PengRobinson EOS in FPROPS

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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 {\bar v}:


p =\frac{{\bar R} T}{{\bar v}-b}-\frac{a(T)}{{\bar v}({\bar v}+b)+b({\bar v}-b)}

where

\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 \\

b &= \frac{0.0778\bar R T_c}{p_c}
\end{align}

It is sometimes more convenient to express the equation as a cubic polynomial in terms of compressibility factor Z


Z^3+(-1+B)Z^2+(A-3B^2-2B)Z-(AB-B^2-B^3)=0

in which


\begin{align}
A &= \frac{a \left(T \right) p}{({\bar R} T)^2} \\
B &= \frac{b p}{{\bar R} T} \\
Z &= \frac{p {\bar v}}{{\bar R} T}
\end{align}

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:


\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}

Clearly to evaluate these functions we need to be able to evaluate \frac{da}{dT} (checked, agrees with Sandler):


\frac{da}{dT}= -0.45724 \frac{{\bar R}^{2} {T_c}^{\frac{3}{2}} }{p_c} \kappa \frac{\sqrt{\alpha} }{ \sqrt{T}}

Comparisons