I am trying to simulate the collision and solidification of liquid iron droplets on a cooled iron surface with a custom coupled Lattice Boltzmann method.
It consists of:

An isothermal model to handle the density and velocity fields (The model by Inamuro et al. is used)

The original work of Yuan/Schaefer couples the thermal model with a different isothermal model, but to me it seems like any two-phase model can be used.

The hot droplets fall through the air onto the surface and are supposed to cool down there very quickly and solidify. From what I understand the droplets should cool down much faster when in contact with the surface than when they are in mid-air. That’s because of the different thermal conductivities of air (0.0262 W/ K m) and iron (80 W/ K m). However the thermal diffusivities of air and iron are nearly the same (around 0.000021 m^2/s). And since the thermal diffusivity is the only parameter of the thermal LBM by Yuan/Schaefer it is impossible for me to simulate the behaviour of quicker thermal transport in metal compared to air. My original idea was to modify the relaxation time tauT of the thermal model depending on the density, but that is not consistent with the equation:

alpha=1/3(tauT-1/2)

where alpha is the thermal diffusivity. This equation can be found in various papers about thermal lattice boltzmann methods.

Putting it all together, my questions are as follows:

Does anybody have an idea why the above equation holds?

Does anybody have an idea how to incorporate different thermal conductivities for the two-phases without modifying tauT?

Has anybody done any successful simulations of two-phase thermal flow?

I know I didn’t go into much detail yet, but maybe you can already answer some questions. If not, I can go more specific

I won't be of great help .. sorry. I tried to think for a while at your problem of different thermal conductivities but equal thermal diffusivity ... but I still don't understand how you could correctly model the behavior of your fluids (my feeling is that you could make it if you rewrite your problem dimensionless units ... but I do not have any idea about the dimensionless number you could use...).

Anyway … I just like to comment to some of your question:

The original work of Yuan/Schaefer couples the thermal model with a different isothermal model, but to me it seems >like any two-phase model can be used.

Yes I agree with you, from my point of view as long as the coupling between the two dynamics (two phase model for binary mixture and passive scalar field) is only one way (forced convection–> you get a value of velocity “u” from your NS solver and you advect the temperature field using the velocity “u” ) or “simplified two ways” (Boussinesq approximation let’s say with a simple thermal buoyancy force term added to the NS dynamic for the binary mixture) … any two phase model can be used.

You could also add some melting/solidification dynamic. See as example:

I extended the algorithm for two-phase model. One of the two phases can become solid or vice-versa.
I use the Shan-Chen method as a model for the binary mixture.

my feeling is that you could make it if you rewrite your problem dimensionless units … but I do not have any idea about the dimensionless number you could use…

Me neither
My newest idea is to use the Prandtl number. I computed that for the liquid phase based on the values in SI units of kinematic viscosity and thermal diffusivity of liquid iron, it is Pr=0.021. The Prandtl-number for the vapor phase (air) is Pr=0.8477. Based on these numbers and the viscosities in the LBM it is possible to compute the thermal diffusivities in LB units for the two phases. If I use the true viscosities of the two phases, then i will get nearly identical thermal diffusivities for the two phases again. But if I use the mean viscosity I will get different thermal diffusivities of a ratio of 40:1. This might work for the simulation but of course it’s not very physical…

Yes I agree with you, from my point of view as long as the coupling between the two dynamics (two phase model for binary mixture and passive scalar field) is only one way (forced convection–> you get a value of velocity “u” from your NS solver and you advect the temperature field using the velocity “u” ) or “simplified two ways” (Boussinesq approximation let’s say with a simple thermal buoyancy force term added to the NS dynamic for the binary mixture) … any two phase model can be used.

OK, that sounds good. I am in the Boussinesq approximation case with “simplified two ways” coupling.

You could also add some melting/solidification dynamic. See as example: [faculty.washington.edu] I extended the algorithm for two-phase model. One of the two phases can become solid or vice-versa. I use the Shan-Chen method as a model for the binary mixture.

I have melting/solidification in there. It is based on this paper by Semma et al. Basically it’s just modifying the streaming step with a solidification factor lambda in the nodes based on local temperature. With lambda=0, the node is liquid and the fluid moves normally, with lambda=1 it is solid and any movement is stopped. Values between 0 and 1 represent nodes in the solidification/melting phase. This technique works quite well, the problem is to have to right temperature in the nodes to compute the lambda from
But I’ll have a look at the paper you suggested. Maybe they have a less heuristic way of doing that.

I am not sure if this is of any help but in the paper M.R. Kamali , S. Sundaresan, H.E.A. Van den Akker, J.J.J. Gillissen, A multi-component two-phase lattice Boltzmann method applied to a 1-D Fischer–Tropsch reactor, Chemical Engineering Journal 207–208 (2012) 587–595 the authors used a function for the relaxation time that was dependent on the density of the two phases (equation 5 in the mentioned paper).

Perhaps you can find a parallel with thermal transport problem.