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>{\bar v}</math>:
- <math>
p =\frac{{\bar R} T}{{\bar v}-b}-\frac{a(T)}{{\bar v}({\bar v}+b)+b({\bar v}-b)} </math> where
- <math>\begin{align}
a(T) &= 0.45724 \frac{{\bar R}^2{T_c}^2}{P_c} \alpha \left(T \right) \\
\alpha &= \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>
Z^3+(-1-B)Z^2+(A-3B^2-2B)Z-AB+B^2+B^3=0 </math> in which
- <math>
\begin{align} A &= \frac{a \left(T \right) p}{({\bar R} T)^2} \\ B &= \frac{b}{{\bar R} T} \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}}&={\bar R} T(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}}&={\bar 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>
\frac{da}{dT}=- 0.45724 \frac{T_c {\bar R}^{2}}{p_c} \kappa \left(\sqrt{\frac{T_c}{T}} - 1\right) </math> Sandler seems to have:
- <math>
\frac{da}{dT}=- 0.45724 \frac{T_c {\bar R}^{2}}{p_c} \kappa \left(\sqrt{\frac{\alpha}{T T_c}} \right) </math>