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Old   September 19, 2018, 07:24
Default Energy equation for multi-component systems
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Andrea Ferrari
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Dear All,

i have a question regarding the energy equation in chtMultiRegionFoam or, in general, the energy equation for multi-component systems. The fluid i am working with is moist air, which is a binary mixture of dry air (a) and water vapor (v). The total heat flux for this multi-components system should be the sum of conductive heat flux (given by Fourier's law) and the heat flux resulting from mass diffusion (see for example [1], Chapter 19):

q_{tot} =q + \sum_{i=1} h_i g_{d,i} 


q = -\lambda\nabla T 

h_i = c_{p,i}(T-T_{ref}) --> enthalpy of component "i"

g_{d,i} = -\rho D_i \nabla Y_i

Y_i = \frac{\rho_i}{\rho}--> mass fraction of component "i
\rho = \rho_a + \rho_v--> density of the mixture
i = a,v
However, if we look at how the energy equation is implemented in (for example) chtMultiRegionFoam, it seems that the contribution of mass diffusion to the heat flux is not there (i am copying from OF 6):

 fvScalarMatrix EEqn
   fvm::ddt(rho, he) + fvm::div(phi, he)
+ fvc::ddt(rho, K) + fvc::div(phi, K)
+ (  == "e"  

      ? fvc::div                                   

              fvc::absolute(phi/fvc::interpolate(rho), U),                                             p,
        : -dpdt
- fvm::laplacian(turbulence.alphaEff(), he)
+ rad.Sh(thermo, he)
+ Qdot
+ fvOptions(rho, he)
I guess the term:

- fvm::laplacian(turbulence.alphaEff(), he)

only accounts for conductive heat flux. So, where is the mass diffusion contribution?



[1]Bird, R. Byron. "Transport phenomena." Applied Mechanics Reviews 55.1 (2002): R1-R4.
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Old   September 20, 2018, 09:24
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Andrea Ferrari
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I just wanted to add that mass diffusivity is correctly accounted for in the specie equation:

fvScalarMatrix YiEqn
      + mvConvection->fvmDiv(phi, Yi)
      - fvm::laplacian(turbulence.muEff(), Yi) 
     + fvOptions(rho, Yi)
Here OF uses the effective viscosity instead of mass diffusivity, which I think is ok for turbulent flows. If I understand correctly, the diffusive mass flux for turbulent flows should be written as:

g_{d,i} = ( \rho D_i + \frac{\mu_t}{Sc_t})\nabla Y_i

In OF this becomes

g_{d,i} = \left (\mu+ \mu_t\right)\nabla Y_i

under the assumption that \rho D_i = \mu and Sc_t=1

It seems to me that this term is then neglected in the Energy equation. The approximation (if it is neglected) should be ok for Lewis number close to 1, where Lewis number for component i is defined as:

Le_i = \frac{\kappa}{\rho c_{p}D_i}

where k is the thermal conductivity. Note that Le=1/Pr following OF implementation.

Are these ideas correct? Is the the heat flux due to mass diffusion neglected in energy equation? Or am i missing something?


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Old   September 24, 2018, 21:19
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Gao Zhengwei
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Hi Andrea,

I have read this part of the code and I think what you said is right. In OF, the unity Sc number assumption is used for specie equation since the differential diffusion model is not available.

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