Peng-Robinson Equation of State

The Peng-Robinson (PR) equation of state (EOS) is a cubic EOS which was initially developed by Peng and Robinson in 1976¹ and was modified in 1978 to enhance the phase behavior predictions of the model². The PR EOS is a common EOS used in many areas of petroleum and chemical engineering.

PR EOS Structure

The PR EOS model is defined by

$\begin{equation}\label{eq:ceos_pr} p = \frac{RT}{v-b} - \frac{a}{v(v+b)+b(v-b)} \end{equation}$

where $a$ and $b$ are the EOS model parameters. The pure component parameters ($a$ and $b$) can be calculated by

$\begin{eqnarray}\label{eq:a_pr} a = \Omega_a \frac{R^2T_c^2}{p_c}\alpha(T) \end{eqnarray}$

$\begin{eqnarray}\label{eq:b_pr} b = \Omega_b \frac{RT_c}{p_c} \end{eqnarray}$

$\begin{eqnarray}\label{eq:alpha} \alpha(T) = [1+m(\omega)(1-\sqrt{\frac{T}{T_c}})]^2 \end{eqnarray}$

In equations \eqref{eq:a_pr}, \eqref{eq:b_pr} and \eqref{eq:alpha} the coefficients $\Omega_a$ and $\Omega_b$ , and the value of $\alpha(T)$ can be calculated by

$\begin{eqnarray} \Omega_a = \frac{8(5X+1)}{49-37X} = 0.45724... \end{eqnarray}$

$\begin{eqnarray} \Omega_b = \frac{X}{X+3} = 0.07780... \end{eqnarray}$

$\begin{eqnarray} X = \frac{-1+(6\sqrt{2}+8)^{1/3}-(6\sqrt{2}-8)^{1/3}}{3} \end{eqnarray}$

$\begin{eqnarray} m(\omega) = 0.37464 + 1.54226\omega - 0.26992\omega^2 , \omega \leq 0.49 \end{eqnarray}$

$\begin{eqnarray} m(\omega) = 0.3796 + 1.485\omega - 0.1644\omega^2 +0.01667\omega^3 , \omega > 0.49 \end{eqnarray}$

Component Properties

The component properties are composed of the critical pressure ($p_c$), critical temperature ($T_c$), accentric factor ($\omega$) and molecular weight ($MW$). Another important component property for cubic EOS models are the volume shift factors descried by Peneloux in 1982³. The volume shifts were introduced as a correction term to the phase behaviour predictions of the molar volume predictions.

Mixing Rules

Note

The main article for cubic EOS models describes the mixing rules with some more detail.

The original mixing rule developed by van der Waals is a quadratic mixing rule. The set of equations describing the mixing are given by

$\begin{equation}\label{eq:a_mix_vdw} a = \sum_{i=1}^N \sum_{i=1}^N u_i u_j a_{ij} \end{equation}$

$\begin{equation}\label{eq:b_mix_vdw} b = \sum_{i=1}^N u_i b_i \end{equation}$

where $a_{ij}=\sqrt{a_ia_j}$, $u_i$ can be the liquid composition ($x_i$), vapor composition ($y_i$) or the total composition ($z_i$) and $a_i$ and $b_i$ are the component EOS parameters for component $i$.

Modifications to the $a$ and $b$ mixing rules were developed to enhance the performance of the EOS models. The most common modified mixing rules are given by

$\begin{equation}\label{eq:a_mix_mod} a = \sum_{i=1}^N \sum_{i=1}^N u_i u_j a_{ij} (1-k_{ij}) \end{equation}$

where $k_{ij}$ is often referred to as the binary interaction parameter (BIP) or sometimes referred to as the binary interaction coefficient (BIC).

D. Y. Peng and D. B. Robinson. A new two-constant equation of state. Industrial & Engineering Chemistry Fundamentals, 15:59–64, 1976. ↩
D. B. Robinson and D. Y. Peng. The characterization of the heptanes and heavier fractions for the GPA Peng-Robinson programs. Gas processors association, 1978. ↩
A. Péneloux, E. Rauzy, and R. Fréze. A consistent correction for redlich-kwong-soave volumes. Fluid phase equilibria, 8:7–23, 1982. doi:https://doi.org/10.1016/0378-3812 $82$ 80002-2. ↩