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.

[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?

[pw0908]

The mass-action equation is generically written as:
$X_{i,a} = (1+\sum_j \rho_j\sum_b n_{j,b}\Delta_{ij,ab}X_{j,b})^{-1}\quad \forall i\in 1,\dots,n_c\quad a\in 1,\dots, n_{sites,i}$
Most (good) SAFT papers should include some details regarding how this equation is obtained / solved.

[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]

Thanks @pw0908 for your explanation.