PengRobinson EOS in FPROPS

This article is about planned development or proposed functionality. Comments welcome.

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{RT}{{\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}}&=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>

\frac{da}{dT}=- 0.45724 \frac{T_c 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 R^{2}}{p_c} \kappa \left(\sqrt{\frac{\alpha}{T T_c}} \right) </math>