Details of the Numerical Algorithm

At axial position Zk, the molar density of reactant A is known at all x, and yj in the flow cross section via the starting profile. The objective is to predict Ca(x, yj,Zk+i) at the next axial step, Zk+i- 

Convective transport in the primary flow direction is evalnated as a second-order-correct first derivative at the fictitious point, x, yj, and z +i/2 midway between Zk and Zk+i  

The alternating direction implicit method (i.e., ADI) is employed to calculate transverse diffusion in the x direction via second-order-correct finite differences for a second derivative using unknown molar densities at Zk+i-Hence, 

At each grid point, x, and yj, in the flow cross section, the mass transfer equation (i.e., PDE) is written as a linear algebraic equation in terms of three unknown molar densities at Zk+i In other words. 

---