Test results for the MSD simulation

[mahanr]

These are some of the test results for the mixture averaged model, having the following governing equation and other necessary formulation

$$ \partial_t\rho_fY_i + \boldsymbol{\nabla}\cdot\boldsymbol{u}_f\rho_fY_i = - \boldsymbol{\nabla}\cdot\left(\rho_f Y_i \boldsymbol{V}_i\right) + S_i $$

The mixture averaged model is implemented in the following manner

$$ \mathcal{D}_i = \frac{1 - Yi}{\sum{j\neq i} \frac{Xj}{\mathcal{D}{ji}}} $$

with the following diffusion drift velocities

$$ \tilde{V}_i = -(\mathcal{D}_i/X_i)\nabla X_i $$

$$ Vc = -\sum{i=1}^{N_g}Y_i\tilde{V}_i $$

Then one may write

$$ V_i =\tilde{V}_i+V_c $$

[mahanr]

The following shows the initial distribution of the different species in the domain

The simulation is isothermal with $T=300K$, Following are the properties of the species

Species	Density ( $kg/m^3$ )	Thermal conductivity ( $W/m^2 K$ )	Specific heat capacity ( $J/Kg-K$ )	Kinematic viscosity ( $m^2/s$ )	Mass diffusivity ( $m^2/s$ )	Molar mass ( $Kg$ )
CH4	12.219165	0.13036146	3989.25671144	1.94675e-6	2.8774413e-06	16.04276e-3
H2	1.5354197	0.36361837	14691.2618807	1.10936e-5	9.5551887e-06	2.01588e-3
Ar	1.748	0.01772	535.329156824	1.201372e-5	4.3671432e-06	38.948e-3

Though it should be noted that all the mass diffusivity values are read from CHEMPROP

[mahanr]

Results Evolution of the mole fraction of CH4

[mahanr]

The results above show substantial differences in the evolution of CH4 as compared to Nilesh's article. Moreover, Nilesh's article considered acoustic scaling of the time $t_{ND} = t/t_s$, with $t_s = 1/\sqrt{\gamma R T}$. Using such a scaling shows that the evolution of the mole fraction of CH4 in the actual time scale is way less diffused in the above simulation than that of Nilesh's article.

[mahanr]

Evolution of the mole fraction of Ar

[mahanr]

The binary diffusivity tensor

As per CHEMPROP at $T=300K$ and $P=1e5 Pa$

- diffusivity CH4 in H2 2.09529e-05
- diffusivity CH4 in Ar 1.58397e-05
- diffusivity H2  in CH4 2.09529e-05
- diffusivity H2  in Ar 1.57869e-05
- diffusivity Ar  in CH4 1.58397e-05
- diffusivity Ar  in H2 1.57869e-05

As per the article

Ar and H2 is 8.14543e-5
Ar and CH4 is 2.17321e-5
CH4 and H2 is 7.37433e-5

[mahanr]

In this equation

$$ \partial_t\rho_fY_i + \boldsymbol{\nabla}\cdot\boldsymbol{u}_f\rho_fY_i = - \boldsymbol{\nabla}\cdot\left(\rho_f Y_i \boldsymbol{V}_i\right) + S_i $$

I have considered the mixture density $\rho_f$ as a constant.

I guess if one uses the following transformation from $Y_i$ to $X_i$,

$$ Y_i = \frac{m_i}{\sum_i m_i} = \frac{n_i M_i}{\sum_i m_i} = \frac{M_i \sum_i n_i}{\sum_i m_i} X_i $$

Here, the prefactor appearing before $X_i$ can not be assumed as a constant as the total number of moles in the domain is not conserved.

[mahanr]

The following results are the mole fraction of CH4 at different $t$, using Nilesh's parameters

The qualitative agreements are much better as states are more diffused here.

[mahanr]

The following results are the mole fraction of Ar at different $t$, using Nilesh's parameters