The Batch Adsorber Problem

The approach we shall take is to teach by example, as we did in the finite Sturm-Liouville integral transform. We start by using the batch adsorber problem to illustrate the procedure. A comprehensive account of the method can be found in the book by Ramkrishna and Amundson (1987). 

To apply Sturm-Liouville transforms to coupled PDEs, we shall need to introduce a new concept function space. We introduce this methodology by way of a practical example batch adsorption in a vessel (Fig. 11.7). 

Let us start with the derivation of the mass balance equations for a batch adsorber containing spherical adsorbent particles. The solute molecules diffuse from the bulk reservoir into the particles through a network of pores within the particle. The diffusion process is generally described by a Fickian type equation 

During the diffusion of solute molecules through the network of pores, some of the solute molecules are adsorbed onto the interior surface of the particle. This process of adsorption is normally very fast relative to the diffusion process, and so we can model it as local equilibrium between the solute in the pore fluid and the solute bound to the interior surface of the particle. This partitioning is often referred to as the adsorption isotherm, which implies constant temperature conditions. When the solute concentration in the pore is low enough, the relationship becomes linear (Henry s law) hence, mathematically, we can write 

We are now in a position to derive the mass balance equation inside the particle. Carrying out the mass balance equation over a small element of thickness Ar at the position r (Fig. 11.8), we obtain for an element of spherical particle 

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The batch adsorber problem will yield a finite steady-state concentration for both the individual particle and also within the well-mixed reservoir. This steady-state concentration can be readily obtained from Eqs. 11.174 as follows. 

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