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A simple CSTR without biomass recycle or biolilm would require a volume of 184 L to achieve the same effluent substrate concentration. Thus, the benefit of the biofilm is apparent.

As with other systems we have studied, biofilm systems also benefit from being housed in a plug-flow or tanks-in-series configuration. The pseudoanalytical approach can also be used to determine the performance of such systems by considering the plug-flow systems to be a series of completely mixed compartments.1'' Although the number of compartments required to adequately simulate a plug-flow system depends on its hydraulic characteristics, considering the bioreactor to contain six is usually adequate.'" Regardless of the situation considered, a number of compartments in series should be assumed, as well as the volume of each compartment. Starting with the last compartment and the desired effluent substrate concentration, it is possible to calculate the flux into the biofilm using the procedure in Example 15.2.3.1. Once that flux is known, it can be used to determine the influent substrate concentration that could be treated with a given media by using the procedure in Example 15.2.3.2. Since the influent to the last compartment is the effluent from the next-to-last compartment, the procedure can be repeated to determine the concentration of organic matter in the influent to the next-to-last compartment, etc. The procedure is repeated until the influent to the first compartment is calculated. If it is equal to the known influent concentration, the system is the correct size. If it is larger than the known concentration, the system is larger than required and the calculations should be repeated with a smaller volume for each compartment. (Alternatively, the excess volume may be acceptable as a factor of safety.) If the calculated influent substrate concentration is less than the known value, the system is too small and the calculations must be repeated with larger compartment volumes.

While the procedure in the preceding paragraph is straightforward, it is tedious because of the need to solve Eq. 15.38 repeatedly. Consequently, Heath et al. ' " have proposed an even simpler method based on normalized loading curves.

Normalized Loading Curves. A graphical representation forms the basis for the normalized loading curve approach to biofilm reactor analysis and design. Heath ct al.1"'"" used the pseudoanalytical approach to solve for the substrate flux associated with various bulk substrate concentrations. This was done for many conditions and the results were presented in generalized form by normalizing the bulk substrate concentration relative to Sshmm and the flux relative to a reference flux, JM(. The curves are called loading curves because the flux to a biofilm is approximately equal to the rate of substrate input per unit of biofilm, i.e., the loading. The reference flux is the minimum flux just required to maintain a steady-state biofilm that is deep. For computational purposes, it is defined as the flux resulting when £ = 0.99 in Eq. 15.31.'" The reference flux depends on the value of the Rittmann number and can be presented in generalized form by a plot of J*K/Ri versus Ri, where JSK is made dimen-sionless in the same way as Js, as indicated in Eq. 15.32.'' Figure 15.12 shows such a plot. ' It can be entered with a known value of the Rittmann number, giving the corresponding value of J*(, which can be put back into the physical domain, i.e., JSK, by using Eq. 15.32.

Normalized loading curves were plotted for fixed values of Ri and k,V" The Rittmann number is used as a parameter because of its fundamental importance, as discussed previously. The dimensionless external mass transfer coefficient, k*, was chosen as the second parameter because it represents the importance of external mass transfer resistance in the performance of the biofilm. As can be seen in Eq. 15.40, if k* is large (>10), external mass transfer resistance is of little importance relative to reaction and internal mass transfer. Conversely, when k* is small (<1) external mass transport is likely to play an important role in biofilm performance. Figures 15.13 through 15.17 present normalized loading curves for Ri values of 0.01, 0.1, 1, 10, and 100, respectively." It should be noted that Ssl,/Sst,,llin is equivalent to S*y Ri and Js/Jsi< is equivalent to J?/J*k-

The use of the normalized loading plots is very straightforward for a biofilm in a completely mixed bioreactor or bioreactor compartment. For a given situation, the value of Ri is calculated with Eq. 15.34 and Figure 15.12 is used to determine the dimensionless reference flux, J*i(. The family of normalized loading curves corresponding most closely to the value of Ri is then used, with the particular curve depending on the value of k*. If no curve corresponds exactly to the Ri and k* values for the system, then interpolation can be used.1" Using the desired bulk substrate concentration, Ssh, the value of Ssh/Ssbmill can be calculated and used to determine JS/JSR (or Ji/Ji,<) from the appropriate curve. Since the value of J*K is known, the value of Js is fixed. Equation 15.41 can then be used to calculate the required biofilm area, A_, and Eq. 15.42 to calculate the associated bioreactor volume. This gives a direct solution for a single CSTR, or allows the iterative procedure discussed previously to be used for a plug-flow system approximated as compartments in series.

"lO4 103 102 10-1 10" 101 102 103 Rittmann Number, Ri

Figure 15.12 Curve for the determination of the dimensionless reference llux, Ji,<. from the Rittmann number. (From P. B. Saez and B. E. Rittmann, Accurate pseudoanalytical solution for steady-state biofilms. Biotechnology and Bioengineering 39:790-793, 1992. Copyright ® John Wiley & Sons; reprinted with permission.)

### Example 15.2.3.3

A synthetic wastewater with a biodegradable COD of 10 mg/L (0.010 mg/cm') is flowing at a rate of 1.0 L/hr (1000 cmVhr) into a CSTR containing a biofilm media with a specific surface area of 90 nr/m" (0.90 cm /cm ). The wastewater, the bioreactor, and the associated biofilm have the characteristics listed in Table E15.2. What total surface area of biofilm would be required to reduce the biodegradable COD to 1.5 mg/L (0.0015 mg/cm')? What bioreactor volume is required to house the media?

The first task is to determine the dimensionless reference flux, JJH from Figure 15.12. This requires calculation of Ri with Eq. 15.34.

Entering Figure 15.12 with Ri = 0.10 gives a value of JiK/Ri of 2.8, thereby fixing J?K at 0.28.

The next task is to determine Js from the appropriate normalized loading curve. This requires calculation of S^/Ssi,,,,,,, and k*. Ssll is given as 1.5 mg COD/ L. Ssb„„„ can be calculated with Eq. 15.22, or by multiplying Ri by Ks (See Eqs.

Table E15.2 Kinetic Parameters, Stoichiometric Coefficients, and System Variables Used in Example 15.2.3.3

Symbol Units Value qH mg substrate COD/(mg biomass COD • hr) 0.2667

Ks mg/L as COD 10

Y„ mg biomass COD/mg substrate COD 0.625

D0 cm/hr 0.02667

15.33 and 15.34), which is 10 mg COD/L, giving Sshn„n = 1.0 mg COD/L. Thus, SslySsl is 1.5. The value of k,* is calculated with Eq. 15.40.

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