in Fig. 2-1^^. This is a convenient arrangement for ^^oderately low falls. Turbines, of this class, of 1500 horne- p(n«rer, having four wheels on the same shaft and working under m head of 25 feet, and making 150 revolutions per minute, have rt^'einly tieen installed by Messrs Escher Wyss at Wangen an der A a re in SwitxcTland, 201. Cone ttirbltie. Another type of inward flow turbine, which is partly axial and ly radial, \b shown in Fig. 241, and is known aa the cone Mne, It has been designed by Messrs Escher Wyss to meet \ie demand for a turbine that can be atlapt-ed t« variable flow^s. l*he example shown has been erected at Cusset near Lyons and ikes 120 revolutions jx?r minute. Tlie wheel is divided into three distinct compartments, the applj' of Waaler being regulated by three cylindrical sluices S, Si S*. The sluices S and 8i are each moved by three vertical ?^uch as A and Aj which carry racks at their upper ends. _- -wo aluic^es move in opposite directions and thus balance ch oilier* The sluice Sa is normally out of actioni the upper 360 HYDRAULICS compartment being closed. At low heads this upper compartrxne is allowed to come into operation. The sluice Sj carries a "ma which engages with a pinion P, connected to the vertical sha. ^ Fig. 240. The shaft T is turned by hand by means of a worm and wheel W. When it is desired to raise the sluice Sg, it is revolved by means of the pinion P until the arms F come between collar* D and E on the spindles carrying the sluice Si, and the sluice S» then rises and falls with Si . The pinion, gearing with racks on A and Ai , is fixed to the shaft M, which is rotated by the rack B gearing with the bevel pinion Q. The rack R is rotated by two connecting rods, one of which C is shown, and which are under the control of the hydraulic governor as described on page 378. The wheel shaft can be adjusted by nuts working on the square-threaded screw shown, and is carried on a special collar bearing supported by the bracket B. The weight of the shaft is partly balanced by the water-pressure piston which has acting underneath it a pressure per unit area equal to that in the supply chamber. The dimensions shown are in millimetres. TDRBINES 36t 362 HTDEAPLICS 202. Effect of changing the direction of the guide bladi|| when altering the flow of inward flow and mixed floi turbine 3« As long as the velocity of a wheel remains constant, fhj backward head impreased on the water by the wheel is the sam^ and the pressure head, at the inlet to the wheel, will remm practically constant as the guides are moved^ The velocity d flow U» through the guides, will, thereforej remain constajit but as the angle ^, which the guide makes with the tangent to the wheel, diminishes the radial component u^ of U, diminishes^ Fig. 242. Let ABC, Fig. 242, he the triangle of velocities for full opariafi . and suppose the inclination of the tip of the blade i.^ made paraM to BC. On turning the guides into the dotted position* tlie incti* nation being «^'), the triangle of %^elocities is ABCi, and the relati« velocity of the water and the periphery of the wheel is now BC^ which is inclined to the vane, and there is, consequently, loss <te to shock. It vnW be seen that in the dotted position the tips of the gtt>«i* blades are some distance from the periphery of the wheel and it ii probable that the stream Hues on leaving the guide blades folio* the dotted curves SS, and if so, the inclination of these ^tre^B lines to the tangent to the wheel will be actually gre-aler thiiD^']» and BCi will then be more nearly parallel to BC. The lt»ss owtf be approximated to as follows : As the water enters the wheel its radial component will remwfi nnaltereilj but its direction will be suddenly changed from flCita BCj and it-a magnitude to BCj; CiCt is drawn parallel to Ai A velocity equal to dCs has therefore to be suddenly impr the water* On page 68 it has been shown that on oertain assumptiuui ^ TURBINES 863 t ^when the velocity of a stream is suddenly changed to 179 is 2g ' it is equal to the head due to the relative velocity of ?«. CiC is the relative velocity of BCi and BCs, and therefore 1 lost at inlet may be taken as a coefficient which may be taken as approximately unity. . Effect of diTniniiihiTig the flow through turbines on Locity of exit. p-ater leaves a wheel radially when the flow is a maximum, lot do so for any other flow. angle of the tip of the blade at exit is unalterable, and if Uo are the radial velocities of flow, at full and part load ively, the triangles of velocity are DBF and DBFi, Fig. 243. part flow, the velocity with which the water leaves the s tt,. K this is greater than u, and the wheel is drowned, 3xhaust takes place into the air, the theoretical hydraulic 3y is less than for full load, but if the discharge is down a tube the velocity with which the water leaves the tube is iu for full flow and the theoretical hydraulic efficiency is for the part flow. The loss of head, by friction in the iue to the relative velocity of the water and the vane, 8 less than at full load, should also be diminished, as also, 3 of head by friction in the supply and exhaust pipes, chanical losses remain practically constant at all loads. Br^T ! i^A Fig. 243. Fig. 244. fact that the efficiency of turbines diminishes at part loads lerefore, in large measure be due to the losses by shock icreased more than the friction losses are diminished, suitably designing the vanes, the greatest efficiency of flow and mixed flow turbines can be obtained at some of full load. 304 HTDBAULU^ 204. Begulaiion of the £ow by cylindriGal gata&. \\ lieu tlie &]>eed of the turbine is adjusted by a gate betw«i the guidefi and the wheel, and the fio\r m less than the nomtftJ, tk velfjtcity XJ with which the water leav^es the gmde is alteral ic mBgnitade bat not in direction* Let ABC be the triangle of velocitiaSj Fig* 244> whem the iiowii nonimL IM tht* How bo diminished until the velocity with wMchlit water leRves the guides is Uoj equal to AD, llien BD is the rsdial velocity of flfi Draw DK pan the VMIB a sodden reiociiyi the water, and there i» a ] To keep the velocity intrmluoed the gate shown connected to the guide bla bladee as well as tlie gat« & ^r of Uo and r, an*i «s ii lil for the water to mmv aii o KD must be im k(KDY equal to % early constant Mr Swain ha 245. The gate g i* npif to adjust the flow tW p\k L The effe<;tive width of i| guides is thereby inarle approximately pmpirtional to thetpiantitf of flow, and the velocity D remains more nearly constant. If the gate is raised, tlie width h of the wheel opening will be greater than t, the widtli of the gate opening, and the radial velocif)'*! Fig. 2iS. SwalQ Ga«. Fig, 246, TURBINES 365 wheel will consequently be less than the radial velocity u guides. If n is assumed constant the relative velocity of )r and the vane will suddenly change from BC to BCi, Or it may be supposed that in the space between the id the wheel the velocity U changes from AC to ACi. k (CC,)^ OS8 of head will now be 2flr The form of the wheel vanes between the inlet and f turbines. form of the vanes between inlet and outlet of turbines »e such, that there is no sudden change in the relative of the water and the wheel. ider the case of an inward flow turbine. Having given 3 the vane and fixed the width between the crowns of the le velocity relative to the wheel at any radius r can be follows. any circumferential section ef at radius r, Pig. 247. Let ? effective width between the crowns, and d the effective " between the vanes, and let q be the flow in cubic feet id between the vanes Ae and B/. RelatiTe Telocity of the water and the Tanes. Fig. 348. 366 HTDRAUUCS The radial velocity through e/is Find by trial a point O near the centre of ef such that a circle drawn with as centre touches the vanes at M and Mi. Suppose the vanes near 6 and / to be struck with arcs of cirdcB. Join to the centres of these circles and draw a curve MCMi touching the radii OM and OMi at M and Mi respectively. Then MCMi will be practically normal to the stream lines through the wheel. The centre of MCMi may not exactly coincide with the centre of e/, but a second trial will probably make it do so. If then, h is the effective width between the crowns at C, h . MMi . ^v = q, MMi can be scaled off the drawing and Vr calculated. The curve of relative velocities for varying radii can then be plotted as shown in the figure. Fig. 249. It will be seen that in this case the curve of relative velocities changes fairly suddenly between c and A. By trial, the vanes should be made so that the variation of velocity is as unifonn as possible. If the vanes could be made involutes of a circle of radius &» ( TURBINES 367 in Fig. 240, and tlie crowns of the wheel parallel, the relative locity of the wheel and the water would remain constant, lis form of vane is however entirely unsuitable for inward >w turbines and could only be used in very special cases for itward flow turbines, as the angles ^ and which the involute akes with the circumferences at A and B are not independent, IT from the figure it is seen that, 8ind = ?5 r nd sin ^ = g^ , ainO R r -T— -T = - . 8m9 r The angle must clearly always be greater than <^. 206. The llmitiTig head for a single stage reaction arbine. Reaction turbines have not yet been made to work under heads ligher than 430 feet, impulse turbines of the types to be presently Lescribed being used for heads greater than this value. Prom the triangle of velocities at inlet of a reaction turbine, .g. Fig. 226, it IB seen that the whirling velocity V cannot be greater than V + u cot ^. Assuming the smallest value for ^ to be 30 degrees, and the oaximum value for u to be 0*25 v 2grH, the general formula 9 becomes, for the limiting case, viv-^2j3s/R)=e.g.R. K t? is assumed to have a limiting value of 100 feet per second, rhich is higher than generally allowed in practice, and e to ye 0*8, then the maximum head H which can be utilised in a one (tage reaction turbine, is given by the equation 25-6H- 346 n/H = 10,000, from which H = 530 feet. 207. Series or multiple stage reaction turbines. Professor Osborne Reynolds has suggested the use of two or more turbines in series, the same water passing through them successively, and a portion of the head being utilised in each. For parallel flow turbines, Reynolds proposed that the wheels S6S HYDRAUUCS and fixed blades be arranged alternately as shown in Fig- S This arrangeinentj although not used in water turbines, is largely caed in 3*eactioii steam turhinos. ^^^^^^^ ^ -^^*==^w J J E^'t^ ^ ^ J\ .^^^si<^^^ Fig. 260. ImihjtdL Figfl, 251, 252. AiIaI Flow Impulse TorlJim. * Taken from Prof, R^yuolda* Scimtijis Papyri, VoL i. TURBINES 369 Fig, 353. L impulse turbines. md turbine. To overcome the difficulty of diimnution of hf with diroinatioii of flow, (intToduced, about 1850, the krrmtion or partial admission flead of the water beitig to the wheel throughout jle circumference as iu the turbines, in the Girard it 18 only allowed to enter ^1 through guide passages I diametrically oppcis^ite ( as shown in Figs, 2*52— tn the first two, the flow is knd in the last radiah [Fig, 252 abo%'e the guide crown are two quadrant-shaped l^nted 2 and 4^ which are made to rotate about a vertical ^Qmns of a toothed wheel. When the gates are over the tJB 2 and 4, all the guide passages are open, and by turning in the direction of the arrow, any desired number of the can be clost^d. In Fig. 2*S4 the variation of flow i^ by means of a cylindrical quadrant-shaped sluice, which, c previous case^ can be made to close any desired number gnide paseages* Several other types of regulators for turbines were introduced by Girard and others, . 25^^ shows a regulator employed by Fontaine. Above the l>lade6, and fixed at the opposite ends of a diameter DD, ro indianibber bands, the other ends of the bands being Sled to two conical rollers. The conical rollers can rotate artialB, formed on the end of the arms which are connected coocbed wheel TVV, A pinion P gears with TW, and by Ig the spindle carrying the pinion P, the rollers can be made irrap, or wT-ap up, the indiarubber band, thus opening or m the guide passages. I the Girard turbine is not kept full of water, the whole of reliable head is converted into velocity before the water the wheel, and the turbine is a pure impulse turbine, prevent loss of head by broken water in the wheel, the air . be freely admitted to the buckets as shown in Figs» 252 A. small heads the wheel must be horizontal but for large it may be verticaL b clase of turbine has the disadvantage that it. c^^^'&.c^'^ 1 370 HYDBAUUCa rati drowned, and tence must alwajB b© plaeod aboTe the water. For low aod variable heads the full head cannot thef©^ bii utilisod, for if the wheel is to be clear of the tail water, amount of hmd equal to half the width of the wheel mast necessity be lost. Fig» 26-t. Girard Hftdial flow Impulse TurWoe, To overGome this difficalty Girard placed tJie wheel tight tube, Fig. 254, the lower end of which is below ihc level, and into which air is pumped hy a small auxiliary the pressure being maintained at the necessary value t* surface of the water in the tube below the wheel. TURBINES an t H be the total head above the tail watar level of the snppl^ ~ the pressure head dae to the atmospheric preasare^ H, lee of the centre of the wheel below the sarface of the i?atery and h^ the distance of the surface of the water in Inbe below the tail water level. Then the air-pressure in tttbe must be W ' the head causing velocity of flow into the wheel is^ therefore, w Xw 7 b that wherever the wheel is placed in the tube below the tail the full fall H is utilised. Ihis system, however, has not foand favour in practice, owing difficulty of preserving the pressure in the tube* The form of the vanee for impulfle turbines, neg- ng fkictloiL Jhe receiving tip of the vane should be parallel to the relative Atf Vr of the water and the edge of the vane, Fig< 255. t exit the relative velocity iv, Fig. 256, neglecting friction, be equal to the relative velocity V,. at inlet, ' the angle a which the tip of the vane at exit makes with lirectiou of Vi is known the triangle of velocities can be drawn, fffeting out DE equal to Vj and EF at an angle a with it and 1 to Vr. Then DF is the velocity with which the water leaves irbeeh W the aidal flow turbine Vi equals t*, and the triangle of ntiee at e3dt is AGB, Fig. 255. t the velocity with which the water leaves the wheel is Ui, iieorotjcal hydraulic eflSciency is E- H "^ IP idependent of the direction of Uj . »iild be observed, however, that in the radial flow turbine of the section of the stream by the circuniference of the for a given flow, will depend upon the radial component of \xd in the axial flow turbine the area of the section of the by a plane perpendicular to the axis will depend upon the I component of Uj, That is, in each case the area ^vill depend the component of Ui perpendicular to Vi. 372 HYDRAULICS Now the section of tlie stream most not fill the outlet area''o{ the wheels and the minininTn area of this outlet so that it is just not filled will clearly be obtained for a given value of XJi when Ui is perpendicular to Vi*, or is radial in the outward flow and axial in the parallel flow turbine. For the parallel flow turbine since BC and B6, Fig. 255, are equal, Ui is clearly perpendicular to Vi when v = ^ = ^'J2gRcosO, and the inclinations a and ^ of the tips of the vanes are equaL Figs. 255, 256. Fig. 257. If R and r are the outer and inner radii of the radial flow turbine respectively, R r * It is often stated that this is the condition for mazimam effieieiiOT bat it ctif is 80, as stated above, for mazimam flow for the given maohine. Tat only depends upon the magnitude of Ui and not apon its direotion. Far Ui ta be radial TURBINES Yr~Vi sec a 373 sec a. Y Y and if u is made equal to ^, Yr from Fig, 255 is equal to ^ sec <^a and therefore, sec ot = ^ «©c 9. 210. Txian^es of velocity for an axial flow Impulsa tur- bine confiidering friction* The* velocity witb which the water leave© the guide passages may be takeu as from 0'94r to 0'97 \^2^H, and the hydraulic losses ia tlie wheel are from 5 to 10 per ceut* If the angle between the jet and the direction of motion of the vane is taken as 30 degrees, and U ia assumed as 0"95 n/2^H, and v ms 0*45 v^2gH, the triangle of veloeitiL^s is ABC, Fig. 257. lUdng 10 per cent, of the head as being lost in the wheel^ the relative velocity tv at ©sit can be obtained from the expression % ^ K now the velocity of exit Ui be taken as 0'22s/2^H, and circles with A and B as centres, and Ui and iv as radii be described, intersecting in D, ABD the triangle of velocities at exit is obtaijiedj and U] is practically axial as shown in the figure. On these assumptions the best velocity for the rim of the wheel is iber<?fore "45 •J2gB. instead of *5 *J2gR. The head lost due to the water leaving the wheel with velocity u i» *M8H^ and the theoretical hydraulic efficiency is therefore ^"2 per cent. The velocity head at entrance is 0*9025H and, therefore, *097H lm& been lost when the water enters the wheel. The efficiency^ neglecting axle friction, will be H - O'lH - 0O48H - O'OQTH e = g = 76 per centt nearly. 21L Imptdse turbine for bigli beads. For high heads Girard introduced a form of impulse turbine, of which the turbine shown in Figs. 258 and 259, is the modem deirelopment. J The water instead of being delivered through guides over an mrc of a circle, is delivered through one or more adjustable nozzles. ^ib TURBINES 876 pie shown, the wheel has a mean diameter of 6*9 feet -revolations per minute; it develops 1600 horse- lead of 1935 feet. pipe is of steel and is 1'312 feet diameter. the orifices has been developed hy experience, and )re is no sadden change in the form of the liquid juently no loss due to shock. of water to the wheel is regulated by the sluices 258, which, as also the axles carrying the same, the orifices, and can consequently be lubricated ae is at work. The sluices are under the control ►vemor and special form of regulator. d of the turbine tends to increase the regulator 11 crank lever and partially closes both the orifices. I speed of the turbine causes the reverse action to igh peripheral speed of the wheel, 205 feet per ?s a high stress in the wheel due to centrifugal ng the weight of a bar of the metal of which the le square inch in section and one foot long as tress per sq. inch in the hoop surrounding the /= 3-36. t;^ 9 = 4400 lbs. per sq. inch. iger of fracture, steel laminated hoops are shrunk ery of the wheel. arrying the blades is made independent of the disc that it may be replaced when the blades become ri entirely new wheel being provided. of the vanes at the inner periphery is 171 feet per herefore, 0*484 v2gH. ity U wth which the water leaves the orifice is 5(7 H, and the angle the jet makes with the tangent K) degrees, the triangle of velocities at entrance is and the angle <f> is 53'5 degrees. 1*1 of the outer edges of the vanes is 205 feet per iming there is a loss of head in the wheel, equal to '^9 2^ -0-06H, = 123*5 ft. per second. 876 HTDRAULIGS If then the angle a is 30 degrees the triangle of velocities ! exit is DBF, Fig. 261. . The velocity with which the water leaves the wheel is th< Ui = 95 feet per sec., and the head lost hy this velocity is 140 fe or -OTSH. Fig. 260. Fig. 261. The head lost in the pipe and nozzle is, on the assumpti( made above, H-(0-97)»H=0-06H, and the total percentage loss of head is, therefore, 6 + 7-3 + 6 = 19-3, and the hydraulic efficiency is 80*7 per cent. Fig. 262. Pelton Wheel. TURBINES 377 The actual efficiency of a similar turbine at full load was found by experiment to be 78 per cent., which allows a mechanical loss of 2*7 per cent. 212. Pelton wheel. A form of impulse turbine now very largely used for high heads is known as the Pelton wheel. A number of cups, as shown in Figs. 262 and 266, is fixed to a wheel which is generally mounted on a horizontal axis. The water is delivered to the wheel through a rectangular shaped nozzle, the opening of which is generally made adjustable, either by means of a hand wheel as in Fig. 262, or automatically by a regulator as in Fig. 266. As shown on page 276, the theoretical efficiency of the wheel is unity and the best velocity for the cups is one-half the velocity of the jet. This is also the velocity generally given to the cups in actual examples. The width of the cups is from 2J to 4 times the thickness of the jet, and the width of the jet is about twice its thickness. The actual efficiency is between 70 and 82 per cent. Table XXXVIII gives the numbers of revolutions per minute, the diameters of the wheels and the nett head at the nozzle in a number of examples. TABLE XXXVIII. Particulars of some actual Pelton wheels. Head infect Diameter of wheel (two wheels) Bevolations per minute r U H. p. 262 89-4" 875 64-5 129 500 *233 7" 2100 64 125 5 *197 20" 650 66-5 112 10 722 89" 650 111 215 167 882 60" 800 79 156 144 ♦289 54" 810 73 186 400 508 90" 200 79 180 300 • Pieard Pictet and Co., the remainder by Eschar Wyss and Co. 213. Oil pressure governor or regulator. The modem applications of turbines to the driving of electrical ^Xiachinery, has made it necessary for particular attention to be X^^d to the regulation of the speed of the turbines. The methods of regulating the flow by cylindrical speed gates ^»^:id moveable guide blades have been described in connection with 378 HTBRAULICS various turbines but the means adopted for moving the gatae i gtiidt?8 have not been discussed, ^M Until recent years some form of differential goven^H almoat entirely used, but these have been almost corople superseded by hydranlic and oil governors. Figs. 26^3 and 264 show an oil governor, aa construct Messrs Escher Wym of Zurich. ~ FigB. 263. Mi. Oil PresBure Begalator for TarbineB. A piston P having a larger diameter at one end than \ other^ and fitted with leathers I and ^i, fits into a double cylbS Ci. Oil under pressure m continuously supplied through app* into the annulus A between the pistons, while at the back of \ iarge piston the pressure of the oil is determined by the refuU TURBINES Fig, 265. ^ppo^ the regulator to be in a definite poBitioiij the spacej »Iiinti the krge piston being full of oil, and the rbine rtinning at its normal speed. The valve Y enlarged diagrammatic section is ehowii in 1 265) will be in such a position that oil cannot or e^icape from the large cylinder, and the sure in the annular ring betfcveen the pistons ill keep the regulator mechanism locked. If the wheel increases in speed, due to a inntioD of load, the balls of the spring loaded remor G move outwards and the sleeve M For the moment, the point D on the lever is fixed, and the lever turns about D as a ftilerum, and tljus raises the valve rod KV, This Howe oil under pressure to enter the large blinder and the piston in consequence moves to ft ' - 1 moves the turbine gates in the manner descnbed later, Hi moves to the right, the rod R, which rests on the ^edge W connected to the piston, falls, and the point D of the Bver MD consequently falls and brings the valve V back to its iginal ptjsition. The piston P thus takes np a new position :irresponding to the required gate opening. The speed of the irbine and of the governor is a little higher than before, the > in speed depending upon the sonsitiverjess of the governor* the other hand, if the speed of the wheel diminialies, the M and also the valve V falls and the oil from behind the irge piston escapes through the exhaust E, the piston moving the left. The wedge W then lifts the fulcrum D, the valve V I automatically brought to its central position^ and the piston P ikes up a new position, consistent with the gate opening being ifficient to supply the necessary water required by the wheeh A hand wheel and screw. Fig* 264, are also pro\^ded, so that gates can be moved by hand when necessary. Tlie piston P is connected by the connecting rod BE to a crank ^f which rotates the vertical shaft T. A double crank KK is OBnected by the two coupling rods shown to a rotating toothed wheel R, Fig. 241^ turning about the vertical shaft of the turbine, ^and the movement, as described on page 360, causes the adjusts &nt of the speed gates. 214. Water pressure regulators for impulse turbines. Fig. 266 shows a water pressure regulator as applied to regulate fche flow to a Pelton wheel, Tlie ar^a of the supply noazle is adjusted by a beak B which TURBINSS 381 : the centre O. The pressare of the water m the icting on this beak tends to lift it and thos to <ipei] The piston P, working in a cylinder C, is also acted mder side, by the pressare of the water in the supply connected to the beak by the connecting rod D£L :he piston is made sofficiently large so that when the {ton is relieved of pressare the pmll on the coDnecting nt to close the orifice. p conveys water nnder the same presBore. Uj the ;h maybe similar to that described in ocmnectKiD with re governor, Fig. 265. rod passes throngh the top of the cylinder.and carries screws on to the square thread cut on the r>d. A 268, which is carried on the fixed f alcnuii <:, iut made 1 the piston. A link /A oonnect£ ef with the lerer M of which moves with the governor d*jei'e and the is connected to the valve rod XV. Tlje vaJie V m neutral position. ■o; "Sleteve (a now the speed of til*: turf^;jj*r Vj iu*:r*ii»*i^. TLe the lever MS tumfc i^'yjvi ti**: f ulcmin The va;v*: V laii*s aiid o^iens the Til*: ;;r<i*«ur»r Oli th*- pist^.iij the U'/ZZ^l^: \VU^ di»iM;iKnTr,p he puRogu nw^ :t L-ft^^ a^aiii tht- A/f aod C:io«* th*: va}Tf T. J^ vveacbefd. If the igteed cf the 382 HTBHAULICS governor decreases the governor sleeve falls, th© valve T and wfLter pressure ia admitted to the top of the piaton, which i then in equihbriuin, and the pressure on the beak B causes it i move upwards and thus open the noaszle* Hydraulic ^^alve for water regulatfjr* Instead of the simple piston valve controlled mechanically, Messi*s Escher Wyss use, fo hi^h heads, a hydraulic double-piston valve Pp, Fig* 260. This piston valve has a small bore through its centre by meansi of which high pressure water which is admitted below^ the valve j can pass to the top of the large piston P, Above the piston is \ small plug valve Y which is opened and closed by the governor. Fig. S69* Hjdranlio valve for autornKtle regalfttbn. If the speed of the governor decreases, the valve V is open^ thus allowing water to escape from above the piston valve, and tif pressure on the lower piston p raises the valve. Pressure water if thus admitted above the regulator piston^ and the prefisiire on tk beak opens the nozzle* Aa the governor falls the vslre V c]n(S«» the exhaust is throttledj and the pressure above the piston P riitf^ "WTien the exhaust through V is throttled to such a degree dm the pressure on P balances the pressure on the under face of tkt piston pf the valve is in equilibrium and the regulator pfetoft tf Joeked* rittHy TUHBINKS 383 the speed of the ^Fernor increases, the valve V is closed^ and tlie excess pressure on th^ tipper face of the pieton valve eaojM^s it to descend, thus connecting tlie regulator cylinder to ejEhatist. The pressure on tlie under face of the regulator piston tbem closes the uozzle. Filter. Between the conduit pipe and the governor valve V, is placed a filter, Figs. 270 and 271, to remove any sand or grit contained in the water. Wit 111 n the cylinder, on a hexagonal fi-anje, is stretched a pn'eee of canvas. Tlie water enters the cylinder by the pipe E, and after passing through the canvas^ enters the central perforated pipe and leaves by the pipe S. FiffK 370, 27L Water Filter for Impuke Turbine Reguktor. To clean the filter while at work, the canvas frame is revolved by me^ns nf the handle shown, and the cock R is opened. Each mde of tlie hexagonal frame is brought in turn opposite the chamber A, and water tlowB outwards through the canvas and ibroa^h the cock R^ carrying away any dirt that may have collected outside the canvas. Af^iliary valve to prei^eni hammer action. When the pipe line ^ long an auxiliary valve is frequently fitted on the pipe near to nozzlet which is anfcomatically opened by means of a catai^ct motion* as the nozzle closes, and when the movement of the nozxle beak is finished^ the valve slow^ly closes again. If no such provision is made a rapid closing of the nozzle ixieans that a large mass of water must have its momentum '^ly changed and very large pressures may be set up, or in : words hammer action is produced, which may cause fracture of the pipe. WTien there is an abundant supply of water, the auxiliary valve is connected to the piston rod of the regulator and opened and elo3*4*d as the piston rod moves, the valve being adjusted so tliat the opening increases by the same amount that the area of tbe orifice dimiiiishes* * See Enffin^ttt VoU ic, p. 265, 384 HYDRAULICS If the load an the wheel does not vmiy through a large m the quantity of water wasted is not large. 215. Hammer blow in a long turbine supply pipe. Let L be the length of the pipe and d its diameter. The weight of water in the pipe is 4 Let the velocity change by an rate of change of momentum is the lower end of the column of w be applied equal to this. Therefore "-}• ^nnt dv in time cL Then &9 -J and on a cross sectioii rf ' in the pipe a force P tmat ft' lepth of the orifice and di it« 1 the centre about which UitS Referring to Fig. 266, let b be width. Then, if r is the distance of 1 beak turns, and r, h the distance ot the elf ising edge oi the ^►ejik from this centre, and if at any moment the velocity of the piston is r» feet per sei^ond, the velocity of closing of the beak ^nll be r In any small element of time ct the amount by which the nozzle will close is BJ- r^Fi Bt Let it 1^ ass^umed that U* the velocity of flow through tis nozzle, remains constant. It will actually vary, due to tk ivs^istances varying with the vel^xnty, hut unless the pipe is very long the errv^r is n*>t gr^^t in neglecting the variation. If then r is the velocity in the pipe at the commencement of this element of time and r - c r at the end of it, and A the area of the pipe, r-A=fr.d,.U ......a) and (r-cr>A-(t-^d(yd^.U Subtracting \2i tnmi (I), r {il ct '' r A .(3). TURBINES 385 If W is the weight of water in the pipe, the force P in pounds it will have to be applied to change the velocity of this water cv in time dt is g of Therefore p^Wr.^o id the pressure per sq. inch produced in the pipe near the ozzle is W r, diUi^o ^ = 7r"A» • Suppose the nozzle to be completely closed in a time t seconds, nd during tlie closing the piston P moves with simple harmonic notion. Then the distance moved by the piston to close the nozzle is br md the time taken to move this distance is t seconds. The maximum velocity of the piston is then vbr ad substituting in (3), the maximum value of -r is, therefore, dv_ vbrridiU ot" 2trirA ' ad the maximum pressure per square inch is ^* 2gtA' 2g.t.A' 2t' gA' here Q is the flow in cubic feet per second before the orifice e^an to close, and v is the velocity in the pipe. ExamvU, A 500 horse-power Pel ton Wheel of 75 per cent, efficiency, and worki ng Bder a head of 260 feet, is sapplied with water by a pipe 1000 feet long and 8" diameter. The load is suddenly taken off, and the time taken by the igtilator to close the nozzle completely is 6 seconds. On the assumption that the nozzle is completely cloned (1) at a uniform rate, id (2) with simple harmonic motion, and that no relief valve is provided, dtermine the pressure produced at the nozzle. The quantity of water delivered to the wheel per second when working at fall Dwer is ^ 500x33,000 „ _ , . , , Q= 260x62-4x'76x60 =^^-^ <="'»* '*«'• The weight of water in the pipe is W = 62-4 X J. (2-25)5x1000 = 250,000 lbs. L. H. ^"^ 386 The Telodty is ^^ = 5-^6 ft. per seo. In case (1) the total pT«sffii]:« acting on the hyvet en^ of thm 4»lixmii nf «il»| the pipe is = 8200 lbs. The presson per sq. inch ia y = - ^14 5 Iba. per sq, iiioh^ w W I? In case (2) p^=^ ^ -^^ =22-1 u per wq. exam: s. (I) Find tteHMMWiic^l horne^pow* diameter, usingt 9XCXXM)00 gailons of m of 25 feet. ! an oversliot watet-wbed S3 &et | • per 24 hours under a IxM had (2) An over^ot water-wheel has i imetez^ of 24 Ceet, and ma^aif^ revolutions per minute. The vekx'ity oi trie water a.^ it enters the bucketi is to be twice that of the wheel" a periphery. If the angle wliicli the water make« with the periphery is to be U degrees, find the fhrection of the tip of the bucket, and the relkfeive velodty of the water and the bucket. (8) The sluiee of an overshot water-wheel 12 feet diameter \& vertioBj above the centre of the wheel. The surface of the water in the shm* channel is 2 feet 6 ineheH above the top of tlje wheel and the centre of Uae sluice o{)ening is B iuclies above the top of thc^ wheel. The velocity of ^ wheel periphery is to be one-half that of the water aa it enters the backets Determine the nuuibt^r of rotationw of the whoeit the point at wludi the water enters the buckets, and tlie direction of the edge of tiie bucket. (4) An overt^iu^t wheel 25 feet diaujeter having a widtJi of 5 feeUMui depth of crowns 12 inches, receiveit 450 cubic feet of water per minute^ ao(i makes 6 revolutions jier minute. There are 64 buckets. The water enters the wheel at 15 degreca from the cro^Ti of the wlieel witli a velocity e^^iial to twice that of the periphery, and at an angle of D degrees with the tan gent to the wheel. Assuming the buckets to be of the form shown in Fig, 180, the lengtb of the radial i>ortiou being one -half tlie length of the outer face of the bucket, tind how much water enters each bucket, and, allowing for oeatn fugal forces, the point at wliich the water l)egins to leave the buckets^ (5) An overi^hot wheel S2 feet diameter has Rhrouds 14 inches deep^ and is required to give 9 horse ■i>ower when making 5 revolutions per micafe- Assuming tlie buckets to be one- third filled with water and of the same form as in the last question, find the width of the wheel, when the tot^l fall is 32 feet and Uie efficiency 60 per cent TUBBINIS 387 I Assodmiig the Telocity of tlie water la the penstock to Ve 1} times that K tl]« wliael's periphciry, and the bottotD of the penstock level with the top m. tba ^wbci&l, find the pomt at which the water entem the wheel. Fmd alao vhere water begins to discharge frona the buckets. ■ (S> A ra^dial blade impnlie wheel of the name width as the channel in vliicb it mud, is 15 feet diameter. The depth of tlie sluice opening i^ ■^ focbaa and the head above the centre of the sluice is H feet. Assuming K coefficient ol vielocity of 0*8 and ttiAt tlie edge ef the sluice is rounded so iliAt there m no contraction, and tlie velocity of the rim of the wlicel is 0'4 pile velocity of flow through the aluicet iind the theoreticid efficiency of lllie 'vrbeeL I (7 1 An overshot wheel has a supply of 80 cubic feet per second on a fall Lf Mfeet I DetienDine tlie probable horse-power of the wheel, and a suitable l^riillii for the wheeL I (8) The water impinges on a Poncelet float at 15° with the tangent to m^UB iHideL, and the velocity of the water is double that of the wheel. Find, Ihj' oorastmciioD, the proper toclination of the tip of the float. I <]d) In a Ponoelet wheel, the direction of tlie jet impinging on tlie floats kwkflfi an angle of W^ with the tangent to the circumference and the tip of nllB floats inakies an angle of W>° with the aame tangent. Supposing tlte ftr^locitj of the jet to be 20 feet per second. And, graphically or otherwise» ■ (1) the proper velocity of ttie edge of the wheel, (2) the height to which the nrater will rise on tlie float above the point of admission, (8) the velocity B ttod direction of motion of the water leaving the float. I 00) Show that the efficiency of a simple reaction wheel increases ■ niyi the speed when frictional resistances are neglected^ but is greatest BMI a fiaiite speed when they are taken into account. m U the speed of the orifices be that due to the head (1) And the efficiency, I 0egl€<:ting friction ; f'2) assuming it to be the speed at maximum efficiency, m ibow that j of the head is lost by friction, and } by final velocity of water, I (ll.> Explain why^ in a vortex turbine, the inner ends of the vanes are I lodined backwards instead of being radial. I ii2< An inward flow turbine wheel has radial blades at tlie outer I F^iphtfry. and at the inner periphery tlje blade makes an angle of 30"" with K Ft m iSbb tangeBt. The total bead is TO feet and ^"^ * Find the velocity of the ■ tUb of (lie wheel if the water discharges radially. Friction neglected. ■ III I The inner and outer diameters of an inward flow turbine wheel ■*& I loot and % feet resfjectively. The water enters the outer circumference ^P^ It with the tangent, and leaves the inner circumference radially. The Br^^ velocity of flow is 6 feet at both circumferences. The wheel makes ^pJ rerqhitions per seconds Determine the angles of the vanes at both V^tnsoferenceaf and the theoretical hydrauhc efficiency of the turbine. V_ (14) Water ia supplied to an inward flow turbine at 44 feet per second, r^ al 10 d^rees to the tangent to the wheel The wheel makes 200 88B HYDRAULICS nmsltitiQtts per miniite^ The inlet radius is 1 foot &iid the ontei nism I % feet. The radiaJ velocity of flow Uirough the wheel is oom&tttut I Find the iudlnation of the vanes at iniet and outlet ol the wheel I Determine the ratio of the kinetic Gmstg^ of the water eiitanflg tbd I wheel per pound to the work done on the whed per pound. I (15) The suppler of water far an inward flow reaction turbine s S(K^ 1 cable feet per minnte and the available head is 40 feet. The vASea sn 1 radial at the inlet, Qxe outer radius is twice the inner, tlie couitisl I velocity of fiow is 4 feet per aecond, and the revolutions are 3S0 pif I minute. Find the velocity of the wheeU the guide and vane angles 1 tlie inner and outer diameters^ and the width of the bucket at inlet ill| outlet. Lomd. Un. 1906. I (16) An inward tiow turbine on 15 feet fall has aji inlet radius of 1 Ml and an outlet radius of 6 inches. Water enters at 15* with the tangent to j the circumference and is discharged radially with a velocity ol S feet per J Be<»ncL The actual velocity of water at inlet is 22 feet per secofid fhem G^^^nnifereutial velocity of the Inlet surface of the wheel m ld| feet pttl second. I Construct the inlet and outlet angles of the turbine vanes* I Determine the theoretical hydrauhc efficiency of the turbine. I If the hydraulic efficiency of the tm-bine is assumed 80 per cent ind tli« I vane angles. I (17) A quantity of water Q cubic feet per second flows throi^ti A j tnrbin&t a^^ ^^^ initial and final directions and velocities are kuavSiJ Apply the principle ol equality of angular impulse and momani di momentum to find tlie couple exerted on the turbine, I (18) The wheel of an inward flow turbine has a peripheral vebcttftf 1 50 feet par second. The velocity^ of whirl of the incoming water ig 4«) tet I per second^ and the radial velocity of flow 5 feet per second. DetenniBB I the vane angle at inlet. I Taking the flow as 20 cubic feet per second and the total losees •* I 20 per cent, ol the available energy, determine the horse *iKjwer of tJnJ turbinoi and the head H. ^^1 If 5 per cent, of the head is lost in friction in the supply pipe, an^H centare of the turbine is 15 feet above the tail race level, find the preaia]i| head at the inlet circumference of the wheel. I (10) An inward flow turbine is required to give 200 harse*pewer is^y a head of 100 feet when running at 500 revolutions per minnte. WM velocity with which the water leaves tlie wheel axiaUy ruay be tall^H 10 feet per secoud^ aod tho wheel is to have a double outlet The diftd^H of the outer circumference may be taken as 1| times tlie inner. Defeei^^l tlio dimensionB of the tiu-bine and the angles of the guide bMay^| vanes of the turbine wheel. The actual efficiency is to be taken as I^H cent, and the hydraulic efficiency as 80 per cent. ^H (20) An outward flow turbine wheel has an internal diameter of fi^Sl feet and an external diameter of 0*25 feet. The head above the turlitflUU 14V5 feet The width of the wheel at inlet is 10 inches, and the iifii|^| I TURBIKES 389 plied per HeconcT m 215 cnbic feei. Asanmmg the hydraulic «?« nrc 211 |ior oent., determine tlie angles of tips o! the Tanes ^a that ^Ixg w«i4^r shall leave the wheel radially, Determiiie the horse -power o£ tiirbme and verify the work done per poand from the triangleB of 121) The total head arailable for an inward -flow ttu-bitie is 100 feet The tnrbine wheel is pl&oed 15 feet above the tail water ley eh When the flow is somud, tliere is a loss of head in the supply pipe of • jmi t i'ut of the head I in tlie goide passages a loss of 5 per cent. ; in the '^'"lieel 9 pi?r cent ; in the down pipe I per cent, i and the velocity of flow ^«^m the wheel and in the supply pipe, and also from the down pipe ib ^ feet iMjr decond- Tljr diameter of the inner circnmference of the wheel is 9^ inches and tht» outer 19 inches, and the water leaves the wheel vanes radially* TtTie wheel haa nnUal vanes at inlet f >et«^nnine tlie tiamber of revolutiona of the wheeL the pretssure head in \ #ye of the wheel* the pressure head at the circumference to the wheel, ^ |3n%amire head at the eu trance tti the guide chamber, and the velocity vliicb thr water Ijaa when it enters tlie wheeL From the data given j^ 2) A horizontal inward flow tnrbine has an internal dXameter of 4 iuchef* and an external diameter of 7 feet The crowns of the I aro parallel and are 8 inches apart The diHerence in level of the i and tail water in 6 feet, and the upper crown of the wheel is jost below I latt water level. Find the angle the guide blade niakea witti the tangent wbnel, when the wheel makea 32 revolutious per minute, and the ■15 cnbic foet per second. Neglecting friction, determine the vane , lite horse -power of the wheel and the theoretical hydranhc efficiency. 9) A parallel flow tnrbine has a mean diameter of 11 feet. The number of revolutions per minute is 15, and the a^dal velocity of is 3'5 feet per second. The velocity of the water along the tips of the . is 15 feet per second. LN^termine tlie inclination of tlie gnide blades and the vane angles that li« water shall enter withont shock and leave the wheel axially, Dc?ieriiiine the work done per pound of water passing thi'ough the wheeL i*M) The diameter of Uie inner crown of a parallel flow pressure turbine f 5 feet atid the diameter of tlie outer crown is B feet. The head over the whie^l iit 11* feett The number of revolutions per minute is 52* The radial Jocitv of flow tlirough the wheel is 4 feet per second. Aaaiiining a hydraulic efficiency of 0*8, determine the guide blade angles 1 rani^ angles at inlet for the three radij 2 feet 6 inches^ 8 feet 8 inches i4Ceet Assrunlng the depth of the wheel is S inches, draw am table sections of » Tanes at the three radii* Find al»o ihe width of the guide blaile in plan, if the up])er and lower are paraUel* and the lower edge makes a constant angle with the HTDEArXICS Huee mt the miidr And the head of 64 iBtLl Uml speed ol ibe whed^l tmliiike has an inner diajneter of 5 leiti] d iftdiBA, ftud makes 4S0 reTolatioiiA^ ] at H lewes tlie nozzles is dotibla the Telodt|*l I Hm wlifiel, and tlie dtreetion of the water iDaJi«« m\ wUli Hie csmttnference ol the wheel. at miei, and the angle of tlie Tane at outlet i mier AmXt learo Ibe idieel nidiaUy. llie tbeocHml bydr&idie efficiency. U 8 per cent, of ^e head! at the mamM^ ia lost in the wheel, find tJie vane angle at exit thoft] Uft leanie imdiaUj. ■0V the l^dnnfo didbncy of the turbine ? P In aA asdal flow Oinrd turbine, kt V be the velocity doe to Die I head. Siqppose the water issiies from the guide blades with Qm 1 O^T, and It dbehazged tnillj with a velocity ^12 V* Lei ^ | iJotily ctf the teoeinigMddiafahaigiiig edges be 0-55 V, ^1^ of the guide liladeB, receiving and dischajrging angles d ■d Iry^snbe effdepey cl the torbine. m^ imi Waier k sn^fiied to an axial flow impulse turbine, having a mma r ef t fibit^ and niakmg 144 revolntioiis per minate, tmder a hmd d \ . Tin aag^ of t^ guide blade at entrance is 30% and the angja tbe le direction of motion at exit is 80"". Eight per eeoi of I the supply pipe and guide. Determine ihe ralAtlvt wtkicHy «i mier end wtieil at entrance, and on the assumption Utat 10 f«f eesi. ci fiie kpt^ hesd is lost in frictfon and shock in tlie wheel, detemunt Hie veitoeily wHh which the water leav^ the wheel. Find the Uydnolk ofthei |W Hke golde bladea ef an inward dow turbine are inclined M SD iliyeui» «Bd the Telocity U tlong tl je tip of the blade is 60 feet pet scooni The i^odtyoC Hie wheel periphery is 55 feet per second. The guide blidei wm trailed an thnt tbey ate inclined at an angle of 15 degrees, the ve C T^mftimng eoasluit. Find the loss of head due to »hock at entranc If the nidtQt of the iimer peripheiy ie one-half that of the outer aad^ tidkU velocity thrangb the wheel ie constant for any flow^ and the inl^ kfl Ibe wheel twlbliy in the first case, &id the direction in which it leaf«« JM liie eeeond om& The inlet radius is twice the outlet radius. 03f) Tile suppler of water to a turbbie is controlled by a speed \ between tlie guides and the wheel. If when tlie gate ia fully opea ^ i^ocity witli which the wmter approaches the wheel is 70 foot pe; seooA^ ^y^ I t f, ! TURBINES 391 m and it TiMkkes an angle of 15 degrees with the tangent to the wheel, find the loss of Ikead by shock when the gate is half closed. The velocity of the inlet periphery of the wheel is 75 feet per second. (81) A Pelton wheel, which may he assumed to have semi-cylindricaJ buckets, is 2 feet diameter. The available pressure at the nozzle when it is doeed is 200 lbs. per sqnare inch, and the supply when the nozzle is open is lOO cuIhc feet per minute. If the revolutions are 600 per minute, estJTnate the horse-power of the wheel and its efficiency. (82) Show that the efficiency of a Pelton wheel is a maximum — neig^ecting fric!tional and other losses — when the velocity of the cups equals half the velocity of the jet. 25 cnbic feet of water are supplied per second to a Pelton wheel through m nozzle, the area of which is 44 square inches. The velocity of the cups is 41 feet per second. Determine the horse-power of the wheel assuming an efficiency of 75 per cent. CHAPTER X. PUMPS. Pumps are machines driven by some prime mover, and used for raising fluids from a lower to a higher level, or for impartmg energy to fluids. For example, when a mine has to be drained the water may be simply raised from the mine to the surface, and work done upon it against gravity. Instead of simply raising the water through a height h, the same pumps might be used to deliver water into pipes, the pressure in which is wh pounds per square foot. A pump can either be a suction pump, a pressure pump, or both. If the pump is placed above the surface of the water in the well or sump, the water has to be first raised by suction; the maximum height through which a pump can draw water, or in other words the maximum vertical distance the pump can be placed above the water in the well, is theoretically 34 feet, but practically the maximum is from 25 to 30 feet. If the pump delivers the water to a height h above the pump, or against a pressure-head A, it is called a force pump. 216. Centriftigal and turbine pumps. Theoretically any reaction turbine could be made to work as a pump by rotating the wheel in the opposite direction to that in which it rotates as a turbine, and supplying it with water at the circumference, with the same velocity, but in the inverse direction to that at which it was discharged when acting as a turbine. Up to the present, only outward flow pumps have been constructed, and, as will be showTi later, difficulty would be experienced in starting parallel flow or inward flow pumps. Several types of centrifugal pumps (outward flow) are shown in Figs. 272 to 276. The principal difference between the several types is in the form of the casing surrounding the wheel, and this form has con- siderable influence upon the efficiency of the pump. The reason PtJMPS 393 this can be easily seen in a general way from the following cnnsideration. The water approaches a turbine wheel with a lii^li velocity and in a direction making a small angle with the directioti of motion of the inlet circumference of the wheel, and Fig, 1^3. Diagrum of Centrifui^al Pump. [thus it has a large velocity of whirl When the water leaves the ] wheel its velocity is small and the velocity of whirl should be zero. jIn the centrifugal pump these conditions are entirely reversed; [the water enters the wheel with a small velocity, and leaves 394 HYDBAUUCS it with a high velocity. If the case surronnding the whfid admits of this velocity being diminished gradually, the Mneiac energy of the water is converted into useful work, but if not, itia destroyed by eddy motions in the casing, and the efficiency of tlid pump is accordingly low. In Fig. 272 a circular casing surrounds the wheel, and prac- tically the whole of the kinetic energy of the water when it leaves the wheel is destroyed ; the efficiency of such pumps is generally much less than 50 per cent. Fig. 273. Centrifugal Pump with spiral casing. The casing of Pig. 273 is made of spiral form, the sectional area increasing uniformly towards the discharge pipe, and thus being proportional to the quantity of water flowing through the section. It may therefore be supposed that the mean velocity of flow through any section is nearly constant, and that the stream lines are continuous. The wheel of Fig. 274 is surrounded by a large whirlpool chamber in which, as shown later, the velocity with which the water rotates round the wheel gradually diminishes, and tie velocity head with which the water leaves the wheel is partly converted into pressure head. The same result is achieved in the pump of Figs. 275 and 276 PUMPS 395 '>; allowing the water as it leaves the wheel to enter giiide p8«sag^, similar to those used in a turbine to direct the water to the wheeL The area of these passages gi-adually increaaeB md a eoiieiderahle portion of the velocity head is thus converted into preeenre head and is available for lifting water. Tliis cliiaB of centrifugal punip is known as the tarbine pump. Pig, 274. EHa^ram of Centrifagftl Pump with Whirlpool Chamber. 21f . Starting centrifugal or turbine pmnps, A centrifngal pump cannot commence delivery unless the wheel, casings and suction pipe are full of water. K the pump is below the water in the well there is no difficulty in starting as the casing will be maintained full of water » WTien the pump is above the water in the well, as in Fig, 272, Ison-retum valve V must be fitted in the suction pipe, to prevent kj^tup when stopped from being drained. If the pump becomes ff or when the pump is first set to work, special meuna have to Ije provided for filling the pump case. In large pumps the air may be expelled by means of steam, which becomes condensed and ihe water rises from the well, or they should be provided with HTBRAUI,ICS i ftir-piimp or ejector as an auxiliaTy to the patdp* StiiaU pmnps geiie?mll>r be easily filled by hand throug^h a pipe sach m \ Aomn at P, Fig, 276. With some classes of pomps, if the pump ha& to commence delivery against full head, a stop valv^e on the rising maiii, , Fig. 296, is closed until the painp has attained the speed necessary to commence delivery*, after which the stop valve is dowlir | opened. It will be seen later that, under special circumstances, other pronsions will have to be made to enable the piimp to commeDOJ dt^liwry. 818. Form of the Tanea of centrlftigal pumps, Tii^^^ conditions to be satisfied by the vanesi of a centrifuia! pump are exactly the aame as for a turbine. At inlet the directioa of the vane shoukl be parallel to the direction of the relativE n^loeity ivf tlie water and the tip of the vane, and the velocity witli which the M-ater leaves the wheel, relative to the pump o»ft \H tht^ vtvtor ffiini of the velocity of the tip of the vane and tl# veUK*ity n*lative to the vane, * Set ptge 4m, PDlfPS 397 Suppoae the wheel and casing of Fig. 272 is full of water^ and the wheel is rotated in the dii'ection of the arrow mth such a velocity that water enters the wheel in a known direction with a Telocity XT, Fig. 277 j not of necessity radial. t^t r he the velocity of the receiving edge of the vane or inlet eirciimferenee of the wheel; Vi the velocity of the discharging circnmfereiice of the wheel ; Ui the absolute velocity of the water &6 it leaves the wheel ; V and Vj the velocities of wlurl at inlet and ontlet respectively; Vr and iv the relative velocities of the water and the vane at inlet and outlet respectively; u and iti the indial velocities at inlet and outlet respectively. The triangle of velocities at inlet is ACD, Fig. 277, and if the vsae at A, Fig* 272, ia made parallel to CD the water will enter the wheel without shock. wangle of vdifoiies Fig. 277. at e*ixt . Fig. 278. The wheel being full of watePj there is continuity of How, and if A and Aj arc5 the circumferential areas of the inner and outer cipDumferences, the radial component of the velocity of exit at the Doter circumference ia Au «i = A," If the direction of the tip of the vane at the outer circnm- ference is known the triangle of velocities at exit, Fig. 278, can be drawn as follows. Set out BG radially and equally to %, and BE equal to v^ Draw GF parallel to BE at a distance from BE equal to ft,, ad EF parallel to the tip of the vane to meet GF in F. Then BF is the vector sum of BE atid EF and is the velocity ith which the water leaves the wheel relative to the fixed casing, 219. Work done on the water by the wheel, Lt*t R and r be the radii of the discharging and receiving enmferences respectively. The change in angular momentum of the water as it passes Ihrgugh the wheel is V,R + Yr/g per pound of flow, the plus sign becing used when V is in the opposite direction to Vj, as in Kgs, 277 and 278, 398 HYDRAULICS Neglecting frictional and other losses, the work done by the wheel on the water i)er pound (see page 275) is g " g ' If n is radial, as in Fig. 272, Y is zero, and the work done on the water by the wheel is -^ foot lbs. per lb. flow. If then H«, Fig. 272, is the total height through which the water is lifted from the sump or well, and Ud is the velocity with which the water is delivered from the delivery pipe, the work done on each pound of water is and therefore, g ^ 2g Let (180* - <^) be the angle which the direction of the vane at exit makes with the direction of motion, and (180" - &) the angle which the vane makes with the direction of motion at inlet. Then ACD is e and BEF is *. In the triangle HEF, HE = HF cot <^, and therefore, Vi " t'l — tti cot ^. The theoretical lift, therefore, is 2g g If Q is the discharge and Ai the peripheral area of the dis- charging circumference, Q Q .^ Vi - Vi -^ cot 4> and H = ^ (1). g If, therefore, the water enters the wheel without shock and all R resistances are neglected, the lift is independent of the ratio — , and dex^ends only on the velocity and inclination of the vane at the discharging circumference. 220. 5B»tioofVitoVi. As in the ca^ of the turbine, for any given head H, Vi and I'l can theoretically' have any values consistent with the product \ PUMPS 399 Vit?i heing equal to grH, the ratio of Vi to Vi simply depending upon the magnitude of the angle ^. The greater the angle ^ is made the less the velocity Vi of the periphery must be for a given lift. Fig. 279. This is shown at once by equation (1), section 219, and is illustrated in Fig. 279. The angle 4^ is given three values, 30 degrees, 90 degrees and 150 degrees, and the product Yv and also the radial velocity of flow tti are kept constant. The theo- retical head and also the discharge for the three cases are there- fore the same. The diagrams are drawn to a common scale, and it can therefore be seen that as ^ increases Vi diminishes, and Ui the velocity with which the water leaves the wheel increases. 221. The kinetic energy of the water at exit from the wheeL Part of the head H impressed upon the water by the wheel increases the pressure head between the inlet and outlet, and the remainder appears as the kinetic energy of the water as it leaves 400 HTD&AULICS the wheel. This kinetic energy is equal to -^ , and can only be utilised to lift the water if the velocity can be gradually diminished so as to convert velocity head into pressure head. This however is not very easily accomplished, without h^ng accomi)anied by a considerable loss by eddy motions. If it be assumed that the same proportion of the head ^ in all cases is converted into useful work, it is clear that the greater Ui, the greater the loss by eddy motions, and the less efficient will be the pump. It is to be ex- pected, therefore, that the less the angle <^, the greater will be the efficiency, and experiment shows that for a given form of casing, the efficiency does increase as <^ is diminished. 222. Gross lift of a centriftigal pomp. Let ha be the actual height through which water is lifted; hg the head lost in the suction pipe ; ha the head lost in the delivery pipe ; and it^ the velocity of flow along the delivery pipe. Any other losses of head in the wheel and casing are incident to the pump, but fe„ ha, and the head 5- should be considered as external losses. The gross lift of a pump is then h=^ha + A, + fcd + o~ > and this is always less than H. 223. Efficiencies of a centrifugal pump. Manometric efficiency. The ratio =^ , or e= ^ vi - vi -? cot 4> Ai is the manometric efficiency of the pump at normal discharge. The reason for specifically defining e as the manometric efficiency at normal discharge is simply that the theoretical lift H has been deduced from consideration of a definite discharge Q, and only for this one discharge can the conditions at the inlet edge be as assumed. A more general definition is, however, generally given to e, and for any discharge Q, therefore, the manometric efficiency maj be taken as the ratio of the gross lift at that discharge to the theoretical head Vi' - Vi A cot 4^ Ai CENTRIFUGAL PUMPS 401 This manometric efficiency of the pump must not be confused with the efficiency obtained by dividing the work done by the pump, by the energy required to do that work, as the latter in many pumps is zero, when the former has its maximum value. Hydraulic efficienSy, The hydraulic efficiency of a pump is the ratio of the work done on the pump wheel to the gross work done by the pump. Let W = the weight of water lifted per second. Let h = the gross head Let E = the work done on the pump wheel in foot pounds per second. Let ek = the hydraulic efficiency. Then W.h The work done on the pump wheel is less than the work done on the pump shaft by the belt or motor which drives the pump, by an amount equal to the energy lost by friction at the bearings of the machine. This generally, in actual machines, can be approximately determined by running the machine without load. jkchuil efficiency. From a commercial point of view, what is g-enerally required is the ratio of the useful work done by the pump, taking it as a whole, to the work done on the pump shaft. Let E, be the energy given to the pump shaft per sec. and em the mechanical efficiency of the pump, then E = E«.em, and the actual efficiency _ W.fea ea- ^ . Gross efficiency of the pump. The gross efficiency of the pump itself, including mechanical as well as fluid losses, is W.h ea = K 224. Experimental determination of the efficiency of a eentriftigal pomp. The actual and gross efficiencies of a pump can be determined directly by experiment, but the hydraulic efficiency can only be determined when at all loads the mechanical efficiency of the pump is known. To find the actual efficiency, it is only necessary to measure the height through which water is lifted, the quantity of water L. H. ^^ 40S HfDRAITLICS dischiirged, and tlie energy E, given to the pump shaft in oj tim©, A. very convenient method of determiniiig £« with a hk degree of accuracy is to drive the pump shaft direct by an electrie motor, the efficiency cun""e* for which at varying loads is kn A better method ia to use some form of transmission dynjuui^ meter t. 225. Design of pump to give a discbarge Q. If a pump is required to give a discliarge Q under a grom lift hf and from pre\^ous experience the probable manom^tnc efliciency e at this discharge is known, the problem of determinTtig suitable dimensions for the wheel of the pump is not diificulL The difficulty really arises in gi\nng a correct value to e and ia making proper allowance for leakage. Tins difficulty w^tl be better appreciated after the losses m various kinds of pumps have been considered. It will then be seen that e depends upon the angle ^, the velocity of the wbeeJ* the dimensions of the wheeU the form of the vanes of the wheet the discharge through the wheel, and upon the form of the casmg surrounding the wheel ; the form of the casing being Just as important, or more important, than the form of the wheel in determining the probable value of e. Design of the tvheel of a ptiffip fftr a given discharge und^fr o given Jiead. If a pump is required to give a discharge Q under an effective head h^^ the gross head h can only be determined if A»i hit and ,j^ f are known. Any suitable value can be given to the velocity tu^ If the pipes are long it should not be nmch greater tlian 5 feet per sec<mi for reasons explained in the chapter on pipes, and the velocity' % in the suction pipe should be equal to or less tlian lij. Tlie velocities ta, and n^t having been settled, the losses h* and h^ can k approximated to and the gross head h found. In the suction pip^^ as explained on page 395, a foot valve is generally fitted, at ubicK at high velocities, a loss of head of several feet may itcenr. The angle <^ is generally made from 10 to 90 degrees. Theoreti- cally, as already stated^ it can be made much greater tim 90 degrees, but the efficiency of ordinary centrifugal pumps Mlglii be very considerably diminished as <f> is increased* The manometric eificiency e varies very considerably ; with radial blades and a circular casing, the efficiency is not generally ' See Etfciriml Engineerinff, Thomilen-Howe, p< 195* t Bee |>apef by Stanlon, Pmc. ItnL Meek Eufft^^ 1909, CENTRIFUGAL PUMPS 403 more than 0*3 to 0*4. With a vortex chamber, or a spiral casing, and the vanes at inlet inclined so that the tip is parallel to the relative velocity of the water and the vane, and i> not greater than 90 degrees, the manometric efficiency e is from 0*5 to 0*75, being greater the less the angle <^, and with properly designed guide blades external to the wheel, e is from 0*6 to '85. The ratio of the diameter of the discharging circumference to the inlet circumference is somewhat arbitrary and is generally made from 2 to 3. Except for the difficulty of starting (see section 226), the ratio might with advantage be made much smidler, as by so doing the frictional losses might be considerably reduced. The radial velocity t^ may be taken from 2 to 10 feet per second. Having given suitable values to u, and to any two of the three quantities, e, v, and <^, the third can be found from the equation 7 e W ~ Vitti cot 0) n = . 9 The internal diameter d of the wheel will generally be settled from consideration of the velocity of flow tt, into the wheel. This may be taken as equal to or about equal to u, but in special cases it may be larger than u. Then if the water is admitted to the wheel at both sides, as in Fig. 273, from which d can be calculated when u^ and Q are known. Let b be the width of the vane at inlet and B at outlet, and D the diameter of the outlet circumference. and B = Then * = ^ , If the water moves toward the vanes at inlet radially, the inclination of the vane that there shall be no shock is sucli that tan ^ = - , and if guide blades are to be provided external to the wheel, as in Fig. 275, the inclination a of the tip of the guide blade with the direction of Vi is found from tan a= -- , The guide passages should be so proportioned that the velocity Ui is gradually diminished to the velocity in the delivery pipe. 404 HYDRAULICS Limitifig velocity of the rim of the whe&L Quite apart from lead lost by friction in tlie wheel due to the relative motion of he water and the wheel, there is also conBiderable loss of energy external to the wheel due to the relative motdon ol the water mi the wheel. Betiyeen the wheel and the casing there is in most pumps a film of water, and between this film and the wheel, frictional forces are set up which are practically proportional to tlie square of the velocity of the wheel periphery and to the ane* of the wheel crowns. An attempt is frequently made to dimitiisli tlris loss by fixing tlit* vanes to a central diaphragm only, the wheel thus being without crowns, the outer casing beiog so fonned that there is but a small clearance between it and ih outer edges of the vanes. At high velocities these frictional rdd/ft- anoes may be considerable. To keep them small the surface d the wheel crowiis and vanes must be made smooth^ and to this eiid many high speed wheels are carefully finished. Until a few years ago the periphei'al velocity of pump wheals twas generally less than 50 feet per second, and the best velocity ms supposed to be about 30 feet per second. They are now, how- ever, run at much higher speeds, and the limiting velocities ar© fixed from consideration of the stresses in the wheel due to centri- fugal forces. Peripheral velocities of nearly 200 feet per second are now frequently used, and Bateau has constructed small pump* with a peripheral velocity of 250 feet per second*. Exatn^U. To hnd the proportions) of a pump with radial bl&d^ at oullH (i.e« ^=±&0°) lo Uft 10 cubio fe«l uf water per »e<]oti(l figainat n head of 50 feet Ai»mnie there are two euotion pipes and that the water ei^tera the vheelfrto both itde«, a^ ti) Fig, S7S, ako that Uie velocity in the auction arid delii^^ ftip!« ftnd the radial velocitj through the wheel are 6 feet per eecondt s^d the manooKtik efficiei^f^y ia 75 per cent. Fir&t to find lu. . ■7fi'-^ = 50, Sinoe ihe bladei aire mdi&l, from which t*! := 46 feet per l^ee, To find Che diameter of the suction pipes. The diseharse is 10 oubk feet per second, therefore 4 from which ii = l*03'^12|". If the radiiiB R of the ^xteroal circ^mferenoe be taken as 2r and r is taken < to the radiua of the auction pipes, then R = 12|", and the namher of p«T second win be Th« felooit; of the inner edge of the vane ia p=2S feel per see. Engineer, 1903* CENTRIFUGAL PUMPS 405 The inolination of the vane at inlet that the water may move on to the vane ^tlioQt abock U tan<?=A, lad the water when it leaves the wheel makes an angle a with v^ such that tana=^. If there are guide Uadea sarronnding the wheel, a gives the inclination of these = •268' The width of the wheel at discharge is Q 10 ».D.6'"». 206x6 =ft| inches about. The width of the n^ieel at inlet =6J inches. 226. The centrifligal head impressed on the water by thm wheeL Head against which a pump vrill commence to discharge. As shown on page 335, the centrifugal head impressed on the water as it passes through the wheel is ^^-2g 2g^ 1>at this is not the lift of the pump. Theoretically it is the head which will be impressed on the water when there is no flow through the wheel, and is accordingly the difference between the pr oso nre at inlet and outlet when the pump is first set in motion ; or it is the statical head which the pump will maintain when ronning at its normal speed. If this is less than — ^-^ , the pump theoretically cannot start lifting against its full head without being speeded up above its normal velocity. The centrifugal head is, however, always greater than 2g 2g' as the water in the eye of the wheel and in the casing surrounding the wheel is made to rotate by friction. For a pump having a wheel seven inches diameter surrounded by a circular casing 20 inches diameter, Stanton* found that, when the discharge was zero and the vanes were radial at exit, he was — s— , and with curved vanes, i> being 30 degrees, he was —^ — . For a pump with a spiral case surrounding the wheel, the centrifugal head he when there is no discharge, cannot be much greater than ^ , as the water surrounding the wheel is prevented from rotating by the casing being brought near to the wheel at one point. ♦ Proceedings Inst, M. £., 1903. 406 HY|>RAriJCS Parsons foimd for a pomp having a wheel 14 inches diameter with radial vanes at outlet, and ninning at 300 reTolatians per minute, that the head maintained without discharge was ^ , and with an Appold* wheel running at 320 revolutions per minitte the statical head was -^ — - . For a pump, with spiral casing, having a rotor 1*54 feet diameter, the least velocity at which it commeTiced to discharge against a head of 14"67 feet wn& 392 revolutions per minute^ and thus he was ^ "^ > ^^^ ^^^ 1^*^ velocity against a head of 17'4 feet was 424 revolutions per minute or K was again ^^—^ . For a pump with circular casing 1 '05t' * larger than the wheel, he was ^ — . For a pump having guide passages surrounding the wheel, and outside the guide passages a circular chamber as in Fig. 275j the centrifugal head may alaa he larger than ^; the mean actual value for tJiis pump found to be 1067^. Stanton found, when the seven inches diameter wheels mentioned above discharged int-o guide passages surrounded by a circuljix chamber *20 inches diameter, that he was ^- when the vanes ot the wheel were radial, and ,-^ ^ when <^ was 30 degrees. That the centrifugal head when the wheel has radial vanes is likely to be greater than when the vanes of the wheel are set hnck is t-o be seen by a consideration of the manner in which the water in the chamber outside the guide passages is probably set m motion. Fig, 2B0. Since there is no discharge, this rotation cauaot be caused by the water passing through the pump, but mu^ he due to internal motions set up in the wheel and casing, Th^ water in the guide chamber cannot obviously n.*tate about die axis 0, but there is a tendency for it to do so, and consequeuily stream line motions, as shown in the figure, are prt)bably «i up. The layer of water nearest the outer circumference of the wheel will no doubt be dragged along by friction in the directs shown by the arrow, and water will flow from the outer casing to take its place ; the stream lines will give motion to the wai/er in ^ the outer casing. * See pftge 4l€. CTNTEIFUOAL PUMPS 407 Wlien the vanes m the wheel are radial and as long as a vane is 'mo^'iTig between any two guide vanes, the straight vane prevents the frictioii between the water outside the wheel and that insidsj from dragging the wat^r backward:^ along the vanej but when the vane is set back and the angle <^ is greater than 90 degrees, there j will bt^ a tendency for the water in the wheel to tnove backwardi while that in the guide chamber moves forward, and conseciuently the velocity of the stream liTiea in the casing will be less in the latter caee than in the former. In either case, the general direction of fiqw of the stream liiieSj in the guide chamber, will b© in the direction of rotation of the wheelj but due to frtction and eddy raotions, even w4th radial vaneSj the velocity of the stream Fig. 2m. ^ will he less than the velocity I'l of the periphery of the wheel. outride the guide chambers the velocity of rotation will b© than i\. In the outer chamber it is to he expected that the rater will rotate as in a free vortex, or itB velocity of whirl will be inversely proportional to the distance from the centre of the TOtor, or will rotate in some manner approximating to this. The liead which a pumpj with a vortes^ chambeTy tcill th^oreti^ llif rnaifUain when the discharge is zero. In this case it is [)bable that as the discharge approaches zero, in addition to the ktcfT in the wheel rotating, the water in the vortei chamber will also rotate because of friction. 408 HYDRAUUCS The centrifagal head dne to the water in the wheel is If R= 2r, this becomes -j ^ . The centrifugal head due to the water in the chamber is, Fig. 281, f^wv^dr Jr^ gn ' To and Vo being the radius and tangential velocity respectively of any ring of water of thickness dr. Fig. 281. If it be assumed that t;oro is a constant, the centrifugal head due to the vortex chamber is g k n' 2g\Tj njJ' The total centrifugal head is then ^'-2g 2g^2g\rJ R«,V ' If rto is 2r and R^ is 2r«„ 2g The conditions here assumed, however, give K too high. In Stanton's experiments he was only — ^ — - . Decouer from experi- ^g CENTRIFUGAL PUMPS 409 ments on a small pomp with a vortex chamber, the diameter being l'3t' * aboat twice the diameter of the wheel, found he to be -rr-^ . Let it be assomed that K is -^ in any pump, and that the lift of the pump when working normally is 7 e Vi Vi e W - Vit^ cot 4^) /i, = — = . Then if fe is greater than v^- \ the pump will not commence to discharge unless speeded up to some velocity Va such that mvj efa*--t;ittiCot<^) ^g ^ g After the discharge has been commenced, however, the speed may be diminished, and the pump will continue to deliver against the given head*. For any given values of m and e the velocity fa at which delivery commences decreases with the angle ^. If ^ is 90 or greater than ©0 degrees, and m is unity, the pump ynW only commence to discharge against the normal head when the velocity is t'l, if the manometric efficiency e is less than 0*5. If <^ is 30 degrees and m is unity, v^ is equal to Vi when e is 0*6, but if <^ is 150 degrees rj is equal to t?i when e is 0*428. Nearly all actual pumps are run at such a speed that the centrifugal head at that speed is greater than the actual head against which the pump works, so that there is never any difficulty in starting the pump. This is accounted for (1) by the low manometric efficiencies of actual pumps, (2) by the angle <t> never being greater than 90 degrees, and (3) by the wheels being surrounded by casings which allow the centrifugal head to be greater than ^. It should be observed that it does not follow, because in many cases the manometric efficiency is small, the actual efficiency of the pump is of necessity low. (See Fig. 286.) 227. Head-velocity curve of a centrifugal pump at zero discharge. For any centrifugal pump a curve showing the head against which it will start pumping at any given speed can easily be determined as follows. On the delivery pipe fit a pressure gauge, and at the top * See pages 411 aud 410. 410 HYDRAULICS of the suction pipe a vacuum gauge. Start the pomp with the delivery valve closed, and observe the pressure on the two gauges for various speeds of the pump. Let p be the absolute pressure per sq. foot in the delivery pipe and pi the absolute pressure per sq. foot at the top of the suction pii)ey then ^ - ^ is the total centrifugal head K- teoo leoo 200Q 2200 RevoUttums per Minute. Fig. 282. 240? A curve may now be plotted similar to that shown in Fig.{282 which has been drawn from data obtained from the pump shown in Fig. 275. When the head is 44 feet, the speed at which delivery would just start is 2000 revolutions per minute. On reference to Fig. 293, which shows the discharge under different heads at various speeds, the discharge at 2000 revolutions per minute when the head is 44 feet is seen to be 12 cubic feet per minute. This means, that if the pump is to discharge againrt this head at this speed it cannot deliver less than 12 cubic to per minute. 228. Variation of the discharge of a oentriftigal puiiV with the head when the speed is kept constant*. Head-discharge curve at constant velocity. If the speed of* centrifugal pump is kept constant and the head varied, the dis- charge varies as shown in Figs. 283, 285, 289, and 292. * See also page 418. CENTRIFUGAL PUMPS 411 The curve No. 2, of Fig. 283, shows the variation of the head with discharge for the pump shown in Fig. 275 when running at 1950 revolutions per minute; and that of Fig. 285 was plotted from experimental data obtained by M. Bateau on a pump ha\nng a wheel ll'S inches diameter. The data for plotting the curve shown in Fig. 289* was obtained from a large centrifugal pump having a spiral chamber. In the case of the dotted curve the head is always less than the centrifugal head when the flow is zero, and the discharge against a given head has only one value. RadilMod^ of How fronhWheA. Fig. 283. Head-discharge curve for Centrifugal Pump. Velocity Constant. Fig. 284. Velocity-discharge curve for Centrifugal Pump. Head Constant. In Fig. 285 the discharge when the head is 80 feet may be either '9 or 3*5 cubic feet per minute. The work required to drive Che pamp will be however very different at the two discharges, teid, as shown by the curves of efficiency, the actual efficiencies for the two discharges are very different. At the given velocity tlierefore and at 80 feet head, the flow is ambiguous and is Unstable, and may suddenly change from one value to the other, ^>r it may actually cease, in which case the pump would not start ^gain without the velocity Vi being increased to 70*7 feet per ^lecond. This value is calculated from the equation • Proeeedingi InsL Mech. Engs., 1903. 412 HYDRAULICS the coefficient m for this pump being 1"02. For the flow to b stable when delivering against a head of 80 feet, the pnmp shool be run with a rim velocity greater than 70*7 feet x)er second, i which case the discharge cannot be less than 4^ cubic feet pe minate, as shown by the velocity-discharge curve of Fig. 2K The method of determining this curve is discussed later. Pump Wheel flScUanv. Rew, per minute 1290. Fig. 2S5. 1 Jl 3 DisduMT^ in, cfL per mJav. Fig. 286. Fig. 287. Example, A oentrifngal pump, when disoharging normally, has a peripher velocity of 50 feet per second. Tbe angle ^ at exit is 30 degrees and the manometric effioienoy is 60 per ceo The radial velocity of flow at exit is '^Jh. Determine the lift h and the velocity of the wheel at which it will start delivei nnder fall head. S _ V=60-(2VA)coBl30 -50-1-nJh. CENTRIFUGAL PUMPS 413 9 from whieh A =87 feet. Ijei Oj be the Telocity of the rim of the wheel at which pumping commences. Then Mraming the centrifogal head, when there is no discharge, is r,=48'6 ft. per sec. 229. Bemonilli's equations applied to centrifagal pumps. Consider the motion of the water in any passage between two consecutive vanes of a wheel. Let p be the pressure head at inlet, pi at outlet and pa the atmospheric pressure per sq. foot. If the wheel is at rest and the water passes through it in the same way as it does when the wheel is in motion, and all losses are neglected, and the wheel is supposed to be horizontal, by Bemouilli's equations (see Figs. 277 and 278), w 2g w 2g ^^^• But since, due to the rotation, a centrifugal head '•=1:1 <^) is impressed on the water between inlet and outlet, therefore, w 2g w 2g 2g 2g ^'*''' **' w w 2g 2g* 2g 2g ^*^- From (3) by substitution as on page 337, w^2g w^2g* g - g •••^^^. and when U is radial and therefore equal to Uy E. + U!«=£^^+Y..^i (6). w 2g w 2g g ^^ If now the velocity Ui is diminished gradually and without shock, so that the water leaves the delivery pipe with a velocity U4, and if frictional losses be neglected, the height to which the water can be lifted above the centre of the pump is, by Bernouilli's equation, w 2g w 2g ^'''• If the centre of the wheel is K feet above the level of the water in the sump or well, and the water in the well is at rest, P^ = K^P^f (8). w w 2g 414 HTDRAUUCS Substituting from (7) and (8) in (6) 9 ^^ = H.+ | = H (9). This result verifies the fundamental equation given on page 888. ^ Further from equation (6) Example. The centre of a centrifngal pmnp is 15 feei above Um level of tlit water in the sump. The total lift U 60 feet and the velocity of dieehafge from flu delivenr pipe is 5 feet per second. The angle at diachaige is 135 degrees, and the radial velocity of flow through the wheel is 6 feet per second. AMnffiipg tfien are no losses, find the pressure head at the inlet and oatlet cironmfiereiioes. At inlet ^=34'-16'-^ tr 64 = 18*6 feet. The radial velocity at outlet is iii=5 feet per second, and y^^.,'..u,.,cot 450^3^^ 9 V 64' . and therefore, ©i' + 5i7j = 1940 (1), from which 17^ = 41*6 feet per second, and V,=46-r, „ „ Then |l' = LLtii>^34feet. The pressure head at outlet is then tr IT 2g = 45 feet. To find the velocity v^ when <p is made 80 degrees. cot ip=»JSt therefore (1) becomes rj' - 6 /^S . Vj = 1940, from which V]=48*6 ft. per sec. and V,=:40 Then 5l = 25-4 feet, and ?^=63-6 feet. 2*7 w 230. Losses in oentrifogal pumps. The losses of head in a centrifugal pump are due to the same causes as the losses in a turbine. L088 of head at exit. The velocity Ui with which the water leaves the wheel is, however, usually much larger than in the case of the turbine, and as it is not an easy matter to diminisli this velocity gradually, there is generally a much larger loss of velocity head at exit from the wheel in the pump than in the turbine. CENTRIFUGAL PUMPS 415 In many of the earlier pumps, which had radial vanes at exit, the whole of the velocity head ^ was lost, no special precautions being taken to diminish it gradually and the efficiency was constantly very low, being less than 40 per cent. The effect of the angle 4^ on the efficiency of the pump. To increase the efficiency Appold suggested that the blade should be set back, the angle 4^ being thus less than 90 degrees. Fig. 272. Theoretically, the effect on the efficiency can be seen by considering the t^iree cases considered in section 220 and illustrated in Fig. 279. When <^ is 90 degrees -^ is '543!, and when <^ is 30 degrees -^ is 'SdH. K, therefore, in these two cases this head is lost, while the other losses remain constant, the efficiency in the second case is 18 per cent, greater than in the first, and the efficiencies cannot be greater than 46 per cent, and 64 per cent. re8i)ectively. In general when there is no precaution taken to utilise the energy of motion at the outlet of the wheel, the theoretical lift is ^'"■7""2^ ^^^' and the maximum possible manometric efficiency is Sabstituting for Vi, i^i - tti cot <^, and for Uj', V,' + tt,', TT V "l" »J. ^' = 2^-2^^^'^^*' , ^ . (vi- u, cot <^)' + th^ 2 W — ^\th cot <^) _ V i^ - Vg^ cosec' 4> 2Vi (Vi - Va cot <^) ' When r, is 30 feet per second, Ux 5 feet per second and <t> 150 degrees, e is 56 per cent, and when <^ is 90 degrees e is 48*5 per cent. Experiments also show that in ordinary pumps for a given lift and discharge the efficiency is greater the smaller the angle <t>. Parsons* found that when i> was 90 degrees the efficiency of a pamp in which the wheel was surrounded by a circular casing was nearly 10 per cent, less than when the angle <t> was made about 165 degrees. • Proceedings Inst, C, E,, Vol. xLvn. p. 272. 416 HYBRA0LICS Stanton found that a pump 7 inches diameter having niiial vanes at discharge had an efficiency of S per cent, less than wto the iiTigle ^ at delivery was 150 degrees. In the first case tht maximum actual efficiency was only 39"6 per cent,, and in thi second vnse 50 i>er cent* It. han been suggested by Dr Stanton that a second reason fo the greater efficiency of the pump having vanee curved hack ^ outlet is to be found in the fact that with these vanes the variatitm of the relative velocity of the water and the wheel is less than L when the vanes are radial at outlet. It has been shown erperi- l mentally that when the section of a stream is diverging, that is the velocity is diminishing and the pressure increasing, there is a tendency for the stream lines to flow backwards towards die sections of least pressure. Tliese return stream lines cause a losi of eiierg^'^ by eddy motions. Now in a pump, when the vanes am radial, there is a greater difference between the relative velcunty of the water and the vane at inlet and outlet than when the angb L ^ is \em than fK) degrees (see Fig. 279), and it is prt^bahle tliere- i fore that there is more loss by eddy motions in the wheel in the former case. Liiss of head at entry. To avoid loss of head at entry the vanu must bo pamnel to the relative velocity of the water and the vane, I Unless guide blades are provided the exact direction in whicl pthe water approaches the edge of the vane is not knoivn. If there were no friction between the water and the eye of the wheel it would be expected that the stream lines, which in the suction pi|K are parallel to tlie sides of the pipe, would be simply turned to approach the vanes radially. It has already been seen that when there is no flow tbe water in the eye of the wheel is made to rotate by friction, and il ii probable that at all flows the water has some rotation in the ep of the wlieel, but as the delivery increases the velocity of rc»tati«i probably diminishes. If the water has rotation in the swb© direction as the wheel, the angle of the vane at inlet will c\mx}f have to be larger for no shock than if the flow is radial. Thi* the water has rotation before it strikes the vanes seems te he indicated by the experiments of Mr Livens on a pumpj the Tanes of which were nearly radial at the inlet edge. (See section 33^.) Tlie efficiencies claimed for this pnmp are so high, that thm conld have been very little loss at inlet* If the pump has to work under variable conditions and the water be assumed to enter the wheel at all discharges in the same direction, the relative velocity of the water and the edge of the CENTRIFUGAL PUMPS 417 »n only be parallel to the tip of the vane for one discharge, ; other discharges in order to make the water move along me a sadden velocity must be impressed upon it, which a loss of energy. b t«s, Pig. 288, be the velocity with which the water enters a and and v the inclination jlocity of the tip of the vane i* - u^ ->l ft respectively. e relative velocity of th and v , the vector difference of u% e radial component of flow fh the opening of the wheel be equal to the radial com- ' j,. ^ • t of Uij and therefore the e velocity of the water along the tip of the vane is Vr. Ui is assumed to be radial, a sudden velocity u, - t? - tta cot us to be given to the water. ih has a component in the direction of rotation u» will be Lshed. has been shown (page 67), on certain assumptions, that if Y of water changes its velocity from Va to Vd suddenly, the ost is ^-^^ — > or is the head due to the change of velocity. this case the change of velocity is u«, and the head lost may lably be taken as -^. K A; is assumed to be unity, the ve work done on the water by the wheel is diminished by u^_ (r - tta cot Oy now this loss takes place in addition to the velocity head lost outside the wheel, and friction losses are neglected, jL_Vi^i tTi' (t?-^,cot^)' = |L',|L'eo8ec'»- ^^"^"^^^)' 2g 2g ^ 2g = — - ^ a cosec' 4^ - (.-gcot^y 2g 2^^^^^^^ ^ 2g v^ u* u,' - . 2t?«, c< = o — oT " o~ cosec' 4> + — s~" 2g 2g 2g 2g H. Vk r,' u* u,' ,..2t?«,cot^ a .a>, ,.v 418 HTDRJl0LtCS Es^Mmpte. The rftdi&l Tdo«itj of flow thfoogh b pump w S f««| per The u^a ^ ii SO degriec and the ADgle i if 15 de^ltem.. Tbe reiodt? oqut cireiunfiertoae U 50 f««t per »ee. mnd the mdliu is twic^ ihas of tae Find Ikw tbtonticy lift on the u«&m|>tiofi ih&l the wtM»l« of Ihe kjnme mag U lo«t &t ezii* = S7*5 feet The theorvtiefti lid &«glecticg aJl Io« ^f&cienej U Iberefore 68 |ier cent. 231. Variation of the head sp«ed of a centriftigal pump. It is of intereet to study by » the variation of the dischargic wbeo fc ta constant, and th( discharge when the relocitj coiui>are the results with experiment. The full curve of Fig. 2 with the discharge when the Tim data for which the cm. the figure. Ml ia 64*9 foet, u^ the naammm with discharge and witb iM ^ of equation (])^ sed^j th the velocity of the ition of the head wiUi e \mmp {» conetant, hid ^ tofll results obtained kam i the ranatioBs of the hai of a wheel is kept fionstot been plotted is indicated to Normal rajdtuML^tlacify^fSlaiviS' \t \2 _ 3 ^ \5 ^'^% b Fi^. 289, Head'diflcharge carve at constant Telocitj. WHion the discharge is zero 2g 2g 10^5 feet. Tlie velocity of flow -f- at outlet haa been assumed equal h^ 2 at inlet. Values of 1, 2, etc. were given to ^ and the correspondiji^ valuiiis of /* found from equation (1). CENTRIFUGAL PUMPS 419 WTieii the discharge is normal, that is, the water enters the thoat shock, ^ is 4 feet and ^ is 14 feet. The theoretical ing no losses is then 28 feet and the manometrio cy is thus SO per cent. For leas or greater values of ^ ^e^atl diminishes and also the efficiency* riie ciir%*e of Fig. 290 shows how the flow varies with the city for a constant value of h^ which is taken as 12 feet. if- 2W1. Rjadinl Vttod^ thrmtph WfieeL VdocitjHiiicharg© curve at cans taut head for Ceatrifugal Pump. It wU be seen that when the velocity Vi is 31*9 feet per second & velocity of discharge may be either zero or 8*2 feet per second, 1^ meatm that if the head is 12 feet, the pnmp^ theoretically, ill only start when the velocity is Zl'Q feet per second and the Aocity of discharge will suddenly become 8"2 feet per second, ' iMJw the velocity v^ is diminished the pump still continues to bdmrge^ and mil do so as long as i?i is great-er than 26*4 feet per Nx»tid. The flow is however tmstable, as at any velocity Vc it may llddenly change from CE to CD, or it may suddenly cease, and it BU not start again until Vj is increased to Sl'O feet per second. 232, The effect of the variation of the centrifugal besul Hd the loss by firiction on the discharge of a piunp* If then the lo.s.ses at. inlet and outlet were as above and were be only lusses^ and the centrifugal head in an actual pump was ^ual to the theoretical centrifugal head, the pump could not be tede to deliver water against the normal head at a small velocity I discharge. In the case of the pump considered in section 281, ^■btd not safely be nm with a rim velocity less than 31*9 ft, PPKec., and at any greater velocity the radial velocity of flow [>iild not be less than 8 feet per second. 490 HTDRAULIGS In mctoal pumpfi, however, it lias been seen that the oentrifagal head at oommenoement is greater than 2^ 2sr- lliere is also loss of head, which at high velocities and in smsH pumps is considerable, dne to friction. These two causes consider- ably modify the head-discharge curve at constant velocity and the Tidocity-discharge curve at constant head, and the centrifugal head at the normal speed of the pump when the discharge is zn% is generally greater than any head under which the pump works, and many actual pumps can deliver variable quantities of wator against the head for which they are designed. The centrifugal head when the flow is zero is m being generally equal to, or greater than unity. As the flow increases, the velocity of whirl in the eye of the wheel and in the casing will climinish and the centrifugal head will there&nre Let it be assumed that when the velocity of flow is u (supposed constant) the centrifugal head is k and n being constants which must be determined by experiment When II is zero and if «• is known Iq can at once be found. Let it further be assumed that the loss by friction* and ed C*tt* melons, apart from the loss at inlet and outlet is -^ . * The loss of besd by frieiioii will no doubt depend not only upon « bat slM apon the Tek>eity r^ of the wheel, and should be written as ^ + ^+etc.. or, as ^"^^'*' ••®*^- If it be snppoeed it ean be expressed by the latter, then the oonvetion if proper Tallies are giTen to &, n^ and Jk|, takes into aoooont the Yariatioo of tki OsntrifUgal head and also the firietion head «,. Lz: \ CENTRIFUGAL PUMPS 421 The gross bead h is then, Vi v' t^ '' = 2^-2^-2^°°^'*^ 2vu cot S -w'cof^ 2ff (kvi — wit)* (^V? 2ff 23 .(2). -^(«*-'> If BOW the head Ai and flow Q be detepmined experimentally, the difference between h as determined from equation (1), page 4 J 7, atiid the experimental value of A, muat be equal to 2g 2^~W^ 'W ~ 2g 2g 2^ ' Iti bein^ equal to (c* — n*). The coefficient fe being known from an experiment when u is sero, two other exx)eriment9 giving corresponding values of k and i» will determine the coefficients n and fcj* The head-discharge curve at constant velocity, for a pump such as the one already considered, would approximate to the dotted Corve of Fig, 289. This curve has been plotted from equation (2), by taking k as 0*5, n as 7*64 and h% as — 38. Substituting \m)ue8 for fc, n, fti, cosec ^ and cot ^j equation (2) beoomea . * = ^*-%H^H.C«' (3), 2ff ■ 2(7 ■ and Ct being new coefficients ; or it may be written ».5^-.^.C.Q. W, Q being the flow in any desired units, the coefficients C^ and Ct 'varying with the units. If equation (4) is of the correct form, three experiments will determine the constants m, Ca and Ci din^ctly, and having given values to any two of the three ^variables h, \\ and Q the third can be found, 233. The effect of the diminution of the centrifugal head and the inoreaa^ of the friction head as the flow increaaes, on %he velocity -discharge curve at constant head* Using the corrected equation (2), section 232, and the given *ir&laes of ^5 W| and h the dotted curve of Fig. 200 has been plotted. Frtjra the dotted curve of Fig> 289 it is seen that u cannot Im.* greati^r than 5 feet when the head is 12 feet, and therefor© the Hew curve of Fig. 290 is only drawn to the point where tt is 5. ITie panip starts delivering when v is 27'7 feet per second and the discharge increases gradually as the velocity increases. MUM 422 HTDRAUUGS The pump wfll deliver, therefore, water under a head of 12 feet at any velocity of flow from zero to 5 feet per secoBd. In sach a pomp the manometric efficiency must have its mayiTnnTn valae when the discharge is zero and it cannot be greater than w fi* — txUx cot ' 9 This is the case with many existing pomps and it explains wliy, when running at constant speed, they can be made to give any discharge varying from zero to a maximum, as the head is diminished. 234. Special arrangements for converting the velocity TP head ^ with which the water leaves the wheel into preasoro head. The methods for converting the velocity head with which tlie water leaves the wheel into pressure head have been indicated on page 3W. They are now discussed in greater detail. TfcowwowV vortex or tthirlpool chamber. Professor James Thomson first suggested that the wheel should be surroimded by a chamber in which the velocity of the water should gradoftlly change from Ui to u^ the velocity of flow in the pipe. Such a chamber is shown in Fig. 274. In this chamber the water forms a frve vortex, so called because no impulse is given to the water while moving in the chamber. Any fluid particle ai, Fig. 281, may be considered as moving in a circle of radius r# with a velocity ro and to have also a radial velocity u outwards. Let it be supposed the chamber is horizontal. If W is the weight of the element in pounds, its momentum Wr perpendicular to the radius is ^ and the moment of mo- Wr r meutum or angular momentum about the centre C is — —, 9 For the momentum of a body to change, a force must act upon it> and for the moment of momentum to change, a couple must act up^^n the body. But since no turning effort, or couple, acts upon the element after leaWng the wheel its moment of momentum must be constant. Therefore, CENTRIFUGAL PUMPS 423 WVqTo 9 is constant or v^Vo = constant. If the sides of the chamber are parallel the peripheral area of the concentric rings is proportional to r©, and the radial velocity of flow t* for any ring will be inversely proportional to Tq, and there- fore, the ratio — is constant, or the direction of motion of any Vq element with its radias Vq is constant, and the stream lines are eqaiAngnlar spirals. If no energy is lost, by friction and eddies, Bernouilli's theorem iwill hold, and, therefore, when the chamber is horizontal 2g 2g w is constant for the stream lines. This is a general property of the free vortex. If 1^ is constant PT" + = constant. 2g w Let the outer radius of the whirlpool chamber be Ru, and the inner radius r^. Let Vr^ and Vr^ be the whirling velocities at the inner and outer radii respectively. Then since VqTo is a constant, and ^' "^ 5!; = constant. w 2g w w 2g 2g w 2g V njJ' WTien R«, = 2r«., w w 4' 2<7 * If the velocity head which the water possesses when it leaves the vortex chamber is supposed to be lost, and i^i is the head of crater ahove the pump and pa the atmospheric pressure, then neglecting friction or w --h,+ Va W ' hr- w 23 w If duoL i, » die be^s of die psnp above the well, the total Tlttrcfore '^ w w 2g 2gV B.V 2g • fiat from eqamdaa (8) paige 419, Tberrfore ***2g g 2g"2i^V K.V- This mi^t hare been wntten down at once from equation (I), section 230. For clearly if there is a gain of pressore head in the Tortex chamber of ^ ( 1 - ^"i), the velocity head to be lost win be leas by this amount than when there is no Yortex chamber. Snbetitnting for Vi and Hi the theoretical lift h is now , r,*-r,ttiCot* tfi' (ri-t*,cot*)' R* r^^. m *= 9 ^ ^— R^' ^^^- J When the discharge or rim velocity is not normal, there is » farther loss of head at entrance equal to 9 and g 2gA^* 2g R.« When there is no discharge Vr» is equal to Vi and "' g 2gR«» 2g- .(2). \ CENTRIFUGAL PUMPS 425 If R = iR«, and i? = ivi, Correcting equation (1) in order to allow for the variation of centrifugal head with the discharge, and the friction losses, »i' - ViUi cot ^ tti' (t'l - Ml cot ^)*R* h = 2 g 2g 2gBJ (v — u cot Oy Vv^ ^ 2nkwvi _ kiu^ 2^ ^ 2g 2g 2g ' ^hich reduces to h=^^'%^^^. 2g 2g 2g The experimental data on the value of the vortex chamber ^09r «c, in increasing the efficiency is very limited. Stanton* showed that for a pump having a rotor 7 inches ^diameter surrounded by a parallel sided vortex chamber 18 inches ^Uameter, the efficiency of the chamber in converting velocity head th> pressure head was about 40 per cent. It is however questionable "whetiher the design of the pump was such as to give the best results IKWsible. So far as the author is aware, centrifugal pumps with vortex chambers are not now being manufactured, but it seems very probable that by the addition of a well-designed chamber small oentrifngal pumps might have their efficiencies considerably in- creased. 235. Turbine pomps. Another method, first suggested by Professor Reynolds, and now largely used, for diminishing the velocity of discharge Ui gradually, is to discharge the water from the wheel into guide passages the sectional area of which should gradually increase from the wheel outwards. Figs. 275 and 276, and the tangents to the tips of the guide blades should be made parallel to the direction ofU,. The number of guide passages in small pumps is generally four or five. If the guide blades are fixed as in Fig. 275, the direction of the tips can only be correct for one discharge of the pump, but except for large pumps, the very large increase in initial cost of the pump, if adjustable guide blades were used, as well as the mechaniical difficulties, would militate against their adoption. Single wheel pumps of this type can be used up to a head of 100 feet wit i excellent results, efficiencies as high as 85 per cent. * Proceedingi Irut. C.E., 1903. / 4dl HnMULULiCS mvaiff l«e& du^cd. They mre now bem^ used to deliver water MKoiiS kaBd§ cf o'VBr dSO feet, mud M. Bateaa lias used a single ^ntf^ ^1^ mdtoK diameter Tmming at 18,000 revelations per snuffle i£> detivvr against a Iiead of 936 feet. iMm ^ Mai mi tke emiramee to ike guide passages. If the i» aie fixed, tiie directicHi of the tips can only be correct c£ xhe pomp. For any other discharge than the ^.^rsBhL i^ dxnKCiGQ cf the water as it leaves the wheel is not piKraljei <o ik^ fixed grnde and there is a loss of head due to Ijb% a lie tk^ indiBatioci of the gnide blade and ^ the vane aaiic^ as ^xii. tiS Ik fce Ae radsal velocity of ^ — 3P^r j *:w. T^KL RE. Fig. 291, is the ^^^^^^^^^'"^ ^N. '^ Twxisy wA which the watm^ leaves o^ ^ ^ ^^ ^ Xii^ radial i^efcciiy with which t^ wastnr e&ters^ the guide passages most be «i and the velocity adco^ ^le jirside isv therefore, BF. TVre is a swiden change of velocity from BE to BF, and on die iBSSit^spdoii that the Ices of head is eqnal to the head due to the TwaitTv Tvicvity FE. the head lost is t T: - Ml cot4~ Hi cot g)^ A: ^rijec tfee kxsjs^ of head is ir-ncot^)* *r5d rise rbev^r^^x-al Hft is ^9 r^* - r\» t c\>t ^ _ (^r ~ « cot BY _ (ri - Ui cot ^ ~ tii cot a)' r^* r* ir^nicota 2nico t^ ^~2g Ux^ (cot * -^ cot o)' tt* cot* ij? 2g 2g 2g 2^ (1)- Tv> ov^r^^t for the dimination of the centrifugal head and to allow fi.>r friciion^ iV_2tr,H.ti, , u^ 2g 2g ^^ 2g' mu$t be addeiU and the lift is then * ^ «\* _ r* ^ 2tM*» cot a 2ri« cotf _ V (cot ^ ■^ cot a)* ig ' 2g^ 2g ^ 2g 2g t^cot^^ tV 2fatriUi fc«i' 2g '^ 2g 2g 2g ' CENTRIFUGAL PUMPS 427 which, since u can always be written as a multiple of Ui, reduces to the form 2gh = mvi^ -^ CuiVi-i^ CiUi'' (2). Equations for the turbine pum/p shown in Fig. 275. Character- istic cwrvea. Taking the data tf = 5 degrees, cot tf = 11-43 <^ = 30 „ cot*= 1-732 o» 3 „ cot a =19-6 D = 2-5d equation (2) above becomes 2^A = '84»i» + 48-3i^t7i-587t^* (3). piat^UMrge^ igv Cubic F^ perlHnu^ O 12 8 4.^ Sndial f VAoeUy aJbBadJb ffxnv Ovt WheA. Te9t par SmcorvdU Fig. 292. Head-disoharge oarres at constant speed for Turbine Pump. From equation (3) taking r, as 50 feet per second, the head- discharge curve No. 1, of Fig. 283, has been drawn, and taking h as 35 feet, the velocity-discharge curve No. 1, of Fig. 284, has been plotted. In Figs. 292 — 4 are shown a as^'ies of head-discharge curves at 428 HTDRAUUGS constant speed, velocity-discharge curves at constant head, and head-velocity curves at constant discharge, respectively. The points shown near to the curves were determined experi- mentally, and the curves, it will be seen, are practically the mean curves drawn through the experimental points. They were how- ever plotted in all cases from the equation 2gh = r087t?i» + 2'26uiVi - 62-lu,*, obtained by substituting for m, C and Ci in equation (2) the values r087, 2*26 and -62*1 respectively. The value of m was obtuned by determining the head A, when the stop valve was closed, for speeds between 1500 and 2500 revolutions per minute. Fig. 282. The values of C and Ci were first obtained, approximately, by taking two values of v^ and Vi respectively from one of the actual velocity-discharge curves near the middle of the series, for which h was known, and from the two quadratic equations thus obtained C and Ci were calculated. By trial C and Ci were then corrected to make the equation more nearly fit the remaining curves. SpmecL —KeyolBUions per liUnuU^. Fig. 293. Velocity-Disoharge corYes at Constant Head. No attempt has been made to draw the actual mean curves in the figures, as in most cases the difPerence between them and the calculated curves drawn, could hardly be distinguished. The reader can observe for himself what discrepancies there are between the mean curves through the points and the calculated curves. It CENTRIFUGAL PUMPS 429 will be seen that for a very wide range of speed, head, and discharge, the agreement between the curves and the observed points is very close, and the equation can therefore be used with confidence for this particular pump to determine its performance under stated conditions. It is interesting to note, that the experiments clearly indicated the unstable condition of the discharge when the head was kept constant and the velocity was diminished below that at which the discharge commenced. Fig. 294. Head-velocity carves at Constant Discharge. 236. Losses in the spiral casings of centrifugal pumps. The spiral case allows the mean velocity of flow toward the discharge pipe to be fairly constant and the results of experiment seem to show that a large percentage of the velocity of the water at the outlet of the wheel is converted into pressure head. Mr Livens* obtained, for a pump having a wheel 19^ inches diameter running at 550 revolutions per minute, an efficiency of 71 per cent, when delivering 1600 gallons per minute against a head of 25 feet. The angle ^ was about 13 degrees and the mean of the angle for the two sides of the vane 81 degrees. For a similar pump 21f inches diameter an efficiency of 82 per cent, was claimed. * Proeeedingi Inst. Mech, Engs.^ 1903. 480 HTDRAULICS Tho author finds th© equation to the head-discharge curreiar the 19i inches diameter purop from Mr Livens' data to be and for thiB 2U inckm diameter pump l'lSvi*-iMiVi = 2gh _. ,_®. The velocitf of rotation of the water round the wheel will h less than the velocity with wliich the i^-ater leaves the wheel sni there will be a losa of head due to the j§udden change in velodty* /!=' Let this losa of head be v* radial velocity of flow at eri wheel nidially, is then trj'-t^iUiCot^ 9 Taking friction and the account, n 5= ~ ^ — - 9 2g which again may hr written I. TTt ^. Tlie head, when u, Utb aasmniBg the water eaten tti^ tjoo of oentrifagal bed its 2g 2^' h = ^g Cuin Cifii' % ^9 The vHluei> of ;«, C and Ci are given for two pumps in eqva&di (1) uml VI). 237. General equation for a centrifugal pump. The CH^ nations for the gross head ft at discharge Q as detennmfti for the seveiTil classes of pumps have been shown to beof thefonn '^ 2ff ^ 2g ^ 2g ' or, if a is the velocity of How from the wheel, , mv^ Quv Ci?i* '^9 2^ 2g in which m varies t jet ween 1 and To. The coefficients Ciandu for any pump will depend upon the unit of discharge. As a further example and illuj^trating the case in whieli ^ certain speeds the How may lie unstable, the curves of Flf^ 285 — 2J^7 may be now considered. When Vi is 60 feet per saw the equation to the head discharge curve is Q being in cubic feet per minute. CENTRIFUGAL PUMPS 431 The velocity-discharge cnrve for a constant head of 80 feet as colated fmiu tluM equation is shown in Fig* 287. Tu stiirt the pump against a head of 80 feet the periphf?ral jdciiy has to be 707 feet per second, at which velocity the ^har^* Q suddenly rises to 4 "3 cubic feet per minute. The curves of actual and nianonietric efficiency are shown in b. 286, the maximum for the two cases occurring at different leharges. 338- Tbe Limiting height to which a single wheel centri- Bil pump can he used to raise water. The maximum height to which a centrifugal pump can raise Iter, depends theoretically upon the niaxiinam %'elocity at which * lim of thc^ wheel can be run. It has already been stated that rim velocities up to 250 feet •rsetoiiil IjavL* been used. Assuming radial vanes and a mano- Btric efficiency of 50 per cent., a pump running at this velocity lift against a head of 9HQ feet, t t}w^' very high velocities, however, the wheel must be of mnterial such as brauKe or cast steel, having considerable nee t^ tensile Rtresses, and ftpecial precautions must be to balance the wheeL The hydraulic losses are also derable, and manometric efficiencies greater than 50 per hardly to be expected. ding to M. Rateai!*, the limiting head against which it is We to raise water by means of a single wheel is about feet, and the maximum demrable velocity of the rim of the i is about 100 feet per second. ngle wheel pumps to lift up to 3oO feet are however being At tliis velocity the stress in a hoop due to centrifugal forces mt 7250 lbs. per sq. incht. 139. The sxictioii of a centrifugal pump. The greatest height through which a centrifugal or other class amp will draw water is about 27 feet. Special precaution has I taken to ensure that all joints on the suction pipe are perfectly ^htj and especially is tliis so when the suction head is greater 15 feet; iJTify under hipecial circumstances is it therefore de* for the suction head to be greater than this amount^ and it ^ advisable to keep the suction head as small as possible^ Centriftiaes,*' ©tc., Itnttetin d* la SociiU de I'Tjidtittrie minfmtfj r. p. aafi, Mnrch, 1902. witijis Sirrnfrih of SJatermlt; WooJ's Strength o/ Strttffural Membert; Turbine^ Stodola. CENTBIFUOAI. PUMPS 433 lO. Series or multi-Btage turbine pomps, has l>eeii stated that the limiting economical head for a single L purap is about 100 feet, and for high heads series pumps low generally used. ^f^^nss^ ^i3i» 296. General Arrangement of Worthington Multi-stage Turbine Pump. By putting several wheels or rotors in series on one shaft, each >tor giving a head varying from 100 to 200 feet, water can be ^ed to practically any height, and such pumps have been U H. "1^ HTOEAt?t.ICS coDJ^tTuct^ to work head of 2O0O feet. The I to work against a beail or W^M3 feet. iHe ntiiDD of rocorsi oo one eimft* m^y be &QfEi one to twelve acconliisi lo II10 total liead* For a given hefid, tlia greater the number 1 rtilors used, the leas the peripheral velocity^, aod within hitiitB the greater the e&CMmej^ FlgB^ 295 and 296 show a longitndixial section and i re gpe ct i%'e ly, of a seried, or muld-stagt^ pump, hf the Worthkigtoii Pnnip Conipaiij-- On the mi* shaft are fixed three pboephor^brcmze rotors^ alternating with I giudeSi which are Tigidl|- oonnected to the outer casings iind I the bearings. The water is drawn in thrL>ugb the pipe at tb ) of the pisp and enters the fir«t wheel axialty. The watar lea^ the first wheel at the outer circmnlerence and paj^ises aloti^ 1 expanding pa^ssagB in which the velocttj' is gradually dtndtiisb at^ enters the second wheel axially. The vane« in the are of hard phosphor-bronze made very smooth to rtnluce fricti^ losses to a minimum. The water parses through the remaintl rotors and guides in a ^miJar manner and is finally discharf into the caaiiig and thence into the delivery pipe. Fig. f97. Siilx«r Malti-sUge Turbine Ptiiiip. The difference in pressure head at the entrances to any tw*> consecutive wheels is the head impressed on the water by *J»«? wheel. If the head is h feetj and there are n wheels tl^ lift is nearly nh feet. The vanes of each wheel and the dirt the guide vanes are detennine<l as ejc plained for the mg\s *Sf> that losses by shock are redxiced 10 a miniuiiitiif mi 's and gnide passages are made smooth so as to reduce the back of each wheel, just above the boes^ *^ ' holes which allow water to get behind part of tlw *he pressure at which it enters the wheel, to balauct? which would otherwise be set up. CENTRIFUGAL PUMPS 435 The pomps can be arranged to work either vertically or rizontally, and to be driven by belt, or directly by any form motor. Fig. 297 shows a multi-stage pump as made by Messrs Sulzer. e rotors are arranged so that the water enters alternately m the left and right and the end thrust is thus balanced. Sciencies as high as 84 per cent, have been claimed for multi- •ge pumps lifting against heads of 1200 feet and upwards. The Worthington Pump Company state that the efficiency DiBishes as the ratio of the head to the quantity increases, the it results being obtained when the number of gallons raised r minute is about equal to the total head. ExampU. A pamp is to be driven by a motor at 1450 revolutions per minute, and •quired to lift 45 oubio feet of water per minute against a head of 320 feet. i&iad the diameter of the suction, and delivery pipes, and the diameter and r of the rotors, assuming a velocity of 5*5 feet per second in the suction and ^ I^pes, and a manometrio efficiency at the given delivery of 50 per cent. one provisionally that tbe diameter of the boss of the wheel is 8 inches. Lei d be the external diameter of the annular opening. Fig. 295. 144 "" 60 X 5-5 * B which d=:6 inches nearly. Fakiog the external diameter D of the wheel as 2d, D is 1 foot. ^ 1450 _^- , Phen, Vi = -g^ X r = 76 feet per sec. (kflBoming radial blades at outlet the head lifted by each wheel is 76^ A=0.5.i|feet = 90 feet. Poor wheels would therefore be required. 24L Advantages of centrifugal pumps. There are several advantages possessed by centrifugal pumps. In the first place, as there are no sliding parts, such as occur in iprocating pumps, dirty water and even water containing coni- •atively large floating bodies can be pumped without greatlj- iangering the pump. Another advantage is that as delivery from the wheel is istant, there is no fluctuation of speed of the water in tlu» tkni or delivery pipes, and consequently there is no necessity air vessels such as are required on the suction and delivery es of reciprocating pumps. There is also considerably less iger of large stress being engendered in the pipe lines by ater hammer*." Another advantage is the impossibility of the pressure in tlu^ * See page 384. 436 HYDRAUMCS in the pipe line it will gfatt [n th€ second case a simikr ^heii the water falls below a hough convenient 15 unooo- Oiing no effective work, ihs %y be more than 50 per ceol ing masimttm discharge. ip may be made to deUwr which water may be takai pump casing riBing above that of the majdmnm head wiiici xh rotor is capable of impressing npon the -w^ter. If the dt'liirry is closed the wheel vn\l rotate witliout any danger of the pfeasuf in the casing becoraing greater than the centrifugal hmd (pa^ 335). This may be of use in those cases where a pump is de- livering into a reservoir or pumping from a reaervoin hi tie iit case a float valve may be fitted, which, when the Tnnter ri^ to a particular height in the reservoir, closei^ the dehverr. Tk pump wheel will continue to rotate but without delivering wat^r, and if the wheel is running at s ^ a velocity that the ct'iitn- fugal head is greater than the hei delivery when the valve is opened valve may be used to stop the flo certain level. This arrangement nomical, as although the pump j power required to drive the pumj. of that required when the pump is It follows that a centrifugal water into a closed pipe system fr regularly, or at intervals, while the punip continues to rotate at a constant velocity. Pump delivering into a long pipe lin€. When a centrifagiil pump or air fun is delivering into a lung pipe line the resistances will vary approximately as the square of the quantity of water delivered by the pump* Let p2 be the absolute pressure per square inch which h^ to be maintaimni at the end of the pipe line, and let the resistances vary ns the square of the velocity r along the pip?. Tlien if the re?iij*tances are equivalent to a head h/^ frr", tk pressure head ] at the pump end tif the delivery pipe must be w w A being the sectional area of the pipe. Let - be the pressure head at the top of the suction pipe, then the gross lift of the pump is u* w w A w If, therefore, a curve, Fig, 298, be plotted having CENTRIFUGAL PUMPS 437 a.s ordinates, and Q as abscissae, it will be a parabola. If on "fclie same figure a cnrve having h as ordinates and Q as abscissae 'be drawn for any given speed, the intersection of these two enures at the point P will give the maximum discharge the pump *will deliver along the pipe at the given speed. JKdfeharffe uv C.Ft,per SecondU Fig. 29S. 242. Parallel flow turbine pump. By reversing the parallel flow turbine a pump is obtained which is similar in some respects to the centrifugal pump, but differs from it in an essential feature, that no head is impressed on the water by centrifugal forces between inlet and outlet. It therefore cannot be called a centrifugal pump. The vanes of such a pump might be arranged as in Fig. 299, the triangles of velocities for inlet and outlet being as shown. The discharge may be allowed to take place into guide passages above or below the wheel, where the velocity can be gradually reduced. Since there is no centrifugal head impressed on the water between inlet and outlet, Bemouilli's equation is tr 2gr K? 2gr * From which, as in the centrifugal pump, XT Vifi pi J) . U* u^ g w w 2g 2g If the wheel has parallel sides as in Fig. 299, the axial velocity of flow will be constant and if the angles <^ and B are properly chosen, Vr and Vr may be equal, in which case the pressure at inlet and outlet of the wheel will be equal. This would have the advantage of stopping the tendency for leakage through the clearance between the wheel and casing. 438 UYDRAULICS Such a pump is airnilar to a reversed impulBe tarbine, tk guide passages tif which are kept full. The velocity witii wM tlie water leavt-B the wheel wuuld however be great and theB above the pamp would depend upon the percentage of the Vt4tjciiy liead that could be converted into preissure head. Since there is no centrifugal d impresgsed upon tie watw, the parallel-flow pump cannot ci vence discharging unless ik water in the pinup is first set in tion by some external ra^i^s but as soon as the flow ia coroine«*.,d through the wheel, tbeM discliarge under full head can be obtained. Fij?. 300. To commence the discharge, the pump would generally have to be placed below the level of the water to be lifted, an auxilian* discliarge pipe being fitted with a discharging valve, and a non- return valve in the discliarge pipe, arranged as in Fig. 300. CENTRIFUGAL PUMPS 439 3 pamp could be started when placed at a height ho above ,ter in the sump, by using an ejector or air pump to exhaust r from the discharge chamber, and thus start the flow :h the wheel. 3. Inward flow turbine pump. :e the parallel flow pump, an inward flow pump if constructed not start pumping unless the water in the wheel were first motion. If the wheel is started with the water at rest ntrifugal head will tend to cause the flow to take place •ds, but if flow can be commenced and the vanes are ly designed, the wheel can be made to deliver water at its periphery. As in the centrifugal and parallel flow pumps, water enters the wheel radially, the total lift is H = Yi^'=2>_£+U'_^ (1). g w w 2g 2g ^ ^ >m the equation £. Xl -'Pi Vl. a^ ^Vl w'^ 2g~w '^2g'^2g 2g' be seen that unless Vr* is greater than 2g 2sr 2g^ less than p, and ^ will then be greater than the total ry special precautions must therefore be made to diminish locity U gradually, or otherwise the efficiency of the pump ) very low. 3 centrifugal head can be made small by making the nee of the inner and outer radii small. 2g^2g 2g le equal to -^ , the pressure at inlet and outlet will be the and if the wheel passages are carefully designed, the re throughout the wheel may be kept constant, and the becomes practically an impulse pump. 3re seems no advantage to be obtained by using either llel flow pump or inward flow pump in place of the centri- pump, and as already suggested there are distinct dis- .ages. I. Reoiprooating pumps. simple form of reciprocating force pump is shown dia- latically in Fig. 301. It consists of a plunger P working in 440 Fig. sola. Vertical Single-acting Reciprocating Pump. RECIPROCATING PUMPS 441 I cylinder C and has two valves Vs and Vd, known as the suction ind delivery valves respectively. A section of an actual pump 8 shown in Pig. 301a. Assume for simplicity the pump to be horizontal, with the jentre of the barrel at a distance h from the level of the water n the well; h may be negative or positive according as the >ump is above or below the surface of the water in the well. Let B be the height of the barometer in inches of mercury, rhe equivalent head H, in feet of water, is H.l?:f-^ = 1133B, which may be called the barometric height in feet of water. When B is 30 inches H is 34 feet. When the plunger is at rest, the valve V© is closed by the head )f water above it, and the water in the suction pipe is sustained by .he atmospheric pressure. Let ha be the pressure head in the cylinder, then At) = H — fe, )r the pressure in pounds per square inch in the cylinder is p = -43(H-fe), cannot become less than the vapour tension of the water. At )rdinary temperatures this is nearly zero, and hn cannot be greater han 34 feet. If now the plunger is moved outwards, very slowly, and there s no air leakage the valve Vg opens, and the atmospheric pressure ^uses water to rise up the suction pipe and into the cylinder, lo remaining practically constant. On the motion of the plunger being reversed, the valve Vs rloses, and the water is forced through Vd into the delivery ripe. In actual pumps if ho is less than from 4 to 9 feet the lissolved gases that are in the water are liberated, and it is there- ore practically impossible to raise water more than from 25 to W) feet. Let A be the area of the plunger in square inches and L the troke in feet. The pressure on the end of the plunger outside the cylinder is equal to the atmospheric pressure, and neglecting he friction between the plunger and the cylinder, the force neces- sary to move the plunger is P = -43 {H - (H - A) } A = -43^ . A lbs., ind the work done by the plunger per stroke is E = -43^ . A . L ft. lbs. pet ffmte ti Ik )fain • "ttAL Hm^ ud tb ■ft i — ,^^" , i.^cAiiL i fci^ ^ vliefc dip waMir larii tk d^mf K bf ilui pliiflftr dnna^tk ■ Ai ■ ■» «iiiv« tki«fc«t * * JS * ^ fool poooflipfrfKmid. iv « ^firi bkmm m Urn pn^ md Am bmiitd Md triwi. mi ikr vehae ^ vuer UM per ai^tioii ftn^b rH WmmbM It ligWr ha tin ^nshuw moml tbi^ b; ^ 1^ ' tr tk viifk U wuer lifted per minate, and K tiF [ viki tkr wilpr is lifted, ^bjtlirpQiiipk W.^ foot poondipff HP= siooo* i j^T - _^.* -t - "icbufe of thfi pomp. Slip. ^ ^-p -i & p)tingier pump is the rite I ' ^u. -4 It ^ l^^i^ P^ £tn)ke multiplied by the nmahsifi | ^ *^m «f ^ fcibigr pir stroke to ihe rolnme digpli<^lffl :,xS'jct hiS^Km d^^ qvBiinei ti called the SKf. ^^i^' tM ««»I &cfcirgr H l» tbi tie tb^retical th0 ( ^ la a ««dr ^^^rkiiig pm 4fai;«]a be le» than five r Tbe c»a«* «>f »««*»"^ ^^kg^pbw will be disc Tnltai luii RECIPROCATING PUMPS 443 246. Diagram of work done by the pomp. Theoretical Diagram. Let a diagram be drawn, Fig. 302, the "dinates representing the pressure in the cylinder and the abscissae le corresponding volume displacements of the plunger. The flumes will clearly be proportional to the displacement of the lunger from the end of its stroke. During the suction stroke, n the assumption made above that the plunger moves very owly and that therefore all frictional resistances, and also the lertia forces, may be neglected, the absolute pressure behind the lunger is constant and efqnsA to H - fe feet of water, or 62*4 (H - h) onnds per square foot, and on the delivery stroke the pressure is 62'4 f Z + H + o- ) pounds per square foot. he effective work done per suction stroke is ABCD which equals I'i.h, V, and during the delivery stroke is EADF which equals 62-4 (Z.^), nd EBCP is the work done per cycle, that is, during one suction ad one delivery stroke. Pressure^ Fig. 302. Theoretical diagram of pressare in a Reciprocating Pump. l/V^ S$ Strokes per nunju/yo -^ Fig. 303. Actual diagram. Fig. 303 shows an actual diagram taken by eans of an indicator from a single acting pump, when running 'j a slow speed. The diagram approximates to the rectangular form and only 444 HYDRAULICS differs from the above in that at any point p in the soction strda^ pq in feet of water is equal to h plus the losses in the suction pipe, including loss at the valve, pins th© head required to accelerate the wat^r in the suction pipe, and qr is the lied required to lift the water and overcome all losses, and to accelerate the water in the deliveiy pipe* The velocity of the plunger beii^ small, these correcting quantities are practically inappreciable. The area of this diagrani represents the actual work done cm the water per cyck^ and is equal to W (Z ^ h}^ together with ^ head due to velocity of discharge i ^, all losses of energy in tiie suction and delivery pipes. It will be seen later that althot at any instant the pressnn in the cylinder i« effected by the ^rtia forces, the total work done in accelerating the water is *o. 247. The accelerationB of t pump piunger and of ^ water in the suction pipe. The theoretical diagramj Fig. 2, has h^u dra^^Ti on die assumption that the velc»city of ' plunger is very small and without reference to the variatioTi f the velocity and of the acceleration of the plunger, but it is now necessary to consider this variation and its effect on the motion of the water in the suction and (lch"vci-y pipes. To realise how the velocity and acceleration of tlic ])hnigcr varj", suppose it to be driven by a crank and connecting rod, as in Fig. 304, and suppose the crank rotates with a uniform angular velocitj' of ^ radians per second. Fig. 304. If r is tlie radius of the crank in feet, the velocity of the crank pin is V = o>r feet per second. For any crank position OC, it is proved in l)ooks on mechanism, that the velocity of the point B is V ()f\ ()\) • ^y luaking BD equal to CK a diagram of velocities EDF is found. When CB is very long compared with CO, OK is equal to OC sin 6^, and the velocity v of the plunger is then Vsin^, and RECIPROCATING PUMPS 445 EDF is a semicircle. The plunger then moves with simple harmonic motion. If now the suction pipe is as in Fig. 300, and there is to be eontiniiity in the column of water in the pipe and cylinder, the velocity of the water in the pipe must vary with the velocity of the plunger. Let V bo the velocity of the plunger at any instant, A and a the cross-sectional areas of the plunger and of the pipe respect- V A ively. Then the velocity in the pipe must be — — . Fig. 305. As the velocity of the plunger is continuously changing, it is continuously being accelerated, either positively or negatively. Let I be the length of the connecting rod in feet. The acceleration* F of the point B in Fig. 305, for any crank angle 0, is approximately F = caV (cos ^ + ^ cos 2^ V Plotting F as BG, Fig. 305, a curve of accelerations MNQ is obtained. When the connecting rod is very long compared with the length of the crank, the motion is simple harmonic, and the acceleration becomes F = wV cos ^, and the diagram of accelerations is then a straight line. Velocity and acceleration of the water in the suction pipe. The velocity and acceleration of the plunger being v and F respectively, for continuity, the velocity of the water in the pipe must be A V — and the acceleration a /.= F.A * 8«e BaUmeing of Enginet, W. E. D«lby. 44*; HYDRAULICS 248. TIM e£Eect of acccleratlofzi of the plnxi^er on Xt% pressore in tiie cylinder during the suction stroke. ^^'!;en the Trfix iiy of ibe plmiig<?r is increaairig, F is poeitiT^, ani :. mcoeleTmlki' Hie wmter in tte suction pipe m foroe P ii T>e«::i:rv-i. Tfce •tTi3->?pheric preasisiie has, thereforej not only U) .'jr, thr- v:»ter an^i oi-en^ouie the remtance in the snetion pipe, V-u: :: h.As mbo to provide the necee^iry force to accelerate tk 'w-^ter, ind the prr--^-'* in tW cylinder is ccinseqnently dimimshed . &5 the Telocity of the plunger decreases, piston has ** exert a reaction upon the relocttyj or pre6sure on the plangvr is :rie ocher rr«tive, an F w-^Tc^r : > diminisl. L«r: L be the * :'h of the m 5^n -„;il arm in t* feet, /» tl :'::- I ::>e at any in^iant in feet per ^c^-xti: f a cohic fuiot c4 water, Tr-vn the masfr of water in the i ;• -^T. 1-, and since by Newton's sec acce ermdng force = ai :. - aCvv'eratinsr f -vv rt^juired is p fr . a, L g T:.- :-\><urv ivr unit arva is :i' 1 :. .^ -;u:vLi\::: head .f water is :7 ■ . F . A tl pipe in feet, a its cross- Dceleration of the i^^ater iu and per second, and u- tke to he accelerated is ii?.a«X law of morion « acceleration, K lbs. ./.lbs., ^. (1 ^i.= LA (7.a ' F arce if any one of the three quantities, L, — , or and other losses the pressure in the v'v'..:..;r r :> ::• w a:;d :V.v h^^ad r>:>:sring the UK^rion of the piston is h + ha, 249. Pressure in the cylinder during the suction stroke when the plunger moves with simple harmonic motion. It ::\o p'ur.c^^r Iv supp^^sed driven by a crank and very long RECIPROCATING PUMPS 447 innecting rod, the crank rotating uniformly with angular velocity radians per second, for any crank displacement ^, F = <»i"rcos^, J , L.A.wV /, nd ha = . cos (f. g.a The pressure in the cylinder is LAui'rcos^ H-A-- ga When B is zero, cos B is unity, and when ^ is 90 degrees, cos B \ zero. For values of B between 90 and 180 degrees, cos^ is e|B^tive. The variation of the pressure in the cylinder is seen in 'ig. 306, which has been drawn for the following data. Fig. 306. Diameter of suction pipe 3| inches, length 12 feet 6 inches. Oiameter of plunger 4 inches, length of stroke 7^ inches. Number of strokes per minute 136. Height of the centre of :he pump above the water in the sump, 8 feet. The plunger is issumed to have simple harmonic motion. The plunger, since its motion is simple harmonic, may be supposed to be driven by a crank 3J inches long, making 68 revo- lutions per minute, and a very long connecting rod. The angular velocity of the crank is u>= ' = 7*1 radians per second. The acceleration at the ends of the stroke is F = a,>.r = 7-rx 0-312 = 15'7 feet per sec. per sec, A / 4 V . a \3- 125/ and K = 12-5. 15-7. 1-63 32 1-63, = 10 feet. 448 HY0EAULICS The preesnre in the cylinder neglecting the wn^tev m xh cylinder at the beginning of the stroke is, therefore, and at the end it is 34-8 + 10 = 36 feet- That is, it is gnatrt than the atmospheric pressure. WTien ^ is 90 degrees, cos is aero, and A^ is therefore jscro, and when is greater than 4K) degrees, eos ^ is negative. The area AEDF is clearly equal to GADH, and the work iam per suction stroke is, therefore, not altjered by the accielemtiiig forces; but the rate at which thr ^^tnger ia working at varii'«i points in the stroke is affected by m, and the force requiivi] t^> move the plunger may be very m^ In the above example, for t move the piston at the commei more than doubled by the ace remaining constant and equal t4 varies from P= 4iJ(8 to P = '43(8 Air i^esseh, Tu quick running t^timpe, or when the leitgtk of the pipe is long, the effects of these accelerating forces tend to become serious, not only in causing a very large increase in the stresses in the parts of the pump, but as will be shown later, under certain circumstances they may cause separation of the water in the pipe, and violi^nt hammer actions may be set up. To reduce the effects of the accelerating forces, air vessels are put on the suction ami delivery pipes. Figs. 310 and 311. 250. Accelerating forces in the delivery pipe of a plunger pump when there is no air vessel. When the plunger commences its return stroke it has not only to lift the water against the head in the delivery pii^e, but, if no air vessel is i)rovided, it has also to accelerate the water in the cylinder and the delivery pipe. Let D be the diameter, a^ the area, and Li the length of the pipe. Neglecting the water in the cylinder, the acceleration head when the acceleration of the piston is F, is L,.A.F nereaiied. ice, the force necessary ki ent of the stroke has be«n ting force, and instead di ,8. A during the stroke, it] )A ha = gcii and neglecting head lost by friction etc., and the water in the cylinder, the head resisting motion is 7j + ha + If F is negative, ha is also negative. '^9' RECIPBOCATINO PUBCPS 449 When the plunger moves with simple harmonic motion the iagram is as shown in Fig. 307, which is drawn for the same ats as for Fig. 306, taking Z as 20 feet, In as 30 feet, and the iameter D as S^ inches. Fig. 307. The total work done on the water in the cylinder is NJKM, rhich is clearly equal to HJKL. If the atmospheric pressure is cting on the outer end of the plunger, as in Fig. 301, the nett rork done on the plunger will be SNRMT, which equals HSTL. 251. Variation of pressure in the cylinder due to firiction irhen there is no air vessel. Head lost by friction in the auction and delivery pipes. If t? is he velocity of the plunger at any instant during the suction troke, d the diameter, and a the area of the suction pipe, the elocity of the water in the pipe, when there is no air vessel, is A — , and the head lost by friction at that velocity is , _4fifJf^L ^^~ 2gda' ' Similarly, if ai, D, and Li are the area, diameter and length espectively of the delivery pipe, the head lost by friction, when he plunger is making the delivery stroke and has a velocity v, is ^^ 2gDa,' ' When the plunger moves with simple harmonic motion, V = <i>r sin ^, nd , _ 4/A'<oVsin'gL ^•^■" 2gda' L. H. 29 430 HVDRAUTJCS If the pump mftkeA n strokes per aeoond, or tli@ nitmber d revoloticiiis el tha cmnk is ^ per second, and 4 is tlia lengtK d the strokei and /, = 2r. Snbvtttatiiig for i>» and r, Plotting values of A/ at vai ' points along the stntke, parabolic cnrv© E>tP, Fig. 808, When ^ i» 90 degree®, ain^ The mean ordinate of the pan head, is then 2^ 3 .amed QJty^ and ^ is a maTiinmru which ia the mean fricti<nial and since the mean frictional head is equal to the energy lost per pound of water, the work done per stroke by friction is 2gda' all dimensions being in feet. foot lbs., r;^ G H '^"^ i N A Sf if n i ! •J I R £ B 1 [ N^-^^ F i S B T i <3-i- Fig. 309. Let Do be the diameter of the plunger in feet. Then and RECIPROCATING PUMPS 451 Therefore, work done by friction per suction stroke, when here is no air vessel on the suction pipe, is The pressure in the cylinder for any position of the plunger .uring the suction stroke is now. Fig. 309, Ao = H — fe — fea~ fe/. At the ends of the stroke h/ is zero, and for simple harmonic lotion hm, is zero at the middle of the stroke. The work done per suction stroke is equal to the area LEMFD, which equals ARSD + EMF = 62-4feV + ^^'P/^^'* . Similarly, during the delivery stroke the work done is 62-4ZV + §^^^g^^*. The friction diagram is HKG, Fig. 309, and the resultant iagram of total work done during the two strokes is EMFGrKH. 252. Air vessel on the suction pipe. Afi remarked above, in quick running pumps, or when the sngths of the pipes are long, the effects of the accelerating forces leeome serious, and air vessels are put on the suction and delivery apes, as shown in Figs. 310 and 311. By this means the velocity Q the part of the suction pipe between the well and the air essel is practically kept constant, the water, which has its elocity continually changing as the velocity of the piston hanges, being practically confined to the water in the pipe between the air vessel and the cylinder. The head required to accelerate the water at any instant is consequently diminished, >nd the friction head also remains nearly constant. Let l^ be the length of the pipe between the air vessel and he cylinder, I the length from the well to the air vessel, a the rosa-sectional area of each of the pipes and d the diameter of the ripe. Let h^ be the pressure head in the air vessel and let the air 'essel be of such a size that the variation of the pressure may for implicity be assumed negligible. Suppose now that water flows from the well up the pipe AB ontinuously and at a uniform velocity. The pump being single icting, while the crank makes one revolution, the quantity of rater which flows along AB must be equal to the volume the )lunger displaces per stroke. 29—2 *:t >^J^\ F^- 31 '- ::.:i-//.:'> :he;rv:i:, including friction and the velocity c> r^rir.a: r.e^'.rvtcd. the atmospheric head - A' «V 4f'A-<uV/ .(1). .1 r.uTth qaannnes of the right-hand part of the :-:-^l'y be very small and hr is practically equal u:: :: >rr>:ke is raking place, the water in the pip^ :l J ':v :r:r ;rvs5j^;irv head at the point B, when the velocity u'^jTt r i> r :\v: ivr sev.vnd, and the acceleration F feet per ><x\ • .1 ;vr ><Ov RECIPROCATING PUMPS 453 Let hf be the loss of head by friction in AB, and h/ the loss in iC. The velocity of flow along BC is — , and the velocity of ow from the air vessel is, therefore, t?.A Aft>r a ira Then considering the pipe AB, .nd from consideratioii of the pressures above B, <vA A«i»r\' ls=K*h-{-2—^^ 2g /• Neglecting losses at the valve, the pressure in the cylinder is hen approximately ,, AkF Tx J. A*o»V , ,, Ali¥ 2gnra' ^ ^ ag A*a)V Neglecting the small quantity ^ 9 « , For a plunger moving with simple harmonic motion I. XT 7, 4/wVAVZ^, . ,.\ AZiiuVcosd ho = lB.-h- -^ — rj-(3 + *i8m*^) . 2ga^d \ir Jag By putting the air vessel near to the cylinder, thus making I small, the acceleration head becomes very small and Ao = H - fe - fe/ nearly, nd for simple harmonic motion The mean velocity in the suction pipe can very readily be etermined as follows. Let Q be the quantity of water lifted per second in cubic feet. Then since the velocity along the suction pipe is practically onstant Vm = — and the friction head is BECIPROOATIKG PUMPS 455 Let k. Fig. 311, be the length of the pipe between the pump and the air. vessel, Id be the length of the whole pipe, and ai and D the area and diameter re8i)ectively of the pipe. Let At be the height of the surface of the water in the air vessel above the centre of the pipe at B, and let Ho be the pressure head in the air vessel. On the assumption that Ho remains constant, the velocity in the part BC of the pipe is practically constant. Let Q be the quantity of water delivered per second. The mean velocity in the part BC of the delivery pipe will be Q :=■ The friction head in this part of the pipe is constant and equal to Considering then the part BC of the delivery pipe, the total head at B required to force the water along the pipe will be But the head at B must be equal to H» + /la nearly, therefore, H. + A, = Z + ^-^ga.-W + H (1). In the part AB of the pipe the velocity of the water will vary with the velocity of the plunger. Let V and F be the velocity and acceleration of the plunger respectively. Neglecting the water in the cylinder, the head Hr resisting the motion of the plunger will be the head at B, plus the head necessary to overcome friction in AB, and to accelerate the water in AB. Therefore, Hr = Ht, + fea + il V^ a + -- For the same total length of the delivery pipe the acceleration head is clearly much smaller than when there is no air vessel. Substituting for H» + /la from (1), If the pump is single acting and the plunger moves with simple harmonic motion and makes n strokes per second, Q = A2rJ, J Arn and tt= . be la. .4^ FA rr^' RECIPBOCATINQ PUMPS 467 X)me below the zero line BC as in Pig. 312 •. The pressure in the blinder then becomes negative; but it is impossible for a fluid k> be in tension and therefore discontinuity in the flow must CKxmrt, In actual pumps the discontinuity will occur, if the curve EFG aJls below the pressure at which the dissolved gases are liberated, >r the pressure head becomes less than from 4 to 10 feet. Fig. 812. At the dead centre the pressure in the cylinder just becomes sero when A + A^ = H, and will become negative when fe + fea > H. Theoretically for no separation at the dead centre, therefore, ha^R-h or ^^^R-h. ga If separation takes place when the pressure head is less than iome head h^y for no separation, /ia ^ H — hm "" fe, md a *^ I Neglecting the water in the cylinder, at any other point in the itroke, the pressure is negative when v^ A * ^ 2g a' h + + fe/+ ^ — a > H. a g ^ 2g a: That is, when And the condition for no separation, therefore, is FA ,(H-/^-/.--;^.fe,) .(2). * See also Fig. 816, pafi^e 459. t Sor&oe tension of flaids at rest is not alladed to. 458 HYDRAULICS 255. Separation during the suction etroke when tlu plunger moves with simple harmonie motion. When the plungt^r is driven by a crank and very long om- necting r<>d, the acceleration for any crank angle is F = w'rcos^, or if the pnmp makea n amgle strokes per second, •1= wn. and I, being the length of the stroke, F is k\ maximiim when ^ is ze place at the end of the stroke if 9 a a nd separab'on will not tata and ^^^ll just not take place whei A , ^ A — <tt'r or ^ .—n a 2 a The minimam area of the si therefore, ^ Aoi^rL ""-giU-h^-h) «ind the maximum number of single strokes per second is I pipe for no separation tSj (3) ^_1 /2g{R-}i^-h)j ''"ttV " A./,.L .(4). Separation actually takes place at the dead centre at a less number of strokes than given by formula (4), due to causes which could not very well be considered in deducing the fornmla. Example. A single acting pump has a 8troke of Ih inches and the plnngeris 4 inches diameter. The diameter of the suction pipe is 3^^ inches, the Ieni?th 12-5 feet, and the height of the centre of the pump above the water in the well is 8 feet. To find the number of strokes per second at wliich separation ^^ill take plact, assuming it to do so when the pressure head falls below 10 feet. H-/i = 26 feet, -:-l-63. and, therefore. _1 / 64x26 ""ir V 1-63 X 7-5 xl2 xl2-5 = 11 = 36 = 210 strokes per minute. Nearly all actual diagrams taken from pumps, Figs. 313—315, have the corner at the commencement of the suction stroke RECIPROCATING PUMPS 459 Toonded off, so that even at very slow speeds slight separation occurs. The two principal causes of this are probably to be found first, in the failure of the valves to open instantaneously, and second, in the elastic jrielding of the air compressed in the water at the end of the delivery stroke. DMA/'cry Zero JPre^suLre/ Fig. 314. .Atnu Line Fig. 815. The diagrams Figs. 303 and 313 — 315, taken from a single-acting pomp, having a stroke of 7^ inches, and a ram 4 inches diameter, illustrate the effect of the rounding of the comer in producing separation at a less speed than that given by equation (4). Even at 69 strokes per minute, Fig. 303, at the dead centre a momentary separation appears to have taken place, and the water has then overtaken the plunger, the hammer action producing vibration of the indicator. In Figs. 313 — 315, the ordinates to the line rs give the theoretical pressures during the suction stroke. The actual pressures are shown by the diagram. At 136 strokes .. .11 If resting to endeavour t sh«Hild overtake the plunger ; \NTiile the piston moves fi 70 degrees, in yts • Tr9 second points the preesure in the c therefore the head available resistances and to accelerate tl The height of the centre o in the samp. The total len, 12*5 feet, and its diameter is Assuming the loss of head a to have a mean valne of 2*5 f et rating the water in the pipe is is, therefore, 20-5x82 12-5 When the piston is at g th behind the piston. Let this c velocity of the water be u f e< takes the crank to torn throng through a distance = 0101tt + i5 = l-2tt + 3-2i The horizontal distance a& ii should be equal tn i*^* RECIPROGATINO PUMPS 461 The frictional resistances, which vary with the velocity, will be small. Assmning the mean frictional head to be '25 foot, the head canning acceleration is 12*55 feet and the mean acceleration of the water in the pipe while the piston moves from rest to g is, therefore, . 12-55x32 ^. ^ fm = — Toic — = ^^ *6®* P©r 86C. per sec. The velocity in the pipe at the end of 0*058 second, should therefore be t? = 32 X -058 = 1*86 feet per sec. and the velocity in the cylinder u = y:^ = ri2 feet per sec. Since the water in the pipe starts from rest the distance it should move in 0'058 second is 12.i32.C058)« = -65in., and the distance it should advance in the cylinder is 0-65 . . , . j;g3ms.= 4m.; so that z is 0*4 in. Then z + 4*2 ins. = 4*6, and V2u + 3'2 ins. = 4'57 ins. The agreement is, therefore, very close, and the assumptions made are apparently justified. 256. Negative slip in a plunger pump. Fig. 315 shows very clearly the momentary increase in the pressure due to the blow, when the water overtakes the plunger, the pressure rising above the delivery pressure, and causing discharge before the end of the stroke is reached. If no separa- tion had taken place, the suction pressure diagram would have approximated to the line rs and the delivery valve would still have opened before the end of the stroke was reached. The coeflBcient of discharge is 1*025, whereas at 59 strokes per minute it is only 0*975. 257. Separation at points in the suction stroke other than at the end of the stroke. The acceleration of the plunger for a crank displacement B mV A is — '- — COS ^, and therefore for no separation at any crank angle B -^costf^^H-fc^^fc--^^^, h,) .(1). •L=A.(l*4^)c«». f 1 -^ ^~) » eqiml to of tJbe cylisdM^ take i^ftae as tbe ikiad . ~:a- :*i_zi^>w rL :^^ - * -lie.T to be less than Ar, and - :- ' * > c 7 ^•T*jv-<^skry Tv onsi-der the condition for nv .^«5. S«c&rs=Dcir vt:^ & -^r^c ^^r Tessel on the snctioii pipe. T_:i vir-^itrr ?trcitr^rj:c: ^«^H lAke plaoe with a large air ^'->t^ .: "^♦r sijT'.c Ti^r^- -' -> -'ii-y neo^ssarr to substitute in r»Mi*~ :- - r^.'o c .v^ jkz>£ o . 4 , section 2o6, A» of Fig. 310 ^ r r. ~ r _. i.-:«i i r r >, Iz. FjC- olO, k_ is negative. y r :« ?^oij~ic:» c ^iie^ i^je ri.:Lz^>rr :> ai the end of the strc'ke ::it T.-.i^m Lzi 4r*f>i . z Z2sT ZLztz r^er^-errr. ihe air vessel and the :^-r fr ^=lf^^- .^ . A - : *. 1 ', section 253, and .^A' 4-"L*VA- * :•! ^> :t:t frticiD.c ':>^fcas^ in the denominator, W .^:c:irikrv\i w-.ii^ < H - n L then. RECIPROCATING PUMPS 463 The maxmmm number of strokes is ^1 / 2g(R-h-K)a A pump can therefore be run at a much greater speed, without of separation, with an air vessel on the suction pipe, than *ithoat one. 259. Separation in the delivery pipe. Consider a pipe as shown in Fig. 316, the centre of CD being at height Z above the centre of AB. Let the pressure head at D be Ho, which, when the pipe ischarges into the atmosphere, becomes H. Let ly 1% and k be the lengths of AB, EC and CD respectively, f^ hf, and A^, the losses of head by friction in these pipes when the lunger has a velocity v, and hm the pressure at which separation ctoally takes place. ' — i/O rw^ ^3• N ( >- I -M fij^y. ; I / Fig. 316. Suppose now the velocity of the plunger is diminishing, and its "etardation is F feet per second per second. If there is to be F A jontinuity, the water in the pipe must be also retarded by — '- — :eet per second per second, and the pressure must always be positive and greater than fe.,. Let Ho be the pressure at C ; then the head due to acceleration in the pipe DC is FAZ. 9 and if the pipe CD is full of water H« = Ho :: — A/, which becomes negative when FAZ, 9 g > Ho ~ h/. alid ar tor for Qo $^t andf, or n. b*^ deihvrv P'aiv at B. RECIPROCATING PUMPS 405 260. Diagram of pressure in the cylinder and work done curing the suction stroke, considering the variable quantity of rater in the cylinder. It is instructive to consider the suction stroke a little more in etail. Let V and F be the velocity and acceleration respectively of :xe piston at any point in the stroke. As the piston moves forward, water will enter the pipe from the ""dl and its velocity will therefore be increased from zero to • — ; the head required to give this velocity is ^'-2^ (1)- On the other hand water that enters the cylinder from the pipe 3 diminished in velocity from — to v, and neglecting any loss due o shock or due to contraction at the valve there is a gain of iiressure head in the cylinder equal to ^-2^-^-2^ ^2). The friction head in the pipe is , 4/Lv'A' ^^=2^^ (3). The head required to accelerate the water in the pipe is ^' = -^ • W- The mass of water to be accelerated in the cyUnder is a variable quantity and will depend upon the plunger displacement. Let the displacement be x feet from the end of the stroke. The mass of water in the cylinder is lbs. and the force required to accelerate it is and the equivalent head is P ^x.F wA g The total acceleration head is therefore 9 \ a / 9 L. H. 30 I 466 I I BTDRACLICS Now hit J be tli£ pressure kead in thte cylinder^ thm „ , «» 4/LAV F/ LAV When the plunger moves with simple harroonic motioii^ and a driven by a crank of radium r rotating uniformly with angnlir velocity % the displacement of the plmnger from the end of tta stroke ife r(l- cos ^), the reiocitv vr ein and ita aceelemtioa ie w'r cos 0. Therefor© i w-r — c ^- - + .,M% 9 ^ g 9 Worh dime during iks swc> ok^* Assuming atmospli^c pressure on the face of the pi the pressnr© per square fool resisting itii motiDn ia (E p. For any small plunger displacement tj, the work done is, therefore, A (H-Ho)w'.aa', and the total work done during the stroke is E= r A(H-Ho)7r.caj. The displacement from the end of the stroke is a* = r (1 - cos ^), and therefore d.r = r sin OdO, and E - I'u' . A (H - Ho) r sin OdO. Jo Substituting for Ho its value from equation (6) , f' 4/LAVr\sin'^ a>Vsin-^ 2gda' 2g a>VcOS^ (oVcOS*^ LA 2 A - n.^ + + tuVcos 0} sm OdS. 9 g 9 (^ i Tlie sum of the integration of the last four quantities of this expression is equal to zero, so that the work done by the accelerating forces is zero, and E-«'Ar r(h+hf)amOde Jo i/LAV 29 . da RECIPROCATING PUMPS 467 Or the work done is that required to lift the water through % height h together with the work done in overcoming the resistance in the pipe. Diagrams of pressn/re in the cylinder and of work done per iCroJke. The resultant pressure in the cylinder, and the head ransting the motion of the piston can be represented diagram- Btttically, by plotting curves the ordinates of which are equal to B^ and H~Ho as calculated from equations (2) and (3). For jdeamess the diagrams corresponding to each of the parts of eqiiation (2) are drawn in Figs. 318 — 321 and in Fig. 317 is shown Ihe combined diagram, any ordinate of which equals H - A - (feZ + cd + ef- gh). t^^^—^-rdS* $h. ^ Figs. 318, 819, 820. w Figs. 821, 822. In Fig. 318 the ordinate cd is equal to and the carve H JK is a parabola, the area of which is 2 4/LA» , ,, -3- W"'^^- 30—2 468 In Fig. 819, the ordinate efis -f"-. and the ordinate gk of Fig. 320 is + COS* B, 9 The areas of the curves are respectively 1 wV and are therefore equal ; and nin the orditiatea are atwa|v d opposite sign the sum of the twc ^eas m K0ro, In Fig. 822, km is equal to and A-Z to — COSE'l JC- J. g \ a / Since cos B is negative betweei )* and 180* the area ^TT is equal to YZU. Fig. 821 has for its ordinate at any p^int of the stroke, tke head H - H,, resisting the niution of the piston. This equals /* + kl i- cd + ef-ghf and the curve NPS is clearly the cnrve GFE, inverted. The area \'XST measured on the proper scale, is the work done per stroke, and is equal to VMHT + HJK. Tlie scale of the diagram can be determined as follows. Since h feet of water = 62"4/i lbs. per square foot, the pressure in pounds resisting the motion of the piston at any point in the stroke is 62-4 . A . /i lbs. If therefiu-e, VXST be measured in square feet the work done per stroke in ft. -lbs. = 62-4 A . YNST. 261. Head lost at the suction valve. In determining the pressure head Ho in the cylinder, no account has been taken of the head lost due to the sudden enlargement from the pij)e into the cylinder, or of the more serious loss of head due to the water passing through the valve. It is probable that the r*A^ whole of the velocity head, ^^ — g, of the water entering the cylinder from the pipe is lost at the valve, in which case the available head H will not only have to give this velocity to the water, but ynW RECIPROCATING PUMPS 469 dso have to give a velocity head ^ to any water entering the sylinder from the pipe. The pressure head H© in the cylinder then becomes „ „ , i;« A" t;* 4/Lt;'A' F/ IA\ 262. yariation of the pressure in hydraulic motors due lo inertia forces. The description of hydraulic motors is reserved for the next chapter, but as these motors are similar to reversed reciprocating pumps, it is convenient here to refer to the effect of the inertia forces in varying the effective pressure on the motor piston. If L is the length of the supply pipe of a hydraulic motor, a the cross-sectional area of the supply, A the cross-sectional area of the piston of the motor, and F the acceleration, the acceleration .FA of the water in the pipe is — - — and the head required to accelerate the water in the pipe is , FAL ga If p is the pressure per square foot at the inlet end of the supply pipe, and hf is equal to the losses of head by friction in the pipe, and at the valve etc., when the velocity of the piston is v, the pressure on the piston per square foot is p« = p - wha - whf. When the velocity of the piston is diminishing, F is negative, ind the inertia of the water in the pipe increases the pressure on ihe piston. Example (1). The stroke of a double acting pump is 15 inches and the number of strokes per minute is 80. The diameter of the plunger is 12 inches and it moves vith simple harmonic motion. The centre of the pump is 13 feet above the water n the well and the length of the suction pipe is 25 feet. To find the diameter of the suction pipe that no separation shall take place, Lssuming it to take place when the pressure head becomes less than 7 feet. As the plunger moves with simple harmonic motion, it may be supposed driven )y a crank of 7^ inches radius and a very long connecting rod, the angular relooity of the crank being 2ir40 radians per minute. The acceleration at the end of the stroke is then 4ir» . 40a . r 60 Therefore, ||Jx40»x|^=84' -20', rom which - = 1*64. a 470 Therefore and d^^r. Ar is clearly lees th&n fjT^ tberefora a^pamtion o&iuiol take pkci mi u^olluo point in the stroke. Example (2). The pump of example (I) deli^rs water into a nsiAg mm 1225 feet long and 5 incbefi diameter, which U fitted with on aii- T^asBi The water is lifte^i thr^anh a total height of 920 feet. Neglecting all lonieft escept frictic*a in the deliTery pipe* deleimine the h«ifr power required to ^ork the pomp. /=0l0i5. Since there is an air vesHel in the delivezj pipe the Teloei^ of flcrw « viD W practically aniform. Let A and a be tlie cross -»eeUo&al areas ively. _ip 10 d " 25 ' a ' 6i The head h lost due to friction is = 1764 fei The total lift is therefore 220 + l76--4=l The weight of water lift^ per minale is A,2r,80 he pnmp (^Under and pipe respot- 2r. 80 ■6 ft, per WG, feafc,. Therefore, . — . 80 X 62-5 lbs. = 4900 lbs. 12 «^= 33,000 ="«■«• Exmuple (3). If in example (2) the air vessel is near the pump and the mean level of the water in the vessel is to be kept at 2 feet above the centre of tlie pump, find the pressure per sq. inch in the air vessel. The head at the junction of the air vessel and the supply pipe is the head necessary to lift the water 207 feet and overcome the friction of the pipe. Th.refore, H, + 2' = 207 + 176-4, H„= 381-4 feet, 381-4 X 62-5 ^=^—144 = 165 lbs. per sq. inch. Kxdtnple (4). A single acting hydraulic motor making 50 strokes per minute has a cylinder 8 inches diameter and the length of the stroke is 12 inches. The diameter of the supply pipe is 3 inches and it is 500 feet long. The motor i- supplied with wat^r from an accumulator, see Fig. 339, at a constant pressure of 300 lbs. per sq. inch. Neglecting the mass of water in the cylinder, and assuming the piston moves with simple harmonic motion, find the pressure on the piston at the beginning and the centre of its stroke. The student should draw a diagram of pressure for one stroke. There are 25 useful strokes per minute and the volume of water supplied per minute is, therefore. 2,-). -rf'^ = 8-725 cubic feet. 4 ,502 At the commencement of the stroke the acceleration is ir^ -— r, and the velocitT oO^ in the supply pipe is zero. RECIPROCATING PUMPS 471 The head required to accelerate the water in the pipe is, therefore, ir».50a.l.8<.500 «~ 60». 2.3^32 ^ tiou xeeiif which is equivalent to 165 lbs. per sq. inch. The effective pressure on the piston is therefore 135 lbs. per sq. inch. At the end of the stroke the effective pressure on the piston is 465 lbs. per sq. inch. At the middle of the stroke the acceleration is zero and the velocity of the piston is II «T= 1 '31 feet per second. The friction head is then •04^1^12^8^ 600' = 15-2 feet. The pressure on the plunger at the middle of the stroke is 300 lbs. - 1^^^^^= 293-4 lbs. per sq. inch. 144 The mean friction head during the stroke is 1. 15*2=10*1 feet, and the mean loss of pressure is 4*4 lbs. per sq. inch. The work lost by friction in the supply pipe per stroke is 4*4 . j . 8^ . 2^ =222 ft. lbs. The work lost per minute = 5500 ft. lbs. The net work done per minute neglecting other losses is (3001b8.-4*4).^.Z..83.25 = 370,317 ft. lbs., and therefore the work lost by friction is comparatively small, being less than 2 per cent. Other causes of loss in this case, are the loss of head due to shock where the water enters the cylinder, and losses due to bends and contraction at the valves. It can safely be asserted, that at any instant a head equal to the velocity head, of the water in the pipe, will be lost by shock at the valves, and a similar quantity at the entrance to the cylinder. These quantities are however always small, and even if there are bends along the pipe, which cause a further loss of head equal to the velocity head, or even some multiple of it, the percentage loss of head will still be small, and the total hydraulic efficiency will be high. This example shows clearly that power can be transmitted hydraulically very efficiently over comparatively long distances. 263. High pressure plunger pump. Fig. 323 shows a section through a high pressure pump suitable for pressures of 700 or 800 lbs. per sq. inch. Suction takes place on the outward stroke of the plunger, and delivery on both strokes. A brass liner is fitted in the cylinder and the plunger which, as shown, is larger in diameter at the right end than at the left, is also made of brass ; the piston rod is of steel. Hemp packing is used to prevent leakage past the piston and also in the gland box. The plunger may have leather packing as in Fig. 324. On the outward stroke neglecting slip the volume of water RECIPEOCATINO PUMP8 473 into th© oylinder is j D/ » L cubic feet, D^ being the dia- leter of the piston and L the length of the stroke. Tlie quantity t water forced into the delivery pipe through the valve V^,> is ■ J (Do' - <f ) L cubic feet, being the diameter of the small part of the "lunger. On the in-stroke, the suction valve is loeed and water is forced through the elivery valve; part of this water enters l» delivery pipe and part flows behind the ^^1 tJirough the port P> ^TThe amount that flowa into the delivery pipe is Fig. 'i2A. ^T^ therefore* {D/^ef) is made equal to (^^ or D^ is V2(i, the Lelivsry^ during each stroke, is ^ Da% cubic f eetj and if there are ces per minute, the delivery is 4245D„%k gallons per J^M^Sl^riUM^ii ^b Fig. 33S. TftDgye Duplex Pump. ^^■L Duplex feed pump. ^^P^ 32-> show^s a section through one pump and steam cylinder rf a Taiigye double-acting pump. 474 HYDRAITUCS There are mro stmm cylinders side hy aide^ one of is afaowD, aad two pump cylinderB in Kne with the srteam cjHm In iho pump the two lower valves an* sucticm valveF and lb two upper delivery valvea. As the pnmp piston P moves to i^ right, the lefl-haad lower valve opens and wat^^r is drawn iski^ pump from the auction chamber C. Daring this stroke the r^ upper valve is open, and vvater is delivered into the delivi^ Ci- ^^^len the piston moves to the left, the water m drawn in thrmigii the lower right valve and delivered through the upper left ^*alTe, The steun engine has douhle j— *- at each end. As the piston appn ka«he« the end at ita stroke t^ earn valve, Fig. 82d, m at tm and e*?veTs the steam port 1 whi to eschauHl. VVTien the piston p enck*!iied in the c>*linder acta as and plunger gradually to rest. inner stej^m port 2 m vspm dia steatn ]:M>rt 2, rlie ^mm ihkm and brings the friisou F\^. 'A'lM. Fig, S27- lji*r tilt* tnw t'ligino anil pump F^hown in ,'^'^ctiuTi be called A aini the otlipr engine nud pmup, not j?huwn, Le called B, As the piston of A moves from right to left, the lever L, Figs. 32o luiil 827, nitates a spindle to the othur end of whicli i^ fixed a cnink ^!, wbii-h moves tlie valve of the cylinder B from left to right :ind ojx^ns the left j>irt of the cylinder B. Just befure tlie piston of A reaches the left end of its strcfko, the piston of B, then*fore, commences its stroke from left t<i rights and bv a lever Li and cniiik Mj lnnvl^s the valve of cyli ruler A also from left lo right, an si the piston of A can then commence its !^*turn i^troke. It should be noted that while the piston of A is moving, that c^f B is practically at rest, and ricr verm. 265. The hydraulic ram. Tilt* bycininlic ratn is a machine which utilises the momentum of a stream of water falling a small height to raise a part of the wat-t^r tti a grt^att^r hciglit. In the iirrruigcment sho\\ni in Fig. 328 water is supplied from a tank, or stream, througli a pipe A into a chamber B, which ha^ rw^o PUMPS +75 V and Yi- Wlien no flow is taking place the valve Y falls seating' and the valve Vj rests on its seating. If water is fed tci flow along the pipe B it vnl\ escape through the open V. The contraction of the jet through the valve opening, 3r as in the case of the plate obstructing the flow in a pipe, 168, causes the pressure to b© ^eater on the under face of ftlve, and when the pressure is sufficiently large the valve x^mmence to close. As it closes the pressure will increase le rate of closing will be continually accelerated. Tlie rapid \g of the valve arrests the motion of the water in the pipe, bere m a sudden riaa in pressare in B, which causes the fVt to open, and a portion of the water passes into the air C. Tlie water in the supply pipe and in the vessel B, after \ brought to rest, recoils, like a ball thrown against a wall, ikB pres.snre in the vessel is again diminished, allowing the to once more escape through the valve V. The cycle of l&cmB is then repeated, more water being forced into the air ber C, in wliich the air is compressed, and water is forced up rfivery pipe to any desired height. Fig, 328, it A be the height the water falls to the ram, H the height to the water is lifted, f W lbs. of water descend the pipe per second, the work ble per second is Wh foot lbs., and if e is the efficiency of the e weight of water lifted through a height H will be H ■ iciency & diminishes as H increases and may be taken as ^ eent* at high heads. j'ig. 329 shows a section through the De Cours hydraulic [the valves of which Brre controlled by sprhigs. The springs w = nOetSt* PUBfPS 471 the fitphon action of the water escaping from the discharge^ ; sir to be drawn in past the spindle of the valve. A cushion air is thus formed in the box B every stroke, and some of thil^H is carritKl into C when the ^^Ive Vj opens. ^M The extreme sirapHcity of the hydraulic ram, together with with which it can be adjusted to work mth vaiying im of water, render it particularly suitable for pumping out-of-the-way places, and for supplying water, for fountains nd domestic purposes j to country houses situated near a stream.* 260. Lifting water by compressed air. A very simple method of raising water from deep wells is b; fis of compressed air. A delivery pipe is sunk into a well^ [the open end of the pip© being placed at a considerable distance [low the surface of the water in the well. J /O — Iw^ WaJLet* f AirTuj^e Fig. 030. Fig. mi. In the arrangement shown in Fig, 330, there is surrounding ths delivery tube a pipe of larger diameter into which air is pumped by a compressor. The air rises up the delivery pipe carrying with it a quantity of water. An alternative arrangement is shown in Fig. dSl. Wbether the air acts as a piston and pushes the water in front it, or forms a mixture with the water, according to Kelly**, lepends very largely upon the rate at which air is supplied to the [pump. In the pump experimented upon by Kelly, at certain rates of I 478 HYDEAUUCS working the discharge was continuoue, the air and the water being mixed together, while at low discharges the action was intermittent and the pump worked in a definite? cycle; the discharge commeneed slowly; the velocity then gradually increased imtil the pipi di.sdiarged full bore; this was followed by a rush of air, after whicli the flow gradually diminished and finally stopped ; after a period of no flow the cycle commenced again. When the rate at which air was supplied w^as further diminished, the water tme up the delivery tube, but not sufficiently high to overflow, and tbe air escaped without doing useful work. Tlie efficiency of these pumps is very low and only in exeeptioial cnsm does it reach 50 per cent. The volume v of air, in cubic feet^ at atuiospheric pressure, required to lift one cubic foot of ifnt^r through a height k depends upon the efficiency. With an ef* ticiency of 30 per cent, it is approximately ^' = 95' and with an efficiency of 40 per cent. *« = He approximately. It is necessary that the lower end of the delivery be at a gresier distance below the surface of the water in the well, than the hriglit of the lift above the free surface, and the well has consequently ^ be made very deep. On the other hand the well is much smaller in diameter than would be required for reciprocating or centrifugal pumps, aiid the initial cost of constructing the well per foot length is considerayj' EXAMPLES, (1) Pind tlie horse-power required to raise 100 cable feet of water p^f minute to s. height of 125 fciet, by a pump whose efficiency is h (2) A centrifugal pump has an inner radius of 4 Laches and an outer radius of 12 inches* The angle the blade makes with the direction ^ motion at exit is 153 degrees. The wheel naakes 545 revolutions per miDnt^* The discharge of the pump is 3 cubic feet per second. The sidej d tlio wheel are jmrallel and 2 inches apart. Determine the inch nation of the tip of the blades at inlet so that tb«^ shall be no shock, the velocity with which the water leayes the wheel ^ the theoretical lift. If the head due to tbe velocity with which the w»t0 leaves tlie wheel ia lost, find the theoretical lift. (3) A centrifugal pump wheel has a diameter of 7 inches and ma^^ 1358 revolutions jier minute. The blades are formed so that the water enters and leaves the wheel witliont shock and t!ie blades are ra^lial at exit The water is lifted bj tlw pump 29 "4 feet* Find the manometric efficiency of the pnmp. PUMPS 479 {41 A cefltrifng&l pump wheel 11 mchee diameter whicli runs at 1308 roToldtiotts per minute ib suiroimded by a Yortex chambet 22 iuchofi di&tiieUT, autl has radi&l blades at exit The prensure head at the circum- lonuice of thti wheel is 23 feot. The water is hfted to a height of 43'5 ImI abov€ the centre of the pomp. Find the efficiency of the whirlpool duutiber, (5> The radial velocity of flow through a pump is 5 feet per second, i Uie felocity of the outer periphery is 60 feet per aeeond. The angle tlie tangent to the blade at outlet makes with the direction of Diotion in 120 degrees. Determine the pressure bead and velocity head wbi?re the water leaves the wheel* assuming the pressure head in tie eye of tlie wheel in atmospheric, and thus determine the theoretical lift. (6) A ctrutrifugal pump with vanes curved back has an outer radius of 10 indieH and an inlet rsdius of 4 inches, the tangents to the vanes at outlet 1>eing Lnelined at 40^ to the tangent at tlie outer periphery. The section of the wheel is ^^ach that the radial velocity of 0ow is constant, 6 feet per second ; and it runs at 700 revolutions per minute. Deteruiipc :— ^1) the angle of the vane at inlet so that tliere shall be no shock, ^S| tlie theoretical lift of tlie pump, (3) Die Yelocity bead of the water as it leaves the wheel* Loud. L1906. (7) A centrifugal pump 4 feet diameter running at 200 revolotions per pumps 5O0O tons of water from a dock in 45 minutes, the meaii| tliellig 00 feet. The area through the wheel periphery is 1200 sqoaroJ \ and the angle of the vanes at outlet is 26^. Determine the hydraulic f Dcy and estimate the average horse -power. B'ind also the lowest i I to start pumping against the head of 20 feet, the inner radius being [ thti outer. Loud. Un. 1906. (8) A centrifugal pump^ delivery 1500 gallons per minute with a lift of I fckot^ has an outer diameter of 16 inches, and the vane angle i» 80^ All \ kinetic energy at dischai*ge is tost, and is equivalent to 50 per cent, of actual Uft^ Find the revolutions per minute and the breadth at the Jet, tlie velocity of whirl being half the velocity of the wheel, Lond. 1906. (9) A centrifugal pump has a rotor 19| inches diameter ; the width of the outer periphery is 3^^^ inches. Using formula {1), section 236, deter- tnine the discbarge of the putnp when the hea<l is 30 feet and t^i is 50. (10) The angle tp at the outlet of the pump of question (9) m IX. Find the velocity witli which the water leaves the wheel, and the ;muin proportion of the velocity head that must be converted into work, tlie other losses are 15 per cent, and the total efficiency 70 per cent* fill The inner diameter of a centrifugal pump is 12 j^ inches^ the outer diameter 21 1 inches. The width of the wheel at outlet is 8g i^chen. Using uation |2k section 236, find the discharge of the pump when the head is *5 f^ot^ and ^e number of revolutions per minute is 440. tofOw BT1ML4ITUCS ntmiiiig at 550 leTola^ msi^B Ihe tip of ih^ vane makes wdgp of tbe vane is 99 de^tes. m& ti^ ^TOCtiOn of motion of the of iiie wheel ui 11| tnehtft and tbe idien a leaves the wheel. is losi by frictioii, aitd that ooe- A& exit s Vsl^ Ibad tlie bead lost at inlet trhd^ I Ibe pvobabfe veldei;^ impcefieod on tbe ^^bsu (U) Deaeoba a foeed TottaL, aad sketch tiia footn of tlici &^ sashcb wbea the aaffolar w^locaty ia conatapt, In a oiBlrifiiiial pmi^ A«fwuliiug bonsoBtaDj imder water, tbe dLimiet^r tie CBBda frf Ibe paMka ^ 1 tac^ aad of tbe oatiide 2 feet, ajid tbe 9 tuiuliua at 400 ivivilotiaiES per aiinute. Find approximittclj bow L tfie «m«ar won^ lie liftad abotv the tail waler level. (14) KTp^aiii llie ac^oB of a oantrifogal pump, and deduce an exprmon r iia ellcieBej. If sacii a paai|i were requiicd to deliver ICXK^ gailosB iB ■r lo a liei^% cl ^ feet, liow wtwld jua. design it? Loud. Un. 190S. (li) Fuid fte i^etd of rotasjoa of a wheel of a centrifu^ pomp wMxt [ lo tin 30O toBB of water 5 feet high in one mmtite ; haviiig ^fiii ' w Ml The velocity of flow through ibe wheel is 4*5 feet pv , aad Ihie vanes are carved backward so that the angle between ilm and a tangent to the cirerunferenoe ts 20 degtees* Loml V^ (16) A oentrifogal ptunp is leqaired to lift 2000 galloiia ef water pf^ minnte ^iroogh 90 feetv The velocity ot liow throogh the wheel hi per second and the efficiency 0-6, The angle the iip-of tbe vane At ootU* makes witli the direction of motion ib 150 degrees. The outer radius d tho wheel is twioe tbe inner. Determine tbe dimensions of the wbeeL (17) A donble-acting phinger pump hss a piston 6 inches cliametor and tbe length of the strokes Is 12 inches. The gross heaxl i& 500 ttsei^ and the pomp makes BO strokes per minute. Assuming no alip, Md tbts discharge and horse-power of the pump. Find also the necessary' diiuaeter for the steam cylinder of an engine driving the pomp diret^ ajssumiiig steam pressure is 100 lbs. per square inch, and the mechanical dficklK| of tbe combination is 85 per cent. (18) A plunger pump is plaoed above a tank containing water tt & temperature of 200' F, Tbe weight of the sncticni valve is 2 lbs. aud iUi diameter 1 j inches. Find the maxim am height abc^ve the ^i^V at the pump may be placed so that it will draw water, tJie barometer standlog at 30 inches and the pump being assumed perfect and ^^itliout clearMCtt (The vapour tension of water at 200" F. is about 11 "6 lbs, per aq. inch/ (19) A pump cylinder is 8 inches diameter and the stroke of the plung* is one foot. Calculate ^e maximum velocity, and the acceleration of 1. PUMPS 481 mtet in the eucUon and delivery pipes, aBBximing tlieir respective diametera > be 7 inchee and 5 inches, the motion of the piston to be simple harroonic, ad the pistcm to make 86 strokes per minute. (20 1 Taking the data of question (19) ealculate the work done on the iciion stroke of the pnmpi (1) neglecting the friction in the suction pipet • (2) indndiog the friction in the suctiou pipe and assuming that the suction pipe is 25 feet long and that/— '01. The height of the centre of the pump above the water in the aump is (21) If the pump in queBtion (20) delivers into a rising main against head of 120 feet, and if the length of the main itself is 250 feet, nd the total mork done per revolution, Assuming the pnmp to be douhl© efeing, find the i.h. p, required to drive the pumpj the efl&ciency being '72 Ii4 no slip in the pump. Find the delivery of the pumpt assuming a slip f 5 per cent (22) The piston of a pump moves with simple harmonic motion, and it ^fives at 40 itrokes per minute. The stroke is one foot. The suction Wk ia 25 feet long, and the suction valve is 19 feet above the surface of the mmt in the sump. Find the ratio between the diameter of the suction lape and the pump cylinder, so that no separation may take place at the bad pointe. Water barometer 84 feet. (23) Two douhle-actiug pumps deliver water into a main without an lir TeflseL Each is driven by an engine with a fly-wheel heavy enough to ^ecp tbe speed of rotation uidf orm, and the connecting rods are very long. Let Q be the mean dehvery of tlie pumps per second, Qi the quantity of Witeif in the main. Find the pressure due to acceleration (a) at the begin - aing of a stroke when one pump h delivering water ^ (&) at the beginning fi the stroke of one of two double-acting pumps driven by cranks at right n^m when both are delivering* When is the acceleration ^ero ? (S4) A double*acting horizontal pump has a piston 6 inches diameter <tlie diameter of the piston rod is neglected^ and tlie stroke is one foot. The water is pumped to a height of 250 feet along a delivery pipe 450 feet kiiig and 4| incheH diameter. An air vessel is put on the dehvery pipe 10 feet from tlie delivery valve. Find the press tire on the pump piston at the two ends of the stroke when the pomp is making 40 strokes per minute^ assuming the piston movea with simple harmonic motion and compare these l^esBiires with the 1 when there is no air vessel. /= *0075. (36) A single acting hydraulic motor makes 160 strokes per minute and E vrith simple harmonic motion. The motor is supplied witli water from an accumulator iu which the presanre is maintained at WO lbs. per square inch. The cylinder is 6 inches diameter and 12 inches stroke. The delivery |iipG IS 200 feet long, and the coefficient, which includes bss at bends, etc, may be taken as/^0'2. ^ UH. ^\ d ifae wiabfe met p^ap m SSiBoi long and 4 infihes r B 6 Bciw mud the stroke 1 foot mmtaEK. 90lBet long, and the Iwad ai t m mo wit wmml en the pomp. The m above the levdd the water in the 2i Bflipie liamionic tno tJon and niskBi cctical dfagrain far the pnnupi qaaattjcl water in the cylinder and 'iT •oL «nzasKiL laks CKiee anywhere in the deliyery pipe of the janm* itas saSA si wsndc. s f^vcn in qaestkm i26), if the pipe first rose 3iiE:»2icaJ.7 ±T yiil ieec aad •hea TcrticaDy for 40t» or rises 40 feet im- Tiwffnapij ^-nxL 2Kf TozzLp aal ^mb mns horiaontally for 50 feet, and KpacssaaL TaLsat ^mx wben ihe pi u mm i head falls below 5 feet? :3Si A jiimn bns ihree smgjLe-acting plungers 29^ inches dianwter .srirniL W ,39aks 13 IJl> decEees wish e«ch o^^ The stroke is 5 feet and 7St> x^mbinr :f scrokes per minase 40. The soctkui is 16 feet and theloigtfa ':t ^re sxkrcara pipe b ±2 &«ec The dehTery pipe is 8 feet diametor and ^j »ii Irc^. Tbe hMa as the dellTerr Talve is 214 feet. Fzzf i z ibe ziizi2.:im dsamecer of the suction pipe so that there is no sep&ruoD:!!. '--Qg— ^^^ do air ressei and that separation takes place when she rr^=sasir^ b^ccoes zenx V The bcrse-power of the pump when there is an air Yeesel on the delirerr vtry zear to the pomp. /^-OOT. ^The ssciient should draw out three cosine curves differing in phase by 12i3 degrees. Then remembering that the pump is single acting, the resultant curve of accelerations will be found to have Tn«.TimniTi poeitiTe •V.A and also negative values of ——-^ every 60 degrees. The maximiun acceleration head is then h^= ± — r-— — • For no separation, therefore, a » '^ . (29) The piston of a double-acting pump is 5 inches in diameter and the stroke is 1 foot. The delivery pipe is 4 inches diameter and 400 feei long and it is fitted with an air vessel 8 feet from the pump cylinder. Tbe water is pumped to a height of 150 feet. Assuming that the motion of the piston is simple harmonic, find the pressure per square inch on the pision at the beginning and middle of its stroke and the horse-power of the pomp when it makes 80 strokes per minute. Neglect the effect of the variaUe quantity of water in the cylinder. Lond. Un. 1906. PUMPS 483 (30) The plunger of a p«mp moves with simple harmonic motioo* Eod the condition that Bepai^ation shall not take place on tlie suction roke and show why the speed of the pump may he increased if an air Nisel Is put in the suction pipe. Sketch an indic^itoT diagram showing jpuc&tioD. Explain ^^ negative slip." Loud. Un. I9()6. I) In & Bingle-acting force pump, the diameter of the plunger ia ties, stroke 6 inches, length of suction pipe 63 feet, diameter of suction pe !2| inches, suction head 0'07 ft. When going at 10 revolutions per innte, it is found that the average loss of head per ntroko between the Letion tank and plunger cy Under is 0*28 ft. Assuming that the friction al WSmB vary as the square of the speed, find the absolute head on the suction da of thJe plunger at the two ends and at the middle of the stroke, the ETolntions being 50 per minute, and the barometric head M feet. Draw a a^Eftm of pressures on the plunger — aimplo hannonic motion being Lond. Un- 1906. (S2) A single-actittg pump without an air vessel has a stroke of \ inches. The diameter of the plunger is 4 inches and of the suction tpe a| inches. The lengtli of the suction pipe is 12 faet, and the centre I ahe pmnp is ^ feet above the level in tlie sum^p. Determine the number of single strokes per second at which theoretl- liiy separation will take place, and explain why separation will actually ike place when the number of strokes is less tlian the calculated value. (38) Hxpiain carefully the use of an air vessel in tlie delivery pipe of a omp. The pump of question (32) makea 100 single strokes per minnte, nd dehvers water to a lieight of 100 feet above the water in the well brough a dehvery pipe 1000 feet long and 2 inches diameter. Large air edeels being put on tlie suction and dehvery pipes near to the pamp. On the assumption that aU losses of head other than by friction m he delivery pip© are neglected, determine the horse-power of the pump, Thefpe is no slip. (34) A pump plunger has an acceleration of 8 feet per second per leoond when at the end of the stroke, and Hie sectional area of the plunger I twice the sectional area of the delivery pipe. The dehvery pipe is 152 bet bug. It runs from the pump horizontally for a lengtli of 45 feet, then rertically for 40 feet^ then rises 5 feet, on a slope of 1 vertical to 8 hori* kttstal, and finally nms in a horizontal direction. Find whether separation will take place, and if so at whiuh section of ihe pipe, if it be assumed tliat separation takes place when the prassnre bead m the pipe becomes 7 feet. |S5) A pump of the duplet kind, Fig, 325, in which the steam piston is Kunected directly to the pump piston, works against a head of h feet of rtter, the head being supphed by a column of watec' in the dehvery pipe« rhe piston area is Aot the pUmger area A, the delivery pipe area a, the angth of the delivery pipe t and the constant steam pressure on the piston ^Jba, per square foot. The hydranho resistance may be represented by r-i V being the Telocity of the plunger and F a ooe^cieut. ^V— ^ 484 HTDRAUUGS' Show thai when the plunger has moTed a diatanoe x from the beguming ol the stroke r»«^(^-*)(l-e-^). Land. Un. 1906. (96) A pomp Talye of Ivass has a specific gravity of 8| with a lift of ^ foot, the stroke ol the piston being 4 feet, the head of water 40 feet and the ratio of the fall valye area to the pistcm area one-fifth. If the Talve is neither assisted nor meets with any reeistanoe to closing, find the time it will take to dose and the ''slip" due to this gradnal closing. 7*6 Timetodoseisgrvod byfennnla, Ss^. /«g;gX82-2. Lond. Un. 1906. CHAPTER XI. HYDRAULIC MACHINES. 267. Joints and packings used in hydraulic work. The high pressures used in hydraulic machinery make it necessary to use special precautions in making joints. Figs. 332 and 333 show methods of connecting two lengths of pipe. The arrangement shown in Fig. 332 is used for small Fig. 832. Fig. 888. Fig. 834. 486 HYDEATJLICS WTOugbt^iroti pipes, no pacldtig being required. In Fig, 333 tho packJDg material is a gutta-percha ring, Pig< 336 siioT^rs an ordinary siK*ket joint for a cast-iron hydraulic main. To maka the joint, a few cords of hemp or tarred rope are driven into* the socket. Clay is then put round the outside of the socket and molten lead run in it* The lead is then jammed into the sockel with a caulking tool. Fig. 33*5 showg various forms of packiag l^rthersj the applications of which will be seen in the esampli given of hydraulic machines. Nffk letUher Ring Utith^r Cup leather Fig. 835. Fig. aa§. Hemp twine, carefully plaited, and dipped in hot taUow, makes a good packing, when used in suitably designed glanda (see Fig. 339) and is also very suitable for pump bucketSi Fig. 323. Metallic packings are also used as shown in Figs* 337 and 338, ^>^. Fig. 337. iig. a-js. 268, The accumulator. The accumulator is a deWce used in connection with hydruulic machinery for storing energy* In the form generally adopted in practice it consists of a long cylinder C^ Fig. 339, in which elides a ram R and into wliich water is delivered from pumps. At the top of the ram is fixed a rigid cross bead which carries, by means of the bolts, a large cylin*ler which can be filled with slag or other heavy material, or it imj be loaded with caet-iron weights as in Fig. 340. The water h HTDRAULIC MACHINES 487 tig. Sd'j. lljdrauUc Acctimulmtor, =1 488 HTDRAtFLICS admitted to the cylinder at any desired pressures through a pipe connected to the cylinder by the flange shown dotted^ and tlie weight is so adjusted that when the pressure per sq. inch in the cylinder is a given amount the ram rises. If d is the diameter of the ram in inches, p the pressure in lbs. per sq. inch, and h the height in feet through which the ram can be lifted, the weight of the ram and its load is W = p/j<flbs., and the energy that can be stored in the accumulator is E = p.|d'./i foot lbs, Tlie principal object of the accumulator is to allow hydraulic machines, or lifta, which are being supplied \rith hydraulic power from the pumps, to work for a short time at a much greater rata than the pumps can supply energy. If the pumps are connect<Mi directly to the machines the rate at which the pumps can supply energy must be equal to the rate at which the machines are working, together with the rate at which energy is being lost hf friction, etc., and the pump must be of such a capacity as to supply energy at the greatest rate required by the machineB, and tJm frietional resistances. If the pump supplies wat^er to an acciuDU- lator, it can be kept working at a steady rate, and during the rinit; when the demand is less than the pump supply, energy can he stored in the accumulator* In addition to acting as a storer of energy, the accumnlator acts as a pressure regulator and as an automatic arrangement for starting and stopping the pumps. When the pumps are delivering into a long main, the demand upon which is varjnng, the sudden cutting off of the whole or a part of the demand may cause such a sudden rise in the pressure as to cause breakage of the pipe line, or damage to the poinp. With an accumulator on the pipe line, unless the ram » descending and is suddenly brought to rest, the pressure camioi rise very much higher than the pressure p which will lift the ram* To start and stop the pump automaticallyj the ram as il approaches the top of its stroke moves a lever connected 1*^ a chain wliich is led to a throttle valve on the steam pipe of tlie pumping engine, and thuB shuts off steam. On the ram agtin falling below a certain level, it again moves the lever and opem the throttle valve* The engine is set in motion, pumping n- commences, and the accumulator rises. HYDRAtTLlC MACHINES 489 ^^am^te. A. bydjanlio er«tie working at & pf«Bimre of 100 Iba. per iq. ineh b&e 30 cwts. &t a rate of SOO feet per minute through a height of 50 feet, onoe J 1| mliiute^. The effioieoaj of the crane ifl IQpet cent, aad aji a<K!i] mala tor is piorided. Fiiiii Ihf volame of the cjlinder af the cranes the minimum horBe^power for the pomp, mud the minimum capacity of the accamulatar, £^ A be the sectional are* of the mm of the orane cylinder m sq^ feet and L \he length of the stroke in feet. Then, I>a44. A. Lx 070- 30x112x60', ! _ 30x112x50 H *^~ 70x144x700 ^P =2-33 eubio feet. Ihe rate of doing work in the lift e^lindtr k 112 :< 30x200 I 07 = 960,000 ft^ Iba. per minute. lli« ve^rk done in lifting 50 feet ia 210,000 ft, lbs. Since thia has to be done •rery one aud hiilf miuuteH, the work the pump munt supply in one and half atee is at ka^st 240,000 ft. lbs., imd the minimym bort^e-power is 240,000 ^ 1*5 = 40,000 ft, lbs. The work done by the pnmp while the crane h lifting is B 240,000x0-25 efoergy stored in the aeoumuktor mn&t be, therefore^ at least 200,000 ft. Ibfk !rhererore^ if T^ it its minimum capacity in cubic feet, V^xTOOx 144 = 20O.0O0» V^, = 2 cubic feet iieaily* 269. DifTeretitial accumulator •• TweddelFa differential accamalator, shown in Fig. 340, has a fixed ram, the lower part, of which is made slightly larger than the upper by forcing a brass liner upon it. A cylinder loaded ifith heavy caet-iron weights elides upon the i*am, water-tight jomtB hein^ made by means of the cup leathers shown. Water is pumped into the cylinder through a pipe, and a passage drilled ajdally along the lower part of the ram. Let p be the pressure in lbs, per sq, inch, d and di the dia- meters of the upper and lower parts of the ram respectively, ^ba weight lifted (neglecting friction) is then r if /t is the lift in feet, the energy stored is E-p.^W-e?)feJootlbs, The difference of the diameters dt and d being small, the pres- sure p can be very great for a comparatively small weight W. The capacity of the accumulator is, however, very smaU. 33us is of advantage when being used in connection with * Procetdingt Imt. Meeh. Engf,, 1874, 490 n" J ^^■d HYDRAULICS V < \ Fig, 340. 1. 11 JJif ;:? -3- X Fig, 841. Hydraulic Inteiisiflflr. HYDRAULIC MACHINES 461 hydraolie rivetera, aB when a demand is made upon the ac- cttmalatorj the ram falls qaickly, hut is suddenly arrested when the ram of the riveter comes to rest, and there m a consequent increase in the pressure in the cylinder of the riveter which clinches the rivet. Mr Tweddell estimatea that when the ac- cumulator is allowed to fall suddenly through a distance of from IS to 24 inches, the pressure is increased by 50 per cent. 270* Air accumolatoir* The air accumulator is simply a vessel partly filled with air and into which the pumps^ which are supplying power to machinery, deliver water while the machinery is not at work* Such an atr vessel has already been considered in connection with reciprocating pumps and an application is shown in connection with a forging press. Fig, 343, If V is the volume of air in the vessel when the pressure is p pounds per sq. inch and a volume v of water ia pumped into the vessel J the volume of air is (V — v), AsBuming the temperature remains constant^ the pressure pi in the vessel will now be p.V If V is the volume of air, and a volume of water v is taken out of the vessel, 271, Intensiflers. It is frequently di-ginible that special machines shall work at a higher pressure than k available from the hydraulic mains. To increase the pressure to the desired amount the intensifier is used. One form is shown in Fig. 34L A large hollow ram works in a fixed cylinder C, the ram being made water-tight by means of a §tafling-box. Connected to the cylinder by strong bolts is a cross head which has a smaller hollow ram projecting fi*om it, and Altering the larger ram^ in the upper part of which is made a Btoffing-box. Water from the mains is admitted into the large cylinder and also into the hollow i*am through the pipe and the lower valve respectively shown in Fig. 342, If p Iba. per aq. inch is the pressure in the main, then on the underside of the large ram there is a total force acting of p 7 D* pounds^ and the pressure inside the hollow ram rises to D* p ^ pounda per sq, inch, D and d being the external diameters of the large ram and the small ram respectively. 492 HYDRAULICS The form of iiiteiiBifier kere ahovm is used in connection with a large flanging press. The cylinder of the press and the tipper part of the intensitier are fiUed with water at 700 Ihs. per sq. inch and the die brought to the work. Water at the same pressore i admitted below the large ram of the intensifier and the preasti in the upper part of the intenaifierj and thus in the press cylinder^ rises to 2000 lbs, per sq* inch, at which preesure the flanging is finished* JbSmaU ptrsq. indL. ^ Tp^ iM 1 Tb Large ^Undjet at IntenrnJUF Nofh Return Vcdvee for Intensi/Ur Fig. 342. ^ 1 aiioon 272, Steam intensifiers. The large cylinder of an intensifier may be supplied wit steam, instead of water, as in Fig* 343^ which shows a steam tenaifier used in ^con junction with a hydraulic forging press, Thes©| intensitiers have also been used on board ship* in connection wit hydraulic steering gears. 273, Hydraulic forging preaa, with steam intensifier and] air accumulator. The application of hydraulic power to forging presses is illu^ I trated in Fig. 343. This press is worked in conjunction with ft] steam intensifier and air accumulator to allow of rapid world^iij The whole is controlled by a single lever K, and the pre« capable of making 80 working strokes per minute. When the lever K is in the mid position everything is at rest j on moving the lever partly to the rights steam is admitted into th^ cylinders D of the press through a valve. On mo\nng the lever w its extreme position, a finger moves the valve M and admits w»ler * Prootedinffi IntU MecK Enff».^ 187i^ HTDRAITLIC MACHIKES 493 relay piston shown at the top of the figure, which opens E at the top of the air vessel* In small presses the valve by levers. The ram B now ascends at the rate of foot per secondj the water in the cylinder c being forced accnmiiiatorf On moving the lever K to the left^ a« soon M passed the central position the valve L is opened to 494 WTDEATJLICS exhauBt, and water from the sir veeael, assisted by gravity, forces! down the ram B, the velocity acqtiired being about 2 feet perl second, until the press head A touches the work. The movementj of the lever K being continuedj a valve situated above the vah J is opened, and steam is admitted to the inteneifier cylinder H;' the valve E closes automatically, and a large pressure is exerted on the work under the press head* K only a very short stroke is requiredj the bye-pass valve L i tamporarily discoimectedj so that steam is supplied eontiiitiousl]^ to the Hfting cylinders D* The lever K is then sim^ply used admit and exhaust steam from the inteneifier H, and no water enters or leaves the accamulator. An automatic controlling ge is also fitted, which opens the valve J sufficiently early to preventj the intensifier from overrunning its proper stroke. BTDRAITLIC MACHINES 495 On© end of a wire rope, or chaiiij is fixed to a lug L on the cylijiderj and the rope ia then passed alternately round the upper and lower pulleys, and finally over the puJley on the Jib of the crane. Fig. 346, In the crane shown there are three pulleys on the ram, and neglecting friction, the pressure on the ram is equally divided among the iix ropes. The weight lifted is therefore one- aixth of the pressure on the ram, but the weight is lifted a distance equal to six times the movement of the ram. Let the number of pulleys on the end of the ram of any crane he J , arranged as in Fig. 347- The movement of the weight wiH then he n times that of tlie ram. Let p be the pressure in lbs. per sq. inch in the cylinder and d the diameter of the ram in inches. The pressure on the i*ara is and the energy supplied to the crane per foot travel of the ram is therefore P foot pounds, kThe energy supplied per unit volume displacement is 144 * p. The actnal weight lifted is W = 6^^pd»lbs., being the efficiency. ^\'hen full load is being lifted e is between 07 and 0^8. For a given lift of the weight, the number of cubic feet of water aaed, and consequently the energy supplied, is the same whatever th© load lifted, and at light loads the efficiency is very small. 275. Double power cranes. To enable a crane designed for Iieavy work to lift light loads iHth reasonable efficiency, two lifting rams of different diameters art* employed, the smaller of which can be used at light loads. A convenient arrangement is as shown in Fig^. 348 and 349, the smaller ram R' working inside the large ram B* ^lien Ught loads are to be lifted j the large ram is prevented from momkg by strong catches 0, and the volume of water used IS only eciual to the diameter of the small mm into the length of the stroke. For large loadsj the catches are released and the two rama move together. Another arrangement is shown in Fig, 350, water being ad- mitted to both faces of the piston when light loads are to be lift^, and to the face A only when heavy loads are to be raised. HTDRAtrtlC MACHINES 49? For a given stroke B of the mm, the energy supplied in the &l«t ease is ■ted in the second case «r|)-{D^-^)ft. lbs., Fi|« 350. AitnitroGg Double -pow«r Hjdrftulio Crftne Cjlindar. 276, Hydraulic crane valves. Figs. 361 and 352 show two forms of lifting and lowering talves tLsed by Armstrong, Whitworth and Co. for hydraulic In the arrangement shown in Fig, 351 there are two inde- pendent valves, the one on the left being the pressure, and that cm the right the exhaust valve. |, 851, Ann stTQQg'Wli It worth Hydraalie Cr&ne ValTe. I u a. Fig. SS2. Armstroog- Whitworth MjriirftuUe Cr&ae SMd« Valve. 4» HTDBAtlUCS tbowu in Fi|f . S52 a single D ^de vake i llv Talvp chest duon^ the prefi&ure puaga in the neiitrml podtion^ If llie Wve enters the cylinder, but if it is rights dbe ejriiixder tkr^u^h die port of Uie slids faydmUlc pnn. Fig. 353 is a section tbroagh kydmnlie preas^ naed for teotui^ springs. is fitted widi m brsas linear, and axiaUf a rod Fr ^* a platan P at thjb irsie end^ il ram is hoUow^ the umsr id fmm the largt? cy lindur m top of the cylinder ADif 1 cylmder ini^de the tsm ii 1]^ a hole drilled along tk the water ts cotitinuooBlf I the annnl^r ring RR. I prerenled hf means d i ' G ^nd 1 rrae If^ther r. and leakage past the --i- i- i ri>c n P by cup leathers L and Lq. «^ -_ :1- vilve >p:r.dle is moved to the right, the port p is -tvcvi ^ ->. :":ir rxhaust, and the ram is forced up by the <->^ : "1- wjkTcr n the annular ring RR. On moving the - -: - V- r- : :he left, pressure water is admitted into the - - ' >" i :r.r r-i::i :> f «n.>e^ down. Immediately the pressure >,-*:•;. ':t -i::: . ni^s baok a^rain. - r " -* n "ir i::i meter of the ram, ti the diameter of the :V - - i:^:v.r:cr of the piston P, and p the water pressure rve acting on the ram is -jT :"r.e ram when pivjsurf is released from the ..f-»: = p^(iy-rf.') lbs., F r - I •1;- - 'Pi lbs. .^" vA.ve spindle S, has a chamber C cast in it, : >:x holes m each ring are drilled through : the chamber. These rings of holes are at .^wX^rt that, when the spindle is moved to the i I ivssite to the exhaust and the other opposite I Nvhti: the spindle is moved to the left, the holes L Fig, a53. HydratUio Pr^aa with Lutlie Valve. Qrdratilic riveter. n through the cylinder and ram of a hydraulic riveter Fig, 354. 600 HYDRAULICS h III 1^ 1 I ■- 1 ^ i Inlft Va/ie Fig. 355. V&lTes for Hydraalio Biveter. HYDRAULIC MACHtJ^ES ^01 e mode of working is exactly the same as that of the small described in section 277. ^ enlarged section of the valves is shown in Fig* 355* On g the lever L to the right, the inlet valve V 18 opened, and \re water is admitted to the large cylinder^ forcing out to- Wlien the lever is in mid position, both valves are , hy the springs S, and on moving the lever to the left, the ftst valve V, is opened, allowing the water to escape from the ier* The pressure acting on the annular ring inside the ram then brings back the ram* Tlie methods of preventing [© are clearly shown in the figures, B, Hydraulic engiBea, rdraulic power is admirably adapted for machines having a CK^ting motion only, especially in those cases where the load Really constant. Fig, 356. Hydraulic Capstati. 501 HTDRAUUCS II hmB masearer been successfully applied to the drivini mdk WB capstaiLS and winches in which a reciprocal i ocMif^ned into a rotarj motion. Tlid kjn&miLtie-eEigine shown in Figs, 356 and 357, haa H fl^liiideri in one castixigi the axes of which meet on the axis of on&k eliaft & Hie motion of the piston P is transmitted to crmnk pin hf short eoDneetin^ rods R. Water is admitted ejchausted tluxmgh m Tmlve V, and ports p. The face of the valve is as shown in Fig. 358, E exhansi port oonnected through the centre of the valve i^ exliaiisl pipe, luid KM the pressure port, connect^ to the msft chamber H hy a small port through the side of the valve. 1 valve seating is generally made of lignnm-vitae, and has A circnlar ports as shown dotted in Fig. 358» The valve receiv^a motion from a small auxiliary crank T, revolved by a project from the cranX pin G. When the piston 1 is at the end of stroke, Fig* 359^ the port p, should be just opening to the preA port, and just closing to the exhaust port E* The port pi shoi be fnDy open to pressure and port pa fully open to exhai When the crank has turned through 60 degrees, piston 3 i A HYDRAULIC MACHINES 503 be at tlie inner end of its stroke, and the edge M of the pressure port ahoold be just closing to the port p^. At the same instant the edg^ 1^ of the exhaust port should be coincident with the lower ed^e of the port pt. The angles QOM, and LON, therefore, ahoald each be 60 degrees. A little lead may be given to the valve i>orts, i.e. they may be made a little longer than shown in the Fig"- 358, so as to ensure full pressure on the piston when commencing its stroke. There is no dead centre, as in whatever position the crank stops one or more of the pistons can exert a turning' moment on the shaft, and the engine will, therefore, start in any position. Fig. 358. Fig. 359. The crank* effort, or turning moment diagram, is shown in Fig. 359, the turning moment for any crank position OK being OM. The turning moment can never be less than ON, which is the magnitude of the moment when any one of the pistons is at the end of its stroke. This tyi)e of hydraulic engine has been largely used for the driving of hauling capstans, and other machinery which works intermittently. It has the disadvantage, already pointed out in connection with hydraulic lifts and cranes, that the amount of ^water supplied is independent of the effective work done by the machine, and at light loads it is consequently very inefRcient. There have been many attempts to overcome this diflSculty, notably as in the Hastie engine t, and Bigg engine. * See text book on Steam Engine. t Proceedings Inst. Mech, Engs,, 1874. lfTDRAlJt.lCS i 280« Bigg hydranUc engine. To adapt the quajitity of wat^sr nsed to the work done*, 1 hafi modified the three cylinder engine by fbdng the crank pin, and aUawmg the cylinders to revolve about it aa centre. The three pistons Pi, Pa and P* are connected to a disc, Fig. 360, by three pins. Thii disc revolves about a fixed ceutre A. The three cylinders rotate about a centre 6, which is capahle d ' being moved nearer or furtlier away from the point A as desired* The stroke of the pietons is twice AG> whether the crank or the cylinders revolve, and since the cylinders, for each stroke* have to be filled with high pressure water, the quantity of water supplied per revolution is clearly proportional to the length AGr. Fig. 360. Higg Hydrnnlio Engine. The alteration of the length of the stroke is effected by of the subsidiary hydraulic engine, shown in Fig. 36L There two cylinders C and d, in which slide a hollow doable ended ram PPi which carries the pin G, Fig. S60. Cast in one piece iMth the ram is a valve box B. R is a fixed ram, and through it wat^r enters the cylinder Ci, in which the pressure is continuously maintained. The difference between the etfectivB areas of P and Pv when water is in the two cylinders, is clearly equal to th e at«» of the ram head Ri . See lilso Engimgrt Tgl. LXxxv» 189S. HrDBAUUC MACHINES From the cylinder Ci the water is led along the passages ^OTfn to the valve V» On opening this valve high-pressure ^ter 19 admitted to the cylinder C. A second valve similar to V, bat not shown, is used to regulate the exhaust from the cylinder C. A?VTien this valve is opened, the ram PPi moves to th left and carries with it the pin (t. Fig* 360. On the exhaust hehig closed and the valve V opened, the full pressure acts upon l»ctth ends of the ranij and since the effective area of P is great th^n Pi it is moved to the right carrying the pin G, If botkj ralves are closed, water cannot escape fron> the cylinder C and the ram is locked in position by the pressure on the two ends* Water Fig. 361. EXAMPLES. ill The ram of & hydraulic crane is 7 inches diameter. Water is flopplied to the crane at 700 Iba. per square inch. By suitable gearing the load is lifted times ob quickly as the raw. Assuming the total efficiency of tb© crane is 70 per eent., find the weight lifted. (*2,t An accumulator has a stroke of 23 feet ; the diameter of the ram is 23 iJM!he<i; the working pressure is 700 lbs. per squaje inch. Find the capacity ol the a^ocumalatoi* in horse -power hour». id) The total weight on the cage of an ammonition hoist is 8250 lbs. The velocity ratio between the cage and the ram is eix, and the extra load QOI ilie cage due to friction may be taken as 30 per cent, of the load on the ea^. The steady spaed of the ram is 6 inches per second and the available i re at the working valre is 700 lbs, per square inch. B^ttimate the loss of head at the entrance to the ram cylinder^ and ling this was to be doe to a smJden enlargement in passing through ^port to the cyhnder^ estimate* on the usual asaumption, tlie area of the -yorl. the ram cylinder being 9| inclies diameter. Lond, Un, 1906, The eflective preaanre p- * 4 506 HTTDRAITLICS Lobs of head Area of port t?= velocity through the ir&lv6i. 4 (4) DeBcribe, with sketche^^ some form of hydraulic Rccumiilatot so able for use in connection with riTstin^. Explain by tl>e aid of dia if pOHsible, the general nature of the curve of pre^gxire on Uie riveter i during the stroke ; and point out tlte re^aaona of the Tariation^. Lond Uij 1905, (See sections 262 and 269,) (5) Describe with sketche*j a hydraulic intensifier. An intejosifler is required to increase the pressure of 700 IbR, per squa inch on the mains to SOOO lbs. per square inch. The stroke of the int fier is to be 4 feet and ita capacity Uiree gallons. Detennine the ( of the rama. Inat. C, E. 1905. (6) Sketch in good proporttou a section through a differential hydra accumulator. What load would be necessary to produce a pressuFe of 1 1 |ier square inch, if tlie diameters of the two rams are 4 incliea and 4| i respectively ? Neglect the Mction of the packing. Give an instance of tlw' use of auch a machine and state why accumulators are used. (7) A Tweddeirs differential accumulator is supplying water to riTetung machines. The diameters of the two rams are 4 inches and 4| icchef respectively, and tlie pressure in the accumulator b 1 ton per square mrJi. Suppoae when the valve is closed Uie accumulator is falling at a vekxity of 5 feet per second, and the time taken to bring it to rest is 2 seconds. tioJ the increase in pressure in the pipe. (8) A hit weighing 12 tons is worked by water pressure, the pn in tlie maiu at the accumulator being 1200 lbs. per square inch ; the lengtiil of tlie supply pipe which is Sj inches in diameter is flOO yards, Wh^i >r| the approximate speed of ascent of this Ijftt on the assumption that lb* I friction of the Btufiing-box, guides, etc. is equal to 6 per cent, of the j load lifted and the ram is 8 inches diameter ? (9) Explain what is meant by the " coefficient of hydraulic resiHtaEoe" as applied to a whole system, and what assumption is usually made rt<gsnl*| ing it? -A direct acting lift having a ram 10 inches diameter i«f '^nppll from an accumulator working under a pressiu^e of 750 lbs. i>er si] When carrying no load the ram moves through a distance of t^^ ' uniform speed, in one minute, the valves being fully open. Estiroatt i coefficient of hydraulic resistance referred to the velocity of the ram, I als^ how long it' would take to move the same distance when tli« i carries a load of 20 tons. Loud. Vn. 1905. ^''^^ead 1 ost = -;^:;^^ . Assumption is made that resistance varies «a •'J 64 62-4 CHAPTER XII. EESISTAJS'CE TO THE MOTION OF BODIES IK WATER. 281, Froude^s* expeiimenta to determine Motional re- istances of tliin boards when propelled in water. It has been shown that the frictional resiatance to the flow of gr along pipes is proportional to the velocity raised to some BF n, which approximates to two, and Mr Froude's classical ents, in connection with the resistance of ahips, show that ince to motion of plane vertical boards when propelled II w^ftter, follows a similar law. ^ft Fig. 303, '^The experiments were carried out near Torquay in a parallel lidted tank 278 feet longj 36 feet broad and 10 feet deep. A light teilway on '* which ran a stout framed truck, suspended from the \xlem of two pairs of wheels," traversed the whole length of the ^ok, about 20 inches above the water level* Tlie truck was pro- lelled by an endless wire rope wound on to a barrel, which could le made to revolve at varjang speeds, so that the truck could llftvirne the length of the tank at any desired velocity between DO and 1000 feet per minute, * Bril. Amu. Rep&tU, li^72-4* 508 HTDRAULIOS Planes of wood, about ^ incli thick, the Bui^ces of widch were covered with variona materials as set out in Table XXXIX, were made of a uniform depth of 19 inches^ and when under experi- ment were placed on edge in the water, the uppefr edge hmg ibont \j inches below the surface. The lengths were varied from ; to 50 feet. T\\e ap^mratus as used hj Fronde is illustrafced and described in the British A^sociution Reports for 1872. A later adaptation of the apparatus as uaed at Haslar for determining the resistance of ships* models is shown in Fig. 361 An arm L is connected to the model and to a frame beam, which is carried on a double knife edge at H, A spring S is attached to a knife edge on the beam and to a fisted knife edge N on the frame of the truck* A link J connects the upper end of the beam to a multiplying lever which moves a pen D over a recordiBg cylinder. This cylinder is made to revolve by means of a worm and wheel, the worm being driven by an endless belt from the ajcle of the truck. The extension of the spring S and thus the move* ment of the pen D is proportional to the resistance of the model, and the rotation of the drum is proportional to the distance mOTedi A pen A actuated by clockwork registers time on the cylinder. The time taken by the truck to move through a given distance can thus be determined. To calibrate the spring Sj weights W are hung from a knife edge, which is exactly at the same distance from H as the pointi of attachment of L and the spring S. From tlie results of these experiments, Mr Frond© mad© the following deductions, (1) The frictional resistance varies very nearly with tlie square of the velocity. (2) Tlie mean resistance per square foot of surface for leugdia up to 50 feet diminishes as the length is increased, but is pmc* tically constant for lengths gr^^at^r than 50 feet, (3) The frictional resistance varies very couidderably wil the roughness of the surface. Expressed algebraically the frictional resistance to the motii of a plane surface of area A wheii moving with a velocity o per second is / being equal to A 10- • KESJSTAiCCE TO THE MOTION OF BODIES IN WATER 509 TABLE XXXIX, Showing the result of Mr Froude'a experiments on the ftnctional ^ (resiivtatice to the motion of thin vertical boards towed through in a direction parallel to its plane. Width of boards 19 inches, thickness fV inch, power or index of speed to which resistance is approxi* ^tely proportionah f^^ tlie mean resistance in pounds per square foot of a surface, length of which is that specified in the heading, when the relocity is 10 feet per second. /i = the resistance per square foot, at a distance from the ing edge of the boards equal to that specified in the heading^ a velocity of 10 feet per second. As an example, the resistance of the tinfoil surface per square at 8 feet from the leading edge of the board, at 10 feet per cond, is esttinated at 0^263 pound per square foot; the mean resistance ia 0"278 pound per square foot. Soffaoe J^-^irea with '^ttmiah Letigth of plameB 2 feet 2^ 2^16 1-93 20 2-0 20 0-41 0*90 0-87 061 0-90 110 0890 0*295 0-725 0-690 0-780 0880 8 feel 20 ie%i 1-99 I '92 2-0 2^0 2*0 0-325 0-278 0-626 0-583 0625 0-714 0-264 0*263 0*504 0-400 0*488 0-520 n f, /j 1-85 1-90 1-89 2-0 20 20 0-278 0-240 0*262 ' 0-244 0*581 0-447 0*480 0-384 0-584 0-465 0-588 0^490 50 feet 1*83 1*88 X-87 2-06 2*00 0-250 0-246 0-474 0*405 0-486 0-226 0-282 0-423 0*887 0-456 The dirainution of the resist^-nce per unit area, with the length, is principally due to the relative velocity of the water and the b<mrd not being constant throughout the whole length, A& the board moves through the water the frictional resistance of the first foot length, say, of the board, imparts momentum to tlia water in contact with it, and the water is given a velocity in til© direction of motion of the board. The second foot length will tlierefore be rubbing against wat^r having a velocity in its own direction, and the frictional resistance w411 be less than for the fir&t foot* The momentum imparted to the water up to a certain point, is accumulative, and the total resistance does not therefore increase proportionally with the length of the board. 510 FTBIlAtJLfCS 282. Stream line theory of the resistance offered to the motion of bodies in water. Rem^tance of ships* In con^deriBg the motion of water along ptpee and channels of uniform section, the water has been aesumed to move in ** stream lines/' which have a relative motion to the sides of the pipe or channel and to each other^ and the readstaiw to the motion of the water has been considered as due t-o the friction between the consecutive stream lines, and between the j water and the surface of the channel, these frictional resistance^ above certain speeds being such as to cause rotational motions the mass of the water. theT Fig. 363. Fig. 8S4. It has also been shown that at any sudden enlargement olj stream, energy is loat due to eddy motions, and if bodies, as are shown in Figs- 110 and 111, be placed in the pipe, there \ a pressure acting on the body in the direction of motion of the water. The origin of the remetance of ships is best realise*! by the "stream line^' theory, in which it i^ assumed that relative to the ship the water is moving in stream lines as shown in Fig*. 363, 364, consecutive stream lines also having relative motion. lEISTANCE TO THE MOTION OF B0DI^1N^^^^»511 Lecordmg to this theory the resistance is divided into three L) Frictional resistance due to the relative motions of con- scutive stream Unesj and of the stream lines and the surface f the ship. (2) Eddy motion resistances dae to the dissipation of the aergy of the stream Hues, all of which are not gradually brought St. J) Wave making resistances due to wave motions set up at i§tirface of the wat^er by the ship, the energy of the waves leing' dissipated in the surrounding water. According to the late Mr Froude, the greater proportion of be resistance is due to frictionj and especially is this so in long h]p<s^ Tftith ftue lines, that is the cross section varies very gradually rtiTu the bow towards midsliipSj and again from the midships DWards the stern. At speeds less than 8 knots, Mr Fronde has huim that the frictional resistance of ships, the full sx>eed of rhich is about 13 knots, is nearly 90 pt^r cent, of the whole ^eBistance, and at full speed it is not much less than 60 per cent, 3e has further shown that it is practically the same as that ■existing the motion of a thin rectangle, the length and area of ihe two sides of which are equal to the length and immersed urea respectively of the ship, and the surface of which has the lame degree of roughness as that of the ship. If A is the area of the immersed surface, / the coefficient of friction, which depends not only upon the roughness but also Upon the length, V the velocity of the ship in feet per second, the _ resistance due to friction is ^| ^5 value of the index n approximating to 2» The eddy resistatice depends upon the bluntness of the stem of Ihe boatu, and can be reduced to a minimum by diminishing the iMtion of the ship gradually, as the stem is approached, and by ^piding a thick stern and stern post. As an extreme case consider a ship of the section shown in JPig, 364j and suppose the stream lines to be as shown in the figare. At the st«rn of the boat a sudden enlargement of the Btn^m lines takes place, and the kinetic energy, which has been g]¥en to the stream lines by the ship, is dissipated. The case is analogous to that of the cylinder, Fig. lU, p* 169, Ehie to the Io6s of energy, or head, there is a resultant pressure acting up<jn the ahip in the direction of flow of the stream lines, and con- leqoently opposing ite motion* 512 HTtlRA0LlCS If the ship baa fine lines towards the stem, as in Fi^, 363, the velocitiei of tlie stream lines are diminished gradually and tk logg of energy by eddy motions becomes very smalL In actiml ships it is probably not more than 8 per cent, of the whde resistance. The wave making resistance depends upon the length and tli6 form of the ship, and especially upon the length of the '^entraiic*'* and ^* run/' By the " entrance *' is meant the front pan of the sliip, which gradually increases in section* until the middle body, which is of uniform section, is reached, and by the *^ nin," the hinder part of the ship, which diminishes in section from tie middle hody to tlie stern post. Beyond a certain speed, called the critical speed, the rate of increase in wave making resistance is very much greater than the rate of increase of speed. Mr Fronde found that for tb S*S, " Merkara " the wave making resistance at 13 kmM, tlie normal speed of the ship, was 17 per cent, of the whole, but at 19 knots it was 60 per cent. The critical speed was about 18 kiiDt** An appro3ciinate formula for the critical speed V in knot^ is L being the length of entrance, and Li the length of the rim b feet. The mode of the formation by the ship of waves can be partly realised aa follows. Suppose the ghip to be moving in smooth water, and the stT^affl lines to be passing the ship as in Fig. 363. As the bow of tb boat strikes the dead water in front there is an increase in pressure, and in the horizontal plane SS the pressure mB k greater at the bow than at some distance in front of it, and consequently the water at the bow is elevated above the nofiBii surface. Now let AA, BE, and CC be three sections of the ship and ^ stream lines. Near the midship section CC the stream lines will be m(0 closely pa<iked together, and the velocity of flow will therefore, than at AA or BB* Assuming there is no 1(» in a stream Hue between AA and BB and applying Hemotiilli'i theorem to any stream line^ w 2g w 2g w 2g* * See Sir W. While's Kami ATcMieeture, Tramaetmm ttf Nm^ Anhint^ 1877 aad 1881, RBSISTAKCE TO THE MOTtOK OF BODIES IN WATEB id since TjL and t^ are less than i'^^ ^ and — are greater than — w w w 513 4 *rhe Burface of the water at AA and BB is therefore higher %n at CC and it takes the form shown in Fig. 363, jTwo sets of waves are thus fomiedj one by the advance of the and the other by the stream hnes at the st^m, and tbesal motions are transmitted to the surroanding water, where energy is dissipated. This energy, as well as that lost in idy motions, must of necessity have been given to the water by^| ■" ship, and a corresponding amount of work has to be done by :ie i^hip's propeller. The propelling force required to do work |t&al to the loss of energy by eddy motions is the eddy resist^^f ice, and the force required to do work equal to the energy of™ waves set up by the ship is the wave resistance. To reduce the wave resistance to a ndnimuin the sliip should Qiade very long, and should have no parallel body, or the sfcire length of the ship should be devoted to the entrance and JH- On the other hand for the frictional resistance to be small, ne area of immersion must be small, so that in any attempt design a ship the resistance of which shall be as small as Dsgible, two conflicting conditions have to be met, and, neglecting^ eddy resistances, the problem resolves itself into making th^| of the frictional and M^ave resistances a minimum. Total rensiance. K R is the total resistance in pounds, T/ the_ ictiotml resistance, n the eddy resistance, and r„ the wave stance, ^^Tf + rt + r^. be frictional resistance r/ can easily be determined when tl; latitre of the surface is known. For painted steel ships / is' [iractieally the same as for the varnished boards, and at 10 feet Br second the frictional resistance is therefore about \ lb. per |tiare foot, and at 20 feet per second 1 lb, per square foot, The^ If satisfactory way to determine r^ and r„ for any ship is tijH lake experiments upon a model, from which, by the principle of ailarity, the corresponding resistances of the ship are deducec be horse-power required to drive the ship at a velocity of V fe sr second is EV the a^v^l th6| HP = 550' To determine the total resistance of the model the apparatus bown in Fig, 362 is used in the same way as in determining the ictional resistance of thin boards. L. a. 33 283. BetemiixLation of the resistance of a Bhlp firotn the resistance of a model of the ship. Tu obtain the resistance of the ship from tiie experimental i^eaifltaiicQ of the model the priociple of sim^Uantyj as stated bf Mr Froudej is used. Let the linear dimensions of the ship be D tunes those of the modeL Cmre^pofuUfig speeds. According to Mr Fronde's theor>% for any speed V„ of the models the speed of the ship at which m resistanoa mnst be compared with that of the model, or th ocMn^ponding speed V* of the ship, is CorrBsponJUng remstances. If lU is the resistance of the model ^ the velocity Vmt and it be assmmed that the coefficients d 'friction for the ship and the model are the same, the redstamoe & of the ship at the corresponding speed V* is Ab an example, snppose a model oue-dxteentb of the siis of the ship; the corresponding' speed of the ship will be fonr timei the speed of the model, and the resistance of the ship at eon^ spondiBg speeds wOl be 1^ or 4006 times the resistance of tJie Comadtbii for the difffsrente of the coeffim^nts of frietimi for HiM mocbl OffM? 0hip. Tlie material of which the immersed surface tlie model is made is not generally the same as that of the ship^ and conseqeentiy R, mnst be corrected to make allowance for the difference of ronghnees of the surfaces. In addition the ship i very much longer than the model, and the coefficient of frictiaii» 0f«ii if the surfaces were of the same degree of roughness, would tbetBfore be less than for the model. let Am he the immersed surfoce of the model and A. the shipw LeC fm be the coefficient of friction for the model and /, for ahipi the values being made to depend not only upon the rougJiD* bat also upon the length. If the resistance is assumed to vaiy V*t Ihe frietioinal resistance of the model at the velocity Y^ is •ai for Iks ship at the corresponding speed Y« the frietioQii r,=/,A.V/. A, = A»I>* But SBBISTANCS TO THE MOnOH OF BODIES IN WATER 515 and, therefore, r,=yiA«V«*D* Then the Tesistance of the ship is R.= (R,-r.)D' + r. = {R.^r.(^-l)}D.. Determination of the cwrve of resistance of the ship from the curve of resistance of the model. From the experiments on the model a curve having resistances as ordinates and velocities as abscissae is drawn as in Fig. 365. If now the coefficients of friction for the ship and the model are the same, this curve, by an alteration of the scales, becomes a curve of resistance for the ship. For example, in the figure the dimensions of the ship are supposed to be sixteen times those of the model. The scale of Telocities for the ship is shown on CD, corresponding velocities being four times as great as the velocity of the model, and the scale of resistances for the ship is shown at EH, corresponding resistances being 4096 times the resistance of the model. H Fig. 365. ^r Froude^s method of correcting the curve for the difference of the €X>€fficienfs of friction for the ship and the model. From the formula rm — Jm-A-m V m , 516 HTBRAUI.ICS the frictional resistance of the model for several values of V, is calculated, and the curve FF plotted on the same scale as aad for the curve RR. The wave and eddy making resistance at an; velcw::ity is the ordinate between FF and RR. At velocities c 200 feet per second for the model and 800 feet per second fi the shipj for example^ the wave and eddy making resistance is bt measured on the scale BG for the model and on the iK^ale EH t the ship. The frictional reaiBtance of the ship is now calculat-ed from form u la r, =/jA«W, and ordinate^ are set down from the c FFj equal to r«j to the scale for ship rpsistance. A third curve thus obtained J and at any velocity the ordinate between this cum and RR is the resistance of the ship at that velocity. For exampl when tlie ship has a velocity of 800 feet per second the resistanc is aCj measured on the scale EH. EXAMPLES. (1) Taking skin Miction to be D'4 lb. per sq^uare foot at 10 faet per aecondt find the skill resietaace of a sMp of 12^000 square feet immeraed surface at 15 kxtots (1 kuot = I'd9 leet per second). Also tind the horse-pQwt*r to drive the ship against tliis resistance. (2) If the skin friction of a ship is 05 of a pound per square foot otf immersed surface at a speed of 6 knota» what horse-power will prob*blj be required to obtain a speed of 14 knota* if the immersed surface is 18,0110 square feet ? You may assume the maximmn speed for which the ship Is designed is 17 knots. (8) The resistance of a vessel is deduced from that of a model |V^ ^ Hnear size. The wetted surface of the model is 20*4 square feet, the ekii friation per square foot^ in fresh water, at 10 feet per Beoond is 0*3 lb., asd the index of velocity is 1'94. Tlie skin friction of tlie vessel in salt i is 60 iba. per 100 square feet at 10 knots, and the index of velocity is ! The total resistance of the model in frei^h water at 200 feet per jimt 1-46 lbs. Estimate the total resistance of the vessel in salt water 4 speed corresponding to 200 feet per minute in ihe model. Lond. Uo. M (4) How from model experiments may the resistance of a inferred ? Point out what corrections have to be made. At a 800 feet per minute in fresh water, a model 10 feet in length with * ' akin of 24 square feet has a total resistance of 2*39 lbs*» 2 lbs. being dws ^ skin resistance^ and -39 lb, to wave-making. What will be the total i ance at the corresponding speed in salt water of a diip 25 tijnes tbe I dimensions of the model, having given that the surface friction per tfp^ foot of the ship at that speed is 1-3 lbs, 7 Lond, Un. 1906* CHAPTEK XIIL STEEAM LINE MOTION. 284. Hele Shaw's experiments on the fLow of thin slieetB of water, l-Vofessor Hele Shaw* has very beautifully shown, on a small e|^le, the fomi of the stream lines in moving masses of water imder varying circumstances^ and has exhibited the change from stream line to sinuous, or rotational flowj by experiments on the flow of water at varying velocities between two parallel glass plnUft^. In some of the exi:)eriinents obstacles of various forma were placed between the plates, past which the water had to flow, and in others, channels of various sections were formed through which tlie water was made to flow. The condition of the water as it Howed between the plates was made visible hy mixing with it a certain quantity of air, or else by allowing thin streams of coloured water to flow between the plates along with the other water. When the velocity of flow was kept sufficiently low, whatever the form of the obstacle in the path of the water, or the form of the channel along which it HoweJj the water persisted ia stream line flow* When the channel between the plates was tnade to enlarge suddenly, as in Fig. 58, or to pass through an orifice, as in Fig. 59, and as long as the flow was in stream lines, •no eddy motions were produced and there were no indications Lof \ij8^ of head. Wien the velocity was sufficiently high for the I flow to become sinuous, the eddy motions were very marked. ^len the motion was sinuous and the water was made to flow l^st, ohstacles similar to those indicated in Figs. 110 and III, the ^ikter immediately in contact with the down-stream face waa *h<mn to be at rest. Similarly the water in contact with the tttiiiulor ring surrounding a sadden enlargement appeared to be '^t TL^i and the assumption made in section 51 was thus justified. * Proc€tdinff» of Naval ArchiUcU, 1S9T &iid IS9B. Engineer, Aug. IS^T and 518 HTDRAUUCS Wlien the Arm was dan^ channels and eiQUOCUS, ilie stnoonglf movm^ iratr-r appeared to be separated from the sides c( llie ckannel bj a thm filiii of water, wbich Profeaoor Hele Sbaw 5a.«g^?>ted was mornig in stream lines, tke velocity of which ia the ilm dii ": as tbe surface of the channel is approached, Thie experi - miao indicaSfMi that a stmilar film surroimded obc^t;kcIe^ of fiu{i4tke wad other forms placed in flowing wata't And :r waa ^ -reA by Pr^jfe^tsr Hele Shaw thatj snrrT^unding a ioip ^s it ^ tkrom^ sdH waler, there ia a thin film moving m. 5t7*?ikiiL b ely to the shi ''i© shearing forces between wiiicii Ami .. : wOodiBS waiei ^ ap eddy motions which .accv:cL:i: fan ^km faid i ci of th lip. 285. C^rrvd abeam Una m< i* Ijtt A &ju9 oC feiil be moriii^ i trred ^ream lineSf and kl JlKFx tr- — --^^Mieoftheato fines. A~ Aaj -t tW rai&a Kfi aiun* of the etiieaiti iiBfi O ci^iie \^. .^ r-iitB«faiB net&i o&dd finrrcrandnig ^B Ic'- '^ :tr -ji-f T^r :x:i: f this element. .: ':»r -^T Tr^s-cLT^ per z.ziiz -^reik on the face hd. - - - > "-!•: 7rf>«?cLre per uziit area on the face ae, ^ t: -.:- - -jitivn -t ine :;ing>rnt to the stream line at c i-'^i * fiw:i :* :~e fiice^s >f and a^. : » ~7 f :i:»e <cr^-az: line at •:, -c^^il:::! jjitf -5 .z i Tem:-^. plane the forces acting W- -^^ ^rix^ "^ r->f -^- icrji^ alzng" the radius away :r^ss^^ r r ictinx aI- n^ :'~^ ra^iins towards the >v>^ --^ i. ..-■^""-w... = '~"-r:-:.r • D. STREAM LINE MOTION 519 TOtind the bend of a river, Oc is horizontal and the component of "W along Oc is zero. Then ^=^^ (2). dr g r ^ ^ Int^prating between the limits R and Bi the difference of p i Oflom pe on any horizontal plane at the radii R and Ri is ^"■^"I/r r^ ^^^' "wliich can be integrated when v can be written as a function of r. Now for any horizontal stream line, applying Bemouilli's equation, or Differentiating — + TT- 18 constant, to 2g w 2g Idp, vdv _ dH w dr gdr ~~ dr .(4). Then .(5). Fig. 367. Free vortex. An important case arises when H is constant for all the stream lines, as when water flows round a river bend, or as in Thomson's vortex chamber. 1 dp_ -vdv w dr~^ gdr Substituting the value of ^ from (5) in (2) — yyo dv_wt^ g dr~ g ' r' from which rdv + vdr = 0, and therefore by integration vr = constant = C 620 HTDBAULIC8 Equation (3) now becomes Pi - p C f^dr w g JR f* = C!/i_j_\ 2g\B? RiV' 2sr Forced vortex. If , 849 in the turbine wheel and centrifugal pump, the angular velocities of all the stream lines are the same, then in equation (3) t? = ciir w g jR and tL^^- ^dr g jR Scou/ring of the banks of a river at the bends. When water runs round a bend in a river the stream lines are practically concentric circles, and since at a little distance from the bend the surface of the water is horizontal, the head H on any horizontal in the bend must be constant, and the stream lines form a free vortex. The velocity of the outer stream lines is therefore less than the inner, while the pressure head increases as the outer bank is approached, and the water is consequently heaped up towards the outer bank. The velocity being greater at the inner bank it might be expected that it will be scoured to a greater extent than the outer. Experience shows that the opi)osite effect takes place. Near the bed of the river the stream lines have a less velocity (see page 209) than in the mass of the fluid, and, as Lord Kelvin has pointed out, the rate of increase of pressure near the bed of the stream, due to the centrifugal forces, will be less than near the surface. The pressure head near the bed of the stream, due to the centrifugal forces, is thus less than near the surface, and this pressure head is consequently unable to balance the pressure head due to the heaping of the surface water, and cross-currents are set up, as indicated in Fig. 367, which cause scouring of the outer bank and deposition at the inner bank. ANSWERS TO EXAMPLES. Chapter I. <1) 8900 lbs. 9872 lbs. (2) 784 lbs. (8) 78*6 tons. (4) 6880 lbs. (5) 17*1 feet. (6) 19800 lbs. (7) P=865,e00 lbs. X=12-6ft. (8) -91 foot. (9) -089 in. <10) 15-95 lbs. per sq. ft. (11) 5400 lbs. (12) 87040 lbs. ^ Chapter II. (1) 85,000 eft (8) 2-98 ft. (4) Depth of C. of B. =21-95 ft. BM= 14*48 ft. (5) 19-1 ft. 69 ft. (6) Less than 18*8 ft. from the bottom. (7) 1*57 ft. (8) 2*8 ins. Chapter m. (1) -945. (2) 14-6 ft. per sec. 18-3 c. ft. per sec. (8) 26-01 ft. (4) 115 ft (5) 58*8 ft. per sec. (6) 63 c. ft. per sec. (7) 44928 ft lbs. 1-36 h. p. 8*84 ft. (8) 86-2 ft. 11-4 ft. per sec. (9) 1048 gallons. Chapter IV, page 78. (1) 80-25. (2) 3906. (8) 37-636. (4) 5 ins. diam. (6) 8-567 ins. (6) -763. (7) 86 ft. per sec. 115 ft. (8) -806. (9) -895. (10) -058. (11) 144-8 ft. per sec. (12) 2-94 ins. (18) *60. (14) 572 gaUons. (15) 22464 lbs. <16) -6206. (17) 5-58 eft. (18) -755. (19) 102 c. ft. (90) -875 ft 186 lbs. per sq. foot 545 ft. lbs. (21) 10-5 ins. 29-85 ins. (22) •688 ft. (23) 4-52 minutes. (24) 17*25 minates. (25) -629 sq. ft. (26) 1-42 hours. Chapter IV, page 110. (1) 18,170 eft. (2) 415 ft (8) 69-9 c ft per sec. 129*8 c. ft. per sec. (4) 2-685. (6) 18-28. (7) 48-3 c. ft. per sec. (8) 1-676 ft. (9) 89-2 ft (10) 2-22 ft. (11) 5-52 ft. (12) 23,500 c. ft. (18) 24,250 eft (14) 105 minutes. (15) 284 h. p. 522 AN8WEB8 TO SXAJfPLES Chapter V. (1) 27*8 ft. (2) 142 ft. (4) -65. (5) 2-888 ft (6) 10-76. 1-4 ft. -88 ft. -782 ft. "0961 ft (8) -61 eft. 28-54 ft. 25-8 ft. 9 ft. (9) 28 per oent (10) 1-97. 21ft. 80 ft. 26 ft 24 ft 15 ft (11) 8-64 eft (12) 8-08 eft. (18) -674 ft •267 ft 7-72 ft (14) 2-1 ft (16) 1-86 c. ft per sec. (16) F=-08181bB. /-•006868. (17) 1-028. (18) -704. (19) 2-9 ft per sec (20) 4*4 c. ft. per sec. (21) If pipe is clean 46 ft (22) 28 ft. 786 ft. (28) Diri^ cast-iron 6'1 feet per mile. (24) 8-18 feet (26) 1 f oot (27) —--A(\ — » ^ = friction per unit area at unit velocity. (28) 108 H. p. (29) 1480 lbs. 1*08 ins. (80) -002825. (81) A:=-004286. n=l-84. (82) (a) 940 ft. (b) 2871 h. p. (88) -045 (84) If d»9^ v^5 ft. per sec., and/=-0066, ;i=102 and Hsl82. (86) 1487x10*. Yes. (86) 68-16 ft (87) lhour48nmL (88) 46,260 gallons. Increase 17 per cent (89) 296*7 feet (40) 6 pipes. 480 lbs. per sq. inch. (42) Velocities 6*18, 6-08, 8-16 ft. per sec. Quantity toB=60c.ftperim: Quantity to C —66*6 c. ft. per min. (46) *468 c. ft per sec (46) Using formula for old cast-iron pipes from page 188, t? =8*62 ft per se (47) 2-91 ft. (48) d=8-8ins. d,=8-4in8. d2=2-9 ins. d3-2'2ini (49) Taking G as 120, first approximation to Q is 14-4 c. ft. per sec. (61) d=4'18 ins. v=20*66 ft. per sec. p=840 lbs. per sq. inch. (68) 7-069 ft. 801ft. C,.=ll-9. C^ for tubes « 6-06. (64) Loss of head by friction ='78 ft. A head equal to ^ will probably be lost at each bend. (66) 43*9 ft. -936 in. (67) 7i=68'. Taking -006 to be/in formula ^=y^ . v=16-6 ft. persec. (68) Vi=8-8 ft. per sec. from A to P. V8=4-96 ft. per sec. from B to P 1^3= 13-75 ft. per sec. from P to C. Chapter VI. (1) 88-6. (2) 1-lft. diam. (8) Value of m when discharge is a maximum is 1-857. «■= 17*62. C=121 Q = 76 c. ft. per sec. (4) -0136. (6) 16,250 c. ft. per sec. (6) 8 ft. (7) Bottom width 15 ft. nearly. (8) Bottom width 10 ft. nearly. (9) 630 c. ft. per sec. (10) 96,000 c. ft. per sec. (11) Depth 7-85 ft. (12) Depth 10*7 ft. (18) Bottom width 76 ft. Slope -00052. (17) C = 87-5. ANSWERS TO EXAMPLES 523 Chapter Vm. 124*8 lbs. -456 H. p. (2) 628 lbs. 104 lbs. 58*7 lbs. 294 ft. lbs. (4) 960 lbs. 261 lbs. 4-7 H. p. (6) 21-8. (7) 57 lbs. (8)^ 194 lbs. Impressed velocity=28-5 ft. per sec. Angle =57'. (10)* 181 lbs. •98. -678. -68. (18) 19-2. YeL into tank —84*8 ft. per sec. Vel. through the orifice =41*6 ft. per sec. Wt. lifted » 10-8 tons. Increased resistance » 2880 lbs. 125 lbs. 8*4 ft. per sec. 1*91 h. p. Work done, 575, 970, 1150, 1940 ft. lbs. Efficiencies ^, '50, |f , 1. 1420 H. p. (18) -9875. (19) 82 H. p. 8666 lbs. 161 H. p. 62 per cent. Chapter IX. 105H.P. (2) <^=29*. V^=7ft.persec 14*8 per min. ll"* from the top of wheel, ^s70^ 1*17 eft (5) 4*14 ft. (8) 82^2'. 10-25 ft. per sec. 1*7 ft. 5*8 ft. per sec. 11° to radios. r= 24-7 ft. per sec. (18) </)=47''80'. a=27"20'. 79' 16'. 19** 26'. 58. 85-6 ft per sec 6'24'. 28^ ms. llfins. 12^89'. 16§ms. 82jins. 99 per cent. <^=78% a=18'. <^=120', a=18°. </)=158'28'. H= 77*64 ft. h. p. =14116. Pressure head =48*58 ft. d=l*22ft. D=2*14ft. Angles 12'' 45', 125*22', 16° 4'. <^=184'58', ^=16'25', a=9n0'. H.p.=2760. 616. Heads by gauge, - 14, 85*6, 81. U=51*5 ft. per sec. <^ = 158"68',a = 25'. H. P. =29*8. Eff. = *957. Blade angle 18° 80'. Vane angle 80° 25'. 8*92 ft. lbs. per lb. At 2' 6" radius, 6^ 10°, <^ = 28° 45', a = 16° 24'. At 8' 8" radius, ^=12° 11' (f> = 78° 47', a = 12° 46'. At 4' radius, $ = 15° 46', (f> = 152° 11', a = 10° 21 . 79° 80'. 21° 40'. 41° 30'. 58° 40'. 86°. 24°. 86*8 per cent. 87 per cent. 12° 45'. 62° 15'. 81° 45'. r=45*85. U=77. V,.=44. v^=86. Ui=28. €=78-75 per cent. '86 ft. 40° to radius. (80) About 22 ft. H. p. =80*8. Eff . = 92*5 per cent. Chapter X. 47*4 H. p. (2) 26°. 68-1 ft. per sec. 94 ft. 60 ft. 55 per cent. (4) 52*5 per cent. ^^ = 106 ft. ^'=61 ft. ^-1=56 ft. g 2g w 11° 86'. 106 ft. 47*4 ft. 60 per cent. 161 h. p. 197 revs, per min. 700 revs, per min. -81 in. Radial velocity 14*2 ft. per sec. 15*6 ft. lbs. per lb. 806 ft. 14 ft. per sec. 624 ANSWERS TO EXAMPLES (16) v=28-64 ft per sec V=ll-8. (16) d=a9im8. D SB 19 ins. Revs, per min. 472 or higher. (17) 15 H. p. 9-6 ins. diam. (18) 6*6 ft (19) Vels. 1-28 and 2-41 ft per sec Max. accel. 2*82 and 4*55 ft per see. per sec (20) 898 ft lbs. Mean friction head=*0268, therefore work doe to frictiaa is very small. (21) 4*61 H. p. 11-91 c ft per min. (22) -888. 4ii^QQi IT (28) p^ — ^^' Acceleration is zero when ^=j(iii+2), m being any integer. (27) Separation in second case. (29) 67-6 and 66*1 lbs. per sq. inch respectively, h. p. sd'14. (81) 7-98 ft 25*8 ft 41*98 ft (82) -648. (88) *6. (84) Separation in the sloping pipe. Chapter XI. (1) 8150 lbs. (2) 8*88 H.P. hours. (5) 4*7 ins. and 9*7 ins. (6) 8*888 tons. (7) 175 lbs. per sq. inch. (8) 2*8 ft. per sec. (9) 4*2 minutes. Chapter Xn. (1) 80,890 lbs. 1425 H. p. (2) 8500 H. p. (8) 4575 lbs. (4) 25,650 lbs. ^ [All numb^i refer to pages.] ^^^^^^^^H Abw>kte Tdoeilj 262 B«ods, loaa of bead due to 140 ^^M Acoeleratiot] io pumps* effect of (ue BemouilU'a theorem 39 ^^^ Eaciprocalmg piuup) appUed to centrifugal pomps 4 IS, 1 Aecumxilatars 423, 437* 439 J Air 4S1 applied to turbineB 334, 349 ^M difleratiluU 482 examples on 46 ^^| bydnulic 486 eipen mental illtiBtrations of 41 ^^| Air gauge, mTeited ext^n^ion of 4B ^^| Air vesflels on pumps 451 , 455 Borda'a mouthpiece 72 ^^B Angtil&r momoutum S73 Boui^diucBq'B theory for discharge of a ■ Angaidir moment uirii rate of change of weir 104 ^J eqoAl to a conply 274 Boyden diffuser 314 ^^B Appold eentrlfu^al pump 415 Brotherhood hyilraulic engine 501 ^^H Aiimedueifl 1, 189, 195 Buoyancy of Hoating bodies 21 ^^M eeclioDft of 216 centre of 23 ^^M Archimedes* principle of 22 ^^H Canal boats, steering of 47 ^^H 497 Capstan, hydraulic aOl ^^U AliDo«phede piretttire 8 Centre of buojanory 23 ^^H Centre of prestjore 13 ^^| Bfteon 1 Centrifugal force, effect of in discbarge ^^^ :B*me« ftnd Goker 129 from water wheel 286 1 Centrifugal head 1 Sftsin's eiperitnents on in oentrifugal pumps 405, 408, 409, 1 eftlih^tion of Piboi tnbe 245 419t 421 ^J diiitribtilioQ of preflsiur^ in Ihe plane m reaction turbtnea 303, 334 ^^M of an ori^ee 59 Ceutrifugul pumpa, see Pumpe ^^H ' distiibatioD of velod^ in the orosa CbanjislB V leoUon of a ahaimel 206 circular, de£^th of flow for maitimum 1 diatributbii of vdoeity in the crous discharge 221 ■ eectiOD of a pipe 144 circular, depth of flow for maiimum 1 iiatribntion of Telocity in the plane of velocity 22U ^J mn orifice 69, 244 coefficienta for, in formnlaa of ^H 1 flow uj chAnoeU 182 Bazin 166, 137 ^H flow OTcr dama 102 Darcy and Basin 183 ^H 1 flow over weirs 89 QanguiUet and Kuttt^r 184 ^H flow through ori^ces 56 coeffioientB for, in logarithmic for- ^f form of the jet from orifioea 63 muloe 300-203 1 3«&iii'fl formulae for ooefficienta, variation of 190 ^J cbAoneLfl 182, 185 curves of veloc i ty and di seharj^ for 222 ^H ori^ccfs aharp' edged 57, 51 dimensions of, for given flow deter* ^^H ¥«loeily al any depth in a Tertical miued by approximation 225-227 V section of a channel 212 diameter of, for given maiimum die* M ▼docity at any point in the orosa charge 224 ^H teetion of a pipe 144 distribution of velocily in croaa i€^ ^^H weir, Hat created 9*J tion of 208 ^H 1 ueir, iharp*creeted 97-99 earthy of trapezoidal form 2111 ^^M L^ weir, sill of amaU thickneas 99 eroAion of earth 216 ^^M t»DEX in of US of I7e lor flow in of SaS lor mx^ SOI, SOT i S3S 1*1 ohetbod c^f determining tlie in 187 im ^ ' bller« 183, 184 It of 231 19t~lSe-9CMI of 119 iJBt, 101. lae. 197. flOQ, i06 183» ISA, 187, 195, 901 l€3, 184, 107. ISS. 195, 197, 1&4, 186, 187, 19a, 1^, 1^. im, 187, 901. 207 18a. 1§4 1^ lift. ISI, 906 18i, 186, 1S7. 2^ r m-196 of, te prcD vdoaty GuiMivifaiar m Gummt meleia tSS «alil»r»tioii of 940 aurlej 938 BiAkdl S40 Curred ttf«uii linia mottcm 51§ CyUndncal mouthpiece 79 Dftms, flow oyer 101 ej^i^zimeDtit on flow in ehonneb M experun«Dt« on flow in pipei 12S fortDQlA for flow in «*haJTLwak 1^ formolA for flow ld plpei 111 Deacon's w&Bte-waler meter ^ Denftitj 3 of gmooline 11 of keroiltie 11 of m«t^ii]j 8 of pare water 4 11 Depth of e^Gtre of pr««»nie 13 Di&griim of prefi£me on a pkne uM lU Diagram of pressure on i vcrSioi] out^ 16 Diagram of work done in a recipiooiUiig pump 413, 459. 467 Diflbrenti&i aooDmalaior 489 IHJfiscBDtial ga.nee 8 Diftdivge cocffii^ient o^ for oriflcei 60 if Orifieefl) ooeflkient of, for Vdntohweicr IS of ft channel ITS {»et Olmnneii} ov«T weirs 83 (<ee Weirs) thrODgh notches S5 i*ct Noichet) throngb oriflceB 50 {Met Orifioei) thiongh pip^ 112 {$€e ¥ipmj DiBtrihotion of ?eloeity on gtom MClka of a channel 206 Dusribataon of veloeitj on crote aeite of a pipe 148 DiTecgeol monthpitoes 7S Dock cuBson 181, 192, 926 Doelis, floftling 31 Diowned nappes of weiis 9fi, 100 Drowned orificee 65 Drowned weirs 9$ appcoKunate fixmink for 301, eoeffictenla for in Beam's fonei^ 187 eoeffieienla for in Duej vid Ba^'i f<»^nJa 183 noeflkientis for in Qanguillel £utter*fi formula 184 smaoa of 216 Elbows, ktea of head due to 140 Ra^itee, ludxanlic mi BnH^eriiood 501 BMlie S03 mm £04 of eextli channels 316 mDEX 527 Ximiapliii, $oltitiotta to wMah iire given in the text — P Boiler, time of omptjing Uirough a moQthpieoe 78 Centrifugal pumpa^ determlaniioa of preftmre heftd at Inl^ and outlet ^^^■Omtril^igal pampB, dimensiooa for a ^M given diaoharge 404 ^FCetitrifugal pumpat seriea, number f of whtida for a givua lift 435 CenCfifogal pi;mp», veiocitj at which deliT«ij start? 412 Clu»iiitle» droukr diameter, for a glvoi majrimiini discharge 224 ^CMsnela, diameter of siphon pipes to given same dboharge m an ac|iitidaot 224 " aun«)a» dimensions of a canal for i gtven flow and slope 2*25, 22t»» 227 dinchargo of an earth 225 flow ini for giren section •nd slope 223 12, 489 P^lotttiog docket height of metacentre of 34 ^ floating doeks, water to be pumped ftom S3 Bad of wa£«r 7 Hjdraolic ma<!hitiery, capacity of aoecunQlator for working a hy- draulic crane 489 Hjdraulic motors variation of the pressure on the plunger 470 IsifMcl on vanesj form of vane for w«t«T to enter wit bout ^hmk and leave in a given direction 271 ItDpat^t on vanest presanrc on a vane when a jet tn contact with ia tamed Ihnmgh a given angle 267 Impaot on vaneci, turbine wheel, form of vanes on 272 Impact o» vanes, turbine wheel, water leaving the vancf; of 269 trnpaot on vanes, work done on a vane 271 Metaoentrei height of, for a floating ■ 34 Eitre, height of, for a ship 36 bpiece, dteeharg^ through^ into [ eondenssr 76 rfehpieoe, time of emptying a boiler by m^ui« of 76 oQlhpleoe^ time of emptying a ttmxww hj means of 78 " ftar of, for a given die- charge 152. 153 I Pipes, discharge along pipe connecting two T titter voir^ Idl, 154 KpipeB m parallel 154 >1^pet| preavore at end of a service pipe 1§1 dock ^tamplea {cunt.) Pontoon^ dimifntionB for given da^ plaeement 29 Pressure on a flap valve 13 PreBsure on a masonry dam 13 Pressure on the end of a pontoon 18 Iteoiprocating pump htted with an air vessel 470 Reciprocating pump, horsepower of, with long delivery pipe 470 Eeciprocating pump, presBure iu an air vessel 470 Eeciprocating pump. Beparation io^ diameter of suction pipe for no 469 Eeciprocating pump, separation in the delivery pipe 464 Eeeiproeating pomp, aeparation in, number of strokes at which sepa- ration takes place 458 Reciprocating pnmp, variation of pleasure in, due to inertia fonies 470 Eeaerroira, time of emptying by weir 108 Beservoirs^ time of emptying through orifice 78 Ship, height of metacentre of 26 Transmisaion of fiuid preasofe 12 Turbine, design of vanes and de- termiDAtion of effitfiency of, con- sidering friction 331 Torbine^ design of vaues and de- termination of eMciency of, frio- tion neglected 322 TurbiDt;, tlimeDMiQEis and form of varies for given horse -power 341 Turbine, double eompartment parallel !iow 349 Turbine, form of vanes for an out- ward flow 311 Turbine, hammer blow in a supply pipe 385 Turbine, velocity of the wheel for a given he^d 321 Venturi meter 46 Water wheel, diameter of breast wheel for given horae-power 290 Weir, CO rr fiction of cooflloient for velocity of approach 1)4 Wtjir, dischargt^ of 94 Weir, disoharge of by approsimation loa Weir, time of emptying reservoir by of 110 4 Fall of free level 51 Fire hose nozzle 73 Flap valve, prefigure on 18 centre of IS Floatliig bodiofl Archimedes, principle of 22 booyanoy of 21 oentre of buoyancj of 23 coaditions of eq.iiilibnTim of 21 cottt&inimg ^at«i-, stubilitj of 29 exomplefl on 34* 516 inetaoentTe of 24 resifiiftnoe to the motion of 507 email di^plaoements of 24 HtAbility of eqoilibrium, ooodltion of 24 stability of flo&iing dock 33 eiabililj of reot&Dgular pontoon 26 stability of veeael i^oQt&miDg wat^r 29 stability of Teaad wholJy imtnerBed 30 weight of fitiid displaced 23 FloAtiBR docks 31 stability of 33 TioKis, doable 237 rod 239 iriirfac« 237 Flow of water deinitions reladng to 36 enetrgy per pound of flowing vater 88 m QpetL obdnnek 176 (k^i* ChannelB) over damB IQl (*« DaniB) over weirs 81 [tite Weirs) tbmugh notches 80 (#«« NotcheF) through orificea 50 {s^e Orifices) through pipes 112 {*ee Pipes) FlnldB (iiquidj^) U ttBt 3-19 exam pies on 19 ooin^resaible 3 deohity of 3 flow of, tkrough orifices 50 iijconipressible 3 in motion 37 presijnre in^ la the same in all direo- tionii 4 pi^sanre on an area in 12 pressnra on a horizontal plane in, is constant 5 Bpecifio gravity of 3 steady motion of 37 stream line motion in 37, 517 trflDfimisaioii of pressure by 11 used in U tubea 9 Tlaeosi^ of 2 Forging press, bydrsnlic 492 Foumoyron turbine 307 Friction coefficients of, for ships* surfaces 509, 615 effect of, on discharge of centrifagal pump 421 e^ect ofp on velocity of exit from Im- pulse Turbine 373 effect of, on velocity of eiit from Poncebt Wheel 297 Froude^a experiments on fimd 507 in centrifugal pumps 400 in cbanuela 180 in pipes 113, 113 Friction (fiwit.) in reciprof!Ating pumps 449 m tnibinea 313, 321, 339, 37ft Ganguillet and Kutter coefficients in formula of 1%, lU experiment* of 183 formula for channels 184 formula for pipe« 124 Oasoline, ffpeciSc granty of 11 Gauges, preeaure differential H inverted air 9 Inverted oil 10 Gauging the flow of water 234 by an orifice 235 by a weir 247 by chemical means 2B8 by floats 239 {*« Floats) by meters 234, 2^1 {tee Met< by Pitot tnbea 241 by weighing 2M examples on 260 in open ehannela 236 in pipes 251 Glaz^ earthenware pipes 136 Gurley*B current meter 238 Hammer blow m a long pip€ 3S4 Haskell's cuirent meter 240 _ Hastie'fi engine 503 Head position 39 preasure 7* 89 velocity 39 High pressure pump 471 Historical devetopment of pip channel formulae 231 Hook gauge 248 Hydraulic aocumulator 4S6 Hydraulic capstan 501 Hydranh'o crane 494 double power 495 valves 497 Hydraulic differential accumulftti Hydraulic engines 501 orank effort diagram for 509 Hydraulic forging press 402 Hydraulic gradient 115 Hydraulic inteusifler 491 Hydraulic machines 485 conditions which vanea of, flatisfy 270 examples on 489, 505 joints for 484 ma^cimum efhclency of 29S packings for 485 Hydraulic mean depth 119 Hydraulic motors, vwiationR of j in, due to inertia forces 469 Hydraulic ram 474 Hydraulic riveter 499 Hydraulics, definition of 1 INDEX' 629 4-19 »r on ▼anes 261 {see Vanes) . in hydranlio motors 469 } in reciprocating pumps ent of 14 f^oge 9 ;auge 9 lydranlic 491 valves for 492 5team 493 turbines 275, 818 {$ee ) n hydraulic work 485 er 255 meter 253 !cific gravity of 11 hydraulic packiugs 486 formulae for flow i 192 5 plottings iB 195 27, 133 499 I 17 ,vity of 8 barometer 7 U tubes 8 leight of 24 aste water 254 55 i , 75, 251 lertia 14 ane of floating body 25 id law of 263 54 of discharge for 73 3 il 71, 76 le 78 of velocity for 71, 73 73 73 3 n 78 h, under constant pressure d at entrance to 70 ptying boiler through 78 MonthpieoM (cofit.) time of emptying reservoir throogh 78 Nappe of a weir 81 adhering 95 depressed 95 drowned or wetted 95 free 95 instability of the form of 97 Newton's second law of motion 268 Notation used iu connection wiUi vanes, turbines and centrifugal pumps 278 Notches coefficients for rectangular {ae Weizs) coefficients for triangular 85 rectangular 80 ($ee Weirs) triangular 80 Nozzle at end of a pipe 159 Nozzle, fire 74 Oil pressure gauge, inverted 10 calibration of 11 Oil pressure regulator for turbines 877 Orlflcea Bazin's coefficients for 57, 61 Bazin's experiments on 56 coefficients of contraction 62, 56 coefficients of discharge 57, 60, 61» 63 coefficients of velocity 54, 57 contraction complete 58, 67 contraction incomplete or aappraiiecl 53, 63 distribution of pressure in plane of 59 distribution of velocity in plane of 59 drowned 65 drowned partially 66 examples on 78 flow of fluids through 50 flow of fluids through,^ under constant pressure 75 force acting on a vessel when water issues frcon 277 form of jet from 63 large rectangular 64 partially drowned 66 pressure in the plane of 59 sharp-edged 52 time of emptying a lock or tank by 76, 77 Torricelli's theorem 51 velocity of approach to 66 velocity of approach to, effect on dis- charge from 67 Packings for hydraulic machines 485 Parallel flow turbine 276, 342, 868 Parallel flow turbiue pump 487 Pelton wheel 276, 377, 880 Piezometer fittings 139 Piezometer tubes 7 1H0KX lomvi C Itt /liii= la '^ for «Mt koa^ ne* cni «M 190^ Itl, 121, IS, ttl lor mml tiraied 121 133, isa B«r ftad old ISS, 137« Il§ i)M» 14s riveted nm, IM VDod l3S« ISB wtohqJu ir^Mi l^'i, las. 13$ n In GingniUfi aud Ku tier's fi^nstUa «tt«t iitm, new atid old 125 for elftsped caithentiriu-e 125 for stod riveted 184 for wood {ripet 125, 164 T&miioii of, vitfa 8erdoe 123 OOiUuciiiig llirie TCKTrmtv I06 oounectiag two r«eeir?oirs 149 oOimfifftJiig two teaer¥oirs, difljueter of for ^ma ditcbarge 152 ootksl velocity in 128 DftrcT'a foimula fo(r 122 deilettmimtioD of the coeffioient C, m giTcn in tables hj logarilhmio pbtiing las dtaiaptcr of, for git«a discharge dianieler for miniitiiiin eont 158 diameter varjing 160 diTided into ivo brmnches 154 elbow4 for 111 empirical fortniila for head lost in 119 empirieal formula fot relocitj gf Sow in lia equation of flow ia 117 examples on flow in 149-162, 170 expert uieoiai detertnmation of loss of head by frictiot] m lid experimentfi on djstributioii of velocity m U4 expetimeny on flow in, Gjitici&ni of 136 fl^tednieiita on loes of head at bends 142 «ipfrimi>nts on loss of head in 122, i2d, m, 131. laa, 136 experimentB on loss of head in, criticism of 138 flow through 112 flow diminishuig at imifonn rate id 157 formula for Ch^^y 119 Darcy 1^2 Fipia {oiut,) fortnnia fqr (^oni.) logarithmic 125, xBi^ las, m4 Beynolds ISl ma&m%ry <4 146 f^docltj'&t anj poiat in 1 oroM section of 143 Mctioo In, loss of bead bf 113 determination of 116 Gangoillet and Enttet*« tmmak kt 124 gmugmg the flow in Ml hAmmer blow in ^1 bead lost St entrance of 70, lU head loet by friclioD m US head loat by friction in, emiukit formula for 119 head loel by Mction Iil, exaaiplii «a 150^163, 170 bea^i loflt by friction in, logimUmiiO formula for ItB^ 133 head rcquinjd lo ^ive tieioaity Vi wai^F m the pip« 146 bead rvij aired to give velodtj to v^^ in the pipe, «{]rproximate ^lu iU hydratilio gfadient lor 113 hydratilii; mean depth of lid joints for 485 law of frictional resistaiioe foi| n^D^ the crjtical velocity 130 law ol 5iotion&l xesistanoe fat, Wjjv the critical velocity 125 Htniting diameteir of 165 logarithmic formula for 135 logarithm it; fomiola for, coeffideiiti m 138 logarithmic formula, use of, foi pii^ tical ealculftiionB 136 logarithinic plotting^ for 136 nozzle at diichafige end of, ares of when energy of jet ia a madmoiD when momentum of jet is a matl* mum 159 pie7,ometer fillings for in^ pre&snre on bt^nds of 160 ptesaure on a cyhnder in 169 preafiure on a plate in 1S8 problems 147 {*^^ Problema| pumping watc^r thjougb long p$»^ diameter of for m inimajn ooet lii reg [stance to motion of Haid In Uf TiBing iibove hydtHulle gndieat 111 short 153 siphon 161 temperBtore, elfeet of, ua flow in 131, 140 tranBrnieeion of power along, h; l^ draulic pressure 162 Taluee of C in the formula r=0%^ for 120, 121 variation of C in the formola r=C^'i for 123 IlfDEX T^rimuon of the diftch«rge of» with strviee 133 veltK^itj of £lo^ ullowable in 102 velocity, heftd riiqiiired to give Telocity to water in I ■IB velo^itj, varifition of, in a crosaMetion of n pipe f43 virtual siope of 115 Pitot tube 241 CAlib ration of 245 Poacelet water wh^l 294 P'oDtooii, pfert«Tire on end of 18 P^tiou hei^ 29 hctsa, forging 493 PfttS, bydraulid 49S, 49B pMinre &t anj point in a 6uid 4 atmonphenc, In feet of w&ter B ginges 8 head 7 mettinred in feet of water 7 on a horizontal piftue in a fluid 5 on A plate in a pipe IBS on pipe bend!4 IHti Principle of ArcliimedeB 19 Principle of siiniliirity 64 Problems, ^olutiooB of which ait given in the t*ixt — chiLnnelH difljueter of, for a given maximnm discharge ^i24 dimennionB of, for a given flow ««rth ditfchmge along, of given di* mensioms and elope 22 \ _ flow in, of given section and Blope ■ n^ ^L filope of^ for tninimttm coat 227 ^m fioltitaons of, b)- approxiination ^ 225^227 , ptpe^ acting as a siphon 161 connecting three Psaervoira 155 connecting two reflen^lri 149 diameter of, for a given diMchar^ 152 divided info two branches 154 head lost in^ wlien flow diminiahea ftt nniform rate 157 lam of hyad in, of varying diameter, leo, lei ptumplng water along, diameter of, for minininro oo»t 1S8 with nozzle at the end 158, 159 E^pnlajon of shipa by water jeta 279 Pmmping water through long pipea 158 Pomp* G«>ntdfngai 392 advantoges of 43y Appold 415 Bemonilli's equation applied to 4U Fiimpi (conL) centrifugal {conL] centrifugal head, effect of variation of on discharge 421 tientrifugal head, impressed on the water by the wheel 405 design of, for given diucharge 402 discharge, effect of the variation of ttiu cen trifilgal head and loaa by friction on 419 diflcharget head ^velocity curve at ^ero 409 diiicharge, varmtion of with the h£ad at oo octant speed 410 discharge, variation of with speed at constant heod 410 effideucies of 401, efEoieneieft of; (experimental de> tennination of 401 examples od 404» 412, 411, 418, 43o, 478 form of vanes 39B friction, effect of on dbchorgt 419, 421 general equation for 421, 425, 428, 4B0 gross lift of 400 head- discharge cMirve at constant velocity 410, 412, 427 head loet in 414 head, variation of with discharge and flpeed 418 bead- velocity curve at constant discharge 429 head -velocity curve at zero dis- charge 409 kinetic energy of water at exit a99 limiting height to which single wheel pump will raiae water 4B1 liuiiting velocity of wheel 404 lo9He£t of head in 414 multi'fltafie 433 »eriea 433 spiral casing for 394, 429 starting of 395 suction at 431 Bulzer series 434 Thomson's vortei chamber 397, 407. 422 triangles of velocities at Inlet and exit 397 vane angle at exit, effect of variation of on the efficiency 415 velocity- discharge curve at conatant heiul 411,412,421.428 velocity, heail-dischargf curve fof at couBtant 410 velocity head, sp^^iil amngement for con%'erting into pressure head 422 velocity, limiting, of rim of wheel 404 532 tKDKX Telocity of whirl, ratio of, to fdocitj of outlet e4g« of vmoe S98 TOftex chfijuber of 397, 407, 423 with whirlpool or vortex ch«imber B97, ^37, in work done on wmler by 9t7 oompTesHed air 477 duplfx 473 exAmpW tm 458, 464, 460» 47S kigh pramtire 472 hjdrftnlic mm 476 {iiacking« for plangera of 47S| 436 reciproeating 439 ftOOGle ration, elfeot of on preasura in oylinder of a 446, 448 acce)er&tioD of the plaoger of 444 acceleration of the water id delivery pipe of 448 acceleration of the water in inotiOD pipe of 445 ur Teaael on dellverj pipe of 454 air veaiel on eactioD pipe of 451 air vessel on taction pipci effect of on a©p«kration 462 eodfflcient of diaaharge of 442 diagram of work done by 443, 4S0, U% 467 diaebftigtt coefficient of 443 di3pl«i 473 examples on 458, 464, 469i 470| 480 frictino, variation of ptesattre in the cylinder due to 449 head lost at auction, valve of 468 head lONt bj Mction in the auotion and deUvery pipes 449 high pre»Bur6 plunger 471 preasufe in cylinder of when the plnngoj move& with simple har- monic motion 446 preaisurt^ in the cylinder, variation of dtie to friction 449 separation in delirerv pipe 463 separation dnring suction atroke 456 separation daring suction stroke when pi anger moves with fiimple harmonic motion 4o8* 461 alip of 442, 451 snetion stroke of 441 suction stroke, separation in 456, 4fll, 463 Tangye duplex HH Tertlcal single acting 440 work done by 441 work done by, diagmm of 445, 459, 467 turbiue 396, 42€ heafl'disoharge currea at oonatant speed 427 lor bine {ei>nt.\ head'Tdocity ctLrv@« discharge 429 inward 6ow 439 multi-i^lage 433 parallel 6ow 437 veloei ly diBcharge carvei si eciif biat head 42B Worthington 432 J work done by 44B H work done by, diagram of (KiS oiprocating pompfi) work done by, series 43$ Heaotion turbines 301 limiting bead for 367 series 367 Beaction wheels 301 efficieney of 304 Beciprocating pumps 4S9 Ue* Bectangular pontoon, etabihtv uf Beotangalar sharp- edged weir §1 Beclangular sluices 65 Rectangular wetr with end oostiM tiona 88 Begnlation of turbines 306, 317, tU 328, 343 Regulators oil pressure, for impulse tnrbine R71 water proisure, for impuife tur^i 379 Relative velocity 265 as a vector 2&% Beservoirs, time of emptying orifioo 76 ReservoirB, time of emptying over « 100 Reaiatance of ^hip 510 Bigg hvdrAulic engine 503 Rivera,' flow of 191, 207* 211 Rivera, acouring biinkB of *%2(> Riveter, hydraulic 500 Scotch turbine 301 Second law of motion 2G3 Sepamtion (*i*f Pumpe) Sbaq>-edKed orificea Bazm'a experiments on 66 di Kiribati on ofveiocity in the ptanj pressure in the plane «f 59 m table of coeffioltmta for, «bfll traction is complete 57, 6^ table of coeffloi«nta for, whtn traction it iuppreased 63 Sharp- edged weir 81 {»et Weirs) Shipa ppopalsion of by water J eta rcBistanoe of 510 resistance of, from model oXJ streani line theory of the of 510 Similarity, principle of M J! 1 ITTOEI S33 forming part of ^ueduct 31G lei pomiM U2t 461 65 gnl&ttng tuirbmes {ite Tarbinefl) »lme 11 !Vien« 11 I, Tftriatton of^ with temperatiire re water 4 iatioi] of, with tempemture 11 r ol % hoij ^ 35 ig dock SI ig ve&st^ QOQtuDiEig water 29 tgttlar pontoon 26 paodoD of fluids 37 nien&ifier 493 line motion 37, 1^3, 517 I SIS Shawns eicperiments oti 384 line thaotj of resistaiioe of m 510 oentrifagal piunp 431 redprooating pnmp 441 Hbe of tttrbine S06 sontnu^tiQn of a cuirent of m lar^ment of a current of 67 inlti-st&ge pamp 434 1 contracttiQH 53 K on dlsdmrge from ori6de \ on diicharge of a w«ir @3 , i«wers and aitnedu^li, par- I of, and values of - in ftU 4=*^; 195 Baofai maxlmtim velooitiea of ^ 15 a and ^ in Bazln's formnla EM of V atid i as deC^^rmmed Ep£rimenlalij and >^ c&leuliit«d oa lopintbniie tarmaia^ lUS, »l-208 tttita for duna 103 for shju^-ed^ed oHfioe, ocmmlete 57. 61 |lbr «narp< edged orifiee, duppre^sed 63 ahftrp^adged wetri 89, for Venturi meters 46 intieli;, v^loeities aboTe which takes place 216 minimum dopei for varjing valuea of the hjdraultc mean depth of brick £)lLivnneb that the veloci^ tnajf not be less than 2 ft, per eeeond 215 momenta of Inertia 15 PeltOD wheels « particutt^m of 377 pipes lead, slope of Rnd velocity of 6ow in 128 reasonable valaes of y and n in the formula ft = ^^ 188 Taloes of^ C in the formula r^Qs^mi 120, 131 valiaea of f in the formula 2ffd vaiiaefl of n in Oangaitlet and Kutt^*s formula IMl iM valuea of n and k in the formula (=itr» 137 resistance to motion of boards in Anids 509 turbines, peripheral Teloeiiiea and heads of Inward and ouc^ard f ow 333 uBefnl data 3 Thomson f centrifagal pump, vortei chamber for 397, 407, 423 principle of simihurity 63 turbine 323 Time of emptying tank or reservoir by an orificp 76 Time of emptyinj;? a tank or reaervoir by a weir 10<J Torricelli^s theorem 1 proof of 51 Total pressure 12 Triaiiji^ular notches 80 dificlifljge through 85 TnrMnes axial flow 276i 343 axial flowt impuUe 368 axial flow, prGHaure or reaction 343 axial fiow, section of tJie vane with the variation of the rftdius 344 Bernuuilli*a equation a far 334 best peripheral velocit> for 329 central vent 320 oentrifuffal head impressed on watar by wheel of 334 eone 'd&\t design of vanes for 346 efficiency of 315, 331 examples on 311. 321, 333, 331, 341, 340, 385, 387 flow through, eflect of diminishing, by means of moreable guide blades 363 flow through, effect of dimiuiabing by means of aluices BM si* INDEX ^^^^^^H Turliiuti {€f>ni.) Turbines {cotit,^ ^^^| Jlow ibitttigbf e£f60t of dttniniBhuig ImB of head in 313, 3*21 ^^M on velooHy O'f exit H63 imixed ^ow 350 ^^^1 FoDlaine, rcigiilAting slaioes 343 form of vaiipi of 355 ^^^| form of Timefl for 308, 347, Bti5 guide blade regulating gesaF for l^oomeyroD B06 352-354 general formula for 31 in open atneam Bi\0 geoeral forToalik, including frietion Swain gate for 374 315 triangle:^ of velocltjea for 3S5— guid^ btadefl for 320, 3S6, 048, 35S, dm 362 wheel of 351 fuidd blades, «fleet c^f ehonging the Hiagaju falln 313 direotton of 362 oil preiiiore regulator for 377 gotde blades, vaHatioti of the angle outward fiow, 275, 306 of, for parallel flow turbil)«ii 344 Beroooilh^B equatini^B for 334« hors« power, to develop a given 339 33y b«st peripheral volooity far, at inlet impnltte 3O0, 369-384 329 AtM flow 363 Boy den 314 diffuser for 314 «Kampleft 387 for high beads 373 doable 316 form of Tanea for 371 examples on 311^ 3B7 Girard 369, 37D, 379 eiperimental deteniiinatioa of iha best velocity for 329 bjdmiilic efRcieiicy of 37l» 378 in airtight chamber 370 Foiirneyrt>n 307 oil preBRure regoUtor for 377 lo{>&6B of head in 313 radiiLl flow 370 Niagara falls 318 tmngles of vetocitiei! for 37*2 Buctjoa tube of 308, 317 triaogleH of velocities tor conaideriog triangles of velocities for 308 friction 373, 376 work done on Iho wheel per lb, d water prii&gnra r^^lator for 379 water 310, 315 w&ter pressure reijulator, hjdraiili« pamllel flow !^76, 34:^ valve for 3#2 adjustable guide bludes for MS water presBiire regulator, water filter Bernouilli'fl rquationa for S4S for 383 deaipi of vanes for 344 work doue on wheel per Ibu of water double compattmtut MB 272, 277» 323 example! on 349, 3B7 incliuation of vanes at inUt of wheel regulation of the flow to 341* 308, 321, 344 triangle of velooitieai for 344 inclinatioo of vanea at outlet of whe^l reaction 301 ms, 321, 345 aiial flow 276-342 in open atream 360 cone 359 ^^H inward iow 275, 318 inward flow 275, 318 ^^M Btroooilli's eqiiatmiii for 334, 339 miied flow 350 be«t peripheral vdooity for, at outward flow 306 inlet 329 paraUel flow 276-342 central vent 320 Scotch 302 examples on 321, 931, 341, mi aeries 368 Batpeiimeotal determination of the regulation of 306, 317. 318, 323, $18, best velocity for 329 S50. 352, 360, 362. BU for low and variable fulls 328 Scotch 301 Franeia 320 Bluicea for 305, 307, 316. 317, 31», borieontal ajcis 337 327, 328, 348, 350, S61, Mi lom&B in S21 auction tube of 306 Themaon 334 Bwain gale for 364 to develop a given horse* power ThomBon's inward flow 323 339 to develop given borae-power 338 tnanglea of veboitiea for 322, 326, trianglea of velocities at inlet an! 332 outlet of impulae 372, 376 work done on the wheel per lb. of triangles of velocities at inlet and water 321 outlet of inward flow 3U« limiting head for reaction turbine triangles of velocitieu at inlet mbi 867 outlet of nii:xed flow 356 INDEX 535 Tiiztiiii08 (ecmt.) triangles of velocities at inlet and outlet of oatward flow 844 triangles of velocities at inlet and outlet of parallel flow 844 types of 800 vanes, form of between inlet and outlet 866 for inward flow 321 for mixed flow 351, 356 for outward flow 311 for parallel flow 344 Velocity of whirl 278, 310 ratio of, to velocity of inlet edge of vane 332 velocity with which water leaves 384 wheels, path of water through 312 wheels, peripheral velocity of 333 Whitelaw 302 work done on per lb. of flow, 275, 304, 815 Turning moment, work done by 273 Tweddell's differential accumulator 489 XJ tubes, fluids used in 9 Undershot water wheels 292 Valves crane 497 hydraulic ram 476 intensifier 492 Luthe 499 pump 470-472 Vanes conditions which vanes of hydraulic machines should satisfy 270 examples on impact on 269, 272, 280 impulse of water on 263 notation used in connection with 272 Pelton wheel 276 pressure on moving 266 work done 266, 271, 272, 275 Vectors definition of 261 difference of two 262 relative velocity defined as vector 266 sum of two 262 Velocities, resultant of two 26 Velocity coefficient of, for orifices 54 bead 89 of approach to orifices 66 of approach to weirs 90 relative 265 Venturi meter 44, 75, 251 Virtual slope 115 Viscosity 2 Water definitions relating to flow of 38 Water (cont,) density of 3 sjpecific gravity of 3 viscosity of 2 Water wheels Breast 288 effect of centrifugal forces on water 286 examples on 290, 886 Impulse 291 Overshot 288 Poncelet 294 Sagebien 290 Undershot, with flat blades 292 Welra Bazin's experiments on 89 Boussinesq's theory of 104 coefficients Bazin's formula for adhering nappe 98 depressed nappe 98 drowned nappe 97 flat crested 99, 100 free nappe 88, 98 Bazin's tables of 89, 93 for flat-crested 99, 100 for sharp-crested 88, 89, 93, 97, 98 for sharp-crested, curve of 90 Rafter's table of 89 Cornell experiments on 89 dams acting as, flow over 101 discharge of, by principle of simi- larity 86 discharge of, when air is not ad- mitted below the nappe 94 drowned, with sharp crests 98 examples on 93, 98, 108, 110 experiments at Cornell 89 experiments of Bazin 89 flat-crested 100 form of, for accurate gauging 104 formula for, derived from that of a large orifice 82 Francis* formula for 83 gaugiug flow of water by 247 nappe of adhering 93, 96 depressed 95, 98, 99 drowned 95, 96, 98 free 88, 95, 98 instability of 97 wetted 95, 96, 99 of various forms 101 principle of similarity applied to 86 rectangular sharp-edged 81 rectangular, with end contractions 82 side contraction, suppression of 82 sill, influence of the height of, on discharge 94 sill of small thickness 99 S36 on>KX (eoMf.) rvqvired to lomr wter in ^doeitj of ftpproaAh, effect of on mai l oil tj meini of 109 dudbaige 90 wi:>si forss of 101 wide llftt-crested 100 w^txXT c< ftnxoeefa, eoneeuoo ol Wbitelaw tozbine 302 eot&eaemt for 9f Whole prenore 12 viuac^T of effTCMdL, eonectiofi of Worthington multi-stage pomp 433 ■pies on M CAvumssx : ruxrsp bt johx cx^t, m.a. at thx unitersitt press. Mr. Edward Arnold's List of Tecbnical & Scientific Publications Electrical Traction. Bv ERNEST WILSON, Whit. Sch. M.LE^E.p F^tjfcssor or t^tectricat Engineering in the Siecncns LAbarAtot^v Ring's. CoUcf^Ci Lciiiciant AND FRANCIS LVDALL, H.A., B.Sc, New Edition. Rewritten and Greatlv Enlarged. Two volumes, sold separately* Demy 8vo,, doth. Vol L| with about 300 Illustrations and Index* 15s. net* Vol. IL, with about 170 Illustrations and Index, 15s. net. In dealing with ihis ever-incrcaaingly important subject ihe authors have divided 4he work into the two branches which are^ for chronological and other reasons, most convenient^ namel}', the utilization of direct and alternating currents respectively. Direct current traction taking ihe first place, the first volume is devoted to electric tramways and direct- current electric railways. In the second volume the application of three-phase alternating currents to electric railway pronlems is considered in detail, and finally the latest developments in single- phase alternating current traction are discussed at length, A Text- Book of Electrical Engineering. BY Dr* ADOLF THOMALEN< Translated by GEORGE W. O. HOWE, M*Sc., Whit. Sch*, A*MJ*E*E*, Lecturer in Elef^trtcn! Enffiwermg »l ibe Ceniral Technical College, Santb Kcnsuigiotn With 454 Illustrations* Royal 8vo., cloth, 15s. net. This translation of the** Kuoe Lehrbuch derEiectrotechnik "' is intended to fill the gap which appears to exist between the elementary icxi-books and the specialiied works on various branches of electrical engineering. It includes additional matter which is to be introduced into the third German edition^ now in preparation. The book is concerned almost exclusively with principles, and does not enter into details of the practical construction of apparatus and machines, aiming rather at laying a thorough foundation which shall make the study of works on the design of machinery more profitable. Only the simplest elements of the higher mathematics are involved. Alternating Currents. A Tesct-Book for Students of En^neenng* BY C, G. LAMB, M.A., B.Sc, Clue Colkg«i, CAinbridgc ; A«a^ocUte Meiitber cf the In&iituUon ef Klectrical Ensineen; AisQCJAtje of tbe City and GuiUji of Londlon, In^iitute. viii + 325 pages. With upwards of 230 inustratjoni* Demy Svo* cloth, IDS, 6d* neL The scope of this book is intended to be such as to cover approximately the range of reading in alternating current machinery and apparatus considered by the author as desirable for a student of general engi- neering in his last year— as ^ for example^ a candidate for the Mechanical Sciences Tripos at Cambridge* LONDON I EDWARD ARNOLD, 41 & 43 MADDOX STREET, W. List 9f attracted so much AttcQUon ig^ Is li^v^ aitaiaed a grtatly qf tte Uuvcraty of London iu ia ^psaeoaixie scunevhit vague ^k&db, so oUcii coQtain FtY^e-Fignre Tables of Mathematical FimctioQS, Br JOBK BORTUW1CK DALE, M.A.* Doay Svo., docile js. 6d. net sdectai for use In the esaniiiialkffls 10 tbe Utssmre of Matkemfttkai irier^ nif fooD of higiker pAiyskal re^eai dt ihis M to iIk vsjirf sv^iiif cinie Alii libour snd . 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Their publication at a low price renders thii conveniciit and rapid method of working nut traverse surveys &ece^ble to a dais of workers from whom it has hitherto been debarred/' — hfiftittg JtmrasL Technical and Scigntific PuUimtions Organic Chemistry for Advanced Students* I Bv JULIUS B. COHEN; Ph.U., B.SCm of Orgmiiic Cbemisiry in (tie Untvcfikj of [.eeds^ and A^ocwt^ ofOweM Demy 8vo*, cloth, 21s. net. book is written for students who have already completed an elementary course of Organic Chemistry, and is intended largely to take the place of the advanced tcKt-book. For it has long been the opinion of the author that, when the principles of classification and ^B synthesis and the properties of fundamental groups have been ^^1 acquired, the object of the teacher should be^ not to multiply ^ facts of a similar kind, but rather to present to the student a broad and general outline of the more important branches of the subject. 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F.L.S., F.G.S., A.S4., '^rofeuor of Botmny axid Lcciurer on Fiufstfy in th^ Cliy of l^ndoti College, and fctrmerly in the HoytLJ A^ncuUura] College^ EW Edition* Revised anu Enlargkd. Demy 8vo., 12s, 6d. net. It Is ji^sl the book that has long been wanted by land agent*^, foresier?, and wood- , aad U should find a place in all technical school libraries," — Fit/d, Manual of Alcoholic Fermentation and the Allied Industries. By CHARLES G. MATTHEWS, FJ.C, F.C.S., ETC XVI+29S pages, with 8 plates, and 40 lllnstrations. Crown 8vo., cloth, 7s. 6d. net. " This js a book worthy of its author p atid well worth perusing by every student* Tlifi student, both old and young, as well as the practical brewer, wilJ hud thh i gives hitn some very useful information.'" — Bnewftri' Guardiart^ ir, Edwmrd Atwc/d's Tec Anita/ <5^ ^tenfifi^ Books Tlie Eiroliitk»ii Tbeory. By Dk. Aogust Weis* «tf HtttaAl Ifirtixj in tbe Unitmity of Abctdeca ; and Two ««1l, «:n-r4t&iad viti^ 396 fvga, whli orct *T^»^|aa >■> aenr tiw ^^falf aad rinmwiiHniin ly efpeanded brfoK : aad iB ^^ iriivie of 1 The Chances of Death and Other Studies in Mxi!>£mi.lAC5 iQ L DiTcr^tj Cc-fW^e^ I^jivioii and lbts^f!y Felloir €# Kmg'i O^legf, Ciabryy. a vok^ %b ^ jSi ad #6& ?■£». witii aiiMicroiis Behaviour. By Professor C. Llovd '■&4-j%« F^e^* ^^l> 26 DliiitntinBi Lu^ crovii Svow, dotb, iqk. 6d. K ^portiac eontiitautm m dici ^sctoaung siib|eci of a"*iifft1 psyc^uK IqO^ covers the vriK^ gniiml fom die beha^^otii of ceUs a|i to that BY THE SAME AiTTBOR, •^ Habit and Instinct ¥111 + 352 pag^, with Photo- ffaemm Fnipiitpeoe. Desy Srck. dorli, t^ ftaia^ ALTttSD RxTSilt. Wjixace :— " Aa ^miobie introdoctioo to ilie sudf ^ a mm^ mtfo^^ mad fanattiic bnttch «i tioloey^ aow for itie first time based ^am m aiibsaBAil fo^^JSMB of cuvidly d h a cf^ed Iftcts sBd IqekHl Indociioti fiooi BY THE SAMB AUTHOR. The Sprini^s of Conduct. Cheaper Edition, Tia^^i; poc^s. Lm^ «ro«s Sixk, clotlio js^ 6d« This voltiaic deals vilh %htSmmBtw^ LlwHf of K»o«ie^ tlie Staiiy of Nature; tlie Evolotioo of dy» nd Mml, Cbokt, Fecluig a»d < , Sody» nd MlBfl, Cbmot] Fecluig a»d Candoci, BY THE SAME AUTHOR. Fsycbology for Teachers, New Edition, entirely An iQtroductton to Child Study. By Dr. W. B. 0*riiM<xxT:k, Cnrnm 9to,. cMk. tit. net. The C Mind : Its Growth and Training. By L^rrrsky of Le«k. Crown 8ro*, cloth, 4s* 6d, net. E LONDON: KOWAKD ARNOLD. 41 ft 43 MADDOX STREET. W. Kt.^- r