Model-based optimization of temperature and feed control strategies for glycerol production by fed-batch culture of osmophilic yeast Candida krusei

Biochemical Engineering Journal 11 (2002) 111–121

Model-based optimization of temperature and feed control strategies for glycerol production by fed-batch culture of osmophilic yeast Candida krusei Dong-Ming Xie, De-Hua Liu∗ , Hao-Li Zhu, Yong-Qiang Liu, Jian-An Zhang Department of Chemical Engineering, Institute of Applied Chemistry, Tsinghua University, Beijing 100084, PR China Received 15 July 2001; accepted after revision 26 November 2001

Abstract In this study, the optimization of temperature and feed control strategies for glycerol production by fed-batch culture of osmophilic yeast Candida krusei was investigated to maximize the final yield whilst to control the residual glucose at a low concentration. For the purposes of convenient control performance and easy numerical solution, the entire fermentation process was proposed being divided into multi-subintervals. In each subinterval, temperature was controlled constantly; while glucose and corn steep liquor were fed in pulse form at each start. Both piecewise-constant temperature (PCT) and discrete-pulse feed (DPF) control strategies were optimized by the complex method of Box based on previous macro-kinetic model and verified experimentally in a 600 ml airlift loop reactor. It was found that, by model-based optimization of only DPF control strategies, the final glycerol yield were significantly improved compared with those by previous empirical strategies. The yield could be improved further by optimization of both PCT and DPF control strategies and by selecting the first half of the whole fermentation process as the control emphasis. The optimization approach proposed appeared promising to solve the multivariable control problem in many fermentation processes. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Glycerol; Fermentation; Temperature; Fed-batch culture; Optimization; Kinetic model

1. Introduction Glycerol production via the osmophilic yeast route has been widely studied especially in China to meet commercial demand [1,2]. In most of previous works [3–6], temperature in the entire culture process for glycerol production was controlled usually at constant 35 ◦ C, since the process at this “optimized” temperature resulted in higher glycerol yield or need less fermentation time than those at other temperatures did. However, the recent works [7,8] found that the temperatures, respectively, suitable for cell growth, nutrients uptake and product accumulation were much different, a dynamic temperature control strategies may be more attractive to practical glycerol fermentation process. Therefore in this study, culture temperature should be selected as the first control variable. Besides that, Fan [3] and Xie [9] had ever found that high glucose concentration for culture of the osmophilic yeast led to significant inhibition in cell growth while low ∗ Corresponding author. Tel.: +86-10-62772130; fax: +86-10-62785475. E-mail addresses: [email protected] (D.-M. Xie), [email protected] (D.-H. Liu).

concentration reduced glycerol productivity, a fed-batch process seemed more attractive for glycerol fermentation. For glycerol fed-batch fermentation, Sun [5] and Yang [6] pointed out that glucose and corn steep liquor (the main phosphorus source) were the two essential nutrients to be fed. They proposed dry glucose powder should be fed in pulse form every certain time interval to maintain glucose concentration within certain range, while corn steep liquor should be fed just based on their experience. However, these feed modes were determined neither by a strict physiological analysis nor by an optimization strategy, thus were difficult to be assured close to an optimal state. To obtain an effective feed strategy, a physiological model and dynamic optimization approaches were usually adopted [10]. However, most of the metabolic mechanisms are very complicated and still difficult to be described, thus the physiological model approach is difficult to obtain the precise control. In fact, a macro-kinetic model for glycerol fermentation had been established in recent study [9], based on which it is possible to determine the dynamic feed strategies by various optimization approaches. Hence, the feed rates or amounts of both glucose and corn steep liquor should be another two control variables to be optimized.

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Nomenclature DPF Eps f1 (T ), f2 (T ), . . . , f6 (T ) fpen J Kdo1 Kdo2 KIS KO KPh1 KPh2 KPh3 KS1 KS2 KSP Kt mt N P PCT PenS PenSf Ph PhFPh PhFS PenVFS S SFS S0 t td T V VFPh VFS Vvap WFS X YP /S YX/Ph YX/S

Greek letters α β µ µmax σ error

discrete-pulse feeding accuracy for iteration calculation temperature functions to modify the kinetic parameters of µmax , mt , YP /S , β, YX/Ph and KS2 , respectively penalty function (–) objective function (g) constant of dissolved oxygen for cell growth (108 cells ml−1 ) constant of dissolved oxygen for glycerol accumulation (108 cells ml−1 ) constant for inhibition of glucose in cell growth ((g (100 ml)−1 )2 ) ratio of oxygen promotion to cell growth (–) saturation constant for phosphorus for cell growth (␮g ml−1 ) saturation constant for phosphorus for glucose consumption (␮g ml−1 ) saturation constant for phosphorus for glycerol production (␮g ml−1 ) contois constant for glucose for cell growth (g (1010 cells)−1 ) saturation constant for glucose for phosphorus consumption ((g (100 ml)−1 )2 ) coefficient of glycerol as a carbon source compared with glucose (–) attenuation constant for culture time for cell growth (h) total maintenance coefficient of glycerol and glucose (g (1010 cells)−1 ) total number of the subintervals (–) mass concentration of glycerol (% (w/v), i.e. g (100 ml)−1 ) piecewise-constant temperature penalty coefficient for constraint on glucose process concentration (–) penalty coefficient for constraint on final glucose concentration (–) concentration of phosphorus (␮g ml−1 ) phosphorus concentration in feed tank of corn steep liquor (␮g ml−1 ) phosphorus concentration in glucose feed tank (␮g ml−1 ) penalty coefficient for constraint on total glucose to be fed (–) mass concentration of glucose (% (w/v), i.e. g (100 ml)−1 ) glucose concentration in glucose feed tank (% (w/v), i.e. g (100 ml)−1 ) initial mass concentration of glucose (% (w/v), i.e. g (100 ml)−1 ) culture time (h) delay time for glucose consumption (h) culture temperature (◦ C) culture volume (ml) feed volume of corn steep liquor at start of each subinterval (ml) feed volume of glucose at star of each subinterval (ml) water evaporation rate in the 600 ml airlift loop reactor (ml h−1 ) feed amount of glucose at start of each subinterval (g) cell numeration (108 cells ml−1 ) yield coefficient of glycerol production for glucose consumption, glycerol per glucose (g g−1 ) yield coefficient of cell production for phosphorus consumption, cell number per phosphorus (108 cells ␮g−1 ) yield coefficient of cell production for glucose consumption, cell number per glucose (1010 cells g−1 ) constant associated with cell growth (g (1010 cells)−1 ) modified constant associated with cell concentration, glycerol per cell number per time (10−8 ml cell−1 h−1 ) specific growth rate (h−1 ) maximum specific growth rate (h−1 ) standard deviation of the relative errors of all discrete simulated data

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113

Subscripts end final state of glucose concentration f final state of the fed-batch fermentation i sequence number of the subintervals (1 ≤ i ≤ N) or the discrete time point low lower limit max maximum value or upper limit up upper limit 0 initial state of the fed-batch fermentation Superscripts (±) the simulated data with one model parameter being changed (0) the simulated data without any changes in the model parameters

To solve the dynamic optimization problem, various gradient-based numerical techniques have ever been attempted based on Pontryagin’s maximum principle [11]. However, this kind of approach usually suffered a singular control problem, which was relatively difficult to obtain a stable numerical solution [12–15]. Additionally, the optimal control profiles obtained in this way are continuous and time-varying, thus difficult to be put into effect since very complicated control equipment and operation are needed. In this study, it was proposed the entire fed-batch process for glycerol fermentation should be divided into multi-subintervals. In each subinterval, temperature was maintained at a constant value while glucose and corn steep liquor were fed in pulse form at each start. By this treatment, both piecewise-constant temperature (PCT) and discrete-pulse feed (DPF) control strategies were optimized conveniently by a general constrained optimization

such key factors as state equations, control/optimization variables, constraint conditions, objective function and numerical optimization approach. 2.1. State equations (bioprocess model) Based on previous works [9,23] and considering the evaporation lose of the culture volume during a long period of fermentation, the macro-kinetic model for glycerol production by batch fermentation with osmophilic yeast Candida krusei was modified as

 t KO dX = f1 (T )µmax exp − 1+ dt kt 1 + X/Kdo1 Vvap S Ph × X+ X 2 V (KS1 X + S)(1 + S /KIS ) KPh1 + Ph (1)

 1 dX S Ph   + f2 (T )mt X   YX/S dt KSP P + S KPh2 + Ph     dS Vvap 1 dP − = + − S  dt f (T )Y dt V 3 P /S        − Vvap S V approach based on previous macro-kinetic model [9], and verified experimentally in a 600 ml airlift internal loop reactor.

−

dPh 1 S2 dX Vvap = − Ph dt f5 (T )YX/Ph f6 (T )KS2 + S 2 dt V

dV = −Vvap dt

(2)

t < td

dP Ph S0 dX =α + f4 (T )β X 2 dt dt (1 + X/Kdo2 ) KPh3 + Ph Vvap KSP P Ph − f2 (T )mt X + P KSP P + S KPh2 + Ph V

2. Formulation of the optimization problem The problem discussed here is to optimize the temperature and pulse feeding amounts in all subintervals to maximize the final glycerol yield, whilst to control the final residual glucose at a low concentration so that glycerol can be efficiently separated from the reaction mixture during the downstream distillation process. Obviously, the approach to be introduced is to solve the dynamic optimization problem by a static optimization approach, which also contains

t ≥ td

(3)

(4) (5)

where X, S and P represent, respectively, the concentrations of cell, glucose and glycerol and Ph is phosphorus content; t, V, Vvap and T denote, respectively, the culture time, working volume, evaporation rate in reactor and culture temperature;

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f1 (T ), f2 (T ), . . . , f6 (T ) are six temperature functions to modify, respectively, the kinetic parameters of µmax , mt , YP /S , β, YX/Ph and KS2 . In Eq. (1), key culture factors including the reductional viability with time, oxygen promotion of growth, glucose and phosphorus limitation and glucose inhibition were considered. By modifying the traditional kinetic models of nutrient uptake and product accumulation, Eqs. (2) and (3) were applied, respectively, to describe the kinetics of glucose uptake and glycerol production, in which phosphorus limitation for both glucose uptake and glycerol production, the energy maintenance supplied by both glucose and glycerol, and the oxygen promotion to glycerol production were considered. Eq. (4) for phosphorus uptake was established considering that phosphorus content is decreased mainly during the growth stage and that higher glucose concentrations lead to greater phosphorus uptake [23]. Values of all kinetic parameters used in this study, as shown in Table 1, were estimated based on experiments in 1.5 l airlift loop reactor under various culture conditions; the six temperature functions [f1 (T ), f2 (T ), . . . , f6 (T )] were expressed as Eqs. (6)–(11). More details about the model building, parameter estimation and experimental verification were described in [9,23].

For the case of DPF-batch fermentation discussed in this study, the following mass conservation equation group (12) must be considered between any two closest subintervals:
 Vf,i−1 , V0,i =Vf,i−1 + VFS,i + VFPh,i , X0,i = Xf,i−1 V0,i
 Vf,i−1 Sf,i−1 + VFS,i SFS Vf,i−1 S0,i = , P0,i = Pf,i−1 , V0,i V0,i Vf,i−1 Phf,i−1 + VFS,i PhFS + VFPh,i PhFPh Ph0,i = (12) V0,i where the subscripts “0” and “f” represent, respectively, the initial and final states, “i” the sequence number of subinterval; VFS and VFPh denote feed volumes of glucose and corn steep liquor, while SFS and PhFS are glucose concentration and phosphorus content in glucose feed tank, respectively, and PhFPh is phosphorus content in feed tank of corn steep liquor. 2.2. Variables to be optimized in PCT and DPF control strategies

f1 (T ) = 2.984 − 0.0409T − 0.000452T 2

(6)

f2 (T ) = −8.575 + 0.564T − 0.00830T 2

(7)

If the process is divided into N subintervals, then total 3N variables including the temperatures (T1 , T2 , . . . , TN ) during every subinterval and the feeding volumes [(VFS,1 , VFS,2 , . . . , VFS,N )+(VFPh,1 , VFPh,2 , . . . , VFPh,N )] at the start of every subinterval should be optimized.

f3 (T ) = 5.667 − 0.300T + 0.00476T 2

(8)

2.3. Constraint conditions

f4 (T ) = −6.335 + 0.383T − 0.00494T 2

(9)

f5 (T ) = −10.597 + 0.595T − 0.00754T
 35 − T f6 (T ) = exp 2.234

2

(10) (11)

Based on previous experience [8], temperature should be controlled within 30–40 ◦ C, since at this range the yeast exhibited good capacity for either cell growth or glycerol production. Therefore, the following constraint was introduced firstly: Tlow ≤ Ti ≤ Tup

Table 1 Values of the kinetic parameters in Eqs. (1)–(4) Parameter

Value

µmax kt KO KS1 KIS YX /S mt KSP YP /S α β YX /Ph KS2 Kdo1 Kdo2 KPh1 KPh2 KPh3

111.0 33.96 2.174 652.3 93.20 1.625 0.05924 0.1920 0.4793 0.02310 1.061E−03 0.05450 152.7 1.998 11.39 163.6 14.38 1.088

(1 ≤ i ≤ N )

(13)

where the subscripts “low” and “up” represent the lower and the upper limit, respectively. Secondly, total glucose to be fed should be limited to assure a suitable working volume for the airlift loop reactor and a great yield rate for more and more glycerol accumulation, therefore  0≤ (VFS,i ) ≤ VFS,max (1 ≤ i ≤ N ) (14) In addition, to avoid significant inhibition in cell growth, the process concentration of glucose (S) should be constrained below an upper limit (Sup ), thus the following path constraint was suggested [9]: 0 ≤ S ≤ Sup

(15)

Finally, since high residual glucose concentration in the final reaction mixture must lead to difficulty in the downstream glycerol recovery process, an endpoint constraint was suggested so that the residual glucose concentration could

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be controlled under an upper limit, Send , which was set as 2% (w/v) in this work. Sf ≤ Send

(16)

2.4. Objective function The objective is to maximize the final glycerol yield subject to the constraints (13)–(16). For many of the constrained global optimization approaches, the constraint (13) for temperature control can be executed directly during the iteration calculus, while Eqs. (14)–(16) for constraints of state variable (S) and terminal states (Vf , Sf ) may be satisfied indirectly by adding a penalty function into the objective function [16,17]. Thus, the following objective function should be minimized: J (T1 , T2 , . . . , TN ), (VFS,1 , VFS,2 , . . . , VFS,N ), (VFPh,1 , VFPh,2 , . . . , VFPh,N ) = −Vf Pf + fpen(VFS ) + fpen(Sf ) +



t0

(NLP) one. In this study, both PCT and DPF control strategies were optimized by the complex method of Box [18,19,23]. The numerical conditions used in this study are: S0 = 30% (w/v), i.e. 30 g (100 ml)−1 , S up = 40 g (100 ml)−1 , X0 = 0.55 × 108 cells ml−1 , P0 = 0 g (100 ml)−1 , Ph0 = 60 ␮g ml−1 ; V0 = 418 ml, V vap = 0.417 ml h−1 , S FS = 180 g (100 ml)−1 (in this study, glucose was assumed also fed in dry powder form [5,6], therefore SFS denotes the density of dry glucose); VFS,max = 80.3 ml (i.e. 144.6 g), which means total 60% (w/v) glucose, based on the culture volume of 450 ml, was used during the whole fed-batch process; PhFS = 0 ␮g ml−1 ; PhFPh = 2000 ␮g ml−1 ; T low = 30 ◦ C, T up = 40 ◦ C; t d = 2 h; t f = 240 h; the convergent accuracy (Eps) is 10−10 .

3. Experimental

tf

fpen(S) dt

115

(17)

3.1. Microorganism and media

The osmophilic yeast C. krusei (ICM-Y-05) used in this where fpen(VFS ), fpen(S) and fpen(Sf ) represent the penalty study, mainly producing glycerol during aerobic cultivation, functions, respectively, for constraint conditions (14), (15) was obtained from State Key Laboratory of Biochemical and (16), they could be determined, respectively, by 

2

N N  pen VFS i=1 VFS,i > VFS,max i=1 VFS,i − VFS,max fpen(VFS ) = (18)

N  0 V ≤ V FS,max i=1 FS,i penSf (Sf − Send )2 Sf > Send Engineering, Institute of Chemical Metallurgy, Academia fpen(Sf ) = (19) 0 Sf ≤ Send Sinica [3–6]. The composition of the media for seed culture was: glucose 10% (w/v), urea 0.3% (w/v), corn steep liquor penS (S − Sup )2 S > Sup or S < 0 (Yiyang Biochemical, Hunan Province, China) 0.3% (w/v); fpen(S) = (20) and for fed-batch culture: glucose 30–40% (w/v), urea 0.3% 0 0 ≤ S ≤ Sup (w/v) and corn steep liquor 0.15–0.45% (w/v). All kinds of Here penVFS , penSf and penS are coefficients for correspondmedia were sterilized at 120 ◦ C for 20 min before any culture ing penalty functions and were set as 0.1, 1 and 1, respecprocesses. tively, based on numerical experience, since at these values stable convergence can be obtained and  the maximum con3.2. Fed-batch culture in airlift internal loop reactor N trol deviation i=1 VFS,i − VFS,max /VFS,max for glucose feed, (S f − S end )/S f for residual glucose and {max[S(t)] − The fermenter was a 600 ml airlift internal loop reactor S up }/S up for process glucose could be assured less than 0.3, (Institute of Chemistry, Academica Sinica), which usually 2 and 0.3%, respectively, which are tolerable to satisfy the contains 400–500 ml culture media. A schematic diagram of constraints of Eqs. (14)–(16) either in industrial or in experthe reactor was shown in Fig. 1. A jacket was installed outimental glycerol fermentation process. side of the external tube, by which the culture temperature could be controlled precisely by heating or cooling the recy2.5. Numerical solution cling water inside. With such a temperature control system, the time needed to increase or reduce the culture temperature within 10 ◦ C was less than 15 min. In many of previous works, fed-batch bioprocesses were optimized usually by indirect dynamic optimization apSeed was pre-cultured aerobically in shaking flask for proach using the maximum principle [11,23]. However in 12 h, and then was inoculated into the culture media in reacthis paper, temperature and feed control schemes for glyctor with a ratio of 10% (v/v). Since pH value within 3.0–5.0 erol fed-batch fermentation process were piecewise and had no significant effect on glycerol fermentation [3,20], discretely treated. Then, the original dynamic optimization only initial pH value was adjusted to 4.5 by addition of 1 N problem was transformed into a non-linear programming HCl. pH value in the later process could be kept by the yeast

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centrifuged at 2000 rpm for 5 min, then 5 ml centrifuged diluted sample was used for glycerol analysis and 4 ml for glucose analysis. Glycerol concentration was determined by the periodate-chromotropic acid analysis and glucose concentration was obtained by Fehling’s test [3,21,22]. Finally, the rest fresh sample was centrifuged at 3000 rpm for 5 min to separate cell for phosphorus content analysis. Phosphorus (mainly in organic form) was converted into inorganic form at first by adding concentrated sulfuric acid and perhydrol, and then its concentration was determined by ammonium molybdophosphate analysis [3,9].

4. Results and discussion 4.1. Analysis of optimal control profiles for glycerol fed-batch fermentation

Fig. 1. A sketch of the 600 ml airlift internal loop reactor.

within 3.0–4.0 without addition of any alkali or acid solution. During fed-batch process, dry glucose powder and diluted corn steep liquor (10%, w/v), based on the numerical optimization results, were fed instantly (i.e. in pulse form) at the start of each subinterval. The culture temperature was controlled at a constant value during each subinterval and shifted to another (if necessary) during the next based on the optimized PCT profile. All experiments at certain culture conditions were conducted duplicately, and the experimental data illustrated in this paper were arithmetic mean values of the duplicate experiments. 3.3. Analytical methods More than 3 ml fermentation broth was sampled during each sampling operation. Firstly, 1 ml fresh sample was diluted to 5–100 ml so that the cell number in diluted sample could be controlled within (0.5–2) × 107 cells/ml, which is convenient for cell numeration analysis. The total cell number was determined by microscope observation using a hemacytometer [3–6]. Next, another 1 ml fresh sample was diluted to 50 ml, about 10 ml of the diluted sample was

To optimize the piecewise control strategies more effectively, the continuous optimal control profiles should be analyzed first, which could be determined by Pontryagin’s maximum principle according to many published works [10–15]. Since the term of feeding rate appears linearly in the Hamiltonian, the optimal feed control problem usually suffered singular control case, which is difficult to solve by general gradient-based techniques [12–15]. In order to convert the singular problem to a non-singular one, Modak and Lim [14] had ever proposed that the culture volume, instead of feed rate, should be selected as the control variable. This non-singular volume-control approach was also adopted and modified further by Xie [9] to determine the optimal feed-rate profiles of both glucose and corn steep liquor for glycerol fed-batch fermentation. A typical numerical result with the same numerical conditions as in Section 2.5 was shown in Fig. 2. It was found that the general temperature profile should start at 30 ◦ C for a shorter period, then gradually increase up to 40 ◦ C and maintain at this point in most of the rest stage (Fig. 2(A)). Corn steep liquor (as the phosphorus source) should be fed at start to meet the great need for the coming exponential cell growth (Fig. 2(B) and (F)). Meanwhile, glucose was fed firstly in maximum feed rate to quickly achieve the upper limit concentration (i.e. 40%, m/v), then maintained within 30–40% with an appropriate feed rate until total 144.6 g glucose was fed (Fig. 2(C)–(E)). In addition, Fig. 2(A)–(C) shows that almost all control performances should be focused at least on the first half of the fermentation process, which was called as the control emphasis in this study. 4.2. Optimization of PCT and DPF control strategies and experimental verification Generally, when the dynamic optimization problem is solved based on the maximum principle, the continuous and time-varying control profile obtained are relatively difficult

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117

Fig. 2. Optimal temperature and feed control profiles ((A)–(C)) determined by the maximum principle and corresponding fermentation process curves ((D)–(F)).

to apply in practical process. Moreover, to obtain such a convergent solution as shown in Fig. 2 is very time-consuming, usually costing more than 1 h even performed in a personal computer with a CPU, PIII/800 MHz. In fact, it seems more attractive if the time-varying profile could be piecewise treated and simplified into multiple discrete subintervals with different simple or steady-state control performances. By this treatment, the dynamic optimization problem could be solved by the static optimization approach introduced in Section 2. Then, the new problems for such treatment are how to use the least subintervals to make the piecewise control strategies approach the continuous optimal control profiles and how to save computation time. For these purposes, two kinds of methods to divide subintervals were investigated. In case 1, the entire fed-batch process was divided equally into multiple subintervals, while in case 2, all subintervals except the last one were divided equally in the first half of the process since analysis

in Section 4.1 proposed it should be the control emphasis. With such two kinds of methods to divide subintervals to optimize PCT and DPF control strategies, the effect of the number of the subintervals on the final glycerol yield was shown in Table 2. It was found that, for either case of the control and optimization emphasis, more subintervals divided must lead to closer glycerol produced to that of optimal control (i.e. N = +∞). Additionally, Fig. 3 also shows that, as the number of the subintervals increased, PCT and DPF control strategies in case 2 approach the optimal control profiles. However, the increased subintervals make the control performance become more complicated and more difficult to perform. More importantly, the computation time needed by the complex method of Box increased exponentially as the variables to be optimized increased [16–18]. Therefore, the subintervals should be divided as few as possible. In fact, Table 2 shows that the final glycerol produced at (N = 4, 6, 8, 10) in case 1 are

Table 2 The effect of the number of subintervals divided (N) on total glycerol produced (Vf Pf , g) when the optimization emphases of both PCT and DPF control strategies were shown on the entire fermentation process (case 1) or on the first half of the fermentation process (case 2) N

Case 1 Case 2

1

2

3

4

5

6

8

65.36

78.27 78.27

80.85 82.20

82.45 83.43

82.78 84.05

83.67 84.70

84.54

9 85.17

10

+∞

84.79

86.18 86.18

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Fig. 3. Comparison between the (PCT + DPF) piecewise control profiles for glycerol fed-batch fermentation with emphasis on the first half of the process ((A) N = 3, V f = 408.95 ml, P f = 20.10%; (B) N = 6, V f = 411.55 ml, P f = 20.58%; and (C) N = 9, V f = 20.67 ml, P f = 412.06%) and the optimal continuous control profiles ((D) N = +∞, V f = 405.92 ml, P f = 21.23%).

only equivalent to those at (N = 3, 4, 5, 6) in case 2, i.e. Nneeded in case 2 =

Nneeded in case 1 +1 2

(21)

On this ground, selecting the first half of the entire fermentation process as the control emphasis and dividing more subintervals in this phase must make the optimization and control more convenient to perform and make the control objective approach optimal. To verify whether the model-based optimization of temperature and feed control strategies proposed in this study could achieve an ideal control objective as expected, four typical temperature and feed control strategies (see Table 3) were compared experimentally in the 600 ml airlift loop reactor, in which the whole fed-batch process was piecewise treated and the temperature profile and feed strategies were determined empirically by Sun [5] and Yang [6] (cases 1 and 2) or numerically by model-based optimization proposed in this study (cases 3 and 4). The experimental results were shown in Fig. 4, while comparisons between the experimental and model-predictive results for cases 3 and 4 were shown in Figs. 5 and 6, respectively. It was found from Fig. 4, the final glycerol yield in case 3 (N = 10, t f = 216 h, V f = 410.7 ml, P f = 18.07%, V f P f = 74.2 g, only with the DPF control strategies optimized) could be improved by 26.4% compared with that in case 1 (t f = 216 h, V f = 418.1 ml,

P f = 14.04%, V f P f = 58.9 g, with feed strategies of Sun [5]) and by 11.9% compared with that in case 2 (t f = 216 h, V f = 415.4 ml, P f = 15.95%, V f P f = 66.3 g, with feed strategies of Yang [5]). If both PCT and DPF control strategies were optimized (case 4: N = 6, t f = 226 h, V f = 417.4 ml, P f = 23.85%, V f P f = 99.5 g) and the first half of the process was selected as the optimization/control emphasis, then it could be further improved by 29.4% compared with case 3 (t f = 226 h, V f = 406.5 ml, P f = 18.92%, V f P f = 76.9 g). However, as for the accuracy of model prediction, Figs. 5 and 6 show that the model-predictive cell growth rate was somewhat greater than the experimental. This was mainly due to the insufficiency of the model accuracy. In fact, the model employed for glycerol fed-batch fermentation in this study just came from the modifications of the previous batch model [9,23] considering certain mass conservations. In addition, even the kinetic parameters of the batch model proposed in [9,23] were derived not strictly from physiological analyses but mainly from numerical fitting. Therefore, although the two fed-batch experiments, as shown in Figs. 5 and 6, suggested cell growth was significantly inhibited when the yeast suffered frequent osmotic pressure shock during glucose pulse-feed operation, the phenomenon could not be reflected in corresponding model predictions. Interestingly, inhibition in cell growth resulted from feed operation does not mean glucose uptake and glycerol

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Table 3 Four typical temperature and feed control strategies, in which the whole fed-batch process was piecewise treated and the temperature profile and feed strategies were determined by experience (cases 1 and 2) or by model-based optimization (cases 3 and 4) 0–24 h

24–48 h

48–72 h 35 ◦ C

Feed strategies of Sun [5] at constant WFS,i (g) 0.0 24.0 24.0 0.00 1.35 1.35 VFPh,i (ml)

72–96 h

96–120 h

120–144 h

144–168 h

168–192 h

192–216 h

216–240 h

(1) 24.0 1.35

24.0 1.35

24.0 1.35

24.0 1.35

0.0 0.68

0.0 0.68

0.0 0.68

Feed strategies of Yang [6] at constant 35 ◦ C (2) WFS,i (g) 0.0 48.0 24.0 36.0 VFPh,i (ml) 0.00 0.00 0.00 1.35

24.0 1.35

12.0 1.35

0.0 1.35

0.0 0.68

0.0 0.68

0.0 0.68

13.5 0.00 426.2

2.8 0.10 417.8

5.1 0.00 410.7

0.0 0.00 400.7

Feed strategies determined by model-based optimization (N = 10, T = 35 ◦ C) (3) 52.5 0.0 22.6 47.9 0.0 0.1 WFS,i (g) 3.00 0.20 0.00 0.00 0.10 0.20 VFPh,i (ml) Vi,f (ml) 439.1 429.3 431.8 448.4 438.5 428.7

Temperature and feed control strategies determined by model-based optimization (N = 6) with emphasis on the first half of the process (4) Ti (◦ C) 30.1 39.2 39.8 39.9 39.9 40 (during 120–240 h) 60.8 0.1 75.3 0.0 0.1 8.6 (at 120 h) WFS,i (g) VFPh,i (ml) 12.86 0.08 0.01 0.00 0.03 0.09 (at 120 h) Vi,0 (ml) 454.6 444.8 476.6 466.6 456.7 411.6 (at 240 h)

accumulation must also be inhibited. Fig. 5 shows the model predictions for both glucose uptake and glycerol production still agreed well with the experimental data. It suggested some key enzymes for nutrient and product metabolism should be activated during feed operation,

Fig. 4. Comparison between the fed-batch process curves with different temperature and feed control strategies. (䉫: T = 35 ◦ C and nutrients were fed by strategies of Sun [5]; 䊐: T = 35 ◦ C and nutrients were fed by strategies of Yang [6]; 䊊: T = 35 ◦ C and feed strategies were determined by model-based optimization of DPF scheme by dividing the whole process into 10 equal subintervals; : both temperature and feed control strategies were determined by model-based optimization of PCT profile and DPF schemes by dividing the whole process into six equal subintervals, and the former five equal subintervals, arranged in the first half of the process, was set as the optimization emphasis).

which counterbalanced inhibition in cell growth. However in Fig. 6, it was observed that most of the experimental data of glucose and glycerol concentrations are relatively higher than the model predictions. This phenomenon was probably caused mainly by error in culture volume calculation since the evaporation rate (Vvap ) used in this study always took the empirical value at constant 35 ◦ C even in a temperature-changing process. However, temperature has

Fig. 5. Comparison between experimental data (䊊: cell, 䊐: glucose, : glycerol) and corresponding model predictions (solid lines) of the fed-batch process, during which T = 35 ◦ C and feed strategies were determined by model-based optimization of DPF scheme by dividing the whole process into 10 equal subintervals.

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Fig. 6. Comparison between experimental data (䊊: cell, 䊐: glucose, : glycerol) and corresponding model predictions (solid lines) of the fed-batch process, during which both temperature and feed control strategies were determined by model-based optimization of PCT profile and DPF schemes by dividing the whole process into six equal subintervals, and the control emphasis was shown on the five equal subintervals in the first half of the process.

significant effect on evaporation rate. Usually the higher temperature leads to the greater evaporation rate. Like case 4 in Table 3, temperature in most of the fermentation cycle is within 39–40 ◦ C, thus the average evaporation rate for this case should be higher than at constant 35 ◦ C and

the practical glucose and glycerol concentrations, as Fig. 6 shows, became relatively higher than the model predictions. Additionally, a relatively simple mathematical model is always favorable either for understanding the basic characteristics of the bioprocess or for simplifying the numerical solution of model-based optimization. To further simplify the model applied in this study and for more convenient application in the future, here a parameter sensitivity analysis was performed by simulating a batch process (35 ◦ C) started at X0 = 0.6 × 108 cells/ml, S0 = 30% (w/v), P0 = 0% (w/v) and Ph0 = 90 ␮g/ml by changing each parameter by ±10%, respectively. The effects of the parameter variations on the process curves were represented by a standard deviation of the relative errors (σ error ) of all simulated data at 10 discrete time points (t = 12, 24, . . . , 108 and 120 h), namely

 (±) (0) 2 (±) (0) 2  n (X −X ) ) + ni=1 (S S−S  i=1 (0) X(0) i i  (±) (0) 2 (±) (0) 2  n (P −P ) (Ph −Ph )  + n + i=1  i=1 P (0) Ph(0) i i σerror = 4n − 1 (22) where n = 10, represents the number of discrete time points; the subscript “i” is the sequence number of the discrete time point; the superscript (±) and (0) represent the simulated data with one parameter changed and without any changes in the parameters, respectively. By this kind of analysis, Table 4 shows that the model Eqs. (1)–(4) could be further simplified since such parameters as α, KSP , etc. have no significant effects on the deviations of the model simulations. Although the macro-kinetic model applied in this study should be further perfected for more convenient and accurate model-predictive control during the future work, the

Table 4 Sensitivity analysis of the model parameters (the parameters KPh2 , KPh3 and Kdo2 are not listed since they are, respectively, proportional to KPh1 and Kdo1 as described in [23])a Kinetic parameters

S.D., σ error

µmax +10 −10

10.40 11.11

+10 −10

KS1 +10 −10

9.92 11.45

KIS +10 −10

YX/S +10 −10

3.15 3.57

YP /S +10 −10 YX /Ph +10 −10 a

Kinetic parameters

S.D., σ error

6.42 7.50

KO +10 −10

4.48 4.70

8.47 9.26

KPh1 +10 −10

8.16 9.36

+10 −10

10.08 14.59

KSP +10 −10

2.97 2.96

7.71 7.02

+10 −10

0.36 0.36

+10 −10

7.32 8.79

19.34 37.07

KS2 +10 −10

5.00 5.70

Kdo1 +10 −10

2.71 2.96

Kinetic parameters kt

mt

α

All the values are in percentage.

S.D., σ error

β

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model-based optimization approach proposed here, however, still appeared very promising to determine the multivariable control strategies for either glycerol fed-batch fermentation or other similar bioprocesses. 5. Conclusions To maximize the final glycerol yield and to deplete the residual glucose during glycerol fed-batch fermentation, optimization of temperature control and feed strategies of both glucose and corn steep liquor were investigated. For purpose of easy solution and convenient application, the control strategies were piecewise or discretely treated by dividing the entire fermentation process into multi-subintervals. In each subinterval, temperature was controlled constantly; while glucose and corn steep liquor were fed in pulse form at each start. Both PCT and DPF control strategies were optimized by the complex method of Box based on the macro-kinetic model. Analysis of the optimal control profiles by the maximum principle showed that all subintervals except the last one should be divided equally in the first half of the fed-batch process, which was viewed as the control and optimization emphasis. Compared with those empirical strategies, the proposed model-predictive PCT and DPF control strategies could improve glycerol yield significantly, which were verified experimentally in a 600 ml airlift loop reactor. The optimization approach proposed appeared very promising to determine the multivariable control strategies for glycerol fed-batch fermentation or other similar bioprocesses Acknowledgements The financial support from China Postdoctoral Science Foundation was gratefully acknowledged. References [1] G.P. Agarwal, Glycerol, Adv. Biochem. Eng. Biotechnol. 41 (1990) 95–128. [2] J. Zhuge, H.-Y. Fang, Progress in the study on glycerol production by fermentation, Food Ferment. Ind. (China) 4 (1994) 65–70. [3] Z.-L. Fan, Process optimization for glycerol production by fermentation with osmophilic yeast, M.Sc. Dissertation, Institute of Chemical Metallurgy, Chinese Academy of Sciences, Beijing, 1996.

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