How does pure association work?

[ianhbell]

I am mystified by the implementation in _X_exact1 for single site association:

Clapeyron.jl/src/models/SAFT/association.jl

Line 328 in 2a3d6ea

function _X_exact1(model,V,T,z,data=nothing)

. It clearly works, as numerical tests comparing against the explicit solutions in Huang&Radosz demonstrate, but I don't follow the logic. Is this taken from a paper, or re-derived? If the former, which paper, and if the latter, how?

[pw0908]

How do you mean? This is just transforming the mass-action equation into a quadratic equation and solving that.
$X_{H} = (1+n_e\rho\Delta_{He} X_e)^{-1}$
$X_{e} = (1+n_H\rho\Delta_{eH} X_H)^{-1}$
Substitute:
$X_{H} = (1+n_e\rho\Delta_{He} (1+n_H\rho\Delta_{eH} X_H)^{-1})^{-1}$
Multiply numerator and denominator by $(1+n_H\rho\Delta_{eH} X_H)$:
$X_{H} = (1+n_H\rho\Delta_{eH} X_H)(1+n_e\rho\Delta_{eH}+n_H\rho\Delta_{He} X_H)^{-1}$
Multiply the denominator across:
$X_{H}(1+n_e\rho\Delta_{He}+n_H\rho\Delta_{eH} X_H)=(1+n_H\rho\Delta_{eH} X_H)$
Re-write as quadratic:
$n_H\rho\Delta_{eH} X_H^2+(1+n_e\rho\Delta_{He}-n_H\rho\Delta_{eH})X_H-1=0$

Other association networks can also be solved analytically, this is just the most-common one. I believe I derived somewhere that you can only solve association networks up to 5 sites analytically but that's not included in Clapeyron.

[pw0908]

Are you talking about deriving the equations or solving the system of equations? For the former, I would recommend Laffite's 2013 paper (still one of the best SAFT papers I've read). For solving... there are lecture slides for the course that were taught by Michelsen somewhere online.

[ianhbell]

Indeed Lafitte has your magical equation:

The simplification approaches all assume that you have only A-B interactions, but no A-A and B-B self association interactions, right. Also, they assume that for mixtures, an A site on molecule i is interchangable with an A site on molecule j? Which is required to do the simplifications in Kraska

[pw0908]

Yes, the ij,ab = ij,ba simplification is something the old SAFT equations used to do. In SAFT-VR (gamma) Mie, they remove this assumption. To me, this makes sense as the water-H-alcohol-e interactions are going to be very different from the water-e-alcohol-H interactions.

[ianhbell]

I'm with you. As no expert in the domain what you wrote does make sense. In Kraska he keeps the non-simplified version where ∆eH ≠ ∆He and now I see the point why

[ianhbell]

I worked this out in sympy:

# Here is the algebraic solutions to the laws of mass-action for the simplified cases
# covered in Huang & Radosz
import sympy as sy

kappa_B, kappa_A, X_A, X_B = sy.symbols('kappa_B, kappa_A, X_A, X_B')

# Definitions of the equations to be solved
# In Eq(), the first arg is the LHS, second is RHS
eq1 = sy.Eq(X_B, 1/(1+kappa_A*X_A))
eq2 = sy.Eq(X_A, 1/(1+kappa_B*X_B))
        
# The solutions
solns = sy.solve([eq1, eq2], [X_A, X_B])
for soln in solns:
    for x in soln:
        display(x)

yielding

of which only one pair should be valid

[ianhbell]

Ah that's cool. I'm no expert in this association stuff, so this is all new to me. Any good reference for the mass-action equations?

[ianhbell]

Ah, ok, now I see it, sorry I guess I was being dense.

[ianhbell]

This discussion arose because we were trying to make sense of the equation in Langenbach and Enders for pure fluids:

[pw0908]

That's a really weird way of writing it...

[ianhbell]

Yeah, and also, incorrect, because you actually need the site fractions rather than the mole fractions where x_j is in that equation.

[ianhbell]

The mass-action equation is generically written as: Xi,a=(1+∑jρj∑bnj,bΔij,abXj,b)−1∀i∈1,…,nca∈1,…,nsites,i Most (good) SAFT papers should include some details regarding how this equation is obtained / solved.

Can you recommend a good SAFT paper? Every one I have read seems to no consider multiple site types, with a single one of each type, rather than multiplicities of sites like you have written

[ianhbell]

Thanks @pw0908 for your explanation.