Optimal temperature and concentration profiles in a cascade of CSTR's performing Michaelis-Menten reactions with first order enzyme deactivation

Bioprocess Engineering

9 Springer-Verlag 1993

Optimal temperature and concentration profiles in a cascade of CSTR's

performing Michaelis-Menten reactions with first order enzyme

deactivation

A. L. Paiva and F. X. Malcata, Porto, Portugal

A b s t r a c t . A necessary condition is found for the intermediate tem- peratures and substrate concentrations in a series of CSTR's per- forming an enzyme-catalyzed reaction which leads to the minimum overall volume of the cascade for given initial and final temperatures and substrate concentrations. The reaction is assumed to occur in a single phase under steady state conditions. The c o m m o n case of Michaelis-Menten kinetics coupled with first order deactivation of the enzyme is considered. This analysis shows t h a t intermediate stream temperatures play as i m p o r t a n t a'role as intermediate sub- strate concentrations when optimizing in the presence of nonisother- mal conditions. The general procedure is applied to a practical example involving a series of two reactors with reasonable values for the relevant five operating parameters. These parameters are defined as dimensionless ratios involving activation energies (or enthalpy changes of reaction), preexponential factors, and initial temperature and substrate concentration. F o r negligible rate of deactivation, the Qptimality condition corresponds to having the ratio of any two consecutive concentrations as a single-parameter increasing func- tion of the previous ratio of consecutive concentrations.

L i s t o f s y m b o l s CE. o mol.m - 3 C~, i mol.m - 3 Cs, o m o l . m - 3 Cs,~ m o l . m - 3 Dai Damin Datot Ea J . m o l - 1 E,, J.mol- x Ev J.mol - 1 i J k ka, i S- 1

Initial concentration of active enzyme Concentration of active enzyme at the outlet of the i-th reactor

Initial concentration of substrate

Concentration of substrate at the outlet of the i-th reactor

D a m k 6 h l e r n u m b e r associated with the i-th reactor ((Vi.kv, o.Ce, o)/(Q.Cs, o) )

M i n i m u m value of the overall Damk6hler n u m b e r

Overall Damk6hler n u m b e r Daj

J

Activation energy of the step of deactivation of the enzyme

Standard enthalpy change of the step of binding of substrate to the enzyme Activation energy of the step of enzymatic transformation of substrate

Integer variable D u m m y integer variable D u m m y integer variable

Kinetic constant associated with the deacti- vation of enzyme in the i-th reactor (ka, o. exp{ - Ea/(R.Ti)})

kd, 0 gm, i Km, o kv. i kv, o N Q R T~ To V, Vmax Xi Xi,opt Yi Yi, opt

s - 1 Preexponential factor of the kinetic constant associated with the deactivation of the en- zyme

mot.m -3 Equilibrium constant associated with the binding of substrate to the enzyme in the i-th reactor, (K,,.0.exp{ - E,,,/(R.Ti)})

m o l . m - a Preexponential factor of the Michaelis-Men- ten constant associated with the binding of substrate to the enzyme

s-1 Kinetic constant associated with the trans- formation of the substrate by the enzyme in the i-th reactor (kv, o .exp{ - E,,/(R.Ti)}) s - x Preexponential factor of the kinetic constant

associated with the transformation of the substrate by the enzyme

- N u m b e r of reactors in the series

m3,s -1 Volumetric flow rate of reacting liquid through the reactor network

J . K - X.mol- 1Ideal gas constant

K Absolute temperature at the outlet of the i-th reactor

K Initial absolute temperature m 3 Volume of the i-th reactor

m o l . m - 3.s- ~ Maximum rate of reaction under saturation conditions of substrate

- Normalized concentration of substrate (Cs. i/Cs. o)

O p t i m u m value of the normalized concen- tration of substrate

- Dimensionless temperature (exp{ - To/Ti})

- O p t i m u m value of the dimensionless temper- ature

Greek symbols O;

fl

q

Dimensionless preexponential factor asso- ciated with the Michaelis-Menten constant (Km, o/Cs, o)

Dimensionless activation energy of the step of enzymatic transformation of substrate (Ev/R.To))

Dimensionless standard enthalpy change of the step of binding of substrate to the en- zyme (Em/(R.To))

Dimensionless activation energy of the step of deactivation of the enzyme (Ed/(R.To)) Dimensionless deactivation preexponential factor ((kd, o.Cs.o)/(kv, o.CE, o))

78

1 Introduction

Continuous stirred tank reactors (CSTR's) have been used to perform biochemical reactions of industrial interest because tfiey are easy to model and control. Furthermore, their operation is economically feasible provided that (i) large throughput feedstocks are handled, (ii) the reac- tion products are subject to narrow market specifications, (iii) the cost of enzyme makeup is not high (which usually implies that the level of purity of the enzyme employed with respect to other inert species is not critical for effi- cient operation), and (iv) the residual enzymatic activity in the effluent stream can be easily destroyed via, e.g., ther- mal treatment.

Theoretical reasonings underlying the design of a series of CSTR's performing chemical reactions which obey power-law [1, 2] and Michaelis-Menten like rate equations [3-8] have been derived previously. The latter analyses all assume a constant activity of the enzyme along the reactor network and throughout time. Unlike most organic and inorganic reactions effected by homo- geneous catalysis, however, the catalyst in enzyme- catalyzed biochemical reactions is often subject to thermal deactivation [9], and this fact must be taken into account in the design of actual reactor networks.

It is the purpos e of this report to obtain the optimum temperature and concentration of substrate in each reac- tor of a series of CSTR's performing an enzyme-catalyzed reaction which obeys Michaelis-Menten kinetics coupled with first order deactivation of enzyme under the assump- tion that such optimum path gives the minimum overall reactor volume for a given final conversion of substrate.

2 Theory

The problem of interest here is to find the minimum overall reactor volume when a nonisothermal cascade of N CSTR's is employed to bring about an enzyme- catalyzed reaction in the presence of enzyme deactivation. The steady-state mass balance to enzyme undergoing first order deactivation in a series of CSTR's with a con- stant rate of addition of fresh enzyme to the first reactor (in order to compensate for the loss of enzyme as a part of the effluent stream from the last reactor in the series) may be written as:

QCE, i - t = Vika, iCl~,i -t- QCE, i, i = 1, 2 . . . . , N , (1)

where Q is the volumetric flow rate of fluid, V is the reactor volume, kd is the deactivation constant, Ce is the concentration of active enzyme, and subscript i denotes the i-th reactor in the cascade. Solution of Eq. (1) with

Bioprocess Engineering 9 (1993) respect to Ce, i coupled with Arrhenius law, one obtains:

C E , i - 1

CE'i= Vi { E~Ti}i = 1, 2 . . . N , (2) 1 + ~ k d , o,exp --

where ka, o is the preexponential factor and Ed is the activation energy associated with ka, T is the absolute temperature, and R is the ideal gas constant. Application of Eq. (2) from i = 1 to i = N finally gives:

C~,i 1

= i , (3)

C~,o y[ (1 + riDajy~) j=l

where the dimensionless volume (or Damk6hler number), Dai, dimensionless temperature, yz, dimensionless activa- tion energy, 6, and dimensionless deactivation preex- ponential factor, t/, are defined as Vi.kv, o.CF, o/(Q.Cs, o), exp{ - To/T~}, Ea/(R.To}, and ka, o.Cs, o/(kv, o.C~,o), re- spectively, and where Cs denotes the concentration of substrate (kv, o will be defined below).

The steady-state mass balance to substrate undergoing consumption according to the Michaelis-Menten rate ex- pression in a series of CSTR's may be written as:

Q C s i - 1 - - v k v ' i C E ' i C s ' i

. . . . + Q Cs i (4)

, i K , . i + C s , i " '

where k~ is the catalytic constant and K,, is the Michaelis- Menten constant. The product k~,~ Ce,~ is often referred to as the maximum rate of reaction under saturation condi- tions of substrate, and is commonly denoted as Vmax. Use of Arrhenius law in Eq. (4), one gets:

k~, o exp - C~, i Cs, i

Q Cs,~- 1 = Vi + Q Cs,~ , (5)

Km, o e x p { - ~ i i } + C s , i

where k~, o and Km, o are the preexponential factors asso- ciated with kv and Kin, respectively, E~ is the activation energy associated with kv, and Em is the standard enthalpy of reaction associated with Kin. Combining Eql (3) with Eq. (5) and algebraically rearranging the result, one ob- tains:

Dai X i Yfli

X i - 1 - - Xl = i , (6)

(ey} + xi) I-[ (1 + tlOajy~) j = l

where the dimensionless substrate concentration, x~, di- mensionless activation energy, fl, dimensionless standard enthalpy change, 7, and dimensionless preexponential fac- tor, cr are defined as Cs, i/Cs, o, E~/(R.To), Em/(R.To), and K I , o~ Cs, o, respectively.

Solving Eq. (6) with respect to Dai for a generic i, and applying the obtained recurrence relation for Da~ with 1 _< j < i in a way similar to that previously employed by

A. L. Paiva and F. X. Malcata: Optimal temperature and concentration in CSTR cascade performing enzymatic reactions. Malcata [-10], one obtains after some manipulation

( X i _ 1 - - Xi)(Ny~i "[- Xi) ye, xg

Da,

=

' ( 6(Xj_l -- xj)(o~y'j -I- xj))

(7)

1 - tl ,~z t Y J

ye~ x,

. "

The intermediate concentrations of substrate and tem- peratures which lead to the minimum value for the overall

reactor dimensionless volume,

Datot,

may then be ob-

tained from the following relationships:

~Datot

j~= l

--= = 0 ~ X i ~ X i x ~, x 2 . . . x i - ~, x l + 1 . . . x ~ , y ~, y2 . . . yi - l, y~, y~ + ~ ... y~, w i t h i = 1,2 ... N - - 1, and 79 The above set of (2N-2) independent equations in (N-l) intermediate concentrations and (N-l) intermediate temperatures can be solved for x~ and yi by a trial-and- error method. In the particular case of a series of 2 CSTR's, the results are plotted in Fig. 1 for various reasonable values of c~, fl, y, 6, and t/.

If the rate of deactivation of the enzyme is negligible compared with the overall rate of the enzyme-catalyzed reaction (i.e., when tt tends to zero), Eqs. (10) and (11)

Oa,o, I

=

aj x, x2 .... xj, y, y2 ...

)

= 0

clYi cGYi x, . . . i - , , x l , x i + , . . . . . y , . y2 . . . y i - , . y i + . . . ~.

w i t h i = 1 , 2 . . . N - 1,

respectively. Combination of Eq. (8) with Eq. (7) yields, after rearrangement:

+

(Nil ~

1

xi)(ay~ i +

Xi)yl6-#)

Xi

~(

~')y )

1 - q (x~-i --

xj)(~y ~" +

j = l \ Xj d }6-fl) yE~(1

_{+ ~Y;7)}

x~_ 1"~ 1 - tl _{j= 1 \} (x j_ 1 - xs)(~ y~ + _{Xj }6 - fl)} = 0 with i-- 1 , 2 , . . . , N - 1. O~Y~+l~7,~---[i+l(l__t](i ( ( X j - l - - X J ) ( ~ x i + , y i + , \ j = , xj x~ I] J ] , ] +

1 - ~ 2

x~)y '6-~''~'~2

j : , ~j J

))

i+ l --[- ~Yi+ l"~, S - i X 2 - - Y i + l X , + t . ) t X i + k - 1 -- Xi+k)(O~y~+k q- Xi+k)

- 2

(

_{i+k (X~_ - xj)(o~y~ + 9}

(

_

))2

j= 1 Xj

A similar combination of Eq. (9) with Eq. (7) gives, after rearrangement:

- , , , -

~ ,( + !~,--'-

~ , ) ( ~ y , " +

",),,.~-~A.

(a(7 --

fl)Y~ - f i x , - q(~zy~(?

,.,

- - : j

. . . .

Yi-(~+') ( x i ~ ' - 1 ) t " "

t l Y ~ ( X ~ ' - - l ) (

c~(6-t-~/-fl)y!'-~)(~y~-t-x~)-t-(a-fl,x'yi-~(~

---~-j

j

(8)

(9)

(lO)

j=, x~ J / /

N - , ~ Yl ~ - ~ - ~ ( x ~ - ~ - xD(x~+k-~ - X~+k)(~ y~+k + X,+k)(<~O / + ,$ _/7)y~' + X,(a --/7))

-'k E XiXi§

k = l

1 - ~ E

i+k (

(x,_, - ~,)(~ y~ + X,)y,~_/~'~

2

j : l Xj J ) )

= 0 with i = 1,2 ... N - - 1. (11)

80 Bioprocess Engineering 9 (1993) 0.950 0.900 E 0.850 D i . a 0.800 j 0.750 J ' ~ ' ' ' ~ ~ ' 0.5 0.7 0.9 1.1 1.3 0.75 0.73 O 3 (.~ 0.71 O ~ 0.69 0.67 1.5 1.10 ii.'a i i i . a I f I / I I I I t I 0.7 0.9 1.3 0.75 0.73 O ~ 0.71 m 0.69 O 0.67 0.65

1.05

, - i i . c - III.C 2 3 4 8 1.500 . . . 1.400 k 1.300 i.d .~_ 1.200 E 1.00 " " (~ 1.100 1.000 0.95 0.900 0.65 0.90 0.5 1.1 1.5 0~ 1.10 0.800 5 0.920 0.900

5

r 0.880 D 0.880 0.840 0.940 . . . . I I I I I I I I 0.820 ~ ' 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.75 . . . 1.10 0.700 ' ' ' . . . ' . . . 0.5 1.0 1.5 2.0 2.5 3.0 3.5 'y 0.75 o 0.73 @ ~ o ~ 0.71 i.d

069

\

0.67 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ?

~~b

ii'b, -

I I I I I I I ~ 1 1 0.73 ~- 0.71 - 0.69 0.67 0.874 1.05 0.873 1.00 "-- 0.872 0.95 0.871 0.65 0.90 0.30 0.32 0.34 0.36 0.38 0.40 0.42 P 0.895 0.890 0.885 0.880 O.875 0.870 0.865 ~ i . c I I I 2 3 4 6

1.05

1.00 --... 0.95 0.90 , , , , , , , , , , ,

I e

0.870 , , , , , 0.06 0.09 0.12 0.15 0.18 i ] 0.75 . . . 1.10 1.10 1.05 ~_o 1.00 "-- S 0.95 0.90 0.73 q 0.71 O ;~ 0.69 0 0.67 - i i . e . i i i . e

1.05

1.00 ~- F 0.65 ' ~ f ~ ~ ' ' ~ ' ~ ' 0.90 0.06 0.09 0.12 0.15 0.18 TI 0.95

A. L. Paiva a n d F. X. Malcata: O p t i m a l t e m p e r a t u r e a n d c o n c e n t r a t i o n in C S T R cascade performing enzymatic reactions. reduce to: 15 ~'

{

X i - l'~ 13 ~Y,:+I + X , + l yi -#~ 1 + ~y~-~-2 ) = 0 (12) 11

Xi+lY#i+l

~

9 X and --- 7 5

k Xi

1

respectively. Equations (12) and (13) can be manipulated to give:

O~Xi_lXi+,y?(yi+l) ~

' \ Y~ / (14) 1 - - Y i + l #

o~Yi+l + Xi+l

\ Yi J / and y l =

#)/

respectively. Combining Eqs. (14) and (15), one finally gets

_ \ x/ ,/ (16)

x-L 1 +

In the absence of deactivation, a local optimum for the temperature exists if (i) Km increases more rapidly with increasing temperature than does kv and (ii) if Km is sufficiently large. The former condition is equivalent to saying that E m > Ev, or, equivalently, 7 > ft. The latter condition implies that the dimensionless preexponential factor associated with the Michaelis-Menten constant is such that e > fl.xi/(7 - fl) (because 0 < Yi < 1 for all reac- tors); since 0 < xi < 1, a conservative constraint reads

c~ > fl/(7 - fl). The variation of the ratio of two consecut-

ive optimal concentrations as a function of the previous ratio of consecutive concentrations is plotted in Fig. 2 for several values of parameter 7/fl.

When fl = ? = 0, Eq. (14) becomes:

X i = N / X i _ I X i + 1 , (17)

which is the result that would also have been obtained if the system were operated under isothermal conditions

Fig. 1. Plots o f (i) the m i n i m u m D a m k 6 h l e r n u m b e r , Damln, (ii) the normalized intermediate substrate concentration, Cs. 1/Cs. o, a n d (iii) normalized intermediate t e m p e r a t u r e , T I / T o , versus (a) c~ (with fl = 0.35, y = 2, 6 = 3, a n d r / = 0.1), (b) fl (with c~ = 1, y = 2, 6 = 3, a n d r / = 0.1), (c) 6 (with ~ = 1, fl = 0.35, 7 = 2, a n d r / = 0.1), (d) 7 (with c~ = 1, fl = 0.35, 6 = 3, a n d ~/= 0.1), a n d (e) r/ (with c~ = 1, fl = 0.35, 6 = 3, a n d ? = 2), under the a s s u m p t i o n that T2 = 300 K a n d xz = 0 . 5

9

/

y/J3=lO.O i / 7/13=4.0 1 o 2 3 4 5 Xi. 1 / X i

Fig. 2. Plot of the ratio of consecutive normalized concentrations, xi/xi+l, versus the previous ratio counterpart, xi_~/xi, for several values of parameter 7/fl

[3-]. This case corresponds to x i / x i + 1 = x i - 1 / x i , which is the equation of the diagonal in Fig. 2.

3 Discussion and conclusions

It should be noted that although parameters e, fi, 6, and t / c a n take only positive values, because activation ener- gies and preexponential factors are always positive, ? is not generally constrained because the binding of substrate to enzyme may be either an endothermic or an exothermic process.

Although Eqs. (10)-(11) correspond to the necessary conditions associated with the minimum overall reactor volume, one or more of the values Xi, opt and Yi, opt thereby calculated can violate the physical constraints 1 _> x~,opt

>- xi+l,opt and 0 < Yi, o p t < 1 for integer values of i com-

prised between 0 and N. In these situations, there is no local optimum, and the global optimum lies on a temper- ature constraint, e.g., the minimum or maximum temper- ature for which operation of the reacting mixture is feas- ible from hydrodynamic or mechanical points of view. It might also be argued that an exhaustive search pattern for the values of xi and Yi which minimize Datot as obtained from Eq. (7) might lead to faster results than use of Eqs. (10)-(11). However, it should be emphasized that as the number of reactors in the series increases, the number of similar calculations using the exhaustive approach in- creases with

n 2N,

where n is the number of search intervals, and so does the C P U time required. In the general case, the system of 2N-2 equations denoted as Eqs. (10)-(11) can be solved by an iterative procedure starting with the much simpler solution for q = 0, i.e., Eqs. (15)-(16).

In the computation of the optimal intermediate con- centrations and temperatures, special care should be exer- cised because, as pointed out elsewhere [10], there is a minimum concentration of substrate at the outlet of each reactor. This observation is due to the existence of two competing processes. Longer residence times lead to larger amounts of substrate transformed per unit of active

82

enzyme, but longer residence times also lead to a larger extent of deactivation of the enzyme.

In the absence of deactivation, 7 must be greater than /? so that Eq. (15) is physically consistent; such condition leads to values of the ratio ~,/fl always above unity. When 7 =/~, this corresponds to having the system operating under isothermal conditions, in which situation the plot of x~/x~ + 1 versus x~_ ~/x~ coincides with the main diagonal in Fig. 2. In this case the optimal intermediate concentration of substrate in every series of two consecutive stirred reac- tors is equal to the geometric mean of the substrate con- centration of the inlet to the first reactor and the substrate concentration of the outlet of the second reactor. In the non-isothermal situations (i.e., 7//? > 1), x~/x~+l is always above xi_~/x~; hence the corresponding curves in Fig. 2 are always located above the main diagonal (the devi- ation with respect to the main diagonal actually increases with increasing 7/]3) and xi is always above (xi-1. xi+ 1) 1/2. The ranges selected for the plots denoted as Fig. 1 were those associated with physically reasonable values, i.e., temperatures between ca. 0 and ca. 60 ~ and inter- mediate concentrations between 0.65 and 0.75, for an initial temperature of 27 ~ and an initial normalized concentration of unity. Inspection of Fig. 1.i. indicates that Da,,~, increases with increasing e,/~, and t/, whereas this behavior is reversed for increasing values of 6 and 7; in the former three situations the variation has a linear trend, whereas in the latter two situations the variation has a decreasing hyperbolic trend. With respect to the nor- malized intermediate concentration, one concludes that it goes through a local m a x i m u m at e ~ 0.7 (see Fig. 1.ii.a) and through two local m a x i m a located at 7 ~ 0.75 and ,-~ 2.5 (see Fig. 1.ii.d), whereas in the remaining situations it monotonically increases (see Fig. 1.ii.b and Fig. 1.ii.c) or decreases (see Fig. 1.ii.e). With respect to the normalized intermediate absolute temperature, one concludes that it goes through a local m a x i m u m at 7 "~ 1.25 (see Fig. 1.iii.d., whereas in the remaining situations it monotonically in- creases (see Fig. 1.iii.a. and Fig. 1.iii.e.) or decreases (see Fig. 1.iii.b. and Fig. 1.iii.c.). In general, the variation of Da,,i, with every p a r a m e t e r is much less sensitive to the size of the increment used in a typical exhaustive search than is the variation of Cz/Co and T~/To versus the same parameter. It can also be concluded that the variation in the optimal operating temperature is much sharper than the variation in the optimal intermediate concentration for the same varying parameter. Although the o p t i m a l intermediate concentration is always comprised between the inlet and outlet concentrations to the first and from

Bioprocess Engineering 9 (1993) the second reactor, respectively, the same does not apply to the intermediate temperature, which m a y lead to heat- ing or cooling with refrigeration requirements. As ex- pected, when t / t e n d s to zero, the effect of 6 on the con- clusions becomes also negligible, as apparent from Eq. (7). The analysis developed above proves that the extra degree of freedom arising from the introduction of temper- ature as a manipulated variable leads to optimal concen- tration profiles along the cascade of CSTR's which are substantially different from those which would have been obtained if isothermal operation were assumed through- out. Hence it might be useful in the predesign steps of biochemical reactors.

References

1. Aris, R.: The optimal design of chemical reactors. New York: Academic Press 1961

2. Levenspiel, O.: Chemical reaction engineering. New York: Wiley 1972

3. Luyben, K. C.; Tramper, J.: Optimal design for continuous stirred tank reactors in series using Michaelis-Menten kinetics. Biotechnol. Bioeng. 24 (1982) 1217-1220

4. Malcata, F. X.: Optimal design on an economic basis for con- tinuous stirred tank reactors in series using Michaelis-Menten kinetics for ping-pong reactions. Can. J. Chem. Eng. 66 (1988) 168-172

5. Malcata, F. X.: CSTR's in biochemical reactions - an optimiza- tion problem. Chem. Eng. Ed. 23 (1989) 112-115, 128 6. Malcata, F. X.: A heuristic approach for the economic optimiza-

tion of a series of CSTR's performing Michaelis-Menten re- actions. Biotechnol. Bioeng. 33 (1989) 251-255

7. Malcata, F. X.; Cameron, D. C.: Optimal design of a series of CSTR's performing reversible reactions catalyzed by soluble enzymes: a theoretical study. Biocatalysis 5 (1992) 233-248 8. Gooijer, C. D.; Hens, H. J. H.; Tramper, J.: Optimum design

for a series of continuous stirred tank reactors containing im- mobilized biocatalyst beads obeying intrinsic Michaelis-Menten kinetics. Bioproc. Eng. 4 (1989) 153-158

9. Creighton, T. E.: Proteins - structures and molecular principles. New York: Freeman 1984

10. Malcata, F. X.: On the maximum conversion of substrate during biochemical reactions performed by a series of CSTRs in the presence of enzyme deactivation. J. Chem. Eng. Japan 23 (1990) 372 375

Received July 2, 1992 A. L. Paiva

F. X. Malcata (corresponding author) Escola Superior de Biotecnologia Universidade Cat61ica Portuguesa Rua Dr. Ant6nio Bernardino de Almeida 4200 Porto