Gradients normal to the flow

For these cases, the conservation statement is made around the outside of the catalyst. In steady-state, everything that is consumed or produced inside the catalyst must go through the outside boundary layer of the fluid surrounding the catalyst. In case of serious selectivity problems with a desired and reactive intermediate, the criterion should be calculated for that component. 

The concentration gradient normal to the outside of the catalyst particle. The rate is expressed on catalyst-filled reactor volume, with e void fraction for this smaller volume the rate must be higher to keep Vrr=Vcr . This is calculated from the continuity requirement that was mentioned above  

Reaction rate equals the mass transfer rate  

To estimate the average gradient, the concentration difference should be divided by the unknown boundary layer depth 5. While this is unknown, the Carberry number (Ca) gives a direct estimate of what concentration fraction drives the transfer rate. The concentration difference tells the concentration at which the reaction is really running. 

Temperature gradient normal to flow. In exothermic reactions, the heat generation rate is q=(-AHr)r. This must be removed to maintain steady-state. For endothermic reactions this much heat must be added. Here the equations deal with exothermic reactions as examples. A criterion can be derived for the temperature difference needed for heat transfer from the catalyst particles to the reacting, flowing fluid. For this, inside heat balance can be measured (Berty 1974) directly, with Pt resistance thermometers. Since this is expensive and complicated, here again the heat generation rate is calculated from the rate of reaction that is derived from the outside material balance, and multiplied by the heat of reaction. 

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Although no real fluid is inviscid, in some instances the fluid may be treated as such, and it is worthwhile to present some of the equations which apply in these circumstances. For example, in the flat-plate problem discussed above, the flow at a sufficiently large distance from the plate will behave as a nonvis-cous flow system. The reason for this behavior is that the velocity gradients normal to the flow direction are very small, and hence the viscous-shear forces are small. 

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