Species Transport

The transport equation for the mass fraction $Y_i$ of a species is given by:

where:

• $t$ is time
• $V$ is volume
• $a$ is the area vector
• $i$ is the component index.
• $\rho$ is the overall density.
• $\mathbf{\text{v}}$ is the velocity.
• $M_i$ is the migration term that is included when modeling the movement of charged species in an electric field.
• ${\mu }_t$ is the turbulent dynamic viscosity.
• ${\sigma }_t$ is the turbulent Schmidt number
• $S_{Y_i}$ is a user specified region source term, or a source term resulting from reactions, for species $i$ .
• $J_i$ is the laminar (or molecular) diffusive flux.
• $C_i$ is the added transport term when modeling concentrated charge migration, to consider the lithium cation migration in the fluid.

Turbulent diffusion is accounted for by the term $\frac{{\mu }_t}{{\sigma }_t}$ . Laminar diffusion is defined by $J_i$ . If species transport is part of a two-way coupled Lagrangian Multiphase or Dispersed Multiphase, or a porous media simulation using the physical velocity approach, Eqn. (1871) is scaled to account for a reduction in available volume through the presence of a dispersed phase or a solid porous phase. For more information on how these phases affect the transport equation, see Volume Partitioning.

Fick's Law

When multi-component diffusion is deactivated, Fick’s Law is used to calculate the diffusive flux $J_i$ , given by:

$$J_i=\rho D_{i,m}\nabla Y_i$$

(1872)

where $D_{i,m}$ is the molecular diffusivity of component $i$ in the mixture.

Multi-Component Diffusion

When multi-component diffusion is activated, the scalar diffusivity of Fick’s Law is replaced by a matrix, and the diffusive flux for component $i$ becomes a function of the mass fractions for all components:

$$J_i=\rho {\sum }_{j=1}^N{D_{i,j}\nabla Y}_j$$

(1873)

$D_{i,j}$ represents the multi-component diffusion coefficients which are calculated using the Maxwell-Stefan equations. These equations define the diffusive flux implicitly as a function of mole fraction gradients, which are given as:

$$\nabla X_i=\frac{M_w}{\rho }{\sum }_{j=1,j\ne i}^N\frac{{X_{i}J}_j}{M_j{\text{D}}_{i,j}}-\frac{{X_{j}J}_i}{M_i{\text{D}}_{i,j}}$$

(1874)

${\mathfrak{\text{D}}}_{i,j}$ represents the binary diffusion coefficient between components $i$ and $j$ . Writing these equations in matrix form gives:

$$\left[B\right]\nabla Y=\left[A\right]J$$

(1875)

where $\left[B\right]$ represents the mapping from mass fraction gradients to mole fraction gradients and $\left[A\right]$ represents the Maxwell-Stefan equations. By inverting $\left[A\right]$ and multiplying by the matrix $\left[B\right]$ , the multi-component diffusion coefficients are calculated:

$$D_{i,j}=\rho {\sum }_{k=1}^N{{A_{i,k}}^{-1}B}_{k,j}$$

(1876)

Soret Effect

When the Soret Effect option is activated, an additional term is added to the diffusive flux $J_i$ for each component.

The Soret flux $S$ is given as:

$$S=\rho \frac{D_{i,t}}{T}\nabla T$$

(1877)

In the case that multi-component diffusion is deactivated, the expression for the diffusive flux becomes [36]:

$$J_i=\rho D_{i,m}\nabla Y_i+S$$

(1878)

and when multi-component diffusion is activated, the expression becomes [35]:

$$J_i=\rho ({\displaystyle \sum _{j=1}^N{D}_{i,j}}\nabla Y_j)+S$$

(1879)

where $D_{i,t}$ is the thermal diffusion coefficient for component $i$ .

Thermal Diffusion

For the Warnatz Model, the thermal diffusion coefficient $D_{i,t}$ in Eqn. (1878) or Eqn. (1879) is computed based on the thermal diffusion ratio. The thermal diffusion ratio between component $i$ and component $j$ is given as [37]:

$$K_{t,ij}=\frac{15}{2}\frac{(2A_{i,j}^{*}+5)(6C_{i,j}^{*}-5)}{A_{i,j}^{*}(16A_{i,j}^{*}-12B_{i,j}^{*}+55)}\frac{M_i-M_j}{M_i+M_j}{X}_i{X}_j$$

(1880)

and:

$$\begin{array}{c}{A}_{i,j}^{*}=\frac{1}{2}\frac{{\mathrm{\Omega }}_{i,j}^{(2,2)}}{{\mathrm{\Omega }}_{i,j}^{(1,1)}}\\ B_{i,j}^{*}=\frac{1}{3}\frac{5{\mathrm{\Omega }}_{i,j}^{(1,2)}-{\mathrm{\Omega }}_{i,j}^{(1,3)}}{{\mathrm{\Omega }}_{i,j}^{(1,1)}}\\ C_{i,j}^{*}=\frac{1}{3}\frac{{\mathrm{\Omega }}_{i,j}^{(1,2)}}{{\mathrm{\Omega }}_{i,j}^{(1,1)}}\end{array}$$

(1881)

where:

• ${\mathrm{\Omega }}_{i,j}$ represents collision integrals between component $i$ and component $j$ .
• $M_i$ is the molecular weight of component $i$

Using the thermal diffusion ratio, the thermal diffusion coefficient for component $i$ is given as:

$$D_{i,t}=D_{i,m}\frac{M_i}{M_w}{\displaystyle \sum _{j=1}^N{K}_{t,ij}}$$

(1882)

where $D_{i,m}$ is the effective molecular diffusivity of component $i$ into the mixture and $M_w$ is the mean molecular weight of the mixture.

For N components, Simcenter STAR-CCM+ solves transport equations for all species and ensures that all mass fractions sum to 1.

Charged Species Effects

Concentrated Electrolyte

The diffusion coefficient $D_{i,m}$ is extended to account for the concentrated solution correction to the ion/salt diffusion ([834]), which is defined as:

• $c_0$ is the solvent concentration.
• $c_i$ is the salt concentration.

Species Transport in Porous Media

In porous media, the diffusive flux is modified by the porosity $\chi$ and tortuosity $\tau$ of the media.

For Fick's Law:

For multi-component diffusion:

For the Soret effect: