Thermodynamic modeling for numerical simulations based on the generalized cubic equation of state

Abstract

We further elaborate on the generalized formulation for cubic equation of state proposed by Cismondi and Mollerup [Fluid Phase Equilib. 232, 74–89 (2005)]. With this formulation, all well-known cubic equations of state can be described with a certain pair of values, which allow for a generic implementation of different equations of state. Based on this generalized formulation, we derive a complete thermodynamic model for computational fluid dynamics simulations by providing the resulting correlations for all required thermodynamic properties. For the transport properties, we employ the Chung correlations. Our generic implementation includes the often used equations of state Soave–Redlich–Kwong and Peng–Robinson and the Redlich–Kwong–Peng–Robinson equation of state. The first two assume a universal critical compressibility factor and are, therefore, only suitable for fluids with a matching critical compressibility. The Redlich–Kwong–Peng–Robinson overcomes this limitation by considering the equation of state parameter as a function of the critical compressibility. We compare the resulting thermodynamic modeling for the three equations of state for selected fluids with each other and CoolProp reference data. Additionally, we provide a Python tool called real gas thermodynamic python library (realtpl). This tool can be used to evaluate and compare the results for a wide range of different fluids. We also provide an implementation of the generalized form in OpenFOAM.

Full text

Phys. Fluids 34, 116126 (2022); https://doi.org/10.1063/5.0122277 34, 116126
© 2022 Author(s).
Thermodynamic modeling for numerical
simulations based on the generalized cubic
equation of state
Cite as: Phys. Fluids 34, 116126 (2022); https://doi.org/10.1063/5.0122277
Submitted: 23 August 2022 • Accepted: 03 November 2022 • Published Online: 16 November 2022
T. Trummler, M. Glatzle, A. Doehring, et al.
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Thermodynamic modeling for numerical
simulations based on the generalized cubic
equation of state
Cite as: Phys. Fluids 34, 116126 (2022); doi: 10.1063/5.0122277
Submitted: 23 August 2022 .Accepted: 3 November 2022 .
Published Online: 16 November 2022
T. Trummler,
1,a)
M. Glatzle,
2
A. Doehring,
1
N. Urban,
1
and M. Klein
1
AFFILIATIONS
1
Institute of Applied Mathematics and Scientific Computing, Bundeswehr University Munich Werner-Heisenberg-Weg 39,
85577 Neubiberg, Germany
2
BooleWorks GmbH, Radlkoferstrasse 2, 81373 Munich, Germany
a)
Author to whom correspondence should be addressed: theresa.trummler@unibw.de
ABSTRACT
We further elaborate on the generalized formulation for cubic equation of state proposed by Cismondi and Mollerup [Fluid Phase Equilib. 232,
74–89 (2005)]. With this formulation, all well-known cubic equations of state can be described with a certain pair of values, which allow for a
generic implementation of different equations of state. Based on this generalized formulation, we derive a complete thermodynamic model for
computational fluid dynamics simulations by providing the resulting correlations for all required thermodynamic properties. For the transport
properties, we employ the Chung correlations. Our generic implementation includes the often used equations of state Soave–Redlich–Kwong and
Peng–Robinson and the Redlich–Kwong–Peng–Robinson equation of state. The first two assume a universal critical compressibility factor and
are, therefore, only suitable for fluids with a matching critical compressibility. The Redlich–Kwong–Peng–Robinson overcomes this limitation by
considering the equation of state parameter as a function of the critical compressibility. We compare the resulting thermodynamic modeling for
the three equations of state for selected fluids with each other and CoolProp reference data. Additionally, we provide a Python tool called real gas
thermodynamic python library (realtpl). This tool can be used to evaluate and compare the results for a wide range of different fluids. We
also provide an implementation of the generalized form in OpenFOAM.
V
C2022 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://
creativecommons.org/licenses/by/4.0/).https://doi.org/10.1063/5.0122277
I. INTRODUCTION
Numerical flow simulations of super- and transcritical conditions
require appropriate thermodynamic models. A state-of-the-art ther-
modynamic model is based on a cubic equation of state (EoS) and
departure function formalism for the evaluation of enthalpy and
energy. This can be found in early
2–5
as well as recent computational
fluid dynamics (CFD) investigations.
6–14
For the cubic EoS, mostly the
Peng–Robinson
15
(PR) or the Soave–Redlich–Kwong EoS
16,17
(SRK) is
employed. Both assume a universal critical compressibility and are,
therefore, only suitable for fluids with a matching critical compressibil-
ity. Volume translation methods
18–21
represent one possible solution
to improve the density prediction for fluids, which are not well
described by SRK or PR. Cismondi and Mollerup
1
suggested the
Redlich–Kwong–Peng–Robinson EoS (RKPR), introducing a third
EoS parameter and formulating all three EoS parameters as a function
of the critical compressibility. Building upon their work, Kim et al.
22
presented a thermodynamic modeling approach based on the RKPR
and demonstrated its advantages and suitability for different fluids.
Despite its advantages, only a few studies
6,23
have employed the RKPR
EoS recently for real gas CFD simulations of n-dodecane injections.
Within the suggestion of the RKPR, Cismondi and Mollerup
1
and ear-
lier Mollerup (see Michelsen and Mollerup
24
) also proposed a general
formulation of the cubic EoS by which all of the well-known cubic EoS
can be described with a particular set of values. Such a formulation
allows for a modularized implementation of all these cubic EoS, thus,
less code duplication and a better readability.
An alternative to a cubic EoS is the PC-SAFT EoS (perturbed-
chain statistical associating fluid theory),
25
which has successfully been
employed by Rodriguez et al.
26
and Rodriguez et al.
27
for two-
dimensional CFD simulations. Another approach is the usage of tabu-
lated reference data, yielding higher accuracy
28
and also a potentially
faster evaluation of the thermodynamic data.
29,30
Recently, also
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artificial neural networks, trained on tabulated data, have been
employed for thermodynamic modeling for real gas CFD simula-
tions.
31,32
However, cubic EoS are still mostly used due to their sim-
plicity and overall good accuracy. In addition to the EoS and the
relations for thermodynamic properties, CFD simulations also require
relations for the transport properties viscosity and thermal conductiv-
ity. Chung et al.
33
proposed correlations for these transport properties,
which are often employed for such real gas simulations.
6–9,21,34
Alternative methods are, for example, the Lucas method
35
for the vis-
cosity and the Stiel–Thodos method
35
for the thermal conductivity.
These methods have recently been employed by Sharan and Bellan
10
for nitrogen and showed a good agreement with NIST reference data.
Alternatively, the residual entropy scaling technique can be used for
the calculation of the thermal conductivity
36
and the viscosity
37
as
done by Koukouvinis et al.
28
In general, it is an important step in CFD
simulations to check the accuracy and suitability of a thermodynamic
model in advance, as included in several studies.
7,8,28
For different flu-
ids, pressure, and temperature ranges, such an evaluation can be com-
plicated and time-consuming. Apart from that, such a verification
requires an already successful implementation of the thermodynamic
model. This is usually not the case during the development or further
development of a CFD solver. The open source library CoolProp
38
provides implementations for the SRK and PR EoS. Bell et al.
39
wrote
a comprehensive thermodynamic library to evaluate chemical proper-
ties specifically targeted for chemical engineering. Our new tool
realtpl, on the other hand, has been specifically designed for appli-
cations in the context of CFD simulations and evaluates the entire
thermodynamic model required for these simulations.
In this paper, we aim to further promote the idea of the generalized
formulation of cubic EoS by Mollerup,
1,24
i.e., one formulation for all
three cubic EoS (PR, SRK, and RKPR). To this end, we describe in detail
how this formulation is solved and present the resulting relations for the
thermodynamic properties. We also outline the overall thermodynamic
model based on this generalized formulation. For the thermodynamic
model, we employ the Chung correlations for the evaluation of the
transport properties. We apply the thermodynamic model to selected
fluids and study its suitability. Therewith, we also demonstrate the good
applicability of the RKPR for all fluids with different critical compress-
ibility factors. In order to test the proposed thermodynamic model and
to apply it to different configurations, we provide an open source
Python tool called realtpl forarealgasthermodynamicpython
library. This tool can be used to evaluate and compare the results for a
wide range of different fluids. Additionally, we also provide the imple-
mentation of the generalized form in OpenFOAM.
The paper is structured as follows: Sec. II presents the thermody-
namic model based on the generalized cubic EoS. Then, in Sec. III,the
applicability and suitability of the thermodynamic model are assessed
for selected fluids comparing the model using SRK, PR, and RKPR.
Section IV contains information about the additionally provided
Python tool realtpl and the validation of the proposed
OpenFOAM implementation based on the generalized formulation.
Finally, the paper is summarized in Sec. V.
II. THERMODYNAMIC MODEL BASED ON A
GENERALIZED CUBIC EQUATION OF STATE
We present a thermodynamic model based on the generalized
cubic EoS. First, we present the EoS and describe in detail how it is
solved. Then, the correlations to evaluate the thermodynamic proper-
ties are presented and, finally, the Chung correlations for the transport
properties are briefly described.
A. Generalized cubic equation of state
We here consider the generalized formulation of a cubic EoS sug-
gested by Cismondi and Mollerup
1
and already earlier by Mollerup
24
pðv;TÞ¼ RT
vbaa
ðvþd1bÞðvþd2bÞ:(1)
The pressure pis a function of the molar volume vand the temperature
T. R denotes the universal gas constant with R¼8314:472 J=ðkmol KÞ.
aand brepresent the two traditional EoS parameters, considering
attractive forces with aand repulsive forces by the effective molecular
volume b. Both are determined by a proportionality factor and the criti-
cal properties p
c
and T
c
of the fluid (see Table I). Furthermore, ais mul-
tiplied by a correction factor athat is a function of reduced temperature
T=Tcand the acentric factor x. It is worth noting that for a¼0and
b¼0, the cubic EoS collapses to the ideal gas law. As a consequence,
mathematically, and also physically, the molar volume vhas to be larger
than the co-volume b(v>b). The common cubic EoS can be described
with special pairs of the values d
1
and d
2
,whered
2
is a supplementary
parameter defined as ð1d1Þ=ð1þd1Þ. Multiplying the denominator
out results in the well-known and often used formulation of
pðv;TÞ¼ RT
vbaa
v2þubv þwb2;(2)
where u¼d1þd2and w¼d1d2. However, the first formulation [Eq.
(1)] yields simpler expressions of the derivations required for evaluat-
ing the thermodynamic properties (see Sec. II B)thanEq.(2). For the
widely used EoS SRK and PR, the proportionality factor in aand bis
constant and d
1
, or, respectively, uand w, are constants with d1¼1
(u¼1, w¼0) for SRK and d1¼1þffiffiffi
2
p(u¼2, w¼1) for PR.
Hence, for SRK and PR, a universal critical compressibility has been
assumed, which is about 0.285 for SRK and 0.263 for PR.
22
Therefore,
these two EoS are only well suited for a certain set of fluids with a cor-
responding similar critical compressibility. To overcome this limita-
tion, Cismondi and Mollerup
1
suggested to evaluate the EoS
parameters as a function of the critical compressibility resulting in the
RKPR EoS. For the detailed evaluation of the EoS parameters, see
Table I. For the RKPR, a different correlation than for SRK and PR is
used to evaluate a. Consequently, also the derivatives by temperature
@a=@Tand @2a=@ T2, required for the evaluation of the thermody-
namic properties, differ for the EoS. Concluding, all three cubic EoS
SRK, PR, and RKPR can be described by Eq. (1), where only the EoS
parameters a,b,andd
1
as well as the evaluation of achanges.
Figure 1 shows a reduced pressure–volume diagram for n-hexane
including cubic isotherms evaluated using the PR EoS. A subcritical
(T<Tc), a critical (T¼T
c
), and a supercritcal (T>Tc) isotherm are
plotted to illustrate the behavior in the different regimes. The subcriti-
cal isotherm crosses the two-phase region, which is bounded by the
bubble-point-line and the dew point-line. In thermodynamic equilib-
rium, the phase change follows the dashed line and the pressure
remains constant (see Fig. 1, dashed brown line). The cubic isotherm
shows a different path within the two-phase region describing meta-
stable states. Note that the part with the positive slope has physically
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no meaning since pressure increases with volume (@p=@vjT).
Therefore, solving cubic EoS at subcritical conditions can be challeng-
ing, as described in more detail below.
In order to solve the cubic EoS [Eq. (1)], the equation is reformu-
lated using the dimensionless compressibility factor as follows:
Z¼pv
RT ;(3)
as well as dimensionless expressions for aand bwith A¼paaðRTÞ2
and B¼pbðRTÞ1. Therewith, one obtains from Eq. (1)
Z¼1
1B=ZA
B
B=Z
ð1þd1B=ZÞð1þd2B=ZÞ:(4)
Recasting results in the cubic form of all considered EoS in terms of Z,
which reads
Z3þa2Z2þa1Zþa0¼0 (5)
with the coefficients
a2¼Bðd1þd21Þ1;(6)
a1¼Aþd1d2B2ðd1þd2ÞBðBþ1Þ;(7)
a0¼Bðd1d2B2þd1d2BþAÞ:(8)
The obtained cubic equation can be solved for real roots, which is well
described in the literature.
34,40,41
A cubic equation has either one real
and two imaginary roots or three real roots (see also Fig. 1). At super-
critical conditions usually only one real root is present. For this reason,
we recommend first checking for the existence of one real root when
evaluations are focused on supercritical conditions. Three real roots
are generally associated with the two-phase region present at subcriti-
cal conditions. As mentioned above, the volume has to be larger than
the co-volume b(v>b) or expressed in terms of the compressibility
factor Z>B. If the physical constraint Z>Bis full-filled, the smallest
root represents the liquid state and the largest root the vapor or gas
state (see also Fig. 1). From a thermodynamic perspective, the center
root is not stable and, therefore, physically meaningless. The correct
root of the two physically stable roots can be identified by comparing
the Gibbs energy (see Sec. II B). An alternative approach is to take
always the largest root, which usually corresponds to the vapor/gaseous
state (exception see below).
FIG. 1. Reduced pressure–volume (p–v) diagram for n-hexane evaluated using the
PR EoS.
TABLE I. Parameters of the EoS adopted from Kim et al.
22
SRK
16
PR
15
RKPR
1
d
1
11þffiffiffi
2
pd2þd1ðd3czZcÞd4þd5ðd3czZcÞd6with cz¼1:168;
d1¼0:428 363;d2¼18:496 215;
d3¼0:338 426;d4¼0:660 000;
d5¼789:723 105;d6¼2:512 392
d
2
01ffiffiffi
2
pð1d1Þ=ð1þd1Þ
A0:427 47 R2T2
c
pc
! 0:457 24 R2T2
c
pc
! 3y2þ3yd þd2þd1
ð3yþd1Þ2
R2T2
c
pc
!
B0:086 64 RTc
pc
 0:0778 RTc
pc
 1
3yþd1
RTc
pc

with d¼1þd2
1
1þd1
y¼1þ½2ð1þd1Þ1
3þ4
1þd1

1
3
Að1þjð1ffiffiffiffiffiffiffiffiffiffi
T=Tc
pÞÞ2ð1þjð1ffiffiffiffiffiffiffiffiffiffi
T=Tc
pÞÞ2
ð3=ð2þT=TcÞÞj
K0:485 08
þ1:551 71x
0:156 13x2
0:374 64
þ1:542 26x
0:269 92x2
ð66:125czZc23:359Þx2
þð40:594czZcþ16:855Þx
þð5:273 45czZc0:258 26Þ
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At very high and low pressures, the smallest root can be
smaller than the co-volume (ZB). In this case, there is only one
physical meaningful root, which is the largest one. At high pres-
sures, this root then corresponds to a liquid state, while at very low
pressures it corresponds to the gaseous/vapor state. It is important
to note that this can also occur in a clearly supercritical regime,
such as for n-dodecane at the ECN-Spray A condition with
p¼8 MPa in the temperature range T¼1097–1500 K us ing PR o r
RKPR. Consequently, a missing check for Z>Bresults in high
deviations for both EoS, while including the check yields moderate
deviations. Figure 4 illustrates the modeling of n-dodecane at the
mentioned conditions and shows that with the included check for
Z>Bonly small deviations occur.
The presented EoS can be extended to model a homogeneous
mixture for an arbitrary number of components. To this end, the EoS
parameter aaand bhave to be evaluated as a function of the mixture.
Details can be found in the literature.
6,8,22,34
B. Thermodynamic properties
In addition to the correlation of density, pressure, and tempera-
ture, also expressions for thermodynamic properties, such as the inter-
nal energy e, entropy s,enthalpyh, and specific heats cpand cv,are
needed for CFD simulations. The evaluation of these quantities
includes several thermodynamic derivatives, which can be solved using
the departure function formalism. For more detailed information, we
refer to Poling et al.
35
and Elliott and Lira,
40
and for details on the for-
mulations for the RKPR to Fathi et al.
6
and Kim et al.
22
The presented
formulations are obtained from Matheis
34
and recast to be valid for
the more generic formulation.
For the internal energy, this can be written as
eðv;TÞ¼e0ðTÞþðv
1
T@p
@Tjvp

dv;(9)
where the subscript 0 refers to the ideal reference state. The solution of
the integral reads
ee0¼aaT@aa
@T

K;(10)
where the term Kcontains the following expression:
K¼1
bðd1d2Þln vþd1b
vþd2b

:(11)
Consequently, the enthalpy his calculated with
hh0¼ee0þpv RT;(12)
resulting in the expression
hh0¼aaT@aa
@T

Kþpv RT:(13)
The entropy sis obtained with
sðv;TÞ¼s0ðTÞþðv
1
@p
@TjvR
v

dvþRln ðZÞ;(14)
resulting in
ss0¼KT @aa
@TþRln 1 b
v

:(15)
Finally, the Gibbs energy gis calculated using Eqs. (12) and (14),
gg0¼hh0Tðss0Þ;(16)
gg0¼aaKþpv RT 1þln 1 b
v

:(17)
As mentioned above, the Gibbs energy can be used to determine the
most stable root out of three real roots. If the smallest root is larger
than B[min(Z)>B], then the smallest root represents the liquid state
[Zl¼minðZÞ] and the largest one the vapor state [Zv¼maxðZÞ].
TherelativedifferencebetweentheGibbsenergyofthetwosolutions
can be evaluated with
dg ¼gvgl
RT ¼A
Bðd1d2Þln ðZlþd1BÞðZvþd2BÞ
ðZlþd2BÞðZvþd1BÞ

ðZlZvÞþln ZlB
ZvB

:(18)
If dg <0(gv<gl), the vapor state is stable. Contrary, if dg >0
(gv>gl), the liquid state is stable.
Theheatcapacityatconstantvolumecvis calculated with
ðcvcv0Þ¼T@2aa
@T2K;(19)
where cv0is evaluated using cv0¼cp0R.cp0, the heat capacity at
constant pressure at ideal reference state, is determined with the 7-
coefficient or the 9-coefficient NASA polynomials. For the corre-
sponding data for the 7-coefficient polynomials, we refer to Goos
et al.,
42
and for the 9-coefficient ones to McBride.
43
Special atten-
tion should be paid to the fact that the polynomials are adapted for
certain temperature ranges and that for a smooth calculation over
several temperature ranges an appropriate implementation has to
be done. Then, the heat capacity at constant pressure cpcan be
evaluated using
cp¼cvT@p
@Tv
!
2@p
@v
T(20)
with
@p
@Tv¼R
vbþ@aa
@T
1
D(21)
and
@p
@vT¼ RT
ðvbÞ2þaað2vþðd1þd2ÞbÞ
D2(22)
with the denominator D
D¼ðvþd1bÞðvþd2bÞ¼v2þðd1þd2Þbv þd1d2b2:(23)
The speed of sound cis calculated using
c¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cp
cv
@p
@vT
v2
M
s:(24)
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C. Transport properties viscosity and heat conductivity
with the Chung correlations
For CFD simulations, also suitable relations for the transport
properties viscosity land heat conductivity kare necessary, where the
correlations by Chung et al.
33
are often employed.
6–9,21,34
Both quanti-
ties are composed of two terms referring to different pressure levels,
l¼lkþlp(25)
and
k¼kkþkp:(26)
The first summand l
k
,andk
k
, respectively, dominates at low pressures
and is based on the Chapman–Enskog theory for diluted gases. The
second term, l
p
and k
p
, dominates at higher pressures and is based on
empirical correlations. The input for the model is composed of the
fluid properties, the temperature, the density, and the heat capacity cv,
where the latter only affects the evaluation of k. For a detailed descrip-
tion, see Chung et al.
33
and Poling et al.
35
III. ASSESSMENT OF THE ACCURACY OF THE
THERMODYNAMIC MODEL
Figure 2 shows the density q,theheatcapacityc
p
, the viscosity l,
and heat conductivity kfor the n-alkane methane (Zc¼0:2863), the
cycloalkane cyclopentane (Zc¼0:2813), the n-alkanes n-hexane
(Zc¼0:2664), and n-dodecane (Zc¼0:2497). All data have been
evaluated at a reduced pressure of p=pc¼1:5. Overall, the thermody-
namic modeling is able to reproduce the non-linear behavior for all
depicted quantities. As expected, the fluids with a critical compressibil-
ity close to 0.285 are well described by SRK, while for n-hexane
with Zc¼0:2664 PR (optimized for Zc¼0:263) gives good results.
FIG. 2. Comparison of the modeled density q, heat capacity c
p
, viscosity l, and heat conductivity kusing the thermodynamic model employing different EoS with reference val-
ues from CoolProp.
38
(a) Methane (Zc¼0:2863), (b) cyclopentane (Zc¼0:2813), (c) n-hexane (Zc¼0:2664), and (d) n-dodecane (Zc¼0:2497). All data have been evalu-
ated at a reduced pressure of p=pc¼1:5.
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For n-dodecane, RKPR yields the best results. In all cases, the density
is modeled very well with the RKPR EoS. It is either comparable to
SRK and PR or much better, if the critical compressibility differs from
the values for which the two were designed.
Figure 3 visualizes the density variation qover the temperature
for these four fluids at different pressures p=pc¼f0:75;1:5;3:0g.
Again, the EoS can reproduce the non-linear density behavior at all
pressure levels. Furthermore, it can be seen that at all depicted pressure
levels, the trend of the suitability of a particular EoS remains about the
same.
In Fig. 2, we have also included other thermodynamic quantities
and in the following we discuss the accuracy of the modeling of these
quantities. The specific heat capacity c
p
is evaluated with Eq. (20).The
peak at the pseudo-boiling is well captured by all EoS, but the maxi-
mum value is underestimated.
The overall behavior of the transport properties (l,k)iswell
described by the Chung correlations. In comparison with the density
evolution for the different cubic EoS, one can see that the error in the
modeling of the density qcorresponds to the error in the modeling of
these two quantities, see PR in Figs. 2(a) and 2(b). This is due to the
fact that the density qis an input parameter for the Chung model,
which directly affects the calculation of the high pressure empirical
terms l
p
and k
p
. Hence, the error of the Chung correlations increases
with an increasing modeling error of the density. Furthermore, for k,
the error of cvaffects the calculation of k
k
. As a consequence, the
Chung correlations yield better results the more accurate qand cvare
modeled. In addition to our findings, we refer to the recent work by
Longmire and Banuti,
32
who also evaluated and discussed the suitabil-
ity of the Chung correlations for modeling transport properties.
IV. PYTHON TOOL AND OPENFOAM IMPLEMENTATION
We provide an open source Python tool called realtpl (real
gas thermodynamic python library) for the presented thermodynamic
model. Additionally, we also provide the implementation of the gener-
alized form in OpenFOAM.
A. Python framework realtpl
Checking the accuracy of a thermodynamic model in advance is
a central step before conducting CFD simulations. For different fluids
as well as different pressure and temperature ranges, such an evalua-
tion can be complicated and especially time consuming. To this end,
we have written an open source Python tool to easily compare the
results obtained with the here described thermodynamic model based
on cubic EoS. The Python tool is called realtpl standing for our
real gas thermodynamic python library. It is directly coupled to the
open source library CoolProp
38
obtaining experimental reference data,
as well as fluid properties, such as, for example, molar mass and critical
properties. In addition, a database was created for the NASA coeffi-
cients, which is also directly coupled to realtpl.Usingrealtpl,
thermodynamic modeling based on the cubic EoS PR, SRK, RKPR can
be compared and also contrasted with the reference data from
CoolProp. The current implementation is designed to evaluate results
over a temperature range (with defined number of temperature steps)
for a given pressure level. The data are displayed graphically and can
FIG. 3. Comparison of the modeled density qusing different EoS with reference values from CoolProp
38
at different pressures. Columns from left to right: (a) methane
(Zc¼0:2863), (b) cyclopentane (Zc¼0:2813), (c) n-hexane (Zc¼0:2664), and (d) n-dodecane (Zc¼0:2497). Rows from top to bottom: p=pc¼0:75;p=pc¼1:5, and
p=pc¼3:0.
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also be exported to a csv file for further processing. Moreover, also
evaluations over temperature and pressure ranges can be done, which
allows for table generation.
28–30
To this end, the ranges and also the
step width can be specified as configuration parameter. Apart from
that, this open source Python tool can serve as an inspiration for
implementing the present model into an internal flow solver.
Table II lists the process time for the evaluation of a representa-
tive configuration to provide estimates for the evaluation times using
realtpl. The data have been evaluated for 10, 10
2
,10
3
,and10
4
tem-
perature steps using a standard laptop (Intel i5, seventh generation).
Startup refers to the reading of the config files and corresponding fluid
properties. Ref. data stands for extracting the reference data from
CoolProp. In the current version, the five quantities density, heat
capacity, speed of sound, viscosity, and heat conductivity are extracted.
This extraction from CoolProp cannot be vectorized and, thus, it has
to be looped over the temperature steps. Therefore, the evaluation
time required scales roughly linearly with the number of temperature
evaluations. This is the main time consumer when about 3 103tem-
perature evaluations are exceeded. For the thermodynamic model, we
have here listed the average of all three cubic EoS named Thermo.
model. To improve performance, the implementation of the thermo-
dynamic models has been recast as vectors, avoiding time-consuming
loops. For this reason, there is no linear scaling of the evaluation pro-
cess. Also at 10
4
temperature evaluations, the time per thermodynamic
model is still about 0.05 s. Here, it has to be noted that five quantities
areevaluated.AmongdifferentcubicEoS,wehaveseenthatitvaries
depending on how often the check for B and the Gibbs evaluation has
to be done. The next contributions are then output and postprocessing
related. Figures refers to the visualization of all fivequantities including
the write out of the figures. For less than 3 103, this is the most time
consuming part. The last part Data-output is the write out of all evalu-
ated data to a csv file and does not consume significant time. For the
entire evaluation at, e.g., 10
3
temperature steps, approxim ately 6 s are
required with writing out of the figures and approximately 2.2 s with-
out. realtpl is available as a PIP Python package and also on
github github.com/ttrummler/realtpl.
B. Generalized Formulation in OpenFOAM
V
R
OpenFOAM is a widely used open source software for simula-
tions, where currently the most recent versions are the foundation ver-
sion OpenFOAM-10
44
and the ESI version OpenFOAM2206.
45
In
both, the PR EoS is available as PengRobinsonGas.Insomein-house
extensions of OpenFOAM,
46–48
the SRK has additionally been imple-
mented. Despite the identical structure of SRK and PR, these two EoS
are often hard coded and, thus, lead to code duplicates. Following the
generalized formulation proposed above, we propose a more general
implementation of cubic EoS to avoid code duplication and to
improve readability. We provide this extension for OpenFOAM under
github.com/ttrummler/realFOAM. In order to keep the traditional
OpenFOAM code structure and to not change the input files, we have
created three separate folders for the different EoS.
Figure 4 shows a validation of our OpenFOAM implementations
comparing the density distribution with that obtained using the Python
tool realtpl. As test configuration, we consider n-dodecane at a
pressure of p¼8 MPa and a temperature range of T¼500–1500 K,
matching the conditions of one operating point of the ECN Spray
A.
8,28
As expected, both implementations lead to nearly identical
results. For RKPR, a very small deviation is visible, which is due to
rounding errors.
V. CONCLUSIONS
We have presented a thermodynamic model for real gas CFD
simulations based on the generalized formulation of cubic EoS. Using
this generalized formulation, all of the well-known cubic EoS can be
described with a particular value for d
1
[d2¼ð1d1Þ=ð1þd1Þ]. We
have provided a detailed presentation of the resulting generalized cubic
equation in Zand practical hints for solving it. To evaluate the ther-
modynamic properties, we presented formulations of the derivatives.
The transport properties are modeled with the Chung correlations.
The thermodynamic model allows for a modularized implementation
of several EoS.
For the cubic EoS, we have considered the well-known formula-
tions SRK and PR. These two are specifically designed for an assumed
critical compressibility factor and, therefore, their suitability is limited.
Additionally, we also considered the RKPR, where the EoS parameters
are functions of the critical compressibility factor. In this study, we
have assessed the applicability of the three EoS for selected fluids and
showed that the RKPR could be a good universally applicable choice
for the EoS. In addition to this, we have demonstrated that overall the
presented thermodynamic model can capture and reproduce the
TABLE II. Overall process time for different numbers of temperature evaluations
(t-ev), for details, see text.
10 t-ev (s) 10
2
t-ev (s) 10
3
t-ev (s) 10
4
t-ev (s)
Startup 0.020 0.020 0.020 0.020
Ref. data 0.037 0.179 1.489 15.894
Thermo. model 0.002 0.002 0.006 0.046
Figures 3.860 3.860 3.860 3.860
Data-output 0.018 0.034 0.146 0.365
Total 3.941 4.099 5.533 20.276
FIG. 4. Comparison of the modeled density for n-dodecane at a pressure of
p¼8 MPa. Comparison of python implementation in realtpl (colored solid
lines) and OpenFOAM implementation (colored dots) with the reference data (black
solid line) taken from CoolProp.
38
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non-linear behavior of relevant thermodynamic quantities with an
acceptable error.
We provide our Python implementation of the generalized EoS
in the form of a ready-to-use open source tool, which can produce
results as those shown in this work for a wide range of fluids.
Additionally, we provide an implementation in OpenFOAM.
ACKNOWLEDGMENTS
This project received funding by dtec.bw—Digitalization and
Technology Research Center of the Bundeswehr—under the project
MaST: Macro/Micro-simulation of Phase Decomposition in the
Transcritical Regime.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Theresa Trummler: Conceptualization (lead); Data curation (equal);
Formal analysis (lead); Investigation (equal); Methodology (equal);
Software (equal); Validation (supporting); Writing – original draft
(lead); Writing – review & editing (lead). Martin Glatzle: Methodology
(equal); Software (equal); Validation (lead); Writing – review & editing
(supporting). Alexander Doehring: Conceptualization (supporting);
Data curation (supporting); Investigation (supporting); Methodology
(supporting); Validation (supporting); Visualization (supporting);
Writing – original draft (supporting); Writing – review & editing
(supporting). Noah Urban: Data curation (supporting); Methodology
(supporting); Visualization (supporting); Writing – original draft (sup-
porting). Markus Klein: Funding acquisition (lead); Writing – original
draft (supporting); Writing – review & editing (supporting).
DATA AVAILABILITY
The data that support the findings of this study are openly avail-
ableinpythontoolrealtpl as a PIP Python package https://
pypi.org/project/realtpl and github https://github.com/ttrummler/
realtpl. The data that support the findings of this study are openly
available in OpenFOAM on github https://github.com/ttrummler/
realFOAM with separate branches for the ESI and Foundation
versions.
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