Cubic Plus Association (CPA)

The combination of a cubic EOS with association with the association term. The sum of the terms goes like:

\[\alpha^{\rm r} = \alpha^{\rm r}_{\rm cub} + \alpha^{\rm r}_{\rm assoc}\]

Cubic part

The residual contribution to \(\alpha\) is expressed as the sum :

\[\alpha^{\rm r}_{\rm cub,rep} +\alpha^{\rm r}_{\rm cub,att}\]

where the cubic parts come from

The repulsive part of the cubic EOS contribution:

\[\alpha^{\rm r}_{\rm cub,rep} = -\ln(1 - b_{\rm mix}\rho)\]

The attractive part of the cubic EOS contribution:

\[\alpha^{\rm r}_{\rm cub,att} = -\frac{a_{\rm mix}}{RT}\dfrac{\ln\left(\frac{\Delta_1 b_{\rm mix}\rho + 1}{\Delta_2b_{\rm mix}\rho + 1}\right)}{b_{\rm mix}\cdot(\Delta_1 - \Delta_2)}\]

with the coefficients depending on the cubic type:

SRK: \(\Delta_1=1\), \(\Delta_2=0\)

PR: \(\Delta_1=1+\sqrt{2}\), \(\Delta_2=1-\sqrt{2}\)

The mixture models used for the \(a_{\rm mix}\) and \(b_{\rm mix}\) are the classical ones:

\[a_{\rm mix} = \sum_i\sum_jx_ix_j(1-k_{ij})a_{ij}(T)\]

with x the mole fraction, \(k_{ij}\) a weighting parameter

\[a_{ij}(T) = \sqrt{a_ia_j}\]

and

\[a_{i}(T) = a_{0i}\left[1+c_{1i}(1-\sqrt{T/T_{{\rm crit},i}})\right]^2\]

and for \(b\):

\[b_{\rm mix} = \sum_ix_ib_i\]

so there are three cubic parameters per fluid that need to be obtained though fitting: \(b_{i}\), \(a_{0i}\), \(c_{1i}\). The value of \(a_{\rm ij}\) depends on temperature while \(b_{\rm mix}\) does not.

Association part

For the association, one must have a solid understanding of the association approach that is being applied. To this end, a short discussion of the general approach is required.