Chapter 12.7, Problem 20P

Interpretation Introduction

Interpretation:

Predict the $T_{xy}$ diagram for the system using the Peng Robinsons equation of state.

Concept Introduction:

Write the expression to calculate the pressure $\left(P\right)$ for the component in liquid phase.

$P=\frac{R{T}^L}{{\underset{\_}{V}}^L-b_m^L}-\frac{a_m^L}{{\underset{\_}{V}}^L\left({\underset{\_}{V}}^L+b_m^L\right)+b_m^L\left({\underset{\_}{V}}^L-b_m^L\right)}$

Here, gas constant is $R$, temperature of component in liquid phase is $T^L$, molar volume of component in liquid phase is ${\underset{\_}{V}}_L$, equation of state parameter for the mixture in liquid phase is $b_m^L$, and equation of state parameter for the mixture in liquid phase is $a_m^L$.

Write the expression to calculate the pressure $\left(P\right)$ for the component in vapor phase.

$P=\frac{R{T}^V}{{\underset{\_}{V}}^V-b_m^V}-\frac{a_m^V}{{\underset{\_}{V}}^V\left({\underset{\_}{V}}^V+b_m^V\right)\left({\underset{\_}{V}}^L-b_m^L\right)}$

Here, equation of state parameter for the mixture in liquid phase is $b_m^L$, equation of state parameter for the mixture in vapor phase is $b_m^V$, temperature of component in vapor phase is $T^V$, molar volume of component in vapor phase is ${\underset{\_}{V}}_V$, equation of state parameter for the mixture in vapor phase is $b_m^V$, and equation of state parameter for the mixture in vapor phase is $a_m^V$.

Write the expression to calculate the equation of state parameter for the mixture in liquid phase $\left(a_m^L\right)$.

$a_m^L=x_1^2{a}_1+x_2^2{a}_2+2x_1{x}_2\sqrt{a_1{a}_2}$

Here, liquid mole fraction of component 1 in binary mixture is $x_1$, equation of state parameter is $a_1\text{and}{a}_2$ respectively, and liquid mole fraction of component 2 in binary mixture is $x_2$.

Write the expression to calculate the equation of state parameter for the mixture in liquid phase $\left(a_m^V\right)$.

$a_m^V=y_1^2{a}_1+y_2^2{a}_2+2y_1{y}_2\sqrt{a_1{a}_2}$

Here, vapor mole fraction of component 1 in binary mixture is $y_1$, and vapor mole fraction of component 2 in binary mixture is $y_2$.

Write the expression to calculate the reduced temperature $\left(T_r\right)$.

$T_r=\frac{T}{T_c}$

Here, critical temperature is $T_c$ and temperature is T.

Write the expression to calculate the parameter used in calculation of a for Peng-Robinson EOS.

$\kappa =0.37464+1.54226\omega -0.26993{\omega }^2$

Here, acentric factor is $\omega$.

Write the expression to calculate $\alpha$ expressed as a function of the reduced temperature.

$\alpha ={\left[1+\kappa \left(1-T_r^{0.5}\right)\right]}^2$

Here, reduced temperature is $T_r$ and parameter of Peng-Robinson EOS is $\kappa$.

Write the expression to calculate the van der Waals parameter $a$.

$a=a_c\alpha$

Write the expression to calculate the equation of state parameter for the mixture in liquid phase $\left(b_m^L\right)$.

$b_m^L=x_1{b}_1+x_2{b}_2$

Here, van der Waals parameters for components 1 and 2 are $b_1\text{and}{b}_2$ respectively.

Write the expression to calculate the equation of state parameter for the mixture in vapor phase $\left(b_m^V\right)$.

$b_m^V=y_1{b}_1+y_2{b}_2$

Write the expression to calculate the van der Waals parameter $b$.

$b=\frac{0.07780R{T}_c}{P_c}$

Here, critical pressure is $P_c$.

Use the fugacity equality for both compounds in both phases:

$\begin{array}{l}{x}_1{\widehat{\phi }}_1^L=y_1{\widehat{\phi }}_1^V\\ \left(1-x_1\right){\widehat{\phi }}_2^L=\left(1-y_1\right){\widehat{\phi }}_2^V\end{array}$

Here, mixture fugacity of coefficient of component 1 in liquid and vapor phase is ${\widehat{\phi }}_1^L,\text{and}{\widehat{\phi }}_1^V$ respectively, and mixture fugacity of coefficient of component 2 in liquid and vapor phase is ${\widehat{\phi }}_2^L,\text{and}{\widehat{\phi }}_2^V$ respectively.

Write the expression to calculate the mixture fugacity coefficient $\left({\widehat{\phi }}_i\right)$.

$\ln \left({\widehat{\phi }}_i\right)=\left\{\begin{array}{l}\frac{b_i}{b_m}\left(Z_m-1\right)-\ln \left(Z_m-\frac{b_{m}P}{RT}\right)\\ -\frac{a_m}{2\sqrt{2}{b}_{m}RT}\left[\frac{2\left(x_1{a}_{i1}+x_2{a}_{i2}\right)}{a_m}-\frac{b_i}{b_m}\right]\ln \left[\frac{Z_m+\left(1+\sqrt{2}\right)\frac{b_{m}P}{RT}}{Z_m+\left(1-\sqrt{2}\right)\frac{b_{m}P}{RT}}\right]\end{array}\right\}$

Here, compressibility factor of mixture is $Z_m$, Van der walls parameter is $b_i$, equation of state parameter for the mixture is $a_m\text{and}{b}_m$ respectively, pressure is $P$, cross parameter for equation of state is $a_{i1}\text{and}{a}_{i2}$ respectively, gas constant is $R$,and temperature is $T$.

Write the expression to calculate the compressibility factor of mixture $\left(Z_m\right)$.

$Z_m=\frac{P\underset{\_}{V}}{RT}$

Here, molar volume is $\underset{\_}{V}$.

Write the expression for temperature difference.

$\begin{array}{l}{T}^V-T^L=0\\ T^V=T^L\end{array}$

Here, temperature of component in vapor phase is $T^V$ and temperature of component in liquid phase is $T^L$.

Write the expression for pressure difference.

$\begin{array}{l}{P}^L-P^V=0\\ P^L=P^V\end{array}$

Here, pressure of component in vapor phase is $P^V$ and pressure of component in liquid phase is $P^L$.

Write the expression to calculate the Antoine equation for the component.

$\begin{array}{l}{\log }_{10}\left(P_i{}^{sat}\right)=A-\frac{B}{T+C}\\ P_i{}^{sat}={10}^{A-\frac{B}{T+C}}\end{array}$

Here, Antoine constants or coefficients are A, B, and C, vapor pressure is $P_i{}^{sat}$, and temperature is T.