!’

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

Printed in the United States of America. Available fromNational Teclinical Information Service

U.S. Department of Commerce5285 Port Royal RoadSprirgtleld, VA 22161

Price: Printed Copy $6.00 Microfiche $3.00

Thi.,epurt -., WWard ● . . . .cwuntd work w.mm,cdbs ah. [’.itcd S(.1- (bwr.mcnt Setther !he United Slamsnor the It. ilrd SI. h-s Know, Resr.rch and Ikv.1.a menl Ad.mi. i.tr.t l... nor . . . of their emplo}tws. nor ..} o?tbelrcvn.trwtw.. .ubcamtr.ctom, o. their empl., res. m.krs .nyw.rr. n!>. .xprrss or implied . . . .swm..n any Ics. I Ii.hilitv orr-rm.s, hilit. r.r the ● -cur..%. .-anpkten”. ... WI.I. ”. .r● ny Informmtiwt, apparatus. product, or processdkrlmed. wrrwr.evl. that it. u.. would mot infrinne prtiv. t.1> OWArishls.

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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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52

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.,*8*

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cc

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urc

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r-lwm

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t-aa1-In?i-8%.3

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<,c

cc

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e,.,.

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lLIL

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rl0

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Na

Nm

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JsmN

ctim

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hlrr-a

mo

0(-

C-.

+IN

C.

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.rmc.

l.

’42

r-a

b-Z0a

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

00No

ol-

-N

CN

Noo

01-o

Iu-

oo1--o-

..

I NN

I%N

SI

++1--

01-N

X?

’mlllo

ml

0I uol----

0

-

+11

om

lz4.

NN

----0

z++

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

.2

23

13

15

i?223456789

10

111213lQ15lb1710192021

22.?32625?6?7?8293031323334353637383 c)40414243464546474849505152535455565758

59

cc

67

10

1010101010

10

lrl

1012

2f!32364447

6163

ER~EkP

59606162

234

.

567

.

234

56

789

101112131415161718lU20’21222324Z52627282930313233

23*56789

101112 *13

.

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

JJIxIENSION flS(4),FP41(bO), &(LO]INTEGER ijtFMTstJS

If)

15

R1819202122?3

:;?627282930313233343536373839404142434445464748495(J515253545556575tl596061

23456789

H

1416

“i 414

2L23232630

4045

46

5667

73104

110

120132

136150

154lb6

172204

2:0210210211

c

c

c

c

c

c

c

c

c

140

150

c

c

c

ccc

cccc

cf,

usLlsbs.

13s

IJo

]I=4xNEA()2)=.31.H[T3)=S~PNIhT

6)=3HuIIT

]zJ

lF(H.EO.RS(l))GO TO 30

CONTINIJEwRTTL(9,/fl)H

~ORPAT (lI+H(l FIO -ti&D AUG= Ab)CALL ExIT(3)lF (N)200s100t200

1.0. *ITM No DATA

(’500s500s 1309140)91

(?, lqo)(FP f(J) ~J=l*KL)500

E (Y0150) (FNT(J)tJ=1sKLl,SIJrl

(12Ah)1.0. nITH OATA

(210022002309.?60)91

kE41) (2,F~T) (A(J) 9J=19N)

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 )

eo Tll 500

CONTihdECOr4TIhUE

NETUhPi

Lt4r)

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)

12131015lb1718197021?22324?5?b27282930313233343s3637

36394041424346454647484950515253se5s56

234567R9

1011121314

.

.

.

.

62

b

1:12

‘i 31/2~21

. .?324263(I34.353b37

~14344

.

uIt{ENsIod C(lo) FRLIJTT

8 L1=C(P) FKL(JTT

9 L2=C(9) FKLllTT10 lF (K)5n,?o.50 FHUUTT

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

FMuUTT100 (,(?)=C(2)+c(6) FNUUTT

102 RC1=KC1*l FlibUTTI1O IF ((C(?) -C(7) )UC(6)) 15(J0150s1Z0 FHbOTTIZo Nc?=l F14~UTT

lJO Uo ~o 200 FHbUTT

150 KcP=z700 L(fl)=El

F14UUTTFKQUTT

?U2 C(9)=E2 FNbUTT?10 KETbRN

LNrlFHQOTTFRbl)TT

C*e****0*9**eoa**o***aa0090*a*.e**0e*0e0*0e0e*0e0e0*0**e*e*00ee**0oo00** ~&p

Ca 3. CALC,JLATIdN SECIIONs ● EtrCaa0***0********eeoe***Qe*e*0**e*0*******0***e*o*@*o*0oe**0o0OoOO40a000o G~;

.-C

cc

c

cc

cccc

:

cc

ccccc

cc

cccccc

c

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

LOS VIA ~flMPOSITIOiN L)EPENDENCL OF Thk

MEAN flSTAR ANO TSTAROlht’UT

P=TPLR(I)lPPLICIT FOR GEM -

T=rhE2(?)

t!Eah’ MSTAN* TSTAM = RSTA=6P(5)9 ~STA=(JP(6)KAL(5) - 1.293 FUN IoEAL9LJu$ KU

ULIPIJTV=VR=GFl(lq) - VOLUNEJMPLIclT FROM GEM -

(+M - STATE PoINT

VAKIAULES~G=G,-1(16) - Calculate (bAS] PNLSSUUL‘JUL. VQU - LOdEFI ANO UPPLU LIMITS Uh lTLuATIOtU v

RcuTlhESGE!(2) - PIJwE-FLUIU LUS A! T9w

FROoTT - ITERATEPLAN

SOLVE ITE~?ATIVELY PG(V,T)=P FUR V(=vttl UY

~ROOT ITERATIoN.tAQIJiBLE, FbNTICN - LOG(V), LuG(P/PG)bEcoNO GIJEss - NEw[cN-RAPtISON ‘ROM oP/oV=Gt4(l~!

15lb17Iti19202122232425262728

::31

3233

234

~

676

1:1112131415lb17lM1920

:;2324?s26272829303132333&3536373&39do

.

cccccccccc

cc

33

3

3c

3

cc

3

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

(hl)TE - cREMlcAL POIENTIAL> IN Xt4U ARE

CALCIJLATELI uY CALLLR)VARIAHLES

LV - l,? - TSTAR >Nn NSTA~ UISPLACLMLNT5

395 - EXPAhS[Uh COEFFICItNiS

C,o’ftfuh ~(qoof))

UI”4E?d SICIJ

1 CAN ( HO) sCU[*T ( 2(J) *EV ( 20)2$Fo% ( 60) ~GM ( 40) sKLN ( bo)

3*KnN ( 6910) oTtiLR ( 50) tAMT ( 30)4*G~(/o)

~fJiJIVALF: ICE

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

LQIJIVALE(JCE1 (CAti(jl ),CG), (CG(d),KCG )2*(f(lNT(2),R)

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 L2=EZ@Y.CGi(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

11511611”7Il&119120

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

1214lb*O

!JU355631

60/0$(J

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

FvPv

PvPvPvPv

PvPv

PvDVPvFVP4

Pv

?VPv

PvPv

PvPvPvPv

PvPvPvPvpv

PvFvPvP ‘JPvPv

Pv

PvPv

Pv

PvPvPvPv

PvPvPvPv

Pv

Pv

135136

2

345

6

7a

1:1112

131415ib

17181$

20212.2

2324?5

26

2728?9

;;

32

3334

3536

37

383940

0142434645

46

4748

49

505152

535455

56

57

.

.

.

.

100

.

1

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

*O5000Io8(J

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 .-.---.., . .

.

CJ

CJCJ

CJ

CJCJcJ

CJCJ

CJCJ

CJCJ

2

34~

67

HY

lG11121310

1516

17ld19

702122

23.26

2s

?627?8

293631

3233

34353b3738394(J

41b?

434445

12

i316

101

cc

c

cc

c

cc

c

c

c

c

L.CN5T. V Ok!. P CALcD kkUf.!Q,UAt!MA

*. k. 11/21/01 -

REvISIUN 2. SAVE LNITIAL L FOR IN1:GRAL PLVtie k. l/3/bd

6b

1030*O

50

CJ

CJCJCdCJ

CJCJ

cJ

CJCJcJ

20) CJ50) CJ

CJCJ

CJCJCJ

CJCJ

CJcJ

CJ

CJCJ

CJ

CJCJ

IfwPuTo UUIPUT LA8EL CJCJ

CJSET UP FRGOTT CJ

CJCJCJcdCJCJ

CJCJ

CJCJCJCJCJCJ

r.JCJCJ

CJCJCJCJ

CJCJ

&CJ

CJCJ

F-ROOT?

15lb1718

192021?2?32425?b2728?930

313233

343536

373839404142

434445

464748

4950515253~~

555657

5b5960

61b~63

6465

66676869767172

7374

.

“

.

102

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

CJ

CJCdCJCJCJ

CJCJ

CJCJCJ

COh(AccN)

757C7778

7980

81

82B384

858687

8M69

90~~9?~3Q&9596579899

100

101!02163]04105106

2

34

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

L-L;>.<.~AT=!.#(l LALL FI,? fqb;JIT,l.3n(ZA6,b~lJ ,Ad) * Y* L* o)Y(I LALL FI(! (sF.iioT,la~(/lx~Ao * bAldo Ab) ~ Y, E, 01

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

F31U iF- (Cn-IS ;(I)) 832CsAJ’3r3,dJ2(l

c

c

28

2’43031

323334

353b37383940

+14.?43

4465

46474b49

505152535455

565758

596061626366

656667he

6970

717?

73747576

77?li

798081

62838*R58667

.

.

.

.

104

.

))1 3U8,JU6*5U!Y

DIP

(JLs

CAI-L uEr.,J (:)Q-l,?.E)N(.lrJ(nt,4)=l

(.ALL RE,ld (11. ~,CA*15QKE))

uO Tu d.)

TIM

fln

R3~u91

929394

95~b

97

9H99

100

101102103100105]06

107108109

!10]~1

112113

Ilu115116117118

119

1201?1l??

1231 ?41?51 2b1?7

1281?9130

131

13213313413513b

137]38

13914016]JIb2

]43144

145] 46147

336370

J?*37(,

3’!7401

+iJb41)-/

c171017<(I

C17>LIlnoo

Tko

—~5011 HRJ !iH (1-T*.2-V,3-S$4-E) / TCOPC/ P-TAo!.E s

7tST

ChEC

.

.

A

.

106

43?441

44.2

4*3445

bb(l

450

451

6’)2+54456

601

462

4b5

46b

.461/brtj

41]

412473

474

50?504505

513515522

52*

525

!526526530532

533

cc

40U0c

:

COJITIAUECALL PEA!)E (7HJ09ExoS

CALL EXITMETUdh

)

LMPYY

SPEC

SPEC1

EMPTIES

CURE

LOAll

E#!PT [ES

FORM

LUAU MUTTUN

212?13

2212?2

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

10

111213

14

15

.

1617le19 -20

212223?6

?52b?7

?$

29303132

333435

3637383940

4j424344454647

4049

50515253

545556

57

5&596061

62636465 k66

.

108

wiiTTk(9,7) (xrJIJPAT (ltJ)?J=lsoNC)

9

90

1011

1516

17

18

19

20

21

22

30

,NPuIAANDoNt’.EU.l

*KFLAb

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

717273

7475

7677-7d79

808182

.936*

85

ah

8?8a

a99091

929394

959b97

9899

1001(71

l(i21!)3

104105100~07l(lij

iOY

110111112113114115116

117

:456

70

709

Ls

=LXPLIM

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(’~@ONF.l)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

87

8889909192

939493

96979899

Ico

101] 02103]04)05106

107l.oe109

110111

112113114115116117

11s119]iiJ

121

122123

124

125

126127128

111

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

\42 .

143144

14s146)47

148

14Y150

151

152153154]55

15b157

156159

160161

162163164

234

56

78

234

5

678

1:11121314 &

t.

112

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

?42526

2728?93(J3132

33

3435

363738

3s40

41

*24344&j

~b474b

495051525354

5556575A5960

61626366

6566676669

234

113

~lJM=J,f)

LJO 1 1=1*N“Ux=x(l?r)uY=Y(lt I)

1 >u~l=sull+l)Y*rlYiJoTPNG=sllt4

tiETIJKfx

~Nn

56

7&

1:111?

:4

5678

1;

1112131015161718

19202122232425202728?930313233

34353b3738

39404]42

fb3444546474$’4950

51

114

10

1515lb?b

40

121

121

56789

101112

234

5678

1;

111213141516

171819202122232425262728

.?930313233343536373a

39404142

4344654667

4$4950

51

115

c!

?

c12

c14?

c>.2

c

12

cIi

c1.2

c

12

c

SUFIHUUTINE i31PIrtiIs IS A Duf.lMY ROUTINE.

HETUrth

krsn

SU~\RUUTTo.:E TEsT

ihTS IS a DIJMMy ROUiIi~E.KETUHA

t.hl)

SUUNOLTTNE SPEC1lrlIS IS A DIJYMv ROUTINE.KETu14N

Lrif)

w.indLTI:.JC 00MltiIS IS a DUMriY +?cIuTINEoRETUrihkrxn

Czz

f’ut’)4y

CuruyguPf4y

f)uPt4y

put’My

ru~MY~u?uvcuPt4y

5253

56234

5

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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