Temperature Effects in Ideal Reactors

Consider the heat transfer occurring at the wall along a small length of external area, dA = 2nRdz, as indicated in the figure. 

Solution of the nonisothermal problem then consists of utilizing equations (4-129) and (4-130) to obtain the relationship among x, T and z along the length of the reactor. 

As an alternative to the procedure above, where the energy balance was written explicitly in terms of the conversion, we may derive the balance in differential terms. 

In commercial operation of fixed-bed reactors in which the catalyst is deactivating, it is often necessary to maintain constant conversion by increasing the temperature of the reactor to compensate for catalyst decay. One can think of this as a constant activity policy in which the bed may be considered isothermal at any given time with the overall temperature level increasing with time of operation. 

Consider the case in which the deactivation rate is taken to be concentration-independent (a common industrial model), i.e., 

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