Batch Process Optimization

Description of the Optimization Problem

This application example deals with optimization of brewery fermentation.

The beer fermentation is a batch process. In the batch and fed-batch

pro-cesses, there is no steady-state, i.e. the states of system change considerably with time. As a consequence of the varying process states, the best operation results can be realized by varying the input variables along optimal trajec-tories during the operation time. This explains why searching for efficient methods for calculating the optimal trajectories has been an important issue for bioreactor control [44]. This example illustrates how the proposed IEC optimization technique can be applied for this kind of problems.

To simulate the beer fermentation process, the model published by Carrillo-Ureta [45] was used. This model was developed from experimental data and shows good results in the aspect of a realistic view of the industrial fermen-tation process. The model takes into account seven components: three com-ponents of the biomass (latent, active, dead), ethanol and sugar, and two important byproducts: ethyl acetate and diacetyl. Most of the process param-eters vary as Arrhenius function of temperature, expect diacetyl appearance and disappearance rate which are constant values. The detailed model of the process can be found in Sect. A.2.

During the beer fermentation, a temperature profile is applied to drive the process. The design of this temperature profile is an optimization problem where the objective is to minimize the operation time, maximize the amount of ethanol, and optimize the quality of beer. In order to apply an optimization algorithm for this problem, it is necessary to design a suitable representation of temperature profile. Temperature profile was represented as a piecewise-linear function with six segments, and every chromosome was composed of the time and temperature values of breakpoints, e.g. Fig. 2.7. Certainly, it is possible to use more than five inner breakpoints, but increasing the number of segments unnecessarily increases the complexity of the optimization problem.

Furthermore, when a large number of segments are used, the optimization may result in a very abrupt profile.

0 50 100 150

7 7.5 8 8.5 9 9.5 10 10.5 11

time [h]

temp. [o C]

Fig. 2.7. Representation of temperature profile x = [30 50 90 130 150 10 9 9 11 11 7]

2.3 Application Examples 29 The temperature profile must obey to certain constraints. The temperature range was set from 8 to 16 degrees centigrade taking into account equipment requirements and temperature profiles used in the beer fermentation indus-try [45]. (Temperature below 8^oC needs too much cold energy, temperature above 16^oC may cause contamination.) In addition, the minimal time differ-ence between a pair of breakpoints was 10 h to constrain the derivative of temperature. The maximum allowed operation time was 160 h. These con-straints were realized at the simulation step, i.e. they were not introduced into the IEC algorithm directly.

Simulation Results

The optimization goal in this experiment is to find a temperature profile that results in short operation time, high ethanol concentration and good quality of beer. The last means that the byproduct concentrations at the end of the fermentation be small, because both byproducts (ethyl acetat and diacetyl) have bad effect on the taste and aroma of beer. Furthermore, it is important to avoid contamination, but this objective was not applied in this application example, because, under industrial conditions (initial concentration of sugar, biomass, etc.), the risk of contamination is not significant if the temperature is below 16^oC [46].

Transforming of these objectives to a quantitative cost function is not a straightforward task, because there is no exact equation that determines the relationship between the byproduct concentrations and the quality. In ad-dition, these objectives are in conflict with each other. For example, if the operation time is short, small amount of ethanol will be produced; very low concentration of byproducts needs long operation time. The great advantage of the proposed IEC approach that a human user is able to balance these objectives easily without any quantitative cost function. Moreover, the user can manually modify the temperature profile during the interactive optimiza-tion. It means that the user is able to introduce the heuristic trial-and-error method into the optimization procedure.

For comparison, the optimal temperature profile mentioned by Carrillo-Ureta et al. [45] was selected as ”reference”. It should be noted that the same model and constraints were used as in [45]. Two experiments were conducted.

The goal of the first experiment was to find a temperature profile that results in similar final ethanol concentration, and lower final byproduct concentra-tions with the same operation time as the reference. The goal of the second experiment was to find a temperature profile that results in similar quality (final ethanol and byproduct concentrations) with shorter operation time as the reference. I obtained the solutions (IEC-1 from the first experiment, and IEC-2 from the second experiment) relatively quickly, after 14 generations in the first experiment and after 16 generations in the second experiment.

Table 2.5 shows some comparisons of the final concentrations of the three

temperature profiles, Figure 2.8, Fig. 2.9 and Fig. 2.10 shows the temperature and concentration trajectories.

Table 2.5. Numerical results of beer fermentation optimization Ceth (g/l)C^acet (mg/l)Cdiacet (mg/l)tend (h)

Reference^a 58.6 8.26 0.0055 160

IEC-1^b 56.2 6.27 0.016 160

IEC-2^c 58.7 8.35 0.0097 144

aFrom the literature [45]

bResult of the first experiment

cResult of the second experiment

With the IEC-1 temperature profile, the final ethyl acetate concentration decreased considerably compared to the reference. It means that the quality improved. At the same time the ethanol concentration decreased to a small degree. The didactyl concentration is significantly bigger, but the tolerance concentration of diacetyl in beer is 0.1 mg/l [47], so this value is acceptable.

The IEC-2 temperature profile improved the operation time with 16 hours (with 10 per cent), moreover it improved the ethanol concentration, even if only to a slight degree. At the same time, the quality decreased slightly, because the ethyl acetate concentration became higher.

The experiments showed that it is possible to increase the quality (this is the goal of the first experiment) or to accelerate the process (this is the goal of the second experiment), but at the cost of productivity (first experiment) or quality (second experiment). It is not surprising because different objectives offer a set of equally good solutions in Pareto’s sense, and the reference profile is a member of this Pareto set. But still the temperature profiles obtained with IEC satisfy the goals that it was aimed at. It supports the conclusion that the proposed IEC framework is useful and flexible tool, and with this tool it is possible to reach different goals relatively quickly and easily.