Analysis of the dynamics of a model anaerobic digester

In this thesis we analyze a model for the process of anaerobic digestion (AD), represented mathematically by an initial value problem in two state variables, substrate and bacteria concentrations, with a nonlinear interaction function. We consider two different forms of the nonlinear interaction term; the Monod and the Haldane function. The initial value problem contains several model parameters that describe the bacteria population, the resident time, and the flow rates. In particular, the model explicitly allows for a non-zero inflow of bacteria as well as an inflow of substrate. This is unlike many other models. While much of the existing literature treats these parameters as constant, we have studied the effect on the model of these parameters being allowed to vary with time. Such variations reflect environmental changes such as variation in temperature and genetic makeup of the bacteria. As a result of nonlinearity and time dependence of model parameters, analytical solutions of our model may not be obtained in general and numerical solution techniques are usually necessary. The major drawback of such numerical solutions is that they are only applicable to specific parameter choices. However, we show how to obtain approximate analytical solutions that valid for all time by implementing a linearization method to overcome the nonlinearity, and extending a multitiming approach to handle the time variation of model parameters. This approach allows for a range of parameter values to be considered, rather than a result obtained by applying a numerical approach to a single choice of parameter values. This thesis can be broken down into four parts, each building upon the previous parts, as detailed below. The first part consists of Chapters 1 - 3. Chapter 1 provides a background on the AD process and its potential benefits, as well as an introduction of some of the mathematical models of AD. The model studied in this thesis is chosen in Chapter 2 to capture the essential behavior of the AD. A literature review of AD models then follows. Chapter 3 begins with an introduction of an AD model with selected parameters varying with time, representing the possible effect of the surrounding environment. Development of a dimensionless form of this model then follows. We then explore the behavior of this model through a study of its physically realistic solutions, that is, those solutions contained in the first quadrant of the phase plane. The second part includes Chapters 4 and 5 and studies details of the systems involving the Monod and Haldane functions and where the model parameters are constant. In these two chapters, we identify fundamental differences in the behavior of the solutions for zero and non-zero bacterial feed. Chapter 4 begins with an analysis of the critical points of the nonlinear system using the Monod function using standard meth- ods. Chapter 5 then follows with an analogous analysis of the nonlinear system using the Haldane function. In both chapters, we identify on a combination of the model parameters that determines the number and stability of critical points. Chapters 6 and 7 comprise the third part, in which the behavior of solutions over time is studied. As in Chapters 4 and 5, the model parameters are constant. Chapter 6 introduces a method by which the system with the Monod function is linearized, making use of the stable critical point found in Chapter 4. The resulting system is linear and can be solved exactly in a form involving elementary functions. This solution is then compared with numerical solutions of the nonlinear system. Exact solutions of the linear system and numerical solutions of the nonlinear system show good agreement for a range of parameter values, where the concept of the term 'good agreement' is made precise through a measure of closeness between two solutions. We establish a necessary condition involving all problem parameter values for solutions to the linearized system to remain in the first quadrant and so be physically realistic. Such a condition was not needed in Chapter 3. Chapter 7 uses the stable critical point from Chapter 5 to linearize the system involving a Haldane interaction function. This critical point is calculated approximately using a perturbation expansion. Again, the solutions of the linearized system comparing favorably to the numerical solutions of the nonlinear system for a range of parameter values. Here, agreement between solutions is measured in the same way as used in Chapter 6. The fourth and final part comprises Chapters 8, 9 and 10. Here, the model parameters are allowed to vary with time, yielding an initial value problem which is both nonlinear and nonautonomous. The time variation of the model parameters is assumed to occur on a second timescale, which is longer than the intrinsic timescale of the evolution of the solutions. This slow variation of the model parameters is introduced to reflect environmental changes in the system. Chapter 8 introduces the multitiming technique, an approximation method that constructs explicit analytic approximations to the solutions of initial value problem involving multiple time scales. Relevant studies are presented and the multitiming technique is illustrated using an example. In Chapter 9, we construct analytic approximations for the slowly varying limiting state solutions to our AD model of Chapter 4 with slowly varying model parameters. In what we believe to be an original approach, our linearization method of Chapter 6, together with a multitiming approach are adapted to construct solutions to the model equations as functions of three timescales. The analytic approximations obtained show good agreement with numerical solutions for a range of parameter values, confirming the validity of the approach. Chapter 10 then applies the method from Chapter 9 to the nonlinear system involving a Haldane interaction function. Finally, we present a conclusion and discuss possible areas of further research in Chapter 11.