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:

$\begin{array}{rcl} \dot{X}&=&-DX+\mu X \\ \dot{S}&=& D(S_{f}-S)-\mu /Y_{xs} X \\ \dot{P}&=&-DP+ (\alpha \mu +\beta) X. \end{array}$

The three states describe the concentration of the biomass ($X$), the substrate ($S$), and the product ($P$) in the reactor. In steady state the feed and outlet are equal and dilute all three concentrations with a ratio $D$. The biomass grows with a rate $\mu$, while it eats up the substrate with the rate $\mu/Y_{xs}$ and produces product at a rate $(\alpha \mu +\beta)$. The rate $\mu$ is given by:

$\mu = \mu_{m}*(1-P/P_{m})*S/(K_m+S+S^2/K_i)$

The fixed parameters (constants) of the model are:

Parameters

Name	Symbol	Value	Unit
Dilution	$D$	0.15	[-]
Dilution	$D$	0.15	[-]
Dilution	$D$	0.15	[-]

const double Ki = 22.0;

   const double Km      = 1.2 ; 
   const double Pm      = 50.0;
   const double Yxs     = 0.4 ;
   const double alpha   = 2.2 ;
   const double beta    = 0.2 ;
   const double mum     = 0.48;

This would result in an right hand side function $f(x,u)$ in this case, because there are no free parameters. Of course one could also specify an implicit DAE system ($F(\dot{x},x,u,p,t)=0$), where it just hast to be said which states are differential and which are algebraic.