Note on the importance of imperfect-gas effects and variation of heat capacities on the isentropic flow of gases

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Title:
Note on the importance of imperfect-gas effects and variation of heat capacities on the isentropic flow of gases
Series Title:
NACA RM
Physical Description:
21 p. : ill. ; 28 cm.
Language:
English
Creator:
Donaldson, Coleman duP
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Isentropic expansion   ( lcsh )
Gases, Real   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Abstract: The errors involved in using the perfect-gas law pv=RT and the assumption of constant heat capacities are evaluated. The basic equations of gas flows taking into account these phenomena separately and at the same time are presented.
Bibliography:
Includes bibliographic references (p. 14).
Statement of Responsibility:
by Coleman duP. Donaldson.
General Note:
"Report date December 10, 1948."

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University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003810749
oclc - 135175507
System ID:
AA00009246:00001


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NACA RM No. LJal4

NATIONAL ADVISORY COMIK'EE FOR AERONAUTICS


RESEARCH MEMORAWNUM


NOTE ON THE IMPORTANCE OF IMPERFECT-GAS EFFECTS AND
VARIATION OF HEAT CAPACITIES ON THE
ISENTROPIC FLOW OF GASES
By Coleman duP. Donaldson


SUMMARY


The errors involved in using the perfect-gas law pv = RT and the
assumption of constant heat capacities are evaluated. The basic equations
of gas flows taking into account these phenomena separately and at the
same time are presented.


INTRODUCTION


The conventional method of obtaining high Mach numbers for
aerodynamic tests is to accelerate the air by means of a pressure
difference so that the random kinetic energy of the molecules of air at
rest is converted into kinetic energy in the test section. For very
high Mach numbers this may occasion high stagnation temperatures and
pressures which introduce effects due to the vibrational heat capacity
and molecular forces and size such that the perfect-gas law pv = RT
and the assumption of constant heat capacities may be no longer
sufficiently accurate to evaluate gas flows.

It is the purpose of this paper to present formulas which are
suitable for handling such problems and to point out the magnitude of
the errors that may be involved in using the perfect-gas law and the
assumption of constant heat capacities.

Tsien (reference 1) has published a theoretical discussion of
this problem in which certain approximations were introduced in order
to obtain solutions that were in a very neat form when the imperfect-
gas effects were moderate. A comparison of Tsien's results with
this work is presented to show the magnitude of these approximations.

In England Goldstein has previously investigated this problem
at moderate temperatures and pressures in order to prove the small
magnitude of imperfect-gas and vibrational-heat-capacity effects in
most supersonic wind tunnels. This report indicates, in general, the
range in which these effects are small but does not present formulas for
handling problems in gas dynamics when these effects are large.






NACA RM No. IBJ14


The present paper is arranged in the following three parts:
temperature effects on perfect-gas flows due to variation of heat
capacities; imperfect-gas effects cn gases without variation of
heat capacities; and gas flows in which both effects are present.
This is done since the formulas in the first part may prove use-
ful to those dealing with the flow of hot exhaust gases and since
it may bring out more clearly the differences between the two effects.


SYMBOLS


a


b


c

Cp

Cv

E ..

M

p
P

R

T

v

V

7


P


Subscripts

o


term in Van der Waals' equation correcting for the
effect of molecular forces

term in Van der Waals' equation correcting for the
effect of molecular size

speed of souni, feet per second

heat capacity at constant pressure

heat capacity at constant volume

energy, foot-pounds

Mach number (v/c)

pressure, pounds per square foot

gas constant

absolute temperature, degrees Fahrenheit

specific volume, cubic feet.per slug

velocity, feet per second

ratio of heat capacities (Cp/CV)

density, slugs per cubic foot

characteristic temperature of molecular vibration



stagnation conditions

critical conditions






NACA PI No.. L814


Errors. Involved in Assuming Constant Specific Heats in the

Presence of High Temperatures in a Perfect Gas

For a perfect gas with constant heat capacities the equation for
.conservation of energy of a steady isentropic process may be written as



CpT + j2 = CpTo
P 2 p 0


If this equation is combined with the equation for the isentropic speed
of sound c2 = 7RT the resulting equation is


=2 2 T 1i) (1)


If the expansion is Isentropic, the pressure and density ratios
corresponding to the Mhch number are





and (2)


0 (T /R


However, if the temperature of the gas is high enough the heat
capacities may not be assumed constant because the vibrational degrees
of freedom of polyatomic molecules are excited. The variation of the
equilibrium value of the heat capacity at constant volume of a perfect
diatomic gas is found from quantum mechanical considerations to be of
the form


C v 5 ) 2 e/T
E 2 T (e/T- 1)2


where e is a constant depending on the gas. The formula may be used
for the mixture air if the value of 0 is placed equal to 5526 when
absolute temperature is measured in degrees Fahrenheit. (See the






4 NACA RM No. I&T14


appendix.) The value of the heat capacity at constant pressure far a
perfect gas is then


p = 7 9 2
R 2 T2


e /T
(e/T 1)2


Figure 1 is a plot of equation (4) and shows that
may not be considered constant above 6000 F absolute.


the heat capacity


When the heat capacity at constant pressure varies according to
equation (4) the energy equation must be written


J/To 2
To


Substituting equation (4) into equation (5) and integrating yields


7RT + 2R w2 = 7RT +
06/T 1


The Mach number is obtained from equation (6)
by 7RT = c2, which gives


-1)


29 1
/1To -- 1


2RO
ee o 1


by dividing through


1 i
- --
e9/T -- 1


Cp/ 7 (ee/T
CV/P 5 (e/T


- 2 + 2) 2e/T


- 12 +


'Tle pressure ratio corresponding to this Mach number is obtained from
tile luentropic equation


where


M2 7 I f
M (T


S2 (e/T






NACA RM No. L8J14


log- = R T
POii


by substituting equation (4) into equation (9) and integrating to give


p 7/2 1 e/To
Po :o 1 -1 ee/T


eO/T
e0 /T-1


e ee/To
To ee/TO_


(10)


Similarly, the density ratio is found to be


P /T 5/2
P-- o


1 e/To
1 -- e/T


ee/T
e0/T-1


e
To


e0/To
e /To_1/


(11)


The differences involved in the use of equations (1) and (2) to predict
the temperature, density, and pressure ratios corresponding to a given
Mach number are given in figures 2(a) and 2(b), n terms of the percentage
differences from the value given by equations (7), (10), and (1) for
stagnation temperatures of 10000 and 20000 F absolute..

It is seen that the assumption of constant heat capacity leads to
appreciable differences in applying the .isentropic law for a perfect gas
if stagnation temperatures above 10000 F absolute are involved.


Errors Involved in the Adsumption of the Perfect-Gas

Law pv = RT for a Gas with Constant Heat Capacities

In order to.evaluate flows in which imperfect-gas effects are present,
an equation of state that takes into account these effects must be chosen.
For the purposes of this paper an equation which takes into account the
effects of molecular forces and size should be sufficient.

A suitable equation is that of Van der Waals



(p -(v-- b = RT (12)
\ v2i






NACA RM No. IBJ14


where b is a term correcting for the volume occupied by the molecules
and a is a term correcting for the effect of molecular forces.

Figure 3 is a graph of Van der Waals' equation in which the quanti-
ties p, v, and T have been made nondimenslonal by dividing by the values
of these quantities at the critical point pc, cs, and Tc, thus making
the graph suitable for any gas. (See reference 2.) The graph may be used
for air if an empirical critical point (pe = 37.2 atm, Tc = 238.50 F abs.,
vc = 0.6438 slugs/ft3) is assigned to that mixture of oxygen and nitrogen.
To give this critical point the values of a, b, and R for air when the
pressure is measured in pounds per square foot, the specific volume in
cubic feet per slug and the absolute temperature in degrees Fahrenheit
are a = 8.78 x 105, b = 0.654, and R = 1716.

The proper equation for an isentropic expansion of a real gas is
(see reference 3)


dE = Cv dT + T- c dv = 0 (13)
[ TTv=Conetant]

which for Van der Waals' equation becomes



dE = Cv dT + p dv + -1 dv = 0 (14)
v2


Equation (14) may be written as


dE = Cv dT + d(pv) v dp + dv = 0


and since -v dp = w dw


dE = Cv dT + d(pv) + -L dv + w dw = 0 (15)
v2


Assuming constant heat capacity at constant volume and integrating
equation (15) gives


E = CT + vp + Constant = Eo (16)
v2-L +






NACA RM No. L8J14


This is then the energy equation for a Van der Waals gas. Dividing
through by the Isentropic speed of sound


c2 = d + T RT
dp CV (v b)2 v


and since


p -. = RT 2a
Sv2 v b v2


then


2 RTo
2CvTo + 2vo v- b
(vo b


(1+


- 2a -2 2CvT 2v
vo2/


( PT _a
v b v2)


S\) v2RT -g
(v -b)2 v


The value of v for an isentropic expansion to be placed in
equation (18) can be formed from equation (14) as follows:


Cv dT + p dv + dv = CT dT+ -- dv=
-2 v -b


then


C dT -
T v-b


and if C, is constant


Tv = (v/ -R
v = (vo b)(0r)


Sb


(20)


From equations (18) and (20), mkowing the stagnation conditions for
an expansion from To to T, the Mach number may be calculated. The
pressure ratio is then found to be


(17)


(18)


(19)






8 NACA RM No. IBJ11


RT a
P = v- v (21)
Po 0o _a
vo b Vo2


and substituting the value of v from equation (20) we obtain


RT a

(Vo b) vo- b)(2 + b
Po RTo a
^-b
Vo b Vo2


Figure 4 shows the conventional pressure ratio and area
ratio pw/(pw)F. l plotted against Mach number for air starting from
stagnation conditions of 5200 F absolute and various pressures compared
with the value obtained using constant ratio of heat capacities and the
perfect-gas law.

Also shown in figure 4 are the values of pw/(pw)?W1 computed by
Tslen's method. It is seen that as the imperfect-gas effects become
large it is no longer possible to simplify the analysis by neglecting
terms containing the squares of and -i although Tsien's results
v pv2'
pV2
are in good agreement at 50 atmospheres when the Van der Waals effect
is moderate.

It is interesting to note that the speed of sound in a Van der Waals
gas



c2 = l + 2- v2PT (17)
dp C.1 (v b)2 v


is not equal to 7PT. The expression for the ratio of specific heats in
a Van der Waals gas is

7 pv2 a (23)
-v pv2 a + ab







NACA RM No. L8J14 9





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NACA RM No. IBJ14




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NACA RM No. I&14


The value of the ratio of heat capacities 7 in this case is


(er2
T})


7 =


ee/T
(e e/- 1)2


5 \2
-+ (-+


pv2 + a
p2 -a b
v


ee/
(e 1) 2


but the speed of sound is found from equation (17) by substituting the
value of Cv/R from equation (3) to be


c2= 1


5 fr
- + '--
2 T)


ee/T
(e/T -)


vT 2a
(v b)2 v


Figure 6 shows the conventional pressure ratio and area ratio plotted
against Mach number for air starting from stagnation conditions of
20000 F absolute and various pressures compared with the value obtained
using constant ratio of heat capacities and the perfect-gas law.


DISCUSSION


The foregoing analyses show that the effects of variation of heat
capacities with temperature do not became important in isentropic expansions
of air until stagnation temperatures of the order of 10000 F absolute are
encountered. Above 10000 F absolute, however, for accurate analysis this
variation must be taken into account. In general, it may be stated that
/0Y 9/T
for diatomic gases these effects are important when ()2 ee /
T (e/T 1)2
becomes appreciable compared to the number 2.5.

The effects of Van der Waals' forces become important when either
the temperature is extremely low for near atmospheric pressures or the
pressure very high for moderate temperatures. These forces must be
taken into account when the value of a/v2 becomes appreciable compared
to the pressure p, or b becomes appreciable compared to v. For air
these effects are unimportant until stagnation pressures of the order
of 50 atmospheres at stagnation temperature of 5200 F absolute are
encountered.


(29)






NACA RM No. 18j14


Talen's method agrees well with the results of this investigation
up to 50 atmospheres in this case, but it appears that it is not
possible to neglect the squared terms of h and -a- when the effects
v v2p
of Van der Waals' forces become appreciable.


CONCLUSIONS


In many cases found in very high Mach number wind tunnels and in
flows of high stagnation temperature or pressure, imperfect-gas effects
and the effects of variation of heat capacities may be present.

For diatomic gases the effect of variation of heat capacities becomes

important when V2 ee-- 2 becomes appreciable compared to 5/2.
vT (es -l
(T 'e 1)
For air these effects become appreciable when stagnation conditions of
10000 F absolute or larger are encountered.

Imperfect-gas effects become important in gas dynamics when a/v2
becomes appreciable compared to the pressure p or b becomes appreciable
compared to v. When air is expanded from a stagnation temperature
of 5200 F absolute these effects become important if the stagnation pressures
are of the order of 50 atmospheres or greater.

Formulas are presented for handling isentropic expansions taking into
account these phenomena both separately and at the same time. Tsien's
method is found to be applicable for small departures from a perfect gas
but is not accurate when the effects of Van der Waals' forces become
appreciable.


Langley Aeronautical Laboratory
National Advisory Committee for Aeronautics
Langley Field, Va.






NACA RM No. IJ14 13


APPENDTI

DERIVATION OF TBE VIBRATIONAL HEAT CAPACITY OF A DIATOMIC GAS


To arrive at the vibrational heat capacity of a diatomic gas, the
individual molecules are treated as linear harmonic oscillators of a
fundamental frequency and Shrbdinger's equation is solved for the
allowable energy states of such an oscillator. These allowable states
are then substituted into the equation for the canonical energy distribu-
tion and the average energy per particle as a function of the absolute
temperature is found. This may be differentiated to obtain the
contribution of the vibrational degrees of freedom of the molecule to
the heat capacity of the gas at any temperature.


The average vibrational energy
(see references 4 and 5)


per particle found in this way is


hV+ heV
E =2 ---V +
2 ehvAkT 1


where


Planck's constant

characteristic frequency of molecular vibration

absolute temperature


Differentiating to obtain the contribution to the heat capacity of this
energy yields


Cvib 1 LE = hY 2
R k 3T \kT/


( hv/kT

(ehv/kT 2


hv
For a particular gas = e is a constant and may be determined from
spectroscopic data. The heat capacity at constant pressure is then


R 2


R 2
R 2


(T2.
\T/


ee/T

(e /T- )2






NACA RM No. I &14


The value of 0 for oxygen is 4010.4 and for nitrogen is 6044.4 far
absolute temperatures measured in degrees Fahrenheit. The value 5526
may be used for air.


REFERENCES


1. Tsien, Hsue-Shen: One-Dimensional Flows of a Gas Characterized by
van der Waals' Equation of State. Jour. Math. and Phys.,
vol. XXV, no. I, Jan. 1947, pp. 301-324.

2. Jeans, James: An Introduction to the Kinetic Theory of Cases.
Cambridge Univ. Press, 1946, pp. 96-98.

3. Epstein, Paul S.: Textbook of Thermodynamics. John Wiley & Sons, Inc.,
1937, pp. 64-65.

4. Frenkel, J.: Wave Mechanics. Elementary Theory. Second ed., Oxford
Univ. Press, 1936, pp. 77-80.

5. Lindsay, Robert Bruce: Introduction to Physical Statistics.
John Wiley & Sons, Inc., 1941, pp. 53-59.







NACA RM No. L8Jl4 15








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NACA RM No. L8J14


P/Pa
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02 1 6 8 0

Mach number, M

(a) To = 10000 F abs.

Figure 2.- Percent error involved in the use of constant-heat-capacity
formulas to obtain T/T p/po, and p/po for air.






NACA RM No. L8Jl&


2 h 6 8 10


Mach number, M

(b) To = 2000 F abs.

Figure 2.- Concluded.


-16





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-8





S0





NACA RM No. L8J14


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Specific volume ratio, L
Figure 3.- Van dev aals' equation in nondenional form.
Figure 3.- Van def Waals' equation in nondinensiona.l form.


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NACA RM No. LJ&T1


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IACA RM No. L&SJI


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Figure 5.- Variation of the ratio of specific heats 7
Waals gas. T = 5200 F abs.


for a Van der


2.2


50 100 150 200






NACA PM No. L8J14 21

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