3.2.4.3. Conservation of Chemical Species

For a material with species $k$ with molar concentration $C_k$ , molar flux $\vector{J}_k$ relative to the mass average velocity and volumetric reaction rate $R_{V,k}$ , letting $b = y_k$ , $\vector{f} = \vector{J}_k$ and $B_\Vol = R_{V,k}$ in equation (3.1) results in the conservation equation for species $k$ ,

(3.15) $\pt{C_k} + \v\bcdot\grad C_k = - \div\vector{J}_k + R_{V,k}.$

For liquid mixtures which are dilute in all species except one, Fick’s law is often used to approximate $\vector{J}_k$ . In this approximation, $\speciesdiffusivity_k$ represents the diffusion coefficient of species $k$ with respect to the concentrated species and it is assumed that the interactions between dilute species is assumed negligible. Again, however, we choose to leave the governing equation in the more general form and require the particular diffusive flux model as user input (see Species Diffusion). Using equation (3.3), the G/FEM residual form is

(3.16) $\symRes_{C_k}^i = \int\limits_\Vol\left[\left(\pt{C_k} + \v\bcdot\grad C_k - R_{V,k}\right)\phi^i - \grad\phi^i\bcdot\vector{J}_k\right]\dV + \int\limits_\Surf q_{n,k}\phi^i\dS = 0$

where $q_{n,k}$ is the mass flux at the boundary. For example, the natural convection boundary condition gives $q_n = k(C_k-C_{\infty,k})$ where $k$ is the mass transfer coefficient and $C_{\infty,k}$ is the bulk concentration away from the surface.

In Aria, each term in (3.16) is specified separately as identified in equation (3.17).

(3.17) $\symRes_{C_k}^i & = \underbrace{\int\limits_\Vol \pt{C_k}\phi^i\dV}_\mathrm{MASS} + \underbrace{\int\limits_\Vol \v\bcdot\grad C_k\phi^i\dV}_\mathrm{ADV} + \underbrace{\int\limits_\Vol C \div \v \phi^i\dV}_\mathrm{DIV} \\ & - \underbrace{\int\limits_\Vol R_{V,k} \phi^i\dV}_\mathrm{SRC} - \underbrace{\int\limits_\Vol \grad\phi^i\bcdot\vector{J}_k\dV}_\mathrm{DIFF} + \int\limits_\Surf q_{n,k}\phi^i\dS = 0$

For a purely incompressible form, we know that $\div\v\equiv 0$ and can remove terms. Equation (3.17) then simplifies to

(3.18) $\symRes_{C_k}^i & = \underbrace{\int\limits_\Vol \pt{C_k}\phi^i\dV}_\mathrm{MASS} + \underbrace{\int\limits_\Vol \v\bcdot\grad C_k\phi^i\dV}_\mathrm{ADV} - \underbrace{\int\limits_\Vol R_{V,k} \phi^i\dV}_\mathrm{SRC} \\ & - \underbrace{\int\limits_\Vol \grad\phi^i\bcdot\vector{J}_k\dV}_\mathrm{DIFF} + \int\limits_\Surf q_{n,k}\phi^i\dS = 0$

Often times it is useful to solve for mass, weight or volume fractions of each species rather than for the concentration directly. In that case, an additional condition exists,

(3.19) $\sum_k C_k = 1$

Using this condition, it is only necessary to solve for $N-1$ species fractions where $N$ is the total number of species present in the problem. The final species, then, is simply given as

(3.20) $C_j = 1 - \sum_{k\ne j} C_k$

This method can be triggered in Aria by specifying the equation term FRACBAL. In this case, the equation for $C_j$ is not included in the system of unknowns but is instead post-processed on the fly. Aria will automatically detect all other species equations and include them in the fraction balance.