CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
CONSTRUCTION OF WIDE-RANGE EQUATIONS OF STATE
THROUGH "MERGING" LOCAL EQUATIONS USING
MIXTURE MODEL
Leonid F. Gudarenko, Vadim G. Kudelkin
Russian Federal Nuclear Center-VNIIEF, Sarov, Russia, 607190
The paper demonstrates abilities of the method for local equation of state "merging" using a mixture
model by the example of construction of a wide-range equation of state of iron. A model used for the
merging and for computation of thermodynamic functions is described in detail. Graphs are plotted
that illustrate behavior of the thermodynamic functions calculated both by the wide-range EOS and by
the merged equations of state.
behavior model and has its own domain of
applicability, and it is local in this sense. Domains
of applicability boundaries for the selected local
EOS are separated on the plane. Each domain is
described by its own EOS. Fig.l conventionally
depicts positions of the domain of applicability for
EOS I and II and transitional domain III in
coordinates "relative compression S=p/p0 temperature T\ T\ is the upper boundary of the
domain of applicability for the EOS describing
domain I, F2 is the lower boundary of applicability
for the EOS describing domain II. (p is current
material density, p0 is material density under
standard conditions).
INTRODUCTION
In some science and engineering areas there is a
permanent need in computation of processes
attended with dramatic changes in material density
and temperature. Several efficient semi-empirical
and theoretical models are known, which allow
construction of quite accurate equations of state
(EOS) in separate regions. However, no unique
model has been developed thus far, which would be
used to derive EOS for computation of material
thermodynamic functions (TDF) in the range of
states from ones close to normal to those
corresponding to ultrahigh energy concentrations.
The typically used technology for construction of
the EOS represents TDF in the tabular form at
reference point grid nodes. The TDF values at the
reference points are calculated by several local
EOS. TDF outside the reference points are
calculated using various interpolation formulas.
The EOS construction method described in this
paper consists in "merging" local EOS using a
mixture model.
II
S=P/Po
FIGURE 1.
To construct the wide-range EOS, it is necessary
to introduce to domain III a function ensuring a
smooth transition to the EOS describing domain I
MERGING METHOD
Each EOS is developed using some material
127
on the EOS describing domain II. The function
form should not depend on the original EOS
representation form. The idea of the proposed
method is that the TDF be calculated in the
transitional domain using the EOS data in the
domain with the mixture model. That is, material in
domain III is represented with mixture of two
constituents, whose TDF is calculated by the
original EOS. All the thermodynamic functions
calculated by the mixture model and given EOS
should therewith coincide on the boundaries.
To calculate the TDF in domain III, it is
necessary to introduce function a, which is equal to
one on the boundary of the first domain and to zero
on that of the second. In domain III, a lies within
0<or<l. With this definition or means concentration
of material, whose properties are described by the
first EOS.
The possibility is analyzed to use two models, i.e.
the thermodynamically equilibrium heterogeneous
mixture model and the model, according to which
both the constituents are of identical density and
temperature and mixture pressure, Pmix, and energy,
Emix, are calculated by simple summation of the
relevant values in accordance with their
concentrations [1]:
equation of state, it is necessary to develop a model
for the TDF calculation in the transitional domain,
which accounts for variation in concentration or as a
function of density and temperature.
To solve this problem, model (1) has been
modified. As a result, the following model for the
TDF calculation in the transitional domain was
obtained:
P(p,T) =
-<*(p,T))P2(p,T) +
(2)
where Ec(p)=E(p,T=Q) is a potential energy
component.
Obtained model (2) allows merging the potential
TDF components with meeting the thermodynamic
consistency condition.
To calculate the constituent concentration, it is
necessary to select a function of two variables that
satisfies the following conditions:
= 0.
PmiX(p, T) = aP!(p, T) + (1-a) P2(p, 7);
(1)
The following method for construction of the
concentration function is proposed. Explain its idea
by an example. Take point C lying in the
intermediate domain (see Fig.l), draw a curve
through it that intersects the domain boundaries at
points A and B. Then the mass concentration of the
mixture constituent, whose TDF is calculated by
EOS 1, can be determined as a ratio of BC curve
length to AB curve length:
EmiJtp, T) = aE}(p, T) + (1-a) E2(p, T).
Hereinafter subscripts denote EOS designed for
calculation in domains I and II, respectively. The
analysis suggests that model (1) allows a smoother
TDF description in the transitional domain,
therefore the model was used later on.
When both the models are used, the resultant
EOS in the transitional domain does not satisfy the
principal thermodynamic identity, which is a
consequence of the second principle of
thermodynamics for equilibrium processes. A
reason for this is the fact that model (1) and the
thermodynamically equilibrium heterogeneous
mixture model satisfy the thermodynamic
consistency condition in the case of constant
mixture constituent concentration. However, in the
case under discussion the concentration is a
function of density and temperature, which is just a
reason for the principal identity unsatisfiability.
Hence, to obtain a thermodynamically consistent
Thus, to determine the a in the entire
intermediate domain, it is sufficient to construct a
continuous family of nonintersecting curves
uniquely mapping boundary Y\ onto boundary F2.
For these families, for example, isochore and
isotherm families can be taken.
Some characteristic features of the function a are
derived from the condition of continuous pressure
128
and energy on the boundaries up to the second
derivatives. In this case on the boundaries the
function a must satisfy the conditions of zero
derivatives with respect to density and temperature
up to the third order. Represent the or as a complex
function:
'\ _ *(~f ~T\\
' ———' "
experimentally studied domain be covered. The
lower boundary F2 of the applicability domain of
the EOS approximating computations by theoretical
Thomas-Fermi model is drawn in accordance with
the Thomas-Fermi model applicability boundary
[5].
Function a is calculated by formulas (3), (4)
along the isotherms. The dependencies for factors k
include 14 adjustment parameters, which are
selected from the condition of minimum TDF
curvature in the transitional domain and the
condition
of
minimum
thermodynamic
inconsistency of the resultant EOS. When selecting
coefficients, the TDF derivative monotonicity and
validity of principal thermodynamic inequalities
„ _ LAC (P>T)
LAB(p,TY
then it is sufficient to construct a function of one
variable satisfying conditions
/(0) = 1;/(1) = 0;-
dx"
dx"
The seventh-degree polynomial is taken for the
function:
f(x)=x4(20x3-70x2+84x-35)+l.
Cp ~Cv
(3)
are followed.
Fig.2 presents continuous iron Hugoniot in
"pressure - relative compression 8* coordinates
computed by WREOS for each of original EOS and
experimental data [6]. In addition, Hugoniots are
depicted that are computed by two theoretical
models: TFPK [7, 8] and KSM [9]. The figure also
shows the transitional domain boundary positions.
With this construction of the function a, the point
corresponding to ce=0.5 lies at equal distance from
the boundaries. However, the practice suggests that
sometimes it is reasonable to displace it to one or
another side. To do this, one more function is
introduced which contains scaling parameter k [2]:
~^_ o-*)*
1Q7P[GPaL
10°
That is, a = f( g(x)). If fc=0.5, g(x)=x.
Preliminary computations showed that at fixed
values of the coefficient k monotonicity of TDF
derivatives is violated in some domains. To avoid
this, the parameter k is taken as a function of
argument x.
10°
104
103
EXAMPLE
102
Below we demonstrate possibilities of the
developed code using construction of the widerange equation of state (WREOS) of iron as an
example. Semi-empirical EOS, which allows state
computations in the domain studied well in shockwave experiments [3], and EOS approximating
computations by theoretical Thomas-Fermi model
and describing the ultrahigh pressure and specific
energy range [4], are taken for the "merging".
The upper boundary Y\ of the semi-empirical
EOS domain of applicability is selected so, that the
7
5=P/P0
1
2
3
4
5
FIGURE 2. Iron Hugoniots (poo=7.87g/cm3) and isentropes:
—————— - Hugoniot, computation by WREOS; — — — Hugoniot, computation by EOS [3]; — — — — - Hugoniot,
computation by EOS [4]; - - - - - - Hugoniot, computation
by TFPK model [7,8]; —————— - Hugoniot, computation by
KSM model [9]; O - experimental data [6]; ———— isentropes computed by WREOS; —— - —— - lower boundary
of the merging domain; —— - - — - upper boundary of the
merging domain.
129
Fig. 3 presents computed porous iron Hugoniots.
The notations are like in Fig.2.
algorithm uses as a few as 14 adjustment
parameters selected from the conditions of
minimum thermodynamic inconsistency and
minimum TDF curvature in the transitional domain.
P [GPa]
10'-
Pnft=2.7g/cmf
REFERENCES
10"-
1. G.I.Kerley. // Sandia Report SAND92-0553. 1992.
104-
2. Tishin A.P., Shinkin G.P.// Zhurnal Vychislitelnoy
Matematiki I Matematicheskoy Fiziki. 1991. V. 31.
No.ll.Pp. 1745-1749.
3. Glushak B.L., Gudarenko L.F., Styazhkin Yu.M.//
VANT. Ser. Mat. Modelir. Fiz. Protsessov. 1991. No.
2. Pp. 57-62.
8=P/P0
0.5
4. Yeliseev G.M., Klinishov G.E. Preprint No. 173.
Moscow, USSR Academy of Sciences IAM, 1982. 21
P-
1.0
1.5
2.0
2.5
3.0
FIGURE 3. Porous iron sample Hugoniots.
When the thermodynamic inconsistency level
was being checked, it turned out that at 8~\ the
inconsistency level was «10%. The maximum value
of ~ 25% is achieved at £-0.03. It is known that in
tabulated EOS the thermodynamic identity is not
strictly fulfilled as well, with this being the case in
the entire domain of EOS definition. For
comparison the inconsistency level was computed
for EOS No. 2140 of US Equation-of-State Library
SESAME [10] also designed for iron TDF
computation. This reached -50% in some domains.
5. Kalitkin N.N. Preprint No.85. Moscow, USSR
Academy of Sciences IAM, 1978. 46 p.
6. M.V.Zhernokletov, R.F.Trunin, L.F.Gudarenko,
V.D.Trushchin,
O.N.Gushchina.
//
Shock
Compression of Condensed Matter - 1997. Edited by
S.C. Schmidt, D.P. Dandekar, J.W. Forbes. The
American Institute of Physics. 1998. P. 51.
7. Kalitkin N.N., Kuz'mina L.V. Preprint No.35.
Moscow, USSR Academy of Sciences IAM, 1975.
8. Kopyshev V.P. Preprint No.59. Moscow, USSR
Academy of Sciences IAM, 1978. 11 c.
CONCLUSION
9. Kalitkin N.N., Kuz'mina L.V. // Matematicheskoye
Modelirovaniye. 1998. V. 10. No 7. Pp. 111-123.
The computed data presented in Figs. 2, 3
demonstrate the feasibility to construct the widerange EOS using the above-discussed technique.
It is authors' view that the proposed method for
construction of wide-range EOS offers the
advantage that it requires no reconstructions in EOS
to be merged. The original EOS are only
complemented with an algorithm and program for
the TDF computation in the transitional domain.
One need not develop phase transition description
algorithm anew, like this is done, for example, for
the tabular representation. The algorithms
introduced to the original equations by their authors
are used. In the considered example, the program
10. S. P. Lyon and J. D. Johnson. // Los Alamos National
Laboratory, Report LA-UR-92-3407.
130