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# 11.1: Peng-Robinson EOS (1976)

The Peng-Robinson EOS has become the most popular equation of state for natural gas systems in the petroleum industry. During the decade of the 1970’s, D. Peng was a PhD student of Prof. D.B. Robinson at the University of University of Alberta (Edmonton, Canada). The Canadian Energy Board sponsored them to develop an EOS specifically focused on natural gas systems. When you compare the performance of the PR EOS and the SRK EOS, they are pretty close to a tie; they are “neck to neck,” except for a slightly better behavior by the PR EOS at the critical point. A slightly better performance around critical conditions makes the PR EOS somewhat better suited to gas/condensate systems.

Peng and Robinson introduced the following modified vdW EOS:

$\left(P+\frac{\alpha a}{\bar{v}^{2}+2 b \bar{v}-b^{2}}\right)(\bar{v}-b)=R T \label{11.1a}$

or explicitly in pressure,

$P=\frac{R T}{\bar{v}-b}-\frac{\alpha a}{\bar{v}^{2}+2 b \bar{v}-b^{2}} \label{11.1b}$

where:

$\alpha=\left[ 1+\left(0.37464+1.54226 \omega-0.26992 \omega^{2}\right)(1-\sqrt{T_{r}})\right]^{2} \label{11.1c}$

Peng and Robinson conserved the temperature dependency of the attractive term and the acentric factor introduced by Soave. However, they presented different fitting parameters to describe this dependency (Equation 4.11c), and further manipulated the denominator of the pressure correction (attractive) term. As we have seen before, coefficients “a” and “b” are made functions of the critical properties by imposing the criticality conditions. This yields:

$a=0.45724 \frac{R^{2} T_{c}^{2}}{P_{c}} \label{11.2a}$

The PR cubic expression in Z becomes:

$b=0.07780 \frac{R T_{c}}{P_{c}} \label{11.2b}$

where:

$Z^{3}-(1-B) Z^{2}+\left(A-2 B-3 B^{2}\right) Z-\left(A B-B^{2}-B^{3}\right)=0 \label{11.3a}$

$A=\frac{\alpha a P}{R^{2} T^{2}} \label{11.3b}$

$B=\frac{b P}{R T} \label{11.3c}$

Similar to SRK, the PR mixing rules are:

$(\alpha a)_{m}=\sum \sum y_{i} y_{j}(\alpha a)_{i j} ;(\alpha a)_{i j}=\sqrt{(\alpha a)_{i}(\alpha a)_{j}}\left(1-k_{i j}\right) \label{11.4a}$

$b_{m}=\sum_{i} y_{i} b_{i} \label{11.4b}$

where binary interaction parameters (kij) again play the important empirical role of helping to better fit experimental data. Due to the empirical character of these interaction parameters, kij’s calculated for PR EOS are unlikely to be the same as the kij’s calculated for SRK EOS for the same pair of molecules.

## Contributors

• Prof. Michael Adewumi (The Pennsylvania State University). Some or all of the content of this module was taken from Penn State's College of Earth and Mineral Sciences' OER Initiative.