- 1.1 Kinetic Model Construction
- 1.1.1 Empirical Kinetic Models
- 1.1.2 Chemical Kinetic Models
- 1.1.3 Biochemical Kinetic Models
- 1.2 Fundamentals of Deterministic Optimisation
- 1.2.1 Formulation and Optimality of Constrained Optimisation Problems
- 1.2.2 Solving Nonlinear Optimisation Problems
- 1.3 Dynamic Optimisation and Parameter Estimation
- 1.3.1 Formulation of Parameter Estimation Problems
- 1.3.2 Discretisation of Dynamic Process Constraints
- 1.4 Conclusion
- References
Chapter 1: Physical Model Construction
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Published:20 Dec 2023
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Special Collection: 2023 ebook collection
F. Vega-Ramon and D. Zhang, in Machine Learning and Hybrid Modelling for Reaction Engineering
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Physical models of reacting systems aim to generalise experimental observations and mechanistic process understanding solely through the application of chemical kinetics principles and without the incorporation of any data-driven approaches. At this chapter’s core are the fundamental concepts of the mass-action law and pseudo-steady-state hypothesis, the application of which is exemplified through the derivation of both chemical and biochemical kinetic models. In virtue of its indispensable role during model development, deterministic optimisation theory is also introduced to illustrate the formulation and solution of constrained optimisation problems, with particular focus on nonlinear parameter estimation problems. This lays the theoretical foundation for model construction methodologies explored in latter chapters.
1.1 Kinetic Model Construction
The construction of an accurate and reliable kinetic model is the foremost task in (bio)chemical reactor design. From an analytical standpoint, the model and its kinetic parameters provide insight on the process by describing the concentration and temperature dependences that control the rate of reaction, potentially allowing us to identify mechanistic information such as inhibiting effects and rate-limiting steps in the underlying reaction network. From a practical application standpoint, the kinetic model provides a mathematical representation of the reaction dynamics, thus linking the reactor performance (e.g. conversion or selectivity) with the operating conditions and feed compositions; this, in turn, makes it an essential tool for reactor optimisation and control.
The complexity and degree of detail of a kinetic model obviously depend on its desired application. In this chapter, we exemplify the physical model construction procedure (from both empirical and mechanistic perspectives) for simple chemical and biochemical reactions. This will serve as an introduction for more complex model development methodologies explored in other chapters.
1.1.1 Empirical Kinetic Models
It must be noted that empirical models are not necessarily void of physical interpretation. Indeed, if the reaction in consideration were an elementary reaction, the power law expression in eqn (1.1) would simply reduce to a mass-action law model where the reaction order of the reactant corresponds to its stoichiometric coefficient and the apparent activation energy corresponds to the real activation energy of the reaction. More often than not, however, power laws are limiting approximations of the true, complex underlying kinetics that govern the reaction. For instance, in transport-limited heterogeneous catalytic reactions the apparent characteristics used to approximate the rate equation can be shown to depend not only on the intrinsic chemical kinetics of the reaction but also on the transport parameters of the system.1 Indeed, the apparent kinetic characteristics of a reaction may vary significantly across different operating ranges (i.e. concentrations, temperatures, and flow regimes, among others), meaning that empirical kinetic models typically exhibit poor extrapolability beyond the conditions on which they have been trained. Moreover, their model structures and parameters do not capture the mechanistic details of the reaction, and so their applicability for tasks such as model-based design of experiments, catalyst development or reactor design is limited. Nonetheless, empirical rate laws are extensively used in industrial process monitoring and control owing to their simplicity, low computational cost, and ease of implementation alongside heat transfer and catalyst deactivation models.2
Although the main focus of this chapter is the development of more rigorous kinetic models, the dynamic parameter estimation techniques described in Section 1.3 are equally applicable to empirical models.
1.1.2 Chemical Kinetic Models
Mechanistic kinetic models aim to capture the concentration and temperature dependences of the rate by considering the sequence of elementary steps that constitute the overall reaction. In this section, we introduce the fundamental concepts of the mass-action law and the pseudo-steady-state hypothesis. Rate expressions for some common reaction mechanisms are also developed, thus setting the basis for the construction of microkinetic models for complex heterogeneous reactions in Chapter 8.
1.1.2.1 Mass-action Law
1.1.2.2 Reaction Mechanisms and Pseudo-steady-state Hypothesis
Now that we have elucidated the application of the mass-action law to the development of rate and (ideal) reactor equations for single elementary reactions, let us consider some reaction mechanisms involving multiple elementary steps. Of particular interest are the series and parallel reaction mechanisms shown in Figure 1.1; albeit these simple schemes are rarely found in practice, they are illustrative examples given that complex reaction mechanisms combine both parallel and series elementary steps.
The mass-action law can once again be applied to develop rate equations for the elementary steps in the mechanisms, after which the net rate of consumption (or formation) of each species can be worked out from a simple material balance. These are summarised in Table 1.1.
Reaction scheme . | Rate equations . | Species molar balance . |
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Series | r1 = k1CA | |
r2 = k2CB | ||
Parallel | r1 = k1CA | |
r2 = k2CA | ||
Series–parallel | r1 = k1CA | |
r2 = k2CB | ||
r3 = k3CB | ||
Reaction scheme . | Rate equations . | Species molar balance . |
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Series | r1 = k1CA | |
r2 = k2CB | ||
Parallel | r1 = k1CA | |
r2 = k2CA | ||
Series–parallel | r1 = k1CA | |
r2 = k2CB | ||
r3 = k3CB | ||
1.1.2.3 Mass Transfer Limitations
Our discussion so far has only concerned intrinsic kinetics whereby the reaction chemistry is the rate-limiting factor. In reality, both homogeneous and heterogeneous reactions encompass other physical phenomena (such as heat and mass transport) that may kinetically control the reaction process. For example, solid-catalysed reactions may be internal mass transfer limited whereby the diffusion of species within the catalyst pores is rate-limiting, or external mass transfer limited whereby interfacial transport between the catalyst surface and the bulk fluid is rate-limiting. In either case, there is a mismatch between the observed reaction rate and the intrinsic rate that would be expected in the absence of mass-transfer limitations.
1.1.3 Biochemical Kinetic Models
In the field of bioprocess kinetics, most modeling methodologies can be classified into structured and unstructured approaches. Structured models are detailed mathematical representations of the metabolism of microorganisms, including information such as microbial structure and physiology to describe cell cultures whose morphology and composition are strongly time-dependent.4 These are particularly useful for metabolic engineering and cellular regulatory process studies.5 On the other hand, unstructured approaches assume the cell culture to be a homogeneous biomass and the dynamics of cell growth, substrate consumption and extracellular product formation are all treated from a macroscopic perspective. These approaches therefore provide a compromise between the mathematical complexity of the model and the incorporation of fundamental microbial kinetics knowledge, thus serving as a powerful tool for the control and optimisation of bioprocesses.6
In this section of the chapter, we will discuss some common unstructured kinetic models and show how these can be mechanistically derived by approximating the cellular metabolism as an enzymatic reaction mechanism. Given the semi-empirical nature of unstructured models, the reader should note that many theoretical derivations have been proposed in the past by incorporating different assumptions related to kinetic, thermodynamic and/or substrate transport characteristics of the biochemical system.7 As such, some of the kinetic parameters that arise in unstructured models do not have a universal physical meaning but rather depend on the assumptions made during model construction.
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The rate constant k2 (h−1) is analogous to the maximum specific biomass growth rate. We notice that, for cultures where the substrate is present in excess , the biomass growth rate equation simply reduces to ; this of course corresponds to a situation where all of the biomass is saturated with the substrate (X = XS).
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The term is typically known as the Monod constant, and it is representative of the affinity of the substrate to the cellular metabolism. We observe that if the assimilation of the substrate is much faster than the desorption of growth products (i.e. k1 ≫ k2), we once again obtain the maximum specific growth rate. This constant is also referred to as the half-saturation constant, given that when the growth rate is half the maximum .
Owing to physiological dissimilarities between different microorganisms, it is ultimately not possible to derive a “master” unstructured model that captures the growth and substrate utilisation dynamics of any microbial population. In some cases, the same microbial culture might even exhibit different dynamic behaviours over a range of conditions due to the influence of metabolic regulations and other intracellular mechanisms.8 Table 1.2 summarises some typical unstructured models in the literature, many of which can also be interpreted as kinetic models for enzymatic reactions.
Description . | Model structure . | Comments and underlying assumptions . |
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Monod model with cellular decay | The rate of endogenous biomass decay is assumed to be first order with respect to biomass concentration. | |
Multiplicative Monod model | Both substrates are required and hence the growth is co-limited. | |
Additive Monod model | Substrates are substitutable and contribute individually to growth. | |
Additive Monod model with competitive substrate inhibition | Can be derived by assuming the substrates compete for the same enzyme intermediate.9 Notice the similarity with Langmuirian model structures. | |
Contois model | Mechanistic derivations considering diffusional barriers and cell flocculation have been proposed.10 | |
Haldane/Aiba/Andrew model for substrate inhibition | High substrate concentrations can inhibit growth; for instance, this structure has been used to model photoinhibition in photosynthetic organisms.11 | |
Tesser model | Analogous to a Monod model with exponential concentration dependences. Can be derived by considering cooperative binding mechanisms.12 | |
Substrate utilisation with maintenance | The rate of maintenance-related substrate utilisation is assumed proportional to biomass concentration. |
Description . | Model structure . | Comments and underlying assumptions . |
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Monod model with cellular decay | The rate of endogenous biomass decay is assumed to be first order with respect to biomass concentration. | |
Multiplicative Monod model | Both substrates are required and hence the growth is co-limited. | |
Additive Monod model | Substrates are substitutable and contribute individually to growth. | |
Additive Monod model with competitive substrate inhibition | Can be derived by assuming the substrates compete for the same enzyme intermediate.9 Notice the similarity with Langmuirian model structures. | |
Contois model | Mechanistic derivations considering diffusional barriers and cell flocculation have been proposed.10 | |
Haldane/Aiba/Andrew model for substrate inhibition | High substrate concentrations can inhibit growth; for instance, this structure has been used to model photoinhibition in photosynthetic organisms.11 | |
Tesser model | Analogous to a Monod model with exponential concentration dependences. Can be derived by considering cooperative binding mechanisms.12 | |
Substrate utilisation with maintenance | The rate of maintenance-related substrate utilisation is assumed proportional to biomass concentration. |
1.2 Fundamentals of Deterministic Optimisation
Optimisation plays a vital role in both development and application of mathematical process models. This section aims to provide an introduction to the formulation and solution of optimisation problems, two recurring themes in this book due to their role in model parameter estimation, model-based design of experiments and reactor optimisation.
1.2.1 Formulation and Optimality of Constrained Optimisation Problems
For most constrained optimisation problems in reaction engineering, the KKT conditions in eqn (1.33)–(1.37) cannot be solved analytically. In such cases, optimisation algorithms must be used, whereby the original optimisation problem is iteratively solved through a sequence of relaxed subproblems, in an attempt to find a solution that satisfies the KKT conditions. To demonstrate this, interior-point optimisation algorithms are discussed in the next subsection.
1.2.2 Solving Nonlinear Optimisation Problems
A wide variety of approaches can be adopted to solve nonlinear programming problems. Here we will deal with deterministic (or gradient-based) optimisation methods, as they guarantee the optimality of the solution and converge satisfactorily for optimisation problems with a relatively low number of decision variables.17 This is of course a very general statement and it should be noted that stochastic methods have also found applications in the chemical engineering field, particularly for optimisation problems involving non-differentiable process models and highly stiff objective functions.18
There are a number of considerations that fall beyond the scope of this discussion. Namely, the careful reader will have already noticed that we have not elucidated how the barrier parameter β is updated after each iteration, or what convergence criteria are used to terminate both the numerical approximation sub-algorithm and the overall optimisation algorithm. Moreover, state-of-the-art interior point optimisation methods such as open source IPOPT incorporate further improvements, like line-search filter methods and second-order corrections, to provide more robust convergence. For a detailed breakdown of interior-point methods, we would guide the reader to ref. 22.
1.3 Dynamic Optimisation and Parameter Estimation
1.3.1 Formulation of Parameter Estimation Problems
More often than not, we do not know a priori the ground truth that comprises the underlying reaction mechanism and its associated rate constants. Moreover, multiple model structures can provide similar fitting accuracy despite incorporating vastly different mechanistic assumptions in their derivations. It is then essential to combine the parameter estimation framework alongside statistical model discrimination methods and model-based design of experiments in order to identify the most plausible model structure. This is further discussed in Chapter 4.
Such modified parameter estimation problems are adopted in Chapter 3 to enforce smoothness of time-varying kinetic parameters by preventing them from changing drastically over short time intervals, or in Chapter 8 where a regularisation term is used to penalise the number of non-zero reaction constants in a complex microkinetic model (thus aiding in model structure identification).
1.3.2 Discretisation of Dynamic Process Constraints
To solve the parameter estimation problem (eqn (1.52)–(1.54)) formulated in the previous section, the differential process constraints in eqn (1.53) must be first converted into algebraic equality constraints that correspond to the time-profile of the state variables. Generally, the analytical solution of the differential constraints is not known, and so numerical discretisation methods must be used. The choice of discretisation algorithm is not trivial as generally there is a trade-off between the numerical accuracy of the approximated derivative and the computational cost of the algorithm.23 In this section, we will discuss orthogonal collocation over finite elements; this discretisation method is well-established and allows for accurate approximation of ODEs with relatively few finite elements, therefore being very favourable for optimisation problems with multiple state variables.24
1.4 Conclusion
The principles of physical model construction for chemical and biochemical reaction processes have been described in this chapter. First, the development of mechanistic kinetic models for simple chemical reaction networks was elucidated through the application of the mass-action law and the pseudo-steady-state hypothesis. We then discussed how some typical biochemical kinetic models can be similarly derived by approximating the cellular metabolism to enzymatic reaction schemes. Finally, the fundamentals of deterministic optimisation were introduced with particular focus on the formulation and solution of constrained optimisation problems.