## Brazilian Journal of Chemical Engineering

*On-line version* ISSN 0104-6632

### Braz. J. Chem. Eng. vol. 14 no. 4 São Paulo Dec. 1997

#### http://dx.doi.org/10.1590/S0104-66321997000400001

**MATHEMATICAL MODELING, AUTOMATION AND CONTROL OF THE BIOCONVERSION OF SORBITOL TO SORBOSE IN THE VITAMIN C PRODUCTION PROCESS I. MATHEMATICAL MODELING**

**A. Bonomi, A.T. Fleury, E.F.P. Augusto, M.N. Mattos and L.R. Magossi**

Agrupamento de Biotecnologia - Divisão de Química - Instituto de Pesquisas Tecnológicas do Estado de São Paulo S/A - Caixa Postal 7141, 01064-970 São Paulo, SP - Brazil

Phone: (011) 268-2211 EXT. 543 - Fax: (011) 869-3566

*(Received:June 11, 1997; Accepted: October 30, 1997 )*

Abstract -In 1990, the Biotechnology and the Control Systems Groups of IPT started developing a system for the control and automation of fermentation processes, applied to the oxidation of sorbitol to sorbose by the bacteriaGluconobacter oxydans, the microbial step of the vitamin C production process, that was chosen as a case study. Initially, a thirteen-parameter model was fitted to represent the batch operation of the system utilizing a nonlinear regression analysis, the flexible polyhedron method. Based on these results, a model for the continuous process (with the same kinetic equations) was constructed and its optimum operating point obtained.Mathematical modeling, control, bioconversion.

Keywords:

**INTRODUCTION**

With the objective of developing a technology for the mathematical modeling, simulation, automation and control of fermentation processes, the Biotechnology Group of the Chemistry Division and the Control System Group of the Mechanical and Electrical Division of IPT established and applied a set of computational tools. This made it possible to implement, on a laboratory scale, a control system applied to the bioxidation of sorbitol to sorbose, the key step in the Reichstein process of vitamin C production, largely employed in the pharmaceutical and food industries (Florent, 1986). In this process, glucose is converted to ascorbic acid through a combination of several stages, including chemical reactions, physicochemical operations and a single biochemical step, the bioconversion of D-sorbitol to L-sorbose by a bacterial strain of *Gluconobacter oxydans* (Boudrant, 1990). This fermentation process was selected as the case study for the proposed development due to its remarkable characteristics of limitation and inhibition by different substrates and product (IPT, 1988).

Several experimental runs were performed in order to study the process, to define the state variables and to identify the most important biological phenomena involved; all this knowledge is important in the formulation of the mathematical model. The proposed unstructured mathematical model was fitted to the experimental data and the adjusted parameters were obtained using a nonlinear regression analysis, namely the flexible polyhedron technique (Himmelblau, 1972).

Economic considerations, with a view to possible industrial application of the process under study, indicated that automation and control would need to be developed to achieve a continuously operating system. For this reason, the model developed for batch operation had to be fitted again to a set of continuous runs performed in the laboratory, with operating variables defined by a performance index that accounts for the most relevant parameters of industrial operation.

**MATERIALS AND METHODS**

**Microorganism**

All the experiments were performed utilizing a strain of the bacteria *Gluconobacter oxydans*, ATCC-621, which was maintained lyophilized in the IPTs Culture Collection.

**Culture Media**

All the culture media utilized (solid to maintain the culture and liquid for flasks or reactor runs) are based on the Mori et al. (1981) formulation.

**Table 1: Range of nominal conditions for the performed batch experiments**

Conditions | Nominal Range |

Sorbitol concentration (initial values) (g.l^{-1}) | 10.0 - 300. |

Yeast extract concentration (initial values) (g.l^{-1}) | 0.0 - 20.0 |

Dissolved O_{2} concentration (controlled values) (mg.l^{-1}) | 0.375 - 6.00 |

K_{L}a (constant values) (h^{-1}) | 60.0 - 300. |

**Table 2: Range of nominal conditions for the performed continuous experiments**

Conditions | Nominal Range |

D (h^{-1}) | 0.0500 - 0.100 |

S_{1e} (g.l^{-1}) | 50.0 - 300. |

S_{2e} (g.l^{-1}) | 2.50 - 5.00 |

K_{L}a (h^{-1}) | 100. - 500. |

O_{2} (controlled values) (mg.l^{-1}) | 1.50 - 3.75 |

**Analytical Methods**

The samples periodically withdrawn from the bioreactor were centrifuged at 15000 rpm and 5^{o} C for 10 min. The cells were resuspended in water to obtain the cell mass concentration by the dry-weight method, using a 0.45 m m Millipore membrane. The sorbitol and sorbose concentrations were determined in the liquid phase by HPLC (Waters System: pump 600E, RI detector 410, integrator 7458, column Shodex SC1011, oven temperature of 72^{o}C, detector temperature of 40^{o}C and with water as eluent - 0.65 ml.h^{-1}). The yeast extract was quantified only during medium preparation.

**Experimental Conditions**

The inoculum was prepared in two steps in a rotary shaker (New Brunswick G-25) at 30^{o}C and 200 rpm during 60 and 36 hours, respectively, in a attempt to adapt the culture to the growth conditions in the bioreactor. The experiments in the bioreactor were performed at a temperature of 30^{o}C and with the pH control at 4.5. Tables 1 and 2 summarize the operational conditions for development of the models for the batch and continuous processes.

**Bioreactors and Instrumentation**

The batch runs were performed in a 7-liter NBS Microferm MF-104 (with a working volume of 3 liters). The continuous runs were performed in a 5-liter Braun Biostat MD (with a working volume of 2 liters) and in a 14-liter Braun Biostat ED (with a working volume of 10 liters). All equipment were provided with the traditional control of the single-loop type for the variables pH, dissolved oxygen and temperature.

**RESULTS AND DISCUSSION**

**Mathematical Model for the Batch Process**

The formulation of the unstructured mathematical model for the bioxidation of sorbitol to sorbose was based on 30 experimental runs (Bonomi et al., 1995).

The major conclusions derived from these experiments, which are reported in details in previous work (Bonomi et al., 1993a), were:

- microbial growth and sorbose production by yeast extract were limited;
- microbial growth and sorbose production by sorbitol were limited and inhibited;
- microbial growth and sorbose production by dissolved oxygen were limited;
- microbial growth and sorbose production by sorbose were inhibited;
- sorbose production is both associated and not associated to the microbial growth;
- severe problems occurred in the adaptation of the microbial population to the growth conditions in the bioreactor.

The unstructured mathematical model for the batch bioxidation of sorbitol to sorbose (Bonomi et al., 1993a), represented by equations (1) to (7), was formulated based on the observations reported.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

The major limitation of the proposed, unstructured mathematical model is the impossibility of representing the adaptation phase, observed in most of the experiments.

The mathematical model described by equations (1) to (7) has 16 parameters, 13 of which need to be adjusted in order to fit the nonlinear model to the set of experimental runs. The other 3 parameters, Y_{PS1}=0.989 g/g, Y_{PO}=0.01125 g/mg and O_{g}=7.5 mg/l, are constant for the present fermentation process. The estimation of the model parameters can be reduced to the solution of an optimization problem, where the performance index to be minimized is a comparison between the experimental and simulated values of the measured state variables. The numerical solution of this problem requires "good" initial guesses for the models parameter values to be estimated. These initial values can be obtained through a detailed analysis of the proposed model, simplifying and linearizing sections of it (Bonomi et al., 1993a, 1993b). The final estimation of the model parameters was performed applying the flexible geometric simplex method proposed by Nelder and Mead and known as the flexible polyhedron search (Himmelblau, 1972). This optimization technique, a derivative-free method of estimation, is especially effective in comparison to derivative methods (such as the Marquardt routine), when the number of parameters to be estimated becomes large (Himmelblau, 1970; Augusto et al., 1994). Table 3 reports the parameter values obtained when fitting the model to the set of batch experiments, while Figures 1 and 2 illustrate the type of fit obtained in two experimental runs.

**Mathematical Model for the Continuous Process**

As mentioned previously, the formulated mathematical model fitted to the batch process was used to simulate the continuous fermentation system, modifying only the balance equations (including the continuous feeding of sorbitol and yeast extract and the continuous withdrawal of the fermented medium). The adapted model and the set of parameters obtained for the batch system were utilized in deriving the optimum operating conditions for the continuous system.

Parameter | Batch System | Continuous System |

(h-1) | 0.735 | 0.696 |

K_{S}1 (g.l^{-1}) | 5.28 | 3.08 |

W (g.l^{-1}) | 222. | 298. |

K_{S}2 (g.l^{-1}) | 0.863 | 0.102 |

K_{O} (mg.l^{-1}) | 0.135 | 0.179 |

K_{P} (l.g^{-1}) | 0.00947 | 0.00651 |

A (g.g^{-1}) | 18.0 | 28.7 |

Bm (g.g^{-1}.h^{-1}) | 8.05 | 9.91 |

K_{bS1} (g.l^{-1}) | 1.75 | 3.78 |

K_{bO} (mg.l^{-1}) | 2.27 | 0.502 |

Y_{XS1} (g.g^{-1}) | 0.530 | 0.235 |

Y_{XS2} (g.g^{-1}) | 0.806 | 0.944 |

Y_{XO} (g.mg^{-1}) | 0.000687 | 0.000132 |

**Figure 1:** Experimental data and prediction of the mathematical model for a batch run with variable dissolved oxygen concentration.

**Figure 2:** Experimental data and prediction of the mathematical model for a batch run with constant dissolved oxygen concentration.

The objective function (FO) to be optimized in order to define the continuous system operating point incorporates, through the weights (w_{i}) defined by a preliminary economic analysis of the process, the productivity, the yield, the final product concentration, the residual concentrations of sorbitol and yeast extract and the supply of oxygen (Fleury et al., 1994).

(8)

The optimum operating point of the continuous system, as well as the resulting steady-state values and the employed weights, utilizing the model and the parameters derived for the batch process, are listed in Table 4.

**Table 4: Optimum operating point **^{(*)}

Batch Model | Continuous Model | |||

Operating Variables | State Variables | Operating Variables | State Variables | |

D = 0.0395 | X = 2.42 | D = 0.0450 | X = 1.96 | |

S_{1e} = 263. | P = 233. | S_{1e} = 438. | P = 366. | |

S_{2e} = 5.09 | S_{1} = 22.4 | S_{2e} = 5.82 | S_{1} = 58.9 | |

K_{L}a = 162. | S_{2} = 2.09 | K_{L}a = 363. | S_{2} = 3.75 | |

O = 1.61 | O = 1.63 | |||

(*) Utilized weights: w_{1} = 30.0; w_{2} = 10.0; w_{3} = 1.2; w_{4} = 1.0; w_{5} = 30.0; w_{6} = 0.5. |

**Figure 3:** Normalized measured and calculated steady-state values for the continuous runs.

Eleven experimental runs were performed; the range of the operating variables (reported in Table 2) was established, including the optimum operating point previously defined. Table 3 lists the set of estimated parameters using the same methodology as that utilized for the batch model, fitting the steady state attained in the experimental runs with the ones simulated by the proposed model. The relation between the normalized steady-state measured and calculated values is presented in Figure 3.

The results show that the proposed, unstructured mathematical model, as well as the parameter values, satisfactorily represents the microbial conversion of sorbitol to sorbose in both the batch and the continuous operation. The comparison of the two fitted parameter sets points out some significant deviations that are confirmed by the difference between the optimum operating point calculated using the parameters fitted for the batch system and for the continuous system (Table 4).

**ACKNOWLEDGMENTS**

The results reported in this paper are part of the project entitled "Technology Development for the Automation and Control of Fermentation Processes," sponsored by FINEP under the PADCT Program (Agreement 4.3.89.0498.00). The authors thank FINEP for its support. One of the authors (AB), also thanks the CNPq for its support in the form of a research fellowship.

**NOMENCLATURE**

A Growth-associated parameter for sorbose production, g.g^{-1 }B_{m }Maximum non-growth associated parameter for sorbose production, g.g^{-1}.h^{-1 }D Specific feed rate, h^{-1 }FO Objective function

K_{bO} Saturation parameter of the product formation rate by the oxygen concentration, mg.l^{-1 }K_{bS1} Saturation parameter of the product formation rate by the sorbitol concentration, g.l^{-1 }K_{L}a Oxygen volumetric transfer coefficient, h^{-1 }K_{O} Saturation parameter of the growth rate by the oxygen concentration, mg.l^{-1 }K_{P} Inhibition parameter of the growth rate by the sorbose concentration, l.g^{-1 }K_{S1} Saturation parameter of the growth rate by the sorbitol concentration, g.l^{-1 }K_{S2} Saturation parameter of the growth rate by the yeast-extract concentration, g.l^{-1 }O Dissolved oxygen concentration, mg.l^{-1 }Og Equilibrium dissolved oxygen concentration, mg.l^{-1 }P Sorbose concentration, g.l^{-1 }S_{1} Sorbitol concentration, g.l^{-1 }S2 Yeast-extract concentration, g.l^{-1 }W Growth inhibition parameter by sorbitol, g.l^{-1 }w_{i} Objective function weights

X Biomass concentration, g.l^{-1 }Y_{PO} Stoichiometric factor for oxygen to sorbose conversion, g.mg^{-1 }Y_{PS1} Stoichiometric factor for sorbitol to sorbose conversion, g.g^{-1 }Y_{XO} Yield factor for oxygen to cell conversion, g.mg^{-1 }Y_{XS1} Yield factor for sorbitol to cell conversion, g.g^{-1 }Y_{XS2} Yield factor for yeast extract to cell conversion, g.g^{-1 } Maximum specific growth rate without inhibition, h^{-1 }m _{P} Specific production rate, g.g^{-1}.h^{-1 }m _{X} Specific growth rate, h^{-1}

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