PengRobinson

Chemical Reactor Design Toolbox Reference Manual

ChemReactorDesign.Basic.Gas.Functions.PengRobinson

The function calculates the values for the state variables according to the cubic EOS after Peng/Robinson (Stanley I. Sandler, 2006).

\begin{equation*} c_{1} \, z^{3} + c_{2} \, z^{2} + c_{3} \, z + c_{4} = 0 \end{equation*}

with the coefficients as (Jürgen Gmehling and Michael Kleiber and Bärbel Kolbe and Jürgen Rarey, 2019)

\begin{align*} c_{1} & = 1 \\ c_{2} & = \frac{b_{mix} \, p}{R \, T} -1 \\ c_{3} & = \frac{a_{mix} \, p}{R^{2} \, T^{2}} - 3 \, \left( \frac{b_{mix} \, p}{R \, T} \right)^{2} - 2 \, \frac{b_{mix} \, p}{R \, T} \\ c_{4} & = \left(\frac{b_{mix} \, p}{R \, T} \right)^{3} + \left(\frac{b_{mix} \, p}{R \, T} \right)^{2}- \frac{a_{mix} \, p}{R^{2} \, T^{2}} \, \frac{b_{mix} \, p}{R \, T} \end{align*}

The volume formulation

\begin{equation*} d_{1} \, V^{3} + d_{2} \, V^{2} + d_{3} \, V + d_{4} = 0 \end{equation*}

can be obtained, by expressing

\begin{equation*} z = \frac{p}{R \, T} \, V \end{equation*}

and inserting it in the equation above. Thus, the following coefficients are obtained.

\begin{align*} c_{1} & = 1 \\ c_{2} & = b_{mix} - R*T/p \\ c_{3} & = \frac{a_{mix}}{p} -3 \, b_{mix}^{2} -2 \, \frac{R \, T}{p} \, b_{mix} \\ c_{4} & = b_{mix}^{3}+\frac{R \, T}{p} \, b_{mix}^{2} - \frac{a_{mix} \, b_{mix}}{p} \end{align*}

The parameters are given as

\begin{equation*} \kappa_{i} = 0.37464 + 1.54226 \, \omega_{i} - 0.26992 \, \omega_{i}^2 \end{equation*}
\begin{equation*} \alpha_{i} = \left[1 + \kappa_{i} \, \left(1 - \sqrt{\frac{T}{T_{c_{i}}}} \right)\right]^{2} \end{equation*}
\begin{equation*} a_{i} = 0.45724 \frac{R^2 \, T_{c_{i}}^2}{p_{c_{i}}} \, \alpha_{i} \end{equation*}
\begin{equation*} b_{i} = 0.07780 \frac{R \, T_{c_{i}}}{p_{c_{i}}} \end{equation*}

For the calculation of the cross coefficient \(a_{ij}\), the geometric mean of the pure component parameters are corrected by a binary parameter \(k_{ij}\) (Jürgen Gmehling and Michael Kleiber and Bärbel Kolbe and Jürgen Rarey, 2019)

\begin{equation*} a_{ij} = \sqrt{a_{ii} \, a_{jj}} \, \left(1-k_{ij}\right) \end{equation*}

Attractive Parameter Eq.(4.99) (Jürgen Gmehling and Michael Kleiber and Bärbel Kolbe and Jürgen Rarey, 2019)

\begin{equation*} a_{mix} = \sum_{i=1}^{n} \sum_{j}^{n} x_{i} \, x_{j} \, a_{ij} \end{equation*}

Van der Waals Co-Volume (4.100) (Jürgen Gmehling and Michael Kleiber and Bärbel Kolbe and Jürgen Rarey, 2019)

\begin{equation*} b_{mix} =\sum_{i}^{n} x_{i} \, b_{i} \end{equation*}

With the compressibility \(z\) the fugacity coefficients are calculated as (Stanley I. Sandler, 2006).

\begin{align*} \varphi_{i} = &\frac{b_{i}}{b_{mix}} \left(z-1\right)- \ln\left(z-\frac{b_{mix} \, p}{R \, T} \right) - \\ \nonumber & \frac{a_{mix}}{2 \, \sqrt{2} \, R \, T} \, \left[\frac{2 \, \sum\limits_{j}^{N} x_{j} \, a_{ij}}{a_{mix}} - \frac{b_{i}}{b_{mix}} \right] \, \left[\frac{z+\left( 1+\sqrt{2} \right) \, \frac{b_{mix} \, p}{R \, T}} {z+\left( 1-\sqrt{2} \right) \, \frac{b_{mix} \, p}{R \, T}} \right] \end{align*}

Nomenclature

\(\Delta H_{res}\) departure enthalpy of the mixture
\(k_{ij}\) binary interaction coefficients
\(N\) total number of species
\(p\) pressure
\(p_{c_{i}}\) critical pressure of species Ai
\(R\) universal gas constant
\(T\) temperature
\(T_{c_{i}}\) critical temperature of species Ai
\(x_{i}\) mole fraction of species Ai
\(z\) compressibility
\(\omega_i\) acentricity factor of species Ai
\(\varphi_{i}\) fugacity coefficient of species Ai

Bibliography

Jürgen Gmehling and Michael Kleiber and Bärbel Kolbe and Jürgen Rarey (2019). Chemical Thermodynamics for Process Simulation, Wiley-VCH.

Stanley I. Sandler (2006). Chemical, Biochemical, and Engineering Thermodynamics, John Wiley and Sons.