PengRobinson EOS in FPROPS: Difference between revisions

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Comments and suggestions are welcome
Comments and suggestions are welcome
==Overview==
==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:
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>
:<math>
P =\frac{RT}{V_m-b}-\frac{a(T)}{V_m(V_m+b)+b(V_m-b)}
p =\frac{{\bar R} T}{{\bar v}-b}-\frac{a(T)}{{\bar v}({\bar v}+b)+b({\bar v}-b)}
</math>
</math>
where
where
:<math>\begin{align}
:<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 \\
a(T) &= 0.45724  \frac{{\bar R}^2{T_c}^2}{p_c} \alpha \left(T \right) \\


\kappa&=0.37464+1.54226\omega - 0.26992\omega^2
\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}
\end{align}
</math>
</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>
It is sometimes more convenient to express the equation as a cubic polynomial in terms of compressibility factor <math>Z</math>
:<math>
:<math>
\begin{align}
Z^3+(-1+B)Z^2+(A-3B^2-2B)Z-(AB-B^2-B^3)=0
Z^3+(-1-B)Z^2+(A-3B^2-2B)Z-AB+B^2+B^3=0
\end{align}
</math>
</math>
in which
in which
:<math>
:<math>
\begin{align}
\begin{align}
A=\frac{aP}{(RT)^2} \\
A &= \frac{a \left(T \right) p}{({\bar R} T)^2} \\
B=\frac{bP}{RT}
B &= \frac{b p}{{\bar R} T} \\
Z &= \frac{p {\bar v}}{{\bar R} T}
\end{align}
\end{align}
</math>
</math>
Line 37: Line 40:
:<math>
:<math>
\begin{align}
\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] \\
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}}&=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]
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}
\end{align}
</math>
</math>
Clearly to evaluate these functions we need to be able to evaluate <math>\frac{da}{dT}</math>: (CHECK THIS)
Clearly to evaluate these functions we need to be able to evaluate <math>\frac{da}{dT}</math> (checked, agrees with Sandler):
:<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>
:<math>
\begin{align}
\frac{da}{dT}= -0.45724 \frac{{\bar R}^{2} {T_c}^{\frac{3}{2}} }{p_c} \kappa \frac{\sqrt{\alpha} }{ \sqrt{T}}
\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>
</math>


Latest revision as of 23:38, 13 January 2013

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{{\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 \\

b &= \frac{0.0778\bar R T_c}{p_c} \end{align} </math>

It is sometimes more convenient to express the equation as a cubic polynomial in terms of compressibility factor <math>Z</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 p}{{\bar R} T} \\ Z &= \frac{p {\bar v}}{{\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> (checked, agrees with Sandler):

<math>

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

Comparisons

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