Steady State DAC Equations

2D Diffusion Equation

$$\frac{\partial Ca}{\partial t}+u_d\frac{\partial Ca}{\partial x}=D_{AB}\left(\frac{{\partial }^{2}Ca}{\partial y^2}\right)$$

Discretize Time with Forward Differerence and Space with Central Difference

$$\frac{C{a}_{i,j}^{(n+1)}-C{a}_{i,j}^{(n)}}{\Delta t}+u_d\left[\frac{C{a}_{i+1,j}^{(n)}-C{a}_{i-1,j}^{(n)}}{2\Delta x}\right]=D_{AB}\left[\frac{C{a}_{i,j+1}^{(n)}-2C{a}_{i,j}^{(n)}+C{a}_{i,j-1}^{(n)}}{(\Delta y{)}^2}\right]$$

Where

• $n\to$ time
• $i\to$ x
• $j\to$ y

Rearranging

$$C{a}_{i,j}^{(n+1)}=C{a}_{i,j}^{(n)}-\Delta t\left(u_d\left[\frac{C{a}_{i+1,j}^{(n)}-C{a}_{i-1,j}^{(n)}}{2\Delta x}\right]-D_{AB}\left[\frac{C{a}_{i,j+1}^{(n)}-2C{a}_{i,j}^{(n)}+C{a}_{i,j-1}^{(n)}}{(\Delta y{)}^2}\right]\right)$$

Source