The need for more efficient heterogeneous reaction units without the usual mechanical agitation and thus erosion is answered by using fluidized beds. In a fluidized bed the agitation and mixing is supplied by the moving fluid which either mix with the particle phase in place, or carry the particles throughout the reactor. The result is a good transfer, heat and mass, operation.

A good example of fluidized beds is circulating fluidized bed or simply CFB, in which the fluid and particles enter from the bottom of the bed and exit from the top. Then the particles, which are catalysts, are collected in different units and re-enter the bed with fresh reactant, which is in the fluid phase. The general pattern in this kind of bed is a core-annular flow. The particles are more at the walls of the bed flowing down forming an annulus, and the gas with less particles are more at the center flowing upward to form the core. This kind of flow was not known to industry till a decade ago when an experiment showed the empty core. Two major drawbacks are backmixing and low contact. The design of inlet and outlet and also gas distributors may improve this pattern.

Design and scale up of these beds are not easy due to the complicated and yet not fully known multiphase flow and also transfer mechanisms. After several commercial applications of these units in oil and so many other industries, the design is still dependent on expensive pilot plant tests. Even though, in some cases the actual unit behaves different from the pilot.

A number of different correlations obtained from experimental studies, are available for these units although as is the case for these equations they are applicable only to certain geometry and ranges. The data for commercial units are not available and so makes it more difficult to verify different theories and correlations.

As in a fluidized bed the two or more phases that are involved in the flow can be all treated as fluid phases, one successful approach is to solve the modified Navier-Stokes' equations. The modification is in two major parts: 1- The interchange of flow properties between phases; and 2- The introduction of a new variable, volume fraction, which is basically the fraction of each phase at a given time and space. A unavoidable fact is that now it is necessary to introduce particulate phase rheological property, viscosity which leads to kinetic theory for this phase.

The solution to these two fluid model is not available analytically and the only way to solve all these non-linear coupled equations is by numerical methods. The numerical method is of a great importance as an improper solution method may give completely wrong answer even for a well proposed physical model.