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For the boundary conditions that dSs

this equation can be solved analytically, giving the following when placed into di-mensionless form::|"

Zero-Order Biofilm. When the substrate concentration at all points within the biofilm is much greater than the half-saturation coefficient, the Monod equation can be approximated as zero-order with respect to substrate concentration (Eq. 3.37). This allows the differential equation describing reaction within a steady-state biofilm to be rewritten as:

Dc d2Ss dx-

Figure 15.18 Conditions under which a limiting-case solution to the steady-state biofilm model differs from the full pseudoanalytical solution by less than 1.0%. Each unshaded area represents a limiting-case solution. The shaded area, labeled Monod-shallow zone, indicates the conditions under which none of the limiting case solutions arc accurate. (From P. B. Saez and B. E. Rittmann, Error analysis of limiting-case solutions to the steady-state-biofilm model. Water Research 24:1181-1185, 1990. Copyright © Elsevier Science Ltd.; reprinted with permission.)

Rittmann Number, Ri

Figure 15.18 Conditions under which a limiting-case solution to the steady-state biofilm model differs from the full pseudoanalytical solution by less than 1.0%. Each unshaded area represents a limiting-case solution. The shaded area, labeled Monod-shallow zone, indicates the conditions under which none of the limiting case solutions arc accurate. (From P. B. Saez and B. E. Rittmann, Error analysis of limiting-case solutions to the steady-state-biofilm model. Water Research 24:1181-1185, 1990. Copyright © Elsevier Science Ltd.; reprinted with permission.)

Solution of this equation with the boundary conditions expressed by Eqs. 15.46 and 15.47 gives the following when placed into dimensionless form:''*

Other Cases. It is theoretically possible for either first-order or zero-order biofilms to be either deep- or fully-penetrated. In such cases, both requirements must be satisfied. A biofilm that is neither first-order nor zero-order has been called a Monod biofilm." A biofilm that is neither deep nor fully-penetrated has been called a shallow biofilm."

Error Analysis. The limiting-case solutions are fine provided the simplifying assumptions are appropriate for the conditions encountered. If they are not, their application can lead to gross errors." Thus, one must be sure a limiting-case solution is appropriate before using it. To help in that assessment, Saez and Rittmann" conducted an error analysis of the limiting-case solutions and prepared Figure 15.18 showing conditions under which a limiting-case solution of the steady-state biofilm model differs from the full pseudoanalytical solution by less than 1.0%. Several important points arise from the figure. First, fully penetrated biofilms are difficult to attain, occurring only when the Rittmann number is greater than 10, i.e., only when the relative growth potential is low. This suggests that the fully-penetrated case is of limited utility. First-order biofilms occur when Ri is small, but Monod-biofilms cover a broad range of Ri values. In fact, the Monod-deep zone could be expanded into the first-order-deep zone, since first-order kinetics is just a limiting case of Monod kinetics and the deep-biofilm equation is straightforward. Perhaps the most important point, however, is that the full pseudoanalytical approach must be used over a broad range of conditions, i.e., the Monod-shallow zone, which covers much of the expected range of Ri values."

In summary, while several limiting-case solutions are available in the literature, most are applicable only under very restricted conditions. The exception to this is the case of the deep biofilm. Thus, Eq. 15.29 represents the most useful limiting-case solution. Nevertheless, for the practical range of Ri values, many problems will require use of the full pseudoanalytical solution.

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