Abstract
Modeling and optimisation of batch processes has been an emerging research field in the recent years. This work introduces the model based optimisation of a batch process, the production of fatty acid methyl esters (biodiesel). The aim of this work is to point out the importance of objective function formulation that greatly affects the optimal values of decision variables.
A transesterification process takes place in a stirred vessel with jacket heating and cooling. The energy flow is manipulated by the flow rate and inlet temperature of the heat transfer liquid. A controller is also included to ensure that the temperature is near the optimal trajectory.
The three studied objective functions are the maximal product concentration (purity of the product), the maximal production rate, and maximal profit generated during production. The most important variables, that may affect the process, are the initial concentrations and the temperature profile during the process. In this work mainly the latter has been studied.
The use of a controller reduces the number of decision variables: instead of numerous heating/cooling energy flow values the most important points of the temperature set-point trajectory are given. They were chosen as follows: the starting high temperature, the time of switching from high to low temperature, and the time of finishing the process. There is no need for separate temperature set-point values for the heating and temperature keeping phases. The controller parameters are also included in the decision variables. All the decision variables have lower and upper limits, and there is a nonlinear constraint that represents the cooling of the reactor under a prescribed temperature at the end of the process.
Numeric solvers were stuck in local minima, thus it became an interesting thought to map the objective function values throughout the whole range of the decision variables, and find a starting point closer to the optimal one. As a result we obtained approximations of the Pareto-fronts, as all the studied objective functions proved to be competing. Also the results show significant differences in the optimal temperature trajectories. The values in the case of the economical objective lie between the values in the cases of the maximal production rate and maximal concentration.
