Process modelling

The key of the optimal control problem is generally regarded as being a reliable and accurate model of the process. For many years, the dynamics of bioprocesses in general have been modelled by a set of first or higher order nonlinear differential equations [14]. These mathematical models can be divided into two different categories: structured models and unstructured models. Structured models represent the processes at the cellular level, whereas unstructured models represent the processes at the population (extracellular) level.

Lei et al. [15] proposed a biochemically structured yeast model, which was a moderately complicated structured model based on Monod-type kinetics. A set of steady-state chemostat experimental data could be described well by the model. However, when applied to a fed-batch cultivation, a relatively large error was observed between model simulation and the experimental data. Another structured model to simulate the growth of baker's yeast in industrial bioreactors was presented by Di Serio et al. [16]. The detailed modelling of regulating processes was replaced by a cybernetic modelling framework, which was based on the hypothesis that microorganisms optimize the utilization of available substrates to maximize their growth rate at all times. From the simulation results that were plotted in the paper, the model prediction agreed reasonably well with both laboratory and industrial fed-batch fermentation data that were adopted in the study. Unfortunately, detailed error analysis neglected to show what degree of accuracy could be achieved by the model. The limitation of the model, as pointed out by the authors, was that the model and it's parameters needed to be further improved for a more general application.

A popular unstructured model for industrial yeast fermenters was reported by Pertev et al. [17]. The kinetics of yeast metabolism, which were considered to build the model, were based on the limited respiratory capacity hypothesis developed by Sonnleitner and Kappeli [18]. The model was tested for two different types of industrial fermentation (batch and fed-batch modes). The results showed that it could predict the behaviors of those industrial scale fermenters with a sufficient accuracy. Later, a study carried on by Berber et al. [19] further showed that by making use of this model, a better profile of substrate feed rate could be obtained to increase the biomass production, while in the mean time, decreasing the ethanol formation. Recent application of the model has been to evaluate various schemes for controlling the glucose feed rate of fed-batch baker's yeast fermentation [20]. Because intracellular state variables (i.e., enzymes) are not involved in unstructured models, it is relatively easy to validate these kinds of models by experiments. This is why unstructured models are more preferable than structured models for optimization and control of fermentation processes. However, unstructured models also suffer the problems of parameter identification and large prediction errors.

The parameters of the model vary from one culture to another. Conventional methods for system parameter identification such as Least Squares, Recursive Least Squares, Maximum Likelihood or Instrument Variable work well for linear systems. Those schemes, however, are in essence local search techniques and often fail in the search for the global optimum if the search space is not differentiable or is nonlinear in parameters.

Though a considerable effort has been made in developing detailed mathematical models, fermentation processes are just too complex to be completely described in this manner. "The proposed models are by no means meant to mirror the complete yeast physiology ..." [15]. From an application point of view, the limitations of mathematical models are:

• Physical and physiological insight and a priori knowledge about fermentation processes are required.

• Only a few metabolites can be included in the models.

• The ability to cope with batch to batch variations is poor .

• These models only work under idea fermentation conditions.

• A high number of differential equations (high order system) and parameters are presented in the models, even for a moderately complicated model.

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