Modelling and optimization of a continuous stirred

Chemical Engineering and Processing 50 (2011) 675–686

Contents lists available at ScienceDirect

Chemical Engineering and Processing: Process Intensification journal homepage: www.elsevier.com/locate/cep

Modelling and optimization of a continuous stirred tank reactor with feedback control and pulse feeding夽 Kaibiao Sun a , Andrzej Kasperski b,∗ , Yuan Tian c , Lansun Chen d a

School of Electronic and Information Engineering, Dalian University of Technology, Dalian 116024, People’s Republic of China Faculty of Mathematics, Computer Science and Econometrics, Bioinformatics Factory, University of Zielona Gora, Szafrana 4a, 65-516 Zielona Gora, Poland School of Information Engineering, Dalian University, Dalian 116622, People’s Republic of China d School of Mathematical Science, Dalian University of Technology, Dalian 116024, People’s Republic of China b c

a r t i c l e

i n f o

Article history: Received 19 February 2010 Received in revised form 5 April 2011 Accepted 7 April 2011 Available online 15 April 2011 Keywords: Biomass yield Continuous stirred tank reactor Feedback control Growth kinetics Periodic solution

a b s t r a c t The work is a presentation of the use of a continuous stirred tank reactor with feedback control to maintain the biomass concentration in desired range. Through analysis of the model’s nonlinear dynamics and numerical performance simulations, conditions are obtained for the existence of the system’s positive period-1 solution. The analysis also indicates that the system is not chaotic. Based on the model’s analysis, it is demonstrated that the selection of suitable operating conditions for the continuous stirred tank reactor can be simplified. This offers the possibility of establishing more general and systematic operations and control strategies that are based on the counteraction of the mechanisms which underlie the adverse effects of the bioreactor dynamics. Moreover, in the article the new objective function is introduced and the aspect of advanced optimization of the biomass productivity is presented. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Chemical and bioprocess engineering play an important role in the production of many chemical products [1]. In particular, bioreactor engineering as a branch of chemical engineering and biotechnology is an active area of research on bioprocesses, including among others development, control and commercialization of new technology [2]. Development of new control strategies requires mathematical models to check behavior and stability of the bioprocesses. The continuous stirred tank reactor belongs to the most important type of steady state bioprocesses [3]. One of the advantages of a continuous bioprocess is the possibility of reaching the highest biomass productivity [4]. The reaching of optimal results and the obtainment of maximal profits, require modern control strategies based on mathematical models or artificial intelligence methods [5]. According to different reactions and differential control technologies, many dynamic models concerning the culture of microorganism in the continuous stirred tank reactor have been established [6–10]. However, there are a lot of

夽 This paper is supported in part by the National Natural Science Foundation of China (Major Research Plan No. 90818025) and the Fundamental Research Funds for the Central Universities. ∗ Corresponding author. Tel.: +48 68 328 2802; fax: +48 68 328 2801. E-mail address: [email protected] (A. Kasperski). 0255-2701/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2011.04.004

factors affecting the growth and reproduction of the microorganisms in the process of bioreacts [11]. Biomass production process requires the control of the substrate concentration in the bioreactor medium due to the substrate inhibition phenomenon. Moreover, the oxygen demand in a bioreactor should be lower than the dissolved oxygen content. The biomass concentration is one of the most important factors which affect the oxygen demand. In order to maintain the dissolved oxygen concentration in an appropriate range, the biomass concentration should not exceed a set level. Therefore it is necessary to control the biomass and substrate concentration in order to obtain the optimal conditions for microorganism growth. It is known that many biological phenomena such as bursting rhythm models in biology and pharmacokinetics or frequency modulated systems exhibit impulsive effects. Impulsive differential equations, which appear as a natural description of observed evolution phenomena of several real world problems, have been introduced in population dynamics by some researchers [12–16]. Tang and Chen [17] introduced a Lotka–Volterra model with statedependent impulsion and analyzed the existence and stability of positive period-1 solution. Jiang et al. [18,19], Smith [20], Zeng et al. [21] have studied the state-dependent models with impulsive state control by applying the Poincaré principle and Poincaré–Bendixson theorem of the impulsive differential equation, respectively. In the microbial process, research on the continuous stirred tank reactor model with impulsive perturbations was undertaken by Sun and

676

K. Sun et al. / Chemical Engineering and Processing 50 (2011) 675–686

Chen [9]. Followed by Guo and Chen [22,23], Tian et al. [24,25], Sun et al. [26], who introduced the state-dependent impulsion into the microorganism culture process, and analyzed the system’s dynamic behavior. However, so far, few papers have discussed the continuous microbial process using nonlinear biomass yield. Recently, Tian et al. [27,28] and Sun et al. [29,30] introduced a sigmoid biomass yield in the batch culture mode, which has high a flexibility to fit any real biomass yield. This paper aims to propose a mathematical model of a continuous stirred tank reactor with feedback control and nonlinear biomass yield, described by the impulsive differential equation, and analyzing the dynamics of the bioreactor. The paper is organized as follows. In Section 2, a continuous stirred tank reactor model with Monod’s growth rate, sigmoid biomass yield and impulsive state feedback control is introduced. In Section 3, the conditions for the existence of positive period-1 solution are obtained by the Poincaré–Bendixson theorem. It also points out that the proposed system is not chaotic according to the analysis of period-2 solution. In Section 4, the numerical simulations are taken to verify the theoretical results, such as the existence of period-1 solutions, and the biochemical essence are discussed. Moreover, in Section 4, after the simulations and discussion, the bioprocess optimization is put forward. Finally in Section 5, the conclusions are given. 2. Model formulation If the microorganisms’ growth proceeds in accordance with Monod-type kinetics, i.e., according to dependence: (S) =

max S , KS + S

which is commonly used to model a large variety of biochemical reaction [4,31–33], then the deterministic model of microbial growth in the continuous stirred tank reactor is of the form [34]:

⎧ dS 1 max S ⎪ ⎪ = D(SF − S) − x ⎪ Y dt ⎪ x/S KS + S ⎨ dx

max S

= x − Dx ⎪ ⎪ KS + S dt ⎪ ⎪ ⎩ + +

(2.1)

S(0 ) = S0 , x(0 ) = x0

where x(t) denotes the biomass concentration, S(t) the substrate concentration, Yx/S the biomass yield, max the maximum specific growth rate, KS the saturation constant, SF the concentration of the feed substrate, D the dilution rate of the continuous stirred tank reactor, t denotes time, x0 and S0 denote the initial biomass concentration and substrate concentration in the bioreactor medium, all parameters are positive. Crooke et al. [35] showed that the biomass yield expression plays an important role for the generation of oscillatory behavior in continuous bioprocess models. Further, Crooke and Tanner [36] have proved that model (2.1) could not exhibit any periodic solution if the biomass yield in the model is constant. Alternatively, in reality, the growth yields have not shown a constant pattern [37–40], the cell characteristics can be approximated with sigmoid function [27–29], which fits well to main models of quantification of maintenance energy in a microorganism growth balance, i.e., the Herbert’s model and the Pirt’s model [41–43]. Thus, in this work the sigmoid function, i.e., Yx/S =

1 , a + exp(−bS)

which has high flexibility to fit any real biomass yield is used, where a = 1/Yx/Smax , b is the cell sensitivity to the substrate under optimal growth conditions (optimal temperature, pH, DOC, and other); a > 1

and b > 0 are the biological constraints [27–29]. Then model (2.1) has the following form:

⎧ dS max S ⎪ = D(SF − S) − (a + exp(−bS)) x ⎪ ⎪ KS + S dt ⎨ dx

max S

(2.2)

= x − Dx ⎪ ⎪ K +S dt ⎪ ⎩ + S +

S(0 ) = S0 , x(0 ) = x0

In particular, when the substrate concentration (S) is high, Yx/S = 1/a = Yx/Smax , and the biomass yield is constant. When the cell sensitivity to the substrate is very high (e.g. b > 100), Yx/S is also practically constant for any substrate concentration differing from zero. According to the design ideas of the bioreactor, the biomass concentration should not exceed a set level. The schematic diagram of the analyzed bioprocess is shown in Fig. 1. The apparatus includes an optical sensing device which continuously monitors the biomass concentration in the bioreactor medium, two electronically controlled valves and switches. These valves and switches work in a synchronous way. A constant dilution rate is set with the use of the valves. Moreover, when the biomass concentration is lower than the set level, switches are closed. In this case the biomass concentration is increasing due to the assimilation of substrate by the cells. When the biomass concentration x(t) in the bioreactor reaches the set level xset (where 0 < xset ≤ xcritical and xcritical is the critical level of biomass concentration in the bioreactor medium), switches are opened, part of the bioreactor medium containing biomass and substrate is discharged, and the next medium portion of a given substrate concentration is inputted. Therefore, system (2.1) can be modified as follows by introducing the impulsive state feedback control:

⎫ ⎧ dS max S ⎪ ⎪ = D(SF − S) − (a + exp(−bS)) x⎬ ⎪ ⎪ KS + S dt ⎪ ⎪ x < xset ⎪ ⎪ dx S ⎪ max ⎪ ⎭ ⎨ = x − Dx KS + S dt ⎪ S = Wf (SF − S) ⎪ ⎪ ⎪ x = xset ⎪ ⎪ x = −Wf x ⎪ ⎪ ⎩ + +

(2.3)

S(0 ) = S0 , x(0 ) = x0

where 0 ≤ Wf < 1 is the part of medium which is removed from the bioreactor in each biomass oscillation cycle. The same part of medium of a given substrate concentration is inputted into the bioreactor in each biomass oscillation cycle. 3. Existence of positive periodic solution Before discussing the dynamics of system (2.3), we must first consider the qualitative characteristic of system (2.2). Clearly, system (2.2) has a boundary equilibrium (SF , 0) and a positive equilibrium (S∗ , x∗ ) if D < max and KS < (max /D − 1)SF where S ∗ = −1 DKS /(max − D), x∗ = (SF − S ∗ )(a + exp(−bS ∗ )) . For the case of KS ≥ (max /D − 1)SF , the equilibrium (SF , 0) is stable. It can be concluded that every solution of (2.2) tends to a stable state (i.e., S(t) = SF and x(t) = 0) if one of the following conditions holds: (i) max ≤ D; (ii) max > D and KS ≥ (max /D − 1)SF . Since x(t) → 0, then the microorganism is not cultured successfully. For the case where D < max and KS < (max /D − 1)SF , the equilibrium (SF , 0) is a saddle point and S∗ > 0, x∗ > 0. Denote

p q

(S ∗ )2 + KS SF −1 D , − b exp(−bS ∗ )(SF − S ∗ )(a + exp(−bS ∗ )) S ∗ (KS + S ∗ ) max KS D(SF − S ∗ ) (KS + S ∗ )2

> 0.


677

Fig. 1. Schematic diagram of the analyzed bioprocess.

(b) b

(SF − S ∗ )S ∗ (KS + S ∗ ) (S ∗ )2

+ KS SF

− 1 exp(−bS ∗ ).

(3.1)

It can be verified that p > 0 when a > (b). We know from the qualitative theorem that the characteristic roots have a negative real part if p > 0 and then the equilibrium (S∗ , x∗ ) (which can be either a node or a focus) is asymptotically stable; else if a = (b), then p = 0 and (S∗ , x∗ ) is a center; else a < (b), then p < 0 and (S∗ , x∗ ) is unstable. Since dS/dt < 0 for S > SF , then any solution starting from the region {(S, x) | S ≥ SF } will enter into the region {(S, x) | S < SF } eventually. Thus, in the following we limit the discussing to the case S0 = S(0) < SF . For the case where the positive equilibrium (S∗ , x∗ ) is stable, all solutions of system (2.3) starting from the region {(S, x) | 0 < S < SF , 0 < x < x∗ } will tend to the equilibrium (S∗ , x∗ ). If the level xset is set higher than x∗ , the control strategy will lose its practical sense. So we focus our attentions on the discussion of the case 0 < xset < x∗ and (S0 , x0 ) ∈ 1 = {(S, x) | 0 < x < xset , 0 < S < SF − (a + exp (− bSF ))x}. For the case where equilibrium (S∗ , x∗ ) is unstable, it can be shown that there exists one limit cycle in with an outer boundary OEFG of the Bendixson annular region, see Fig. 2, where FG is a segment on the line l1 passing the point (SF , 0) with the slope −(a + e−bSF ), where the derivative of l1 along with system (2.2) dl1 /dt | (2.2) < 0: l1 :

x+

SF S − = 0. a + exp(−bSF ) a + exp(−bSF )

(3.2)

In this case, if no control strategy is applied, all trajectories starting from the region will tend to the limit cycle. When the

control strategy is adopted, the dynamics of system (2.2) can be altered. The discussion regarding this case can be divided into two cases: Case 1: 0 < xset < x∗ and (S0 , x0 ) ∈ 1 ; Case 2: xset ≥ x∗ and (S0 , x0 ) ∈ 2 = {(S, x) | 0 < x < x∗ , 0 < S < SF − (a + exp (− bSF ))x}. 3.1. Existence of period-1 solution 3.1.1. Existence of period-1 solution for xset < x∗ In order to apply the existence criteria of period-1 solution, i.e., Theorem A.2, we need to construct a closed region P so that all the solutions of system (2.3) enter into and be retained. These ideas will be illustrated as follows by using Fig. 3. As shown in Fig. 3, the line x = xset intersects the isoclinal line dx/dt = 0 at the point A(S∗ , xset ) and intersects the line l1 at the point B(SB , xset ). The line x = (1 − Wf )xset intersects l1 at the point H(SH , (1 − Wf )xset ). Denote C = C(SC , (1 − Wf )xset ) and D = D(SD , (1 − Wf )xset ), where SC = (1 − Wf )S∗ + Wf SF > S∗ and SD = (1 − Wf )SB + Wf SF > SB . The impulsive set M ⊆ AB = {(S, x)|S ∗ ≤ S ≤ SB , x = xset }, and N = I(M) ⊆ CD = {(S, x) | SC ≤ S ≤ SD , x = (1 − Wf )xset }. Whether the equilibrium (S∗ , x∗ ) is stable or not, the following theorem holds. Theorem 1. Suppose that D < max , KS < (max /D − 1)SF and 0 < xset < x∗ . Then system (2.3) has a period-1 solution with (S0 , x0 ) ∈ 1 . 3.1.2. Existence of period-1 solution for xset ≥ x∗ For the case where (S∗ , x∗ ) is unstable, we know that there exists a limit cycle around (S∗ , x∗ ). Let be the limit cycle, x = xupper be the line tangent with the trajectory starting from the initial point K(SK , x∗ ) and x = xmax (xmax > x∗ ) and x = xmin (xmin < x∗ ) be the lines tangent with (see Fig. 4), where SK = SF − (a + exp (− bSF ))x∗ . Then we have the following theorem. Theorem 2. Suppose that D < max , KS < (max /D − 1)SF , a < (b) and xset ≥ xupper , where (b) is determined by Eq. (3.1). Then system (2.3) has no period-1 solution and all trajectories tend to the limit cycle. Note that the line l1 intersects the isoclinal line dx/dt = 0 at the point F(S∗ , (SF − S∗ )/(a + exp (− bSF ))). Since dl1 /dt | (2.3) < 0 for the segment FG, then we have xupper < (SF − S∗ )/(a + exp (− bSF )). Thus

Fig. 2. The case for (S∗ , x∗ ) is unstable and system (2.2) has a limit cycle.

Corollary 1. Suppose that D < max , KS < (max /D − 1)SF , a < (b) and xset ≥ (SF − S∗ )/(a + exp (− bSF )), where (b) is determined by Eq. (3.1). Then system (2.3) has no period-1 solution and all trajectories tend to the limit cycle.

678


Fig. 3. Illustration of system (2.3) for 0 < xset < x∗ .

Fig. 4. Illustration of system (2.3) for the case where xmax < xset < xupper .

For the case where xmax < xset < xupper , we have the following theorem.

Suppose that D < max , KS < (max /D − 1)SF and Theorem 5. 0 < xset < x∗ . Then system (2.3) has no period-k (k ≥ 2) solution.

Theorem 3. Suppose that D < max , KS < (max /D − 1)SF and a < (b). Then there exists 0 < x < xmin and xmax < x¯ < xupper such that for xmax ≤ xset ≤ x¯ , system (2.3) has a period-1 solution with (S0 , x0 ) ∈ 2 .

4. Simulations and optimization

For the case where x∗ ≤ xset ≤ xmax (see Fig. 5), we have the following theorem. Theorem 4. Suppose that D < max , KS < (max /D − 1)SF , a < (b) and x∗ ≤ xset ≤ xmax . Then for Wf ≥ 1 − xK /xset , system (2.3) has a period-1 solution with (S0 , x0 ) ∈ 2 .

We have analyzed theoretically the feedback control for microorganisms of any biomass yield and Monod-type kinetics. These results are important, they not only provide the possibility of a check of system dynamic property, but also provide the possibility of making simulation of real process according to the mathematical models determined in the article. 4.1. Numerical simulations

3.2. Existence of period-k (k ≥ 2) solution In the last two subsections, we have analyzed the existence of period-1 solution of system (2.3). Next, we will discuss the existence of the period-2 solution.

To verify the obtained results, the numerical simulations of system (2.3) are presented. In view of the large practical importance, the case where (S∗ , x∗ ) is stable is presented and discussed. Moreover the microorganisms with Yx/Smax = 0.5[g/g] (i.e., a =

Fig. 5. Illustration of system (2.3) for the case where x∗ ≤ xset ≤ xmax .


679

Secondly, we set xset = 12[g/l]. The time series and phase diagram for system (2.3) with Wf = 0.2 is presented in Fig. 8, from which we can see that the trajectory tends to be periodic. Thirdly, we change the value of Wf . The phase diagram for system (2.3) with different values of Wf is displayed in Fig. 9. From Fig. 9 we can see that the existence of period-1 solution does not depend on the value of Wf in this case, but the value of Wf affects the position and tendency of the period-1 solution. Therefore, the numerical simulations are consistent with the theorems obtained and presented in Section 3. In the following section, we discuss the aspects of the bioprocess optimization. 4.2. Optimization of the bioprocess In Section 3, it is shown that if D < max , KS < (max /D − 1)SF and 0 < xset < U(D, SF ), then system (2.3) has a unique period-1 solution ((t), (t)), where Fig. 6. Illustration of vector graph of system (2.2) when (S∗ , x∗ ) is a stable focus.

U(D, SF ) = 1/Yx/Smax = 2), b = 1.5, max = 0.5[1/h] and KS = 0.5[g/l] are used for the demonstration of system behavior. In order to ensure the existence of positive equilibrium we set SF = 30[g/l] and D = 0.3[1/h]. Then we have S∗ = 0.75[g/l] and (b) 0.53. Because (b) < a = 2, (S∗ , x∗ ) = (0.75, 12.6) is stable, which can be seen in Fig. 6. Next we check and show the influence of xset and Wf changes on the existence of period-1 solution. Firstly, we set xset = 13[g/l]. The time series and phase diagram is presented in Fig. 7. It can be easily seen from Fig. 7 that no impulse occurs when xset > x∗ = 12.6[g/l] and the trajectory tends to the stable focus (0.75, 12.6).

b

3

4.2.1. The objective function and the constraints 1. The objective function POUT (Wf , D, SF , xset ) =

Wf xset T

c

13

12

12

2

11

11

10

10

1

9

9

20

30

8 0

40

T

Dxdt,

(4.1)

0

x(t)

1.5

10

13

2.5

0.5 0

1 + T

where POUT – the biomass productivity (i.e., the received biomass in the biomass oscillation cycle); the unit of POUT is [(g/l)/h] = [g/(lh)], T – the period of the biomass oscillation.

x(t)

S(t)

a

SF − (DKS /(max − D)) . a + exp(−b(DKS /(max − D)))

10

20

30

dS/dt=0

dx/dt=0

8 0.5

40

1

1.5

2

2.5

3

S(t)

t (h)

t (h)

Fig. 7. The time series and phase diagram for system (2.3) starting from initial point (S0 , x0 ) = (1, 8) for Yx/Smax = 0.5[g/g] (i.e., a = 2), b = 1.5, max = 0.5[1/h], KS = 0.5[g/l], D = 0.3[1/h], SF = 30[g/l] and xset = 13[g/l].

a

11

11

10.5

10.5

x(t)

x(t)

6

12 11.5

11.5

8

S(t)

c

12 10

10

10 9.5

9.5 4

2 0

10

20

30

t (h)

40

9

9

8.5

8.5

8 0

10

20

30

t (h)

40

8

2

4

6

8

10

S(t)

Fig. 8. The time series and phase diagram for system (2.3) starting from initial point (S0 , x0 ) = (1, 8) for Yx/Smax = 0.5[g/g] (i.e., a = 2), b = 1.5, max = 0.5[1/h], KS = 0.5[g/l], D = 0.3[1/h], SF = 30[g/l], xset = 12[g/l] and Wf = 0.2.

680


b

12

11.5

11.5

12 11.5

11

11

11

10.5

10.5

10.5

x(t)

x(t)

c

12

10

x(t)

a

10

10

9.5

9.5

9

9

9

8.5

8.5

9.5

Wf = 1/20

8 1

2

3

4

5

6

Wf = 1/10

8

7

2

4

6

8.5 8

8

Wf = 1/4 2

4

6

S(t)

S(t)

8

10

12

S(t)

Fig. 9. The phase diagram for system (2.3) starting from initial point (S0 , x0 ) = (1, 1) for Yx/Smax = 0.5[g/g] (i.e., a = 2), b = 1.5, max = 0.5[1/h], KS = 0.5[g/l], D = 0.3[1/h], SF = 30[g/l], xset = 12[g/l] and Wf = 0.05, 0.1, 0.25.

2. The constraints

B. Case 2. For D ≥ Dmin and Wf > 0, a continuous bioprocess is obtained,

(a) 0 ≤ Wf ≤ Wfmax < 1, Wf – the part of medium which is removed from the bioreactor in each biomass oscillation cycle, Wfmax – the maximal part of medium which is removed from the bioreactor in each biomass oscillation cycle, (b) SFmin ≤ SF ≤ SFmax , SF – the concentration of the dosaged substrate, SFmin – the minimal concentration of the dosaged substrate, SFmax – the maximal concentration of the dosaged substrate (for example SFmax = 50[g/l], because dosaged substrate concentration cannot be too high. The dosage of the substrate of high concentration may cause, e.g. substrate inhibition), (c) 0 < Dmin ≤ D ≤ Dmax = max SF /(SF + KS ), D – the dilution rate, Dmin – the minimal dilution rate. If SF ≥ SFmin and D ≥ Dmin (where SF ≤ SFmax and D ≤ Dmax ), there is Yx/S = Yx/Smax = 1/a, max – the maximum specific growth rate, KS – the substrate saturation constant, (d) 0 < xset < U(D, SF ), xset – the set level of the biomass concentration in the bioreactor medium. There are two cases of the optimization: A. Case 1. For D ≥ Dmin and Wf = 0, a simple continuous bioprocess is obtained, i.e.,

⎧ 1 max S dS ⎪ x ⎨ dt = D(SF − S) − 1/a KS + S

⎪ ⎩ dx = max S − D x

1 T →0 T

(1)

steady

max (1 − (1)

POUT

OPT

state

T

Dxdt = Dx, 0

reaches

its

maximum

when

(1)

Wf xset T

+

1 T

T

Dxdt, 0

and the constraints are 0 < Wf ≤ Wfmax , SFmin ≤ SF ≤ SFmax , Dmin ≤ D ≤ Dmax = max SF /(SF + KS ) and 0 < xset < U(D, SF ). Proposition 1. For given constant Wf , D, SF and xset , the period T of the biomass oscillation can be determined by the following equation

T = −B ln(1 − Wf ) + A ln

SF − DKS /(max − D) − (1 − Wf )axset

SF − DKS /(max − D) − axset

,

(4.4)

max KS , (max − D)[(max − D)SF − DKS ]

B=

KS + SF . (max − D)SF − DKS

Proposition 2. For the given constant D, SF and xset , the biomass pro(2) ductivity POUT achieves its maximum when Wf → 0, and the maximal biomass productivity is (2)

POUT

OPT

=

(max − D)(SF − (DKS /(max − D)))(SF − (DKS /(max − D)) − axset ) + D xset . (SF + KS )(SF − (DKS /(max − D)) − axset ) + (max KS axset )/(max − D) (4.5)

(1)

→ max x.

(2)

POUT (Wf , D, SF , xset ) =

KS /(KS + SFmax ))x.

Moreover, for SFmax KS , DOPT → max and x → SFmax /a and (1) POUT OPT

x = xset

The objective function is determined by Eq. (4.1), i.e.,

(1)

D = DOPT =

KS /(KS + SFmax )). The biomass productivity is

= DOPT x = max (1 −

(4.3)

According to Proposition 1 and Eq. (4.1), for a given constant (2) a, Wf , SF and xset , POUT is a function of D (see Fig. 10). To ensure a ¯ max = max (SF − axset )/(KS + positive T, the maximal value of D is D SF − axset ).

constraints are SFmin ≤ SF ≤ SFmax and the Dmin ≤ D ≤ Dmax = max SF /(SF + KS ). For the steady state, i.e., dS/dt = 0, dx/dt = 0, we have max S/(KS + S) = D, which implies that S = KS D/(max − D), (1) x = (SF − KS D/(max − D))/a. The biomass productivity POUT in

x − Dx

x = −Wf x

A=

and the objective function is equal: POUT (D, SF ) = lim

=

KS + S dt ⎪ ⎪ ⎪ ⎪ S = W f (SF − S) ⎪ ⎪ ⎩

where

KS + S

dt

⎧ ⎫ dS 1 max S ⎪ ⎪ = D(S − S) − x ⎪ ⎬ F ⎪ dt 1/a KS + S ⎪ ⎪ x < xset ⎪ ⎨ dx max S ⎪ ⎭

(4.2)

From Fig. 11 it can be easily seen that for given constant D, SF (2) and xset , the biomass productivity POUT achieves its maximum when Wf → 0.


b

2.427

c

2.3185

2.135

(2)

POUT

(2)

2.3175

POUT

(2)

POUT

2.318

2.3165

0.1

0.2

2.125

W = 1/10

W = 1/4

f

f

0

2.13

2.317

W = 1/100 2.4269

681

0.3

0.4

2.316

0.5

0

0.1

f

0.2

D [1/h]

0.3

0.4

0.5

0

0.1

0.2

D [1/h]

0.3

0.4

0.5

D [1/h]

(2)

Fig. 10. Dependence of the biomass productivity POUT on D for Yx/Smax = 0.5[g/g] (i.e., a = 2), max = 0.5[1/h], KS = 0.5[g/l], SF = 30[g/l], xset = 5[g/l] and Wf = 0.01, 0.1, 0.25.

b

2.5

c

2.5

2.5 2

1.5

1.5

1.5

(2)

(2)

POUT

2

POUT

2

(2)

POUT

a

1

1

1

0.5

0.5

0.5

D = 0.2[1/h]

D = 0.01[1/h] 0

0

0.2

0.4

0.6

0.8

0

0

1

0.2

D = 0.4[1/h]

0.4

0.6

0.8

1

Wf

Wf

0

0

0.2

0.4

0.6

0.8

1

Wf

(2)

Fig. 11. Dependence of the biomass productivity POUT on Wf for Yx/Smax = 0.5[g/g] (i.e., a = 2), max = 0.5[1/h], KS = 0.5[g/l], SF = 30[g/l], xset = 5[g/l] and D = 0.01[1/h], 0.2[1/h], 0.4[1/h].

Proposition 3. For the given constant D, the biomass productivity (2) POUT defined in Eq. (4.5) achieves its maximum when SF → SFmax OPT

and xset → U(D, SFmax ). According to Proposition 3, for predefined xset , the maximal biomass productivity is (2)

POUT

OPTmax

=

((max −D)SFmax −DKS )(SFmax −(DKS /(max − D))−axset ) (SFmax +KS )(SFmax − (DKS /(max − D)) − axset )+((max KS axset )/(max −D))

+D

× xset ,

which is a function of D. Then optimizing the bioprocess is to (2) find the optimal DOPT such that POUT achieves it maximum. OPTmax

Moreover,

for

SFmax KS ,

KS /(SF + KS )) → max and

(2) POUT OPT

(xset )

= max (1 −

KS ) SFmax KS + SFmax

(1)

D = DOPT = max (1 −

1−

KS

KS + SF +

KS

to be periodic. The key to biomass production is a certain suitable feedback state (i.e., xset ) and the control parameters (i.e., D and Wf ) according to practice. In addition, the appropriate initial concentrations of microorganisms and substrate should also be considered so that the culture will achieve the periodic state as soon as possible. Additionally, bioprocess optimization was also covered in this work. In the first case, the optimization of a simple continuous bioprocess (i.e., bioprocess in which D ≥ Dmin and Wf = 0) was presented. In the second case, it was demonstrated that theoretically it is possible to obtain almost optimal biomass productivity without lost of possible synchronization of simultaneously performed bioprocesses (Formulas (4.2) and (4.6)). This is possible, when the substrate is dosed by both continuous (i.e., D ≥ Dmin ) and pulse (i.e., Wf > 0) method. In this case the biomass productivity mainly depends on D and Wf (see Figs. 10 and 11), and the synchronization of simultaneously performed bioprocesses depends on Wf . Moreover it was shown that improving the biomass productivity and synchronization of simultaneously performed bioprocesses are conflict criteria (see Fig. 11).

(4.6)

→ max xset ,

where xset < SFmax /a. 5. Conclusions The article discussed usage of the mathematical model of the continuous stirred tank reactor with feedback control and nonlinear biomass yield to maintain the biomass concentration in a desired range. It was shown that the stability of the bioprocess (i.e., the existence of the positive period-1 solution), depends on both the biomass yield and the microorganisms growth rate. Furthermore, the conditions for the existence of the system’s period-1 solution were obtained (i.e., Theorem 1, Theorem 3 and Theorem 4). It also pointed out that the system (2.3) is not chaotic according to an analysis of period-2 solution existence (i.e., Theorem 5). According to the theoretical results, the biomass production tends

Appendix A. Consider the following general autonomous impulsive differential equations:

⎧ S˙ = P(S, x) ⎪ ⎪ ⎪ ⎨ x˙ = R(S, x)

⎪ S = I1 (S, x) ⎪ ⎪ ⎩ x = I2 (S, x)

(S, x) ∈ / Mimp

(A.1) (S, x) ∈ Mimp

where (S, x) ∈ R2 , P, R, I1 , I2 are all functions mapping R2 into R, Mimp ⊂ R2 is the set of impulse, and we assume: (H1) P(S, x) and R(S, x) are all continuous with respect to (S, x) ∈ R2 ; (H2) Mimp ⊂ R2 is a line, I1 (S, x) and I2 (S, x) are linear functions of S and x.

682


Fig. A.1. Illustration of successor function. Fig. A.2. Illustration of Bendixson field.

For each point A(S, x) ∈ Mimp , we define I : R2 → R2 : I(A) A+ = (S + , x+ ) ∈ R2 , S + = S + I1 (S, x), x+ = x + I2 (S, x) We can observe easily that Npha = I(Mimp ) is also a line of R2 or a subset of a line, and we assume throughout this section that Npha ∩ Mimp = ∅. We call I the impulsive function. Definition 1. [44] ((t), (t)) is said to be period-1 solution if in a minimal cycle time, there is one and only one impulse effect. Similarly, ((t), (t)) is said to be period-2 solution if in a minimal cycle time, there are two and only two impulse effects. Definition 1 adapted to the proposed system in its simple way is as follows. Definition 1 . [30] The system (2.3) has period-1 solution (i.e., ((t), (t))) if in the biomass oscillation cycle only one impulse effect occurs. Similarly, the systemhas period-2 solution if in the biomass oscillation cycle exactly two impulsive effects occur. Definition 2. (Successor function) Suppose that the impulse set Mimp and the phase set Npha are both line, as shown in Fig. A.1. Define the coordinate in the phase set Npha as follows: denote the point of intersection Q between Npha and x-axis as O, then the coordinate of any point A in Npha is defined as the distance between A and Q, and denoted by a. Let C denote the point of intersection between the trajectory starting from A and the impulse set Mimp , and B denote the phase point of C after impulse with coordinate b. Then we define B as the successor point of A, and the successor function of A by F(A) = b − a. Lemma 1. tinuous.

The successor function defined in Definition A.2 is con-

Proof. As shown in Fig. A.1, there exists a Poincaré map:

(A) → C ∈ Mimp , and I(C) = B ∈ Npha . On the one hand, from the continuity of the impulsive function I, for any ε > 0, there exists ı1 > 0 such that for any point C in the neighborhood U(C, ı1 ), there is |I(C ) − I(C) | < ε. On the other, by the continuity of the Poincaré map to the initial value, for above ı1 > 0, there exists ı > 0, for any point A in the neighborhood U(A, ı), there is | (A ) − (A) | < ı1 . Thus, for any A1 ∈ Npha ∩ U(A, ı), let (A1 ) → C1 ∈ Mimp ∩ U(C, ı1 ), I(C1 ) = B1 ∈ Npha , then there is |B1 − B | = | I(C1 ) − I(C) | = | I( (A1 )) − I( (A)) | < ε. Therefore, the successor function defined in Definition A.3 is continuous. Lemma 2. Suppose that model (A.1) satisfies the assumption H1 and H2 . Also assume that there is a bounded closed region ABCDA. The boundaries A D and B C are the no cut-arcs of model (A.1), and on AD and BC, the direction of the orientation field determined by model (A.1)

is pointing to the interior of ABCDA, as shown in the following figure. In addition, there is no singularity in the interior of ABCDA and the boundary. One of the boundary CD is the impulse set of the model (A.1), the corresponding phase set satisfies I(CD) ⊂ AB, AB is also the no cut-arcs of model (A.1), and on AB, the direction of the orientation field determined by model (A.1) is pointing to the interior of ABCDA. Then there must exist at least one period-1 solution of model (A.1) in region ABCDA.

D and B C by Proof. Denote ABCDA = G, and denote the no cut-arcs A 1 and 2 , respectively (see Fig. A.2). Then consider the trajectory

(C, t) of model (A.1) starting from the point C, the trajectory (C, t) must enter into G when t is increasing. Since 1 , 2 and impulse set AB are all no cut-arcs of model (A.1), there is no singularity in the interior of G and the direction of the orientation field determined by model (A.1) is pointing to the interior of G, then when t is increasing ¯ Let the more, (C, t) will intersect line segment AB at the point C. phase point of C¯ be C1 ∈ CD. If C1 = C, then (C, t) is the period-1 solution; If C1 = / C, we define the coordinate in the phase set CD as follows: denote the coordinate of C by O, and define the coordinate of the other point in CD according to the distance of the point to C, and denote the coordinate of C1 by c1 , then the successor function of C is F(C) = c1 > 0. Similarly, we consider the trajectory (D, t) of model (A.1) starting from the point D, the trajectory (D, t) must enter into G when t is increasing. When t is increasing more, (D, t) will intersect line ¯ Let the phase point of D ¯ be D1 ∈ CD. segment AB at the point D. / C, then the If D1 = D, then (C, t) is the period-1 solution; If C1 = successor function of C is F(D) = d1 − d < 0. By Lemma A.1, i.e., the continuity of the successor function, there exists at least one point E between C and D such that F(E) = 0. Therefore, there exists a period-1 solution of model (A.1) in region G. Definition 3. (Corless et al. [45]). LambertW function W is defined as multiple valued inverse function of f : y → yey = x. Then we have W(x)eW(x) = x, and its derivative satisfies: x(1 + W (x))W (x) = W (x) when x = / 0 and x = / − 1/e. W(x) has two branches when x > − 1/e, here we define W(0, x) as principal branch satisfying W(0, x) ≥ − 1 and another branch as W(− 1, x) satisfying W(− 1, x) ≤ − 1 (see Fig. A.3). The proof of Theorem 1 Proof. Firstly, it can be easily shown that all the trajectories of system (2.3) starting from the region 1 must interest with segment AB and then jump to segment CD. Next, we construct the closed


683

where

˛1

D(a + exp(−bS ∗ )) D(a + exp(−bS ∗ )) , ˛2 , max a max (a + 1)

˛3

Fig. A.3. The property of LambertW function.

region P ⊂ 1 such that all the solutions of system (2.3) starting from 1 enter into and retain in P . By Eq. (3.2) there is

SB = SF − (a + exp(−bSF ))xset , SH = SF − (a + exp(−bSF ))(1 − Wf )xset .

Then SD = (1 − Wf )SB + Wf SF = (1 − Wf )(SF − (a + exp(−bSF ))xset ) + Wf SF = SH . Let l2 be the line passing through A and C: l2 :

(SF − S ∗ )x + xset (S − SF ) = 0.

⎧ ⎨ (SF − S∗ )x + xset (S − SF ) = 0,

⎩ D(SF − S) − (a + exp(−bS)) max S x = 0, KS + S

there is (SF − S)[D(SF − S ∗ )(KS + S) − max xset S(a + exp(−bS))] = 0.

(A.2)

Let ϕ(S) = D(SF − S∗ )(KS + S) − max xset S(a + exp (− bS)). Then ϕ(S∗ ) = D(SF − S∗ )(KS + S∗ )(1 − xset /x∗ ) > 0 for S∗ < SF and xset < x∗ . Take the first and second derivatives of ϕ with respect to S then we have ϕ (S) = D(SF − S ∗ ) − max xset [a + (1 − bS) exp(−bS)], ϕ (S) = max xset b exp(−bS)(2 − bS). Denote

(xset )

Dx∗ (a + exp(−bS ∗ )) − a exp(1). max xset

Then (xset ) satisfies that

⎧

(xset ) ≥ exp(1), ⎪ ⎪ ⎪ ⎨

0 ≤ (xset ) < exp(1),

0 < xset ≤ ˛1 x∗ , ˛1 x∗ < xset ≤ ˛2 x∗ ,

⎪ −1/ exp(1) ≤ (xset ) < 0, ˛2 ⎪ ⎪ ⎩

(xset ) < −1/ exp(1),

x∗

If xset ≤ ˛1 x∗ , then ϕ (S) > 0 and ϕ(S) > ϕ(S∗ ) > 0 for S∗ ≤ S ≤ SC . In this case, Eq. (A.2) has no root in the interval [S∗ , SC ], which also implies that the line l2 does not intersect the isoclinal line dS/dt = 0 for S∗ ≤ S ≤ SC ; if ˛1 x∗ < xset ≤ ˛2 x∗ , then we have ϕ (S) = 0 if and only if S = S¯ = (1 − W (0, (xset )))/b, where the function W is defined in Definition A.3 and 0 < S¯ ≤ 1/b. In this case, Eq. (A.2) has, at most, two roots in the interval (S∗ , SF ), that is there are at most two intersection points between the line l2 and the isoclinal line dS/dt = 0 for S∗ ≤ S ≤ SC ; if ˛2 x∗ < xset ≤ ˛3 x∗ , then we have ϕ (S) = 0 if and only if S = S¯ i = (1 − W (i − 1, (xset )))/b, where 1/b < S¯ 1 ≤ 2/b and S¯ 2 ≥ 2/b. In this case, Eq. (A.2) has at most three roots in the interval [S∗ , SC ], that is there are at most three intersection points between the line l2 and the isoclinal line dS/dt = 0 for S∗ ≤ S ≤ SC ; if xset > ˛3 x∗ , then ϕ (S) < 0 and ϕ(S) < ϕ(S∗ ) for S∗ ≤ S ≤ SC . In this case, Eq. (A.2) has at most one root in the interval [S∗ , SC ], that is there are at most one intersection point between the line l2 and the isoclinal line dS/dt = 0 for S∗ ≤ S ≤ SC . Next, we construct the closed region P according to the number of the intersection points between line l2 and the isoclinal line dS/dt = 0 for S∗ ≤ S ≤ SC . Let k denote the number of the intersection points between line l2 and the isoclinal line dS/dt = 0 for S∗ ≤ S ≤ SC . Then k ∈ {0, 1, 2, 3}. Also denote the j th intersection point by Ej , j ≤ k. Case 1: k = 0, that is there is no intersection point between line l2 and the isoclinal line dS/dt = 0 (see Fig. 3a)). From the qualitative characteristic of system (2.3), we know that dl1 /dt | (2.3) < 0 for the segment DB, dl2 /dt | (2.3) > 0 for the segment AC and dx/dt | (2.3) > 0 for

It can be easily obtained that l2 is also passing through the point G(SF , 0). From

D(a + exp(−bS ∗ )) . max (a − exp(−2))

< xset ≤ ˛3

xset > ˛3 x∗ ,

x∗ ,

CD. Hence, the closed region P can be constructed with boundary consisting of AC, CD, DB and BA. Case 2: k = 1, that is there is one intersection point E1 between line l2 and the isoclinal line dS/dt = 0. Let E2 be the intersection point between x = (1 − Wf )xset and the isoclinal line dS/dt = 0 (see Fig. 3b)). From the qualitative characteristic of system (2.3), we know that dl1 /dt | (2.3) < 0 for DB, dl2 /dt | (2.3) > 0 for AE1 , dS/dt | (2.3) = 0 and dx/dt | (2.3) > 0 for E 1 E2 , dx/dt | (2.3) > 0 for E2 D, where E1 E2 is the arc of isoclinal line dS/dt = 0 between E1 and E2 . Hence, we construct a closed region P , the boundary of which consists of AE1 , E 1 E2 , E2 D, DB and BA. Case 3: k = 2, that is there are two intersection points E1 and E2 between line l2 and the isoclinal line dS/dt = 0. In this case, we construct a closed region P , the boundary of which consists of AE1 , E 1 E2 , E2 C, CD, DB and BA. Case 4: k = 3, that is there are three intersection points E1 , E2 and E3 between the line l2 and the isoclinal line dS/dt = 0. In this case, let E4 be the intersection point between x = (1 − Wf )xset and the isoclinal line dS/dt = 0. Then we construct a closed region P , the boundary of which consists of AE1 , E 1 E2 , E2 E3 , E3 E4 , E4 D, DB and BA. Then by Lemma A.2 system (2.3) has a period-1 solution. This completes the proof of Theorem 1. The proof of Theorem 2 Proof. It can be easily shown that all trajectories starting from the region 2 will not intersect the line x = xset and tend to the limit cycle eventually.

684


The proof of Theorem 3 Proof. As stated earlier, if a < (b), then (S∗ , x∗ ) is unstable and there exists one limit cycle . It follows from x(t) = 0 and S(t) = SF − (SF − S0 )e−Dt for t ∈ (0, + ∞) is a solution of system (2.3) that does not intersect line x = 0. Then there exists 0 < x < xmin such that the trajectory l3 starting from the initial point E (S ∗ , x) intersects the line S = S∗ at the point E (S ∗ , x¯ ), where xmax < x¯ < xupper . For xmax ≤ xset ≤ x¯ , let A be the point of the trajectory l3 intersecting with the line x = xset and C be the point of A after impulsive function I. As shown in Fig. 4, we should consider the following two cases according to the value of x and (1 − Wf )xset : Case 1: (1 − Wf )xset ≥ x. In this case, let E be the intersection point of line (1 − Wf )xset and the trajectory l3 . Since dx/dt | (2.3) > 0 for the segment ED, then we can construct the closed region P , the E, ED, DB and BA , where A E is the boundary of which consists of A part of the trajectory between A and E, then by Lemma A.2 system (2.3) has a period-1 solution. Case 2: (1 − Wf )xset < x. In this case, let E be the intersection point of the line (1 − Wf )xset and S = S∗ . Since dS/dt | (2.3) > 0 for the segment E E, dx/dt | (2.3) > 0 for the segment ED, then we can construct the closed region P , the boundary of which consists of E , E E, ED, DB and BA , where A E is the part of the trajectory A between A and E . Then by Lemma A.2 system (2.3) has a period-1 solution. Therefore, we complete the proof of Theorem 3. The proof of Theorem 4 Proof. Suppose that D < max , KS < (max /D − 1)SF , a < (b) and x∗ ≤ xset ≤ xmax . Let E and E be the points of the limit cycle intersecting the line S = S∗ . Let A and A be the points of the limit cycle intersecting with the line x = xset , C and C be the point of A and A after impulsive function I. Denote K (K ) as the intersection point between the line passing through A and C (A and C ) and the limit cycle for SA < S < SC (SA < S < SC ). If this K does not exist, then let K denote the point A . Let E be the intersection point between the line x = (1 − Wf )xset and the limit cycle . If the line x = (1 − Wf )xset does not intersect the limit cycle , then denote E as the intersection point between the line x = (1 − Wf )xset and the isoclinal line dx/dt = 0. If Wf ≥ 1 − xK /xset , then (1 − Wf )xset ≤ xK (see Fig. 5). Case 1, when Wf > 1 − x/xset , we have (1 − Wf )xset ≤ x. In this case, we can construct the closed region P , the boundary of E , E E, ED, DB and BA , where A E is the part of which consists of A the limit cycle between A and E . Case 2, when 1 − xK /xset ≤ Wf < 1 − x/xset , we have x < (1 − Wf )xset ≤ xK . In this case the boundary

E, ED, DB and BA , where A E is the part of the of P consists of A limit cycle between A and E, then by Lemma A.2 system (2.3) has a period-1 solution.

the point (S1+ , (1 − Wf )xset ) from the point (S1 , xset ), we can know that there is one and only one of the following sequences: S1 ≤ S2 ≤ S3 ≤ · · · ≤ SB ; S1 ≥ S2 ≥ S3 ≥ · · · ≥ S∗ . Since the sequences in Case (i) and Case (ii) are monotone and bounded, then there exists a limit for them respectively. Let S˜ = limn→∞ Sn . It can be shown that S˜ = 1 by Lemma A.2, that is S1 ≤ S2 ≤ S3 ≤ · · · ≤ 1 or S1 ≥ S2 ≥ S3 ≥ · · · ≥ 1 . It follows by Definition A.1 and the proof of Proposition 3.3 in [18] that there is no period-k solution (k ≥ 3) in system (2.3) and the system is not chaotic. This completes the proof. The proof of Proposition 1 Proof. Let ((t), (t)) be the period-1 solution. Denote (t) = (t) + a(t), then (t) is also T-periodic. By the first two equations of (4.3), we have d = D(SF − ). dt Then we have (t) = SF + ((0+ ) − SF ) exp(−Dt). particular, for t = T, In (T) = SF (1 − exp (− DT)) + (0+ )exp (− DT). Since (0+ ) = (0+ ) + a(0+ )

we

have

= (Wf SF + (1 − Wf )(T )) + a(1 − Wf )(T ) = Wf SF + (1 − Wf )((T ) + a(T )) = Wf SF + (1 − Wf )(T ).

Therefore, (T ) = SF + ((0) − SF ) exp(−DT ) = SF + [Wf SF + (1 − Wf )(T ) − SF ] exp(−DT ), which implies that (T) = SF , so we have (0) = Wf SF + (1 − Wf )(T) = SF or (t) = (t) + a(t) = SF . Following from the second equation of the system (4.3) that 1 KS +SF − a d = d. [(max − D)(SF − a)−DKS ] ((max )/(KS + ) − D)

dt =

Therefore, travelling along (t) = SF − a(t) from the point P0 ( 0 , (1 − Wf )xset ), with t = tP0 to the point P1 ( 1 , xset ), with t = tP1 in the counterclockwise direction yields the period T.

xset

=

T

(1−Wf )xset

KS + SF − a 1 d (max − D)(SF − a) − DKS

= −B ln(1 − Wf )+A ln

The proof of Theorem 5 Proof. Suppose that (, ) is a period-1 solution of system (2.3). Then we have (0 , 0 ) ∈ N ⊆ CD and (1 , 1 ) ∈ M ⊆ AB, where 0 = (1 − Wf ) 1 + Wf SF . Let (S, x) be an arbitrary solution of system (2.3). From the third equation of (2.3), we know that all trajectories enter the region {(S, x) | S ≥ S∗ } by impulse. Denote the first intersection point of the trajectory to the impulsive set M by (S1 , xset ), and the corresponding consecutive points are (S2 , xset ), (S3 , xset ), · · ·, respectively. Consequently, under the effect of impulsive function I, the corresponding points after pulse are (S1+ , (1 − Wf )xset ), (S2+ , (1 − Wf )xset ), (S3+ , (1 − Wf )xset ), · · ·. By the qualitative analysis of system (2.3), we know that x˙ > 0 for S > S∗ and x˙ < 0 for S < S∗ . Since 0 = (1 − Wf ) 1 + Wf SF > S∗ , then the period-1 solution (, ) lies in the region {(S, x) | S∗ < S < SF , (1 − Wf )xset ≤ x ≤ xset }. By the dynamics of system (2.3), for the trajectory of the solution (S, x) jumps to

(A.3)

SF −DKS /(max − D) − (1 − Wf )axset

SF − DKS /(max − D) − axset

where A=

max KS , (max − D)[(max − D)SF − DKS ]

The proof of Proposition 2 Proof.

It can be easily shown that

lim T = 0,

Wf →0

lim T = ∞.

Wf →1

Let (Wf , D, SF , xset ) =

T . Wf

B=

KS + SF . (max − D)SF − DKS

,


To take the first derivative of with respect to Wf , there is ∂ 1 = ∂Wf Wf2

∂T Wf − T ∂Wf

Because

.

(2)

∂POUT

OPTmax

∂xset

Denote ∂ ∂T = Wf − T. ∂Wf ∂Wf

=

∂ T 2

∂Wf2

= Wf

∂Wf

B (1 − Wf )

2

− A(axset )2

Wf

(max − D)SF − DKS

1 [SF − (DKS /(max − D)) − (1 − Wf )axset ]

2

KS + SF (1−Wf )

2

2

−

max KS (axset ) /(max −D) [SF −(DKS /(max −D))−(1−Wf )axset ]

2

for xset < U(D, SF ). Therefore, we have ∂/∂Wf ≥ 0, which also implies that (Wf , D, SF , xset ) = T/Wf ≥ (0, D, SF , xset ) = B + Aaxset . On the other hand,

T

Dxdt ≤ Dx(T ) = Dxset 0

1 = lim T →0 T

0

T

1 Dxdt = lim Wf →0 T

T

Dxdt. 0

Therefore, (2) POUT (Wf , D, SF , xset )

=

(2)

≤ POUT (0, D, SF , xset )

(SF − (DKS /(max − D)) − axset )xset

B(SF − (DKS /(max − D)) − axset ) + Aaxset

+ Dxset

(max − D)((max − D)SF − DKS )(SF − (DKS /(max − D)) − axset ) + D xset . (max − D)(SF + KS )(SF − (DKS /(max − D)) − axset ) + max KS axset

=

The proof of Proposition 3 Proof.

Let

(D, SF , xset )

(max − D)(SF + KS )(SF − (DKS /(max − D)) − axset ) + max KS axset (max − D)((max − D)SF − DKS )(SF − (DKS /(max − D)) − axset ) (SF +KS )(SF −(DKS /(max −D))−axset )+((max KS axset )/(max − D)) = (max −D)(SF − (DKS /(max −D)))(SF −(DKS /(max −D))−axset ) =

= 1 (D, SF , xset ) + 2 (D, SF , xset ),

where 1 (D, SF , xset )

2 (D, SF , xset )

(SF + KS )(SF − (DKS /(max − D)) − axset ) (max − D)(SF − (DKS /(max −D)))(SF −(DKS /(max −D))−axset ) SF + KS , = (max − D)(SF − (DKS /(max − D))) max KS axset . = 2 (max − D) (SF − (DKS /(max −D)))(SF −(DKS /(max −D))−axset ) =

Since ∂2 /∂SF < 0 and 1 KS + DKS /( − max −D) ∂1 =− < 0, max − D (SF − (DKS /(max − D)))2 ∂SF (2)

then we have (D, SF , xset ) is decreasing on SF . Thus POUT

OPT

is

increasing on SF and tends to be maximal when SF → SFmax . There(2)

fore, for constant D and xset , the maximal productivity POUT

OPTmax

is (2)

POUT

OPTmax

=

((max −D)SFmax −DKS )(SFmax −(DKS /(max −D))−axset ) (SFmax +KS )(SFmax −(DKS /(max −D))−axset )+((max KS axset )/(max − D))

× xset .

2

−max KS (axset )2

2

≥0 (2)

for xset < U(D, SFmax ). Then POUT

OPTmax

is increasing on xset . There(2)

fore, for constant D, the biomass productivity POUT achieves its maximum when Wf → 0, SF → SFmax and xset → U(D, SFmax ). References

≥ 0,

1 T

∂ (B/(1 − Wf )) + Aaxset (1/(SF − (DKS /(max − D)) − (1 − Wf )axset ))

= Wf

=

(SFmax + KS )(SFmax − (DKS /(max − D)) − axset ) + (max KS axset )/(max − D)

Then (0, D, SF , xset ) = 0. To take the first derivative of with respect to Wf , we have = Wf

−D

[(max −D)SF − DKS ] (SF + KS ) SF −(DKS /(max −D))−axset

(Wf , D, SF , xset ) = Wf2

∂ ∂Wf

685

+D

[1] D. Krishnaiah, Chemical and bioprocess engineering: a special issue of applied sciences, J. Appl. Sci. 7 (15) (2007) 1989–1990. [2] Y. Zhao, S. Skogestad, Comparison of various control configurations for continuous bioreactors, Ind. Eng. Chem. Res. 36 (1997) 697–705. [3] J.E. Bailey, D.F. Ollis, Biochemical Engineering Fundamentals, 2nd edition, McGraw-Hill, New York, 1986. [4] K. Schugerl, K.H. Bellgardt (Eds.), Bioreaction Engineering: Modeling and Control, Springer-Verlag, Berlin, Heidelberg, 2000. [5] A. Kasperski, T. Miskiewicz, Optimization of pulsed feeding in a Baker’s yeast process with dissolved oxygen concentration as a control parameter, Biochem. Eng. J. 40 (2008) 321–327. [6] S. Koga, A.E. Humphrey, Study of the dynamic behavior of the chemostat system, Biotechnol. Bioeng. 9 (1967) 375–386. [7] Y. Kuang, Limit cycles in a Chemostat-related model, SIAM J. Appl. Math. 49 (1989) 1759–1767. [8] D.F. Fu, W.B. Ma, S.G. Ruan, Qualitative analysis of a chemostat model with inhibitory exponential substrate uptake, Chaos Soliton. Fract. 23 (2005) 873–886. [9] S.L. Sun, L.S. Chen, Dynamic behaviors of Monod type chemostat model with impulsive perturbation on the nutrient concentration, J. Math. Chem. 42 (2007) 837–847. [10] P. De Leenheer, H. Simth, Feedback control for chemostat models, J. Math. Biol. 46 (2003) 48–70. [11] A. Kasperski, Modelling of cells bioenergetics, Acta Biotheor. 56 (2008) 233–247. [12] X.N. Liu, L.S. Chen, Complex dynamics of Holling type II Lotka–Volterra predator–prey system with impulsive perturbations on the predator, Chaos Soliton. Fract. 16 (2003) 311–320. [13] S.L. Sun, L.S. Chen, Permanence and complexity of the Eco-Epidemio-logical model with impulsive perturbation, Int. J. Biomath. 1 (2008) 121–132. [14] X.Z. Meng, L.S. Chen, Permanence and global stability in an impulsive Lotka–Volterra n-species competitive system with both discrete delays and continuous delays, Int. J. Biomath. 1 (2008) 179–196. [15] S.Y. Tang, L.S. Chen, Density-dependent birth rate, birth pulses and their population dynamic consequences, J. Math. Biol. 44 (2002) 185–199. [16] S.Y. Tang, L.S. Chen, The effect of seasonal harvesting on stage-structured population models, J. Math. Biol. 48 (2004) 357–374. [17] S.Y. Tang, L.S. Chen, Modelling and analysis of integrated pest management strategy, Discrete Contin. Dyn. Syst. Ser. B 4 (2004) 759–768. [18] G.R. Jiang, Q.S. Lu, L.N. Qian, Chaos and its control in an impulsive differential system, Chaos Soliton. Fract. 34 (2007) 1135–1147. [19] G.R. Jiang, Q.S. Lu, L.N. Qian, Complex dynamics of a Holling type II prey–predator system with state feedback control, Chaos Soliton. Fract. 31 (2007) 448–461. [20] R. Smith, Impulsive differential equations with applications to self-cycling fermentation, Thesis for the degree doctor of Philosophy, McMaster University, 2001. [21] G.Z. Zeng, L.S. Chen, L.H. Sun, Existence of periodic solution of order one of planar impulsive autonomous system, J. Comput. Appl. Math. 186 (2006) 466–481. [22] H.J. Guo, L.S. Chen, Periodic solution of a turbidostat system with impulsive state feedback control, J. Math. Chem. 46 (2009) 1074–1086. [23] H.J. Guo, L.S. Chen, Periodic solution of a chemostat model with Monod growth rate and impulsive state feedback control, J. Theor. Biol. 260 (2009) 502–509. [24] Y. Tian, L.S. Chen, A. Kasperski, Modelling and simulation of a continuous process with feedback control and pulse feeding, Comput. Chem. Eng. 34 (2010) 976–984. [25] Y. Tian, A. Kasperski, K.B. Sun, L.S. Chen, Theoretical approach to modelling and analysis of the bioprocess with product inhibition and impulse effect, BioSystems 104 (2011) 77–86. [26] K.B. Sun, A. Kasperski, Y. Tian, L.S. Chen, New approach to the nonlinear analysis of a chemostat with impulsive state feedback control, Int. J. Chem. Reactor. Eng. 8 (2010) 99. [27] Y. Tian, K.B. Sun, L.S. Chen, A. Kasperski, Studies on the dynamics of a continuous bioprocess with impulsive state feedback control, Chem. Eng. J. 157 (2010) 558–567.

686


[28] Y. Tian, K.B. Sun, A. Kasperski, L.S. Chen, Nonlinear modelling and qualitative analysis of a real chemostat with pulse feeding, Discrete Dyn. Nat. Soc., 2010 (Article ID: 640594). [29] K.B. Sun, Y. Tian, L.S. Chen, A. Kasperski, Nonlinear modelling of a synchronized chemostat with impulsive state feedback control, Math. Comput. Modell. 52 (2010) 227–240. [30] K.B. Sun, Y. Tian, L.S. Chen, A. Kasperski, Universal modelling and qualitative analysis of an impulsive bioprocess, Comput. Chem. Eng. 35 (3) (2011) 492–501. [31] H.J. Rehm, G. Reed, Microbial Fundamentals, Verlag Chemie, Weinheim, 1981. [32] J.R. Lobry, J.P. Flandrois, G. Carret, A. Pavemonod’s, Bacterial growth model revisited, Bull. Math. Biol. 54 (1992) 117–122. [33] J. Alvarez-Ramirez, J. Alvarez, A. Velasco, On the existence of sustained oscillations in a class of bioreactors, Comput. Chem. Eng. 33 (2009) 4–9. [34] A. Novick, L. Szilard, Description of the chemostat, Science 112 (1950) 715–716. [35] P.S. Crooke, C.J. Wei, R.D. Tanner, The effect of specific growth rate and yield expression on the existence of oscillatory behavior of a continuous fermentation model, Comput. Eng. Commun. 6 (1980) 333–342. [36] P.S. Crooke, R.D. Tanner, Hopf bifurcations for a variable yield continuous fermentation model, Int. J. Eng. Sci. 20 (1982) 439–443.

[37] A. Apuilera, T. Benitez, Relationship between growth, fermentation, and respiration rates in Saccaromvces cerevisiae: a study based on the analysis of the yield YPS , Biotechnol. Bioeng. 32 (1988) 240–244. [38] G. Andrews, Estimating cell and product yields, Biotechnol. Bioeng. 33 (1989) 256–265. [39] D.W. Tempest, O.M. Neijssel, The status of YATP and maintenance energy as biologically interpretable phenomena, Ann. Rev. Microbial. 38 (1984) 459–486. [40] S.P. Tsai, Y.H. Lee, A model for energy-sufficient culture growth, Biotechnol. Bioeng. 35 (1990) 138–145. [41] F. Menkel, A.J. Knights, A biological approach on modelling a variable biomass yield, Process Biochem. 30 (1995) 485–495. [42] H.J. Rehm, J. Reed, Biotechnology, Microbial Fundamentals, Verlag Chemie, Weinheim, 1981. [43] K. Schugerl, K.H. Bellgardt, Modeling and Control, Bioreaction Engineering, Springer-Verlag, Berlin, Heidelberg, 2000. [44] D. Bainov, P. Simeonov, Impulsive differential equations: periodic solutions and applications, in: Pitman Monographs and Surveys in Pure and Applied Mathematics, 1993. [45] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, On the LambertW function, Adv. Comput. Math. 5 (1996) 329–359.