Radcliffe et al. (1982), while presenting a method for determining the external film mass transfer coefficient k, by the analysis of initial breakthrough concentrations, stated that external film mass transfer dominates mass transfer behavior during the early portion of breakthrough curves obtained from fixed-bed adsorbers. It is important, however, to define the conditions under which their assumptions and their method are valid, as will be discussed in this note.
DevelopmentThe solute continuity equation for a fixed-bed adsorber, with provision for axial dispersion, is for "dilute" systems (i.e., systems for which u is essentially constant). Assuming that only external film mass transfer is of importance during the early part of the breakthrough curve, Radcliffe et al. employed the rate law ( c -c,) at where c, is the interfacial fluid-phase solute concentration. They stated that "initially the fluid-phase concentration at the particle surface is zero." Hence, setting c,= 0, substituting Eq. 2 into Eq. 1, and letting (ac/at), be zero based on the physical argument that it is small (this assumption is valid if the solute distribution ratio, qo/co, is large), we get The analogous solution for D,=O (which is often the case, especially when the fluid phase is a liquid) is:Again, a plot of c/co vs. 0, extrapolated back to d = 1, allows one to determine k,, if E , a, L , and u are known. I Frequently, "a" is unknown and hence is lumped together with k,, and the combined quantity k g is determined. This procedure is acceptable because the product k,a, and not either one alone, is what really matters in determining the fixed-bed behavior).The question arises as to whether the assumption that c, = 0 at O= 1 is indeed correct, thereby validating this method. Consideration of the simple limiting case where the solid phase has no adsorption capacity whatsoever is sufficient t o prove that c, > 0 can easily exist near O = 1 . When fluid containing solute first contacts the solid phase, diffusion starts to occur across the fluid boundary layer. After a relatively short time (on the