Mathematical Modelling and Activated Sludge Systems

Content Table

• Basic Definitions
• Historical Aspects of Modelling Activated Sludge Systems
• Model Building for an Activated Sludge System
• Complex Biokinetic Models
• Organizing a Simulation Study
• Practical Model Applications
• Related Articles
• Resources
• References

Basic Definitions

Originally, the term “simulate” meant to imitate or feign something. More recent definitions of simulation are more precise and refer to using a model to predict the performance of a system under different conditions, exploring the effects of changing conditions on the behaviour of a real system, or designing a model of a system and conducting experiments with that model. Thus a simulation involves a well defined object of interest in the real world (system) and its description (model) used to understand and predict certain behavioural aspects of the system, or evaluate operational strategies for the system (Figure 1).

Figure 1. Modelling and simulation concept of a real system (Vangheluve et al. 2002)

It should be realized that a perfect model is never built and it is always a simplification of reality. In general, the simplest model should be preferred as it provides an adequate description of a given system. Simplicity often leads to clarity in thinking and evaluation. Moreover, it allows to avoid errors caused by a lack of understanding of the fundamental cause-and-effect relationships in the system. A simulation model can be:

• A mental conception (a conceptual model) - represents an understanding of the cause-effect relationships between components of that system,
• A physical model - e.g. a pilot plant, these models are usually relatively expensive to build and unwieldy to move,
• A mathematical model - quantitatively describes certain aspects of a system, such as effectiveness, performance or technical attributes, and cost. In engineering applications, many types of mathematical models can be applied and these models can be categorized in several different ways depending upon specific attributes considered, e.g. deterministic vs. stochastic, mechanistic vs. empirical, steady-state vs. dynamic, lumped- parameter vs. distributed-parameter,
• A combination of all of these.

Most problems of interest in the real world are usually too complex that a simple mathematical model can not be constructed and it is not possible to solve the equation(s) analytically. In such cases, the problems can be evaluated using a computer simulation which is defined as the use of a computer software to predict the performance of a real system under different conditions. The development of a mathematical model and subsequent computer simulations with the model have many advantages compared to experimenting with a real system (McHaney 1991):

• Experimentation conducted without disruptions to existing systems (testing of new ideas may be difficult, costly or otherwise impossible in systems  that already exist);
• Testing a concept prior to installation, which may reveal unforeseen design flaws and improve the design concept;
• Detection of unforeseen problems or bugs, which may exist in the system’s design (debugging time and rework costs can be avoided) or operation (improvements to system operation may be discovered);
• Gaining in system knowledge, which might be dispersed at the beginning;
• Much greater speed in analysis (simulation permits “time compression” to fractions of seconds or minutes representing minutes, hours, days, or even years of system time;
• Forcing system definition in order to produce a valid working model of a system;
• Enhancing creativity which can be exercised without the risk of failure.

Historical Aspects of Modelling Activated Sludge Systems

During almost 100 years of their history (since 1914), activated sludge systems have been expanding their capabilities from BOD removal to the integrated nitrogen and phosphorus removal. Moreover, several sidestream treatment technologies have been developed in the last decade to stabilize and enhance the efficiency of nitrogen removal. The process development has been accompanied by improvements in the design methods, a better understanding of the process involved (including their capabilities and limitations), and ways to optimize operation. In these areas, mathematical modelling and computer simulation are now considered to be valuable tools. Metcalf and Eddy (2003) emphasized that “computer modeling provides the tool to incorporate the large number of components and reactions to evaluate activated sludge performance under both dynamic and steady-state conditions, and to easily design multiple staged reactors as well as a single-stage complete-mix reactor”.

The history of modelling activated sludge systems can be divided into three periods:

• I period - The period lasting from the process discovery until the early 1950’s can be called “empirical design, piloting and guesswork”. The initial  design methods of activated sludge tanks were simple and entirely empirical in nature.
• II period – This period (1950s – 1980s) can be characterized as the formal application of chemical reaction type kinetics to relate (at steady-state) microbial growth and organic substrate utilization under aerobic conditions. Most of the initial kinetic studies originated from chemical kinetics and  were focused on the solution of particular, well-defined problems, such as the microbial growth under steady-state and fully controlled environmental conditions. This resulted in the domination of models which had a narrow range of applicability and were incapable of predicting the diverse adaptive reactions under changeable (dynamic) environmental conditions.
• III period – This period (1980s- ) can be characterized as the application of reactor engineering principles in combination with large matrixes of kinetic expressions and stoichiometric constants. A great complexity of the modern models resulted from the ability of defining the behaviour of influent wastewater components (i.e., organic, nitrogen and phosphorus fractions) and biomass components (i.e., heterotrophs, nitrifiers and PAOs with further subdivisions) and the reaction stoichiometry and kinetics.

Model Building for an Activated Sludge System

The starting point for any model development is the description of submodels that make up a complete model of the activated sludge system (Figure 2). The submodels include a hydraulic configuration model, influent wastewater characterization model, bioreactor model and sedimentation tank (clarifier) model. The hydraulic configuration model describes tank volumes and the liquid flow rates between tanks. The influent characterization model calculates the influent concentrations of state variables (COD, nitrogen and phosphorus fractions) in terms of the physical state and biodegradability. The bioreactor model couples a hydrodynamic mixing (flow pattern) model with a biokinetic model (describing biochemical conversions in the activated sludge) or other source terms that describe environmental conditions, e.g. process temperature and oxygen transfer. The flow patterns in a bioreactor are described at the extremes as plug flow or completely mixed. A value of the dispersion number indicates which of the two patterns is approached. The process temperature can be calculated from a heat balance over the bioreactor accounting for such components as solar radiation, atmospheric radiation, conduction and convection, evaporation, aeration, mechanical energy from mixing and biological processes. The clarifier models are described in terms of various degree of complexity ranging from simple ideal point settlers to two-dimensional (2-D) hydrodynamic models.

Figure 2. Schematic representation of a complete model of an activated sludge system

Complex Biokinetic Models

A biokinetic (activated sludge) model is the most important part of the overall model of an activate sludge system. Complex biokinetic models describe a range biochemical (and sometimes chemical) processes occurring simultaneously in the system. Modelling activated sludge processes became a discipline in the mid-1960’s (Henze et al. 2000). The authors implicitly referred to the study of Downing et al. (1964) who developed a simple model for ammonia oxidation to nitrate (as a one-step conversion) by nitrifying bacteria in the activated sludge process. In the late 1970s, modelling activated sludge processes was reaching the most advanced level at the University of Cape Town (Republic of South Africa) by Marais’ research group. This research group developed a model providing a consistent framework for the description of biological processes in the activated sludge process, including carbon oxidation, nitrification, and denitrification. That model was identified of special significance by the IAWPRC (former name of IWA) Task Group who developed the Activated Sludge Model No. 1 (ASM1) in 1987 (Henze et al. 2000). The ASM1 was also an inspiration for further studies in modelling other processes occurring in activated sludge systems, especially enhanced biological P removal (Figure 3). The introduction of the IWA-type Activated Sludge Models (ASM1 and later models, such as ASM2, ASM2d and ASM3) constituted the most significant contribution in the field of modelling biological processes of municipal wastewater treatment in the past 20 years. These models have received a widespread acceptance, first in the research community, and later among practitioners. The uniform structure of these models constituted a convenient base for further development of model concepts and incorporation of additional processes (e.g. growth of GAOs, filaments and anammox bacteria).

Figure 3. Development of the complex biokinetic models for biological nutrient removal activated sludge systems

A convenient way of the description of complex biokinetic models is a matrix format (so called the Petersen matrix). Such a notation ensures a transparent presentation of the model structure, i.e. the choices of model constituents and the parameters occurring in the expressions describing the interactions between model components. The component symbols are listed across the top of the table heading and the considered processes are listed down the left side of the table. The elements within the matrix comprise the stoichiometric coefficients, n_i,j, whereas the process kinetic rates ,r_j, are placed down the right column adjacent to the stoichiometric matrix. An example of the Petersen matrix for a simple biokinetic model of two-step nitrification (growth of ammonia and nitrite oxidizing bacteria, AOB and NOB) is presented in Table 1 (stoichiometric part) and Table 2 (kinetic part).

Organizing a Simulation Study

In order to develop a credible model for both science (for inter-comparability of the results) and decision making, any simulation study should be performed in accordance with a good, disciplined methodology. For example, Jakeman et al. (2006) proposed ten basic (iterative) steps in development and evaluation of environmental models (Figure 4). According to the authors, the main constituents of good model-development practice are a clear statement of modelling objectives, adequate setting out of model assumptions and their implications, and reporting of model results.

Figure 4. Iterative relationship between ten steps in development and evaluation of environmental models (Jakeman et al. 2006)

A number of model applications in activated sludge systems has been growing very rapidly in the recent years and a more standardized use of dynamic simulations is thus particularly essential. Several systematic protocols (guidelines) have been developed for conducting simulation studies, including BIOMATH (Belgium), STOWA (Holland), WERF (USA), HSG (German speaking countries), JS (Japan) and IWA Task Group. These protocols in principle follow the general procedure proposed by Jakeman et al. (2006) for environmental models. Most of these protocols emphasized the importance of several elements, such as quality of the available plant data, determination of mixing conditions (flow pattern) in the bioreactors and clarifiers, characterization of wastewater and biomass as well as parameter estimation in the biokinetic and settling models.

Quality of the available plant data should be first evaluated using a continuity check for flow rates and mass balance calculations for oxygen demand, solids, nitrogen and phosphorus. This procedure is outlined including also the use of error diagnostics and data reconciliation techniques. Results of tracer studies in bioreactors and clarifiers are used to calculate a value of the dispersion coefficient (alternatively, several empirical formulae can be used). The STOWA or WERF guidelines for wastewater characterization provide procedures for a complete organic and nitrogen fractionation in the influent wastewater. The most important, readily biodegradable organic fraction can be determined using either physio-chemical (coagulation-flocculation) or biological (respirometric) methods. Laboratory scale experiments are considered a valuable tool for determining stoichiometric and kinetic parameters in continuous procedures and batch tests, respectively. These experiments can be classified as direct methods, which focus on specific parameters that can directly be evaluated from the measured data, and optimization methods, which involve a procedure of fitting model predictions to measured data.

It is not realistic to expect that a model works perfectly, and therefore, uncertainty in model predictions becomes an inherent property of modelling. The uncertainty analysis is necessary to determine the confidence (reliability) with which the model can be used. This is essential for decision making and allows to recognize the model constrains and avoid incorrect interpretations of the model predictions. The effects of parameter uncertainty on the confidence of model predictions can be determined using two well-known approaches, such as uncertainty and sensitivity analysis. With the first approach, propagation of the various sources of uncertainty to the model output is evaluated which, however, requires substantial computational efforts. The uncertainty analysis provides probability distributions of model outputs, which are subsequently used to derive the mean, variance and quantiles of model predictions. The second approach (sensitivity analysis) essentially comprises perturbing input parameters in a defined range of values and observing the effect of the perturbation on model predictions. Sensitivity analysis can be used to investigate if the parameters that were modified during model calibration are indeed influencing the model outputs significantly. For newly developed models, sensitivity analysis also provides useful information about the model response to changes in the values of specific parameters.

Practical Model Applications

Practical applications of the complex biokinetic models can generally be classified under the following categories (Figure 5):

• Optimization of the performance of existing plants – models can assist in evaluating scenarios of alternative operating strategies and physical  modifications to the plant configuration. Optimization of an existing facility represents the best opportunity for use of mathematical modeling  approaches since the proposed model can be calibrated and validated using actual data from the facility;
• Upgrade of existing plants – model can be used to evaluate a number of plant upgrading options including formulation of different process retrofitting and expansion options as well as making best use of the available infrastructure;
• Design of new facilities - models can assist in identifying and quantifying the key design parameters;
• Development of new treatment concepts - models can be used to test hypotheses in a consistent and integrated manner, which results in a better  understanding of the fundamental behavioural patterns controlling the system response;
• Legal regulation – models can allow judgments to be made about the impact of new effluent requirements on treatment system design and cost;
• Educational purposes (teaching and staff training) - models can be used as an active tool to increase process understanding, exploring new ideas and developing a general conception of the system.

Figure 5. Potential practical applications of mathematical modelling and computer simulation of activated sludge systems

Hauduc et al. (2009) presented the results of a world-wide survey on the use of activated sludge models. The main objectives identified for building and using a model were: optimisation (59%), design (42%) and prediction of future operations (21%), but the distribution of modelling tasks varied in terms of the organisation type using the model. It was emphasized that the majority of North-American and European modellers are using models in different ways. In Europe, models are primarily used by researchers for optimisation purposes, while most modellers in North America are employed by private companies and carry out design studies.

Different types of a computer software can be used to implement mathematical models and subsequently predict the performance of WWTPs. These include spreadsheets (useful primarily for steady-state simulations and mass balancing), low level programming languages, general-purpose simulators (e.g. MATLAB/Simulink, ACSL, Maple, Mathematica, Stella) and specific WWTP simulator environments (or simulation platforms). The latter offer a high level of flexibility with minimal user training and without the requirement of knowledge of programming. The first specific simulator using ASM1 was SSSP (stands for Simulation of Single Sludge Processes), developed at Clemson University, South Carolina (USA). All the common commercial simulation platforms were developed in the 1990s. The latest versions are modular, multi-purpose programs in which the studied process configuration can easily be laid out with a graphical user interface with a great variety of output options (Figure 6). The programs contain extended libraries of predefined process unit models. The model libraries are usually provided in an open format which allows to modify and extend the existing models using so-called model editors. Advanced tools for model calibration and process control are also available. The most popular commercial simulators include ASIM (Switzerland), BioWin (Canada), GPS-X (Canada), SIMBA (Germany), STOAT (UK) and WEST (Belgium).

Figure 6. A main window of the modern simulator environment

Related Articles

• Computational Methods in the Management of Hydro-Environmental Systems
• Data Modelling
• Decision Support Systems
• Systems Modelling
• Numerical Analysis
• Physically Based Simulation Modelling
• Activated Sludge Process

Resources

This issues in this article are covered in the book, Mathematical Modelling and Computer Simulation of Activated Sludge Systems, by Jacek Makinia.The international, comprehensive guide to modeling and simulation studies in activated sludge systems leads the reader through the entire modeling process – from building a mechanistic model to applying the model in practice.

References

Jacek Makinia, Mathematical Modelling and Computer Simulation Activated Sludge Systems, IWA Publishing: 2010