Abstract
Optimal enzyme feed rate profiles have been calculated, based on a model for a fed-batch simultaneous enzymatic and microbial reaction (SEMR) process. The model parameters corresponded to a relatively slow citric acid fermentation. The profiles were calculated using an iterative algorithm based on the minimum principle. Penalty functions were used to enforce inequality constraints on the enzyme feed rate. Significant improvements in the objective function relative to that for the best constant enzyme feed rate were found. The effect on the optimal profiles of changes in the parameters of the model and the objective function were investigated, as was the effect of introducing the stationary state assumption to eliminate glucose concentration as a state variable. Major differences between bang-bang control variable profiles and singular arcs were found, with the singular arc solution slightly better than the optimal bang-bang control.
Similar content being viewed by others
Abbreviations
- a :
-
N-vector of initial conditions
- b 1–b10 :
-
parameters defined in Table 2
- c :
-
vector of cost parameters
- c 1–c6 :
-
penalty function parameters
- E :
-
enzyme concentration (U/l)
- f :
-
N-vector of functions
- F :
-
enzyme feed rate (U/l-h)
- g :
-
N-vector of functions
- G :
-
glucose concentration (g/l)
- H :
-
Hamiltonian
- J :
-
objective function
- J * :
-
modified objective function
- L :
-
number of integration steps per time interval
- L :
-
number of control variables
- M :
-
number of time intervals
- n :
-
iteration index
- N :
-
number of state variables
- P :
-
product concentration (g/l)
- r 1 :
-
glucose formation rate (g/l-h)
- r 2 :
-
product formation rate (g/l-h)
- t :
-
time (h)
- T :
-
final time (h)
- u :
-
L-vector of control variables
- x :
-
N-vector of state variables
- z :
-
N-vector of adjoint variables
- Z :
-
total enzyme fed (U/l)
- β :
-
convergence parameter
References
Asenjo, J.A.;Sun, W-H.;Spencer, J.L.: Optimization of batch processes involving simultaneous enzymatic and microbial reactions. Biotech. Bioeng. 37 (1991) 1087–1094
Constantinides, A.;Spencer, J.L.;Gaden, E.L.: Optimization of batch fermentation processes: Development of models for batch penicillin fermentations. Biotech. Bioeng. 12 (1970) 803–830
Cazzador, L.: On the optimal control of fed-batch reactors with substrate-inhibited kinetics. Biotech. Bioeng. 31 (1988) 670–674
San, K.Y.;Stephanopoulos, G.: A note on the optimality criteria maximum biomass production in a fed-batch fermentor. Biotech. Bioeng. 26 (1984) 1261–1264
San, K.Y.;Stephanopoulos, G.: Optimization of fed-batch penicillin fermentation: A case of singular optimal control. Biotech. Bioeng. 28 (1986) 72–78
Cuthrell, J.E.;Biegler, L.T.: Simultaneous optimization and solution methods for batch reactor control profiles. Comp. Chem. Engg. 13(1/2) (1989) 49–62
Lim, H.C.;Tayeb, Y.J.;Modak, J.M.;Bonte, P.: Computational algorithms for optimal feed rates for a class of fed-batch fermentations: Numerical results for penicillin and cell mass production. Biotech. Bioeng. 28 (1986) 1408–1420
Kurtanjek Z.: Optimal nonsingular control of fed-batch fermentation. Biotech. Bioeng. 37 (1991) 814–823
Luus, J.: Optimization of fed-batch fermentors by iterative dynamic programming. Communication to the Editor, Biotech. Bioeng. 41 (1993) 599–602
Lee, J.;Ramirez, W.F.: Optimal fed-batch control of induced foreign protein production by recombinant bacteria. AIChE J. 40 (1994) 899–907
Denn, M.: Optimization by variational methods. New York: McGraw-Hill 1969
Lewis, F.L.: Optimal Control. New York: John Wiley and Sons 1986
Bryson, A.E.;Ho, Y-C.: Applied Optimal Control. New York: Hemisphere Publishing 1986
Sun, W-H.: Optimization of batch simultaneous enzymatic and microbial reaction processes. Doctoral dissertation, Columbia University, 1989
Author information
Authors and Affiliations
Additional information
The support of one of the authors by the National Science Foundation (Grant CBT-84-20552) is gratefully acknowledged.
Rights and permissions
About this article
Cite this article
Asenjo, J.A., Sun, W.H. & Spencer, J.L. Optimal control of batch processes involving simultaneous enzymatic and microbial reactions. Bioprocess Engineering 14, 323–329 (1996). https://doi.org/10.1007/BF00369477
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00369477