Chemical Reactor with Radial Dispersion - Summary of the Problem

Consider a packed bed reactor: a cylinder filled with catalyst pellets.

The fluid flows past the catalyst pellets, on which reaction takes place. Due to the heat of reaction, the temperature can change; in the case of an exothermic reaction the temperature increases. The energy is removed at the wall. Energy released near the wall is easily removed, but energy released in the center of the cylinder must get from there to the wall, using essentially two pathways. One pathway is heat conduction through the fluid and solid, and another is due to mixing as the flows around adjacent pellets combine downstream. This mechanism is called dispersion. The rate of reaction depends on temperature. If the temperature rise is small, then the temperature is the same at different radial postions in the bed. If no mass is removed at the wall, then there is no mechanism to create concentration variations in the radial direction, provided the concentration is uniform upstream. In that case, a radial dispersion model is not needed. If the temperature rise is large, however, then the temperature will be different at the center and wall of the cylinder, highest at the center for exothermic reactors. In that case, the rate of reaction will also vary in the radial direction, and a radial dispersion model is needed. That model is developed in detail elsewhere.

A typical problem is one for a single reaction; here we assume the system is dilute, or the number of moles does not change under reaction. The nondimensional equations are

There are two basic problems solved in this chapter. Both use a first-order irreversible reaction.

The parameters for the first problem are

The parameters for the second problem are the same except for

The difference between the problems is illustrated by the overall heat transfer coefficient and its dependence on the Biot number.

Rearranged, this gives

When the Biot number is small, then the overall heat transfer coefficient is almost entirely the heat transfer coefficient at thewall. This is the first case, where the ratio is 0.75. When the Biot number is large, then the overall heat transfer coefficient has a significant component due to the packed bed. This is the second case, where the ratio is 0.13. In this case, significant temperatur variations can arise radially if the heat of reaction is large enough, since there is a significant resistance to heat transfer through the packed bed, compared with the resistance at the wall. Thus, by comparing problems one and two, we can see the impact of this important parameter, the Biot number.

The two problems are solved using the orthogonal collocation method and the finite difference method.