Continuous adsorption columns are complex in that the adsorbent fills with time, and thus the material being adsorbed can break through the bed after a given time. At this time the column need to be replaced. The adsorption columns can be designed using a simplified set of differential equations. Assuming the fluid and solid phase are isothermal, no pressure drops along the column, no chemical reaction occurring in the bed, radial dispersion and axial diffusion are negligible, linear driving force approximation used to describe mass transfer rate between fluid and solid phase, and the equilibrium of adsorption is described by a linear isotherm, the equations are,
$$\frac{\partial C_i}{\partial t}=-\frac{\partial (uC_i)}{\partial z}+\left(\frac{1-\varepsilon}{\varepsilon}\right)\rho_a\frac{\partial q_i}{\partial t}$$
$$\frac{\partial q_i}{\partial t}=\frac{3k}{r_p}(C_i-C^*_i)$$
$$q^*_i=K_DC^*_i$$
where $\rho_a$ is the adsorbent density and $\varepsilon$ is the bed porosity.
The graph below shows how the key variables change the concentration in the vapour phase in the adsorption column (left) and the concentration breaking though the column with time (right).