PengRobinson EOS in FPROPS: Difference between revisions

From ASCEND
Jump to navigation Jump to search
← Older edit
 
(20 intermediate revisions by 2 users not shown)
Line 5: Line 5:
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>
\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_m-b}-\frac{a(T)}{V_m(V_m+b)+b(V_m-b)}
\end{align}
</math>
</math>
It is sometimes more convenient to express the equation in terms of compressibility factor:
where
:<math>
:<math>\begin{align}
\begin{align}
 
Z^3+(-1-B)Z^2+(A-3B^2-2B)Z-AB+B^2+B^3=0
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}
\end{align}
</math>
</math>
The various fluid-dependant constants and variables in the above are defined as:
 
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
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>
in which
:<math>
:<math>
\begin{align}
\begin{align}
A=\frac{aP}{(RT)^2}
A &= \frac{a \left(T \right) p}{({\bar R} T)^2} \\
\end{align}
B &= \frac{b p}{{\bar R} T} \\
</math>
Z &= \frac{p {\bar v}}{{\bar R} T}
:<math>
\begin{align}
B=\frac{bP}{RT}
\end{align}
</math>
By definition:
:<math>
\begin{align}
Z=\frac{PV_m}{RT}
\end{align}
\end{align}
</math>
</math>
==Departure Functions==
==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.
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.
Line 49: Line 40:
:<math>
:<math>
\begin{align}
\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]
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}
\end{align}
</math>
</math>
Clearly to evaluate these functions we need to be able to evaluate <math>\frac{da}{dT}</math> (checked, agrees with Sandler):
:<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}}
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>
</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

Retrieved from "https://ascend4.org/index.php?title=PengRobinson_EOS_in_FPROPS&oldid=4102"