INTERACTION OF HYDRODYNAMIC ENVIRONMENT ON PERFORMANCE OF HOMOGENEOUS BIOREACTORS WITH ENZYME KINETIC MODELS

Cuent Stuies of Biotechnology Volume II. - Envionment INTERACTION OF HYDRODYNAMIC ENVIRONMENT ON PERFORMANCE OF HOMOGENEOUS BIOREACTORS WITH ENZYME KINETIC MODELS ŽELIMIR KURTANJEK * Faculty of Foo Technology an Biotechnology, Univesity of Zageb, Zageb, Coatia Abstact Inteaction of hyoynamic effects on enzyme eactions in homogeneous non-ieal inustial bioeactos is in the wok fom theoetical an computational point of view analyze. The esults ae extene to the case of unstuctue moel of a chemostat with an extacellula pouct synthesis. Consiee ae the cases of two limiting hyoynamic states: 1) total segegation o macomixing; 2) complete o mico mixing. The effects of hyoynamic conitions ae epesente by assume expeimentally etemine esience time istibution cuve. Fo the case of complete molecula mixing Zwieteing's life expectancy function is applie. Fo macomixing state convesions ae etemine by convolution of esience time istibution function with convesion functions fom batch kinetics. Stuie ae effects of egee of mixing on efficiency of Michaelis- Menten kinetics without inhibition an with substate inhibition (negative oe kinetics). Chemostat is moele with unstuctue Mono base kinetics fo biomass gowth an substate consumption. Theoetical evolution of the esience time istibutions is eive fom vaious combinations of pefectly stie vessels an ieal plug flow in tubes. The aim of the wok is to establish pactical guiance fo estimation of lowe an uppe bouns fo bioconvesions in nonieal inustial eactos. Key wos: micomixing, esience time istibution, enzyme kinetics Intouction Moelling complex hyoynamic behaviou is one of the most ifficult numeical poblems, but at the same time it is of funamental impotance to many aspects of biochemical engineeing. It equies solution of Navie-Stokes equation with supplie heological an/o tubulence moels couple with heat an mass balances. A numeical solution is attainable by CFD - computational flui ynamic methos that ae base on extensive space an time meshing an use of finite element appoximation of solution of Navie-Stokes equation. Fom eacto engineeing aspect, hyoynamic effects in a eacto ae ue to flow, pumping an impelle otation in vessels, an fo * Coesponing autho aess: Želimi Kutanjek, Faculty of Foo Technology an Biotechnology, Pieottijeva ul. 6, 1 Zageb, Coatia 89

ŽELIMIR KURTANJEK continuous bioeactos ae usually escibe on macoscopic scale by pobability esience time ensity function RTD, (1,2,3). Howeve, hyoynamic effects on chemical an biochemical eaction on micoscopic scale an knowlege of RTD function oes not suffice fo evaluation of a bioeacto pefomance (4). In this wok a numeical metho base on mass balances is applie fo the bounay cases of the ealiest an the latest micomixing. The bounay cases of micomixing coespon to maximum mixiness an segegate flow espectively. On macoscopic scale, the moel of "tanks in seies" is applie fo numeical evaluation of RTD function. Mathematical moel Moelling of hyoynamics effects base on RTD expeiments can be simplifie in view of assume flow pattens in a nonieal (inustial) bioeacto. Numbe of moel paametes vaies fom zeo to a case with seveal ajustable moel paametes subject to statistical fitting poceue with ata fom a tace impulse esponse measuement an theoetical moel. The zeo paamete moels ae the limiting cases coesponing to segegate an maximum mixiness flow. Moels with one ajustable paamete ae eive fom visualisation of a flow patten like in a tank in seies o as a ispesion flow in a tubula eacto. Moels with moe ajustable paametes ae constucte by assuming a complex flow as in system of inteconnecte vessels. Most commonly is applie RTD function E(θ) of the tank in seies of equal volume pefectly mixe tanks is obtaine by the invese of Laplace tansfom of the pouct of N iniviual tansfe functions: 1 i= N 1 E( θ ) = L (1) i= 1 1 s + 1 N In Eq. (1). RTD is expesse as function of imensionless time θ efine as the atio of eal time an the aveage esience time, which equals to the atio of a vessel volume an volumetic flow ate though the vessel. The function is moel fo a vey boa spectum of RTD functions anging fom ieally continuous stie tank eactos (CSTR), vaious packe be eactos, an in the limiting case fo N it coespons to a esponse of a plug flow in a tubula eacto. In Fig. 1. ae epicte typical RTD pofiles obtaine by Eq. (1). Fo given RTD function uniquely is efine the life expectancy function LEF as function of the expectancy vaiable (also name "intensity function"), Λ (λ), given by: Λ ( λ) ( λ) E = λ 1 E( θ ) θ (2) 9

INTERACTION OF HYDRODYNAMIC ENVIRONMENT ON PERFORMANCE OF HOMOGENEOUS BIOREACTORS WITH ENZYME KINETIC MODELS RTD.8.6.4.2 1 2 3 4 LEF 1.75 1.5 1.25 1.75.5.25 1 2 3 4 Figue 1. Resience time istibutions RTD an Life expectancy function LEF fo tank in seies moel fo the following vaiance of esience times:.1;.2; an.5. Mass balances fo eacting species in a bioeacto une batch opeating conitions ae expesse in the matix fom: θ T c = Da α () c (3) Stoichiometic matix fo a set of eactions is given by α, an volumetic eactions ae given as components of the vecto given by genealise Michaels-Menten kinetic expessions. Applie is the elative time scale leaing to intouction of Damköhle imensionless numbe efine as the atio of maximum ate of eaction an maximum ate of convection though eacto. The balances (3) apply fo intacellula an extacellula species eive fom homogeneous an/o stuctue moels of micooganism metabolism. Howeve, it is moe meaningful when volumetic ates ae eplace by specific ates v expesse pe unit y mass of cells: = v () c c X (4) The balances ( 3 ) ae integate fo given initial conitions c() = c an a specifie Da, yieling the solutions c( θ, Da ). The initial concentations fo batch balances (3) ae ientical to concentations in a feeing steam in case of continuous flow though eacto. Effects of micomixing on bioeacto pefomance ae elate to egee of eaction of enzyme ates. When enzyme eaction ate is inhibite by substate, oe of eaction ate changes fom 1-st oe (n=1) in the concentation ange bellow satuation constant c s << K s to negative 1-st oe (n = -1) fo high concentations c s >> K s (Fig. 2). Degee of mixiness effects the eaction ate only in the concentation egion B whee the oe of eaction is -1 < n < 1. In the egion A egee of mixing has no effect on eaction ate, while in egion B incease in egee of mixiness esults in incease of eaction ate. Fo given expeimentally etemine RTD function effect of egee of mixiness on enzymatic ates can be backete into the egion efine by the bouns evaluate fo zeo mixiness (segegate flow) an maximum mixiness. Due to the ange of oe of eaction, the uppe boun on convesion is evaluate by the maximum mixiness case, an the lowe boun of convesion coespons to segegate flow. The bouns ae calculate base on solution of balances fo eacting species une batch conitions. The balances ( 3 ) ae integate fo given initial conitions c() = c an a specifie Da, yieling the solutions c( θ, Da ). The initial con- 91

ŽELIMIR KURTANJEK centations fo batch balances (3) ae ientical to concentations in a feeing steam in case of continuous flow though eacto. Fo the limiting case of completely segegate flow, which fo given RTD istibution can be intepete as the case of the latest mixing, mass balance in steay state is given by the convolution of RTD an concentation pofile: c = E, ( θ ) c( θ Da) θ (5) n 1 n n -1 v(c s ).6.5.4.3.2.1 2 4 6 8 1 A B c s Figue 2. Change of oe n of enzyme eaction ate with substate inhibition coesponing to substate concentation egions The opposite bounay case is efine by assumption of the ealiest mixing, when the maximum mixiness is achieve. In this case the mass balances involve the life expectancy function an ae evaluate by a set of nonlinea iffeential equations: c λ ( c α ) (6) T ( ) ( c Da ) ( ) c λ = + Λ λ ( λ) The asymptotic ight han bounay conition is applie: λ c ( λ) = λ= (7) The solution of the steay state poblem is given by the initial conition: c S = c( λ =, Da) (8) 92

INTERACTION OF HYDRODYNAMIC ENVIRONMENT ON PERFORMANCE OF HOMOGENEOUS BIOREACTORS WITH ENZYME KINETIC MODELS Integation of (6) has to be evaluate in the evese iection (fom ight to left) with the initial conition etemine fom the nonlinea algebaic equation: T = Da α ( c, λ = ) + Λ( ) ( c( λ = ) c ) (9) In view of the elative time scale, when mean esience time is equal to 1, the limiting value fo λ is eplace by a numbe fo which RTD is pactically negligible, fo example fo λ = 4 ( o 8 ). The nonlinea poblem (9) equies an iteative poceue such as Newton algoithm, an since system of iffeential equation is "stiff" an equies backwa integation, it has to be execute with a obust ODE integato, such as LSODE. Results The moel fo effects of egee of mixing on pefomance of continuously opeate bioeactos is analyse. The poblem is motivate by the impotance of evaluation of bounaies of convesion (o pouctivity) base only on RTD infomation without pio knowlege of actual egee of mixiness, o the nee fo solution of Navie-Stokes equation by CFD softwae. The maginal eo is etemine fo a ange of Damköhle numbes in a pactical egion, fom the washout state when convesion is appoaching zeo, to the uppe values of Damköhle numbe fo which convesion is appoaching 1 %. In view of applicability of simulation esults it is essential to have the evaluation in the full space of paametes, which is accomplishe by ening the poblem in imensionless fom. The imensionless fom of the moel equations also impoves numeical stability of integation algoithm. Investigation of RTD on pefomance of bioeactos has been of a continuous inteest to biochemical enginees woking with non-newtonian liquis in eactos of lage volumes (5,6), in cases of complicate flow pattens occuing in open containes fo bioegaation of pollute wates (7), o in multiphase aeate bioeactos (8,9,1,11). In this wok pefomance of chemostat, i.e. continuously an isothemally opeate liqui phase bioeacto with suspene micooganisms, is investigate. The balances ae accounte by the ate limiting substate ( y ) an biomass ( z ). Dimensionless balances in the batch moe of opeation, coesponing to Eq. (3), ae: θ y = β Da µ ( y) z (1) θ z = Da µ ( y) z (11) µ ( y) y = (12) 2 y γ S + y + γ I 93

ŽELIMIR KURTANJEK The initial conitions ae povie by the imensionless values of the feeing concentations y() = 1 z() = 1. The pocess kinetics is given by the imensionless gowth function (13) fo which the kinetics oe changes in the ange fom + 1 to -1 with incease of substate concentation y. This infomation is sufficient fo poving that maximum mixiness will povie the uppe boun fo substate convesion, while the segegate flow coespon to the lowe boun. In Fig. 3 ae pesente esults obtaine fo the following set of paametes: γs =1, γ I =1, β =,8, an vaiance of RTD σ 2 ( θ ) =,5. The softwae Mathematica v. 4..(12) has been use fo numeical evaluation of the substate an biomass balances (5-6). 1 9 8 7 6 5 C SUBSTRATE CONVERSION A B D 2 4 6 8 1 Da.5.4.3.2.1 C BIOMASS D B A 2 4 6 8 1 Figue 3. Substate an biomass concentations as function of Damköhle numbe at following mixing conitions: A) maximum mixiness; B) tue (test case). C) ieally mixe; D) segegate flow. The batch moe balance (3) is evaluate numeically by use of NDSolve with Da as a vaiable paamete. Fo the segegate balance (5) numeical integation by aaptive Gauss quaatue is applie as povie by the NIntegate pogam. The nonlinea equation fo etemination of the missing ight han bounay value is calculate by Newton algoithm as given in NSolve. The citical step is the backwa integation of the nonlinea set of stiff equations (6). Applie is the poweful aaptive integato NDSolve which encompasses LSODE algoithm. In oe to povie analysis of egee of mixiness at a ange of Da numbe, the integals ae efine as a function Da. On Fig. 3- ae also shown esults obtaine fom the test poblem (exact solution), an the case fo assume ieally mixing (ieal chemostat). The eos fo the uppe an lowe boun with espect to the exact solution ae also given in Table 1. Table 1. Compaison of eos in substate convesion fo the uppe an lowe boun of mixiness Pecentage eo in substate convesion Da uppe boun eo maximum mixiness lowe boun eo segegate flow 2 2,77-9,477 4 1,49-7,71 6,444-4,926 8,223-3,33 1,128-2.345 The esults pove that fo mixing in bioeactos efine by the test moel of "tanks in seies" the maximum mixiness moel will povie vey accuate uppe boun on substate convesion fo the whole pactical ange of Damköhle numbe. Maximum eo of the uppe boun is +2 % 94

INTERACTION OF HYDRODYNAMIC ENVIRONMENT ON PERFORMANCE OF HOMOGENEOUS BIOREACTORS WITH ENZYME KINETIC MODELS at the point close to "washout", an with futhe incease of Da the eo eceases to,1 % when convesion is pactically 1%. Eos of the lowe boun ae fo a egee off oe lage. Conclusions Evaluation of convesions in a nonieal biochemical eacto equies solution of the numeically intensive poblems of mixing in bioeactos base on Navie-Stokes equation an application of the computational flui ynamics (CFD). The poblem can be effectively appoximate by evaluation of uppe an lowe bouns of convesions base on the assumptions of maximum mixiness an segegation. The maximum mixiness povies the uppe boun while the segegate yiels the lowe boun fo substate convesions. Howeve, fo a chemostat without inflow of biomass the lowe boun calculate by segegate flow is zeo (plug flow chemostat oes not sustain biomass). Fo teste moels of RTD eive fom tanks in seies moel the uppe boun gives small eo < +2 %, an is eceasing with incease of Da value. Maximum eo fo the lowe boun is < -1 %, an is also eceasing with incease of Da value. Refeences 1. Danckwets PV. Chem Eng Sci 1958; 8:93. 2. Zwieteing TN. Chem Eng Sci 1959; 11:1. 3. Kames H, Westetep KR. Elements of Chemical Reacto Design an Opeation, Acaemic Pess, New Yok 1963. 4. Plasai E, Davi R, Villemaux J. Micomixing Phenomena in Continuous Stie Reactos Using a Michaels- Menten Reaction in the Liqui Phase. Chemical Reaction Engineing-Houston, Weekman VW, Luss D.(es) ACS Symposium Seies 1978; 65:126-139. 5. Menishe T, Metghalchi M, Gutoff EB. Mixing Stuies in Bioeactos. Biopocess Eng 2; 22:115-12. 6. Kelly WJ, Humphey AE. Computational Flui Dynamics Moel fo Peicting Flow of Viscous Fluis in a Lage Femento with Hyofoil Flow Impelles an Intenal Cooling Coils. Biotechnol Pog 1998; 14:248-258. 7. Newell G, Bailey J, Islam A, Hopkins L, Lant P. Chaacteising Bioeacto Mixing With Resience Time Distibution (RTD) Tests.Wat Sci Tech 1998;37(12): 43-47. 8. Polla DJ, Ison P, Shamlou A, Lilly MD. Influence of a Popelle on Sacchaomyces ceevisiae Fementations in a Pilot Scale Ailift Bioeacto. Biopocess Eng 1997; 16:273-281. 9. Gavilescu M, Tuose RZ. Resience Time Distibution of Liqui Phase in an Extenal-loop Ailift Bioeacto. Biopocess Eng 1986; 14:183-193. 1. Ruffe HM, Wan L, Lubbet A, Schugel K. Intepetation of Gas Resience Time Distibution in Lage Ailift Towe Loop Reactos. Biopocess Eng 1994; 11:153-159. 11. Ruffe HM, Petho A., Schugel K., Lubbet A., Ross A., Deckwe WD. Intepetation of the Gas Resience Time Distibution in Lage Stie Tank Reactos. Biopocess Eng 1994; 11:145-152. 12. Wolfam S. The Mathematica Book. Cambige Univesity Pess, Cambige 1996. 95