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By Robert C. Johnson
NAT ONAL AERONAUTICS AND SPACE ADM NISTRATION "
NASA SP- 3046
REAL-GAS EFFECTS IN CRITICAL FLO'_
PROPERTIES OF NITROGEN AND HELIU/_
NEWTONS PER S()UARE METEI
(APPROX. 300 ATM)
By Robert C. Johnson
Lewis Research Center
Cleveland, Ohio
e_ A
Scientific and Technical Information Division
OFFICE OF TECHNOLOGY UTILIZATION 1968
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Washington, D.C.
SUMMARY
A critical-flow factor for gaseous nitrogenand helium flowing through critical-flow
nozzles has been calculated. This factor provides a convenient means for determining
the mass-flow rate of these gases through critical-flow nozzles. In addition, the results
include the nozzle throat velocity, the compressibility factor, the entropy, the enthalpy,
the specific heat, the specific-heat ratio, the speed of sound, and the critical pressure,
density, and temperature ratios. These results are tabulated as functions of the plenum
pressure and temperature. The pressure range is to 300x105 newtons per square meter
(approx. 300 atm). For nitrogen, the temperature range is from 100 to 400 K. For he-
lium, the range is from 15 to 400 K.
The FORTRAN IV subroutines used to calculate these results are also included.
These routines permit three different sets of independent variables. In addition to the
plenum pressure and temperature, the other independent variable is either the nozzle
exit pressure, the nozzle exit Mach number, or the nozzle exit temperature.
INTRODUCTION
In recent years, a number of methods have been developed for calculating the mass-
flow rate of nonideal gases through nozzles. These methods assume the flow to be one-
dimensional and isentropic. In reference 1, the authors present tables of the compres-
sible flow functions for the one-dimensional flow of air. Reference 2, using the state
equations of reference 3, presents tables for air, nitrogen, oxygen, normal hydrogen,
parahydrogen, and steam that permit the calculation of the isentropic mass-flow rate of
these gases through critical-flow nozzles. A critical-flow nozzle is one that operates
with a throat Mach number of 1. Reference 2 refers to a number of other reports that
describe other methodsfor making this type of calculation.Recently there has beena needto calculate the mass-flow rate of helium andnitro-
gen at pressures greater than havebeenpreviously reported. This needarises from theuse of these as driver gasesfor the propellant feed in vernier-control space thrusters.These gasesare commonly stored at pressures of the order of 250×105newtonspersquare meter. Nitrogen is one of the gasestreated in reference 2, but the pressurerange is only to 100xl05 newtonsper square meter. Helium is one of the gasestreatedin reference 4, but again the pressure range is only to 100×105newtonsper square meter.
This report presents a critical-flow factor that permits the isentropic mass-flowrate of nitrogen andhelium through critical-flow nozzles to be calculated from plenumconditions. For the caseof nitrogen, the temperature range is from condensationto400 K. For helium, the temperature range is 15to 400 K. For both gases, the pressurerange is from 0 to 300×105newtonsper square meter.
As a result of these calculations, two groups of quantities are tabulated. The firstgroup consists of quantities that dependonboth the plenum conditions andthe conditionsin the nozzlethroat, where the Mach number is 1. Thesequantities are
(1) Critical flow factor(2) Nozzle throat velocity(3) Ratio of throat to plenumpressure(4) Ratio of throat to plenum density(5) Ratio of throat to plenumtemperatureThe secondgroup consists of thermodynamic point functions that dependonly on
plenum conditions. Theseare(1) Compressibility factor(2) Enthalpy(3) Entropy(4) Specific heatat constantpressure(5) Specific-heat ratio(6) Speedof soundIn addition to these tabulations, a description of the calculation procedure is given
in appendixB. Appendix C presents a description of the FORTRANIV subroutines thatwere usedto make these calculations. The listing of these subroutines is presented inappendixD. All symbols are defined in appendixA.
ANALYSlS
Basic Equations
The calculations in this report make use of three basic relations. The first relation
describes the pressure-temperature-density behavior ef the gas. This is represented by
a compressibility factor Z which is a function of density and temperature and is defined
as
Z = Z(p, T) - p (1)pRT
The second relation describes the specific heat of the gas at vanishing density where the
compressibility factor equals 1. This specific heat is a function of temperature and is
represented by
= ~ TCv Cv (T) = Cp( ) - R (2)
(In this report, an ideal gas is defined as one whose compressibility factor is 1 and
whose specific heat is constant. ) The third relation describes the saturated vapor pres-
sure as a function of temperature. This is represented by
Psat = Psat (T) T _ T c (3)
For temperatures greater than critical, the fluid is always a gas for the pressures in-
volved. (For T _ Tc, the saturation pressure can be considered to be infinite. ) Since
this report is only concerned with the gaseous phase, equation (3) is merely used to con-
firm that the fluid is a gas. This condition is represented by
P < Psat (4)
Working Equations
Nitrogen. - The equations that follow were developed by the National Bureau of
Standards (NBS) cryogenic laboratories in Boulder, Colorado and are given in refer-
ence 5. The equation for the compressibility factor is equation (5) in reference 5 modi-
fied by dividing both sides of the equation by pRT. The equation is further modified by
changing the density units from gram-moles per liter to kilograms per cubic meter.
The following equation results from these modifications:
Z(P,T) = I + i+_+_+_+ O+ + 02
T T 2 T 3 . T5 / 6+ B8p3 +
T
where
+ p2(B..._12 B13+ ----._+
\T 3 T 4
p5
A = -7.135xi0-6
B 1 = 1.2034917×10-3
B 2 = -2.5107891xi0-I
B 3 = -4.9681584×i01
B 4 = 3.7073373xi02
B 5 = 1.496473xi06
B 6 = 2.1027719xi0-6
B 7 = -2.4516046×10-4
B 6 = 2.3102822x10-9
B 9 = 4.9666462
B10 = 1.6771266×103
BII = -1.656225×I05
BI2 = -6.5374809×10-5
B13 = 2.4209106×10-2
BI4 = -I.126389
B15 = 1.1829604X10-12
The equation for specific heat at vanishing density Cv is equation (4) in reference 5
modified by dividing by R and subtracting 1 from the result. The modified equation isas follows:
where T is inKand
V
_ =0_i+R °12T + (_3 T2 + °t4T3 + _5 T4
(6)
(5)
_1 = 2.501146
_2 = -9"720581xi0 -5
_3 = -1"036056×10 -6
_4 = "4"437256×10 -9
_5 = 6"825596×10 "12
4
The relation between the saturated vapor pressure and temperature is equation (1) in
reference 5 modified by changing the units of pressure from atmospheres to newtons per
square meter. The following equation results:
J2
l°gl0Psat = Jl + _ + J3T + J4T2 + J5 T3 + J6T4 + J7 T5
where
Jl = 5.5335216
J2 = -3"0507339xi02
J3 = 1"6441101xi0-I
J4 = -3"1389205xi0-3
J5 = 2"9857103xi0-5
J6 = -1"4238458xi0-7
J7 = 2"7375282xi0-I0
(7)
Helium. - The equation for the compressibility factor is that developed by the NBS
cryogenic laboratories in Boulder, Colorado and is equation (1) in reference 6. This
equation is modified by dividing both sides by pRT. The equation is further modified by
changing the density units from gram-moles per liter to kilograms per cubic meter.
The equation that results from these modifications is as follows:
B2B3B4Z(p,T)= I+ 1 +- +- +- + P + +__ p2 B8p3 B15p4 B16p5
T T2 T3 T5 / 6 + --T + --T + --T
where
p2 BF(] BI0 BII_+p2(BI2+BI3 BI4._IeAp2/T
+ L\T3 + T'_-+ T5/ \-_ T-"_"+ --_-/j
A = -4.057xi0 -4
B 1 = 4.0665013xi0 -3
B 2 = -I.1267764xi0 -I
B 3 = 2.3039266xi0 -2
B 4 = -5.7468818xi0 -2
(8)
5
B 5 = 1.3691368×10 -1
B 6 = 9.7390626xi0 -6
B 7 = 7.0543876×10 -4
B 8 = -5.3854984×10 -6
B 9 = -3.8053762xi0 -3
BI0 = 2.625179xi0 -2
Bll = 7.6742661x10 -2
Bl2 = -8.7904911xi0 -7
Bl3 = 1.9960611xi0 -6
B14 = -8.1300167xi0 -6
B15 = 3.6743583x10 -8
B16 = -3.4049435xi0 -II
Since helium is monatomic, the specific heat at vanishing density is given by
Cv_ 1.5
R(9)
And since the temperatures involved for these calculations are always greater than criti-
cal, the helium is always in the gaseous phase for the pressures involved.
Calculations of Plenum Thermodynamic Functions
These functions are the compressibility factor, enthalpy, entropy, specific heat at
constant pressure, specific-heat ratio, and speed of sound. The basic equations that are
used to evaluate these functions follow. The working equations are listed in appendix B.
Compressibility factor. - This function is evaluated by use of equations (5) or (8).
Enthalpy and entropy. - These equations, which are derived from equations (6) and
(7) in reference 5, are
H dT + - (I0)
S dT in p - - 1 + T(-_--_Z + K S (11)
R W aW/ J
The temperature integrals in equations (10) and (11) are indefinite integrals whose con-
stants of integration are included in K H and KS, respectively. The values of KH and
K S depend on the choice of the fluid reference state; K H and K S are chosen such that
when the fluid is at this reference state the values of enthalpy and entropy are zero. The
reference state for nitrogen is the triple point (i. e., T = 63. 156 K, P = 0. 1253x105
N/m2), which is that used in reference 5. The reference state for helium is the liquid at
a temperature of 0 K and a pressure of 0 newtons per square meter. This reference
state is the same as that used in reference 6.
Specific heat at constant pressure. - This function is given by
(12)
Specific-heat ratio. - This function is given by
CpV- -I+
C v
(13)
Speed of sound. - This function is given by
(14)
where k is the isentropic exponent and is defined by
k--P- op _-t + ap aTP S p p T
(15)
Calculation of Nozzle Throat Thermodynamic Functions
These quantities are the critical-flow factor, nozzle throat velocity, ratio of throat
to plenum pressures, ratio of throat to plenum densities, and ratio of throat to plenum
temperatures.
Critical-flow factor.
tion:
- The critical-flow factor C* is defined by the following equa-
c*-Gt R_° (16)
PO
The mass-flow rate per unit area Gt is determined by the methods of reference 2. The
assumptions involved in this calculation are
(1) The flow from the plenum to the nozzle throat is isentropic.
(2) The flow is one-dimensional.
(3) The Mach number in the nozzle throat is 1 (i. e., the nozzle is choked).
If, in addition to these assumptions, the specific heat of the gas is constant and the
compressibility factor of the gas is 1 (i. e., the gas is ideal), the ideal-gas, critical-flow
* is constant for a given gas and is represented byfactor Ci
* [7 _. 2 ._(7i+1)/(_i-1)11/2 (17)
* has a value of 0.6848 for nitrogen where 7i = 1.4 and 0. 7262 for heliumwhere C i
where T i = 5/3.Nozzle throat velocity. - The nozzle throat velocity is also equal to the speed of
sound in the nozzle throat since the Mach number is 1. This is evaluated by use of
equation (14).
Throat to plenum pressure, density, and temperature ratios. - These ratios are
directly calculated through knowledge of the pressure-density-temperature state of the
gas at both the plenum and the nozzle throat locations.
Although these functions are calculated by the methods of reference 2, the iteration
procedures for calculating the plenum density, the nozzle throat density, and the nozzle
throat temperature are different. A description of these procedures is given in appen-
dix B. The procedures in this report permit calculation at pressures close to those that
cause condensation. This calculation could not be done by the methods of reference 2.
RESULTS AND DISCUSSION
Calculations were performed for nitrogen and helium.
tions yielded
For both gases, the calcula-
8
(6)
(7)
(8)
(9)
(lO)
(11)
(1) Critical-flow factor, C* - Gt R_f_°
Po
(2) Nozzle throat velocity, v t in m/sec
(3) Critical pressure ratio, pt/Po
(4) Critical density ratio, pt/Po
(5) Critical temperature ratio, Tt/T o
Compressibility factor, Z o = po/PoRTo
Enthalpy, Ho/R in K
Entropy, So/R
Specific heat, Cp, o/R
Specific-heat ratio, ro = Cp, o/Cv, o
Speed of sound, a o in m/sec
The values of the specific gas constant R is 296. 774 square meters per square sec-
ond per K for nitrogen and 2077.15 square meters per square second per K for helium.
The tables for the critical-flow f actor permit the calculation of the isentropic mass-
flow rates per unit area by means of equation (16). This factor is plotted in figure 1 for
1.1
1.0
.g
"y
.7.__%- .....
"- Ideal-gas
f
_r
value
Temperature,K
175
2OO
250
3OO
40O
5O i00 150 200Pressure, N/m2
Figure 1. - Critical-flow factor for nitrogen.
250 _OxlO.s
9
the case of nitrogen to indicate the extent of deviation of this factor from the value it
would have if nitrogen were an ideal gas with a specific-heat ratio of 1.4. The actual
mass-flow rate _n of a gas through a critical-flow nozzle of geometric throat area A t
is given by
l:n = CDAtG t (18)
The discharge coefficient C D mainly represents the effects of the nonisentropic and
non-one-dimensional flow in the boundary layer of the nozzle. The discharge coefficient
is generally determined by a nozzle calibration and is usually plotted as a function of
Reynolds number. Typical values of CD are between 0.96 and 1. The results for the
nozzle used in reference 7 indicate that C D is independent of Mach number in the range
from 0.2 to 1. Since this indicates that compressibility effects on the discharge coeffi-
cient are negligible fer flows up to critical, the real-gas effects on the discharge coeffi-
cient should also be negligible for flows up to critical.
The maximum nozzle throat pressure to maintain critical flow in the nozzle can be
easily determined by multiplying the plenum pressure by the tabulated values of the
critical pressure ratio. The pressure-temperature-density state of the gas at the noz-
zle throat can be determined from the plenum conditions by use of the tables of the criti-
cal density and temperature ratios. The speed of sound in the nozzle throat is the same
as the tabulated value of the nozzle throat velocity.
All the tables in this report contain at least one nonsignificant figure, which aids in
tabular interpolation.
The results for the two gases are discussed in the following paragraphs.
Nitrogen
The results for nitrogen are presented in tables I(a) to (k) for pressures up to
300x105 newtons per square meter and temperatures that range from 100 to 400 K. The
state equation (eq. (5)) used in this report was developed in reference 5 from pressure-
volume-temperature data representing pressures up to 300x105 newtons per square
meter and temperatures up to 350 K. The root-mean-square error in calculating the
compressibility factor is quoted as 0.2 percent in reference 5. This error would, of
course, be larger if only regions near saturation were considered. The errors in de-
rived functions such as specific heat are estimated to be under 5 percent.
Since the state equation (ref. 5) used for the calculations in this report does not rep-
resent data above 350 K, the functions calculated from this equation in the region from
350 to 400 K represent an extrapolation. A comparison of this data with that tabulated
10
in reference 3 indicates that this extrapolation is goodto within 0.3 percent in both thecompressibility factor and the specific heat at pressures up to 100xl05 newtonspersquare meter. Becauseof the form of the state equationandthe fact that the extrapola-tion is toward higher temperatures, the extrapolation shouldbe valid to 300x105newtonsper square meter.
Helium
The results for helium are given in tables II(a) to (k) for pressures up to 300x105
newtons per square meter and temperatures that range from 15 to 400 K. The state
equation in this report was developed in reference 6 from data representing pressures
up to 100xl05 newtons per square meter and temperatures that ranged from 2.5 to 570 K.
In the temperature range from 10 to 300 K, reference 6 estimates that the calculated
values of enthalpy and entropy are accurate to within 3 percent, and the specific heat is
accurate to within 5 percent.
Since the state equation (ref. 6) used for the calculations in this report represents
data whose pressures extend only to 100xl05 newtons per square meter, the functions
calculated from this equation at pressures greater than 100x105 newtons per square
meter represent an extrapolation. For temperatures greater than 20 K, reference 8
states that this extrapolation is good to 1300x105 newtons per square meter.
A comparison of the compressibility factor calculated from the experimental data
in reference 8 with the compressibility factor calculated from equation (8) indicates that
this extrapolation is valid for temperatures greater than 15 K and pressures up to
300x105 newtons per square meter.
Subroutines
In addition to the tables presented in this report, the FORTRAN IV subroutines used
to compute these tables are described in appendix C and are presented in appendix D.
While the tables in this report are for critical flow, the subroutines used to compute
these tables are more versatile. In fact, they can be used to compute either subsonic
or supersonic isentropic flow functions for three different sets of independent variables.
All three sets of these variables include the plenum pressure and the plenum tempera-
ture. The third independent variable is one of the following:
(1) Nozzle exit Mach number
(2) Nozzle exit pressure
(3) Nozzle exit temperature
11
It shouldbenotedthat for critical flow, the downstreamnozzle reference station is al-ways the nozzlethroat. For supersonic flow this would not be true. Thus, in the de-scriptions of the calculation procedures in appendixesB and C, the downstreamnozzlereference station is referred to as the nozzle exit.
CONCLUDING REMARKS
The tables of the critical-flow factor in this report provide a means for calculating
the one-dimensional isentropic mass-flow rate of nitrogen and helium through critical-
flow nozzles. For nitrogen, this critical-flow factor ranges from its ideal-gas value of
0.648 to a value of 1.196 at 155 K and 140x105 newtons per square meter. For helium
the range is from 0. 483 at 15 K and 300×105 newtons per square meter to a value of
0. 814 at 15 K and 30x105 newtons per square meter. The ideal-gas, critical-flow factor
for helium is 0.726. This indicates that significant errors would occur if the ideal-gas
values of the critical-flow factor were used in mass-flow calculations.
The .subroutines in this report have been used to reduce calibration data for criticai-
flow nozzles. The design of these subroutines permits easy modification for other gases.
Lewis Research Center,
National Aeronautics and Space Administration,
Cleveland, Ohio, August 22, 1968,
128-31-06-77-22.
12
APPENDIXA
a
C*
C D
Cp
Cp
C v
Cv
G
KS
k
M
P
SYMBOLS
nozzle throat area, m 2
speed of sound, m/sec
Psat
critical-flow factor
discharge coefficient
specific heat at constant
pressure, J/(kg)(K)
specific heat at constant
pressure for a gas at
vanishing density,
J/(kg)(K)
specific heat at constant
volume, J/(kg)(K)
specific heat at constant
volume for a gas at
vanishing density,
J/(kg)(K)
mass-flow rate per unit
area, kg/(m2)(sec)
enthalpy, J/kg
constant in enthalpy
equation, K
constant in entropy
equation
isentropic exponent
Mach number
R
S
T
T c
Tsat
V
Z
Z I, • . . , ZVI
Y
P
Subscripts:
e
i
O
t
mass-flow rate, kg/sec
pressure, N/m 2I,...,n-l,n
minimum pressure at
which condensation oc-
curs for given temper-
ature, N/m 2
gas constant, m2/(sec 2) (K)
entropy, J/(kg)(K)
temperature, K
critical temperature, K
maximum temperature at
which condensation oc-
curs for fixed pres-
sure, K
velocity, m/sec
compressibility factor
functions of compress-
ibility factor as defined
in appendix B
specific-heat ratio
density, kg/m 3
nozzle exit conditions
ideal gas
plenum conditions
nozzle throat conditions
when Mach number is 1
estimate in an iteration
process
13
APPENDIX B
CALCULATIONS
The following functions of the compressibility factor are used in the calculations:
ZI = Z(p,T) - ppRT
(B1)
ZII=Z+ TfaZ_l\a._.T]p-_ a(__)pm2).
ZIII= Z +p a(._p) _ 1 a(._p)T RT W
(B3)
(B4)
Z V f0 __(_ ®m
kaT]p p
(B5)
C v - Cv
R(B6)
In terms of these functions, equations (10) to (13) and (15) become
H=/Cv dT+ T(Z IRZ v) + K H (B7)
where K H is 508.31 K for nitrogen and 6. 98973 K for helium.
f_s.. dT ZIv+KsR JR T
(B8)
14
I
where K S is 0. 77124 for nitrogen and 4. 75063 for helium.
¢p Cv z 2_ IIR R ZVI +
ZIII
(z7 - - - III+ '
E z,. Zv,/
(B9)
(B10)
(Bll)
The calculation of the isentropic flow functions from the plenum to the nozzle exit
involves the following equations:
PO =
PO
ZI(P o, To)RT o
(B12)
2
ae = PeR k(Pe' Te)
Pe
(B13)
2v e
R
So- Se f T° Cv dT
=0=j% -_¥-
PO
In -- - Ziv(Po, To) + ZIV(Pe, Te)Pe
(B14)
If T° Cv=2
R
_'T edT+ TO [ZI(Po, TO)- Zv(Po, TO)_- Te[Zi(Pe , Te)- Zv(Pe , Te)]t
(B15)
15
The independentvariables for the plenum conditions are the pressure and tempera-
ture. The calculation of the plenum thermodynamic functions involves density. It is,
therefore, necessary to solve equation (]312) for density. Since this equation involves
density implicitly, an iterative procedure is necessary for solution. A description of
this procedure follows:
First estimate of plenum density:
Succeeding estimates:
Po, n
Po (B16)Po, 1 -
RT o
= Po, n-1 + O(_P)T (Po - Po, n-1 ) (B17)
where
T RToZIH(Po, n-I' To)
When the last two density estimates agree to within one part per million, the compu-
tation is considered complete. For certain cases of helium at high pressures and low
temperatures, this procedure failed to converge. For these cases, the iteration proce-
dure is restarted with the following initial estimate:
_ Po (B19)Po, 1
3RT o
This restart permitted convergence for all tabulated cases.
If the nozzle exit independent variable is the temperature, the nozzle exit density
can be determined by solving equation (B 14) for Pe" Since Pe is involved implicitly in
equation (B14), an iterative procedure for solution is used. This procedure is as follows:
First estimate of nozzle exit density:
16
lnPe, 1 = InPo-/_e T° Cv dTR T
T
(B20)
Succeedingestimates:
r S(in Pe ) ]
In Pe, n = In Pe, n-1 + _(So _ Se)jT (So - Se) (B21)
where
_. a(ln pe ) I = 1 (B22)
__(S--o --SeiJT Zii(Pe, n_l, Te)
and
A(So_ Se )=__Te T°CvR dTT+ ZIV(Po' To) - ZIV(Pe, n-1' Te) (B23)
When the last two density estimates agree to within one part per million, the computation
is considered complete.
For physically valid solutions for either Po or Pe' ZI' ZII' and ZII I must be
positive. That is, the density has to be positive, the pressure has to increase with tem-
perature at constant density, and the pressure has to increase with density at constant
temperature. These conditions were verified for all tabulated cases.
Once the thermodynamic state of the gas is known at both the plenum and the nozzle
exit, isentropic flow quantities can be calculated. For example, the nozzle exit Mach
number can be calculated from equations (B13) and (B15), and the nozzle exit mass-flow
rate per unit area from the nozzle exit density and equation (B15).
If the nozzle exit Independent variable is either pressure or Mach number rather
than temperature, a nozzle exit temperature has to be estimated such that the calculated
pressure or Mach number agrees with the prescribed pressure or Mach number. This
temperature estimate is then used in equations (B20) to (B23) to calculate the nozzle exit
density. (This procedure always assures that the nozzle exit entropy is equal to the
plenum entropy. ) The procedures for these two cases are discussed separately.
If the nozzle exit pressure is the independent variable, the first estimate of the noz-
zle exit temperature is less than the plenum temperature and either greater than the
saturation temperature for the case of nitrogen or greater than the critical temperature
for the case of helium. These conditions take precedence over the following equation
for the first temperature estimate. This estimate represents the nozzle exit tempera-
ture that would exist if the gas were ideal.
17
';i-=T "e
Te, 1 o0
(B24)
The second temperature estimate is given by
Te, 2 (Pe - Pe, 1) (B25)
where
\ ?'i ]\Pe, l]
(B26)
The other estimates are given by
(Te, n-1 - Te, n-2_ (pe _
Te'n= Te'n-l+ \Pe, n-1 Pe, n-2/Pe, n- 1) (B27)
For all estimates, a check is made to determine that the temperature is either
greater than saturation for the case of nitrogen or greater than critical for the case of
helium. When the calculated nozzle exit pressure Pe, n agrees with the prescribed noz-
zle exit pressure Pe to within one part per million, the nozzle exit temperature is con-sidered to be known.
If the nozzle exit Mach number is the independent variable, the first estimate of the
nozzle exit temperature is made on the basis of the gas being ideal. This estimate is
W o= (B28)
Te, 1 Ti - 1 21+_
2 Me
The second estimate is given by
(M e - Me, 1) (B29)
18
where
taT : (_i- 1)Me
i To /
(B30)
The succeeding estimates are given by
fTe, n_ - Te, n_2_
Te, n = Te, n_ I + \Me, n-I Me, n-2/ (M e - Me, n_ 1) (B31)
For all estimates, a check is made to determine that the temperature is either above
saturation for the case of nitrogen or above critical for the case of helium. When the
calculated nozzle exit Mach number agrees with the prescribed nozzle exit Mach number
to within one part in ten thousand, the nozzle exit temperature is considered to be known.
19
APPENDIX C
DESCRIPTION OF FORTRAN IV SUBROUTINES
The subroutine used to calculate the thermodynamic properties of nitrogen is refer-
enced by the following statement:
CALL CNIT (KKK, PA, TA, AM, PB, TB, FLOW, KODE)
For a valid computation, the following conditions must be satisfied:
55 K<T<501K (c1)
P < Psat (c2)
p < 351×105 N/m 2 (C3)
The subroutine used to calculate the thermodynamic properties of helium is refer-
enced by the following statement:
CALL CHEL (KKK, PA, TA, AM, PB, TB, FLOW, KODE)
For a valid computation, the following conditions have to be satisfied:
5.4 K< T < 501K
p < 305x105 N/m 2
(C4)
(C5)
For both subroutines, certain variables are returned through labeled common.
These are referenced by the following statement:
COMMON/OUTPUT/OUX(15), Z (6, 2), KODI(5)
The following symbol definitions apply to both subroutines:
KKK Controls entry to and exit from the subroutine. If KKK=0, just the plenum proper-
ties are calculated. If KKK=2, both the plenum and the nozzle exit properties are
calculated. If KKK=l, just the nozzle exit properties are calculated. For a given
set of plenum conditions, at least one reference has to be made for KKK=0 or 2
before a reference can be made for KKK=I.
20
PA
TA
AM
PB
FLOW
KODE
OUX(1)
oux(2)
oux(3)
oux(4)
oux(5)
oux(6)
oux(7)
Plenum pressure, Po' N/m2
Plenum temperature, To, K
Nozzle exit Mach number, M e
Nozzle exit pressure, Pe' N/m2
Nozzle exit mass-flow rate per unit area, Ge, kg/(m2)(sec)
Indicates the independent variables to the subroutine. If KODE=I, the inde-
pendent variables are PA, TA, and PB. If KODE=2, the independent varia-
bles are PA, TA, and AM. If KODE=3, the independent variables are PA,
TA, and TB.
Actual mass-flow rate Ge divided by ideal mass-flow rate Ge, i" The idealmass-flow rate is defined as follows:
2 (pe_2/Yi _ :Pe_(Yi- 1)/Y_ 1/2
Ge'i _i -1 RTo \_oo,] \_oo: _J (C6)
for Me _ 1
_y : 2 ._(Yi+l)6:i-l)ll/2 Uo for Me=Mt=l
Ge' i = [ i\_-_/ ]
where 7i = 7/5 for nitrogen and yi
Nozzle exit specific heat, Cp, e/R
Nozzle exit specific-heat ratio, Ye
Nozzle exit isentropic exponent, k e
Plenum enthalpy, Ho/R , K
Plenum specific heat, Cp, o/R
Plenum specific-heat ratio, 7o
= 5/3 for helium.
(C7)
21
ocx(8)
oux(9)
oux(10)
OUX(ll)
OUX(12)
OUX(13)
Plenum isentropic exponent, k o
Plenum pressure as calculated from plenum density and temperature, N/m 2
For KODE=I, this is the nozzle exit pressure in newtons per square meter
as calculated from nozzle exit density and temperature. For KODE=2, this
is the nozzle exit Mach number as calculated from nozzle exit thermodynamic
gas state. For KODE=3, this is set equal to zero.
Indicates the degree of convergence achieved in calculating the plenum den-
sity and is defined by
OUX(II) = 1 Po, n (C8)
Po, n- 1
Indicatesthe degree of convergence achieved in calculationof nozzle exit
density and is defined by
OUX(12) = ln{-e'n-tl_/P '\ Pe-A'n--1 1
\ Pe, n / Pe, n
Ratio of plenum pressure to saturation pressure, or
(C9)
OUX(13)- Po
Psat
(CIO)
OUX(14) Ratio of nozzle exit pressure to saturation pressure, or
OUX(14)- Pe (Cll)
OUX(15) Plenum entropy, So/R
Psat
The following symbols refer to functions of the compressibility factor.
tions are defined in appendix B.
These func-
22
Z(1, 1) Z(1, 1) = ZI(Po, To)
Z(2, 1) Z(2, 1) = ZII(Po, To)
Z(3, 1) Z(3, 1) = ZIII(P o, T o)
Z (4, 1) Z(4, 1) = ZlV_ o, To)
Z(5, 1) Z(5, 1) = Zv(Po, To)
Z(6,I) Z(6,I)= ZVI(Po,To)
Z(1,2) Z(1,2)= Zi(Pe,Te)
Z(2,2) Z(2,2)= Zii(Pe,Te)
Z(3,2) Z(3,2)= Zill(P,Te)
Z(4,2) Z(4,2)= ZlV(Pe,Te)
Z(5,2) Z(5,2): Zv_Oe,Te)
Z(6,2) Z(6,2)= Zvi(Pe,Te)
The following symbols represent integers that are used to indicate if the calculation
is valid. If all these integers equal zero, a valid calculation has been performed. If
these are not zero, the conditions are as follows:
KODI(1) Equals 1 if the plenum conditions are out of range in either temperature or
pressure. A value of 1 terminates the calculation.
KODI(2) Equals 1 if the nozzle exit conditions are out of range in either temperature
or pressure. A value of 1 terminates the calculation. For the case of nitro-
gen, the computation is permitted to continue if pe/Psa t is between 1 and 3.
If this is the case, KODI(2) is set equal to 2.
KODI(3) If KODE=I, this quantity equals 1 if the calculated nozzle exit pressure fails
to converge to the prescribed nozzle exit pressure. If KODE=2, this quan-
tity equals 1 if the calculated nozzle exit Mach number fails to converge to
the prescribed nozzle exit Mach number.
KODI(4) Equals 1 if the iteration procedure for the calculation of the plenum density
fails to converge.
KODI(5) Equals 1 if the iteration procedure for the calculation of the nozzle exit den-
sity fails to converge.
23
APPENDIX D
FORTRAN IV SUBROUTINES
$1RFTC
C
IO
11
CNITS LIST,DECK
SUBROUTINE CNIT (KKK,PA,TA,AM,PB,TBtFLOW,KOI)E)
EQUIVALENCE (R,RR)
COMMQN /I)UTPUT/ OUX(15),Z(6,2),K[]Dl(5)
COMEON /CONV/ MMgM,NN
I)ATA AI,A2,A3,A4,A5/Z.50II46,-9.720581E-5,1.O36056E-6,-4.437258E-9
1,6.82559hE-12/DATA R,GAMA,GAMB,GAMC,GAME/296.774,.2R5714286,.2,1.42857143,7-O/
CP(S)=AI+(A2+(A3+(A4+AS*S)*S)*S)*S
CS(S)=AI_ALQG(S)+(A2+(A3/2.0+(A4/3.0+AS_S/4.0)*S)*S)_S
CH(S)=(AI+(A2/2.0+(A3/3.O+(A4/k.O+AS_S/5.0)_S)_S)=S)*S
IF (KKK.EO.I) GO TO ]0
I)U I N=I,5
KUDI(N)=O
DO 2 N=I,12
OUX(N)=O.O
OUX( lb)=O.O
[)[/ 3 NX=I,2
O0 3 N=I,6
Z(N,NX)=O.O
CALL LEJGIC (PA,TA,OUX(I3),KOOI(I))
IF (KODI(1).EO.2) Kr)Dl(1)=t
IF (KLII)I(I).EO.I) RETURN
THE ITERATION PROCESS FOR CALCULATING THE PLENUM DENSITY FOLLOWS.
A=PA/(R*TA)
RHIJA=A
CALL ZETA (I,RHOA,TA,ZtI)
DO 7 MM=I,50
OUX(II)=(RHOA-PA/IZ(I,I)mR*TA))/RHOA
IE (ABSIOUX(II)).LT.I.OE-6) GO TO 8
AAA=(Z(I,I)-A/RHOA)/Z(3,1)
IF (I.O-AAA) 5,5,6
AAA=AAA/2.0
GO TO 4
RHOA=RHDA*(I.O-AAA)
CALL ZETA (I,RHOAgTA,Z,I)
KOOI(4)=I
CALL ZETA (3,RHOA,TA,Z,I)
IF ((Z(I,I).GT.O.).ANO.(Z(?,I).GT.O_).ANO.(Z(B,I).GT.O-)) 60 TO 9
KOD1 ( I ) = t
RET[JRN
THE PLENUM THERMOL)YNAMIC FUNCTIONS ARE CALCULAIED BY THE FOLLUWING
STATEMENTS
CV=CP(TA)-Z(h,I)
GA=Z(3,] )+Z(2,1)*_2/CV
OUX(7)=GA/Z(3,I)
OUX(_)=GAIZ(I,I}
OUX(6)=CV_L)OX(7)
t]UX(5)=CH(TA)+TA*(Z(1,I)-Z(5,I))+50_.31
OUX( tb )=CS( TA )-AL 06 (PHOA)-Z (4_ t ) +0o 77t24
OUX(9)=Z( l, I )_:TA:;:R_H{JA
IF (KKK.EO.O) R_T{JRN
(;O TI) (II,IS,I6),KOE)E
AM=().(_
B4
12
13
14
15
C
C
C
C
C
16
17
18
19
2O
Pl
27
73
24
C
THE INITIAL ESTIMATE OF T:HE NOZZLE EXIT TEMPERATURE WHEN THE NOZZLE
EXIT PRESSURE IS GIVEN IS MADE BY THE FOLLOWING STATEMENTS,
I B:TA*(PB/PA)**GAMA
IF (TB.GE.126.36) GO TO 14
IF (PB-3.39S4E6) 12,12,13
X=ALOGIO(PB)-5.0057166
TBI=77.A635+(19.5407+(5.33082+(I.41895+.309IO6*X)*X)*X)*X
IF ((TB.LT.TBI).ANO.(TA.GT.I.OOI*TBI)) TB=TBI
GO TO 14
IF (TA.GT.126.48) TB=126.36
TBI=TB
IF (PA,GToPB) GO TO 17
KODI(2)=I
RETURN
PB=PA*(I.O+GAMB*AM**2)*m(-3.5)
THE INITIAL ESTIMATE OF THE NOZZLE EXIT TEMPERATURE WHEN THE NOZZLE
EXIT MACH NUMBER IS GIVEN IS MADE BY THE FOLLOWING STATEMENTS.
TB=TA/(I.O+GAMB_AM**2)
TBI=TB
GO TO 17
PB=PA=(TB/TA)**(3.5)
IF (TA.GT.TB) GO TO 17
KODI(2)=I
RETURN
KODI(3)=O
NN=I
DO IR N=I,4
OUX(N)=O.O
OUX(IO)=O.O
OUX(12)=O.O
DO 19 N=I,6
Z(N,2)=O.O
FLOW=O.O
KODI(5)=O
CALL LOGIC (PB,TB,OIJX(I4),KODI(2))
IF (KOOI(2).EO.I) RETURN
THE ITERATION PROCESS FOR CALCULATING THE NOZZLE EXIT DENSITY
FOLLOWS.
AL=ALOG(RHOA)+Z(4,1)+CS(TB)-CS(TA)
ALA=AL-Z(4,1)
CALL ZETA (2,EXP(ALA),TB,Z,2}
DO 21 M=I,50
OUX(12)=ALA-AL+Z(4,2)
IF (ABS(OUX(12)).LT.I.0E-6) GO TO 22
ALA=ALA-[)UX(12)/Z(2,2)
CALL ZETA (2,EXP(ALA),TB,Z,2)
K001(5)=I
RHQB=EXP(ALA)
IF (RHI)A-RHOR) 23,23,24
KOOI(2)=I
REIURN
CALL ZETA (3,RHUB,1B,Z,2)
25
C THE THERMODYNAMIC FUNCTIONS AT THE NOZZLE EXIT C()NDITIONS ARE
C CALCULATED BY THE F(}LLOWING STATEMENTS.
CVV=2.0_(CH(TA)-CH(TB)+TA_(Z(I,I)-Z(5,I))-TB_(Z(1,2)-Z(5,2))}
CV=CP(TB)-Z(6,2)
GA=Z(3,2)+Z(2,2)_:_Z/CV
OUX(3)=GA/Z(3,2)
OUXi4)=GA/Z(I,2)
OUXiZ)=CV_OUX(3)
C
25
GO TO (25,29,33),KODE
AM:ASORT(VV/(Z(1,2)*OUX(4)*TB))
IF (NN.NE.1) BI=OUX(IO)
DUX(IO):RHOB*Z(I,2)*R*TB
PERR=PB/OUX(IO)-I.O
IF (A6S(PERR).LT.I.OE-6) GO TO 34
IF (NN.GT.2O) GO TO 28
NN=NN+I
C
C THE SUCCEEDING ESTIMATES OF THE NOZZLE EXIT TEMPERATURE ARE MADE
C BY THE FOLLOWING STATEMENTS FOR THE CASE OF A GIVEN NOZZLE EXIT
C PRESSURE.C
IF iNN-2) 27,26,27
26 TB=TB*(I.O+GAMA*PERR)
IF (TB.GE.TA) TB=.999_TA
TB2=TB
(;0 TO 20
27 TB=TB+(TB2-TBI)_(PB-OUX(IO))/(OUX(IO)-BI)
TBI=TB2
TB2=TB
C
GO TO 20
2R KODI(3)=I
GO TO 34
29 PB=Z(1,2)*TB_:R_RHOB
IF (NN.NE.I) BI=OUX(IO)
OUX(IO)=ASORT(VV/(Z(1,2)*TB_OUX(4)))
IF (ABS(I.O-OUX(IO)/AM).LT.I.OE-#) GO TO 34
IF (NN.GT.20) GO TO 32
NN=NN+I
CC THE SUCCEEDING ESTIMATES OF THE N_ZZLE EXIT TEMPERATURE ARE MADE
C BY THE FOLLOWING STATEMENTS FOR THE CASE OF A GIVEN NE]ZZLE EXIT
C MACH NUMBER.
C
IF (NN-2) 3t,30,31
30 IB=TB_(I.O-O.4OO_TB*AM_(AM-OIJX(IO))/TA)
IF (TB.GE.TA) TB=.999_TA
TB2=TB
GQ TO 20
31 1B=TB+(TB2-TBI)_(AM-()UX(IO))/(OUX(IO)-BI)
IBI=TB2
TB2=TB
C
G[) lO 2O
32 K[II)I ( 3 ) = 1
(;[) lO 34
33 AM:ASORT (VV/ (7 ( l ,2 )_;OUX(4);: TB) )
Pr_: 7 ( ] , 2 ) :::_ =::_',HI)_,::: T R
26
34
C
C
C
C
35
36
37
C
OUX(IO)=O.O
CALL L()GIC (PB,TB,EHIX(I4),KODI(2))
IF ((VV.GT.O.).AND.(Z(2,2).GT.O.).ANO.(Z(3,2).GT.O.))KUOI(2)=I
RETURN
GO T_)
THE ISENTR[)PIC FLOW PROPERTIES ARE CALCULATEI) BY THE FOLLOWINGSTATEMEN1S.
FLOW=PB*SORT(VV/RB)/(Z(I,2)*TB)
TBF=(PB/PA)*_:GAMA
IF ((AN.EO°I°O).AND.(KODE.EO.2)) GO TO 36
FLOWI=PA*SORI(GAME*(PB/PA)*_GAMC*(I.O-T_F)/(RB,TA))GO T(} 37
FLOWI=PA*SORT(I.b7985E-3/TA)
OUX(1)=FLOW/FL[)W|
RE tURN
END
35
$IBF1C NZET LIS[,OECK
C
C THE FUNCTIONS OF THE COMPRESSIBILITY FACTOR ARE CALCULATED IN THEC FOLLOWING SURRL]UT [NE °
C
SUBROUTIf_E ZETA (K,P,T,Z,J)
OIMENSIQN Z(6,2)
DATA A,B1,B2,B3,B4,B5,B6,B7,BS,B9,BIO,BlI,BI2,B13,BI4,BIS/-7.135E-
IE,I.2034917E-3,-2oSIO789E-I,-4.9681584EI,3.7073373E2,1o496473EE,
22-IO27719E-6,-Z.4516046E-4,2°3tO2822E-9,4°9866482,1o6771286E3,
3-I°656225ES,-6°5374809E-5,2.420910RE-2,-I°I26389,1°]g29604E-]2/CO(X,Y,Z)=X*S3+Y*S4+Z*S5
SI=t.O/T
S2=SI*S]
S3=SI*S2
$4=$1_$3
SS=SI*S4
P2=P*P
EXP[]=EXP(A*P2)
AA=I.O/(2.0*A)
BB=I.O/A
CI=CO(Bg,BIO,_LI)
C2=CO(R]2,B13,BI4)
IF (K.EO.2) GO f(] 1
BAI=BI+R2*SI+B3*S2+B4*S3+_5*S5
BA2=BE+B7*SI
ZA=I.O+(BAI+(BA2+(BS+B15*SI*P2)¢P)*P)*P
ZB=EXPO_:P2*(CI+C2*P2)
Z(1,J)=ZA+Z_
ZA=I.O+(2.0*BAI+(3.O*BA2+(4.0*I_8+6.0*BIS_SI*P2)_P)_P)*p
ZB=P2*EXPt)*(3.0*CI+(2.0_A*CI+5.0*C2+2.0_C2*P2*A)*P2)
Z(3,,I)=ZA+Z_
IF (K.EO.I) REFtJRN
_T=B 1-B3_$2-2,0*B4*S3-4.0*Bb*S5
CIP=-CO(3.0:_9,4.0*_I(),b.O*Bll)
C2 P=-C(t( 3.0*B 12,4. O_B I 3,5.0*B 14)CC I=C ] +CIP
CC2=C2+C2P
ZA=I.O+(_T+(_6+BR*P)*P)_P
27
ZB=P2_EXPO*(CCI+CC2_P2)
Z(2,J)=ZA+ZB
ZA=(BT+IB6/2.0+BS*P/3.0)_P)*P
ZB=AA*(EXPO*(CCI+IP2-BB)*CC2)+BB*CC2-CCI)
Z(4,J)=ZA+ZB
IF IK.EO.2) RETURN
CIPP=CCI(g.O_B9,16.0*_LO,25.0_BII)
C2PP=CU(9.D_BI2,16.0*BI3,25.0mbI4)
ZA:-((B2*SI+2.0*B3*S2+3.0#B4*S3+5.0*B5_SS)+(B7mSI/2.0+BI5*SI*P**3/
Ib.O)*P)mP
ZB=AA*(EXPO*(CIP+(P2-BBI*C2P)+BB_C2P-CIP)
Z(5,J)=ZA+ZB
CCIP=CIP+CIPP
CC2P=C2P+C2PP
ZA:(2.O*B3*S2+6.0*B4*S3+20.O*BS*S5)*P
ZB=AA*(EXPO*(CCIP+IP2-BB)_CC2P)+CC2P_B-CCIP)
Z(6,J)=ZA+ZB
RETURN
END
$IBFTC NLOG LIST,DECK
C
C THE FOLLOWING SUBROUTINE IS USED TU DETERMINE WHETHER OR NUT THE
C FLUID IS A GAS, AN[) WHETHER OR NOT IHE PRESSURE ANO TEMPERATURF LIES
C WITHIN THE RANGE OF THE STATE EOUATION.
C
SUBROUIINE L{IGIC (P,T,R,J)
LUGICAL TNH,PNH
TNH=T.GT.501.O
PNH=P.GT.351.0E5
J=O
R=O°O
IF((T.GT.126.26).ANO..NOT.TNH.AND°.NOT.PNH) RETURN
J=!
IF((T.LT.55.0).OR.TNH.[)R.PNH) RETURN
XLIM=IO.O**(-305.O7339/T+5.5335216+(.1644110I+(-3.1389205E-3+(2.98
157103E-5+(-I.423B45BE-7+2.7375282E-ID_T)_T)*T)_T)*T)
R=P/XLIM
IF (R.GT.3.O) RETURN
J=O
IF (R.LE.I.O) RETURN
J=2
RETURN
END
28
$1RFTC
4
5
6
7
R
9
C
C
C
C
I(3
C
II
12
C
CHELS LIST,DECK
S_)_RNIITINE CHEL (KKK_,PA,TA,Am,P_,TF_,FLOW,KODE)
EOUIVALFNCF (R,RS)
C(-)_If"i[)r't /OUTPIJT/ OtJX(15),Z(6,2),KOO1(5)
COM_'iON /CONV/ MM,M,NN
f_ATA R, C,AMA, GA_F_,GAh_C,GA_E/2077.15, .4, . 333333333, I .2,5.01
IF (KKK.EO.I) GO TO II
DO I ,,o=],5
KOf)l (Jx_) =0
DO 2 N=I,12OHX(N)=O.O
OIIX( 15 )=0,0
DO 3 NX=I,2
_)0 3 N=I,6
Z(N,NX)=O.O
CALL LF]GIC (PA_,TA,,O(IX(13),KO!31(1))
IF (KF]!)I(]).EO.]) RETI!RN
THE ITERATION PROCESS FOR CALCIILATIIqC, THE PLENIIM DENSITY FOLLOWS.
A=PA/( R,X:TA )
RH(]A=A
DO 9 NX=I._2
K[]OI (z+) =0
CALL ZETA (I,RHOA,TA,Z,I)
OUX(II)=(RI-_OA-PA/(Z(I,I)_RmTA) )/RHDA
IF (AF_S(qUX(II)).LT.I.OE-6) GO TO R
AAA={7(I,I)-A/RHOA)/Z(3,1)
IF (I.O-AAA) 5,5,6
AAA=AAA/2.0
G_} TO 4
RM[)A =RN{]AX= { I .O-AAA )
CALL ZETA (I._RH(]A,TA._Z_,I)
KOF)I (z,-)=1
CALL ZETA (3,,RHOA,TA,Z,I)
IF { (Z{I,I).GT.O.).AND.{Z(2,1).C_T.O.).AND.IZ(3,1).GT.O.))
RHt-IA=AI 3.0
KOF)I (I)=I
RETIJRN
THE PLENIPM THERMODYNAMIC FUNCTIONS ARE CALCtlLATFD BY THE
STATEMENTS
CV=I.5-Z(6,1)
GA=Z ( 3, i )+Z (2, L )*_2/CV
F)UX( 7)=GA/Z ( 3,,i )
l_IJX(R) =GA/Z ( !, i )
O_)X (6) :CV_-OUX (7)
(I_!X(5)=TAX:(I.5+Z(1,1)-Z(5_,I})+6.qR973
OUX( 15 )=I .5_ALOG( TA )-AI_OG( RHOA)-Z (4_, i )+4. 75063
OIJX(g) =Z ( 1, I )*TA_RX-'RHOA
IF (KKK.EO.O) RETIIRN
GO Tn (12,13,14),KOI)E
A_=O.O
GO TO
FElL LOW ING
I0
29
C
C
C
13
C
E
C
C
C
1A
17
C
c
(.
] ()
21
22
C
C
c
c
"IHF T,'ITTTAL t"%1 T'_ATt- tiF IHI-: I,q]7ZLl- FXTI TF',.PFI_nTIIRF ,,iHFi,t Till- _\kFIZZLF
I-_ ! I- PRpS%I!I.41- IS GI\/P"I", IS r,ihi/F FrY [HF FflLClll4lt\,f:, STAIF-i,'Fi\tlS.
1I_ = TA;:-" ( P14/PA ) -",-";:-'(:, l\ I,+ h
IF ((Tk. LT.'-J.4_)} ).i_t,il).(T/_.ql.5..':,.f1A)) I-_',=h.4()1.
Tt-- (I->A.(:T.PH,) f_li [ti ]_J
Klit)l (2):]
RI- |liRki
PR = PA ;'.: ( I . ()+G Af'ti_,;'.: A,'_::_';: 2 ) ;;:-':: ( -2 ._ )
TH_ I,\IIT)AL ,c:.C:,T1f,_ATF IIF IHF i\lfiZZI F F×!T TFr. PF_AlltPf- Wl-tPi',l THE r,lrlZZI F
I-Xf[ ,.IhCFt _,,:lli,iHbq t_ (:,!\/P,xt l'_ ,<_AIIP H,y It-ii-: PllLL(i;41lixif; <-,IATI-i<,Fr,,TS.
T H,: I A / ( 1 . ()+(:,A :.i, _,;:: A,< ;::;: 2 )
i P,1 = 1 ix
G(I lit 1 5
P P,= t)A;i: ( I vii TA ) .i:;:: ( 2.5 )
IF (IA.C:,T.Tv',) Gii ti_ l'fl
Klllil (2)=l
Klll)} (4) :,)
_Hr,!= 1
I!fi !;-> ,i:'l , A
tlIIX ( ix, ) :(; .(1
fI!IX( l(i)=(i.(/
IIIIX ( ]2 i =i;.ft
Fill } / "=l ,pt:,
/ (,,i,2)=r;,()
i-I_i ll.I--( , oil
ChLL LItC. T(: (P_,lP:,ltliX(],'_),Kiit)i(7))
l;- (Killil (2)°t-,i.l) _Ti-{ll_,,i
K!!l)] (h ) ={)
rHF IIF_;_A-fIr],\I PPlICFR, (-, FliP, CAI_CIILAIIr, u-_ ll-IF f\if]7ZL l: FXIT I_Ft\ISITY
bill. I ill,,I,c:,.
AI =A!., v; ( _ i_l I/I)+/ ( 4-, l. ) + } . Li;::At_I IG.
AI_A=AI -7 (L_, I )
f:t\ll_ 7_-i-h (2,pb'(F'(nl_i\),Tl4,/,2)
I)li 1 c-) I.,=l_.h(t
I'ii_7( } 2 ) =ALM-t/I +7 (;_,2)
IlL ( h _-,<q ( !)1]7 !7))ol .1 .()F--_) f:
ill i'l= 1! I. A--I I! I_ "!2)/i( ,2)
Cr,.l_l /t ] A ( ,,-{u(A A),T,4,7,2)
P hi ti',=l 7_'( AI )
Iv ( _'i411A--k't411,* ) )J , I • _")
K!,nl (::):I
_'_ P I I II. >, ,
!'ALl ?t IA ("_._+'4_1.-_,_ ",,7._2)
[_,/ IA )
1 r! ;r;
TI'!: itIF,e',dl,lYn, M,,ll. FI,I, CTlll,,Ic, :\-I- ll-IF i\q/771 P _-)'<l] (;ll,'l II lilt 'xi<': Ai<l--
;:/',1C,ILA I_-Ii tV I [_ l-_il t_ti'.ll"U:, _ I/_/b.i',-,,,I c-,.
uW--2.,,;;: ( fll;:( I .'_+I ( 1 . 1 )-/ ( b. 1 ) )-i_:;:( 1 .':-,+7 ( 1 , 2 )-7r,j= 1 .,,_;(_,._ )
,:.,_: 7 ( 4. :-) ) + 7 i .-' . .-' ) ;;:;:. /_.
iiii'x (4) =r:.t 7 (4.))
,,,_)))
30
24
25
C
2_
P7
pR
2q
C
-_
31
li_I×(I,)=AA/Z( 1._,2
fit,× ( 2 ) =CV=',:tll I_ ( 3
A,,I=A%0_ I ( V\/l ( / ( i , 2 ]:',-'!]II×(4) =_I'_ ) )
TF (k,l,i.i,4_.]) _}=[HIX(]O)
rlli×( ] _)) =P,Hi _4:'.-'Z( i, 2 );'.:P_','-T _,
P_Rq:PH,/(ilIX ( ] (i)-] ,(i
IF (8i_S(PERP).LT.].()e-6) Cd) 1II _,2
IF (u,U\I.AT.2()) G() lli 2_
KIi\_= i',h'4+ I
IHF S!!CCI-FOIAH:, FSFT_'4ATFS F)F TI-4F I,,JI7ZLF #XIT TE_,,PFRAFI_RF AR# MAI)F
BY THE FF_LIFIWII,_G STATF,'_#k, TS F{IR T_4F CASF nF A C_TVFN i_q-IZZLF I:XIT
P# F-SSIIRF.
IF (i,!_'+-2) 2b,24,25
TB=TR;:-'(] .n+AAr._A=I-'PF-PR)
IF (TR.C_F.TA) F_=.999;;:TA
T P.2=TP,
G(_ T_) 1
II4=TP,+ (TP,2-TR 1 );;:( PP--I HIX ( 10 ) ) I ( (il!X( ] 0 )-B } )
TBt=IB2
TK2=TR
C_FI T() I R
KI)!)I ( 3 ) =i
r,lj Ttl 32
PB=Z ( ].,2 )='.:TB=;:q;:-'R_I-]H
IF (I,,k_.N_F.]) P,I=_!IX(I())
DIIX(IO)=ASnRT(VV/(7(I,2):;:TR=:-'[HIX(4-) ) )
IF (ABS(I_oO-I-,!X(]n)/Ar_).I_T.].OE-z_} C_F. Tii 32
TF (_\:,._.C_T.2r)) G[) TIJ 3P.
_\IN = N i_+ 1
T_II: SIICCEFI)Tf,,,C_ E:STT,:_AFFS QF TI4F NF/7Z[_F PXTT TF_PFRATf_RE: ARF ,-_Af)E
BY T_I_ E:t}l_Lf)WTr,_C_ S.TATF,_IENTS E:flR THE: CASE FiF A GIVFN _,.IQZZI_F fSXTTi,,_AC,M N!IMRFR,
IF (k,i,_-2) 2c).2_,29
T B: T K;',:( ]..0-0.887 ;:-"T _4=;:A,',i-';:(Ai4-fH IX ( tO
IF (TR.C_E.TA) T_,=.999,'.-'TA
TBZ=TR
Gll TF) 1 £
I'B=TB+ ( 1 BZ-T, _,t ) ;',: ( l_,_,-f/LIX ( [(1 ) ) / ( I1l IX
TRI=TF{2
TRZ=TR
)ITA)
In )-_1 )
r_(1 T[! ]R
KOn] ( 3)=}_
Au,=A S(:_R-f (VVl (Z ( 1,2 ) ;','-!1._× (4 )=;:T6) )
P#,=7 ( i ,2 );:-'_;',:Pwl ]_=;:TF_
Iit_X(] A)=O.A
CALl_ LF)GTC, (P;4,TI4,illlX([6),t<lll)}(2)
IF ((vv.r_T.n.).A,,,_.(Z(2,2).{;T.n.).z_r,_n.(7(_,.2).r-l-.().)) r_!_ T(! 33
K_]n] (P)=I
P hlllV i,_
TW# TSEU, IUQPIC I-Li/',i PRi_PFRTTPS ARe CAI_Ciil AIFI) Bv It4F I.:_ILL_IIWT,,IA
31
C
C
33
B4
B5
C
STATEMENTS.
FL[)W=PB:;=S()RTIVV/RF_)/(Z(I ,2); TB)
TBF= ( Pn/PA );'¢GAMA
IF ((A_,I.E().I.O).Ar,IF).(Kr}I/E.FCl.2)) G(I T_} 34
FLOW I =PAx:SORT (GAME=_ ( PR/P A ) ;:_:GA_v,C;= ( 1.0-TB F ) / ( RR;:TA ) )
G(] T[/ 35
FLOW [ = PA;XSOR T ( 2.53£79F-4/TA )
OiIX(1 )=FL(]U/FLDWT
RFTtlRN
E N F)
$IBFTC HZFT LIST,F)FCK
C
C THF FUNCTIFINS file T_4E CCImPRESSIBILITY FACT{1R ARE CALCllLATEI_ IN THE
C FFILLOW I NG SUBRQ!IT I_;E.
C
SUBRF)UTINE ZETA (K,P,T,Z,J)
DIMENSION Z(6,2)
DATA A,BI ,B2 ,RB,B4,BS,B6,B7,BR,Bg,BIO,RII ,BI2 ,B13,BI4,BI_,RI6/
l--4. 057 E-4,4.066_0 [ 3F-3 ,- i • 1267764F-[ , 2. 3039__66E-2 _ -5. 746RR l RE-2,
21.369136_E-I ,9. 7390626 E-A, 7.0543R76F-4,-5. B£54gR4E-6 ,-3 • £053762 E-B
3,2.625179E-2,7.67426AE-2,-£.790491IE-7,I-9960611F-69-R.1300167E-6P
43. 67435R3 E-£ ,-3. 4049435 F-11 /
C[](X _Y _Z ) =X;_SB+Y;:%4+Z_;-'S5
Sl=I .n/T
$2=S1¢S]
S3=S2X:S1
$4=$3;_$1
S 5 = S 4;',-"S t
AA=A'_S l
P2 = P;',:P
EXP()=FXP ( AA ;',:P2 )
CI=CCI(Bg,BIO,B11 )
C2=C(}( BI2 ,e_13 ,R14 )
IF (K.EO.2) GO Til l
RA 1 :R I+R2_S 1 +B3;_S2+B4_ $3+B5_$5
RA2=B6+B7_S1
ZA= 1.0+ ( BA 1+ ( BA2+ (B£:;_S l+ ( B l 5:4:SI+R 16_S I x:p ) _P ) ;',:P)_:P) x,p
ZR= (CI+C2*P2);_PP_EXP(I
Z(1, I)=ZA+ZB
_A=_+(2_BA_+(3_X_BA_+(4_X_S_+_5_B_5_S_+6_=;_B_6_`:`S_*_'_*_
I ),:-'P) _P
ZB=( 3. nx:C 1 + ( 2. O*AAX, C i.5.0.C2+2. OX,AA*C 2;_ P2 ) ;_P2 ) ==P2,_F XPO
Z( 3,_,J ) =7 A+ZR
IF (K.EC_.I) RETI)Rr,_
RT=R 1-R 3_,_$P-2. ():',q4 4_ S 3-4. O;r-B5_4:S 5
CIP=-CD(3.Ox:Rg_4.0_Bl(I_5.0_KII)
C2 P=-C.q ( 3. O:;:R 1 2 , 4. OX:B I 3, _. O;-'R 14 )C[ i=(2.0X_CI+ClP)/AA
C22= (3. ():;:C2+C 2 P ) iAA
ZA=I .0+ ( BT +R 6_::P) ;',-'P
Z H=P2,XF × Pf1:;,(C I+C t P+ (-AAX:C 1+C2_+C2 P-AA_;-'C2 X:P2_)=',_P2_)
ZiZ,,I)=/A+ZB
7A=(RT+H6xPZP.O);_P
7B=C).5_ ( EXP[F (C, 11-C22/AA+(-C1+C22-C2 :P2 )';P2 )-CI 1+C22/AA )
7 ( 4, J ) =Z A+ZB
IF (K°Ffl.2) RFIIIRN
32
CII=(CI+CIP)/AA
C22=(2.0*C2+C2P)/AA
ZA:(-(B2*SI+2.0*B3*S2+3.0*B4*S3+5.0*B5_S5)+(-B7*SI/2.0+(-BR,SI/3.O
I+(-BI5*SI/4.0-BI6*SI*P/5.0)*P)*p),p),p
ZB=O.5*(EXPOm(CII-C22/AA+(-CI+C22-C2*P2)*P2)-CII+C22/AA)
Z(5,J)=ZA+ZB
ClPP=CO(9.0_B9,16.0*BIO,25.0*BII)
C2PP=CO(9.0*BI2,16.0*BI3,25.0*_I6)
CII=(2.0*CI+3.O*CIP+CIPP)/AA
C22=(6.0*C2+5.0*C2P+C2PP)/AA
ZA:(2.0*B3_S2+6.0*B4_S3+2O.*BS_SS)_p
ZB:O.5*(EXPO*(CII-C22/AA+(-2.0*(CI+CIP)+C22+(AA*CI-3.0*C2-2.0*C2P+IAA*C2*P2)*P2)*P2)-CII+C22/AA)
Z(6,J)=ZA+ZB
RETIJRN
END
$IBFTC HLOG LIST,OECK
C
C THE FOLLOWING SIlBROUT.INE IS IJSED TO DETERMINE WHETHER OR N[)T THE
C FLLIID IS A GAS, AN[) WHETHER OR NflT THE PRESSURE AND TEMPERATURE LIES
C WITHIN THE RANGE OF THE STATE EOIJATIflN.
C
SIIBR[]IITINE Li]GIC (P,T,R,J)
J=O
R=O.O
IF ((P.LT.305.0ES).AND.(T.GT.5.4).AND.(T.LT.501.O))J=l
RETURN
END
RETURN
33
34
REFERENCES
I. Jordan: Duane P. ; and Mintz, Michael D. : Air Tables. McGraw-Hill Book Co., Inc.,
1965.
2. Johnson, Robert C. : Real-Gas Effects in Critical Flow Through Nozzles and Tabu-
lated Thermodynamic Properties. NASA TN D-2565, 1965.
3. Hilsenrath, Joseph, et al. : Tables of Thermodynamic and Transport Properties of
Air, Argon, Carbon Dioxide, Carbon Monoxide, Hydrogen, Nitrogen, Oxygen, and
Steam. Pergamon Press, 1960.
4. Johnson, Robert C. : Calculations of Real-Gas Effects in Flow Through Critical-Flow
Nozzles. J. Basic Eng., vol. 86, no. 3, Sept. 1964, pp. 519-526.
5. Strobridge, Thomas R. : The Thermodynamic Properties of Nitrogen from 64 to
300 ° K between 0. I and 200 Atmospheres. Tech. Note 129, National Bureau of
Standards, Jan. 1962.
6. Mann, Douglas B. : The Thermodynamic Properties of Helium From 3 to 300 ° K
Between 0.5 and 100 Atmospheres. Tech. Note 154, National Bureau of Standards,
Jan. 1962.
7. Dudzinski, Thomas J. ; Johnson, Robert C. ; and Krause, Lloyd N. : Performance of
a Venturi Meter with Separable Diffuser. NASA TM X-1570, 1968.
8. Glassford, A. P. M. ; and Smith, J. L., Jr.: Pressure-Volume-Temperature and
Internal Energy Data for Helium from 4.2 to 20 ° K Between 100 and 1300 atm.
Cryogenics, vol. 6, no. 4, Aug. 1966, pp. 193-206.
35
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NASA-Langley, 1968 _ 12 E-4512 201
