Enthalpy equation for multicomponent mixtures
 

Hello everybody,

I would like to write a laminar, steady-state solver for multicomponent, reacting mixtures. I have some questions about the enthalpy equation which is used in several solvers for multicomponent mixtures (in steady state conditions, SIMPLE algorithm):

fvm::div(phi, h)
- fvm::Sp(fvc::div(phi), h)
- fvm::laplacian(turbulence->alphaEff(), h)
==
fvc::div(phi/fvc::interpolate(rho)*fvc::interpolate(p, "div(U,p)"))
- p*fvc::div(phi/fvc::interpolate(rho))

1. Thermal diffusivity
In laminar conditions the turbulence->alphaEff() term should become k/Cp (where k is the thermal conductivity). Is it correct?

2. Pressure term
I cannot understand why the term associated to the pressure is written as:

div(p*phi/rho)-p*div(phi/rho)

I expected just the first term:

div(p*phi/rho)

3. Temperature gradient
In the enthalpy equation the div(k grad(T)) term is written as laplacian(k/Cp grad(h)) since grad(h)=Cp grad(T) .
However for multicomponent mixtures we have:

grad(h)=Cp grad(T) + sum h_i grad(Y_i)

where i is the species index and the sum is over all the species.Therefore:

grad(T) = 1/Cp ( grad(h) - sum h_i grad(Y_i) )

So, I expected this term in the enthalpy equation:

- fvm::laplacian(k/Cp, h - sum ( h_i grad(Y_i) ) )


Thank you very much.
Alberto

Concerning your second point:

Isn't basically phi=rho*U at the faces?
Then I would understand to be phi/rho = U at the faces.
Then p*div(phi/rho) = p* div(U) which would be the effect of compression...

Regards
Dominik