Setting Up Partial Differential Equations

The mass balances considered up to this point are confined to cases involving a single independent variable, time or distance. Suppose, now, that diffusion takes place from an external medium into a porous cylindrical or spherical particle, which is initially devoid of diffusing spears. Concentrations will then vary both with time and radial distance and in the case of the cylindrical particle, with axial distance as well. This is a system that is distributed in both time and distance and that consequently leads to a PDE (see Table 2.1). 

Let us assume that the variations are with respect to one distance variable x and to time t. We previously considered, at the ODE level, distributions in time only, or in distance only. This led to the schemes represented by Equation 2.1 and Equation 2.2. To deal with simultaneous variations in both time and distance, we superpose the two expressions — that is, we write 

On occasion, mass will enter at r + Ar and leave at r, in which case the scheme is adjusted accordingly. 

This is the formulation that has to be used when the system is distributed over distance r and time t. When variations occur in more than one direction, we merely add appropriate terms to the left side of Equation 2.17, for example, Rate of mass in at y and over Ay and so on. Note that the time derivative in Equation 2.17 is now a partial derivative, because we are dealing with more than one independent variable. The following illustration provides an example of the application of Equation 2.17. 

Diffusion into a slab (a) development of concentration profiles and (b) difference element for the mass balance. 

Illustration 2.6 Unsteady Diffusion in One Direction Pick s Equation 

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The "rate" theory takes into account the physical processes such as convection and diffusion, and sets up partial differential equations to describe them. The solution of these equations (in the adsorbate concentrations and their derivatives, with time and axial distance along the column as independent variables) is then essayed. 

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