8.9.2 Full Multicomponent Diffusion

A careful treatment of chemical species diffusion in the species transport and energy equations is important when details of the molecular transport processes are significant (e.g., in diffusion-dominated laminar flows). As one of the laminar-flow diffusion models, ANSYS FLUENT has the ability to model full multicomponent species transport.

General Theory

For multicomponent systems it is not possible, in general, to derive relations for the diffusion fluxes containing the gradient of only one component (as described in Section  8.9.1). Here, the Maxwell-Stefan equations will be used to obtain the diffusive mass flux. This will lead to the definition of generalized Fick's law diffusion coefficients [ 85]. This method is preferred over computing the multicomponent diffusion coefficients since their evaluation requires the computation of $N^2$ co-factor determinants of size $(N-1)\times(N-1)$, and one determinant of size $N \times N$ [ 79], where $N$ is the number of chemical species.

Maxwell-Stefan Equations

From Merk [ 51], the Maxwell-Stefan equations can be written as

where $X$ is the mole fraction, $\vec{V}$ is the diffusion velocity, ${\cal D}_{ij}$ is the binary mass diffusion coefficient, and $D_T$ is the thermal diffusion coefficient.

For an ideal gas the Maxwell diffusion coefficients are equal to the binary diffusion coefficients. If the external force is assumed to be the same on all species and that pressure diffusion is negligible, then $\vec{d}_i = \nabla X_i$. Since the diffusive mass flux vector is $\vec{J}_i = \rho_i \vec{V}_i$, the above equation can be written as

After some mathematical manipulations, the diffusive mass flux vector, $\vec{J}_i$, can be obtained from

where $Y_j$ is the mass fraction of species $j$. Other terms are defined as follows:

$$D_{ij} = \left[ D \right] = \left[A\right]^{-1} \left[B\right]$$ (8.9-8)

$$A_{ii} = - \left(\frac{X_i}{{\cal D}_{iN}} \frac{M_w}{M_{w,N}} +\sum^{N}_{\stackrel{j=1}{j \neq i}} \frac{X_j}{{\cal D}_{ij}} \frac{M_w}{M_{w,i}} \right)$$ (8.9-9)

$$A_{ij} = X_i \left(\frac{1}{{\cal D}_{ij}} \frac{M_w}{M_{w,j}} - \frac{1}{{\cal D}_{iN}} \frac{M_w}{M_{w,N}} \right)$$ (8.9-10)

$$B_{ii} = - \left(X_i \frac{M_w}{M_{w,N}} + \left(1 - X_i \right) \frac{M_w}{M_{w,i}} \right)$$ (8.9-11)

$$B_{ij} = X_i \left(\frac{M_w}{M_{w,j}} - \frac{M_w}{M_{w,N}} \right)$$ (8.9-12)

where $\left[A\right]$ and $\left[B\right]$ are $(N-1)\times(N-1)$ matrices and $\left[D\right]$ is an $(N-1)\times(N-1)$ matrix of the generalized Fick's law diffusion coefficients $D_{ij}$ [ 85].