Peng-Robinson Equation of State for Mixtures

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This simulation uses the Peng-Robinson equation of state for mixtures to plot isotherms for n-butane(1)/n-octane(2) mixtures on a log pressure versus log volume graph. Use the sliders to select the temperature and the mole fraction $x_1$ of the liquid phase (blue isotherm). The green isotherm is for the vapor phase (mole fraction $y_1$) that is in equilibrium with the liquid. The values of $y_1$ and the equilibrium pressure in the upper-right corner are determined from Raoult's law. The dashed black line connects the liquid and vapor compositions that are in equilibrium.

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The Peng-Robinson equation of state for mixtures is used to plot pressure P versus volume V on a log-log graph: $$ P = \frac{RT}{V-b_m} - \frac{a_m}{V^2 - 2Vb_m - b_m^2} $$ where R is the gas constant ([cm$^3$ MPa] / [mol K]), T is temperature (K), P is in MPa, and V is in cm$^3$ / mol.

$a_m$ is the attraction parameter and $b_m$ is the repulsion parameter for the mixture: $$ a_m = \Sigma_{i=1}^2 \Sigma_{j=1}^2 z_i z_j (1-k_{ij} \sqrt{a_i a_j}) $$ $$ b_m = \Sigma_{i=1}^2 z_i b_i $$ where z$_i$ is the component mole fraction, $k_{ij}$ is the binary interaction parameter, and a and b are the attraction and repulsion parameters for a pure component.

The binary interaction parameter can be calculated [1]: $$ k_{ij} = 1 = \frac{1}{2} \frac{b_2}{b_1} \sqrt{\frac{a_1}{a_2}} - \frac{1}{2} \frac{b_1}{b_2} \sqrt{\frac{a_2}{a_1}} + \frac{1}{2} \frac{b_2 RT}{\sqrt{a_1 a_2}} \frac{\theta_1}{(T/T_{c,1})^{\theta_2}} $$ for an alkane/alkane mixture $\theta_1 = 0.22806$ and $\theta_2 = 0.18772$.

The attraction and repulsion parameters for a pure component are: $$ a = 0.457 \frac{R^2 T_c^2}{P_c}(1 + \kappa (1 - \sqrt{T/T_c}))^2 $$ $$ b = 0.00778 \frac{RT_c}{P_c} $$ where $T_c$ is the critical temperature (K) and $P_c$ is the critical pressure (MPa).

$$\kappa = 0.37464 + 1.54226 \omega - 0.26992 \omega^2$$ where $\kappa$ is a simplification term and $\omega$ is the acentric factor.

Raoult's law is used to calculate the pressure of the mixture at VLE: $$P_{vle} = x_1 P_1^{sat} + (1 - x_1) P_2^{sat}$$ and $P_{vle}$ is used to determine the vapor mole fraction: $$y_1 = x_1 P_1^{sat} / P_{vle}$$ where $x_1$ and $y_1$ are the liquid and vapor mole fractions of hexane, and $P_i^{sat}$ is the saturation pressure that is calculated using the Antoine equation: $$P_i^{sat} = 10^{A_i - \frac{B_i}{T + C_i}}$$ where $A_i$, $B_i$ and $C_i$ are Antoine constants.

Reference

[1] A. O. Elnabawy, S. K. Fateen and M. M. Khalil, "Semi-empirical Correlation for Binary Interaction Parameters of the Peng-Robinson Equation of State with the van der Waals Mixing Rules for the Prediction of High-Pressure Vapor-Liquid Equilibrium," Journal of Advanced Research, 4(2), 2013 pp. 137\[Dash]145. doi:10.1016/j.jare.2012.03.004.

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