PengRobinson EOS in FPROPS: Difference between revisions

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

@@ Line 5: / Line 5: @@
 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 one of several equations of state that can be expressed in the form
+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>
-\begin{align}
+p =\frac{{\bar R} T}{{\bar v}-b}-\frac{a(T)}{{\bar v}({\bar v}+b)+b({\bar v}-b)}
-P=\frac{RT}{V-b}-\frac{a}{V^2+ubV+wb^2}
-\end{align}
-</math>
-or
-:<math>
-\begin{align}
-Z^3-(1+B^*-uB^*)Z^2+(A^*+wB^{*2}-uB^*-uB^{*2})Z-A^*B^*-wB^{*2}-wB^{*3}=0
-\end{align}
-</math>
-Where
-:<math>
-\begin{align}
-A^*=\frac{aP}{R^2T^2}
-\end{align}
 </math>
-:<math>
+where
-\begin{align}
+:<math>\begin{align}
-B^*=\frac{bP}{RT}
-\end{align}
+a(T) &= 0.45724  \frac{{\bar R}^2{T_c}^2}{p_c} \alpha \left(T \right) \\
-</math>
-In the Peng-Robinson values for <math>u,</math> <math>w</math>, <math>b</math> and <math>a</math> are set as
+\alpha &= \left( 1+\kappa \left( 1-\sqrt{\frac{T}{T_c}} \right) \right)^2 \\
-:<math>
-\begin{align}
+\kappa &= 0.37464+1.54226\omega - 0.26992\omega^2 \\
-u=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>
-\begin{align}
+Z^3+(-1+B)Z^2+(A-3B^2-2B)Z-(AB-B^2-B^3)=0
-w=-1
-\end{align}
 </math>
+in which
 :<math>
 \begin{align}
-b=\frac{0.7780RT_c}{P_c}
+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}
-a=\frac{0.45724R^2T_c^2}{P_c}[1+f\omega(1-T_r^{\frac{1}{2}})]^2
+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>
-\begin{align}
+\frac{da}{dT}= -0.45724 \frac{{\bar R}^{2} {T_c}^{\frac{3}{2}} }{p_c} \kappa \frac{\sqrt{\alpha} }{ \sqrt{T}}
-f\omega=0.37464+1.54226\omega-0.26992\omega^2
-\end{align}
 </math>
-For any given pair of P,V and T we can solve for the other. Other thermodynamic properties can be obtained from the departure functions.
 ==Comparisons==
 [[Category:Development]]

PengRobinson EOS in FPROPS: Difference between revisions

Latest revision as of 23:38, 13 January 2013

Overview

Departure Functions

Comparisons

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