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8.9.1 Fickian Diffusion

Mass diffusion coefficients are required whenever you are solving species transport equations in multi-component flows. Mass diffusion coefficients are used to compute the diffusion flux of a chemical species in a laminar flow using (by default) Fick's law:

 J_i = - \rho D_{i,m} \nabla Y_i - D_{T,i}\frac{\nabla T}{T} (8.9-1)

where $D_{i,m}$ is the mass diffusion coefficient for species $i$ in the mixture and $D_{T,i}$ is the thermal (Soret) diffusion coefficient.

Equation  8.9-1 is strictly valid when the mixture composition is not changing, or when $D_{i,m}$ is independent of composition. This is an acceptable approximation in dilute mixtures when $Y_i << 1$, for all $i$ except the carrier gas. FLUENT can also compute the transport of non-dilute mixtures in laminar flows by treating such mixtures as multicomponent systems. Within FLUENT, $D_{i,m}$ can be specified in a variety of ways, including by specifying ${\cal D}_{ij}$, the binary mass diffusion coefficient of component $i$ in component $j$. ${\cal D}_{ij}$ is not used directly, however; instead, the diffusion coefficient in the mixture, $D_{i,m}$, is computed as

 D_{i,m} = \frac{1 - X_i}{\displaystyle\sum_{j,j \neq i} (X_j/{\cal D}_{ij})} (8.9-2)

where $X_i$ is the mole fraction of species $i$. You can input $D_{i,m}$ or ${\cal D}_{ij}$ for each chemical species, as described in Section  8.9.4.

In turbulent flows, Equation  8.9-1 is replaced with the following form:

 J_i = -(\rho D_{i,m} + \frac{\mu_t}{{\rm Sc}_t}) \; \nabla Y_i - D_{T,i}\frac{\nabla T}{T} (8.9-3)

where ${\rm Sc}_t$ is the effective Schmidt number for the turbulent flow:

 {\rm Sc}_t = \frac{\mu_t}{\rho D_t} (8.9-4)

and $D_t$ is the effective mass diffusion coefficient due to turbulence.

In turbulent flows your mass diffusion coefficient inputs consist of defining the molecular contribution to diffusion $D_{i,m}$ using the same methods available for the laminar case, with the added option to alter the default settings for the turbulent Schmidt number. As seen from Equation  8.9-4, this parameter relates the effective mass diffusion coefficient due to turbulence with the eddy viscosity ${\mu_t}$. As discussed in Section  8.9.5, the turbulent diffusion coefficient normally overwhelms the laminar diffusion coefficient, so the default constant value for the laminar diffusion coefficient is usually acceptable.

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