Modelling packed bed adsorption

For a project I am trying to make a model that fits with some experimental results of a fixed/packed bed reactor of the adsorption of CO2. My equation is:

e* dC/dt = -v* dC/dz - kc*a*C

I managed to discretize the problem and so far I am able to obtain just the general result of an advection equation (a starting concentration at x=0 that travels through the length of the reactor), however, I am not able to model the adsorption part, where the initial concentration still travels through the reactor, but decreases. However, over time, the concentration at x>0 should increase, since there is new CO2 coming in. Do you know any way to model this? So not just an initial input going through the length, but basically a constant input for the advection equation?