Optimal design of two interconnected enzymatic bioreactors

Journal of Process Control 14 (2004) 785–794

www.elsevier.com/locate/jprocont

Optimal design of two interconnected enzymatic bioreactors

J�erome Harmand a,*, Alain Rapaport b, Abdou Kh. Dram�e b

a Laboratoire de Biotechnologie de l’Environnement, Institut National de la Recherche Agronomique, Avenue des �etangs, 11100 Narbonne, Franceb Laboratoire d’Analyse des Syst�emes et Biom�etrie, Institut National de la Recherche Agronomique, 2, place Viala, 34060 Montpellier cedex 1, France

Abstract

This paper deals with the optimal design of two interconnected continuous stirred bioreactors in which a single enzymatic

reaction occurs. The term ‘‘optimal’’ should be understood here as the minimum of the total volume of the reactors required to

perform a given conversion rate, given a quantity of matter to be converted per time unit. In determining the optimal volume, it is

considered that the input flow may be distributed among the tanks and also that a recirculation loop may be used. The optimal

design problem is solved for a wide class of kinetics functions including, in particular, the well-known Michaelis–Menten kinetics

function. The analysis of the optimal configurations is investigated, and it is shown that the concept of ‘‘Steady State Equivalent

Biological System’’ (SSEBS) first introduced by Harmand et al. [AIChE J., in press] for microbial reactions only applies to enzy-

matic systems which have non-monotonic kinetics. In addition, a stability as well as a sensitivity analysis of the optimal configu-

rations are performed.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Optimal design; Reactors in series; Enzymatic reactions; Interconnected systems; Optimization

1. Introduction

The question of optimally designing chemical orbiochemical systems has been assessed by several au-

thors these last 30 years. An important effort has been

made by the chemical engineering community to devel-

op general and systematic theories for optimizing pro-

cess synthesis (see for instance the paper by Feinberg

[2]). This task turns out to be much more complex for

the case of biological systems. One reason for that is the

difficulty of finding a simple and yet accurate model torepresent all the important dynamics of living organisms

interacting in a biosystem. Furthermore, it should be

noticed that most of the existing results are limited to

the optimal design (in a sense to be clarified later on) of

a series (cascade connection) of N continuous stirred

tank (bio)-reactors (CSTR).

Biological processes can usually be classified into

two classes of systems: microbiological and enzymaticreactions. In simple terms, microbiological-based reac-

*Corresponding author. Tel.: +33-46-84-25-159; fax: +33-46-84-25-

160.

E-mail address: [email protected] (J. Harmand).

0959-1524/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jprocont.2003.12.003

tions define (bio)reactions where a substrate degrada-

tion is associated with the growth of certain organisms

while the second, the enzymatic reaction, may beviewed as a chemical reaction with specific kinetics

functions.

Given a model of a series of CSTR, representing

enzyme or microbiological reactions, and a flow rate to

be treated Q, the problem of determining optimal con-

ditions for steady-state operation is of great interest. For

instance, conditions have been proposed to minimize the

total retention time (TRT) required to attain a givenconversion rate ð1� SN

S0Þ (here S0 and SN denote respec-

tively the input and output substrate concentrations).

Observe that this specific problem is equivalent to the

minimization of the total required volume (TRV) when

Q is constant (as it is the case at steady state). The main

motivation for using a series of continuous stirred tanks

reactors is that their industrial operating conditions and

reliability are better if compared to a plug flow reactor(PFR). Thus, the problem of finding conditions under

which the performance of a series of CSTR is compa-

rable to that one obtained with a PFR is of high prac-

tical interest.

When the objective is to optimize a series of biore-

actors, existing results of the literature can be classified

mail to: [email protected]

1V 2V

01 , SQQ α= 02 ,)1( SQQ α−=

22,, SXQQR β=

22,, SXQ

Fig. 1. The two tank system.

786 J. Harmand et al. / Journal of Process Control 14 (2004) 785–794

into two distinct classes: those dealing with mono-fed

systems composed of N bioreactors with different vol-

umes (cf. for example Luyben and Tramper [5] for

enzymatic reactions and Hill and Robinson [4] formicrobial reactions) and those dealing with multi-fed

systems composed of N equal size reactors (cf. for in-

stance Dourado et al. [1]). From the authors knowledge,

the determination of the optimal volumes of the tanks

(not necessarily equal) in the case where the input flow is

distributed among the units of the series has never been

investigated.

Recently, Harmand et al. [3] have proposed new re-sults about the optimal design of two interconnected

bioreactors in which both a recirculation loop and the

distribution of the input flow rate between the two

reactors were allowed. It was shown that the optimal

solution of the problem of minimizing the TRV depends

on the ratio S01=S2 where S0

1 is one of the two roots of a

particular second order equation whose coefficients are

determined from the system data and S2 is the outputsubstrate concentration to be obtained. More specifi-

cally, if S01 > S2, the optimal design is obtained by con-

sidering the feeding only into the first reactor while no

recirculation is needed. At one end, one has S01 ¼ S2

corresponding to the case where only one tank is nee-

ded. For the case in which S01 < S2, it was shown that the

TRV can be significantly decreased (when compared to

the existing design procedures found in the literature) bydistributing the input flow rate and introducing a re-

circulation loop with a certain flow rate. In such a case,

the optimal solution is not unique and the concept of

‘‘Steady State Equivalent Biological Systems’’ (SSEBS)

was introduced to help the designer to choose one

among all the possible equivalent TRV-optimal solu-

tions. This new concept is interesting because it allows

the designer to take into account additional propertiesof each one of the TRV-optimal solution, such as

observability or controllability, and then to decide

which one is the ‘‘best’’ with respect to these additional

criteria.

In this paper, the SSEBS principle is extended for

enzymatic biological reactions. The problem to be

solved is formulated as follows: ‘‘Given the model of a

biological process and a flow rate Q to be treated pertime unit at steady state operation, determine the min-

imum TRV to attain a specified conversion rate

ð1� S2S0Þ.’’

This paper is organized as follows: First, the process

under interest is presented and its model is given. Then,

the optimal design problem is proposed and solved

using two different approaches. In the next section, the

stability and the sensitivity of the optimal configura-tions are provided. The results are discussed and an

illustrative example is solved in simulation and ana-

lyzed. Finally, some conclusions and perspectives are

drawn.

2. Modeling

Assuming the same single enzymatic biological reac-

tion in each of the two reactors, the dynamical equations

of the general system represented in Fig. 1 are given by

_S1 ¼ a QV1S0 þ b Q

V1S2 � ðaþ bÞ Q

V1S1 � lðS1Þ;

_S2 ¼ ð1� aÞ QV2S0 þ ðaþ bÞ Q

V2S1

�ð1þ bÞ QV2S2 � lðS2Þ;

8><>: ð1Þ

where S1 and S2 are the substrate concentrations in

the first and the second reactor, respectively; aQ and

ð1� aÞQ are the input flow rates in the first and second

reactors, respectively while bQ is the recirculation flow

rate; lðSÞ is the kinetics function; V1 and V2 are the

volumes of the reactors 1 and 2.

Note that the system (1) does not describe the massbalance on the enzymes. Here, they have to be consid-

ered as catalysts. As such, they are not consumed by the

reaction and only their presence in the reactor––and not

their concentration––is important to determine the rate

of the bioreaction.

3. Design problem

Considering the equilibrium of system (1), one has

V1 ¼ Q aS0þbS2�ðaþbÞS1lðS1Þ

� �;

V2 ¼ Q ð1�aÞS0þðaþbÞS1�ð1þbÞS2lðS2Þ

� �:

8<: ð2Þ

The system (1) at steady state is then described by the

system of equations (2), which has a physical signifi-

cance when the following constraints are satisfied:

a 2 ½0; 1� and bP 0; ð3ÞlðS1ÞV1 P 0 ) aS0 � ðaþ bÞS1 þ bS2 P 0; ð4ÞlðS2ÞV2 P 0 ) ð1� aÞS0 þ ðaþ bÞS1 � ð1þ bÞS2 P 0;

ð5Þ

J. Harmand et al. / Journal of Process Control 14 (2004) 785–794 787

S0 > S1 P 0; ð6Þ

S0 > S2 P 0; ð7Þfrom which it is easy to establish that (4)–(7) imply the

following inequalities:

06 aþ bS2 � S1S0 � S1

6S0 � S2S0 � S1

: ð8Þ

Given Q ¼ Q > 0, S2 ¼ S2 > 0 and S0 ¼ S0 > S2, it is

straightforward to show from Eqs. (2) that the totalvolume V to be minimized can be expressed as a func-

tion of a, b and S1:

V ða; b; S1Þ ¼ V1 þ V2

¼ QaS0 � ðaþ bÞS1 þ bS2

lðS1Þ

�

þ ð1� aÞS0 þ ðaþ bÞS1 � ð1þ bÞS2

lðS2Þ

�

¼ Q ðS0

�� S1Þ

1

lðS1Þ

�� 1

lðS2Þ

�

� a

�þ b

S2 � S1S0 � S1

��þ Q

S0 � S2

lðS2Þ

� �: ð9Þ

Since QðS0�S2lðS2Þ

Þ is constant, we are led to the following

optimization problem:

Optimization problem ðPÞGiven numbers Q, S0 and S2 such that Q > 0,

S0 > S2 > 0, and the function lð�ÞP 0, minimize the

objective function

Jða; b; S1Þ ¼ aðS1Þ a

�þ b

S2 � S1S0 � S1

�; ð10Þ

where

aðS1Þ ¼ ðS0 � S1Þ1

lðS1Þ

�� 1

lðS2Þ

�ð11Þ

under the constraints (3), (6) and (8).

Remark. If a ¼ 1, b ¼ 0 and lð Þ is the Michaelis–

Menten kinetics, note that we get the same expression as

the one first proposed in Luyben and Tramper [5] for

N ¼ 2.

4. Optimal solutions

Consider the following technical hypothesis, where l0

denotes dldS and f ðSÞ ¼ 1

lðSÞ � 1

lðS2Þ:

Hypothesis H1. The kinetics function lð�Þ has the fol-

lowing properties:

(1) lð0Þ ¼ 0,(2) lðSÞ > 0 8S > 0,

(3) the function 1lðSÞ is strictly convex.

Remark 1. This hypothesis means that the optimization

problem to be solved in the following is not only valid

for a specific kinetics function (such as Michaelis–

Menten,. . .) but for any kinetics function which verifiesS ! l0ðSÞ ¼ lðSÞ2 decreasing on ð0; S0�.

Remark 2. Hypothesis H1(3) entails that there exists,at most, only one value of S such that l0ðSÞ ¼ 0.

5. General solution to the optimization problem

Proposition P1. Let Hypothesis H1 be verified. Then theoptimal design of the system (1) is given by:

1. If lðS2Þ6 lðS0Þ, then a� ¼ 1, b� ¼ 0 and S�1 is the un-

ique solution of

f ðS�1Þ ¼ ðS�

1 � S0Þl0ðS�

1Þl2ðS�

1Þ

� �ð12Þ

which belongs to ðS2; S0Þ.2. If lðS2Þ > lðS0Þ then

• If l0ðS2Þ > 0, then a� ¼ 1, b� ¼ 0 and S�1 is the un-

ique solution of

f ðS�1Þ ¼ ðS�

1 � S0Þl0ðS�

1Þl2ðS�

1Þ

� �ð13Þ

which belongs to ðS2; ~SÞ where ~S 2 ½S2; S0� is such thatf ð~SÞ ¼ 0.• If l0ðS2Þ < 0, then there exists an infinity of optima

ða�; b�Þ that lie on the line

aþ bS2 � S�

1

S0 � S�1

� S0 � S2

S0 � S�1

¼ 0; a 2 ½0; 1�; bP 0;

ð14Þ

with S�1 the unique solution of

l0ðS�1Þ ¼ 0 ð15Þ

which belongs to ð0; S2�.• If l0ðS2Þ ¼ 0 then S�

1 ¼ S2 and a� ¼ 1, bP 0.

Proof. See Appendix A. h

6. Design analysis

6.1. Stability analysis

In this section, a sufficient condition for the stability

of the optimal configuration is given:

Proposition P2. When a and b are not simultaneouslynull, the equilibrium point ðSe

1; Se2Þ 2 R2

þ of the system (1)is locally asymptotically stable if the following conditionsare fulfilled:


ðaþ bÞ QV1þ ð1þ bÞ Q

V2þ l0ðSe

1Þ þ l0ðSe2Þ > 0;

ðaþ bÞ Q2

V1V2þ ðaþ bÞ Q

V1l0ðSe

2Þ þ ð1þ bÞ QV2l0ðSe

1Þþl0ðSe

1Þl0ðSe2Þ > 0:

8><>: ð16Þ


6.2. Sensitivity analysis

In this section, the sensitivity of the optimal design

with respect to the parameters of a specific kinetics

function is analyzed. The kinetics function under inter-est is written as

lðSÞ ¼ lmaxSS2=Ki þ S þ Ks

: ð17Þ

Remark 1. Note that this kinetics function verifies H1.

Remark 2. The study of this specific kinetics function

has the advantage of being equivalent to the Michaelis–

Menten kinetics function when Ki tends to infinity.

In the following, the sensitivity functions of V1 and V2with respect to Ks and Ki are derived. Consider the re-

sults given under the form of Proposition P2 (thisproposition can be used since Hypothesis H1 is verified).

For each of the three considered cases, the sensitivity

functions of interest can be derived as follows.

Proposition P3. Let the kinetics function to be defined as(17) and let S0

1 and S2 be the two roots (possibly identical)of the function f ð Þ on R�

þ. Introduce the followingfunctions:

BðpÞ ¼ S1ðpÞ2

Kiþ S1ðpÞ þ Ks;

CðpÞ ¼2S1ðpÞ oS1ðpÞ

op

� �Ki

;

DðpÞ ¼ aS0 � ðaþ bÞS1ðpÞ þ bS2:

1. For cases 1 and 2.1 of Proposition P1, the sensitivityfunctions of an optimal configuration can be computed as

oV1oKs

¼ �ðaþ bÞA1BðKsÞlmaxS1ðKsÞ

� A1BðKsÞDðKsÞlmaxS1ðKsÞ2

þ1þ A1 þ 2S1ðKsÞA1

Ki

� �DðKsÞ

lmaxS1ðKsÞ; ð18Þ

oV2oKs

¼ �ðaþ bÞA1

S2

2

Kiþ S2 þ Ks

� �lmaxS2

þ S0 � S2 � DðKsÞlmaxS2

; ð19Þ

oV1oKi

¼ �ðaþ bÞA2BðKiÞlmaxS1ðKiÞ

� A2BðKiÞDðKiÞlmaxS1ðKiÞ2

þ� S1ðKiÞ2

Kiþ A2 þ 2S1ðKiÞA2

Ki

� �DðKiÞ

lmaxS1ðKiÞ; ð20Þ

oV2oKi

¼ðaþ bÞA2

S2

2

Kiþ S2 þ Ks

� �lmaxS2

� S0 � S2 � DðKiÞlmaxK

2i

ð21Þ

with

A1 ¼oS1ðKsÞoKs

¼ � KiS1ðKsÞ2 � S0S2Ki

3S1ðKsÞ2ð�2S2 þ S2

2 þ KsKi þ S0S2Þ;

A2 ¼oS1ðKiÞoKi

¼ � KsS1ðKiÞ2 � S0S2Ks

3S1ðKiÞ2ð�2S2 þ S2

2 þ KsKi þ S0S2Þ:

2. For case 2.2 of Proposition P1, one has

oV1oKs

¼ ½E1ðS0 � S1ðKsÞÞ � aðKsÞA3�BðKsÞlmaxS1ðKsÞ

� A3BðKsÞF ðKsÞlmaxS1ðKsÞ2

þF ðKsÞ 2S1ðKsÞA3

Kiþ A3 þ 1

� �lmaxS1ðKsÞ

;

ð22Þ

oV2oKs

¼½E1ðS1ðKsÞ � S0Þ þ aðKsÞA3� S

2

2

Ki þ S2 þ Ks

� �lmaxS2

þ S0 � S2 � F ðKsÞlmaxS2

; ð23Þ

oV1oKi

¼ ½E2ðS0 � S1ðKiÞÞ � aðKiÞA4�BðKiÞlmaxS1ðKiÞ

� A4BðKiÞF ðKiÞlmaxS1ðKiÞ2

þF ðKiÞ 2S1ðKiÞA4

Kiþ A4 � S1ðKiÞ2

K2i

� �lmaxS1ðKiÞ

; ð24Þ

oV2oKi

¼½E2ðS1ðKiÞ � S0Þ þ aðKiÞA4� S

2

2

Kiþ S2 þ Ks

� �lmaxS2

� ðS0 � S2 � F ðKiÞÞS2

lmaxK2i

ð25Þwith S1ðKsÞ ¼

ffiffiffiffiffiffiffiffiffiKsKi

p,

A3 ¼oS1ðKsÞoKs

¼ Ki=ð2ffiffiffiffiffiffiffiffiffiKiKs

pÞ;


A4 ¼oS1ðKiÞoKi

¼ Ks=ð2ffiffiffiffiffiffiffiffiffiKiKs

pÞ;

E1 ¼oaðKsÞoKs

¼ ðS0 � S2ÞKi

2ffiffiffiffiffiffiffiffiffiKsKi

pðS0 �


pÞ2;

E2 ¼oaðKiÞoKi

¼ ðS0 � S2ÞKs

2ffiffiffiffiffiffiffiffiffiKsKi

pðS0 �


pÞ2:


7. Discussion of the results

Compared to the existing results of the literature, the

present study can be viewed as a generalization when

only two enzymatic bioreactors are used. Indeed, whenlðS2Þ6 lðS0Þ in Proposition P1, the present results are

exactly equivalent to those first proposed by Luyben and

Tramper [5] for Michaelis–Menten system or those ob-

tained by Malcata [6]. The present proposition extends

these results in the sense they are not only valid for

Michaelis–Menten kinetics but also for any kinetics that

satisfies Hypothesis H1.

The case lðS2Þ > lðS0Þ and l0ðS2Þ > 0 of PropositionP1 has never been proposed for enzymatic bioreactions.

However, a similar result has recently been established

in the framework of microbial reactions and the concept

of SSEBS was proposed. In fact, this principle only

holds for enzymatic biological reactions which have

non-monotonic kinetics as it is shown hereafter. Here is

a very important point of the present study: What was

found for every microbial reactions verifying HypothesisH1 does not systematically hold for all ‘‘enzymatic ki-

netic functions’’. In particular, considering the results

obtained under the form of Proposition P1, it is

straightforward to establish that, for Michaelis–Menten

kinetics, the inequality lðS2Þ < lðS0Þ always holds. In

other terms, lðS2Þ > lðS0Þ never occurs. Thus, for

Michaelis–Menten systems, and whatever the data of the

problem are, there does exist only one unique optimalconfiguration allowing the user to perform its objectives

from an input-output point of view. However, note that

this so-called SSEBS concept holds for any other non-

monotonic kinetics function that verifies Hypothesis H1.

In addition, a sufficient condition for the local

asymptotic stability of the optimal configuration has

been proposed. Concerning this last result, it is inter-

esting to note that, although the system under interest inthis study significantly differs from the one studied in

Harmand et al. [3], the conditions under which the

system is locally asymptotically stable are exactly the

same. In particular, from the above Proposition P2 it

follows that if both conditions l0ðSe1ÞP 0 and l0ðSe

0ÞP 0

are satisfied, then the local asymptotic stability of the

optimal designed system is guaranteed. From the au-

thors knowledge, it is the first time that such results are

provided within the framework of an optimal design

problem of an enzymatic bioreactor.

Concerning the sensitivity analysis, no qualitativeresults can be formulated since the relative variation of

the volumes with respect to a given parameter depends

on the specific data of the problem. An example of some

quantitative information that can be used in practice

from a specific problem––in particular for parameter

identification purposes––is given in the next section.

Finally, to summarize this section, it is of particular

interest to point out the following general qualitativeresults (valid whatever the kinetics function may be

provided that it fulfills Hypothesis H1):

• On the one hand, given the data of a specific problem

to be solved, if the expected performances are ‘‘tight

enough’’ (lðS2Þ < lðS0Þ in Proposition P1), neither

the implementation of a recirculation loop nor the

‘‘distribution’’ of the input flow rate between thetwo tanks does allow any improvement with respect

to the results existing in the literature.

• At the opposite of what was found for microbial reac-

tions, when an enzymatic reaction is considered the

kinetics of which is monotonic (as in the case of the

so-called Michaelis–Menten reaction), there does

not exit an infinity of equivalent configurations,

whatever the data of the problem are. In this case,the ‘‘SSEBS’’ concept does simply not apply.

• On the other hand (when a non-monotonic kinetics

function that verifies Hypothesis H1 is used), if

lðS2Þ > lðS0Þ and l0ðS2Þ > 0 in Proposition P1, then

there exists an infinity of optimal configurations hav-

ing the same TRV. It allows us to verify that the con-

cept of SSEBS still holds for a class of enzymatic

biological reactions. In this case, additional consider-ations such as economical or structural (observabil-

ity, controllability, etc.) can be used to globally

optimize the desired process, cf. for instance [3]. Be-

cause they include the cases which can be found in

the literature, these results may be viewed as a gener-

alization (for the case of two tanks) of the original re-

sults proposed by other authors, see e.g. [5] or [6].

• Finally, sufficient conditions for the local asymptoticstability of the designed system around a given equi-

librium point as well as a study of the sensitivity of

the optimal configuration with respect to kinetics

parameters are given.

8. Application

8.1. The Haldane kinetics function: modeling

Since the present results are not new relatively to the

Michaelis–Menten model (in particular, the SSEBS does

0 10 20 30 40 50 60 70 80 90 1000.2

0

0.2

0.4

0.6

0.8

1

1.2

S2

f(S

1), (

norm

aliz

ed)

Fig. 3. The function f ðS1Þ for a Haldane model and different values

of Ks.

0.6

0.8

1


not apply), the use of the kinetics function described by

(17) is preferred in this application section. First, it is

easy to verify that this kinetics function verifies

Hypothesis H1. Considering the results of PropositionP1, one can compute, for different values of S2 the

optimal design. As S2 varies from 0þ to S0, the com-

putation of f ðSÞ and aðSÞ are reported in Figs. 3 and 4

for different values of Ks.

In order to illustrate the results, let us consider the

following model parameters:

The influence of these parameters on the shape of the

kinetics function is represented in Fig. 2.In the following, the optimal configurations are

investigated when S2 varies from 0þ to S0 for each value

of Ks indicated above. (Since the optimal configuration

is less influenced by Ki, a nominal value for this

parameter is chosen for the simulations: Ki ¼ 30.)

l0 Ks Ki S0 Q

0.045

h�1

0.1, 0.5, 1, 5,

10, 20, 30, 50

or 100 mg l�1

30

mg l�1

100

mg l�1

1

l h�1

0 10 20 30 40 50 60 70 80 90 100-0.4

-0.2

0

0.2

0.4

S2

a(S

1), (

norm

aliz

ed)

Fig. 4. The function aðS1Þ for a Haldane model and different values

of Ks.

9. The Haldane kinetics function: discussion of the

numerical results

The optimal volumes obtained with the actual ap-proach are represented in Fig. 5 as a function of S2 whenS2 varies from 0þ to S0 (equivalently when the conver-

sion rate q ¼ 1� S2S0

varies from 1 to 0þ). The corre-

sponding values of a� are pictured in Fig. 7. In order to

simplify the presentation, only the optimal configura-

tions corresponding to b� ¼ 0 have been chosen and

only a� is used to fulfill the conditions of Proposition P1.

In order to compare our results with those alreadyproposed in the literature, a 3D representation is given

0 10 20 30 40 50 60 70 80 90 1000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

S (mg/l)

µ(S

) (1

/h)

µ with KS=0.1 mg/l

µ with KS=100 mg/l

Fig. 2. The l function (Haldane) for different Ks values.

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5x 104

Act

ual a

ppro

ach

: V1* e

t V2*

S2

V1* with Ks=100

V2* with Ks=100

V2* with Ks=0.1

V1* with Ks=0.1

Fig. 5. Optimal values of V1 and V2 using the actual approach.

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

S2

Act

ual a

ppro

ach

: α* (

β*=

0)

Ks=0.1

Ks=100

Fig. 7. Optimal values of a.

0 20 40 60 80 100 020

4060

801000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x 104

KsS2

Vto

tal

Total required volume with Luyben

Total required volume : actual approach

Fig. 6. Comparison of the total required volumes: actual and Luyben

and Tramper approaches.

00.2

0.40.6

0.81

20 30 40 50 60 70 80 90 100

0

20

40

60

80

100

120

α*

S2

β*

Each "S2–line" verifies equation (14)

Fig. 8. Optimal values of the pairs ½a�;b��, Ks ¼ 20, Ki ¼ 30.

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

300

350

S2

dV1/dKs

dV/dKs

dV2/dKs

Fig. 9. Sensitivity of the optimal volumes with respect to Ks (Ks ¼ 20,

Ki ¼ 30).


in Fig. 6. This figure shows the TRV computed when

using the approach by Luyben and Tramper (the surface

of the top) and the actual one (the surface of the bot-

tom) as a function of S2 and for each value of Ks con-

sidered in the above Table. The two surfaces are

identical until the conversion rate is sufficiently small (or

S2 sufficiently high) for the present approach to be

attractive in terms of TRV. When S2 increases from 0þ

to S0 (i.e. q decreases from 1 to 0þ), it can be seen that

after having reached a minimum (corresponding to S01),

the TRV (when using the approach by Luyben and

Tramper) increases again while it continues to decrease

when following the present approach. This is due to that

the inhibition by the substrate leads to a decrease in the

reaction rate. In our approach, the fact to allow a dis-

tribution of the flow rate (or the implementation of arecirculation with a specific flow rate) leads to con-

stantly decrease the TRV.

To better illustrate the SSEBS concept, we have also

computed, for Ks ¼ 20, different values for a� and b� as a

function of S2 (with S0 ¼ 100 mg/l). In other words, for

a given S2, we have computed the set of optimal pairs

(a�, b�) that verify Eq. (14) and therefore define, for agiven conversion rate, a SSEBS (cf. Fig. 8).

10. Sensitivity analysis

Following Proposition P3, for S2 from 0þ to S0, it is

easy to compute the sensitivity of the optimal volumes

with respect to Ks and Ki. The computation was realized

with nominal values Ks ¼ 20 and Ki ¼ 30. The numerical

results are proposed in Figs. 9 and 10. From these re-

sults, the most important conclusion that can be drawnis that the optimal configuration is more influenced by

the parameter Ks than by Ki. Another information is

that this influence is more important when S2 is small (qis large) than when S2 is large (q is small). To better

understand this concept, consider the following exam-

ple. Consider the following data of a fictious optimal

design problem: S2 ¼ 1, S0 ¼ 100, lmax ¼ 0:045, Q ¼ 1,

0 10 20 30 40 50 60 70 80 90 100–30

–25

–20

–15

–10

–5

0

5

S2

dV/dKi

dV1/dKi

dV2/dKi

Fig. 10. Sensitivity of the optimal volumes with respect to Ki (Ks ¼ 20,

Ki ¼ 30).


Ks ¼ 20 and Ki ¼ 30. The optimal design is given byV �1 ¼ 6900 and V �

2 ¼ 3872. Using Proposition P3, it is

easy to compute the following sensitivity functions:oV1oKs

¼ 211 and oV2oKs

¼ 184. Now, for any practical reason,

instead of working around S2 ¼ 1, consider the objective

is to work around S2 ¼ 30. In such a case, the optimal

configuration is given V �1 ¼ 4136 and V �

2 ¼ 0. Again, it is

easy to compute the following sensitivity functions:oV1oKs

¼ 64 and oV2oKs

¼ 0. Now, consider the real value for Ks

is not 20 but 25, that is 25% deviation from the nominal

value which can be considered as a rather large error.

The expected deviations dV1 and dV2 from their optimal

values V �1 and V �

2 are

• for S2 ¼ 1, dV1 ¼ ð25� 20Þ oV1oKs

¼ 5� 211 ¼ þ1055

and dV2 ¼ ð25� 20Þ oV2oKs

¼ 5� 184 ¼ þ920;

• for S2 ¼ 30, dV1 ¼ ð25� 20Þ oV1oKs

¼ 5� 211 ¼ þ320and dV2 ¼ ð25� 20Þ oV2

oKs¼ 5� 0 ¼ 0.

The real values of the optimal volumes if Ks ¼ 25

instead of 20 are

• for S2 ¼ 1, V �1 ¼ 7894 and V �

2 ¼ 4860, that are respec-

tively deviations dV1 ¼ 7894� 6900 ¼ 994 and dV2 ¼4860� 3872 ¼ 988 (which are in accordance with theabove computed expected deviations);

• for S2 ¼ 30, V �1 ¼ 4440 and V �

2 ¼ 0, that are respec-

tively deviations dV1 ¼ 4440� 4136 ¼ 304 and dV2 ¼0 (which are, again, in accordance with the above

computed expected deviations). The correspond-

ing relative deviations are 6.85% for V1 and 0%

for V2.

The correct interpretation of these sensitivity func-

tions appears as soon as the relative deviation from a

‘‘nominal optimal design’’ are computed. For S2 ¼ 1,

the relative deviations are 12.59% for V1 and 20.33% for

V2 while, for S2 ¼ 30, the corresponding relative devia-

tions are 6.85% for V1 and 0% for V2. This is the practicalsense to give to the curves 9 and 10. Indeed, it means

that––under these specific conditions––it is crucial toidentify with a better accuracy Ks than Ki since this last

parameter has a rather small relative influence on the

optimal configuration when compared to the influence

of Ks. Thus, the identification effort should be put on

this specific parameter during the identification step.

11. Conclusion

In this paper, the optimal design (in the sense of the

minimization of the TRT or equivalently, at steady

state, of the TRV) of two interconnected biologicalreactors has been revisited. It is supposed that a single

enzymatic biological reaction occurs and that the

kinetics function satisfies some mild technical assump-

tions (Hypothesis H1) that include, in particular the

well-known Michaelis–Menten function. The problem

was formulated as follows: ‘‘Given the model of the

biological process and a flow rate Q to be treated per

time unit at steady state operation, determine the min-imum TRV to attain a specified conversion rate

ð1� S2S0Þ.’’

In determining the optimal solution, the following

situations have been investigated:

ii(i) different volumes for the tanks,

i(ii) distribution of the input flow rate among the tanks,

(iii) presence of a recirculation loop.

In particular, Proposition P1 for solving the problem

has been proposed. The present approach is interesting

since only the comparison between lðS2Þ and lðS0Þ is

needed to get a qualitative idea of the optimal design.

This approach has never been proposed in the literature.

Furthermore, the results are valid not only for a given

kinetics function but for a quite large class of kinetics

functions. In addition, it has been shown that the SSEBSprinciple proposed in the case of microbial reactions

whatever the growth rate (provided it verifies Hypoth-

esis H1), does not apply in the case of enzymatic systems

with monotonic kinetics.

Under some conditions depending on the data of the

problem and when non-monotonic kinetics are consid-

ered, it was shown that the TRV can be significantly

decreased by distributing the input flow rate and intro-ducing a recirculation loop with a certain recirculation

flow rate. In such a case, the optimal solution is not

unique and the concept of SSEBS is introduced for

enzymatic biological systems to help the designer to

choose one among all the possible TRV-optimal solu-

tions. This concept is interesting because it allows the

designer to take into account additional properties of


each one of the TRV-optimal solution, such as observ-

ability or controllability, and then to decide which one is

the ‘‘best’’ with respect to these additional criteria.

Furthermore, it is important to note that the stabilityof the new proposed family of processes is analyzed.

Finally, the sensitivity of optimal volumes with re-

spect to the parameters of a specific kinetics function

(exhibiting an inhibition by the substrate) has been

analyzed. In particular, it has been shown that the

optimal solution is more sensitive to the affinity coeffi-

cient than to the inhibition coefficient of the Haldane

kinetics function.Among the mathematical results of the paper, the

question of explaining why what was established for

microbial reactions is not true for enzymatic biosystems

remains an open question. In any case, the experimental

validation of these results has to be searched for.

Future research concerns more complex biological

processes such as multi-component reactions in which

more than one species reacts. The extension of the re-sults to the design of more than two reactors with many

biological reactions occurring in each one is also under

investigation. Another exciting research topic is the use

of a larger class of kinetics functions and, in particular,

those involving a product inhibition.

Acknowledgements

The author would like to thank Nicolas Bernet and

Renaud Escudi�e, LBE Narbonne, France for their

fruitful comments.

Appendix A

Proof of Proposition P1. Since ðS0 � S1ÞP 0 8S1 2½0; S0�, one has signðaðS1ÞÞ ¼ signðf ðS1ÞÞ 8S1 2 ð0; S0�.

1. First, assume lðS2Þ6 lðS0Þ. Under Hypothesis

H1(2), 1

lðS2ÞP 1

lðS0Þand (i) f ðS0Þ6 0. Furthermore, one

has (ii) f ðS2Þ ¼ 0 and, by Hypothesis H1(1) (iii)

limS1!0 f ðS1Þ ¼ þ1 > 0. The fact that f is strictly

convex (guaranteed by Hypothesis H1(3)) together

with (i)–(iii) entails that f ðS1Þ > 0 for S1 2 ð0; S2Þ,f ðS1Þ < 0 for S1 2 ðS2; S0Þ and f 0ðS2Þ < 0. Noting that

aðS2Þ ¼ aðS0Þ ¼ 0, one can conclude that aðSÞ > 0 onð0; S2Þ and aðS1Þ < 0 on ðS2; S0Þ. Since a is continu-

ous, it is bounded and its minimum is attained in

ðS2; S0Þ.To prove the uniqueness of the minimum, let S�

1 be

such that a0ðS�1Þ ¼ 0. One has a0ðS�

1Þ ¼ 0 ¼ �f ðS�1Þþ

ðS0 � S�1Þf 0ðS�

1Þ. It follows that f 0ðS�1Þ < 0 and there

exists eS 2 ðS2; S0� such that f 0ðSÞ < 0 for S 2 ðS2; eSÞ,f 0ðeSÞ ¼ 0 and f 0ðSÞ > 0 for S 2 ðeS ; S0Þ or f 0ðSÞ < 0

for any S 2 ðS2; S0�. Then the minimum S�1 is attained in

ðS2; eSÞ. Furthermore, a00¼�2f 0ðS1ÞþðS0�S1Þf 00ðS1Þ>0

for S12½S2;eS �. Thus, the restriction of a on ½S2;eS � is

strictly convex which is sufficient to prove the unicity ofthe minimum of a on ðS2;S0Þ.

Since aðS1Þ < 0 8S1 2 ðS2; S0Þ, aþ b S2�S1S0�S1

must be

maximized. Since S�1 > S2 then

S2�S�1

S0�S�1

b� < 0. The opti-

mal values for a and b are given by a� ¼ maxðaÞ ¼ 1and b� ¼ minðbÞ ¼ 0. Finally, S�

1 is computed by

solving oaoS1

¼ 0 which has only one solution belong-

ing to ðS2; S0Þ. This proves the case 1 of the propo-

sition.

2. Now, assume lðS2Þ > lðS0Þ. As above, it is then

easy to establish the following: (i) f ðS0Þ > 0, (ii)f ðS2Þ ¼ 0 and, by Hypothesis H1(1) (iii) limS1!0 f ðS1Þ ¼þ1 > 0. Noting that f is strictly convex, two cases can

occur: The minimum of f is attained for S1 2 ð0; S2Þ orfor S1 2 ðS2; S0Þ. Since signðl0ðS1ÞÞ ¼ signðf 0ðS1ÞÞ, the

sign of l0ðS2Þ allows us to discriminate between the two

above cases:

• If l0ðS2Þ > 0, then f 0ðS2Þ < 0 and the minimum off ðS1Þ is obtained for S1 2 ðS2; S0Þ. Furthermore,

f ðS0Þ > 0. Thus, there does exist eS 2 ðS2; S0Þ=f ðeSÞ ¼ 0. Using the fact that f is strictly convex, it

is straightforward to establish that such a eS is unique

and one has f ðS1Þ < 0 8S1 2 ðS2; eSÞ and f ðS1Þ >0 8S1 2 ð0; S2Þ [ ðeS ; S0Þ. Thus, we can conclude that

aðS1Þ < 0 8S1 2 ðS2; eSÞ and, because a is continuous,

its minimum is attained on this interval. Finally, thecomputation and the unicity of the minimum of aon ðS2; eSÞ can be proved using the same reasoning

than previously and, again, the optimal design is ob-

tained for a� ¼ 1, b� ¼ 0 and S�1 is computed as

above. This proves the case 2.1 of Proposition P1.

• If l0ðS2Þ < 0 then f 0ðS2Þ > 0. Then, there exists eSsuch that f ðS1Þ < 0 8S1 2 ðeS ; S2Þ and f ðS1Þ >0 8S1 2 ð0; eSÞ [ ðS2; S0Þ. Again, the sign of a is di-rectly deduced from the above and the existence

and the unicity of its minimum that lies in ðeS ; S2Þare proved following the same reasoning than previ-

ously. However, note now that S�1 < S2. As a conse-

quence,S2�S�

1

S0�S�1

> 0 and one has to choose any a and

b such that aþ bS2�S�

1

S0�S�1

� S0�S2S0�S�

1

¼ 0, a 2 ½0; 1�, bP 0

and S�1 satisfies l0ðS�

1Þ ¼ 0. This proves the case 2.2

of Proposition P1 when l0ðS2Þ < 0.

• For the specific case where l0ðS2Þ ¼ 0 then f 0ðS2Þ ¼ 0.

Since f is convex, one has f > 0 8S1 2 ½0; S0� � fS2g.The same conclusion can be formulated about aðS1Þthe minimum of which on ð0; S0� is obtained for

S�1 ¼ S2. The only solution for ða; bÞ such that the

constraint (8) is satisfied is then a� ¼ 1 and b� P 0.

This proves the case 2.3 of Proposition P1 when

l0ðS2Þ ¼ 0. h


Proof of Proposition P2. Denote, for simplicity, D1 ¼Q=V1 and D2 ¼ Q=V2. The dynamics of (1) is given by

d

dtS1S2

� �¼ �ðaþ bÞD1 bD1

ðaþ bÞD2 �ð1þ bÞD2

� �S1S2

� �

� lðS1ÞlðS2Þ

� �þ aD1S0

ð1� aÞD2S0

� �: ðA:1Þ

The linearization of the system ðS1; S2Þ at the equilib-

rium point ðSe1; S

e2Þ is locally asymptotically stable if the

eigenvalues ðk1; k2Þ of the Jacobian matrix M of the

system (26) computed at the equilibrium point ðSe1; S

e2Þ

are such that

k1 þ k2 ¼ trðMÞ¼ �ðaþ bÞD1 � ð1þ bÞD2 � l0ðSe

1Þ � l0ðSe2Þ < 0;

k1k2 ¼ detðMÞ¼ ððaþ bÞD1 þ l0ðSe

1ÞÞðð1þ bÞD2 þ l0ðSe2ÞÞ

� ðaþ bÞbD1D2 > 0;

which is exactly the condition (16). Thus, if the above

condition holds, the local asymptotic stability of the

system is guaranteed. h

About Proposition P3

The computation of oVioKs

and oVioKi

is straightforward. Thesensitivity of the total volume to any of the two

parameters under interest is simply computed asoVoKs

¼ oV1oKs

þ oV2oKs

and oVoKi

¼ oV1oKi

þ oV2oKi. The only delicate point

is the computation of the sensitivity of the roots of a

polynomial equation. (When Haldane model is consid-

ered, the solution of f ðS�1Þ ¼ ðS�

1 � S0Þðl0ðS�

1Þ

l2ðS�1ÞÞ in P1 and

P2, is a third order equation.)

In order to keep the following development tractable,the sensitivity of the roots of a second order equation to

a parameter is analyzed. The idea is easily extendable to

a more complex case (higher equation degree or a

greater number of parameters). Consider the second

order equation

aðpÞx2 þ bðpÞxþ cðpÞ ¼ 0; ðA:2Þ

where p is a parameter, aðpÞ, bðpÞ and cðpÞ are the

coefficients of the equation.

When p ¼ p, one gets three roots x ¼ fxi; i ¼ 1; 2gand we have

aðpÞx2 þ bðpÞxþ cðpÞ ¼ 0: ðA:3Þ

Consider a small variation dp of p around p. The newvalues of the root vector xþ dx verifies

aðp þ dpÞðxþ dxÞ2 þ bðp þ dpÞðxþ dxÞ þ cðp þ dpÞ ¼ 0:

ðA:4Þ

Considering a first order development of coefficients

of this polynom as ðaðp þ dpÞ � aðpÞ þ dp oaop ðpÞÞ, ðbðpþ

dpÞ � bðpÞ þ dp obop ðpÞÞ, ðcðpþ dpÞ � cðpÞ þ dp oc

op ðpÞÞ, andafter some simplifications and manipulations, one has

oxop

¼ dxdp

¼ � a0ðpÞx2 þ b0ðpÞxþ c0ðpÞ2xaðpÞ þ bðpÞ ðA:5Þ

with a0 ¼ oaop, b

0 ¼ obop and c0 ¼ oc

op.

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[2] M. Feinberg, Toward a theory of process synthesis, Industrial and

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[3] J. Harmand, A. Rapaport, A. Trofino, 2003. Optimal design of

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Optimal design of two interconnected enzymatic bioreactors