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This is to certify that the

thesis entitled

'L Study of Thermodynamic Irreversibility in a
Fluid kpanding Behind a. Moving Piston.”

presented by

Ralph M. Rot ty

has been accepted towards fulfillment
of the requirements for

m— degree ill—la]...—

MflM/W ,

Major professor V

 

A STUDY OF THE THERMODYNAMIC IRREVEHSIEILITY IN A FLUID

EXPfiNDING BEHIND A MOVING PISTON

By
Ralph M. Rotty

F“

A THESIS

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1953

fit-16515

AC KN (IN LED L‘x EMISNTS

The author expresses his sincere thanks to the members of the Guidance
Committee: Professor C. H. Pesterfield, Chainnan, ProfesSor L. C. Price,
Professor C. C. Dehitt, Dean T. H. Osgood, and Professor C. P. Wells, for
their assistance and comments during the period of this research work.

In addition, the author wishes to express his appreciation to the
many students and other staff members who have assisted with various phases
of the work. Contributions of time and effort made by the following deserve
special thanks: Dr. J. T. Anderson for his several suggestions, especially
in regard to heat exchanges between the fluid and the cylinder walls;

Dr. H. J. Jeffries for his assistance in the many instrumentation problems;
Mr. P. J. DeKoning for his suggestions regarding the use of SR—h strain
gauges; Messrs. S. Mercer, Jr. and D. J. ienwick for their criticisms and
suggestions; and Messrs. D. W. Seble and C. M. Hedman for their assistance
in the modification of the Russell Engine and general mechanical help.
Very special acknowledgement is due to Dr. N. A. Hall of the University of
Minnesota for his interest and fundamental suggestions.

Finally, the author wishes to acknowledge the following who also con-
tributed: Dr. H. Parkus, Dr. L. L. Otto, Dr. C. 0. Harris, Messrs.

J. Hemmye, S. Bergner, S. Peck, 8. Sterling, and H. H. Barnum.

31127.

0"’

Ralph McGee Hotty
candidate for the degree of

Doctor of Philosophy

Final examination, May 19, 1953, 9:00 A.M., Room 105, Olds Hall

Dissertation: A Study of the Thermodynamic Irreversibility in a Fluid
Expanding behind a Moving Piston.

Outline of Studies

Major subject: Mechanical Engineering (Thermodynamics)
Minor subjects: Physics, mathematics

Biographical Items
Born, August 1, 1923, St. Louis, Missouri

Undergraduate Studies, University of Missouri, l9h0—h3
State University of Iowa, l9h6—h7
(8.5. in 3.3., 19h?)

Graduate Studies, California Institute of Technology l9h7—h8
(M.S. in Meteorology, l9h8)

California InStitute of Technology l9h8-h9
(M.S. in Mech. Eng., l9h9)
~Michigan State College, l9h9-53

Experience: Weather Officer, United States Army Air Forces,
l9h3-h6; Graduate Assistant, California Institute of
Technology, l9h7-h8; Research Meteorologist, American
Institute of Aerological Research, l9h8—h9; Instructor
in mechanical Engineering, Michigan State College,
l9h9-53.

Member of Tau Beta Pi, Pi Mu Epsilon, Eta Kappa Nu; Associate of
Society of the Sigma Xi.

 

c--..--

A STUDY OF THE THERMODYNATJC IRILEVERSIBILITY IN A FLUID

EXPANDING BEHIND A MOVING PISTON

By

Ralph M. Rotty

AN ABSTRACT

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

Year 1953

/ ,,/f\J, ‘#7Z: / y . F

ABSThACT

In this study, the concept of entropy production is fundamental.
The rate at which entropy is produced is a direct measure of the irre—
versibility of a process. In order to determine the energy degraded
during a given process, it is necessary simply to integrate with respect
to time the product of the rate of entropy production and the absolute
temperature.

The first major hurdle in such a determination is the development of
an analytical expression for the rate of entropy production. For the
case of a viscous fluid, it has been found convenient to use a fluid
dynamics approach. Beginning with the statement of Newton's Second Law
and the equation of continuity, it is shown that the rate of energy de-

gradation in an element of a viscous fluid is given by:
—-b
->
To=-9-. T+ - 'v
/K> 1? :7 (.(Lij ‘57)

‘where: 9 = rate of entropy production per unit mass
T = absolute temperature

/0 = fluid density

75': rate of heat flow into the fluid element
f lij = viscous stress tensor

4

V = velocity of the fluid element

For the case of a fluid expanding in the cylinder of an engine, the
terms on the right hand side of the equation were simplified by taxing

averages of velocity and temperature in cylindrical fluid elements. The

equation was then reducei to

 

2
foTe=£(§I)2+2(T-Ts)+y_ (9132

The term,‘5 C§§)2, represents the rate of energy degradation in a

unit volume of fluid as a result of heat exchanges with other fluid

h(T-T)2
elements. This is called the d1 effect. The term, E.______§__.,
f
c

represents the rate of energy degradation as a result of heat exchanges

between the fluid and the cylinder walls, and is called the a2 effect.

dV 2

The final term,-%/L(a; , called the 5 effect, is the energy degradation

as a result of the viscous forces.

'The procedure required in the evaluation of these terms was demon-
strated on the old Russell Steam Engine in the Power Laboratory. The
engine was modified to run on compressed air rather than steam, and the
fluid (air) volume in the cylinder was divided into seven elemental
volumes. Pressures and temperatures were measured instantaneously at
many points within the expanding air by employing a system of sliding
brass tubes which passed through the cylinder head and piston. Forty
gauge thermocouple wire was used in the measurement of temperature and
an SR—h strain gauge mounted on a thin diaphragm was used to measure
pressure. The physical measurement of some quantities supplemented by
reasonable estimations of others, made possible the calculation of a1, a2,

and B effects. Data are presented on each effect with the Russell Engine

running at 60, 120, 180, and 2hO revolutions per minute.

‘11-"

In the case of reciprocating engines the a2 effect is by far the
largest. Variation of B with engine speed is shown to be linear, while
a1 and a2 although increasing with speed in this range may not continue

to do so at still higher speeds because of shorter time duration for the

flow of heat.

 

TABLE OF CONTENTS

INTRODUCTION . . . . . . . . .
OI‘ISAGER THEORY . . . . . . . . .
THEORETICAL ANALYSIS . . . . . . .
APPARATUS AND METHODOLOGY . . . . . .
MEASUREMtNT OF TEmPERATURE . . . . .
RECORDING OF PISTON POSITION . . . .
LEASUILEMENT OF PRESSURE . . . . . .
DETEIUEINATION OF II‘ESTANTANEOUS FLUID VELOCITY
PPESENTATIOI‘J OF DATA Aim CALCULATED RESULTS. .
S'UIELARY AND CONCLUSIONS . . . . . .
APPENDIX . . . . . . . . . .

BELICII’IMPIIY . O O O O C O O O

10
19
22
2h
27
29
3h
us
So
88

INTROD ucriou

According to the first law of thermodynamics, energy can be neither
created nor destroyed but may be transformed from one form to another.
This might lead one to believe that if two units of energy are supplied
to an engine, the engine should.be able to deliver two units of useful
work. Experience shows this to be untrue, as a certain portion of
the energy supplied must be rejected as low grade energy. This effect
is expressed in general terms by the second law of thermodynamics, which
states that for any exchange of energy from one form to another, or.from
one body to another, a certain portion of the energy is "degraded" or made
unavailable. Thus it is often said that the second law of thermodynamics
is a limitation on the first law of thermodynamics.

In the absence of all the effects which cause energy to be degraded,
one may have an ideally reversible process. In such a process all of the
energy transfers could be reversed and all matter returned to the same
state or condition which it occupied prior to the process.

In elementary thermodynamics it is shown for such a reversible process

that the work which a fluid is capable of producing may be expressed as:
'W = SSpdV for a non-flow process
W = - S‘Vdp for a steady-flow process

These expressions are widely used by engineers in computing the work pro-

duced by processes which approximate "reversible" processes.

S.
A better approximation (especially for processes which do not approx—
imate "reversible" processes very closely) sometimes given to students in

undergraduate thermodynamics courses, may be expressed as:

'4
‘4:
ll

SFUV - Qf for a non-flow process

a.
ll

- SVdp - Qf for a steady-flow process

Here Or is the energy degraded or lost as available sources of work. Qf
is usually in the form of low grade heat energy.

The quantity Qf is sometimes called 52135 since this is the amount
of energy which the process "shirks" delivering simply because it is not
a reversible process. The determination of the magnitude of the shirk
is not covered in most thermodynamics work, and it is this determination
‘with which the major portion of this treatise deals.

Before advancing to the heart of the subject, it seems appropriate
to make a few remarks regarding entropy changes in reversible and irb
reversible processes. In a given system undergoing any process, the entropy
change may be positive, or negative, or zero depending on the process.
However, for a reversible process the entropy changes of the system must
be exactly balanced by an opposite change in the entropy of the sur-
roundings. In such a case, if the entropy of the system increases, the
entropy of the surroundings must decrease by exactly the same amount; that
is, the sum of the entropy of any system and its surroundings is constant
for any reversible process.

Similarly for an irreversible process the entropy change of the

system may be positive, negative, or zero, but in contrast to the revers-

6.
ible processes, the summation of the entropy of the system and its sur-
roundings must increase for irreversible processes. If this line of
thought is followed further, making use of the fact that all processes
are at least slightly irreversible, the conclusion is reached that the
entropy of the universe is constantly increasing and approaches a maximum.
For the purposes of this discussion, however, it is far more important to

emphasize the pgoduction or creation of entropy. The rate at which entropy

 

is created during a given process may be taken as a measure of the irre-
versibility of the process.

This notion of entropy creatigg_leads to a new statement of the

 

second law of thermodynamics: "Entropy may be created, but never destroyed."

ONSAGER THEORY

Recently the thermodynamics of irreversible processes has become
very important. This is expecially true in connection with diffusion
processes. Mbst of the work analyzing these processes is based on
Onsager's theorem, which regards the processes of diffusion and the flow
of heat and electricity as representable in terms of a flow and a ther-
modynamic force.

Each flow is directly proportional to the corresponding force. For

example the flow of electricity (current) is proportional to the EMF:

Similarly, the heat flow is proportional to temperature gradient:

—-,

q (x: grad T

-> —->

In general, if J is a rate of flow and X is a force, there is an

empirical relationship:

-o- ->

J a L X
L is a scalar quantity and is of the nature of a conductance or the re-
ciprocal of a resistance. If two or'more of these transport processes
take place simultaneously each flow is taken as being proportional not
only to its own force but also on the other forces as well. This is more
general than to assume each flow depends on only one force.

For example, in the case of thermal diffusion from one vessel to

--'h —-O-
another, there is an energy flow, J1, and a mass flow, J

2, and

--> —-i- -o-
Jl’Lnxl +L12X2

-—> —>
J2='L'21x1 +L22 X2

Here L11 and L22 are related to the ordinany thermal conductance and
diffusion coefficient, respectively. The coefficients L12 and L2]- rep-
resent the possible coupling between the two flows. Thus L12 represents
the tendency of the diffusion force to give rise to a flow of energy.

In general a set of linear equations describe a set of coupled flows

in a physical problem:

Ji '2 Lik Xk
k

Such equations are usually called the phenomenological relations
(called thermodynamic equations of motion by Eckart). The "coupling"
phenomenological coefficients, Lik’ i“ k, may or may not be zero in a
particular case.

Onsager's Theorem may be stated as follows: Provided g proper choize

 

 

‘33 made fog the flows, J1, and forces, X the matrix pf phenomological

i)

 

 

 

 

coefficients, Lik i§.§ymmetric, i.e., Lik = Lki' The "proper choice" is
assured by first writing the rate of entropy production and choosing the

flows and forces in such a manner that
2"” "
/09=' Ji'Xi

i .

Where/f: is mass density and 9 is the rate of entropy production per unit
mass.
The solution of a problem involving an irreversible thermodynamic

change can then be obtained by first employing various physical laws which

9.
apply to the situation to obtain an expression for the rate of entropy
production. The selection of the forces and flows then makes it possible
to write a set of phenomenological relations for which the matrix of the
coefficients is symmetric. The phenomenological coefficients which do not
involve "coupling", L11, have been evaluated for many transport processes.
This leaves the "coupled" coefficients, Lik’ to be determined. Many of
these may be evaluated from physical boundary conditions, and the use of
the Onsager Theorem will double the number of known "coupled" coefficients.
That is, if Lik is determined from physical conditions, then Lki is known
to be the same by Onsager's Theorem. Finally then the solution may be

written in the form:

fG'ZZ Ink-{f};
i k

lO.

THEORETICAL ANALYSIS

In proceeding to analyze the thermodynamic irreversibility in a
single fluid during its expansion, consider a small elemental volume.
From Newton's second law, the equation of motion of a unit element of
volume can be written (in the absence of external gravitational and elec-

trical forces):
V- ’T’ij (1)

Where: is mass density
d
V is the vector velocity of the element
t is time

'7".J is the stress tensor given as follows:

 

 

 

 

     

.. 1* BVX 2 __z 2 ”x .. 3:2: BzV

_P _———. -_ -—

+ (3 ax 3 «9y 3 32 y .3?) (-3-— + 0X 2)
bvx 3V 14 3V 2 3V 2 3"z

1/13 ( y ax / (3 3y 3 Ex 3 32 (3: 3y

( 3V3; _ .
f 52 5;— fl dz 8y 3 az ' 3 5x 3 c>y

 

 

or, shortening the notation:

+ aq + 3q 2 . ""
71:). .9513, ”La—.3. 5—33 -_3_(le V) (5-3] (2)

l

where: P is the pressure

/u is the viscosity

+
qi,qj are components of velocity V

ll.

)\
1:

ii“:
=O,i¥j
Multiplying equation (I) by the vector V

lei-33 .V.(v.7’ij) (3)

ct

01“

2
dV 2 7’ -’ 7/ V '3
f j— =3 V0 ( ij 0 V) — ( ij 0 > o J (h)

This is a general energy equation. In order to recognize the tenns
in this equation and compare it with the general energy equation frequently
used in thermodynamics, apply the divergence theorem to the first tenn on
the right hand side of the equation. The integral over any elemental

volume
SV°(7I;;° “d7” = S '73 ‘d" (5)
'r a-

The right hand side of equation (5) can be recognized as the work
done per unit time moving the fluid without changing the size or shape of
the fluid. Note that in the simple case of a non—viscous fluid flowing in
a duct with uniform velocity parallel to the axis of the duct, this surface
integral becomes

Top) . I =- -Pv

where v is volume flow per unit time. In this simple case the external
work done is zero, but if the velocity is not always parallel to the re-
stricting boundaries, the integral over the entire surface sounding the
fluid element will give a force moving the boundaries back, for example a
fluid impinging on a turbine blade. Thus the first integral represents the

negative of what is usually referred to in elementary thermodynamics as rate

12.
of change of flow work plus the external work done by the fluid.
In regard to the second term, again take the simple case of a non-
» -—>
Viscous fluid so that 4713 ' V) - V reduces to Simply P V' J.

From the continuity equation:
—p
.Pfl.+V-/ov=o (6)
at

Since the total derivative is related to the partial derivative by

 

———-—-d( ) = 3( ) WWW >, (7)
dt 6t

31: 3 [355; + '3’. <Z>D (8)
dt a‘t
or,
if "V'f’? + V‘V/O= 70‘}? ‘ (9)
Therefore,
v. ?=_}.9fi.’o—.4&-d1 (10)
/° dt dt
and
-’ d l
pv v—det (n)

This is the work of expansion per unit volume and therefore must be equal
to the rate of heat addition minus the rate of increase in internal energy.

Writing the usual energy equation of elementary thermodynamics:

Q_ = - - 9;, -.9_ ‘ +
dt (KE) W dt (F.W.) dt (L) Q (12)

‘where W’ is the time rate of work done by the fluid
Q is the time rate of heat added to the fluid

Then by analogy

_ . - d ,r.
v- (7’1j v > = - w ‘fa‘t (Funk)

4713. V) - ‘v’w ~ng (2)

Considering the heat flow vector, at
v-Ir- - Q
Then

9?: +Vo‘q’-<?' .V).‘v’

dt id
However, from (2),

(nj-Vw v --PV- v+<nij -V>- v

la<1- 3 .,
. a [ l + q 2 .
fllj fl _3 Ti -—3 (le V) 513]

Therefore,

dE+ 0-.4- .-’= . 5-,
/"EE; ‘57 <1 P ‘7. V (I113 ‘57) V

As shown above the continuity equation leads to the relation:

Then

[/3CEE + P.Ea%éEL) +‘KJ7. '3’. (IqLij .VQ?) . ‘7'

13.

(13)

(lb)

(15)

(16)

(17)

(18)

(19)

(2O)

or,

FT§§+Vo7€=(n13 ~V>oV <21)

Defining an entropy flow vector,

-’ .5
s 8% (22)
V'g'i2[TV°?'?'VT] (23)

T ,

fromwhich

v-quv°?+%'VT (2h)

Therefore (21) becomes

fT§E+Tv. ?.-?.VT+(flij OV)0?

(25)

Examine the terms on the left side of this equation. 3-: is the
. . . ds ,
entropy change ( per unit mass) per unit time. Then /0 a: 15 the entropy

change (in the elemental unit volume) per unit time. V' .8: the
divergence of entropy, is the amount of entropy which leaves (flows out)

the unit volume. Therefore, /° 93 + V- ‘5’ is the-rate at which entropy
dt

is being produced. In other words, this describes the strength of the

entropy source and has a minimum value of zero for reversible processes.

 

,oe= §E+V-‘é’ao (26)

Here 0 equals the strength of the entropy source per unit mass.

15.

ds
The term T( — + - ‘5) = T9 is then the rate at which
Pdt V /°
energy is being degraded, and equation (25) becomes:
—> d
,oT9=-so§7T+(Dij-V)- v (27)

Dividing equation (27) by the absolute temperature, T:

'4

/og=-.T‘1.VT+%(nij°V)° V (28)

This entropy production rate must be divided into forces and flows
in order to apply Onsager's technique, Certainly from the first term, it
can be seen that

Jl = T? = energy flow

"* -172

.9
l T2

but the selection of a flow and a force which give the second term is very
difficult. Because of the tensorial character of the force involved, this
term can have no influence on the vectorial flow, T's: Therefore, Onsager's
technique leads only to the conclusion that there is no coupling between
the two flows, since by Curie's theorem a force of tensorial character
cannot give rise to a flow of vectorial character (in linear equation).
Some other technique must then be used to put equation (27) in a usable form.
Before proceeding it is proper to mention here that equation (28) for
the rate of production of entrOpy is general in that it applies to any
single fluid undergoing expansions or compressions in any piece of ap-
paratus. For example, it is valid for steam flowing through a turbine.

The remainder of this discussion will be aimed at the special case of the

16 .
expansion of a fluid in the cylinder of an engine.

. 1 -" . .
Since the term #1115) . V) . V is difficult to evaluate, it was

found necessary to assume that in the cylinder all velocities are directed
toward the moving piston, and further that the velocity is uniform across

any cross section. That is, V a V2 - O,all partial derivatives of Vy and

Y
Vz are zero, and 3.135. =93}, - 0. Then
3y Sz
V 2
. . . v.5 ____a x 2
(5113 V) 3/( x) ( 9)
‘s’
The term - T - VT can be simplified for the case of a fluid in a

cylinder with‘less drastic assumptions.

‘3'. 3'
T
Therefore
‘3’ T
—-OVT=--q*ol (30)
T T2

The quantity represented here by 21* is the heat flow vector leaving the
elemental volume. Think in terms of slices through the cylinder as shown

in Figure 1. The heat flow may then be divided into the sum of two parts.

4

 

 

17.
These are the flow of heat within the fluid along the axis of the cylinder,
and the flow of heat from the fluid to the cylinder wall or vice-verse.
Similarly the temperature gradient VT may be considered as the sum of
the component along the axis of the cylinder plus the component directed
into the cylinder wall. Then:

fe=-%(g-E)-EE(VT)I.+ 3“};- .._2£V)2 (31)
T T

Since qx is the flOW'of heat within the fluid,
qx a):

and the expression for entropy production, which may be written:

f0 k (332- ELMVT) +EMST (33)

T2 25x T2
shows the production of entropy to be a result of three separate phenomena.
They are:
(l) The exchange of heat between fluid elements
(2) The exchange of heat between the fluid and the cylinder walls.
(3) The viscous drag.
The rate at which energy is degraded may be obtained by multiplying

by the absolute temperature.
a .15 2 _ S. . it an 2 '
flTO ("3" :) T (VT)r + 3/43?) (314)

The quantities most difficult to obtain for use in this expression are, q,

the heat exchanged with the cylinder walls and (V7T)r, the radial temper-

l8.

ature gradient. As an approximation the temperature of the fluid element

shall be taken (cylindrical slice) as uniform and then q = h (T - TS).

Here Ts is the temperature of the surface of the cylinder wall, and h is

an estimated film coefficient. Similarly the radial temperature gradient
Ts - T

may be estimated by using . The configuration
configuration factor

 

factor is taken as the volume of the fluid element divided by the area of
metal with which the fluid is in contact.

Finally, then, an expression is obtained which will estimate the rate
of energy degradation and involve only quantities which can be measured
(or approximated).

Qf - Z/oTQ (35)

(where the summation means summation over both volume and time.)

=.3591_'£2+h('1’-'I‘) dv2
fire T (dx) EWS" -3-/-l(-a—x- -—-) (36)

Note that each term in this equation has units of energy divided by the

product of time times volume.

l9.

APPARATUS AND METHODOLOGY

To determine the energy degraded in a fluid expanding behind a
moving piston, the previous section suggests that the volume of fluid be
divided into a number of small volumes; the energy degraded in each small
volume computed; and then all added together to give the total energy
degraded.

To illustrate the use of such a technique, data were taken on the
Russell Steam Engine in the Mechanical Engineering Power Laboratory.

This engine was formerly run as an undergraduate experiment, but since
about l9h5 this test had been omitted from the laboratory procedure.

The engine was due to be dismantled and removed from the laboratory when
it became apparent that it might have some value in a study of this type.
The single cylinder of this engine, because of its size (nine inch bore
by fourteen inch stroke), lends itself very nicely to measurements within
the fluid inside the engine. Therefore this engine was chosen to demon-
strate the technique suggested here for measuring degradation of energy
due to irreversibility.

Some simple engine modifications were found desirable. For
simplicity in calculating the results, it appeared desirable to run the
engine on compressed air rather than steam. The Russell Engine was
therefore connected to the parallel supply of the college air line and
the Joy Air Compressor of the Power Laboratory. Since measurements were
to be made within the air, it was necessary to place a measuring element
in the fluid, obtain a measurement, and remove the element before the

piston crashed into it on the return stroke. After-considerable thought

20.

the scheme shown in Figure 2 was adopted.

 

 

 

 

 

 

 

 

 

 

 

 

 

i- Jl=j a-
K Le

/

v: %
e i g

_ O

 

 

 

Figure 2

This scheme, which made it possible to easily obtain data in the fluid

at the crank end, consisted simply of a system of brass tubes sliding on
each other. One set of tubes had an one-eighth inch diameter hole through
which the smaller tube was inserted. The larger tubes were fixed to the
piston in such a manner that an eighth inch hole ran completely through
the piston, enabling the smaller tube to pass through the piston. The
larger tube was made long enough so that it always extended out through

a hole in the cylinder head (even when the piston was at the bottom dead
center position). The smaller tubes were made fourteen inches longer than
the larger ones and were held in a given position by clamps. A pressure
or temperature element could be located at the end of the smaller tube and
positioned at any point within the cylinder. The larger tube then slid
over the measuring element and "uncovered" it when the piston was in a
position beyond the measuring element. That is, the element was

"covered" by the larger tube until the piston passed and the element was

"thrust" into the fluid volume.

21.
Holes were drilled through the piston and through the head for three
such sets of tubes. The locations of these sets of tubes are best under-

stood by referring to Figure 3.“__

 

 

Figure 3

A final modification made on the engine was that it was made to run
single-acting instead of double-acting. The engine was manufactured with
D-slide valve type of valve mechanism, and it was therefore possible to
clamp the D for the head end in a position such that the chamber on that
end of the piston was always open to the exhaust line. This made it pos-
sible to have the holes through the head considerably larger than the
outside diameter of the large tubes. No packing was necessary at this
point since atmospheric pressure was present at all times on both sides of
the head.

The volume of air in the crank end of the cylinder was divided into

a number of small volumes as follows (See Figure h):

22.

 

 

 

 

 

 

 

 

 

 

 

 

@112 B; C LI D 5 E 6 F 7 G 81-19
I I I ' I I I
I: I i 1 . ; ; z
I l : : . I .
: : I I : : : :
I ' ' I | '
. I : ; . I I I
I: It: LII‘ IV V VI vi]
a. . : ; . : .
I ' I , I ' I
I ' l g I l I
. I ' I I I ' I
! .; i : : ' ' '
Ll. ' . . r L :
Figure h

When the piston is at position A, the fluid is taken as having a
single small volume a. Similarly when the piston is at B the fluid has a
single small volume I (note that I includes volume a), but when the piston
is at C the fluid is made up of two small volumes I, II. At D it is made
up of three small volumes, I, II, and III, and so on until at H it is made
up of seven small volumes, I, II, III, IV, V, VI, and VII. The true
significance of these volumes and the reasons for choosing them in this

manner will be apparent in the discussion of the method of calculations.

Measurement of Temperature

The measurement of the temperature at a given point within the air
at the exact instant the piston is in a given position requires a temper-
ature element which must react very very rapidly to any changes. For that
reason the temperature element chosen in this work was a very fine wire
thermocouple. If the thermocouple is made of very fine wire, the mass of
metal which.must be heated for the thermocouple bead to assume the temper-
ature of the surroundings is very small, and therefore the amount of heat

required is small and the time required for the heat to flow is small.

23.

In this case, Number £0 gauge copper-constantan thennocouple wire was
used. Number to wire has a diameter of 0.033l inches and a time constant of
the order of 0.05 second. Especially at the higher engine speeds, this time
constant is not sufficiently short to insure accurate temperature readings.
The gradients as measured and shown in the Appendix were in part due to lag
in the response of the thermocouple. Number to gauge wire is the smallest
thermocouple wire which is commercially available, and was satisfactory for
the purpose of demonstrating the effect of temperature non-uniformity on
the irreversibility.

When the Russell Engine was run on compressed air, the temperatures which
occurred in the cylinder were of the order of forty to eightydegrees Fahrenheit
and the maximum variation encountered within the cylinder was of the order of
forty Fahrenheit degrees. In this temperature range, the potential difference
developed by a copper-constantan thermocouple is 0.023 millivolts per degree
Fahrenheit. Therefore, several stages of amplification and very careful and
accurate recording are required. Since amplification of a direct current
potential of such small magnitude requires very special equipment, a chopper
circuit which makes it possible to use alternating current amplifiers was used.
The thermocouple was run directly to an Ellis BA—l Chopper Amplifier where the
signal was chopped and then passed through three stages of amplification.

The output of the Ellis Amplifier was fed into a cathode ray oscilloscope
where the signal was further amplified before being placed on the plates of

the cathode ray tube. This circuit is shown schematically in Figure 5.

 

 

 

 

, 3 Ellis
Thermocouple in Tube Chopper Oscilloscope
JEEEEEEEizzzzii - -
a Amplifier Ell—1 '

 

 

 

 

 

 

 

Figure 5

2h.

By using a dual-beam oscilloscope, it was possible to make the temper-
ature measurements in this manner and correlate them with piston position
by using the other beam to pick up a signal for piston position as described
below.

Figure 6 shows a sample of the temperature data as recorded on the
dual-beam oscilloscope for two positions of the thermocouple. The temper-
ature is inversely proportional to the distance between the two lines in
the upper part of each picture. The lower pattern on each shows variation
in piston position which will be discussed below.

The temperature measuring circuit was calibrated by holding the
thermocouple in a water bath of known temperature and noting the number of
lines deflection on the oscilloscope. Such data, taken at several tempera-
tures of the water hath, made it easy to prepare a calibration curve of

temperature versus lines deflection.

Recording of Piston Position

The lower portions of the pictures in Figure 6 show an envelope of a
high frequency signal in which the amplitude of the signal varies with
piston position. This was accomplished through the use of a differential
transfo rmer .

A differential transformer is one in which the magnetic core is
movable, thereby changing the magnetic linkage between the primary and
secondary coils of the transformer. The change in magnetic linkage in turn
causes a variation in the potential induced in the secondary.

In this work an "Atcotran" Type 6205 differential transformer was ex-
cited by a 2000 cycles per second signal from an oscillator. The magnetic

core was attached to the pantograph reducing mechanism already on the

 

 

 

Figure 6. Oscillograms of Temperature and
Piston Position with Russell Engine
Running at 180 RPM. Thermocouple
at Position B (above) and
at Position 1 (below) .

25.

26.
engine so that the fourteen inch piston travel was reduced (in the
measurement system) to approximately 3.5 inches, a length which the
"Atcotran" type 6205 is capable of handling. The secondary of the trans-
former was connected directly to the oscilloscope in the case of position
reference for temperature measurements, but it was necessary to connect
through the Brush Amplifier to record position on the Brush Recorder in
the case of position reference for pressure measurements. These systems

are shown diagrammatically in Figure 7.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Oscillatoxl
k
Brush
D-C
Amplifier
h__(.)sc o cope . I
(a) . (b)
(a) Piston position recording circuit for use with temperature

measurement.
(b) Piston position recording circuit for use with pressure
measurement .
Figure 7
The calibration of lines deflection on the recorder for each piston
position was accomplished by the direct means of positioning the piston

at the various positions and noting the number of lines deflection on the

recorder .

27.
Eeasurement of Pressure
Considerable difficulty was encountered in the measurement of
pressure at the various pointS'within the fluid.
The stationary tubes which pass through the piston were used to trans—
mit the pressure from the point in question to a strain gauge type pressure
pick-up located externally in reference to the cylinder. This arrangement

is shown in Figure 8.

WA U ‘ Strain gauge

 

 

 

 

l _T pic k—up
Pressure to
be measured [—1 Pressure Transmitted
Here 'ston Head Through Tube to Here
Figure 8

The attenuation and time lag of a pressure signal transmitted through a
tube of constant cross-section has been theoretically discussed by

Arthur S. Iberall in Research Paper RP 2115, Journal of Research of the
National Bureau of Standards, Volume h5, July 1950. Iberall has presented
a series of curves from which the attenuation and phase lag may be deterh
mined for various amounts of damping.

In the measurements made on the Russell Engine, a pressure trans-
mission tube of l/32 inch inside diameter was used. ‘This very definitely
resulted in large damping of the pressure wave as it is transmitted through
the tube. However, by making the instrument volume (acting on the diaphragm
of the pressure pick—up) small enough, the ratio of the amplitudes of the
pressure wave at the two ends of the tube could be kept very close to
1.000 for the speed range used with this engine (60 to 2&0 revolutions

per minute, or 1 to h cycles per second). This condition also resulted in

28.
phase lags of less than 3 degrees. As a check, a tube of larger diameter
was used and was found to give essentially the same amplitude of pressure
variation at the diaphragm. A conventional indicator card was taken and
this also indicated that the pressure pick—up at the end of the tube was
measuring correctly.

A sketch of the pressure pick-up is shown in Figure 9.

Clamping nut

  
  
  

Active Strain Gauge
- Brass Diaphragm
0.011" Steel Washer

ting Strain Gauge

Figure 9

A flat thin (0.010 inches thick) disc of brass was used as a diaphragm. A
type 0-19 SR-h strain gauge was mounted on one side of the diaphragm,and
pieces of 0.011 steel with l/h inch diameter holes were used to provide a
backing and clamping device for the diaphragnas well as to provide for a
small instrument volume.

A "dummy" or temperature compensating gauge was mounted on the side
of the clamping nut and the two SRph gauges were connected as two arms of
a bridge circuit. The remaining two arms of the bridge were provided by
the Brush Bridge Amplifier. A deflection of the diaphnmmzresulted in a
change in the resistance of the active gauge and therefore an unbalance in
the bridge. The unbalance was amplified and then recorded on the Brush

Recorder. This electrical system is shown diagrammatically in Figure 10.

29.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ii “11
Active
Brush $rush
Dummy 4 Bridge -
__j Amplifier Recorder
1?
Figure 10

The largest trouble spot in obtaining a satisfactory pressure pick-
up was in the clamping of the edges of the diaphragm. In order that the
pressure pick-up give repeatable data, the diaphngmlmust deflect equal
amounts for equal pressures. This requires that the edges of the diaphragm
be held rigidly at all times, so that true diaphn¥m1action can result from a
pulsating pressure wave. The best clamping technique was that of tighten-
ing down a nut on a suitable fitting soldered on the end of the pressure
transmission tube. Considerable force was required on the wrench in order
to insure satisfactory clamping of the diaphragm edges.

The pressure pick-up was calibrated statically by running up the pressure
in the cylinder to any desired value with the engine flywheel blocked to
keep it from running. The number of lines deflection on the Brush Recorder
was noted for each value of pressure in the cylinder and a calibration
curve prepared.

A sample of the record of pressure variation with piston position

reference as recorded with the Brush apparatus is shown in Figure 11.

Determination of Instantaneous Fluid Velocity
Since the pressure patterns as measured by the technique described
above gave almost no variation in pressure within the cylinder at a given

instant, it was assumed that on the average, the velocity distribution of

 

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Figure 11. Record of Pressure and Piston Position
with Russell Engine Running at 120 RPM. Pressure
Pick-up at Position F. (Position on left,
Pressure on right).

30.

31.

the fluid within the cylinder was linear. That is, the velocity of a
fluid "particle" at the piston was moving with the velocity of the piston
while the velocity of a fluid "particle" at point 1 (Refer to Figure h)
'was zero, and "particles" elsewhere had velocities proportional to the
distance from the end of the cylinder. This technique is evident in the
velocity distribution patterns plotted as part of the data in the Appendix.

A curve of piston position plotted against time is presented in
Figure 12. In the case of the Russell Engine the length of the crank, R,
was 7 inches, and the length of the connecting rod, L, was h2.5 inches.
From simple trigonometry, the distance the piston has moved from bottom
dead center is:

2
R

x = L [1 -'-§ sinzml/2 - R cos fl - (L - R)
L

R
where fl is the crank angle. Note for small'i ratios, this reduces to

R - R cos fl, (simple harmonic motion). The true position curve is plotted
as the solid curve in Figure 12. The curve indicated by the broken line

is the position curve for simple harmonic motion and is shown for compara—
tive purposes. This would occur if a Scotch Yoke mechanism (or effectively

zer0Ǥ ratio) were used on the engine instead of a simple crank.

The velocity curves also show the small deviation of the actual vel-
ocity from the velocity in simple harmonic motion. From a velocity polygon

it is easily seen that piston velocity may be expressed:
R .
'E Sin.¢ cos ¢ d;

V = sin fl - dt
V1 - 0% sin m2

 

 

33.

 

R . . .3 . ,5? .
However, for small-E, as in this case, 1 - (L Sin fl) is very close
to unity, so then

d,
Vfi[sin¢-%sin¢cos @1233

The velocity curve plotted from this equation is the solid line, and

that of simple harmonic motion is the broken curve.

,-

The average value of the sine curve indicating velocity is 5 times

 

n
its maximum value while the average value of the approximate true velocity
curve is 2'02 times the maximum value of the sine curve. The proximity of

n

the curves for pure harmonic motion to the actual curves in Figure 12 as
well as the near equality of the average values indicates that only slight
errors in the calculated irreversibility will result by assuming harmonic
motion.

When the engine is running at 60 revolutions per minute, the average

velocity of the piston is found to be éQ x.§§ = 2.33 feet per second. On

60 12

the assumption of harmonic motion the maximum velocity is therefore 3.66
feet per second. For simplicity of calculations, the velocities pre-
sented in Table I are based on harmonic motion.

TABLE I
PISTON VELOCITY (FEET PER SECOND) AT VARIOUS POSITIONS

 

 

 

Piston Position 60-rpm 120,rpm 180-rpm 2hO_rpm
A 1:3h7 2:69 hzoh 5:39
B 2 :56 5:12 7:69 10 :25
G 3:31 6:62 9:9h 13:25
D 3:625 7;25 10:88 1b:50
5 3:66 7:32 10:98 lb:6h
E 3:625 7:25 10.88 1h:50
F 3 :31 6 :62 9 :91; 13 ".25
G 2:56 5:12 7:69 10:25
H 1.3h7 2.69 h.oh 5.39

 

3’.

PRESENTATION OF DATA.AND CALCULATED RESULTS

In the section of this thesis dealing with the mathematical development,
equation (36) was shown to be valid for the energy degraded in a fluid ele-

ment behind a moving piston.

EidTZ h(T-T)2 .11 dV2
T9 =- (_..) + - S + (- (36)
/° T dx T £0 3 fl dx

2
¥ (53 ,:which describes the energy degraded because of heat exchanged be-
x

2
T - T
tween the fluid elements, may be called the al effect; 2-S—fE——§l , which
T
0

describes the energy degraded because of heat exchanged between the fluid

and the cylinder walls, may be called the a effect; and gh/u(§¥)2, which

2

describes the energy degraded as a result of the viscous drag in the fluid,
may be called the B effect.

By using the techniques described in the section on Apparatus and
Methodology, the variations in pressure, temperature, and velocity were
recorded with the Russell Engine running at 60, 120, 180 and 2h0 revolutions
per minute. These data are presented in graphical form in the Appendix.
For each speed, variations in pressure, temperature, and velocity are shown
for eight piston positions.

Knowing the temperature and temperature gradient of each fluid element
(from data presented in the Appendix) the al effect was easily calculated
for each element. Similarly, knowing the temperature and the velocity

gradient, the B effect was easy to obtain for each element. The determin-

35.
ation of the a2 effect was the most difficult and also the least accurate.
The determination of the a2 effect requires the knowledge of the

magnitude of the film coefficient, h. Researchers in the field of con-
vective heat transfer have developed various empirical formulas for special
cases of heat transfer by convection. None of the cases quite fits the
situation of a fluid expanding inside a metal container. For a fluid

flowing through a horizontal tube McAdams has given the expression

h = 0.023 If. He's Pr'h
D
‘where: k is the thennal conductivity of the fluid

D is the tube diameter
Re is the Reynolds number

Pr is the Prandtl number, gtL
k

Many of the other expressions given for h (for ducts of circular
cross-section) are modifications of McAdams' formula. For example, in
flow through short tubes the flow pattern at the entry to the tube results
in an effective increase in the turbulence and therefore an increase in
the heat transferred. This has led to the use of "shortness factors".
"Shortness factors", always greater than one and usually less than two,
are multiplied by the film coefficient as determined by McAdams' formula
in order to obtain a coefficient for the short tube.

Although it was not developed for the situation of a fluid expanding
in a cylinder, McAdams' formula was used to estimate the film coefficient
for the situation encountered in the Russell Engine. This technique pro-
vides only an estimate but the coefficient determined in this manner

should be of the right order of magnitude. However, any "shortness factor"

36.
or other corrective factor to McAdams' formula will apply equally well
to all engine speeds and the values for film coefficient and therefore
the values for the a2 effect will still have the same relative magnitudes
at the various speeds.

Tables II, III, IV and V present a tabulation of the al, 02 and B
effects for each speed. The total of each effect was obtained by multi-
plying the effect in each elemental volume by that volume; adding these
together and then multiplying by the effective time the piston "was at that
position". The effective "time at a position" was taken as the time
elapsed during the piston travel from a point midway between the position
in question and the last lettered position, to a point midway between the
piston position in question and the next lettered position. For example,
(referring again to Figure h), the "time at position C" was taken to be
the time which elapsed while the piston was moving from point 3 to point b.
Table VI presented in the Appendix gives a summary of times at each
position for each speed.

A complete tabulation of the calculations is also presented in the

Appendix in Tables VII, VIII, IX, and X.

37.

 

 

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OIOH x O0.0 OIOH x OO.H OnOH x H0.0 H O
:OH x O0.0 .OH x H0.0 OH x O0.0 nOH x OH.O nOH x OH.O H x Om.O H>
OH . muoH x O0.0 On muOH x OO.H O wan x mm.O H
OIOH x OOO OnOH x OH.H OuOH x OmO >H
OIOH x OO.O OIOH x OH.H OIOH x Om.O HHH
OIOH x HH.O m1OH x OH.H OuOH x mm.O HH
OIOH x OH.H O.OH x H0.0 O1OH x H0.0 H O
OHO 0mm HH\OHO OHO com OHHOHO OHO ommmpH\OHO OOOMMHO
m Hmpoa pcoemmm Hog a we Hmpoe names 0 Hog me _He fiance pcoEmHo Hog He COHMHmom

 

 

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OHuOH x HO.H HIOH x OO.H m:OH x HH.H OIOH x mHO.H OuOH x OO.: OnOH x OH.O H
HuOH x OO.H OIOH x mm.m OnOH x mH.O >H
HuOH x mOO.H OsOH x HO.H OuOH x mH.O HHH
H.OH x OO.H :aOH x H0.0 OIOH x OH.O HH
H:OH x mOO.H muOH x OO.m OuOH x H0.0 H O
mica x MAO.H NIOH K AH.M mIOH x HNJ.H NICH x NJ.H mIOH N 40.4 mIOH N 0:.H >H
HIOH x O0.0 m1OH x OO.H muOH x OO.H HHH
HaOH x H0.0 O:OH x mm.m muOH x HO.H HH
HuOH x OO.O OIOH x Om.O muOH x HO.H H O
OnOH x HOO.H HuOH x Om.m OuOH x OO.H OIOH x O0.0 OIOH x O0.0 m.OH x OO.H HHH
HuOH x HO.O O:OH x OOH.H muOH x mOO.H HH
HnOH x Om.m O:OH x OH.H m..OH x HO.H H o
O..OH x OHO.H OncH x OO.H OIOH x OO.H OaOH x H0.0 O:OH x O0.0 mnOH x OOO.H HH
OIOH x HO.H OaOH x OO.H muOH x OOO.H H O
OaOH x O0.0 OIOH x O0.0 HnOH x H0.0 OIOH x H0.0 OuOH x OH.H OIOH x OOH.O H O
OuOH x OO.O muOH x OOO.H OIOH x OO.O O.OH x H0.0 O:OH x Om.O mnOH x OO.H a <
OHO OOO OO\OHO OHO 0mm OH\OHO OHO com HHxOHO HOOEOHO
0 H969 £850 0 Hon a NO 13.09 pcosmmm Hog me He H.309 #:memcmm .Hmm Ho QOHMMMOOH

 

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OaOH x OO.H msOH x OO.H HnOH x HO.O OOOHOO HOO HOHOH
OH-OH x HHm.H O:OH x mmm.H muOH x OO.H OIOH x OO.O O.OH x H0.0 OuOH a Om.m HHH
OuOH x OO.H m..OH x OOH.H OIOH x mm.m H>
OuOH x HHm.H OnOH x O0.0 OuOH x H0.0 >
O.OH x OO.H O.OH x H0.0 OIOH a Om.m OH
OuOH x «OO.H OrOH x H0.0 OuOH x Om.m HHH
OuOH x HH:.H OIOH x HO.O O:OH x Om.m HH
OaOH x mmO.H OnOH x OH.H OIOH x Om.m H O
OHIOH x O.m OnOH x HO.H muOH x OH.H N.OH x OOH.H O1OH x Om.m OuOH x O0.0 HH
O.OH x O0.0 O.OH x OH.O OIOH x N0.0 >
OuOH x O0.0 m..OH x OO.H OIOH x O0.0 >H
OIOH x H0.0 O:OH x Om.O OuOH x OO.O HHH
OuOH x HH.O OIOH x OO.H OuOH x H0.0 HH
OuOH x O0.0 muOH x O0.0 OuOH x O0.0 H O
OHO com OHxOHO OHO OOOOHH\OHO OHO OOOOHH\OHOH HOOMMHO
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hS.

SUMOXRY AND CONCLUSIONS

From Tables II, III, IV, and V of the previous section, the results
may be best summarized in graphical form, as in Figures 13, lb, and 15.

Figure 13 shows the variation in a1 effect with engine speed. The
point for 120 revolutions per minute is definitely high, but when it is
kept in mind that the al effect depends on the square of the temperature
gradient this might be attributed to experimental error. The al effect
need not increase with engine speed since a speed may be reached where the
duration time decreases faster than the square of the temperature gradient
increases. Even at the comparatively low speeds possible with the Russell
Engine the flattening of the curve of “1 effect is apparent.

Figure lb shows the variation in 02 effect with engine speed. In this
case the point at 180 revolutions per'minute appears to be in error. In-
vestigation showed, however, that the temperature level of the air in the
cylinder during the 180 revolutions per minute run was a few degrees lower
than during the other runs. Since the 02 effect is that due to transfer
of heat from the fluid to the cylinder walls it is evident that a lowering
of fluid temperature will affect the magnitude of 02. Thus, in an ideal
turbine 02 becomes zero. Here as in the case of al the magnitude of the
effect need not continue to increase with speed, since a2 describes heat
flow between.fluid and cylinder walls, and higher speed means shorter time
.for the heat to flow.

The p effect is shown in Figure 15. This effect should continue to

inicrease at still higher speeds since the velocity gradient squared in—

 

 

h9.
creases faster than the duration time decreases. The fact that the E
effect resulted in a straight line, serves as a check for the calculations
since this is consistent with the mathematics.

From the summary of the results as presented in these graphs, it is
evident that in a reciprocating engine, unless care is taken to keep the
cylinder walls and fluid at nearly the same temperature, the 02 effect is
much larger than the other effects and dominates the irreversibility
picture.

As fluids of greater thermal conductivity or of greater viscosity are
used, the irreversibility will be increased.

Finally, it must be emphasized that the values presented here are
relative and approximate, and only valid for this particular engine. For
higher speeds the temperature measurement becomes more difficult, and as
seen at the higher engine speeds vsed in this work (while still compara—
tively low speeds), response lag in the thermocouple contributed to the
apparent temperature gradients. For engines with a smaller L/R ratio, the
assumption of simple harmonic motion may introduce an appreciable error.
Thus, techniques used will necessarily be refined as the theory is applied
to other engines, but they gave sufficiently accurate results for this

general study of the effects which cause degradation of energy in an

engine cylinder.

APPENDIX

50.

 

 

 

 

TABLE VI

"TIMES AT EACH POSITION"

(TIME IN SECONDS)

 

Ii

J
J

 

 

Speed RPM
Position

60 120 180 2h0
A .085 .0u25 .0283 .0212
B .070 .0350 .0233 .0175
c .050 .0250 .0167 .0125
D .0h5 .0225 .0150 .0112
E .0h5 .0225 .0150 .0112
F .050 .0250 .0167 .0125
G .070 .0350 .0233 .0175
H .085 .0h25 .0283 .0212

83.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-'\
SABLE 711 8h'
TABULaTIOEJ 3F CALCULATIONS FOR 60 RPM (cyLljJDEh 3315977371; nggpggtgypgug = 58°F)
Positior P Tl V I 1" it’ll/‘82 TV— 7dr\°l‘ r I 1 " j ‘3’ Volume Time at Total 0 Total (1 To+al B
L . \ v70: 5 weft/x.“ M ‘- ‘ " “ ’ ' l ”
p51. 0 fl / c 1 FTP” 3 n . 3 3 POSition , , , 2 .
Elfiiint abs. 0? ft/sec BTU/ftheC (ft/5'39 fwfi (°F/ft)2 ft BTU/ftBSec H ”it see BTU/ft sec ft sec BTU BTU BTU
1 a 19.0 62 0.670 8.28 x 10‘5 1005 100 0.01875 1.008 x 10‘51H35 x lo"4 6.71 x 10‘7 0.0180 0.085 1.60 x 10“9 2.11 x 10"7 1.05 x 10‘9
-1' _
B 1 23.3 71 1.28 1.59 x 10"}4 236 - 100 0.060 1.001 x 10-6 8.05 x 10 4 1.153x10 7 0.0737 0.070 5.37 x 10'9 0.35 x 10‘6 5.96 x 10‘lo
C 1 31.6 77 0.8 1.1 x 10-9 98.5 100 ‘0.0880 1.035 x 10‘66-SH X 10"h'6-02 X 10‘: 0-0737 1
11 33.5 75 2.5 3.73 x 10-4 8.5 100 0.088 1.00 x 10752.28 X 10-3 6.02.x 10‘ 0.0737 0.050 7.03 x 10‘9 1.08 x 10-5 0.73 x 10‘ O
D I 30.5 80 0.6 1.09 x 10-34 52 .6 36 0.0880 2.6 x 1041.11, x 1073 3.51 x 10'? 0.0737 1
11 30.7 79 1.8 2.785 x 10"4 52.6 36 0.1875 2.6 x 10-71.215 x 10:3 3.5 x 10‘ 0.0737 9 Si
111 36.2 78 3.0 0.00 x 10"ll 52.6 36 0.0880 2 6 x 10-73.67 x 10 3.5 x 10‘ 0.0737 0.005 2.58 x 10' 1.982 x 10‘ 3.08 x 10‘lo
E 1 32.0 69 0.5 9.6 x 10‘5 29.5 27.6 0.0880 2.0 x 10-12.50 x 10‘fi 1.92 x 10*; 0.0737
11 32.0 70 1.0 2.22 x 10“? 29.5 27.6 0.1875 2.0 x 10—7p.21 x 10—0 1.92 x 10‘ 0.0737
111 32.7 71 2.3 3.29 x 10-4 29.5 27.6 0.1875 2.0 x 10-35.59 x 10- 1.93 x 10" 0.0737 6 1
1 33,2 72 3.2 (0.25 x 10~h 29.5 27.6 0.0880 2.0 x 10-71.775 x 10‘3 1.93 x 10‘ 0.0737 0.005 2.65 x 10'9 9.65 x 10‘ 2.55 x 03‘ 3
F 1 21.0 62 0.33 70.88 x 10-5 15.8 100 0.0880 1.005 x 10:21.69 x 10-5 1.01 x 10-8‘0.0737 ,
11 20.7 60 1.00 1.26 x 10-h 15.8 100 0.1875 1.005 x 10 0.62 x 10-§ 1.02 x 10-8 0.0737 1
111 21.3 66 1.65 1.79 x 10-h 15.8 100 0.1875 1.005 x 10‘ 1.16 x 10‘fi 1.02 x 10‘3 0.0737
IV 21.7 68 2.32 2.53 x 10.0 15.8 100 0.1875 1.003 x 10-62.55 x 10“7 1.02 x 10- 0.0737 ‘ 8 _6
7 22,3 70 2.98 3,06 x 10-9 15.8 100 0.0880 1.001 x 10'69.0 x 10"4 1.03 x 10‘ 0.0737 0.050 1.925 x 10‘ 5.06 x 10 1.88 x 10‘lo
G 1 13.0 51 0.210 2.05 x 10-5 6.55 320 0.0880 2.36 x 10-5266 x 10:2 0.12 x 10"9 0.0737
I 12.0 50 0.60 5.39 x 10‘5 6.55 320 0.1875 2.36 x 10759.0 x 10 0.10 x 10‘; 0.0737
111 11.1 57 1.065 7.69 x 10'5 6.55 320 ‘ 0.1875 2.36 x 10” 7-93 X 10‘7 0.16 X 10’ 0.0737
:7 10.8 60 1.09 9.9 x 10-5 6.55 320 0.1875 2.35 x law—50.06 x 10 5 0.18 x 10-9 0.0737
V 11.2 63 1.92 1.26 x 10-5 6.55 320 0.1875 2.35 x 10-53.2l x lO-h 0.21 x 10‘9 0.0737 8 6 10
VI 11.9 66 2.35 1.525 x 10-'4 6.55 320 0.0880 2.35 x 10 2.1 x 10‘ 0.25 x 10‘9 0.0737 0.070 7.29 x 10' 1.058 x 10‘ 1.295 x 10‘
H 1 10,5 52 0,1 1.01 x 10-5 1.00 196 0.0880 1.03 x 1031.123 x 10"5 9.05 x 10110 0-0737
:1 10.5 55 0.3 3.02 x 10-5 1.00 196 0.1875 1.03 x 10H63'19 x 10 9.11 x:url‘ 0.0737
111 10.5 57 0.5 5.10 x 10-5 1.00 196 0.1875 1.025 x 10_65.26 X 10‘; 9.15 xlfl'lg 0.0737
IV 10.5 59 0.7 6.73 x 10—5 1.00 196 0.1875 1.025 x 10 66.92 x 10‘ 9.18 x:rrlo 0.0737
V 11,5 62 0.9 8.38 x 173-5 1,170 196 0.1875 1.3123 x 10:51.37 X 10"; 9.22 x 10:10 0.0737
VI 10.0 60 1.1 9.66 x 10-5 1.00 196 0.1875 1.02 x 10 6 Sb x 10‘h 9.27 x10_100.0737 8 _6 '11
711 10.3 66 1.27 1.075 x 10—0 1.00 196 0.075 1.02 x 10‘ .70 x 10‘ 9.32 2:03 0.0552 0.085 6.01 x 10* 1.22 x 10 3.88 x 10-
1.719 x 10-7 5.26 x 10-5 3.08 x 10'9

 

 

 

 

   

‘1‘
\‘

1
TABLE VIII ? 85.

TABULATIDN OF CALCULATIDIQS FOR 120 RPM I.'(.CYLINDE1{ SURFACE TE.‘»‘1'PERATU1E = 57°F)

 

    
 

 

 

 

 

 

 

 

   

Position T V h (dV/dx)2 (uiT/dx)? {'C 0.1 .7 02 [3 Volume 31:11:66 T Total (1.1 Total 0.2 | Total {3
and . . \ ‘ - _ 051' 10.0 :
Element Egg: °F ft/sec BTU/£82566 (ft/sec ft) (°F/1t)2. ft BTU/ft3sec BTU/ftBSeC BTU/ft3sec ft3 sec BTU BTU 1 BTJ
. j ‘ I ;
A a 20.5 07 . 1.35 1.52 x 10-8 0180 i 100 10.01875 1.051 x 10:j-l.6 x 10‘3 2.61 x 10—6 0.0181 0.0125 8.25 x 10‘lo 1.25 x 10"6 2.00 x 10-9
l .
B 1 25.0 62 2.56 2.93 x 10~h1 911 g 110 10.060 1.015 x 10 9 2.33 x 10'8 6.00 x 10-7 0.0737 0.035 2.70 x 10-9 6.00 x 10-7 1.5533(10-9
I. . i f —)
0 1 36.6 66 1.65 32.77 x 10% 390 1 576 10.0880 0.17 x 10:;é 0.85 x 1031 2.55 x 10-; 0.0737 8 6 10
11 37.7 70 0.96 '6.87 x 10“4= 390 7 576 70.0880 0.16 x 10 ~ 2.08 x 10‘ 2.56 x 10‘ 0.0737 0.025 1.533 210- 5.05 x 10* 9.0 x 10‘
. 7 _, _
D 1 36.5 68 1.21 2.16 x 10*? 210 5 000 {0.0880 2.90 x 10-2 5.61 x 10‘; 1.365 x 10 3 0.0737
11 36.7 71 3.63 5.3 x 10‘4 210 000 10.1875 2.90 x 10‘. }.90 x 10_3 1.37 x 10‘7 0.0737 _8 _5 10
111 38.5 75 6.05 8.11 x 10‘h; 210 000 .0.0881 2.89 x 10—9 9.26 x 10 1.38 x 11' 10.0737 0.0225 1.005 x 10 1.19 x 10 6.81 x 10'
l _)
E 1 28.9 61 0.91 1.05 x 10-0 118 729 0.0880 5 29 x 10-6 1.5353:10 Z 7.60 x 10:8 0.0737
11 28.7 69 2.72 3.38 x 10-4 118 729 0.1875 5 28 x 10"6 0.91 x 10‘ 7.68 x 10 8 0.0737
111 28.9 73 0.53 5.11 x 10-h 118 729 0.1875 5 26 x 10-6 1.31 x 10-3 7.75 x 10* 0.0737 8 5 10
1V 29,5 78 6.30 6.96 X 10-0 118 729 0.0381 5 25 X 10.6 6.07 x 10—3 7.80 x 10‘ 0.0737 0.0225 3.50 x 10' 1.398 x 10“ 5.11 x 10‘
F 1 20.9 58 0.66 8.55 x 10-5 63.2 900 0.0880 6.55 x 10-6 1.87 x 10'6 1.03 x 10‘2 0.0737
11 20.9 63 1.98 2.10 x 10-h 63.2 900 0.1875 6.50 x 10-5 7.70 x 10-§ 0.07 x 10— 0.0737
111 21.2 68 3.31 3.16 x 10‘11 63.2 900 0.1875 6.52 x 10-é 3.87 x 10*; 0.11 x 10-8 0.0737 _
IV 21.8 73 0.63 0.20 x 10‘? 63.2 900 0.1875 6.50 x 10-0 1 3773210- 0.15 x 10‘ 0.0737 _8 _5 10 ‘
V 22.9 78 5.97 5.35 x 10“4 63.2 900 0.0880 5,19 x 10—6 0 97 x 10-3 0.18 x 10‘ 0.0737 0.025 6.00 x 10 1.198 x 10 3.77 x 10'
0 I 10.7 55 0.03 1.66 x 10‘5 26.2 801 0.0881 6.10 x 10-5:0.I# x 10:2 1.660 x 10‘8 0.0737
11 10.1 60 1.28 1.065 x 10‘h 26.2 801 0.1875 6.12 x 10"6 9.85 x 10 , 1.677 x 10-8 0.0737
111 10.3 65 2.10 1.63 x 10'8 26.2 801 0.1875 6.11 x 10 1.06 x 10‘: 1.695 x 10' 0.0737
IV 10.8 70 2.99 2.17 x 10-11 26.2 801 10.1875 6.08 x 10 3.69 x 10—0 1.705 x 10‘8 0.0737
V 15.7 75 3.85 2.77 x 10’“ 26.2 811 ‘0.1875 6.07 x 10‘ 8.95 x 10‘ 1.723 x 10‘8 0.0737 8 5 1
VI 17.3 80 0.70 3.60 x 10"11 26.2 801 0.0880 6.06 x 10* 3.99 x 1073 1.70 x 10~ 0.0737 0.035 9.01 x 10‘ 1.385 x 10‘ 2.63 x 10‘ O
H 1 10.9 09 0.20 2.52 .x 10"5 5.71 676 0.0880 0.95 x 10‘ 3 59 x 10‘5 3 59 x 10‘9 0.0737 '
11 10.1 53 0.60 5.79 x 10"5 5.71 676 0.1875 0.93 x 10' 9.65 x 10 3.61 x 10*9 0.0737
111 13.5 58 1.00 8.32 x 10"5 5.71 676 0.1875 0.92 x 13 8 57 x 1017 3.65 x 10‘9 0.0737
1v 12.9 62 1.39 1.005 x lo‘h 5.71 676 0.1875 0.91 x 10‘ 2.67 x 10‘5 3.67 x 10‘9 0.0737
7 12.5 67 1.79 1.265 x 104L 5.71 676 0.1875 0.90 x 10- 1 28 x 10-9 3.71 x 10-9 0.0737
71 12.0 71 2.19 1.075 x 10.1 5.71 676 0.1875 0.89 x 1 2.90 x 10-9 3.70 x 10‘9 0.0737 11
~711 12.7 75 2.50 1.71 x 10"11 5.71 676 0.075 0.88 x 1 1.3823(10‘3 3.76 x 10-9 0.0552 0.0025 1.038 x 10‘7 0.78 x 10'6 7.75 x 10‘
3.26 x 10-7 6.38 x 10-5 6.10 x 10-9

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

TABLE IX

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11881111011 OF 0111301111018 FOR 180 RPM(CYLII‘DE1+. 581711013 1811181011881: = 50°F)
. . , 2 ‘1"- t
130::Zlon pgi .T V h 2 \dV/dx) JMT/dfld fc al 02 B 1Volume 1310:3313?“ Total <11 Total 02 Total B
Element abs: 7 ft/sec BTU/ft sec (ft/sec ft) (°F/ft) ft BTU/ft3sec BTU/183666 810/113sec 113 sec BTU BTU BTU
A a 25.0 00 2.0 2.00 x 10—h 9000 576 0.01875 0.22 x 10-6 2.58 x 10*3 5.85 x 10‘6 0.0180 0-0283 2.20 X 10‘9 1-387 X 10'6 3'03 X 10_9
B I 37.5 50 3.80 5.60 x 10‘LL 2125 001 0.060 3.22 x 10-5‘ 0 1.3058~10"6 0~0737 0-0233 5-52 X 10"? O 2'31 X 10‘?
C I 39 .8 56 2.5 b.15 X lO—L! 888. 1296 0.08811 915 X 10‘6 3.15th 10": 5.6lL X 10"}: 0.0737 8 __7 .0
II 39.8 62 7.5 1.005 x 10-3 888 1296 0.0880 9.11 x 10"6 1.395x:10‘/ 5.70 x 10" 0.0737 0.0167 2.32 x 10' 2.16 x 10 1.3953(10 .
0 1 35.2 57 1.8 3.01 x 104* 075 780 0.0881 5.7 x 10—6 5913 X 10': 3-03 X 10“; 09737
:1 31.0 62 5.15 6.80 x 10-0 075 780 0.1875 5.69 x 115 Mm X 10‘. 3-05 X 10‘ 0.0737 8 6 -9
111 30.5 67 9.1 1.055 x 10—3 075 780 0.0880 5.68 x 10-5‘3.83 x 10" 3.08 x 10-7 0.0737 0.0150 1.8853110“ 0.79 x 10* 1.013x 10 /
E I 23.3 57 1.36 1.71 x 10-h 266 900 0.0880 6.55 x 10‘6 3~§7 X 10'E 1'69 X 10‘; 0'0737
11 23.0 62 0.08 3.98 x 10-8 266 900 0.1875 6.50 x 10-6 2‘26 X 10'. l~7l X 10‘7 0'0737
111 23.3 67 6.81 6.07 x 10—1 266 900 0.1875 6.53 x 10‘6 1-09 X 10" 1-73 X 10‘ 03737 8 -10
17 20.0 72 9.53 8.25 x 10‘5 266 900 0.0880 6.52 x 10 5.67 x 10‘3 1.70 x 10‘7 0.0737 0.0150 2.89 x 10‘ 7.73 x 10"6 7.60 x 10
F 1 18.5 50 0.99 1.055 x 10—0 102 830 0.0880 6-05 X ”'61 O c 8'99 X 10% OM37
11 18.5 59 2.98 2.60 x 10-L 112 830 0.1875 6.00 x 10:3 6-67 x 10:3 9.0 x 10‘8 0-0737
111 18.5 61 0.97 3.88 x 101 102 830 0.1875 6.03 x 10 61 3-95 X 10 '2 9-15 X 10: mm
17 19.0 69 6.96 5.29 x 10-4 102 830 0.1875 6.02 x 10‘. 1.20 X 10‘3 9-25 x 10 8 0-0737 8 ‘ —6 10
V 20.5 70 8.95 6.82 x 1075 112 830 0.0881 6.01 x 10-5 5.77 x 10' 9.32 x 10‘ 0.0737 0.0167 3.71 x 10‘ 9.15 x 10 5-63 X 10'
0 1 11.5 52 0.60 5.1 x 10"5 59.2 900 0.0880 6.57 x 10'5 h-Sl x 10‘: 3.73 x 10‘8 0.0737
11 10.7 57 1.92 1.205 x 10-h 59.2 900 0.1875 6.55 x 10"6 1-15 X 10"h 3.77 x 10‘8 0.0737
111 10.7 62 3.20 1.78 x 10-h 59.2 900 0.1875 6.50 x 10:? 1~16 X 10:1 3'80 x 10-8 0'0737
17 11.1 67 1.18 2.03 x 1041 59 .2 900 0.1875 6.53 x 10 6: [“16 X 10 6 3-8“ X 10—8 03737
7 11.9 72 5.77 3.10 x 10-' 59 .2 900 0.1875 6.52 x 10‘ ‘ 1-02 X 10:3 3-88 X 10‘. 0'0737 8 —6 -10
VI 12.7 77 7.05 3.70 x 10-h 59.2 900 0.0880 6.50 x 10-5‘ 0.18 X 10 3.91 X 10' 0.0737 0~0233 5-78 X 10‘ 9'88 X 10 3'9h x 10
. _l1’
H 1 12.2 08 0.30 2.95 x 10-5 12.9 900 0.0880 6.57 x 10"6 1d88 X 10 5 8'08 X 10—9 0-0737
11 10.9 53 0.90 6.61 x 10'5 12.9 900 0.1875 6.56 x 10:2 6'88 X 10‘; 8°15 X 10_9 0'0737
111 10.6 58 1.50 9.68 x 10-5 12.9 900 0.1875 6.55 x 10.6, 1'59 X 10'h 8-2h X 10'9 0-0737
17 10.7 63 2.09 1.31 x 10‘ 12.9 900 0.1875 6.53 x 10 1-08 X 10-u 8-31 X 10"Z 0-0737
V 10.8 68 2.69 1.58 X lO—h 12.9 900 0.1875 6.51 x 10‘6 3.1h X lO—h 8.h0 X 10-6 0.0737
71 11.2 73 3.29 1.91 x 1041 12 .9 900 0.1875 6.50 x 10‘6 6-90 X 10‘ 8118 X 10" 0-0737 8 6 -10
711 11.8 77 3.82 2.25 x 10-0 12.9 900 0.075 6.09 x 10‘6; 2'96 X 10‘3 8.52 x 10‘9 0.0552 0.0283 8.91 x 10- 7.03 .x 10‘ 1.17 X 10
2.72 x 10-7 0.01 x 10-5 9.60 x 10—9

 

 

 

 

 

IL... I.

i

TAFLE X. 87.

 

 

(CYLINDER SUiFACE TEIVPEIQLTURE = 52 °F)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.5011T100 3: 011051171313 301 200 851
Position‘ P T V (”V d‘)2 (dT/dx)21' f ‘ a Time at
and psi h 2 Q / 1 2 2 c 7 “l . 2 B VOlume Position Total 01 Total 82 Total 8
Element abs: °F ft/sec BTU/ft sec (ft/sec ft) (°F/ft) ft ‘ BTU/ft’sec BTU/ft3sec BTU/ft3sec ft3 sec BTU BTU BTU
‘ i - ’ ”J *7 “)4 .-
0 a 05.5 50 2.7 0.90 x 104L 5 16700 2300 0.01875§1.68 x 10" 2.01 X 10 1.008 x 10 5 0.0180 0.0212 6.58 x 10‘9 8.05 x 10‘8 1.08 x 10—9
1 i _.
B I 06.0 55 5.1 8.29 x 10—h i 3780 2920 0.060 $2.125:(10‘5 2~6l X 10 L 2.39 x 10‘6 0.0737 0.0175 2.70 x 10‘8 3.37 x 10‘7 3.08 x 10‘9
G I 00.0 57 3.32 5.19 x 10-9 7 1585 1765 0.0880 f=1.2833«.:10"57§-’3Ll X 10:% 1.01 x 10:: 0.0737
11 00.0 60 9.95 1.07 x 10‘3 f 1585 1765 0.0880 £1.2803110- 7 ~33 X 10 1.02 x 10 0.0737 0.0125 2.36 x 10‘8 3.31 X 10—6 1.873x;10‘9
- _ 1 i ‘1.16 x 10-b 5 31 10—7
D I 32.1 56 2 02 3.37 x 10 h 1 801 1935 0.0880 11.01 x 10-5 _ . x 0.0737
11 31.7 60 7.26 8.08 x 10-8 ‘ 801 1935 10.1875 1.0053(10-5 é-lBS X lO_2 5.02 x 10'7 0.0737
111 32.0 71 12.1 1.08 x 10‘3 801 1935 .0.0881 1.00 x 10-5 -30 X 10 5 5.50 x 10-7 0.0737 0.0112 3.08 x 10-8 7,92 X 10—6 1.3013:10-9
E 1 25.2 55 7 1.81 2.28 x 10-h 070 2020 30.0880 1.07 x 10‘: g-50 X 10:E 3.00 X 10‘7 0.0737
11 21.6 62 7 5.00 5.23 x lO‘h 070 2020 50.1875 1.07 x 10‘ -35 X 10_3 3.00 x 10- 0.0737
111 20.0 70 ’ 9.06 7.83 x lO‘g 070 2020 20.1875 1.06 x 10‘5 2-56 X 10_2 3.08 x 10-7 0.0737
17 21.5 78 :12,7 1,03 X 10-2 171 2020 :0.0880 1.16 x 10-5 1-97 X 10 3-10 X 10“7 0.0737 0.0112 0.81 x 10-8 1.1713(10—5 1 013,.10-9
F 1 19.5 56 1.32 1.05 x lo-h 253 1210 0.0880 8.81 x 10': 5-39 X 10:5 1.605 x 10'7 0.0737
11 19.0 62 3.98 3.30 x 10-h 253 1210 0.1875 8.79 x 10:6 3- l X 10 3 1.62, x 10—730.0737
111 19.0 68 6.63 5.06 x 10- 253 1210 0.1875 8.75 x 10 6 1-31 X 10:3 1.605 x 10"7 0.0737
17 20.0 70 9.28 6.93 x 10-0 253 1210 0.1875 8.75 x 10‘, 3~35 X lO_2 1.66 x 10'750.0737
7 20.8 80 11.91 8.60 x 1041 253 1210 0.0880 8.72 x 10—0 1-015 X 10 1.68 x 10-a 0.0737 0.0125 1.01 X 10-8 1.77 x lo.5 7 57 X 10.10
0 1 10.0 57 0.85 27.78 x 10—5 105 960 0.0880 6.98 x 10:210.25 X loji 6.68 x 10"8 0.0737 '
11 11.0 62 2.56 1.88 x 10‘ 105 960 0.1875 6.97 x 10 6 6.92 x 10—0 6.70 x 10‘950,0737
111 11.6 67 0.27 2.85 x 10—9 105 960 0.1875 6.96 x 10:611.go x 10_3 6.81 x 10-§10.0737
IV 15.8 72 5.98 3.99 x 10‘11 105 960 0.1875 6.95 x 10_6; - 0 X 10_3 6.88 x 10'510.0737
V 16.0 77 7.69 5.03 x 10-11 105 960 0.1875 6.92 x 10 673-13 X 10_2 6.90 x 10' '0.0737
71 16.5 82 9.00 5.92 x lo‘h 105 960 0.0880 6.90 x 10- 71-135 X 10 7.01 x 10- l0.0737 0.0175 5.38 X 10-3 2 19 x 10-5 5 3 X 10‘10
1 ' '
H 1 13.2 55 0.00 .3.90 x 10-5 23 729 0.0880 5.30 x 10*? g.78 X 10:2 1.055 x 10‘270.0737
11 12.0 60 1.20 ,9.17 x 10'5 23 729 0.1875 5.30 x 10-_-6 .02 x 10-0 1.071 x 10‘v 0.0737
111 12.0 60 2.00 1.3652c10-h 23 729 0.1875 5.29 x 10 6 $.01 X 10_ 1.082 x 10‘8 0.0737
IV 12.7 69 2.80 1.9o5ac10‘h 23 729 0.1875 5.28 x 10:6 ~57 X 10-0 1.50 x 10-8 0,0737
7 12.9 73 3.60 2.10 x 10-11 7 23 729 0.1875 5.27 x 10 6 5.755 x 10__3 1.511 x 10'? 0.0737
71 12.9 78 0.00 2.68 x 10‘“ 23 729 0.1875 5.25 x 10:6 6' g X lO_3 1.52 x 10‘ 0.0737
711 13.1 82 5.10 3.12 x lo-h 23 729 0.075 5.20 x 10 -9 X 10 1.535 x 10-8 0.0552 0.0212 5.57 x 10"8 1 36 x 10—5 1 5771(10-10
. ‘.___—__J2.91 x 10‘7 7.96 x 10‘5 1.28 x 10‘8

 

 

 

     

 

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BIBLIOGRAPHY

Brown, Aubrey I. and Marco, Salvatore M. Introduction to Heat
Transfer, 2d ed. MbGraw-Hill Book Company, New York, N .Y., “1951

 

Brown, Charles L; Basic Thermodynamics. McGraw—Hill Book Company,
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deGroot, S. R. Thermodynamics of Irreversible Processes. Inter-
science Publishers Inc.,N Neerork, N .Y., 1951

 

Denbigh, K. G. The Thermodynamics 2£ the Steady State. methuen's
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Eckart, Carl. The Thermodynamics of Irreversible Processes.
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Iberall,'Arthur S. Attenuation of Oscillatory Pressures in Instrument
Lines. Research Paper RP 2115, Journal of Research of the National
Bureau of Standards Vol. 05, July 1950, p. 85 -108

 

Joos, Georg. Theoretical Physics. 2d ed. Hafner Publishing Company,
New York, N:Y.,41950

 

Keenan; Joseph H. Thermodynamics. John Wiley and Sons, New York,
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Leaf, Boris. Phenomenological Theory of Transport Processes in Fluids.

Physical RevieW, Vol. 70, 1906, p. 708

marks, Lionel 5. Mechanical Engineers' Handbook, 0th ed. MoGraw-
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MbAdams, William H. Heat Transmission, 2d ed. McGrawn-Hill Book
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Porter, A. W. Thermodynamics. Methuen's monography on Physical
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Streeter, Victor L. Fluid Dynamics. McGraw-Hill Book Company, New
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