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In the initial model verification study, a set of steady state, first order analytical solutions were applied for preliminary comparison with numerical model simulations. For

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the initial model verification procedure, we only included first order biodegradation, no dispersion, and advection transport and reaction processes (Equations 27 and 28).

0 = −𝑣𝜕𝐶

𝜕𝑥− 𝐾1(𝐶) − 𝐾2(𝐶) (27)

𝐶(𝑥) = 𝐶₀𝑒(−𝐾1−𝐾2𝑣 )𝑥 (28) Where K1 and K2 correspond to the following (Equations 29 and 30):

𝐾1 = 𝜇𝑚𝑎𝑥 𝑌 ∗ 𝑋𝑎 (29) 𝐾2 =𝜇𝑚𝑎𝑥 𝑌 ∗ 𝜌𝑏𝑢𝑙𝑘 ∅ ∗ 𝐼𝑏𝑖𝑜∗ 𝑋𝑠 (30)

In Equations 29 and 30, Xa and Xs are vectors (in the vertical direction of the filter) and are specified by some important initial conditions. For simplicity, we assumed that Xs was uniformly distributed with depth (x) and Xa was constant at the inlet, with no initial presence throughout the depth of the biofilter.

Table 6 summarizes the distribution of model parameters incorporated in this initial round of model verification study. The hydraulic loading rates (HLRs) considered represent the full range of HLRs encountered in practice (min, median, and max) (Evans et al. 2013a, 2013b). Microbial growth associated parameters were also selected from median reported values from studies examining the microbial growth characteristics of biofilm communities degrading organic carbon substrates in laboratory scale filtration units (Rittmann et al. 1986). Similarly, media characteristics were chosen based on the similarity to those expected in full scale practice (Evans et al. 2013a, 2013b). The comparisons between numerical and analytical models were made at three different Courant (𝐶𝑟 = 𝑣∆𝑡

∆𝑥 , where v is the interstitial pore water velocity, ∆𝑡 the time step, and ∆𝑥 the grid size) numbers to assess the degree of numerical dispersion of the analytical solution for different hydraulic

145 loading rates. Peclet numbers (𝑃𝑒 = 𝑣∆𝑥

𝐷 , where ∆𝑥 is the grid size, v the interstitial pore water velocity, and D is the dispersion coefficient) were not listed here given that the initial comparisons did not include the effects of dispersion.

Table 6- Summary of Values used in the Initial Numerical/Analytical Solution Comparisons

Variable Name Symbol Nominal Value Units

Filter length x 1 m Number x computational points nxpoints 100 unitless Initial concentration C0 10 µg/L Aqueous biomass

concentration Xa 6E-05 Kg cells/m3

Solid biomass

concentration Xs 8E-06 Kg cells/Kg grains

Cell Maximum

Growth Rate 𝜇𝑚𝑎𝑥 8.3E-03 1/second

Half Saturation

Constant Ks 1.2E-03 Kg MCLR/m3

Yield coefficient 𝑌 0.2 Kg cells/Kg MCLR

Bulk density of sand 𝜌𝑏𝑢𝑙𝑘 _{1.6E03 Kg/m3}

Porosity of sand ∅ 0.3291 m3 _voids/m3 _total

Hydraulic

Conductivity of Sand K 4.69 m/hr

HLR q 0.0061,6.10,50.10 m/hr

Interstitial velocity

of fluid 𝑣 0.01 m/sec

Courant Number Cr 0.3710,370.7, 3,448 unitless

Peclet Number Pe N/A unitless

Overall, for first order degradation scenarios, the numerical approximation to the analytical solution were valid for all three Cr number scenarios with corresponding sum of squared residuals (SSRs) (L2 norm) ranging on the order of magnitude from 1E-15 to 1E- 18 (Figure 21). Simulations with the lowest flow rate and Cr number demonstrated the highest SSR (worst accuracy), whereas simulations with the highest flow rate were

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improved (best accuracy) (Figure 21). Generally, higher Cr numbers resulted in a higher magnitude of numerical dispersion due to the fact that the upwind differencing scheme was only first order accurate. However, since the Cr number was changing in magnitude with the increase or decrease in velocity (not grid spacing), the poor accuracy of the numerical model was likely affected by the high magnitude of first order degradation for simulations with low interstitial pore water velocities.

The magnitude of the microbial growth rates was then altered to investigate to what extent the above observation was true, in that higher maximum growth rates result in higher discrepancies between the numerical approximation and the analytical solution. The magnitude of the maximum growth rates was changed from 8.3E-05, to 8.3E-03, to 8.3E-01 per second, while all other growth conditions were kept constant (i.e., half saturation constant and yield coefficient). In general, the numerical approximations to the analytical solutions were valid for all cases, with SSRs ranging from E-20 to E-13 (Figure 22).

However, higher growth rates (8.3E-01) resulted in higher discrepancies between the numerical approximation and analytical solution (under-prediction of the exponential decay). Therefore, the accuracy of the numerical solution seems to be directly proportional to the magnitude of the growth rate and indirectly proportional to the hydraulic loading rates, where the most discrepancy can be expected at low interstitial velocities and high microbial growth rates.

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Figure 21. Numerical approximation (red line) to analytical solution (blue *) comparisons and SSR for different hydraulic loading rates, a) Cr = 0.3707 b) Cr = 370.7 and c) Cr = 3,448.

Figure 22. Numerical approximation (red line) to analytical solution (blue *) comparisons and SSR for different microbial growth rates, a) μmax = 8.3E-05 b) μmax= 8.3E-03 c) μmax =

8.3E-01.

An analytical solution was then obtained for the steady state dispersion, advection, and first order reaction equation (Equations 31 and 32). This particular analytical solution was derived assuming that the form of the second order ordinary differential equation can be reduced to a quadratic equation in which the roots can be solved for analytically

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(2010), where the resulting analytical solution is presented below (Equation 32). Here Knet refers to the combination of K1 and K2, representing the contributions from both aqueous and solid biomass to the overall first order biodegradation rate.

0 = 𝐷𝜕2𝐶 𝜕𝑥2− 𝑣 𝜕𝐶 𝜕𝑥− 𝐾1(𝐶) − 𝐾2(𝐶) (31) 𝐶(𝑥) = 𝐶₀𝑒 𝑣 2𝐷𝑥(1−√1+ 4𝐾𝑛𝑒𝑡𝐷 𝑣2 ) (32)

The agreement between numerical approximations of the steady state ADRE and the analytical solution derived above (Equation 32) will now be reviewed for different

combinations of the Cr and Pe numbers. When varying the Cr numbers, the hydraulic loading rates were changed with a constant dispersivity value (α = 0.0103 m). Contrarily, when varying the Pe numbers, the magnitude of the dispersivity (αmin = 2.7E-04 m, αmed = 2.11E-03 m αmax = 0.2 m) was changed while holding the hydraulic loading rate constant (HLR = 6.10 m/hr). The values of the min, median, and maximum dispersivities were

summarized from the literature for clean bed, uniform sand column experiments with length scales at or below 3 m (determined using a conservative tracer). These dispersivity values were considered most relevant for the expected laboratory and field conditions for validating the model output. The corresponding Pe numbers from the dispersivity values and constant HLR were 3,704, 474, and 5 for the verification study. Under both of these schemes (i.e., varying Cr or Pe), the effect of increasing and decreasing the maximum growth rates will be discussed. Identical conditions for the model input for the previous verification study were adopted in this analysis (Table 6).

Changing the Cr numbers with dispersion included (at a realistic Peclet number) had little effect on the resulting microcystin transport across the filter compared to without dispersion (Figures 21 and 23). The distribution of microcystin-LR across the filter and the

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agreement of the numerical simulation and the analytical solution for simulations with dispersion included are very similar to results without dispersion at a realistic Pe number to full scale biofiltration systems (Pe = 370.70). Changing the maximum growth rate at the different Cr numbers with dispersivity fixed also has little effect on the resulting

distribution of microcystin-LR across the filter and the agreement of the numerical and the analytical solutions as compared to the results with no dispersion included.

Increasing the dispersivity values (low Pe number, ~5) had a noticeable effect on the transport of microcystin across the simulated filter compared to without dispersion (Figures 21 and Figure 24). Including dispersion with the first order reaction term increased the removal efficiency across the filter as opposed to without dispersion. This result was in agreement with what was expected conceptually as dispersion increases the mixing potential and availability of microcystin substrate for the degrading bacterial community. The overall agreement of the numerical and analytical solutions was excellent, with a very low SSR observed for all Pe numbers and either advection or dispersion

dominated systems (Figure 24).

Some discrepancies were observed between the numerical and analytical solutions at much lower Pe numbers that are unrealistic for the system under this study (Pe < 0.01, corresponding dispersivity values > 1 m). These discrepancies between the numerical approximation and analytical solutions were most likely due to the handling of the boundary condition at the filter effluent. For the numerical simulations, the solution assumes that the gradient at the previous grid block is equivalent to the gradient at the boundary, which may result in an under prediction of the effluent concentration, especially at steep gradients near the boundary (brought on by dispersion). This under prediction at

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the boundary affects the resulting distribution of predicted microcystin concentration across the filter, resulting in a slightly altered numerical approximation.

Figure 23. Numerical approximation (red line) to analytical solution (blue *) comparisons and SSR for different hydraulic loading rates holding dispersivity constant, a) Cr = 0.3707 b)

Cr = 370.7 and c) Cr = 3,448.

Figure 24. Numerical approximation (red line) to analytical solution (blue *) comparisons and SSR for different dispersion coefficients holding HLR constant, a) Pe = 3,704 b) Pe = 474

and c) Pe = 5.