Cubic plus association

The following CPA EoS are available in TP-Cloud:

CPA-SRK
CPA-PR

Cubic plus association (CPA) EoS were developed to describe hydrogen-bonds, also referred to as association. Water, methanol and ethanol are fluids where association is important to capture in order represent thermodynamic properties accurately. The Helmholtz energy of CPA can, as the name suggests, be formulated as ¹:

A_{\text{CPA}} = A_{\text{cubic}} + A_{\text{Ass}}

where $A_{\text{ass}}$ is the Helmholtz energy from the association contribution. In principle, any cubic EoS can be used for the first term. However, Peng-Robinson (PR) or Soave Redlich Kwong (SRK) are used in practice, and parameters for fluid interactions are available in the literature mainly for these choices. TP-Cloud has CPA-SRK and CPA-PR implemented. The association contribution is ²:

A_{\text{Ass}} = RT\sum_iN_i\sum_{A_i}\left(\ln{\left(X_{A_i}\right)}-\frac{X_{A_i}}{2}+\frac{1}{2}\right)

where $N_i$ is the mole numbers, and $X_{A_i}$ is the fraction of molecules not bonded at site $A_i$ in molecule $i$ . $X_{A_i}$ is given by the implicit equation:

\frac{1}{X_{A_i}} = 1+\left(\frac{1}{V}\right)\sum_jN_j\sum_{B_i}X_{B_j}\Delta^{A_iB_j}

where $B_j$ refers to a site, and $\Delta^{A_iB_j}$ is the association strength, given by:

\Delta^{A_iB_j} = g_{ij}\phi_{ij}\kappa^{A_iB_j}\left[\exp{\left(\frac{\varepsilon^{A_iB_j}}{RT}\right)}-1\right]

where $\phi_{ij}, \kappa^{A_iB_j}$ and $\varepsilon^{A_iB_j}$ are binary parameters obtained from pure-component parameters by use of mixing rules. Furthermore,

\phi_{ij}=\frac{b_i+b_j}{2}

and the radial distribution function is:

g_{ij}=\frac{1-0.5y}{\left(1-y\right)^3}

and

y=\frac{b}{4v}

where $b$ is the co-volume of the mixture:

b=\sum_i\sum_jx_ix_j\left[\frac{b_i+b_j}{2}\right]

To use CPA, it is necessary to also specify association scheme for the molecular interactions. Popular choices of association schemes for molecules are the 2B scheme (1 donor and 1 acceptor), the 4C scheme (2 donors and 2 acceptors), and the trivial 0 association scheme (0 donors and acceptors) where the molecule does not take part in association. Which association scheme that works best depends on which components that are interacting, and this choice is often motivated by the actual bonding between the molecules. TP-Cloud has incorporated the association schemes that give the best accuracy. CPA can also be combined with Peneloux volume shift to improve prediction of liquid-phase densities.

References

Kontogeorgis, Georgios M., et al. "An equation of state for associating fluids." Industrial & engineering chemistry research 35.11 (1996): 4310-4318. ↩
Aasen, Ailo, et al. "Thermodynamic models to accurately describe the PVTxy-behavior of water/carbon dioxide mixtures." Fluid Phase Equilibria 442 (2017): 125-139. ↩

Cubic plus association

References

Footnotes