Long, Cylindrically Shaped Catalysts

The homogeneous diffusion model is slightly more complex in cyUndrical coordinates relative to the model described above in rectangular coordinates. Additional complexity arises because the radial term of the Laplacian operator (V V = V ) accounts for the fact that the surface area across which radial diffusion occurs increases linearly with dimensionless coordinate r/ as one moves radially outward. Basic information for = f(t]) is obtained by integrating the dimensionless mass balance with radial diffusion and chemical reaction  

The first integration step produces the following result for the concentratiou gradient  

The second integration generates the molar density profile  

2 Critical Dimensionless Radii and Intrapellet Damkohler Numbers 

These critical values are obtained directly from the reactant molar density profile by implementing the condition that = 0 at = / critical- Hence, the critical dimensionless spatial coordinate is calculated by solving for the appropriate root of the following nonlinear algebraic equation  

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If equation (16-16) is evaluated when ( critical = 0, then, by definition, one finds that Acnticai = 2, which agrees with the first entry in Table 16-1 for long cylindrically shaped catalysts. L Hopital s rule confirms that... 

Without any prove it is stated here that the geometry factor T falls between the two extremes of 2/3 for the infinitely long slab and of 6/s for the sphere for almost all practical cases. Thus T is almost always close to unity. This holds for any catalyst geometry, hence also for catalyst geometry s commonly found in industry, for example ring-shaped or cylindrical catalyst pellets. For this type of pellet it can be shown (Appendix C) that the geometry factor T equals ... 

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