CC BY
25
UDC 536.77
DOI: 10.18698/0236-3941-2021-3-108-121
THERMODYNAMIC PROPERTY CALCULATION IN VAPOR-LIQUID EQUILIBRIUM FOR MULTICOMPONENT MIXTURES USING HIGHLY ACCURATE HELMHOLTZ FREE ENERGY EQUATION OF STATE
T. Luo
A.Yu. Chirkov
luotaiming@yandex.ru chirkov@bmstu.ru
Bauman Moscow State Technical University, Moscow, Russian Federation
Abstract
Thermodynamic properties of multicomponent mixtures in phase equilibrium were studied. The tangent plane criterion was used for stability analysis, and the Gibbs energy minimization was employed for phase equilibrium calculation when the successive substitution didn't converge. Thermodynamic properties of a 12-component natural gas mixture in vapor-liquid equilibrium were calculated with highly accurate Helmholtz free energy equation of state GERG-2008, simplified GERG-2008 and common cubic Peng — Robinson (PR) equation of state. Results show that in vapor-liquid equilibrium, GERG-2008 has high accuracy and works better than simplified GERG-2008 and PR-equation of state. Simplified GERG-2008 and PR-equation of state both work unsatisfactorily in vapor-liquid equilibrium calculation, especially near the saturation zone. The deviation function in GERG-2008 can significantly affect the accuracy of GERG-2008 when calculating thermodynamic properties of mixtures in vapor-liquid equilibrium
Keywords
Vapor-liquid equilibrium, stability analysis, tangent plane criterion, Helmholtz free energy equation of state, PR-equation of state
Received 26.02.2021 Accepted 18.03.2021 © Author(s), 2021
Introduction. Accurate description of thermodynamic properties of natural gas mixtures in vapor-liquid equilibrium is of great significance for many applications involving natural gas, such as natural gas transportation, storage and liquefaction. Equations of state are commonly used to describe phase equilibrium over a wide range of pressure, temperature and mixture composition. Based on equations of state, Michelsen and Mollerup [1] proposed a combined stability analysis and phase equilibrium method to solve the vapor-liquid equilibrium problem for multicomponent mixtures.
The stability analysis is based on the principle that the total Gibbs energy of a mixture at given temperature, pressure and overall composition has to be at the global minimum when the mixture is in the state of thermodynamic stability. Based on this principle, the Gibbs' tangent plane criterion was proposed by
Baker et al. [2] to test thermodynamic stability. However, he didn't give a computational approach. Michelsen [3] proposed a number of numerical methods for stability analysis based on Gibbs' tangent plane criterion. Besides, the use of stability analysis to generate initial estimates for phase equilibrium calculation was also suggested by Michelsen [4]. In phase equilibrium calculation, successive substitution is a simple and major approach to solve the system of phase equilibrium equations. In order to improve convergence efficiency of successive substitution, Orbach and Crowe [5] proposed the dominant eigenvalue method (DEM), and Crowe and Nishio [6] proposed the general dominant eigenvalue method (GDEM), which is more effective than DEM. Successive substitution is convergent in most practical cases; however, in the vicinity of critical point, its convergence rate is very low. In order to solve this problem, Michelsen
[4] and Michelsen and Mollerup [1] proposed Gibbs energy minimization by the second order method. This method is also based on the principle that the total Gibbs energy of a mixture at given temperature, pressure and overall composition is at the global minimum. In this method, cubic equations of state are mostly used models, but multi-parameter equations of state explicit in Helmholtz free energy are rarely used as models.
In this work, GERG-2008, which is a highly accurate Helmholtz free energy equation of state, was used to test phase stability of a 12-component natural gas mixture and calculate its thermodynamic properties in vapor-liquid equilibrium. In GERG-2008, a deviation function is used to model mixture behavior with higher accuracy. The influence of this deviation function on accuracy of GERG-2008 was investigated in phase equilibrium calculation. Besides, Peng — Robinson equation of state (PR-EOS) was compared with GERG-2008 in calculating thermodynamic properties of this multicomponent natural gas mixture in vapor-liquid equilibrium.
Equation of state for mixtures. This work involves one highly accurate Helmholtz free energy equation of state GERG-2008 and one cubic PR-EOS. The PR-equation of state for mixtures is described in [7], and the binary interaction coefficient is obtained from [8]. The GERG-2008 equation of state is briefly described below.
The GERG-2008 equation of state is the dimensionless form of the Helmholtz free energy which includes an ideal gas part and an empirically residual part [9]:
a(5, t, x) = a°(p, T, x) + ar(S, t, x), (1)
where x is the molar composition in a mixture, 5 is the reduced mixture density and x is the inverse reduced mixture temperature.
The reduced mixture variables 5 and x are defined as:
5 = -^- and T = Tr(X), (2)
pr (x) T
where pr and Tr are the composition-dependent functions for the mixture density and temperature, respectively. The following mixing rules are used:
1 2 Nr-1 £ 2 R Xj + Xj 1
/—ч ~ ^ Xj + L L 2XjXjPv, jjУV, jj 2 Pr (X) j = 1 Pa, j j = 1 j = j + 1 ß2, jjXj + Xj 8
f \ 1 :
3
vpa/3 pa/3, j
(3)
N N-1 N X. + X.
ж_, ^ __ ж_, ж_, _ л y^j \ .Л- j
,jj rT
Tr(x) = X xT i + X X 2x-xjPT,jYT, ij -}— (Tc, iT, j)05. (4)
i = 1 i = 1 j = i + 1 Pt, ijxi ^ xj
The binary parameters pv, j and yv, j in equation (3), pT, j and yT, j
in equation (4), and the critical parameters pc, i and Tc, i of pure components
are given by Kunz and Wagner [9].
The ideal-gas part a0 is given by
N
a0 (p, T, x) = X xi [aO (p, T) + ln xi ], (5)
i = 1
where a°oi is the reduced Helmholtz free energy in the ideal-gas state of the pure component i.
The residual part ar is
N
ar (5, t, x) = X xiaroi (5, x) + Aar (5, x, x), (6)
i = 1
where aroi is the residual part of the reduced Helmholtz free energy of the pure
component i and Aar is a deviation function used to model complex mixture behavior.
The simplified GERG-2008, i.e., GERG-2008 without the deviation function (DF), means that equation (6) doesn't include the deviation function Aar.
For PR-EOS, the method in [10] can be used to calculate thermodynamic properties of a mixture, for example fugacity coefficients and their derivatives, which are needed in the stability analysis and phase equilibrium calculation. Since GERG-2008 is a Helmholtz free energy equation of state, all thermodynamic properties of a mixture can be obtained by combining various derivatives of equation (1), which is described in detail in [9, 11].
Stability analysis. The stability analysis is based on the tangent plane criterion, which is first introduced by Baker et al. [2]. It can be used to test whether the original phase of a mixture is stable if overall composition z(z1, z2,..., zn), temperature T, and pressure p of the mixture are given. If the original phase of the mixture is tested to be unstable, the tangent plane criterion also provides the composition of a new phase which can be split to decrease the Gibbs free energy of the mixture. A computational approach based on minimizing the tangent plane distance was suggested by Michelsen [3].
For a n-component mixture of composition z(z1, z2,..., zn) at specified temperature T and pressure p, the necessary and sufficient condition for stability of this mixture is that the reduced tangent plane distance is non-negative for any trial phase composition w (w1, w2,..., wn) [3]. The reduced tangent plane distance is:
tpd(W) = X Wi (ln Wi + ln tyi (w) - ln zi - ln q>i (z ) ) =
i
= X Wi (ln Wi + ln 9i(W) - di ), (7)
i
where is the fugacity coefficient of component i and di = ln zi + ln^-(z).
By using the Gibbs — Duhem relation [12], straightforward differentiation of tpd(W) with respect to the (n-1) independent mole fractions yields the sta-tionarity condition:
ln Wi + ln (W) - di = k, i = 1, 2,..., n, (8)
where k is independent of the component index i.
Introducing new variables Wi = exp (-k) Wi, a different but equivalent criterion can be formulated as [3]
tm(W) = 1 + £ Wi (ln Wi + ln (W) - di - 1 ) > 0, (9)
i
where no constraints on the Wi except Wi > 0 are required. The stationary points of tm(W) satisfy
f d tm(W ) ^
dWi
= ln Wi + ln (W) - di = 0, (10)
Jt, p, Wj, j Ф i
where when deriving the of equation (10), the Gibbs — Duhem relation [12] is also used.
Finally the modified tangent plane function tm(W) can be written as
tm (W) = 1 + Ws X wi (ln Ws + ln wi + ln (W) - di - 1) =
i
= (1 - Ws + Ws ln Ws) + Wstpd (W), (11)
where Ws = X Wi > 0.
i
Since the term (1 - Ws + Ws ln Ws) in the equation (11) is always nonnegative, tpd(W) will be negative at all points where tm(W) is negative; thus a
negative tm(W) implies that the system is unstable.
In order to verify stability, we need to determine the global minimum of the tangent plane distance, i.e., locate all local minima of tm(W). However, this is a very time-consuming task in the general case. Michelsen [1] presented a practical computational approach in which the search for local minima is performed with a number of selected initial estimates in the hope that these searches will uncover at least all negative minima.
The Dominant Eigenvalue Method [5] can be used to solve equation (10). In the vicinity of the critical point, a second order minimization of tm(W) [1] is used instead of the successive substitution method.
Phase equilibrium calculation. For the common vapor-liquid equilibrium of a multicomponent mixture, the following phase equilibrium conditions must be satisfied:
- equality of temperature
Tl = Tv = T; (12)
- equality of pressure
pl = pv = p; (13)
- equality of chemical potentials
l^l (i = 1,2,..., n), (14)
where l denotes the liquid phase and v denotes the vapor phase.
The General Dominant Eigenvalue Method (GDEM) [6] an be used to solve the system of equations (12), (13) and (14). If the efficiency of the GDEM decreases, it is recommended to use the Gibbs energy minimization by the second order method [1, 11]. At specified T, p and z, the vapor-liquid equilibrium calculation can be formulated as
minG(T, p, v, z - v), (15)
where G is the Gibbs energy and v is vapor amounts (per mole of mixture) .
For the convenience of calculation, G/(RT) is used as the objective function in the unconstrained minimization. Newton's method for solving the minimization problem calculates correction vector from:
H Av + g = 0. (16)
The gradient vector of G/(RT) is obtained as:
8 ] = gi = ln fV - ln fl, (17)
dvi ^ RT,
where fiv and fil are the fugacities of component i in the vapor and liquid phases respectively.
The elements Hj of the Hessian matrix H are calculated according to
= SgL =mf + aif U8)
dvj dvj dlj
where lj is liquid amount (per mole of mixture).
If the current estimate is sufficiently close to the solution, the Hessian matrix is positive definite and second order convergence is obtained. However, if the Hessian matrix is not positive definite, the iterations may fail, which is frequently the case in early iterations. In order to guarantee convergence, a trust region method [13] was suggested by Michelsen and Mollerup [1].
Results. In order to study thermodynamic properties of the multicompo-nent mixture in vapor-liquid equilibrium, we select a 12-component natural gas named as N75 from the literature [11]. The molar fractions of components in the natural gas N75 are as follows: methane 85.9284 %, nitrogen 0.9617 %, carbon dioxide 1.5021 %, ethane 8.4563 %, propane 2.3022 %, rc-butane 0.4604 %, isobutane 0.2381 %, rc-pentane 0.0630 %, isopentane 0.0588 %, rc-hexane 0.0228 %, rc-heptane 0.0057 %, rc-octane 0.0005 %.
Thermodynamic properties of the natural gas N75 in vapor-liquid equilibrium were calculated with the help of PR-EOS, GERG-2008 without DF and GERG-2008. The tangent plane criterion is first used to find out whether the mixture is in vapor-liquid equilibrium. If the mixture is in vapor-liquid equilibrium, the estimates obtained from the tangent plane criterion are used for solving the system of phase equilibrium equations. The successive substitution is first used to solve the system of phase equilibrium equations. If the successive substitution can't converge, the Gibbs energy minimization by the second order method is used. In stability analysis and vapor-liquid equilibrium calculations the input variables are T, p and overall (or phase) composition x, while in GERG-2008 the input variables are p, T and composition x, so a density
solver is required. The density solver [14, 15], which is modified from [16], can provide reliable density solutions although it is relatively time-consuming.
Figs. 1-5 show thermodynamic property predictions of the natural gas N75 in vapor-liquid equilibrium with the help of PR-EOS, GERG-2008 without DF and GERG-2008. Their relative deviations from NIST (National Institute of Standards and Technology) database were also calculated. Table shows comparison of thermodynamic property predictions in vapor-liquid equilibrium with the help of PR-EOS, GERG-2008 without DF and GERG-2008 for the natural gas N75.
Comparison of thermodynamic property predictions in vapor-liquid equilibrium with the help of PR-EOS, GERG-2008 without DF and GERG-2008 within 5-60 bar for the natural gas N75
Thermodynamic property PR-EOS GERG-2008 without DF GERG-2008
Av. (%)* Max. (%)** Av. (%) Max. (%) Av. (%) Max. (%)
Density (p) 2.34 14.33 0.77 19.90 0.02 0.28
Isobaric enthalpy difference (Ah) 4.25 92.84 1.42 22.66 0.03 0.37
Vapor fraction 3.99 144.63 1.64 31.05 0.03 0.51
Molar fraction of methane in liquid phase 3.56 11.47 7.96 30.35 0.06 0.26
Molar fraction of methane in vapor phase 0.14 0.45 0.12 0.69 0.00 0.01
Molar fraction of n-octane in liquid phase 19.17 213.88 6.89 24.78 0.28 2.47
Molar fraction of n-octane in vapor phase 35.98 272.10 10.05 34.20 0.40 2.31
N * Average relative deviation, (100 %/ N) ^ i = 1 ** Maximum relative deviation, max |l00 % (xi - xnist)/ xnist|; (xi - xnist)/ xnist |} .
Fig. 1 shows that GERG-2008 has high accuracy in predicting molar density of the natural gas N75 in vapor-liquid equilibrium. Table shows that the average
p, kmol/m"
Г,К
PR-EOS
GERG-2008 without DF
GERG-2008
Equation of state c d
Fig. 1. Density versus temperature and its average relative deviation (d) from reference data for the natural gas N75 in vapor-liquid equilibrium with the help of PR-EOS, GERG-2008 without DF and GERG-2008 at different pressures:
a) 5 bar; b) 30 bar; c) 60 bar
relative deviation of molar density predicted by GERG-2008 is 0.02 %. In prediction of molar density in vapor-liquid equilibrium GERG-2008 without DF works better than PR-EOS in general, but much worse than GERG-2008. However, near the saturation zone where density varies sharply, GERG-2008 without DF and PR-EOS both work unsatisfactorily in predicting molar density. Table shows that the maximum relative deviations of predicted molar density by PR-EOS and GERG-2008 without DF are 14.33 and 19.90 % respectively.
Fig. 2 shows that PR-EOS works worse than GERG-2008 and GERG-2008 without DF in predicting isobaric enthalpy difference of the natural gas N75
Ah, kJ/mol
170 190 210 T, К a
176 192 208 224 240 T, К b
GERG-2008
GERG-2008 without DF
Equation of state c d
Fig. 2. Isobaric enthalpy difference versus temperature and its average relative deviation (d) from reference data for the natural gas N75 in vapor-liquid equilibrium with the help of PR-EOS, GERG-2008 without DF and GERG-2008 at different pressures: a) 5 bar; b) 30 bar; c) 60 bar
in vapor-liquid equilibrium. Table shows that the average relative deviation of isobaric enthalpy difference predicted by PR-EOS is 4.25 %. GERG-2008 works much better than GERG-2008 without DF in predicting isobaric enthalpy difference in vapor-liquid equilibrium. Table shows that the average relative deviations of predicted isobaric enthalpy difference by GERG-2008 and GEGR-2008 without DF are 0.03 and 1.42 % respectively.
Fig. 3 shows that GERG-2008 has high accuracy in predicting vapor fraction of the natural gas N75 in vapor-liquid equilibrium. Table shows that the average relative deviation of vapor fraction predicted by GERG-2008 is 0.03 %.
170 190 Temperature, К a
0.9 0.8 0.7 0.6 0.5 0.4 0.3
230
0.1 0 176
/ : / / 1 - 1 1
- Г : f -PREOS ---GtRG-ZWB Cf GERG-ZOOB NIST
192
208 224 Temperature, К b
240
220 230 240 Temperature, К
PR-EOS
GERG-2008 without DF
Equation of state
d
GERG-2008
Fig. 3. Vapor fraction versus temperature and its average relative deviation (d) from reference data for the natural gas N75 in vapor-liquid equilibrium with the help of PR-EOS, GERG-2008 without DF and GERG-2008 at different pressures:
a) 5 bar; b) 30 bar; c) 60 bar
In prediction of vapor fraction in vapor-liquid equilibrium, GERG-2008 without DF works better than PR-EOS, but much worse than GERG-2008. Table shows that the average relative deviations of predicted vapor fraction by PR-EOS and GERG-2008 without DF are 3.99 and 1.64 % respectively. However, near the bubble point, PR-EOS and GERG-2008 without DF both work unsatisfactorily in predicting vapor fraction. Table shows that the maximum relative deviations in predicting vapor fraction are 144.63 % for PR-EOS and 31.05 % for GERG-2008 without DF.
Fig. 4 shows that PR-EOS works worse than GERG-2008 and GERG-2008 without DF in predicting molar fraction of rc-octane in liquid phase. Table shows that the average relative deviation of molar fraction of rc-octane in liquid
130 150 170 190 210 230 Temperature, К a
176 192 208 224 240 Temperature, К b
220 230 240 Temperature, К
s >
<D T3 <D
1
<L>
Й U
PR-EOS GERG-2008 GERG-2008 without DF Equation of state d
Fig. 4. Molar fraction of n-octane in liquid phase versus temperature and its average relative deviation (d) from reference data for the natural gas N75 in vapor-liquid equilibrium with the help of PR-EOS, GERG-2008 without DF and GERG-2008 at different pressures: a) 5 bar; b) 30 bar; c) 60 bar
phase predicted by PR-EOS is 19.17 %. GERG-2008 works much better than GERG-2008 without DF in predicting molar fraction of n-octane in liquid phase. Table shows that the average relative deviations of predicted molar fraction of n-octane in liquid phase by GERG-2008 and GEGR-2008 without DF are 0.28 and 6.89 % respectively.
Fig. 5 shows that GERG-2008 has high accuracy in predicting molar fraction of n-octane in vapor phase. Table shows that the average relative deviation of molar fraction of n-octane in vapor phase predicted by GERG-2008 is 0.40 %. In predicting molar fraction of n-octane in vapor phase, PR-EOS works worse than GERG-2008 without DF. Table shows that the average
PR-EOS
GERG-2008 without DF Equation of state d
GERG-2008
220 230 240 250
Temperature, K c
Fig. 5. Molar fraction of n-octane in vapor phase versus temperature and its average relative deviation (d) from reference data for the natural gas N75 in vapor-liquid equilibrium with the help of PR-EOS, GERG-2008 without DF and GERG-2008 at different pressures:
a) 5 bar; b) 30 bar; c) 60 bar
relative deviations of predicted molar fraction of n-octane in vapor phase by PR-EOS and GERG-2008 without DF are 35.98 and 10.05 % respectively.
Conclusion. We compared GERG-2008 with PR-EOS and simplified GERG-2008 (GERG-2008 without DF) in calculation of thermodynamic properties of the natural gas N75 in vapor-liquid equilibrium.
In vapor-liquid equilibrium calculation, GERG-2008 is superior to PR-EOS and GERG-2008 without DF. Both GERG-2008 without DF and PR-EOS work unsatisfactorily in vapor-liquid equilibrium calculation, especially near the saturation zone, but in general, GERG-2008 without DF works better than PR-EOS. For the component with a very small molar fraction in a mixture,
PR-EOS and GERG-2008 without DF have very large deviation in predicting molar fractions of this component in liquid and vapor phases. In addition, calculations show that the deviation function in GERG-2008 (see equation (6)) can significantly affect the accuracy of GERG-2008 when calculating thermodynamic properties of mixtures in vapor-liquid equilibrium.
The tangent plane criterion can be successfully used to find whether the original phase of a mixture is stable or not. If the original phase is unstable, the estimates, which are produced by the tangent plane criterion, can be considered as relatively accurate estimates to solve the system of vapor-liquid equilibrium equations.
Translated by Authors
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Luo T. — Post-Graduate Student, Department of Thermal Physics, Bauman Moscow State Technical University (2-ya Baumanskaya ul. 5/1, Moscow, 105005 Russian Federation).
Chirkov A.Yu. — Dr. Sc. (Phys.-Math.), Head of Department of Thermal Physics, Bauman Moscow State Technical University (2-ya Baumanskaya ul. 5/1, Moscow, 105005 Russian Federation).
Please cite this article as:
Luo T., Chirkov A.Yu. Thermodynamic property calculation in vapor-liquid equilibrium for multicomponent mixtures using highly accurate Helmholtz free energy equation of state. Herald of the Bauman Moscow State Technical University, Series Mechanical Engineering, 2021, no. 3 (138), pp. 108-121. DOI: https://doi.org/10.18698/0236-3941-2021-3-108-121
