The dispersion model is often used to analyze the performance of non-ideal plug-flow reactors. In essence, the plug-flow reactor model is augmented by the addition of a diffusion-like term involving the second derivative of concentration or mole fraction as a function of axial position and a dispersion constant. Consider such a case without chemical reaction. The differential equation is:

\[k_{D} \frac{\partial^{2} c_{a}}{\partial z^{2}}-v_{m} \frac{\partial c_{a}}{\partial z}=\frac{\partial c_{a}}{\partial t}\]

a. Why do we not have an explicit expression for the changes in \(c_{a}\) as a function of radial position?

b. Why do we not need to consider \(v=f(r)\) ?

c. Before entering the vessel, we have a vat containing a uniform concentration of reactant, \(c_{a o}\). At the entrance, dispersion and convection occur. At the reactor exit, we assume a perfectly mixed solution with no concentration gradients. Initially, the reactor contains no \(a\). Show, based on flux balances, that the boundary conditions are:

\[z=0 \quad c_{a o}=-\frac{k_{D}}{v_{m}} \frac{\partial c_{a}}{\partial z}+c_{a}(z=0) \quad z=L \quad \frac{\partial c_{a}}{\partial z}=0\]

These are the famous Danckwerts boundary conditions.

d. Solve the equation in the steady state using the Danckwerts boundary conditions and compare with a solution specifying that \(c_{a}=c_{a o}\) at \(z=0\).