Shape Factors for Particles in Packed Bed Exchange

Example 18. Shape Factors for Particles in Packed Bed Exchange 

In Reprint C in Chapter 7, the behavior of a tracer pulse in a stream flowing through a packed bed and exchanging heat or matter with the particles is studied. It is shown that the diffusion in the particles makes a contribution to the apparent dispersion coefficient that is proportional to v2 fi/D. The constant of proportionality has one part that is a function of the kinematic wave speed fi, but otherwise only a factor that depends on the shape of the particle (see p. 145 and in equation (42) ignore all except the last term and even the suffixes of this e, being unsuitable as special notation, will be replaced by A. e is defined in the middle of p. 143 of Chapter 7). In this equation, we should not be surprised to find a term of the same form as the Taylor dispersion coefficient, for it is diffusion across streams of different speeds that causes the dispersion in that case just as it is the diffusion into stationary particles that causes the dispersion in this.7 What is surprising is that the isothermal diffusion and reaction equation should come up, for A is defined by 

7 Howard Brenner has generalized the method to a whole class of phenomena in his magisterial paper, A general theory of Taylor dispersion phenomena. Physicochem. Hydrodyn. 1, 91-123 (1980). 

The prescription for this shape factor given in Eq. (237) requires only that we find the first coefficient in the expansion and suggests that we might try a solution of the form 

Ui satisfies div(grad iq) = - 1 in S and U = 0 on dS. We are sure of getting a positive constant, because 

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