Bioreactor
The bioreactor example is an easy bioreactor with an substrate that is converted to a product by the biomass in the reactor. It has three states and a control that is describing the feed concentration of substrate. It is taken from the examples folder of the ACADO toolkit described in:
Houska, Boris, Hans Joachim Ferreau, and Moritz Diehl. "ACADO toolkit—An open‐source framework for automatic control and dynamic optimization." Optimal Control Applications and Methods 32.3 (2011): 298-312.
Originally the problem seems to be motivated by:
VERSYCK, KARINA J., and JAN F. VAN IMPE. "Feed rate optimization for fed-batch bioreactors: From optimal process performance to optimal parameter estimation." Chemical Engineering Communications 172.1 (1999): 107-124.
Model Formulation
The dynamic model is an ODE model:
The three states describe the concentration of the biomass (), the substrate (
), and the product (
) in the reactor. In steady state the feed and outlet are equal and dilute all three concentrations with a ratio
. The biomass grows with a rate
, while it eats up the substrate with the rate
and produces product at a rate
. The rate
is given by:
The fixed parameters (constants) of the model are:
Name | Symbol | Value | Unit |
Dilution | ![]() |
0.15 | [-] |
Rate coefficient | ![]() |
22 | [-] |
Rate coefficient | ![]() |
1.2 | [-] |
Rate coefficient | ![]() |
50 | [-] |
Substrate to Biomass rate | ![]() |
0.4 | [-] |
Linear slope | ![]() |
2.2 | [-] |
Linear intercept | ![]() |
0.2 | [-] |
Maximal growth rate | ![]() |
0.48 | [-] |
This would result in an right hand side function in this case, because there are no free parameters. Of course one could also specify an implicit DAE system (
), where it just hast to be said which states are differential and which are algebraic.