Two different strategies for baker s yeast fermentation process simulation

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1 Two diffeent stategies fo bake s yeast fementation pocess simulation CLINA P. LÃ 1, FILMNA. AR 2 1 Dept. of Poduction ngineeing, 2 Dept. of Industial lectonics Univesity of Minho Campus de Azuém, Guimaães PRTUGAL cpl@dps.uminho.pt, Filomena.oaes@dei.uminho.pt Abstact The simulation pocedue epesents an impotant tool to undestand clealy the bake s yeast fementation pocess. This wok compaes two diffeent stategies to solve a mathematical model in ode to pedict the behaviou of the concentation of the state vaiables ove a 2 h time peiod, in a well-mixed eacto. Mass balances witten fo all the components constitute a system of initial value poblem. Consideing the kinetics and the gas tansfe ates elations, algebaic euations, as pat of the diffeential system, a diffeential-algebaic system is defined. Two FRTRAN9-based simulatos wee employed fo studies concening the bake s yeast fed-batch fementation. Both the simulation packages wee seen to be an efficient tool fo the simulation and tests of the non-linea pocess. Keywods: bake s yeast fementation, simulation, DAs, Ds. 1 Intoduction The building and use of mathematical models based on obseved data is fo long accepted as a basic scientific methodology. Models may be of a moe o less fomal chaacte, but they have the basic featue that they attempt to link obsevations togethe into some patten [1]. With the pogess in digital technology, and thinking of binging the theoy into pactice, computational modelling and model-based applications have emeged and ae ecognized as aeas of geat pioity [2]. The conventional appoach fo pocess modelling is based on the balance euations fo mass, enegy, and, if necessay, momentum and population. This fom of modelling euies futhe knowledge about eaction kinetics, themodynamic, tanspot and physical popeties. Real pocesses in the chemical, biochemical and food industy ae in thei vast majoity non-linea MIM systems (Multiple Input Multiple utput). Thei dynamics and contol ae difficult to study both fo theoetical and pactical easons. In many instances expeiments with eal industial pocesses ae not caied out fo easons of economy and safety. ften on-line measuements ae not available o simply they ae too expensive. The simulation pocedue epesents an impotant tool to undestand clealy the bake s yeast fementation pocess. The simulation consists on the integation of a set of non-linea diffeential euations, in ode to the state vaiables, by consideing o not as pat of the system, algebaic euations, concening mass tansfe elations and kinetics laws. The diffeent methods of integation used ae listed in the liteatue [3, 4]. In this aticle two diffeent stategies ae compaed. 2 Bake s yeast fementation - Modelling appoach The simulation model consists of a set of algebaic euations, elated to the kinetic models of bake s yeast gowth and a set of five diffeential euations, in ode to the state vaiables. The latte esults fom a mass balance of the femente. Fo moe details, see [5]. Next sections descibe the models implemented in the simulation pogams Kinetic model east gowth is chaacteized by thee metabolic pathways C6 H126 + a2 + bnx [ NH3] bclh HXX N NX + cc2 + dh2 (espiatoy gowth on glucose) (1) C6 H126 + gnx [ NH3] gcl H HXX N NX + hc2 + ih2 + jc2h 6 (fementative gowth on glucose) (2)

2 C2 H6 + k2 + lnx [ NH3] lclh HXX NNX + mc2 + nh2 (espiatoy gowth on ethanol) (3) whee epesents glucose; oxygen; X biomass; ethanol; C cabon dioxide and,, : specific gowth ates fo the thee pathways. In the seuel X,,,, C mean concentations. The metabolic pathways of fementative gowth on glucose and oxidative gowth on ethanol ae competitive. This competition is govened by the espiatoy capacity of the cells. If the instantaneous oxygen uptake capacity exceeds the oxygen need fo total espiatoy glucose uptake, then, all suga uptakes follows the espiatoy pathway (1) with the emaining oxygen being spent on ethanol espiatoy uptake (3). thewise, if the instantaneous oxygen uptake capacity is not enough, then, pat of glucose uptake follows the espiatoy pathway (1) while the emaining follows the fementative pathway (2). The kinetics euations fo bake s yeast gowth, consideed as Monod euations, ae detemined as follows. The total specific gowth ate, t, is the sum of the gowth ates fo the thee pathways t + = +. (4) The specific gowth ates, i, can be elated to the coesponding substate fluxes,, and yield coefficients (Table 1),, by t X X X = + + (5) whee X and X epesent the yield coefficients of biomass in glucose in the oxidative and fementative phases, espectively; X is the yield coefficient of biomass in ethanol in the oxidative phase in ethanol. As ethanol uptake is influenced by the pioity of glucose uptake, which functions as an inhibito, the specific gowth ate on ethanol can be descibed as K i = (6) + K + K i whee is the imal specific gowth ate, K i is the inhibition paamete and K is the satuation paamete. Howeve, this euation holds tue only if thee is an available espiatoy capacity of the cells. Table 1. ield coefficient values (poposed by [6]) Coefficient Value.49 g biomass/g X glucose X.5 g biomass/g glucose.1 g biomass/g X ethanol.72 g biomass/g X ethanol 1.2 g biomass/g X oxygen.64 g biomass/g oxygen.81 g biomass/g cabon dioxide.11 g biomass/g cabon dioxide 1.11 biomass/g cabon dioxide The glucose uptake,, is slightly diffeent because it follows two metabolic pathways: oxidative and fementative = +. (7) The glucose,, and oxygen,, uptake follow a Monod kinetics, espectively whee = (8) + K = (9) + K is the imal specific glucose uptake ate, K and K ae satuation paametes and is the imal specific oxygen uptake ate. By (1), it can be seen that the oxidative glucose uptake depends on the availability of dissolved oxygen, and may be defined as = (1) a whee a is the stoichiometic coefficient of the oxygen in the espiatoy pathway of glucose and is the oxygen uptake on glucose. Two situations may occu: excess of oxygen that

3 implies no fementative gowth of biomass; lack of oxygen and conseuently excess of glucose that implies no espiatoy gowth on ethanol. Table 2 esumes mathematically what has been said; only two of the thee metabolic pathways coexist. a Respiative Regime a > Respio- Fementative Regime Table 2. Bake s yeast kinetics X =. (11) = (12) = min(, ) (13) 1 2 (14) = X. a (15) = X. a = (16) Two auxiliay euations, (17) and (18), must be added to the euations listed in Table 2, fo the estimation of the specific gowth ate on ethanol, defined as: K i 1 = (17) + K + Ki and X = ( a ). (18) 2 X The elevant kinetic data wee taken fom onnleitne and Käppeli [7] (Table 3). Paamete Table 3. Kinetic paametes Value 3.5 g gluc -1 g biom -1 h g -1 2 g biom -1 h h -1 K.1 gl -1 K i.1 gl -1 K s.2 gl -1 K o.1 mgl Mechanistic model The mechanistic model fo the fed-batch fementation is obtained fom mass balances fo all components, consideing that the eacto is well mixed. Futhemoe it is assumed that the yield coefficients ( s) ae constant and the dynamics of the gas phase can be neglected. Then the set of diffeential model euations is: mass balance fo the biomass dx d = ( + + D)X (19) mass balance fo the suga = X X + ( f )D (2) whee f is the substate concentation in the feed, d d dc mass balance fo the ethanol = X X D mass balance fo the oxygen = X X D + TR mass balance fo the cabon dioxide = X C + + DC CTR (21) (22) (23) The accumulation of the woking volume duing the fed-batch pocess is epesented by dv = DV. (24) The dilution ate (atio feed ate/volume), D, is defined by F D =. (25) V The gas tansfe ates ae given by - TR = K L C a ( ) ( C ) (26) CTR = K L a C (27)

4 whee K i L a ae oveall mass tansfe coefficients fo oxygen and cabon dioxide and * and C* ae the coesponding euilibium concentations. 3 Numeical methods fo integation The set of model euations that descibes the mechanistic fo the fed-batch fementation, diffeential euations (19-24) plus the algebaic euations (11-16, 25-27), foms a system of diffeential-algebaic euations. Diffeential-algebaic (DA) systems ae diffeent fom odinay diffeential (D) systems in that, while they include D systems as a special case, they also include poblems that ae uite diffeent fom Ds, [3]. ome of these systems can cause sevee difficulties fo numeical methods. Conseuently, the numeical solution of these systems is a vey active aea of eseach. The index of a DA system is a measue of its degee of singulaity. Roughly speaking, D systems, y '= f ( t, y), have index zeo. Diffeential euations coupled with algebaic euations, y '= f ( y, z), = g ( y, z), have index one if g =, can be solved fo z given y (that is, if g z is nonsingula) and othewise have an index highe than one. The system in study is of index one so, the highe ode index system would not be hee discussed. The initial conditions fo the DA system in contast to the D system, must be consistent, in the sense that they must satisfy the constaints of the system and possibly also some of the deivatives of the constaints. DAL is a package feely available on the Intenet via NetLib. DAL uses a fixed-leading-coefficient BDF methods fo index-one DAs, and teats the linea systems as full o banded, but in vaious details it addesses the issues of DA poblems diectly. Moe details on the algoithm ae descibed in [3]. The eo toleances, ATL and RTL (absolute and elative eo toleances, espectively), wee taken to be 1-4. This value was obtained afte seveal simulation uns whee no significant diffeences wee found in the final esults. When consideing bake s yeast gowth model as an D system, the algebaic euations ae calculated pio to the set of diffeential euations. The method employed is a vaiable step algoithm, based on embedded aafayan method, involving the Butche s fomulas of 4 th and 5 th ode [4], INTG outine. The tuncation eo is foced to be less than a limit defined by the use [8]. All computations epoted wee pefomed in double pecision on a Pentium 5 MHz compute. 4 imulation pocedue The simulation pogams of bake's yeast poduction wee developed in FRTRAN9 wokspace. Figue 1 descibes, in a schematic fom, the MAIN pogam developed fo the fed-batch fementation. The pocedues inside the block dashed line, depend on the stategy used: DA/D systems. The only diffeence in the D based integation method is that, as aleady efeed, the algebaic euations ae calculated befoe the definition of the five diffeential euations outine, FN. FUN outine defines the diffeential-algebaic system. Call INTG FN Input initial conditions Input initial paametes and coefficients es D system t=t end Call DAL Pint / ave esults nd DA system FUN BNCK Figue 1. Flowchat of the MAIN pogam outine. Figues 2 and 3 pesent, espectively, the flowchats fo the outine whee the diffeential and algebaic system euations ae defined, FUN, and the outine whee the switch between the espiative and the espio-fementative models is veified, BNCK. DAL integato automatically calls these two outines, defined by the use. The uns wee taken at the same conditions, namely- Initial values: X()=.3 g/l, ()=1.3 g/l, ()=.8 g/l ()=.66 g/l, C()=.2 g/l, V()=2 g/l,

5 f =3 g/l Final volume: 5 l Manipulated vaiable: Constant Glucose feed-ate, F=.12 l/h. Algebaic euations (14), (15), (16), (26), (27) Call BNCK R? es Algebaic euations (11), (12), (13), (26), (27) 5 Results The state vaiables, biomass, X, suga,, ethanol,, oxygen,, and cabon dioxide, C, concentation pofiles, duing a 2 hous un, do not diffe significantly fo both diffeent stategies. Figues 4 and 5 show, espectively, the X, and, and concentation pofiles. These esults egad to the DA-based system. The cabon dioxide was kept constant duing all uns. Nevetheless the D-based method shows some small oscillations, paticulaly in biomass, X, and oxygen,, pofiles, possibly due to numeical poblems. These oscillations ae eflected on the elative eo behaviou. 1 8 X Diffeential euations (19 24) X,, (g/l) 6 4 Figue 2. Flowchat of the FUN outine whee the diffeential algebaic system is defined. R efes to the espiative egime. > * C > C* (8) (9) a es es es Respio-Fementative Regime, F = * C = C* Respiative Regime, R Figue 3. Flowchat of the BNCK outine t(h) Figue 4. Biomass, X, suga,, and ethanol,, concentation pofiles on a 2h simulation un, unde the defined initial conditions (g/l) t (h) Figue 5. xygen,, concentation pofile on a 2h simulation un, unde the defined initial conditions Figues 6 and 7 pesent the elative eos (defined, in pecentage, as the diffeence between the values

6 obtained in DA-based method and the D-based pocedue, divided by the fist) fo the fou state vaiables. Both oxygen,, and suga,, have small concentation values. Howeve, the oxygen has highe elative eo, namely afte 8 h un; small vaiations in small concentation values incease the elative eos. This is also the eason fo the behaviou fo the ethanol,, elative eo afte 18 h un. This behaviou indicates that one of the integato is moe sensitive to descibes the decay of the concentation duing the time peiod. elative eo (%) X t (h) Figue 6. Relative eo (%) between the two model stategies fo the biomass and ethanol concentations elative eo (%) t (h) Figue 7. Relative eo (%) between the two model stategies fo the suga and oxygen concentations 6 Conclusions Two simulatos wee developed and compaed fo studies of fed-batch bake s yeast fementation. The pocess was implemented in open-loop mode at constant feed ate of glucose substate (F=.12 l/h). Fo the same initial conditions, both stategies seem to chaacteize the case study. In the DA-based method, the initial state should be consistent and well defined. The diffeences in the final esults obtained by using the two methods wee measued in tems of concentations elative eo (%). ome diffeences could be found, namely fo some values of un time peiod whee small concentations of the state vaiables wee obtained. The DA based method seems to be moe obust and less sensitive to numeical poblems. 7 Acknowledgements The authos ae gateful to Algoitmi Reseach Cente fo intenal funding suppot of the poject. Thanks ae also due to dite Fenandes (Minho Univesity) fo helpful discussions. Refeences [l] Ljung, L., ystem Identification - Theoy fo the Use, Pentice-Hall, New Jesey, [2] dga, T. F., Modelling and Contol - Back to the Futue, Pat I, CAT Communications, 19 (1), 1996, pp [3] Benan, K.., Campbell,. L., Petzold, L. D., Numeical olution of Initial-Value Poblems in Diffeential-Algebaic uations, lsevie, New ok, [4] Lapidus, L., einfeld, J. H., Numeical olution of Ds, Academic Pess, New ok, [5] liveia oaes, F., Monitoização e contolo de fementadoes Aplicação ao femento de padeio, Ph.D. Thesis, Univesidade do Poto., 1997 (in potuguese). [6] Pomeleau,., Modélisation et Contôle d'un Pocédé Fed-Batch de Cultue des Levues à Pain (acchaomyces ceevisiae). PhD Thesis, Univesité de Monteal,199. [7] onnleitne, B., Käppeli,., Gowth of acchaomyces ceevisiae is contolled by its Limited Respiatoy Capacity: Fomulation and Veification of a Hypothesis, Biotech. Bioneg., 28, pp [8] Feyo de Azevedo,., ubotinas em Fotan paa utilização no sistema WangV 8 da FUP, Publicação intena DQ, FUP, Poto, 1987 (in Potuguese.