GERED - a code for equation-of-state calculations at 1

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a wire e in c e
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a tº ice ra.
-
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GE-------->
FNGR LIBRARY
JAN 23
1969
UNIV. OF WASH.
IS
QN)
O
UN
O
O
- UNIVERSITY
1
|
Of
V
E
R M
CAL | FORN
0 R
||
E
iii.
UNIVERSITY OF MICH
3 9015 09522.400 5
TID-4500
UC-34 Physics
In a vurr era c e
IR, a
diatio ra.
UN | WE R S | T W 0 F
In alb coratory
C A 1 ||F0 R N | A
| | WE R M0 RE
UCRL-50500
— A CODE FOR EQUATION-OF-STATE
CALCULATIONS AT MODERATE PRESSURES
GERED
F.
J.
Rogers
September
9,
1968
Contents
ABSTRACT
I. INTRODUCTION
A.
Nomenclature
II. BASIC EQUATIONS
.
III. FORMULATION OF EQUATION
A.
Hugoniot Calculations
Zero-Degree-Isotherm Pressure
º
Hugoniot Temperature
E.
Adiabatic Temperature
and Energy
and Energy Along Adiabats
the Hugoniot
Pressure
That Intersect
-
-
F.
Isothermal Pressure and Energy
Hugoniot Relations for a Second Shock
IV. CODING PROCEDURE
G.
A.
Hugoniot Calculations
B.
Zero-Degree-Isotherm Calculations
C.
Hugoniot Temperature
D.
Adiabatic Calculations
V. GENERAL DESCRIPTION AND USE OF THE CODE
A. General Description
Flow Diagram
B. Instructions for Setting up the Problem
C. Sample Problem—Calculation of Us(Up) for
APPENDIX I. Derivations of Y(Vorſ) and Y'(VoIP)
APPENDIX II. Fortran Listing of the Code
-iii
-
-
-
2024 Aluminum
GERED — A CODE FOR EQUATION-OF-STATE
CALCULATIONS AT MODERATE PRESSURES
Abstract
This report describes
a
Fortran
code which
is used
to convert shock wave data
into equation-of-state information through the Dugdale–MacDonald, Slater, or free
volume gamma relationship and the Mie-Grüneisen equation of state. The code is
written to allow flexibility in the calculation of the particle velocity associated with a
given free-surface velocity measured in a shock experiment.
I. Introduction
GERED is
a
versatile Fortran
code
2.
from shock Hugoniot data:
Calculation of pressure, energy,
Calculation of pressure, energy,
3.
Calculation
for making
the following
equation-of-state
calculations
1.
4.
and temperature along the Hugoniot.
and gamma along the 0°K isotherm.
of pressure, energy, and temperature along adiabats and or
therms that intersect the Hugoniot.
Calculation of the relationship of shock velocity to particle velocity.
The report discusses the theoretical
detailed instructions for its use.
of what the code does may go
and mathematical basis of the code and gives
The reader who is interested in a general description
directly
A.
to Section V.
NOMEN CLATURE
Subscript Notation.
A
Adiabatic
H
I
K
0H
OK
iso
Hugoniot
Isothermal
Zero degrees Kelvin
Refers to the foot of the Hugoniot (initial conditions)
Refers to point on the zero-degree isotherm where Pi—K
=
0.
Variable Definitions
A.
polynomial fit coefficients
Us(Up)
Specific
CV
heat at constant volume
E Energy
EP Specifies termination condition for iteration
P Pressure
t
on Us(Up)
Defined in text
T Temperature
where an adiabat intersects the Hugoniot
THo Temperature
V
Volume
VAA Volume
at which an adiabat
intersects
the Hugoniot
Minimum volume attained in the experiment
Vs Volume XVOH at which gamma is
Free-surface velocity
known from thermal
data
Un Particle velocity
Uf Velocity of
Us
rarefaction wave
the
Shock velocity
Maximum shock velocity used in the experiment
Us
"Y
0D
Grüneisen gamma
Debye temperature
3.
II. Basic Equations
The following equations provide the foundation for development of the code:
Hugoniot Relations
U
p
Pºr
H
-
EH
=
-
U
-v.)
Voh
V
(
S
Pop
+
OH
=
"sºp
–F#–1-,
(2)
Vori
1
E0H
(1)
2.
+ 3
(VOH
- V) (PH
+
(3)
PoH).
& (E
- Ek).
(4)
H.
R.
Christian, Phys. Rev.
Walsh and
1544 (1955).
Rice, McQueen, and Yarger, Phys. Rev. 108, 196 (1957).
–2
—
*Walsh,
=
97,
ºy
P - PR
M.
'J.
Mie-Grüneisen Equation of State
Grüneisen Gamma
*
Y(V)
=
- (#3 -
#
3
-º (P
dy2 \ K vº)
2
d
jº
(PkV
( 5)
2t/3
)
where
Rarefaction
t
-
t
-
t
-
produces the Slater gamma,
0
1
2
produces the Dugdale-MacDonald gamma,
produces the free-volume gamma.
Velocity
Ur
-
V
y
8P 1/2
( #)
III.
This section outlines
dV.
(6)
Formulation of Equations
of the equations the code
the derivation
will
be
required
to
Solve.
A.
In
a shock experiment,
HUGONIOT CALCULATIONS
the two
velocities
value of
required by the Hugoniot relations
Up
following relationship:
Urs - Up
that are measured are Us and Ufs.
The
in Eqs. (1) - (3) is obtained from the
(7)
"Ur.
Substitution of Eq. (6) gives
U
e
fS
=
U
p
W.
V
H
B.
ºr)".
ôV
V.
(8)
S
ZERO-DEGREE-ISOTHERM
PRESSURE AND ENERGY
Hugoniot data provides a reference for the Mie-Grüneisen equation which, to
gether with the Y relationship of Eq. (5), produces the following differential equation:
(10)
(9) can now be written
-
E.
3V
(*#)(En
Fº
2te.
series,
-
(EH
thermodynamics,
[from Eq.
(11)
(4)]
becomes
(13)
Eq. (13) may be written
-
c.
T. \8
(#)
Porſ)
PH (VoII
dT.
*H
H
V
-
0.
V
H
\
-*
V
av
-
(14)
|
(PH
+
dEH
+
|
–
|
Eq. (3),
= =
3 1
to
According
EK)
dv
-
law
H
-
c. dT. T. (#)
0T
P.H
requiring solution
dS equation of thermodynamics
ds.
‘‘’H
dE.
H
(PH
(12)
of
first
Ek):
1
Ek).
Along the Hugoniot, the first
Using the
|
HUGONIOT TEMPERATURE
C.
T.H
|#
the equation
T
a
&
PH
=
FK
+
is
Y
expressed as
| 2
=
K
-
+
K |P H
x-y-K
When
(9)
. 0K Pk dV.
2E.
+
= 0e
9W
V
-
FK, Eq.
(E., H
Ek)
+
is
E.K
Of
+
gives
jū
In terms
z =
*k
*H
\E
*
E.
law of thermodynamics
3
(Pa
first
2.
av
-
the
*k
V
-
v.
= 0
\,
2 r
3V
K\EH - Ek
T
At T
"K.,
*H
2P.
V
P.K
dV.
(15)
Y
(16)
V
C
V
V
(V)
-
PH
-
PoH
|.
(VOH
-
2Cy
+
—— P'
-
Y(V)
+++ TH
V)
Eqs. (15) and (16) into Eq. (14) and division by Cy dV gives
|Pă
TH
ðT/.
(#)
=
,
Now substitution
of
V
V
=2(V)
8T
(#)
=
Equation (4) provides the following relationship:
(17)
1
0H
-
l
V)
+
w)
Cºw)
(Voh
-
ŽToº
=
T
1
V
Voh results in
|r,
b(V)
-
at
Toh
\
Tori
|
Tr(V)
+
Use of the initial condition
-
yº,
(
f(V) exp
=
=
TH
form
a solution of the
V
This equation has
PH
**)
(18)
w)
#
-
exp
=
q(V)
(.
V
Where
0H
PRESSURE AND ENERGY ALONG ADIABATS
THAT INTERSECT THE HUGONIOT
S
VOH
PA
“A
dV
E
-
?'y
(PA
-
13
PH),
(21)
—
(22)
-ī.
%).
V+
º
*—
PA(;
-
Ph
I'
E!,
# Y
#)\
-
Y
-
(;1
Pº
'
=
º
Y
.
Eqs. (21) and (22) into Eq. (19) and rearrangement
gives
(23)
FA
- -
V
Eq. (19) gives
Ş.
of
They are:
V
PH
y? -
-
--
(20)
Eq. (19) can be obtained from Eq. (20).
;Y
in
'
*
Prº
H (V)|.
8P
PA
Integration
subtracted from the
the result, after rearrangement,
v
Substitution
EH
|P(V)
the Hugoniot
of
(#
#)
(#).
=
8E
F
required
The derivatives
2}=
Y(V)
+
E.(v)
H\'
=
E(V)
' +
to
If
Mie-Grüneisen equation referenced
Mie-Grüneisen equation referenced
and P,
the
(19)
is
-(BE)
8P
is
(£)
ôV
to
dV
V
-
--
A
A.
an adiabatic process
p.
For
V
Wrmin
§
Case
1:
D.
H
PA(V) dV.
(24)
Case
2;
V
> WOH
In this region,
the Hugoniot
is
is necessary to find a different
The required reference state is
not known,
and it
state for the Mie-Grüneisen equation.
obtained from thermal data. The energy in this case is given by
'Y
Po
1076
and substituting
them into
gives
E
I
Eq. (19), with the approximation
(9E/9P),
0,
and
(°E/ove,
s
0
Evaluating the derivatives
(25)
0
zkº,
(V) [P(V) -
+
=
E(v) - E,(v)
Pol
J
"
reference
=
T
:
the code.
bP.
will
be
TEMPERATURE
dS equation of thermodynamics
TA
They
(#).
reduces
to
in
V
-
Cy dTA
+
ſh
0
dº
Rearrangement
=
Cº
TA
first
(26)
this region are included
ADIABATIC
E.
Along an adiabat, the
"0
in
!
Several alternatives for defining
discussed in the next section.
Y
1
:
(x
1.
--=
PA'
-º)
Y'V)
dV.
(27)
gives
* --4-(#),
V
V
TA
-
*Ho
when
VA
-
VAA'
6
."IT
.
E.
E.
-
D
(30)
9
c.(?)",
is
PRESSURE AND ENERGY
It
ISOTHERMAL
The isothermal energy can be obtained from the Debye theory.
Tº
yields
d
%
(
F.
(29)
using the
condition
:ſ
X
e
-
dV.
-
V
Y(V).
Eq. (29),
HT.T
of
-
dT
...A. TA
Integration
(28)
Eq. (16) gives
O
Substitution
of
A
av.
dT
where
V
0B
The isothermal
P.I
=
"Don
a
ºy
V
0H
*)
pressure is
=
P,
G.
If
( J.
exp
K
- E.)
K’ “
(E,
+%
V v-I
HUGONIOT RELATIONS FOR A SECOND SHOCK
material under
a
pressure
Poh
and moving with a velocity Up due to a
first
shock wave is hit with a second shock wave, the conservation equations for the second
wave may be referenced to a coordinate system moving with the first wave. The
Hugoniot relations in this case are the same as Eqs. (1) - (3) except Us and Up 3.1°e Il OW
in a coordinate system moving with the particle velocity of the first shock.
IV. Coding Procedure
This section describes
how
numerical solutions
to the equations of Section
III
are
obtained.
A.
HUGONIOT CALCULATIONS
Inspection of Eqs. (23) and (26) reveals that the is entropic pressure required in
the solution of Eq. (8) cannot be calculated until the Grüneisen gamma is known. To
(note that
calculate y from Eq. (4) it is first necessary to solve Eq. (11) for
-
Ek
and
PK
Eq. (11) requires a knowledge of
PH and PH which can only be
PK -Ek).
–
In the
by solving Eq. (8) for
obtained from a previous knowledge of Us
Up:
f(U) or
and
iterate
until
the
necessary
guess
latter case it is
to make an initial
for Us(Up)
system produces a stationary solution for Us(Up).
At low shock pressures, a good approximation to make is
However,
Substitution of this approximation into the Us(Urs) fit (supplied as input data) gives an
initial guess for Us(Up). To provide flexibility, the code is written so that the order of
accomplished
the
may be different than the order of the
Us(Urs) fit. This is
Us(Up) fit
least-squares
points
fitting
by generating 60 evenly spaced (Us, Up)
and
them to a
polynomial.
The value of
on each cycle,
volumes
for
Ur
a specified
These
WAA.
will emerge later) is calculated,
(the steps leading to this value
set of adiabats which intersect the Hugoniot at the
depend on the
Us(Up)
Ur values
–
fit.
If
-
fit does not correctly
the
then the
obtained
or equivalently the Ufs/ Up values
Ur values
f(Up),
will not be the same as for the previous cycle, and the current set of (Us, Up) points
al
is used to generate a new set of Ai. The iterative process is
terminated after
specified number of iterations or when, for the adiabat that lies highest,
represent
1–
Us
EP
(;r - tº)
r
r
ſu:-
3 0.
In this analysis volume is being treated as the independent variable; thus, it is
necessary to calculate Us at each Vi to obtain the Hugoniot values required in the
solution of Eq. (11). The relationship between Us and V is obtained by substitution of
Eq. (1) into the polynomial fit to
The result, after transposing Us, is
Us(Up).
Ao2 U
º'S
(
V
- ++–
1
+
A., U 2
WOH
3
:(
1
is
The Newton–Raphson method
used on Eq. (31)
involves repeated use of the formula
F
(31)
solve for Us,
each Vi.
This
(**)
(º)
F'
i
si
si
- U_.
+ . . .
i
- U n-1 -
U
Il
y \?
-
S
+
at
1
l
A,
wº)
= 0 =
to
F(U.)
S
correction term becomes <10-8 the process
terminated.
necessary
guess
This guess
obtained from
iteration
U
a
linear approxi
(32)
make an initial guess
each Vi:
…)
*i-1
–
U
+
(
=
Si
Si-1
(
1
U
the
-
The code proceeds by using the following equation
UI
To start
at
vº)
to
-
A2
(
-
*/
-
is
l U.
.
to
is
Eq. (31):
º,
mation
to
it
is
When the
(33)
where for convenience
s1
-
will
be used throughout this
report
to
-
The Superscript
number.
U
1
0.99
-
S-1
,
=
-
n
>|<
U
-
--
-
-
indicate an iteration
For
linear fit
Us is calculated directly from Eq. (33). The formula for calculating
PH, from Us and V values, is obtained by eliminating Up from Eqs. (1) and (2). The
result is
a
P..
=
H
P.A.,
OH
+
- .2
1
U
's
Voti
V
-
(
)
WOH
(34)
-
ZERO-DEGREE-ISOTHERM
B.
CALCULATIONS
Equation (11) may be reduced to three first-order differential equations in the
variables R, Q, and S by means of the substitutions:
dependent
V
R(W)
=
W
V
!
R
=
Q
=
dV
Prºſv)
-EK (for convenience only),
=
(35)
0H
-E.
!I
-
=
(36)
Prº,
t
-
(37)
With these substitutions and rearrangement
S
1
2t
--|3|#
(EH
+
Eq. (11) produces
4t
R)
+
-
# (Pu-o]
as
(2 + t)
3V
(EH
+
R)
+
PH
-
s|}/*-*.
(38)
The pressure and energy are obtained by application of the fourth-order Runge-Kutta
method” to the three simultaneous Eqs. (36) - (38). (The code was obtained from the
CIC library.)
The initial conditions required
R(Vok)
=
-Ek(Voz)
Q(Vorº)
=
Pr(Vok)
°s.
------
(38) are
= 0,
(39)
0,
(40)
(Vok)
Phºvok)
-?
the solution of Eq.
tw.
(Vok) EH (Vok)
|Pººvoº
-?'
Gill, Cambridge Phil. Soc. Proc.
woºl
96
-
P
start
47,
S(Vok) ·
=
to
(1951).
->
(41)
where the subscript "OK" refers
to zero degrees
Equation (41) was obtained by differentiating
from Eq. (3). It is
'
1
=
!
|| –
Kelvin
Eq. (4).
and zero
pressure.
The derivative
of Err
H is obtained
-
The integration is terminated when Us(V) 2. Us
.
When Y is expressed as a
polynomial in V, the fourth-order Runge -Kutta"nethod is used directly on Eq. (12).
and ^
required by Eq. (41) must be determined
The values Of
VOK. 7(VOR),
(VOK)
SO
will lie near to
before the integration of Eq. (38) can proceed. In general
Vok
that, if there are no phase transitions in this area, the following approximation
pertains:
*(Von)
+
'(Von)(V - Von).
A means of approximating Y(VOH)
and Y'
obtained are:
*
At
v.
2
1
Vog' -
|A:
I
is outlined in
the appendix.
The results
-(+++)
2A2
=
(43)
(44)
‘A.(
4
-
5
(*;
3 + 2
-6A,A,
-
*).
7(VoIP
(VoIP)
1)
=
3
*(v)
Voh
(45)
V
equation produces the following equation which defines the
0K the Mie-Grüneisen
value of
+
0
G(VOR)
=
Vok:
Vok PH (Vok)
-
(46)
Y(Vok) EH(Vok).
of
is
y'
0.
-
to
of
in
be
n
(VOR)
to
at
of
at
is
of
Eq. (46)
G(V),
equally spaced points
The root
isolated by testing the sign
Newton–Raphson method
decreasing V, starting
next used
reduce the
Voh. The
107°.
uncertainty (volume)
the location
the root
an absolute magnitude of
is still undertermined.
Eq.
However,
can now
calculated from
(43).
(Vok)
-
-
!
y
is
is
Y
is
to
-
substituted for
Voff,
(VoIP)
Since Vok
near
(Vok).
The Grüneisen gamma
calculated directly from Eq. (4).
C.
HUGONIOT TEMPERATURE
of
–
is
of
l
of
10
–
in
is
l
of
of
Eq. (11) has provided the set
values Vi, Yi, and PH. This
integration
Eq.
Simpson's one
information
utilized
the
(18) by application
third rule, for integration of evenly spaced points. The Ph. values required in the
integration are obtained by differentiation
Eq. (34). The result
Solution
I
P..
=
H
(47)
VOH (1
where
-
dus
*
B
(PH - Pori) (x + 1/3)
- B)
-
Hu,
-
1
-
=
A2
2
+
2A3Us3
+
3A4(Us?)
•** >
V
VOH
The code is written so that the user may elect to input a constant value for Cy or to input
the molecular weight and Debye temperature and have the code calculate Cy from the
following fit to the Debye equation:
C, ,
V
=
–4
2.494.293 × 10_`
C2
(1.0 – 3.22 X 10-4 X
x 10° x* -1.7829
-
4.8772
+
3.04973 × 10^* x* -4.65022 × 10^*x”
+
–5 6
2.30082 X 10 ” x”)
× 10^* x*
(48)
where
molecular weight,
C2
=
C
1
- Debye temperature -=
x
=
"Pon
exp
V
-
(
W
Voh
7.
V
*)
-
C1/T.
The error in the fit is less than 0.002% for x
< 5.
To start the integration of Eq. (18), the interval (V OH” Vok) is broken into
20 segments and the trapezoid rule is used to calculate Tok (=T1) and T2 using the
approximation CV (T,) 2C Cy (Tori). At subsequent points, an estimate of the
l
temperature required to calculate Cv; (T,) is obtained from a linear extrapolation of
the previously
calculated temperatures
D.
Ti- 1
and
Ti 2’
ADIABATIC CALCULATIONS
Pressure along
an adiabat is obtained by applying the fourth-order Runge-Kutta
Eqs.
method to
(23) and (26) as required.
FOr
gamma is obtained by making a least-squares polynomial
sº V S.
fit to
'min
Voh
a specified number of the (Yi,
V)
points obtained from the solution of Eq. (4).
The points to be used in the fit are selected according to the following formula:
-11–
i
=
Integer Part
| 1 +
J(M1/N2)]
where
M
total number of points,
1
N 2 - approximate number of points to be used in the fit,
J
=
0,
To these points are
1,
2,
. . . .
-
(N2
1)
added the point ‘Yoh. Voff),
already been selected.
In the region V >
VOH it
“M,” VM.)
if it
to supply the code with values of
has not
E,(V).
polynomial fit (quadratic or less) to
provide
for the case when Eb(V) is not well
Eð(V) determined from thermal data. To
behaved (due to a phase transition), the code is written so that the region V > VOH
may be divided into as many as five intervals, and a polynomial fit to Eb(V) is specified
for each interval.
This is
done by providing
is necessary
and the point
coefficients
of
The following options for defining
a
Y (V)
in the region V
>
Voh are built into
the
code:
1.
2.
A linear extrapolation according to Eq. (43).
Use of thermal data, as for E,(V), to define
polynomial of order two or less.
required as input data.)
3.
A quadratic interpolation which fits Y(VoIP), y' (VoIP), and a value of gamma
>
at SOme
determined from thermal data. The coefficients of the
Vs
WOH
interpolating
C1
C2
=
polynomial are found to be:
- ~'
2
Y(VOH) * C3 VoII - Y (VOH) VoIP
— .."
= Y'
(Vorſ)
-
2C3 V OH"
3
(Vs
4.
in up to five intervals as a
(The coefficients of the polynomial are
Y
-
VoIP
2
A quadratic interpolation to Vs. as in
(3),
then use of a
fit (quadratic or less)
To allow for discontinuities in Y " (V),
may
the Y fit (as mentioned above)
be broken into as many as five intervals.
However, since
PA has been assumed continuous, discontinuities in Y cannot
be handled directly. A discontinuity in Y can be approximated by smoothing
to Y (determined from thermal data).
Y-vs-V curve and taking small step sizes in
No provision is made in the code to allow for the fact that
the
dependent.
- 12 —
the solution of Eq. (26).
the phase
lines are pressure
If
the adiabats are being extended
to zero
pressure,
the solution of Eq. (26)
is
terminated when PA becomes <107°. The value of U, at PA = 0 is obtained from a
linear interpolation (extrapolation) of the
Vi) points and the (U r; V) points.
(PAP
l
Energy and temperature along the adiabats and the velocity of the rarefaction
wave are obtained by using Simpson's one-third rule to integrate Eqs. (24), (30), and
(6), respectively.
V. General Description and Use of the Code
This section gives
a general description
of the code and detailed instructions
for
its use.
A.
GENERAL DESCRIPTION
The flow diagram in Fig.
shows the various options available within the code.
Any path that leads from START to END outlines an acceptable problem.
The box
numbers refer to the sections labeled below which give a brief description of the
operations being performed.
1
The code is constructed so that it can make calculations from any of the
following gamma relationships:
(a) Dugdale-MacDonald,
(b) Slater, (c) free volume,
having
up
(d) a series fit to Y
to 10 coefficients.
The initial values of Y and º' are
1.
required in the calculations.
They may be input or calculated by the code.
(Normally
they should be calculated within the code.)
The Cy, which is used in the code to compute temperatures, may be input
as a constant, or it may be calculated internally from the Debye equation. Poh may
also be calculated from the Debye theory.
2.
The basic problem being attacked in GERED is the determination of pressure,
energy, and gamma along the zero-degree isotherm from the Mie-Grüneisen equation
3.
of state and the particular definition for gamma selected in item
above.
This section
These calculations
1
of the code will
for PK, FK, PH, PH,
and TH:
may also be made for a second shock produced by a reflected or double shock.
4. The first law of thermodynamics and the Mie-Grüneisen equation of state
output values
together with
PA, EA,
a
fit
ºy,
to the gamma values in
3
(just preceding) allows the calculation of
TA/ To along adiabats that intersect the Hugoniot (the volume
must be specified).
These calculations may be made in either
Hugoniot
from the intersection with the
as long as they remain in the range
considered in 3 (otherwise the gamma fit is questionable). Pl and EI along
and
intersection
at the
direction
of volume
isotherms
in the same volume range may be calculated from the Debye theory (this part of the
code
will also
output
9D).
equation is referenced to thermal data,
5. In the region V >
Voh the Mie-Grüneisen
rather than to the Hugoniot. In this region it is necessary to supply the code with
– 13
GºD
GENERATE A
FIT FROM A
U.(U.)
U, U
i.)
6
|
DATA FIT
3
-
ISOTHERM
--
*
W
CALCULATION
FOR SECOND SHOCK
EXIT
l
AND For
EXIT
2
INPUT OR CALCULATE
Cy
2
SELECT GAMMA
*
MAKE ZERO-DEGREE
ISOTHERM
3
EXIT
-
EXIT
4
ISOTHERMAL
|_s.
CALCULATIONS
=Vot
Yº
EXIT
6
EXIT
7
|
|
FIT
of
Flow diagram
14
–
Fig.
–
|
U.(U.)
W
TO OBTAIN
1.
6
ITERATE
YMIN
|
Ҽ
GERED options.
EXIT
5
CONTINUE
DIABATIC
CALCULATION UNTIL
A
4
CALCULATION TO
=
0
CONTINUE
ADIABATIC
PA
5
|
To
4
MAKE ADIABATIC
CALCULATION DOWN
V
4
W
CALCULATION
T
information
E,(V) and y(V). To provide the E6(V) information, coefficients for
quadratic fits to
Eð(V) in up to five intervals may be supplied as input data. (This
gives freedom to deal with discontinuities in E,(V) due to phase transitions.)
To pro
vide the required information on Y (V), one may require the code to: (a) Make a linear
on
extrapolation of
for V
coefficients
VoII. (b) Use up
quadratic fit to gamma in
>
up to four separate quadratic
to five intervals
to define y.
The
for a
each interval must be supplied as input
data.
(c) Fit a quadratic to 7 OH" won and a value of Y at some V > Voh determined
from thermal data. (d) Extrapolate Y, as in (a) or (c), then use input coefficients for
Y
(V) to cover the rest of the necessary range.
6.
The adiabatic calculations provide a means of calculating Us(Up). This
involves an iterative process for which coefficients of a Us(Urs) fit must be supplied
optional. The interation
as input data. The order of the resultant
Us(U) fit is
continues for a specified number of iterations or until, for the adiabat that lies highest,
-
tº-
fits
to
y
than some specified value. If the problem is not completed in
time,
may
the allotted
it
be restarted by using the last set of Us(Up) coefficients
|U.
generated.
|/U. is less
(These coefficients
B.
are punched out on cards for convenience.)
INSTRUCTIONS FOR SETTING UP THE PROBLEM
The code is written so that the Fortran used is acceptable to the 7030, 3600, and
6600 compilers at LRL. However, the control cards, which must accompany the
Fortran deck, are not the same for the 7030 as for the 3600 and 6600 machines. In
addition, because of storage considerations,
for the 3600 it is necessary for the
statement BANK RESULT (1) to appear directly after the COMMON statement named
RESULT, which appears in the subroutine named COMGEN. (This statement–BANK
RESULT (1)—is illegal on the 7030.)
Except for the value of the machine-dependent variable MAC, the data cards used
all three computers are identical. (MAC is used to allow for the fact that
stores an array row-wise, whereas other computers store column-wise.)
on
The following units are assumed in the code output:
Volume (cc/g),
Energy (Mb-cc/g),
Pressure (Mb),
Velocity (cm/usec),
Temperature
(*K),
Specific heat (Mb-cc/g-"K).
– 15 —
the 7030
CARD
1:
MAC
MAC
FORMAT I5
10 on the 7030
=
on the 3600 and the 6600.
= 100
CARD
2:
WOH
VOH, EOH, TOH, USMAX, DELV
=
VOH
VOH
EOH
=
initial volume.
=
VOK (initial).
initial energy.
For
FORMAT F10.6
a reflected
shock set
-
If
E0H = 0., the code will
calculate E0H from the Debye theory. Isothermal
calculations will be inconsistent if this option is
EoPI
=
not used.
TOH
=
USMAX
=
DELV
=
Toti
=
initial temperature.
maximum shock velocity.
volume increment used to solve Eqs. (36)–(38). It
must be chosen so that at most 1000 incremental
"smax
7
steps are taken.
CARD
3:
A(I)
FORMAT F10.6
A(1)
=
of the Us(Up) fit. There may be up to
Seven coefficients.
If the Us(Up) fit coefficients are
being calculated internally from a fit to Us(Ufs),
coefficients
this card
CARD
-
may be left blank.
CV, C1, C2, GAMOH, GAMP0H
CV = specific heat at constant volume if
FORMAT F10.6
4:
= 0.
C1 and C2
equations
If CV = 0. the Debye
will be used to
calculate the specific heat at constant volume. In
this case:
C1
C2
GAMOH -
GAMP0H
CARD
Ty
OH 7
Debye temperature at V
molecular weight.
initial value
=
VOH,
of gamma.
initial value of gamma prime.
If GAMOH = GAMP0H = 0. gamma and gamma prime
will be calculated internally (this is the normal case).
YOH
=
P0H, PK, EK, DPK, VOK
P0H
0. for a single shock (this is a necessary condition
5:
for
start the calculation)
pressure
along Hugoniot for a double or
initial
the code to
reflected shock.
-1
6
FORMAT F10.6
PK
=
initial pressure
It is
EK
=
DPK
=
VOK
=
0.
#
for
on the
zero-degree isotherm.
a single shock.
initial energy on the zero-degree isotherm. It is
= 0. for a single shock.
0. for a single shock.
For a double or reflected
Shock, it is the initial derivative of
Pk(V).
a single shock starting at VOH.
for
0.
For
con
a
tinued single-shock calculation or for a reflected
shock, VOK = VOK.
CARD
6:
NFIT, II, NY, NOIT, NIT
N2, N3,
N2
=
=
0
if
no adiabatic calculations
are to be made
number of points to be used in the gamma fit if
adiabatic calculations are to be made.
To insure
sufficient accuracy,
It
40.
N3
FORMAT I5
this number should be at least
may not exceed 98.
if a Us(Up) fit is to be generated from an input
U s(Ufs) fit
0
-
NFIT
II
NY
if
Us(Up) fit
Number of
1
a
is
Us(U)
to be input.
fit coefficients.
Slater gamma is to
1
if
the
2
if
the Dugdale-MacDonald
4
if
if
gamma is to be used
gamma
the free-volume
is to be used
gamma is to be input as a series.
0
if
II
3
+
4
number of coefficients
II
NOIT
0
=
be used
=
if
-1
4.
NY
if
in the input gamma fit
may not exceed
10.
no adiabatic calculations
are to be made
if adiabatic calculations are to be
calculations are to be made
made but no
Us(U)
=
maximum number of iterations to be allowed.
iteration on Ur may be started either from a
fit
NIT
If
=
Or a
Us(Up) fit.
This allows
The
Us(Ufs)
the problem to be
restarted at any point in the iteration.
number of isotherms to be calculated.
no adiabatic calculations
are to be made,
go
to CARD
16.
-
NNN, NFITORDER, NX, NDP
NNN
1 if the adiabatic calculations
CARD
7:
V
=
are to terminate at
VOH
if the adiabatic calculations are
to zero pressure.
0
– 17
to be continued down
FORMAT I5
NFITORDER
number of coefficients
fit.
NX
1
if
It
to be used in the gamma
may not exceed 10.
the adiabatic calculations
are not to be
continued to the left of the Hugoniot (in the
plane).
-
2
if
the adiabatic calculations
are to be
P-V
con
tinued to the left of the Hugoniot.
of high pressure points, for which
the value of
already known, that are to be
Up is
NDP
GO
to CARD
CARD
8:
10
if
the number
added to the set of (Us, Up) points to be generated
fit.
by the code from an input
Us(Urs)
no Us(Up) calculations
are to be made.
UFSMIN, UFSMAX
UFSMIN
=
FORMAT F10.6
minimum free-surface velocity to be used in
normally
generating the
Us(Up) fit. (It should
be 0.)
UFSMAX
FORMAT F10.6
Us(Urs)
There may
coefficients.
up
D(I)
to
D(I)
be
9:
fit
CARD
maximum free surface velocity to be used in
fit.
generating the
Us(Up)
Severl.
NUMADS, DELVV,
EP
be calculated.
It
number
adiabats
not exceed 30.
may
solving
volume increment
be used
Eqs. (23) and (26). No more than 1000 incremental
steps are allowed.
in
DELV.V
FORMAT I5, 2F10.5
to
NUMADS
EP
to
10:
of
CARD
maximum allowed value of the ratio
|U.
-
wº- |/U.
which will terminate the iterative process.
This
condition must be satisfied for the adiabat that
volumes at which the selected
diabatics
inter
five cards.
values.
They should be listed
18
decreasing
to
of
sect the Hugoniot. The number
volumes listed
require up
equal
may
must be
NUMADS.
This
in
V.A.A.(I)
FORMAT 6F10.6
NUMADS
a
VAA(I), I
–
11.
to
CARD
=
1,
lies highest.
the adiabats are not to be extended
>
a
the end
the calculated gamma and
to
If
is
the code
be read
will
below (see
>
(V
of
5
each segment
VOH) used
FORMAT 3F10.6
3*NA
I=1,
gamma determined from thermal
of
=
GAMMA(I)
Vof
FORMAT F10.6
volume
the end
to express Eb(V).
GAMMA(I),
14:
VS
the
range where
14).
at
=
VBX(I)
I=1,
VS
the interpolated
thermal data begins.
use the first gamma fit
CARD
be made,
in
at
volume
of
the thermal gamma
VBX(I),
FORMAT I5, F10.6
If
between
2
is
interpolation
13:
16.
VOH.
volume greater than VOH to which the gamma
quadratic
be extrapolated (linear).
=
to
VS
CARD
CARD
fit
range V
CARD
go to
NA, VS
NA = number of segments into which the thermal
energy derivative is to be broken in the volume
12:
to
CARD
pressure,
to zero
is
If
coefficients
There will be as many cards as there are
segments called out by NA. Each card
for
gamma fit
the successive intervals.
There may
in
a
is
data.
from
VOH the coefficients for the first segment
gamma
be calculated by the code.
be
X(I),
to
is
and
I=1,
a
in
GAMMA(2)
GAMMA(3).
FORMAT 3F10.6
3°NA
coefficients
the thermal energy derivative.
layout
The
of the cards
the same as for CARD
except there may be no dummy coefficients.
There may be
10
18.
19
gamma as
a
to
of these.
—
CARD
polynomial fit
–
0
NDP
=
If
of V.
go to
17.)
a
CARD
coefficients
14
FORMAT 4E20. 10
of
DU(I)
1,
4,
go
NY
to
I=
=
DU(I),
+
(Note:
16:
II
CARD
If
is
CX(I)
-1. and leave blank spaces for
GAMMA(1)
of
15:
=
CARD
this region just insert
blank card.
quadratic interpolation
be made set
=
a
If
extrapolated
is
If
will
on each card.
to
Vs
one to three coefficients
2
If
be
function
CARD
17:
USH(I)
=
UPH(I)
=
If NIT
=
CARD
18:
19:
19
I=1,
There
TITLE-You
may punch a title, up to
identify the problem.
may be as many
32
spaces long, to
SAMPLE PROBLEM-CALCULATION
u,(U)
This problem was set
FOR
2024
OF
ALUMINUM
up as follows:
1
It was decided
CARD
FORMAT 7F10.6
NIT
isotherm temperatures.
as 20.
C.
CARD
FORMAT 2F10.6
NDP
Up values that correspond to the USH(I) values.
TISO(I),
=
I= 1,
shock velocities for which the corresponding
Up values are already known (see CARD 7).
CARD
0 go to
TIS0(I)
CARD
UPH(I),
USH(I),
to use the 3600 machine,
so MAC
=
100.
2
VOH
=
1/DENSITY
-
EOPH > H
TOH
~
**TOH
=
300.0
HA
O
=
=
.3592
*OH
c p dT
=
0.00167
-
O
(Determined from experimental data.)
Experimentation with several different materials indicates that a step size
DELV 3 0.003 VOH (=0.0012) will give at least four-place accuracy in the
US
MAX
=
.98
solution of Eq. (11). The accuracy of the solution can be determined by varying
the step size.
For convenicence DELV was set equal to 0.001.
CARD
3
Since the
CARD
generated internally, a blank card was inserted.
Us(Up) fit was to be
4
It was decided
CV
to have the code calculate
= 0.
C1
= 37.5.0
C2
=
27.0
GAMOH
=
GAMP0H
0.
=
0.
–20
Cy, *OH:
and
'oh; therefore:
CARD
5
Since all the values
CARD
On
this card are 0., a blank card was inserted.
6
N2
=
70
N3
= 0
This is
a large enough sample to give a good gamma fit.
A cubic fit to
Us(U)
Was Selected.
The Dugdale-MacDonald
It
has been found that the iteration. On
No isotherms
CARD
gamma was selected.
Ur converges rapidly.
are required for this calculation.
7
NNN
It was necessary
= 0
that the adiabatic calculations
be continued
problem
could
be
calculated.
so that Ur
PA = 0 for this
high-order
The
fit was selected to insure a good fit to gamma.
Calculations to the left of the Hugoniot were not needed for
to
NFITORDER
NX = 1
= 10
this problem.
They were obtained after the iteration was
completed.
NDP
CARD
high-pressure points were already known.
Up values at four
= 4
8
UFSMIN
=
UFSMAX
CARD
0.
=
.75
Determined from Shock data.
9.
The
Us(Urs) fit coefficients
The results were:
D(1)
CARD
=
were determined from
0.5282
D(2)
=
0.74607
a
quadratic least-squares
D(3)
=
fit.
0.16432
10
NUMADS
DELVV
EP
=
= 21
This number was felt
to be a reasonable compromise
between
accuracy of the
Us(Up) fit—since Us is nearly a linear
required machine time.
function of
Up —and the
=
.001
.001
Chosen equal to DELV for convenience.
The iterative process is found to be rapidly convergent
down to a level determined by machine roundoff and by
interactions
level,
within the code.
If EP is less
than the noise
iteration is likely to continue for NOIT cycles even
–21–
in the Us(Up) fit can be obtained.
this reason it is a good policy to always make NOIT a
though no better accuracy
For
small number
restart
and
the
iteration if convergence is
not achieved.
CARD
11
The volumes at which the adiabats intersect the Hugoniot were chosen to give a
higher weight to small
slope and sound velocity would be
Up values so that the
reproduced with good accuracy.
The points used were:
VAA(I)
CARD
=
.35,
.3475,
.345,
.3425,
.34,
.3475,
.335,
.3275,
.325,
.30,
.2925,
.285,
.2775,
.27,
.2625,
.255,
.2475,
.24,
.315,
.3075,
.2325.
12
The normal melting point of aluminum
is
932°K.
Upon melting,
the
specific
from 0.3796 cc/g to 0.4097 cc/g. Consequently, for V > Voh
different fit to the thermal energy derivative Eð(V) was made in each of the
volume changes
regions 0.3592
<
0.5. In the region
Y was obtained by having the code obtain a quadratic fit to
V 3 0.3796,
0.3592 < V 3 0.3796,
*OH' Yoh. and Y(0.3796).
NA = 3
VS
CARD
0.3796
×
V
×
0.4097,
0.4097
×
V
×
Therefore
= . 3796
13
VBX(1)
VBX(2)
VBX(3)
CARD
. 37.96
- .4097
- 5
14
In
the mixed-phase
region, Y was determined by making use of the appropriate
cycle and the Clausius-Clapeyron equation. The value obtained
thermodynamic
was:
Y
=
3.126 at V
=
For V
0.3796.
>
0.3796 Y was assumed to be a constant.
Therefore
GAMMA(1)
GAMMA(4)
GAMMA(7)
CARD
al
=
-1.0
=
3.126,
GAMMA(2)
GAMMA(5)
=
3.126,
GAMMA(8)
=
GAMMA(3)
GAMMA(6)
=
=
GAMMA(9)
=
=
=
.0
15
the volume range 0.3592
expression
In
I
Eoſ V)
=
C
<
V
×
0.3796, Eð(V) was calculated from the
P.
Vo.
"
–22
The following fit was obtained:
=
Eð(v)
-10. 5766
V - 91.917.5 v2.
+ 63.46.67
In the volume range 0.3796
Eð(V) Was calculated from the heat of
fusion. The value obtained was 0.1591.
In the volume range 0.4097 × V × 0.5,
again calculated from
was
Eð(V)
CP/Va. At V = 0.4097 the value obtained was
0.229. For V > 0.4097, E,(V) was assumed to be constant. Therefore
CARD
CARD
-10.5766, CX(2)
CX(1)
=
CX(4)
= .
CX(7)
=
1591, CX(5)
.299,
CX(8)
II
=
=
=
2, this card was
=
63.4667,
CX(6)
CX(9)
=
-91.9.175,
0.,
=
=
CX(3)
-
0.
onitted.
17.
USH (1)
= 1. 195
UPH (1)
=
.513
USH(2)
=
1.309
UPH(2)
=
.638
USH(3)
=
1.326
UPH(3)
=
.673
USH(4)
=
1.367
UPH(4)
=
.702
18
Name required since NIT
CARD
V 3 0.4097,
16
Since
CARD
×
=
0.
19.
The title used was:
AL/D-M GAMMA/Ufs-QUAD/Up-CUBIC
This problem converged
in four cycles.
highest
three lowest and the
adiabat were:
The (Urs/Up)" values obtained for the
Cycle
(Ufs/ Up)" values for WAA value
35
0.345
34.75
of:
0.2325
1
2.00 1638
2.004448
2.001314
2.0834.66
2
2.00 1635
2.004440
2.001315
2. 09.1860
3
2.00 1636
2.004 443
2.001315
2.092 127
4
2.00 1637
2.0.04443
2. 001315
2.092 129
These results indicate that random errors occur in the sixth decimal place. However,
inspection of the three low adiabats indicates that the actual accuracy of Urs/Up is
approximately 0.003.
(The error in Urs/Up is due to the interpolation used to obtain Ur
at PA
= 0.
It
may be reduced by reducing DELV.V.)
–23–
The
Us(Up)
Us
fit obtained was
=
0.52816 -- 1.4952
Up
– 0.6704
2
U"
O
+
3
0.4081 U ’.
p
The fit obtained with the high pressure points included was:
U-S
=
0.5314 + 1.3812 U
p
-
0.0531 U*
O
-
0.3257 U".
The card layout for this problem is on the next page.
–24–
p
PROGRAMMER
C-G-For
COMMENT
STATEMENT
NUMBER
PROBLEM
10
||
9
||
8
||
7
||
6
||
5
||
4
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58||59[60|61||6
Appendix I.
Derivations of
and
Y'(VoII)
Y(VOH)
Underlying
and
PK(VoH)
-
!
!
!I
||
f
.
.
)
->
v
2
y-l
PH
-1
)ºv
-
(§
I
2t
2
...
Prºv
+
it
2t
Pl:
+
V
111
-
(49
OH
P.
*
H
P"
V=V
H
H
y-l
PH
PH
+
P.
H
4t
H
*
P"
2P
f
t
\
(2
-
=
#)-(−4
;-(−
3
1
-(#
4t
tº
(PHV
)
=
-
t
Y
+;
11
(VoIP):
*(vog)
(Von)
into Eq. (5) yields the following expressions
-
.
of (d)
Y(VoIP) and
111
P}(VoIP).
the above approximations
+
for
III
P.(Vorſ)
Yl
Substitution
0,
~
PH(VoIP)
*
3 4t
(a)
are the four assumptions:
these derivations
P'
–2
V=Vo H
(50)
Before Y(VoIP)
is
-
-
ti
it
Y'
equations,
necessary
(VoIP) can be calculated from the above
III
relationship
and
of PKCVOH) to
PH(VoIP)
PH (VOH) and to determine the
and
t
to evaluate
Elimination of Us from Eqs.
3
is
...)
(51)
obtained after rearrangement:
–26–
series
is
V(PH)
a
Up
expressed as
given by Eq. (51) yields
squared, the following expression for
in
1/2
=
is
If
the
resulting equation
"Asº
the
(1) and (2) gives
-
of the series in
A
first substituting
V
Reversion
+
(AU
obtained by
Up
0H
p
-
1
-
W
F
=
PH
at:2
The functional relationship between PH and
Eq. (34) with the result
Us(Up) fit into
V
Hugoniot quantities.
PH.
2
V = V Arr
0H
If
Voh
— —#:
Prº
H
2 *
A.
|| 1 —
—--- PrºH
2VOHA2
2
2
+|
4
A.
Eq. (4) is differentiated
. 2
5VoIIA3
—ºtº-t-
–
2A3\
–º
|
2
A.
Af
P.
2
-- . . .
* H
.
(53)
three times with respect to V and the approximations
resulting equation,
(a) through (c) are substituted in the
the following relationship
is
obtained:
^
111
P k (VoIP) - PH(VoIP
(Vor,)
OH
--vi-
Pk(VoH) - BH(VoIP)|
Equation (5) gives the following relationship
"oh!
. .
WOH
for 7(VoIP/Voff
v Ph'VoIP
+
(* 3V !)
2
-
FHVoº)
(55)
Substituting Eq. (55) into Eq. (54) and taking the energy derivatives
rearrangement,
-
111
Pk (VOH)
=
|PH
2
P"
-
H.
ſ
4P
-
(2
+
6V
gives, after
II
0P
V -=V
Equation (54) gives the following expressions
V
(54)
:
for
(56)
OH
the
pressure derivatives evaluated
at
= V2,...:
OH
dP
dV | \,
V-Vof
d”P
--
d°P
1
(57)
---,
4A-A*
-
-->2-1
—3– |x,
(58)
OH
V-VOH
-
H
—a
dV
-a-,
Vori
H
dVT
A%
-
H
-
| y,
—-------.
-18A*A*
2 **1
12A. A.
3 **1
(59)
V
V
Substitution of Eqs. (57) -(59) into Eqs. (49) and (50) gives finally
7(VoIP)
I
"Y
(VoIP)
- (2
=
-
-(
–-
1
Vof
+
3
t
-
)
+
2
A2
(60)
2A2,
+
A2 (
4
- 5t) -
3
)
–27
-
6A3A1
+
3 +
(*#
2t
)
-
(61)
Appendix
II.
Fortran Listing of the Code
D0
till
1
is 1,50
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ENjëiíčić
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ENDCLICH:
CLICHE COM5
COMMON/EO/NA.CX (15), WBX [5].VS, GAMMA (15)
ENDCLICH:
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COMMON/PRES/P0H.DU(10), NY
ENDCLICH:
II
ºn
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Fºniſtry, ºffix,&#ima,Tx,2HPK,11x,2}{K,
"Yº
x-Ušºf
11X,2HPH,11X,2{H,
iii.
His
4??
J-MA, NHA*2)
intysiſ; 12.6.3Hºws, F12.6.1H)//
3,408, (CXIJ),
406 WRITE OUTPUT
iriº,
Wººyº
DIFs. [18X
IF
1275
$5%
1./WQH
s
RHOOH
(WOK) 894, 1275,094
sº
GO TO 394
CALCULATE WOK FROM
695 CALL WOKCALC
E0
C
Csassssssssssssssssssssssssssssssssssssssssssssssssssaaaaaaaaaaaaaaaaaat
(46)
Cassassasssssssssssssssssssssssssssssssssssssssssssssaaaaaaaaaaaaaaasses
994 CONTINUE
Csasssssssssssssassssssssssssssssssssssssssssssssssssssaaaaaaaaaaaaaaaat
IF
s
s
C
E0
CALCULATE INITIAL CONDITIONS FOR
(11)
PHU (1) spHUG (WOK)
EHU(1) s.5epHU(1)
(WOH-WOX)+EOH
DPHU(1) s-PHU(1) alſº RM (WOK)
DEHUs.5s (DPHU(1) (WOH-WOK) -PHU(1))
(PK) 897,696,997
997
TO 898
896
(NY) 176, 177.176
IF
:*:::::
176
SQR
sº
(?)
"ºn-antºwn
Ns?
170
ºur
1)
•
.
+
[
]
&
&
GO TO
GAME}{8
-GAMPÖHeº
090 CONTINUE
SQR (2)
SQR (3) s-EK
HU(1))
sº
C C
N=3
88.8888.88888.8888.888&68888&8888.888888388888388
SOLVE EQ
Dºlys-Dºly
s.
170
(11)
23WQH
r
CALL R&G (N.W.0%. VEND, DELW.SQR)
-
?
Cºë86&&&.388888&&&888&&$888&68&868&6&888&@8888.8888888.888&6888&888&ºtº888&
M1 s??
WALS".
1,111
giftſ)/w011)
#if:{i}
G
5:
DC
:
s
31
)
:
C
Qassassessesssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
NTEGRATE
GAMMA (W)/W
(SE: EQ (10)
X-y0ſººty
...iiiºliiºliſhiii!:
MNX1
EC
sh:
$888&
3888&8888.8888.8883
(10)
WDIF's WQH-W0K
tº-3AM"Hºy!HegAMP0H
C:
Y#Jewſkis'ſ
stºº
*H
)
)
(1)
)*PHU(1)-POH)
–33
ºf
C
(GAMP0HsVDIF)
–
st!/AD7C
JH
PH;
TC
(AD
(DPHU
)
sºxPF
RAY
; ::
; )
) -
AC7.
(;
M!
Cº
;
-1,
EC C!
suyº
35
JWvis (WGH/WQX'ssCOEs:XPF
53
.
.
888668888.88888.888386.88836&9888.888338
NY:CRATE
(1)
SNTS•0.1.
if
(PK) 1315, 1316, 1315
1315 Hºſſ (1) aſ GH
Tº sº. 3
GO TO 1317
1516 WSTEPsyDIF/26.0
&
!
###"...is
######,
Pispºſº
(WDCT)
PD1-Pisſºl
(WDCT)
WNºſewºº-WDCT
pºſs twäH/V5CT) as:0EaÉXPF (GAMP0HsVNET)
1314
GMTGaGMTG+YHT
(WQY)
Piațºjº
episTºº!
PD1
(WOK]
gºrge-VSTEPs (CNTC+.5s (PO1s WDIF4F1)/UVN)
Trisºn?
G/CW
1377 CONTINUE
Hºſſ (1) shºe (1.4.5&T'ſ 1/70H)
GMTG2açNTC+.5e (RAT (1) +8AT (2) 1808 LV
TT2sºn ſºº/CW
Hºſſ (2) s?hu/ADTCſ2)s (1.4.5s,772/70H)
C1
sºut!)
C1
sººj (2)
CW1 stwººt (HUT (1)
CV2sCWOFX
(HUT
)
i2)]
Iss, MNX?
D0 36
tº-pººjſij
tº;
Tºsz.8%Uſ (1-1)-Hºſſ (1-2)
CWSeCWQFT (TG)
Hºnºran wºotºut-ºutinſºm
TT2-Ty
CW1scW2
CW2*CW3
aſſºwanctiºn...sºm
&gggggs&sesssssssssssssssssssssssss
º:
Is?.
00 280
ſºrtin-wół:/yºh
Is?. M!
ºffij-i./iºrſ
00 2.5i
Lºgº.º.º.º.
XYAs?HU (#1)
wº
6)
i.i.
º
riºtzhai.
(I), CAMM(I),PKK (I), EKKII),PHU(1),
•
‘w
9
-
tilt
QUTPUT
º
Hºlſ.
88.8888.88888.88888.8888.8888.8888.8888.88888.8888.888
Rºsul TS
TAPE_3,102, (RAT
AND PLO'ſ
tº r
QUT
.
CHPRESSURE
########.
kºtº'ºs-isomon
*ºtºqf
kºi
AAU. AAW. AAW. AAP
ºf Pºisſtäki.
stipchº.º.º.º.5%)
Cºll_201N1
XYAsºſ
!!!,3;
A
;
Zºli
ºf
4,
J., XYA,0,10HPRESSUREIMEGABARS),3)
(MEGABARS),3
1,0,5HV/WWH,
*###..º.º.º.º.w....
A8
Ç
C
251
•
250
S8888.88888.8888.8888
M1
(1)
$6
fººl,M1,1,1)
(1H8, AD7C,PKK.M1,1,1)
(Mi)
ºRID (14,9,1,1,3,5HV/WQH,4,0.,
CALL SETCH (30, ,5%.
XYA,0,23HENERGY (MEGABARS-CC/GRA
,0,0,1,0)
-34
931
, ICYC, AAU, AAY, Ali,Mº
ºf'."
§º.º.º.º.
#
jºin
#################
I4/4A0/
ºwrºat
1GY
/ins-isºmon be
tº jºinićtiº,M1,1,1]
CALL SETPCH(0,0,1,0,500)
#
#
#;
S:
CH!
*
,0,5HGAMMA,4)
-
jūīºf riffitºšicyc,
C
e
AAU, AAV, AAM,AAP
§:##1.”
“...fºllºw-unminuºus"
#:
!
1509
70
till
tº
išoikºsti,TIsott))
IF (N2) 70,70,71
RETURN
POINTS
Ks ()
C
TO BE USED
TO MAKE
A LEAST SQUARES
FIT
TO CAMMA
AM1 shi
AN2*M2
UNUMsAM1/AN2
WXY's 1.-UNUM
DO 58
Kºkº'ſ
is 1, N2
WXY sy)(Y+UNUM
Js WXY
WAD (K,2) =WO
(J)
58 GAMAD (K) • GAMM (J)
IF (M1-J) 141,141,140
141
NFITPTSsk
140
WAD (K+1,2) =WO(M1)
GAWAD (K+1} sº AMM(M1)
GO TO
66
NF: TPTS=k*1
GO TO 66
65 DO 39 l =1, M1
WAD
39
(I.2) swo (I)
ºffirginiſt1)
NFITPTS-M1
66 NFITPTS=NFITPTS+1
WAD (NFITPTS,2) syſ)H
twº ſºiffsj-tingh
Cºssass8888.888.888.888.888.8888.888388.88888.8888.8888.8888.8888.8888.88888.88888.8888
Dºlys-DELV
IF (NNOIT) 20,109,20
Cºssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
C MAXE LEAST
SQUARES
28 CALL GAMF. T
FIT
TO
GAMMA
Cassssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
C MAKE
ADJA8ATIC CALCULATIOS
CALL ADI A8
C88388.8888.8888.888sssssssssssssssssssssssssssssssssssssssssssssssssssssss
09:
IF (NNOI7+1) tº 1,109,091
IMMs (MM+!
URA (IMM)
sºu
(M1)
USMAXs 1.7.1.1 sushAX
GC 70 44
:29 F (NDP) $65,865,864
864 CC 063 s!,NDP
Ksh;CPYS+:
JS K sºjSH!!)
-35 —
SUCROUTINE WOXCALC
C&CCCCCCC866888.808888&6888.8888.8888.8888.8888.8888.88888.8888.88888.8888.8888.8888
CALCULATES WOK BY SOLVING EQ. (46)
C THIS SUBRIUTINE
,
C8&8888888.8888.88888.88888.8888.8888.8888.8888.8888.8888.8888.8888.8888.8888.8888.8888
BY ITERATIVE SOLUTI
THIS SUBROUTINE CALCULATES VOX, MHERE PK • EX •
(W) s ()
OF EQUATION: Wsph (W) - GAMMA (W)
C
C
sº
COM!
USE
WW2 s WOH
#
"...wº-tº
Dºls.
01&WOH
D0 5 K-1, 15
WW1swV2
WW2sWW2-DEL
G1s02
C2•C (WW2)
IF (C1862) $5,21,3
5 CONTINUE
35
sI º
!º-ºwn-wn/iº-ºn
CALL EXIT
50 CCsC (WOt.)
IF (G1sqG) 55,21,34
55 WW2-vOL
MºsVW1-WW2
MºsVW2-WW1
G! •CC
56 IF (ABSF (G2) -.
57 WT =WOL4.0000
IF (G(WT) scG)
21
WOKswCL
RETURN
END
*iºliiliº
*
> THIS
38888.8888.8888.8888.88888.888ssssssssssssssssssssssssssss
SUBROUTINE CALCULATES US 9W SOLVING EQ (31)
:****************assassssssssssssssssssssssssssssssssssssssssssssssssss
USHIN, USMAX - ESTIMATED RANCE OF SHOCK VELOCITY WILUES
US: COM!
USE
COM5
–36–
ºh".”
XX (1)
axºDIF
XX (2) a 1.00018XX (1)
USMs, 958USMIN
,
Fºirºtºzºro
10
!
2.
OF US,
WALUE
FUNCfióN
CHECK
ºnzº.”.”
AT THIS
BY HAND,
WALUE
MAY NEED
OF B DOES NOT
WIELD
UNIQUE
LOWER USMAX)
RETURN
20
35
IF (NFIT-2)22,25,22
Y. Tij} (7:it:35
RETURN
22 Z1 =7ERO (8, XX (1))
Z2-7:RO (8, XX (2) )
iſ 5 T-2.5|
inckxx-??sixx (1)-xx(1-1))/(22-21)
XX (1+1)
IF (ABSF
syx (
) -8NCRXX
(3NCRXX) -.00000001
31
21 =72
1
fºurtigh
impf
ºff ºffijfND'Souffon
5 Z2*7:R0 (8, XXII +1).)
16;
5.
) 50, 50,51
101
fo ZEROtx)
= 0 WITHIN
50
ITERATIONS)
****
CALL EXIT
50 DIFs). X (1+1)-X
Xs)(X
(I +1)
RETURN
END
SUBROUTINE RKGIN, XS, XF.H, Y)
Cºasts&sasagaaaaaass8888.888&ssssssssssssssssssssssssssssssssssssssssssss
*::::::
CTHIS
SU880UTINE
!
USE
COM2
USE
COM6
gº
USES THE RUNGA KUTTA METHOD TO SOLVE EQ.
CºON/RUNGA/E (4), F(4), R(4)
disj.Yū), Yºſë
Eſ 2) s. 29.20952.1081345
Eſ 5) s 1.707106781 18655
Eſ 4) s. 16666666666666666666
Fſ 1) s2.
F
F
F
( 2)
s?.
3) s?.
( 4) s2.
Rſ 1) a .5
R (2) s. 29.20952.108.1545
R{5} =1.707106781 18655
R(4)
s.5
MsABSF
M! st
(
[XC-XF}/H]+.5
#:#;
{5}*.5s (Y (4) +POH) s (WOH-A1)+ECH
+:
DC 5 K-1, M
M’ sh;14
PHU (K}sy (4)
!? (Y-.9suSMAX)21,21,24
–37 –
(11)
OR
(12)
24 RETURN
21 WO(X) •A"
-- (PHU(K)-P0H) alſº RM (A1)
sy (5)
Gºitº) sais (PHUIK) -Y (2)) / (ERUIK) *Y (3)]
Pºº (Y) sy (2)
Exx{K} =-Y (3)
00 5 Ja 1,4
Dººjtk)
ºutk)
tº fo tiš,11,15, 11),
11 A1*A1+A2
J
DERIV (A1, Y.YP)
15 CALL
D0 5 1 =1, N
(Jia (YP (I) sh-F (J) sq (1))
YII) syll) +TEMP
TEMPs:
5 Qii) sqt!)+3.87 ERP-R(J) syp (I) sh
Mºſuº
SUCROUTINE DERIW (WOL,Y,DERSOR)
Cººeeeeeeeeeeeeeeeeeeeeeeººººººººººººººº. 88.8888.08888.8888.88888.8888.8888.8888
C THIS SUCROUTINE CALCULATES S PRIME (E0. 50)
Caeseºeeeeeeeeeeeeeeeeeeeges&8888.888.8888.8888.8888.8888.8888.88888.8888.8888.888
),DERSOR(0 )
DIMENSION
US:
COM!
Y(
Pººjº (WOL)
E}{s.5s (Płłłł'OH)2 (WOH-WOL)
Y (4)
Y (5) sº H
sº
tº OH
Zsſ. WWOL
IF (NY)4,5,4
4 XPs.I/WOL
CAMWs).0
D0 6_1 =1. NY
YUs)(Psw(\L
CAWs.
AMW+}\}{1}a}{U
6 MPs)(U
tº
Rſ1}
Y (3) s-Y (1)
*-ºsº (1)
=-PH-SAMWs
(Y (1)-EH)
*
#.
UºN
5 km ſººty (5) jaz
DERSCR (1) -- (Y (2) sza (CCIII.4) sºlº CC (11,5]s [PH-Y (2)))
!
+2.sy (1) s (CCIII.1) shºpH-Y (2))) / (EH-Y (3) )
Q? (2) sy (1)
(3) sy (2)
EN)
SUCROUTINE
*::::::::::::::::::::
3.15.5ºlº.
SIMPSONS 1/3 RuLE *::::::
C*******assasssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
88.8888.88888.88888.88888
WSES,
D!
###!!
10N.W.A.
(1000),Y (1000)
}/H; +1.5
####..s.
*WALS+.56%
(Y (1)) +Y (2) )
is 3.M
' Wiſſi-Wittſ-24.333333333333styū-21-4.syſi-1},ytinsh
50
ſ
–38–
Rºſvºn
EN)
SUCROUTINE GAMFIT
86888088888.880880888.888 S8888.8888.8888.8888.8888.8888.8888
C8888.8888.08868880888
THIS SUCROUTINE MAKES A LEAST SQUARES FIT TO CAMMA
Ceeeeeeeeeeeeeeeeeessessesssssssssssssssssssssssssssst
C
USE
&8888.8888.8888.88888
COM!
COMMON/AAZ/MAC
l l
D0
= 1,
NFITPTS
WAD (1,1) =1.0
= 2, NFITORDER
D0 1
Pºitº'ºtrišf shºts fir'fö ºn
foºiriſhi.
tuft
RRR,M,www.
/
his BEEN MADE
1NT ºf stºirs
TAP; 5,101, IWAD (1,2), GAMADI1), RRR(1), Isl, NEITPTS)
ºff ºfficifºr;
i2ha
1
HPITE_OUTPUT
fºrf7/4&pººfs
jśń"in
l
101
WAD,
GAMMA
######!
ºf
º'fit º'ºsińuits
(2715.6,
RESIDUAL
//
100
J
Jºjº Wapſi,z)
ºitºitR,
GAMA),C,Dee,
ºf ſºirºfs,
jºičijj.I.T.ſºſſº
Wipii, Jy-Winfi,
W
1
7744h
E15.6].)
RETURN
END
SUCROUTINE ADIMB
Cººessessessssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
THIS SUBROUTINE COMPUTES PRESSURE, ENERGY, PARTICLE VELOCITY. AND
"Alon;
DIABATIC ENERGY
DEFINITIONSassº(X(1)
s.
Ajºſs' INTºº-Wºliº.
C
C C
Tºfu:
PKXII)
sho! ABATIC PRESSURE
PHU(1) shºjº,0NIOT PRESSURE
EHU(1) shuCONIOT ENERGY
i
USE
RATIOslº A/70
),
WL
(0
888&8888.8888.8888.8888.8888.8888.8888.8888.8888.8888.88888.888
Y
ſºlº
(I) JºA
(I) slºpg|RATURE
sº
G
AMM
ADTC
(2)
,DELLL (2)
COME5
CºN/TITLE/AAU,
AAW, AAW, AAP, ICYC,NX
WL
=
1./WOH
R}00H
WL (1) syſ)H
(2) s/MOTC (M1) syſ)H
lºſs ADTC
(M1)
XYAs?HU (M1)
º;;
stºß,6;
OUTPUT Tip:
FORMAT (12HCYCLE
XYA,0,1 CHPRE SSUREIMEGABARS
(MEGABARS),5)
,
w
ºil
MºITE
AAW AAM.AAP
1, 7,
•
!
#
i.
#####,
**
00
|
ššuº'Along Apiagars
4/4A0/2 $ºsºvº'ſſong
0,0)
CALL SEICRI (ADTC (1),PHU(1)
i.i)
CALL POINTsūjić, pºſſ,
98.031.W.W
NUMADS
WAsRHOOH
—39
–
-
8096
1.
WA*WAA (1)
s
15
WG
######$!!...vii.º.º.
W/WOH, 4,0.
)
IFINA) 47,47,40
47 WGZs 1.05
GO TO 49
*
tºº)
Xsa (1) / (1,-A(2)
XGUESsuSMIN+.98 (USMAX-USMIN)
If tº-XGUES)46,46,45
45 XaXGUES
46 XXYY ex
DIFs,
§ #.
0.18X
USSTART-SQRTF (WJHePHUG (WA)/8088)
T
USSTART,
MRITE
su
&
G
i##"
###########
102
UPSTART
NUMBER,15/4X,5HUSs,
F15.6/4X,5HUP-,F15.6)
Dºl\}(X=0&LW
Dºlwe0&LWW
D0 13 IZ-1, NX
Y (1) s?HA2
Ne!
Xs):YYY
Ceeeeeeeeeeeessessesssssssssssssssssssssssssssssssssssssssssssssssssssss
C INTEGRATION
OF EQ. (25)
CALL RECAIN, WA, WL (IZ ),00LLL (17), Y)
Ceeeeeeessesssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
*italiº-waw".ºut.
XFe2.swC (M1)
Y (1) spºk (M1)
IF (NNN)63,64,65
ifiºſºziºs
64
C INTEGRATION
88.8888.8888.8888.88.8888.8888.8888.88888.8888.8888.888
OF EQ.
(26)
CALL ADIAS2IN, W0 (M1), XF, DELV,Y)
21
*::::: Jºl,M
flºwntºw
º
D0_50
30
Cºasessessessessesssssssssssssssssssssssssssssssssssssssssssssssssssssss
C INTEGRATION
OF CAMMA (W)/W
(SEE EO 30)
CALL SIMP (ADTC, RAT, WALS,08:LLL (17), WA, W0 (M))
88.888883&SC38888.8888.8888.8888.8888.8888.8888.8888.8888.8888
D0_52 Jel,M
32 ADTC (J) ºf:YPF (-AOTC
(J)
)
Cassasses asssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
C INTEGRATION
OF EQ (6)
CALL SIMP (EMU,0PHU, WALS,DELLL (12), WA, W0 (M))
838888.8888.8888.8888.8888.8888.88888.0888.8888.8888.88.08888.8888.8888
WALSs:0}}+.5s (WOH-WA) s?HA2
Caesaaeedassssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
C INTEGRATION
OF EQ (24)
CALL SIMP(EKK,PKK, WALS, DEL, WA, W0 (M))
- 88.8888.8888.8888.8888.88888.8888.8888.888&sssssssssssssssssssssssssss
IF iſ Z-1)22 25,22
WS-Upsilºſºſin):
tººlſ M-1))
1 s (EHU(M)
RATIOisufs
22 00_41,
41
..?]
115
...
:75
ARY
Js 1,
*ºft.§:%mº
ºf
Wiſ (J) syO(J)/WOH
00
42
(PKKIM-1)/IPKKIM-1)-PKKIM))-1.0)
s]
.
If [Z-1126,27.26
J
fºr
º.º.º. 3.;
3,113,5,110
fºirſ/74%. Tºš/Wºlfº.6%
fºlſ
(63,5M€7A,70X,2HPA,
11X,2RPH,11x,2HEA,9x,5HGAMMA,
–40
–
25
Fº łºśījj,
5x,5HTA/70,7x
10X,5HW/WQH,9X, IHW//)
Riº Olimpiji
"ijtjäſuºjičij.
Vijjīn)
ºf
º, º:
fºiſt
Ž;
TAPE 3,29
ºwn.º.º.
fêtioWING IS
AN EXTENSION
US
1,
iFúz.;;
OF THE ADIASAT.10
THE
BY THE KNOHN VALUE.0F.GAMMA //)
4,545.6, F12.6,F15.6)
1)
J),
º: fiº.3,
roºfiñº,
fiº.
25
26104
1High PRESSURE LIHIT/6x,35HALLOWED
WRITE
, Jº
!
ADTC (J), EHU (J
CALL POINTS (HUT, PKK, M1,
ºut.J),EKKt.J), GAHºtJ),
3.35.1%
(i+1) =D(i)+UFSs (DI2]+UFSs (DI3]+UFSs
+UFSA (DI5]+UFS
s
1
()
(6) +UFSs) (7) )))))
UPP (1+1,2) suPSTART
(4)
GO TO 28
26 WRITE OUTPUT
ID
!
US (1)
(1)
=
0
15
tonfiniſt
UPP[1,2) =0.0
NOPTS=NUMADS41
UPP (1,1)
1,
NOPTS
=
l
=
40
DO
DELW's DELWXX
1.0
J-2, NFIT
40
D0
ºft.*.**i.”
*:::::*
40
RETURN
END
C
TO
Y)
SUBROUTINE REGA (N, X0, XF.H,
C888&séssassssssssssssssssssssssssssssssssssssssssssssssssssssssst
THIS SUBROUTINE USES THE RUNGE KUTTA METHOD
SOLVE EQ. 25
(4), F(4), R(4)
Qiqi, Yiği, Yºtº
COMMON/RUNGA/E
jºnsſøn
ſ
0. = H (
(I)
=
0
8
MsABSF [XO-XF)/H]+1.000001
H2s.58
1,N
D0.8
sy
M
sy
K-1,
GAMM (K)
PHU (K)
W0 (K)s
A1
5
DO
Als)(0
CALL DERIWA (A1, Y.YP)
24!}
RETURH
)
::
)
-
=0 Y
(I) sh-F (J)
sq.(1))
+TEMP
(!) +3.87EMP-R (J) syp (1)
END
DERIWA (WOL,
OPA)
-41
-
SUBROJTINE
Y,
5
Y!:
J
1,N
TEMP-E {J}s (YP
sh
L
=
15,
ºf 70 5
!!
15
DC
5
OC
(YP (1)) 1.2.2
5PHÚſkj-5057f7-YP (1))
Ja 1.4
11),
(+5.11,
Als th2
CALL DERIWA (A1, Y.YP)
ºf
2
If
(2)
(3)
PKK (K) sy (1)
288.888
C888.888.888&8888.8888.888.8888.888.8888.8888.8888.8888.8888.8888.8888.88888.888
:
THIS SUBROUTINE CALCULATES THE ADIABATIC
sessess
PRESSURE DERIVATIVETEO.
23)
C88888.8888.888&8888.8888.8888.8888.8888.8888.88888.8888.8883.38888.8888.8888.88888.888
USE
COM!
DIMENSION Y (6
), DPA
(8 )
WA*WOL
PAsPHUG (WA)
Y (3) spa
GAs CAMºU (WA)
MPA (2) “GA
GAMART
sºlº
U(WA)/GA
DPH--PAs TERM (WA)
*
DEHe .5s (DPH's (WOH-WA) -PA)
-GAWs):H&DPH+Y
DPA (1) spas (RHO-GAMART)
(?)s
SUCROUTINE ADIAS2IN, X0, XF,H,\)
Cºººººººººººººººººººººººººººººººººººººººººººººººº
C
THIS SUCROUTINE USES THE RUNGA-KUTTA
METHOD
(CAMART-RHO-GAW)
&8888.888.8888.8888.8888.888
TO SOLVE EQ. (26)
Cºººººººººººººººººººººººººººººººº.388.8888.8888.8888.8888.8888.8888.8888.8888.8888
US:
US:
COM2
COME5
(4), F (4), R(4)
COMMON/RUNGA/E
jñºsión disj. Yiği, whº
Fºsſtºj}#}.5
H2s.5sh
Y (7) sº Aº (M1)
Y (0) skſ
Y (2) sy (T)
YP (1) =-1.080pHUIM1)
D05 K-1.M
DPH1 (M1) =SQRTF
=0.0
sp?HU(M1)
(-YP (1))
PH.U.M1+1)
Pºk (H1) sy (1)
W0 (Mi) sº?
GAMM(M1) sy (2)
If [Al-W8x (NA))33,35,1
35 IFIPKK (M1)-.000001)
131
ºf TURN
31
is 1,151,132
132 M1 shit!
00 5 Ja 1.4
§
70
tiš,11,15, 11), J
11 A1*A1+H2
15 CALL SERIVB (A1,
1,N
D0 5
is
TEMPs: (Jia
s
(YP
Y.YP)
(I) sh-F (J) sq.(1))
######prº
*Q (I) +5.87EMP-R
#"
......Sººº...!!NE,
(J) syP
(1)
I sh
DERIVB W, Y, YP)
*************sassasssssssssssssssssssssssssssssssssssssssssssssssssssss
-42–
{};
THIS SUCROUTINE CALCULATES THE ADI
iLSö täLCULATESTGAMMA FOR WoWOHISECTION
if
Č
C
SSUAE
DERIWATIVE (EQ 26)
IV-0)
COMS
Y
DIMENSION
YP10
)
COM!
),
USE
USE
(0
Casaaaaaaaaaaaaaaaaaaaaaaaassssssssssssssssssstaaaaaaaaaaaaaaaaaaaaaaaas
WA*W
R}{0s!.
IF
WWA
1,1,2
50'5"ri.Ni
(WA-W8X (1))
N=1 +58 (I-1)
(I-1)
GAA=GAMMA(N)*WAs
(GAMMA (N+1)*CAMMA
CAMPsGAMMA (N+1)+2.8 WA86AMMA (Nº.2)
(2) sº AM
gº”
/GAM) syſ 1}-GAMER}{08EOPFNC
(2) acAM
(GAM+1.-W.As
(N+2) aWA)
GAMP0H/CAM)
sy
RETURN
CAMs) (7) +GAMP0H2 (WA-W (0)
Y
1
6
ºw”,”
Ne4+58
Y
6 3
GO TO
4,4,5
(WA)
)
4
IF
2
(WA-WS)
(1) •GAMs?!!08:0PFNC (WA)
SUCROUTINE ISOTHERMSIN0,7)
Cºººººººººººººººººººººººººººººººººººººººº
&
8888.8888.8888.88888.8888.88888.8888
THIS SUBROUTINE CALCULATES PRESSURE
C
AND ENERGY ALONG ISOTHERMS
C86888.8888&8888.888888.8888.88888.8888.8888.8888.8888.888.8888.8888.8888.8888.8888.888
©
USE COM!
US:
COM2
101
MRITE_OUTPUT TAPE 3,101,NO.7
formiſſiºi, 20x, ºf$öß'Number 12,52x,12HTEMPERATURE-FM.2/
120Y. 10H------------------ ,52X,21H--------------------//)
Mºſt jumpur TAPE 5,102
vol.,5x, ºpressure, 10x,&ENERGY
102
fºurt;,&Völunt, ižºisºlative
ſījiàº'àº.77)
1.727.5%iºi,
pā’īf.º.º.2
ti-ſphuiſ)
EDIFs:THM (T)
(I) •PKK (I)
*
+EDIF
+GAMM(I) /W0 (I) agolf
(!), ADTC
TAPE 5,105,
(1),WT (I),EMU(1),
CAMM(I),
I
HUſ
W0
'ºn
stºk (1)
E}{U(1)
105
FCºAI
(F11.5,0X, F10.5,0x,F11.6,7X,F11.6,0x,FC.4,10x.F9.2)
FUNCTION PHUG (WOLUME)
E0
PH
Cºassassssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
THIS FUNCTION CALCULATES
FROM
(54)
*:::::::: *intinuuuuuuuuuuuuuuu
WQL*WO-UM:
80s!.-WOLs?HOOH
CALL JSCALC (88)
PHUGsks KsKHOOHs38+P0H
RETRI
l
!
US: COM6
ENC
-43
*::::::::::::::inuuuuuuuuuuuunrººf":
tº
ſºlº...?
FUNCTION CALGULATES THE SPECIFIC HEAT FROM THE
Essassassessssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
DIMENSION
US:
DD (10)
COM)
IFIC!)6,7,6
7 CWOFT's CW
Rºſvºn
6 CONTINUE
DD (1) = 1.0
s-5.223-4
s-4.0772E-2
DD (4) e-1.70293-5
DD (5) sy.04975:-5
DD (6) s-4.650222-4
DD (7) s2.300022-5
DD (2)
DD (3)
XODeC1/TEMP
FODs
1.0
Us)(0)
D0
1
1s2.7
fön-fúDºğütl)su
1. Usue)(OD
-
CWOFTsz4.94.295:-58FOD/C2
RETURN
END
“rºutinium"
THM (T)
FUNCTION
Cººººººººººººººººººººººººººs
C
88.8888.8888.8888.8888.8888.888.888.8888.8888.8888.8888
THIS FUNCTION CALCULATES THE
THERMAL
ENERGY
YsC1/T
AL00sL00F (X)
ETMMs. 00024.94295/C2s (T +C1s (-.574002+5.22E-48ALOG
1
+Xs (4.0772:-24x8
(.091453-54xa (-1.016502-54x8 (1.16256E-4-.46016&-58)())))))
FUNCTION G (WOLUME)
Cººeeeeessesssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
CTHIS
FUNCTION CALCULATES G (W)-DEFINED
*:::::::::COM!
BY
EQ.
(46)
8888.88888.8888.8888.8888.8888.8888.8888.8888.8888.8888.8888.8888.8888
WOLs.WOLUM:
CAMsCAM0%GAMP0Hs
(WOL)
(WOL-WOH)
PłłłsºuG
0, 0
E}}{s.58PHHs (WOH-W ôLjith
glºw-tº
tº ###"
!
'ºuï', w
#(ºst
WQX
LIES
IE QUTSIDE
SIDE
SUSROUTINE WOKCALC)
CALL EXIT
EM)
–44
THE
W
WOLUME R
RANGE
OF
º
FUNCTION EOPFNC (WDUM)
Cassaaaaaaaaaaaaaaassassassssssssstaaaaaaaaaaaaaaaaaaaaaaaaaaaaassessess
CTHIS FUNCTION CALCULATES THE THERMAL ENERGY DERIVATIVE (SECTION IV-D)
Cassassassssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
USE
COM!
USE COM5
Nº. 1
IF (NA-1)5,4,5
5 DO 5 Isl, NA-1
IF (WDUM-W8X(1)).2.2.5
2 N=1 +38 (I-1)
GO TO 4
3 Ne4+58 (I-1)
4 EV-FNC's CX [N]+WDUMs (CX (N+1)+CX (N+2) a WDUM)
#!"
º
ãº"minimummim”
FUNCTION TERM (WOLUME)
#
C8888.88.888&8888.8888.888.8888.8888.8888.8888.8888.8888.888.888.8888.8888.8888.8888.8888
CALCULATES THE HUGONIOT
PRESSURE DERIWATIVE/PRESSURE
COM!
USE
COM3
WAs WOLUM:
DUS's A (2)
8 is 1.-WA/WOH
UPs)(88).
ZXsUP
IF (NFIT-2)3,4,5
5
50'ſ i-2,
8X8s I
ºff:
DUS*DUS+8)(387Xs
1 ZX-ZXs UP
4 TERMs
RETURN
A (1 +1)
(DUS+1./8H)
/ (WOHs (1.-Bºis DUS)]
END
º:
FUNCTION ZERO (B.DDD)
Cºssaaaaassasssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
CTHIS
CALCULATES F (US)-DEFINED
BY EQ. 31
USE COMS
7:R0s A (1)
9Cs5800D
XDUMsgC
DC :
!
1s2,NFIT
ŽRQ-ZER}+\{1}
(DUMs), DUM28C
axDUM
ZEROszERO-ODD
RETURN
ENC
fiſh C7:
Oh GAMFU (WDUM)
**************ssessesssssssssssssssssssssssssssssssssssssssssssssssssss
–45 –
:
THIS FUNCT! ON CALCULATES GAMMA(V) FROM THE LEAST SQUARES FIT TO GAMMA
Catecasaasaaaaaaaaassessessessesses&8888.8888.88888.8888.8888.8888.8888.8888.888
XDUM*WDUM
CQ 1 i = 2, NFITORDER
GAMFU's SAMFU+C
ºxDUM
(I)
!
XDUMexDUMeWDUM
RETURN
END
FUNCTION
C8888&0&0&0&0&0
CAMPFU (WOUM)
&0&0&0&0&0&0&08&888888.8888.88888.8888.8888.8888.8888.888.8888.888
CTHIS FUNCTION DIFFERENTIATES
THE LEAST SQUARES FIT TO CAMMA
C88888&688&6888.88888.888.8888.8888.8888.888.8888.8888.8888.888.8888.8888.8888.8888.888USE
COM!
CAMPFUsC (2)
XOUMsWDUM
#
}
jºritain
CAMPFUsGAMPFU+GIs C (I) syDUM
!
XDUMs)(DUMs WDUM
Rºſſuſºn
END
-46