An invariant manifold approach for CSTR model reduction in the presence of multi-step biochemical reaction schemes. Application to anaerobic digestion

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Abstract

A systematic methodology for CSTR model reduction has been developed for multi-step biochemical reaction schemes. The proposed method neglects the dynamics of the fast steps by projecting the overall system dynamics on the slow-motion invariant manifold of the system. In particular, using reaction invariants in the description of the overall system dynamics, the slow-motion invariant manifold is calculated by solving the pertinent invariance equation via series-solution or singular perturbation techniques. The proposed method is an alternative to the quasi-steady state approximation which does not rely on a priori physical information. The proposed approach is applied to two model reduction problems arising in anaerobic digestion. The results provide a rigorous answer on how to properly eliminate the fast dynamics of the acidogenesis step.

Introduction

Multi-step biochemical processes involve a variety of reaction schemes that relate with life. In every living organism, numerous reactions evolve simultaneously or sequentially with the aim of producing energy for production of new cells and for maintenance. From an engineering point of view, the dynamics of biochemical process schemes involved, either at a single cell level or at the level of pure or mixed microbial cultures, are particularly interesting. In particular, mixed microbial processes are quite common in numerous applications in biochemical engineering and environmental technology. In such complex microbial environments as in the activated sludge process or the anaerobic digestion process, the interactions between different microbial groups are complicated and extremely diverse (mutualistic, competitive, amensalistic, prey–predator) [1]. There are a variety of mathematical structured and unstructured models of varying complexity, aiming at describing the microbial interactions and the dependence of the process performance on the bioreactor feed characteristics and the prevailing operating conditions.

Recognizing the importance and general applicability of some of these bioprocesses, such as activated sludge and anaerobic digestion, the International Water Association (IWA) formed task groups that developed typical generic models such as the activated sludge model (ASM) [2] and the anaerobic digestion model (ADM) [3]. The IWA model for anaerobic digestion [3] defines a framework for modelling a complicated process, based on the previous kinetic and modelling studies of various researchers [4], [5], [6], [7]. The models developed by the task groups reflect the current understanding of the key biological and physicochemical processes that take place in such complex environments. In that regard, they are quite useful as process simulators, while they also offer excellent educational tools. Since these models are sufficiently structured they must always be properly modified and calibrated in order to adequately reflect a particular situation at hand.

The use of complex models such as ASM and ADM, although valuable for general process simulation, has severe shortcomings if they are intended to be used for process control and optimisation [8]. This is because there are difficulties in determining the numerous model parameters (non-identifiability of parameters), while manipulating a large number of equations limits the applicability for the dynamic analysis, process simulation and control. In addition, although the model assumptions reflect quite well our current understanding of the physical processes involved, many of the individual steps may actually be so fast so that they do not influence the overall process dynamics. Simpler models are needed, that adequately describe the dynamics of the key measured variables. Such a reduction is meaningful from an intuitive point of view, since (a) some of the processes may have no or small impact on the measured process variables of interest, i.e. they may be unobservable, or they may be negligible (b) some processes may be lumped to fewer processes without again any appreciable loss of information, (c) some processes exhibit faster transients over a given time scale than others and, therefore, can be assumed to undergo instantaneous changes (d) for the specific operating conditions, stoichiometric constraints between the state variables may be valid at all times and as a result may be used to reduce the number of state variables. In complex multivariable systems such as biochemical systems, however, this intuitive basis of simplification relying on physical arguments may be risky, since the individual dynamics of the variables may be interrelated and the distinction between slow and fast often becomes obscure.

An excellent review of the existing mathematical methods for model reduction in chemical reaction systems has recently been made by Okino and Mavrovouniotis [9]. There, three general types of model reduction methods are reviewed: lumping, time-scale methods and invariant manifold methods. Such methods have been applied to reduce models for complex chemical systems [10], [11], [12], [13], [14], as well as complex biochemical pathways [15], [16], [17], [18]. The Quasi Steady State (QSS) approximation has widely been used in (bio)chemical engineering [19], [20], [21] as well. Actually, the QSS approximation can be justified mathematically by applying perturbation analysis techniques, through which the dynamics of some states of the system (e.g. products with low solubility in the liquid phase or substrates involved in fast reactions) can be neglected [8], [22].

These methods may be used to justify or reject a particular model reduction that has been made based on phenomenological arguments. However, to our knowledge, only limited amount of work has been done on developing mathematically solid model reduction for complex environmental processes that contain mixed cultures of microbial species.

Such systems contain a significant amount of biomass, maintained in the system through immobilisation or recycle, such as biofilm systems and the activated sludge process respectively, and are characterised by relatively slow changes in the biomass in comparison with the various soluble substrates and metabolic intermediates that are rapidly adjusted to reflect the feed characteristics and the mixed culture composition. Thus, it is common knowledge that, in such systems the overall biomass concentration and the individual microbe composition vary with time constants that are often several orders of magnitude higher than the hydraulic retention time (inverse dilution rate). However, even in cases of suspended growth systems, in which particulate biomass and substrates are removed at the same rate from the reaction system, there may be additional valid basis for model reduction. In particular, the basis for model reduction in such systems arises from the existence of fast and slow bioconversion steps (which is often the result of wide differences in specific growth rates of the involved species). In addition, the microbial network structure may often allow for approximate decoupling of the fast steps from the slow steps, allowing for further model simplification.

In this paper, a systematic methodology is proposed for the reduction of dynamic models for multi-step biochemical reaction schemes in a CSTR. The proposed methodology is based on an invariant manifold formulation of the model reduction problem, and in particular, on projecting the CSTR dynamics on the slow-motion invariant manifold. To be able to compute the slow-motion invariant manifold, the proposed method makes use of reaction invariants [10], [12], [23], [24] in the description of CSTR dynamics, and subsequently the invariance equation is solved via series solution or singular perturbation techniques [13], [14].

Once the general methodology is presented, it is applied to an important mixed culture multi-step biochemical process, the anaerobic digestion process. The importance of this process results from the fact that it finds wide application in municipal sludge treatment, in the treatment of high organic strength industrial wastewaters, in the treatment of the organic fraction of municipal solid waste and in the recent years, in the exploitation of energy crops for the production of biogas [25]. In this process, a complex microbial consortium degrades complex organic material, generating biogas (a methane and carbon dioxide mixture) which is a useful renewable energy source. The application of the proposed method results in reducing the model order to a system consisting only of what is called in chemical engineering “rate limiting steps”, namely, hydrolysis of particulate matter and acetoclastic methanogenesis [26], [27], [28], [29]. In this way, model reduction is justified through a rigorous mathematical and systematic methodology, which is complementary to the QSS approximation methodology (as applied by Perrier and Dochain [8] and Dochain and Vanlrolleghem [22].

It should be stressed, however, that although the proposed methodology has been developed for mixed microbial processes, it can also be applied to other non microbial biochemical systems, such as metabolic pathways, as well as to general chemical reaction systems, following minor modifications.

Section snippets

General model for a continuous stirred tank biochemical reactor

The biochemical reactions by mixed microbial cultures, involve numerous chemical species consumed (substrates) and produced (intermediate or final metabolic products) and microbial groups mainly grown. Chemical species produced by a microbial group are often the substrate for the growth of other microbial groups, making the whole process a sequence of individual process steps in a scheme where the preceding steps may be independent of those that follow.

Assuming that biochemical reactions,

A very simple motivating example

To illustrate the ideas outlined in the previous section, a very simple example is discussed. Consider a chemostat described by the following mathematical modeldXdt=DX+YrdSdt=DS+Frwhere X and S are the biomass and substrate concentrations respectively, D is the dilution rate, Y is the biomass yield, F = D·So is the feed rate of the substrate, So is the substrate concentration in the feed, r=1/YμmaxS/(KS+S)X is the reaction rate, μmax is the maximum specific growth rate constant of the

Proposed approach for model reduction

The goal of this section is to develop a systematic model reduction methodology for mixed culture bioprocesses described by a model of the form (2) or (3), based on projecting the system dynamics on the slow-motion invariant manifold, in the sense described qualitatively in the previous section.

Following the theoretical results of Kazantzis and Kravaris [14], the proposed methodology involves two steps:

  • (1)

    identifying and computing the appropriate slow-motion attractive invariant manifold for the

Application to anaerobic digestion

The model reduction methodology outlined in the previous section will now be applied to two cases of anaerobic digestion. In the first case, a soluble substrate is fed to a chemostat and the processes taking place are acidogenesis and methanogenesis. In the second case, a particular substrate is fed, with a hydrolysis step preceding acidogenesis and methanogenesis. The fast dynamics is the first process step in the first case, while in the second case it lies between two slower steps. As a

Conclusions

Biochemical reaction systems are multistep processes. The rates of some individual steps may differ significantly, while the rates of other steps may vary at similar levels. In this work, we have developed a systematic methodology for the reduction of the order of the general system considered, based on the calculation of the slow-motion invariant manifold, on which the process dynamics is projected. Further reduction in the system order may be achieved if certain slow modes have negligible

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