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ASTEM-A COLLECTION OF FORTRAN SUBROUTINES TO EVALUATE THE 1967 ASME EQUATIONS OF STATE FOR WATERISTEAM AND DERIVATIVES OF THESE EQUATIONS
by
K.V. Moore
nero~et nuclear Company NATIONAL REACTOR TESTING STATION
ldaho Falls, ldaho - 83401
DATE PUBLISHED-OCTOBER 1971
PREPARED FOR THE . - ,;i U. S. ATOMIC ENERGY COMMISSION
IDAHO OPERATIONS OFFICE UhlnED CAhlTDArT A T f l n - 1 1 - i ~ '
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' ANCR-1026 Mathematics and Computers
TID-4500
ASTEM - A COLLECTION OF FORTRAN SUBROUTINES
TO EVALUATE THE 1 9 6 7 ASME EQUATIONS OF STATE FOR
WATER/STEAM A N D DERIVATIVES OF THESE EQUATIONS
by b '
K. V. Moore
' . .NU I I - C t
This .report was prepared as .an account of work sponsored by 'the United States Government ... Neither
. the United States.nor the,United States Atomic Energy Commission, nor a n y of their employees, nor any of their. contractors, subcontractors, or their employees, makes any warranty, express or-implied, or assumes any legal liability or responsibility .for the accuracy, com- pleteness .or. usefulness of any information, apparatus, product o r process disclosed, o r represents that , i t s use would not infringe privately owned.rights.
L
AEROJ E T NUCLEAR COMPANY
. D a t e P u b 1 i s h e d - O c t o b e r . j 9 7 1
PREPARED FOR THE U. S. ATOMIC ENERGY COMMISSION IDAHO OPERATIONS O F F I C E
UNDER CONTRACT NO. A T ( 1 0 - 1 ) - 1 3 7 5
plSlRlBlsTUlW OF THIS CNGUPlee.MT IS UNLIMITED 1
ACKNOWLEDGMENTS
This work i s a por t ion of a more general p ro jec t t o develop an
e f f i c i e n t computer program of water proper t ies f o r use i n f u tu r e reac to r
sa fe ty analys is codes. Special thanks i s given t o D r . L. J. Ybarrondo
and M r . H. D. Curet of t h e Nuclear Safety Division f o r t h e i r sponsorsh5p
and he lp fu l support of t h i s phase of t h e overa l l r eac to r sa fe ty ana lys i s
e f f o r t within Aerojet Nuclear Company and t h e U . S. Atomic Energy Com-
mission.
ABSTRACT
ASTEM i s a modular s e t of FORTRAN I V subroutines t o evaluate t h e
s t a t e equations of l i qu id water and steam as published by t h e American
Society of Mechanical ~ n g i n e e r s (1967). Any thermodynamic quanti ty in- - cluding der ivat ive proper t ies can be obtained from these rout ines by a
user supplied main program.
As provided t h e ASTEM package includes a sample main program and
requires 6 8 ~ bytes of core storage (double precision) on t h e IBM 360175.
iii
CONTENTS
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . ; . iii
. . . . . . . . . . . . . . . . . . . . . . . . . . I INTRODUCTION 1
I1 . SYMBOL DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . 2
. . . . . . . . . . . . . . . . . . . . . . . . . I11 ASME SUBREGIONS 3
. . . . . . . . . . . . . . . . . . . . . . . . . IV THERMODYNAMICS 5
V . ASTEM ROUTINES AND COMMON BLOCKS . . . . . . . . . . . . . . . 8
1.1 CONA . 1.2 COW . 1.3 GIBBAB 1.4 HELMCU 1.5 IASME . 1.6 INDEX . 1.7 PSATK . 1.8 PSATL . 1.9 ROOT . 1.10 UNITS . 1.11 WR1T.U
2 . INTERNAL ROUTINES . . . . . . . . . . . . . . . . . . . . . 14 .
2.1 BINOMX . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 EDER . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 FUNX . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 POLATT . . . . . . . . . . . . . . . . . . . . . . . . . 15 .. 2.5 POLYN . . . . . . . . . . . . . . . . . . . . . . . . 15
. . . . . . . . . . . . . . . . . . . . . . COMMON BLOCKS 16 3 . 1 /ASMCON/ . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 /ASMCOX/ . . . . . . . . . . . . . . . . . . . . . . 16 r" . . . . . . . . . . . . . . . . . . . . . . . 3.3 ASM ME^/ 16 . . . . . . . . . . . . . . . . . . . . . . . 3.4 AS ME^^/ 16 3.5 /ASMESL/ . . . . . . . . . . . . . . . . . . . . . . . 16 . . . . . . . . . . . . . . . . . . . . . . 3.6 IBINFACI 16 3 ~. 7 / D ~ Z Z Z / . . . . . . . . . . . . . . . . . . . . . . . 16 3.8 /ERRPVT/ . . . . . . . . . . . . . . . . . . . . . . . 17 3.9 /EUNITS/ CRP . CRT . CRV . CRH . CRS .TO. JC . GC . SQJC . SONIC2 . . 17 3.10 /MLJNITs/ CRPMyCRTM.CRVMyCRSM.TOMy JCM,GCM,SQJC.M,SONICM 17 . . . . . . . . . . . . . . . . . . . . . . 3.11 /ROOTLM/ 18
. . . . . . . . . . . . . . . . . . . . . . 3.12 /SATLIN/ 18
. . . . . . . . . . . . . . . . . . . . . . VI . . SAMPLE PROBLEMS ; 20
V I I . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . 24
L
F I G U R E S
1. A S M E S u b r e g i o n s . . . . . . . . . . . . . . . . . . . . . . . . 3
. TABLES
1. SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2. A S M E S T A T E E Q U A T I O N F O R M S . . . . . . . . . . . . . . . . . . . 4
3. ASMF: BOUNDARY VALUES FOR SUBREGIONS . . . . . . . . . . . . . . 5
4. THERMODYNAMIC VARIABLES EXPRESSED I N TERMS O F g , f , and P ' . . . 7
5 . SUBROUTINE ROOT ARGUMENTS . . . . . . . . . . . . . . . . . . . 12 6 . CROSS-REFERENCE O F SUBROUTINES, FUNCTIONS AND NAME COMMON
B L O C K S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
ASTEM - A COLLECTION,OF FORTRAN SUBROUTINES.
TO EVALUATE THE 1967 ASME EQUATIONS.OF STATE FOR
WATER/STEAM AND DERIVAT IVES OF THESE EQUATIONS
.I. INTRODUCTION
This co l lec t ion of subroutines ca l led ASTEM (ASME Steam ~ a b l e s ) f o m s
a self-contained modular s e t t o evaluate t h e Gibbs , Helmholtz and satura-
t i o n 1ine.f 'unctions a s published by t h e American Society of Mechanical
Engineers[''. Any thermodynamic quanti ty can be computed by these rout ines
from a main program supplied by t h e user . The a b i l i t y t o obta in der ivat ives
of any order makes ASTEM somewhat'unique.
This package i s wr i t t en i n FORTRAN I B f o r t h e IBM 360/75 and requires
6 8 ~ bytes of storage. As progr&med,, all var iab les a r e double precisioned
for computing accuracy on t h e 360 system. Conversions t o other computers
and conversion t o s ing le precision can be easidy accomplished by t h e user .
Fundamental t o t h i s repor t i s t h e ASME publication "~hermodynamic and
Transport Propert ies of Steam" (1967 ASME Steam Tables) , t h e American
Society of Mechanical Engineers, 1967. The equations and constants from
t h i s book a r e not reproduced nor s t a t ed within t h i s report. because of t h e i r
ra ther complex form and t he pos s ib i l i t y of typographical e r ro rs . Also, a l l
symbols used i n t h i s repor t a r e t h e more standard English symbols fo r ther-
modynamic quan t i t i es whereas t h e . ASME uses a s e t of Greek symbols fo r t h e
reduc.ed dimensionless var iables . I n t h i s repor t a symbol i s used f o r a
par t i cu la r physical quanti ty imma.t e r i a l of u n i t s . Unless spec i f i c a l l y noted
or when numbers a r e quoted without u n i t s , they a r e assumed t o be "reduced
dimensionless" a s used by t h e ASME..
This report assumes t ha t t h e reader has a working knowledge of the r -
modynamics and t h e i r der ivat ive re la t ionships . A good c l a s s i ca l thermo-
dynamics textbook and t h e ASME Steam Tables a r e necessary references fo r
use i n conjunction with t h i s document.
11. SYMBOL DEFINITIONS
The following t a b l e gives the symbol def in i t ions a s used i n t h i s
repor t a s well a s t h e corresponding ASME symbol.
TABLE 1
SYMBOLS
This Report ASME Quantity
P B Pressure
T 8 Temperature
v X Spec i f i c volume
h E Spec i f i c enthalpy
u - Specific in te rna l energy
s CI Specific entropy
x - Mass qual i ty
g 5 Gibbs function
f '4 Helmholtz mnct ion
B _ . Coeff f cient of thermal expansion
C - Specific heat capacity a t constant pressure P
( ah/aTIp
Cv - Specific heat capacity a t constant volume . .
( a u / a ~ lv
- $ ($)T isothermal c m p r e s s i b i l i t y
- a Isentropic sonic 'velocity S
- . d - - - ( 3 ~ 1 3 ~ ) ~
J
Y ' ' - Ratio of spec i f i c heat capaci t ies Cp/Cv
111. ASME SUBREGIONS
The published ASME equations a r e def ined , fo r l imi ted ranges of pres-
sure and temperature. A l l thermodynamic quan t i t i es i n these equations
a r e expressed i n a normalized (reduced dimensionless) form such t h a t a t
t h e c r i t i c a l point pressure, spec i f i c volume and temperature a r e unity.'
The subregions f o r each equation i s shown i n Figure 1, t h e equation forms
a r e given i n Table 2 and boundary values are l i s t e d . in Table 3 .
Figure 1. ASME subregions.
i
TABLE 2
ASME STATE EQUATION FORMS
2 (steam) 0 5 P < P ~ ( T ) , Tt 5 T I TI
o 5 P 5 P ~ ( T ) , T~ < T < T~
3 ( s t e a m ) pL < P < P ~ ( T ) , T~ < T < 1
5 ( t w o - p h a s e )
6 ( t w o - ~ h a s e )
B o u n d a r y b e t w e e n 2 a n d 3
'TABU 3
ASME BOUNDARY VALUES FOR SUBREGIONS
Quanti ty ASME 'Normal i z ed Value Spec i f i c Value
T ( t r i p l e , point ) 27316/64730 z 0.421999 -32 .018~ t
I
P, ( t r i p l e point ) P ~ ( T ~ ) z 2.7633 x -0.08865 ps ia
1 pKO1 ' 0 -747519 -2398.2 p s i a
PC ( c r i t i c a l point) 1 22120000~/m~, -3208.23 p s i a
P2 U000/2212 = 4.52 14503.77 ps ia
I V . THERMODYNAMICS
Many excellent references ex i s t i n . t h e f i e l d of thermodynamics and
it i s not t h e purpose of t h i s repor t t o t abu l a t e a comprehensive review
of well known mater ia l . Within t h i s framework we s h a l l t abu la te several
of t h e more usefu.1 r e l a t i ons h d t h e i r der ivat ives . The four fundamental d i f f e r e n t i a l equations a r e :
du = Tds - Pdv
dh = Tds + vdP
and i n two-phase h -h d P = I u dT T v -v
g f
and t h e general r e l a t i ons between these thermodynamic var iables a re :
Thermodynamic der ivat ives can be expressed i n terms of .any th ree
independent der ivat ives and an appropriate s e t of s t a t e proper t ies . The
generally accepted standard form i s t o express the . der ivat ives 5n terms
of B,, K , and C along with P, v , and T i n s ingle phase. For homogeneous P
two-phase mater ia l , ~ r i d g m a n ' ~ ] uses dP/dT P , Cv , vf , vg , s , v , T, P,
and 6 ( x , 1-x ) where
and
(subscr ipts g and f r e f e r t o gas and f l u i d , respect ively . )
Various thermodynamic quan t i t i es and standard der ivat ives a r e l i s t e d
i n Table 4 i n terms of ~ ( P , T ) , f ( v ,T ) , and t h e sa turat ion pressure P(T) .
This t ab l e along with der ivat ives a s tabulated by ~ r i d ~ m a n [ ~ ] form a
working se t with which any thermodynamic -quant i t y can be. calculated
' algebracia l ly .
To simplify wri t ing, a shorthand notation of t h e . exp l i c i t p a r t i a l ' . .
der ivat ives i s used i n Table 4. For instance, define g , t o be an exp l ic i t . function of (P ,T) , . .
then gp - ' and gpTE$[(%)P]+ * '
TABLE 4.
THERMODYNAMIC VARIABLES EXPRESSED I N TERMS OF g , f , and P'
Single-Phase: @(P,T) and f ( v , ~ 1.; Two-Phase P(T)
Helmholtz Function
not defined '
not defined
not defined
not defined
* Subscripts g and' f mean t h a t t h e quant i ty ' i s evaluated on t h e gas and l i q u i d s i d e , respect ive ly .
V. ASTEM ROUTINES AND COMMON BLOCKS
ASTEM. i s a package of 16 rout ines and 12 common blocks. Ten of t h e
common blocks a r e block data and t h e other two transmit information
between subroutines. Unless redimensioned by t h e user , t h e highest de-
r i v a t i v e allowed i s order 9 .
Since a l l a r rays within ASTEM a r e one-dimensional t o save s torage
area , t h e user must understand t h e equivalence of an "e f fec t ive two-
dimensional" a r ray a s explained i n INDEX.
1. USER ROUTINES
,
The rout ines normally ava i l ab le t o users a r e described i n alphabet-
i c a l order: CONA, COW, GIBBAB, HELMCD, IASME, INDEX, PSATK, PSATL, ROOT,
UNITS, and WRITEA.
1.1 CONA - Converts u n i t s of ar ray AA.
,Usage: CALL CONA(AA,IOA,A,IB,JB.,CA,.CI,CJ)
Input : AA, a s ing le dimensioned a r r ay equivalent t o M ( L ,L)
where L = IOA+1
I B and J B a r e index biases
IOA i s t he order of a r r ay AA
CA i s u n i t s conversion f o r numerator
C I i s u n i t s conversion f o r denominator I1
C J i s u n i t s conversion f o r denominator JJ
Output: Single dimensional a r r ay equivalent t o A(LA,LA)
where LA = IOA+l-IB-JB
Example: Let AA be a 4 x 4 a r ray of t h e Gibbs function and i t s I
(P ,T) der ivat ives t o t h i r d order.- Set up t h e v a r ray
t o second order.
CALL CONA (AA.,~,A,~,O,CRV,CRP,CRT)
Output: ~ ( 1 ) = M ( ~ ) * C R V = v
, . ~ ( 2 ) = M ( ~ ) * C R V / C ~ P = (E) T
e t c . 1.2 corn -
Converts un i t s of up t o 10 var iables
Usage: CALL COW ( J ,C , ~ 1 , 1 1 , ~ 2 , 1 2 ,X3,13. . .~10 ,110)
J = Number of X arguments
C = Array of s i x conversion fac tors
~ ( 1 ) = c r i t i c a l pressure
~ ( 2 ) = c r i t i c a l zbsolute temperature
~ ( 3 ) = c r i t i c a l specif ic volume
~ ( 4 ) = enthalpy conversion fac tor
~ ( 5 ) = entropy conversion f ac to r
~ ( 6 ) = "zero" temperature
The array C can be obtained by a c a l l t o UNITS.
X's and 1 ' s a r e input var iables t o be converted where i f
I = 1, X = pressure
I = 2, X = change i n temperature
I = 3, X = spec i f ic volume
I = 4 , X = enthalpy, spec i f i c i n t e rna l energy, Gibbs o r Helmholtz
I = 5, X = entropy
I = 6, x = nonincrementd temperature.
I f I i s posi t ive , convert from reduced u n i t s t o u n i t s
defined by t h e C ar ray.
For I > 0, X i s returned a f t e r multiplying by ~ ( 1 ) .
I f I i s negative, convert from C ar ray un i t s t o reduced
un i t s .
For I < 0, X i s returned a f t e r dividing by C ( [ I ( ) . 1 . 3 GIBBAB
Gfpen (P,T) compute t h e Gibbs function and a l l der ivat ives t o order N.
Usage: CALL GIBBAB(G,P,T , N , I )
Input: P = pressure
T = temperature
N = der ixat ives t o order N , 0 I N I 9
I = ASME subregion e i t he r 1 or 2.
Output : G array containing t h e Gibbs function and i t s der ivat ives
evaluated a t (P ,T) '
Dimension. G a t l e a s t ( ~ + 1 ) ( ~ + 2 ) / 2
(See INDEX f o r arrangement of output array. )
1 . 4 HELMCD
Given (v ,T) compute t h e Helmholtz function and all der ivat ives t o
order N , .
Usage: CALL HEIMCD(F,V,T,N,I)
Input: V = spec i f ic volume
T = temperature
N = der ivat ives t o order N , 0 5'N 2 9
Output: F array containing t h e Helmholtz function and i t s deriva-
t ives evaluated a t ( v ,T )
Dimension F a t l e a s t ( ~ + 1 ) ( ~ + 2 ) /2
( s ee INDEX f o r t h e arrangement of t h e output array. )
1 .5 IASME
I n i t i a l condition subroutine t h a t .must be ca l led once t o s e t values
of binomial coeff ic i .ents and f a c t o r i a l s into. /BINFAC/ and t o i n i t i a l i z e
computer dependent res idua l s u o and ul ['I i n /ASMCON/ . Usage: CALL IAsME(1)
where I i s t he maximum order of der ivat ives t o be calculated.
L i m i t : 1 1 1 5 9
1 .6 INDEX
Computes t h e FORTRAN locat ion index of a packed single-dimensioned
a r ray t h a t corresponds t o a double-dimensioned t r i angula r a r ray A ( I , J ) .
In .ASTEM, two-dimensional ( i;, functions of two independent var iab les )
quan t i t i es a r e used within t h i s program as packed, single-dimensioned
quan t i t i es . For example, l e t % b e t h e FORTRAN symbol t o denote t he Gibbs
function and i t s P and T der ivat ives , then i f we wish t o compute t h e
Gibbs der ivat ives t o order n, G would have t o be dimensioned (n+l ) (n+2) /2
f o r a packed array. For a numerical example say t h a t 'we require der i -
va t ives t o t h e t h i r d order , then (3+1)(4+1)/2 = 10.
When calculated. by a c a l l t.o .GIBBAB, t h e a r ray G i s returned with
t h e following values,:
Rather than sketch t h i s type of a r ray t o obta in t h e equivalence, a
c a l l t o I F E X w i l l .compute t h e proper index value when supplied with t h e
a r r ay s i z e and t h e double FORTRAN index. For example, compute t h e index,
L, f o r g PPT
where 1 = 3
I M = JM-= 4
then L = I N D E X ( ~ , ~ , ~ , ~ , I D U M M Y ) = 7
and IDUMMY = IP = maximum number of elements i n t h e J t h Pow = 3.
1.7 PSATK
Computes sa tu ra t ion pressure and der iva t ives t o order n a s a funct ion
of temperature.
Usage: CALL PSATK(P,T,N)
Input: T = temperature .
N = der ivat ives t o order N , 0 6 N I 9
Output: ~ ( 1 ) = Psat
~ ( 2 ) = P' ( d ~ / d ~ )
~ ( 3 ) = P" (d2p/d~?.)
e t c . Dimension P a r ray a t l e a s t (N+l) .
1.8 PSATL
Computes t h e pressure boundary, PL, between ASME regions 2 and 3
and i t s der ivat ives t o order n as a funct ion of T.
Usage: CALL PSATL(P ,T , N )
Input : T = temperature
N = der ivat ives t o order N , 0 < N 5 9
( ~ o t e : Derivatives of order 3 or higher a r e zero.)
( ~ o t e : Derivatives of order 3 o r higher a r e zero.)
output: ~ ( 1 ) = PL
1.9 ROOT
Computes i t e r a t i v e l y a roo t of t he Gihhs, Helmholtz, o r s a t w a t i o n
pressure function.
where I = type of ' root and NOGO = .TRUE. i f i t e r a t i o n f a i l u r e occurs.
I I I = ASME subregion number
TABLE 5
SUBROUTINE ROOT ARGUMENTS
1.10 UNITS
A c a l l t o UNITS r e t u r n s t o t h e use r t h e conversion constants f o r
English o r metric (MKS) u n i t s .
Usage: CALL UNITS(C , I)
Input: I = 1, English constants from /EUNITS/
I = 2, MKS constants from /MUNITS/
Output,: Array C dimensioned by 10.
I 1.11 WRITEA
Write t h e a r ray ( o r por t ions t h e r e o f ) i n t r i a n g u l a r form when A i s
single-dimensioned.
Usage: CALL WRITE(A,I,J,IPAGE)
Input : ar ray A
1,J e f f e c t i v e dimensions of A
IPAGE i s t h e current page number (both input and ou tpu t ) .
I f 'negat ive , no paginating .
*
Output example: I = J = 4; I-B = J B = 0
~ ( 1 ) A ( 2 ) ~ ( 3 ) ~ ( 4 )
~ ( 5 ) A(6) ~ ( 7 )
~ ( 8 ) ~ ( 9 )
~ ( 1 0 )
c (1)
~ ( 2 )
c ( 3)
~ ( 4 )
c ( 5 )
~ ( 6 )
c (7)
~ ( 8 )
C(9)
English I Metric
C r i t i c a l pressure
C r i t i c a l temperature
C r i t i c a 1 spec i f i c volume
Enthalpy conversion
.Entropy conversion
"Zero" temperature
Mechanical equivalent t o heat energy
Gravi ta t ional conver- s iona l f a c t o r
Pressure conversion
CRP, l b f / i n z
CRT ,R
CRV, f t 3/lbm
CRH, Btu/lbm ( CRH=CRV*CRP*SQJC )
CRS , Btu/lbm-R ( CRS=CRH/CRT )
To, 459.67 R
J C , f t - lbf /Btu
GC, ft-lbm/sec2-lbf
, SQJC , ( i n 2 / f t 2 ) / ( f i - l b f / ~ t d .
CRPM, ~ / m ~
CRTM, K
CRVM, m3/kg
CRHM, J/kg ( CRHM=CRVM*CRPM )
CRSM, ~ / k g - K (CRSM=SRHM/CRTM)
~ 0 ~ ~ 2 7 4 . 1 5 K
J C M , Unity
GCM, m/sec2 ,
SQJCM, Unity b
' SONICM, Unity ~ ( 1 0 ) Conversion f o r normalized sonic v e l o c i t y squared
(SQJC = 1 4 4 / ~ c )
SONIC2, ( f t 2 / s e c 2 ) / ( l b f - f t 3/in2-lbm)
Example: I = 4 , J = -4; I B = J B = 0
~ ( 1 )
~ ( 2 ) ~ ( 3 )
~ ( 4 ) ~ ( 5 ) A ( 6 )
~ ( 7 ) ~ ( 8 ) ~ ( 1 0 )
(1f J < 0 , t h e binomial a r ray i s wr i t t en . )
2. INTERNAL ROUTINES
The following subroutines are normally - not ca l l ed by t h e user:
Let ~ ( x , ~ ) H(x,y) = F(x ,y) , then given ( a ) H ( x , Y ) , F(x ,y) ; compute
G ( X , Y ) and i t s der ivat ives or (b ) H(x,y), ~ ( x , ~ ) ; compute F ( X , Y ) and i t s
der ivat ives . Usage: CALL B I N O M X ( G , I G X , I G Y , H , I H X , I H Y , F , I F X , I F Y , J I ~ , M G Y ,
MKX ,MHY ,MFX ,MFY ,LTYPE)
Input : A. LTYPE = 0, Compute G i\
H = ar ray i n packed s ing le dimensional form
equivalent t o double dimensions IHX , I H Y
~ ( 1 ) = H ( 1 , l ) = H
~ ( 2 ) = ~ ( 2 ~ 1 ) = aH/ax
~ ( 1 , 2 ) = aH/ay
e t c . F = ar ray F i n packed singTe dimension
F, aF/ax, aF/ay . . . . equivalent a r ray F(IHX,IHY)
JI = order of i n i t i a l computation
(JO = 0, compute G = F/H)
JF = highest order
( ~ x a m ~ l e : If JF = 2, compute der iva t ives t o
2nd order) MGX,MGY: MHX,MHY: MFX,MFY a r e
index biases f o r G , H , and F.
Example: For t h e a r ray F, elements a r e biased
F ( I+MFX , J+WY ) . B. LTYPE = 0, Compute F given G and H
(Input descr ipt ion s imilar t o A. )
2.2 EDER
-Calculate t h& der ivat ives with respect t o t of
I Usage: CALL EDER(z,x,B,BL,M,M,Io)
I Input : I 0 = . highest order f o r der ivat ives ~ 5
. M = number of terms i n sum
(1 I. M 2 7 )
BL = A
X = .e A (1-t
N = n i (dimension M )
B = bi (dimension M )
Output: Z a r ray (dimension 10+1)
2.3 FUNX
Defines and computes t h e functions f o r subroutine ROOT described
i n Section 1'.9.
2.4 . POLATT
Interpola tes t h e s t r a igh t l i n e s i n a t a b l e of Y , X .
Usage: . Function POLATT (XY ,XX ,N , K )
Input: XY = Array of Y,X: ~ ( 1 ) ~ ~ ( l ) , ~ ( 2 ) ~ ~ ( 2 ) . . . . . where.X(N+l) > X ( N )
XX = input value of X .. . . t
N = number of pa i r s of X,Y i n XY a r ray
K = memory index
Output : Interpolated va lue of Y arid K., memory index.
. . - 2.5 POLYN
,. .
' Compute t he der ivat ives with respect t o X of .
Usage: CALL POLYN(X,Z ,A,NI ,M,L) '
Input: X = value of main var iab le
' 1 5
M = number of terms i n sum
(1 I M 1 2 0 )
A = ai dimensioned M
N I = in teger niy dimensioned M
L = dimension of Z
(de r iva t ives computed t c ~ order L-1)
Output : Z ar ray .
COMMON BLOCKS
/ASMCON/ mst be i n i t i a l i z e d by a c a l l t o IASME.
(Note: /ASMCON/ appears i n block da ta DA12 with approximate values
which a r e reca lcu la ted i n IASME using p rec i se values from /ASMCOX/.)
3.2 ]ASMCOX/
Set i n block d a t a D A l l wi th values t o c a l c u l a t e /ASMCON/.
3.3 /ASMF;L/
Set i n block d a t a DAT2; contains constants f o r t h e Gibbs functions.
3.4 ASM ME^^/ Set i n block da ta DAT3; conta'ins t h e constants f o r t h e Helmholtz
funct ions .
3.5 /ASMESL/
Set i n block d a t a D A T ~ ; contains t h e constants f o r sa tu ra t ion pres-
sure l i n e and PL l i n e separat ing subregions 2 and 3.
3.6 /BINFAC/
Contains an a r ray of binomial c o e f f i c i e n t s and an a r r a y of f a c t o r i a l s
which a r e ca lcu la ted i n IASME.
( ~ o t e : The dimensions of t h i s a r r a y l i m i t t h e de r iva t ive ca lcula-
t i o n s t o 9 th order o r l e s s . ) -
3.7 /DUMZZZ/
An intermediate s torage area.
( ~ o t e : The dimensions of t h i s a r r a y l i m i t t h e de r iva t ive calcula-
t i o n s t o 9 t h order o r l e s s . ) I
3.8 /ERRPVT/ i
Contains a r rays of it e ra t ion e r r o r requirements and search l i m i t s
f o r subroutine ROOT. Values s e t i n block da ta DAT7.
3.9 /EUNITS/ CRP ,CRT ,CRV,CRH ,CRS ,TO , J C ,GC ,SQJC ,SONIC2
Contains conversion constants s e t i n block d a t a f o r English u n i t s .
- This common block i s ava i l ab le t o use r s by c a l l i n g UNITS.
CRP = 3208 p s i a
P (reduced) = P ( ~ s ~ ~ ) / c R P
CRT = 1165.14 R
T (reduced) = [T(F)+TO]/CRT
CRV = 0.0578 ft 3/lbm
v (reduced) = v ( f t 3 /lbm) /CRV
CRH = 30.14 Btu/lbm .
g , f , h , o r u (reduced) = [h o r u ( ~ t u / l b m ) ] / ~ ~ ~
CRS = sent~opy norma.li zat ion
s (reduced) = s (~tu/ lbm-F) /CRS
TO = 459.67 R (d i f f e rence between absolute temperature)
.Scdle ( R ) and Fahrenheit (F)
.J,C ',7a .16 f t- lbf/Btu (mechanical equivalent t o heat energy)
GC = 32.174 ft-lbm/sec2-lbf
SQJC = 1 4 4 / ~ ~ , pressure conversion from l b f / i n 2 t o ~ t u / f t ~
SONIC2 = conversion f o r normalized sonic v e l o c i t y squared
a 2 ( f t 2 / s e c 2 ) = a: ( n o r m a l i z e d ) * ~ 0 ~ 1 ~ 2 S
~ . ~ O . / M U N I T S / CRPM,CRTM,CRVM,CRSM,TOM,JCM,GCM,SQJCM,SONICM
Contains conversion f o r metr ic (MKS) u n i t s . Available by c a l l i n g
UNITS.
CRPM = 2.212 x l o 7 ~ / m ~
CRTM = 647.3 K
CRVM = 0..00317 m3/kg
CRHM =' 70120.4 J /kg
CRSM = 108.3275 ~ / k g - ~
TOM = 273.15 K
JCM = 1. N-m2/5
GCM = 9.80665 m/sec2
SQJC = 1. N-m2/5
3.11 /ROOTLM/
Contains p a i r s of ( v , ~ ) which def ine t h e approximate boundaries of
ASME subregions 3 and 4 o r f o r subroutine ROOT. Values s e t i n block d a t a
DAT 8.
3.12 /SATLIN/
Contains approximate values of Tsat a s 'a' funct ion of P f o r sub-
r o u t i n e ROOT. Values s'et i n block d a t a DAT9.
The following t a b l e i s a cross-reference Yor subruutlnes , . huc t iona +
and name common blocks within ASTEM.
' TABLE 6
CROSS-REFERENCE OF SUBROUTINES, FUNCTIONS AND NAME COMMON BLOCKS
* Contains s. wri te statement and format.
** Contains dimensions t h a t r e s t r i c t maximum order .
*** Contains dimensions t h a t l i m i t number of summation terms.
V I . SAMPLE PROBLEM
The following sample problem i s designed t o t e s t a l l rout ines i n
t h e ASTEM package.
3 P = ~ O O ' p s i a , T=150F
LEVEL 1 8 ( SEPT 5 9 I 0 S / 3 5 5 FOPTRAN H
COMPILER OPTIONS - NAqE= ~ A I N ~ O P T = O 2 ~ L I Y E 3 N I = 5 8 t S I Z E = O D O O K ~ S O U R C E t E B C O I C t N O L I S T ~ N O 0 E C K ~ L O A O ~ M A P ~ N O E O I T ~ N ~ I D ~ X R E F
I S N 3 0 0 2 I M P L I C I T R E A L * 8 ( A - H . 0 - L I - I S N 0 0 0 3 OIMFYSION G 1 5 5 ) r G V ( 4 5 I I S V 0 0 0 4 OIMEYSION ~ ~ ( 1 0 ) . c ~ ( l 0 1 I S N 3 0 0 5 L O G I 2 4 L NOGO
C C I N I T I A L I Z E
1SN 0 0 0 6 CALL 14SME( 9 ) I S Y 0 0 0 7 CALL U V I T S ( C E I ~ ) I S N 0 0 0 8 CALL U V I T S ( C M 1 2 ) I S N 0 0 0 9 JPAGE = 1 I S Y 0 0 1 0 W R I T E ( 6 1 1 0 1 ) JPAGE 1SN 3 0 1 1 1 3 1 F O R W 4 T ~ l H l r 2 3 H A S T E ~ / M J I ) O 2 t O 6 / 2 5 / 7 l K V ~ ~ 1 O X l 0 i S A M P L E PROBLEMm
1 '70Xv5 iPAGE 1 1 3 ) I S N 0 0 1 2 U P I T E ( b v l 0 2 ) I S V 3 0 1 3 1 3 2 F O R W A T ( l H O ~ 6 X v 8 H P ( P S 1 4 ) 1 1 3 X 1 5 i r ( F ) * B X s l O H V ( F I 3 t L 3 ) l l 5 X l l H X 1
1 1 0 x 1 1 4 H I P I D T ( P S I A / F J r 5 X r 9 H P L I P S I A J J c C - CASE 1
I S U 0 0 1 4 L = 1 I S N 0 0 1 5 T = 250.00 I S V 0 0 1 6 X = .3DO I S N 0 0 1 7 PL = 0.00 I S N 0 0 1 8 5 CALL C O N U ( ~ ~ C E I T ~ - ~ ) I S V 0 0 1 9 CALL P S A T K ( G s l s 1 ) I S N 0 0 2 0 P = ; ( I ) I S N 0 0 2 1 DPDT = G ( Z ) * C E ( l ) / C E ( Z ) I S Y 3 0 2 2 I F ( L.EQ.2 I GO TO 1 0 I S N 0 0 2 4 CALL GIBBAB(G,P1T11,1) I S Y 0 0 2 5 -VF = G ( 2 ) I S N 0 0 2 6 CALL G I B B P B ( G , P ~ T 1 1 1 2 ) I S N 0 0 2 7 VG = G ( 2 ) I S V o o z e GO TJ 1s I S N 0 0 2 9 1 3 CALL R O O T ( V F , P V T ~ U F ~ S F ~ Q ~ N O G O ) I S Y 0 0 3 0 CALL R O O r ~ V G ~ P ~ T ~ U G ~ S G ~ 3 t N O G O I I S Y 0 0 3 1 CALL PSATL~PLITIO) I S N 0 0 3 2 1 5 V = X*(VG-VF) + VF I S V 0 0 3 3 CALL C O N U ( 4 1 C E t P t l r T r 6 1 V ~ 3 t P L t l l I S N 9 0 3 4 W R I T E ( 5 1 1 0 3 ) P ~ T ~ V ~ X I O P O T * P L I S N 0 0 3 5 1 0 3 FORMAT(1H r6E18.8) I S N 0 0 3 6 GO T I ( 2 0 1 3 0 ) l L
c C CASE 2
I S V 0 0 3 7 2 0 L . 2 I S N 0 0 3 8 I S M 0 0 3 9
I S V 0 0 0 0 1SN 0 0 4 1 I S q 0 0 4 2 I S Y 0 0 4 3 I S N 0 0 4 4 I S N 0 0 4 5 I S N 0 0 4 6
. I S N 0 0 4 7
. c C CASE 3
3 0 P = 900.00 T = 153.00 CALL C O N U ( Z ~ C E * P ~ - I s T s - 6 ) CALL G I B B A B ( G s P ~ T , 9 1 1 ) G.1 = G ( 1 ) v = ; (2 ) GK = - ~ ( 3 ) / ~ ( 2 ) I = I N O E X ( l 1 1 0 1 Z 1 1 0 1 I O )
I S N 0 0 4 8 I S Y 0 0 4 9 I S N 0 0 5 0 I S V 0 0 5 1 I S V 0 0 5 2 I S M 3 0 5 3 I S N 0 0 5 4 I S N 0 0 5 5
I S N 0 0 5 6 I S N 0 0 5 7 I S N 0 0 5 8 I S Y 0 0 5 9 I S N 00'50 I SY 0 0 6 1 I S N 0 0 6 2 I S M 0 0 6 3 I SW 0 0 6 4 I S N 0 0 6 5 I S N 0 0 5 6 1SY 0 0 5 7 I S N 0 0 6 8 I S Y 0 0 5 9 I S Y 0 0 7 0 I S N 0 0 7 1 I S V 0 0 7 2
S = - G ( I ) €3 = ; ( I + l ) / G ( 2 ) I = INDEX( l r l O r 3 r l O r I D ) CP = - T * G ( I AS = DSQRTt CP*CE( lO) / (ZP*GK/V - T * B * B ) ) CALL CONU(8rCErGIr4rVr3rGKr-lrSr 5 r B r - 2 r C P r 5 r P r 1 9 T r 6 1 W R l T E ( 6 r 1 0 4 ) P r T r G I r V r G K r S r B r C P r A S
1 3 5 FORHAT( ~'HOI~HP = r E 1 8 o 8 9 5 H P S I A ~ 1 3 X p 3 4 1 ' = r E l $ o B r 2 H FpL6X*34 ; =r 1 E l 8 . B r 7H B T U / L B / l H r 3 H V = rE19 .8 r 7 H F T 3 / L B r l l X r 3 H < =rE18.81 2 7H P S I A - l r l l X r 3 H S = , E 1 8 * 8 r 9 H B T U / L B - F / l H 93HB = r E l 8 - 8 , 4 H , F - l r 3 ~ Q X , ~ H C P Z ~ ~ 1 8 . 8 ~ 9 ~ BTU/LB-Fr 9 x 1 3H4S=r E 18.8r 7Y F r f S E C )
1 0 5 FORMAT( l H O r l O X r 3 3 H G I B B S ARRAY I N ASME REDUCED U N I T S ) W R I T E ( 5 r 1 0 5 ) CALL WRITEA(Gr 1 0 1 1 0 1 - 1 )
1 0 6 FO%MbT( l H O r l O X r 4 3 H S P E C l F l C VOLUME ARRAY I N ASME REDJCED UY I T S ) U R I T E ( b r l O 6 ) CALL C O N A ( G r 9 ~ G V r l r O r l o D O r l o D ~ r 1 - 0 0 ) CALL W Q I T E A ( S V r S r 9 r - 1 )
1 3 7 FORMAT(lHOrlOXr38HSPECIFIC VOLJNE AQ%AY I N E N Z L I S i U h I T S ) W R I T E ( 6 r 1 0 7 ) CALL C O N A ( G r 9 r G V r l r O r C E ( 3 ) p C E I 1 J r C E ( 2 ) 1 CALL W R I T E A ( G V r 9 r 9 r - 1 )
1 0 8 FORM4T(liOrlOXr34HSPECIFIC VOLUME ARRAY I N MKS U N I T S 1 W R I l E ( 6 r 1 0 8 ) CALL C O N A I G v 9 r G V r l r O r C M ( 3 ) r C H ( 2 ) ) CALL W R I T E A ( G V r 9 r 9 r - 1 ) STOP E NO
ASrEMIMOD02~06/25/71KVM SAMPLE PROBLEM
P = 0.40000000D 03 PSIA V = 0.163223530-01 FT3 ILB 8 = 0.311602050-03 F-1
1 = 0.150000000 0 3 F . G = -0 .119389880 02 BTUlLB K = 0.31038922D-0'5 PSTA-1 S = 0.21463864D DO RTUIL8-F CP= 0.998720700 0 0 BTUILB-F AS= 0.507957790 0 4 FTISEC
G I B l S AaRAY I N ASME PEOUCED UNITS J I 1 1 2 3 4 5 5 7 0 9 1 0 1 -3.9bO39D-01 3.21442)-01-3.207110-03 2.722289-09-6.01519D-05 1.857690-05-7.704890-05 4.050260-06-2.587710-06 1.955810-06 2 -8 .295579 00 1.16703)-31-5.334170-03 1.267481-03-3.737123-04 1.407300-00-7.291000-55 4.615333-05-3.469650-05 3 -7 .376880 01 8.51533)-01-1.031200-01 1.294670-02-4.555260-03 2.493000-03-1.429900-IYJ 9-957230-04 4 1 .349450 02-2.91838) 0 0 5 .857610-01 1.595299-01-8.944270-02 3.145010-02-2.294680-02 5 -6 .156115 02 5.99705) 31-2.664500 0 1 1.438860 0 1 3.132200-01 0.385140-01 6 2 .418580 03-3.24786) 0 2 4.717410 0 2 3.214160 01-8.948240 0 1 7 -5 .131970 09 3 .923943 '04 -1 .307200 0 9 1.201240 03 8 2.830380 06-6.897181 0 6 2.278880 05 9 -7 .493870 07 5.79291) 0 8
10 3 . 9 0 1 4 2 0 0 9
SPECIFIC VOLUME 4 R R A Y I N ASME REDUCED UNITS J f I 1 2 3 4 5 5 7 9 9 1 3.21442)-01-3.20711)-03 2.722280-04-6.015190-35 1.857690-05-7.704890-06 4.050260-06-2.587711-06 1.955810-06 2 1.16703)-01-5.33417)-03 1.267480-03-3.707120-39 1.43730D-04-7,29103D-05 4.61593D-05-3.469650-05 3 , 8.515330-01-1.031200-01 1.294670-02-4.555260-03 2.490800-03-1.42990D-03 9.957230-0a 4 -2.91838) 0 0 5 .857611-01 1.595290-01-9.864270-02 3.145010-02-2.29458D-32 5 6.997050 01-2.664509 0 1 1.438860 0 0 3.192200-01 8.385140-01 6 -3 .247850 02 4.717413 0 2 3.21415D 01-9.948240 0 1 7 3.923940 04-1.30728) 0 4 1 .201240 03 8 -5 .891180 06 2.278883 05 9 5 .792910 08 t
SPECIFIZ VOLUME 4RRAY I N ENGLISH UNITS I 1 2 3 4 5 6 7 8 9
1.632240-02-5.07608):08 1.343010-12-9.249770-17 0.914ObD-21-1.15111D-24 1.886100-20-3.756060-32 8.848650-36 5.08b090-05-7.24607)-11 5.366730-15-0.892600-19 5.789270-23-9.348050-27 1.844070-30-4.322330-34 3.185120-08-1.202263-12 4.704900-17-5.159870-21 8.734240-25-1.573620-28 3.415590-32
-9.3688bD-11 5.861383-15 4.975690-19-8.598240-23 9.530230-27-2.157390-30 1.927890-12-2.288330-16 3.851120-21 2.663540-25 2.100790-28
-7 .680460-15 3.477181-19 7.38457D-23-6.408090-25 7.964050-16-8.270161-20 2.368690-24
-1.201450-15 1 .?37343-21 8.660690- 18
SPECIFIC VOLUME 4RRAY I N MKS UNIYS 1 1 2 3 4 5 6 7 8 9
1.018970-03-0.596099-13 1.763693-21-1.761790-29 2.459750-37-Q.617110-45 1.09605D-52-3.16577)-60 1 .081690-67 5.715240-07-1.18096)-15 1 . 2 6 8 6 0 0 - 2 3 - 1 . 6 7 7 3 Q M 2.878720-39-6.74241D-47 1.929760-54-6.557570-62 5.44242D-03-3.5269Q)-17 2.001070-25-3.18424D-33 7.871310-41-2.042810-48 6.430960-5b
-3 .411010-11 3.09512D-19 3.810760-27-9.55101D-35 1.535410-42-5.06452D-50 1.263430-12-2.175043-20 5.309890-29 5.32564D-37 6.324220-44
-9.060020-15 5.949083-22 1.832440-30-2.306290-37 1.691020-15-2.596883-23 1.05800D-31
-4.59189D-16 6.858943-25 5.958150-17
VII. REFERENCES
1. C. A. Meyer, R . B. McClintock, G. J . S i l v e s t r i , R. C . Spencer, J r . , 1967 ASME Steam ~ a b l e s -- Thermodynamic and Transport Proper t ies of Steam, New York: The American Society of Mechanical Engineers (1967).
2. P. W. Bridgmen, The Thermodynamics of E l e c t r i c a l Phenomena i n Metals and a Condensed Collect ion of Thermodynamic Formulas, New York: Dover Publ ica t ions , Inc. (1961).