F Diffusion in a Sphere with Fast Reaction - Singular Perturbation Theory

DIFFUSION IN A SPHERE WITH FAST REACTION - SINGULAR PERTURBATION THEORY  

In the previous sections we have seen several examples of transport problems that are amenable to analysis by the method of regular perturbation theory. As we shall see later in this book, however, most transport problems require the use of singular-perturbation methods. The high-frequency limit of flow in a tube with a periodic pressure oscillation provided one example, which was illustrative of the most common type of singular-perturbation problem involving a boundary layer near the tube wall. Here we consider another example in which there is a boundary-layer structure that we can analyze by using the method of matched asymptotic expansions. 

Although this book is almost exclusively about fluid mechanics and heat transfer processes, the problem considered here is the transport of a reactant into a catalyst pellet within which there is a very fast reaction. This is a very convenient problem to illustrate the principles of the matched asymptotic method, and it also has very important practical applications in the field of chemical reaction engineering. 

In reality, a typical catalyst pellet will be a porous solid that may be quite complicated or even irregular in shape with a large number of catalytic reaction sites distributed throughout. However, to simplify the problem for present purposes, the catalyst pellet will be approximated as being spherical in shape. Furthermore, we will assume that the catalyst pellet is uniform in constitution. Thus we assume that it can be characterized by an effective reaction-rate constant kef that has the same value at every point inside the pellet. In addition, we assume that the transport of reactant within the pellet can be modeled as pure diffusion with a spatially uniform effective diffusivity To Author simplify the problem, we assume that the transport of product out of the pellet is decoupled from the transport of reactant into the pellet. Finally, the concentration of reactant in the bulk-phase fluid (usually 

We have not formally derived a species transport equation. However, in the present case we have assumed that transport is strictly by an effective diffusion process. Hence the concentration will be spherically symmetric and governed by a radial diffusion equation with a reaction rateR  

---

F. Diffusion in a Sphere with Fast Reaction - Singular Perturbation Theory ... 

---