PengRobinson EOS in FPROPS
Work on this is went on as a part of GSoC 2010 (Project Ankit) and continues as part of GSoC 2011 (Richard Towers). See also FPROPS.
Comments and suggestions are welcome
Overview
The Peng-Robinson EOS is a cubic equation of state in that it contains volume terms to the third power. It is usually expressed to give Pressure in terms of Temperature and Molar Volume:
- <math>
P =\frac{RT}{V_m-b}-\frac{a(T)}{V_m(V_m+b)+b(V_m-b)} </math> where
- <math>\begin{align}
a(T)&=0.45724 \frac{R^2{T_c}^2}{P_c} \left(1+\kappa \left(1-\sqrt{\frac{T}{T_c}} \right) \right)^2 \\
\kappa&=0.37464+1.54226\omega - 0.26992\omega^2 \end{align} </math>
It is sometimes more convenient to express the equation as a cubic polynomial in terms of compressibility factor <math>Z=\frac{PV_m}{RT}</math>
- <math>
\begin{align} Z^3+(-1-B)Z^2+(A-3B^2-2B)Z-AB+B^2+B^3=0 \end{align} </math> in which
- <math>
\begin{align} A=\frac{aP}{(RT)^2} \\ B=\frac{bP}{RT} \end{align} </math>
Departure Functions
Departure functions represent the departure of the real properties from the ideal properties - i.e the properties of a fluid at zero pressure or infinite molar volume. The departure functions of the Peng-Robinson equation of state are as follows:
- <math>
\begin{align} H_{m}-H_{m}^{\text{ideal}}&=RT(Z-1)+\frac{T\left(\frac{da}{dT}\right)-a}{2\sqrt{2}b}\ln\left[\frac{Z+(1+\sqrt{2})B}{Z+(1-\sqrt{2})B}\right] \\
S_{m}-S_{m}^{\text{ideal}}&=R\ln (Z-B)+\frac{\frac{da}{dT}}{2\sqrt{2}b}\ln\left[\frac{Z+(1+\sqrt{2})B}{Z+(1-\sqrt{2})B}\right] \end{align} </math> Clearly to evaluate these functions we need to be able to evaluate <math>\frac{da}{dT}</math>: (CHECK THIS)
- <math>
\begin{align} \frac{da}{dT}=- 0.45724 \frac{T_c R^{2}}{P_c} \kappa \left(\sqrt{\frac{T_c}{T}} - 1\right) \end{align} </math> Sandler seems to have:
- <math>
\begin{align} \frac{da}{dT}=- 0.45724 \frac{T_c R^{2}}{P_c} \kappa \left(\sqrt{\frac{\alpha}{T T_c}} \right) \end{align} </math>
- <math>
\begin{align} \alpha=\left( 1+\kappa \left( 1-\sqrt{\frac{T}{T_c}} \right) \right)^2 \end{align} </math>