Let us rethink advanced mixing rules for cubic equations of state

One of the most challenging aspects of using a Van der Waals type equation of state for mixtures is determining the appropriate expressions for the coefficients $a$ (attractive parameter) and $b$ (covolume) involved in this equation. It has been 45 years since Huron and Vidal first proposed the “EoS/$g^E$” advanced mixing rules. By equating under infinite reference pressure, the mathematical expression of the excess Gibbs energy $g^E$ derived from an equation of state, to the same quantity issued from an explicit activity coefficient model, they deduced an expression for the ratio $a/b$. A decade after Huron and Vidal’s initial proposal, building upon the original proposal, Michelsen subsequently derived the “zero reference pressure” (ZRP) approach and proposed the approximate ZRP mixing rules MHV1 and MHV2. Throughout the 1990′s and 2000′s, the Huron-Vidal and ZRP approaches were subject, often empirically, to multiple revisions in order to remedy some of their well identified shortcomings. It would appear that the debates surrounding advanced mixing rules are now over, with the latest conclusions proposed in the 2000s enjoying a degree of consensus.

The objective of this article is to reopen the debate in light of the scientific insights gained from our recent research on advanced mixing rules for cubic equations of state. The concept of deriving mixing rules by equating the excess Gibbs energy expressed from an equation of state to the same quantity expressed from an activity coefficient model (this equality is called a “matching equation”) was undoubtedly an appealing one. However, experience has shown that such a matching equation is not without its limitations, particularly due to the lack of sufficient constraints. We have reached the conclusion that the only way to derive advanced mixing rules that are free from shortcomings is to ensure that not only are the complete expressions of $g^E$ from an equation of state and from an activity coefficient model equal but also that the three separate contributions that make it up (i.e., combinatorial, residual and the product of the pressure by the excess volume $P·v^E$) are equal. As discussed in this paper, achieving this objective is challenging and we conclude that the best and unique solution for developing safe mixing rules is to modify the matching equation proposed by Huron-Vidal and Michelsen and to only equate the residual contributions.

Based on this observation, we demonstrate how the demonstrations of the ZRP and HV mixing rules can be reworked to arrive at a unique and universal (independent of the reference pressure) mixing rule, called UHVM (Unified Huron Vidal Michelsen) mixing rule. We hope that this result will encourage new thinking on mixing rules for cubic equations.