An improved one-dimensional (1-D) model for the secondary clarifier, i.e. the Bürger-Diehl model, was recently presented. The decisive difference to traditional layer models is that every detail of the implementation is in accordance with the theory of partial differential equations. The Bürger-Diehl model allows accounting for hindered and compressive settling as well as inlet dispersion. In this contribution, the impact of specific features of the Bürger-Diehl model on settler underflow concentration predictions, plant sludge inventory and mixed liquor suspended solids based control actions are investigated by using the benchmark simulation model no. 1. The numerical results show that the Bürger-Diehl model allows for more realistic predictions of the underflow sludge concentration, which is essential for more accurate wet weather modelling and sludge waste predictions. The choice of secondary settler model clearly has a profound impact on the operation and control of the entire treatment plant and it is recommended to use the Bürger-Diehl model as of now in any wastewater treatment plant modelling effort.

## INTRODUCTION

The performance of secondary settling tanks (SSTs) significantly affects the effluent quality as well as the biomass inventory (i.e. the biomass distribution in the bioreactor and clarifier) in a wastewater treatment plant (WWTP). As biomass is the driving force for the biokinetic conversion processes, SST operation will affect the performance of the entire treatment plant – in reality and when modelling WWTPs.

Traditional SST layer models available in most commercial simulation platforms, e.g. the 10-layer model by Takács *et al.* (1991), do not capture the settling dynamics in sufficient detail (Li & Stenstrom 2014b). Under normal dry weather operating conditions, this model may behave reasonably. However, its predictions under anomalous operating conditions (e.g. peak flows due to rain events) lose realism and could lead to erroneous results.

Several shortcomings that cause this lack of predictive power have been reported (Jeppsson & Diehl 1996; Plósz *et al.* 2011; Bürger *et al.* 2011, 2012; Li & Stenstrom 2014a). First, the numerical flux between two adjacent layers has to be chosen in accordance with the theory for the governing partial differential equation (PDE). Jeppsson & Diehl (1996) found that the numerical solutions of Takács’ model do not converge to the exact solution of the PDE when the number of layers is increased. One reason is that the numerical flux function in the Takács model contains an empirical constant *X*_{T} that is not present in the governing PDE. This is in conflict with one of the fundamental principles for any consistent modelling methodology (Bürger *et al.* 2011) which states that each model parameter should be present in the governing equations and not be introduced in the numerical method itself. Jeppsson & Diehl (1996) proposed the Godunov scheme as an alternative. This scheme has since then been used in a number of studies (Diehl & Jeppsson 1998; Plósz *et al.* 2007; Bürger *et al.* 2013) and is mathematically sound (Bürger *et al.* 2011). Another suitable numerical flux has recently been introduced by Li & Stenstrom (2014a).

A second important shortcoming of both the traditional layer models, e.g. the one by Takács *et al.* (1991), and the recent one by Li & Stenstrom (2014a) is that they only account for convective flow and gravity settling. Other phenomena such as compression settling, turbulent diffusivity and dispersion are not included. Takács’ model compensates for these missing effects by applying a coarse discretization (10 layers) which introduces significant numerical dispersion. Although this artificial smoothening produces a more realistic profile, it is not able to describe the true dynamics of an SST, causing modellers to make unrealistic calibrations of the settling parameters or even to change the number of layers, not realizing that this impacts the numerical dispersion. Attempts have been made to handle this shortcoming of Takács’ model by accounting for compression through additional terms in the settling flux function (Stricker *et al.* 2007). By this approach, the governing PDE still has only first-order derivatives. It is, however, commonly known that compression necessarily involves also the gradient of the concentration leading to a second-order PDE (De Clercq *et al.* 2008).

More recent one-dimensional (1-D) models explicitly account for dispersion and compression by a second-order term in the governing PDE (Hamilton *et al.* 1992; Watts *et al.* 1996; Lee *et al.* 2006; Plósz *et al.* 2007). For instance, Plósz *et al.* (2007) showed that by introducing a second-order term dispersion can be modelled as a separate phenomenon and not by numerical dispersion introduced through the discretization. The drawback of this approach is, however, that all previously unmodelled phenomena (such as compression settling and inlet dispersion) are now lumped into one single term. This is too coarse to sufficiently capture the true settling dynamics, creating the risk of compensating for uncaptured dynamics by either unrealistic calibration of the settling parameters or by introducing an extra parameter in the numerical solution (for example the reduction factor *η*_{c} in the model by Plósz *et al.* (2007) which is not present in the original governing PDE). As mentioned above, such procedures are not consistent with good modelling practice.

A new 1-D model which allows improved and more realistic simulations of secondary clarifiers has recently been presented (Bürger *et al.* 2011). All implementation details can be found in Bürger *et al.* (2013). This new simulation model, called the Bürger-Diehl model, consists of a method-of-lines formulation, i.e. a set of time-dependent ordinary differential equations (ODEs), obtained by a spatial discretization of the governing PDE by appropriate methods. A physically correct numerical solution is obtained as the ODEs are integrated in time. Either a fixed time-step solver can be used with a maximal step given by the CFL (Courant–Friedrichs–Lewy) condition (Bürger *et al.* 2013) or an adaptive solver. Furthermore, the Bürger-Diehl model allows the modeller to account for several phenomena in a modular way, making it very flexible in its application.

The purpose of this study is to investigate the qualitative effect of the Bürger-Diehl model on the operation and control of a WWTP. Moreover, we elucidate the specific added value of the settler model's features on the predictions of biomass concentrations throughout the system and the development of mixed liquor suspended solids (MLSS) based control actions. The results obtained with the Bürger-Diehl model are compared to the Takács model since this is historically the most commonly used one in the WWTP community. Note that it is not the scope of this work to present a calibrated model but to illustrate when and why traditional layer models fail to capture the true dynamics of a WWTP and how this can be improved by the specific features of the Bürger-Diehl model. For this purpose, simulations are performed with the COST/IWA benchmark simulation model no.1 (BSM1) (Copp 2002; Gernaey *et al.* 2014). Finally, some guidelines for the application of this new settler model in WWTP modelling in general are also provided.

## MATERIALS AND METHODS

### Benchmark simulation model no.1

The BSM1 is a standardized simulation procedure for the design and evaluation of control strategies of conventional WWTPs in terms of effluent quality and operational costs, comprising a detailed description of plant layout, models, inputs and evaluation criteria; see Figure 1.

The results shown in this contribution were obtained with the BSM1 with two different settler models (Bürger-Diehl and Takács) in the modelling and simulation platform WEST (http://www.mikebydhi.com, Denmark; Vanhooren *et al.* 2003). The fourth-order Runge–Kutta numerical integration method with variable time step (RK4ASC) was used for the numerical integration of the ODE system. All simulations were performed with the standard input file for storm weather conditions (i.e. 100 days of steady state followed by 21 days of dry weather and 7 days of storm weather). The input flow rate (*Q*_{in}) and the incoming concentrations of readily biodegradable substrate (*S*_{S}) and ammonium (*S*_{NH}) during 7 days of storm weather are shown in Figure 2.

### Bürger-Diehl settler model

On the right-hand side, the first term represents convective transport (due to feed flow, underflow and overflow) as well as particle settling due to gravity. The second term includes a compression function (*d*_{comp}) and a dispersion function (*d*_{disp}). The last term is a singular source term modelling the feed mechanism (*Q*_{f} is volumetric inflow rate to the settler, *A* is the constant cross-sectional area).

To numerically solve this PDE, it is discretized by dividing the tank into a user-defined number of layers (*n*). The flux *F*(*X*, *z*, *t*) is replaced by numerical fluxes containing the Godunov flux. Moreover, a specific implementation of the compression term is needed to ensure that as the number of layers increases, the numerical solution converges to the physically correct solution of the PDE (Bürger *et al.* 2013). In the Bürger-Diehl model, the number of layers *n* can thus be set by the user, depending on the desired accuracy and on the computational time and resources available. Some guidance is provided in the section on practical implications.

To calculate the correct effluent and underflow concentrations, two extra layers are added at the top and bottom of the tank, respectively. It is important to emphasize that the effluent and underflow concentrations are generally not the same as the concentrations in the top and bottom layers within the settler (Jeppsson & Diehl 1996). This is not taken into account in traditional layer models. The extra layers in the underflow region can be interpreted as the start of the outlet pipe. The underflow concentration *X*_{u} can be defined as the concentration in any of these two layers.

Important features of the Bürger-Diehl model include the optional inclusion of compressive settling and flow-rate-dependent inlet mixing phenomena. This is achieved by specific terms for compression and dispersion in the 1-D PDE. The model is very flexible since the compression and dispersion terms can be switched on or off to meet the user's needs without affecting the solvability of the model. Hence, in its simplest form (no compression and no inlet dispersion), the Bürger-Diehl model reduces to the one by Diehl & Jeppsson (1998) and is a reliable alternative for current models.

*et al.*(2005). Despite accurate measurements of batch tests of activated sludge (De Clercq

*et al.*2005) and different approaches to solve the inverse problem (De Clercq

*et al.*2008; Diehl in press), it is difficult to obtain an appropriate compression function. The logarithmic function obtained by De Clercq

*et al.*(2008) contains three parameters that are difficult to identify. We choose here the following simpler function with similar features but with only two parameters (

*γ*and

*X*

_{crit}):

Here, *ρ*_{s} and *ρ*_{f} are the densities of the solids and the fluid, respectively, and *g* is the constant of gravity. This function has an exact primitive, which implies a simplified implementation. Note that we do not propose this function as the ultimate approach to model compression but merely aim to illustrate the added value of extending a settler model with this phenomenon in a modular way. Owing to the modular structure of the numerical scheme presented by Bürger *et al.* (2013), the constitutive function can easily be updated or replaced whenever future research provides further insight into the compression phenomenon.

The dispersion function *d*_{disp} is often set as the product of the fluid velocity and a continuous function of the depth. The continuous function has its maximum at the feed level and is zero some distance away from the inlet (Bürger *et al.* 2013). This allows modelling a region of higher turbulence around the feed inlet at increased hydraulic loading. Since the goal of this contribution is to investigate the impact of the choice in settler model on the sludge inventory and related control actions, the focus will be on the effect of adding compression settling. The impact of the dispersion function is therefore outside the scope of this paper.

The parameter values used for the different simulations throughout this work are summarized in Table 1.

Parameter | Value |
---|---|

Hindered settling | |

V_{0} [m/d] | 474 |

r_{h} [l/g] | 0.576 |

r_{p} [l/g] | 2.86 |

Compression | |

X_{crit} [g/l] | 5 |

γ [m²/s²] | 1.2 |

Parameter | Value |
---|---|

Hindered settling | |

V_{0} [m/d] | 474 |

r_{h} [l/g] | 0.576 |

r_{p} [l/g] | 2.86 |

Compression | |

X_{crit} [g/l] | 5 |

γ [m²/s²] | 1.2 |

### Case study: the Eindhoven WWTP

To better understand the added value of the specific features of the Bürger-Diehl model, it is helpful to first consider the behaviour of a full-scale SST under both dry and wet weather conditions. Therefore, online measurements of both underflow concentration (*X*_{u}) and sludge blanket height (SBH) during a 2-day period of dry weather followed by a 2-day storm event at the WWTP of Eindhoven (The Netherlands) are provided in Figure 3. The total height of the tank is 4 m and the feed inlet is located at the top of the tank. The underflow rate in this WWTP is controlled as a fixed ratio (0.65) of the incoming flow rate.

From these data, an important distinction can be made between the operational state and consequent control requirements during, respectively, dry and wet weather. During dry weather, no sludge blanket is detected, signifying that under these conditions the settler is overdesigned and operating at only a fraction of its potential capacity. This is unfortunately the case for many WWTPs worldwide and indicates that during dry weather, the system's efficiency could be increased significantly, for example by operating the bioreactors at a higher sludge concentration. In contrast, when a storm peak hits the WWTP, a sludge blanket of almost 2 m is formed and care needs to be taken to avoid the loss of sludge from the system. The underflow concentration on the other hand does not undergo any large variations during the storm event. Only a small dilution effect can be observed. Hence, for this case, the main impact of a storm weather event can be found in the SBH variations*.*

The SBH can thus be a crucial operation and control variable during a storm weather event or when imposing higher solids loads to, for example, operate the bioreactors at a higher sludge concentration even during dry weather conditions (implying higher conversion rates and, hence, more flexibility in operation). It is important to develop improved insight (which also implies good quality predictions) to judge when process control should focus on keeping sludge in the system and safeguarding effluent quality or on maximizing biokinetic conversions.

## RESULTS AND DISCUSSION

### Impact of compression settling on predicted concentration profiles in the SST

To illustrate the effect of compression settling on the SST performance, open-loop simulations (with a fixed underflow rate *Q*_{u} = 18,831 m³/d) are performed (1) with the model of Takács and (2) with the Bürger-Diehl model with the compression function activated. Figure 4 shows the differences in SBH and underflow concentration predictions between both models. The SBH is defined as the height of the first layer with a concentration that exceeds the threshold value of 0.9 g/l. An increased flow rate to the SST in the simulations with the Bürger-Diehl model will cause the sludge blanket level to rise significantly and result in only a modest increase in the underflow concentration. This contrasts with a very drastic increase in the underflow concentration and a moderate effect on the SBH in the Takács model. By only considering hindered settling, the sludge in the Takács model will settle unrealistically fast resulting in a highly concentrated bottom layer. By including compression, settling will be slowed down at higher concentrations due to a compressive force. Hence, the inclusion of compression settling creates a dampening effect on the underflow concentration resulting in smaller variations on the underflow but a pronounced increase of the SBH.

To compare the behaviour of the models to the measured trends observed in Figure 3, ‘closed-loop’ simulations were performed with *Q*_{u} = 0.65 *Q*_{in} (as in the WWTP of Eindhoven). As the waste flow rate is kept at a constant value of 385 m³/d, only the recycle flow rate to the biological reactors will vary. The results are shown in Figure 5. In the case of the Takács model, an increased loading to the clarifier during a storm peak results in a significant dilution of the underflow concentration and only a very short elevation of the sludge blanket. In contrast, the simulations with the Bürger-Diehl model (with compression) predict only a slight dilution effect during the storm peaks. Here, the elevation in sludge blanket is clearly the dominant effect, which corresponds to the observations made in reality. It thus becomes clear that by accounting for compressive settling, the dynamics of SBH and underflow concentrations during storm weather can be modelled in a much more realistic way.

To further illustrate the differences in behaviour between the two settler models, the complete concentration profiles for the open-loop simulations at times *t* = 0.1 d (before the storm event) and *t* = 4.2 d (when the maximum flow rate hits the SST) are shown in Figure 6. From these concentration profiles, the importance of another feature of the Bürger-Diehl model becomes evident. The inclusion of additional layers at the outlet boundaries not only allows more realistic predictions of the underflow concentrations but also influences the entire concentration profile including the SBH.

As could be seen from Figure 4, the SBH in Takács’ model never drops below a value of 0.4 m, corresponding to the thickness of the bottom layer. This minimum SBH is inherent to the structure of the Takács’ model: sludge settles to the bottom of the tank and accumulates in the last layer. (Adding additional layers to reduce the thickness of the bottom layer is not to be done for the Takács’ model as one changes the numerical dispersion and hence the dispersion of the settler.) Unless the settler would be operated at extremely dilute circumstances, Takács’ model will never be able to predict a sludge blanket of 0 m. However, in a full-scale treatment plant, the absence of a sludge blanket is often encountered during dry weather operation. Consequently, the Takács model predicts a persistent error of almost half a metre. The reason for this behaviour is the common but erroneous assumptions that the underflow concentration is always equal to the one in the bottom layer. In the Bürger-Diehl model, this problem does not occur due to the existence of additional layers below the bottom, which represent the underflow region. Thus, by explicitly modelling the underflow region as well as extending the settling behaviour with a compression function, a more advanced 1-D model provides more realistic predictions of the SBH behaviour which would allow operation of the SST in a more efficient way.

### Impact of compression settling on the performance of the biological reactors

The previous section illustrates that sediment compressibility notably influences the SBH and the underflow concentration. Whereas the SBH is mainly important with respect to the performance of the SST, the impact range of the underflow concentration stretches out much further, as a large part of the underflow is recycled to the bioreactor. Hence, compression settling will also influence the biosolids concentration in the bioreactors. Figure 7 shows the predictions for the MLSS concentration in the first activated sludge unit (ASU 1) for both the Takács and the Bürger-Diehl model with compression. The latter model increases the predicted effect of a storm peak on the MLSS concentration (i.e. dilution directly after the peak and increased concentration during the recovery phase after a peak event). Owing to the dampening effect of compression on *X*_{u}, less sludge is instantaneously returned to the bioreactor when a storm hits compared to the Takács model. This results in less recovery and a more pronounced effect of the storm peak on the bioreactor performance. Thus, traditional settler models, which do not account for compressive settling, might severely underestimate the effect of a storm event due to an underestimation of the biomass dilution effect in the bioreactor and hence instantaneous severely reduced conversion rates. The latter will result in underprediction of potential peaks in the effluent chemical oxygen demand (COD) and NH_{4}. When using such a model for developing mitigation strategies under wet weather, one risks taking insufficient action.

*r = μX*, where

*μ*is a growth kinetics function. As an example, the nitrification rate in the first aerated activated sludge unit and the relative differences in predicted nitrification rate between the two models are shown in Figure 8. The relative differences are calculated as follows (with T for Takács, BD for Bürger-Diehl):

From the figure, it becomes clear that the effect of compression settling influences the performance of the entire plant. The observed differences in underflow concentration between the models result in a maximal difference of almost 20% for the predicted nitrification rate. Hence, when only hindered settling is considered (as is the case for the traditional layer models), this might ‘force’ modellers to calibrate kinetic parameters for the wrong reasons. If *X* is wrongly predicted by the model, the degrees of freedom in *μ* (e.g. *μ*_{max}, affinities) will be used to obtain the correct value for the total conversion rate *r*. Figure 8 further demonstrates that slight differences in conversion rates already exist during dry weather operation. The relative error also needs significant time to reduce to lower values after a storm event.

### Impact of compression settling on the development of control strategies

Since the sludge inventory is the driving force behind the performance of a WWTP, a pronounced difference in the predictions of the biomass concentrations will also influence plant-wide control strategies. Therefore, we investigate the impact of the two settler models on the implementation of a control strategy that aims to maintain the MLSS concentration in a desired range through manipulation of the underflow rate *Q*_{u} (see Figure 1).

As a first step, the very simple control strategy of Figure 5 was adopted (*Q*_{u} = 0.65 *Q*_{in}). This control strategy should reduce the plunge in the MLSS concentration during a storm event since more sludge will be recycled to the biological tanks. However, during highly dilute conditions (which usually occur during storm weather), this control strategy can become insufficient. Therefore, the constant-ratio controller is extended with a proportional-integral (PI) controller which controls the MLSS concentration in ASU 1 at a setpoint of 2,800 g/m³ by adapting the underflow rate. The control strategy is implemented in ASU 1 since this is the first location where disturbances in the incoming flow will be perceived. If the dilution effect is counteracted here, it will also counteract the effect in the downstream bioreactors. The PI controller serves as an auxiliary control strategy and will therefore only become active if the MLSS concentration drops below 2,500 g/m³. Once the MLSS concentration surpasses an upper threshold (MLSS > 2,850 g/m³), the PI controller is switched off. The limits for manipulation of the underflow rate are set to 0.33 and 1.5 times *Q*_{in} (Tchobanoglous *et al.* 2003).

As both settler models predict different concentrations in the underflow of the SST, a similar manipulation in *Q*_{u} will lead to different responses in the MLSS concentration. Hence, both models require different tuning of the control parameters. To further understand the nature of each model's requirements, the responses to a step increase in the volumetric underflow flow rate *Q*_{u} are examined. To mimic the behaviour under high flow conditions, a step increase in *Q*_{u} from 18,831 to 20,000 m³/d was applied under a constant incoming flow of 30,000 m³/d (and corresponding incoming concentrations selected from the standard BSM1 input file). (Note that investigating the step response during actual storm weather (60,000 m³/d) is not feasible since simulations with a long period of such increased flow conditions would upset the system too much, making it no longer representative of realistic operating conditions.)

Figure 9 shows the resulting step responses for both models. On the left-hand side, the absolute values of the MLSS concentrations are depicted; on the right-hand side, the net step-response values are shown with respect to the steady-state value of each model before the step increase. For both models, the MLSS concentration shows an initial steep increase followed by a much slower further increase until a new steady state is reached. The initial steep increase can be related to a period where the sludge that is present at the bottom of the clarifier is simply recycled at a higher rate. However, this sludge will not be replenished at the same velocity as it is pumped away, causing the underflow concentration to drop and a switch to a second period where the system response slows down until a new steady-state value for the MLSS concentration is reached.

The initial steep increase in MLSS concentration is 30% larger for the Bürger-Diehl model compared to the Takács model. Owing to a combined effect of compression settling and the additional layers in the underflow region, more sludge is present in the SST when using the Bürger-Diehl model. Consequently, as more sludge is readily available to be recycled when *Q*_{u} increases, it will take somewhat longer before the underflow concentration will be affected and the settling sludge flux to the bottom layers becomes limiting. Owing to the underprediction of the SBH in the Takács model during storm weather, the response to the control action will be wrongly predicted, which may result in a poorly tuned controller. Hence, developing a control strategy based on Takács’ model poses a risk of not producing the desired system behaviour under closed-loop conditions in practice.

As the Bürger-Diehl model (which has been shown to predict a more realistic system behaviour) predicts a larger response for the same increase in *Q*_{u}, it will require a more conservative tuning of the parameters in the PI controller to ensure a stable system response. Moreover, the control strategy developed for the Bürger-Diehl model requires a higher ratio in the constant-ratio controller. The explanation for this lies in the much lower underflow concentration predicted by the Bürger-Diehl model during storm weather as was illustrated in Figure 4.

Figure 10 shows the manipulations in underflow concentration and the MLSS concentration when a control action with a constant ratio and a PI controller is tuned and implemented for both models. The control parameters for the Takács model are *K*_{p} = 10 and *τ*_{I} = 1 and a constant ratio of 0.65. In the Bürger-Diehl model a constant ratio of 0.75 and a more conservative PI controller with parameter values *K*_{p} = 1 and *τ*_{I} = 5 are applied. The large dilution in MLSS concentration that was observed in the open-loop simulation results of Figure 7 is successfully counteracted by the applied control strategies. Owing to the lower PI settings in the Bürger-Diehl model, much lower control actions for the underflow concentration are now applied during the storm event. This will not only influence the recovery period of the MLSS concentration but also affect the cost calculations in the BSM model.

These results show that the choice of settler model can notably influence the evaluation of proposed control schemes. Since a real SST typically undergoes a significant increase in the SBH during storm weather, thereby storing additional sludge in the system, its response to an increase in the underflow rate can be more extreme than would be predicted by the Takács model as the latter underpredicts the elevation in the SBH. Consequently, operating a real SST calls for more conservative control parameters than would be suggested by the Takács model. Hence, switching to more advanced settler models can potentially benefit the development of many future operation and control strategies. Moreover, this would allow for more advanced control strategies to be developed (for example control on the SBH in order to operate the system at higher sludge concentrations).

### Practical implications of switching to a more advanced settler model

As the Takács model depends on the numerical dispersion (which is inherent in the model structure) for its simulation results, it should only be used with a 10-layer discretization (as this approximately mimics dispersion under dry weather). However, Jeppsson & Diehl (1996) demonstrated that a discretization with 10 layers is too coarse an approximation to capture the detailed dynamic behaviour of the settler. By applying the Godunov scheme for the settling flux and handling the compression term in a mathematically sound way, the Bürger-Diehl model ensures that increasing the number of layers will result in a more accurate approximation of the governing PDE, thus producing smaller errors in the underflow concentration. Note that for reasons of comparison, all simulations of the Bürger-Diehl model in this contribution have been performed with the same coarse discretization level as is required for the Takács model. Their accuracy could easily be improved by a finer discretization.

*t*as follows: with

_{i}*X*

_{u,n}the concentration in the underflow,

*n*the discretization level and

*X*

_{u,360}the underflow concentration of the reference simulation with 360 layers.

Unlike the underflow error, the error in the effluent concentration will not propagate throughout the system. Furthermore, accurate predictions of effluent concentrations in a 1-D model are currently still troublesome as 1-D models do not include discrete settling behaviour. Therefore, the numerical error on the effluent concentration is not shown here.

Figure 11 shows the numerical errors in *X*_{u} and SBH for simulations with the Bürger-Diehl model with compression. The grey boxes indicate the time intervals over which the numerical errors were calculated. During dry weather, the errors in the underflow are quite small, even for the 10-layer discretization. However, during wet weather, when the underflow concentration increases rapidly, the error for the 10-layer discretization augments up to an average of approximately 10%. By using a 30-layer discretization this error is reduced to less than 5%. Note that the numerical error increases without compression settling since compression somewhat dampens the variations in the underflow concentration.

Also for the SBH a 10-layer discretization (with a minimum variation of 40 cm) is clearly quite coarse to describe the dynamic behaviour. The numerical error reduces significantly when a 30-layer discretization is applied.

However, more accurate predictions will inevitably come at a cost of increased simulation time. More layers imply more computations at each time step. Furthermore, the maximum allowed time step of the numerical scheme to ensure a stable and correct solution becomes smaller as the number of layers increases. For explicit fixed-step solvers (such as Euler or RK4), the maximum allowed time step is restricted by the so-called CFL condition, which unfortunately restricts the time step substantially when compression or dispersion is included. We refer to Bürger *et al.* (2013) for the details. In recent work, Diehl *et al.* (submitted) compared different ODE solvers with respect to their efficiency for the simulation of BSM1 with the Bürger-Diehl model under storm weather conditions. Moreover, they introduced a semi-implicit time-discretization method for which the simulation time with a 30-layer discretization was shown to be approximately seven times faster than a standard explicit solver such as Euler (placing it in the same range of computational effort as a 10-layer simulation with an explicit solver).

These results show that the currently used Takács model can be replaced by the Bürger-Diehl model, providing a reliable alternative without having to make too many sacrifices with respect to simulation time. Consequently, from the results presented in this section, it is recommended to use a discretization with at least 30 layers for simulations where the SST is coupled to one or more biological reactors. With a discretization of 30 layers, the relative errors on the underflow concentration and SBH are reduced significantly while the simulation time is still acceptable. If more detailed simulation results are required from the modelling study or when the settler is modelled as a stand-alone system, the number of layers is recommended to be increased in order to have more accurate predictions.

## CONCLUSIONS AND PERSPECTIVES

In this contribution, the impact of the new Bürger-Diehl settler model on operation and control of a WWTP in comparison to the traditional Takács model is investigated by using the BSM1.

Open- and closed-loop simulations were performed with both settler models during storm weather conditions, and the simulated underflow concentration (

*X*_{u}) and SBH were qualitatively compared to online settler data of the WWTP of Eindhoven. It was shown that the Takács model overpredicts the variations in*X*_{u}and underpredicts the SBH elevation whereas the Bürger-Diehl model provides more realistic predictions of*X*_{u}and SBH behaviour by accounting for compression settling and explicitly modelling the underflow region.The impact of the two settler models on the sludge inventory and conversion rates in the bioreactors was investigated, as poor predictions of the recycled biomass force modellers to calibrate kinetic parameters for the wrong reasons. A difference of almost 20% for the predicted nitrification rate was observed between the two settler models during storm weather indicating that the choice of settler model influences the simulation results of the entire treatment plant.

An MLSS-based control strategy was implemented. Simulation results showed that operating an SST calls for more conservative control parameters than would be suggested by the Takács model due to the underprediction of the SBH elevation in this model. To improve operation and control of WWTPs, we need to step away from traditional layer models towards more sophisticated models such as the Bürger-Diehl model.

Although the Bürger-Diehl model is not associated with a fixed number of layers, we have found that a discretization of the model with 30 layers provides an acceptable trade-off between model accuracy and the required simulation time.

## ACKNOWLEDGEMENTS

RB is supported by Fondecyt project 1130154; Conicyt project Anillo ACT1118 (ANANUM); Red Doctoral REDOC.CTA, MINEDUC project UCO1202; BASAL project CMM, Universidad de Chile and Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción; and Centro CRHIAM Proyecto Conicyt Fondap 15130015. We kindly acknowledge Waterboard De Dommel for providing the online data of the WWTP at Eindhoven.