Thermal Gradients Inside Catalyst Pellets

The final complication that must be introduced into the discussion is the fact that thermal conductivity limitations may cause temperature gradients in addition to concentration gradients within the pellet. To analyze these, the combined 

a-4 can be used to eliminate either C, or T, from one of the differential equations with the result that in general, only one (nonlinear) with one dependent variable must be solved. 

The maximum temperature difference in a particle (without further complications of external mass and heat transfer resistances) is for complete reaction, C, = 0. as pointed out by Prater [111]  

This result is actually true for any particle geometry, under steady-state conditions. 

The steady-state solution win then only be a function of the modulus evaluated at the surface conditions, and also P and y. A full set of computations was performed by Weisz and Hicks [112], and Fig. 3.7.a-l shows some results for a spherical pellet with y = 20 they also presented graphs for other values of y. 

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When a fast reaction is highly exothermic or endothermic and, additionally, the effective thermal conductivity of the catalyst is poor, then significant temperature gradients across the pellet are likely to occur. In this case the mass balance (eq 32) and the enthalpy balance (eq 33) must be simultaneously solved using the corresponding boundary conditions (eqs 34-37), to obtain the concentration profile of the reactant and the temperature profile inside the catalyst pellet. The exponential dependence of the reaction rate on the temperature thereby imposes a nonlinear character on the differential equations which rules out an exact analytical treatment. Approximate analytical solutions [83, 99] as well as numerical solutions [13, 100, 110] of eqs 32-37 have been reported by various authors. 

Catalyst supports such as silica and alumina have low thermal conductivities so that temperature gradients within catalyst particles are likely in all but the finely ground powders used for infrinsic kinetic studies. There may also be a film resisfance fo heaf fransfer af fhe exfemal surface of the catalyst. Thus the internal temperatures in a catalyst pellet may be substantially different than the bulk gas temperature. The definition of the effectiveness factor, Equation 10.23, is unchanged, but an exothermic reaction can have reaction rates inside the pellet that are higher than would be predicted using the bulk gas temperature. In the absence of a diffusion limitation, rj > 1 would be expected for an exothermic reaction. (The case > 1 is also possible for some isothermal reactions with weird kinetics.) Mass transfer limitations may have a larger... 

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