Following the verification of the first order reaction analytical solutions, non-linear reaction kinetics were considered to approximate both microbial growth and substrate utilization (of microcystin) according to the Monod model. We first begin the derivation of the steady state analytical solution using the transport equation for microcystin (including
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advection and dispersion), considering growth of both aqueous and solid biomass as reaction components, where Monod’s nonlinear mathematical relationship between substrate and bacterial growth rate is considered (Equation 33).
0 = 𝐷𝜕2𝐶 𝜕𝑥2− 𝑣 𝜕𝐶 𝜕𝑥− 𝜇𝑚𝑎𝑥 𝑌 𝑋𝑎( 𝐶 𝐾𝑠+𝐶) − 𝜇𝑚𝑎𝑥 𝑌 𝜌𝑏𝑢𝑙𝑘 ∅ 𝐼𝑏𝑖𝑜𝑋𝑠( 𝐶 𝐾𝑠+𝐶) (33) To develop useful analytical solutions to the nonlinear terms in Equation 33, we first considered the maximum and minimum ranges of the hyperbolic function describing
bacterial growth rate as a function of substrate concentration. In addition, we considered situations in which dispersion was neglected, since most current biofiltration units are moderately advectively dominated systems (Pe = 97.10). Finally, the concentration of degrading microorganisms remained constant for the development of the first analytical solution and the growth of aqueous biomass in the filter was only considered (solid biomass was neglected).
At low substrate concentrations (C<<Ks), the relation between bacterial growth rate and substrate concentration was linear and first order. This resulted in a first order
approximation for the nonlinear reaction terms in Equation 33 (Equation 34). However, at high substrate concentrations (C>>Ks), the bacterial growth rates were constant and not related to substrate concentration (zero order). The resulting expression for the nonlinear reaction terms is now zero order at high substrate concentrations (Equation 35).
𝑑𝐶 𝑑𝑡 = −𝑣𝑚 𝐾𝑠 (𝐶) (34) 𝑑𝐶 𝑑𝑡 = −𝑣𝑚 (35) Where, 𝑣𝑚 =𝜇𝑚𝑎𝑥 𝑌 𝑋𝑎.
Modifying Equation 33 with Equations 34 or 35 (and neglecting solid biomass and dispersion terms) and then integrating the resulting equations results in the following
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analytical solutions (Equations 36 and 37) for low and high substrate (microcystin) concentrations, respectively. 𝐶 = 𝐶0𝑒−𝑘1𝑥 (36) 𝐶 = 𝐶0− 𝑘0𝑥 (37) Where, 𝑘0 = 𝑣𝑚 𝑣 and 𝑘1 = 𝑣𝑚 𝑣∗𝐾𝑠.
Another analytical solution was applied for intermediate ranges in substrate concentration where the Monod kinetic model is more non-linear than at either extreme low or high substrate concentrations. Parlange et al. (1984) detailed an analytical solution to Equation 33 at steady state (neglecting dispersion and solid biomass growth) that includes nonlinear reaction kinetics for a constant concentration inlet boundary condition (Equation 38). Both Bekins et al. (1998) and Goudar and Strevett (2000) further described a Lambert W solution to iteratively solve for S as a function of vm (µmax), Ks, S0 (initial susbtrate concentration), and distance (x) (Equation 38).
𝐶 = 𝐾𝑠∗ 𝑊 {𝐶0
𝐾𝑠exp (
𝐶0−𝑣𝑚𝑥
𝐾𝑠 )} (38)
Unlike the preliminary first order degradation verification studies above, the non- linear comparison studies required a more detailed investigation of the expected range in growth characteristics of microcystin degrading bacteria. Thus, microbial growth
parameters relevant to bacteria native to surface waters were first compiled and used in the absence of data for growth parameters of microcystin-degrading bacteria. The minimum, median, and maximum ranges for each of these growth parameters are
presented and were varied in this verification study, ranging from a difference of three to five orders of magnitude (Table 7). Lastly, the biomass concentration (using dry weight) of the aqueous bacteria entering the filter was determined from median values of carbon
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contents of bacteria cells presented in the literature as well as an average influent cell concentration of 1E09 cells/mL (Table 7).
Table 7- Summary of Microbial Growth Parameters Used in Verification Studies for Non- Linear Monod Kinetics
Variable Name Symbol Units Nominal Value
Cell Maximum Growth
Rate 𝜇𝑚𝑎𝑥 1/second
1.74E-06 3.89E-05 3.83E-04 Half Saturation Constant Ks Kg MCLR/m3
5.0E-07 1.68E-03 1.21E-01 Yield coefficient 𝑌 Kg cells/Kg MCLR
0.13 0.371 0.545 Inlet Aq. Biomass
Concentration Xa Kg cells/m3 0.01
The rate of substrate utilization by a fixed concentration of microorganisms (𝑣𝑚(
𝐶
𝐾𝑠+𝐶)) was plotted as a function of substrate concentration (C) to determine the appropriate ranges in substrate concentrations where the above analytical solutions (Equations 36, 37 and 38) would be valid for the min, med, and max ranges in growth parameters (Figure 25). The resulting solutions demonstrated relatively limited ranges where the above analytical solutions would be valid for both first and zero order
approximations of the nonlinear substrate utilization kinetics (for all ranges in the growth parameters, min, med, and max) (Figure 25B and 25C). From inspection of both subfigures, it is evident that the first order approximation threshold is well below 50 µg/L of
microcystin, somewhere in the range between 0-5 µg/L, whereas the zero-order
approximation threshold is approximately 0.5E07-1E07 µg/L of microcystin. The initial concentration ranges for first, non-linear, and zero order approximations were determined
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to be 0.01, 100, and 1,000-10,000,000 for all ranges in growth parameters based on the trends observed in Figure 25. The initial concentration ranges were varied for the high concentration of substrate due to the distinct differences in substrate locations where the zero-order approximation was valid (Figure 25C).
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Figure 25. Dimensionless Monod curves for a) substrate utilization rate as a function of microcystin concentration, b) determining the first order approximation threshold (red dotted lines are linear approximations), and c) determining the zero-order approximation
threshold.
The analytical and nonlinear numerical solutions were first compared at low
substrate concentrations (C << Ks) using the analytical solution described in Equation 36 at
A
)
B
)
C
)
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a singular Courant (Cr) number reflecting a minimum operational HLR of a full scale biofiltration system (HLRmin = 0.3 m/hr) (Figure 26). At low substrate concentrations, the highest removal efficiency of microcystin was achieved for the lowest magnitude bacterial growth parameter combinations followed by the median and maximum value combinations (Figure 26). Agreement between the analytical solutions and the numerical approximations was excellent for each case (SSRs above E-18). The removal of microcystin was generally faster at lower magnitude combinations of growth parameters since the rate of bacterial growth (at low substrate concentrations) was directly proportional to the µmax/Ks ratio, where this ratio was largest for the parameter combinations that were the smallest in magnitude (Table 7).
Figure 26. Comparison of analytical and nonlinear numerical solutions for a) minimum, b) median, and c), maximum bacterial growth parameters at a minimum HLR (0.3 m/hr) and
low substrate concentration (C<<Ks).
At high substrate concentrations (C >> Ks), a reduction in the removal efficiency of microcystin was observed as compared to at lower substrate concentrations (Figure 27). This result was due to the fact that at higher concentrations, the substrate utilization kinetics are not dependent on the concentration of substrate, rather the maximum growth
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rate. Therefore, as the substrate concentration increases, the rate of utilization will not increase (as maximum growth rate is fixed), which will result in less removal at higher and higher concentrations above the “high” concentration threshold (where C >> Ks).
The best removal efficiency at high concentrations of microcystin was demonstrated by the simulation with the lowest maximum growth rate (Figure 27). This result was due to the fact that the initial substrate concentration was much less for this simulation than the simulation with the highest maximum growth rate (on the order of 1,000 compared to 10,000,000). Comparably, at lower HLRs, the removal efficiency increased as there was an increase in contact time for degradation to occur across the filter between the substrate and suspended microorganisms. The relatively good agreement between all numerical approximations and analytical solutions also indicated that the nonlinear approximation of the growth kinetics was valid. The relatively high values of the SSR compared to previous experiments (on the order of 1E-05 vs. 1E-18) were due to differences in the scales of the substrate concentrations that were under comparison (g/L vs. µg/L) as opposed to actual discrepancies between the numerical model simulations and the analytical solutions.
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Figure 27. Comparison of analytical and nonlinear numerical solutions for a) minimum, b) median, and c), maximum bacterial growth parameters at a minimum HLR (0.3 m/hr) and high substrate concentration (C>>Ks). Initial substrate concentrations were varied between
1000, 100,000, and 10,000,000 µg/L, respectively.
The numerical approximation of the Monod equation at intermediate values of substrate concentration (100 µg/L) was very similar to the analytical solution using the Lambert W method (Figure 28). These results indicated that the numerical approximation using the Picard iteration was robust enough to successfully approximate even the most nonlinear portions of the Monod hyperbolic equation. The resulting microcystin removal efficiencies across the filters were almost equivalent to the removal efficiencies at low microcystin concentrations, but were slightly lower in magnitude (Figure 28). This result suggests that the concentration tested for moderate and maximum growth conditions (100 µg/L) may not have been in the distinct nonlinear portion of the Monod model, and may be closer to the linear approximation threshold. Nonetheless, the good agreement with the Lambert W solution demonstrated that the numerical solution to the nonlinear model was still accurate, with low SSR values observed for each simulation at moderate/high growth conditions (Figure 28).
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Figure 28. Comparison of analytical and nonlinear numerical solutions for a) minimum, b) median, and c), maximum bacterial growth parameters at a minimum HLR (0.3 m/hr) and
moderate substrate concentrations (C~Ks).