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 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>
:<math>
\begin{align}
\begin{align}
P=\frac{RT}{V-b}-\frac{a}{V^2+ubV+wb^2}
P=\frac{RT}{V_m-b}-\frac{a(T)}{V_m(V_m+b)+b(V_m-b)}
\end{align}
\end{align}
</math>
</math>
or
It is sometimes more convenient to express the equation in terms of compressibility factor:
:<math>
:<math>
\begin{align}
\begin{align}
Z^3-(1+B^*-uB^*)Z^2+(A^*+wB^{*2}-uB^*-uB^{*2})Z-A^*B^*-wB^{*2}-wB^{*3}=0
Z^3+(-1-B)Z^2+(A-3B^2-2B)Z-AB+B^2+B^3=0
\end{align}
\end{align}
</math>
</math>
Where
The various fluid-dependant constants and variables in the above are defined as:
:<math>
:<math>
\begin{align}
\begin{align}
A^*=\frac{aP}{R^2T^2}
a(T)=0.45724\times \frac{R^2{T_c}^2}{P_c} \left(1+\kappa \left(1-\sqrt{\frac{T}{T_c}} \right) \right)^2
\end{align}
</math>
::<math>
\begin{align}
\kappa=0.37464+1.54226\omega - 0.26992\omega^2
\end{align}
\end{align}
</math>
</math>
:<math>
:<math>
\begin{align}
\begin{align}
B^*=\frac{bP}{RT}
A=\frac{aP}{(RT)^2}
\end{align}
\end{align}
</math>
</math>
In the Peng-Robinson values for <math>u,</math> <math>w</math>, <math>b</math> and <math>a</math> are set as
:<math>
:<math>
\begin{align}
\begin{align}
u=2
B=\frac{bP}{RT}
\end{align}
\end{align}
</math>
</math>
By definition:
:<math>
:<math>
\begin{align}
\begin{align}
w=-1
Z=\frac{PV_m}{RT}
\end{align}
\end{align}
</math>
</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>
:<math>
\begin{align}
\begin{align}
b=\frac{0.7780RT_c}{P_c}
H_{m}-H_{m}^{\text{Ideal Gas}}=RT(Z-1)+\frac{T\left(\frac{da}{dT}\right)-a}{2\sqrt{2b}}\ln\left[\frac{Z+(1+\sqrt{2})B}{Z+(1-\sqrt{2})B}\right]
\end{align}
\end{align}
</math>
</math>
:<math>
:<math>
\begin{align}
\begin{align}
a=\frac{0.45724R^2T_c^2}{P_c}[1+f\omega(1-T_r^{\frac{1}{2}})]^2
S_{m}-S_{m}^{\text{Ideal Gas}}=R\ln (Z-B)+\frac{\frac{da}{dT}}{2\sqrt{2b}}\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>:
:<math>
:<math>
\begin{align}
\begin{align}
f\omega=0.37464+1.54226\omega-0.26992\omega^2
\frac{da}{dT}=...
\end{align}
\end{align}
</math>
</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==
==Comparisons==


[[Category:Development]]
[[Category:Development]]

Revision as of 10:50, 14 July 2011

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>

\begin{align} P=\frac{RT}{V_m-b}-\frac{a(T)}{V_m(V_m+b)+b(V_m-b)} \end{align} </math> It is sometimes more convenient to express the equation in terms of compressibility factor:

<math>

\begin{align} Z^3+(-1-B)Z^2+(A-3B^2-2B)Z-AB+B^2+B^3=0 \end{align} </math> The various fluid-dependant constants and variables in the above are defined as:

<math>

\begin{align} a(T)=0.45724\times \frac{R^2{T_c}^2}{P_c} \left(1+\kappa \left(1-\sqrt{\frac{T}{T_c}} \right) \right)^2 \end{align} </math>

<math>

\begin{align} \kappa=0.37464+1.54226\omega - 0.26992\omega^2 \end{align} </math>

<math>

\begin{align} A=\frac{aP}{(RT)^2} \end{align} </math>

<math>

\begin{align} B=\frac{bP}{RT} \end{align} </math> By definition:

<math>

\begin{align} Z=\frac{PV_m}{RT} \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 Gas}}=RT(Z-1)+\frac{T\left(\frac{da}{dT}\right)-a}{2\sqrt{2b}}\ln\left[\frac{Z+(1+\sqrt{2})B}{Z+(1-\sqrt{2})B}\right] \end{align} </math>

<math>

\begin{align} S_{m}-S_{m}^{\text{Ideal Gas}}=R\ln (Z-B)+\frac{\frac{da}{dT}}{2\sqrt{2b}}\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>:

<math>

\begin{align} \frac{da}{dT}=... \end{align} </math>

Comparisons

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