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Deep learning based combining rule for the estimation of vapor–liquid equilibrium

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Abstract

Vapor–liquid equilibrium (VLE) data plays a vital role in the design, modeling and control of process equipment. In this study, to estimate the VLE data of binary systems, a deep neural network (DNN)-based combining rule was proposed based on the cross-term parameter \((a_{ij})\) in the two-parameter Peng–Robinson cubic equation of state (PR-EoS) combined with the one-parameter classical van der Waals mixing and combining rule (1PVDW). Experimental VLE data of alternative binary refrigerant systems selected from the literature were calculated using both the PR + 1PVDW and the DNN-based model. Vapor phase mole fractions \((y_i)\) and equilibrium pressures (P) obtained from the proposed DNN-based and PR + 1PVDW models were compared in the terms of average percent deviations. For the DNN-based model, the vapor phase mole fractions give at least as good results as the models in the literature, and also it has been shown that a much better estimate of the equilibrium pressure (P) is obtained when compared with that of the literature. Results obtained using the proposed DNN-based model are presented with tables and graphs. For the equilibrium pressure, while the average percent deviation errors \((\Delta P/P \%)\) calculated in the literature are less than 7.739, the errors obtained with the proposed DNN-based model are smaller than 3.455. And also, for vapor phase mole fractions, while the maximum error \((\Delta y_1/y_1 \%)\) in the literature is obtained as 6.142, the largest error calculated with DNN-based model is 3.545. It has been seen that the proposed DNN-based model makes more practical and less error-prone estimations than the methods in the literature.

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Data availability

All data are cited in the Table 1 and Reference List.

Abbreviations

EoS:

Equation of state

DNN:

Deep neural network

GA:

Genetic algorithm

OF:

Objective function

PR:

Peng–Robinson

ReLU:

Rectified linear unit

vdW:

Van der Waals

VLE:

Vapor–liquid equilibrium

1PVDW:

One-parameter classical van der Waals

a(T):

Energy parameter in the EoS (Eq. (4))

\(a_{i}\) :

Energy parameter in the EoS of pure component i

\(a_{j}\) :

Energy parameter in the EoS of pure component j

\(a_{ij}\) :

Cross-term parameter

\(a_{m}\) :

Energy parameter for the mixture

b :

Co-volume parameter in the EoS (Eq. (4))

\(b_{i}\) :

Co-volume parameter in the EoS of pure component i

\(b_{m}\) :

Co-volume parameter for the mixture

\(f_i\) :

Fugacity for component i

\(K_i\) :

Equilibrium ratio

\(k_{ij}\) :

Binary interaction parameter

N :

Number of experimental data points

\(N_C\) :

Number of components

P :

Pressure (MPa)

\(P_c\) :

Critical pressure

R :

Universal gas constant

T :

Temperature (K)

\(T_c\) :

Critical temperature (K)

\(T_r\) :

Reduced temperature

w :

Weight coefficient of artificial neuron

\(x_i\) :

Liquid phase mole fraction of component i

\(y_i\) :

Vapor phase mole fraction of component i

\(\alpha (T)\) :

Temperature-dependent function

\(\Delta\) :

Average deviation

\(\varphi _{i}\) :

Fugacity coefficient of component i

\(\omega\) :

Acentric factor

\(\kappa\) :

A function of acentric factor

\(\mu _{i}\) :

Chemical potential of component i

calc:

Calculated data

exp:

Experimental data

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Author notes

  1. Dilek Özmen and Atilla Özmen contributed equally to this work.

Authors and Affiliations

  1. Department of Chemical Engineering, Istanbul University-Cerrahpaşa, Istanbul, Turkey

    Sezin Bekri

  2. Department of Chemical Engineering, Yalova University, Yalova, Turkey

    Dilek Özmen

  3. Department of Electrical-Electronics Engineering, Kadir Has University, Istanbul, Turkey

    Atilla Özmen

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  1. Sezin Bekri
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Correspondence to Sezin Bekri.

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Appendix

Appendix

The weight coefficients and bias values obtained from the proposed DNN model are given below.

For Propilen(1) + R23(2) binary system

$$\begin{aligned} {\textbf{W}}^{(1)}= & {} \left[ \begin{array}{lllllllll} 4.61 &{} 1.94 &{} -1.87 &{} -3.37 &{} -12.48 &{} -9.08 &{} -6.87 &{} -9.67 &{} 9.08 \\ -20.55 &{} -11.11 &{} 7.39 &{} -0.11 &{} -1.71 &{} -0.10 &{} 12.55 &{} -2.05 &{} -3.34 \\ -3.72 &{} -6.23 &{} 3.16 &{} 2.90 &{} -3.00 &{} -2.47 &{} -17.05 &{} 10.69 &{} -7.09 \\ \end{array}\right] \\ {\textbf{W}}^{(2)}= & {} \left[ \begin{array}{lllllllll} 10.93 &{} 31.69 &{} -7.37 &{} -9.84 &{} 8.47 &{} 1.55 &{} 2.07 &{} 3.58 &{} 2.53 \\ 3.21 &{} 4.36 &{} 4.89 &{} 4.40 &{} -14.35 &{} 8.34 &{} -1.21 &{} -3.47 &{} -16.73 \\ 8.42 &{} 6.00 &{} -0.21 &{} -4.07 &{} -3.72 &{} -10.27 &{} -4.97 &{} 1.39 &{} 3.18 \\ 0.69 &{} 7.61 &{} -8.84 &{} 17.59 &{} 37.83 &{} 12.09 &{} -7.35 &{} -4.85 &{} 3.09 \\ -5.74 &{} 3.38 &{} -6.81 &{} 9.46 &{} -0.12 &{} -10.64 &{} 0.97 &{} -0.44 &{} -2.81 \\ -0.37 &{} -20.93 &{} 2.82 &{} 2.79 &{} -0.03 &{} -10.02 &{} -5.25 &{} -2.37 &{} -7.61 \\ -5.32 &{} -8.74 &{} 5.91 &{} 1.09 &{} 6.58 &{} 5.56 &{} -2.57 &{} 3.61 &{} -7.27 \\ -5.75 &{} -8.09 &{} -14.55 &{} -4.34 &{} -19.20 &{} -1.56 &{} 3.13 &{} -8.62 &{} -11.01 \\ 10.10 &{} -3.92 &{} 15.19 &{} 4.17 &{} 2.48 &{} -22.19 &{} -0.66 &{} 0.40 &{} 5.47 \\ \end{array}\right] \\ {\textbf{W}}^{{(3)T}}= & {} \left[ \begin{array}{lllllllll} 1.05 &{} 2.96 &{} 23.52 &{} 11.78 &{} -7.69 &{} -0.66 &{} -7.31 &{} 11.87 &{} 23.14 \\ \end{array}\right] \\ {\textbf{b}}^{{(1)T}}= & {} \left[ \begin{array}{lllllllll} 4.64 &{} -5.82 &{} 4.26 &{} 13.64 &{} -3.46 &{} 16.38 &{} 2.67 &{} 4.57 &{} 14.74 \\ \end{array}\right] \\ {\textbf{b}}^{{(2)T}}= & {} \left[ \begin{array}{lllllllll} -8.66 &{} -3.03 &{} 7.98 &{} -5.91 &{} -1.88 &{} -6.09 &{} 7.31 &{} 4.86 &{} 12.69 \\ \end{array}\right] \\ {\textbf{b}}^{{(3)}}= & {} \left[ \begin{array}{l} 0.73 \\ \end{array}\right] . \end{aligned}$$

For R125(1) + R600a(2) binary system

$$\begin{aligned} {\textbf{W}}^{(1)}= & {} \left[ \begin{array}{lllllllll} -7.38 &{} 10.52 &{} -3.93 &{} 5.40 &{} 1.25 &{} -4.61 &{} -3.62 &{} 3.95 &{} 19.51 \\ -6.88 &{} 3.76 &{} 5.84 &{} 2.12 &{} -20.06 &{} 5.10 &{} -4.64 &{} 2.26 &{} -5.58 \\ -6.18 &{} -5.23 &{} -13.28 &{} -2.23 &{} -36.29 &{} -4.89 &{} 0.99 &{} -13.97 &{} 8.49 \\ \end{array}\right] \\ {\textbf{W}}^{(2)}= & {} \left[ \begin{array}{lllllllll} 19.25 &{} 5.20 &{} 8.21 &{} 13.64 &{} 1.67 &{} 4.86 &{} -1.27 &{} -17.51 &{} -6.25 \\ 4.28 &{} 2.07 &{} 1.89 &{} -5.16 &{} 19.83 &{} -18.15 &{} -5.04 &{} -4.42 &{} 3.59 \\ 1.70 &{} -7.93 &{} 12.74 &{} 0.21 &{} -3.23 &{} -12.55 &{} 8.83 &{} 28.62 &{} -2.51 \\ 5.46 &{} 4.69 &{} 0.93 &{} -16.69 &{} 2.59 &{} -8.81 &{} 11.22 &{} -5.75 &{} 7.17 \\ -2.50 &{} 6.40 &{} -5.68 &{} -8.07 &{} -2.73 &{} -12.03 &{} -2.40 &{} 11.74 &{} 6.79 \\ -20.58 &{} 2.41 &{} -17.38 &{} -8.15 &{} -2.49 &{} -3.05 &{} 12.42 &{} 7.27 &{} -4.06 \\ -6.28 &{} 10.40 &{} 2.44 &{} -6.85 &{} 7.60 &{} -8.71 &{} 1.57 &{} -6.37 &{} -3.50 \\ -42.97 &{} 10.50 &{} -4.90 &{} -21.84 &{} 6.58 &{} 4.48 &{} -6.24 &{} 6.64 &{} -11.37 \\ -4.38 &{} -14.30 &{} -41.55 &{} -26.61 &{} -5.89 &{} -9.56 &{} -34.14 &{} -9.93 &{} -6.52 \\ \end{array}\right] \\ {\textbf{W}}^{{(3)T}}= & {} \left[ \begin{array}{lllllllll} 1.71 &{} 7.62 &{} -11.56 &{} -15.43 &{} 0.26 &{} 3.63 &{} -0.22 &{} -5.53 &{} -29.46 \\ \end{array}\right] \\ {\textbf{b}}^{{(1)T}}= & {} \left[ \begin{array}{lllllllll} 1.77 &{} -6.30 &{} -32.50 &{} -12.13 &{} -8.35 &{} 10.21 &{} 0.92 &{} 24.61 &{} 3.36 \\ \end{array}\right] \\ {\textbf{b}}^{{(2)T}}= & {} \left[ \begin{array}{lllllllll} -1.53 &{} -9.27 &{} 0.92 &{} 6.31 &{} 3.33 &{} -5.30 &{} -4.80 &{} 1.32 &{} 9.01 \\ \end{array}\right] \\ {\textbf{b}}^{{(3)}}= & {} \left[ \begin{array}{l} 1.17 \\ \end{array}\right] . \end{aligned}$$

For R134a(1) + R290(2) binary system

$$\begin{aligned} {\textbf{W}}^{(1)}= & {} \left[ \begin{array}{lllllllll} -0.59 &{} 7.43 &{} 1.55 &{} 26.60 &{} 33.32 &{} 25.21 &{} 5.54 &{} 6.64 &{} 5.12 \\ -2.00 &{} -8.54 &{} -27.44 &{} -9.45 &{} -4.75 &{} -12.98 &{} -9.61 &{} 17.60 &{} -3.57 \\ -6.54 &{} 33.00 &{} 18.11 &{} -28.33 &{} -9.10 &{} -4.56 &{} -19.02 &{} -6.68 &{} -14.90 \\ \end{array}\right] \\ {\textbf{W}}^{(2)}= & {} \left[ \begin{array}{lllllllll} 9.59 &{} 4.18 &{} 16.51 &{} 6.29 &{} -18.29 &{} 6.53 &{} 6.68 &{} -21.75 &{} -6.51 \\ -2.61 &{} 3.68 &{} -6.64 &{} -19.09 &{} -2.34 &{} -9.12 &{} -3.06 &{} -15.20 &{} -7.07 \\ -17.99 &{} -27.69 &{} -24.26 &{} -30.43 &{} 3.22 &{} -1.11 &{} -9.88 &{} 12.48 &{} -0.03 \\ -12.58 &{} 0.18 &{} -15.84 &{} -12.30 &{} 20.06 &{} -9.62 &{} -12.37 &{} -10.00 &{} -15.86 \\ 44.74 &{} -48.53 &{} -3.23 &{} 1.68 &{} 38.14 &{} 7.75 &{} -10.25 &{} 15.69 &{} -2.05 \\ 8.72 &{} 18.64 &{} 1.54 &{} 31.15 &{} 0.01 &{} 17.69 &{} -3.13 &{} 7.84 &{} -3.60 \\ 0.82 &{} 5.80 &{} -21.17 &{} -4.42 &{} -5.27 &{} 7.63 &{} -1.90 &{} 5.25 &{} 7.90 \\ 14.29 &{} -4.97 &{} -12.89 &{} 2.56 &{} -21.81 &{} -13.70 &{} 0.65 &{} -1.64 &{} 20.18 \\ -11.59 &{} -22.32 &{} -8.74 &{} 13.50 &{} 10.92 &{} -14.66 &{} 16.81 &{} 7.56 &{} -12.36\\ \end{array}\right] \\ {\textbf{W}}^{{(3)T}}= & {} \left[ \begin{array}{lllllllll} -1.73 &{} -6.02 &{} -3.69 &{} -7.47 &{} -4.82 &{} -11.05 &{} -12.73 &{} -13.34 &{} 14.99 \\ \end{array}\right] \\ {\textbf{b}}^{{(1)T}}= & {} \left[ \begin{array}{lllllllll} 16.92 &{} -5.96 &{} 45.22 &{} -32.69 &{} 0.16 &{} 7.18 &{} 21.59 &{} 3.67 &{} 5.92 \\ \end{array}\right] \\ {\textbf{b}}^{{(2)T}}= & {} \left[ \begin{array}{lllllllll} -2.84 &{} 11.54 &{} 10.01 &{} -18.94 &{} 27.95 &{} -0.02 &{} 8.68 &{} 2.43 &{} -5.36 \\ \end{array}\right] \\ {\textbf{b}}^{{(3)}}= & {} \left[ \begin{array}{l} 1.02 \\ \end{array}\right] . \end{aligned}$$

For R134a(1) + R600a(2) binary system

$$\begin{aligned} {\textbf{W}}^{(1)}= & {} \left[ \begin{array}{lllllllll} -11.87 &{} 3.64 &{} -0.32 &{} -0.61 &{} -3.60 &{} -3.60 &{} -9.65 &{} 0.01 &{} 9.15 \\ -19.89 &{} 1.67 &{} 5.16 &{} -9.92 &{} -11.32 &{} 7.44 &{} -2.42 &{} -6.52 &{} -2.14 \\ -2.01 &{} -2.58 &{} -2.82 &{} -6.94 &{} -12.51 &{} -2.30 &{} -2.25 &{} -0.56 &{} -3.28 \\ \end{array}\right] \\ {\textbf{W}}^{(2)}= & {} \left[ \begin{array}{lllllllll} -8.29 &{} 8.27 &{} 0.88 &{} -6.68 &{} 5.27 &{} 5.35 &{} -3.43 &{} -1.19 &{} -10.83 \\ 1.86 &{} -3.98 &{} -7.73 &{} -4.97 &{} -0.09 &{} -6.38 &{} 3.05 &{} -5.12 &{} 4.45 \\ 71.32 &{} 4.57 &{} 0.30 &{} -9.80 &{} -4.15 &{} -8.81 &{} 1.71 &{} 11.32 &{} -4.36 \\ 15.67 &{} 7.11 &{} -1.05 &{} -1.64 &{} -5.31 &{} -9.91 &{} 2.10 &{} -0.38 &{} 2.58 \\ 0.91 &{} -14.32 &{} 0.23 &{} 5.42 &{} 12.14 &{} 8.84 &{} 3.10 &{} 2.82 &{} 6.48 \\ -7.91 &{} -2.72 &{} -3.31 &{} -7.28 &{} -3.59 &{} 7.78 &{} 5.46 &{} 9.14 &{} 6.77 \\ 14.20 &{} -26.99 &{} -6.51 &{} -6.15 &{} -9.07 &{} -4.21 &{} -1.44 &{} 13.70 &{} -5.86 \\ 11.80 &{} -6.86 &{} -8.10 &{} -3.73 &{} -4.49 &{} -5.46 &{} 6.98 &{} -4.61 &{} 0.73 \\ -7.10 &{} 9.87 &{} 12.66 &{} -7.32 &{} 8.51 &{} -4.04 &{} -13.55 &{} -6.92 &{} 1.30\\ \end{array}\right] \\ {\textbf{W}}^{{(3)T}}= & {} \left[ \begin{array}{lllllllll} 1.15 &{} 9.49 &{} 1.29 &{} 4.17 &{} 9.70 &{} 5.30 &{} 3.63 &{} -6.81 &{} -8.81 \\ \end{array}\right] \\ {\textbf{b}}^{{(1)T}}= & {} \left[ \begin{array}{lllllllll} 0.25 &{} -6.29 &{} -7.80 &{} 17.74 &{} 10.74 &{} 8.26 &{} -24.09 &{} -13.22 &{} 8.57 \\ \end{array}\right] \\ {\textbf{b}}^{{(2)T}}= & {} \left[ \begin{array}{lllllllll} -4.11 &{} -4.10 &{} -0.93 &{} -7.33 &{} -7.90 &{} -4.94 &{} -7.78 &{} -28.50 &{} -5.31 \\ \end{array}\right] \\ {\textbf{b}}^{{(3)}}= & {} \left[ \begin{array}{l} 1.24\\ \end{array}\right] . \end{aligned}$$

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Bekri, S., Özmen, D. & Özmen, A. Deep learning based combining rule for the estimation of vapor–liquid equilibrium. Braz. J. Chem. Eng. 41, 613–629 (2024). https://doi.org/10.1007/s43153-023-00377-0

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