J< OU_1 58339 >[g
THE
ENERGY CHART
PRACTICAL APPLICATIONS TO
RECIPROCATING STEAM-ENGINES
BY
CAPTAIN H. RIALL SANKEY, R.E. (ret.)
Member of the Institution of Civil Engineers,
Member of the Institution of Mechanical Engineers,
etc,
RUGBY:
ALBERT FROST AND SONS,
WARWICK STREET,
1905
ALBERT FROST AND SONS,
PRINTERS,
RUGBY.
PREFACE.
THE preparation of this book was begun about nine years ago,
but it is only recently that the Author has had time to complete
it for the press. As the title implies, the scope of the book is
confined to the application of the energy chart to reciprocating
steam engines, and special attention has been paid to the explana-
tion of practical methods. It is true that in Chapter II. the
thermodynamics of a perfect gas are explained in an elementary
manner by means of the chart, but this is done as an introduc-
tion to its application to the steam engine.
The energy chart for a steam engine is drawn for I Ib. of H 2 0,
the chemical symbol being used to express the fact that in a steam
engine a mixture of steam and water is in reality the working
fluid; the word steam is thus used in the sense of dry satu-
rated or else superheated steam ; strictly therefore its use should
be limited to reversible cycles in which the weight of the substance
is constant, but by the use of the "quality line," a concep-
tion described for the first time in this book, the chart can be
used for any weight of H 2 desired. Little progress can be made
with the practical application of the chart without this concep-
tion, combined with the convention that, except during expansion,
the chart shall only represent the pressure, temperature, and
volume, but not the heat (i.e., the entropy), for the reason that
in an actual engine the weight of H a O is constantly varying,
except during expansion and compression, and then only if there
are no leaks.
In Chapter^ X. and XL a method of designing the cylinders
and obtaining the economy of an engine by means of the energy
chart is developed, and this the Author believes will be found
superior to the ordinary or p v method. Attention is also called
IV. PREFACE.
to the method of obtaining the cylinder ratios, as illustrated by
the case of a quadruple expansion marine engine.
It is hopfcd that the short chapter on the SO 2 engine will
prove of interest. This chapter, together with Appendix I., was
contributed by Mr. G. P. Mair, to whom the Author owes thanks
for revising the proofs, and for making many of the drawings and
calculations. He also wishes to express his indebtedness to Mr.
C. Hole, who drew most of the figures, to Mr. A. E. Reynolds for
revising the calculations, and to Mr. C. H. Wingfield for many
valuable hints. The energy chart, Plate I., was drawn in 1893
by Mr. Guy E. Lloyd, under the Author's direction, and was
published for the first time in its complete form in Professor
Ripper's book on the Steam Engine.
Great pains have been taken to obviate clerical errors, but
undoubtedly some must exist, and the Author will be much
obliged if any reader who finds such errors, will communicate
with him.
The Author has ventured to re-name the chart, and call it
the energy chart. The name temperature- entropy chart strictly
applies only when the temperature and entropy alone of
the substance at all state points are given. When, however,
volume lines are added, work as well as heat is represented by
areas on the chart, and it then becomes in reality an energy
chart.
H. RIALL SANKEY.
7 DEAN'S YARD,
WESTMINSTER,
September, 1905.
CONTENTS.
CHAPTER I.
PAGE.
INTRODUCTORY . . . . . . . . . . . . . . . . i
Various Methods of Graphic representation of Thermodynamics
p. v. Method, or Pressure- Volume Method 6 <f> Method, or
Entropy-Temperature Method Entropy (Thermodynamic
Function) Elementary Principles Constant Volume Line
for a Gas Constant Pressure Line for a Gas State-Point.
CHAPTER II.
THERMODYNAMICS OF A PERFECT GAS EXPRESSED ON AN ENERGY
CHART . . . . . . . . . . . . . . . . 7
Representation of Internal Energy Representation of External
Work Done Transformations at Constant Temperature, Heat.
Volume and Pressure Complete Cycle (Illustrated by Stirling
Hot Air Engine) Carnot Cycle Constant Volume Cycle.
CHAPTER III.
ENERGY CHART FOR H 2 O (MIXTURE OF STEAM AND WATER) . . 20
Water Line and Saturation Line Constant Volume Lines
Constant Pressure Lines Dryness Fraction Lines Chart for
Superheated Steam Determination of the Internal Energy at
any Point Representation on the Energy Chart Scale of
Internal Energy Lines of Equal Internal Energy Lines of
Equal Total Heat.
CHAPTER IV.
ELEMENTARY THERMODYNAMICS OF H 3 O EXPRESSED ON THE
ENERGY CHART . . ... ... . . . . .,.28
Comparison of <f> and p. v. Diagrams Representation of External
Work, and Heat Changes Transformations in an Ideal Steam
Cylinder Transformation at Constant Pressure Adiabatic
Expansion Transformation at Constant Volume Transfor-
mation along any Line Measurement of External Work Done
Measurement of Heat Supply.
. CONTENTS.
CHAPTER V.
PAGE.
APPLICATION TO A RECIPROCATING STEAM ENGINE. PRINCIPLES 41
Comparison of Ideal and Actual Steam Engine Cylinders Effect
of Feed-Water Temperature Effect of Initial Condensation
Effect of Clearance Quality Line Proportional Water Line
Compression Pressure less than Boiler Pressure Expansion
Line with Initial Condensation Equivalent Weight of Water
and Theoretical Re-evaporation Line Expansion Line with
Leaky Admission Valve Expansion Line with Leaky Exhaust
Valve Expansion Line with Leaky Admission and Exhaust
Valves Expansion Line with Jacket and Leaky Admission
Valve Exhaust Line : Incomplete Expansion Summary
To Draw the Quality Line Corresponding to any Given Trans-
formation Line To Draw the Transformation Line for Less
than lib. of Steam, given the Quality Line To find the Tem-
perature and Pressure at the end of a given Number of Expan-
sions under Adiabatic Conditions Determination of Point
of Cut-off Economy of the Rankine Engine Economy of an
Actual Engine Steam Consumption of an Engine Mean
Pressure of Rankine Engine Mean Pressure of Actual Engine
Equivalent Feed.
CHAPTER VI.
<j> DIAGRAMS OF STEAM ENGINESS DERIVED FROM THEIR INDICA-
TOR DIAGRAMS . . . . . . . . . . . . . . 69
Location of Initial Point Corresponding <p Engine and Volume
Factor Plotting of the 6 <f> Diagram (Numerical Example)
Energy Retained in Cylinder (" Play " Energy) Heat Energy
per 6 <f> Diagram <f> Cylinder Feed Comparison with
Rankine Cycle Location of Losses- Loss due to Throt-
tling Loss due to Initial Condensation, Leakage and Radia-
tion Loss due to Incomplete Expansion Loss due to Throt-
tling through Exhaust Ports Loss due to Compression and
Clearance Leakage of Admission Valve past Cylinder, direct
into the Exhaust Losses in a Locomotive Cylinder due to
Throttling at Admission and at Exhaust Combining the
Forward and Back-end Indicator Diagrams of a Cylinder.
CHAPTER VII.
<j> DIAGRAM OF A SIMPLE JACKETTED ENGINE . . . . 84
Plotting the 6 <f> Diagram (Numerical Example) Heat per <f>
Diagram Comparison with Non- Jacket ted Engine and with
Rankine Cycle.
CHAPTER VIII.
$ DIAGRAMS OF COMPOUND ENGINES . . . . . . . . 87
Rankine Cycle for Compound Engine (Numerical Example)
Effect of Leak from H.P. Cylinder Effect of Clearance
Initial Condensation in L.P. Cylinder Expansion in L.P,
Cylinder Exhaust Line of L.P. Cylinder Other Assumptions.
CONTENTS.
CHAPTER IX.
PAGE.
COMPOUND ENGINES. TRANSFER OF INDICATOR DIAGRAMS TO THE
ENERGY CHART . . . . . . . . . . 97
Volume Factor for Each Cylinder.
Example I. Compound Condensing Engine (Non- Jacket ted).
Example II. Compound Condensing Engine.
Example III. Jacket ted Compound Condensing Engine with
Reheater between the Cylinders.
Example IV. Compound Condensing Engine, Large Cylinder
Ratio, with Reheater.
Example V. Triple-Condensing Engine.
Example VI. Triple-Condensing Jacketted Engine, with and
without Reheater and Jackets.
CHAPTER X.
USE OF THE ENERGY CHART IN DESIGNING STEAM ENGINES . . 1 1 1
Pre-Determination of 6 <f> Diagram Example, Simple Engine
Sketching in <j> Diagram Adjustment of Mean Pressure
Effect of Clearance Results Obtained Comparisons with
Standard Effect of changing the assumptions made
Change in Admission Line Effect of Adding a Jacket In-
crease in Clearance Leakage past Admission Valve direct
into Exhaust.
CHAPTER XI.
DESIGN OF COMPOUND STEAM ENGINES . . . . . . . . 121
Standard of Comparison (Numerical Example) Sketching
Diagrams of proposed Engine Adjustment of Work Done in
the Cylinders Determination of Ratio of Cylinders Altera-
tion of Cut-Off in H.P. (to reduce Mean Pressures) Equaliza-
tion of Work in the Cylinders Alteration of Cut-Off to Increase
Mean Pressure Cylinder ratios for a quadruple expansion
Marine Engine.
CHAPTER XII.
SUPERHEATED STEAM . . . . . . . . . . . . . . 1 33
Constant Pressure and Constant Volume Lines, with old and new
valves of Specific Heat Internal Energy Superheating at
Constant Pressure Rankine Cycle for Superheated Steam
Comparison with Saturated Steam Engine Using same range
of Pressures Diagrams of Actual Engines working with
Superheated Steam Example I. : Compound Condensing
Engine with Reheater (moderate ecomony) Example II. :
Compound Condensing Engine with Reheater (very good
economy).
VU1. CONTENTS.
CHAPTER XIII.
PAGE.
EXPANSION OF STEAM WITHOUT DOING EXTERNAL WORK. . . . 144
Two Numerical Examples Expansion after formation at Con-
stant Pressure ; Superheating by Throttling Velocity of
issuing Steam Recovery of Motion Energy.
CHAPTER XIV.
APPLICATION OF THE ENERGY CHART TO OTHER SUBSTANCES . . 1 50
Chart for Sulphur Dioxide (SO 2 ) Principles of Binary Engines
Example of a Combined Steam and SO a Engine.
APPENDIX I.
PLOTTING # DIAGRAMS BY MEANS OF THE SLIDE- RULE. . . 156
APPENDIX II.
USEFUL FORMULAE IN CONNECTION WITH THE ENERGY CHART
FOR H 2 O, ILLUSTRATED BY MEANS OF NUMERICAL EXAMPLES 157
BIBLIOGRAPHY 165
SPECIAL TERMS.
REFERENCES TO THOSE USED IN THIS BOOK . . . . . . 166
THE ENERGY CHART
PRACTICAL APPLICATION TO RECIPROCATING
STEAM ENGINES.
CHAPTER I.
INTRODUCTORY.
OF late years it has become more and more the custom to use a
graphic method in making thermodynamic investigations in connec-
tion with heat engines, and this method has many advantages over
the purely algebraical.
In dealing with the thermodynamic relations of a substance it
is necessary, in order to define the state of the substance, to consider
the temperature and pressure as well as the volume occupied for a
given weight ; and in the case of a transformation of the substance
from one state to another the quantity of heat to be added or sub-
tracted, and also the work that has to be done either by or on the
substance, have to be determined.
The graphic representation should therefore exhibit in a conven-
ient manner all the above particulars.
Several methods were described by Willard Gibbs in 1872, but
for practical use there are two of these methods which offer special
advantages. The first is that in conformity with which the indicator
diagrams of an engine are drawn, and in which the pressures are taken
as ordinates and the volumes as abscissae ; the temperature of the
substance could be exhibited by a series of curves, and heat and work
by areas ; this is the pressure- volume or " p v" method. In the
second method the temperature of the substance is taken as the
ordinate, and for the abscissa a certain function of the heat supply
known as " entropy " is plotted.
I
2 THE ENERGY CHART.
It was Clausius who gave the name of "entropy " to this function,
and it is usually denoted by </>. Rankine, however, called it the
" thermodynamic function." Maxwell denoted absolute temperature
by 0, and J. McFarlane Gray, who was the first* to introduce the
method into this country gave the name of <f> chart to a diagram
prepared on these principles. This method is therefore known as the
temperature-entropy or < method. Pressures and volumes can be
exhibited by curves, heat and work by areas, as will presently be
shown. A chart so drawn is an energy chart, as explained at page 15.
There has been a great deal of discussion as to the precise
meaning of the term " entropy/' and there appears to be a con-
siderable amount of misconception. Strictly, " change of entropy "
is the conception to be considered, and it can only be applied
in relation to the state of the substance in one condition to
its state in another condition, when the change of state is so
produced that it could be reversed step by step or, as it is generally
expressed when the change is reversible. In practical heat engines
(either gas or steam) the majority of changes produced in the working
substance during the stroke are not reversible, and the term "entropy"
is therefore not strictly applicable to practical cases. Hence in the
practical use of the method, the abscissae of the diagram only represent
the changes of entropy of the working substance for certain portions
of the stroke, as will be seen in the sequel. It is only by limiting the
entropy scale on the chart to certain portions of the cycle of
operations, that practical use can be made of the method.
The Author has now for several years past studied the 9 <t>
method with a view of applying it practically in the drawing office,
not only for designing engines, but also as a means of analysing the
indicator diagrams of engines in order to discover the thermo-
dynamical defects that may exist. He is satisfied that the method
offers advantages in these directions far exceeding those of the p v
method when used alone, and the object of this book is to develop
these practical applications.
Elementary Principles. To commence with a simple case, as is
well-known i Ib. of water at 60 F. requires the addition of one British
* "On the Rationalization of Regnault's Steam Experiments" by J.
McFarlane Gray. Paris Meeting of the Institution Mechanical Engineers. 1 880.
ELEMENTARY PRINCIPLES. 3
Thermal Unit to raise its temperature i F., in other words the
specific heat of water at 60 F. is one B.Th.U. or 778 foot-lbs. (778 is
now very generally accepted as Joule's equivalent instead of 772).
To represent this fact graphically, let an ordinate of temperature
p P (Fig. i) be measured from absolute zero to represent 461 +
3TF-
lAft&JLfwo.
rfF
py
B
ABS. ZERO
FlG. 2.
FIG. 3.
FIG. 4.
60 F., and let the vertical strip P'p represent one B.Th.U. by its
area, then, since the difference in the ordinates at p and p' is
one degree,
(461 + ^-f
= i B.TLU.
= 778 foot-lbs.
4 THE ENERGY CHART.
If now one more heat unit is added to the Ib. of water, the graphic
representation would be the strip p' P" (Fig. 2), and since the mean
height of this strip is greater than that of the former, obviously
P P ** P P
from which it follows that the inclination of PP' to the axis of x is less
than the inclination of P'P". Continuing this process it is clear that
the successive additions of one B.Th.U. to the Ib. of water can be
represented graphically as in Fig. 3.
If instead of the temperature rising one degree at a time the incre-
ments were infinitely small, the polygon P P' P" . .P IV - would become
a curve, and it is this curve which has to be found. The assumption
will be made that the specific heat of water is constant, which it is not
quite, but for practical purposes it can be considered as such.*
Let A B (Fig. 4) be this curve, let 6, the absolute temperature at
the point P be taken as the ordinate, and let the " entropy " <f> be the
abscissa of this point measured from some arbitrarily chosen origin.
The increment of heat to be added to raise the temperature of the
i Ib. of water d is d 6 because the specific heat of water is taken as
unity, and this increment of heat is represented graphically by the
area of the strip below P P 1 , shaded horizontally. This area is equal
to B d <f>.
Hence d 6 = d <
7 f\
or d <t> = -a- whence < = log + const.
u
In the case of the so-called perfect gases, the specific heat when
the volume is unchanged is practically a constant, at any rate at
moderate temperatures, and so likewise is the specific heat for change
of condition under a constant pressure. The question of the specific
heat of gases at high temperatures is at the present moment sub
judice. The experiments of Mallard and Le Chatelier seem to prove
that the specific heat increases considerably with temperature ; but
Dugald Clerk asserts that these experiments have been wrongly
interpreted. At present the specific heat of atmospheric air at
constant volume is taken as 0.168 ; water being unity.
* See " Variable and absolute specific heats of water," by J. M. IVtcFarlane
Gray. Proceedings Institution Civil Engineers. Vol. CXLVII., page 347.
CONSTANT VOLUME AND PRESSURE LINES. 5
Constant Volume Line for a Gas. By the reasoning adopted in the
case of water, the graphic representation on an energy chart for a gas
receiving heat at constant volume is C v d 6 d <, where C v is the
specific heat at constant volume. Hence < = C v log + const.
is the equation to a constant volume line for a gas. To determine the
integration constant it is to be noted that the origin has been chosen
arbitrarily. It can therefore be assumed that < is zero when the
temperature of the substance is 9^ In other words is the intercept
on the axis of Y of the particular constant volume line under con-
sideration ; that is the length A a in Fig. 4.
-400'F
Hence <f> = C v log ^
As an example the constant volume line for
7 cubic feet of atmospheric air has been
drawn, as shown in Fig. 5, arranging the
origin in such a position that ;
0^ = 215 + 461.
To draw this curve, points in the curve
* = 0.168 to g< 2I5 + 46l
have to be computed and plotted. A
constant volume line for any other volume
can be similarly drawn, and a chart can
be prepared giving a number of such
volume lines, as in Fig. 6.
Constant Pressure Line for a Gas.
If the change in the condition of
a gas is effected by the addition or
subtraction of heat at constant pressure, the change can be
followed on an energy chart by means of lines at constant
pressure. Such a line will have the same general form as a
constant volume line. Thus for atmospheric air the specific
heat at constant pressure is 0.238, so that the equation to the
constant pressure line for one atmosphere is
lOCf
-04 O -04
SCALE OF ENTROPY
FIG. 5.
= 0.238 log ~
THE ENERGY CHART.
-400
300
F
200"
100*
z
CUBIC FEET
4.
z
1
7
/
2
10 --05
FIG. 6.
A series of constant pressure lines can be drawn, as has been done for
atmospheric air in Fig. 7.
3OO
200*
IOO
ZOO
/
IQO L.BS PER I
50
IO
10 05 o -o-*
FIG. 7.
Obviously the constant volume and constant pressure lines can be
combined together on one chart. Any point on such a chart expresses
a definite state of the substance and is called a slate-point. A contin-
uous succession of state-points is a transformation line.
CHAPTER II.
THERMODYNAMICS OF A PERFECT GAS EXHIBITED ON AN
ENERGY CHART.
Representation of Internal Energy. If the change of state in a gas
due to the addition of heat is effected at constant volume, no external
work is clone by the gas. In other words, the whole of the heat added
is expended in increasing the internal energy of the gas. In Fig. 8
a constant volume line has been
drawn through the state point P,
and a transformation at con-
stant volume is effected until the
state point Q is reached. The
heat it is necessary to supply is
represented by the area shaded
by vertical lines, and this area
also represents the amount by
which the internal energy at Q
is greater than it is at P. During
this transformation the gas is
supposed to be contained in a
closed vessel of unchangeable
volume.
Returning to the state-point P, it will be supposed that heat is
abstracted from the closed vessel until finally the absolute zero of
temperature is reached. During this process the state-point will
travel along the constant volume line, and obviously therefore, the
area included between the constant volume line through P, the
vertical through P, and the horizontal line representing the absolute
zero of temperature, an area which is shaded with diagonal lines in
Fig. 8, represents the total internal energy of the gas under con-
sideration when in the condition represented by the state-point P.
To repeat ; the diagonally-shaded area represents the total internal
ADS. ZERO
THE ENERGY CHART.
energy of the gas at the point P, and the vertically-shaded area
represents the increase of internal energy from P to Q.
In Fig. 9 let the original state of the gas be represented by the
point P, from which it will be seen that the volume is V, the pressure
/>, the temperature 0, and the internal energy of the gas is equal to the
area comprised between the constant volume line V, the vertical
drawn through P, and the axis of x, which is at absolute zero. Let
the gas be now expanded and heat be added to it at the same time at
such a rate that the temperature remains constant. This process will
be graphically represented by the transformation or change of state
FIG. 10.
taking place along the horizontal line P Q, and the amount of heat
required to reach the state Q is represented by the rectangle PQqp.
The internal energy of the gas at Q is represented by the area con-
tained by the constant volume line V \ 9 the vertical through Q, and
the axis of x. In the imaginary case of a gas remaining perfect at all
temperatures, even at the absolute zero, the specific heat for change
at constant volume would be constant, and then all the constant
volume lines are the same curves simply displaced horizontally.
It follows, therefore, that the internal energy at P is equal to what it is
EXTERNAL WORK. 9
at Q, and further that the heat supply and the work done are equal.
If, however, the gas is not perfect, the specific heat will vary,
and the internal energy at P will not be equal to what it is at Q.
Representation of External Work Done. On referring to Fig. 6, it
will be seen that the volume at Q is greater than it is at P, and the exter-
nal work done by the gas in expanding from P to Q is represented by the
vertically-shaded area included between the transformation line at
constant temperature, and the two constant volume lines V and F t
continued to infinity to meet the horizontal line representing absolute
zero.
.. _p
zi :iU'
dC
FIG. ii.
P
FIG. 12.
Fig. 10 is the p v diagram corresponding to Fig. 9. P Q is here
represented by a curve P Q marked " isothermal." The verticals
P p and Q q are represented by the adiabatic curves through P and
Q t and the constant volume lines V and V 19 by the verticals P p
and Q q. The correspondingly shaded areas in the two figures
are equal, if the scale of foot-lbs. per square inch employed are the
same in both. It is interesting to note that in the <f> diagram
See page 4.
IO
THE ENERGY CHART.
the heat area is bounded by finite lines and the work area by
infinite lines. The reverse is the case in the p v diagram.
If the transformation had taken place from Q to P,
Fig. 9, it would be necessary to abstract heat represented
by the rectangle P Q p q, and at the same time to do work
in compressing the gas ; this work is represented by the area in-
cluded between the constant volume lines, V and I / 1 , and P Q.
Figs, ii and 12 show the case of adiabatic expansion, that is to
say, no heat is either added or abstracted. The external work done by
FIG. 14.
the gas is given by the area included between the constant volume lines
and is shaded by vertical lines both in Fig. 11 and in Fig. 12, and
obviously the internal energy at Q is less than it is at P by the
amount of the work done in expanding.
Figs. 13 and 14 show a transformation at constant volume, that
is to say, the point Q is situated on the same volume line as P. No
external work is done, although the pressure of the gas increases (see
Fig. 7) from p to p 9 but internal work is done equal to the heat
added, and is given by the horizontally-shaded area.
VARIOUS TRANSFORMATIONS.
II
I
p q
FIG. 15. FlG - l6 -
Figs. 15 and 16 show a transformation at constant pressure from
P to Q. It will be seen that the gas is expanded and therefore does
work, and that heat has to be added.
The following transformations have
up to the present been shown :
(i) At constant temperature, Figs. 9
and 10.
At constant heat, Figs, n and 12.
At constant volume, Figs. 13 and 14.
At constant pressure, Figs. 15 and
16.
If through any state-point P (Fig. 17)
the constant volume line and the adiabatic
be drawn, the heat chart will be divided
into four unequal portions or zones, which
are marked L, II., III., and IV. If the
transformation line lies wholly in L, then,
as shown in Fig. 18, work has to be done
on the gas to compress it, and at the same FIG. 17.
(2)
(3)
(4)
IV
APS. ZERO
12
THE ENERGY CHART.
time heat has to be supplied, and the addition of this work and heat
to the internal energy of the gas at P gives the internal energy of the
gas at Q. The amount of heat supplied is shown by the horizontally-
shaded area, and the external work done on or by the gas by the vertically -
shaded area. This shading of areas has been adopted throughout the
book, and applies whether the heat is added or abstracted, or whether
the work is done on or by the gas. Fig. 19 is the p v diagram corres-
ponding to Fig. 18.
FIG. 19.
If P Q lies in II. (Figs. 20 and 21), heat has to be supplied and
work is done by the gas. A portion of the heat supplied is, so to
speak, directly converted into work, as shown by the double shading,
and the internal energy at Q is equal to the internal energy at P,
less the remainder of the work done by the gas, plus the remainder of
the heat.
When P Q lies in III, (Figs. 22 and 23), work is done by the gas
and heat has to be removed. The internal energy of the gas at Q
is equal to that at P, less the work done and less the heat removed.
VARIOUS TRANSFORMATIONS.
AB3. ZERO
p q
FIG. 20.
FIG. 21.
/
V;
A
&
AB^.ZERO
m
q p
FIG. 22.
THE ENERGY CHART.
Should Q lie in IV. (Figs. 24 and 25), work has to be done on the
gas and heat has to be abstracted. In this case a certain portion
of the heat abstracted is, so to speak, directly converted into the
work of compressing the gas (compare Fig. 20), and the internal
energy at Q is equal to that at P plus the remainder of the work done
on the gas, less the remainder of the heat abstracted.
f
'
k BS. ZERO
FIG. 24.
FIG. 25.
It is obvious from the above that, whatever change takes place
in a gas, the amount of work done on or by the gas, and the amount
of heat required to be supplied or abstracted in order to produce
the change, can be exhibited graphically on the chart. The
condition of the gas after the application of a given amount of work
or heat is, however, indeterminate, because it depends on the
path followed by the transformation, as the following will explain :
If two states of the gas are given at P and Q, the state Q can
be reached from P by various paths. The state of the gas and its
internal energy at Q are however independent of the paths, in
accordance with the second law of thermodynamics, but the path
VARIOUS TRANSFORMATIONS.
followed depends upon the amount of the heat and of the external
work, and on the manner of their application. Thus in Fig. 26, if
the path P n Q is followed, the heat to be supplied is greater than
for the path P m Q, but the external work is less, by the exact
amount that the heat is greater.
It will be seen from the above that the heat added or abstracted
is always represented by the area included by the transformation line
and the adiabatics drawn through the initial and final state points;
and the work done by or on the gas is represented by the area included
FIG. 26.
p q
FIG. 27.
between the transformation line and the constant volume lines drawn
through the initial and final state points. This rule is shown in Fig.
27 by the areas shaded by horizontal lines for the heat supply and
by vertical lines for the work done.
This conclusion is in reality true for any substance, therefore
a 6 <f> chart provided with constant volume lines not only exhibits
graphically the heat changes that are necessary to effect the trans-
formation of a substance from one state to another, but also the
work that has to be done by the substance or on the substance.
Such a <f> chart is therefore an energy chart not merely a heat
chart.
i6
THE ENERGY CHART.
Complete Cycle. If the gas is subject to a series of transforma-
tions and finally returns to the initial state, it is said to have com-
pleted a closed cycle and the process can be exhibited graphically
by combining the four figures 18, 20, 22, 24, as follows :
A B C D (Fig. 29), is the indicator diagram of a hot-air
engine with a closed cycle. Fig. 28 shows the corresponding </>
diagram. During the transformation D A, the air receives heat at
constant volume, and there is a further addition of heat, at a slightly
diminishing temperature, during A B^B. At the point B begins
the abstraction of heat at constant volume along the transformation
line B C, and during C D D heat is further abstracted at a slightly
FIG. 28.
FIG. 29.
rising temperature. It will be seen that :
A BI B is an example of Fig. 20
B C 22
CD^D 24
DA 18
The algebraical sum of the work done by and on the gas is the
area A B^BC D^D, and the difference between the heat added and
the heat abstracted during the cycle is equal to the same area.
The thermal efficiency is the ratio of the heat utilised as work to
the total heat supply, which is obviously equal to
Area A B 1 B C D D
Heat supply.
The apparent heat supply is shown in Fig. 28 by the area edged
with horizontal lines, but these diagrams are those of a Stirling
STIRLING HOT AIR ENGINE. 17
hot-air engine, an engine in which the regenerative principle is
applied that is to say, the heat abstracted during the transform-
ation B C is stored in the regenerator and is part of the heat added
at the next stroke during the transformation D A. By the use of the
regenerator, therefore, the actual heat supply is reduced to the heat
represented by the area d D A B B C c, and the actual heat re-
jected to the heat abstracted during the transformation C D D.
The thermal efficiency, when a regenerator is used, is therefore
much increased, and is :
Area A BCD
Area d D A B C c
IOO-
FIG. 30. FIG. 31.
Ideally the transformations A B and C D should be carried out
at constant temperature, as shown in Figs. 30 and 31. It will be seen
that the area A B C D is equal to the rectangle A C 1} therefore the
thermal efficiency of the ideal Stirling cycle with regenerator is :
Rectangle A C t
Rectangle A b
a
which is obviously equal to-^ - or the same as that of the Carnot
cycle (see below). *
As a general rule the cycles that have practically to be con-
sidered consist of a transformation during which heat is supplied
(admission period), followed by a transformation at increasing
volume and diminishing pressure (expansion period), after which
heat is abstracted or rejected (exhaust period), and finally the
initial state is reached by a transformation at diminishing volume
and increasing pressure (compression period).
2
i8
THE ENERGY CHART.
Carnot Cycle. A cycle consisting of two isothermal and two
adiabatic transformations is known as the Carnot cycle, and it can be
shown to be the most efficient cycle that can be arranged for a
given highest temperature and a given lowest temperature. The
general proof is more readily effected analytically, but any special
case can be proved with great simplicity by means of the chart.
Thus, for instance, A B C D (Fig. 32) is a Carnot cycle, inasmuch
as A B and C D are transformations at constant temperature
(isothermal), and B C and D A are transformations at constant heat
(adiabatic). The cycle A J5 1 C 1 D differs only in that the trans-
B
ABS.ZERO
\B'
-C'
trfer
FIG. 32.
FIG. 33.
formation A B} r is not at constant temperature, and matters have
been so arranged that the heat supply is equal to that of the Carnot
cycle, i.e., the area A J3 1 ft 1 a is equal to the area A B b a. In the
Carnot cycle the heat rejected is the rectangle D b, and in the other
cycle the heat rejected is the rectangle D 6 1 , which is greater by the
rectangle C 6 1 . Since the heat supplied is the same in both cases,
it is obvious that the second cycle is not so efficient as the Carnot.
It will be observed that the highest and lowest temperatures are the
same in both cycles, namely X and 2 . As before stated, the thermal
CONSTANT VOLUME CYCLE. IQ
efficiency of a cycle is obtained by dividing the amount of heat
utilised as work by the heat supplied. In the case of the Carnot
cycle, the heat utilised is represented by the rectangle A C, and the
heat supplied by the rectangle A b. The areas of these rectangles
are proportional to t 2 and respectively. Hence the thermal
f\ f\
efficiency of the Carnot cycle = -~n ?
^i
Constant Volume Cycle. A cycle may be limited by other
considerations than that of temperature. For instance, the trans-
formation during the supply of heat may be at constant volume,
as is the case for instance in an ideal gas engine, because the piston
theoretically does not move until the explosion is complete. Or
the supply of heat may be partially at constant volume and partially
at constant pressure, as in an actual gas engine, because the piston
begins to move before the explosion is completed. The rejection
of heat may also be at constant volume. Fig. 33 shows a cycle in
which the heat supply is at constant volume and the heat rejection
is also at constant volume, and such a cycle is obviously far less
efficient than the Carnot cycle for the same total range of temperature
as is shown by the dotted lines. The development of the application
of the <j> method to these constant volume and constant pressure
cycles belongs to the domain of the internal combustion engine,
and cannot therefore be proceeded with in this book.
CHAPTER III.
ENERGY CHART FOR H 2 O.
(MIXTURE OF STEAM AND WATER)
Water Line and Saturation Line. Returning to page 4, it
was seen that the </> line for water is a logarithmic curve. Take
any point A (Fig. 34) on this curve at temperature 0, and let the
pressure remain constant whilst heat is still being added to the
water. As is well known, steam will be formed, and when sufficient
heat has been added (namely, the latent heat) the whole of the
water will have been converted into steam. Further, so long as any
water is present, both the pressure and the temperature remain
constant. This physical fact is expressed on the chart by making
the point representing the state of the substance (H 2 O) move along
the horizontal straight line drawn through the point A. The heat
added from the moment steam commences to be formed to the
moment all the water has been converted into steam is represented by
a rectangle whose height is 0, and whose width is such that the area,
A B x 0, is equal to the latent heat of steam at the temperature 0.
It will be noted that the point B represents on the chart the
condition of i Ib. of dry saturated steam at a temperature 0. Evi-
dently for each point A on the water line a point can be found corres-
ponding to B. All these points lie on the " Saturation line."
Constant Volume Lines. The next step is to draw the constant
volume lines. When half the latent heat has been applied, one
half of the Ib. of water will have been converted into steam, and the
state point will have obviously travelled half way from A to B, to the
point M (Fig. 34). The volume of the steam is, of course, one half
what it is at B, therefore the volume of the mixture ( Ib. of water
and i Ib. of steam) at M (neglecting the water volume) is very
approximately one half the volume at B. Similarly the volume at
ENERGY CHART FOR H 2 O.
21
the point N, where A N equals ~ A B is approximately -th of the
volume at B.
As stated, the water volume has been neglected, and a slight
correction is required in this respect ; to make this clear, let Fig. 35
represent a cylinder containing i Ib. of water at temperature and
provided with a tight-fitting piston, weighted to correspond with
the saturated steam pressure at the temperature 0. The volume
M
e
h
Z
kl
J
,11:
FIG. 34.
O'OIO CUB. FOOT
FIG. 35.
FIG. 36.
below the piston is 0.016 cubic foot very approximately at any
temperature that need practically be considered. Now let half the
latent heat be added, the volume of the steam will be half the steam
volume at B, and to this must be added the volume of i Ib. of water,
as represented in Fig. 36. The water present is represented by the
thick black line in Figs. 35 and 36. Similarly at the point N (Fig.
34) the volume of the mixture is equal to - th volume of steam at
B plus (n -th) volume of i Ib. of water.
22
THE ENERGY CHART.
As a practical matter, the easiest way to find the constant
volume lines is to first find the points on the saturation curve where
the volume is I, 2, 3, 4, etc., cubic feet, which can readily be done
by means of a steam table such, for instance, as that published in
Professor Ewing's book on the Steam Engine.*
The next step is to divide the horizontal intercepts, A B (Fig. 34),
drawn through these volume points, into a number of equal spaces,
so as to find the positions of volumes i, 2, 3, 4, etc., on each of them.
The correction for water volume can then be applied if desired, but
except at high temperatures it
is practically negligible. Points
of equal volume can then be
joined together by a fair curve.
This process is illustrated in Fig.
37, where the volume line for
two cubic feet is shown.
Constant Pressure Lines.
As regards the constant pressure
lines, they are evidently horizon-
tal during the period of steam
formation, and can be plotted
by means of a table giving
pressures and temperatures.
Thus finally the chart for
i Ib. of H 2 O, as given in Plate
i, is obtained.
Dryness Fraction Lines.
This energy chart also gives lines
of constant "dryness fraction." They are similar curves to the satura-
tion line, and the method of plotting them is obvious.
In order to obtain diagrams of suitable size for this book, two
other energy charts for H 2 O have been used. One of them (Fig. 38),
is used in those cases where the absolute zero is shown, the temperature
scale is 300 F. per inch, and the heat scale is 300 B.Th.U. per square
inch. (For example see Fig. 91). In the second chart (Fig. 39)
__ _ I2O
FIG. 37
* The Steam Engine and other Heat Engines, by J. A. Ewing, M.A.. B.Sc..
F.R.S., M.Inst.C.E.
ENERGY CHART FOR H 2 0.
23
4.00 F;
(see, for instance, Fig. 85), the temperature scale is 75 F. per inch,
and the heat scale is 60 B.Th.U. per square inch.
Chart for Superheated
Steam. The energy chart
for superheated steam is
also given in Plate i. The
constant pressure and
volume lines are logarith-
mic curves, similar to those
given in Figs. 6 and 7
for air.
IOO -
FIG. 38.
They were originally drawn several years ago, when there
was not sufficient evi-
dence to depart from
the then accepted value
of the specific heat
of superheated steam,
namely, 0.37 at con-
stant volume, and 0.48
at constant pressure.
These lines were there-
fore drawn according
to the equations :
= 0.48
and
A
100
<}> = 0.37
It will be noticed
that only a few con-
stant volume lines are
given, but that a pres-
sure and a volume
scale are marked. By
means of these scales
he pressure or volume
at any point of the
superheated field can
be read off the chart. FlG - 39-
These scales are based on a property of these logarithmic curves, that
2*0
24 THE ENERGY CHART.
the horizontal intercept between any pair of them (either two
constant pressure or two constant volume lines), is constant. The
scales themselves are logarithmically divided.
A new chart for the superheated field, based on the latest de-
terminations, has, however, now been drawn, and is given in Fig.
135-
FIG. 40.
FIG. 41.
Determination of the Internal Energy at any Point. In Fig.
40, let P be any point on the chart. Through this point
draw the constant volume line. If this constant volume line could be
continued down to absolute zero, then the area bounded by this line,
by the vertical through the point P, and the horizontal line repres-
enting absolute zero would represent the internal energy of I Ib. of
H a O in the condition defined by the point P. As, however, this
curve cannot be drawn, another manner of procedure must be
adopted.
Representation on the Energy Chart. Let E be the unknown inter-
nal energy in I Ib. of water at 32 F., this will be the internal energy at
the point A. Let steam be made at the pressure corresponding to
32 F., namely at 0.089 Ibs. per square inch. When the volume of
INTERNAL ENERGY. 25
the point P has been reached at a the heat supplied will be repres-
ented by the rectangular area below A a. The heat added will have
been expended both in doing work against the pressure at A, and in
adding to the internal energy of the substance. The work done is
equal to the volume at P multiplied by the pressure per square foot
at A, and the heat units added are equal to A a multiplied by the
absolute temperature at A, namely 493 F. But A a is equal to :
_ Volume at P. ___ x the entropy of saturated steam
Volume of saturated steam at 32 F.
at 32 F.
and the volume of saturated steam at 32 F. can be taken as 3400
cubic feet per lb., and the entropy of saturated steam at 32 F.,
as 2.2. Hence the heat units added are :
volume at P 01.
493 x - lI bT x 2 ' 2 = -3 18 x volume atP "
The pressure per square foot at A is 12.8 Ibs., so that the work
done expressed in B.Th.U. is :
~g x volume at P = 0.016 x volume at P.
Hence the addition made to the internal energy by shifting the state
point from A to a is :
(0.318 0.016) volume at P = 0.302 x volume at P.
The internal energy at the point P is F B.Th.U. greater than
the internal energy at A, where F is the number of heat units repre-
sented by the shaded area in Fig. 40. Thus finally the internal
energy at P is :
E + 0.302 x vol. at P + F.
where E is the internal energy of water at A, i.e., at 32 F.
To take a numerical example, let P be taken in the position
shown in Fig. 40. Reading from the chart, it will be seen that the
volume at P is 4.5 cubic feet, and by measuring the shaded area, F
is found to be equal to 622 B.Th.U. Thus the internal energy at
this point is :
E + 0.302 x 4.5 + 622 == E + 636.4 B.Th.U.
Generally the internal energy is stated as from 32 F., therefore
the term E is omitted.
Scale of Internal Energy. Practically it is inconvenient to
measure the area F when it extends so far down as 32 F. But if
26
THE ENERGY CHART.
the internal energy atPis known, that for anyother point Q on the same
volume line can readily be found by determining the area F' (Fig. 41).
To carry out this idea, the internal energy at various points along the
constant temperature line of 200 F., has been calculated and is
given on the chart (Fig. 43), and it will be observed that a
scale of equal divisions is thus formed. The use of this scale of
internal energy can best be illustrated by means of a numerical
example.
Required to find
the internal energy at
the point P (Fig. 42) :
The volume at the
point P is 10 cubic
feet. The internal en-
ergy for this volume
at 200 F. is seen to
be 436 B.Th.U. To
this must be added
the B.Th.U.'s repre-
sented by the area
F . This area is com-
posed of a rectangle
and an approximate
triangle. The approx-
imate triangle is equal
to the triangle a b c,
which is equal to the
FIG. 42. rectangle aba. It
will thus be seen that the area F is equal to the rectangle a p,
and the B.Th.U. represented by this area are equal to the
absolute temperature at a, namely, 228 + 461 = 689 F. multiplied
by a b, measured on the entropy scale. This works out to :
689 x 0.488 = 336 B.Th.U.
Hence the internal energy at P (reckoned from 32 F.), is
436 + 336 = 772 B.Th.U.
It will be observed that this internal energy is that of i Ib. of H 2 O,
in the condition represented by the point P (Fig. 42), where the
INTERNAL ENERGY.
JOO
dryness fraction is 0.743, and is made up of the internal energy of
0.743 Ibs. of steam at 250.5 F., and of 0.257 Ibs. of water at the
same temperature.
Another Method. This indicates another way of determining the
internal energy at any point. On the chart (Plate i), the internal
energy for i Ib. of water at varying temperature, and likewise the
internal energy for i Ib. of saturated steam is given by means of the
internal energy scales. From these scales it will be seen that the
internal energy of i
Ib. of water at the
temperature of the
point P (250.5) is
220 B.Th.U., and of
i Ib. of steam at the
same temperature,
963.5 B.Th.U.
Hence the internal
energy at P is :
963.5 x 0.743 +
220 X 0.257 =
771.5 B.Th.U. as
before.
Lines of Equal
Internal Energy.
From the preceding
there is obviously
no difficulty in ob-
taining a series of
points on the chart
100-
200*
300 CUB: FT
FIG. 43-
at which the internal energy will have the same value, and if such
points be joined, curves of equal internal energy will be obtained.
A few of these lines are shown on Fig. 43.
Lines of equal total heat of formation at constant pressure can
obviously be drawn in a similar way. Such lines are desirable when
using the chart to calculate the blading of steam turbines, but are
not helpful for reciprocating steam engines, and are therefore not
given here.
CHAPTER IV.
ELEMENTARY THERMODYNAMICS OF H 2 O EXPRESSED ON THE
ENERGY CHART.
THE energy chart for H 2 O can be used precisely in the same way
as the chart for a gas, and the heat added or abstracted, and
the work done by the H 2 O, or done on it can be represented graphic-
ally in the same manner. One example will suffice. Suppose the
original state point is P (Fig. 44), and that the path P Q is followed
to reach the state point Q, then the area shaded vertically contained
between the constant volume lines drawn through P and Q respec-
tively, is the external work done by the expanding H 2 O, and the area
shaded horizontally is the heat required to be supplied. The con-
dition of the H 2 at any point during the transformation can be
read off the chart, thus at X (Fig. 44), there is :
Pressure 28.0 Ibs. per square inch (abs.)
Temperature 246.3 F.
Volume 10.0 cubic feet.
Dryness fraction 0.69
At P the volume is 4 cubic feet, and at Q it is 15 cubic feet, so that
the steam has been expanded as it obviously must have been since
it has done external work.*
Comparison of <f> and p v Diagrams. The expansion line of
the corresponding p v diagram can readily be plotted by reading
off the volume and pressure at a number of points from the chart and
the expansion line P Q (Fig. 45) is thus obtained. The converse is,
however, not possible without further knowledge ; that is to say, if
P Q is given on the p v diagram it cannot be located on the chart,
because the dryness is not known. P Q might, for instance,
be the expansion line of the indicator diagram of a simple non-
* It is recommended that Fig. 44 be plotted on Plate I. and the above values
read off.
ELEMENTARY THERMODYNAMICS OF H 2 O. 2<)
condensing steam engine as shown in Fig. 46. The volumes marked
are obtained by measurement of the engine cylinder. At the
point x, for instance, the volume is 10 cubic feet, and this is the
volume of steam present in the cylinder at the point under consider-
ation. The indicator diagram does not tell, however, how much
water is present in the cylinder at the same point, and thus the
dryness cannot be determined, and consequently the point X cannot
be located on the chart. All that is known is that it must be
l<4 CUB: FT
somewhere on the constant pressure line equal to the pressure at
x, viz., 28 Ibs. per square inch absolute. It will, however, be
shown in the sequel (page 70) how the point X can be fixed on
the chart, when the weight of feed per stroke is known.
It is important to note that the work done by the steam in ex-
panding from the state P to the state Q (Fig. 46), is equal to the area
P Q q p, and that this area is independent of the quantity of
water present. It follows that the work area on the chart
included between the constant volume line through P, the trans-
formation line P Q, and the constant volume line through Q, is
30 THE ENERGY CHART.
constant, no matter what the dryness fraction may be, so long as the
pressures are the same and the number of expansions are the same.
This is shown in Fig. 47 by a comparison of two expansion lines P Q
and P x Q . On the other hand, the heat required to be added does
depend on the dryness fraction, and the greater the dryness fraction
(or, in other words, the greater the proportion of steam), the greater
is the quantity of heat to be added. Fig. 48 will make this clear ;
the various expansion lines have the same number of expansions, as
FIG. 47-
FIG. 48.
can be ascertained by replotting on Plate i, and since the slope of these
lines becomes greater and greater as the dryness increases, it follows
that the vertical strip representing the heat supply also increases, or,
in other words, the heat to be added is greater the greater the dryness.
In fact, when there is very little steam present, as in the case of the
expansion line P'" Q'", heat has to be abstracted. The p v diagram
gives no information on the heat supply, so that the 9 <#> diagram is
more complete in this respect. It will be seen, in fact, that the 6 $
diagram gives all the information obtainable from the p v diagram,
and in addition shows the quantities of heat to be added or ab-
stracted to obtain a given transformation. At the same time, it
VARIOUS TRANSFORMATIONS OF H 2 O. 31
would be a great mistake to abandon the p v diagram in favour of
the cf> diagram ; on the contrary, they should be employed con-
jointly, and it must not be forgotten that practically the start
must be made from the p v diagram, inasmuch as this is the diagram
given by the steam engine indicator.
Transformations in an Ideal Steam Cylinder. Let it be supposed
that there is a cylinder fitted with a steam-tight piston, which can
be weighted so as to produce any desired pressure per square inch
in the cylinder. Let it also be supposed that there are means to
enable heat to be intro- *
duced into the cylinder or
abstracted from it in any
desired manner. Let the
cylinder contain i Ib. of
H 2 O, and let it further be
assumed that when oper-
ations are commenced the
pressure is 100 Ibs. abso-
lute per square inch and
that the volume is 2 cubic
feet. These data locate
the state point on the
chart at the intersec-
tion of the constant pres-
sure line 100 and the con-
stant volume line 2,
and it will be seen by
reference to Plate i that
pr
AB3. ZERO
FIG. 49.
the dryness fraction is 0.45, that is, the cylinder contains 0.45 Ib.
of steam and 0.55 Ib. of water, amounting, of course, to I Ib. of H 2 0.
The internal energy of the Ib. of H 2 is represented by the vertically-
shaded area in Fig. 49. The method of measuring this area and
finding the corresponding heat units has been given in Chapter III.
In this case the internal energy is 660 B.Th.U. A graphic repre-
sentation of all the data concerning the i Ib. of H 2 contained
in the cylinder is thus obtained, namely, the pressure (100 Ibs., per
square inch), the volume (2 cubic feet), the temperature (327.5 F.),
32 THE ENERGY CHART.
the proportion of steam to water (0.45), and the internal energy
(660 B.Th.U.)
Transformation at Constant Pressure. Now let a transforma-
tion be effected in the cylinder by adding 210 B.Th.U. and main-
taining a constant pressure on the piston. In these circumstances
the state point will move along the horizontal constant pressure
line until the point Q (Fig. 50) is reached, the position of Q is deter-
mined by making
PQ x (327.5 + 461) = 210 B.Th.U.
p - =- 266
The distance 0.266 is to be measured on the entropy scale.
The state of the H a O in the vessel after the transformation
has taken place is given by the position of the point Q on the chart,
and is read off as follows :
Pressure .......... 100 Ibs. per square inch abs.
Temperature ....... 327.5 Fahr.
Volume ........... 3 . 05 cubic feet.
Dryness fraction . . 0.69.
Although the volume of the steam has been increased from 2 cubic feet
to 3.05 cubic feet by the transformation, it has not been " expanded "
because the increase in volume is due to evaporation of some of the
water. External work has been done in pushing back the piston,
and this work is represented on the chart by the vertically-shaded
area which is equal to the pressure per square foot, multiplied
by the increase of volume, divided by Joule's equivalent :
100 X X '<* = 19-4 B.Th.U.
On comparing Fig. 50 with Fig. 9, which is the corresponding case for a
gas, it will be noticed that the external work done is much less than the
heat supplied, and not equal as it is in Fig. 9. The reason is that
in the present case a large proportion of the heat is required to
evaporate the water in the cylinder, i.e., to do internal work.
Adiabatic Expansion. Returning to the initial state, let a trans-
formation take place by allowing the steam to expand by reducing
the pressure on the piston, but without adding or deducting any
heat. The state point will in this case obviously follow a vertical line
VARIOUS TRANSFORMATIONS OF H 2 O.
33
(adiabatic expansion), as shown in Fig. 51. If, for instance, the
steam is allowed to expand four times, i.e., until the volume =
2x4 = 8 cubic feet before the transformation ceases, the position
of Q on the chart will be determined by the intersection of the ver-
tical through P and the constant volume line 8. The state of the
H 2 O at point Q as read off the chart (Plate i), will be found to be ;
Pressure 23.4 Ibs. per square inch abs.
Temperature 236.5 F.
Volume 8 cubic feet.
Dryness fraction . . 0.47
P
q
FIG. 51.
The dryness fraction is thus slightly greater than at the point P,
where it was 0.45 : the steam is therefore drier after expansion than
before, although no heat has been added. If, however, the point P
had been situated more to the right, as at P 1 , for instance, where the
dryness fraction is 0.7, then at Q L , the dryness fraction would be
0.66, so that the steam would be wetter at the end of the adiabatic
expansion than at the beginning. The general rule is that the steam
will be wetter, at the end of the adiabatic expansion than at the
3
34
THE ENERGY CHART.
beginning, when the dryness fraction line through the initial point
slopes from left to right, and will be drier when it slopes from right to
left. This is an important point to observe, because it is not in-
frequently stated, in a general way, that water is formed by adiabatic
expansion.
The internal energy of the steam at Q is less than at P by the
vertically-shaded area, Fig. 51 ; that is, by the amount of work
done, or, in other words, the energy converted into work is derived
ABS. ZERO
P <T
FIG. 53.
solely from the heat stored in the H a O, which is obviously correct
since no heat has been added.
Transformation at Constant Volume. Next let a transformation
at constant volume, produced by abstracting heat from the steam,,
be considered (Fig. 52). The state point will in this case follow the
constant volume line from right to left until, for instance, a pressure
of 50 Ibs. per square inch is reached, at the point Q, when heat
represented by the horizontally-shaded area, equal to 212 B.Th.U.,.
VARIOUS TRANSFORMATIONS OF H a O.
35
IV
jn
ABS. ZERO
will have been abstracted*, and the state of the I Ib. of H 2 can be
read off the chart (Plate i) as follows :
Pressure 50 Ibs. per square inch abs.
Temperature 281 F.
Volume 2 cubic feet
Dryness fraction .. 0.24
The internal energy is less at Q than at P by an amount represented
by the horizontally-shaded area = 212 B.Th.U., which is obviously
correct since no external work has been done.
Transformation at constant vol-
ume produced by adding heat is
clearly the reverse of the above.
Transformation along any Line.
Now let the point Q representing the
end of the transformation be situated
as in Fig. 53, and let the transform-
ation take place along the line P Q.
The external work done by the ex-
panding steam is given by the verti-
cally-shaded area, and the heat which
has to be added is shown by the
horizontally-shaded area. The inter-
nal energy at Q is, by the law of con-
servation of energy, equal to the in-
ternal energy at P, less the work
done, plus the heat added, and a
moment's consideration will show FlG - 54-
that the graphic representation on the chart agrees with this.
In a manner similar to that shown in Fig. 17, the chart
for H 2 can be divided into four zones, as shown in Fig. 54, around
the point P, as marked I., II., III., and IV. If the point Q repre-
senting the end of the transformation lies in Zone L, the following
differences in the state at Q and P occur :
* The B.Th.U. represented by the shaded area aie easily calculated as
follows : The average temperature between P and Q is 3O4.2 F t and the differ-
rence of the entropy between P and Q is 0.277. Hence the B.Th.U. re-
quired = (304.2 -|- 461) x 0.277 = 212. This calculation assumes that P Q
is a straight line : the curvature of this line will add about 0.1 B.Th.U.
THE ENERGY CHART.
The volume is less.
The pressure is greater,
External work has to be done on the H 2 O, and heat added to it,
and the internal energy is greater by the sum of these two.
The dryness fraction may be greater or less according to the
relative positions of P and Q.
This is shown in Fig. 55, and Fig. 56 is the corresponding p v
diagram.
FIG 55-
FIG. 56.
If Q lies in the Zone II.:
The volume is greater.
The pressure may be greater or less according to the relative
positions of P and Q.
External work is done by the expanding H a O, and heat
has to be added.
The internal energy and the dryness fraction may be greater
or less according to the relative position of P and Q.
This transformation is illustrated in Fig. 57, and Fig. 58 is the
corresponding p v diagram.
VARIOUS TRANSFORMATIONS OF H a O.
37
FIG. 57.
FIG. 59.
FIG. 60.
THE ENERGY CHART.
It Q lies in Zone III.:
The volume is greater.
The pressure is less.
External work is done by the expanding H 2 O, and heat has to
be abstracted.
The internal energy at Q is less than that at P by the sum
of the external work done and the heat abstracted.
The dryness fraction may be greater or less according to the
relative positions of P and Q.
This transformation is illustrated in Fig. 59, and Fig. 60 is
the corresponding p v diagram.
Q -J?
FIG. 61. F J G. 62.
If Q lies in Zone IV.:
The volume is less.
The pressure may be greater or less according to the relative
positions of P and Q.
The internal energy may be greater or less according to the
relative positions of P and Q.
External work is done in compressing the H 2 O, and heat
has to be abstracted.
This transformation is illustrated in Fig. 61, and Fig. 62 gives
the corresponding p v diagram.
MEASUREMENT OF WORK AND HEAT.
39
240 -J
Measurement of ExternalWorkDone. It will be seen from the pre-
ceding that any transformation that can be effected in the condition
of the i Ib. of H 2 contained in the cylinder can be represented graphi-
cally on the chart, and further that the amount of work done by or on
the H 2 and the amount of heat supplied or abstracted, can be
obtained by the simple measurement of areas. As an illustration,
let the transformation given in Fig.
63 be considered to show how
these areas can be readily ascertain-
ed. The first step is to draw a
horizontal straight line Q b a.
The area representing the work
done is thus divided into two
portions. The area between the
constant volume lines and a Q can
be obtained by calculation, as it
is the work done at the constant
pressure at Q (28 Ibs. per square
inch in this case) by the change
of volume from that at P to
that at Q, that is, in this case
from 2 cubic feet to 10 cubic feet.
This portion of the work is thus
equal to :
14 D
X W
28 x 144 (10 2) foot-lbs. =
32256
778
FIG. 63.
= 41.5 B.Th.U.
The remainder of the work done is found by measuring the area
a P Q, which is 3.21 square inches on Plate i. The heat scale
is such that i square inch represents 10 B.Th.U. Therefore this
area represents 32.1 B.Th.U On the whole, therefore, the work
done by the expanding H 2 O is 41.5 + 32.1 = 73.6 B.Th.U.
Measurement of Heat Supply. As regards the heat added it is
equal to the area b P Q, plus the rectangle below b Q. The former
area measures 1.21 square inches, so that it represents 12.1 B.Th.U.,
and the latter is found by measuring b Q on the entropy scale, namely
0.3, and multiplying this figure by the absolute temperature at Q
or 246.5 + 461 = 707.5. This rectangle, therefore, contains
212.2 B.Th.U., and altogether the heat added is 224. 3 B.Th.U.
40 THE ENERGY CHART.
All other cases can be similarly treated, noticing, however, that
the horizontal line b a should always be drawn through whichever
of the two points P or Q is the lower.
CHAPTER V.
APPLICATION TO A RECIPROCATING STEAM ENGINE :
PRINCIPLES.
IT is most important to observe, and it is the essence of the whole
reasoning of the preceding chapter, that the cylinder at all times
contains i Ib. of H 2 O. In an actual steam engine cylinder, how-
ever, it is only during the expansion period and the compression
period that no change takes place in the weight of H 2 contained in
the cylinder, and this is only true if no leakage in or out of the
cylinder takes place during the expansion or the compression, and
moreover during the latter period the weight is far less than during
the former. During the admission period the weight is continually
being increased, and during the exhaust diminished. By making
certain reservations, it is, however, possible to apply the results ob-
tained in Chapter IV. to the case of an actual steam engine cylinder,
as will now be shown.
Comparison of Ideal and Actual Steam Engine Cylinder. Fig. 65
shows two cylinders, one of them represents the ideal cylinder,
which contains at all times i Ib. of H 2 O, and which in this
chapter will be called the vessel, the other is an engine cylinder
connected to a boiler in which a constant pressure is maintained of
say 150 Ibs. per square inch absolute, the temperature of which is
358.2 F. The weight of H 2 O in this cylinder varies from zero
at the beginning of admission (supposing there is no clearance) to
i Ib. at cut-off, remains constant during expansion (supposing there
are no leaks), after which it diminishes to zero again at the moment
the revolution is completed.
The admission of steam from the boiler to the cylinder corres-
ponds to the supply of heat to the H 2 in the closed vessel, and
when the admission is complete the piston in the cylinder will have
moved through a volume equal to the volume of i Ib. of saturated
42 THE ENERGY CHART.
steam at 358.2 F.> viz., 2.97 cubic feet, and the external work
done will, therefore, be 2 . 97 x 150 x 144 foot-lbs. But, i Ib. of
water has been taken out of the boiler, and in order to maintain the
pressure in it, i Ib. of water must be pumped in against a pres-
sure of 150 Ibs. per square inch. The volume of this i Ib. of water
is 0.016 cubic foot, and the work expended is therefore 0.016 x 150
x 144 foot-lbs.
o =J
3SB-2 r
-LATENT HEAT-
661 B.TH.U
ABS. ZERO
FIG. 64.
u g STEAM ENGINE CYLINDER
INITIAL POSITION
POSITION! AT END OF ADMISSION
OF STEAM
" IDEAL. CLOSED VESSEL
> INITIAL POSITION
I
POSITION AT END OF ADMISSION
OF HEAT
FIG. 65.
The net work done is therefore
(2.97 0.016) x 150 x 144 foot-lbs.
In the vessel the space occupied behind the piston is also 2 . 97
cubic feet so soon as all the water is evaporated, but the piston did not
start from the end of the vessel, there was the volume of i Ib. of water
behind it, namely 0.016 cubic foot, so that the work done in this
case is also (2.97 0.016) x 150 x 144 foot-lbs.
It will be seen, therefore, that in both cases the work done is
the same. Referring to the chart, the "waterline" represents the
APPLICATION TO RECIPROCATING STEAM ENGINE. 43
volume o . 016, as has already been mentioned, and thus the vertically-
shaded area in Fig. 64 represents the work done in either case.
If it is assumed that the temperature of the water in the
vessel at the beginning of the admission period is the same as that of
steam at 150 Ibs. pressure, namely 358.2 F., the amount of
heat to be introduced into the vessel to evaporate the water is the
latent heat of i Ib. of steam at 358.2 F., namely 861 B.Th.U. In
the case of the steam engine cylinder it is clear that this is also the
amount of heat that has to be added to maintain the pressure in
the boiler, if it is supposed that the boiler feed is at 358 . 2 F. This
amount of heat is represented on the chart (Fig. 64) by the
rectangle A B b a. Under these conditions, therefore, the
chart gives both the work done and the heat required for the whole
transformation A B, whether the I Ib. of water be contained in a closed
vessel, or introduced into a steam cylinder from a boiler.
An important difference, however, exists between the vessel and
the steam cylinder, so long as the transformation is incomplete.
When for instance, the state point has arrived at C (Fig. 64)
the proportion of water and steam in the closed vessel can be read
off the chart ; but, in the case of the steam cylinder, the information
A C
is that at the point C there is , -g-lbs. of steam in the steam cylinder,
but the amount of water y if any, is not known. In the ideal steam
engine, for instance, there would be no water.
Effect of Feed-water Temperature. It will now be supposed that
the feed-water enters the boiler at a lower temperature, say at
100 F., and that the water in the vessel is also at this temperature
at the beginning of admission. This obviously makes no difference
in the work done, neglecting the quite secondary consideration of
the very small difference in the water volume. A greater amount
of heat, must, however, be added to the vessel to raise the tempera-
ture of the water from 100 to 358.2 F., namely* :
330.8 68.4 = 262.4 B.Th.U. additional.
The total heat required is shown on the chart (Fig. 66), by the
horizontally-shaded area. In the engine cylinder the I Ib. of feed
enters at 100 F., and to maintain the pressure and to raise the
temperature of the feed to 358.2 F., 262.4 B.Th.U. have to be
* See Scale of Water Heat, Plate I.
44
THE ENERGY CHART.
added. The heat required is, therefore, the same both in the vessel
and in the cylinder, namely :
262 .4 + 861 = 1123.4 B.Th.U
Effect of Initial Condensation. -Let itnext be supposed that as the
steam enters the steam cylinder a certain proportion of it is con-
densed, say 10%. When the valve closes, therefore, the steam
cylinder contains i Ib. of H 2 0, composed of ^thlb, of water
and T yh Ib. of steam. To make the comparison, the supply of
pr \so
B
100*1
ABS. ZERO
FIG. 66.
FIG. 67.
heat to the closed vessel must be stopped when ^ths of a Ib. of
steam have been evaporated, that is, when the state point C (Fig. 67)
is reached. In the case of the steam engine cylinder, the amount of
heat introduced, so to speak, with the steam into the cylinder, is equa
to the larger area shaded horizontally in Fig. 66, but the difference
has been abstracted by the cylinder walls, etc., thus causing the initial
condensation. Therefore the total amount of heat introduced is the
same in both cases. The two cases would, however, be more strictly
comparable if it were supposed that the walls of the closed vessel
were capable of storing up heat: in the case under consideration
EFFECT OF CLEARANCE. 45
the storage would be ^th of the latent heat, and is represented
by the rectangle under C B (Fig. 67).
As regards the work done in the vessel, it is
_~ x 144(2.97 0.016) 150 foot-lbs.
In the steam engine cylinder, the volume swept through as regards
the steam is T 9 ^ x 2.97 = 2.67 cubic feet,* together with ^ x 0.016
cubic foot in respect of the water produced by condensation ; but
i Ib. of feed water has to be introduced, so that the net work done is
Q
r x 144 (2.97 0.016) 150 foot-lbs.
or the same as in the vessel. Fig. 67 therefore represents the state
of things both in the vessel and in the cylinder, as regards work
done and heat supply, and, at the state point C the condition of
the steam, both in the cylinder and in the vessel as regards, pressure,
volume, and dryness fraction, can be read off the chart.
Effect of Clearance. So far it has been assumed that there is no
clearance in the vessel or in the cylinder. It will now be shown how
clearance can be exhibited on the chart, supposing, in the first
instance, that the steam in the clearance is compressed in such a
manner that it reaches boiler pressure at the moment the admission
valve opens. Under these circumstances no steam will be required
to fill up the clearance in the cylinder to boiler pressure. Dealing
with a numerical example in which the admission temperature is
358.2 F. and the exhaust temperature 257.5 F., let it be
supposed that the clearance in the cylinder and in the vessel is 0.2
cubic foot and that at the moment of closing of the exhaust there
is 2.0 cubic feet in the cylinder. The condition of the i Ib. of H 2 O in
the vessel corresponding to these two points will therefore be given by
the state points G and F on Fig. 68, and the transformation may
take place along any line joining these two points, such as the line
shown in the figure. In order to effect this transformation in the
vessel, heat has to be abstracted from the Ib. of H 2 0, as shown by
the horizontally - shaded area included between the adiabatics
through G and F, and the transformation line F G. Work has to be
done in compressing the steam in the clearance, as given by the
* This is also the steam volume at the point C, see the chart, Plate I.
4 6
THE ENERGY CHART.
vertically-shaded area between the constant volume lines. The
p v diagram of F G is given in Fig. 69, and in this figure the work
done is represented by the vertically-shaded area, and this figure will
obviously also represent the compression portion of the p v diagram
of the engine cylinder. Therefore the work done in compressing
the steam that remains in the cylinder at the moment of closing the
exhaust is given by the vertically-shaded area on the chart
(Fig. 68). It is, however, otherwise with the heat required to be
abstracted from the steam in the engine cylinder ; it is not equal to
or iso i
257*3,
VOL 12
Z7o~cu.fr
FIG. 69.
the horizontally-shaded area in Fig. 68, because in the cylinder
there must be considerably less than I Ib. of H 2 0. It is to be
observed that the weight of steam is the same both in the vessel and
in the cylinder at all times, and in the numerical example under
consideration at the moment corresponding to the closing of the ex-
haust valve, the vessel contains 2 cubic feet of steam at the exhaust
temperature 257. 5 F. ; on referring to the chart (Plate i), it will be
seen that at this temperature saturated steam has a volume of 12 cubic
feet, hence the weight of the 2 cubic feet of steam will be th Ib. In
the vessel there is, therefore, fib. of water at the moment the
QUALITY LINE. 47
exhaust closes, but there is no means of telling how much water there
is in the cylinder. It will, therefore, be assumed that all the water
was swept out by the exhaust or else evaporated, so that at the
moment the exhaust valve closes the cylinder contains dry steam,
the weight of which, as shown above, is th Ib.
Quality Line. The amount of heat abstracted from the cylinder
during the compression period can be readily determined by means
of the chart. Let it be imagined that there is another cylinder
just so much bigger than the engine cylinder that it contains i Ib. of
dry steam at the moment of closing of its exhaust valve, and let
it further be assumed that the quality of the steam in the large
cylinder is the same as in the original cylinder at corresponding
points. The transformation line in the large cylinder will start from
/ on the saturation line (Fig. 68) in accordance with the assumption
that the steam is dry at the corresponding point F, and the remainder
of the transformation line / g, can be easily plotted, since at corres-
ponding points the volumes must be in the proportion of I to 6 (in the
numerical example under consideration). It will be seen that the
transformation line / g gives the quality of the steam in the original
engine cylinder at all points during the compression, and it will
therefore be called the " quality line." Thus at the point G (Fig. 68)
the quality of the steam, as represented by the ' quality point " g, has
a dryness fraction of o . 4 nearly. The heat it is necessary to abstract
from the large cylinder to make the i Ib. of H 2 O in it follow the trans-
formation line / g, is given in Fig. 68 by the area below g / down to
absolute zero, and the heat required to be abstracted from the original
engine cylinder will evidently be this area divided in proportion to the
volumes of the two cylinders, that is, in the present numerical
example in the proportion of i to 6. If g / is a curve drawn
proportionately to g/ in the above ratio, it is clear that the
area g/ 1 w 1 w 1 is the heat required to be abstracted from
the engine cylinder during the transformation F G. Further, the heat
change during any portion y x of the transformation F G is equal to
the area bounded by the portion y x of the curve g / x and the dotted
verticals drawn down to absolute zero through the points # and y x .
To make this point clear, a portion of Fig. 68 has been re-drawn in Fig.
70, in which the temperature scale is larger, and the area giving
THE ENERGY CHART.
the heat required to be abstracted from the engine cylinder during
the portion F % of the compression is shown shaded horizontally.
Fig. 71 gives a case of a quality line in which the compression
line F G is vertical, so that as far as the closed vessel is concerned it is an
adiabatic, that is to say, no heat change takes place in the vessel.
In the case of the engine cylinder, however, heat represented in
amount by the horizontally-shaded area (carried down to absolute
zero) has to be abstracted. On measurement it is found that this
amount of heat is 92.5 B.Th.U.
250-3
FIG. 70.
FIG. 71.
It has been assumed so far that the compression starts at F with
dry saturated steam. There may, however, be a certain proportion
of water remaining in the engine cylinder, at the point F, say,
for instance, 0.03 Ib. ; so that dealing with the previous numerical
example the engine cylinder at the moment the exhaust valve
closes, will contain lb. of steam and 0.03 of water. The dryness
fraction at the point F is therefore
r+ o.Q3 = ' 4
and the state point of the steam on the " quality " line is,
therefore, given by the point / in Fig. 72. The curve f g can be
PROPORTIONAL WATER LINE.
49
determined from the curve F G, as was done in the case given in Fig.
68, and further the curve g / can be obtained in a like manner.
From the above it will be seen that even in the case of a
cylinder containing less than i Ib. of H 2 O, the work done and the
heat changes due to any transformation can be graphically shown
on the chart.
Proportional Water Line. In the preceding, only the heat
change required to effect the transformation of the fraction of the
i Ib. of H 2 O in the cylinder itself has been considered, but obviously
during this transformation the balance of the i Ib. (in the form of water)
!-"^"ir 1
> - JL -* >
FIG. 72.
FIG. 73.
must be raised, in the boiler, from the exhaust temperature at F to the
admission temperature at G. In the numerical example given in
Fig. 72 there was + 0.03 = 0.197 Ib. of H 2 in the steam
cylinder at the point F, in fact, during the whole of the transforma-
tion F G\ therefore, in this case, 0.803 lb- of feed water had to be
raised in temperature in the boiler. A curve G K can be drawn, as
shown in Fig. 72, based on the water line of the chart, which
may be called the " proportional water line," and the area contained
by this curve and the verticals through K and G down to absolute
zero, and shaded horizontally in Fig. 72, gives the heat necessary to
raise the temperature of the o . 803 Ib. of feed water in the boiler,
from the temperature at F to the temperature at G. Fig. 72 there-
fore shows graphically, by means of two separate areas, the heat
change in the cylinder itself, and also the heat supply required to
raise the temperature of the feed water. In the closed vessel the
4
50 THE ENERGY CHART.
heat required to be abstracted during the transformation F G is
shown on the chart by the horizontally-shaded area below F G
(Fig. 68), and the two areas referred to above when added together,
having regard to their sign, total up to this area, as appears from the
following. In Fig. 72 it is found by measurement that the area
below g / is 0.123 + x m / 1? where is absolute temperature at
/ , and the area below G K is o . 059 + x K I ; the former area is to
be taken with a negative sign, because the heat is abstracted. Hence
added together the two areas makeup 0.06 4 + (m /i K I) 0. It
is also found by measurement, that the area below F G (Fig. 68)
is 0.064 + x g' /', and that : g' /' = m / x K I. A general
proof of this proposition is complicated, and it is suggested that
it be verified by working out several numerical examples.
Compression Pressure less than Boiler Pressure. It will now be
supposed that the cylinder has not only clearance, but, further, that
the compression does not reach the boiler pressure. In the vessel this
state of things would be represented by the transformation line fol-
lowing a constant volume line G H, as shown in Fig. 73, as soon as the
compression ceased. It is not necessary to consider the portion F G
as that has already been done in the previous investigation. No
external work is done during the transformation G H, because the
volume does not change and this is shown on the chart, but an
amount of heat represented by the horizontally-shaded area below
G H has to be supplied. In the steam cylinder matters are, however,
somewhat different. Let g represent the quality of the steam in the
steam cylinder at the point G just before the admission from the
boiler opens ; immediately afterwards the communication is estab-
lished and steam rushes in to fill up the clearance to boiler pressure.
The question arises how much steam has thus to be admitted. To
simplify matters let it be supposed that the cylinder walls are non-
conducting. If the boiler is large enough not to appreciably drop
in pressure when admission takes place, the steam that fills the clear-
ance will have been evaporated at constant pressure. (In the
vessel, it will be observed, the steam is formed at constant volume
during the transformation G H ). The energy due to the velocity
of the in-rushing steam is derived from the steam itself: this velocity
will at first cause eddies which in time will disappear, the energy
COMPRESSION LINE. 51
again appearing as steam energy. Let it be supposed that sufficient
time elapses for this process to be completed before the condition
of the steam at H is considered, and that it is then dry saturated
steam. The question how much steam has been admitted from the
boiler to produce this result, under the limitation assumed, can most
readily be answered by calculation. At G the quality of the steam
is that represented by g, and, using the method described on page
27, the internal energy of I Ib. of H 2 O whose state is represented
by g is equal to 936.4 B.Th.U. But from Fig. 73 there is only
T? lb " - ^1 lb ' 0{ H *
in the steam cylinder, so that the internal energy at G is
936.4 x 1*. = 46.2 B.Th.U.
At H there is 0.2 cubic foot of dry steam at 150 Ibs. pressure and
since i lb of steam at this pressure has a volume of 2 . 97 cubic feet
and the internal energy is 1109 B.Th.U. per lb. (see Plate i), it follows
that the internal energy in the steam cylinder at His 74. 7 B.Th.U.
Clearly, therefore, the boiler has had to provide
74.7 46.2 = 28.5 B.Th.U.
Now the total heat at 150 Ibs. pressure is 1191 B.Th.U. Hence
28.5
lb. of steam has to be introduced from the boiler to carry out
the transformation G H y bearing in mind the limitations that have
been assumed, namely that the walls are non-conducting, that the
cylinder at the point G contains steam, whose dryness fraction is
0.8, and that all the eddies induced by the in-rush of the steam
on entering have disappeared.
If the steam at G is wetter than assumed, more heat will obviously
be required, and the amount could be ascertained as shown above
if the quality of the steam were known. More steam from the boiler
will be required if the walls are conducting in order to make up for
initial condensation. It is to be observed that the condensation
here considered is that occurring during the period of making up the
pressure in the cylinder to that in the boiler (during " pre-admission "
in fact). The condensation which takes place during admission,
is not included.
THE ENERGY CHART.
pr 130
Expansion Line, with Initial Condensation. A transformation
line, such as that shown in Fig. 74, due to the expansion in a steam
engine cylinder will now be considered. It will first be assumed
that both the admission valve and the exhaust valve are absolutely
tight. It is obvious that in the case given in Fig. 74 heat has to be
supplied to the i Ib. of H 2 O contained in the cylinder, during
expansion, and the total amount so supplied from C to D is repres-
ented by the area below C D bounded by the adiabatics through C
and D and shaded horizontally. On the assumption that the valves
are tight, this heat can only be sup-
plied, in an actual engine, from heat
due to initial condensation and stored
in the cylinder walls or in the water
produced up to the point of cut-off, or
by heat transmitted through the
cylinder walls from jackets.
The case in which the heat
derived from initial condensation is
stored in the cylinder walls, etc.,
will be considered more closely.
On the supposition that the cylinder
has no clearance, the cylinder
receives i Ib. of steam per stroke,
arid the heat supply per stroke is
thus represented by the area
below A^AB (Fig. 75), a portion
of this heat is transferred to the
FlG< 74 cylinder walls by condensation, and
the'state of the steam at cut-off is represented by the point C. The
work done up to this point is represented by the area bounded by
A A C, and the constant volume line through C; the remainder
of the heat supplied per stroke is contained in the steam in the
cylinder, in the cylinder walls, in eddies, and in the water produced
by condensation. The heat stored in the cylinder walls and con-
tained in eddies is evidently equal to the latent heat of the steam
which has been condensed, and this amount of heat is shown on the
chart by the rectangular area below C B. But the heat contained
EXPANSION LINE.
53
C B
in the water produced by condensation is clearly the fraction -s-js of
the heat represented by area below A t A, i.e., the water heat per Ib.
Further, the law of conservation of energy requires that the energy
remaining in the steam in the cylinder shall be equal to the total heat
supplied less the work done, less the heat in the water produced by
condensation, less the heat stored in the cylinder walls. Suppose
the steam is produced at a constant pressure of 150 Ibs. per square
c\,
150
FIG. 75-
FIG. 76.
inch absolute, then the total heat supply per Ib. is 1191 B.Th.U. (see
Plate i). If E is the internal energy in i Ib. of water at 32 F.
measured from absolute zero, then the total energy to be considered
per Ib. is 1191 + E. Let the point C be situated on volume line 2,
2
the dryness fraction will therefore be and the cylinder
contains 0.326 Ib. of water and 0.674 Ib. of steam. The work done
is obviously 150 x 2 x 144 = 43200 foot-lbs. = 55.6 B.Th.U, and
it may be noted that if no condensation had taken place the work
done up to cut-off would have been
x 55-6, = 82.6 B.Th.U.
54 THE ENERGY CHART.
The work areas are also shown on the p v diagram (Fig. 76). The heat
stored in the cylinder walls is equal to 0.326 of the latent heat per
lb., at 150 Ibs. pressure which is 861 B.Th.U. Thus 281 B.Th.U. are
stored in the cylinder walls and in eddies.*
The heat in the water produced by condensation is 0.326 of
the water heat of i lb. of water at 150 Ibs. pressure, and is there-
fore 0.326 (330 + E), and can be shown on the chart (see Fig. 75) by
drawing a " proportional water line " for 0.326 lb. of water. Hence
Work done - 55.6 B.Th.U.
Heat in walls and in eddies = 281 B.Th.U.
Heat in the water in the cylinder =0.326 (330 + E) B.Th.U.
Total = 443 + 0.326 E B.Th.U.
The internal energy in the steam in the cylinder at the point C (Fig.
75) must therefore be : 1191 443 + 0.674 E = 748 + 0.674 &
But the internal energy of 0.674 of I lb. of steam at 150 Ibs. pressure
is 0.674 ( II0 9 + ) = 748 + 0.674 E, which agrees with the above
result.
Equivalent Weight of Water and Theoretical Re- evaporation Line.
An "equivalent" weight of waterf can be calculated such that the heat
in the cylinder walls and in eddies is just sufficient to raise its temper-
ature from that at the point A to that at the point A (Fig. 75). This
water would then act as a store for the heat represented by the
rectangle below C B. In the numerical example under considera-
tion this heat is 281 B.Th.U. and the difference of temperature is
108 F., hence the weight of the " equivalent " water in this case is
281
-^58 = 2. 59 Ibs.,
which, added to the 0.326 of a lb. of water produced by
condensation gives 2.92 Ibs. of water at a temperature 358.2
F. If this water cools with the steam as the expansion proceeds,
giving up heat to the steam, it is clear that when the temperature
of 250.2 F. is reached 2.92 x 108 = 315-3 B.Th.U. will have
been transferred from the " equivalent " water to the steam in the
* From the chart the area below C B is equal to C B x 819.2, but by
measurement CB = 0.343, hence heat = 0.343 x 819 = 281 B.Th.U., or
the same number.
f The idea of this " equivalent " weight of water is due to P. W. Willans
See Min. Proc., C.E. f Vol. CXIV., page 35.
THEORETICAL RE-EVAPORATION LINE.
55
cylinder. The expansion line will therefore be placed to the right
of the adiabatic through C as shown by C R (Fig. 77) ; the point R
must be so situated that the area below C R is 315 .3 B.Th.U. The
expansion line C R (Fig. 77) shows the greatest amount of heat that
can be restored, from the heat in the equivalent water, to the steaifc
in the cylinder of an engine, when the admission and exhaust valves
are quite tight and there are no jackets, and this expansion line C R
can be called the " theoretical re- evaporation line." In an actual
engine the expansion always falls considerably short of C R., unless
the admission valve is leaking
B
ABS. 2BCRO
ABS. ZERC
br
FIG. 77-
FIG. 78.
This theoretical re-evaporation line is important, and it will,
therefore, be shown how to obtain it generally, not merely by a
numerical example. In Fig. 78, the heat stored in the " equivalent "
weight of water is equal to the area of the rectangle below C B,
which is = C B x X , and since the temperature through which the
" equivalent" weight of water is raised is t a , its weight is
C B x
56 THE ENERGY CHART.
To this imist be added the weight of water produced by conden-
C B
sation equal to j g lb., and which at the beginning of expansion is
at the temperature 0. The supposition is that as the expansion
proceeds, and the temperature of the steam drops, the temperature of
these two weights of water will keep exact pace, and that the
resulting heat is transmitted to the steam. The amount of heat thus
transmitted when the expansion is complete at the point R is :
(*i *.) + C B x B.Th.U,
and this heat is represented on the chart by the area below C R.
(ft \ f) \
~2/
Hence K R - ^ (^ + .,)
It is to be observed that K R, C B, and A B, can be measured on the
chart on the entropy scale, thus in the case of the numerical
example previously considered the values are :
C B = 0.343 and A B = 1.05,
and since X 358.2 + 461 and 2 = 250.2 + 461
,, , v D 0.686 / 108 f \
therefore KR = yil . g + 8l9 . g (^ + 358.2 + 461 )
0.686
== I530~ ( I02 * 7 + 8l 9- 2 ) = 0.413 entropy units.
The two areas which total up to make up the area below
C R (Fig. 77) are shown graphically on the chart as follows
(see Fig. 75): The heat stored in the walls of the cylinder
and in eddies is the area below C B shaded horizontally.
The heat in the water produced by condensation is equal to
C B
j-g x the water heat at A, and is equal to the area shaded by dots
obtained by drawing a curve A a, deduced from the water curve by
proportioning the horizontal distances to the vertical through A
C B
in the ratio ^rg- This area is in fact the heat required to raise
C B
lb. of water from temperature to temperature a . Further
the loss of p v due to condensation is shown by the area between the
constant volume lines through C and B, and is shaded vertically.
EXPANSION LINE WITH LEAKS.
57
Expansion Line with Leaky Admission Valve. Let it now be sup-
posed that the inclination of the expansion line is not due to heat re-
covered from the walls or from the condensed steam* or to the effect of
a jacket, but is solely caused by leakage from the admission valve, and
that the exhaust valve is absolutely tight. This being the case it must
be considered that at the point D (Fig. 79) the cylinder contains I lb.
of H 2 0, and consequently the state of the steam in the cylinder at
that point can be read off the chart. There will, however,
be less than i lb. of H 2 at C, and at all other points between C
and D. The question is, what amount of steam has leaked into the
(B
FIG. 79.
FIG. 80.
cylinder between the points C and D. This question cannot be
answered definitely because the weight of water present in the
cylinder at the point C is not known, although the weight of steam
C A
present is known, being equal to -j-g lb. Some idea can, however,
be formed if an assumption is made as to the weight of water present
at C. Suppose, for instance, that the dryness fraction is 0.9. The
* This is a purely imaginary case, as there must always be some heat
returned to the steam by the walls during expansion.
THE ENERGY CHART.
prjso
state of the H 2 O is, therefore, represented by the point c. If
steam of this quality were expanded adiabatically it would follow
the line c d, and reading from the chart (Plate i), the volume of
steam at d would be n.i cubic feet. At c the volume is 2.67,
thus the steam would have expanded -g- or 4.16 times. Hence,
if in the actual cylinder there were no leakage, and consequently
under the assumed conditions the expansion were adiabatic, the
volume of the steam at C, equal to 2 . o cubic feet, would have in-
creased to 4.16
x 2 = 8.32
cubic feet at
the end of the
expansion. In
the case under
consideration,
however, there
are 12 cubic
feet of steam
in the cylinder
I 51 | at the point
Z), so that ev-
idently 12
8.32 = 3.68
cubic feet, at
the pressure 30 Ibs. per square inch, have leaked past the ad-
mission valve into the cylinder during the expansion ;
but the volume of saturated steam at the pressure of the
CUB. FEE TT
FIG. 81.
~~
= 0.27 Ib. of steam has
point D is 13.5 cubic feet, so that
leaked into the cylinder. Strictly, a correction is needed to the above,
because the argument used implies that the steam leaking into the
cylinder was formed at the pressure at which it leaked in, that is at
varying pressures dropping from the pressure at C to the pressure
at D. In practice, however, the steam would be produced at the
boiler pressure, which is certainly somewhat higher than the pressure
at C. Eddies will therefore be formed by the steam leaking in, and in
so far as they are re-converted into heat before the point D is
EXPANSION LINE WITH LEAKS. 59
reached, a corresponding weight of water will be evaporated. The
calculation, therefore, shows a slightly too great an amount of
of leakage. The expansion line C D has been plotted on the p v
diagram (Fig. 81), together with the corresponding saturated steam
line.
Expansion Line with Leaky Exhaust Valve. If the admission
valve is tight, the expansion line will, due to the leakage of the
exhaust valve, fall far short of the theoretical expansion line, even
to the extent of sloping to the left of the adiabatic at the beginning
of the expansion, where, owing to the greater difference of pressure,
the leakage through the exhaust valve is greatest.
Expansion Line with Leaky Admission and Exhaust Valves.
It is obviously difficult to disentangle the effects of both these
leakages with any accuracy, but from the above it will be evident
that the expansion line will be nearly adiabatic at the beginning,
or even slope to the left, and at the end of the expansion will slope
considerably to the right. Fig. 112 is an example.
Expansion Line with Jacket and Leaky Admission Valve. Lastly,
let the expansion line C D be due, partly to heat added during the
expansion by means of a jacket and partly to leakage past the ad-
mission valve, the exhaust valve being supposed to be absolutely
tight. Since by supposition the admission valve leaks the cylinder
will contain the maximum weight of H 2 O at the point D, and this
weight will be weight of the feed less the weight of steam passing
through the jacket per stroke ; the latter can be determined experi-
mentally and for a numerical example let it be supposed to be tVth
of the feed. The cylinder thus contains ifths of the feed at
the point D, but, since the chart (Plate i) is drawn for I Ib. of
H 2 O, it must be considered that the cylinder contains I Ib. of
H 2 O at the point D, hence the feed per stroke is $% of a Ib., and
the jacket steam is T Vth of a Ib. per stroke.
Since the cylinder is jacketed the condensation ought to be
small, and therefore let it be assumed that the dryness fraction of the
steam in the cylinder at C is 0.9, and it is to be remarked that there
is less than i Ib. of H 2 O at C. The quality of the steam at C is there-
fore represented by the point c on the chart (Fig. 80). Let
another and larger cylinder containing i Ib. of H a O in the state
6O THE ENERGY CHART.
represented by c be considered whose admission valve is, however,
absolutely tight, and let it be supposed that this cylinder is provided
with a jacket supplying more heat than the previous one in the
proportion of the volume at c to the volume at C, namely in the pro-
portion of 2.67 to 2. The heat given up by the second jacket per
2.67
stroke is that due to the condensation of~ of ^th of a Ib. of steam
and by the resulting water falling in temperature from 358 . 2 F.
to 250.3 F., and this amount of heat can be found by proportion
from the corresponding amount for i Ib. of steam, as represented on
the chart by the area below A' A B. Thus the line c d has been so
2.67
drawn that the horizontally-shaded area in equal to ~ x ^th of
the area below A' A B. It follows that c d would be the expansion
line in the larger cylinder if the whole of the heat in the jacket were
transmitted to that cylinder. Further, if the whole of the heat due
to condensation were recovered the expansion line in the larger
cylinder would be shifted to c d', the point d' being found as previously
explained. In practice the whole of the heat stored in the walls
due to condensation cannot be recovered, nor can all the heat ex-
pended in the jackets be transferred to the steam in the cylinder.
Hence c d! is the limiting expansion line due to all the heat from the
jacket, and from the walls of the cylinder, when there is no leak past
the admission valve. From the chart it is seen that the volume at
d' is 13.4 cubic feet, obviously, therefore, the corresponding volume
2
in the smaller cylinder would be r-g- x 13.4 = 10 cubic feet,
which gives the point D L ; that is to say if there were no leak, and all
the heat in the jacket were transferred, and all the heat stored in
the cylinder walls, etc., were recovered, the expansion line in the
smaller cylinder would be C D, instead of C D. The volume at D
is, however, 12 cubic feet, so that if all the jacket heat and all the
heat in the walls is recovered, the leakage is 12 10 = 2 cubic
feet, and since i Ib. of saturated steam at 250.3 F., has a volume
of 13.5 cubic feet, it follows that the leakage per stroke is
= 0.148 ib.
This is the minimum leakage, because it has been obtained
EXHAUST LINE. 6l
on the supposition that the whole of the jacket heat is transferred
and all heat stored in the walls of the cylinder is recovered. If,
however, frds of the jacket heat is transferred and a similar pro-
portion recovered from the cylinder walls, then the point d' will
be shifted to d," by making the area below c d" frds the area below
c d'. The volume at d" is found to be 12.6 cubic feet, so that the
corresponding volume in the smaller cylinder is
2
12.6 x gg 9.42 cubic feet,
and the leakage is 12 9 . 42 = 2 . 68 cubic feet, the weight of which is
gf = 0.198 1*
On comparing Figs. 79 and 80, it will be seen that the lines C D
are the same ; in the former case it was supposed that the whole
effect was due to leakage, and it was found that this leakage was
0.27 Ib. of steam per stroke. In the second case the effect was due
partly to leakage and partly to a jacket and to heat recovery from
the walls, and it has been shown that then the minimum leakage is
0.148 Ib. per stroke. The result has thus been established that the
minimum leakage is o.i481b., and the maximum leakage o.27lb.
Limits to the leakage are thus fixed.
Exhaust Line : Incomplete Expansion. An exhaust line at con-
stant volume will next be considered as represented by D E in Fig.
82. In the closed vessel which contains i Ib. of H 2 O it will be seen
that at D the state of the steam is
Pressure 30 Ibs. per square inch
Volume 10 cubic feet
Temperature 250.3 Fahr.
Dryness fraction 0.74
The transformation line D E is obtained by abstracting heat as
shown by the area below D E shaded horizontally and there is no
internal work done.
At E the state of the i Ib. of H 2 is :
Pressure 15 Ibs. per square inch
Volume 10 cubic feet
Temperature 213 . 2 Fahr.
Dryness fraction 0.385
Internal energy 626 B.Th.U.
62
THE ENERGY CHART.
ZOO'
In the steam engine cylinder, however, matters are somewhat
different. At the point D the state of the H 3 O is by assumption the
same, both in the closed vessel and in the cylinder, but at E the
cylinder contains only steam, except a little water that may
cling to the walls and the piston body. There is no means of knowing
what this amount of water is, and it depends much on the design of
the engine as regards drainage. From the chart it will be seen
that at the point E the cylinder contains 0.385 Ib. of steam, at a
pressure of 15 Ibs., and at
a temperature of 213 . 2 F.
The internal energy of this
steam is 0.385 times that
of i Ib. of saturated steam
at 15 Ib. pressure, which is
1075 B.Th.U., as read on
the scale of internal energy
on the chart (Plate i).
This energy is therefore
414 B.Th.U., and is shown
by the vertically-shaded
area. It will be observed
that this energy is less
than the internal energy
of the i Ib. of H 2 O at the
corresponding point in the
closed vessel, which is rep-
resented by the area below
the dotted constant volume
line. The internal energy
of i Ib. of saturated steam at the pressure of 15 Ibs. per square inch
is also shown in Fig.Jte by the area below the constant volume line
for 26 cubic feet drawn through the point B 19 and the dotted line
drawn through the same point is such that the horizontal intercepts
to the adiabatic through B^ are in the ratio 0.385 to i, for instance
no Into = 0.385/1. The area below this dotted line represents there-
fore the internal energy of the steam present at E, and is equal to
the vertically-shaded area.
FIG. 82.
VARIOUS PROBLEMS. 63
Summary. Reviewing the various cases of transformation
lines that have been considered in detail, it will be seen that in every
case the external work done by or on the steam in the cylinder is cor-
rectly shown on the chart, but that the heat added or abstracted is not,
in many cases, shown directly on the chart, but that it can always
be obtained in these cases by various geometrical constructions, or
by simple calculations.
A variety of useful problems will now be considered.
To draw the Quality Line corresponding to any given Transforma-
4on Line. Let P H (Fig. 83) be a transformation line for less
than i Ib. of steam, the quality of the steam at the point P is
given, as represented by the point p. It is desired to draw the
quality line. The first step is to find the weight of H 2 O present
during the transformation P H. The weight of steam present at P
will evidently be less than i Ib. in the proportion of the volume
of saturated steam at the pressure of P to the volume at P, and
the weight of water at P will be in proportion to the dryness
fraction at p. Thus the weight of H 2 O at P can be readily found,
and, from the conditions of the problem, this is the weight of H 2 O
throughout the transformation P H. The next step is to find the
" quality " point for any other point Q, situated on P H. The
weight of steam at Q is less than i Ib. in the proportion of the
volume of saturated steam at the pressure of Q, to the volume at Q.
This weight, deducted from the weight of H 2 O previously found,
gives the weight of water at Q. The dryness fraction at Q is there-
fore known, and the position of the point q can be fixed. The
quality line p h can therefore be drawn. The following numerical
example (given in Fig. 83) will further illustrate the matter :
The weight of steam at P is
2
i x : Ib. = o.i481b.,
and since the dryness at p is 0.9, the weight of water at P is
2
(i 0.9) Ib. = 0.0148 Ib.
The weight of H 2 O is therefore 0.163 Ib., and this is the weight
during the transformation P H.
At Q the weight of steam is i x -~ = 0.157 Ib. Hence the
6 4
THE ENERGY CHART.
weight of water at Q is o . 163 o . 157 = o . 006 lb., and the dryness
fraction is 0.157/0.163 = 0.965, the position of q is therefore
located. Any other point on p h can be similarly found.
At the point Q i9 the weight of steam present is
xx^f =0.163,
which is the total weight of H 2 0, hence there is no water present,
and <7 t is on the saturation line.
At the point H the apparent weight of steam is
0.873
= 0.2OI lb.,
FIG. 83.
FIG. 84.
and as this is greater than 0.163, it follows that the steam must
be superheated at H. The volume of i lb. of steam in the con-
dition of the 0.163 lb. of steam at H must obviously be
0.873/0.163 = 5.4 cubic feet, and therefore the position of h is
found at the intersection, in the superheated field, of the constant
volume line for 5.4 cubic feet, and the constant pressure line for
100 Ibs. per square inch.
To draw the Transformation Line for less than I lb. of Steam given
the Quality Line. This problem is obviously the reverse of the previous
VARIOUS PROBLEMS.
one, and in Fig. 84 a numerical example is given showing how to
draw an adiabatic through the point H for ^ = o. 125 lb. of steam ;
the steam at H being saturated. At g the dryness is seen from
the chart to be o . 9, hence the weight of steam at G is o . 125 x o . 9 =*=
o.i 125 lb. The- volume
of 'saturated steam at the
pressure of G is 13 . 5 cubic
feet, hence the volume of
steam at G is 13.5 x
0.1125 = 1.5 cubic foot.
G is therefore located, and
any other point can be
located in the same way.
To find the Tempera-
ture and Pressure at the
end of a given number of
Expansions under Adia-
batic Conditions. Let C
(Fig. 85) be the state point
of the beginning of the ex-
pansion. The volume at
this point can be read off
the chart. Multiply this
volume by the number of
expansions and find the
intersection D of the adi-
abatic through C, with the
FIG. 85.
volume line for the expand-
ed steam. The tempera-
ture and pressure can be read off the chart. In Fig. 85 the number of
expansions has been taken as 12. Hence the volume at D is 2.55
x 12 = 30 . 6 cubic feet, and as read off the chart :
Pressure at D = 7.5 Ibs. per square inch abs.
Temperature ,, = 180 F.
It is interesting to note that Rankine, in his " Steam Engine and
Prime Movers," says at page 392 that the above problem can only
be worked out (by means of thermodynamic formulae) by " a tedious
5
66
THE ENERGY CHART.
process of trial and error " (see also Appendix IL, Steam engine
trials, by P. W. Willans, M.Inst.C.E. Vol. CXIV., Min. Proc.
Inst. C.E.).*
The problem could be equally easily solved by means of the
chart if the expansion C D were not adiabatic.
Determination of point of
Cut Off. Referring to Fig.
86, ABDA L is the <f> dia-
gram for a perfect steam
engine (Rankine cycle), work-
ing between the temperatures
of 359 F. and 212 F., and it
will be seen that the volume
of the steam at release in the
cylinder is 22.6 cubic feet,
and that at cut-off it is 2.95
cubic feet. The cut-off, there-
fore, obviously takes place at
2 Q^
= 0.13 of the stroke,
FIG. 86.
*I2
and the steam is expanded
22.6
2-95
= 7.66 times.
In the case of an actual engine, account must be taken of the
clearance volume, and a moment's consideration will show that the
point of cut-off can be obtained from the formula :
Point of __ volume at cut-off volume in clearance,
cut-off ~~ volume at release "volume in clearance.
Economy of the Rankine Engine. The area of the 6 <f> diagram
of the perfect engine (Fig. 86) is found on measurement to be equal
to 17.2 square inches when drawn on Plate i, and, since the heat
scale is 10 B.Th.U. per square inch, it represents 172 B.Th.U.
This perfect steam engine therefore produces 172 x 778 foot-lbs.
of work for every Ib. of feed water. Further, the number of
* Willans' construction applies only to the case when the steam initially,
at the point C, is dry saturated.
VARIOUS FORMULA. 67
B.Th.lL* required by the engine per Ib. of feed water is total the heat
at 359 F -> les s the water heat at 212 F. = 1010 B.Th.U. There-
fore an expenditure of 1010 B.Th.U. is required in order to obtain
172 B.Th.U. of work per stroke. But i horse-power per minute
is equal to
= 42-4 B.Th.U,
so that this perfect steam engine can do
172
-^ = 4.06 horse-power per Ib. of feed, per minute,
that is for an expenditure of 1010 B.Th.U. Hence, the economy
of this engine, as defined by the Thermal Efficiency Committee of
the Inst. Civil Engineers, is 252 B.Th.U. per I.H.P. per min. The
various operations just performed are included in the formula :
r? t . Heat supply per stroke.
Economy of engine 42.4 ^r 1\ . ^ , ,* -
j o T T jj ea t represented by <f> diagram.
/expressed as B.Th.U. \
\ per I.H.P. per min. )
= 43-4
Thermal efficiency.
Economy of an Actual Engine. This formula applies also in the
case of an actual engine ; thus, for example, referring to Fig. 91, it
will be seen that the vertically-shaded area of the 6 <f> diagram
(representing the work done) is 65.5 B.Th.U., and the heat supply
per stroke is represented by the area whose contour is dotted and is
equal to 821 B.Th.U. (as found on page 75). Using the
above formula :
Economy of actual engine = 42.4 x ^ = 533 B.Th.U. per I.H.P.
to 65.5 per mm.
Steam Consumption of an Engine. One H.P. is equal to
42.4 x 60 = 2545 B.Th.U. per hour.
Hence the feed water can be obtained by dividing 2545 by the
number of B.Th.U. utilized as work per Ib. of 6 </> cylinder feed,f
* ThefB.Th. U/utilised by the Rankine engine can be found with sufficient
accuracy thus : The mean temperature is
359 I = 285 F..
and at this temperature the horizontal distance between the water line and the
adiabatic B D, measured on the entropy scale (Plate i), is 1.17 and the differ-
ence of temperature is 359 212 = 147 F. Hence, very approximately,
the area, A A B D represents = 1.17 x 147 = 172 B.Th.U.
t For definition see page 73.
t>8 THE ENERGY CHART.
as given by the area of the <f> diagram or diagrams. The formula
is therefore
Lbs. of feed water per | = / B.Th.U. represented by <f> dia-
I.H.P. per hour J 545 f grams per Ib. of </> cylinder feed.
Mean Pressure of Rankine Engine. Referring again to the per-
fect steam engine of Fig. 86, it is obvious that the work represented
by the <f> diagram, namely 172 B.Th.U., must be equal to the
mean pressure in the cylinder x the volume swept by the piston.
From the figure it will be seen that this volume is 22.6 cubic feet.
Hence the mean pressure is
172 X 778
^g = 5920 Ibs. per square foot = 41 . i Ibs. per square inch.
Mean Pressure of Actual Engine. In the actual engine of Fig.
91, the work done is 65.5 B.Th.U., and the volume in the cylinder
at release is 12.3 cubic feet, but the volume in the clearance is 0.8
cubic foot, so that the volume swept by the piston, being the differ-
ence between these two volumes, is n .5 cubic feet. Hence the mean
pressure is
65.5 x 778
11.5 x 144 = 30 -7 Ibs. per square inch,
which is the same as the mean pressure obtained from the p v dia-
gram (Fig. 87), see page 73. The general formula is :
M F B.Th.U. represented by 9 <f> diagram
</> Volume at release 6 < Volume of clearance.
"Equivalent" feed. As explained under Line 131, Report of
the Committee of the Institution of Civil Engineers, on Steam Engine
and Boiler trials, the equivalent feed is obtained by the formula :
Equivalent feed = Lbs ; f < eed w ter P er x Heat supply per Ib.
n l.H.r. per hour noo
The next step is to apply these various cases and constructions
to the complete indicator diagram of a steam engine.
CHAPTER VI.
<j> DIAGRAMS OF STEAM ENGINES DERIVED FROM THEIR INDICATOR
DIAGRAMS.
THE indicator diagram given in Fig. 87 will be taken as a first
example. Further data relating to this diagram are as follows :
Boiler pressure : 115 Ibs. per sq. in. absolute (100 Ibs. gauge)
Pressure at engine stop-valve : no Ibs. per sq. in. absolute
Exhaust pressure : 14.7 Ibs. per sq. in. absolute
Area of piston : 155 square inches.
Stroke : 6 inches. Therefore the volume swept by the piston
is 0.538 cubic foot.
Weight of feed per stroke, i.e., per diagram, corrected* for
leakage past the cylinder into the exhaust is: 0.0382 Ib.
Location of
Initial Point.
Selecting M
on the expan-
sion line at a
point before
the exhaust
valve opens,
it will be
seen from Fig.
87 that at
this point the
volume of steam
-T
FIG. 87.
in the cylinder is 0.35 cubic foot, and
that the pressure is 40 Ib. per square inch absolute. Referring now
to the chart, it will be seen that the volume of saturated steam
at this pressure is 10 . 3 cubic feet per Ib. Thus obviously the cylinder
0.35
contains ^ = o.O34olb. of saturated steam. A portion of this
* The engine was fitted with a slide valve, 5/ was therefore deducted from
the actual feed. See also page 120.
70 THE ENERGY CHART.
steam, however, is due to that retained in the clearance each stroke.
Let it be assumed that at the point N, about half-way up the com-
pression line, the cylinder contains only saturated steam ; at this
point the volume in the engine cylinder is 0.125 cubic foot and the
pressure 27 Ibs. per square inch absolute as read off the p v diagram.
At this pressure saturated steam has a volume of 14.9 cubic feet
per Ib. Hence the weight of steam in the cylinder at the point N is
0.125
14.9
= 0.0084 Ib., and this is the weight of steam retained in the
FIG.
FIG. 89.
clearance on the assumption made. Deducting this weight from
the weight of steam at the point M, namely o . 034, it is found that the
diagram accounts for 0.0256 Ib. of steam passing through the cylinder
per stroke. According to the data, however, the weight of feed per
stroke is 0.0382 Ib., and the difference, namely
0.0382 0.0256 = 0.0126 Ib.,
must obviously be in the cylinder at the point M in the form of
water, since the correction for leakage direct into the exhaust has
been made. Thus at this point there is 0.034 Ib. of steam and
<f> DIAGRAM OF AN ACTUAL ENGINE.
0.0126 Ib. of water.
Hence the dryness fraction is :
0.034
__ ZLJL _ - -
__ _
0.034 + 0.126 7>
The point M can therefore be located on the chart (see Fig,
88), as the point on the 4olbs. per square inch constant pressure
line at which the dryness fraction is 0.73.
Corresponding 8 < Engine and Volume Factor. In the actual
engine there was present at the point M, o.04661b. of H 2 0, con-
sisting of 0.034 Mb. of
steam and of 0.0126 Ib.
of water, occupying a
volume of 0.35 cubic
foot. At the point M
on the chart there is,
however, i Ib. of H 2 O,
consisting of 0.73 Ib. of
steam and of 0.27 Ib.
of water, occupying a
volume of
feet. The
7 . 5 cubic
engine con-
templated by the chart is, therefore
FIG. 90.
- = 21.4 times larger
0-35
than the .actual engine, all volumes relating to the actual engine must
therefore be multiplied by this factor to obtain the transfer of the
p v diagram to the chart. The engine whose diagram is drawn
on the chart can conveniently be called the " corresponding
6 <jf> engine/' and the factor the "volume factor.' 1 * An equivalent way
of looking at the matter is to apply a new volume scale to the p v
diagram of the actual engine as shown in Fig. 87.
Plotting of the 6 $ Diagram. The next step is to take a number
of points on the contour of the p v diagram marked I, 2, 3, in Fig. 90,
and read off the pressure and volume for each point. The volumes
are then to be multiplied by the volume factor. The following
table is thus established :
* This term " volume factor " was first used by the Author in 1894 see
Proceedings of the Institution of Mechanical Engineers, 1st and 2nd Feb.,
1894. The volume factor there mentioned is, however, the inverse of that of
the text.
THE ENERGY CHART.
TABLE.
I
2
3
4
Reference
Number of Point.
Volumes.
Abs. Pressure
per square inch.
p V
e <
I
037
0.80
95
2
.125
2.67
87
3
-155
3-32
77
4
.225
4.81
59
5
275
5.88
50
M
350
7-49
40
6
450
9-63
32
7
.525
11.22
23
8
575
I2.3O
18
9
.200
4.28
18
10
.I5O
3.21
21
N
.125
2.68
27
ii
.075
1.61
44
12
-037
0.80
66
The point on the chart corresponding to point I on the p'v
diagram is obviously the intersection of the constant pressure line
for 95 Ibs. per square inch (Col. 4) with the constant volume line for
0.80 cubic foot (Col. 3), and similarly for all the other points. The
<t> diagram given in Fig. 89 is thus obtained. No pressure and
volume lines are shown in this figure, and it is suggested that columns
3 and 4 in the table be plotted on tracing paper placed over the
chart (Plate I). A more rapid method is gived in Appendix I.
The shaded area in Fig. 89 is the work done by the "correspond-
ing $ engine " expressed in B.Th.U., and is therefore equal to
the work done by the actual engine multiplied by the volume factor,
viz., 21.4. The area* of the < diagram when plotted on Plate I,
* Fig. 73 is drawn to too small a scale to be able to measure this area with
any accuracy. The figure given in the text was obtained by plotting on
Plate I. This remark applies to all subsequent diagrams of the same kind.
HEAT ENERGY PER DIAGRAM. 73
will be found on measurement to be 6.55 square inches, and since
the heat scale of this chart is 10 B.Th.U. per square inch, the
work done by the " corresponding 0< engine " is 6.55 x 10 = 65.5
B.Th.U.
The mean pressure of the actual engine obtained in the usual way
from the p v diagram (Fig. 87) is 30.6 Ibs. per square inch, so that
the work done per stroke by the actual engine is : Mean pressure
per square foot x volume swept by piston = 30.6 x 144 x 0.54
foot-lbs.; or equal to g =3.06 B.Th.U. Multiplying by the
volume factor 21.4, gives 65.2 B.Th.U. as the work done by the 6 <f>
engine. The agreement of this figure with that obtained from the
$ chart is a check on the accuracy of the plotting.
Energy retained in Cylinder ("play" energy). From Fig. 89 it will
be seen that the admission valve opens at the point 12 and at this
point the < engine has a cylinder volume of 0.8 cubic foot. What
is the internal energy of the H 2 O contained at this point in the
cylinder, just before the admission valve opens ? By drawing
the quality line as explained at page 63, the dryness fraction is found
to be 0.695, and the pressure is 66 Ibs. per square inch, and at this
point it is found that the internal energy is 835 B.Th.U. per Ib.
(see Fig. 43). At 12 there is, as already seen o.oo84lb. of H a O.
Hence the internal energy in the cylinder at 12 is 7.0 B.Th.U. This
amount of energy is retained in the engine clearance each stroke
and might be called the " play " energy, corresponding to the
" play " steam.
Heat Energy per 6 <f> diagram. The next investigation will be to
show graphically on the chart the amount of heat energy introduced
per diagram into the $ cylinder. The first step is to find the feed
water per diagram, or the " < cylinder feed." From the given data
the feed water per stroke of the actual engine is 0.0382 Ib., and since
the 6 < engine is 21.4 times larger, the feed water per diagram for
it is 0.818 Ib. Another way of obtaining this figure is as follows :
At the point M (Fig. 89) the <f> engine contains i Ib. of H 2 O, but
as previously shown the weight of steam remaining in the clearance is
0.0084 x 21.4 = 0.180 Ib.
It follows that the feed of the <j> engine is i o . 180 = o. 820 Ib. per
diagram or practically the same figure as before. The area whose con-
74
THE ENERGY CHART.
tour is dotted in Fig. 91, which is Fig. 89 reproduced on a smaller scale,
is, therefore, the representation of the heat supply per stroke,
and it can thus be seen at a glance what a small proportion
of the heat supply has been utilised as work. This proportion is the
" thermal efficiency of the actual engine.''*
Comparison with Rankine Cycle. The question arises what pro-
portion would a perfect steam engine (Rankine cycle) have been able
to utilise under the circumstances. In the first place it is to be
observed that the pressure at the engine stop- valve is less than in the
boiler owing to losses in the steam
pipes, so that the pressure at the
engine stop- valve is no Ibs. per
square inch, as given in the data.
This must be considered as the
pressure the perfect steam engine
has to work with, corresponding
to a higher temperature limit of
334.5 F., and the loss due to the
drop of pressure must be laid to the
account of the steam pipes. Fur-
ther, the exhaust pressure is 14 . 7 Ibs.
per square inch which corresponds
to a temperature of 212 F.,
and this is the lower limit of
temperature for the perfect steam
engine of comparison. It will be
observed that the feed tempera-
ture is lower, but this is due to de-
fective feed arrangements, and the
consequent loss must be placed to their account, since theoretically
the feed can be raised to the exhaust temperature. It follows from
the above that the heat supplied per stroke is the latent heat of
0.818 Ibs. of steam evaporated at a pressure of no Ibs. per square
inch absolute, added to the water heat of the same weight of water
raised from 212 F. to 334 . 5 F. If the engine has no clearance, then the
diagram of the Rankine cycle is shown by the area A t A B /?, and
* Subject to the feed having been corrected for leakage past the cylinder
direct into the exhaust.
FIG. 91.
6 <f> CYLINDER FEED. 75
this is for an engine using i Ib. of feed. The <j> engine under dis-
cussion, however, uses 0.818 Ibs. of H 2 O per stroke, so that for com-
parison the Rankine cycle should be drawn for 0.818 Ib. of H 2 O, or
in other words, should contain 0.818 of the heat units in the area
A i A B p. In order to show this graphically, the point a t is found
such that -2~g == 0.818, and similarly other points are found
along the curve a a x . Then the area a t a B ft represents the
work done by the Rankine engine using 0.818 Ib. of dry steam. The
heat supplied per diagram, and the manner of supplying it, is, there-
fore, shown in Fig. 91, by the area whose contour is shaded by dots.
This is the amount of heat supplied per diagram both to the <f>
engine corresponding to the actual and to the perfect steam engine,
and on measurement is found to be equal to 821 B.Th.U. Hence, the
thermal efficiency of the actual engine is -g^ = 0.08. As regards
the Rankine engine, it rejects the heat represented by the rectangle
a x X, and therefore converts into work the remainder of the area,
as shown by oblique shading. It must be borne in mind that the
perfect engine has no clearance, although Fig. 91 might make
it appear that it had, but the weight of steam supply to the perfect
engine is 0.818 Ib. per diagram, and it will be seen therefore that
a a is really the water line of the chart drawn for 0.818 Ib. of
water ; it follows also that neither the steam line nor the volume
lines nor the dryness fraction lines of the chart apply to the perfect
steam engine of the dimensions now being considered. The
obliquely shaded area therefore represents only the heat utilised by
the perfect steam engine, and it is found by measurement that it is
equal to 118.4 B.Th.U.*
Location of Losses. It can be judged from Fig. 91 to what extent
the actual engine fails, and also where the failure takes place. It has
* This figure can be obtained by calculation, with ample accuracy, thus :
The range of temperature of the Rankine cycle is from 334. 5 to 212 F. The
mean of these temperatures is 273, and the horizontal distance between the
water line and the adiabatic drawn through the point B (Fig. 91) is 1.18
measured on the entropy scale. The range of temperature is 334. 5 212 =
122 . 5. Hence the B.Th.U. converted into work by the Rankine cycle for i Ib.
of steam =1.18 x 122.5 = 144.6. Therefore for a cylinder feed of o.8i8lb.
the heat utilized is : 144.6 x 0.818 = 118.3 B.Th.U.
THE ENERGY CHART.
76
already been seen that the heat utilised per diagram by the 6 <f>
engine is equal to 65 . 5 B.Th.U. The total loss is therefore the differ-
ence, namely 49.2 B.Th.U. and the " efficiency ratio "* is
-
118.4
- o
~ *
t STEAM PIPE PKESSURC
m
^awiAWMVra^UPiqkll
EXHAUST PRESSURE D
This considerable loss is due to a variety of causes, as follows :
(1) Wire drawing of steam at
admission.
(2 ) Condensation of admitted
steam due to cylinder
walls, and water pres-
ent in cylinder at ad-
mission.
(3) Radiation to external ob-
jects.
(4) Conduction by the cylin-
der walls, piston, etc.,
to the body of the en-
gine and to surround-
ing objects.
(5) Leakage of admission
valve into the cylin-
der.f
(6) Leakage of exhaust valve.
(y) Leakage past the piston
rings.
Incomplete expansion.
ABS: ZERO
c s
FIG. 92. (g)
(9) Wetness of steam at admission.
(10) Increase of back pressure due to exhaust passages and
valve (exhaust wiredrawing),
(n) Compression in clearance.
These losses can be localized on the chart and their magnitude
exhibited as will now be shown.
Loss due to Throttling. In Fig. 92 the steam pipe pressure, the
See recommendations of the Committee on the Thermal Efficiency of
Steam Engines. Proceedings Institution of Civil Engineers, Vol. CXXXIV.
t The leakage direct into exhaust has already been allowed for, see page
69.
LOCATION OF LOSSES. 77
admission and exhaust pressures are shown. It would appear at
first sight that the loss due to throttling is represented by the area
A A B C. It is not, however, quite so great as this, because the
energy represented by this area is still present in the steam in the
form of velocity of the whole mass, a velocity which is, at
any rate in part, arrested when the steam enters the cylinder,
so that it re-appears as heat, and is thus able to do work in
the engine cylinder. The theoretical amount of this possible
work must, therefore, be deducted from the work represented
by the area A^A B C, in order to find the theoretical loss due
to throttling. The matter may be regarded in this way : One
Ib. of steam, in the condition of the state-point C, receives an
amount of heat represented by the area A A B C, and thus attains
to the state point S in the superheated field. B^S is a constant
pressure line, and the area below C B^S is equal to the area
A^A B C. The triangular area below B^S is so small that it may
be neglected, and therefore the value of D K, which determines the
position of 5 can be found, with all needful accuracy, from the follow-
ing equation :
DK x 6 C = A -ZA x (BA _o Ai )
The lengths A B and A C can be measured from the chart
(Plate i), and the temperatures can be read off it. It is clear that of
the heat thus added to the steam the portion represented by
C B^S K D can theoretically be converted into work, and this
portion should be deducted from the area AA B C to obtain the
true theoretical loss due to wire-drawing from the pressure at A to
the pressure at A . If, therefore, a line L M be drawn such that the
area L A B M is equal to the area C B^S K D, then the area
M C shaded thus
represents the loss due to wire-
drawing. It is obvious that :
AL/A A = CD 1C c
The pointLL can therefore easily be found, since A A lt C D and C c can
be read off the chart (Plate I).
Fig. 92 is not drawn to scale in order to better exhibit the
7 8
THE ENERGY CHART.
FIG. 93.
areas under discus-
sion. Fig. 93 is,
however, drawn to
scale, and represents
the case in which
the steam is wire-
drawn from 170 to
150 Ibs. per square
inch, and the ex-
haust is 2 Ibs. per
square inch. The
calculation is as
follows :
From Plate I it
is found that
4^ = 10.2 F.
C = 231.8 F.
Cc = 358.2 +
463 = 819.2 F.
Hence
AL =
231.8 X 10.2
FIG. 94.
819.2
= 2.9 F.
The mean width of the area
A L M C as measured on the
entropy scale is 1.033. Hence
the loss due to wire-drawing in
this case is (10.2 2.9) 1.033
= 7.5 B.Th.U. per Ib. of feed.
Loss due to initial Con-
densation, Leakage and Radia-
tion. In Fig. 94 the line C D
represents the expansion line.
If the engine had been
LOCATION OF LOSSES.
79
perfect the exhaust line would have been the adiabatic through B.
The loss is therefore represented by the area included between the
adiabatic through B and the line C D. As explained on page 74, to
take account of the effect of clearance, the Rankine cycle diagram
must be drawn so that the expansion line falls on the adiabatic
through B, and the water line falls on a a t ; this is the
Rankine cycle for an engine using the " </> cylinder feed "
per stroke. The losses under consideration, per diagram,
are therefore represented by the area shaded thus,
and the B.Th.U.'s meas-
ured off the chart
must be increased in the
proportion of
i
"0 </> cylinder feed"
to obtain the losses per
i Ib. of steam passed
through the cylinder.
But, on the other hand,
the ratio of the actual
areas show the percent-
age losses.
Loss due to Incomplete Ex-
pansion. In Fig. 95, D E
represents the release at con-
stant volume, and if an adiabatic
be drawn through D, the ap-
proximately triangular area
shaded thus
FIG. 96.
represents the loss due to incomplete expansion. It will be seen that
this loss is considerable in the case shown in Fig. 95, which is that of
a condensing engine. For a non-condensing engine the loss is much
less, as will be seen by examining Fig. 96, in which the drop of
pressure is the same as in Fig. 95, namely 6 Ibs. per square inch,
lass due to Throttling through Exhaust Ports. In Fig. 97, E F
pr
FIG. 95.
pr 21
8o
THE ENERGY CHART.
represents the pressure in the cylinder during exhaust, and A is at
the pressure in the exhaust. The difference between these two pres-
sures is the pressure required to
drive the steam through the
exhaust ports and the area
shaded thus
is
l : \*\*\\W'\\\\\vE'
FIG. 97.
obviously the loss due to the
pressure in the cylinder at ex-
haust being greater than the
exhaust pressure.
Loss due to Compression
and Clearance. It was shown
on page 75 that the water line
for an engine cylinder having
a clearance represented by the volume at G, is a a x (Fig. 98), and
if F G is the compression line and G H the constant volume line of
the clearance, it will be
seen that the area includ-
ed between the two lines
a a i and F G H, shaded
a H
thus
repre-
FIG. 98.
sents the loss due to com-
pression and clearance
per " <f> cylinder feed "
which must be increased
as before to obtain the
loss for i Ib. of steam
passed through the en-
gine.
The principal losses
occurring in a reciprocat-
ing steam engine have
now been located on the
chart, and the results obtained have been collected together
in Fig. 99. The area a x a B 8 represents the work that a Rankine
LOCATION OF LOSSES.
81
engine, whose temperature limits are 350 and 130 F. would do, the
feed per stroke being taken as 0.818 lb., and this area divided into
the area H C D E F G representing the work done by the actual
engine is the " Efficiency ratio. 11
Leakage of Admis-
sion Valve past Cylinder
Direct into the Exhaust.
In the example worked
out in this chapter it
was stated that the
feed had been corrected
for the direct leak-
age into the exhaust.
There does not appear
to be any particular
advantage in showing
this loss graphically on
the chart, although
this could be done by
an area added on the
right hand side of the
saturation line. Account
should, however, be taken
of this leakage in
calculating the thermal
efficiency and the effic-
iency ratio. Thus, if
the direct leakage into FI G. 99-
the exhaust is o . 004 lb. per stroke, the real feed, in the example,
will be 0.0382 + 0.004 ~ o.0422lb. per stroke and the thermal
efficiency becomes o. 08 x ~ = 0.072, and the efficiency ratio
0.0382
=
Some of the losses tabulated at page 76 are inherent to the
conditions under which the engine is working, and the material
used in its construction. Others have to be incurred in order to
save greater losses in other directions, such, for instance, is in-
6
82
THE ENERGY CHART.
pr 2 g : o
complete expansion, wire-drawing at admission, and the loss due to
clearance. None of these losses can be absolutely expunged although,
with suitable arrangements, they can be reduced very materially from
what they are in the case under consideration. What these arrange-
ments are is not in the province
of this book to consider, but
their thermodynamic advantage
can easily be tested by means of
the chart, as is illustrated by
the following example :
Losses in a Locomotive
Cylinder due to Throttling at
Admission and at Exhaust.
As an example, the results
of a trial on engine No. 3001
of the Paris-Orleans Railway
can be taken. The indicator
cards are given on page 382,
Proceedings, Institution Mech-
anical Engineers, February,
1904. From these cards it is
seen that at a speed of 36.6
miles per hour :
228 Ibs per sq. in. abs,
216
// /////////////
* **-* "' A A. A Ml-tSj?^
26-
pr 14 r '
FIG. 100.
Boiler pressure
Admission
Exhaust ,,
= 22
and at a speed of 68.0 miles per hour :
Boiler pressure = 228 Ibs. per sq. in. abs
Admission = 195
Exhaust = 26
The data available is insufficient to enable the $ diagrams to be
plotted, but the effect of the throttling of the steam can readily be
seen by the following approximate method. On the chart (Plate I.)
the pressures are set off, and an approximate expansion curve is
sketched in as in Fig. 100. Then the losses are shown, when the
engine is running at the slower speed, by the areas of which the con-
LOSSES IN LOCOMOTIVE CYLINDER. 83
tours are dotted, and when running at the higher speed, by the
shaded areas.
Strictly speaking, the admission loss is not as great as here
shown, but should be corrected by the method described on page
77. Nevertheless this approximation shows the exhaust loss to be
very much larger than the admission loss both at the slower and
higher speeds, and it is therefore evident that a valve gear that will
produce a free exhaust, so long as there is sufficient pressure for the
blast, is of greater economical value than fancy gears for improving
the admission.
Indicator diagrams of the L.N.W.R. locomotive " Precursor "
are given in the Proceedings of the Mechanical Engineers referred to
above (page 82), and it will be found on plotting them, as just
described, that the same inference can be drawn.
Combining the Forward and Back End Indicator Diagrams of
a Cylinder. In a double-acting engine, the feed is distributed
between the two ends of the cylinder, probably not exactly equally.
If it is assumed that the efficiency ratio of each end of the cylinder is
the same (which is probably very nearly true for a horizontal engine,
but only approximately true for a vertical engine owing to the differ-
ence in drainage of the two ends), an approximation, sufficiently
close for practical purposes, will be obtained by combining the two
diagrams and taking the arithmetic mean of the pressures at the
same volume.
CHAPTER VII.
< DIAGRAM OF A SIMPLE JACKETTED ENGINE.
THE </> diagrams 01 a simple jacketted engine will next be considered.
In this case the steam going through the cylinder and that going
through the jackets must be separated, and having done so the
plotting of the </> diagram is effected exactly in the same manner
as described for the non- jacketted engine. The comparison with the
corresponding Rankine engine is, however, somewhat different. The
400 j t matter will be illustrated by
considering the following num-
erical example. The p v dia-
gram is shown in Fig. 101,
which gives the absolute pres-
sures and volumes in the
actual engine. From experi-
ment it was found that o . 0321
Ibs. of feed were required per
diagram, of which 0.0023 Ibs.
passed through the jacket,
leaving 0.0298 Ibs. as the
cylinder feed per diagram.
The temperature of the water
drained from the jacket was
240 F.
Plotting the 6 < Diagram.
Carrying out the method describ-
ed on pages 69 to 72, it is found
that the dryness fraction of the
H a O in the cylinder at release is 0.87, and that the volume factor is
32 . 15. The <f> diagram can now be plotted, as in Fig. 101, as well as
the diagra m for the Rankine steam engine having the same weight
SIMPLE JACKETTED ENGINE. 85
passing through the cylinder. On the same figure is shown the
p v diagram and </> diagram for the engine when working between
the same temperature limits, but without the jackets in use. A direct
comparison can be made between the conditions of the steam during
expansion by drawing the theoretical re-evaporation lines by the
method given on page 54. It will be noticed that in the case of the
non-jacketted engine the theoretical re-evaporation line falls outside
the diagram, while in the jacketted engine it falls inside. This
improvement is effected by the jacket by reducing the condensa-
tion.
Heat per < Diagram. In order to ascertain the actual value of
the jacket, it is desirable to show graphically the amount of heat
supplied by that portion of the boiler feed which is used for the
purpose of warming the jackets. In the actual engine o.oO23lbs.
passed through the jacket per diagram, and since the volume
factor is 32 . 15 the weight passing through the jacket of the <j>
engine is o.074lb. Prolong A B to / (Fig. 101) so that
B J _ jacket feed
a B cylinder feed
then the rectangle below B J will represent the latent heat in the
steam passing into the jacket per diagram. Then draw through / a
curve / S, proportionate to the water line, down to a temperature of
240 F. which is the temperature at which the water leaves the
jacket. The area below B J 8 is the heat per stroke in the jacket,
so that altogether the area whose contour is dotted represents the
heat converted into work per diagram for the perfect jacketted <f>
engine. This area is found by measurement to represent 143
B.Th.U., and likewise the heat represented by the < diagram
of the jacketted engine is 85.1 B.Th.U. ; hence its "Efficiency
Ratio" is
The <f> diagram for the engine without jackets measures 74.1
B.Th.U., but in this case the heat supply is smaller, namely 133
B.Th.U. Hence the " Efficiency Ratio is
" = 0.556;
86 THE ENERGY CHART.
which is less, so that in this case there is a gain by using jackets.
The theoretical diagram thus found does not show the losses in the
cylinder due to incomplete expansion, etc., but these are given by
comparing the actual expansion curve with the adiabatic of the
Rankine engine drawn through the point B.
CHAPTER VIII.
4 DIAGRAMS OF COMPOUND ENGINES.
THE diagram of the H.P cylinder can obviously be treated exactly
as if it were the diagram of a simple engine. The L.P. diagram, how-
ever, requires some special consideration because it receives its
steam not from a boiler, but either from the H.P. cylinder direct or
through the intermediary of a receiver, and in some cases the steam
in the receiver is " re-heated." Further, owing to leaks, the L.P.
cylinder may receive either more or less steam than the H.P.
cylinder. If, therefore, the feed is measured into the engine there
is no certain knowledge as to the amount of feed per diagram in the
L.P. cylinder. If, on the other hand, the consumption of the engine
is measured by the condenser method, then the feed passing through
the L.P. per diagram is known, apart from the direct leakage past
this cylinder into the exhaust, but there is doubt about the feed of
the H.P. cylinder, a doubt which cannot be removed without further
data.
Rankine Cycle for Compound, Engine. Before considering the
case of an actual engine, that of the perfect compound steam engine
(Rankine cycle) will be dealt with. In Fig. 102 let O a be the admis-
sion temperature, Ob the exhaust temperature of the H.P. cylinder
and also the admission temperature of the L.P. cylinder, and O c the
exhaust temperature of the L.P. cylinder. Consider two separate
closed vessels each fitted with a piston and each containing I Ib. of
H 2 O, and let it be supposed that heat can be introduced into or
abstracted from these vessels in any desired manner. Let these
vessels be called I. and II., and to commence the cycle, let vessel I.
contain i Ib. of water at a temperature of 0*,. The state point is
A in Fig. 102. Heat is applied to vessel I. to raise the temperature
of the water to B at so as to reach the state point A, and then
further heat is applied to evaporate at constant pressure from A to J3.
88
THE ENERGY CHART.
The steam is then allowed to expand adiabatically to the point C.
At this stage let vessel II. be considered and suppose that it contains
I Ib. of water at the temperature Ob state point A x , and let matters
be so adjusted that the heat abstracted from vessel I., in order to
follow the transformation line C A it can be introduced into vessel
II., in such a way that the i Ib. of water it contains shall follow the
transformation line A^C. It
is clear from an inspection of
Fig. 102 that the heat rejected
by vessel I. during the trans-
formation C A i is exactly equal
to heat required by vessel II.
to follow the transformation
A^C, hence at the moment
the i Ib. of H 2 O in vessel I.
becomes water at A (thus
completing the cycle in this
vessel), the i Ib. of H a O in
vessel II. will have reached the
state point C, and the heat
transfer from vessel I. to vessel
II. is complete. It will be no-
ticed that the H 2 in vessel II.
is not fully evaporated, but it is
now allowed to expand adiabati-
cally until the state point D is
reached, after which heat is abstracted and work done by the piston on
the H 2 O in such a manner as to obtain the transformation line
D A 2 . At ~A 2 , vessel II. contains i Ib. of water at temperature Ct
and to complete the cycle in this vessel heat has to be introduced
into it to raise the temperature of the water to Ob. The cycles in
both vessels have thus been completed and the initial conditions again
obtained. It will be observed that the heat utilised per Ib. of feed
is the same as that of the perfect steam engine (Rankine cycle)
working between the extreme limits of temperature, namely O a and O c .
The two vessels I. and II. represent therefore the perfect compound
steam engine. The matter has been considered as if it were a transfer
FIG 1 02.
COMPOUND ENGINE: EFFECT OF LEAKS.
of heat from vessel I. to vessel II. during the transformation C A^
but obv ously as regards vessel II. the result would be the same
if the actual I Ib. of H 2 contained in vessel I. at the point C were
transferred in its then condition to vessel II. Vessel I. can, therefore,
be regarded as representing the H.P. cylinder and vessel II. as repre-
senting the L.P. cylinder of a perfect compound engine. The
arguments and results obtained at page 41 et seq., by com-
paring a closed vessel with a steam engine cylinder, apply equally
to each of the above vessels and their corresponding steam engine
cylinders. A triple, or quadruple, expansion engine can obviously
be treated in the same way.
pr,so A
P r 30 A,
P r2 A
D D
FIG. 104.
FIG, 103.
Effect of Leak from H.P. Cylinder. Let it now be supposed
that there is a leak, so that only T %ths of the steam present at C is
transferred from the H.P. cylinder to the L.P. cylinder, the re-
maining j^th being lost so far as the engine is concerned. The
volume of steam in the L.P. cylinder at the point corresponding to C
will in this case be T 9 ^ths of the volume at C ; this determines the
point C 1 (Fig. 104). The quality of the steam will still, however, be
represented by the point C if the leakage past the L.P. cylinder
consists of water and steam in such proportion that the quality of
the steam introduced into the L.P. cylinder is the same as in that
THE ENERGY CHART.
of the Rankine engine and if the expansion is adiabatic the quality of
the steam during expansion is given by the straight line C D, from
which the line C 1 D 1 is deduced in the manner explained on page 64.
Fig. 103 gives the p v diagram of this compound engine with a
leak past the L.P. cylinder, deduced from its <f> diagram (Fig. 104) ;
the line C D shows
what the L.P. p v dia-
gram would have been
had there been no leak.
As regards the quality
of the steam in the
L.P. cylinder, two ex-
treme cases may occur.
In the first case the
whole of the water
present in the H.P.
cylinder may leak away
together with ^th of the
steam present at the
point C ; the steam in
the L.P. cylinder would
then be dry saturated
steam at the point C ,
and the quality line
would be C-L^-L (Fig.
-oradD'Jr A 105), assumed adiabatic,
from which the expan-
sion line C' D is deduc-
FIG< IOS ' ed. The other extreme
case is when none of the water present at C in the H.P. cylinder
leaks away, i.e., it is all transferred to the L.P. cylinder together with
T %ths of the steam. The weight of the water at C (Fig. 102 or 104) is
found from the chart to be o.o97lb. for the numerical data
given in Fig. 102, and this weight of water mixed with ^ths of the
steam at C, namely,
(i 0.097) y 9 ^ = o.8i3lb. of steam,
gives o.giolb. of H 2 0, and the dryness fraction works out to
EFFECT OF CLEARANCE.
B
0.894, a state which is represented by the point C 2 (Fig. 105).
Again, assuming adiabatic expansion, the quality line C 2 d, is drawn
from which the expansion line C' 2 D 2 is deduced.
The effect which a
leak, occurring between
the H.P. and the L.P.
cylinder, has on the <
diagram can thus be judg-
ed. The effect of a leak
from the steam chest past
the H.P. cylinder into the
L.P. cylinder could simi-
larly be ascertained.
Effect of Clearance.
The effect which will
be produced on the
6 <f> diagrams if the
cylinders have clearance
but no compression, and
with proportionally the
same leak as above
past the L.P. cylinder,
will next be studied.
To better show the
effect, somewhat large
clearances will be as-
sumed, namely 0.5
cubic foot in the H.P.
cylinder and 3 cubic feet in the L.P. cylinder. Referring to Fig. 106,
the portion H B D of the < diagram of the H.P. cylinder is easily
drawn. At D there are 12.2 cubic feet of steam, and by supposition
T ^th of this steam leaks past the L.P. cylinder and is lost to the engine.
For simplicity it will be assumed that the whole of this leak takes place
at the point Z), and not during expansion or admission. This assump-
tion is expressed, by the point B being on the saturation line, and by
the expansion line B D being vertical. It will also be supposed that all
the water present at D in the H.P. cylinder leaks away at the same
FIG. 106.
92 THE ENERGY CHART.
time. On this assumption there are : 12 . 2 (i o. i) = n . 16 cubic
feet of steam at 30 Ibs. pressure, whose weight is - ~^ = o . 828 lb., to
3 * i
put into the L.P. cylinder, which at that moment contains steam in
the condition, as regards pressure and volume, represented by the
point F (the point F t can be located because the back pressure
is 2 Ibs. abs., and the clearance volume is 3 cubic feet). If it is
assumed that there is no water present at F , the quality in the
L.P. cylinder at this point is represented by / on the saturation
line.
Initial Condensation in L.P. Cylinder. A portion of the steam
from the H.P. cylinder on entering the L.P. will be condensed, and its
latent heat will disappear into the cylinder walls, etc. Let it be as-
sumed that 10% is condensed then ^ x 0.828 lb. of water will be
produced. On the assumption already made that there is no water
in the L.P. cylinder at the point F , the weight of steam present at
that point will be in the ratio of the volumes at F and / 1? or :
Altogether, therefore there will be & x 0.828 + 0.017 = 0.762 lb.
of steam, and 0.083 lb. of water present in the two cylinders immedi-
ately after communication has been established, which gives a dryness
fraction of 0.9. The combined volume is 12.2 + 3.0 = 15.2 cubic
feet, so that the volume per lb. is
I^ 2
,- =20.0 cubic feet,
and on reference to Plate I., it will be seen that the intersection
of this volume line with the 0.9 dryness fraction line occurs at a
pressure of 17. 7 Ibs. per square inch therefore, this must be the
pressure established in the L.P. cylinder when the communica-
tion is opened between it and the H.P. cylinder. The intersection of
this pressure line with the volume line for 3 cubic feet gives the point
H of the L.P. cylinder (Fig. 106), and the intersection with the 12.2
cubic foot volume line the point E of the H.P. cylinder. The quality
of the H 2 is = 0.9, and the " quality point " e is thus located. A
property of the quality line is that the ratio of the volume at any point
on this line to the volume of the steam it represents is a constant.
In the case under consideration the steam is contained in two com-
EXPANSION IN L.P. CYLINDER. 93
municating cylinders. Hence the ratio of the volume on the quality
line to the sum of the volumes of the two cylinders will be a constant
for any pressure line, and in the example this ratio is : the sum of
the volumes at H and E (= 15.2) to the volume at e (= 20.0),
The ratio is therefore I 5- 2 -
20.0.
Expansion in L.P. Cylinder. This engine has no receiver,
and therefore the same valve acts as the exhaust valve of the H.P.
cylinder and as the cut-off valve of the L.P. cylinder. Until the valve
closes the two cylinders are in communication, and the steam expands
in them as if they were one vessel. Let it be supposed that this ex-
pansion is adiabatic, and that none of the heat stored in the cylinder
walls is returned to the steam. The quality of the mixture at the be-
ginning of the expansion is given by e, so that the quality line of the
mixture is the adiabatic e d . At the moment the valve closes the
volume in the H.P. cylinder is, by the assumption made as to clear-
ance, 0.5 cubic foot, and the time required for describing E F is
obviously that required for one stroke. During this time the L.P.
piston will also have moved one stroke, that is to say, the L.P. exhaust
valve will be on the point of opening. At this stage it is necessary to
know the volume swept by the L.P. piston ; and inasmuch as the
clearance was not specified as a percentage of the volume swept, any
reasonable volume can be assumed for the purpose of Ms example,
sa Y 33 cubic feet. Therefore at the moment the valve closes there
are 33.0 + 3.0 + 0.5 cubic feet in the two cylinders, and the
steam has therefore expanded from 15.2 to 36.5 cubic feet or 2.4
times. The volume at e for i Ib. of steam is seen from the chart
to be 20 cubic feet, so that with 2.4 expansions the volume at the
point in the quality line corresponding to the point F will be
20 x 2 .4 = 48 cubic feet, and it will be seen from the chart that this
point on the quality line is at the pressure 6 . 5 Ibs. per square inch.
The point F in the H.P. cylinder is therefore found by the inter-
section of the 6 . 5 pressure line with the o . 5 cubic foot volume line,
and the point D it in the L.P. cylinder is at the intersection of the
same pressure line with the 33.0 + 3-0 = 36.0 cubic foot volume
line.
In the above, the quality line e d was assumed to be adiabatic,
94
THE ENERGY CHART.
and it will be interesting to find the effect of a different assumption
on the position of the points F and Z) . Fig. 107 is a reproduction
of a portion of Fig. 106. First, let it be assumed that the quality
line is e d\. As previously, the point d\ (corresponding to D )
will lie on the 48 cubic foot volume line, and is therefore determined
as shown in Fig. 107 whence D\ and F' are found by drawing the
pressure line through d\. The same construction holds if the
" quality " line
found.
is e d'\, and the points D'\ 9 and F" are thus
L.P
H.P
FIG. 107.
FIG. 1 08.
To fix other points in the lines H^ D l and F E it is first necessary
to find the relation that exists between the volumes in the two
cylinders at different points of the stroke. As the case under
investigation is purely ideal, it will be assumed that the ratio of
the connecting rod to the crank is infinity. The point D in Fig. 106
corresponds to point D in Fig. 108, and the cranks being assumed
opposite, the simultaneous point for the H.P. cylinder is F lf both in
Fig. 106 and Fig. 108. This latter figure shows the condition of things
after an angle ^ has been described by the crank pins. The volume
cos (2 TT \l/) \ + 0.5 cubic feet.
EXPANSION IN L.P. CYLINDER. 95
swept by the L.P. piston up to the point P is*
( r r cos (2 TT i/') j L
where L is area of L.P. piston. But 2 r L plus the clearance is
obviously equal to the volume in the cylinder at the point Z) (Fig. 106),
which is seen to be 36 cubic feet. Hence the volume in this cylinder
when the crank has described an angle of \f/ is
( i cos (2 TT i/') J +3 cubic feet.
Similarly the volume in the H.P. cylinder at the same moment is
12.2 0.5
2
The sum of these volumes at the point P under consideration is
therefore.
22.35 f 1 cos (2 ^ ^)) + 3-5 cubic feet.
Referring to Fig. 106, suppose it is desired to find for each cylinder
the point on the respective <f> diagram when the pressure is u Ibs.
absolute. The volume on the quality line at this pressure is seen to
be 30.5 cubic feet, and since the volume on the quality line at e is
20.4 cubic feet, and the sum of the volumes in the two cylinders at
HI and E, is 15 . 2 cubic feet, therefore the sum of the cylinder volumes
at P and P is
15.2
^ x 30.5 = 23.2.
Equating this volume to the expression found above, the
following equation is obtained :
23.2 = 22.35 ( i cos (2 TT i/') j + 3.5
whence i cos (2 TT \f>) = o . 882
and finally
Volume in H.P. = 5. 85 x 0.882+0. 5 = 5. 66 cubic feet
L.P. 16.5 x 0.882 + 3.0 = 17. 55 cubic feet.
The points on the n-lb. pressure line can thus be located at P and P 1 .
Other points can similarly be obtained, and thus the complete
exhaust line in the H.P. cylinder, and the admission line in the L.P.
cylinder can be drawn.
* On the assumption just made, that the connecting rod is infinite,
although not so shown in Fig. 108.
Q6 THE ENERGY CHART.
Exhaust Line of L.P. Cylinder. At D the exhaust of the L.P.
opens, and the diagram of this cylinder follows the constant volume
line drawn through D until the back pressure line at 2 Ibs. is
reached at E, then it follows the pressure line up to the point F^.
Other Assumptions. To obtain these < diagrams a great num-
ber of assumptions have had to be made. Any other assumptions
could, however, have been dealt with in a similar manner, such, for
instance, as initial condensation in the H.P. cylinder, leakage past
the piston rings, drop of pressure through the exhaust valve of the
H.P. cylinder, etc., and the corresponding <f> diagrams could have
been drawn.
The above example is not intended to represent the case of an
actual compound steam engine, but to exhibit various thermo-
dynamic transformations on the chart, and to show the com-
parative ease with which such problems can be solved by this graphic
method. It will be probably admitted that such problems are
practically insoluble by a purely algebraical method.
CHAPTER IX.
COMPOUND ENGINES.
TRANSFER OF INDICATOR DIAGRAMS TO THE ENERGY CHART.
IN this chapter several examples of compound and triple-expan-
sion engines have been placed on the chart, but before describing
them it is necessary to settle whether the volume factors should be
taken the same for each cylinder or not.
Volume Factor. On referring to page 71 it will be seen that
the volume factor depends on the total volume in the steam cylinder
which is the sum of the clearance volume and the volume swept by
the piston. Therefore, if in a compound or triple-expansion engine
each cylinder is treated separately, the volume factor will not be the
same for each cylinder, unless the weight of play steam in each cylin-
der is the same. This method of treating each cylinder separately
was originally adopted by the Author, as will be seen by referring to
the remarks he made at a meeting of the Institution of Mechanical
Engineers (February, 1894), an d has the advantage of showing the
quality of the steam during the expansion in each cylinder, provided,
however, there are no leaks out of or into the engine ; that is to say,
the flow of H 2 O in each cylinder is the same. There is, however, the
disadvantage that the areas of the </> diagrams are not proportional
to the work done in each cylinder, as they would be if each cylinder
had the same volume factor. On the whole, however, it is better to
usj different volume factors, as by this method the various calcula-
tions are considerably simplified, and the graphic representation is
more easily grasped, and therefore it has been adopted in the
following examples : *
EXAMPLE I.
Compound Condensing Engine (non-jacketted). The indicator
diagrams for this engine are given in Fig. 109, and the following
particulars of the engine are needed to draw the <f> diagrams :
* Although the references are not given, the data for these examples are
taken from actual tests.
THE ENERGY CHART.
SO-
SO-
10-
O J
FIG. 109.
FIG. no.
Diameters :
H.P. cylinder = 15^1 ins.
L.P. cylinder 3i ins.
Stroke = 48 ins.
Revs per min. =75
Clearances :
H.P. cylinder = 8. 3%
L.P. cylinder =5.6%
Cylinder feed per
stroke 0.612 Ibs.
The same process is used
as for the example given
on page 69, a different
volume factor being ob-
tained for each cylinder,
which works out as
follows :
H.P. cylinder 1.35
L.P. cylinder ^1.44
The <$> diagrams have
been plotted from the
indicator cards, using the
method given in Appen-
dix I., and are given in
Fig. no; it will now be
shown how the economy,
thermal efficiency, effici-
ency ratio, etc., can be
arrived at from these
diagrams.
The areas of the </>
diagrams are found to
represent in the H.P.
cylinder 79.7 B.Th.lL,
and in the L.P. cylinder
74.5 B.Th.U.
COMPOUND ENGINE EXAMPLE. 99
The 6 <j> cylinder feed of the H.P. cylinder is found by multiplying
the H.P. cylinder feed by the corresponding volume factor, and is
therefore equal to 0.85 Ib.
The Rankine cycle corresponding to this <f> feed has been drawn
and is shown by the area whose contour is dotted. Since, however,
the L.P. volume factor is larger than that of the H.P. cylinder, its
</> feed will also be larger in proportion of the volume factors.*
Hence the 74.5 B.Th.U. shown by the L.P. diagram must be reduced
to agree with the smaller H.P. <f> cylinder feed so that the corrected
value is :
74-5 x ^g = 69.9 B.Th.U.
Efficiency Ratio. The area of the Rankine engine cycle repre-
sents 234 B.Th.U., and the sum of the H.P. and L.P. diagrams of the
actual engine =79.7 + 69.9 = 149.6 B.Th.U. Therefore the
140. 6
Efficiency ratio = - = 0.64.
Economy of the Engine. As was proved in Chapter VIII.
Heat supply per stroke
Economy of engine - 42.4 * H eat represented by ^ diagram
/expressed as B.Th.UA
\ per I. H.P. per min. J
The heat supply per stroke is found in the following manner :
Total heat of steam at 353 = 1189.6 B.Th.U.
Less water heat at 126 = 94.2
Nett heat supply = 1095.4
These values are read off the chart (Plate i), and are for i Ib. of
steam, but the actual engine works with 0.85 Ibs. of < cylinder feed.
Therefore the nett heat supply per stroke for this engine is equal to
1095.4 x 0.85 =931 B.Th.U. Thus:
Economy of engine = 42 . 4 x jg = 264 B.Th.U. per I.H.P. per min.
Thermal Efficiency. The heat converted into work has just
been found to be 149.6 B.Th.U. Hence :
Thermal efficiency = -^~ = 0.161
y3
* This accounts for the compression line of the L.P. cylinder lying to the
left of the Rankine cycle water line.
100 THE ENERGY CHART.
Steam Consumption of Engine. This is obtained by the formula
given at page 67, and in this case the consumption is :
14 x Ot8s = J 4-4 6 lbs - P er I-H.P. per hour.
The ' 'equivalent feed" will be smaller than this in the proportion of the
nett heat supply per stroke (viz., 1095.4) to uoo B.Th.U.* or :
Equivalent feed 14.46 x "~ 14. 40 lbs. per hour.
Mean Pressure. The formula as given in Chapter VIII. is :
_ Heat represented by 6 <f> diagram.
M.fc.P. = 5.4 * 0- ^ VoTj at rei^se '^T$ Vol. of clearance.
In the case of the H.P. cylinder this works out as follows :
7Q 7
M.E.P. =5.4 x . Q - A = 5Q.8 lbs. per square inch.
y . O - O . O
The M.E.P. referred to the L.P. cylinder is calculated in a similar
manner, but here it is necessary to alter the volume in the clearance
and at the release by the ratio of volume factors.
Thus M.E.P. - 5.4 x ?9-7 + 6 9-9 _
i.^
33-5 x 4^-0.6
= 26. 3 lbs. per square inch.
Cut-Off in Cylinders. The cut off is given by the formula :
Volume at cut off clearance volume
Point of cut off = volume swept by the piston. -
Hence for the H.P. cylinder
^ O - r\ r\
Point of cut off = r^g-zr^g = 28% of the stroke.
and for the L.P. cylinder
Point of cut off = *3*Zi'.77 ^ 3 8 - 6 % of the st ke.
Number of Expansions. This is given by the formula :
, f . Total volume in L.P. cylinder.
Number of expansions = TT -^ - , ^ ^ . TT -/ r . 3
F Volume at C.O. in H.P. cylinder.
i.44
= 12. i
2.6
See line 131, page 18, Report of the Committee on Steam Engine and
Boiler Trials. Institution of Civil Engineers.
COMPOUND ENGINE EXAMPLE. IOI
It is clear that the total volume has to be adjusted proportionately
to the volume factors.
Cylinder Ratio. The clearance and release volumes for both
cylinders are read off the chart, and those of the L.P. cylinder
must be adjusted by multiplying by the ratio of the volume factors
thus :
H.P. 6 <f> Clearance volume = 0.6 cubic feet
H.P. , Release = 7.8
L.P. Clearance = 1.77 x 0.937 = 1.6
L.P. Release = 33.5 x 0.937 = 31.4
Then < volume swept by H.P. piston =7.2
L.P. = 29.8
7.2
Hence the cylinder ratio = ^g = I to 4.14
Proportion of Work Done in Each Cylinder. The work done in
each cylinder is in the proportion of the areas of the diagrams adjusted
for the volume factors, thus :
Work done in H.P. cylinder 79.7 i
,, L.P. 69.9 0.88
EXAMPLE II.
Compound Condensing Engine. The p v diagrams for this engine
are given in Fig. in, and it requires 25.5 Ibs. of steam per I. H.P.
per hour. The $ diagrams are given in Fig. 112.
A comparison of the p v diagrams of this engine with those of the
last example gives no indication that the engine is far less economi-
cal, although working under the same conditions, but a
comparison of the < diagrams brings out this fact in
a clear manner. The theoretical expansion line has been
drawn through the point of cut-off in the H.P. cylinder, and
the actual expansion line falling so far inside this line indicates
that a serious leak takes place in this cylinder, namely past the
piston and through the exhaust ports and into the L.P. cylinder.
The effect of the additional weight of steam flowing through the L.P.
cylinder is well exhibited by the increased size of its < diagram.
102
I_SS. ABS.
ISO-
THE ENERGY CHART.
The following par-
ticulars relating to this
engine have been cal-
culated in the same
manner as given in
Example I.
Volume factors :
H.P. cylinder =1.29
L.P. cylinder 1.37
Heat supplied per
min. per I. H.P.
= 438 B.Th.U.
Equivalent feed
= 24.3
Efficiency ratio
= 0.48
Thermal efficiency
= 0.097
EXAMPLE III.
Jacketted Compound
Condensing Engine, and
with Reheater between the
Cylinders.
Pressure at stop valve
= 171 Ibs. per square
inch, abs.
Superheat at stop valve
= 81.5 F.
The p v diagrams for this
engine are given (re-heater
in action), in Fig. 113, and
the </> diagrams in Fig.
114.
COMPOUND ENGINE EXAMPLE.
103
Lbs. of steam used per
I.H.P. per hour =
11.24
The 6 < diagrams have
been drawn with the fol-
lowing volume factor ratio :
H.P. cylinder _ i
L.P. cylinder 1.05
It will be noticed that
the steam in the L.P. cyl-
inder is nearly dry when
the re-heater is in use,
but is somewhat wet,
as shown by the dotted
expansion line, when the
re-heater is out of action.
It will also be noticed that
the steam at cut off in the
H.P. diagram is very dry
owing to the use of super-
heated steam, but the in-
itial superheat is not suffi-
cient to keep the steam
superheated up to the
point of cut off. The
following are the econo-
mic results deduced for
this engine :
Heat supplied per min.
per I.H.P. = 214
B.Th.U.
Equivalent feed
Efficiency ratio
= 0-7I5
Thermal efficiency
= 0.198
UBS
I BO-
FIG. 113.
-4-00*,
FIG. 114.
104 THE ENERGY CHART.
A close comparison of the indicator diagrams given in Figs, no,
in, 113, and their respective </> diagrams is recommended, and
attention is called to the manner in which the latter exhibit the
great difference in economy that exists between these three engines,
a difference which is not shown by the indicator cards.
EXAMPLE IV.
Horizontal Four-valve Fleming Compound Engine, with Reheater
between the Cylinders. The indicator diagrams for a test made at
about the full rated load (500 I.H.P.) are given in Fig. 115, and the
further particulars needed to transfer the diagrams to the energy
chart are as follows :
High pressure cylinder diameter . . . . 15 inches.
Low .... 40.5
Stroke .... 27
Diameter, piston rod, H.P .......
j Crank end .. 4!
' * 'Headend ..
Clearance, H.P. cylinder ...... 3.95 %
L.P. ...... 4-67%
Feed water per I.H.P. per hour passing
through the cylinders ........ 12.8 Ibs.
Reheater steam (5% of cylinder steam) .. o.7lb.
Leakage past admission valve into the
exhaust (3% of cylinder steam) . . 0.4 Ib.
The ^diagrams should be drawn for a feed of 12.4 Ibs., and are
given in Fig 116. The stop- valve pressure, 167 Ibs. per square inch
absolute, and the condenser pressure 1.9 Ib. per square inch abso-
lute, have been marked. On measuring the diagrams it is found
that the H.P. diagram represents 91.5 B.Th.U., and the L.P.
diagram 91.1 B.Th.U., and since the percentage clearance h nearly
the same in both cylinders, the difference in the volume factors can
be neglected.
The following calculations are given to show the degree of
accuracy that may be expected from <f> diagrams.
The point a on the 167 Ib. pressure line is found* to be situated
See page 74.
COMPOUND ENGINE EXAMPLE.
105
at the volume 0.22 cubic foot, and since the volume of saturated
steam at this pressure is 2.68, the
a i v j / j 2.6l '22 n,
<J> cylinder feed is -gg = 0.9210.
But the steam going through the re-heater is 5%, and 3% has to
be added for the direct leakage. Hence the total feed of the cor-
responding (f> engine per stroke is
0.92 (i + 0.05 + 0.031 = 0.994 Ib.
from which the feed per I.H.P. of the actual engine works out to
2545
91.5 + 91.1
x 0.994 = 13.8 Ibs.
instead of: 13.9
as given by the
data.
The mean
pressure of the
H.P. cylinder is
91.5
5.4 x
7.2 - 0.28
= 70.8 Ibs. per
square inch, and
the figure given
in the Paperf is
69.9. The mean
pressure of the
L.P. cylinder is
5.4 x ._$L*__
52 - 2.4
= 9.9 Ibs. per
square inch,
x
Ibs.'obs.
/
- 120
s^^^ /
- 80
r
~~""^ /
* t ^
- o
7
1 1 1 1 I [
e s 4 3 z 10
66
VOL. CUB. FT.
' n lbs.abs.
. 20
FIG. 115.
whereas the figure given in the Paper is 9.7.
The ratio of the cylinders is
7.2 0.28 __ _j
52 2.4 ~~ 7".^
whereas in the Paper the ratio is given as i : 7.33.
f The data for this example are taken from a Paper read before the
American Society of Mechanical Engineers (Vol. XXV.)
106 THE ENERGY CHART.
The total heat per Ib. of steam for the admission temperature
of 367 F., and the exhaust* temperature of 130 F. will be found to
be, on reference to Plate I., 1193.8 98.3 = 1095.5 B.Th.U.
Hence the thermal efficiency as regards the steam passing through
the cylinders of the engine is
91.5 + 91.1 g
1095.5 x 0.92
But this figure must be reduced to take account of the direct
leakage and of there-heater steam, in the proportion of 0.994 to 0.92
(see above). Thus the thermal efficiency of the engine is
o o-Q 2 a
o.ioi x - , = o.ioo
0.994 ^
The heat converted into work by the corresponding Rankine
engine is i . 19 x 243 = 289 B.Th.U.
Hence the thermal efficiency of the Rankine engine is
289
and the efficiency ratio is ~ ^4- Lastly, the economy of
the engine is ^^gt = 251 B.Th.U. per I.H.P. per minute.
It will be observed that the cylinder ratio is very high for a
compound engine, but the cut-off in the L.P. cylinder has been
arranged so as to equalize the work done in each cylinder, namely,
91.5 and 91.1 B.Th.U. A very considerable toe is thus produced
in the H.P. diagram, as is seen both on the < and on the indicator
diagram ; it is, however, more conspicuous on the former. Mr.
Rockwood, who is the author of compound engines with large
cylinder ratios, maintains that this drop tends to dry the cylinder
walls, and thus reduces the initial condensation. It may be added
that Mr. Willans was of the same opinion, and the quality of the
steam during the expansion, as shown by the 6 <f> diagram of the
H.P. cylinder confirms this view. The shape of the L.P. expansion
line points to a leak (see page 59) in the admission valve of that
The temperature at the exhaust of the engine is not given in the data,
but it may reasonably assumed to be isoF., and this is the temperature to
take as the lower limit for the Standard of Comparison of the Institution of
Civil Engineers. The Superheat has also been neglected.
TRIPLE-CONDENSING ENGINE EXAMPLE.
107
367
cylinder, and if this leak had not existed, the expansion line would
have followed approximately the chain dotted line, and the drying
effect of the re-heater would have become more obvious (compare
Fig. no). Points
have been marked
along the perimeter
of the indicator
diagrams at equal
intervals of time,
namely $th of a
revolution, and since
the cylinders are
placed tandem
fashion, the instant
i will be at the
beginning of the
stroke in both cyl-
inders. These
points have also
been marked on the
6 < diagrams.
ST
FIG. 1 1 6.
EXAMPLE V.
Triple-Condens-
ing Engine. In Fig.
117 are given the
p v diagrams for
this engine and in Fig. 118 are shown the corresponding
< diagrams. There is nothing specially noticeable about this
engine, and it can be classed as of an average type, but the economy
is very good.
The following are the economic results for this engine:
Heat supplied per min. per I.H.P. = 217
Lbs. of steam used per hour per I.H.P. = n .8
Equivalent feed = 11.7
io8
THE ENERGY CHART.
Efficiency ratio
Thermal efficiency
0.764
0.197
26-
20-1
FIG. 117.
FIG, 1 1 8,
EXAMPLE VI.
Triple-Condensing JackeUed Engine With and Without Reheater
and Jackets. Fig. 119 shows the p v diagrams for this engine with the
reheater in use, and Fig. 120 with the reheater out of action. Fig.
121 shows the $ diagrams for the two cases ; the full lines being
those for the engine when using the reheater and jackets, and the
dotted lines without the reheater and jackets. The two diagrams
plotted on the chart show in a very clear manner how much drier
the steam is throughout the expansion when the engine is using
the reheater and jackets.
TRIPLE EXPANSION ENGINE EXAMPLE.
ICQ
FIG. 119. FIG. 120.
The following are the economic results for this engine :
With
Re-heater
Without
Re-heater
Heat supplied per min. per I.H.P. . .
249.7
264.4
Lbs. of steam used per hour per I.H.P.
13-5*
13-9
Equivalent feed
13-7
I4.I
Efficiency ratio
0.610*
0.575
Thermal efficiency
0.171*
0.161
* The steam flowing through the cylinders was 1 1 . 3 Ibs. per I.H.P. per hour ;
the difference, 2. 2 Ibs., was used by the re-heater and jackets. The efficiency
of the steam in the cylinder is therefore 0.73, and its thermal efficiency is o. 204.
no
THE ENERGY CHART.
On page 81 (Fig.
99) it was shown how
the amount of the various
losses due to initial con-
densation and leakage,
etc., incomplete expan-
sion, back pressure, etc.,
could be graphically
shown on the chart,
and this process can be
applied in a precisely
similar manner to the
preceding examples. This
is a matter left to the
student.
FIG. 121.
CHAPTER X.
USE OF THE ENERGY CHART IN DESIGNING STEAM ENGINES.
So far the < diagrams have in general been obtained from the
indicator diagrams and the dimensions of the engine. It is now pro-
posed to reverse the process, that is to say, it is required to draw
the <j> diagrams, and from them determine the proportions of the
engine and the indicator diagrams.
Pre-determination of <f> Diagram. From the various formulae
given at the end of Chapter V. it will be seen that if the <
diagram of a simple engine is given, the mean pressure, the point
of cut-off, and the economy of the engine can be calculated very
easily, and obviously the p v diagram can be obtained by simply
plotting the pressures and corresponding volumes as read off the
chart. The question thus arises can the 6 < diagram of a steam
engine be pre-determined. The answer is, it can be done with
fair accuracy in any case, and if there is some previous knowledge
of the type of engine under consideration, its valve motions, etc.,
etc., the 6 <f> diagrams can be laid down with a considerable degree
of accuracy. It is certain that the design of an engine can be
worked out, as regards its thermo-dynamics, far more easily by this
method that by the usual p v method, and with a greater degree of
accuracy. This statement will be illustrated by means of the
two following numerical examples.
Example Simple Engine. Determine approximately the </>
diagram of a non-condensing, non-jacketted simple engine, when
working at the best point of economy, with a stop- valve pressure
of 100 Ibs. per square inch absolute. Speed about 150 r.p.m.
Since no details are given, any reasonable assumptions are ad-
missible. Some of these assumptions will afterwards be varied to
see the effect on the general result.
Sketching in 6 <f> Diagram.* The admission line must first be dealt
* It is suggested that the diagram be sketched on tracing paper placed over
the chart, Plate I.
112
THE ENERGY CHART.
with, and it will be assumed that there is 5 Ibs. drop in pressure at the
beginning of admission, gradually increasing to 10 Ibs. at the point of
cut-off. Since the engine is non-jacketted and the speed is moderate,
it is a matter of general knowledge that there will be considerable
initial condensation ; let it therefore be assumed that the dryness
fraction at cut-off is only o . 65. On these assumptions the admission
line H C can be drawn as shown in Fig. 122, neglecting for the
moment the clearance in the cylinder. The theoretical re-evapor-
ation line through C is now drawn by the method explained on page
54, as shown by the chain-dotted line, and it is known by experience
FIG. 122.
FIG. 123.
that the actual expansion line falls somewhat short of this, it can
therefore be drawn in approximately as shown by the full line C D.
Turning to the exhaust line, let it be assumed that the opening of
the exhaust valve is such that a difference of i Ib. per square inch in
pressure is required to drive the steam out of the cylinder, that is to
say, since the back pressure is 14.7, the exhaust pressure must be
15.7 Ibs. per square inch absolute. The exhaust line F D of the
diagram can therefore be drawn in. The intersection D of this
exhaust line with the expansion line C D occurs where the volume
is 22 cubic feet, and the area of the diagram included between the
DESIGN OF A SIMPLE ENGINE. 113
lines H C D F is found on measurement* to be 10.0 square inches,
representing 10 x 10 B.Th.U., so that the mean pressure is
100 x 778
= 22 X 144 == 24 ' 5 lbS ' Pe
Adjustment of Mean Pressure. From many trials it
is known that for a non-condensing engine the best economy is
obtained with a mean pressure between 40 and 45 Ibs. per square
inch ; the release volume must therefore be reduced approximately
in the proportion of 24.5 to 45, that is to say, it ought to be about
12 cubic feet. The release at constant volume ought, therefore,
to take place approximately along the 12 . o cubic foot constant volume
line, as shown in Fig. 122. The area of the 6 <j> diagram is thus
reduced to 9.42 square inches, and the mean pressure becomes
04.2
5.4 x - - = 45.0 Ibs. per square inch.
12
which is within the limits assigned above.
Effect of Clearance. Let it now be assumed that the clearance is
6% of the total volume of the cylinder, that is of the volume swept
by the piston, then the clearance volume must be
> x 12 = 0.68 cubic foot,
and thus the compression line F H can be drawn (Fig. 123), on
the supposition that the exhaust closes at such a point that
the compression pressure will just reach stop- valve pressure
at the moment the admission valve opens. Judging from the 6 <
diagrams given in Figs, no to 121, no serious error will be made
if F H (Fig. 123) be taken as the compression line of the <f> diagram.
The volume swept by the piston is, however, diminished by the
clearance volume and becomes 12 0.68 = 11.32 cubic feet.
The area of the <f> diagram is thus somewhat reduced, in fact
to 7.3 square inches, so that the mean pressure is reduced to
34. 6 Ibs. per square inch. This is a somewhat low mean pressure
for economy under the conditions imposed, the release volume must
therefore again be reduced to, say 9.5 cubic feet, which alters the
clearance volume to o . 55 cubic foot. The area of the </> diagram
* When laid down on the larger energy chart, Plate I. This note applies
to all the subsequent areas given.
8
114 THE ENERGY CHART.
now becomes 6.87 square inches, so that the mean pressure is
68 7
5.4 x 7 /re =42.0 Ibs. per square inch.
-/ -----
V 9 ' 5 "^o
which is within the limits specified above, and thus the final <
diagram is as shown in Fig. 125.
Results Obtained. The point of cut-off is obtained from the
formula given at page 66, so that inserting the numerical values
of the various volumes as read off the chart (see also Fig. 124) :
Point of cut-off = y/Zolss = ' 29
The economy of the engine can be obtained by applying the
formulae given at page 67, or else as follows : From Fig. 125 it
will be seen that 4 ' 3 ~^2 >55 = 0.87 of a Ib. of steam is admitted
4.36*
to the cylinder per
0.55 3.15 C 9 B 's T stroke. The tempera-
4
-I
VOL
OL-ATCUTOFF-*-
AT RELEASE
ture of admission is
327.5 F., and the
temperature of the ex-
haust steam is 212 F.
TTon^A iViA "R Th TT Qiin-
VOL. IN CLEARANCE
plied per Ib. of steam is
FIG ' 124 ' 1182 - 180 = 1002
B.Th.U. per Ib. Hence the heat supplied per stroke is 0.87 x
1002 = 872 B.Th.U. The work produced is represented by the
area of the </> diagram (Fig. 125), which has already been found
to be equal to 68.7 B.Th.U., therefore the thermal efficiency of the
engine is -o- 1 - = 0.079, and the economy of the engine
= 4 2 -4 = Z-ZA B.Th.U. per I.H.P. per minute.
0.079
4.36 cubic feet is the volume of i Ib. of saturated steam at the stop
valve pressure, viz.: icolbs. per square inch abs., which is the pressure the
engine has to account for. 0.55 is sensibly the volume at the point a of the
"proportional " water line of the Rankine cycle corresponding the <f> cylinder
feed.
DESIGN OF A SIMPLE ENGINE. 115
Comparison with Standard. The Institution of Civil Engineers
standard steam engine of comparison working under the same
temperature conditions and supplied with 872 B.Th.U. per stroke,
produces the work, represented in Fig. 125 by the area whose
contour is shaded by dots, containing 12.02 square inches. Hence
the thermal efficiency of the corresponding standard steam engine is
120.2
= 0.138,
and the " efficiency ratio " is ~ = 0.57.
FIG. 125.
FIG. 126.
On reference to Fig. 3 of the Report of the Thermal Efficiency
Committee of the Institution of Civil Engineers, it will be found
that the standard engine of comparison working between the
temperature limits of 327.5 F. and 212 F., requires 310 B.Th.U.
310
per I.H.P. per minute. Hence the efficiency ratio = ~ =0.57
JT'T'
or the same result as before
The indicator diagram corresponding to the <f> diagram thus
obtained should be plotted, as a final check, but this is left to the
student.
Il6 THE ENERGY CHART.
Effect of Changing the Assumptions made. The shape of the <
diagram just obtained depends on the assumptions made, which are
tabulated below for reference :
Admission line : 5 Ibs. drop at admission, increasing to TO Ibs.
drop at cut-off.
Expansion line : Dryness fraction of steam present at cut-off
0.65. Expansion line sketched in, using theoretical
re-evaporation line as a guide.
Release : Adjusted to give between 40 and 45 Ibs. mean
pressure.
Exhaust line : I Ib. loss of pressure due to ports.
Compression line : Such as to give admission pressure at the
moment of admission.
Clearance in cylinder : 6%.
The effect on the diagram of varying some of these assumptions
will now be made.
Change in Admission Line. The admission line will first
be varied by supposing that the engine is fitted with a better
admission valve, and that in this way the drop at the beginning of
the stroke is reduced to i Ib., and the drop at cut-off to 2 Ibs. The
<t> diagram given in Fig. 126 is thus obtained. It will be seen that
the volume at cut-off is less than in Fig. 125, and therefore, since
the point of cut-off has not been altered, the release volume must be
diminished. Referring to the formula given on page 66, it will be
found by transposition that :
V. = V <~ V * +Vk
C
where V r = Volume at release
V c = cut-off
Vit = in clearance
and c = Point of cut-off
If k is the percentage clearance, then
V k = k (Vr V k )
* =
DESIGN OF A SIMPLE ENGINE. 117
Hence
*
v t = v c i + k
,
"*"
i + k
v ^ i f &
_(' + ,- " -^-< +k '
and in the numerical example
__ _JE + 0.06
*" /^w ^-
0.29 + 0.06
= 9.0 cubic feet approximately,
which is the volume at release as shown in Fig. 126. The area of
this $ diagram is found to be 7 . 15 square inches. Hence the mean
pressure is
5-4 x (9 Q ^1 Q 5) " = 45-4lbs. per square inch,
or, as might be expected, higher than before, but still not too high
for economy. The heat supplied per stroke is, however, somewhat
larger than in the former case, because the clearance volume is
smaller. Thus, the weight of feed per stroke is
4,36- 0,5- =
and since the total heat per Ib. is the same as before, namely, 1002
B.Th.U., the heat supply per stroke is 885 B.Th.U.
Thus the Thermal efficiency is
"88*5 = - 8 >
and the Efficiency ratio is
0.08
~o7i 3 8 = 0-59-
Finally the economy of the engine is
^J = 530 B.Th.U. per I.H.P. per minute,
or nearly the same figure as before. The reason that there is no
sensible thermo-dynamic improvement is that the cut-off has been
kept the same if it were made slightly earlier a better economy
would be obtained as can easily be verified by assuming a release
volume of, say 10.0 cubic feet, and re-calculating. As already
observed, there is a considerable increase in the M.E.P., so that the
economy per B.H.P. would be improved.
n8
THE ENERGY CHART.
Effect of Adding a Jacket. The next alteration made will be to
reduce the initial condensation by adding a jacket, and let it be
assumed that 0.8 is the dryness fraction at cut-off, so that the volume
.at cut-off is 3.65 cubic feet. The expansion line will slope more to
the right owing to the heat supplied by the jacket during expansion,
us shown in Fig. 127, and if the cut-off remains as before, namely, at
0.29 the release volume will be found to be n.8 cubic feet
as shown in Fig. 127, and the 6 <f> diagram can be completed as
shown. The actual clearance volume has been kept the same as in
Fig. 126, i.e., 0.51 cubic foot, so that the percentage clearance will
be reduced. The mean pressure of this diagram will be found to
be 45 Ibs. per square inch.
B
FIG. 127. FIG. 128.
The work done per stroke is equal to 94 B.Th.U., and this
work is produced by the expenditure of the heat in the cylinder steam
and in the jacket steam. The weight of cylinder feed per stroke is
4.36 0.51
- 4<36 * = o.883lb.
so that the heat required for the cylinder steam is 885 B.Th.U.
per stroke, and if it is assumed that the jacket steam is ^th of the
cylinder steam, a usual proportion, 88 B.Th.U. have to be
added, so that altogether the heat required by the engine per stroke
See footnote on page 114.
DESIGN OF A SIMPLE ENGINE.
is 973 B.Th.U. The thermal efficiency is therefore
Q4
and the economy of the engine is
_4 2 -4 __. B.Th.U. per I.H.P. per minute.
O.OQO ^ r r
Thus, in this case, the jackets, combined with the reduced clear-
ance, effect an improvement of 16.8 %.
Increase in Clearance. As a last example the original assump-
tions tabulated at page 116 will be reverted to, except that the clear-
ance will be increased to 12 % of the volume swept by the piston ;
and it will be further assumed that the compression only reaches
50 Ibs. per square inch absolute. This reduction in compression
will cause additional condensation, so that the dryness fraction of
the steam at cut-off will be less than in Fig. 126, say 0.6.
The expansion line can be drawn in as before, but if the same
mean pressure is to be maintained the release volume will have to be
modified (only a little, however, because the increased clearance re-
duces the volume swept by the piston), say, as a first trial, to 10.0
cubic feet. The admission line, the expansion line, and the exhaust
line can thus be sketched in, as shown in Fig. 128. There is no
reason to suppose that the compression will materially differ from
the lower portion of that shown in Fig. 125. Therefore the com-
pression line can be drawn in as far as the intersection with the
volume line representing the clearance, which is approximately I . I
cubic foot. The diagram *s closed by the intersection of this constant
volume line with the admission line. The area of the <j> diagram
thus drawn is found to be 6.76 square inches, so that the heat
utilised is 67.6 B.Th.U., and the mean pressure is
5.4 x '-- = 41 Ibs. per square inch.
10 " i . i
which is approximately the same as for Fig. 123.
To find the weight of feed per stroke it must be noticed that
steam has to be supplied first to fill the clearance from 50 Ibs. to
admission pressure, and afterwards to follow up the piston to the
point of cut-off. As shown on page 74 and Fig. 91, the heat units
required per stroke are shown by the area below a t a B (Fig. 128),
which is found to be 882 B.Th.U. ; or by calculation as follows :
120 THE ENERGY CHART.
The weight of feed per stroke is
4-36 0.53
4.36 =o.881b,
and, as before the heat supply per Ib. is 1002 B.Th.U., thus the
B.Th.U. per stroke are 882. Hence thermal efficiency is
67.6
88^" = =
and the economy of the engine is
4 0765 = 552 B.Th.U. per minute,
from which it appears that the increased clearance results in a
reduction of economy at about 3.4 %.
Leakage past Admission Valve Direct into Exhaust. In the
above calculations no account has been taken of the direct leakage
into the exhaust. The amount of this leakage depends in a very large
measure on the type of admission valve, and is certainly far greater
with slide valves than it is with piston valves fitted with rings and
springs. f A correction should, therefore, be made in the efficiency and
economy figures obtained. Suppose that the engine is fitted with
a slide valve, then since the speed is 150 r.p.m. it would appear from
Professor Capper's report to the Steam Engine Research Committee
of the Institution of Mechanical Engineers, that the leakage in question
is about 5% of the cylinder feed.J The figures previously obtained
must therefore be corrected in this proportion. Thus, in the last case
of the unjacketted engine (Fig. 125), the economy of the engine would
be altered to 534 x 1.05 = 560 B.Th.U. per minute per I.H.P., and
the Efficiency ratio would be reduced to 9 -= 0.56.
In the case of engines having piston valves fitted with rings and
springs, the leakage in question is very small and no practical cor-
rection is needed ; but if not fitted with rings and springs this leakage
will be serious, anything from 5 to 20% of the cylinder feed. At
present there is little or no experimental data for Corliss and drop
valves, an allowance of from i to 3% may be made however.
* 0.53 is the volume at the point corresponding to the point a in Fig. 91.
t Probably from five to ten times greater.
t See leakage for trial CC 3 , Proceedings Institution Mechanical Engineers,
March, 1905.
See the Author's remarks on Prof. Capper's paper on " Steam Research."
Proceedings Institution Mechanical Engineers, March, 1905.
CHAPTER XL
DESIGN OF COMPOUND STEAM ENGINES.
THE following numerical example illustrating the use of the energy
chart in designing a compound steam engine will be considered.
Find the approximate <f> diagrams of a condensing steam
engine working at 28 Ibs. mean pressure referred to the L.P.
cylinder, the stop valve pressure being 140 Ib. absolute, and the
condenser pressure 2 Ibs. absolute.
Standard of Comparison. The < diagrams of the perfect
compound steam engine (Rankine cycle), working under these
conditions are given in Fig. 129, assuming equal division between
the cylinders of the total temperature range (353 to 126.5). For
the sake of comparison with the engine being designed, the following
figures in connection with this " perfect " compound steam engine
are tabulated below :
The heat supplied per stroke is,
1189.5 94.5 = 1095 B.Th.U. per Ib.
H.P. Cylinder.
Work done 127.5 B.Th.U.
Economy 327 B.Th.U. per I.H.P. per inin.
Mean pressure 48 . i Ibs. per square inch.
Point of cut-off 0.222 of stroke.
Dryness fraction at exhaust 0.895
L.P. Cylinder.
Work done 145 B.Th.U.
Economy 312 B.Th.U. per I.H.P. per min.
Mean pressure 5 . 76 Ibs. per square inch.
Point of cut-off o. 105.
Dryness fraction at exhaust . . o . 800
Ratio of L.P. cylinder to H.P.
cylinder i to 9 .5.
122
THE ENERGY CHART.
Both Cylinders combined.
Thermal efficiency 0.248
Economy 172 B.Th.U. per LH.P. per min.
Equivalent feed 9.4 Ibs. per LH.P. per hour
Mean pressure referred to L.P. cylinder. .6.85 Ibs. per sq. inch
In reckoning the
economy of the L.P.
cylinder, it has been
considered that it is
supplied with steam of
the quality represented
by the point D (Fig.
129).
Sketching < Dia*
grams of proposed Engine.
For the actual engine
it will be assumed that
the quality of the steam
at cut-off in the H.P.
cylinder is 0.80, and
further, the following
assumptions have been
made :
H,P. cylinder clear-
ance 5.25 %
L.P. cylinder clear-
ance 4-7%
Drop of pressure be-
tween H.P. and L.P.
FlG . I29 . cylinder . . 2 Ibs. per
square inch.
Back pressure in L.P. cylinder . . I Ib. per square inch.
As in the case of the simple engine, the 6 <t> diagrams shown in Fig.
130 can be sketched in a preliminary manner. The area of both
these diagrams together is found to represent 172 . 4 B.Th.U., and with
the release volume shown (40 cubic feet) a m^an pressure of 23 . 6 Ibs.
It is suggested that the diagrams be sketched on tracing paper over the
Energy chart, Plate I.
DESIGN OF A COMPOUND ENGINE.
123
per square inch is obtained. To increase this to 28 Ibs. (as required
by the example), the release volume must be diminished to about
33 cubic feet, as shown by the dotted release line.
Adjustment of Work Done in the Cylinders. It will be seen that
the work done in the H.P., as represented by its < diagram, is
greater than that done in the L.P. cylinder. If it is desired that,
when the mean pressure is 28 Ibs. the work done in each cylinder
should be equal, the exhaust of the H.P. must be raised somewhat.
The areas, when the
diagrams are laid down
on Plate I. are found
to be 9.12 and 8.12
square inches respect-
ively, and half the
difference should be
deducted from the H.P.
diagram, and since the
length of the exhaust
line (F E) of the H.P.
diagram is 1.7 inches,
the amount to raise the
exhaust of the H.P. and
the admission of the L.P.
is j- (9.12 8.12)
- = 0,294 inch. On
thus altering the dia-
grams, it is found that ,
the L.P. diagram is a
little larger than the H.P.
A further correction can
be made if deemed FlG - 13 *
necessary, and finally the </> diagrams given in Fig. 131 are obtained,
from which the following results are deduced :
Results: $ feed for each cylinder: o.goslb.
H.P. Cylinder.
Work done 82.7 B.Th.U.
Economy 502 B.Th.U. per I.H.P. per minute.
124
THE ENERGY CHART.
Mean pressure 47 - lbs - P er square inch.
Point of cut-off 0.226 of stroke.
L.P. Cylinder.
Work done 84.0 B.Th.U.
Economy 489 B.Th.U, per I.H.P. per minute.
Mean pressure 14-6 lbs. per square inch.
Point of cut-off 0.285 of stroke
Ratio of L.P. to H.P. cylinder i to 3.2.
Both Cylinders combined.
Thermal efficiency o. 168
Economy 254.5 B.Th.U. per I.H.P. per min.
Equivalent feed 13 -9 lb s. per I.H.P. per hour.
Efficiency ratio 0.660
Mean pressure referred
to L.P 27.6 lbs. per square inch.
Determination of Ratio
of Cylinders, The method
of determining the ratio
of the L.P. to the H.P.
cylinder requires some ex-
planation. The volume of
the steam at release in
the H.P. cylinder is
seen (Fig. 131) to be
10 cubic feet, but the
volume in the clearance
is 0.50 cubic foot, so
that the volume swept
by the piston is 9.5
cubic feet. In the same
way the volume swept
by the L.P. piston is
33.0 1.9 = 31-1
cubic feet. Hence the
ratio between the volumes
of the two cylinders is
I : 3-2.
Fro ni.
The manner of deter-
DESIGN OF A COMPOUND ENGINE. 125
mining the points of cut-off was explained on page 66.
The diagrams have been drawn on the supposition that the 6 <f>
feed is the same in both cylinders, and this assumption is only
true, as was explained at page 97, if the weight of play steam is
the same in both cylinders.
Assuming that the steam in the clearance of the L.P. cylinder is
saturated at the point G the weight of H 2 O in the clearance will be
Vol. at G,
1 x Vol. at B 3 ,
where B 3 is the point on the saturation curve at the same tempera-
ture as G. In the case under consideration this is
Hence the total steam = 0.905 + 0.055 = 0.960 Ib. Therefore,
in order that the expansion line of the L.P. diagram shall represent
the quality of the steam on the chart, which is drawn for i Ib.
of H 2 O, the volumes of the L.P. <f> diagram must be increased in
the ratio of 0.96 to i, and the volume factor must be increased in the
same proportion, and becomes 1.04.
The corrected volumes in the L.P. cylinder are thus :
Clearance 1.9 x 1.04 1.97 cubic feet
Cut-off 10.75 x 1.04 = ii. 2
Release 33.0 x 1.04 = 34.3
and the L.P. diagram can be re-drawn for these values.
It is thus seen that no serious error is introduced in assuming
the L.P. diagram to remain as drawn, and in general it will be found
that the volume factors are practically equal to one another except
in the case of a large difference between the percentage clearances of
the two cylinders.
Alteration of Cut-off in H.P. The effect of altering the cut-off
in the H.P. cylinder, without altering either the cut-off in the L.P.
or the ratio of the cylinders will now be considered. As an example,
let the cut-off be changed to o . i. If this change in the cut-off did not
affect the initial condensation, the volume at cut-off would still be
2.65 cubic feet (see Fig. 131), but the volume at release would be
increased to 21.4, as shown by the dotted line in Fig. 132. The
126
THE ENERGY CHART.
range of temperature in the H.P. cylinder would thus be increased
from 103 F. to 133 F., and although the real increase is not so great
as this (as will be seen from Fig. 132), the initial condensation will be
greater, or the quality of the steam at cut-off will be reduced to say
0.68, and on referring to the chart (Plate i), it will be seen
that the volume at cut-off is thus 2 . 15 cubic feet. The volume in
/ I the clearance in the H.P.
cylinder can best be
determined by trial and
error. Assume that it
is 0.8 cubic foot, then
the volume at release =
353
(2.15 0.8)
= 13-5
O.I
cubic feet, and this would
make the volume in the
clearance
13-5 x T&J = 0.675.
Take 0.7 cubic foot
as a second approxim-
ation, the release volume
then becomes 14.5 cubic
feet, and the clearance
volume is o . 725. A third
trial will show that 0.75
-PC 2. \ is near enough for pract-
ical purposes, and this
gives a release volume of
FlG ' * 32 ' 14.0 cubic feet, and the
<t> diagram of the H.P. cylinder can be drawn in as shown in Fig. 132.
Since the ratio of the L.P. to the H.P. clyinder is i to 3.2, it follows
that the release volume in the L.P. will be 14.0 x 3.2 = 45 cubic
feet, and since the cut-off in this cylinder by supposition remains
unchanged at 0.285, the volume at cut-off will be
45 x 0.285 + 2 = 14.8 cubic feet.
The 6 <f> diagram of the L.P. cylinder can thus be sketched in as
DESIGN OF A COMPOUND ENGINE.
127
353
shown in Fig. 132, and the following results are obtained in the
manner previously described :
Results. Heat supplied per stroke : 1095 x 85 = 930 B.Th.U.
H.P. Cylinder.
Work done 84.5 B.Th.U.
Economy 470 B.Th.U. per I.H.P. per minute.
Mean pressure 34.3 Ibs. per square inch.
L.P. Cylinder.
Work done 66. o B.Th.U.
Economy 601 B.Th.U. per I.H.P. per minute.
Mean pressure 8.5 Ibs. per square inch.
Both Cylinders combined.
Economy 266 B.Th.U. per I.H.P. per minute.
Efficiency ratio 0.632
Mean pressure referred
to L.P. .. 18.7 Ibs.
per square inch.
Equalization of Work
in the Cylinders. It will
be seen that the work
done in the H.P. cylinder
is now considerably great-
er than in the L.P. Can
the work be equalised by
altering the cut-off in the
L.P.? Suppose, for in-
stance, the L.P. cut-off
is made 0.2 instead of
0.285, then, since the re-
lease volume of the L.P.
remains the same, the
admission volume will be
reduced to 10.4 cubic
feet. This approximate-
ly determines the ex-
haust pressure of the
H.P. cylinder, and since FIG. 133.
-PJ.
22S]
128 THE ENERGY CHART.
the number of expansions in this cylinder are not altered there
must be a loop at the end of the expansion as shown in Fig.
133. The change made in the point of cut-off of the L.P. has had
the effect of approximately equalising the work in the two cylinders.
The I.H.P. developed by the engine, if the revolutions remain
constant, will be proportional to the mean pressure Deferred to the
L.P. cylinder. Hence the change oi cut-off in the H.P. cylinder from
0.226 to o.io reduces the I.H.P. in the ratio of 27.6 to 18.7.
Alteration of Cut-off to increase Mean Pressure. If it were desired
to find the cut-off in the H.P- cylinder that would give a greater
mean pressure referred to the L.P. of say 40 Ibs. per square inch, the
release volume in the L.P. would have to be diminished, and from
Fig. 131, and the results obtained above, it will be seen that the
27 . 6
release volume will have to be somewhat less than ^ x 33 = 23
cubic feet. The release volume of the H.P. will then become 2 ^
3-2
= 7 cubic feet. The volume at cut-off in the H.P. will, however, be
greater than in Fig. 131 because the initial condensation will be re-
duced. The drawing in of the </> diagrams from these particulars is
left to the student.
Assuming that the volume at cut-off is 2 . 9 cubic feet, correspond-
ing to 0.9 dryness fraction, the cut-off will be 2 0.41.
Cylinder Ratios for a Quadruple Expansion Marine Engine.
The following example is intended to show how easily the ratio of
cylinders can be determined by means of the chart. This ratio
depends not only on the admission and exhaust pressures, but
also on the conditions under which the engine is working. The
engine is supposed to be a four-crank marine engine, quadruple
expansion, the cylinders being placed side by side, and giving equal
turning efforts at the economical load. The engine is intended for
a cargo vessel, and therefore the steam economy is of importance.
Let 13.5 Ibs. per I.H.P. be aimed at. Let the admission pressure
be 160 Ibs. per square inch absolute, and the exhaust pressure
in the L.P. cylinder be 3 Ibs. per square inch absolute. Under
these circumstances, experience teaches that a mean pressure of
CYLINDER RATIOS.
129
30 Ibs. per square inch is suitable. It will be found that the
Rankine engine for the pressures given, converts 255 B.Th.U.
into work, and that its release volume is 95 cubic feet, as shown
in Fig. 134, at the point 8. The mean pressure is therefore
255
5.4 x - = 14.5 Ibs. per square inch,
FIG. 134.
so that for 30 Ibs. mean pressure the release volume must be
reduced to something less than 45 cubic feet. By plotting on the
chart, it will be found that the shaded area e D8, when the
release volume is 43 cubic feet, is 14 B.Th.U., so that the mean
pressure becomes
- -
5.4 x
= 30.3 Ibs. per square inch,
130 THE ENERGY CHART.
which is near enough. The area AA B De represents the </>
diagram of an ideal engine without clearance, or any losses except
that due to cutting off the toe. Let this area be divided into four
equal parts each containing 60 B.Th.U., so as to conform to the
condition that each cylinder is to give an equal turning effort.
This division is made by the lines AJ)^ A< 2 D 2 ^and A 3 D Q) and
the volumes at the points D D 2 D 3 and Z) 4 are the release volumes
of the respective cylinders. As read off the chart they are :
H.P. ist I.P. 2nd I.P. L.P.
5 . 65 12,2 26 . o 43 . o cubic feet.
So that the ratios are:
i : 2.15 : 4.6 : 7.6
Since these are the ratios for an engine without losses, they might
be regarded as those to be aimed at, but it is desirable to find
what these ratios would be in an actual engine with the usual
clearances, which for a marine may be taken as :
H.P. ist I.P. 2nd I.P. L.P.
12% 15% 10% 15%
and on this basis, and by the method previously described in this
Chapter, the </> diagrams for the actual engine have been sketched
in as shown in Fig. 134. Taking the volume factor of the H.P.
cylinder as unity, the volume factors of the other cylinders are
found to be as given in the figure. Further, the <f> feed of the
H.P. cylinders is o.84lb. Hence the work done in the various
cylinders per Ib. of steam is as follows :
H.P.
40.1
ist. I.P.
39-5
2nd. I.P.
42.0
L.P.
39-8
0.84
equal to :
48.0
0.84 x 0.97
48.2
0.84 x i. 02
49.0
0.84 x 0.97
48.5 B.Th.U.
Altogether, therefore, the work done by the engine per Ib. of steam
is 193 . 7 B.Th.U., so that the economy is 2545/193 . 7= 13 i Ibs.
of steam, or somewhat better than aimed at. From Fig. 134 it will
be seen that the volume swept by the L.P. piston is 34 . o 5 .2
= 28 .8 for the </> feed of the L.P. cylinder, which corresponds to
o.84 28 x'o.Q7 =35-4 cubic feet per Ib. of feed.
CYLINDER RATIOS. 131
Hence the mean pressure is
5.4 x ^T T ~ 2 9 6 Ibs. per square inch.
O>J * T"
which practically agrees with the conditions laid down. The
volumes swept by the pistons in the various cylinders, adjusted for
the volume factors are as follows :
H.P.
4 5 o . 54
= 3-96
ist I.P.
10 I -5
2nd I.P.
21 . 2 .1
L.P
34.05.2
0.97
8.78
I . O2
I8. 5
0.97
29.7
The cylinder ratios are therefore :
i : 2.2 : 4.7 : 7.5
which, it will be seen, are practically the same as those for the ideal
engine. Thus it is only necessary to lay down the ideal engine,
A A B De. if the object is to obtain the ratio of the cylinders
It is worth noting that in the Quadruple Expansion Rankine
engine, that is, when the expansion is carried down to the point
(Fig. 134), it is only the L.P. cylinder which will be very much
larger than that of the actual engine, because the lines AD^ A Z D 2
and A Q D^ will only be slightly lowered, to make each cylinder of
the Rankine engine account for Jth of the area e Z) 4 8 which
was found to be equal to 14 B.Th.U. Hence A^D^ will be lowered
3 . 4 F. ; A 2 D 12 6 . 2 F., and A 3 D S 8 .8 F. On working this out
it will be found that the cylinder ratios for the quadruple Rankine
engine are :
i : 2 . 32 : 5.8 : 16 . 2
In the < diagrams, as sketched, the expansion has been
carried down to the exhaust pressure in the first three cylinders.
Judging from Example No. IV. (page 104), and from general
experience, it is probable that a better economy would be obtained
by releasing the steam somewhat earlier in each of these cylinders,
so as to get the drying effect of the " toe." Suppose, for instance,
that a 4 Ib. drop is allowed for in each cylinder, then a small
(approximately) triangular area corresponding to this drop will
have to be deducted at the points D l D 2 and Z> 3 . These de-
ductions will slightly affect the positions of the lines A B A^B*
132 THE ENERGY CHART.
and A 8 B 3 (but to no practical extent) if it is desired to have equal
work in the four cylinders, the release volumes will, however, be
reduced as will be seen by plotting on the chart, Plate I., thus :
H.P. ist LP. 2nd I.P. L.P.
5-40 10.8 22.5 43.0
so that the cylinder ratios are:
i : 2.0 : 4.2 : 8.0
As before, these are the ratios for the ideal engine; it is
suggested, as an exercise, to sketch in the <j> diagrams of the
proposed actual engine to see to what extent the ratios will vary
from those just obtained.
CHAPTER XII.
SUPERHEATED STEAM.
Constant Pressure and Constant Volume Lines. In the case oi
superheated steam the constant volume and pressure lines will
conform to those of a gas, as shown in Plate i. This energy chart was
drawn many years ago, and for want of authentic information the
specific heat at constant pressure was taken as a constant and
equal to 0.48, and the specific heat at constant volume as 0.37.
Referring to page 23, it will be seen that the constant pressure curves
are therefore drawn according to the equation
A
< = 0.37 tog j
v i
and the constant volume lines according to the equation
&
= 0.48 hf ^
Recently, however, many determinations of the specific heat of
superheated steam have been made which show that it is not
constant, and that the value is higher than given above. These
determinations still need confirmation, but for practical purposes
it would appear that, for the range within which superheated steam
is used, Cp can be taken as 0.6, and C v as 0.46.
These new values have been taken for working out the examples,
and a chart has been drawn for them in Fig. 135, the old values
being shown by dotted lines.
The entropy at any point in the superheated field is given by the
general equation
<k = ^ + c (log e s hg e ) t
where <#> t = entropy of saturated steam at the absolute temperature
, and O s = absolute temperature of superheat.
Thus the increase of entropy of steam in the superheated field above
the entropy at the same pressure on the saturation line varies in
134
THE ENERGY CHART.
direct proportion to the value taken for Cp, and the existing chart
in Plate i can be used, making the correction for the particular value
-aocfl
40O
5
FIG. 135-
of Cp, that may be accepted when definite data are available. The
same correction can be made with respect to the constant volume
lines.
SUPERHEATED STEAM INTERNAL ENERGY,
135
The following numerical examples were originally worked out on
the chart for superheated steam (Plate i), and the results then obtain-
ed are given as well as those derived from the new data, in this way
it is easy to note the difference made.
FIG. 136.
Two constant pressure and constant volume lines are shown
in Fig. 136, and it will be noticed how very nearly vertical they are
owing to the temperature and entropy scales chosen for this figure.
In Fig. 137 the temperature scale has been reduced, and the entropy
scale has been increased. In this way the curves become much
flatter, and the intersections of the volume and pressure lines are
somewhat less acute.
Internal Energy. To find the internal energy at any point of the
superheated field, it is only necessary to add the heat required to
superheat at constant volume to the internal energy of saturated
steam at the same volume. Thus in the case of point P (Fig. 137),
at which the volume is 5 . o cubic feet it will be seen from the chart
(Plate i) that i Ib. of saturated steam of this volume has an
136 THE ENERGY CHART.
internal energy of 1099 B.Th.U., and the heat required to superheat
at constant volume from B to P is represented by the shaded area in
Fig. 137. This area is more readily obtained by calculation than by
measurement if the specific heat at constant volume is assumed con-
stant. The temperature at B is 317 F., and at P it is 500 F., and
FIG.
FIG 139.
taking the specific heat as 0.37 the heat required to superheat from
317 F. to 500 F., at constant volume is
(500 317) x 0.37 67.6 B.Th.U.
The internal energy at P is therefore
1099 + 67.6 = 1166.6 B.Th.U.,
and is represented by the shaded area in Fig. 138.
If, in accordance with the latter determinations, the specific heat
at constant volume is taken at 0.46, the heat required to superheat
from 317 to 500 at constant volume is
(500 317) x 0.46 = 84 B.Th.U.
RANKINE CYCLE FOR SUPERHEATED STEAM. 137
With the new value, therefore, the shaded area in Fig. 137 repre-
sents 84.0 B.Th.U., and thus the internal energy at P is found to be
1099 + 84 = 1183 B.Th.U.,
or 16.4 B.Th.U. more than with the old value.
Superheating at Con-
stant Pressure. When a I0 o
superheater is used the
steam is superheated at
constant pressure, that
is, it flows from the 50
boiler through the super-
heater at the boiler
pressure, at any rate,
theoretically it is sup-
posed to do so. Practi-
cally, however, there is
a drop of pressure due to the
resistance of the pipes.
Rankine Cycle for Super-
heated Steam. Let the case of
an ideally perfect steam engine
be considered, supplied with
i 1 1
B 10 15
CUB. FEET PE R
FIG. 140.
FIG. 141.
superheated steam produced at constant pressure. Starting with the
feed water at the exhaust temperature, saturated steam is first formed
according to the transformation line A^AB (Fig. 139), and the cor-
responding points are shown on the p v diagram (Fig. 140) and on the
ideal closed vessel (Fig. 141). The steam is then superheated at con-
stant pressure along the line B 5, and there will now be i Ib. of
superheated steam behind the piston. The heat supplied is the area
below the transformation line A A B S, and the area shaded by
vertical lines is the work done on the piston, reckoned to the
back pressure, in moving it from A to 5 (Fig. 141), and is also
represented on the p v diagram by the area shaded by vertical
lines. The internal energy of the steam at the point 5, is repre-
sented by the area in Fig. 139, shaded by lines sloping from left
to right.
THE ENERGY CHART.
At S the heat supply is stopped and the steam is expanded adia-
batically until the exhaust temperature is reached, after which the
heat remaining in the cylinder is rejected at constant pressure, thus
completing the cycle as shown in Fig. 142. The figure whose area
is shaded with dots is the </> diagram of the ideal superheated steam
engine giving the maximum utilisation as work, and the p v diagram
of this engine is easily derived,
and is given ^in Fig. 140.
From the chart it will be seen
that in the case of the num-
erical example chosen, the
volume of the steam at S is
5.0 cubic feet, and at D it is
24.5 cubic feet. Hence the
number of expansions is 4.9.
FIG. 142.
The following results are readily obtained by using the methods
described in chapter V. :
Rankine Cycle for Super-
heated Steam.
Specific
Heat
C # = o. 4 8.
New value
of
Specific Heat
= 0.6.
Heat supplied
1082
1102
Work done
157-0
159.6
Thermal efficiency
O.I45O
0.1448
Economy
294-5
295-0
Mean pressure
34- 6
35-i
B.Th.U. per Ib.
B.Th.U.
B.Th.U. per I.H.P. per min.
Lbs. per square inch.
Comparison with Saturated Steam Engine using same Range of
Pressures. It will be interesting to compare this ideal superheated
steam engine, with an ideal saturated steam engine, working between
the same pressures. The <f> diagram of the ideal saturated steam
engines is given in Fig. 143. The only conditions are that the
pressure of supply shall be the same as in the case of the super-
heated steam engine and that the exhaust pressure shall also be the
same. The results obtained are as follows :
SUPERHEATED STEAM ENGINE EXAMPLE.
139
Heat supplied 1001 B.Th.U. per Ib.
Work done 139 B.Th.U.
Thermal efficiency .... 0.139
Economy 307 B.Th.U. per I.H.P. per min.
Mean pressure 32.6 Lbs. per square inch.
Ratio of expansion 5.54
<$> Diagrams of Actual
Engines Working with Super-
heated Steam. The transfer
of the p v diagrams to the
chart is effected in pre-
cisely the same manner as
for engines using saturated
steam, it is only necessary
therefore to give a numerical
example as follows :
EXAMPLE I.
Compound Condensing En-
gine with Reheater. In Fig.
145 are shown p v diagrams
of an engine using super-
heated steam, the H.P. and
L.P. diagrams being combined
in the usual way in the
proportion of the cylinder
volumes, and the saturation curve has been added. The actual feed
was io.361bs. per I.H.P. per hour, and the following temperatures
were observed:
Temperature in superheater 750 F.
at stop valve 635 F.
in reheater 320 F.
Fig. 144 shows the corresponding < diagrams. It is to be
noticed that the steam is superheated at the point of
cut-off, the temperature being 490 F., and further that the super-
heat is maintained throughout the expansion in the H.P. cylinder.
In the L.P. cylinder the steam is practically saturated at cut-off
APS. ZERO
FIG. 143.
140 THE ENERGY CHART.
owing both to the steam at the H.P. release being slightly super-
100
FIG. 145.
SUPERHEATED STEAM ENGINE EXAMPLE. 141
heated, and also to the effects of the reheater. The economic
results obtained are as follows : *
Economy : 206 (200) B.Th.U/s per I.H.P. per minute.
Or ii. 8 (11.5) Ibs. " equivalent feed. 1 '
Thermal efficiency . . . 0.198 (0.204)
Efficiency ratio 0.790 (0.812)
EXAMPLE II.
Compound Engine using Superheated Steam compared with the
same Engine using Saturated Steam. The data for this example are
taken from a paper by Professor D. S. Jacobus, read before the
American Society of Mechanical Engineers, in December, 1903 (Vol.
XXV.)
The indicator diagrams for Test No. 2 with superheated steam,
are given in Fig. 146, and for Test No. 4 with saturated steam, in
the same figure (dotted lines). The particulars of these tests are as
follows :
Test No. 2. Test. No. 4.
Horse Power (indicated) .. .. 420.4 .. 406.7
Feed Water, per I.H.P 9. 56 Ibs. .. 13.84
Equivalent Feed .. .. .. 11.4 .. 13.7
Steam Pressure at Engine .. .. 157.1 abs. .. 159.8 abs.
Superheat at Throttle .. .. 374.5 F ..
Vacuum at Engine .. .. .. 25. 8 ins. .. 24.47
Cylinders H.P. . . .. .. 16.37 inches.
L.P 28.03
Stroke . . . . . . . . 42
Piston Rods, H.P 3.0
L.P 3-5
Clearances H.P... .. .. 4.1%
L.P 5.8%
By means of the above data the 6 <f> diagrams given in Fig 147
have been plotted, those for the superheated steam engine being
* Taking Cp = 0.6. ; the figures in the brackets are based on Cp
= 0.48.
142 THE ENERGY CHART.
shown in full lines, and those for the saturated steam engine in
eoo F.
Fro. 147
SUPERHEATED STEAM ENGINE EXAMPLE. 143
dotted lines. The great increase in the area of the former will be
noticed, corresponding to the actual feed per I.H.P. ; but the true
economy is not in proportion to these areas because the heat supply
per Ib. of feed is considerably greater with the superheated steam,
as shown in Fig. 139. In the paper Professor Jacobus gives tne
economy of the two tests as 205.0 and 248.2 B.Th.U. per I.H.P.
per minute, figures which' are proportional to the "equivalent"
feeds.
CHAPTER XIII.
EXPANSION OF STEAM WITHOUT DOING EXTERNAL WORK.
LET it be supposed that there is an adiabatic vessel containing V cubic
feet, divided by a diaphragm D D into two parts, whose volumes are
FIG. 148.
FIG. 149.
V ^ and V 2 respectively (Fig. 148), and let the portion whose volume is
V contain i tb. of H 2 in the state represented by the point P on
the chart (Fig. 149), and let the portion whose volume is F 2 be at
absolute zero of pressure. Now let the diaphragm be removed.;
in what condition will the Ib. of H 2 O be, so soon as any eddies, that
may have been formed, have disappeared ?
It is clear that the internal energy in the new condition will be
the same as in the old, as by supposition there has been neither gain
nor loss of energy, the state point must therefore lie at the inter-
section of the volume line V and the curve of equal internal energy
drawn through P. The point Q in Fig. 149 is thus obtained.
EXPANSION WITHOUT EXTERNAL WORK.
145
Example No. i. To illustrate this two numerical examples will
be taken and it will be assumed in the first that P is at 150 Ibs. per
square inch, F t = 2.7 cubic feet and K 2 =5.0 cubic feet, so that V
= 7.7 cubic feet.
On referring to the chart giving lines of internal energy
(Fig. 43), it will be seen that the internal energy at the point P
is 1040 B.Th.U., and from Fig. 149, that the intersection of the
1040 B.Th.U. line of internal energy and the 7.7 cubic foot volume
line is at Q on the 51 Ibs. per square inch pressure line. The two
points P and Q can be plotted on the chart (Plate I.), and it
pr igo
FIG. 150.
FIG. 151.
will then be seen that the dryness fraction at P is 0.91 and at Q
0.935. Hence the following result has been obtained: If steam
at 150 Ibs. per square inch absolute pressure and of dryness 0.91
expands 7.7/2.7 2.85 times without doing external work, its
condition will be defined by 51 Ibs, per square inch absolute pressure
and 0.935 dryness.
Example No. 2. In the second example the pressure will be taken
as before at 150 Ibs. per square inch, but the volume V will be
increased to 2.95 cubic feet, so that P lies on the saturation line ;
V 3 will be taken at 5.48 so that V = 8.44, and the number of
expansions will be the same as before. On referring to Fig. 43,
10
146 THE ENERGY CHART.
it will be found that the internal energy at P is 1109 B.Th.U., and
that the intersection of this internal energy line with the 8.44
volume line lies in the superheated field at the point Q (Fig. 150),
whose temperature is 332 F. and pressure 52.5 Ibs. per square inch.
On referring to Plate i, it will be seen that the temperature of sat-
urated steam at 52.5 Ibs. pressure is 284 F. Hence the steam has
been superheated 48 F. by expanding without doing work.
Expansion after Formation at Constant Pressure. Superheating
by Throttling. It will be noticed that in the above the i lb, of H 2 O
was assumed to be in the state represented by P, without any refer*
ence as to the manner in which the steam was formed, A case in
which steam is formed at constant pressure, and then expanded by
w throttling to a
lower pressure,
will now be con*
sidered. In Fig,
152 is shown a
diagrammatic re-
presentation of a
boiler connected
to a cylinder by
FIG ' I52 ' a pipe fitted with
a valve M, by means of which valve the steam pressure can
be throttled before it reaches the piston. The piston is shown
at the point of cut-off, and it is supposed that the cylinder
at this moment contains i lb. of steam at the reduced or throttled
pressure. This steam will obviously be superheated, and the
problem is to find the quality of the steam at this state point on
the chart. The first step is to find the velocity of the steam.
Velocity of Issuing Steam. When the steam rushes into the cylin-
der under the effect of the difference of pressure existing between
the boiler and the cylinder, a portion of its energy will be expended
in giving velocity to the steam, in other words in producing motion
energy, and the equivalent of heat energy disappears, In the
cylinder this motion energy will take the form of u eddies " in
the steam, and these eddies will gradually die away, and as they
do so, the energy in question will re-appear as heat. The steam a*
VELOCITY OF STEAM JET. 147
it leaves the boiler is represented, as regards its condition, by the
point P (Fig. 151), and assuming no gain or loss of energy from or
to external objects the expansion through the valve M will be
adiabatic, and hence the condition of the i Ib. of H 2 (which, after
passing the valve is moving with a high velocity V), is represented by
the point Q ; a moment's consideration will show that the heat
energy converted into motion energy is represented by the shaded
area of the Rankine cycle (Fig. 151). If R is taken to denote the
number of B.Th.U. in this area, then obviously
y*
Y- x ~ g = 778 x VR
or V = 223 V R feet per second.
Example. As a numerical example if P is at 150 Ibs., and Q at
100 Ibs. per square inch, it will be found by measurement that R =
32 B.Th.U. Hence V = 223 A/32 = 1280 feet per second.
Recovery of Motion Energy. The matter will be further dis-
cussed by means of the same numerical example (Fig. 151) ; the
boiler pressure being 150 Ibs. absolute, and the pressure in the
cylinder 100 Ibs. absolute. The internal energy of the steam in the
cylinder at the moment the Ib. of H 2 O has been introduced will be
equal to the energy required to produce i Ib. of saturated steam
in the condition represented by the point P, less the work done on the
piston up to this point, and less whatever portion of the motion energy
still remains in the form of eddies. There is no means of telling what
this portion is, but limits can be fixed. Thus the conversion of the
motion energy in the form of eddies back into heat may be exceed-
ingly rapid so that no eddies (and therefore no motion energy),
remain at the point of cut-off. The other limit would occur if the
transformation of motion energy into heat energy is a slow process,
in comparison with the piston speed, so that the whole of the motion
energy is rejected from the engine in the exhaust.
Each limit will be considered separately.
In the first case, when all the motion energy is re-converted
into heat, the condition of the steam at the moment of closing the
admission valve will evidently be found by determining the inter-
section of the constant pressure line through P with the constant
internal energy line for the total heat of steam at pressure P, less the
148
THE ENERGY CHART.
work done at the pressure of P. In the numerical example the total
heat is 1191 B.Th.U., but the volume V is not known, and thus the
value of p v cannot be found, and a process of approximation must
be resorted to.
Assuming, as a first approximation, 5 cubic feet as a value for V
(which it will be observed, is rather more than the volume of saturated
steam at the cylinder pressure, viz., 100 Ibs. per square inch) the work
done up to the point of cut-off is
IPO x 144 x 5
778
= 93 B.Th.U.
FIG. 153.
FIG. 154.
The internal energy remaining will therefore be 1191 93 = 1098
B.Th.U. On reference to Fig. 43 it will be seen that the inter-
section of the internal energy line for 1098 B.Th.U., and the pressure
line for 100 Ibs. per square inch, occurs in the saturated field, and the
volume at the point of intersection S (Fig. 153) is 4.33 cubic feet.
Taking this volume for a second approximation the point S (Fig.
153), which occurs in the superheated field, is determined, and at this
point the volume is 4.45 cubic feet.
Repeating the above operation twice more, the volume at the
point of cut-off is found to be 4.38 cubic feet, and 5 3 is the point
representing the condition of the steam in the cylinder. The cycle
is continued by adiabatic expansion and release at constant pressure
as shown in Fig. 153.
RECOVERY OF MOTION ENERGY. 149
The second limiting case assumes that the whole energy of
motion imparted to the steam in entering the cylinder remains as
such up to the point of cut-off. The internal energy of the steam
at the point of cut-off is therefore less than in the first case by, the
amount of the motion energy, which latter can be found by using
the formula given on page 147, and is 32 B.Th.U. The internal
energy of the steam at cut-off is therefore (1191 32) less the work
done. As before, the volume V at cut-off is not known, but by re-
peating a similar process, a series of approximations marked S, S 1? 5 a ,
and S 3 , in Fig. 154 are obtained, and the point S 3 represents the
state of the steam at cut-off with sufficient accuracy, and is found
to be at a volume 4.22 cubic feet. The limits are, therefore, 4.22
and 4.38 cubic feet, and as a practical matter the volume after
the expansion by throttling can be taken as 4.3 cubic feet.
CHAPTER XIV.
APPLICATION OF THE ENERGY CHART TO OTHER SUBSTANCES.
So far the chart has been drawn, and its use shown for the sub-
stance H 2 O, and in Chapter II. some particulars were given as to
its adaptation to air and other gases. The energy chart can, however,
be drawn for any other substance, if the necessary physical properties
are known, in exactly the same manner as that for H 2 0; for instance,
for such substances as ammonia (NH 3 .) 5 carbon dioxide (CO 2 ), and
sulphur dioxide (SO 2 ), all of which are used in refrigerating machines,
and the indicator cards from such engines can be transferred to
their respective charts.
The energy chart for sulphur dioxide is given in Fig. 155, and, as an
example of its use, the case of a binary engine will be considered, The
use of the secondary or binary engine operated in conjunction with a
primary steam engine has long been known and has from time to time
been advocated, but the first difficulty was to find a suitable sub-
stance, that is to say, a substance such, that with the ranges of
temperature available, will give a reasonable mean pressure, say
from 40 to 50 Ibs. per square inch, and whose vapour pressure
at the exhaust temperature of the primary steam engine is not
too great for constructional reasons, say 300 Ib. per square inch.
Sulphur dioxide meets these requirements with a condensing steam
engine. The second difficulty, which has only been recently overcome,
is to arrange the stuffing boxes, valves and joints so that there will
be no leaks, so as to guard against the harmful effects of such leakage,
and to obviate the loss of a costly substance. These difficulties
have been solved by Prof. Josse, of Berlin, and there are many S0 a
engines running in Germany, the largest of which indicates 400 H.P.,
but the question is still undecided whether the undoubted large
thermodynamic gain is sufficiently great to show a commercia
ENERGY CHART FOR SO a .
151
6$ CHART FOR S
aocTFahr-
profit, after the increased depreciation, interest on capital, etc., has
been allowed for.
The actual SO 2
engine from which
the data for the
following example
has been obtained is
working in a spin-
ning mill in Ger-
many, and during
the trial was devel-
oping 327 H.P.
The indicator
diagram taken from
the engine does not
materially differ in
appearance from
that of a steam
engine with the ex-
ception that the
cylinder clearance
of the SO 2 engine
is very much larger
owing to larger
valves being used
on account of the
great viscosity of
the vapour. In
this particular en-
gine the clearance
is as much as
29.5%.
Fig. 156 shows
the <f> diagram of
the engine, and it
will be noted how
a*o
S C AL. K
NTRO P V
FIG. 155.
the large clearance affects the position of the diagram on the chart.
152
THE ENERGY CHART.
On the same figure is given the exhaust temperature of the L.P.
cylinder of the steam engine, viz., 161.4 F., and the lower part
of its diagram is represented by the area whose contour is shaded with
dots. Before the S0 2
engine was fitted the
steam engine exhausted
at this temperature,
and the exhaust steam
was condensed by
coming into contact
(in a jet condenser)
with condensing water
at62.iF. This range
of temperature 161 . 4
62.1 99.3 F.
was entirely lost for the
expansion of the steam,
and it is the function
of the binary engine to
utilise this range.
With this object
in view, the exhaust
steam of the steam
engine is condensed by
means of liquid S0 3
(instead of as usual by
water) in a surface con-
denser which is both
the condenser of the
steam engine and the
boiler of the SO 2 engine.
In this way a vacuum
corresponding to 159 . 4
is maintained in the
exhaust pipe of the
FIG. 156.
steam engine. The S0 2 is evaporated and reaches the engine stop
valve at a temperature of 144.5 F., corresponding to a pressure of
169 Ibs. per square inch absolute. It passes through the S0 2 engine,
SO a WASTE HEAT ENGINE EXAMPLE. 153
expanding and doing work, and is then exhausted into a surface
condenser at a pressure of 61.5 Ibs. per square inch, where it is con-
densed by cooling water whose temperature is 62 . i F. The Rankine
cycle for an engine using the same <j> feed, namely, 0.545 lb. of SO 2 ,
is shown in Fig. 156, and the various losses are easily localised,
and are given as percentages of the heat supply due to the exhaust
steam.
The shaded area between the temperature 159.4 and 144.5,
representing 2.2%, is the loss in the boiler of the S0 2 engine due to
radiation, loss by transmission of heat, etc.
The SO 2 engine works between the temperature limits 144.5
and 81.1, converting 7.4% of the heat supply into work, while the
area shaded by dots represents the engine's losses, namely 2.1%.
These losses include the loss due to condensation, incomplete
expansion, throttling through the admission and exhaust ports,
and to clearance, etc.
The area shaded between the temperature 81.1 and 62.1,
namely 3.1%, represents the loss due to the inefficiency of the con-
denser, and this can be subdivided into a portion due to the trans-
mission of heat through the tubes, namely, between the temperatures
81.1 and 68.8, and a second portion due to the rise in temperature
of the condensing water, namely between the temperature 68.8
and 62.1. The remainder of the heat supply is rejected to the
exhaust and this amounts to 85.2%.
The economic results of this SO 2 engine obtained from the chart
are as follows :
Heat supplied per I.H.P. per minute = 577 B.Th.U.
Flow of SO 2 per I.H.P. per hour = 208.5 Ibs.
Thermal efficiency . . . . . . = o . 074
Efficiency ratio . . . . . . = o . 778
M.E.P. .. .. .. .. =46.1 Ibs. per sq. in.
Number of expansions . . . . = 2 . 22
Cut off = 48% of stroke
i Ib. of exhaust steam from the H 2 O engine evaporates 5. 6 Ibs. of
SO 2 in the boiler, so that the H 2 O used per I.H.P. per hour for
the secondary engine
= 37-3
154
THE ENERGY CHART.
It must be borne in mind, however, that the heat in this " feed "
water is recovered from the waste heat of the primary steam engine,
or to put it in another way, 327 I.H.P. are developed by the aid of
the S0 2 engine without burning any additional coal. It is desir-
able to know the thermal efficiency and the economy of the H 2
and SO 2 engines combined, and this can be done as follows, the
data being obtained from the trial referred to aboyp. There were
two steam engine supplying exhaust steam to the SO 2 boiler, one of
which was economical and the other not.
Feed-Water
B.Th.U.
I.H.P
Ibs.
per hour.
per
minute.
No. i steam engine
505
15-1
276
No. 2
209.6
21.9
409
Both steam engines combined..
714.6
17.1
315
SO 2 engine alone
327.
577
The total I.H.P. is therefore 1041.6, and the total feed water is
714.6 x 17.1 = 12, 220 Ibs. per hour. Hence the feed water per
I.H.P. is 11.7 Ibs. per hour. By a similar calculation it will be
found that the B.Th.U. per I.H.P. per minute are 216, so that
the gain in comparison with the two steam engines combined is
315 216
216
100 X
= 45-8%.
It is to be observed that this considerable gain is in part due to
the steam engines not being economical. Had they been economical
the percentage gain would have been less. Thus if the primary
steam engine required 12 Ibs. of " equivalent " feed per I.H.P. hour
the thermodynamic gain due to the addition of an S0 2 engine would
be about 28 %, so that the " equivalent " feed water for the com-
bined H 2 O and SO 2 engines would be reduced to 9.3 Ibs.
Or
B.Th.U. per I.H P. per min. for steam engine = 220
combined H 2 O and
SO 2 engine
= 170
SO 2 WASTE HEAT ENGINE EXAMPLE. 155
As a comparison, it may be stated that a steam engine using
superheated steam, and requiring the very small actual feed of
9.3 Ibs., would need about io.4lbs. of " equivalent " feed, corres-
ponding to 190 B.Th.U. per I.H.P. per minute.
APPENDIX I.
PLOTTING < DIAGRAMS BY MEANS OF THE SLIDE RULE.
The method of transferring an Indicator Diagram to the Energy
Chart is given in extenso in Chapter VI., but in actual practic<
the process can be much shortened by the help of the Slide Rule
as follows :
At any point M, Fig. 157 (which is a partial reproduction ol
Fig. 87) the dryness fraction is ascertained by the method given
at page 70. Then, knowing the pressure and dryness fraction, the
point M can be located on the chart, and in the example giver
it will be found that this point falls on volume line 7.49.
If K is any other point on the Indicator Diagram then :
(/> volume at M
< volume at K =
A M
x BK
FIG. 157.
The term:
4> volume at M
A~M
can be set as a constant on the
slide rule, and when multiplied by the length of B K for different
positions of K t measured on the same scale of equal divisions as
A M, will give the <J> volume at the various points. Obviously
the positions of K would be chosen at pressures suitable for easy
plotting on, the chart, such for instance as 20, 40, 60, 80, and 90
in the example given.
APPENDIX II.
USEFUL FORMULAE IN CONNECTION WITH THE ENERGY CHART
FOR H 2 O.
ILLUSTRATED BY MEANS OF NUMERICAL EXAMPLES.
For ease of reference the various formulae used in this book
have been collected together.
Notation. The following Notation has been used :
H Total heat expressed in B.Th.U., reckoned from 32 F.
h Water heat expressed in B.Th.U., reckoned from 32 F.
R B.Th.U. utilized by Rankine cycle.
c Point of cut-off.
k Percentage clearance calculated on piston displacement.
V c Volume in cylinder at point of cut-off.
V r Volume in cylinder at point of release.
Vk Volume in clearance.
9 Absolute temperature.
t Initial temperature of steam
m
/ 2 Exhaust temperature of steam J ln
t s Temperature of superheated steam.
Point of Cut-off. The point of cut-off can be found from
the following equation:
_ Vc-Vk
~ VrVk
Example No. i (see Fig. 125).
Vc 3-15, V h = 0.55, Vr = 9-5 cubic feet.
158 APPENDIX,
Release Volume. The release volume can be found from
either of the following equations :
Vc - Vk
v
Vr =
+
* - c + k '
Example No. 2 (see Fig. 125).
V k = 0.55, V L = 3.15 cubic feet.
c = 0.29, k = 0.06
Vr = 3>I5 Q^ 9 ' 55 + -55 = 9-5 cubic feet.
r: v - J + - 6 u- < .
Kr - cTo6~+~^729 x 3-15 = 9*5 cubic feet.
Number of Expansions. The number of expansions in any
cylinder are equal to pA
Example No. 3 (see Fig. 125).
Vr = 9.5. V c = 3 '15 cubic feet.
Number of expansions = = 3.01
In a compound or triple-expansion engine the total number of
expansions is
V'r x volume factor of H. P. cylinder.
V c x volume factor of L. P. cylinder.
where V' r is the release volume of the low pressure cylinder and
V c is the volume at cut-off in the H. P. cylinder.
Example No. 4 (see Fig. no).
V c = 2.6, V'r = 33.5 cubic feet.
Volume factors . . H.P. cylinder, 1.35; L.P. do., 1.44.
oo < X I "3*5
Number of expansions = ^ - ^~~r = s 12.1.
2 * o x .L . 44
Volume swept by Piston. The volume swept by the piston
of the 6 $ engine is V r V k . To obtain the volume swept by
the piston of the actual engine, divide by the volume factor.
Example. ,No. 5 (see Figs. 87 and 8q).
Vk = 0.8, Vr = 12.30, Volume factor = 21.4.
APPENDIX. 159
Volume swept by piston of actual engine
_. L^ -- L_
== 0.538 cubic foot.
Cylinder Ratio. The cylinder ratio is equal to the volume
swept by the H. P. piston divided by the volume swept by the
L. P. piston or
. _ V r Vk Volume factor of L.P. cylinder.
Cylinder ratio = (y ^ _ y ^ ) x Volume factorof H.P. cylinder.
Example No. 6 (see Fig. no).
Volume factors : H.P. cylinder, 1.35; L.P. cylinder, 1.44.
V k = 0.6, V' k = 1.77, Vr = 7-8, V, = 33-5-
~ ,. , x . 7-8 0.6 1.44 i
Cylinder ratio = ^T^ - ^ = ^
Heat Supply. As defined by ithe Thermal Efficiency Com-
mittee of the Institution of Civil Engineers, the heat supply is
equal to the total heat of the steam, at the pressure and tem-
perature of formation, less the water heat at the temperature of
the exhaust. The total heat and the water heat can be obtained
from the energy chart, Plate I.
Example No. 7 (see Figs. 87 and 89). Saturated Steam.
Pressure, 115 Ibs. per square inch abs.
Exhaust temperature, 212 R
H = 1183.6 B.Th.U., h = 180 B.Th.U.
Heat supply per Ib. = 1003.6 B.Th.U.
Example No. 8 (see Figs. 144 and 145). Superheated Steam.
Pressure, 140 Ibs. per square inch.
Temperature of superheat, 635^ F.
Exhaust Temperature, 140" F.
H = 1189.5 + (635 353) 0.6 = 1358.7- * = 108.0.
Heat supplied per Ib. = 1250.7 B.Th.U.
Heat Utilization of the Rankine Cycle. The heat utili-
zation of the Rankine Cycle can be found by means of the 6 <f>
chart, by the following very approximate formula :
R = (t^ t 2 ) x width of Rankine cycle diagram at the
mean temperature L l 2 measured on the entropy scale.
160 APPENDIX.
Example No. 9 (see Fig. 91).
<i = 334.5 F.
Width of diagram at the mean temperature 273 is 1.18
Heat utilized by Rankine cycle :
R = (334-5 212) 1.18 = 144.6 B.Th.U.
For superheated steam add to the above : - x additional
width of diagram due to superheating, measured on the entropy
scale.
Example No. 10 (see example No. i, page 139).
Temperature of superheat t s = 635 F. ; other temperatures as
in example 9.
Additional width of diagram 0.16 entropy units.
Additional heat units utilized
6?< 3<Q
== ~~~-" x 0.16 = 22 B.Th.U.
Total heat units utilized by Rankine cycle (superheated steam)
R = 144.6 + 22 = 166.6 B.Th.U.
6 < Cylinder Feed. The <j> cylinder feed is equal to the
actual feed multiplied by the volume factor. The volume factor
can be obtained as shown on page 71.
Example No. n (see Fig. 91).
Actual feed, 0.0382 Ib. per minute.
Volume factor, 21.4.
6 <t> cylinder feed = 0.0382 x 21.4 0.818 Ib.
The 6 </> cylinder feed is also equal to - c 7^~ -, where V a
is the volume at admission of the proportional water line of the
Rankine cycle for the 6 < cylinder feed (for example volume at the
point a Fig. 91). If the compression in the engine reaches
admission pressure, as in Fig. 123, V u is equal to the clearance
volume Vh, If the compression is less than the admission pressure,
V a can be found by drawing a line parallel to the water line through
the state point representing the end of compression (see Fig. 128).
Example No. 12 (see Fig. 125).
V a = 0.55, V c = 9.5 cubic feet.
<f> cylinder feed = (9.5 0.55) = 0.895 Ib.
APPENDIX. l6l
Heat Utilization of an Actual Engine. The heat utilized
by the actual engine corresponding to the < engine can in the
case of a simple engine be obtained by measuring the area of
the $ diagram in square inches and multiplying j by the heat
scale.
Example No. 13 (see Fig. 89).
Area of 6 < diagram . . 6 . 55 square inches.
Heat scale . . . . 10 B.Th.U. per square inch.
Hence :
Heat utilized = 6.55 x 10 = 65.5 B.Th.U.
In a compound or triple-expansion engine the area of the
<f> diagrams must be adjusted for the different values of the
volume factors. If the feed is measured into the boiler, then
the volume factor of the H.P. cylinder should be taken as the
basis, and the areas of the <f> diagram of the other cylinders should
be multiplied by their volume factor, divided by the volume factor
of the H.P. cylinder. If, however, the steam supply to the engine
is measured by the condenser method, then the volume factor of the
L.P. cylinder should be taken as the basis.
Example No. 14 (see Fig. no).
Area of H.P. cyJinder </> diagram . . 7.97 square ins.
L.P. -.7-45
Volume factor H.P. cylinder .. .. 1.35
,, L.P. .. ..1.44
Heat scale . . . . = 10 B.Th.U. per square inch.
Feed measured into the boiler
Heat utilized by the engine = 7.97 x 10 + 7.45 ^ x 10.
= 149.6 B.Th.U.
Heat Supply per Stroke. The heat supply per diagram or
per stroke is equal to : 6 <f> cylinder feed x heat supply per Ib.
Example No. 14 (see Fig. 91).
<f> cylinder feed : 0.818 (the same as in example No. n).
Heat supply per Ib. : 1003.6 ( , No. 7).
Heat supply per stroke = 1003.6 x 0.818 = 821 B.Th.U.
l62 APPENDIX.
Thermal Efficiency of Rankine Cycle. The thermal
r>
efficiency of the Rankine cycle is equal to ^ ^
Example No. 16 (see Fig. 91).
R = 144.6 (From example No. 9).
H = 1183.6
i (From example No. 7).
h = 180. )
144.6
Thermal efficiency of Rankine engine = r == X 4-
Thermal Efficiency of an Actual Engine. The thermal
efficiency of an actual engine is
_ Area of <f> diagram x heat scale.
== (H h) x 6$ cylinder feed.
Example No. 17 (see Fig. 91).
Area of <t> diagram .. 6.55 square inches.
Heat scale . . 10 B.Th.U. per square inch.
H .. .. .. 1183.6 B.Th.U.
h ...... 180 B.Th.U.
<f> cylinder feed .. o.8i81b. per hour.
Thermal Efficiency = i = ' 8 '
Efficiency Ratio. The efficiency ratio is:
Thermal Efficiency of actual Engine.
Thermal Efficiency of Rankine cycle,
and can also be found from :
area of <f> diagram.
area of Rankine cycle x </> cylinder feed.
Example No. 18 (see Fig. 91).
Taking the same data as for examples 16 and 17 :
Efficiency ratio = - = 0.56.
or with the data of examples 9, n, and 13 :
65.5 _ -A
"" 144.6 x 0.818 ~~ ' 5 *
Economy. The economy of the engine when expressed in
B.Th.U., as recommended by the Theimal Efficiency Committee
of the Institution of Civil Engineers, is
APPENDIX. 163
Thermal Efficient Actual Engine BTh - U - P er LH ' P - P er min '
Example No. 19 (see Fig. 91).
Thermal Efficiency = 0.08 (as in example No. 17).
42.4
Economy = - 3 = 530 B.Th.U. per I.H.P. per minute.
If the economy is to be expressed in terms of the actual feed water
it can be found by the following formula :
Ibs. of feed water per _ /B.Th.U. represented by 6 < diagrams
I.H.P. per hour 2 545/ p er jj^ o f cylinder feed.
Example No. 20 (see Fig. 91).
B.Th.U. represented by <f> diagram . . 65.5.
<f> cylinder feed ...... 0.818 Ib.
Feed water = _-- = 31.6 Ibs. per I.H.P. per hour.
Example No. 21 (see Fig. 144).
B.Th.U. represented by <j> diagrams. . 206.
< cylinder feed . . . . . . o . 85 Ib.
Feed water = 2 " 45 2 * ' 8 ^ ==10.36 Ibs. per I.H.P. per hour.
Equivalent Feed. The " equivalent " feed is equal to the
actual feed divided by noo and multiplied by the heat supply
per Ib. (see page 89, Report of the Committee on Steam Engine
and Boiler Trials, Institution Civil Engineers).
Example No. 22 (see Fig. 91).
In example No. 7 the heat supply is 1003.6 B.Th.U., and from
example No. 20 the actual feed is 31. 6 Ibs. Hence the equivalent
feed is:
= 3I ' 6 x^r 3 ' 6 = *8 Ibs. per I.H.P. per hour.
Example No. 23 (see Fig. 144);
In example No. 8 the heat supply is 1250.7 B.Th.U., and if
the actual feed is 10.36 (example No. 21), the equivalent feed is
= 10.36 x "5? =11. 8 Ibs. per I.H.P. per hour.
164 APPENDIX.
Ratio of Work done in Cylinders of a Compound
Engine. The ratio of the work done in the cylinders is equal
to the ratio of the B.Th.U. represented by the <f> diagrams of
the respective cylinders adjusted for the volume factors.
Example No. 24 (see Fig. no).
B.Th.U. represented by < diagram of H.P. cylinder 79.7
L.P. -, 74.5
Volume factor .. .. .. H.P. 1.35
L.P. 1.44
Ratio of work done = ~-^ x -~~ ^ &Q
74.5 1.44 o.oo
Mean Pressure. The mean pressure in a cylinder is equal to
B.Th.U. represented by the <f> diagram.
Tf ~\7
Example No. 25 (see Fig. no).
B.Th.U. represented by </> diagram = 79.7
V r = 7.8 cubic feet
Vit = 0.6
Mean pressure in cylinder
79-7
= 5.4 x g g = 598 Ibs. per square inch.
The mean pressure referred to the L.P. cylinder is
B.Th.U., represented by all the <f> diagrams.
""" 5 ' 4 X V\ x Ratio of volume factors V k .
where V' r is the release volume in the L.P. cylinder and Vk is
the clearance volume in the H.P. cylinder.
Example No. 26 (see Fig. no).
B.Th.U., represented by H.P diagram = 79.7
>l J-/.X. ,,
(adjusted for volume factor) =' 69.9
V' r = 33.5 cubic feet.
V k = 0.6
Ratio of H.P. volume factor to L.P. volume factor
1.44
M.E.P. referred to} _ 70.7 + 69.9
L.P. cylinder .J $-4 x ^3.5 x 0.937 0.6
26.3 Ibs. per square inch.
BIBLIOGRAPHY.
The following is a list of the principal books dealing with the
subject of the entropy- temperature or energy chart :
Transactions of the Connecticut Academy of Science. Vol. ii.,
page 309. 1873.
Zeitschrift des Vereines Deut, Ing. M. Schroeter. 1883.
Rationalization of Regnault's Experiments. Proc. Inst. Mech.,
Engineers, 1889. J. Macfarlane Gray.
Steam Engine Trials. Proceedings of Inst. C.E. Vol. xciii., 1889,
and vol. cxiv., 1893. P. W. Willans.
Superheated Steam Engine Trials. Proc. Inst. C.E. Vol. cxxviii.
1897. Wm. Ripper.
The Theta-Phi Diagram. Henry A. Golding. 1898. Technical
Publishing Co., Manchester.
Revue de Mecanique. Professor Boulvin. June, 1897.
A new graphical method of constructing the Entropy Temperature
Diagram of a Gas or Oil Engine from its Indicator Card. H. T.
Eddy. Vol. xxi. Trans, of American Soc. of Mech. Engineers.
1900.
Cours. de Mecanique applique aux machines. 3 me fascicule
theories des machine thermiques. J. Boulvin, Paris. 1893.
The Steam Engine and other Heat Engines, by J. A. Ewing, M.A.,
B.Sc., F.R.S., M. Inst. C.E. 1894.
The Entropy Temperature Analysis of Steam Engine Efficiencies.
Sydney A. Reeve. 1897.
The Entropy Diagram and its Uses. H. B. Brydon. Power. June
and Juiy, 1904.
Entropy in S team Engine Practice (serial) . The Mechanical Engineer,
August 27th, 1904.
i66
BIBLIOGRAPHY.
Steam Engine Theory and Practice. W. Ripper. 1899.
Entropy, or Thermo-dynamics from an Engineer's standpoint, and
the Reversibility of Thermo-dynamics. Jas. Swinburne. 1904.
The Theory of Heat Engines. An address delivered to the Students of
the Institution of Civil Engineers, I7th January, 1902. By Capt.
H. Riall Sankey, R.E. (ret.) M. Inst. C.E.
The Temperature-Entropy diagram. G. James Wells. Proc. Man-
chester Society of Engineers.
REFERENCES TO SPECIAL TERMS
USED IN THIS BOOK.
PAGE.
Constant pressure line .. 5, 22
Constant volume line ... 5, 20
Corresponding < engine 71
Dryness fraction line ... 22
Energy chart 2, 15
Entropy 2
Equivalent feed 68
Heat energy per 6 <f>
diagram 73
Heat scale i, 22
Initial point 69
Lines of equal internal
energy 27
PAGE.
Proportional water line ... 49
Quality line 47
Saturation line 20
State point 6
Theoretical re-evaporation
line 54
9 < cylinder feed 73
9 <f> diagram 28
Transformation line 6
Volume factor 71
Water line 20
INDEX.
PAGE.
Adiabatic expansion ... ... ... 32
,, ,, Temperature and pres-
sure at the end of ... ... ... 65
Admission line, simple engine ... ... 116
Admission valve, Leakage of ... 57 59 76
,, ,, ,, direct into exhaust 81 120
,, Wetness of steam at ... ... 76
,, Wire drawing at ... ... 76
Air engine, Stirling's ... ... ... 16
Binarv engine ... ... ... ... 150
B. Th. U. utilised as work ... ... 67 159
Capper, Professor (on leakage) ... ... 120
Carnot cycle ... ... ... 17 18
,, ,, Thermal efficiency of... ... 19
Chart for superheated steam ... 23 134
Combining forward and back end indicator
diagram ... ... .. ... 83
Comparison of ideal and actual steam engine
cylinder ... ... ... 41
,, with Rankine cycle, I. C. E.
standard ... ... 74 115
Comparison of 6 <f> and p. v. diagrams ... 28
Complete cycle ... ... ... ... 16
Compound engines, Examples of 97 101 102 104
i2i 139 141
M ,, expansion in L.P. cylinder 93
,, ,, with large cylinder ratio ... 104
,, ,, ,, medium ,, ,, ... 97
leak trom H P. cylinder 89 101
Rankine cycle of ... ... 87
6 <J> diagrams of ... 87 97-108
123-128 139
,, ,, Sketching B <f> diagrams of 122
,, ,, Standard of comparison ... 121
i, ,, Exhaust line of ... ... 96
,, transfer of indicator dia-
grams to the energy chart 97 139
Compound engines volume factor ... 97 103
Compression in clearance ... ... 76 80 113
Compression pressure ... ... ... 50
Condensation initial ... ... 52 76 78
,, during pre-admission ... 51
Effect of initial ... ... 44
Water produced by ... ... 53
,, compound engine superheated
with reheater ... 139 142
Constant pressure cycles ... ... ... 19
M ,, for a gas 5, for H a O 22
Superheating at ... 137
., Transformation at ix 32
Constant volume cycles ... ... ... 19
line ii, for a gas 5, for H 2 O 20
,, Transformation at 10 34
Corresponding <p engine ... ... 71
,, Work done by ... 73
Clausius ... ... ... ... ... 2
Clearance, Compression in
Effect of
76 80 113
45 91
,, Weight of steam remaining in ... 73
Work done in compressing the
steam in ... ... ... 45
Cut off, Determination of point of ... ... 66
,, ,, in cylinders ... ... ... 100
,, ,, ,, Alterations of, in H.P.
cylinder ... ... 125
Cycle of operations in Carnot 's engine ... 18
,, ,, ,, steam engines ... 41
Cylinder feed, <f> ... ... 79 160
Cylinders, Cut off in ... 66 100 125 128
,, Equalisation of work in ... ...127
,, ratio ... ... 101 121 124 159
,, walls, Heat stored in ... ... 52
Design of compound steam engine ... ... 121
,, ,, simple engine ... ... ... in
Designing steam engines by means of energy
chart ... ... ... in 121
Determination of point of cut off ... 66 100
Division of energy chart into four zones n 35
Dryness fraction ... ... ... ... 71
,, ,, lines ... ... ... 22
Dugald Clark (specific heat of gases) ... 4
Economy of actual steam engine ... 6799162
,, Rankine engine ... ... 66 121
Eddies due to leaks ... ... ... 58
,, Effect of ... ... ... ... 51
Efficiency ratio ... ... ... 99 162
,, ,, of compound engine ... 115
,, M of jacketted engine ... 85
Elementary Thermodynamics of H fl O ... 28
Energy chart ... ... ... 21520
applied to other substances ... 150
Division of, into four cones ... 35
forH a O 20
for SO a ... ... ... 150
Representation of internal
energy on ... ... ... 24
Energy chart, Superheated steam, new values
of specific heat ... ... ... 134
Energy, Internal ... ... 7 24 37 53 6s
168
INDEX.
PAGE.
Energy retained in cylinder ... ... 73
Engine, Actual economy of ... ... 67
,, mean pressure of ... 68 124
steam consumption of ... 67
Stirling's... ... ... ... 16
Binary ... ... ... ... 150
Compound, examples of 101 102 104 139
volume factor . . 97
Example of jacketted ... 102
,, non-jacketted 97-102
,, with reheater ... 104
with large cylincer ratio 104
Sketching $6 diagrams of 122
Theoretical 87
Condensing, superheated compound
with reheater ... ... 139 142
Corresponding <p ... ... 71
Designing, by means of energy
chart ... ... ... in 121
Economy of ... ... 67 99 162
Jacketted, compound condensing
with reheater ... ... ... 102
Jacketted, efficiency ratio of ... 85
,, Loss of heat in engine ... 76 et seq.
Rankine, economy of ... 66 121 159
,, ,, Mean pressure of 68 121
tl ,, for superhr atfd steam ... 137
simple admission line ... ... 116
,, efiVct of increased clearance 119
point of cut off ... ... 114
Steam consumption of ... ... 100
Thermal efficiency of ... 74 162
triple condensing, Examples of 108-110
jacketted with
re-heater and jackets 108
,, Waste heat, numerical example of... 151
Entropy ... ... ... ... ... i 2
Equivalent feed ... ... ... 68 163
Definition of (Inst. Civil
Engineers) ... ... 68
Equalisation of work in the cylinders ... 127
Equivalent weight of water ... ... 54
Examples. See table of contents, Chapters
VII., VIII., IX. and XII.
Exhaust line ... ... ... 61 79
of L.P. cylinder ... ... 96
loss 79
locomotive cylinder ... ... 82
ports, Throttling through... ... 79
valve, Leaky ... ... ... 59
wire drawing ... ... 76 79
Expansion, Adiabatic ... 32
,, Incomplete ... ... 61 76 79
Isosthe-mal ... ... ... 8
line 52 57 59 79
in L.P. cylinder ... ... 93
Expansions of steam without doing external
work ... ... ... ... 144
Expansions, Number of ... ... zoo 158
External work, Measurement of ... ... 39
,, Representation of ... ... 9
Feed, Equivalent ... ... ... 68 163
temperature, Effect of ... ... 43
PAGE.
Feed, 8 cylinder ... ... ... 79
Factor, Volume ... ... ... ... 7 1
Gas engine ... ... ... ... 19
cycle ... ... 19
perfect ... ... 4
Specific heat of ... ... ... 4
Graphic representation of energy chart ... 3
Gray, J. McFarlane ... ... ... 2 4
H 2 O, Elementary thermodynamics of ... 28
Heat energy per 6 diagram ... 73 161
per <f> diagram jacketted engine ... 85
scale... ... ... ... ... 22
stored in the cylinder walls ... ... 52
M supply, Measurement of ... 39 159
,, utilized by actual engine ... 73 161
,, ,, Rankine engine ... 74 159
H.P. cylinder effect of alteration of cut off 125 128
,, Leak from ... ... 89 101
Incomplete expansion ... ... 61 76 79
Increase in clearance, Effect of, in simple
engines ... ... ... ... 119
Indicator diagram, combined forward and
backward end ... ... ... 83
Indicator diagram, compound engines, trans-
fer of, to energy chart ... ... 97
Indicator diagram, simple engines, transfer
of. to energy chart ... ... ... 69
Indicator diagram, transfer to energy chart
with superheated steam ... ... 739
Initial condensation ... ... ... 52
Effect of ... ... 44
,, in L.P. cylinder ... 92
Initial point, Location of ... ... ... 69
Internal energy ... ... ... 7245362
,, Lines of eqnal ... ... 27
,, ,, representation on energy chart 24
,, of steam ... ... ... 34
superheated steam... ... 135
Isothermal expansion ... .. ... 8
Isothermal lines for steam... ... ... 20
Jacket, Effect of adding ... ... ... 118
Jacketted engine, compound,condensing,with
reheater ... . ... ... 102
Jacketted engine, Efficiency ratio of .. 85
,, heat per <f> diagram ... 85
Jacket and leaky admission valve ... ... 59
Joule's equivalent of heat ... ... ... 3
Leakage, Maximum ... ... ... 58
,, Minimum ... ... ... 60
Admission valve ... 57 59 76 78 120
,, direct into exhaust 81 120
,, of exhaust valve ... ... 59 76
past cylinder into the exhaust ... 69
,, past piston rings ... ... ... 76
Leaks, Eddies due to ... ... ... 58
Lines of equal nternal energy ... ... 27
total heat ... ... ... 27
Location of losses... ... ... 7576
of initial point ... ... ... 69
Locomotive cylinder, exhaust loss ... ... 82
,, losses due to throttling
at admission and at exhaust ... ... 82
Losses due to throttling ... ... 76 82
INDEX.
I6 9
PAGE.
Losses, List of ... ... ... ... 76
Location of, on energy chart 75 76
L.P. cylinder, Initial condensation in ... 92
McFarlane Gray ... ... ... ... 2 4
Maximum leakage... ... ... ... 58
Mean pressure ... ... ... 68 100 164
,, Ajustmentof ... 113123
,, ,, Increase of: by altering cut off 128
,, ,, of an actual engine determined
from 6 <f> diagram 68 100
,, ,, of Rankine engine ... 68121
Measurement of external work done ... 39
,, of heat supply ... ... 39
Mechanical equivalent of heat ... ... 3
Minimum leakage ... ... ... ... 60
Motion energy, Recovery of ... ... 147
Non-condensing engine ... ... ... 69
Number of expansions, Calculation of ... 100
Perfect gas ... ... ... ... 4
,, Thermodynamics of ... ... 7
heat engine (Cai not cycle) ... ... 18
Piston rings, Leakage past... ... ... 76
Piston, Volume swept by ... ... ... 158
Play energy ... ... ... ... 73
M steam ... ... ... ... 73
Point of cut off, simple engine ... 100 114
M ,, compound engine ... 100 125
Pre-admission, Condensation during ... 51
Pre-determination of 6 <f> diagran ... m et seq.
Pressure, compression ... ... ... 50
constant, Transformation at 1132
,, lines, Constant... ... 5 6 n 22
,, Superheated steam ... 133
,, Mean effective ... ... 68 100 164
p. v. diagrams, comparison with 6 </>
diagrams ... ... ... 28 69 104
p. v. method ... ... ... ... i
Prof. Josse ... .. ... ... 150
Proportional water line ... ... ... 49
Proportion of work done in each cylinder 101 123
Plotting the 6 <j> diagram ... ... 71 84 97
Quality line ... ... ... 47 73 93
,, corresponding to any given trans-
formation line ... ... ... 63
Quality point ... ... ... 63 92
Rankine ... ... ... ... ... 2
,, engine, Comparison with... 74 115 122
for compound engines 87 122
,, Economy of ... 66 121 159
,, ,, Mean pressure of ... 68 121
for superheated steam ... 137
,, ,, Thermal efficiency of 74 161
Ratio, Cylinder ... ... ... 101 124 159
,, Efficiency ... ... ... 99 162
Re-evaporation line, Theoretical ... ... 54
Reheater compound condensing engine (super-
heated steam) ... ... ... 139
Reheater compound condensing engine
jacketted ... ... ... ... 102
Release volume ... ... ... 79 158
Reversibility of heat engines ... ... 2
Representation of external work ... ...929
PAGE.
Saturated steam engine compared with super-
heated steam engine ... ... 138
Saturation line ... ... ... ... 20
Saturated steam ... ... ... ... 21
Scale of internal energy ... ... ... 25
Simple jacketted engine ... ... ... 84
Specific heat of gases ... ... ... 4
of superheated steam 23 134
Standard of comparison for compound
engine ... ... ... ... 121
State point ... ... ... ... 6 7
Steam, Chart for superheated ... 23134
consumption of an engine ... 67 100
cylinder, Transformation in an ideal 31
in clearance. Work done in com-
pressing in ... ... ... 45
engines, see under engine
formation under constant pressure... 20
Internal energy of ... ... ... 24
Superheated ... ... ... 133
Transformation line for less than
one Ib. of ... ... ... 64
,, velocity of issuing ... ... ... 146
,, Work done by expanding ... ... 29
Stirling hot air engine ... ... ... 17
Superheated steam ... ... 23 133
M M Chart for ... 23 134
,, engine, Compound condensing,
with rf heater ... ... 139 143
Superheated steam constant pressure line 133
n >. volume line 133
,, energy chart, new values
ot specific heat ... 134
Superheated steam engines compared with
saturated steam engine ... ... 138
Superheated steam engine compared with
combined steam ar d waste neat engine 155
Superheated steam, internal energy ... 135
,, Rankine cycle for ... 137
transfer of indicator dia-
grams to energy chart 139
Superheating at constant pressure... ... 137
steam by expansion without
doing work (by throttling) ... 146
SO 3 , Energy chart for ... ... ...150
engine ... ... ... ... 150
combined with steam engine,
numerical example ... ... 151
Temperature entropy method ... ... 2
Temperature and pressure at end of adia-
batic expansion ... ... ... 65
Theoretical re-evaporation line ... ... 54
Thermal efficiency ... ... 17 99
of an actual engine 74 162
of Carnot cycle ... 19
of Rankine engine 121 161
Thermodynamics of a perfect gas ... ... 7
H a O 28
Total heat, Lines of equal ... ... ... 27
Transformation along any line ... ... 35
,, at constant pressure n 32
,, at constant temperature ...8 33
,, at constant volume zo 34
,, line for less than one pound
of steam ... ... 64
170
INDEX.
PAGE.
Transformation in an ideal steam cylinder ... 31
line, Quality line correspond-
ing to any ... ... ... ... 63
Throttling through exhaust ports ... ... 79
Triple condensing engine ... ... ... 107
,, jacketted engine with re-
heater and jackets ... ... ... 108
<f> Chart... ... ... ... ... 2
cylinder feed... ... ... 79 160
diagrams of compound engines 87 97 et seq^
Heat energy in ... ... 73
Plotting of ... ... 84 97
,, Pre-determination of in 121
,, Sketching of .compound steam
engine ... ... ... ... 122
,, diagrams, Sketching of, simple steam
engine ... ... ... ... in
engine, Corresponding ... ... 71
Useful formulae ... ... ... ... 157
Valve leakage ... ... ... ... 120
Velocity of issuing steam ... ... ... 146
Volume factor ... ... ... ... 71
Compound engine ... ... 97
line, Superheated steam ... ... 133
PAGE.
Volume swept by piston ... ... ... 158
Waste heat engine compared with super-
heated steam engine ... ... *55
Waste heat engine, Numerical example of ... 151
Water line ... ... ... ... *o
Proportional ... ... ... 49
present at exhaust ... ... ... 62
produced by condensation ... ... 53
Weight of steam remaining in the clearance 73
Wetness of steam at admission ... ... 76
Willans * 54
WillardGibbs i
Wire drawing at admission ... ... 76
Exhaust ... ... 76 79
Work done in compressing the steam in the
clearance ... ... ... ... 45
done by the corresponding 0$ engine 73
in cylinders, Adjustment of ... 123
,, by expanding steam ... ... 29
,, ,, Proportion of, in each cylinder
ioi 163
, r external, Measurement of ... ... 39
External representation of ... ... 9
Internal ... ... ... ... 10
