The governing equation for a Diffusion-Reaction equation is given by:
\begin{equation}
\frac{\partial C_i}{\partial t} + \nabla \cdot N_i = R_i,
\end{equation}
where:
$C_i$ is the concentration of species (i) (in moles per unit volume),
$N_i$ is the flux of species (i) (in moles per unit area per unit time),
$R_i$ is the reaction rate of species (i) (in moles per unit volume per unit time).
$N_i = D*\nabla{C}$ by Fick's law\
\begin{equation} \frac{\partial C_i}{\partial t} + \nabla \cdot ({D*\nabla{C}}) = R_i, \end{equation}
Now imagine this to be happening in a porous medium, I was struggling to derive the equations through volume-avergaing:
For instance, the volume-averaged concentration of a species (say, $\bar{C}$) is related to concentration of the species in the pore channel (say, C):
\begin{equation} \bar{C_i}= {\epsilon*C_i}, \end{equation}
$\epsilon$ is the porosity fraction
I understand that the first term changes to $\frac{\partial (\epsilon*\bar{C_i})}{\partial t}$
I am unsure how the other two terms change, should I just replace C by $\bar{C}$ in the flux term ?
