A Modeling and Simulation Study of the Role of Suspended Microbial Populations in Nitrification in a Biofilm Reactor

Abstract

Many biological wastewater treatment processes are based on bacterial biofilms, i.e. layered aggregates of microbial populations deposited on surfaces. Detachment and (re-)attachment leads to an exchange of biomass between the biofilm and the surrounding aqueous phase. Traditionally, mathematical models of biofilm processes do not take the contribution of the suspended, non-attached bacteria into account, implicitly assuming that these are negligible due to the relatively small amount of suspended biomass compared to biofilm biomass. In this paper, we present a model for a nitrifying biofilm reactor that explicitly includes both types of biomass. The model is derived by coupling a reactor mass balance for suspended populations and substrates with a full one-dimensional Wanner–Gujer type biofilm model. The complexity of this model, both with respect to mathematical structure and number of parameters, prevents a rigorous analysis of its dynamics, wherefore we study the model numerically.

Our investigations show that suspended biomass needs to be considered explicitly in the model if the interests of the study are the details of the nitrification process and its intermediate steps and compounds. However, suspended biomass may be neglected if the primary interests are the overall reactor performance criteria, such as removal rates. Furthermore, it can be expected that changes in the biofilm area, attachment, detachment, and dilution rates are more likely to affect the variables primarily associated with the second step of nitrification, while the variables associated with the first step tend to be more robust.

Appendices

Appendix A: Growth and Reaction Rates

The growth rates μ _A , μ _N , μ _I are calculated from Table 1 through

$$ \mu_i=\sum_{j=1}^6 \frac{v_{ij}\cdot P_j}{f_i\cdot\rho},\quad i\in I_b, $$

(21)

where

$$\begin{aligned} \mu_A(\bar{S}) &{}= \mu_{A,\max}\cdot \frac{S_\text {O2}}{K_{A,\text {O2}}+S_\text {O2}} \biggl(\frac{S_\text {NH4}}{K_{\text {NH4}}+S_\text {NH4}}-(1+\eta)\cdot b_{A} \biggr), \end{aligned}$$

(22)

$$\begin{aligned} \mu_N(\bar{S}) &{}= \mu_{N,\max}\cdot \frac{S_\text {O2}}{K_{N,\text {O2}}+S_\text {O2}} \biggl(\frac{S_\text {NO2}}{K_{\text {NO2}}+S_\text {NO2}}-(1+\eta)\cdot b_{N} \biggr), \end{aligned}$$

(23)

$$\begin{aligned} \mu_I(\bar{S}) &{}= \frac{1}{f_I}\cdot (f_{XI}+\eta) \biggl(f_A\cdot b_{A}\cdot\frac{S_\text {O2}}{K_{A,\text {O2}}+S_\text {O2}}+ f_N\cdot b_{N}\cdot\frac{S_\text {O2}}{K_{N,\text {O2}}+S_\text {O2}} \biggr). \end{aligned}$$

(24)

Similarly, the reaction rates r _O2, r _NH4, r _NO2, r _NO3 are calculated through

$$ r_k = \sum_{j=1}^6 v_{kj}\cdot P_j,\quad k\in I_S, $$

(25)

where

$$\begin{aligned} r_\text {O2}(f_A\rho,f_N\rho,\bar{S}) =& \frac{3.43-Y_A}{Y_A}\cdot\mu_{A,\max}\cdot \frac{S_\text {O2}}{K_{A,\text {O2}}+S_\text {O2}} \cdot \frac{S_\text {NH4}}{K_{\text {NH4}}+S_\text {NH4}}\cdot f_A\rho \\ &{}+ (1-f_{XI})\cdot b_{A}\cdot\frac{S_\text {O2}}{K_{A,\text {O2}}+S_\text {O2}}\cdot f_A\rho \\ &{}+ \frac{1.14-Y_N}{Y_N}\cdot\mu_{N,\max}\cdot \frac{S_\text {O2}}{K_{N,\text {O2}}+S_\text {O2}} \cdot \frac{S_\text {NO2}}{K_{\text {NO2}}+S_\text {NO2}}\cdot f_N\rho \\ &{}+ (1-f_{XI})\cdot b_{N}\cdot\frac{S_\text {O2}}{K_{N,\text {O2}}+S_\text {O2}}\cdot f_N\rho, \end{aligned}$$

(26)

$$\begin{aligned} r_\text {NH4}(f_A\rho,f_N\rho,\bar{S}) =& \biggl( \frac{1}{Y_A}+i_A\biggr)\cdot\mu_{A,\max}\cdot \frac{S_\text {O2}}{K_{A,\text {O2}}+S_\text {O2}} \cdot \frac{S_\text {NH4}}{K_{\text {NH4}}+S_\text {NH4}}\cdot f_A\rho \\ &{}- (i_A-i_If_{XI})\cdot b_{A} \cdot\frac{S_\text {O2}}{K_{A,\text {O2}}+S_\text {O2}}\cdot f_A\rho \\ &{}+ i_A\cdot\mu_{N,\max}\cdot \frac{S_\text {O2}}{K_{N,\text {O2}}+S_\text {O2}} \cdot \frac{S_\text {NO2}}{K_{\text {NO2}}+S_\text {NO2}}\cdot f_N\rho \\ &{}- (i_A-i_If_{XI})\cdot b_{N} \cdot\frac{S_\text {O2}}{K_{N,\text {O2}}+S_\text {O2}}\cdot f_N\rho, \end{aligned}$$

(27)

$$\begin{aligned} r_\text {NO2}(f_A\rho,f_N\rho,\bar{S}) =& \biggl(- \frac{1}{Y_A}\biggr)\cdot\mu_{A,\max}\cdot \frac{S_\text {O2}}{K_{A,\text {O2}}+S_\text {O2}} \cdot \frac{S_\text {NH4}}{K_{\text {NH4}}+S_\text {NH4}}\cdot f_A\rho \\ &{}+ \frac{1}{Y_N}\cdot\mu_{N,\max}\cdot \frac{S_\text {O2}}{K_{N,\text {O2}}+S_\text {O2}} \cdot \frac{S_\text {NO2}}{K_{\text {NO2}}+S_\text {NO2}}\cdot f_N\rho, \end{aligned}$$

(28)

$$\begin{aligned} r_\text {NO3}(f_N\rho,\bar{S}) =& \biggl(-\frac{1}{Y_N} \biggr)\cdot\mu_{N,\max}\cdot \frac{S_\text {O2}}{K_{N,\text {O2}}+S_\text {O2}} \cdot \frac{S_\text {NO2}}{K_{\text {NO2}}+S_\text {NO2}}f_N \rho. \end{aligned}$$

(29)

Note that the growth rates as well as the reaction rates throughout the paper use both $\bar{C}(z)$ and $\bar{S}(t)$ as arguments. Furthermore, the reaction rates use f _A (z,t)ρ, f _N (z,t)ρ, and u _A (t), u _N (t) as arguments.

Appendix B: Remarks on Equilibria

2.1 B.1 Washout Equilibrium

It can be verified that the trivial equilibrium with λ=u _A =u _N =u _I =0 and bulk substrate concentrations attaining inflow concentrations always exists. To investigate the stability of this equilibrium in dependence of model parameters with analytical techniques turns out to be rather difficult because of the involved hybrid structure of the mathematical model. For the simpler single-species model of a biofilm reactor with suspended bacteria, it was found in Mašić and Eberl (2012) that two conditions need to be satisfied for the trivial or washout equilibrium to be stable: (i) The dilution rate needs to be larger than the growth rate of the suspended bacteria. This is the standard criterion for washout in completely mixed continuous reactors. (ii) In order to avoid also that a biofilm can establish itself, the bulk substrate concentration must be small enough so that the diffusive flux of substrate into the biofilm is smaller than the decay rate. Both conditions together lead to the observation that a high flow rate alone is not sufficient to lead to washout of the microbiology from the reactor. This also explains to some extent why bacterial biofilms are very difficult to eradicate and to prevent in situations where they are unwanted.

In the absence of analytical results, numerical simulations are carried out to investigate whether the observations from the single species biofilm model carry over to the essentially more involved multi-species model. The simulations start with initial data that are small perturbations of the trivial equilibrium. In particular, we introduce a small amount of biomass in the system, both suspended and sessile, namely

The simulation parameters are as in Tables 3 and 2. We set the dilution rate to D=5/day, larger than the maximum specific growth rate of both species. The colonizable surface area is set to A=A _reactor, i.e. we consider a reactor without added biofilm carriers. In our illustrative studies, we investigate the stability of the system with respect to the inflow concentration of ammonium $S_{\text {NH4}}^{0}$, assuming that neither nitrite nor nitrate are present in the influent, $S_{\text {NO2}}^{0}=S_{\text {NO3}}^{0}=0~\mbox{g/m}^{3}$. The bulk oxygen concentration is kept at S _O2=5 g/m³.

We increase the inflow concentration of ammonium, as the growth controlling substrate, in small increments, starting from $S_{\text {NH4}}^{0}=0$. In all cases, the system reaches steady state. In Fig. 8, we plot the amount of suspended biomass and the biofilm thickness at steady state. We find that the trivial equilibrium is stable for $S_{\text {NH4}}^{0}<0.04~\mbox{g/m}^{3}$, i.e. no biomass can be established for very small substrate concentrations. When the inflow concentration of ammonium is further increased, the trivial equilibrium becomes unstable and both, biofilm and suspended populations establish themselves. The observed critical value at which the trivial equilibrium becomes unstable is much smaller than the half-saturation coefficient for ammonium K _NH4=0.169 g/m³, which implies that severe substrate limitation is required for stability of washout. Initially, the biofilm biomass increases faster than the suspended biomass, but both are always present in accordance with Remark 2.2.

Fig. 8

Steady state values for varying the initial ammonium concentration $\mathrm{S}_{\text {NH4}}^{0}$ for total amount of suspended biomass [g] (solid, left y-axis) and biofilm thickness [m] (dashed, right y-axis) in the hybrid model at D=5/day and A=A _reactor (Color figure online)

In conclusion, the stability of the trivial equilibrium depends on the parameters and operating conditions. A biofilm cannot be established if the influent substrate concentration is too small.

We compare these findings for the nitrification model with the mathematically simpler single species single substrate model, for which the stability of the trivial equilibrium could be investigated with analytical techniques. First, we point out the qualitative similarity of Fig. 8 with Fig. 5 of Mašić and Eberl (2012). For a more quantitative comparison of the dual species system with four substrates with the single species, single substrate model, we recall that in Proposition 3.5 of Mašić and Eberl (2012) a necessary and sufficient condition for the stability of the trivial equilibrium was found. Derived from this in Corollary 3.6 were easier to use sufficient conditions for stability and for instability. Each of these two sufficient conditions consists of a pair of inequalities for the model parameters, one stemming from the suspended biomass and one from the biofilm. In essence, it was found for the single species, single substrate model that the trivial equilibrium is stable if growth of suspended biomass is dominated by washout, decay and attachment, and if production of new biomass in the biofilm is dominated by lysis. In particular, it was found that the stability of the trivial equilibrium is independent of the detachment coefficient E, in accordance with the findings of Abbas et al. (2012) for a biofilm reactor without suspended biomass.

The simple stability criterion of Mašić and Eberl (2012) cannot be directly applied to or derived with rigorous techniques for the multi-species, multi-substrate case. Nevertheless, based on the biological interpretation of the stability criterion we can suggest an easy to evaluate “approximate” ad hoc criterion for the multi-species case, where we focus on the persistence of AOB as the main player in the system in dependence of ammonium and oxygen, as its primary substrates that are required for growth. Since the oxygen concentration in the aqueous phase is fixed, this ad hoc criterion is cast as a condition on the bulk concentration of ammonium. An important further adaptation is required due to the fact that in the simpler model of Mašić and Eberl (2012) biomass decay was described as a simple lysis model, while in the current nitrification model decay of active biomass is due to endogenous respiration and inactivation. Incorporating this, we adapt from Mašić and Eberl (2012) the following non-rigorous ad hoc criterion for instability of the trivial equilibrium by comparing growth and loss of AOB:

$$\mu_{A,\max} \frac{S_\text {NH4}^0}{K_\text {NH4}+S_\text {NH4}^0} \frac{S_\text {O2}^0}{K_{A,\text {O2}}+S_\text {O2}^0} > D + \alpha + \frac{S_\text {O2}^0}{K_{A,\text {O2}}+S_\text {O2}^0} b_A (1+\eta) $$

or

$$\frac{S_\text {NH4}^0}{K_\text {NH4}+S_\text {NH4}^0} > \frac {b_A (1+\eta)}{\mu_A}, $$

where the first inequality stems from the persistence of suspended biomass and the second one from persistence of the biofilm. From this double inequality and the parameters in Tables 3 and 2, we can obtain the critical inflow concentration $S_{\text {NH4}}^{0}$ for which stability of the trivial equilibrium is lost as $S_{\text {NH4}}^{0}\approx 0.04$, in very good quantitative agreement with the more accurate simulation documented in Fig. 8. While this finding is not a formal proof, it suggests strongly that (i) the fate of AOB is critically determining the overall fate of the system, and (ii) that the qualitative behavior of the multi-species, multi-substrate system can be well approximated by the qualitative behavior of the simpler and mathematically easier accessible single species, single substrate model.

2.2 B.2 Persistence Equilibrium

For the much simpler Freter model that describes persistence of a single species depending on a single resource in a CSTR with wall attachment, it could be shown that at least one asymptotically stable non-trivial equilibrium exists (Jones et al. 2003). For the single species, single substrate biofilm model in Mašić and Eberl (2012) it was not possible to handle this question analytically, due to the essentially more complicated algebraic computations that would need to be performed. Nevertheless, it was confirmed in numerous numerical simulations that the system attains a non-trivial equilibrium if the trivial equilibrium is unstable. The current model is not only algebraically, but also structurally much more involved than the single species model and, therefore, we cannot expect to be able to investigate the question of existence and stability of non-trivial equilibria in dependence of model parameters mathematically. For the present study, the results of which are documented in the subsequent sections, extensive numerical simulations of the model have been conducted. In these simulations, the biological parameters were kept constant at values validated against experimental data and the parameters describing operating conditions for the reactor were varied. The model converged to an equilibrium in all cases.