Axial Mixing of Liquid in a MWPB

The liquid flowing inside a MWPB can be described with a one-parameter dispersion flow model. As we show in Section 3.3, the axial mixing coefficient or, more correctly, the axial dispersion coefficient is the specific parameter for this model. Relation (3.112) contains the link between the variance of the residence time of liquid elements and the Peclet number. We can rewrite this relation so as to particularize it to the case of a MWPB. Here, we have the possibility to compute the variance of the residence time of the liquid through the stochastic model for the liquid flow developed previously in order to obtain the value of the axial dispersion coefficient  

If we carefully observe the expression of the characteristic function of the residence time distribution for the evolution of a liquid element (q (s,Hd), relation (4.201)), we can notice that it is difficult to compute the expressions of the derivatives cp (0, Hd) and (p (0, H ). Using the expansion of the hyperbolic sine and cosine respectively as multiplication series, we obtain the following simplified expression for the characteristic function  

The analytical computation for the first derivative of the characteristic function gives relation (4.226) where the functions ai(H(j), i = 1,6 are written with the relations (4.227)-(4.232)  

For the second derivative q (0, Hj) at point zero, we use the definition formula coupled with the I Hospital rule for the elimination of the non-determination of 0/0 type. The result is  

Considering relations (4.220) and (4.221), we can observe that we have all the required elements to compute the axial dispersion coefficient. The theoretical computed values for the axial mixing coefficient for the case where the bed height has a practical importance are shown in Table 4.5. For the cases when the selection Vx = 1 dm/s is not justified by the operational conditions, we replace FIj by H. We can introduce Hj through equation (4.210)  

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