Approximating mixture model to pure model

[Sush1090]

Hello,

I wanted to know if it is possible to extract a pure model approximation from a mixture model.

For example, say I have a model with two components of cPR which are build over that parameters mentioned in the .cvs and the adjoining idealmodel . The mole fraction is also known.

Can I approximate the behavior of this model with a pure pseudo-fluid - which doesn't really exist? So in other works find the parameters of this pseudo-fluid that fit the description of cPR parameters and its corresponding idealmodel.

This might be a question that is not correctly asked as I have not really specified what "approximating the behaviour" really means. This is because I am not sure what metric seems apt for such a statement.

[longemen3000]

I don't think you can do this in general, For example, even for cubics, the Soave alpha function has a minimum at $T_r = \frac{1}{m^2}$, the mixture value of a $= \sum {a_{ii}a_{jj} \sqrt{\alpha_i \alpha_j}}$ will have some kinks caused by the presence of those minimas. For cPR, probably the situation is better (the c in cPR stands for consistent after all), but the parameters of the pseudo-pure model will need to be calculated via optimization vs a formula, and the criteria of optimization will select which properties of the pseudo pure will be similar to that of the mixture, and which properties will diverge.

For ideal models, the problem simplifies. in particular, i think that if you use the reid ideal model, then, for the pseudo pure ideal model, the parameters are just a linear combination of the parameters for each substance ($a = \sum{a_i x_i}$).

if we restrict our residual model to cubics, then we have two options

• we create a pseudo-fluid that fits the mechanical critical point of the mixture

For the first approach, we can do this by doing the following steps:

1. find the mechanical critical point (Tc,Pc,Vc) at the selected composition (i have that function in the fast_tp_property branch
2. find the alpha function. In the case of a soave alpha, we need to find the edge pressure at T = 0.7Tc, and calculate the acentric factor. For other alpha functions, the procedure may be more complicated and may result in inconsistent parameters according to Le Guennec's criteria). This model probably will match saturation pressures but could differ on enthalpies.

The second approach is just finding some coordinates of Pc,Tc that minimize the difference in properties. this model will match properties but could differ in saturation pressures.

There is a series of papers that talk about the tradeoffs of fitting a cubic, depending on what parameters you allow to vary:

• An accurate direct technique for parameterizing cubic equations of state: Part I. Determining the cohesion temperature function in the low-temperature range: https://doi.org/10.1016/j.fluid.2008.01.003
• An accurate direct technique for parameterizing cubic equations of state: Part II. Specializing models for predicting vapor pressures and phase densities https://doi.org/10.1016/j.fluid.2008.01.013

[Sush1090]

Thank you for this detailed answer.

I understand the difficulties now. I will take a look at the papers mentioned.