OPTIMAL MODE OF OPERATION OF BIOREACTOR FOR FERMENTATION PROCESSES

Transcription

1 Chemical Engineering Scimce, Vol. 47. No. 15/16, pp wa9-2509p Printed in Great Britain PergPmon Pre Ltd OPTIMAL MODE OF OPERATION OF BIOREACTOR FOR FERMENTATION PROCESSES JAYANT M. MODAK Department of Chemical Engineering, Indian Intitute of Science, Bangalore , India and HENRY C. LIM Biochemical Engineering Program, Univerity of California, Irvine, CA 92707, U.S.A. (Received 15 April 1991; accepted fir publication 2 Jonunry 1992) Abtract-A ytematic analyi of the problem of maximizing metabolite yield and bioreactor productivity for fermentation procee i preented in thi paper. The periodic operation of a variable-volume tirred tank bioreactor i examined in the framework of optimal control theory for multiple control variable. The feaible combination of inlet and outlet flow rate are identified. The candidate for optimal mode of operation are hown to be. either teady-tate continuou, repeated batch or repeated fed-batch operation. The optimal mode of operation among thee three candidate i analytically determined for four type of fermentation procee. A numerical example i pravided to illutrate the analytical reult. 1. INTRODUCTION Batch, continuou and fed-batch (emibatch) are the mot commonly ued mode of operation of a bioreactor for production of a number of valuable metabolite by fermentation procee. In addition, repeated batch and repeated fed-batch are two cyclic mode operation which are commonly employed in reearch a well a in fermentation indutry. In recent year, a number of tudie have been reported for determining optimal operating policie for batch, continuou and fed-batch (emibatch) bioreactor. Thee include determining optimal temperature or ph policie for batch procee (Contantinide and Rai, 1974), and the rate of addition of ubtrate for fedbatch procee (Modak et al., 1986; Park and Ramirez, 1988; San and Stephanopoulo, 1989). A few tudie have alo been reported in which optimal policie for more than one operating variable have been determined (Chu and Contantinide, 1988; Modak and Lim, 1989). However, in all of the above tudie, the mode of operation of a bioreactor, namely, batch, continuou or fed-batch, i fixed a priori and the optimal policie determined accordingly. An important and challenging problem which ha received very little attention i that of determining the mode of operation of a bioreactor which i optimal for achieving the deired objective of either maximizing the metabolite yield or the bioreactor productivity. The procee tudied in thi regard in the previou invetigation have been cell ma production with either contant (Weigand, 1981; Matubura et al., 1985) or variable (Weigand et al., 1979) cellular yield and metabolite production with Ludenking- Piret kinetic (Haegawa et al., 1987). However, the tudie cited above are retrictive in cope due to everal aumption regarding the fermentation procee. A comprehenive and ytematic analyi of determining the optimal mode of operation for complex fermentation procee i lacking thu far in the literature. In the preent work, we preent uch an analyi, utilizing variable-volume emibatch reactor framework propoed earlier for chemical reaction (Waghmare and Lim, 1981; Parulekar et az ). In thi framework, the problem of determining the optimal mode of operation of a bioreactor i poed a an optimal bontrol problem with the inlet and outlet flow rate of the bioreactor a control variable. In the firt part of thi work, feaible mode of operation of a bioreactor are identified by examining the combination of inlet and outlet flow rate within the framework of optimal control theory for multiple control variable (Goh, 1964, Gabaov and Kirillova, 1972). In the econd part of the work, the optimal performance of each of the feaible mode of operation i compared and the optimal mode of operation i identified for two different objective, namely, maximizing the metabolite yield and the bioreactor productivity. 2. PROBLEM FORMULATION We conider a fermentation proce in which a microorganim X produce a metabolite P while utilizing a ubtrate S. The proce i occurring in a tirred tank bioreactor in which the inlet and outlet flow rate are not necearily equal. The dynamic ma balance equation decribing uch a proce in a variable-volume emibatch reactor can be written a follow: i =/4x -FIX/v B = F,(S, 6 = 7cx - FIPIV ti=ff,-ff2 - )/V - ax (1) 3869

2 3870 JAYANT M. MODAK and HENRY C. LIM where x,, and p denote the concentration of cell ma, ubtrate and metabolite, repectively. V i the bioreactor volume and F, and F, are the inlet and outlet flow rate, repectively. The inlet tream i aumed to contain only ubtrate at the concentration S,. /A, x, and c are the pecific rate of cell growth, metabolite production and ubtrate conumption, repectively, and are aumed to be function of the concentration of cell ma, ubtrate and metabolite. The flow rate are aumed to be bounded [eq. (2a)] and the bioreactor volume i contrained [eq. (2b)], i.e. By chooing appropriate combination of F 1 and F,, the ma balance equation (1) can imulate batch, fedbatch (emibatch), and continuou operation. For cyclic operation of a variable-volume emibatch bioreactor, the bioreactor volume and tlie coticentration of cell ma, ubtrate and metabolite atify cyclic boundary condition, i.e. V(O) = Wfh (0) = (t/)* 40) = W/h P(O) = p(t_f) where tf i the period of operation. We conider two different objective function; maximizing metabolite yield, IPI, and maximizing bioreactor productivity, IP,, by manipulating the inlet and outlet flow rate, F, and F,. (3) F,P dt yjt ZPi = (44 F, SF dt max IP, = Ft.Fz r tf Jo F,F dt *f V max dt 0 W) with the final time, tf, being unpecified. The ma balance equation for the metabolite in eq. (1) can be rearranged a follow: (pri) = XXV - F,p. (5) In view of the cyclic boundary condition (3) and eq. (S), the objective function in eq (4a) and (4b) can now be expreed a follow: Note that S, and V,, tr nxvdt IP, = O,, 0 i IP, = O *, F, dt If nxvdt (W. (6b) dt 0 are aumed to be contant, and therefore are eliminated from the objective function ZP, and ZP,, repectively. In order to repreent the optimization problem in tate-variable notation, we define the tate variable, Xl = x, x2 =, xg = p, and x4 = V, and three auxiliary tate variable, x5, x6, and x,, which atify the relation xg = 7cXIX.$, x,(o) = 0 ii-, = F,, X6(0) = 0 (7) 2, = 1, x,(o) = 0. In term of the newly defined tate variable, the objective function IP, and ZP, are x,lt_f) ZP, =- X60/) X5@/1 IP, =- x,(t_f) and dynamic equation (1) for the fermentation proce and the boundary condition (3) can be expreed a where i=f, +g,f,+g,f, Xi(O) = Xi@/), i = 1,2,3,4 x = (XI,X2,X3,X4.X5,X6.X,) f1 = (CUEI, - ux,,~x~,o,~x~xq,o,1)~ T -- x1 (F-x2) g1 =, 9 -E, l,o,o,o [ x4 x4 1 g, =(O,O,O, - l,o,l,o)t. The optimal control which maximize the performance index alo maximize the Hamiltonian H defined in eq. (10): H=AT(f, +g,f,+ g,f,) = JI + +,FI + +2F2. The adjoint variable A atify ii = -g,.&(o) = a&,), i = 1,2,3,4 I ik = 0 k = 5,6,7. (8) (9) (10) (11) The adjoint variable, A,, A,, A,, atify different boundary condition depending on which of the objective function (ZP, or IP,) i ued. Thu IP,: l,(t,) = -L &At,) 1 IP,: &(t,) = - XT@,) X&f),@_f) = - xz(t/) 9 n6(t,) = 0, &(t,) = 0 (12a) &(t,) = - 2. x tf) (12b) x,(t,) The Hamiltonian H [eq. (lo)] i linear in both control variable F, and F,, and therefore the optimization problem i a ingular control problem (Bryon and Ho, 1975). The maximization of the Hamiltonian depend on the ign of two witching function, 4, and

3 Optimal mode of operation of bioreactor for fermentation procee i, a follow: D E F F libax* #i > O Fi = Fi.8, qji = 0 i= 1,2 (13) Fmin = 0, C#J~ < 0 t where ubcript min, max and refer to minimum, maximum and ingular feed rate, repectively. The interval during which the witching function q& and/or & are & are identically zero over a finite interval of time i referred to a a ingular interval (Bryon and Ho, 1975). Pontryagin maximum principle (Pontryagin et al., 1962) fail to yield any information regarding the feed rate during a ingular interval. The ingular interval feed rate (F,.,) can be obtained by etting 9i and the time derivative of & to zero (Bryon and Ho, 1975): ( > ( > i = 1,2. +!?+g gi =...= _$!!!& =o, The differentiation of +* i continued till control variable F, appear explicitly for the firt time in the time derivative. It ha been hown previouly (Modak et al., 1986) that, for optimal control problem of fedbatch fermentation, Fi appear explicitly in the econd derivative of 4~~, i.e. for k = 2. We will now adopt a modal analyi propoed earlier for analyzing a variable-volume chemical reactor operation (Waghmare and Lim, 1981; Parulekar et al., 1988). The inlet and outlet flow rate can take on maximum, ingular (intermediate) or minimum feed rate and the digit 1, 2, artd 3 are ued to refer to the bioreactor operation with maximum, ingular and minimum feed rate. Thu, there are nine poible combination of the flow rate F, and F,. Each combination i aociated with a combination of the gradient of the Hamiltonian, Figure l(a) how the diviion of the entire phae plane, F,-F,, into nine ubpace according to the direction of the gradient. Each of the ubpace i identified in Table 1. In the following ection, the optimal control theory for multiple control variable will be ued to eliminate the infeaible combination c- l,g B i A (a) (b) Fig. 1. (a) Control pace illutration with contraint on feed rate and poible direction of Hamiltonian gradient, (b) poible tranition between four feaible combination of inlet and outlet flow rate. of inlet and outlet flow rate. The feaible mode of operation will be determined from the remaining combination of inlet and outlet flow rate. 3. IDENTIFICATION OF FEASIBLE MODES OF OPERATION The optimal control theory for multiple control variable which i ued to identify the feaible combination of feed rate i outlined in the Appendix. We begin our analyi by examining thoe combination Table 1. Nine poible combination of inlet (F,) and outlet (F2) flow rate = 0 42 < 0 (F, = F,.,) (Fz = F2.r) (Fz = Fmin = 0) 41 z=o F(11) 312) H(l3) (F, = F-2 91 =o E(=) I(221 ~(231 (F, = FL,) 41 <O D(31) ~(32) B(33) (Fi = Fmin = 0)

4 3872 JAYANT M. MODAK.which involve withdrawal of the bioreactor content at controlled (intermediate) rate, that i, mode 1(22), G(12), and C(32). Model I(22) In thi mode of operation, both the feed rate F, and F, take on the intermediate value (+i = & = 0), that i, ubtrate addition, a well a withdrawal of the bioreactor content i at controlled rate. A continuou reactor operation (F, = F2), either with contant or with time-varying feed rate, involve thi mode of operation. Therefore, analyi of mode I yield the information regarding the feaibility of the continuou operation a an optimal operation. Mode I require that both the witching function 4, and & be identically zero. Therefore, eq. (10) yield and HENRY C. LIM I can be a part of the optimal operating mode. Thu, continuou operation i a feaible mode of operation if the objective i to maximize the bioreactor productivity. It wa reported earlier for a contant-yield cell ma growth model (Matubura et al., 1985) that the cellular yield obtained in a repeated fed-batch mode of operation i higher a compared to that in the continuou operation. It mut be emphaized that the reult obtained in the preent tudy are applicable for a general kinetic model without any aumption on the kinetic of the fermentation proce. We now analyze feaibility of mode 1(22) for maximizing the productivity of the bioreactor. Since both the witching function +i and & are identically zero, their time derivative mut alo vanih during thi interval. Subtitution of 4i = 0, I& = 0 into & =-0 yield In order that thi mode be optimal, an additional equality type condition, eq. (All), developed for a general ingular control problem (Appendix), ha to be atified. The required matrice, Qz and B,, are [ a a a. C&x, _r &c&--x,)+ ~,x,l T I 47 = ~4 ~4 ~4 which yield il,nx, = 0. (17) Since pecific metabolite production rate (IL) and cell concentration (x1) are nonzero, eq. (17) can be atified if and only if 1, = 0. From the definition of the adjoint variable, eq (11) and (12b), eq. (17) require that l/t, = 0. Thu, incluion of mode 1(22) in the optimal mode of operation require that the period of operation t, be infinite, or that the operation i tationary. Thu, we can conclude from the above () 0 0 QzBz = 1 x: 0 The condition (A8) that the matrix Q2BI hould be ymmetric require &Xl - &(SF - x2) + A,% = 0. x4 (15) The two neceary condition (14) and (15) can be atified imultaneouly if and only if A,@,) = 0. (16) We now examine the condition given by eq. (16) for two different objective function ZP, and IP,. If the objective of the fermentation i to maximize the metabolite yield (ZP,), then condition (16) can never be atified in view of the terminal condition on &(t,) [eq. (12a)]. Therefore, mode I (22) which involve the operation with intermediate value of both feed rate i not part of the optimal mode of operation. Thu, continuou operation which involve mode I cannot be optimal if the objective i to maximize the metabolite yield. If the objective i to maximize the bioreactor productivity (ZP,), then condition (16) i trivially atified [adjoint variable given by eq. (12b)] and mode x: 0 analyi that the continuou bioreactor operation i not optimal for maximizing the yield of the metabolite, but i one of the likely candidate for maximizing the bioreactor productivity. Furthermore, continuou mode of operation i eentially a tationary or teady-tate operation. Mode G(12) or C(32) The witching function & i identically zero for thee two mode of operation, while 4, i either poitive (12). or negative (32). Setting & and it time derivative, &, to zero, yield & = [AXI - ~2cG - * x2) + L%lFi + A 5 Kx 1 = 0. XP (18) The Legendre-Clebch ality require that a mdtz- convexity condition of optim- d2+= o. [ 1 (19)

5 Optimal mode of operation of bioreactor for fermentation procee 3873 Differentiating eq. (18) and collecting the term involving F, together, condition (19) can be repreented a Equation (18) and (20) together imply that &(#cjci < 0. (21) In view of the terminal condition, [eq (12a) and (12b)], 2, i poitive for both the objective function. Since the pecific metabolite production rate (x) and the cell concentration (xi) are nonnegative, the Legendre-Clebch convexity condition of optimality, eq. (I9), i not atified. Therefore, mode C and G are not part of the optimal operating mode. In other word, a controlled withdrawal of the bioreactor medium i not a feaible mode of operation in the optimal configuration. Whenever the medium ha to be withdrawn from the bioreactor, it ha to be withdrawn at the maximum rate. Mode E(21) and F(11) Since the bioreactor content have to be withdrawn at the maximum rate, the quetion which need to be anwered i when to withdraw? In mode E and F, the inlet flow rate i nonzero which implie that the bioreactor content are being withdrawn at the maximum rate before the bioreactor volume reache it maximum limit, V,,. We will how that thee two mode are not permitted in optimal operation by deriving a contradiction. Let u aume that the optimal operation involve mode E or F before the bioreactor volume reache it maximum limit. We denote the optimal inlet and outlet flow rate by Fr and Ft, repectively, and the objective function by ZP:. We chooe an interval [ti, t,], 0 G t, d t < t, =G t,, uch that the outlet flow rate in the aumed optimal operation (Ff) i maximum in the interval [tl, t2] and the bioreactor volume V doe not reach it maximum limit, Vnur, during thi interval. We now conider another operation with inlet and outlet flow rate, F, and F,, uch that F F- -A_=1 *, O~t~tf. x4 x4 In view of the aumption (22) on the inlet and outlet feed rate and the ma balance equation (I), it can be conciuded that xi(t) = x?(t). 0 =S t < t/ for i = 1,2,3 \ I (22) mut be le than ZP:, i.e. *j *r xx1 x4 dt x*x:x: ZP, -ZPf = O 0 - If However, from eq. (23), I lf 0 if F, dt Ff dt 0 0 ff xx,x4 dt = ff 0 0?I dt < 0. rc*x:x,dt > n*x:xx dt and, from eq. (22), F2 dt < 11 Fz dt. Thi implie that ZP, > ZP:, which contra- dict our aumption that ZP: i the maximum value of the objective function. In other word, our aumption that the optimal operation involve withdrawing the bioreactor content at the time when the ubtrate i till being added to the bioreactor i incorrect. Therefore, mode E(21) and F(l1) are not feaible mode of operation. The bioreactor content have to be withdrawn only after the bioreactor volume reache it maximum limit at which time the inlet flow ha to be hut off. Earlier, we have etablihed that withdrawing the content at the controlled rate i not optimal, and therefore mode O(31) i the only feaible mode for withdrawing the bioreactor content. Mode A(23) In thi mode, the outlet feed rate F, i zero and the inlet feed rate F, take on intermediate value. Thi mode of operation correpond to a ingular interval of a ingle-cycle fed-batch operation. During the ingular interval, the witching function &, and it time derivative are et to zero to compute the ingular interval feed rate F,.,. The exitence of the ingular interval, which depend on the kinetic of the fermentation proce, ha been invetigated in detail in our previou tudy (Modak et al., 1986), and therefore thi analyi i not repeated here. Thu, out of the total nine combination of inlet and outlet flow rate poible, four combination, namely, mode C(32), E(21), F(ll), and G(12), are not feaible combination in optimal operation of a bioreactor. Mode 1(22) i eentially a teady-tate continuou operating mode and i not optimal for maximizing the metabolite yield. But it i a poible mode of operation for maximizing the bioreactor productivity. Figure l(b) how the poible tranition between the remaining four combination, namely, mode A(23), B(33), D(31), and H(13). It i aumed that the tranition between the mode i poible due to the ign change of only one of the witching function, +r or &, at any given time, but not of both imultaneouly. Four different feaible cyclic operating mode can be identified from the tranition chart hown in Figure l(b). Thee are 0 xq = xz, O<t<t, (23) =- xt, t, 6 t < t/. If ZPT i indeed optimal, a we have aumed, then ZP, reulting from any F, and F, uch a that given in eq. mode I mode II mode III mode IV HBD HABD I-IBABD I 22. cr 47:15/10-o

6 3874 JAYANT M. MODAK and HENRY C. LIM Mode I ( ) i a repeated batch operating mode coniting of feeding the ubtrate at the maximum rate (13) followed by batch (33) and then terminating the operation by rapid withdrawal (31). Mode II ( ) i a repeated fed-batch operating mode coniting of period of feeding the ubtrate at the maximum rate (13), controlled addition of ubtrate (23), batch (33) and rapid withdrawal (31). Mode III ( ) i alo a repeated fedbatch operating mode imilar to mode II, but an additional batch (33) interval exit between rapid filling (13) and controlled addition (23). Mode IV i a continuou operating mode. The ingle-cycle batch and fed-batch operation are limiting cae of repeated batch and repeated fed-batch operation, repectively, in which the entire bioreactor content are withdrawn at the end of the operation. It hould be noted that the identification of feaible mode of operation in thi ection i achieved without any retriction on the kinetic of the fermentation proce. It remain valid even if the number of pecie involved in the fermentation proce i higher than that conidered here. Thi i poible becaue, even if the number of ma balance equation increae, the baic nature of the dynamic model a expreed in eq. (9) and ubequent analyi remain unchanged. In the following ection, we compare the performance of each of the four feaible mode of operation in order to identify the optimal mode among the feaible mode of operation of the biorcactor. 11 c F2SF dt = (V,, - V&S, and Jo tf &P df = (v,, - voh+ (24) 0 In view of eq. (24), the objective function for maximizing the metabeilite yield, eq. (4a), and the bioreactor productivity, eq. (4b), can be implified to yield metabolite yield: bioreactor productivity: IP, =(I METABOLITE,P/ zp l (2W F -$.-)y YIELD Mode IV (22~ontinrrou teady-tate operation (C) For continuou teady-tate operation, the ma balance equation, eq. (l), reduce to p-d=0 D(S, - ) - ux = 0 (26) xx-dp=o where D i the dilution rate, D = F,/ V = F2/ V. The objective function for maximizing the metabolite yield can be expreed a follow: 4. DETERMINATION OF OPTIMAL OPERATING MODE OF BIOREACTOR The feaible mode of operation are identified in the previou ection without any aumption regarding the pecific rate of the fermentation proce. However, the kinetic of the fermentation proce play a crucial role, a hown later, in determining the optimal mode of operation. We, therefore, focu our attention on thoe fermentation procee in which the pecific cell growth (p), metabolite production (n) and ubtrate conumption (a) are influenced by only the ubtrate concentration in the bioreactor. Our main objective i to gather a much information a poible analytically without reorting to numerical computation. With thi objective, we relax the contraint on the maximum flow rate, i.e. we aume that the ubtrate addition to and withdrawal from the bioreactor at the maximum rate i intantaneou. The intantaneou addition and withdrawal implie that there i very little reaction occurring during the period of addition and withdrawal at the maximum rate. Thi aumption i valid a long a the rate of addition and withdrawal at the maximum rate are much greater than the rate of fermentation which i a reaonable aumption ince fermentation procee involving microorganim are inherently low procee. The intantaneou withdrawal of the bioreactor content implie Thu, the optimal continuou teady-tate operation i achieved when the ubtrate concentration in the outlet tream i maintained at the level, which maximize the objective function given by eq. (27), i.e. The optimal dilution rate and the concentration of the cell ma and metabolite can be computed from eq. (26). Mode I ( jrepeafed batch operation (RB) Each cycle of repeated batch operation conit of rapidly filling the bioreactor (13) with the ubtrate to increae the volume from V, to V,,,, followed by batch operation (33) and terminating the cycle by rapid withdrawal of bioreactor content to reduce the volume from V,, to V,. In view of the cyclic boundary condition (3), the concentration of cell ma, ubtrate and metabolite (xbr,, and pb) at the end of the rapid addition (13) or at the beginning of batch period are related to the concentration at the end of the operation (x, a/, and p/) by the following ma balance equation: xb = tlx/, b = t1 - tl)f + q/. pb = VP/ cw

7 Optimal mode of operation of bioreaotor for fermentation procee 3875 where tl = V,/ V_,. The dynamic ma balance equation for the metabolite and the ubtrate [eq. (l)] for the batch operation (33) can be rearranged to yield dp d= -_orpf-p= I =*?I -dd. Sf = (29) During the rapid withdrawal mode (3 l), which occur at the end of fermentation, the concentration of cell ma, metabolite and ubtrate remain unchanged. Subtituting eq (28) and (29) into eq. (25a), we get C=%l IpFB = P/ _ 1 J., = -- SF SF (1 - cd. (30) It can be een from eq. (30) that the metabolite yield i a function of the ratio of initial to maximum bioreactor volume (q) and the final ubtrate concentration (So). Thu, the optimal value of q and X which reult in maximum metabolite yield can be obtained by etting the partial derivative of eq. (30) with repect to r7 and / to zero, For a given value of the final ubtrate concentration (,), the.+l which maximize the metabolite yield mut atify eq. (31a), while, for a fixed q, the final ubtrate concentration (f) which maximize the metabolite yield mut atify eq. (31b). It can allo be een from eq. (30) that the final metabolite yield i determined by the intantaneou metabolite yield, which i defined a the ratio of the pecific metabolite production rate (7~) and the ubtrate conumption rate (a). We, therefore, claify fermentation procee into four type according to the dependence of the intantaneou metabolite yield (x//a) on the ubtrate concentration. Thee four type, chematically hown in Fig. 2 are type A type B type c yield increae monotonically with increae in the ubtrate concentration yield decreae monotonically with increae in the ubtrate concentration yield undergoe a maximum with increae in the ubtrate concentration Y Sb Subtrate concentration f Subtrate concentration Sb f =b Subtrate concentration e A Subtrate concentration Sb Fig. 2. Variation of intantaneou metabolite yield x/v with ubtrate concentration: (a)-(d) repreent four type of fermentation procee (identified a A-D in text).

8 3876 JAYANT M. MODAK and HENRY C. LIM yield i independent of the ubtrate concentration. We now tudy the condition for realizing an optimal value of tl for a fixed value of,, eq. (3 la), for four type of procee. The numerator of eq. (31a) can be expreed a a difference between area encloed by the curve ABCDA [ = c (rc/c)d] and ABCEDA [= (n/a)&, -,)] a hown in Fig. 2(a). It i clear B* from Fig. 2(a) that (x/a)d < (x/a),, (, - /) or f a(zpf )/arl < 0. Thi &plie that, for any value of, the metabolite yield for type A fermentation procee decreae a tl increae, or the metabolite yield i maximum at the lower limit of tf, i.e. of = 0. A rl tend to 0 (, + S,), the repeated batch operation approache a ingle-cycle batch operation in which the entire bioreactor content are withdrawn at the end of the operation and the metabolite yield [eq. (30)] i given by (ZP:B),=, = IPf = + SF!! d. P I a, Q (32) It can alo be concluded from eq. (32) that the yield will be maximum when J alo tend to zero, i.e. ubtrate i completely exhauted. Thu, for type A fermentation procee, repeated batch operation with zero final ubtrate concentration and complete withdrawal of the bioreactor content i the optimal repeated batch operation. Such an operation may be phyically unrealizable in practice becaue complete exhaution of ubtrate may require very large operating period. Nonethele, the analyi give u the maximum yield that i theoretically poible and alo provide the guideline for operating the bioreactor. For example, for type A procee, repeated batch operation with a low a final ubtrate concentration and a near a complete withdrawal of bioreactor content a permitted within other operating contraint would reult in a high metabolite yield. Similar analyi for type B fermentation procee reveal [Fig. 2(b)] that, for any value of,-, a(zpt )/&) > 0. In other word, the metabolite yield i maximized at the upper limit of r), i.e. t) = 1, for type B fermentation procee. A q tend to unity (b + l). the repeated batch operation approache a continuou teadytate operation and the metabolite yield [eq. (30)7 can be obtained uing l H6pital rule a follow: limzp~=zp~=$[~(s,-_31==,. (33) 1-i A expected, the metabolite yield given by eq. (33) i identical to that of the continuou teady operation given by eq. (27). Furthermore, the metabolite yield given by eq. (33) i maximized when / = 0, or when the ubtrate i completely conumed. The optimal operation i phyically unrealizable, but give u the maximum yield poible and alo ugget that continuou operation with a very low dilution rate would reult in a high metabolite yield. For type C fermentation procee [Fig. 2(c)], we can conclude that, for a given value of,, there exit an ~(0 < rl < 1) which atifie eq. (31a). Equation (31a) along with eq. (31b) will reult in optimal value of tl and /, which maxirnize the metabolite yield for type C procee. Thu, repeated batch operation with partial withdrawal of the bioreactor content i an optimal repeated batch operation for type C procee. For a type D proce, eq. (31a) i trivially atified, and therefore the metabolite yield i independent of r~ [Fig. 2(d)]. Thi i an expected reult becaue in type D fermentation procee, the intantaneou metabolite yield i independent of the ubtrate concentration, and therefore the final metabolite yield i independent of the type of operation. Since the intantaneou yield i cot&ant (n/a = Y,), the metabolite yield [e. (30)] i given by ZIy = Y&Y, - SJ)_ Therefore, for any value of q, the metabolite yield will be maximized when z = 0, i.e. the ubtrate i completely exhauted. Mode II ( ) or III ( j repeated fed-batch operation (RFB) Both repeated fed-batch operation (mode II and III) conit of an interval (23) in which the ubtrate i fed to the bioreactor at a controlled (intermediate) rate and the outlet flow rate i zero. Thi interval i referred to a a ingular interval in which the witching function 4, i identically zero. The general characteritic of the ingular interval in fed-batch operation for maximizing the final metabolite yield have been extenively tudied (Modak et al., 1986). It ha been hown (Modak et al., 1986; Staniki and Levaauka, 1984) that, during the ingular interval, the ubtrate concentration hould be maintained at a contant level,,, which maximize the intantaneou metabolite yield, x,/c, i.e. d d 7c 0 a,_,= It ha alo been hown (Modak and Lim, 1987) that the ingular interval exit if and only if eq. (34) i atified. Since the intantaneou yield i monotonic for type A and type B procee, while it i contant for type D procee, eq. (34) cannot be atified for thee three type of procee. In other word, the ingular interval (23) doe not exit for type A, B and D procee, and therefore repeated batch operation i the only feaible mode of operation. For a type C proce a ingular interval exit, and therefore a repeated fed-batch operation i a feaible mode of operation. Each cycle of the repeated fed-batch operation (mode II), i initiated by filling the bioreactor rapidly (13) to increae the bioreactor volume from V, to V,, uch that the ubtrate concentration increae from, to,,,. Thi i followed by the ingular interval (23) in which the ubtrate i added 0.

9 Optimal mode of operation of bioreactor for fermentation procee 3877 at a rate which maintain the ubtrate concentration contant at, and the bioreactor volume increae from V, to it maximum value, V,,,. The batch operation (33) follow until the metabolite yield i maximized. The cycle i terminated by rapid withdrawal (31) of the bioreactor content. In view of the cyclic boundary condition (3), the concentration at the end of the rapid addition (13) or the beginning of the ingular interval (23) (x,,,, p,) are related to the concentration at the end of the operation (x,,,-, p,) by the following ma balance equation: rx, = V, 9 e, =(5-v)&+ rlq,rp=rlp/ (35) where t = VJ V_, and q = V,,/V_, _ During the ingular (23) interval, the ubtrate concentration i contant at,, and the ingular interval feed rate F,., can be determined by etting the time derivative of the ubtrate concentration to zero: or d -=FF,(S,-)/V-ox=0 dt (34) Subtituting eq. (36) into eq. (1) and rearranging, we get or which yield d(pv)=[a(sp--)]_dv (37) Pb - tp - [~G-qJl-5) (38). where ubcript b denote the concentration at the end of the ingular interval (23) or the beginning of the batch interval (33). During the batch interval, the metabolite and ubtrate concentration dynamic i decribed by eq. (29) which yield Pf - Pb = -Ed. Sf d Combining eq (35), (38) and (39), the objective function for maximizing the metabolite yield, eq. (25a), can be expreed a = [5(I-S)],(1-e)+~m~d (40) SR (1 - tt) Subtituting eq. (35) into eq. (40), the metabolite yield can alo be expreed a We note that the metabolite yield in repeated fedbatch operation [mode II ( )] i alo a function of r~ and,. The repeated fed-batch operation (mode III) i identical to mode II except that an additional batch (33) period exit after rapid addition (13) and the ingular interval begin after the batch period when the ubtrate concentration reache,,,. Since, during the batch operation, the ubtrate concentration decreae, the ubtrate concentration at the beginning of the batch period (b) mut be higher than that at the end of the batch operation (,) or beginning of the ingular interval. During the rapid addition, the bioreactor volume increae from V,, to V:, and therefore the concentration at the end of the rapid addition (13) or at the beginning of the batch period (33) (~6. k, pi) are related to the concentration at the end of operation by the following ma balance equation: 6 xb = Ix,, e ; = (C - q)s, +?S/, C'P; = tlpf (42) where 5 = V~V_, and q = V,,/V,,,. The concentration of ubtrate and metabolite (,,,, p.) at the end of the batch operation or at the beginning of the ingular interval are related to thoe at the beginning of the batch operation by batch operation dynamic [eq. (29)] a follow: r'(p -Pie) = <' ;;d. (43) Sb. Since the ingular interval (23) operation in mode II ( ) and III ( ) i identical, we can ue the relation (38) developed earlier to relate the concentration of the ubtrate and metabolite at the end of the ingular interval (b and pb) to thoe at the beginning of the ingular interval (, and P3 Pb - c p; = [:(F - SQl- el>. (44) For the batch period which follow the ingular interval, we can ue eq. (29) to relate concentration of ubtrate and metabolite at the beginning of the batch period to thoe at the end of the batch peribd. Combining eq (42)-(44), the metabolite yield in repeated fed-batch operation can be expreed a follow: (IP:PB),, = $?r d x 0 - =f ct - cu -w,+wf-,l + - %I.$ (1 - rl). (41) Comparion of the metabolite yield obtained by repeated fed-batch operation [mode II, eq. (40)] and [mode III, eq. (45)]

10 3878 JAYANT M. MODAK and HENRY C. LIM yield WI - %n)- + =ml (1 -VI. (46) Since (n/a)=, i the maximum value of (Z/U), the RHS of eq. (46) i alway poitive. In other word, the metabolite yield obtained in repeated fed-batch operation (mode II) i alway greater than that obtained in repeated fed-batch operation (mode III). We now compare the metabolite yield obtained in repeated fed-batch operation with that obtained in repeated batch operation for type C procee. It wa alo hown earlier that, for type C procee, there i a unique et of q and / (?*, f), which reult in maximum metabolite yield in repeated batch operation. Let u conider a repeated fed-batch operation with q = q* and = ST, which reult in optimal performance for a repeated batch, but uboptimal for a repeated fed-batch operation. Uing eq. (41) for metabolite yield for a repeated fed-batch operation (mode II) and eq. (30) for a repeated batch operation, we get 7t (b*-,) - ad = 1 0 0% Em S, (l-v*). CO - l*wp + v*q - %I1 (47) Since (Z/D),, i the maximum value of (E/Q), the RHS of eq. (47) i alway poitive. In other word, the metabolite yield obtained in uboptimal repeated fedbatch operation (mode II) i alway greater than that obtained in optimal repeated batch operation (mode I). Thu, it i hown that for type C fermentation procee, even uboptimal repeated fed-batch operation reult in higher metabolite yield a compared to that obtained in the optimal repeated batch operation. A pointed out earlier, the metabolite yield [eq. (41)] in a repeated fed-batch operation depend on q and /. Therefore, optimal value q and J which reult in the maximum metabolite yield can be determined by etting the partial derivative of the metabolite yield given by eq. (41) to zero, i.e. VW We firt examine eq. (48a). Since (x/c), i the maximum value of (Z/G), the RHS of eq. (48a) i alway negative. In other word, for any, fixed value of,, metabolite yield decreae a tl increae. Therefore, the metabolite yield i maximized when q = 0, i.e. when the bioreactor content are completely withdrawn. A PJ + 0, repeated fed-batch operation approache a ingle-cycle fed-batch operation. Furthermore, if t,~ i zero, eq. (48b) how that metabolite yield_ i maximized when alo approache zero, i.e. the ubtrate i completely conumed. Thu, a ingle-cycle fed-batch operation with complete conumption of the ubtrate i the optimal mode of operation for maximizing the metabolite yield for type C fermentation procee. Thu, the optimal mode of operation of a bioreactor for the four type of fermentation procee are: (i) type A-ingle-cycle batch operation (q = 0), (ii) type B--continuou teady-tate operation (q = l), (iii) type C-ingle-cycle fed-batch operation (9 = 0), and (iv) type D-any repeated batch operation. An example given below numerically illutrate the analytical reult developed in thi ection. The pecific rate expreion choen for the tudy are /J = 0.125, _ LX= -kk,p2+k2p, a=&+k_ (49) By chooing appropriate value of contant kl, k, and k, we can reproduce all four type of fermentation procee decribed earlier. The mot intereting cae i that of type C fermentation procee which i oberved for everal amino acid, antibiotic and enzyme production. The model choen i imilar qualitatively to that propoed for production of lyine, u- amylae and penicillin, but not quantitatively. The reult of the computation are preented in Table 2 and 3. For a fixed value of q, the optimal value of f which reult in maximum yield i determined for repeated batch operation from eq. (31b) and for repeated fed-batch operation from eq. (48b). It can be een from Table 2 that, for type A procee, the metabolite yield decreae a tf increae or yield i maximized a q approache zero. Furthermore, a q approache zero, the optimal final ubtrate concentration alo approache zero_ Thi confirm our earlier reult that repeated batch operation with complete exhaution of ubtrate followed by complete withdrawal of bioreactor content i the optimal mode of operation for type A procee. For type B procee, the metabolite yield increae a q increae and for each q the optimal value of f i zero. Thu, continuou operation with zero ubtrate in the reactor i optimal for type B procee_ The reult in Table 3 how that, for type C procee, the yield i maximized at intermediate value of q for repeated batch operation. Both ingle-cycle batch (q = 0) and

11 Optimal mode of operation of bioreactor for fermentation procee 3879 Table 2. Variation of metabolite yield with the ratio of initial to maximum bioreactor volume, 1 Repeated batch operation 5~ A Type R (k, = 0, k2 = 3, k3 = 0.03) (k, = 4.8, k2 = 3, ka = 0) rl r Yield r Yield 3.1 x x 10-S x 10-a O Table 3. Variation of metabolite yield with the ratio of initial to maximum bioreactor volume, q TYP C Type D (k, = 4.8, k, = 3, k, = 0.03) (k, = 0, k, = 3, k, = 0) Repeated batch Repeated fed-batch Repeated batch rl a/ Yield 31 Yield r Yield x lo x lo x 10-b X 10-S x lo x lo x lo E E% :o a : Optimal, for repeated fed-batch operation. *Optimal r for repeated batch operation. continuou proce (V = 1) reult in lower metabolite yield a compared to repeated batch operation with partial withdrawal (q* = 0.89, f = 0.099). The metabolite yield obtained in repeated fed-batch operation with q* and i i higher than that obtained in the optimal repeated batch operation. The maximum yield i obtained in repeated fed-batch operation when rf approache zero for type C fermentation procee. For type D procee, the metabolite yield i independent of q. BIOREACI OR PRODUCIWITY Bioreactor productivity, a defined by eq. (25b), depend on the concentration of the metabolite a well a on the period of operation. The concentration of metabolite i determined by the ratio of pecific metabolite production to ubtrate conumption rate, while the period of operation i determined by the rate at which microorganim can grow, that i the pecific growth rate. Therefore, the problem of max- imizing bioreactor productivity i much more complex than the problem of maximizing metabolite yield. The analytical reult uch a thoe obtained for maximizing metabolite yield are difficult to obtain for maximizing the productivity for a general fermentation model. In order to gain ome inight, we make an additional aumption that the pecific ubtrate conumption rate i proportional to the pecific growth rate, i.e. p//o = Y, where Y, i a contant. The material balance for ccl1 ma and ubtrate in eq. (1)

12 3880 JAYANT M. MODAK and HENRY C. LIM require that x = Y,(S, - ), 0 < t < t, (50) if eq. (50) i atified at t = 0. For the ake of implicity we aume that eq. (50) i atified at the initial time. The dynamic ma balance for ubtrate and metabolite in eq. (1) then implify to The optimal value of q and J can be obtained by etting the partial derivative of ZP, with repect to q and / to zero, i.e. i = F,(S, - )/V - 0 Y,(S, - ) 6 = xy,(s, - ) - F,p/V. (51) Thu, the fermentation model i completely pecified in term of concentration of ubtrate and product only. Mode IV (22jcontinuou teady-tate operation A dicued earlier, continuou teady-tate operation i one of the feaible operation for maximizing the bioreactor productivity. The bioreactor productivity can be expreed uing the teady-tate ma balance equation from eq. (51) a follow: IP$ = Dp = 7t Y,(S* - ). Thu, the maximum productivity that can be achieved in continuou operation i (~J%n,, = Cn Y.&G - 419, (52) where, i the ubtrate concentration reactor uch that $ Cn YX (S, - )l, = 0. in the bio- Mode I ( krepeated batch operation The repeated batch operation conit of period of rapid filling (13), batch interval (33) and rapid withdrawal (31). The detail of the operation have been decribed earlier. Utilizing eq (28) and (29) the final metabolite concentration can be expreed a (1 - tt)p, = SJJ I$ 62 (53) where 9 = VO/ V,,, Since the addition (13) and withdrawal (31) are intantaneou, the period of operation tf i the ame a that of the batch operation, which can be expreed uing batch interval dynamic given by eq. (51) with F, = 0, i.e. Zf = * 1 8, ayx&-) d. (54) Combining eq (53) and (54), the bioreactor productivity in repeated batch operation can be expreed a ZPf - =b a ;d =I f-s* 1 (55) d Thu, bioreactor productivity eq. (55) i alo a function of the ratio of initial to maximum bioreactor volume, q. and the final ubtrate concentration, So. (53 We now compare the productivity in the repreated batch culture [eq. (55)] with that of the maximum obtained in continuou operation [eq. (52)]: db;d ZPfB - (ZP:),, = b = 1 S, ayx@, - 4 d - cm Yx(& - )I, b x YASF CT YJSF - 4l, & = =r u y&g - 9 I Sb 1, c y&% - ) d. (57) Since [n Y,(S, - )], i the maximum value of [x Y,(S, - )], the numerator in the RHS of eq. (57) i alway negative. In other word, the productivity in repeated batch operation i alway le than the maximum productivity obtained by continuou operation. Mode II ( ) or III ( j repeated fed-batch operation The repeated fed-batch operation ha a ingular interval (23). It ha been etablihed earlier (Modak et al., 1986) that maintaining the ubtrate concentration contant, though optimal for maximizing the metabolite yield, i not optimal for maximizing the productivity. Thu, for the problem for maximizing the productivity, the ubtrate and metabolite conoentration at the beginning of the ingular interval (, and p,) are not known a priori. A outlined in Section 2, the inlet flow rate during the ingular interval, F1,,, i calculated by etting the time derivative of witching function #B, to zero. Uing the defi&tion of adjoint variable [eq. (1 l)] and the witching function [eq. (lo)], the ingular interval flow rate, FImS, for the implified model can be expreed a follow: F _== dv 1.S = dt Q Yx(& - ) 1 - (58)

13 Optimal mode of operation of bioreactor for fermentation procee 3881 Rearranging eq. (58), we get d cr (lc b - a 7t) t= (&b - aail ) = g(). Thu, the variation of ubtrate concentration during the ingular interval i known, but the ubtrate concentration at the beginning of the ingular interval (,) i not known. In the following analyi, we will aume that the concentration of ubtrate and metabolite at the beginning of the ingular interval are : and P:, repectively, and how that the repeated fed-batch operation reult in lower bioreactor productivity than the optimal continuou operation. We conider the repeated fed-batch operation in which the operation i initiated by rapidly filling (13) the bioreactor to increae the bioreactor volume from V, to V. The concentration of the ubtrate and metabolite at the beginning of the ingular interval (23), (:, p:), are related to the concentration at the end of the operation, (So, p,), by material balance given below: T,* = (r - fi)s, + VS/, TP.* = tpf (60) where e = VJV,, and q = V,,/V,,,ax. During the ingular interval, the ubtrate dynamic i given by eq. (59) while the metabolite balance can be rearranged to yield d(pv) dt = 7c Y,(S, - ) K (61) The ingular interval continue till the bioreactor volume increae from V, to V,,,,,. Equation (59) and (61) can be coupled to relate the concentration of ubtrate and metabolite (b and pb) at the end of the ingular interval or at the beginning of the batch period (33) to,* and p:, i.e. pb - tp: = ab Y,(S, -) v d 4?() vnl, (62) where g() i given by eq. (59) and the bioreactor volume V i alo a function of ubtrate concentration [eq. (%)I. The operating time t, for the ingular interval i given by eq. (59), i.e. (63) The concentration of ubtrate and metabolite at the beginning, (b, pb), and at the end, (/, pf), of the batch period (33) which follow the ingular interval are related by eq. (29) and the operating time rb for the batch interval i given by eq. (54). Combining eq (29), (60), and (62), we get (1 - VIP, = ny,(s, - ) v d + SF g() v,, and the total period of operation tf i d. (64) (6% Thu, the bioreactor productivity in repeated fedbatch operation i (66) We now compare the productivity of repeated fedbatch with that of the continuou operation: ZPfFB-(ZP$),,, a%)-)~* = + = 7c -d : max =b u - c7t KM, - m, -5 =b YA&- - 41, vnl,, * : g() vlna, IL ya% - ) v - cw J - &+ -- d J* ay*(sp -) = 7l YXCJF CR YAS, - )l, & 0 Yx(S, =* ab 1 -d + = d : g() Sb CYX(S, -) (67) Since [n Y,(S, - )], i the maximum value of Cn YX(% - )l,, and V i le than V,,, the numerator of both term in the RHS of eq. (67) are negative, which implie that productivity in repeated fed-batch i le than the maximum productivity that can be achieved in continuou operation. Similar analyi can alo be done for repeated fed-batch operation to how that the productivity in continuou operation i the maximum. Thu, for maximizing the productivity, continuou mode of operation i the optimal mode among the four feaible operating mode. The reult of thi ection are illutrated with an example. The kinetic model choen for the tudy i the ame a that decribed earlier [eq. (49)] and the reult are preented in Table 4. For the model parameter choen, the maximum productivity in continuou operation i achieved at the ubtrate concentration, = For the fixed value of final ubtrate concentration J, the value of rf which reult in maximum productivity i computed for repeated batch and repeated fed-batch operation. Among the three mode of operation, the maximum productivity i obtained in continuou operation with, =,. Thu, continuou operation i the optimal mode of operation for maximizing the bioreactor productivity. However, the continuou operation at r =, may not alway be poible becaue of ome operating contraint uch a poor controllability or iflow ubtrate concentration in the outlet tream i deirable.

14 3882 JAYANT M. MODAK and HENRY C. LIM Table 4. Variation of bioreactor productivity with the final ubtrate concentration,, k, = 4.8, kz = 3, k, = 0 Continuou IPEON Repeated batch Repeated fed-batch x 1O x x IO- * I Table 4 compare the three mode of operation for different value of the final ubtrate concentration. For / below,, the repeated fed-batch operation reult in the highet productivity among the three operation. The advantage of repeated fed-batch over repeated batch and continuou operation diminihe a the final ubtrate concentration approache,. The optimal q increae a, increae. At J =,, the optimal value of TV i unity for repeated batch a well a for repeated fed-batch operation, i.e. both operation become continuou operation. For / greater than,, the bioreactor productivity in continuou operation decreae a, increae. Furthermore, the optimal value of q i unity for both repeated batch and fed-batch operation. It can be concluded that continuou operation i the optimal mode of operation for maximizing bioreactor productivity for, 2,. For / le than,, the repeated fed-batch operation with partial withdrawal i the optimal operation for maximizing the bioreactor productivity. 5. CONCLUSIONS In thi paper, the problem of determining the optimal mode of operation of a bioreactor for maximizing the metabolite yield and bioreactor productivity ha been analyzed. A general variable-volume emibatch bioreactor model i formulated and feaible combination of inlet and outlet flow rate are identified uing the framework of optimal control theory for multiple control variable. It i hown that the withdrawal of the bioreactor content at intermediate rate i not optimal, and therefore the optimal trategy for removal of the bioreactor content i to withdraw at the maximum rate. Furthermore, in a emibatch mode of operation, imultaneou addition and withdrawal are not permitted. The withdrawal ha to be done after the bioreactor volume reache it maximum limit. The feaible mode of operation of a bioreactor have been hown to be: (i) repeated batch operation ( ), (ii) repeated fed-batch operation ( or ), and (iii) continuou teady-tate operation (22). The continuou mode of operation i not optimal for maximizing the metabolite yield, wherea it i one of the candidate for maximizing the bioreactor productivity. The feaible mode of operation are identified without any aumption on the kinetic of the fermentation proce. However, the optimal mode of operation among the four feaible mode i hown to depend on the kinetic of the fermentation proce. The metabolite yield i hown to depend on the final ubtrate concentration in the bioreactor (J) and the ratio of initial and maximum bioreactor volume (q). The optimal mode of operation of a bioreactor for maximizing the metabolite yield i determined by the dependence of the intantaneou metabolite yield, defined a the ratio of peciftc metabolite production rate (7~) and pecific ubtrate conumption rate (a), on the ubtrate concentration. The fermentation procee are claified into four type A, B, C, and D depending on whether intantaneou yield (x/u) increae, decreae, goe through maxima or remain contant with increae in the ubtrate concentration. The optimal mode of operation for the four type of fermentation procee i: (i) ingle-cycle batch for type A, (ii) continuou for type B, (iii) ingle-cycle fed-batch for type C, and (iv) any repeated batch operation for type D procee. In ome of the cae, the optimal operation i phyically unrealizable, and the optimal reult hould be viewed a theoretical limit on the performance of the bioreactor. For example, a true optimal operation for type C procee, a cyclic fed-batch operation with complete conumption of ubtrate (So = 0) and then complete withdrawal of bioreactor content (II = 0), i not poible. In uch a cae, the fed-batch operation with a low a final ubtrate concentration and a near a complete withdrawal of the bioreactor content a i permitted within other operating contraint would reult in high metabolite yield. The continuou mode of operation of a bioreactor i hown to be uperior than either repeated batch or repeated fed-batch operation for maximizing the bioreactor productivity. NOTATION F, inlet feed rate F2 outlet feed rate IP,, IP, objective function defined by eq. (4)