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Cubic equations of state

Redlich Kwong

The Redlich Kwong (RK) equation is used in the Chao-Seader method of computing thermodynamic properties.

\begin{displaymath} P = {R T \over V - b} - {a \over \sqrt{T} V (V + b)} \end{displaymath}

with

$\displaystyle a_i$ $\textstyle =$ $\displaystyle {\Omega_a R^2 T^{2.5}_{ci} \over P_{ci}}$
$\displaystyle \Omega_a$ $\textstyle =$ $\displaystyle 0.42748$
$\displaystyle b_i$ $\textstyle =$ $\displaystyle {\Omega_b R T_{ci} \over P_{ci}}$
$\displaystyle \Omega_b$ $\textstyle =$ $\displaystyle 0.08664$

and the mixing rules:

$\displaystyle a$ $\textstyle =$ $\displaystyle \sum^c_{i=1} \sum^c_{j=1} y_i y_j a_{ij}$
$\displaystyle a_{ij}$ $\textstyle =$ $\displaystyle (1 - k_{ij}) \sqrt { a_i a_j }$
$\displaystyle b$ $\textstyle =$ $\displaystyle \sum^c_{i=1} y_i b_i$

where kij is a binary interaction parameter; original RK:

Soave Redlich Kwong

Soave's modification of the Redlich Kwong (SRK) EOS is one of the most widely used methods of computing thermodynamic properties. The SRK EOS is most suitable for computing properties of hydrocarbon mixtures.

\begin{displaymath} P = {R T \over V - b} - {a \over V (V + b)} \end{displaymath}

with

$\displaystyle a_i$ $\textstyle =$ $\displaystyle a_i(T_{ci}) \alpha(T_{ri},\omega_i)$
$\displaystyle a_i(T_{ci})$ $\textstyle =$ $\displaystyle {\Omega_a R^2 T^{2}_{ci} \over P_{ci}}$
$\displaystyle \Omega_a$ $\textstyle =$ $\displaystyle 0.42747$
$\displaystyle \alpha(T_{ri},\omega_i)$ $\textstyle =$ $\displaystyle \left[ 1 + (0.480 + 1.574 \omega_i -0.176 \omega^2_i) (1-\sqrt{T_{ri}}) \right]^2$
$\displaystyle b_i$ $\textstyle =$ $\displaystyle {\Omega_b R T_{ci} \over P_{ci}}$
$\displaystyle \Omega_b$ $\textstyle =$ $\displaystyle 0.08664$

and the mixing rules:

$\displaystyle a$ $\textstyle =$ $\displaystyle \sum^c_{i=1} \sum^c_{j=1} y_i y_j a_{ij}$
$\displaystyle a_{ij}$ $\textstyle =$ $\displaystyle (1 - k_{ij}) \sqrt { a_i a_j }$
$\displaystyle b$ $\textstyle =$ $\displaystyle \sum^c_{i=1} y_i b_i$

API SRK EOS

Graboski and Daubert have modified the coefficients in the SRK EOS and provided a special relation for hydrogen. This modification of the SRK EOS has been recommended by the American Petroleum Institute (API), hence the name of this menu option. It uses the same equations as the SRK except for the α:

\begin{displaymath}\alpha(T_{ri},\omega_i) =\left[ 1 + (0.48508 + 1.55171 \omega_i -0.15613 \omega^2_i) (1 - \sqrt{T_{ri}}) \right]^2\end{displaymath}

and specially for hydrogen:

\begin{displaymath}\alpha(T_{ri},\omega_i) = 1.202 e^{-0.30288 T_{ri}}\end{displaymath}

Peng Robinson EOS

The Peng-Robinson equation is another cubic EOS that owes its origins to the RK and SRK EOS. The PR EOS, however, gives improved predictions of liquid phase densities.

\begin{displaymath} P = {R T \over V - b} - {a \over V (V + b) + b (V - b)}\end{displaymath}

with

$\displaystyle a_i$ $\textstyle =$ $\displaystyle a_i(T_{ci}) \alpha(T_{ri},\omega_i)$
$\displaystyle a_i(T_{ci})$ $\textstyle =$ $\displaystyle {\Omega_a R^2 T^{2}_{ci} \over P_{ci}}$
$\displaystyle \Omega_a$ $\textstyle =$ $\displaystyle 0.45724$
$\displaystyle \alpha(T_{ri},\omega_i)$ $\textstyle =$ $\displaystyle \left[ 1 + (0.37464 + 1.5422 \omega_i -0.26992 \omega^2_i) (1 - \sqrt{T_r}) \right]^2$
$\displaystyle b_i$ $\textstyle =$ $\displaystyle {\Omega_b R T_{ci} \over P_{ci}}$
$\displaystyle \Omega_b$ $\textstyle =$ $\displaystyle 0.07880$

and the mixing rules:

$\displaystyle a$ $\textstyle =$ $\displaystyle \sum^c_{i=1} \sum^c_{j=1} y_i y_j a_{ij}$
$\displaystyle a_{ij}$ $\textstyle =$ $\displaystyle (1 - k_{ij}) \sqrt { a_i a_j }$
$\displaystyle b$ $\textstyle =$ $\displaystyle \sum^c_{i=1} y_i b_i$