## Info

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Figure 15.15 Normalized loading curves for Ri = 1. (From M. S. Heath, S. A. Wirtel, B. E. Rittmann, and D. R. Noguera, Closure to discussion of 'Simplified design of biofilm processes using normalized loading curves'. Research Journal, Water Pollution Control Federation 63:91-92, 1991. Copyright © Water Environment Federation; reprinted with permission.)

growing as a biofilm. Two techniques are available, one based on steady-state experiments45 and the other on transient-state experiments.2" Space does not allow us to discuss those procedures, but the reader should be aware of their existence, should the need to measure biofilm kinetics arise.

15.2.4 Modeling Transport and Reaction: Limiting-Case Solutions

Although the pseudoanalytical approach to modeling transport and reaction has greatly simplified the computation of the flux into a steady-state biofilm, it still does not lead to closed form solutions to bioreactor mass balances. Many circumstances exist, however, in which direct calculation of the bulk substrate concentration in a biofilm reactor would be advantageous. Consequently, several investigators have proposed simplifying assumptions for limiting cases that allow closed analytical solutions. There are four of them for steady-state biofilms.11

Deep Biofilm. As we saw in Section 15.2.3, it is possible to solve directly for the flux into a deep biofilm, which is one in which the substrate concentration at the solid-biofilm interface is zero. Under that condition, the dimensionless flux is given by Eq. 15.29, which was presented earlier.1"'1'' Its use results in a different flux for each liquid-biofilm interface substrate concentration.

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Figure 15.16 Normalized loading curves for Ri = 10. (From M. S. Healh, S. A. Wirtel, B. E. Rittman, and D. R. Noguera, Closure to discussion of "Simplified design of biofilm processes using normalized loading curves." Research Journal, Water Pollution Control Federation 63:91-92, 1991. Copyright © Water Environment Federation; reprinted with permission.)

Fully Penetrated Biofilm. A fully penetrated biofilm is one in which the change in the substrate concentration with depth within the biofilm is negligible. In other words, the entire biofilm contains substrate at almost the same concentration as the liquid-biofilm interface. By assuming that Ss, is equal to Ss, throughout the biofilm, an analytical solution can be obtained.16

First-Order Biofilm. When the substrate concentration at all points within the biofilm is much less than the half-saturation coefficient, the Monod equation can be approximated as first-order with respect to substrate concentration (Eq. 3.38). This allows the differential equation describing reaction within a steady-state biofilm to be rewritten as:

where L,* is given by

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Figure 15.17 Normalized loading curves for Ri = 100. (From M. S. Heath, S. A. Wirtel, B. E. Rittmann, and D. R. Noguera, Closure to discussion of "Simplified design of biofilm processes using normalized loading curves." Research Journal, Water Pollution Control Federation 63:91-92, 1991. Copyright © Water Environment Federation; reprinted with permission.)

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