Simplified Forms of the General Continuity Equation

As already mentioned, the form of the fundamental continuity equations is usually too complex to be conveniently solved for practical application to reactor design. If one or more terms are dropped from Eq. 7.2.a-6 and or integral averages over the spatial directions are considered, the continuity equation for each component reduces to that of an ideal, basic reactor type, as outlined in the introduction. In these cases, it is often easier to apply Eq. 7.1.a-l directly to a volume element of the reactor. This will be done in the next chapters, dealing with basic or specific reactor types. In the present chapter, however, it will be shown how the simplified equations can be obtained from the fundamental ones. 

It is very common in reactors to have flow predominantly in one direction, say z (e.g., think of tubular reactors). The major gradients then occur in that direction, under isothermal conditions at least. For many cases then, the cross-sectional average values of concentration (or conversion) and temperature might be used in the equations instead of radial point values. The former are obtained from  

In this case, the approximation would clearly be best for highly turbulent flow, for which the velocity profiles are relatively flat. The discrepancies actually enter into the effective transport coefficients, which have to be empirically measured in any event. Another approximation concerns the reaction rate term  

a-6 becomes after integration over the cross section  

This last equation is simply integrated to give  

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