Overview of Mechanical Engineering for Non-MEs Part 1: Statics. 4 Rigid Bodies II: Equilibrium.
The MES Code: Chemical-Equilibrium Detonation-Product ...
Transcript of The MES Code: Chemical-Equilibrium Detonation-Product ...
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LA-6250 CIC-14 REPORT COMCTION
(L3REPRODUCTION
● COPY
UC-45
Issued: December 1976
The MES Code: Chemical-Equilibrium
Detonation-Product States of Condensed Explosives
by
Wildon Fickett
-—--II
@
:
10s alamosscientific laboratory
of the University of CaliforniaLOS ALAMOS, NEW MEXICO 87545
An Affirmative Action/Equal Opportunity Employer
uNITED sTATES
ENERGY RESEARCH AND DEVELOPMENT ADMINISTRATION
CONTRACT W-740S-ENG. 36
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CONTENTS
Units and Energy Zero ● 00.0.0.0.0 ● ..........● ,.....00....000...... ● .....0.00.
Terms ..,,..,.0. .,0,,...., .............................0.....................
Symbol S . *,...**.. . .,.0..,,. .,, . . . ...0 . . . . . . . . . . . . . . . . . . . . . ..0...... . . . . ...0.
I.
II,
III.
INTRODUCTION ..0.,..s,. ● 0.,....0, ................0... ..............0...
INPUT .............*...*...* .0..0.0000. ..,0...,0.. ..000...... ...*......
OVERVIEW .............0......00 ● ...s000000...........● ,,*....**. ....*..
A. State Point .......0.....................................● .........
B. Components of the State-Point Calculation .............*...... .....
10 Ideal Thermodynamic Functions .0.00...,. ● ,0.0.000, ............02. Solid Equation of State ............0.......00.....0..... ● ....*
Gas Equation of State ..................00...0.....0 ● ....0.... .:: Mixture ...............0...0 ..............00.... ● .......0......5. Chemical Equilibrium ......................................0...6. Derivatives ● 00....0... ● ...000.... ...0....0.. .......*... .......
c. Controls for Particular Loci .......................0.00... ........
1. Contours of Constant T,v, s, ore ............................2. Detonation Hugoniot .0..0..0... ....00..... ● ....................3* Chapman-Jouguet Locus .0.,...,... ....00...0. .........0.....0..0
EQUATIONS ................0......00s.....0.... ● .......0.,00............
A.
B.
r
7 ‘— E.
F.
Ideal Thermodynamic Functions ● .0....... .....*.... .................
Solid Equation of State .0..0,.0.0 ● 0.0...... .....0...0 .,00..00.. ...
Gas Equation of State ● ..............0.... .........................
1. Ideal Gas ..............0..0.....0... ...............0.. ● ....0..2. LJD (Lennard-Jones-Devonshire Cell Theory) ...*.......,........3. KW (Kistiakowsky-Wilson) ........................● .....● .... ● ..
Mixture .......,,..● ● ,..,.,.● .● ..0● ,● .● ,● ,...,● ● ● ● .,● ● .............
1. Ideal Mixing ● ● o,● ..0.● ...,● .● ● .● ...● ● ..,..,● .● ● ....*,..● ..● ...2. LH Mixing ...,,● ● .● ● ..● 0● ,● ● *,....● ..● ● ...● .* ● ...● .....0......●
CS Mixing ● .....● ● ,....● ● ...........0.................● ● ....● ..:: One-Fluid Mixing ....● .......● *...● .● .● ...● ● ..● ● ....,.● ..● .● 0..
Chemical Equilibrium .,,...................● ..................● ....
1. Initial Calculation ....● ..................................,.*.2, Main Calculation ...........● ......● *....● .,...,,● .● ...* ● ● ....●
Miscellaneous ...,.0.0,,.● ● ● ...● ● ● ● .*..0....,,● ,........● ....● ● ● ● o●
v. PROGRAM ...........● .● ..............● .● ....● ● .....,..● ..● ......0.......
A. Equilibrium Iteration ..,.● ...● ● ,.● ,..0● .*.....,................● ..
B. Iterations ....● ...............0● .....● ● ● ..0● ● O.0.● ● .● ● *.,......*. ●
iv
iv
v
1
2
7
7
8
8810111113
13
131414
14
14
16
19
191926
27
27272829
30
3032
34
35
35
38
...777
CONTENTS (contd)
c. LJD Gas EOS Integration ,.0..0.,00. .00.0.0,0,, SOO,.*.O.S. ● ..00.00..
D. FROOT-Solve f(x) =0 by Iteration .................................
VI. SAMPLE IIsIPUT/OUTPUTAND TEST ● ...● ,● ● ● ● a● ● ..● .,● ..s...● c.● ● .● ● s.● ● ● ..s.
APPENDIX: A. CHEMICAL EQUILIBRIUM EXAMPLE .*.● ....,.........● ..............
B. QUASI-STATIC WORK .● .s..● s.● ..6s.,..● ,● .,● ● ....● .● ● ...● ....● ..
c. PROGRAM ..● .,.....● ,● ● .....● .● ,..*..0.● ...● .,..● ..● 0.● ..● ..● ..
40
41
45
54
56
59
D. COMMON STORE .,.● ,..● ● ..● ..● ......● ..s.● ,...● .s● ..● .● ● .● ● ....● 117
iv
Units and Energy Zero
The primary units are centimeters, grams, microseconds,K (pressure in Mbar, specific energy in Mbar - cm3/g), butthe mole and the system mass M. (specified by the user) arealso used as the mass unit. The energy unit kcal, prevalentin the literature at the time the code was made, is used fora few input and output quantities.
Energies are relative to elements in their standardstates (as defined by the NBS) at T = O. For elements whichare gaseous under ordinary conditions, such as oxygen, thetypical standard state is the molecular form (02) in the hy-pothetical ideal gas state at a pressure of one atmosphere.For elements which are solid under ordinary conditions, suchas carbon, the typical standard state is the most commoncrystalline form (graphite) at zero pressure.
Terms
Contour - Locus of constant T, e, v, or s
CJ - Chapman-Jouguet
Cs - Nonformal -solution mixture rule
EOS - Equation of state
Ideal (part) - Translation plus internal part of the EOS(Sees. 111.B and IV.C)
Imperfection (part) - Configurational part of the EOS (Sees.111.B and IV.C)
KW - Kistiakowsky-Wilson (EOS)
LJD - Lennard-Jones-Devonshire (EOS)
LH - Longuet-Higgins mixture rule
Lattice (in LJD) - The cold (T= O) state with all moleculesfixed on their lattice sites
NBS - National Bureau of Standards
Pure-Fluid - Pure species (gas EOS)
Unreacted Material - The unreacted (in general metastable,as in an explosive) material in theinitial state (po, To)
v
Symbol S
Intensive thermodynamic functions
p, T, v - pressure, temperature, chemical potential
Extensive thermodynamic functions
V, E, H, A,F,S,C Cv-P’
v, e, h, a, f, s, c Cv -P’
Composition
molar volume, internal energy, en-thalpy, Helmholtz free energy,Gibbs free energy, entropy, constant-pressure and constant-volume heatcapacities
corresponding s ecific quantitiesf(per unit mass) equal to (n/Mo)
times molar]
n- total number of moles per !10grams of system
‘9- number of moles of gas per !10grams of system
ns - number of moles of solid per !10grams of system
n. -1 number of moles of species i per No grams of system
Xi - mole fraction of species i in the gas phase = ni/n9
xs - ns/n
Intermolecular potential
r*, T* - minimum-energy radius and well depth
v* = (N/fi)(r*)3 - molar volume of fcc lattice with moleculesat distance r*
n,m- repulsive and attractive indices
Chemical equilibrium calculation
c-
s-
y
cY.-~
~-
;-
Other
D-
A-
number of elements
number of species
empirical-formula coefficients
species-formulae coefficients
renumbering
supersaturation index
detonation velocity
AHf (NBs)
vi
Symbols (cent)
Other (cent)
F- “free energy” for equilibriumgas: Fi = pi-l?T Ln xi (Sec. ~
y - (-~ !Lnp/~ Rn v)s - adiabatic
r- v(~p/~e)v --Gri.ineisencoeffic
fi-[Ho(T) - H~j (NBs)
MO - system mass
constants; solid: ~s = Fs;V.E)
exponent
ent
N-
cl-
R-
P-
z-
Avogadro’s number
heat of reaction, Sec. IV.F
molar gas constant
density
sum over all species
‘9- sum over gas species
u-- particle velocity
z- pV/RT
Superscript or over
i-
8
*-
-..
ideal
imperfection
NBS tabular function (translation plus internal)
average
(1) function (as distinguished from a variable) e.g. ~(T,v),used only when needed to make this distinction
(2) decoration, as in ;
Subscript or under
i-
o-
9-
s-
!L-
H-
species index
unreacted material in the initial state
gas
solid
lattice (cold curve) function for the LJD EOS
Hugoniot
other subscripts - partial derivative
-- vector
matrix2
vii
THE MES CODE: CHEMICAL-EQUILIBRIUM
DETONATION-PRODUCT STATES OF CONDENSED EXPLOSIVES
by
Wildon Fickett
ABSTRACT
The MES code calculates states of the reaction products ofdetonation in gaseous or condensed explosives under the assump-tion of thermal and chemical equilibrium. The products may con-sist of any number of gaseous species and one solid species.The condition of equilibrium includes the number of phases: thesolid may or may not be present depending on the current state.In addition to the primary calculation of the Chapman-Jouguetstate at a specified set of initial densities, the detonationHugoniot, and contours of constant temperature, density, energy,or entropy, each at a specified set of pressures may be obtain-ed. All of the first derivatives (e.g., sound speed and heatcapacity) are calculated at each point.
The solid equation of state is one constructed from a givenshock Hugoniot under the assumption of constant Gruneisen coef-ficient. For the gas, either the ideal gas, the Kistiakowsky-Wilson, or the LJD (Lennard-,Jones-Devonshire cell theory) equa-tion of state may be used. Several choices of “mixture rules”for extending the last one, a pure-fluid equation of state, tomixtures are offered. For LJD, the input data are the parame-ters defining
1. INTRODUCTION
the intermolecular potentials of the species.
The MES code, which performs the calculations described in Refs. 1 and 2,
was made some years ago, became dormant when the IBM 7094 became obsolete, and
has just been reactivated “as is” in response to a request. I make no apology
for things I would now do differently. The only significant changes made in the
code are: (1) the replacement of the original machine-language equilibrium rou-
tine with an equivalent FORTRAN version, (2) the replacement of several relatively
small machine-language 1/0 routines by approximately equivalent FORTRAN versions,
and (3) a new scheme of diagnostic printing.1
II. INPUT
The data are entered in packs, each pack preceeded by a CON card. The CON
cards have 2A6 format; only the first two fields are data; the user may enter
comments on the rest of the card. The first two fields, each left justified, are
the word CON and the pack name. Each pack consists of one or more strings, num- -
bered sequentially in the write-up; each string consists of one or more cards.
Strings are described by listing their fields; alphameric items are underlined.
A few strings, such as the table of initial densities for the CJ locus, and ini-k
tial pressures for the Hugoniot and contours, are of indefinite length; these
must be terminated by a zero. Most of the formats are 6E12.7. A few are 1216 or
12A6; these are marked (I) or (A).
Data for a job (a batch-type submission) are divided into runs, Each run
begins with a CON, PAS card and ends with a CON, REND card. The job ends with a—— ——CON, JEND card following the last CON, REND. Each run has two parts. Part 1——(preliminary), beginning with CON,=S, generally enters parameters of the con-——stitutive relations. In Part 2, beginning with the first CON, SAN card, the——packs CJ, TED, and PV each specify a complete calculation to be performed immedi-—_ —ately after the pack is read; the user may often want to have more than one of
these packs within a run, as well as additional SAM packs. With a few obvious
exceptions, the packs of either part may be entered in any order.
The input program reads one pack at a time. It expects to find a fixed num-
ber of strings in each pack (even though, under some options, not all of them are
used). If the first card of a pack is not a CON card (usually because the user
has gotten the wrong number of strings in the previous pack), the program reports
an error and skips to the next CON, PAS card and begins reading there. In de-——scribing the input data, the phrase “not used” means that the data of the field
or string in question is not-used by the pro~ram under the specified conditions.
It will, nevertheless, be read by the program; the user may enter anything, but a
blank field(s) is suggested. The only exception is that if an entire pack is not
used, it may be omitted.
.Remarks
1. Alphameric constants in the input are given in caps, underlined.
2. Except for the equilibrium calculation, which has its own special order, the
first slot of all composition arrays is the number of moles of solid.
3. There are no prestored defaults. Many standard values must be supplied by
the user under CON, FOB; the standard pack is given in the sample input,——
2
Sec. VI. CON, WIT, g.~., does have a mechanism for preserving previously——entered items when it reappears.
A. Part 1●CON, PAS, run label - begin Part 1. The run label becomes part of the standard——output header.
. ●CON, blank, comment.——.CON, SWIT - switches.——
u (1) cliff,fixdiff, gas, solid, mix, eq, CJ, PV, PVC.
cliff - 0 -
1-
fixdiff - 0 -
1-
gas -o-
1-
9-
solid - 0-
1-
mix -o-
1-
2-
3-
4 -
eq -o-
1-
CJ -o-
1-
no action
calculate equilibrium-composition derivatives
at each point
no action
fixed-composition derivatives similarly
ideal-gas EOS
LJD EOSKMEOS
incompressible
Gruneisen EOS
none (pure fluid); omit XIP pack
ideal mixing
LH mixing
CS (conformal solution) mixing
One-Fluid mixing
fixed composition
equilibrium composition
equilibrium CJ condition
frozen CJ condition
Pv - 1, 2, 3, 4 for T, v, s, e constant on contours
(see CON, PV)——Pvc -0 - no action
1- under CON, PV_, use last T, p from previous cal-.
culation instead of input Tc, pc
Two of the switches are set (or reset) elsewhere: the mix switch under CON,*
XIP and the PVC switch under CON, PV.——
This pack may also be entered one or more times in Part 2 to change switch
settings between calculations. A negative item means: “don’t store this item”
[use currently stored (last previously entered)] value.
●CON, TIp - ideal thermodynamic functions——1. (I) number of species, degree of fit n,
2. Tmin, Tmax,
3. ao, al, a2, .... an, d, AH;, AHf(To), [Ho(To)-H~]/RTo.
Polynomial fit coefficients for [HO(T)-H~]/RT, enthalpy integration constant
d, heats of formation, and enthalpy at To, One such string for each species.
.CON, SEp - solid E@S——1. r, Cp/R, a, Vo, To, Eo/RTo.
Gruneisen coefficient, heat capacity, thermal-expansion coefficient (K-l),
initial volume (cm3/mole), initial temperature (K), initial energy. For in-
compressible solid, only V. is used.
2. blank, co, c1’ C2’ C3’ C4”Hugoniot fit coefficients: ‘H(v) ‘~ci(v/vo)i
i=0
●CON, GEP - gas EOS (omit pack for ideal gas), LJD EOS——1;(1) potential index (1, 2, 3 for LJ, MCM, MR),
2. n, m, An, Am, r*, T*.
repulsive/attractive exponents, multipliers, well radius r* (A = 10-lo
m),*
well depth T* (K); r* and T not used for a mixture.
●KijEOS——1. (1)9.
20 a, B, e.
●CON, XIP - Mixture (omit if SWITmix = O), LJD EOS.— ——1. (1) number of (gas) species, type.
type - 1, 2, 3, 4 for ideal, LH, CS, One-Fluid;
2. Sr,**
ST, rr, Tr, n, m.
S,sr T - scale factors: multiply all input r* by Sr and all input T* by ST.
r~, T; - reference r*, T* (LH only).
n,m- potential n and m (one-Fluid only); ordinarily same as n, m in
CON, GEP [omit if SWITgas = O (ideal 9as)]—.3. X* - one for each (gas) species.
4“ L*- one for each (gas) species.
.
k-
4
●KI! EOS——1. (I) 9.
2, s;, ‘, K,
3. ~.
Here ki ~ [r~/(N/~)]1’3, where ki is the usual KN covolume and N is
Avogadro’s number.
4. blank.
●CON, EQP - equilibrium (omit pack for fixed composition)——1. (I)
c-
s-
P-
P’-
@-
2* ~-
3. (A)
c, S9 Ps -3 P: 4*
number of elements
number of species
number of phases minus 1; used only if @ = O
first guess for pwhen @ = 1
0: fixed number of phases (p + 1)
1: equilibrium number of phases (one or two);
system empirical-formula coefficients (number of gram-atoms of each
element).
blank, element symbols, ~, AO.—A print label; all fields right-justified.
4* & - species-formulae coefficients.
This consists of s strings, one for each species, so that each string is a
row of% The first field, format A6, is the right-justified element symbol,
and the remaining fields, format 16, are the coefficients. (For example,
with elements C, H, O, and N, the string for carbon dioxide is: C02, 1, 0,
2, o).
5. (1)~1 - species renumbering for two-phase system (@=l) or given system (@=O).
6. (I) ~“ - species renumbering for one-phase system (@=l); not used for $=0.
Dimensions are all fixed once c and s are given:
(J(C), CJ(s x c), J(c), ~“(c) .
In~the species may be listed in any order. Thea’s specify renumbering. If k
is the number of a species as originally entered in~ then its new number is ak.
The ak must be chosen so that after renumbering the following conditions are sat-
isfied: the formula coefficients of the first c species must be linearly
5
independent; the danger of convergence failure will be minimized by choosing for
them those species expected to be present in largest amounts. Two special re-
quirements simplify the program: For @=O, p=l (two phases), the solid species
must be number c. For $=1, the user must supply two possible systems: Two phase
(solid present) andone phase (solid absent); the program chooses the correct one
at each T and p, For the two-phase system the solid must be number c. For the
one-phase system it must be (nominally) present as number s; here the program
assigns it a large free energy so that its calculated mole fraction is negligibly
small .
For details and examples see Sees. IV.E and VI, and Appendix A.
.CON, FOB - knobs——1. FROOT E’S,
2. FROOT r’s.
3. 1/2 A M p, 1/2 AT, Souter, Einner.
4. FROOT bounds,
The FROOT items are in the order given in Sec. V.B, Table III; the bounds are
in pairs (rein,max). The A’s are the displacements for the numerical differenti-
ation, Sec. 11.B. The c’s in string 3 are for the outer and inner equilibrium
iterations, Sec. V.A.
.CON, DBIJG- print store——
Do the standard error print (mainly the entire common store) at this point.
B. Part 2.CON, SAM, material label—. - initial state and begin Part 2
1. Po(9/cm), Po(Mbar), TO(K), Me(g), AHf(To) (kcal/mcl).
F’!. - system mass; must agree with empirical formula under CON, EQP.——
AHf(To) - enthalpy of formation of unreacted material at po, To relative to
elements in standard states at To.
2. &- number of moles of each species (for system of !10grams). First field is
for solid. For equilibrium composition, these are guesses for the first
iteration.
We have picked To for the heat of formation because AHf(To) is the quantity
usually listed for explosives, and because it is needed for the calculation of
the heat of reaction q as usually defined (Sec. IV.F). An alternative, which may
be more convenient for some cryogenic materials, is to enter in place of AHf(To)
the enthalpy of formation at To from elements at T=O. If this is done, the CON,
~ input must also be changed by entering AH? in place of AHf(To) and setting
.
v
6
[HO(To) - H~]/RTo for each species to zero. This has the advantage of making
the TIP input simpler and independent of To. The disadvantage is that the value
of q printed out will have a small error (which does not affect on any other cal-
culated quantity).
.CON, TED - detonation Hugoniot at given~—.
~ - pressure table.
●CON, pv - contour of constant T, v, s, or e at given p——1. (I) k, PVC.
k - 1, 2, 3, 4 for constant - T, v, s, e
Pvc - PVC switch (see CON, SWIT)
2. Tc, pc - initial point~ot used if PVC # O).
3, ~- pressure table.
First calculate the point (Tc, PC), then use the value of T, v, s, or e from
this point as the constant value for the locus. Use T and p from last previously
calculated point for Tc, pc instead of input values if PVC # O,
●C(IN,CJ - CJ locus at given~o——
&o - initial-density table.
.CON, REND - end of run——
.COJ{,JEND - end of job——
111. OVERVIEW
In this section we give an overview of the problem and program, including
the principal equations. Some of these equations
definitions; Sec. IV gives the detailed equations
Section V, together with comments in the program,
scription of the program itself.
are schematic or just serve as
implemented by the program.
provides a more detailed de-
We define the term state point and some related symbols in Sec. A, the main
components of the state-point calculation in Sec. B, and the higher level part of
the program which uses the state-point calculation in Sec. C. Principal routine
names are given in parentheses with some of the section headings.
A. State Point (MEs)state point is the usual set of thermodynamic variables V, E, H, A, F, S,
and some of their derivatives, the chemical potentials pi, the mole fractions xi,
and the total number of moles n (per system mass Mo) at given T and p. Recal1
that we have at most one solid species, and the convention that it is the first
1isted.
7
The mole numbers and mole fractions are given by (see symbol sheet)
n =x n. = n +n
1 s 9
z‘9 = 9 ‘i
‘1 = ‘s ‘ns’n
‘9= rig/n
Xi = ni/ng, i>l(gas) .
Note that xi, i>l is the mole fraction in the gas phase, The composition may be.—either fixed (specified) or equilibrium (recalculated at each state point).
Because the equations defining the equilibrium state are implicit and com-
plicated, we obtain derivatives by numerical centered differencing over carefully
chosen intervals, rather than attempting to use the very lengthy analytic expres-
sions.
Routine MES calculates a state point at given T and p, using the five pack-
ages whose generic names are given in the subheadings of the next section. These
packages constitute the bulk of the program.
B. Components of the State-Point CalculationAn extensive quantity for the system is the linear mole fraction sum of those
for the two phases, e.g.,
E = xSES+XE99 “
It is convenient to separate functions like Eg and Es into ideal and imperfection
parts. The ideal part, superscript i, represents translation plus the internal
partition function; the imperfection part, denoted by a prime, represents the
configuration integral.
Descriptions of the five main components of the EOS calculation follow.
1. Ideal Thermodynamic Functions (TIM). We use superscript * to denote the
portion of the ideal part that represents the internal partition function; most
of the work is in getting this number (for the solid it is the whole value).
These functions are tabulated by the NBS and others; they are represented to the
program by polynomial fits, The tabulations refer each species to itself at
T=O, The program adds the heats of formation at T = O to refer all to the same
reference, namely elements in their standard states at T = O. The physical state
corresponding to the tabulation is the given T and p = p* (1 atm for a gas and
zero for a solid), with the stipulation
tical ideal-gas state at T and p*. For
as that for an ideal gas at the same T,
that a gaseous species is in the hypothe-
the gas phase , we define the ideal part
p, and x, so that
Ei (T, p, X) = X9 Xi E;(T) ,
.
s’ (T, p, X) = z g xi S;(T) - R Rn p/p* + R~gxi %n xi , and
~; (T, p, Xi) = F;(T) + RT In p/p* + RT klnxi .
2. Solid Equation of State (SEM). For the extensive thermodynamic functions
we have
E (T, p) =Ei(T, p=O)+E’(T, p) , etc.
With the equation of state in the often-used form p(T, V), the imperfection quan-
tities may be defined as the integrals along the isotherm:
1V(T, p)
E’ (T, p) = (T pT - p) dv ,
V(T, p = O)
1V(T, P)
S’ (T, p) = pT dV , and
V(T, p = O)
9
P’ (T, p) s F’ = E’ + pv - TS’ ,
The EOS used takes the form of equations for T and p along an isentrope through
an unknown point Tl, VI on p = O, and incorporates simple approximations to the
ideal functions on p = O, The equations can be put into a form such that for
given Tand p the point T,, VI at the foot of the isentrope can be eliminated
and the complete EOS at T, p can be obtained by iterative solution of one equa-
tion in one unknown. After this is done, the approximate ideal part is subtract-
ed to give the imperfection part (with the correct ideal part, calculated by TIM,
added later).
3. Gas Equation of State (GEM). Here we have
E (T, p, X) = Ei (T, p, X) + E’ (T, p, X) , etc.
For the usual form p(T, V) the imperfection functions are given by the integrals
along the constant-composition isotherm
1V(p, T)
E’ (T, p, X) = (T PT - p) dV and
RT/p*
JV(p, T)
S’ (T, p, X) = (PT - R/V) dV ,
RT/p*
in which the term R/V in the second integral subtracts off the ideal gas part.
AlSO
H’ = E’ + (z-l) ,
A’ =E’-TS’ ,and
F’ = ~1 -TS’ ,
10
For the chemical potentials we have
x = (nF’)n n=, ,i
where the partial derivative is at constant T, p, and n., j#i, of the gas phase.
Both of our imperfect-gas equations of state, the ~JD and the KW, have the
form p (T, V, x), and are not explicitly invertible to V (T, p, x), At given T,
p, x we must then first solve the equation
~(T, V,x) = p
(where jYon the left distinguishes the pressure
pressure on the right) ?or’V and then calculate
given T and this V.
function from the given value of
the imperfection functions at the
The KW EOS is a simple one, and includes the composition dependence. The
LJD EOS is much more complicated, requiring for its calculation the numerical
evaluation of several definite integrals. In its original form it applies to
only a single, pure species. Here it is extended to apply to the gas mixture
through one of several mixture rules, described next.
4. Plixture (XIM). The program offers several options for describing the
gas mixture. The first is the general one of ideal mixing, for which each mix-
ture property is a linear mole-fraction sum of those of the individual species
E’ (T, p, X) =G
xi E; (T, p) , etc.
This may, of course, be applied to any (pure-species) EOS.
The other mixture rules assume that the pure-species EOS is based on an in-
termolecular potential function and express the mixture properties as expansions
in the potential functions or potential-function parameters of the individual
species. In some cases the outcome of this expansion is that the mixture is rep-
resented by a fictitious fluid with a certain mean potential whose parameters de-
pend on the composition,
5. Chemical Equilibrium (EQM). The composition of the system can be ex-
pressed in terms of the progress variables of J independent reactions (ordinarily
11
J is the number of species minus the number of elements). Ne symbolize these
reactions by
zv X.=0, j=l, ....J ,ij 1
i
where Xi represents one mole of species i, so that v.. is the (molar) stoichio-lJ
metric coefficient of species i in reaction j. The equilibrium composition is
the solution of
E xi aik = Qk k 1,= .... ki
x ‘ij pi(T, p,nj)=O j=l, ..0, Ji
with the aik, the chemical formula coefficients (number of moles of element k
one mole of species i), and the ok, the empirical-formula coefficients of the
in
system (total number of moles of element k in the system; total of k elements).
The first set of equations represents mass conservation, one equation for each
element; the second represents the usual “equilibrium-constant” relations for the
reactions, The second equations are put in the form
x !Lnxi = - E Vij ~i (T, lI,&)/RT, j = 1,‘ij J
i9. . . .
with ~i defined as
Fi = pi - RT Ln xi for a gas species and
~i = pi = Fs for the solid.
The advantage of this procedure is that, for the ideal gas, the~i are independent
of x, so the x-dependence is confined to the Ln xi terms. For the real gas the~i
12
do depend on~, but this dependence is small
solved by a direct iteration method based on
gas problem: Guess x, calculate?’i for this
fixed at this value, recalculate the ~i from.
gence.
enough so that the equations can be
a procedure that solves the ideal-
.x, find the ideal gas x for the ~i
the new ~, and repeat to conver-
6. Derivatives (GAMM). The first partial derivatives (for the complete* system) are calculated by centered difference from symmetrical displacements in
!Lnp and T of carefully chosen size. Three derivatives are approximated by cen-
tered differences
CP= (ah/aT)p = Ah/AT ,
(a An p/a !tnV)T = A !Lnp/A fin v ,
(a l.nv/aT)p = A finv/AT ,
where A denotes a difference between the two symmetrically displaced points in p
or T. The remaining derivatives are then obtained from these
c /cpv = 1 + (pv/cpT) (-a Rn p/a !tnv)T [T(a !tnv/aT)p]2 ,
Y = (Cp/Cv) (-a In p/a M V)T , and
r =Y (pv/cpT) T (a En v/aT)p .
.
c. Controls for Particular LociThe rest of the program uses the (T, p) state-point routine (MES) to calcu-
late points on various thermodynamic loci. The routines are given in the section
titles.
by
is
1. Contours of Constant T, v, s, or e (PV, MESC). The locus is specified
a given value of the desired variable. Thus for constant e, for example, it
the solution of
?Y(T, p)=ec ,
with ec the given value of e, specified either as the
which the program is to calculate ec, or taken from a
Under control of PV, the program calculates points on
set of specified values of p, using MESC to calculate
values of Tc and pc at
previously calculated point,
the locus for each of the
each point.
2. Detonation Hugoniot (TED, HUG). This procedure works the same way, with
TED the control and HUG the Hugoniot-point calculator. In this case the equation
solved is the Hugoniot equation
h- ho =+(P-Po) (vO-v) ●
3. Chapman-Jouget Locus (CJ). A CJ point is located by iterative solution
of a form of the CJ condition
U+C = D ,
with u and D given by the shock conservation relations, over a set of points on
the Hugoniot, with each Hugoniot point calculated by HUG. The CJ points are de-
termined for the specified set of initial densities,
Iv. EQUATIONS
In this section we give the remaining equations in essentially the form used
by the program,
A. Ideal Thermodynamic Functions (TIM)he system 1s deflned by a matrix of constants, one row for each species.
Each row contains
ao 1,a, .... ah, d, AO, A(TO), fi(To)/RTo .
Here we have defined, in terms of the IIBSnotation
AO = AH; = AHf(0), A(TO) =AHf(To) ,
;(T) = Ho(T) - H: ,
14
where AH; and AHf(T) are the heats of formation from the elements in their stand-
ard states at T = O and T = T. The standard state is defined as the standard
form at the temperature of interest and pressure p* (1 atm for a gas, O atm for
a solid), with a gaseous form in the hypothetical ideal gas state. For elements
carbon, hydrogen. S oxygen, and nitrogen (CHON), the standard forms are solid
graphite, and gaseous H~, 02, and N~, respectively. The symbol Ho(T) - H: de-
notes the enthalpy at T relative to that of the same substance at T = O. Thew
constants ao through an are the coefficients of a polynomial fit of degree n to
8(T)/RT, and d is the integration constant for the entropy. Recalling our con-
vention that the solid always be species 1, we define
6.=011 if i # 1 and
6.=111 ifi = 1 .
With i the row (species) index, and j the column index, we have
H~/RT = ~ aijTj + A~/RT ,
j=o
</RT=~(j+l)a Tji.i 9
j =0
nS~/R = aio lnT+~ [(j+l)/j]aijTj+di - (ail - 1) fm p/p* , and
j=l
F;/RT = H:/RT - S~/R .
15
All these
temperature is
The ideal
quantities are relative to elements at T = Of (entropy at this
zero for all species).
functions for the gas mixture are then
Hj/RT = ~ xi H~/RT9
S;/R = ~gxi S~/R -~ xi In xi, etc.9
The mixture heat of formation from elements at To is
A9(TO) = X9 Xi Ai(To) .
The total free energy of the gas is not needed, but the F; of the individual spe-
cies are used in calculating those Fi that determine the equilibrium composition.
The program uses the fits only over the range of Tmin to Tmax from the input.
Outside this range, the results are an extrapolation using the assumption that the
heat capacity is constant at its boundary value.
B. Solid Equation of State (SEM)The solid EOS is constructed from a reference curve and the assumption of a
constant Griineisen coefficient to get off the curve, Defining y = V/V. (with V.
here the normal volume of the solid at p = O), the reference curve (see Fig. 1)
is the shock Hugoniot pH(y) for y < 1 and the p = O line for Y > 1. Ne assume a
simple form for the ideal functions on p = O, use it to calculate the complete
‘The Hugoniot calculation uses the enthalpy relative to elements at T ; with thisreference state indicated by explicit inclusion of the reference temperature Toas a parameter (together with argument T), the quantity supplied by TIM for thiscalculation is
nH~(T;To)/RT = ~ aijTj - (To/T) [fii(To)/RTo] + Ai(To)/RTo .
j=0
Note that for the alternate SAM input (See.11.B), this becomes identical to H~/RTabove, i.e., the reference t~erature becomes zero.
16
.
EOS, and then subtract it to get the imperfection part. Using superscript I todenote the isentrope through point Y1 on P = U and superscript o to denote func-
tions on p = O, we have for this isentrope
P1 = P1 (Y; Yll and
T1[ 1=TrY; YIS TO(Y1) .
On the reference curve p = O we take
To = To (Y) = (Y - 1)/~
EO = EO (To) = Cp (T” - To)
with constant heat capacity Cp and thermal expansion coefficient a. blithconstant
Grtineisen coefficient r we find
E/V. = (w) P + 9 (YI ,
9(Y) = P~ (Y) [1/2(1 -Y) -m] +Eo fory<l, and
9(Y) = ~-’Cp (Y - 1) fory>l .
For pl and T1 we find
P1(Y; YJ -r= ~y-(T+l) ‘y(r+l) g’(y)dy and
‘1
T1(Y; Y,) = TO(Y1) (Y,/Y)r .
17
If we represent the reference Hugoniot by a power series
n
E
.pH(y) = ai Y1
i=0
with coefficients chosen such that
n
i=o
(to make the result simpler), the integral can be done analytically and we find
P1 = PI(Y) = (Cp/do) [r/(r-l ] [(Y1/Y)r+’ - 1] fory>l,
P1 = p2(y)+p1 (l) fory~l ,
P*(Y) = - (1’/y)g(y)+ (r2/yr+l) [I(y) - I(1)] , and
II(Y) = yr-lg(y) ~y
= i [; g+ (y 1‘+i-l ) - (++;) (*T) (yr+i+l-l ) ,i=o
To get the EOS for given p and T, wemust eliminate (by iterative solution)
y, from pl =pandT1=T.
.Computation
Given p, T, the function defining y is
T1(Y)/T - 1=0.
18
T1(y) is calculated (fory< 1) by first calculating P2(Y), then solving Pi(Y) =
p foryl to get
v(r+l) A = (Cp/a Vo) [r/(r+l)]Y, = p + (P-P2(Y))/A] ,
.
and then calculating T1 from the equation above. When y, is close to 1 (the*
usual case) the I?HSis expanded in a binomial expansion
Y1 = l+kx+@(-1))(2+..o
k = l/(r+l)
x ‘ [@4’A●
Fory>l, yl is
Y1 = Y (1 +P/A)l/(r+l) ,
with expansion
Y1 [=yl+kx+;k(k-l)xz +...1
x= p/A .
c. Gas Equation of State (GES)
1. Ideal Gas. The EOS is
. V = RT/p ,
+ and the imperfection quantities are all zero.
2. LJD (Lennard-Jones-Devonshire Cell Theory). This EOS is based on an in-
termolecular potential u(r) (see Fig. 2) with well depth kT* (k is Boltzmann’s
constant) and radius r*. We define a reduced temperature and volume
19
P
i Y,Y
Fig. 1. Solid Hugoniot and isentrope.
u
.
Fig. 2. Intermolecular potential fOrLJD EOS.
0 = T/T*, -r= v/v*, v* = 2-1/2 ~(r*)3 ,
with V* the volume of an fcc lattice with intermolecular se~aration r* (N is
Avogadro’s number). The EOS is a sum of two parts: A lattice (T = O) part, with
all molecules fixed on the sites of a regular fcc lattice, and a thermal part,
for which the partition function is approximated by a cell integral, in which one
molecule moves while all its neighbors remain fixed on their lattice sites.
We first define the lattice and cell-integral functions. We define x as
minus the reduced lattice energy
x= -+Z u (T1’3 r*)
with Z the coordination number of the lattice for which we use 12, the fcc
value, The cell-integral function G(y) is defined as
Jb*
-W(T,X)/f3 ~x, b*9(Y) = yx2e = 0.55267
.
.
0
G(y) = g(y)/g(l) .
20
Here the integration variable x is the distance from the cell center in units of
the nearest neighbor distance at the given T, b* is the distance of the cell
boundary from the center, and Idis the cell potential relative to its value at
the cell center, the potential of a molecule at a given distance from the cell
center, in the field of its 12 nearest neighbors smeared into a uniform spherical
shell (at the nearest neighbor distance). The cell potential is given by the
spherical-smoothing integral
W(x,-r) = )xX1[U(T1/3 r* x,) - ~(T1/3 r*)] dx, .
For
the
1-x
the two types of potential-function terms we
smoothing integral can be done analytically,
consider (power and exponential)
but the cell integral cannot.
The cell integrals come from the canonical ensemble, with independent vari-
ables T and V. The natural imperfection quantities are thus those relative to
ideal gas at the same T and V. Elsewhere, we have used imperfection quantities
relative to the ideal gas at the same T and p. The two are the same for all
except the entropy and free energies, which are related by
S’(T,p)/R = S’(T,V) + !Lnz ,
A’(T,p)/RT = A’(T,V)/RT - !tnz, and
F’(T,p)/RT = F’(T,V)/RT - Ln z ,
with the argument p or V denoting imperfection with respect to ideal gas at the
same p or V, respectively. In the following, we mark A’ with respect to V by
argument V (we will refer to it later); all quantities without an argument are
with respect to p.
The imperfection thermodynamic functions are
z = pV/RT=l+e ‘1 [T~T - G(TWT)] Y
E’/RT = e-’[-x+ G(W)] ,
21
H’/RT = E’/RT+ (Z-1) ,
A’(T,V)/RT = 1 - X/e - log 2m2“2g(l) ,
A’/RT = A’(T,V)/RT - Pm z ,
F’/RT = A’/RT+ z-l , and
S’/R = E’/RT - A’/RT .
The derivatives, with partialswritteri for p(T,V), E’(T,V) [and z(T,13), E’(-r,13)]
are
(V/R)PT = (Ze)e = l-e-2 ~(w~w=) - G(W) G (TWT)]s
(V2/RT)pv = TZT - z = ~(@T/e
+e H)‘2 G2 N= g(1)- “FwT) - ‘(PWT)T)] ‘ and
C~/R ~ R-l E’T = (E’/RT)e = e-2 [G(W2) - G2(w)] .
The potential functions may be written in the form
[ 1u(r)/kT* = (n-m)-l mf(r,n) + nf(r,m)
f(r,q) = (r*/r)q or eq(l-r/r*), (q = n or m) .
The three potential forms are
Potential Repulsive Term Attractive Term
LJ power power
MCM exponential exponential
MR exponential power
22
.
.
.
Y
(The symbols sand (3have been previously used instead of n andm for the expo-
nential form). To write x and W, define coefficients a for X, and b for W for
repulsive and attractive terms as follows:
Coefficients Repulsive Attractive
a = -1/2 AnZn/(n-m) -1/2 AmZm/(n-m)
b = Zn/(n-m) 6Zm/(n-m)
6= 0, lforAm=O, Am#0
Here the constants An and Am are the Madelung lattice constants for a power-law
term and may be regarded as an empirical multiplier for an exponential term.
They may also be regarded as switches: An # O, Am = O gives a one-term (repul-
sive-only) potential. For a power-law term, the proper value of An or Am takes
into exact account the contributions of all neighbors for the lattice contribu-
tion; to do this for the exponential form a volume-dependent function would be
required. For the b-coefficients, used in the cell integrals, only nearest neigh-
bors are considered, so that the A’s do not appear.
The functions x and W and their derivatives, are, like the potential, a sum
of a repulsive and an attractive term. Using the coefficients a and b defined
above, we write the expressions for one term for each of the two forms.
a. Power Form.
x= at-q
-CXT = -(q/S) x
T(T)(T)T= (q2/9)x
w = bt-q I(x,q)
TWT = -(q/3) w
23
T(TW)TT
= (q2/9) w
!Z(x,q) = [*(q-*, X]-l [(,-X)-(W2) - (,+x)-(q-Z)] -,
m
= E c. x2ii=l ‘
c.1
= (q-2+2i)!/[(q-2)! (2i+l)!]
Ci+l = ci[(q+2i)(q+2i-1)]/[(2i+3)(2i+2)]
The series is used for computation.
b. Exponential Form.
1/3s = q-r
% =-(s/3) x
dwJT = [(s2-s)/9]x
W = beq-s fl(s,x)
‘rW== (b/3)eq-s f2(s,X)
T(TWT) = (b/9) eq-s f3(s,x)
where the functions f,, f2, and f all have the form3
24
.
.
.
.
fi(s,x) =ai(s) (sX)-’ sinh(sx) + Bi(s) sx sinh(sx) + yi(s,x) cosh(sx) + di(s)
with the coefficients ai, Bi, yi> ~i given in Table I.●Computation of the g-integral
The 16-point Gauss method is used:
*b
J
161*f(x) dxaZ b ~ ai f(xi)
o i=
xi ‘;b*(”
I
‘Yi) ,
where the ai and yi are the weights and arguments of the method. These are sym-
metric about the center of the interval (O,b*)
Y1 = y,6=o.99
.
.●
al = a16s0”03
.
.●
a8 = a9s0”1g “
TABLE I
COEFFICIENTS FOR THE EXPONENTIAL FORM
i ai(s) f3i(s) yi(S,x) 6i(s)
1 1+s-’ o-1
-s -1
2 -(s+2+2s-’ ) -1 2(1+s-’) s
3 S2+2S+4+4S-1
2+3s -(3S+4+4S-’+SX2) S-S2
In the region of interest, the integrands often effectively vanish for x signifi-
cantly less than b*. The program allows for this by, in effect, choosing the
numerically appropriate upper limit each time, as described in Sec. V.C.
3. KW (Kistiakowsky-Wilson). The KW EOS has its own built-in mixture
rule in the form of a linear mole-fraction sum of covolumes ki
Its reduced variable x:
x= k/V(T+f3)a
for 6 = O, would be that corresponding to a repulsive-only potential with
k= (r*)3(T*)a .
(actually f3is fixed at 400 K). The equations are
z E pV/RT = l+xeex 3
E’/RT = [aT/(T+e)] (z-l) ,
F’/RT = (e6x-1)/~+ z - 1 - !?n z , and
P;/RT = (e6x-1)/6 - Ln z + (Kki/k)(z-l) .
The derivatives are, with partials of p (T,V) and E’ (T,V)
(V/R)PT = z - (l+Bx)E’/RT ,
(V2/RT)pv= - (l+Bx)(z-1) - z , and
C~/R = {2 - [l+a(l+Bx)]T/(T+6)}EI/RT .
26
D. Mixture (XIM)All the following forms except ideal mixing require the
tential functions for all binary interactions. To get these
tentials for each species, we use the combining rules for r*
*
‘ij= ~(r~ + r;) and
T:. =lJ
(T; T~)l/2 ,
intermolecular
from the given
and T*
and assume the same functional form for all (including common values of the
pulsive and attractive exponents n and m).
po-
po-
re-
All of the sums in this section, of course, extend only over the gas species.
10 Ideal Mixinq. The properties of the mixture are just linear mole fraction
sums of the properties of the components,
given Tand p. Thus
V(T,p) = E xi Vi(T,p) ,
and similarly for the other extensive var.
component is just equal to its Gibbs free
P~(T,P)/RT = F~(T,P)/RT .
each calculated as a pure fluid at the
ables. The chemical potential of each
energy at the given T and p
2. LH Mixing. This is based on an expansion about the properties of a fixed
specified reference fluid (subscript r) in powers of r: - r; and T; - T~. All the
pure species are assumed to be described by the same reduced EOS p (Ti, ei), Ti =
V/V~, e = T/T~. The partial derivatives are those of the reference-fluid func-. tions with T and V as independent variables: p (T,V), E’ (T,V). The equations
are
f . ~Xjxkfjk
j,k
27
9, = E ‘j ‘k gjk
j,k
f. =Jk r? /r*Jk r
z = ‘r /’ - [Tiv(-p$V) + (~)ipvl“-1)+3[1+(Y’)/pvl‘g-’)\H’/RT = ~H/RT+ [~,RT - ~i\R - R-l (TpT-p)(-pT~v)],f-l)}r
F’/RT = [F’/RT + (E’/RT)(f-l) + ‘(z-l )(g-l)]r
p{/RT ={ I (Z ‘j ‘~~~ -1, + (f-l]F’/RT+ E’/RT 2
j
[( ) 01)+ 3(2-1) 2X Xjr~/r~-l + g-l r ,
j
with all of the thermodynamic functions and their derivatives on the right evalu-
ated for the reference fluid at the given T and V.
3. CS Mixing. This is an improvement over the LH form, with a composition-
dependent reference fluid chosen to make f = g = 1. The same assumptions apply.
The reference fluid is defined by
*r; = F* = z Xi ‘j r..
lJand
i~j.
.
28
The
the
def-
.
.
thermodynamic functions are al
chemical potentials, which are
nedhereandf=g=l:
u;/RT ={F’/RT+ E’/RT
[
n/?*(
~7*/ani = 2X Xj
j
n/?*(
aF*/at-ii = 2~ Xj
j
These partials are at constant n<,
just those of this reference fluid except for
the same as those for LH but with r; and T; as
1)T:. 7* - 1lJ
, and
j#i.
4. One-Fluid Mixing. This is similar to the CS form, except that the expan-
sion variable is taken to be the potential function itself instead of its param-
eters r; and T;. It yields a tractable result only for a power-law potential.
The reference fluid is defined by the mean parameters
‘~ = E xi Xj T~j (r~j)q, q = n or m ,
i,j
-*r = (Sn/Sm)l/(n-m) , and
-*T = +/(n-m)/#(n-m) .
Again the thermodynamic
potentials are those of.
(n/T*) a?*/ani
(n/;*) a;*/ani●
sqi
functions are those of the reference fluid. The chemical
the CS form but with
= [2/(n-m)l ~ni~n - ‘mi~m) >
= ~+ ~(n-m)l ~sni~n - nsmi~m) Y and
=E Xj T~j (r~j)q , q=norm .
j
29
E. Chemical EquilibriumRecall the definitions given in the input description:
number of independent species
number of elements
total number of species
Number of phases (p=O: gas only, p=l: gas + solid)
empirical-formula coefficients
species-formulae coefficients ,
and that, for a complete equilibrium calculation, the user
systems: a one-phase system with the solid only nominally
number s, and a two-phase system with the solid present as
Symbols newly defined here are
supplies two possible
present as species
species number c.
q=
v=m
‘k =
ns =n9 =
Xi =
i(k =
number of moles of each species for a system preparedfrom independent species only,
stoichiometric (chemical reaction) coefficients of thedependent species (independent species have coefficient1),
C-P
T ‘kj = change in number of moles of gas in reaction k,J=number of moles of solid,
number of moles of gas,
(gas) mole fraction ni/ng except xc = ns/ng for solidpresent, and
equilibrium constant for reaction k.
We change to superscript s and g here for solid and gas to avoid confusion with
indicial subscripts.
1. Initial Calculation. The initial calculation generates constants for
the main calculation. Mathematically, it consists of a change of basis from
(moles of) elements to (moles of) independent species. In the new basis each de-
pendent species is expressed as a linear combination of independent species, with
coefficients that are just the independent-species stoichiometric (chemical reac-
tion) coefficients for the reaction producing one mole of the given dependent
species from the independent ones. The system could, of course, be prepared from
qi moles of each independent species i (i = 1 to c) instead of Qi moles of each
element.
30
We now define some index conventions to simplify writing the equations.
Divide the a-matrix into independent (c by c) and dependent (s - c by c) parts.
Number the complete columns, and the rows of the independent part, with i or j.
Number the rows of the dependent part with k. (See Fig. 3.) The ranges are.
i andj: ltoc
.k: C+l to s,
and denote by Xi or .Zja sum over i or j from 1 to c and by Zk
C+l to s. Then a and v are solutions of the linear equations
a sum over k from
x aij ‘i = Qj, j=ltocandi
x aij ‘ki = akj’j= ltoc, k=c+ltos.
i
The dimensions of~are the same as the dependent part of~and we retain them
same usage and range of indices for it. (See Fig. 4.) -
ELEMENTSI ciori~
.
.
INDEPENDENT
&
.- ——— -- -- —
DEPENDENT
INDEPENDENTSPECIES
I c
C+l k
Fig. 3. The a-matrix. Fig, 4. The v-matrix.
2. Main Calculation.
now in all sums except that
i andj: 1 to c-p
and we use primes to denote
We retain the above index conventions except that
in the definition of K the ranges of i and j arek
Y
sums or products over this range.
The equations can be reduced to a set of c-p equations in c-p unknowns; the
remaining mole fractions are determined in the process of solving these or are
calculated afterward. The equations are obtained by expressing the dependent
species in terms of the independent ones , writing out the first derivatives, and
applying the Newton-Raphson method of iteration. Letting superscript (n) number
the iteration step, the result is
X(n+l) =i (1 +hi(n) )x(n), i =1 to e-p
i
(with super (n) understood in the following equations). The increments h!n) areJ
the solutions of the linear set
x ‘ A.. h = ~i, i =ltoc-pj
lJ j
Aij = Xiti..+
lJ EX~ V~j
k ‘ki
Fi = -~i + xi +x kxk ‘ki
‘ki = ‘ki - ‘Vk- 1) ~i
EI
‘k = j ‘kj
32
I ‘ki‘k = ‘k ‘j ‘i
.
. The first equation is a linear system for the hj. In the second equation dij is
the Kronecker delta, the xi’s are the independent-species mole fractions, and
the Xk’s are the dependent-species mole fractions, expressed in terms of the
‘mole fractions of the first c - p species through the next-to-the-last equation,
which is just the set of equilibrium-constant relations for the reactions. Fi-
nally ~i is the normalized qi and Vk the (gas) mole change in reaction k. The
Kk and ~i are independent of the x’s and are calculated at the beginning.
The Xk’s, k = c+l to S, are found in the process of solving this system.
The number of moles of gas is
When the solid is present (p = 1)
c- p equations because, although
general all of the first c - p xi
the c th equation is decoupled from the first—each of the first c - p equations contains in
(by virtue of the presence of the xi in the
equilibrium-constant relations), none contain xc (by virtue of its absence from
the same). Thus, in this case, only the first c - p equations are solved itera-
tively, and then xc is determined from the c th equation evaluated for this—solution.
As mentioned earlier, in the complete equilibrium case, when a solid may or
may not be present, the program carries two possible systems, one with and one
without the solid. It begins by solving the case found to be correct on the
previous entry, then tests to see if it has the correct one and switches to the.
other if necessary. If the two-phase system is solved, the test is the sign of
the number of moles of solid; if it is positive, the choice was the correct one.●
If the one-phase system is solved, the test is as follows. Find the first reac-
tion in the two-phase system that involves the solid. Call this the test reac-
tion. In the two-phase system the mole fractions involved in it would have to
33
satisfy
‘ki‘k =x/lI:x. o
kJl
(with Kand thev’s, of course, from the two-phase system). Define a saturation
index ~ for the test reaction as
A
s = (xk/IT;xi‘ki)/Kk
and evaluate ~ for the x’s (of the spec
If ~ c 1 the system is unsaturated with
one-phase choice was the correct one.
F. Miscellaneous
es involved) from the one-phase solution.
respect to deposition of solid, and the
The actual Hugoniot and CJ iteration functions are given in Sec. V.B. (Table
III). Note that the Hugoniot function uses enthalpy relative to To and that ho
is for the unreacted material. t
The “heat of reaction” q is defined, for products at given T, p, as the
energy released by reaction of the unreacted material in the initial state (To,
PO) to products at the same temperature and pressure (but having the composition
calculated at the given T, p)
Q = [XgAg(h) ( -+ ‘s ‘s ‘o)]products [()]- AT - RTOo unreacted
q = (n/Mo)Q .
The particle velocity u and detonation velocity D are given by
J = (P-PO)(VO+O
‘If the alternative procedure (heat of formation at T = O entered under CON, SAM,Sec. II) is followed, then the Hugoniot energies will be relative to T-. mevalue of Q will be that at T = O, with the term (-RTO) incorrectly subtracted.
34
p~D2 = (P-Po)/(vo-v) ●
.
.
.
.
On the isentrope, the particle velocity u, which would require evaluation of the
Riemann integral, is not calculated. For the isentrope whose initial point is a
CJ point, the quasi-static work done by the explosive in expansion of the prod-
ucts to the given pressure, described in Appendix B,
W(p) = ej - ~ U; - ei(p)
is printed in the u-slot.
v. PROGRAM
The program listing, Appendix C, contains a list of all routines and common
data blocks, with a one-line description of each. It also contains, with each
routine, comments giving the routine’s specifications. In most cases, the logic
is simple enough that suitably placed internal comments suffice to describe it.
The exceptions are collected here, in order of decreasing logical level in the
program. A detailed list of the contents of all common data blocks is given in
Appendix D.
A. Equilibrium Iteration - EQMTo calculate a state point at given T and p, MES calls TIM for the ideal
f~ee energies (F; at the given T), and then SEM for the solid free energy p; ~
Fs(T, p). It then calls EQM, which calls XIM for the gas mixture EOS in the
process of finding the equilibrium composition. Finally it calls TIM for the
complete set of ideal functions at the new composition, and then COUT for the
complete state.
Because of the multiplicity of EOS and mixture-rule options, EQM and XIM are
the most complex routines of the program, EQM is outlined in Table II. Sub-
script i means all species; for example, ;i + xi means save all gas mole frac-
tions. Also, only the most important input and output items of routines XIMand
EQMS are indicated. In words, the solid~ is calculated once at the start be-
cause it depends only on T and p, which are here fixed. First, the mixture rou-
tine XIM(l) is called for the imperfection chemical potentials. These are then
added to the ideal ones (calculated earlier by TIM) to get the ~i which are the
input to the (ideal , i.e., constant-~i) equilibrium routine EQMS. The current
35
a p;, r* + XIM(l)
A
xi + xi
TABLE II
THE EQUILIBRIUM-EOSROUTINEEQM
solid ~, depends on T, p on-
gas p; (r* for LH & One-Flu
(F; from TIM earlier)
save old xi
new xi at constant Fi
Y
d only)
A
if (Xlxi -xil <c): go toy done?
Axi + 1/2 (xi+ xi) next guess: mean of new and old
Fi, F* -+XIM(2); go to 6 (CS or One-Fluid only)
Y if [(C$ o~ One-Fluid) and(lr-F I < c)]: go toa outer done?
(gas state) +-XIM(3) final state
composition is saved for later use in the convergence test and in getting the
next guess, and then EQMS is called to calculate the composition implied by the
current pi and ~s. If the resulting composition change is small enough, this
finishes the iteration for other than the CS or One-Fluid mixture rules (used
only with LJD EOS). If not, the next guess for the composition is taken as the
mean of the new and old values, new pi are computed by XIM(2), and the cycle is
repeated starting at (3. For the CS and One-Fluid mixture rules the completion
of the above (inner) iteration is one step of an outer iteration whose conver-
gence test is that the value of F* from X1M(2) after the inner iteration is com-
plete be close enough to its value furnished by XIM(l) at the start at a. This
is discussed in more detail below.
We next describe the action of XIM, which is different for the different
equations of state and mixture rules. Here GEP is the gas EOS routine, which
36
gives the ideal-gas, LJD, or KW equations of state (according to the gas switch
from CON, SWIT) and XIMchooses the mixture rule from this switch and the mix——switch, which we here call k, from CON, SWITor CON, XIP. For the LH, CS, and—— ——One-Fluid mixture rules, XIM calls XIMSfor the detailed computations. The no-
. tation XMT+ GM indicates that the pure-fluid state in the GM array is moved to
the mixture state in the XMT array. Also, T* and V* are to be understood in
addition wherever r* is indicated, The numbers (l), (2), and (3) refer to action.
taken at the XIM(i) calls, i = 1, 2, 3. The options are as follows:
●KW EOS
Recall that GEP calls HKW, which calculates the complete (mixture) EOS.
(l), (2), (3) HKWvia GEP supplies the state, including thep~.
●k=o : No-mix [normally used only for a pure fluid (single s~ecies)]
(1) GEP supplies, the pure-fluid EOS for the input r*. The~i are all set to the
pure-fluid F9’
and XMT + GM,
(2) No action,
(3) XMT +-GM.
●k=l : Ideal mix
(1) For ~ach species, set r* to r; , call GEP for the pure-fluid EOS, and set Vi
toF.
(2) No a~tiono
(3) Calculate all
those for the
●k=z : LH mix
(1) Set r* to the
EOS, and call
(2) Call XIMSfor
mixture imperfection quantities as linear mole-fraction sums of
individual species.
input reference-fluid value, then call GEP for the pure-fluid
XIMSfor the p~.
the p;.
(3) Call GEP for the final state, then XMT+ GM. (Why any action is needed here
is no longer clear to me.)
●k=3 : CS mix, and k=4: One-Fluid mix
(1) Call XIMS to get F*, set r* to F* and call GEP for the EOS, then XIMS for. the p;. (XIMScalculates both F* and the p;, using the reference-fluid
state. Here two calls on it are necessary, the first gets r*, which is
. needed for the reference-fluid state. There are actually two entries to
XIMS that are not distinguished here; the first calculates only ~*, the
second both F* and the pi.)
37
(2) Call XIMS for thep~ and F*.
(3) No action.
The outer iteration described earlier is necessitated by a time-saving de-
vice introduced in GEP. A second entry GEP(2) calculates the approximate gas
EOS (for LJD) by the quick route of the LH expansion from the reference state
currently stored. This time-saving entry is used by XIMS to get the gas state
required for the calculation of the Vi. Where the mean r* depends on composition
as for the CS and One-Fluid mixture rules, the validity of this expansion is en-
sured by the outer iteration.
B. IterationsThe iterations are tabulated in Table III. All except that for the equilib-
rium composition described above can be written as a function of a single vari-
able and are controlled by FROOT. Note that the gas and solid EOS iterations,
independent and on the same level, must both be completed as part of the equilib-
rium iteration, which must in turn be completed as part of the calculation of
the function for the Hugoniot or constant-v, s, or e iterations. Finally the
Hugoniot iteration must be completed as part of the calculation of the function
for the CJ iteration. This stacking requires careful adjustment of the conver-
gence criteria for a
and their magnitudes
Hugoniot function is
CON, FOB for several——criterion E, (2) The
reliable system. The functions chosen are not too nonlinear
and slopes are reasonable size (the factor of 20 in the
introduced for this reason). In general there are slots in
constants for each FROOT iteration: (1) The convergence
“guess-constant” r, in most cases the ratio of the second
to first guess, and (3) upper and lower bounds for the iteration variable or the
related physical quantity. Not all of (2) and (3) are used in every case. Under
“Bounds”, subscripts and max denote (3), other bounds are computed as indi-
cated. Finally, we use subscripts 1 and 2 in the “Guesses” column to denote it-
eration steps 1 and 2. Remarks on some of the iterations follow.
●CJ—
For the second guess, a constant-y isentrope is a sufficiently good approx-
imation to the CJ 10CUS. For the lower bound, we use the constant-y approximation
to the constant-volume detonation pressure. In the function, y is evaluated for
either fixed or equilibrium composition according to the setting of the input
switches; thus either a frozen or equilibrium CJ point may be obtained.
38
‘0
t‘m1
+~
1.--N
1
;1
‘7
1-1
-
v---%--
1
<---3-
-n+
-LnS
1x-n
Nx
x-
LIIXN
*-nx
-11
mu(u
>
II11
ux-
c-ul-x
1=
8vvv
..
39
.Hugoniot
The value of f’ for the second guess is that calculated by numerical differ-
ence on the last step of the iteration for the previous point. On the first
time through it is the input guess-constant r.
.Gas
The derivative f; for the second guess is calculated from the derivative
(ap/aVg)T furnished by the gas EOS routine GEM..S01id
The function, given in Sec. IV.B, is based on the isentrope relation, The
quantity a in the upper bound is the thermal-expansion coefficient.
co LJD Gas EOS Integration (In GES)As shown in Fig. 5 the problem is to evaluate to prescribed accuracy for
any u an integral
*
r f(x;ci)dx ,
0
with the integrand depending on a in such a way that for some a it effectively
vanishes at some x appreciably less than b*. lieuse the 16-point Gauss approxi-
mation (Sec. IV.C); its straightforward application to this case would waste
those points lying in the region of f(x) sO. (Actually all seven integrals are
done at the same time; for simplicity the description here is for one.)
The method is to check for the vanishing of the integrand and to perform
the integration in segments if necessary. Because a changes little between most
of the entries, this turns out to be reasonably efficient. The part of GES that
does the integration consists of a control code, which chooses and adds the seg-
ments, and a procedure that integrates over a segment and reports how soon the
integrand becomes negligible, if at all. All segment lengths are b*/2n, n inte-
gral . b2The segment integration procedural’ (b,, b2) evaluates ~ f (x;a)dx by the
bl16-point Gauss method and reports three conditions for the si~e of the integrand
near the end of the interval, Let i (i = 1 to 16) be the Gauss point at which
the integrand first becomes negligibly small. Then I reports that the upper
limit b2 is
“too small” for no such i (LD = 1),
40
“just right” for 14 < i < 16 (LD = 3), or
“too large” for i < 14 (LD = 2),
LD being the Fortran variabje that reports the condition. (The actual test is
. that the condition lf(x.)/~ aj f(xj)l < c be satisfied for each of the seven.1 j=l
integrals). The algorlthm which evaluates the complete integral using the seg-
. ment procedure ~ (b,, b2) is given in Table IV.D. FROOT-Solve f(x) = O by Iteration
Number the successive steps in the iteration 1, 2, 3, .*., n, ..O with nth—(function) argument Xn and function fn ; f(xn). The complete current state of
the iteration is contained in FROOT’S argument array~
&l & - convergence: Ifn] < E
2 Xn
3 fn
4 Xn-’
5 fn-l
6 Xn-2
7 fn-2
8n - step number
9k - branch index:
1 - finished (Ifnl < S)
2 - continue
3 - error 1:~n
= Xn (see belOW)
4- error 2: fn = fn-l .
Each time it is called, FROOT bumps n by 1, finds the new Xn and returns to the.
user, who calculates fn = f(xn) and calls FROOT again. The user exits from this
loop on convergence via the branch number k. The prototype program is (E pre-
. stored )
n+o, )&)?, Xn-l 1+x
9 9
TABLE IV
ALGORITHM FOR THE CELL INTEGRAL
ab 2 + b’, bl + b’/2
I +~ (O,bl)
if (bl just right): done
if (bl too large): <b’ i-b*/2, go to U>
if (b, too small):
<BI+I~~(b 1, bz)
if (bz too large or right): done
if (b2 too small): < b’ + 2b’,bz+b’, bl ~b’/2, go to B>>
b’ is upper limit fromlast entry
first segment
halve and start again
add 2nd segment
double
The upper limit b’ (not shown is that b’ is bounded ~ b*) is saved
each time and used as
cessive segments look
bl too large
the first guess on the next entry. The suc-
like
b, too small
Io bl
Ib2
I Io bl b2
.
.
.
.
42
a CALL FROOT
GO TO (6, y, 61, ~z) k
y calculate fn = f (xn)
GO TO a
6,, 62 error handling
B done, proceed
Here n, Xn , etc., refer to slots in & as indicated above. The user starts the
iteration by setting the step index n to zero and supplying x and x:, the first9
two guesses for x. The iteration then proceeds through the a-y loop to conver-
gence. Because all the current state is in~ (and none of it is stored in FROOT),
one copy of FROOT can simultaneously control any number of interdependent itera-
tions (i.e., the calculation of f(x) for some iteration may itself require an
iterative solution involving FROOT).
The algorithm is the secant method
x xn-1=
n- fn-l (xn-xn-l )/(fn-fn-l ) ,
with a refinement wherein ‘Xn is a provisional value of Xn subject to modifica-
tion, In the normal case it is accepted and the store is stepped down as fol-
1Ows :
(x,f)n-2 + (x,f)n-’ ,
(x,f)n-’ + (X,f)n ,
xn+~n ●
43
The modification replaces ‘Xn if, roughly, it does not lie in the range of previous
x’s and if two f’s of opposite sign are on hand, Precisely, if
(1)3?” is not between Xn and X“+z, and.
(2) (sign fn = sign fn-l) and (sign fn # sign fn-2)
.
then%n is recalculated (before the step-down) with point (n-2) instead of (n-1)
~n = Xn - fn (xn-xn-2)/(fn-fn-2) .
Also, saving of old points is done in such a way that once two f’s of differ-
ent sign are on hand, the step-down will never result in three f’s of the same
sign. Precisely, if condition (2) is satisfied, then the step-down is preceded by
(x,f)n-’ + (x,f)n-* .
We remark that, because ~ specifies the state completely, the user, knowing
the program’s algorithm, may wish to change it in flight. A common case is that
a lower bound x* is known for x, and FROOT at some step supplies Xn < x*. The
user might replace Xn by x* whenever this happens.
As an example (see Fig. 6), we write the code to solve f(x) = x1/2-A=()*
The secant recipe can easily give a negative x as the next guess; we prevent this
by introducing a fixed lower bound x = x*, x* small.
DATA C(l)/1.O E-8/
DIMENSION C(9), KC(2)
EQUIVALENCE (KC, C(8))
KC(1) = O
c(4) = 1.
c(2) =1.2
a CALL FROOT (C)
GO TO (6, y, 6, 6) KC(2)
.
44
a=cz,
.
.
f
/
b*x
o
Fig. 5. Integrandsintegral,
ox
for the LJD cell Fig. 6. Iterative solution of f(x) =xl/2 - A = O by FROOT.
y IF (C(3) < X*) C(3) = X*
c(3) = SQRT (C(2)) - A
GO TO (Y,
6 CALL ERR
B CONTINUE
VI. SAMPLE INPUT/OUTPUT AND TEST
Tables V and VI are a key to the output labels and a sample calculation,
which also serves as a fairly complete test of the program. The calculation is a
chemical equilibrium calculation for the explosive RDX with the LJD EOS and the
CS mixture rule. Part 2 of the input defines the following:
(1)
(2)
(3)
(4)
A state point atT = 2000, p = 0.3.
Detonation Hugoniot points at p = 0.3 and 0.2.
CJ points for p. = 1.6 and 1.8.
CJ isentrope points at p = PCJS 0.3, and 0.2.
A CON, DBUG print follows item (l). The point printed here is the last slightly——displaced one used in the finite-difference calculation of the derivatives.
45
TABLE V
OUTPUT KEY
Line
1
2
3
4
5
6
7
Notes
1.
2.
3.
4.
5,
P v/v.
v e
v E/RT
v9
E~/RT
Vr E~/RT
Vs E:/RT
F* ?*
T
s
S/R
1
Sr/R
S:/R
-*v
uorW D P. q
‘9ns n
z ‘1 ‘2 ‘3I
‘9 ‘4 ‘5 ‘6
z;‘7 ‘8 ‘9
zs ‘1 o ‘11 ‘12
Lines 1-3 are for the complete system. For lines 4-6 the first four itemsare for the gas phase, the gas-phase reference fluid, and the solid phase,respectively. The reference fluid differs from the gas phase only for theLH mixing rule (Sec. IV.D.2).
The last item in line 2 gives the state of super- or undersaturation of thesystem with respect to the solid phase; Rn ~ (the saturation index, Sec. IV.E)is printed i~ a single-phase system was specified and it wants to precipitatesolid, and -n (the number of added moles of solid which would just saturatethe system) ii printed if a two-phase system was specified and no solid ispresent.
All mole numbers are moles per mole of system (one mole of system = M. grams,with M. from the SAM input pack).
The “u or W“ slot in line 1 is the particle velocity for the Hugoniot or CJpoint, the quasi-static expansion work (Sec. IV.F and Appendix B) for the CJisentrope, and meaningless otherwise.
All ot~er symbols are defined in the symbol list. Recall that z ~ pV/RT(and z = pV/RT - 1) and that a prime denotes an imperfection quantity (Sees.111.B.I-3). Molar quantities for the system are per.mole of system and forthe phases are per mole of phase. All quantities divided by RT are dimen-sionless.
46
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53
ACKNOWLEDGMENTS
James D, Kershner, T-4, and Jack D. Jacobson, T-4, helped with the program-
ming required to reactivate the code. The equilibrium routine was redone earlier
in FORTRAN by Paul Bird, L-3.
.
.APPENDIX A
CHEMICAL EQUILIBRIUM EXAMPLE
We give in Table A-I an example of a system with an equilibrium number of
phases (solid carbon may be present or absent). The user must specify two sys-
tems, a two-phase system and a one-phase system, and make an appropriate choice
of independent species for each. He does this, after defining the species and
empirical formula via ~and ~, by giving the two renumbering 91 and ~“, Recal1
the definition of ai: the (original) ith species becomes the aim. Take, for—example, i = 4, a
;= 1; CO, originally the fourth species, becomes the first in
this the two-phase system. Recall that the solid must be numberc_ in the two-
phase system, and number ~ in the one-phase (where the program assigns it a large
F to make its mole fraction negligibly small). The horizontal dashed lines divide
the a matrices into independent and dependent parts. The corresponding &and q
are given below each a , and the reactions are written out in full below the &’s.
The saturation test f~r the one-phase system is made on the two-phase system
reaction
C()+H2.
with the saturat
A
s =
evaluated for
C(S) ~H20
on index ~,
( )/‘H20/xCxH ‘4
2
the mole fractions
s
from the one-phase solution. If ~ f 1 the SyS-
tem is unsaturated with respect to deposition of solid carbon, and the one-phase
.
.
system is the correct choice.
54
..
.
I ON
I-O
NI-O
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ON
+11
NS
’8o
u{000vrJ~
55
APPENDIX B
QUASI-STATIC WORK
Summarized here is what might be loosely termed the Carnot cycle for explo-
sives; plus a numerical example. The object is to calculate the maximum energy
that can be extracted quasi-statically from an explosive by detonating it. The
quasi-static process is that given by Jacobs.3
Confine the explosive in an upright cylinder of unit length and cross sec-
tion, closed at the top by a rigid cap and at the bottom by a movable piston.
Assume that all confining materials including the piston are rigid massless non-
conductors of heat. Move the piston into the cylinder with constant velocity U1
greater than or equal to the Chapman-Jouguet (CJ) particle velocity. As the pis-
ton begins to move, instantaneously initiate the detonation at the piston surface.
The detonation front will then move upward with complete-reaction wave velocity
D1 determined by U1. The detonation products (reacted material) will be in a uni-
form state with pressure p, and particle velocity U1. When the detonation wave
reaches the upper end of the cylinder, attach the piston to the cylinder at its
position at that instant of time and remove the driving force on the piston. Al-
low the cylinder to move upward under gravity deceleration until its velocity is
reduced to zero. Extract work reversibly from the cylinder of product gases in
this position by first lowering it slowly to its original position, and then re-
leasing the piston and allowing the products to expand adiabatically and revers-
ibly to some final pressure.
Calculate the net work done on the surroundings in this process. The cylin-
der has unit volume and contains p. grams of material. Take work and energy per
unit mass of material. The piston moves a distance ul/D1 with force p,, so the
work Wp done by it is
wP
= P,u,/PoD, ,
or, using the conservation relation p = Po”lDl (neglecting po)~
56
The kinetic energy K of the reaction products is
K= U;lz ,
●
The work done by the system in expanding to pressure p is
.
‘i(P)I(p) ={ pi(V) dv ,
VI
where pi(v) is the isentrope through the product state (PIs vi)” Summin9 the
contributions gives for the net work W(p) on the surroundings for expansion to
pressure p
w(p) = - WP+K+ I(p)
= I(p) -~U;
12=e 1 -Z”l - ei(P) Q
.
We remark that W(po) differs little from the heat of reaction q as conven-
tionally defined -- the energy change in the surroundings for reaction to products
at To, po. To see this and show how the various energy changes enter, we have
prepared Fig. B-1, which shows the closed cycle in the p-v plane, and Table B-I,
which lists the steps in clockwise traversal beginning at point O, the unreacted
explosive. The numerical example consists of the calculated values for Comp. B
from Ref. 2, and is for Tc = 300 K, P. = 1 atm. For state 2, the calculated tem-
perature is 518 K and the mean y is 1.25.
.
57
I
-ab-llx
2(n
kQ
P. - ----- _____ ____
o ; REACTION AT To,fIo: COOLING TO ToI P~ODUCTS TO AT P.
SPECI:C VOLUME (V)
Fig. B-1 , Closed cycle of energy changes in the p-v plane.
TABLE B-1
ENERGY STEPS IN CLOCKWISE TRAVERSAL OF FIG. B-1
Comp, B
Process Energy Change (mm/ S)2 = Mj/kg
Detonation 12‘1 - ‘o = ; pl(V. -vl)=~ul 1.80
Adiabatic expansion‘2 - ‘1 = - l(PO) -7.56
Cooling at p.‘3 - ‘2
= Cp (T. - T2) -0,28
Reverse reaction‘o - ‘3 ‘q 6.04
Thus W(po) = e. - e2 = q - Cp (T2 - To) 5.76
58
APPENDIX C
PROGRAM
I.D. LP-0730
22
2
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23
13
15
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10
111213lQ15lb1710192021
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60
II
CALL POLT(3)CALL PI?l-I (?ALcAQ - cc/cti/c~/LG/cs/cL 9 600 cAR)CA(.L PRTlJ(31.11ER *l(I~nLK)
(.ALL. PRIl{(4LCONT ?l~.CONr)CALL PNfid(3LfiMG0 lo,tMu)
GALL PRIrI(3LEF\14q10,FMJ*)CALL PR1,J(31.FMx, lJ*$:MX)
CALL PM11~(3(.FML), ifloFMu)CALL PNI.J(ZLFt.J, III,Pk)LALL PN1fj(3LFObt+l, ?FOti)
C~I.L PR~(.1(i?Ll;M,311,bw)CA(,L FRIIi(zL~;P.lfI1bP)
CALL PRIIJ(pLtiEo ;,tsbF)
CALL PRIN(2Li\s.5osds)CALL P141,., (?(.EA08,1*Lfi)CALL PRT l(2LEV,10ttv)CALL PRIN(al.FLAL? .lu,F”LAd)
CALL Ff/T,l(3LG’4TJlo*cMT)CALL PR1,4(.)[.KF.v* 1O*KEV)
CALL PR1 l(ql.sPc*lo*~Pc)
CALL Pdl:~(3LK4L*l/$KAL)CALL PRItJ(3LKdN?40?KuN )
CALL P4TId(31.KEN~?ll*<LiY)LALL PRld(3LKI M$llj?KlFl)CALL PRI i(2LpTC10~PT)CALL PRTiJ(41.RlioT oIu,NnOT)C4LL PRI,il?LSM,ZI)!~P)
cALL pK[(J(?L,s~,2i)t>P]CALL PRl,j(3LSl)r,.l,J?SUc)
LALL PRT I(4LS{JCG. lU.SLICG)CALL PPIiq(4LTHEh *i0.TMER)CALL PNI,d(qLT:4G~10!l,46)CALL PRI,q[3LTMS,70,TMS)CALL PR~,J(~LTP. ?iO? lP1CALL PRTIJ{sLX’4T,?O$~ MT)CALL F’RIN(ILXMU9 L09XMU)CALL PR1’4(3LxPF, I0,xPF)CALL PRIsJ(3LXPGJ 10tXPa)(.ALL PRI,J(3LXPR, I0,XPK)CALL P~ld(3LXPT,109XP1)
t4ET(JrJh
E.Nn
SUqCXCIUTINE FIO (8*PMT*N*A??(L)
I$=PEAO .)411 9iJRlhT *WUIrMT=PoRw4TIW=NIJMBER OF .40RCS UF ARRAY A F(M I.(J.
FOR N=o,F.wT IS hULLEi41Th ARGUMLNT &AStL,<L IS T*E ❑o141) LLkGTFI UP TIIL LAMELo
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If)
15
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CONTINIJEwRTTL(9,/fl)H
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1.0. *ITM No DATA
(’500s500s 1309140)91
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(12Ah)1.0. nITH OATA
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tio To 5(n)
READ(10*FMT) (A(J) $J=l*N)bO TU 500
wRITE (?.PMT) (A(J) 9J=1?N)bo TU 500
WRITE (99FMT)(A(J)9J= lsN )
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CONTihdECOr4TIhUE
NETUhPi
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SURRWTINE FROOTT(C*K)FNOCT FUNLTION
c- FNouT AN6C(5) - xi)C(6) - UELTA X
c(?) = XN
F(JR K=OsIILRAlt
TABuLATOR
, AH$?AY
FOR K=l~l@ULATL. PROVII)E FRO(J! WITh VALUESxO?XO+UEL!A X9.,,,AN-THEN LXIT
L13LJI’4ALE,NcE (E10Kc1)s(E2~Ac2)
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ZU LALL FRc),)T (C) FRuUTTjo bO TU 21,.! FNLUTT
Ft+~dTT50 1F(KC1)1OO,6O,1OObO C(?)=C(5)/0 bO To If)?
FMbUTT
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102 RC1=KC1*l FlibUTTI1O IF ((C(?) -C(7) )UC(6)) 15(J0150s1Z0 FHbOTTIZo Nc?=l F14~UTT
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NEGULAR CALCULATION OF CA~ EO? F~K PuNE
SPEC1E5 AT GIVEN TtP NY IIERAIIVE SOLUTIONUF P(T,V]=P, dITti P(T9V) UIVLN MY ULM.FOP cs CR ~-FLUIO MIX, TMIS CAN 13k A MIXTURE
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P=TPLR(I)lPPLICIT FOR GEM -
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t!Eah’ MSTAN* TSTAM = RSTA=6P(5)9 ~STA=(JP(6)KAL(5) - 1.293 FUN IoEAL9LJu$ KU
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(+M - STATE PoINT
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~ROOT ITERATIoN.tAQIJiBLE, FbNTICN - LOG(V), LuG(P/PG)bEcoNO GIJEss - NEw[cN-RAPtISON ‘ROM oP/oV=Gt4(l~!
15lb17Iti19202122232425262728
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l*P=rhEHi~), (3)GM - STATE POINT FHC14 PRLVIOUS K=l CALCULATION
(GF)1(2) OUTPU(ILt4x(/...) - (GAs) MCLL FNACTIU;JS
OLIP(JTAMT - N[XTURC sTcT: POINT
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CALCIJLATELI uY CALLLR)VARIAHLES
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1 (?( 921) *CAR ) ,(Z( +60) 9CONr ) 9(Z( 6ao) *LVz?(,’( 1410) oFOH ) 9(Z( 1490)$OM ) s(Z( 162U)SKLN3$(7( 16qt))*KON ) ●(Z( l$20)*ThLN ) Q(z( ~~lO)*AM14*(z(1470),~p)
UItlt:wSI(!N cG(10)$KCG(2I
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3? (FOn(2~) OVRU), (FOH(19) ,V~L)4s(!;M( !5), JR)t(6P(10) *PG)5, (ThkH(l ),P), (ThE~(2 ),T)(3* (GP(5),MSTS ),(GP(6)*TSTA)7t;x~{l (16),RsTAT), (AwT (17) $TSTAT)
c ---------------------------- ---------- K=l - REGULAR ------------------- CL?lF (VN) 20,1.?*EO GE*vR=ll).()CG(4)=A[.{IG(VR)
CALL COUT (\HGEP.1)
KCG( 1)=9
CAI.L FR(?OTT (CG,KOl~(4,8))
fiExIr=tic(>(2)~0 Tu (15n,rIl*80 s7U)9KExI!CAI-L O!31,G (s14GEP,1)UO T() 15.)CALL CBIJ1> (~dGEP,/)(>0 r,) l’j, }
IF (NON(Q.8)) qZ,9(J,~t?lF (KCG(l)-Z)92,%I $92
CG(2)=CG(4)-CG(3)*~G/ (VR*bt4(ld))VI?= EXP(CG(?))
IF (VRU-VR)q4J94~90VRavl.lu
UO T(J 9nlF (VI?-VA[.) 97,97,98!’R=VNLKEN(ll)=KEN(ll)*l
41424344454647464950 “51525354 -
5556
575859
60616?
b3
6465
66b768697071727376757ci77
7M798il81az838485868788899091929394959e979899
100 “
.
64
.
c ----------------- -------- --------- ----- K=d - LH LXPAF4sfU:t -------------- GLP?uO Lv(l)=T$fnT/TsTA-Ioo
qlu t_V(,?l=RSTAT/R%TA-l .0c Ado
c 610 k cOMPUlk lnLtiM(l UUTP(JT
412 Lv(31=ti~(4) oEV(l)+S. 00GM(lo)oLVIZ)Ldu E’i(4)=-(l@GM(l*)+P/b~l (lc?))”LV(ll
1 ● “OU0(bp(15)*V/bM (J<) )OLV(Z)4J0 EV(5)=(bF (ti)+(l°b:\ (13)-P) fJ(b14(16)/14))okv[l)
1 ~“ieu~(PQtiM(14)/F4- l.lJ)O~lj(.f)
Soo xh’T(l) =uP(15)+t.v (4)xMT
XI.T~uT
xt.’TxPTXPT
xvT
~)=(,N(+)+EV(3)-(P/(KO l) J*tV(4)-EV(5)
7)=~*X;4T(l)/(ROl)-I .u3)=K~T(d]+A~T(7j
5)=uN(51*Ev (3)
4)=A~T(>)-X~4T(/)
fl)=(j~(ll)-t. v(blR)=XMT(l)-HeT/P
lN
4P+(ITE - 7=PV/!?l
I*P
LMXut4 1
234 ERP/*r5 FRP/*T
67f4 1) EHP/LJ 141
10 7i<P11 sl+p/H
12 l) P/u v<13 1) p/J I14 1) VMZI) T
15 Vti
16
17 XPU Mu(.jP/KT
101
102
103104105Iof’i10710H109
110111112113114)15llb]17l]lj11912U1211221?3124!?5126:271?81?913013113,2]33134
)3313h
137
13ti139140141142
143~44
~&5
]66I&-tIfbti~by
150151]52! 53154~ 55
15b157
1 Sti
159160
.
65
161
SU}\RUIJTT:JE (icM (K) (2LP.
c PURF: GAS F(!S ?IT GIVEN T*V FEf’c ---------------------------------------------------------------------- (,~fi
c K=) PRE1-l.Jo REAC IWPUI PACKS ANL~ PUN LJLJ 6Er,
c CALL GES (l . ...) FCN P~LLI~iNA~Y P~OCEbSING (Sklb UP GP). GL*c CALLED flY CCN
c
~~h
tf=7t NA14. CALCLJLAIF STAl~ P(ltN14 KAL(3]= ~EF
c (J - [OF.Al. GaSi LLJCAI.c
~~$
9- K!4 EOS; CALL HKvJc
~~fi
OThEd - LJC EUsl CALL GEs(200..) (-j~?4
c ----..--...” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ----------------- GL14
cc
cc
cccc
c
ccccc
ccc
ccc
MAIN (A=2) SPkC5 15LnINPUT
l=THF.I?(?)V=Vl~=GN(IS)TSTAVSTA=CP( 6),(7) - TsTAI’4? V?TA14.
CO,!L1OSITION UEPLNOLNT FUK Lti
FAI-(3) - FOS CHi)lCL. SEE ABOVLllL lPIJT
bt4 - STSTE POINT. btJ IS utANKANtiEV
FILLER LJUT hE~E*tuC!lES
( 20) 9GM ( 40)( 6*1U) 9KON ( 6,101
( 2(J)
c
3- LQUIVALE,.!CE (GP(6i, TSTA)9[6p((),Vs~A)
1 ,(GM[i) OTbu), (Gr4(2) *TbEl A) 9(GM(16) oPti)
2 9(THLR(3) *T), (ctiNT ( ),?4)
3c
s(GM_(15)*VR)
j EQt!IuALE:JcE (17M(~5),fjAM) *(GM[z6)tALPH) s(GM(27)*dkl)
1 *(GM (db)$CbAM) *(GM (4Y)$CPR) S(GM(30)9ULVU! 1cc
3 100 60 TO (Iof)oq2000)sKc 094C 9$6 PKELIMINA14Y
?34567 .8~
1011
.
;:1415
Hltl192(-1?12223.?425?b27?b?9303132333435
3b373d
3940414243
44
454647
4649
50515253
:;Sb5?58 .
.
.
.
.
11
142(J.21
22
2426
303133
3536434750
5153
55
566 .j65
6771.r ~
7476
1O(J
102
100I(lb
111
112
;:~
12312b1.27131133135141
143
15[)
~5.31’55
157161
1 $-~*FE)CALL NEAP(o,tj,GP)
I(I1O KE(”J)=oln12 KE(L)=O102U fic (5)=KOM(1*4)]oZ2 hE (6)=K,)N(2,Jj)lnZ6 IF (Kt(I)-9) lo30s1025s10d0
ln25 KAL(3)=91026 V!5TA=1,010<7 rSTA=lo.’I
IIIZ8 L(I T(J 3000103o CALL GES(l,KE~GP,tiM)ln+(J CALL PRTIY(4HV*S ,19vSTA)
ln50 KAL(j)=l1060 bO Tu 3f)o0
clq’44
CIq$6 MAItuclqYLl
~000 KE~J(I)=KEIf(l)+l2002 El = vo
C2n04C2n06
.?n10 IF (KAL(3)) 2200,2iJ50*220uZ050 00 2(160 1=3,15ZnbU bM(I)={).(JZ070 b.Y(3)=l,n
z(l~u 1$I’!(9)=100~q90 bM(14)=l.n
?lOu GO T(J 23J0C21Y4C21Y6
2200 TA,J=VA/VSTA
Z?iu lHET4=T/rSTA2PZU KE(5)=KCI?+(I,4)~7d0 KE(6)=f(IlN(2,4)
cz>d4 IF (KAL(3)-~) 2240s2236s2<60
2736 CALL HKx(l)27J8 bO TCI 2301)
c~760 CALL GES (2.KE9GPoGv)
c
2700 CJ!4(?2)=G4(1S)~alo Ut.l(?i)=fi!(lo)2-420 bM(70)=l; 1(6)
2730 bM(l’+)=Gd(l.])~q+u bM(ld)=GY(12)2750 bM(17)=(;hl(5]z352 VP = ElzabU PG=GM(~)O\/@T/VRZs?o uM(lS)=VQ
2?d0 bM(12)=(14*T/VR+*21*(GM (10)-GM(3))
239u bM(13)=(J/vR)eGP(9)2600 ~M(14)=-6’~(13)/6P(12)2410 L1=GM(ll)2412 E2=GM(7)2&ZU bM(ll)=GM(8)
UEAUKANGE AND FINISH (JM
59696162636465666768697071727374
7576
77
76798081828384
858487
8689909192
939*95
96
;:
1;;
101
102103
]04105106107108) 09
110111112
113114
115llb117118
.
67
pot,21121s21722523i?~35
240
246245
12
ccccccccccc
- 10
LJM(7)=Gw(IO)
bM(b)=Gt.I(Q)
GM(11))=GM(31-1.ObM(V)=VU-R*T/PG
et.I(I!)=EI
bP(s)=E?
kEN(l*4\=KE(3)K~r4(z,4)=KE(4)
cGL~ =-vR*Gt4(~2)/?Gcvn=l Ps(~)-1.o+GN(d)
uLvl)T=GP(14)/vRbAl!=c~Av@(lo~+ccdMo~M (3) 0( !*ULVL)T )Oo~/cvH)
HET=l.o/(cGA’!tiGP(s) ~TOOLvuL/cvH)
ALPH=GJ’1M~dET-1 ,o
LP2sCVR*GAM/CG~tA
CALL DOUT (3HGEYol)
mETUNhENI)
SUBROUTINE GES (KoL,GPOGM)
LJD cELL THEORY G45 Ebt4. ~TATt bUMROUIIfWE
ftEVIslcN I. LOA!4LS1 HlhdR tRNf3k -IJOINTS 14 AND lb USE A FOR 13 ANU 15
he F. 1/5/62
REVISIu\ 2.Ch~NfiL utiS. CALL ON NEG. KAP PHINTUI.F. 2/<6/bd
bIMENSIOA A(R),Y(Ll),s (7)9w(5)sW 1(5) swc(5)s1 C-(A)sGM~15),Gl15)S~P( 7)*Ltb)~E(10)
12 EQ(JIVALENCE (6 (+) $~AU), (G(2) t~tikTA) ,(13(15)$B)s(usA1C 96
c 90 PidELe-!+AIN BRANCh12 1(JO bO T(.I (2[IIJ910001SK20 ?Uu y’(l) =oo~?.i9400fJ3
22 >tlz Y(?) =O.’-J6L575I2.224 >04 Y(3)=Oo@5c:6?l?o2$1 7U6 Y(L)=0,75R4 )4413 (J >Ou Y(S)=006178762632 ?10 Yi4)=0.L5~O167834 ?12 Y(7)=0.?~16,]75536 ?14 Y(R)=O.O4SO1?51O40 ?.2U A(1)=0,0)71S245S4;? ?i’2 A(2) =11.f!6?25352444 ?2+ A(3)=U.O.)%15H51246 ?<6 A(&)=Oe12462R9750 ?dti A(5)=0,1b959599s? >30 A(6)=0.IoQ1565254 732 A(7)=0.IAz6034256 234 A(H)=0013’265061b@ P40 u=0,27633562 >~~ bf4A%=0.?7~335
2345678
1:11121314
:21718192021
::?k252627?8293031323334
H37
68
.
103104
lob116112114122124132133
153156lb417%174?Oi205207
30+306
312317
244 uUP=l.0661420746 bOowN=O.97778543248 LPs=18.fi?00 UP (71 =004~60120GP(5) **3~du cALL lI~(l,L,GP)
310 GO TO 5nr)oc q9Ll
C 9’)2 MAIh LNTRYc 994 ANITIAL
lnOO lAIJ=hP (I)101O lhFT&=GP(?)ln12 L(3)=L(3)+1ln~o l~(TAU) 1040,1040,1G30ln30 IF (TflkT1i )lo40,1040c105010$(1 CALL Ct3Ui;(3@GESc;)
ln>u Ch(l)=TA(Jlnbo CALL *R (2,L,cti)10Io NEI=O
lnMO IF (L(s)) 4000t120U,4000
C119(I FINU I.lMLTl?UO *l=YO(l.O*Y(l))121O CALL WR (7JL,til)
lZZO lF ((*l (.3)- ABS(CK(z)))/TnETA-tpS) 1400*1400*1250C1?J8C1?40 ChECK AhD LLIWLR
1>~() w2=ti~(l.0+Y(3))
176u CALL WR (3,L,w2)1?7U IF ((n2(.l)- A8S(CH(2)l)/TnETA-kPS) 160001bOOo1300l~oo KE1=NE1+l
1310 Do 1320 1=1,s1?20 W1(I)=W2(I)13J0 Ll=Fl*Hu04td
la~O IF (L(5)) 4100,125~04100C1794
C17%6 I?AIs.E14G0 IF (d-R’~AX )1450,1410s1410“]61O IV2=14*(1. II*Y(3))
142u cALL WR (39Lsw:)15J0 bO TO 1600lk~o lJO 1*6o 1=195
l&bo *2(I) =w1(I)lLIO tl=.l*rJuP
16?2 IF (L (5) )4150,1476,41501474 KEI=KE1+l
lAdU IF (n-BNdX) 152001520s14YU]~Y() E?=1314Ax1<(JO W2=U~(leO+y(3))1<10 CALL WR (3sLs#$)15.20 *l=ij*(loo*y(l))1%30 CALL WR(.3,Ltdl)1’540 lF ((wl (3)- At3s(CIi(2)))/ThETA-liPS) 155091600,16001%50 IF (B-BMAX) 1450*ltoos160u
CI%96 FINlsfllAOO KE?=KE1.KE21610 L(4)=KE2/L(3)
C1096
C1OY8 ~FJTE(iRATE
20U0 DO 2010 1=1s72nlU S(I)=O.o
C20*8
383940414243644546474849505152535455565758596061626364656667686970717273747576777879808182838485868788899091929394959697
69
3613643703?2
373375
4076214234.?!Y427431433
435442
447450
563
~q50 [)0 2550 [0=1,16Z060 lF([b-14) 2100s2070s2100
2n10 00 i!O~O 1=195?ou(l *(I)=w2(I]2f)d2 KE=ll-IG
20$0 (Jo T(J 2400z]OO IF (ILI-16) ?150c2110*2150
2110 Uo 21<0 1=1052~l?o W([)=hl([)2122 KE=17-IG2130 bO TU 24OO
c2150 IF(Iti-i3) 2160s2160~2200~lb(l XaaO(loo-y(I(3))2]/0 KE=IU2180 Go To 2300
CZ1Y8ZpIJO KE=17-:G7710 x=R*(!.O+Y(KE))
CZ?Y8
22[Jo CALL ‘#R(~,L,nl)CZ3h8 CALCt lhT$GRANDS
24Uo E(l)=A(KE)@w(2)* ExP(-u(3)/Tti~~A)z61O L(?)=w(4)*E(1)zrizo L(?)=w(3)*E(I)
2&30 L(f$)=iu(Q)QE(2)2L40 t(5)=X(3~0E(3)~dbo L(6)xw(7)o~(~)
24b0 E(7)=w(5)oE(I)c2&98
2%00 00 2510 1=197251u s(I)=E(I)+s(I)
c ---NoTE - SU# S(I) IS 1/2 IN!LGRALC2T38
/54G IF (L(5)) 4200c2550s42002550 CONTINUE
C;994C2QY6 CALC UUTPUTc~q%8 lNTE~RAL:
3(’)UO 00 3010 1=2*7
3nlo s(I)=?(I)/s(l)3n2U lH=l./T&ETA2flJo s(l)= 2.ncRe’3(1)Sn.12 b(I?)=T’/ocH(3)
30+0 b(7) =l.!)+f; (Iz)-TM*S(2)20:}2 bo:j)=-T’!OCh(~)s(ISO b(4) =G(ll)*Tl<*s(3)
30b2 u(14)=8.df157662@S (l)3000 G(6) =lo,l+!3( 13)-ALJb(s(14 ))3f)~ll 1~, (3 (3) ):]o~o,3~80,SLoo
3080 CaLL PNId(14A6ES NkG. KAP s01tG(3)1CALL PRrJ(4hTAU5tl*T AU)
2fi90 bO TO 31103]00 6(6) =S(6)-ALOG(G(3))
3I1O L2=TM*TY~120 b(Q) =l,n-F2*(S(6)-S (3)0s(2!))313u u(lo)=Tf.I.@CH( 4)*EZ+(S(4) -S(2) *~Z-THElA*S(7))3140 G(ll)=E7* (s(5)-s(3)+*2)
c3)50 G(5)=13(4)+G(3)-1 .J
98
99
100
101102103lo~105106107108109110111112
.)
.
i13114115116117
118119120]21122123I 24! ?5IZ612712M129130131132]33134]35]36j37138135}40141}42143144~b5146\67148149150151]52153154!55156157 .
.
70
567573
5756.026Ub
61nb~h62062262463?
633fj&]
642
650(j51
6bl6636676“11
b?..j
700
70b707
712
71772Il
10
17212325
2635
~b3740
*14?
43Q4
c4qso
S(loo
c?0
ti(7)=G [(, )* G(3) -l*o6( Q) =6(4)-G(6)
Uo 3210 T=3Q15W4(I)=G (I)
IF( L(5)) 4300,50 uo~4300
DIAGWSTI$ PRINT
CALL ?RII’4 (28H GES f!IAG. ‘TAUO!hLIA$t$ SSO*B)k(l)=TAuL(?)=TtiETAL(:3)=B
CALL PRIN (?H ss3sti)
bO To 1?00CALL Fl?ld (2tiRSsl*d)
bO T() 125{)CALL PRIN (2ilBstl!d)
60 10 1474UO 421O 1=1,7
L(I)=E(I)/A(KE)L(II)=IGL(Q)=XL(lfl)=ti(?)CALL PI?JN(7mI,X*W S,3*E(8))
CALL pRIN(7hINTS 5,?*E)
bo T(J 2550CALL PRIN(?PG(oU71 $t15JG)
CALL FRIV(7hS(IhTlS~S)
HETUqh
Lh(l
SUllROIJTIetE wR(K*L*LI)
Gks CLLL POIENIIALbIMEl~SIOtJ CC (6) *C(6),WN(61sWC(b) SW{O)
1 ,KF-Ps(2) $hMAX(2)lf (0) ~A(bO)tAL(3010D (7)oL(6)30 cQuIvALEtJcE (wN,cc(7) 1,(KuNE,Uf*L19 (Kr~Y)
1 ,(A(31) *AL)
?oO bO TO (1ooU,2OOOC3OOO),Kc OYUc 094 PRELIMo EN~hY
c 996c QYe StT POT. ANO EQulVo AnGS
]s00 IJO 1004 I=Is6.-
]nUZ *(T)=O.171004 L(I)=O.OIOU6 M=30
101O bO TO (1020,1040,11)60), LlnZO KA=IlnZ2 hR=l
103u bO To 1100104O AA=~ln42 I(R=2
1050 ~o 70 1100lnbO KA=llf)t’2 KR=2
Cl(l?oc
158
159]60161
162163164
165
166!6716f3169170
171j72173
174
175176
177~ 78
179180181182
183184~ 85
186
187188
189
234567
89
101112
;:151617
181920
2122
23
242526
27
C11Y4)?UO IF (AN) l;100,1800ti21(l
cb=FNbi3. FMA(j=AhI
KB=KR
NQ=l
bO T(J (1.ISO-1650)OKR
J=(K(J-l)*M
KE~=J+MA(KE1)=G P(G -1,0)/6.0L=A(KEl)/+O!}
L1=l.OE-7*E
CALCC
LJ
L=c-4*k/f4*lllF(E. L-E) I4I3O*148O*151O
CO’iTIhlJC
CALL FIO (3MdOT,
1 53fi (1-I0,12XS32H A-IIIMo TCIO SMALL* A30*l/4**30= 1PE15.71s
2i$AfJ*l) *o)
hMAX(K12)=l6EPS(KQ) =-ALOG(l,OE-7*A (Kcl)/A[KE))/oo6931
J=IK J-1)03
Cc(J*l)=AACC(J.Z)=-SAQG/30fI
CC(J+3)=-CC(J+2)*G/~oowh (J+l) =,l$t
*?4(J*Z) =-413*(j/3.fJ#Pi(J+3)=-wN(J+2)0(j/3 co
00 TO 17mfI
EXPJ=(KU-1)03CC(J+l)=AA
CC(J*2)=-AA/3.OCC(J+~)=AA/qoo~N(J+~)=13sMN(J+2)=-lti /3.0wN(J+3)=1313 /9.0
2829303132
3334
35363738
39404142434445
4b47
464950
5.1
52535655
565758
596061
626364
65666768bY
7071
7273
747576
7778
79
80818?83(I485
8b87
.
.
72
AITNACllVk PART
. 303
30530 f
311313
31+CALC.
LJ3?23L7332334?
3263*i344
EXP~.+5
~!l 13!3.33b23bt3
3b 7
3“?13t4377
4r12
f+oil41Z414
.$lh
42042343(I435
446
45(J456
.
C17~6
1750 bO TO (Itir)0,]9001~K0
C] 796
C]7’J8lnllo IF’ (AH) 5000!5000$1810lRlu b=FM
]n12 bB=Frw1R20 AG=AM]Rdo KH=KA
ln40 KQ=i?
lRbo bO TO 130(1CIQ’+4
10UO lF (L(6) )6000,500(I*4000
CIQYOC1QY2c] Q94 MAIN lNITIAL tNTHY
c]qY6C]QYH REPULSIVE PART
ZOUO lF (AN) 2500s2500~2010
201O b=FP4zrIZO Kd=tiii.2nJu f((l=l
C>n44
c~f146ZObO 60 T(J (?06092150)9KR
C2f154c.2n513
2nho F.=i)(l)*~[-G/3.0)ZOtO J=(K(J-1)0’I~r)du Do 21?0 1=193ZOYO hE=J*Iz1(JU C(US)=CC(KE)*E
ZI1O WC(P,L)=WN(KE)SE
:~.?u Go TO 2451)c21+4CZ146
Zl>cl I=D[l I**’7.33333333c~l 76
z1}Io S=(;OT~~>lj ~= EAF(G-$)2?00 J=(KIJ-l)u”\
?210 Uo 22~o [=1,3Z?ZO KE=I+J
Z7JLI C(Kf)=CC(<E)*C2240 NC(KE)=AN(KE)~E2250 C(.J+2)=C(J+2)*S27fI0 C(.J+3) =C(J+3)0(S@S-S)
C~796zaUO J=(KIJ-1)*42a10 A(,)*i)=S
ZatU Sl=i.O/Sz?~~ A(.J*~)=Sl2~40 A(J.J)=l,I)+sl~abo A(.J*4)=- (S+2.13*2.09S1)~qbo A(J+’%)=/.rJ~(l.~*Sl )~q{(l A(.j+ti) =s0s+200~s+4011 +4000>1z?~o A(J+7)=?.0+3.O*Siq9u A(J+3)=- (?.0*S*4.0+4.O*S1),26(JO A(J+9)=s-sOs
c2644
&i+
fl$J
90SJl~:
9394Q59097989Y
100
101102103104105106lo”?108109110111112113114115llb11711811912’0121]2i123124125120127128129130
13]132133
134135136
13713fj
13?160141142143
144
14514b147
73
461
46747147347s470
477502
505
510
512513515517521
522
5305.J’25345(,1506552
2<<0 A8=KAzqJU nQ=2
Z%*O ho Tll 205fYCZG’94
CZqYb2600 U(?)=C(l)-C(4)~Alu U(3)=C(21-C(5)
26.?0 U(4)=C(3)-C(6)ZA30 IF (L (6) )4100s5000$41U0
Czclwu
c?QYi
ATTRAC1lVL PART
MAIN kNTRY FUR CtLL POTLNTIAL.
CALC.
LJ
6?5f13fl633636641
16814%150151152153154155156 .
157]58159160 .16116216316e165166]67168169170171172173174175176177]78179180181)e2183184
185186187
l$h18919i)191192193194193)9b]97198{99200201202203.?G4?05206
●
207
.
74
657661fjbj665
67Q672
676
701
7Q~704
70671172a7.3( I
743
76575(I753
75h
7647b57hi77U
77;774777
1002
1004]007101+10ibIoc!l1026
iozr103210371~4]l~J.+1051
10521055I06Z
1063
33709327~*3450
/6.o+E2/12u.(J
ATTNAC]lVk PART
L
75
Lfkn .Vn
>u,\uuiJTI,4r rKW(~l
c
*K*
KlsTIAK(l#SKY-dILS(lN EuuATION UF SIATt
c
bKn
------ ----- ----- ------ ----- ----- --- ~~n
c h=l: EOS, CALLED FNoM GEM(2)c
~&m
K=?: PUS, CALLEO FROM xIM(2) hK*
c I(=1
c
bK*INPuT
c
p~a
]$V=rAUtTHETAXGP (l). (i?). (FUN KW~VSiAh=fs]AN=l) hK*
c XI=Ei4X - (GAS) POLt FN4CTI044S -
c
hK*
KI=4PG - COVOLUMES. NOTE: THIS ARRAY 1S COvOLUMEsc
p~*
KR=KIP(I) - N(J. sPtcILs
c OLTPIJT
p~n
c
hK*
AE - X*EXP(\~ETA*X )=7-1.LUCALS USLU UY K=2 ENTRY.
c
~Kh
c
rf(n
FoR KW$ kSTAtiIJ FOR LJU. ~Kh
c K=2
c
hKd
INPuT (SEE dEFINITIONS A(30VL)
c AE FdG)! K=l
~~*
c
t.K*ht.! Fi+c@l Kcl ~~n
c NI=XPG
c
~1(*
N:IESc
~~n1. ONLIKE LJi), FCli *nIc@ MIX PA*T 1> tiONt IN xIM-AIPsc
c
K?.-
FOR K+I EVEREVERYTMING 1S UUNL tlkkk.
c
~~n--..,----- . . . . . . . . . . . . . . . . ...----.”.- .----” --------------------- -------- ).~”
cL
bCJ TU (100,500)!<”
c --------------------------- ---------- K=lS ElJS -------- -------- --------
liJO r!K=O110 UO 120 ~=i*KPlZO t\K=rnK*EPX (I*])*XPG(I)
clbO X=hK/(VG(T+Tti)**JL~)160 t= F.AP(@ET*x)110 AE=X*E
c?00 bM(3)=XF+l.o?10 tiM(4)=ALP*T*XE/(r+rh)
212 6M(6)=(E-1.01/RET-ALOb(GM (3))
~
345b7 .
89
:: .12131415lb1 ‘fIn192(J21?2i?J26i?b2627i?8293031323334353637~~
3940414263444546674b4950515253545!l565758 “
76
.
b?
6!73
105yor111
113
114123
131132
6b
b4
13
2(T21
22
2327
c ..,--.------ . . ...-...”. . . . . . . . . --------- f(=~, Nus ------------------------ HI(I!
~~m
hR*
HI(*~&*
~~n
pgh
~Km
MJC3RUU71NE sm (K9L)
cc SULIL) EO. UF STATE AIJAPIOH
c-.
c NLVISION 1 - FOR ktm SEMS COOt
c w.!=. 10/61L
cc
10?02020
1 (7( 16?()~9(7( 461Jat(z( 16,1.74*(7( 1861-I5J(Z( 1920
ccc
—tKEIY ) ,(Z( 1690) 0KuN ) *(2( 920) sLAK 1,CONT ) ,(Z( 680) SLV ) *(2( 141O)*?LJH )QKAL 1 ,(L( YO(J) !KLV I $(Z( 1R40)*SP ).*SPC ) .(L( lu20)*s~ ) ●(2( 1990]*rM> )oTHER )
LOCAL LQ
kQUIvALf.4CE (CAR (51) oCE)*(CUWl(2) QK)
1 ,(THEd(ll,P) *(THEA (J) *T) s(SP.(l)*VS)LJO TU (lonO,lO)oK
c qY4C qYb PRELIRINANYC QY8
10UU CALL REAP (
IoU~ hAL(*)=]
lno2 1F(sP(2) )1oo4sIoo3s1oo41003 KAL(4)=0
c1004 CALL SE*S(l)
1006 bM(9 )=0.8
1:2020
596061bd63646S66676H697071727374
234sb78
1:1112131*15lb17
le192021222324?526
27i?tl293031323334353637383940414243
.
77
L
ln10 CALL oOuT (3HSEP*1)ltlzo Go T(J 300
ClnyHcc SkS tlAIN
10 AEN(d)=KE.J(z)+l
<0 if (KAL(+)) 4O,1OO*4O
c
40 CALL SEWS(2)c
50bLI TU 200
cc
100 SM(9)=I.Q11J2 Srf(lu)=l.fl~u4 SM(ll=SP (4)
Iu6 SM(?)=O.fl1(J8 SM(31=P*J5/(R*T)
]10 SM(41=0.O
112 SM[S)=S!’(?)114 SM(+l=o.fl116 st4(7)=sf.4(’1)]18 SM(R)=Ppuo CALL oour (3tiSEkf92)
q(lIJ ?4ETURA
CALCO IdCUMPRLSsltILk SOLID OUIPUT
c Al?ij$i I(=I9 P~EL.
c K=2* MAIN
c L=l, lSCTnEI?wc L=2s I>FNTRCPCc
c >PECsc IN~{lTcccc
cccc
SP SEPS IrwUT (StE StPS)
SPC “SEPS CUIPUT (SLL SLPS)T lEMP
Vs ‘SOLID Vi)L
446!346
67
48&9
505152 .
535455
5657 -58596061
62636465
66
6768
69
707172737675747778798081(+2838Q858b87
ccccccccc We F. -19/61cc
NEW SES hu~Ii?uuTIr~t*lTIi SIh(jLL IILRAIIUN FOR V(P,T)
lhPUT -P*[UUTPIJT - V ANd 5LS IMP. TnERHO kNblf=l FOR PriLLK.z FCN M411N
3 COMMON Z(4000)
2345b7t!9
H12131415 “
78
.
.
3
33
ccc
3
ccc
3
ccc
3cc
LQIJIVALE”JCE
1 (S~~(l) OG)$(SP(31,AL~H) $(SP(4) ?VU)r(5P(5)OTO)
2 0(SU(8)OC)s(SP (13) sA)o(sP(14\soI)
3 s(sQ(l%) *(j219(sP(lb) *CONSI)4 Q(sP(18) *ch19(sP(20) 9cb)*(sP(Jo)?c61)~ *(SM(QI*Y)*(SP(1O) *Yl)*(SM(l!)~Tc)6 O(SM(l%) ST1)O(SM(13) sPH),O(SM (15) *WLLG)
7 *(S’J(16) 9P2)C(SM(17) 9Y14)$(>fl(113) 9x)
8 0(5.4 (19),YU)
20 GO TO (lOfJ.1000),KIJRELIM
11 -1110 H=Q03142JF.,513 1 )2 A=(<*SP(?)/[ALPI++$VO))~ (C/(G+l:U))22 110 (11=1.0/(,;+1.0)2s ~<o e2=().5*1.()/~
c31 150 LN(lI=GI33 160 UO 1*O T=?,435 Ilo PI=Idb 15(J LN(I)= Cd(I-l) O(Gl-(F1-IOU)) /~1
c45 700 Uo ?jo 1=),s47 ?10 Fx=[-152 ?c’U C(i(I)=C(I)/(G*FI)56 P30 CGl(I)=C(I)/(G+FI*l,Ol
cO* >50 Ll=n.oa!! ?60 L2=0.Obb ?“/0 00 ~+f) T=\957 L) 7uU LI=E1+cG(I)“r3 >yO L2=E.?+CG1 (II76 3U0 CONS r=O.5*El-G2~E2
c102 60 T() 20(J0
c
. c MAIh
1617Ifi1920212223242526?7?829303132
333 t,
35ah373839404142
434445
66
474849
5U515253
545556575859606162
63646566676569707172737475
79
206212
216
231
c1000 CS(*I=YInZU (,S(2)=Y~FOflI!n.10 Yu=i.O+.$LOHe (T-TO )In+l) fic$(i)=q
c
c IILKAIL ON Y=V/VOIluo c4L1. FRoI)TT (CS,NUNI)
llio hEXIT=KcS(2)11<(J WG TU (l%(jG. ]i50*li30, i14U)*K~XlT
i~dO CALL dF].J,; (eHsEWS,i]1132 1,0 Tl) is,)o
Ii*! CALL Cti(lli (4HSEPSQ1)]l+c? Go TiJ i~tlfj
c FuNCTIUhl]bU If (CS(Z)-YU) lll!osIldo~l160 -
llbo CS(2)=Y!Ill)U GO TO 1200
i~80 If (YL-cS(~)) 1200~lZOOOl190ilYu Us(?)=yL
c1700 Y=cs(.z)
CALCC P2(Y)Ll>in rti=c(5)~>21J Li=cG(5)~2jo L2=CU1 (%)1250 [)U 1.?90 1=1,4iabo fiE=b-Iis/U VH=PH*Y*C(KE)
i7su Li=El*Y+cG(Kz)l>~o [email protected](KE)
c1200 wEEG=(o.5-G2@Y) *PIi1I1O YG=Y**G
c13d0 r2=- (G/Y)uwF.EG+G*(b/Y)*
1 (0.5@Ll-b2*Y*E2-CIJN3T/YG)cc CALC Yl~ Yll, SLPARAIE oh Y
~q~O lF (Y-i.o) 1700sifluO$lt100
c
:]400 CS(3)=Z.!IE-6* (TC-T)
c-1470 CALL UOIJT (4tlSENS,l)If,du bO TO 11OO
c
CALCULATE OUTPUTsM(i)=Y*vfl
SM(7)=P*5~4(l)/(R*f)
>14(”i)=(!.,l+i.O/6)*Sw (7)*’40*UEtG/(K*T)-sP (L)*(l.o-10/T)SM(6)=S@(2)~ALOC(Ti/T)
.
>M(5)=S!1(3) -S!4(6)
st.l{2) =s!4[3)-sM(7)
SM(4)=SP(5)-SM(7 )GO T() 2non
CALC: FN. Ful? Y LESS IHAN 1
76
777879806182838b85 .
868788H9 .
;;92Q39495‘+697
9899
too101
102
jo3104105106
107lGa
109Iioiil]i.?li3114Iis116li71181191201211221?31?41?5i?b127i?n12913013113,3133134135 ●
.
80
jn ]
3103133’153173k’1
32/,
.331334340
.34134334b35(]352X54361365
370373400
401
c17U0 ~=(p-i72)*YG~Y/A1710 1G1=1oO*X172U Y11=CN(4)OX
1730 Do 1/40 1=1,31YJ2 KE=4-I1T41J yll=(y:l+~N(~~))*x
17~2 Yl=Yl~*l.Oc
17’JO Tl=Y1l/ALpH*TO17hU TC=Tl*(Y,i!/Yl)/YG
:770 bO 10 1400
cc
lR\)O A=o/A
]Rlo Y61=IOO*X]q<O Yl]=Ch(4)~X1P3U Uo lmso 1=1,3
in’+u hE=4-I
lI?bO fll=(Yll*CNIKE))OdIR60 fll=Y-l.[)*Y*Ylilq62 Y1=Y1:+l.9
c
IR70 il=Yli/ALPH*TOlqmo lc=T1*YGl*Y/(Yll*l.nl
1a% bO To 1400.
402
c
:cccc
33
33
c3
cc
CALC. T
CALCO t’h. FUR Y OVt.N 1. .
CALct T
LNr)
sUHROUTTNE TIP (K)
ITF AUAPIUN
REVISION 1. FIX TIM FoR TIMs KLvIsIoN IM.F. DkC. 61
IIM
COM/~Uh Z(40(?0)l)If!Ei#SIO.J
1 CONT ( ?n) sEA ( dO(J) QLMXz,li~ ( 10) ,KEN ( 60) SKIM~sTMER ( qo) OTM~ ( 20) QTMb4*TP (?11910)
UIMENSIf?14 KEITH
EQIJIVAI-FIICE
1 (7( .$ho) 9cONT ) O(Z( 4aO) *LA2?(7( 15qo),*E ) .(1( 1b20)*KkN
3s(z( 19?,II,THER ) O(Z( 1990) *TMS49(Z( 20101*Tp )
LOCAL E.J,LJI~
( 20)
( 10)I 20)
EQUIVALENCE (KIP(2), ,KS~$(KXM;4)SKN) S(CONT(3)$A1M1
1 ,(THER( l)*P)9[lmEi-! (3)oT)
98
3 100 GO 10 (1ooo*2OOO),K
c 9Y6C 998 PkLLIMIhANY
l~b
137!36139]40141]42\&3144145llbb147lad]69150i5115?153154155156157158159lbO
:61162163
164
2
3L56789
10
::13141516
H19202’1222324252627?82930
.
81
~(,‘2(1
6674
100
lnuO CALL nEAP (
1 60HOX KS,KN/ T tlCUlvOS/ FIT COLr~So-Al IC At{tDsIJLL,hFstlR. o.S:.*-2!KE)
CALL tiEA~(o.~,Th(4))
101o KS=fiE(l)lnzo NN=KE(2]ln30 KEl=KrN.~ln40 uo 1{170 I=l,Ks
ln>() CALL ifEh# (o,KE1oEA)106O UO 1070 J=l,KE1
111/O lP(I,J)=Es (J)tiO 10 J(3OO
C]Q96
C199HZOUO KE14(.I) =KE?.1(3)+1ZO1O lh(l)=T
2060 KE(l)=r(szfl~~ hE(/)=~&
2nu0 lH(2]=P/ATM2090 lh(31=tiE(~)
MAIN
cc
1(-)1
cccc.cccccccc
c
ccc
cc
cc
c
c
1616
16
Ltwn
SURI?CJUTINE TIMs (KsTM?A*X*G*F)
luEAL GAS TnENrIIJ fNS SUdRCUTINkREIJISIfIN l.-ccNsT. LP LATLft51uhs
.4. F. OEC. 61
K(I)=RS, NO. UP bPECIkSK(?)=KN, UkGWt~ U* }11Tri(I)=T IN LILbNLks A.T?!(.2]=PINATM,
Tli(3)=T Sud ZtNulfi{4)=TMIfJTt-(5)=T~AX
A- CUEFFICILN1 MATRIX (SEL tiHITE-UP)
X - AULE ~uACTii)hS (%(1) FUN bbLio)
fOEAL 63S Tt.EP.:-lGoVNaMIC FUi4CTlUhSI:\PUT
obTPuTG - TUT,AL lmEkMU kuNcTIOhs FOR bAStSOLIC~
- FhEL LNLw61LS AT i$PKLLATIVL 10 ELEMLNTS AT O KELvIN
i I\~s
COHVON 7(001)())uIMENSIOF4 K(2),1H(5)9A (2Us10) sX(2U)*6(~O)sF (20)
1,EA(50), C(20), H(.?O), S(20 )sGI(dO)LQIJIvALF:4cE
1 (7( 4dIl) sEA )
1 9(EA(1) *c)9(EA(dl)om) s(kA(41)*Sl2*(EA(61) ,61)
313.?33343536373839404]4243444546474849505152535455565758
2345b78
1:1112131415
:;lb19202122?3242s262728293031
.
.
.
.
82
..162 !1
2123
2534
36
36
61
4b
::
60
6166
66f’i7
71
7.3
7577
1(1S
1U7
1151?4126
127Is?
135140
155
15/165
200
213
215~17
730?3{)234244
251257
301
304
3i2
ccc
c
c
c
c
c
c
2k3 @=~.’?87l9L-34 KS=K(l)
5 N=K($)6 T=TH(l)
i’ PLOG=ALOU(TH(2) ),B Tq=T~(3)
lPyh=TH(L)TM4X=TH(5)
9 I(E)10 no 20 I=l*a(J2U EA(I)=O.U
tll)UhP 11 !U FIT KANGL30 lF(T-TMA<) 60s60*4040 I1=T,MAX50 bO TU 1106(I lF(T-TMI”+) 70~70*9il/0 ll=TMIN
bO GO T(j 11o%0 T~=T
]00 1
110 Uo 2+0 T=I*KS)12 (1(E))120 J=rJ
l.icl FJ=JJ*U AIJ=A(I*.J+l)]50 ti(T)=(ri (I)*JIJ)*Tl160 C(I)=(C(~)+iFJ ●1.O)*AIJ)*T1)70 s{I)=(S(I)+((FJ*l.U)/} J)*AIJ)~Tl
180 J=.J-1l% IF (J)13U,21O,1JO>,)0 k Al)U FlhS! IEN.MS
?10 AIJ=A(l$l)?<0 H(I) =K(l)*ALJ730 C(I)=CII)*AIJ?4u s(I) =s(l)+AIJ@ALCG(Tl )+A(IsN*~)-PLUb
c AIJLl CUNST CP FUNCTIUttS CUTSIUk?’Jo IF (T-T1) 2h0,~90,<A0ZOO tl(T)=( llotI(I)*C(I) *(T-T 1) )/1~?o S(T) =S(I)+C(I)*ALOb (1/11) -
c?Y(J k(Tl=H( I)-s(I)+A(ItN*J) /(N*T)
c 70U I(EI MJxTuRE SIJMS?10 Dc 360 I=?,As
712 X1=X(I)a.?U bI(2)=GI(2)+XI*(H( l)*A(Iti*+3)/(hOT))730 ti!(3)=GI (3)*X1°L(Il
FXI=O.OlF(XI.GT. o.fl)FXI=XlOALOG (J.1)
Gl(~)=Gi (+)*XI~>(I)-P XIGI(5)=[;I (5)*X10A(ISN+4 )
6) =GI(b)*XI~(n(1)-( TO/L)*A(I?N*5 ●A(l*N*4)/(ROl))
323334353(!37
3ti
3940
4142
&344
45464748
49505152535455565758596(J616.?63646566676869-/071727374?s7677787980818283868586878889go91
SIJI\N(JIJTIJ[ XIII (K~L)
c GAS (PIXTIIRE) ECUSIIUN CP STAIL Al TO ps X1
c I(=1 PEAC II*FuI OA!A (r~cL.)
c K=? L!AIN CALCULAT1ON -
c L sPEcIFIEs PCRrICN OF tQ CUl)k PI?(IM. ‘.4tiIcH AIM IS LALL~U (StL *RITLuP)
: Ihrbr
c I*P=TbEk(3) Q(l)
c AI - (GhS} POLL FRACT10N5c (ILTPu1
c AMT - M1ATLJ14E STATL
c AMII - MU*S
c 14cuTT:,ES
c %.IM> - CE741LED CALCULATIONS ~o~ LH* CS~ l-FLUIOCc CJLLS GEP(.2) IN PiuDLE FUN NLF. bTATL FUR MU*Sc uEP(I/~) - PURE STATE pcIrIT (NkbuLAR/LH kAPANsXUN)
c --------------------------- --------------------------- ------------------
ccc
ccc
~
6
b
a
cc
c6
W.F. Y/ol
Altf
cos4~~uN 7(&ooo)
LJI~ElwSIUYI CONT ( .201,EA ( 200),EMx (z*GP ( ?o),CM ( *o)cG+T ( 39~,K4L ( ZO)SI(IM ( 1O)$THEN (4*XIT ( 3ol*xHlJ ( iio)*xlJF ( 20*SVXPG ( .209 20)*XPK ( 201*XPT (e , KLh(k. lo)
01’!LNSION KE(2),E1 (6)
1!SXC(20),SXF (20)LUUIVALF../CE
1 (7( 66))9C9NT ) *(L( w80) JLA ) ,(2( 117(J)?LMX )2?(7( l+?())9c~ ) Q(.Z( 1’+Y))?U* ) *(.Z( 15301 *bMl )
3,[7( lb.lol*KAL ) *(2( 1080) *KIN ) ●(Z( 192U)*TMCR )4?(7( 221n),xMT ) $(Z( Zd4d)~XMU ) 9(1( 226u)cAPF I5?(?( 266,\).XPG ) ,(Z( 3u69)*APR 1 9(Z( 3i-lf3u)9APl )
6 *(Z [I02(})SKEN)
LOCAL EO,llId
LQ!lIVALE.iCE1 (COINT(?),R) *(GP(5),RSTA) *(GP(o) *Ts[A)* (eP(7)*v51A)2* (<IM(l),KR), (ThEN(l) *P) 9(TMLR(J)9T)
3? (XMT(16),RSTAT) *(AMT( 17) QTSTAT) *(XMT(16)$VSTA r)
92939695
9697
98
1::\ol
2
3-4
5h
78
1:11121316151617lb19202122232425?627
2629303]32
333e
35
363738
3940
414243
b4454667
4e49
84
.
.
(1
14
1722
:;343641J43
].L6~31)134
140
141
175l“!?201203265297211213215
4*(11 A(21)t Sx(j). (E A(41)~SXF 1 XSf’
c ~IP
tIO bO T() (1 OO,NOO)SK ~IPC ---’ ----------- ‘----p~tLIY ------- ‘--- xII’
~00 CA~,L HEAP ( ~IP,l!J4~11~A KG,NAL/ SC@*SCT*RoRkF*T*KtF*N~M/ (R”)/ (To) S xl?
1 *-7*KE) XIP
CALI. NEA(~(O*6*El) XIP
CALL KEJP(ooKEoX~14) XIP
CALL IJEAP(o,KE.XPT) XIP.
I1O KR=KE(l) XII’.
1?0 RAI.(5)=KE(2) x~*
130 LAO I’JO I=I,KR XIP
l~o AP~(I)=XP$? (I)@El (l) XIP.
150 APT (I)=xPT([)oE1(2) x~h
C 16U XIP.
170 IJO ?00 T=I,KR X1l’.
lc!O LJO ?!)0 J=l,KR xl!’
,~0 APG(I,J) =(XPR(I)+XPA(J) )/<.0 XIM
?,)(A APF(I*J)= XIRSQRT (XP1(I)*XF’I (J)!
c ?dtlP1O AMT(ll)=E! (3)
XIP)(1*
720 AMT(12)=rI (t,) AIR
>30 AMT(13)=E.l (S) xII”?+o AMT(14)=EI (6) xI*
?5U IF (KAL($.)-$) 3000~60s300 Xxt’
76d uo ?/0 I=l,KR xl?’.?/0 APn(I)=El(3)0c@hT (b)*XPR(l)**~ Xlr’
quO bo TO 6nO!l xIh
c .- --------- ------------------- -.D------ MAIN ---------------------------- xIP
suo KE=KAL(5)+I XIP.
~u2 KF-’1(4) =<F:F4(4)+1 x:x
c CriECK FOR KIASIUWSKY-MILSON XIR
Ri14 IF (KAL(%]-9) 81000ftb*810 XIR
a06 CAI.L GEP(I) XIP.
pOd CALL PK4(2) XIM
c XI!’!
R09 GO TO lIO(J )!lM
c XIM
R1O bO TO (l,100s2000 *300Ut40UU SOOUOISKL XIP
c 0$0 NU PIA (ZLRU)c qvtl
]q,JO GO TU (] o?O,6OOO,11OO)$Lc~lllo N() MIX.-ONL
IO.?(I LALL GEp(I)
In3Li IJO 1040 I=I,KR164(J AMII(I)=5.Y(5]ln50 GO Tti 6000
cln98 N() HIX-Tr+MkL
c MIXTURL=RCF. FLUIDlIUO AMT(l)=G1-~(lS)
1]10 AMT(z)=’<.1(4)11~0 XMT(3)=G 4(17)ll~u AN!T(*)=G.4(20)
1140 XMY(’J)=S ~(5)~]SO AMT(0)=G4(11)l)oil AMT(7)=G14(1O)117(J )WT(H)=(jr+(Q)lldO 60 lo 6000
50
51525356555657585960616263646566676ti697(J7172737475767-778798081828384858687888990919293949596979899
10CI101102103104105106107108109
85
.
c . . . . . . . . - . . . . . . . . . . . . . . . --------- . - . ” ------------- ----------------------- XIP
lDtAL -(JNL
IJO ?110 Ta~*Kl?rsTJ=xPT([)HSTA=XPN(l)
VSTL=COkT(51*i/ST~*03LALL EEP(]I
UF’T(iOI)=GMIIS)bMT(z*I)=GM(17)
bMT(3*I)=Gt4(%lAMII(I)=G4(5)UO Tu 61?no
lUEAL -THKELxMT(l)=n,n
AP.T(.3)=0,0AMT(5)=~oIIIJO P/:0 !s1,KR
ANT(l) =C.!A(I+l)O(,M! (lsI)*AKIT(l)
AMT(3)=F.1>.(I+l)O~j~d I (~sI)*A!4T(~)AMT(5)=EAX (I+l)*bMl (3$I)*Af4T(3)
c -------- .--. . . . . . . . . . . . . ------------------------ ----------------------- XIF.
c:Qby Lh (TwO)
C2Q98
Xltlx~r’J
30U0 IJO To (lI)lo,3100t3400),L xIRc~,’)lJ4 Xluc~fil,(l Lh-UNE XI*.
201O RSTA=XMT(lI) XIP,
3nc!U lSTJ=X14T(I?) xII’
3nj0 VST.i=CONT (5)0RsTA**3 yjmS040 CALL GEP(I) XII.
c xIR
c LFI-lwO )(1P
~IdO CALL XIPS (1) XII’SIZU bo T~) 600n
c33Yti
~Iw
LM-THNLL XIP$3400 CALL GEP(2) XJP
bO TU 6!)J,) x~P,
c -------- ---------------- ------------------------ ----------------------- X1*
c CS (lhtiLE) ANO ONE tLUIC (FOUN) XIP
4nO0 bU TO (~I~10,4100,6uOO)OL XIP
C4n04
c4fibb
xIh
CSO l-tiLIJID - OhE :IP
6n10 CALL xI’~:i (0) xIn
4.120 TSTA=TSTAT X!P.
4030 RSTL=ASTAT XIP
4n40 vSTA=V5T6T ~JP!
4050 CALL EEp(l) XJF
c XIE
4100 CALL XIrJs (1) XIRCss l-1’l.uio - Two
110]11112113114115
116117
118119120
]?1J2?
123] 24
125
126l??
128]pc)
13V
131132] 33
134135136137
1 3a139140
141142]4314k14514614?14&149]50
151
152153154
15515b151
158159160
161162] 63]6416516b167
168169
86
405 bO TIJ 6nO0
c407 etIUO CALL 00UT(3hXIM*l)
414 efil(l HETJNN
c
41s ktwn
. SUNNOUTIiE XI(4S (K)
C ThIS ROIJTINE hAU HAO ~WCMES lN WL 73s WWWJ FOR PhulObTUNE
ccc
c
c
33
333
c33
cc
3“c
3
cc
PERFORM [)ETAILEc cA~cULATIONS FUN XIM
AIMS
co~’4oN 7(4000)UIMEINSIdFJ
1 ZPF (20*?I))9 X~G (2Ut2019 CONI ( 20)
2,Ea ( 200) tE’qA ( 20) Stitl ( ●o)
z9G~ ( %n) ,KAL ( 20) *KIM ( 10)4$X41(J ( 20) ?~~l ( 30) 9LV ( 20)
UI$tkNS[tllq F11LK(15)UIM}.NSIC+ SXti(20),sX~ (20) *s~h(2U) *S1N(2U)EQUIVALENCE
1 (7( 226L)),XPF ) ,(L( 2060) tAPu ) 9(Z( 4fiO)cCCJP.T )
2?(7[ 48,J)ttA ) ,(1( l17u)*Lf4A ) $(Z( lf090)*bH )39(Z( 147))9GP ) ,(.Z( loOO)SKAL ) 9(Z( 168U)QKLM )4$(7( 224,1),XMIj ) ,(Z( 2Clo)SXPT ) 9(Z( 680) *LV )
LGIIIVALE,ICE (KIP(l) QKR). (K[M(<)$KS) s(KIM(3) *Kcl, (KlM(6) $Kh)
LQllIvALEo~cE (xMT(lo),NSTAI )$ IXMT(1 7) ~lSIAl) S(xMT(lul,VSTAl )
1? (GP(5),RSTA), [b~(o)trSTA) 0(GP(7), vSTA)
“/04 ‘TO 7ilq0 (N,”PFdk O(WE F1.UIIJ)LQUIVtILFi~CE (XMT(1)*81LK(<))
EQIJIvALEwCEI(EA(?l),SXG) O(EA(41)C>XF )s(E4(61) tSNN)9(EA(81) sSTN)
90 CALCULATE MAR- TO*RO / HOo II.*
3s
71?13
14lb
100 LJo 130 1=1*KRIZO >XF(I)=O.(1130 kIxG(I)=[).fy140 rsTlir=oo,l
150 HS.TAI=O.Oc 190 L-BRANCh- CS OR 1-FLUIO
PUO P.(2=<AL (51
?10 UO To (1500.300sjou,euo) tRQc 7s0 CS(Ot? Lh)
-aUo Uo 350 1=1*KR~lo 00 330 J=l*KQR2U 5XF(I)=ft~X(w*l)~XPP (I*J)*5XF (1)qJO SXG(I)=EMi (J*l)*K#b( l,J)*hXG(l)q*O lSTAr=E~{ (I*l)*SXF(I )*TSTAT350 NSTAT=EM<( l*~)*SK(j(I )*1-!STAT
~bO UO TLI 7flJc ?90
4U0 Lv(ll)=ll.o410 LV(12)=0.O
ONE-FLUIO
170171172]73174
175
2
3
456
789
1011121314151617;8192021?223242526?72829303132333U353637
3639404142
~344f+s
66474$369
50515253
87
1170 bO TU 1500Cll$u C5
1300 00 122n T=l*KR
l?IU STN(i)=?.n*(sxF(Il/TSTAT-l .0)1220 SRN(ll=7..:l*(SXG(I)/tiSTAT-l .u)]?iU b(j Tu 1400
Cl?$o OIWE-FLUIUlauo Uo 1320 I=~,Ka
1310 STN(I)= -200-2.09EJ*lnILK (15)0SAF(l)/LV( 11)1 -HILK(l*)*SXG(I1/EV(lZ ))
.1?20 sRrJ(l)= 2.o*E3*(sx*(I)/Evlll)-sAb (1)/cv(lzl)C1’?$o UCJOIN ton MU CALC.
1.AUO IJO 1*1O I=i*t(RI&l(l AMiI(I) =x~T(5)*300*ANl(7) *S~.l(l) *XM~(2)QSTN(I)1500 RETURA
LNn
12*60
)
!245556
515859606162 .
63
64
6566
67.
6869
7071-72737475
76777679
808182
8386
8586
87EiFl99909]
9293
94
2
34
56
789
10111213
::
:;18
.
88
2,(?( 1350),EPC ) .(Z( 1.1601 ?LP\J ) ,(L( 31ou],?LAR )
3*(7( 1600) *K&L ) ,(Z( lb80) *KIM ) ,(f( 16qu)*hUlW )
cLQUIVALS!4CE (KIP(21,KS) t(hIM(3)tKC)
2 *(EpA*KEPA)t(EPAl cKtPAl) i(LPALtKEPAL)3
cs (EPc*KEPc)
L(JllIvALFNcE (FLAH(49) *L)
c
c80 hAL(6)=l
100 CALI. HEAP (
1 loriox C*SSPt-OPPJPtiI / CAP Q S2 s-6,KCPC)
C21L RE.’ILI (o,KEPC,EPC)110 ~c=Kt.Pc(l)
1?0 fis=KtPC(2)1S0 HLAI)(l(J! IGO ) (Ktl(I)*I=l$l~)
1“0 *ofJ~AT (1?,!6)150 Urj lbo ~=l,Ks160 HEAI)(lO,I~O ) L(l) *(KLPAL (I*d)*J=loKcJ1~() fOfJf~Ar fAh.1116)
I’dU REAll(10./o0 ) (KLPA(llsI=IsKS)I%(J XEAI)(10,?!IO )?Uo FOPVAT (1215)
c~Llo WRITE(9*310 )
310 FOR~&T (61.o720 U() 3.30 I=19KS730 *RrTE(q,.{%rr )
1 ,<EPA140 FOPMAT (6H
c &’$(J6(JO Uo %1(J 1=1910~lo LPc(I)=KE~cfI)520 On b:(l I=l,Ks%30 t-p&(I)=KEIJA(I)<40 LPA1(I)=<FPA1(I;55U do s7fJ 1=1,)(s5~0 LJO 570 J=l,Kc<Io LPAL(I?,J) =KEPAL(I,J)qo(J KA~(6)=l
c %*O
6U0 cALL EQPS(KUN (ls8)9EPC*EPU~EPAL~LPAL (lQd)cFPAsCP4L)c ~vo
l(joo KETUNh
cLkr)
(KLPA1(I; sI=l$KS)
(KE1(l)$I=l$Ks)
sl<A6/)
L(I) s(KLPAL (L*JI$J=l*KC)
I)cKEPA1 (l)s&b,ll16)
cc
ccc
c
ccc
PsT=TPEQ(I)Q(:j). INFLICIT FOR GLP VIA AAM-XIMS5H - STATF GF SCLIIJIMG - Ir)EfiL STATE
OUTP(JTXI=EMX - (GAS) POLt FRACTIONSAM~ - GAS (MIXTU*E) STATE
VANIAI+LES
1s
2021
?22324252627?629303132333.353637383940414243444s46474.849
5051525354555657Sci5960616263646566
2345fJ789
1011
89
ccc
cccc
11
1
c111
c
l?c)*E’tx (~6),fjM (
69 10),KLY (O* 10)*SH (
20),XMT (
) S(Z( 117(J) *CMX
) 9(L( l*qu)tuf4) ,(Z( Caoo) 91fkv) C(z( ld2u)9sM
) *(2( Z21O)9AMT
LQlJIV4LEi!CE (F08(15) QEPS)9(I(IM(2) *KS)LIJllIvOLE!JCE (GP(5)!USTA) 9(XHT(16) 9RS1AT),., .. .. . .- —— - . .
1 100c
3 ?006 ?10
c1(1 3,J0
c
13 G(JO15 %lU20 %20
c26 ~i)o
c25 R,Jo
27 21(J31 R2033 R30
c3 lb Qbrl
c34 930
CALL CO(JT (.jriEQN*ll
I-Jo S/0 I=19KSEMR(l+l )=r%G(I+l)+~PU (1)LA(I)=EPX(I)
if (AAL(+)) RO0920U09800
KE:l(L3)=KFN(l~)+l
CALL EQPS (KFV,tMXsEMN9tNu)
60 ln20 LV=EV* .tIt4S(E~4X(I)-C4( I))e4 103o lF (KEN!131-1) lo’Yu.iu5001040
In40 EM<(I)=(EMX(I)+EA(l))/2. O;; 10hfJ CONTINUE
d.lCK MLkE FUR INNER (ALLI
SKIP FUN FX. COt4Po
tQUIL, XI Al ~IXkIl P-l?LUt (tPG)
121314
15
lb
17
10
;;
21
2223?6252627
2825
3031
3233
34
3536
37383940
4142
434465464748
49505152
53565s56
575t959
6U
616263
666566
67
2:
7071
.
.
.
.
76
101~cl+~117
?10114~.?l
123126
131
132
11
1
111116
1012141517
2022
CALI. Cour (3 HEo P,3)
IN:WLR C(JNVEHGLNCElF (EV-F,IH(16)) 13(lo*1300~lllu
CALL xIv(z92)bO T(J S()()
UuTtti cUNVCntiLRcLlF ((KAL(S) -3)O(KAL(5)-4) ) .?OUU01J1LJ*20U0
IF ( A8~(NSTA-RS rAl)-LPS) 1.32u9dOU92Uuv14=x..tT (I)
CALC FlhAL THLRMUCALL XIW(2?3)CALL CouT (.3HEoMQ3)
.
ccc~
cc
cccc
cc
c
c
c
c
SIJHRUlJTI-iE YES
MIXTUhE EQUATION UP SrATL CUNIRUL
CALCULATE EQuArIOh uF SIAIL A1GIVEN T AN[) P
ULPLALE (LALL SCM
IF SL> NLUULRLS 1INPUT
led, - PST (IMPLIc~Y)O(JTPUT - sEE cOUT
tiEq
cc’4:~uh 7(6000)“Ih\~f”sI~l~ KE$4(60)
LQJIVALF4CE1 (7( 10?q),KEN 1
KE”I(ll)=oheld=,)
hEN( 13)=!)KE’4(h)=t<F’1 (6)+1fiE’J(l?l =hEN(14)*l
CAl_L COUT (3HM:S*1)
CALL TI$A(?)CALL SE~ (2.1)CALL EQi+
CALL TIv(2)CALL COUT
CALL COLJT (3HMESSZ)KETURA
7273
747576
77
7879
80
81
82
8384858687
888990
234
5
6
789
101112131+151617lH192021L’2232625262728?93U31323334
35
3b
373A394U
>4
c
ccccccccccc
:c
PIJTtNIIALs)
( z(l)[ 20)( 10)( 20)
) 9(Z( 103O)9CMN )) ,(1( i3qO)*FN )1 9(Z( lb130)*KKM )) S(2( 1990) *INS )
2345b
7b
1: “
11121314 .15
1617
1s19202122?3242!J?b?7282930313233343536373839404142434+45.%6.474849505152535455565758
.
92
cccccc
140
cccccccc
ccc
cccccc
33
3
c3
5PECS
INP(JTPsTsROUTINE OuTPU!s
o~TPuT
THER-E CJE}.STATE 1S LLE?41mTS AT O K.
tlOLE NUWdENSkhn
SUnRWTIIIE POUT(K)
PRI+sT OUTPUT
$(=1 PRINT LAF3ZL:
K=2 MAIN PNLNT
I(=3 UIFFEGLhTIAIION PNINT
REVISIUN 1. Abo PUNCFI wlru~ (KAL(lI) ~!~)u. F. lu/16/bl
rJolJT
?o),FN ( 20),GM (10),KAL ( iir))9Kk& (ZU),TnLR ( bo)sX~T (
8) ,61r4(iu)
EQUIVALENCE (KItA(2).Ks)
5 ‘+
606162
636465
666768
69
707172
7376
75
7677
7Lr7980818283
81+
234
567
8
1:
11121314ib16171819
202122.232425
26?728?93(J3i
:2
33
93
3
1217
243il
3?
33so
55
0572
7375
124
219%i~
215
216
271
272
301304
306
307
326
c10 o(I To (lor)$lfiot300),K
c ‘j 6 LAI+EL~
c Yti100 *l?ITE.(9,7~oll)
120 *RTTE(~*”7120)
1’+0 wR1rc(~,713fl)lF(o.EO.0) 60 TO 4U0
c --- SKIP Purwcm142 IF (KAL(lI)) 40091bI)0400
c,S0 wRTTE (7,7150) ((FL~b(Js Il~J=l$12)sI=l *4)IbO IV~[TL(9, ?i611)1(,2 *RITL[9,7162) (FLAti( Ig5)?iziJKS)lb~ wR11L(9,7164)]/u ULI To 40{1
Cl/b RE(;,PNI141Cllkj
l&10 AE>I(9)=K}.0!( 9)+1200 *I?ITF. (9,7JOI)) KEIJ(Y)
1 , (TIiER( IlsI=1,5)o!{C(L)? Thcrn(?)
z , TrFQ(e) *l*LR(9),THLN( lJ)9[tP:d (i) 9L=l*310LMr{ (71
?~O wt?TTt(9,7~lfl) IHER(12)*THLH (16)0]rhFR(zo)t7HFfl (~l)~(FN(I)?I=l ~31~Xtil(ll*AMl (-2).ZART(h),X4T(/),(Fh (lltl=4*b)sbM(lb) ;QM(4)*bM(lll$
3uM(lI) l, (Fo!(I),I=l*Y) *5t4(l) ?SM(2)*SM(0),5M (7), (FN(lls I=Lo*:2)
4 * (~~T(T*15 ), 1=1,3)?40 CMI.L cuur (*,4POL1*I)
iF(f).EIJ. )) GO TC I+UUc --- Sttrp >LJfqcli
?42 IF (KALIII)) 400*25fI0400.
4* (FN(I)QT=1,12)UO Tu 400
OiFF. PRIN’
*RITt:(9,7”10i]) (TflEiiCALL 00IJT (4}{PouI,z)lF((l.EiJ.(1] (;fl TO 4UU
‘--SRIP PIINCHlF (h~L(ll)) 400,320tc+o0
wHTTP-(9t/~21))
4), IHLI?(5)(lT)!ThER(7)HLH(15)
i)sA=22,i?5)
34
35
3b
373839604142434445
Ab67
4849
5051‘i.?535*5s56
57565960Al
b?
636465
6667
6&6971171
72737*75“?b
777t47~
80818.?
S38485
8687Ril89
9091
9293
.
.
.
94
.
327
ccc
ccc
c
cc
ccc
ccccc
c
66
6c
6
c
6
Lkn
>UIIRUUTT.IE PESc(KG$AsUc)
CALCULATE MIXTURt CCUATIUh OF S!ATLAT GIVEN T AhC
P FUN KG=lv FCN KG=<
s FCR KE=d
E ticR KG=*
ITEUATE oh T UNUER LONTKUL U* FNOUTO
USING MFs FOH FUNCTiON LALCtJLAIION
ALL USt sAt4k FRO(JT LALL*BkAhcH oN PUNCTION.
)
~il95
9(Y979699
100
101102103
10410!J106107JON109
110
111112
11311%
115116117118
23456789
1011!2
1316
151617la19?0
?122?3?*?5?6
.272d?9
303132
33
3a
95
1
11
cc
111
LQUTVALENCE (THE14( l) tP)t[!HEI+(3)tT) t(THkN14)tU)kQuIVALEIJCE (THER(5),u), (1HEN(6) *V) ,(TtiLR(14)9HH)&QIJIVALE14cE (HE (.?) sPO)D(HL(6)tVO) 9(ML(7)9HO)
2>3637
3h
39404142
4344 .
45464748 .49
50515253
5455
56575h5’46U61
626364
6566
2:6970
71
72-/3
?3
;67a9
10111213141s1617
181920 .
96
1
1
..
1.?46
1!1314
1721
3134
3540
41424447
5154Sb
576?
b465
75104~(1(1
12112412513013’0
135
cc
ccc
cc
46
81012i!u
3040
4251J
60627(Jtl?21576&l
~.?~3
t!4d!y86
90Y296$3
lUO
llIJ\co150151~oo
cc
cccc
cc.
:ccc
c
LQL)IVALFNCE (FO@(21) $TL)*(FOd(2d) tlu)
LO(JIVALEIICE (CAR (ll)sCH) f(Ct4(M)QKCti)
CALCLILATE HuGONICT PUINT AT blVt.N Ps
ITE~ATE ON T LJN!JER LONIH(JL Up F140uT,USI(JG FIEs IN FUNCTIUN cALCULAllUN
T1. AND TU ARE f30uNuS ON IIEHAIION T
riErj(14)=q
REII(15)=KEN (15)*1
hEN(7)=KFh(7)*llF (1)21’l,lz,zo
T=?OOO.OCH(4)=T/1000.ttcl{(l)=ll
CALL FRoI)TT (CH$KON(2s8))
hEXIT=KCd(21bO To (17n,”12,70,60) 9KExIl
C41.I. 1.)F311.; (JHHUGOI)GO TU 15,)CALL 081.II; (3HHUG,?)00 TU 151)
IF (Kopd (2,8)) 81?75,81lF (hcr{(~)-%) ljl,7fJ,dlCH(zl=Cn(4)-CH(5)/~Cd (8)I=cti($)*!oof!o
lF (lU-T) ~39R3J65i=lu{>0 TU 90lF (l-TL) 86,90s90I=TL
LALL PES
U=VQ* SCPT[(P-Po)/(VO-V))u= s~JNT((P-Po) *(uo-v))(:ri(;)=r/~:]oo,
CH(j)=20.*P*((Hb-IiO)/( (P-Po)*vOl-O~b*(loO+V/Vo) )
CALL of)lJT (3HHLJG,1JbO 10 40LF (KCt4(I)-Z) 200,200$151FOR(S)=(Cti(7)-CH(3))/(Cb (b)-CH(2))NET1.JKh
SURRWTT:IE GAMM(K1,K2)
CALCULATF. EdlJATION CF STAIE l)LNIVATIV~stlYN(JMEPIcAL UIFFERL,~CIriLJ UF ~ Ah(J TUS[NG ).!Es FOR EQ. U; STATk PUINls
INPuT
(1) Kl=l - OiFFkRENIIATt *IT14 CURi4kNT coldoI!iaN(FiAEO UR EWLLIW41UPI CUWOSITIUN)
(~) 1(2=1 - OIFFLREN]IATL A! ~IxtD CCJpIjJOSITiUN
(3) ThER (1) p cEhTLN P(JINT (~tJANrIllLS
7!1
22232425?6
27?82930313?,3334353b37
3d39
40414243444566
474849
50
5152
53
54555657585960
6162
636465
234
56
789
10
;;
1314
97
cc
ccc
cc,.“
cccccc~
c
ccc
c
cc
cccccc
ccc
c
cc
c
3
6
6
c+
ch
1~
12!4
lbcc
2 J‘J 1
c~~
24
(Q) EPN (~)
OUTPUT
Suclj [J)
(2)(31(4)(5)
:.s}
(7)(!1)
AE!OVE OUTPUT IS
c (>OUNU SPLLU) rjAPMC.j t,J,wcl I(JN ‘ 6AP.t.l=((V/VUi/(UAM/(b.$W*l-P/PO) ) )*@[-GAM)-l GAFMCAP GAMMA (I>ulhLKf4AL)cP/K
(C LN V/U T] (CCIJ>TANT P)
—OR i3UTH CWWI!IUI+ ANO ~LACkl) INAPPi?UPN~AIE inEd LOCATIONS
(u)tf-
SAVE NC
1 0E4 ( 10) *E:~h ( 20) *FOB ( 60);*tiF ( 1,7) ,KAL ( 20) ,THLR ( 50)?*SIICG ( 20)
tQ(JIvALtNCE
I (7( loon),oER ) ,(Z( lu3111*L~N 1 $(Z( 14)lJ) *run~*(7( 15co)*hE ) ,(/( 10GO)*KAL ) ,[Z( l~20)*IhtRs,(Z( 191}o),51)CG )
LOIJ:VJLENCE {ThEd(l )*P)*(lHER(31$ T)*(TMLR(8) *V)Iu sAVt CLNTLU VALbLs20 [lEa(!)=&3U L;EO(.?)=V
+0 uCD(J)=Tau uEl>(.&)=rHER(211
b(l L2=EMN(3)98 M&Ih COUL /lLST Ak(jS.
100 lF [fiZ) 11O*4IJO*11UI1O lF (KAL(6)) 2UO~40~*200~%u UIFF. AT ~IXo COMPO?00 LM=KAL (6)~lIJ hA1. :b)=O
1!516
171819
202122232425?6272829
3031
32
3334353537383940414.?
4344&5
46
4748
4950515253545s5657
58
5960
616263
646S66676ti
697\)71
727374
98
n=]btl TU lol)fl
THFti(26)=stJcG
ltiFH(31J)=SUCGlhER(29)=$UCrjTHEFI{28)=SUCG
NAL(b)=LM
StT OU~PUl1)
2)4)5)
SLT OUIPUi
CALL LOUT (4HGAPw,i)
lF (K1)410,6CO,41O
K=?60 TO lnoo
IREH[22)=SUCG(I)lHril(2-i) =s{JcG(2)lHFN(zL)=\uc.G(3)iliER(25)=SllCG (4)THFI?(27)=5uc6(5)CALL COUT (4HGAPM*A)
KETuNr\
DIFFE*ENCC SUdRUUIIINE‘l)tLIA P
P=Fotj(i3)@DER(I)
r=l)Lti(3)
CALL vEs“ER(5)=VP=nLK(l)”/FOR(13)
CALL PES(JEn(ti)=vsu(-l3(b)=- ?.0*ALOG(*CB(13))/ALUb (UkR(b)/UER(6))
OtLTA TP=nEti(~)l=nL#(3) *(l.n*Foti(lQ) )
CALL NEsLJE2(5)=VUE2(7)=T/~ER(]7)0T~twN (3)
T=aLw(3) *(loo-t-c5(L6) )
CALL rJEsuEP16)=vuER(3)=T.iER (17)*T~LPN (3)>UCG(I) =(CEH(7)-JER (S))/(Z.0°U&K(~)0PG9 (14)0E2)SUCG[~)=ALOG(DkFi(5)/llt.R (6))/(C!0001JLN(3)0FUtl (14))
CALC. (jlJTPuTI=r)LA(s)W=,IER(2)t.~=r;ER (-)oTOsuc6;8] /SUCG(t)
bbrG(l)=sllc(;(6)/(i* ,l->ucb (6)~Ll*T*>ucb (d))
surG(3)=l.0/(SuCG[i )~tl)SUCG(2) =S,ICG(1)*SUC(; (3)-100>LJCG(4)= sqNT(stiCu (l)ODEW(l)@ULR (i?))
SUCG(5)=(V/mE(6)l/(SUCG [1)/(SuCb(l)*l.0-rit(2) /UEN(l)))
>lJcG(5) =sucG(5)~*(-sucG (11,)-1:0
GO T() (3iJ0,50U)9K
7576
7778
7980818283
84858687
888990919.?9396
95
969798
99100101102
10310410s10b107108109110111112
113114
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1211 ?2i231?4I 251?61271?81?9
130131132
133134
99
c2!J:1
c CALCULATE ISOTHEkw ~r5UcPUHE, 1>LNTR0Pt0c OR COh’5TAhl-CkkH{iY 1c Ar vAL(JES GF o IN p
c (F!)J KAL(13)=1,2*3,4c I~ruT
c Sur(l)~ (z) - TC,PC (INITc AA].(1), K.IL(~) - (JIFF’ swII
c KAI.(d) - 1,293,4 *UG (.OASc PT - IJRE5SUNE TAALL
c ourPi]?
c PRII.TED ?cIhT oUfPU7
c -------- .-
C
c,.
c
c
c
kEvIsION 10LEAVE P cOI+RECi ON txIl
FOR KE>TAuT UPTIUNt. 3;l/btZ
( 60) *PT ( 513)
( 5(J)
Z( lb20)$KLN ) 9(Z 1750)*P1 )~*(?( ltlfj~)ssljc ) ,(2( lY20)*ThER )
8
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LG{livALF.icE (THEN (1) !P), (IHER(3)* T)LGUIVALF.{CE (THEh(8) *v)* (!HkN(9) tt)*(Tr!LI?
KEll(16)=9
CALL POUT (1)[=<:Jc(l)/=\lJc(Z)
LAI.I. YES
CALL PouT(2)>UC(3)=VSUC(4)=<
>UC[>I=EIF (KAL(l)+KAL(2)! 70$112$70CALL GA~’f (Kf,L(1),KbL(2))
cALL POlJT(3)112 1=1
11+ If (PT(I)) 200,200?116]16 P=PT(I)I.?O CALL PESC (KAL(8),5L’C)
Id2 CALL POUT(2)124 IF (KAL(I)+KAL(2)) 130t14Ut13U130 CALL 6AW’4 (KAL(1)*KAL(2))
134 CALL Pour(3)140 1=1*1142 bO TU 114
?O(l KEThRNkh~
13),s)
r=v
PvPvPv
PvpvPvPvPv
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c
c C.ILCULATE OLI. MUGO poI~lb Al ~ VALUESc FC,J’,[) IrJ P1 AEN&Y
cINPUT
: KAL(l), KAL(2) - IJIFF Ski
c PT - PRESSURE TAdLtS
c OUIPIT
c PRrOjrEo wxhl OUTpUT
c -----------
c
c hEVISION I.LEAVE P cOkREC
c FOR KE~TART OyTIUN
c *. t. s/l/hi
LG(IIVALE?JCE (THEH(l),P)
122JJJ o
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lUOl!JOldo?UI)
SET TblJES> UN FIRST LhT14Y
CALL PObT(IJ1=1
lF (@T(I)) 200*200,*0#=\)T(I)
1=1+1CALL huG
CALL POUT(Z)IF (KAL(1)+KAL(2)) 10IJ,3091OLJcALL GAMM (f(AL(l)QKAL(2))LALL PObT(3)bC T,-I 3,1KE7JJtih
SUNROJJTINE c.J
cc CALCULATE CJ LoCUS AT VALIJES UF RnO LLRU
c I.w I/or TABLF.
c
c ITE@ATE oh P ALONG hUGONI(JT UNIJ~c CJ COt.OITIGh IS SATISP”It.U
c I~lJIJT
c KAI,.(l)! K&L(2) - EIIFF S*i!C~E~
c NOT - INITIAL-DENSITY TAdLEc OLTPIJT
c PRINIEI) PolhT CJIJTPUT
c .-.---.., . .
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cc(?)=o.~,el?l-ol$(jo(t -1,(J)
P=CC(2)KL+J(15)=I)CALL buG
IF (KAL(7)) 320,2s’u,3zo
CALL G4w:.4(1 .0)
CC(”l)=TflEN(2/)L7CI T’J 3~oCALL 6AUM(0,1)
cco)=ThEH(2d)
CAI.I- UOUT (2tIcJ,l)G() To 120
I12+()*C*(;o=c~(~)
CALL FIJG
tlE(g)=T}.E4 (q)-TbZti (4)~*2/eoO
CALL ~OUT(2)CALL GA’~P(KAL (l),KAL (2))
C2LL POUT(3)E+c(G1)
1=1+1
IF (RcT(I)) &o$470*60RE TtJRh
OUTPUT P’01)4T
SAvk lNITIAL E FUR lSt INTk6KAL
CJCJcd
CdCJcJ
CJ
CJCJCJCJ
CJCJCJCJCJCJcJ
CJ
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100
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i
c -------------------------------------- 1. Si(.AN 10 REw[) ------------------20 LAl_L FI(’ (3’.121T,6H( IzM6),12,E, U]
IF (,,L(f,,.}J.iEr.R)) bc 10 ZU
c -“-------- --------------------------- . do tJLGIN HUN --------------------10 ~or,llhdf
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i* (Iwc(C..l,i+C(lFJ])
1 C4LL Ea4(3LcoN, 14Ltian Ir%fJuT PACR$L)cc >tA~Cn F(JH (,UN ijtiANVlE-?IJO Uo ~1~.?o [=1026
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LMPYY
SPEC
SPEC1
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CURE
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FORM
LUAU MUTTUN
212?13
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707
Uo 1/ I=\.2!’lHVECT(I$l)=EPA(I )-vEcl(I,/)=EPA~( [)
12 Lo*JT1htJF’
C nLkINZ CllATtidL c~NSTA~iS ANG )NC=C5F(I)ltis=c~p ())
r+p>{I=c~o(~),~o=ks-rjcl*~l=,,lJ+~r4p~i[l=t.lPriI+llF(,\#P [.EIl,l.ANO.KUP .tO.O
IWP=C>P (2)13 Kv=l14 l\cl.~P=l\c- Jp
:F(I’JCPP.LF.O) CALL ilf3U6(2)
Do I 1=1*NSNAVF:C( I) =.IVEcT(I,KV)
l+AV=iAVrC(l)
Uo 1 “=l,P:C1 KAI-$IIA (N IVQJ)=ALPMA(IQJ)
urJ 2 d=l.>c
2 RAI.l’ffA i~.S+ICJ)=(;fiTOM( J)
EURULM ALPHA ACCOROING IO A-VLCTOG
Go LO 30
9
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9
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30
,NPuIAANDoNt’.EU.l
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wRITE(?,8)I*(XNLMA l(IoJ)*J=ltNC)
bO 10 (10,15lF(NPhI.Elj.I bo Tu 11
kETUUhlNP=f)
KV=2
bO TO 14GO TU (lh,lfl
Uo 17 I=l,rl[;ANl15([)=xf~(J([)
UO 1? J=l.NcAh(Il~tiTs (I,J)=xNIJtiAl( I,J)#.NIIs(NDI)=r]P
UO 19 I=l,NSNAVECS(l)=NAVEC (I)
>ll~4tJs=slJ.4i]UO 19 J=I,NC
~.’BAf~b(J)=2BAR(J)nFLr,[:=?GO Tu (P,),%l)?NPSI>P=IJ
tiv.?
UO To 22r.p=l
Kv=lAh,l(XUl)=14peO TO 16NP=cbP(~)IWps=iip+l(>0 10 (31,33),NPs14P=1f#V=iAFLAu=l
VO TU 1614P=()AV=?
(JO To 32
ENr)
676a
6970
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8?8a
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117
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2 .*L~JKI (Il=i~hKI II)+AKU,:AT (l?J)QRPUAG(J)
c
c cALc~~LA7= L;. xJ3 LJc & J=\,kcr.’P4 ALrJXJ( J)=AL!15(RXcOMP (J))
C CALCLJLATL d,JLE FRAcTIONS OF IJCPENLJLN7 +tJLcltSlJo 6 I=I,N()
AL~t<l=AL~KI (I)lJ~ 5 J=l .kc~~
5 AL,JK1=XL IAI.XNUPA1(l!J )*XLNXJ(J)
lF(xl.NxT .GT,o,o)xLhxI=O. oIF (XI-NXI .LT.-EXPL IM)XLNXI=-EXPL1M
6 l<XC1.)MP (l+.,c)=Exp(xLNxl )
C sET UP IIER,\TIOi., LOCPlFIIIE1/c1).E,;.o)c(I lo 11lF(IrE2r.).GT, ITEHcYc )60 :U 20
C TL>T FOP Coe.VFRtiENcEUO 7 J=l ,:icPPlF(.\tiS (* (J) ).6T.L#SILON) Gu TO 11
7 CO’lTih(JF
C PAsSLO C,JAVEL15EKCE TESIcC cALcLILflTt Nl!~.lEl{ OF WLJLES
HN\3d=l. ]L/o H 1=1, ,.[)
8 Hr,~d=Hk(;:j.RXCOf.iF (I*AC) O(Xhu(IJ-i.0)
cM[)L ( ~) =S)]M~)/l<Nc.jC CALCWLATL Xc ANO NOLES CF SOLI’J F_uH NP=l
F.!4nL(2)=,},o
lF(’[email protected])GO TO H1Ac=,;+AR(,~c)
LJO tif) I=],NCtio Xc=XC-(XNUMAY (I,!NC)-OB&R (h,C)*[XhU (l)-l.O)l~NXCOPP (i*NC)
1;
11121314IS
ltJ192021 .
2223
24?52b
27282930
313233
343536
373a3Y604142434445
464748.49505152535.55
;;58
5s6061626364656667 .
68
110
hxco14P(A~)=x~
k.Ml~L(2)=<C*t40L (l)C nLSTURF’ ,,IOLF FR.jcTION UGUER
u! Uo ~ I=~,(”s
[uAV=,VAVEC(l)9 AcnVPII)=$/xcOMP(kAV)
lF(XPhI.E’)0))60 TO <2
C FXIT WITH co!~UTEO E&uLLltiRIUf’+ COM~USL~lUrII
10 kF.TLHN
cC ~h~hy T() CA1.CIIL?JTE COI?HECTIOINS TO M(JLL fMAcTIONb
1! (JO 1.3 J=I,NCNPSU$4F=0.O
l~U ]/ I=ltNR12! Sb)~F=Sd.F.(Xl/lJMAl(l* J)-GuAR13 F(J) =GAA4(J) -RxcoMJJ(J) -suMF
Uo ],+ J=l,NC!tPLJo 1’+ JP=l,hCMP/.(.l,.JP)=,)eo
lF(J.EQ.Ju) A(J,JP)=QxCOPP (Jl~G 1+ I=l,N!)
14 A(,I,JP) =4(J,JP)+(XNUMAT (IsJIAtNIIUAT (l, 1P)
CAI.L LSS(,.CPP, 1,6,A,F,0ET)IF (IJLT.Fu.11.I))GO jC dO
IJO 17 J=l,Nc’4P
J)*(x~LJ
-UUAR (J
1)-1.0) )OF+XCUMP(r*NC)
O(XNU(I) -I.O)l*~XCOMP (I+NC)~
HXr[J,<P (.J)=NXCOMP(J) *(l,O*fI(Jl)
6’370717273747576
7778798081
828384~~
86
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121
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124
125
126127128
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lF(ALNS.LT,O.O)tiO Ifl 10
F.ACPXNI,F. FUR ?-PIiAsE SL.LUTXON
EXCbANGE A-VECTORS, ETC., FCK CrlANuL IN Fw(JMuLR OF PIJASLS.3U l\P},FLA(;=/
LJ(J 31 I=lorvoU(J 31 J=1oNC.AN{IPSV=X i!j$IAT$(IsJ)ANll~4$iTS (I,Jl=XNUMA[ (I*J)
31 XN1l’~AT (l..l)=Y.NUPSVU(J 3,> .1=1 ,NCtiHAM:>V=:rlO!RS (J)tiHAdS(J)=,.8~R (J)
32 Wij$N(J)=j~AhSV00 :!.~ I=l,N’JNAv=NAVF.C’; ( [ )
liAVELs (I)=N4vEc(I)
33 hAVEC(I)=t,AJ
u(I 3+ I=l*l~l!lAh.[lSAV=~’IIJS ([l
AtNtlS(I)=X’,lJ (I)34 ANll( [)=< I,ISAV
SUIIIJSAV=S!,MLS>U*IIJ>=SIJ f ;>UO{(:=sUl~,)SAVr,p=x~du(fi II)
bO T(J 10U1
TtST fi)~{ SATISFACTORY L-PHASE SOLUTION
+0 IF (.xCoGT,’I,~I)GO TO 10>Ipl. LO REPLACED dY Thk FCLLOWING TwfJ STATEMENTS.
KENS,4NL$I ?/10/7640 LMC)L(7)=XC
lF(xc.(;T.r300)G0 TO 10NU SuLIo PHIiSE. sO To 1-PHASL SaLUTIUN
(JO 10 3,)LNII
SUN140UTI.JE LSS (N,P,I*A,S*DETI
C M~DIt IEII FOR FOIiTRAN IV HUG
16 lJI..iE,wSIC(~ A(I?N]! U(I*M)
16 UUllHLk P;/EclSIOh Sl,SZ,CUIPiiU
lb (,N=N
lb r.ir4=fA
16 SN=l .
21 Uo O J=],N~j
?3 L=.I-1
26 lF (J.EO.r,N) Go TO 73: T=.$E~(A(JsJ))3; b\l=J
31 b12=J+L
:?’+130131132133136135136]37 -138]3’4]60
141
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143144
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148
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151
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56
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LIO 1 K=*J7,NF+
A=AHS(A(<,J))
1P (X.LF,T) GO TU 11=XMl=<COI/lIhlJFlF (M1.rq.J) GO TO 4
Uo 2 K=l,t:rlI=A(J,K)
A(J,h)=A(t!l,K)A(MIQK)=T
s N =-srl
iF (MP.l F.0) GO TO 4IJO j K=I,IIM
7=II(J,K)u(,J,A)=il:t,l,to
b(..!!,r.)=T
lF (O{J,J)OEQ.O.) tiC TO 1SUO 6 K=P>,Nh
>1=0.S2=0.
IF (L.E{J.n) GO ?0 5
Sl=i)UTPJil(L.d(J.])$ I,A(l,n),l}A(,j.K)=(A(.J.K)-S!)/A (J,J) “b2=l;UrPU!( JcA(K$l)* 1.A(10M2)$i)
~(h~$’21 =.’I(K,!4?)-S?IF (hP.LE.0) (;o 10 QlF (4(J,$J).EfJ.Oo) bC TO 13L!(J w K=I,!~M51=(I.
iF (L.E,;.q) GO T(I MS1=,l!JTP:<,) (L,A(J$l),l ,u(l,K),l)ti(.I,fi)=(.!(JtI;)-Sl)/A(JsJICO..IYIN1KuET=A(l,I)@SN
IF (L)ET.EIJOn.) GO !o 13
iF (N.EC.1) (;0 TO +5(10 10 J=Z,NN
IJET=05T*A(J,J)lF (:;ET.Eo.(J.) GO ?13 13
IF (NP’.E).O) GO TO 15M3=IvN-1
1;(I jz J=l,MM
UO 11 L=1*M3
lfl=hf.+-~
>1=(:.
IPIZ=M1*l
h=tlN-P/*1
S1=,!UTPNO (K,A(M1OM2)* It@(M29J) cl)u($il,J)=P4(Ml,J)-slcoi4rIN(JEG(J Ttj lb
13 dET=OoOKETURhLrYn
15
161?181920212223
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33
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LJO 1 1=1*N“Ux=x(l?r)uY=Y(lt I)
1 >u~l=sull+l)Y*rlYiJoTPNG=sllt4
tiETIJKfx
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56
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SUFIHUUTINE i31PIrtiIs IS A Duf.lMY ROUTINE.
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6s
7
8
.
1:
1112
13
I*15
16
17
181920
21222324
2526272d
?9
303132
33343536
2
116
APPENDIX D
COMMON STORE
The contents of the common arrays are listed in Table D-I.
.General Notes
(1)
(2)
(3)
(4)
(5)
(6)
If there is a principal routine dealing with an array, its name is given in
parentheses at the end of the description line.
A few important equivalent names are given; these are denoted by a preceding
equals sign.
A single subscript indicates a one-dimensional array (for example xi); a
double subscript indicates a two-dimensional array.
One-dimensional arrays containing species properties or
entire system list the solid as the first species. The
solid is X5 = ns/n.
compositions for the
mole fraction of the
Unless otherwise stated, the units are cm-g-ps or cm-mol-ps,
Unless otherwise stated, all derivatives for the system are at chemical equi-
librium.
.Particular Notes
(1) GM - the prime here denotes imperfection quantities with respect to ideal gas
at the same temperature and volume. See Sec.
(2) THER - the subscript o on quantities near the
zen-composition derivative.
TABLE D-I
COMMON STORE
CAR - matrix of FROOT arrays (columns)
1. CC-CJ2. CH - Hugoniot
- constant-v, s, e contours;: :; - gas EOS5. Cs - solid EOS
CONT - constants
1. 1.98719 R(cal)2. 831439 x 10-5 R(Mbar-cm3/9)3. 1.01325 x 10-6 atm to Mtjar4. 0.04184 kcal to Mbar-cm5. 0.426012 (N/@) X 10-24
DER - d-
1. p
2. v3. T4. v+5. v_6. H+7. H-
IV.C.
end of the array denotes a fro-
fferentation (GAMM)
EMG -~i, “free energies” for equilib-rium constants (EQMS)
117
EMN - phase mole numbers
1. ‘92. n~
n=n + ns;: Xg = i /n5. xs = n~/n6. n/MQ(moles/gram)7. !tns (saturation index) or -ns
(all solid evaporated)
EMX - xi, mole fractions
FMU - p;, imperfection chemical poten-tials
FN - ni, mole numbers
FOB - knobs
1-6. FROOT E’S
7-12. FROOT r’s13. Al = A kn p (GAMM)
A2 = AT (GAMM)& C, equilibrium outer (EQP)16. c, equilibrium inner (EQP)17,18. ---19-36. FROOT Xmin, Xmax in pairs
GM - pure-fluid gas state (sub g) (GEM)
10 T= v/v*2. e = T/T*
z,= pV/RT:: E /RT5. F’/RT6. (ze)~7. ~;=8. Cv/R9. z - RT/p100 z-1
S’/R;:: (ap/aV)T];. (ap/aT)V
(aV/aT)p15: v16. p17. H’/RT18. (Z;l)~ = TxT/e
19. W~;T)R= ‘X/e20.
24. ---25.26. ;-1 (aE/av)p27. I/r28. -’(a !Lnp/a LnV)T29. C’/R30. (~kn V/aT)p
GP - gas Eos (GEM)
1. nm
$: An
4“ ‘~ . RsTA5. r6. T* = TSTA7. V* = VSTA
L
.
HE - initial (unreacted) state (CON)
1.
2.3.4.5.
6.7.8.
KAL
1.
::4.5.
6.
7.8.
9.
10.11.
KEN
PO (9/crns)
p. (Mbar)To (K)M. (g/mole)AHfo (kcal/mol
at To)e relative to elements
Vo = l/po-
!O ~TojToj = (n/Me) AHfo
j ~“j
- option switches, see CON, SWIT——
equilibrium differentiationfixed composition differentiationgas (0/9/other: ideal/KW/LJD)solid (O for incompressible)mix O no-mix 3 - CS
1 ideal 4 One-Fluid2 LH
composition (0/1 for fixed, equilib-rium)
CJ (0/1 for equilibrium/frozen)contour (1, 2, 3, 4 for constant-T,
v, s, e)
Tc, pc choice (0/1 for input/previ-ous)
z- 1 .,
punch output if # O
entry and iteration counts t
21, VfjV =~~$g(l) (Vf = free vol-
22. b/2 (integration limit)23. ---
118
KIM - sizes -n#-
1. r = KR number of gas species2* s = KS total number of species
c = KC number of elements:: n = KN degree of ideal-function fit
J
KON - triggers and diagnostic switches
&: &’!n v/a !?mT)p
THER - system state (COUT)
1. p
2. v/v.3. T
PT - input pressure table (PV, TED).
ROT - input PO table (CJ)
SM- solid state (sub S) (SEMS)
10 v2. E’/RT3. H’/RT4. A’/RT
F:/RT:: S /R7. z = pV/RT8, P
v/v. = Yh. T11,
‘1
SP - solid Eos (sEMs)
1, r2. Cp/R3. a4. Vo5. To6. Eo/RTo. ---
;:12. CO-C4 - Hugoniot fit13-20, working store
Suc - contour initial state (PV’)
10 Tc
:: ;:4. Sc5. Ec
K
SUCG- derivatives working store (GAMM)
1.3 2. ~;~ (ae/av)p
3.
4. u5. D6.7.
q (Mbar - cm3/g)q (kcal/g)
8. v9.10. i11. a120 f13. s14. H(T;Tn)/RT - relative to elements
15016.17.18.19.20.21.22.23.24.25.26,27.28.
vE/RTH/RTA/RTF/RTS/Rz = pV/RTYp-l(ae/av)p
l/r
at To
cyo (frozen)J(p) equilibriumj(p) frozenwhere j(p) is the
{[j(p) = VIVO Y~l
CJ function:
- Po/P)/Y]}-y
co (frozen)::: p-l (ae/av)p frozen
TMG - F: - ideal free energies
4. c“5. CJ function j(p) [see THER (28)]6. - (a !?.np/a 8nv)T
719
TMs- ideal functions (super i) (TIMs)
1. Eg/RT2. Hg/RT3. Cg/R4. Sg/R5. ~(To)
}relative to
6, Hg(T;To)/RT elements at To7. H</RT8. C;/R9. Ss/R
Fs/RT;:: As(To)
}relative to
12, Hs(T;To)/RT elements at To
TP - CON, TIP strings 3 .... one row~ s~ies
XMT - gas mixture state (sub g) (XIM)
1. v2. E;/RT3. H1/RT4. A1/RT5. F /RT6. S’/R
z-1;: V-RT/p ~9. ~ xi xj T~j/T~ (for LH m~x)100 ~*xi xj rjj/rr (for LH mix)11.
‘E(for LH mix)
12. Tr (for LH mix)
130 n (for One-Fluid mix)14. m (for One-Fluid mix)
v; [for One-Fluid mix)::: F = RSTAT (from XIMS)17. T* = TSTAT (from XIMS)18. V* = VSTAT (from XIMS)
XMUP~/RT, gas species only (XIM)
xw - T~j (xIM)
xPG- rjj (XIM)
xPR - r: (XIM)
xpT - T: (xIM)
REFERENCES
1. W. Fickett, “Calculation of theDetonation Properties of CondensedExplosives,” Phys. Fluids 6_, 997-1006 (1963).
L2. W. Fickett, “Detonation Properties
of Condensed Explosives Calculatedwith an Equation of State Based onIntermolecular Potentials,” Los
.
Alamos Scientific Laboratory reportLA-2712 (December 1962).
30 S. J. Jacobs, “Energy of Detona-tion,” Naval Ordnance Laboratoryreport NAVORD 4366 (September1956).
I
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