University of Groningen

Modelling of gas-liquid reactors - stability and dynamic behaviour of gas-liquid mass transferaccompanied by irreversible reactionElk, E.P. van; Borman, P.C.; Kuipers, J.A.M.; Versteeg, Geert

Published in:Chemical Engineering Science

DOI:10.1016/S0009-2509(99)00207-9

IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.

Document VersionPublisher's PDF, also known as Version of record

Publication date:1999

Link to publication in University of Groningen/UMCG research database

Citation for published version (APA):Elk, E. P. V., Borman, P. C., Kuipers, J. A. M., & Versteeg, G. F. (1999). Modelling of gas-liquid reactors -stability and dynamic behaviour of gas-liquid mass transfer accompanied by irreversible reaction. ChemicalEngineering Science, 54(21), 4869-4879. DOI: 10.1016/S0009-2509(99)00207-9

CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).

Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.

Download date: 06-10-2018

Chemical Engineering Science 54 (1999) 4869}4879

Modelling of gas}liquid reactors } stability and dynamic behaviourof gas}liquid mass transfer accompanied by irreversible reaction

E.P. Van Elk!,*, P.C. Borman", J.A.M. Kuipers#, G.F. Versteeg#!Procede Twente BV, P.O. Box 217, 7500 AE Enschede, The Netherlands

"DSM Research, P.O. Box 18, 6160 MD Geleen, The Netherlands#Department of Chemical Engineering, Twente University of Technology, P.O. Box 217, 7500 AE Enschede, The Netherlands

Abstract

The dynamic behaviour and stability of single-phase reacting systems has been investigated thoroughly in the past and design rulesfor stable operation are available from literature. The dynamic behaviour of gas}liquid processes is considerably more complex andhas received relatively little attention. General design rules for stable operation are not available. A rigorous gas}liquid reactor modelis used to demonstrate the possible existence of dynamic instability (limit cycles) in gas}liquid processes. The model is also used todemonstrate that the design rules of Vleeschhouwer, Garton and Fortuin, Chemical Engineering Science, 47, 1992, 2547}2552, arerestricted to a speci"c limit case. A new approximate model is presented which after implementation in bifurcation software packagescan be used to obtain general applicable design rules for stable operation of ideally stirred gas}liquid reactors. The rigorous reactormodel and the approximate design rules cover the whole range from kinetics controlled to mass transfer controlled systems and arepowerful tools for designing gas}liquid reactors. ( 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Gas}liquid reactor model; Dynamic stability; Limit cycle; Perturbation analysis; Hopf bifurcation; Design rules

1. Introduction

1.1. Single-phase systems

Multiplicity, stability and dynamic behaviour ofsingle-phase reacting systems have been investigated in-tensively in the past. During the last 50 years a lot ofpapers have been devoted to this subject. Most papers(Bilous & Amundson, 1955; Aris & Amundson, 1958;Uppal, Ray & Poore, 1974,1976; Olsen & Epstein, 1993)deal with purely theoretical treatment of multiplicityand stability in single-phase reactors, while other papers(Baccaro, Gaitonde & Douglas, 1970; Vejtasa & Schmitz,1970; Vleeschhouwer & Fortuin, 1990; Heiszwolf & For-tuin, 1997) discuss both experimental and theoreticalwork.

From all these papers one general and important con-clusion can be drawn: whenever a system can be de-scribed su$ciently accurate by two ordinary di!erential

*Corresponding author. Tel.: 00-31-53-4894480/00-31-53-4894337;Fax: 00-31-53-4894774.

E-mail address: edwin.vanelk@procede.nl (E.P.V. Elk)

equations (ODEs) with respect to time (one heat andone material balance, like an irreversible single-phase"rst-order reaction in a CISTR reactor), the occur-rence of instability is easily predicted by analysing thecharacteristics of the linearised system of equations forprocess conditions in the neighbourhood of the steadystate.

1.2. Gas}liquid systems

Prediction of the dynamic behaviour of gas}liquid twophase reactors is usually more complex, since these sys-tems involve: (1) more than two component balances, i.e.ODEs and (2) mass transfer between the gas and theliquid phase.

In literature only a few papers (Ho!man, Sharma& Luss, 1975; Sharma, Ho!man & Luss, 1976; Huang& Varma, 1981a,b; Singh & Shah, 1982; Vleeschhouwer,Garton & Fortuin, 1992) dealt with the (dynamic) behav-iour of gas}liquid processes. In none of these papers isa general applicable methodology presented to predictthe dynamic behaviour of gas}liquid reactors.

Huang and Varma (1981a) treat the dynamic behav-iour of gas}liquid reactions in non-adiabatic stirred tank

0009-2509/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 2 0 7 - 9

reactors. Unfortunately, this paper is only applicable forthe speci"c case of pseudo-"rst-order gas}liquid reac-tions in the so called fast reaction regime. For this speci"ccase the key features of the system can be described bytwo ODEs (a gas-phase material and an overall heatbalance) and a technique similar to that for single-phasereactors can be applied to investigate the dynamic behav-iour. Vleeschhouwer et al. (1992) treat the analysis of thedynamic behaviour of an industrial oxo reactor. How-ever, this gas}liquid process is also described by only onematerial balance of the liquid-phase component anda heat balance of the liquid-phase. The gas-phase compo-nents are not taken into account at all. This meansthat the process is simpli"ed to a pseudo-"rst-ordersingle phase reacting system. This implies constantliquid-phase concentrations of the gas-phase compo-nents which is strictly only valid for systems with a com-pletely saturated liquid-phase (slow reaction regime andnot too large Hinterland ratios (see Westerterp, vanSwaaij & Beenackers, 1990)).

In the current paper two new models are introduced:(1) a rigorous model that can accurately describe thedynamic behaviour of gas}liquid reactors over a widerange of conditions and (2) an approximate model thatcan be used to obtain design rules for stable operation ofgas}liquid reactors.

The rigorous model simultaneously solves the Higbiepenetration model (partial di!erential equations) and thegas-phase and liquid-phase material balances (ODEs) forall components. Moreover, the heat balances for bothphases are taken into account on macro as well as onmicro scale. The model is an improved (non-isothermal)version of the model presented and validated in detailelsewhere (van Elk, Borman, Kuipers & Versteeg, 1999).

Using the rigorous model it is shown that the phenom-enon limit cycle (dynamic instability) found for singlephase reactors can also exist in gas}liquid reactors, evenif mass transfer limitations are important. Generallythese sustained oscillations have to be avoided, becausethey may adversely a!ect product quality and down-stream operations and can lead to serious di$culty inprocess control and to unsafe operations. It is shown thatthe design rules of Vleeschhouwer et al. (1992) are notgenerally applicable for gas}liquid reacting systems. Therigorous model has one major disadvantage: it is notsuited to create a so-called stability map that character-ises the dynamic behaviour of the system as a function ofcertain selected system parameters.

The approximate prediction is however suited to cre-ate such stability maps from which design rules can beobtained. Our approximate method is more general thanthe design rules presented by Huang and Varma (1981) orVleeschhouwer et al. (1992). The rigorous model, whichtakes into account all relevant phenomena, is still re-quired in order to check the approximate results fora chosen set of operating conditions.

2. Theory

2.1. Introduction

The problem considered is a dynamic gas}liquid reac-tor with mass transfer followed by an irreversible secondorder chemical reaction:

A(g)#cbB(l)Pc

cC(l)#c

dD(l) (1)

with the following overall reaction rate equation:

Ra"k

R0,1,1e~E!#5@RT[A][B]. (2)

The mathematical models used are based on the follow-ing assumptions:

1. The mass transfer in the gas phase is described withthe stagnant "lm model.

2. The mass transfer in the liquid phase is described withthe penetration model.

3. The contact time according to the penetration modelis small compared to the liquid-phase residence time.

4. Both the gas and the liquid-phase are assumed to beperfectly mixed (i.e. CISTRs).

5. The reaction takes place in the liquid-phase only.6. The liquid-phase components (B,C and D) are non-

volatile.

2.2. Rigorous model

For the penetration model the phenomenon of masstransfer accompanied by a chemical reaction is governedby the following equations for mass and heat:

L[A]

Lt"D

a

L2[A]

Lx2!R

a, (3)

L[B]

Lt"D

b

L2[B]

Lx2!c

bR

a, (4)

L¹Lt

"

jolC

P,l

L2¹Lx2

#R*H

RolC

P,l

. (5)

To permit an unique solution of the non-linear partialdi!erential equations (3)}(5) one initial Eq. (6) and twoboundary conditions (7) and (8) are required

t"0 and x*0: [A]"[A]l,"6-,

,[B]"[B]l,"6-,

,

¹"¹l,"6-,

, (6)

t'0 and x"dp: [A]"[A]

l,"6-,,[B]"[B]

l,"6-,,

¹"¹l,"6-,

, (7)

4870 E.P. Van Elk et al. / Chemical Engineering Science 54 (1999) 4869}4879

Ja"!D

aAL[A]

Lx Bx/0

"kgA[A]

g,"6-,!

[A]x/0

maB,

AL[B]

Lx Bx/0

"0, (8)

JT"!jA

L¹LxB

x/0

"hg(¹

g,"6-,!¹

x/0).

The material and energy balances describing the systemon macro scale are

d[A]g,"6-,

dt"

[A]g,*/

![A]g,"6-,

qg

!

Jaa

eg

, (9)

d¹g

dt"

¹g,*/

!¹g

qg

!

JTa

ogC

P,geg

, (10)

d[A]l,"6-,

dt"

[A]l,*/

![A]l,"6-,

ql

#

Jaa

el

!Ra,"6-,

, (11)

d[B]l,"6-,

dt"

[B]l,*/

![B]l,"6-,

ql

!cbR

a,"6-,, (12)

d¹l

dt"

¹l,*/

!¹l

ql

#

JTa

olC

P,lel

!

;A(¹l!¹

#00-)

olC

P,lel<R

!

R*HR

olC

P,l

.

(13)

The overall rigorous mathematical model (see Fig. 1)combines the micro and macro model equations (3)}(13)by simultaneously solving the micro and macro model.The macro model is coupled to the micro model by theboundary conditions. The isothermal version of themodel is described and validated in detail elsewhere (vanElk et al., 1999).

According to the penetration model a liquid element isexposed at the gas}liquid interface for a period h duringwhich mass transfer and accompanying chemical trans-formation takes place. Subsequently, the element is in-stantaneously swept to the ideally mixed liquid bulk andreplaced by a new fresh one. The dimensions of the liquidelement are assumed to be in"nite compared to thepenetration depth d

pand therefore no direct mass and

heat transport to the liquid bulk via the liquid elementoccurs

d[A]l,"6-,

dt"

[A]l,*/

![A]l,"6-,

ql

!Ra,"6-,

. (14)

The liquid-phase concentration directly after the elementhas been swept to the ideally mixed liquid bulk is:

[i]l"

Ni,%-%.

#Ni,"6-,

el<R

. (15)

The numerator represents the amount of moles pres-ent in the liquid phase after the contact time h (seeFig. 1):

Ni,%-%.

#Ni,"6-,

"[i]"6-,

el<

R#P

dp

0

([i]![i]"6-,

) dxa<R.

(16)

For the heat balance a similar procedure is used.

2.3. Simple model (2 ODEs) and perturbation analysis

The question arises whether or not the stability ofgas}liquid reactors can be predicted without having tocompletely solve the complicated and time-consumingrigorous reactor model presented in the previous section.This is interesting for two reasons: (1) the rigorous reac-tor model requires much computational e!orts and (2)the rigorous reactor model investigates only one set ofconditions at a time, while predictive methods (like the socalled perturbation analysis) result in a stability map thatindicates the dynamic behaviour as a function of twochosen parameters. So, if a su$ciently accurate approx-imate prediction method could be derived this would bevery attractive.

Vleeschhouwer et al. (1992) successfully predicted thedynamic system behaviour and the transition froma limit cycle to a point-stable steady state of a commer-cial scale gas}liquid oxo reactor using the so-calledperturbation analysis. Analytical solution of the per-turbation analysis is restricted to systems that can bedescribed by two ODEs. The analysis involves linearisa-tion of the governing non-linear ODEs in the neighbour-hood of the steady state

d

dtCLx

1Lx

2D"AC

Lx1

Lx2D with A"C

Lf1

Lx1

Lf1

Lx2

Lf2

Lx1

Lf2

Lx2D

x1/x1sx2/x2s

,

(17)

where x1

and x2

represent the perturbed concentrationand temperature, respectively, in the case study of Vlees-chhouwer et al. (1992). The functions f

1and f

2represent

the right-hand sides of the mass and heat balance, wherex1s

and x2s

represent the steady-state values of the con-centration and temperature, respectively. A system isconsidered point-stable if, after a su$ciently small per-turbation from the steady state, the system returns to itsinitial state.

A su$cient and necessary condition for a point-stablesteady state is that the slope condition (det A'0) as wellas the dynamic condition (trace A(0) are ful"lled (infact, this means that both eigenvalues of the linearisedsystem must have negative real parts). If the slopecondition is violated (det A(0) the system shows static

E.P. Van Elk et al. / Chemical Engineering Science 54 (1999) 4869}4879 4871

Fig. 1. The rigorous reactor model.

instability (extinction or ignition to a static stable point).If the dynamic condition is violated (trace A'0) thesystem shows dynamic instability (limit cycle).

If a plot (stability map) is made with S-shaped curvesof the loci of points representing the steady states of thesystem as a function of a certain parameter (for examplethe coolant temperature), two additional curves can bedrawn to divide the "gure in three distinct regions, eachwith a characteristic dynamic behaviour. The two curvesare the fold bifurcation curve (det A"0) and the Hopfbifurcation curve (trace A"0). The fold bifurcationcurve is also known as the saddle-node bifurcation orslope curve. The three distinct regions are: I. region withpoint-stable steady states; II. region with dynamic insta-

bility (also called orbitally stable region or limit cycleregion); III. region with static instability.

Thus, the perturbation analysis requires simpli"cationof the overall reactor model to a system with only twoODEs. Most obvious is to reduce the system descriptionto a system of equations that only includes one heatand one material balance. Vleeschhouwer et al. (1992)reduced their system to the following set:

d[B]l,"6-,

dt"

[B]l,*/

![B]l,"6-,

ql

!cbR

a,"6-,, (18)

d¹l

dt"

¹l,*/

!¹l

ql

!

;A(¹l!¹

#00-)

olC

P,lel<

R

!

R*HR

olC

P,l

(19)

4872 E.P. Van Elk et al. / Chemical Engineering Science 54 (1999) 4869}4879

which implies that a constant value for [A]l,"6-,

had to beassumed to enable calculation of the reaction rate. Itseems reasonable to choose the steady-state value forthis.

2.4. Approximate model (3 ODEs) and bifurcation analysis

The perturbation analysis presented by Vlees-chhouwer et al. (1992), as described in Section 2.3, fails forsystems in which mass transfer limitations are important.In this section an extension of their theory to thesesystems will be described. This new approximate methodrequires simpli"cation of the rigorous reactor model toa system of ODEs and algebraic equations (AEs) only.The proposed system of ODEs is:

d[A]l,"6-,

dt"

![A]l,"6-,

ql

#

klEaa

el

(ma[A]

g![A]

l,"6-,)

!Ra,"6-,

, (20)

d[B]l,"6-,

dt"

[B]l,*/

![B]l,"6-,

ql

!cbR

a,"6-,, (21)

d¹l

dt"

¹l,*/

!¹l

ql

!

;A(¹l!¹

#00-)

olC

P,lel<

R

!

R*HR

olC

P,l

. (22)

Depending on the required accuracy and the reactorsystem considered, more ODEs can be added, forexample if the reaction is reversible or if the gas phase hasto be taken into account, too.

The model requires an algebraic expression (AE) forthe enhancement factor to replace the micro model. Thefact that the micro and macro balances are no longersolved simultaneously and that the mass and heat bal-ances are decoupled on micro scale makes the model anapproximate model. Van Swaaij and Versteeg (1992) con-cluded in their review that no generally valid approxim-ate expressions are available to cover all gas}liquidprocesses accompanied with complex (reversible) reac-tions. However, for some asymptotic situations theseexpressions are available. For the present study weassumed that the reaction is irreversible and obeys "rst-order kinetics with respect to A and B. Then the follow-ing approximate relation can be used for estimating theenhancement factor:

Ea"

E2a,PS

2(Ea=!1)GC1#4

(Ea=!1)

E2a,PS

D0.5

!1H, (23)

where

Ea="S

Da

Db

#SD

bD

a

[B]l,"6-,

cb[A]i

l

,

Ea,PS

"HaCG1#n

8Ha2HerfCS4Ha2

n D#

1

2HaexpA

4Ha2

n BD, (24)

Ha"Jk

R[B]

l,"6-,D

akl

.

For systems for which no approximate expression of theenhancement factor is available, a polynomial "t of dataobtained by separate calculations of the rigorous reactormodel can be implemented.

Creating a stability map of the system described byEqs. (20)}(24) by the analytical perturbation analysis isnot possible. However, bifurcation software packageslike LOCBIF or AUTO can create a stability map for thesystem using a numerical bifurcation technique. Imple-mentation of Eqs. (20)}(24) in bifurcation software (LOC-BIF) results in a new and general prediction method.This method is very powerful for attaining design rulesfor stable operation of gas}liquid reactors.

LOCBIF is a software package that has the numericalroutines to explore the existence and stability of equilib-ria in dynamical models with limited e!orts.

3. Results

3.1. Introduction

To demonstrate the applicability of the models de-scribed in the previous sections, six "ctitious but realisticcases were used. The values of important parametersused in the simulations are presented in Tables 1 and 2.

It is assumed that the gas-phase concentration remainsconstant and additionally that the contribution of thegas phase to the overall heat balance can be neglected.For systems where these additional assumptions arenot valid, gas-phase conservation equations should besupplemented to the approximate model.

The odd cases (1, 3 and 5) refer to dynamically unstableand the even cases (2, 4 and 6) refer to dynamically stableconditions. Cases 1 and 2 are mainly controlled by therate of the reaction kinetics (Ha"0.24), while cases 3 and4 (Ha"2.9) and cases 5 and 6 (Ha"63.3) are controlledby mass transfer processes.

Including the work of Vleeschhouwer et al. (1992) thesecases cover the whole range from pre-mixed feed bulkreaction characterised by (Al!1)Ha2;1 till fast "lmreaction with (Al!1)Ha2<1. The physical meaning of(Al!1)Ha2 is the ratio of the maximum conversion inthe liquid bulk to the maximum transport through themass transfer "lm (Westerterp et al., 1990). Vleeschhouweret al. (1992) had a system with (Al!1)Ha2;1, forcases 1 and 2 we have a system with (Al!1)Ha2+1, for

E.P. Van Elk et al. / Chemical Engineering Science 54 (1999) 4869}4879 4873

Table 2Operating conditions and initial values used and the corresponding steady state values (approximate model)

Case Cases 1 and 2 Cases 3 and 4 Cases 5 and 6

[A]g,"6-,

100 mol m~3 100 mol m~3 5.0 mol m~3

[B]l,*/

5000 mol m~3 25,000 mol m~3 25,000 mol m~3

¹#00-

425 and 441 K 469 and 507 K 673 and 715 K;A 35,000 and 55,000 W K~1 150,000 and 250,000 W K~1 75,000 and 100,000 W K~1

Regime Kinetics controlled/intermediate Mass transfer controlled Mass transfer controlledHa

s0.24 2.9 63.3

(Al!1)Ha2 0.94 (+1.0) 140 ('1.0) 66,074 (<1.0)¹

s468 K 565 K 842 K

¹0

488 K (step disturbance 203C) 575 K (step disturbance 103C) 852 K (step disturbance 103C)[A]

l,"6-,,s50.1 mol m~3 2.0 mol m~3 0.004 mol m~3

[B]l,"6-,,s

1558 mol m~3 4354 mol m~3 3760 mol m~3

;A$%4*'/ 36-%

'50,418 W/K (Fig. 5)! ' 214,848 W/K (Fig. 7)! ' 98,934 W/K (Fig. 10)!¹

#00-, $%4*'/ 36-%' 438 K (Fig. 5)H ' 509 K (Fig. 7)! ' 693 K (Fig. 10)!

!If one of these two design rules is ful"lled the system will show point stable steady states.

Table 1Fixed parameters used for all simulations

Case A(g)PA(l), A(l)#B(l)PP(l)R

a"k

R,1,1[A][B]

kR0,1,1

500,000E!#5

90,000 J mol~1

R 8.314 J mol~1K~1

kl

3.5]10~5m s~1

kg

100 m s~1(no gas resistance)D

a10~9m2s~1

Db

10~9m2s~1

ma

1.0<

R10 m3

el

0.5a 1000 m2m~3

Ul,*/@065

0.005 m3s~1

ol

800 kg m~3

CP,-

2000 J kg~1K~1

j 0.02 W m~1K~1

*HR

!160,000 J mol~1

¹l,*/

303 K

cases 3 and 4 we have (Al!1)Ha2"140 and for cases5 and 6 "nally (Al!1)Ha2<1.

3.2. Case 1 and 2 (rigorous model)

Solving the rigorous reactor model presented in Sec-tion 2.2 requires simultaneous solution of a set of couplednon-linear ordinary and partial di!erential equations (seeFig. 1) and results in the change of temperature andconcentration in time. The change of the reactor temper-ature ¹

lin time for the parameter values of cases 1 and

case 2 given in Tables 1 and 2 is shown in Fig. 2. Case1 results in a limit cycle (dynamic unstable steady state)whereas case 2 results in a point-stable steady state.

Fig. 2. Solution of the rigorous reactor model. Ha"0.24,(Al!1)Ha2"0.94, ¹

s"468 K. Case 1: ;A"35,000 and case 2:

;A"55,000.

3.3. Cases 1 and 2 (simple model)

A stability map, obtained by a perturbation analysis ofEqs. (18) and (19) using the data of cases 1 and 2 and[A]

l,"6-,"50.1 mol m~3 (the steady-state value, see

Table 2) is given in Fig. 3. From the stability map it canbe seen that for both cases 1 and 2 a limit cycle ispredicted (region II). From Fig. 2 it can however be seenthat this does not agree with the results obtained fromthe rigorous reactor model developed in the presentstudy. In Fig. 4 the solution of the simple model is givenand it can be seen that this does indeed show a limit cyclefor case 2, as predicted.

It can thus be concluded that the perturbation analysisdoes agree with the results of the simple model, butapparently the simple model fails for complex processes.This means that the perturbation analysis as presentedby Vleeschhouwer et al. (1992) is not generally applicable

4874 E.P. Van Elk et al. / Chemical Engineering Science 54 (1999) 4869}4879

Fig. 3. Stability map of the simple reactor model (2 ODEs). The steadystate temperature is plotted as a function of the cooling temperatureand the heat transfer parameter UA. The points 1 and 2 refer to cases1 and 2 from Table 2. Regions I, II and III refer to point stable, limitcycle and static unstable conditions, respectively.

Fig. 4. Solution of the simple reactor model (2 ODEs). Ha"0.24,(Al!1)Ha2"0.94, ¹

s"468 K. Case 2: ;A"55,000.

and only valid for the speci"c case of systems with a com-pletely saturated liquid phase ((Al!1)Ha2;1, seeWesterterp et al., 1990).

3.4. Cases 1 and 2 (approximate model)

A stability map, obtained by a bifurcation analysis ofEqs. (20)}(24) was created using the LOCBIF bifurcationsoftware package. From the stability map (Fig. 5) it ispredicted that case 1 is a limit cycle and case 2 is point-stable, which corresponds with the results of the rigorousmodel. From the stability map the following design ruleis obtained for this speci"c system: as long as either UA islarger than 50,418 W/K or ¹

#00-is larger than 438 K the

steady states are point stable (region I).

Fig. 5. Stability map of the approximate reactor model (3 ODEs). Thesteady state temperature is plotted as a function of the cooling temper-ature and the heat transfer parameter UA. Points 1 and 2 refer to cases1 and 2 from Table 2. Regions I, II and III refer to point stable, limitcycle and static unstable conditions, respectively.

Fig. 6. Solution of the approximate reactor model (3 ODEs). Ha"0.24, (Al!1)Ha2"0.94, ¹

s"468 K. Case 1: ;A"35,000 and case 2:

;A"55,000.

The solution of the system of Eqs. (20)}(24) for cases1 and 2 is shown in Fig. 6. Comparing Fig. 2 (rigorousreactor model) and Fig. 6 (approximate model) showsthat the approximate model with 3 ODEs (20)}(22) and1 AE (Eq. (24) substituted in Eq. (23)) gives reasonableresults.

3.5. Cases 3 and 4 (rigorous and approximate model)

Cases 1 and 2 used for the calculations presented in theprevious sections correspond to systems that are mainlycontrolled by the reaction kinetics (Ha"0.24). In thissection it will be shown that the bifurcation analysisbased on the approximate (3 ODEs) model is also applic-able in the mass transfer controlled regime (Ha'2.0).

E.P. Van Elk et al. / Chemical Engineering Science 54 (1999) 4869}4879 4875

Fig. 7. Stability map of the approximate reactor model (3 ODEs). Thesteady state temperature is plotted as a function of the cooling temper-ature and the heat transfer parameter UA. The points 3 and 4 refer tocases 3 and 4 from Table 2. Regions I, II and III refer to point stable,limit cycle and static unstable conditions, respectively.

Fig. 8. Solution of the rigorous reactor model. Ha"2.9,(Al!1)Ha2"140, ¹

s"565 K. Case 3: ;A"150,000 and case 4:

;A"250,000.

This is done by appropriate modi"cation of the condi-tions (cases 3 and 4 de"ned in Table 2).

A stability map, obtained by a bifurcation analysisusing LOCBIF and the approximate model, is shown inFig. 7. It is predicted that case 3 is a limit cycle and case4 is point-stable. From Fig. 8 it can be seen that thiscorresponds with the results obtained from the rigorousmodel. The exact design rules for point stable steadystates are given in Table 2.

Fig. 9 shows the solution of the approximate model.Comparing Figs. 8 and 9 shows some discrepancy be-tween the approximate and the rigorous model. This ismost probably caused by the fact that the approximatemodel does not take the temperature pro"le on micro

Fig. 9. Solution of the approximate reactor model (3 ODEs). Ha"2.9,(Al!1)Ha2"140, ¹

s"565 K. Case 3: ;A"150,000 and case 4:

;A"250,000.

Fig. 10. Stability map of the approximate reactor model (3 ODEs). Thesteady state temperature is plotted as a function of the cooling temper-ature and the heat transfer parameter UA. The points 5 and 6 refer tocases 5 and 6 from Table 2. Regions I, II and III refer to point stable,limit cycle and static unstable conditions, respectively.

scale into account (Vas Bhat et al., 1997). Moreover, forthese cases the heat of reaction is produced mainly in themass transfer "lm near the gas}liquid interface, contraryto the approximate rigorous method which assumes bulkheat generation.

3.6. Cases 5 and 6 (rigorous and approximate model)

Cases 3 and 4 are mainly controlled by mass transfer(Ha"2.9, (Al!1)Ha2"140), but the concentration ofthe gas-phase component A in the liquid bulk is not yetcompletely zero ([A]

l,"6-,"2.0 mol m~3). Cases 5 and

6 are characterised by a very fast reaction (Ha"63.3,(Al!1)Ha2<1), so that the liquid bulk is fully depleted

4876 E.P. Van Elk et al. / Chemical Engineering Science 54 (1999) 4869}4879

Table 3Overview of the results.

Case ¹#00-

;A ¹s

Rigorous Simple model (2 ODE) Approximate model (3 ODE)

(K) (W/K) (K) model Solution of Perturbation Solution of Bifurcationmodel analysis model analysis

1 425 35,000 468 Limit cycle Limit cycle Limit cycle Limit cycle Limit cycle2 441 55,000 468 Point stable Limit cycle Limit cycle Point stable Point stable

3 469 150,000 565 Limit cycle Limit cycle Limit cycle4 507 250,000 565 Point stable Point stable Point stable

5 673 75,000 842 Limit cycle Limit cycle Limit cycle6 715 100,000 842 Point stable Point stable Point stable

ConclusionP Exact solution Simple model fails Approximate model succeeds

Fig. 11. Solution of the rigorous reactor model. Ha"63.3,(Al!1)Ha2"66074, ¹

s"842 K. Case 5: ;A"75,000 and case 6:

;A"100,000.

Fig. 12. Solution of the approximate reactor model (3 ODEs). Ha"63.3, (Al!1)Ha2"66074, ¹

s"842 K. Case 5: ;A"75,000 and case

6: ;A"100,000.

([A]l,"6-,

"0.004 mol m~3). This is achieved by appro-priate modi"cation of the conditions (Table 2).

A stability map, obtained by using the approximatemodel is shown in Fig. 10. It is predicted that case 5 isa limit cycle and case 6 is point-stable. From Figs. 11 and12 it can be seen that this corresponds with the results

obtained from the rigorous model and the approximatemodel, respectively. The exact design rules for pointstable steady states are given in Table 2. Comparing Figs.11 and 12 shows again some discrepancy between theapproximate and the rigorous model.

4. Conclusions

A rigorous model is presented that can accuratelydescribe the dynamic behaviour of ideally stirred gas}liquid reactors over a wide range of conditions. Themodel is used to demonstrate the possible existence oflimit cycles in gas}liquid processes.

It is concluded (see Table 3, simple model) that theperturbation analysis from Vleeschhouwer et al. (1992) isnot generally applicable for prediction of the dynamicbehaviour of ideally stirred gas}liquid processes.Application of these design rules is restricted to systemswith a fully saturated liquid phase ((Al!1)Ha2;1, seeWesterterp et al. (1990)).

The new approximate model presented in this papergives more general applicable results and covers thecomplete region from pre-mixed feed bulk reaction((Al!1)Ha2;1) to fast "lm reaction ((Al!1)Ha2;1).A bifurcation analysis based on this model is a powerfuland general tool to obtain design rules for stable opera-tion of ideally stirred gas}liquid reactors.

The decoupling of mass and heat balances on microscale increases the error of the approximate model andthe corresponding bifurcation analysis (especially in thefast reaction regime). Therefore, the rigorous reactormodel should be used to check the obtained design rulesand to investigate the dynamic system behaviour in moredetail.

Notation

a speci"c surface area, m2 m~3

A heat transfer area, m2

E.P. Van Elk et al. / Chemical Engineering Science 54 (1999) 4869}4879 4877

[A] concentration of component AAl Hinterland ratio (de"ned by e

lkl/aD

o),1

[B] concentration of component B[C] concentration of component CC

Pheat capacity, J kg~1 K~1

[D] concentration of component DD

46"4#3*15di!usivity, m2 s~1

E!#5

activation energy, J mol~1

E46"4#3*15

enhancement factor, 1hg

gas-phase heat transfer coe$cient,W m~2 K~1

*HR

heat of reaction based on R, J mol~1

Ha Hatta number de"ned as J(kR[B]D

a/k

l2),1

J46"4#3*15

molar #ux, mol m~2 s~1

JT

heat #ux, W m~2

kg

gas-phase mass transfer coe$cient, m s~1

kl

liquid-phase mass transfer coe$cient,m s~1

kR,m,n

reaction rate constant,m3(m`n~1) mol~(m`n~1) s~1

m reaction order, 1m

46"4#3*15gas}liquid partition coe$cient, 1

n reaction order, 1N counter with start value 1 at t"0, 1N

46"4#3*15number of moles, mol

R46"4#3*15

reaction rate, mol m~3 s~1

R'!4

ideal gas constant, J mol~1 K~1

t simulation time variable, s¹ temperature, K; heat transfer coe$cient, W m~2 K~1

<R

reactor volume, m3

x place variable, mx$*.%/4*0/-%44

place variable de"ned as x/J(4Dah),1

[] concentration, mol m~3

[]46"4#3*15

concentration, mol m~3

Greek letters

c46"4#3*15

stoichiometry number, 1dp

assumed thickness of liquid element, meg

gas-phase hold-up, 1el

liquid-phase hold-up, 1j thermal conductivity, W m~1 K~1

o density, kg m~3

h contact time according to penetrationmodel (de"ned by 4D

a/pk2

l), s

ql

liquid-phase residence time, s

Subscripts

0 initial valuea component Ab component Bbulk at bulk conditionsc component C

cool cooling mediumd component Dg gas phasei interfacei species iin at inlet conditionsl liquid phases steady-state value¹ temperature

Acknowledgements

These investigations were supported by DSM Re-search Geleen.

References

Aris, R., & Amundson, N. R. (1958). An analysis of chemical reactorstability and control* parts I}III. Chemical Engineering Science, 7,121}155.

Baccaro, G. P., Gaitonde, N. Y., & Douglas, J. M. (1970). Anexperimental study of oscillating reactors. A.I.Ch.E. Journal, 16,249}254.

Bilous, O., & Amundson, N. R. (1955). Chemical reactor stability andsensitivity. A.I.Ch.E. Journal, 1, 513}521.

Elk, E.P., van, Borman, P.C., Kuipers, J.A.M., & Versteeg, G.F. (1999).Modelling of gas}liquid reactors* implementation of the penetra-tion model in dynamic modelling of gas}liquid processes, submittedfor publication.

Heiszwolf, J. J., & Fortuin, M. H. (1997). Design procedure for stableoperations of "rst-order reaction systems in a CSTR. A.I.Ch.E.Journal, 43, 1060}1068.

Ho!man, L. A., Sharma, S., & Luss, D. (1975). Steady state multiplicityof adiabatic gas}liquid reactors: I. The single reactor case. A.I.Ch.EJournal, 21, 318}326.

Huang, D. T. -J., & Varma, A. (1981a). Steady-state and dynamicbehavior of fast gas}liquid reactions in non-adiabatic continuousstirred tank reactors. Chemical Engineering Journal, 21, 47}57.

Huang, D. T. -J., & Varma, A. (1981b). Steady-state uniqueness andmultiplicity of nonadiabatic gas}liquid CSTRs. A.I.Ch.E. Journal,27, 481}489.

Olsen, R. J., & Epstein, I. R. (1993). Bifurcation analysis of chemicalreaction mechanisms. II. Hopf bifurcation analysis. Journal ofChemical Physics, 98, 2805}2822.

Sharma, S., Ho!man, L. A., & Luss, D. (1976). Steady state multiplicityof adiabatic gas}liquid reactors: II. The two consecutive reactionscase. A.I.Ch.E. Journal, 22, 324 331.

Singh, C. P. P., & Shah, Y. T. (1982). The e!ect of gas feed temperatureon the steady state multiplicity of an adiabatic CSTR with a fastpseudo-"rst-order reaction. Chemical Engineering Journal, 23,101}104.

Uppal, A., Ray, W. H., & Poore, A. B. (1974). On the dynamic behaviorof continuous stirred tank reactors. Chemical Engineering Science,29, 967}985.

Uppal, A., Ray, W. H., & Poore, A. B. (1976). The classi"cation of thedynamic behavior of continuous sirred tank reactors* In#uence ofreactor residence time. Chemical Engineering Science, 31, 205}214.

Van Swaaij, W. P. M., & Versteeg, G. F. (1992). Mass transfer accom-panied with complex reversible chemical reaction in gas}liquid systems: an overview. Chemical Engineering Science, 47,3181}3195.

4878 E.P. Van Elk et al. / Chemical Engineering Science 54 (1999) 4869}4879

Vas Bhat, R. D., van Swaaij, W. P. M., Benes, N. E., Kuipers, J. A. M.,& Versteeg, G. F. (1997). Non-isothermal gas absorption with revers-ible chemical reaction. Chemical Engineering Science, 52, 4079}4094.

Vejtasa, S. A., & Schmitz, R. A. (1970). An experimental study of steadystate multiplicity and stability in an adiabatic stirred reactor.A.I.Ch.E. Journal, 16, 410}419.

Vleeschhouwer, P. H. M., & Fortuin, J. M. H. (1990). Theory andexperiments concerning the stability of a reacting system in a CSTR.A.I.Ch.E. Journal, 36, 961}965.

Vleeschhouwer, P. H. M., Garton, R. D., & Fortuin, J. M. H. (1992).Analysis of limit cycles in an industrial oxo reactor. ChemicalEngineering Science, 47, 2547}2552.

Westerterp, K.R., van Swaaij, W.P.M., & Beenackers, A.A.C.M. (1990).Chemical Reactor Design and Operation, New York: Wiley.

E.P. Van Elk et al. / Chemical Engineering Science 54 (1999) 4869}4879 4879