Dynamic analysis of enzyme reactors exhibiting substrate inhibition multiplicity

Chemrcd

Engmeenng

SEwwe

Shorter

1977 Vol

32 pp

557-559

Pecgamon

Pms

Pnnted

m Great

Bntam

Comtnun~cations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..*.........................*.......... Dymumc

analysis

of enzyme

reactors

(Recerued 17 September

exhibiting

1 INTRODUCTION

10 February

1976)

B’y = l/(1 + a/x,, +x,/KS)-(llE)(dy/dt)

1 + G/K.’

6

where v 1s an effectiveness factor introduced in order to take into account the effect of convective and dlffuslve resistances It depends on bulk con&tions and system parameters Figure 1 depicts the effectiveness factor vs Thlele modulus with the Sherwood number as parameter It 1s seen that for small values of 9 the effectiveness factor tends to 1 0 Such condltlons are approximated in the case of fine suspensions and very high substrate ddTuslvlty For such sltuatlons the pseudo homogeneous reactlon rate based on a unit volume of the hquld phase grven by eqn (2) 1s approximately valid 1

5 4 3 ‘1 2

(2)

1 •t IQ& f x,/t

where xb = G/C., and C,, IS a reference concentration Mixmg in a CSTR is intense and hence the region of multiplrcity extends over a wide range[7, 83 This means that such reactors are hable to mstabddy problems when the optimal steady-state design lies in the nelghbourhood of the multlphcity re@on For such cases steady-state analysis is not sufficient for design An unsteady-state substrate matenal balance on the reactor shown m Fig 2 assummg the pseudo-homogeneous rate expression of eqn (2) grves x,

(5)

The steady state solution of eqn (4) 1s shown m Fig 3 It IS clear that for the product free feed the multlphclty remon extends over the range 0 33 < 0’G 0 66 The choice of the optlmal combmatlon of reactor volume and conversaon level depends on the relative weights of the reactor construction and operatmg costs, the cost of the feed stream and the value of the product When such an economic balance corresponds to a value of 0’ lymg within the multlphclty range, for example 0’ = 0 5, it IS seen from Fig 3 that tigh conversion and low conversion stable steady states are possible The abdlty of the reactor to attam the desued Hugh conversion steady state depends on the uutml conditions and transient behavlour The numerical solubon of eqns (4) and (5) for different uutlal con&tlons are displayed on the phase plane of Fig 4 The trajectones indicate that the region of asymptotic stablhty of steady-state (c) LSrelatively small so that a disturbance may force the system to the domam of influence of steady-state (a) at which the system remams after the &sturbance is removed The cost of the control system requed to mamtam the h& conversion steady-state has to be provided for in the reactor design econonuc balance This may favour the choice of a larger reactor volume, a cascaded or a compartmental configuration resultmg m a umque steady-state solution

2 PERFECTLY MIXRD REACTOR

r, = ACE 1-e

multiplicity

free sves

For a substrate mhiblted reaction catalyzed by permeable enzyme-carrymg solids, the reactlon rate per urut volume of sohds surrounded by a hqmd of composition Ca may be expressed as [41

I i K/C,

mhbition

1975, accepted

The chmce of a reactor contiguratlon to conduct a particular reachon depends on the form of the rate expresslon[l,2] For simple reactions the plug flow reactor mves the h&est conversIon per unit volume However for complex rate expressions the situation ISmore involved due to the phenomenon of multlphcrty Pseudo-homogeneous modelhng of substrate mhlbrted Immobdlzed enzyme reactlons IS known to give nse to multiple steady states m CSTR reactors[3] It has also been shown that this type of modellmg results m unique steady states m the case of plug flow reactors [4] The first part of this work mscusses the performance of a CSTR after stating the comhtions vahdatmg the pseudo homogeneous assumption The repons of asymptotic stability of the steady states are generated through the transient analysis of the reactor operatmg I* the multiphcity regon In the second part of ths communication a pseudohomogeneous tubular reactor dynanuc model mvolvmg axml dispersion 1s developed The range of the system’s physlco-chenucal and nuxmg parameters resulting m steady state multlphclty are mvestlgated usmg the mapping method [6]

r=kCB

substrate

q=qx,+V~l(l+~/x~+x~/a)+Vdx,/dt

i2

Fig

3

1 Effectiveness

4

’

5

”

6

7

“1

8

= l/(1 + a/x,

+ x,lt)+(l/E)

(3) (d&/dr)

(4)

A simdar balance on the product assuming that the feed 1s product CES

Vol

32

No

5--G

Fig 2 Matenal balance on a CSTR 557

10

factor plot (R =0 1, I? =OOl)

or W(x, -x,)

9

Shorter Commumcatlons

558

demonstrated the numerical mstablhty of the solution of this type of equation when the mtegratlon proceeds in the forward direction (from z = 0 to z = I) and recommended backward Integration for handling such problems Hlavacek and Hofmann [6] developed an efficient method for mappmg the re@on of parameters wthm which non-umque solutmns are possible The mappmg method consists m findmg a smtable transformation which enables the calculation of the value of one of the parameters for whrch the deliberately asstgned values of the other parameter and exit concentration xbO satisfy the system’s equations The mtroductlon of the substitution 5 = Pe(l -z) mto eqns (6)-(S) aves respectively

Fig 3 Steady-state

multlphclty

at

m CSTR (x~ = 1 0)

and at

where

6

6

10

Ag 4 Phase plane analysis m CSTR (KS = 0 1, R = 0 01, f = 10) 3 TUBULAR REACTOR WITH AXIAL DISPERSION Assunung pseudo homogeneous kmebcs, an unsteady state substrate matenal balance over a differential element of the reactor shown m Fig 5 gves 1 a*.% ----Pe 82

ax&,

82

Da

a&

1+ iZl& + xt.lK* = s,

where concentration C,, has concentration The Dankwert’s model are

(6)

been taken as the reference boundary condrtlons for this

For a gwen choice of A and xbO,eqn (9) IS Integrated with respect to & startmg at 6 = 0 The value of [ at which eqn (10) LSsatisfied LS numencally equal to the Peclet number The correspondmg Damkohler number 1s then calculated from eqn (12) 3 2 Results

Steady state The above procedure has been repeated for suitable ranges of xbOand A m order to generate the Da - xb,, - A plot of Fig 6 It 1s clear that multiphclty of the steady-state does occur for certain values of Da and Pe Figure 7 shows the concentration profiles along the reactor for a case with multiple steady states (dashed curves, s,, s2, So)

atz=O (7) and at z = 1

Fig 6 Regions of multlphclty for axial dtsperslon model It is known that this axial dlsperslon model represents a plug flow reactor when Pe tends to mlimty and a CSTR when Pe tends to zero 3 1 The mappmg method The steady state equation LSobtamed by settmg (ax, /a~) = 0 111 eqn (6) This gives a two-pomt boundary value dlfferentml equation the solution of which consists m numerical Integration and boundary con&tlon lteratlon Raymond and Amundson[5]

v Cb

_

NA-

.-

+N,+nN,

I”

ro

z -dme&onless h

h+ah

Fig 5 Mass balance on the reactor

axial

m&d&e

Fig 7 Transient response of the tubular reactor (axml &sperslon model) for a step mcrease m substrate feed concentration

559 Trunsrent behauwur The transient behavlour of the system 1s described by eqns (6)-(S) together with the nutial con&tion, &(0,2)=x90(2)

Chemical Engmeenng Department CLWO University, Cairo, Egypt

S

S

ELNASHAIE A H GABER M A EL-RIFAI

(13)

The systems equations have been solved for gwen m&al condltlons usmg the Crank-Nicolson method[9] The set of system parameters used m the computations resulted m the multiple steady state profiles presented m Fig 7 In such cases It IS important to generate mformatlon about the transient behavlour of the system both for the purpose of studymg start-up and for the understandmg of the effect of various kmds of disturbances on the system’s response Start-up Inmal condmons that he completely below the nuddle steady state (s2) lead to the high conversion steady state (s3) On the other-hand uututl condltlons that he above the nuddle steady state lead to the low conversion steady state (s,) Efiect of feed drsturbances The sohd hnes of Fig 7 show the effect of mcreasmg the feed concentration from x, = 1 0 to x, = 1 4 on the response of the h& converslon steady state (sa) The effect IS an mcrease m the substrate concentration along the length of the reactor When the duration of the &sturbance 1s long enough, 7d > 10, the concentration profile moves above that of the nuddle steady state (s2) and the reaction IS mhlblted such that when the disturbance 1s removed the system does not return to its ongmal steady state (So), Instead it settles at the low conversion steady state (sl) Figure 8 shows the effect of decreasmg the feed concentration from x, = 1 0 to x, = 0 8 on the response of the high conversion steady state (sl) The effect IS a decrease m the substrate concentration along the length of the reactor No mhbltlon occurs and when the &sturbance 1s removed the system returns to its ongmal steady state (s3)

NOTATION

Da /Pe bulk phase substrate concentration substrate concentration in the feed substrate concentration m reactor effluent enzyme concentration per unit volume of sohds bulk product concentration reference concentration k&H l Damkohler number = v~,t G D. h H k R K K K= K, Pe 4 Ii ; Xb Xf

Y z

drffusaon coefficient axial distance coordmate total packed length of reactor rate constant L.-E kC 1 - E c..* system constant K/C.,* system constant K./C,, Peclet number = vH/D. volumetnc feed rate reaction rate per umt volume of solids particle radms mterstdlal thud velocity m packed bed hquld hold up volume of a CSTR C, /C,, or C, I& C,,lC,,, c.X,, &menslonless length = h/H

Greek symbols c fractional solrd catalyst hold up volume e v/q 4 Tbele modulus 11 eff@iveness factor 0’ qlkV 7 tie

REFERENCES

8 10 6 2 4 0 r,dlmenslonless axtal coordinate FM 8 Transient response of the tubular reactor (axml &sperslon model) for a step decrease m substrate feed concentration 4

CONCLUSION

The concentration mulhphcrty phenomena m pseudo homogeneous tubular reactors discussed m ths study anse from axial nuxmg Plug tlow reactors are also liable to steady-state multiphcity when the homogeneous assumphon does not hold [4] In any case the well xmxed reactor etiblts concentration multiplicity over a wider range of parameters Transient computations for CSTR have shown the importance of dynanuc studies m the region of multlphcrty Trausrent computations for the axial dispersion model have shown that mcrease m the feed concentratlon can cause an mhlbltlon and an lrreverslble decrease m the reactor’s productivity

[l] Ldly M D and Sharp A K , J Chem Engng London 1968 215 CE12 [2] Denbigh K G and Turner J C R , Chemical Reactor Theory Cambridge Umverslty Press 1971 [3] O’Nedl S P , Lay M D and Rowe P N , Chem Engng Scr 1971 26 173 [4] El-ma M A, Elnashale S S and Gaber A H , Proc Symp Analysts and Control of Immobdlzed Enzyme Systems Compi&ne 1975 [S] Raymond L R and Amundson N R , Can J Chem Engng 1964 42 173 [6] Hlavacek V and Hofmann F , Chem Engng Scr 1970 25 173 [7] Elnashaie S S and Yates J G , Chem Engng Scr 1973 28 515 181 Elnashale S S and Cresswell D L , Int Conf for Flulduatlon and Its Applrcatrons Toulouse 1973 [9] Lapldus L , Digdal Computatwns for Chemrcal Engmeers McGraw-Hdl, New York 1962