Averaged area logarithmic-mean

The filtration equation (5.3-21) must be adopted if a centrifuge is used for separation. During centrifugation, the surface of the filter cake accessible for flow decreases since the cake builds up toward the axis of the centrifuge. Therefore, must be substituted by the product AcAim where is the average flow area and Aim is the logarithmic mean area through the cake. This yields ... 

The average area Am is given by the average of both surface areas A t = A(rx) and j = A(r2). One gets the arithmetic mean for the flat plate, the logarithmic mean for the cylinder and the geometric mean for the sphere. It is known that... 

If the ratio of inlet and exit driving forees is less than 2, an arithmetic average can be used with little error. For larger driving-force ratios the logarithmic mean could be used, though examples show that this slightly overestimates the required area. 

Figure 11 Plot of the mean square jump length versus the average logarithm of the distance of the measurement area to the steps. The graph shows that measurements that are made further away from a step yield larger jump lengths. All distances were measured in atomic spacings.

Fig. 10.47. Temporal evolution of mean grain size in phase field model of grain growth (adapted from Chen and Yang (1994)). Plots are of logarithm of average grain area as a function of time, with the two curves corresponding to four (crosses) and thirty-six (circles) different order parameter fields.

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