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1 M odelling Principles

1.1 FundamentalsofModelling1.1.1 Use of Models for Understanding, Design andOptimization of BioreactorsAn investigation of bioreactor performance might conventionally be carriedou t in an almost entirely empirical man ner. In this approach, the bioreactorbehavior wou ld be studied u nd er practically all com bination s of p ossibleconditions of operation and the results then expressed as a series ofcorrelations, from which the result ing performance might hopefully beestimated for any given set of new operating cond itions. This em piricalprocedure can be carried out in a very routine way and requires relatively littlethought concerning the actualdetail of the process. While this m igh t seem tobe rather convenient, the procedure has actually many disadvantages, since verylittle real understanding of the process would be obtained. Also very manyexperiments wouldbe required in order to obtain correlations that would coverevery process eventuality.Compared to this, the modelling approach attempts to describe both actualand probable bioreactor performance, by means of well-established theory,which wh en described in ma thema tical terms, represents a work ing m odel forthe process. In carrying out a modelling exercise, the modeller is forced toconsider the nature of all the important parameters of the process, their effecton the process and how each parameter can be defined in quantitative terms,i.e., the modeller must identify the important variables and their separateeffects, which,in practice, may have a very highly interactive combined effecton the overall process. Th us the very act of mo delling is one that forces abetter understanding of the process, since all the relevant theory must becritically assessed. In addition, the task of formulating theory into terms ofmathematical equations is also a very positive factor that forces a clearformulation of basic concepts.Once formulated, the model can be solved and the behavior predicted by themodel compared with experimental data. A ny differences in performance may

Biological Reaction Engineering, Second Edition.I. J.D unn ,E.Heinzle,J. Ingham , J. E. PfenosilCopyright 2003W I L E Y - V C H Verlag GmbH& Co. KGaA,WeinheimISBN:3-527-30759-1

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10 1 Modelling Principles

then be used to further redefine or refine the model until good agreement isobtained. Once the mod el is established it can then be used, with reasonableconfidence, to predict performance under differing process conditions, and itcan also beused for such purposes asprocess design, optimization and control.An inpu t of plant or expe rimen tal data is, of course, required in order toestablish or validate the model, but the quantity of experimental data required,as compared to that of the emp irical approach is con siderably reduced. A pa rtfrom this, the major advantage obtained, however, is the increasedunderstanding of the process that one obtains simply by carrying out themodelling exercise.These ideas are summarized below.Empirical Approach: Measure productivity for all combinations of reactoroperating conditions, and make correlations.- A dvantage: Little thoug htisnecessary- Disadvantage: M any experiments arerequired.ModellingApproach:Establish amodel, and design experiments to determinethe model parameters. Compare the model behavior with the experimentalmeasurements. Use the model for rational design, control and optimization.- A dvan tage: Fewer experimen tsarerequired,andgreater un derstand ingisobtained.- Disadvantage: Some strenuo us thinkingmay benecessary.

1.1.2 General Aspects of the M odelling App roachAn essential stage in the development of any model, is the formulation of theappropriate mass and energy balance equations (Russell and Denn, 1972). Tothese must be added appropriate kinetic equations fo r rates of cell growth,substrate consumption and product formation, equations representing rates ofheat and mass transfer and equations representing system property changes,equilibrium relationships, and process control (Blanch and Dunn, 1973). Thecombination of these relationships provides a basis for the quantitativedescription of the process and comprises the basic mathematical model. Theresulting model can range from a very simple case of relatively few equationsto models of very great complexity.Simple models are often very useful, since they can be used to determine thenu merical values for m an y impo rtant process parameters. For examp le, amodel based on a simple Monod kinetics can be used to determine basicparameter values such as the specific growth rate ( J L I ) , saturation constant (Ks),biomass yield coefficient (Yx/s)and maintenance coefficient (m). This basickinetic data can be supplemented by additional kinetic factors, such aso x y g e ntransfer rate (OTR), carbon dioxide p roduction rate (CPR), respiration qu otien t(RQ) based on off-gas analysis an d related quantities, such as specific oxygen

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1.1 Fundamentals ofModelling 11

uptake rate (qo2)> specific carbon dioxide production rate (qco2)> may also bederived and used to provide a complete kinetic description of, say, a simplebatch fermentation.For complex fermentations, involving product formation, the specificproduct production rate (qp) is often correlated as a complex function offermentation conditions, e.g., stirrer speed, air flow rate, pH, dissolved oxygencontent and substrate concentration. In other cases, simple kinetic models canalso be used to describe the functional dependence of productivity on celldensity andcell growth rate.A more detailed "structured kinetic model" may be required to give anadequate description of the process, since cell composition may change inresponse tochanges in thelocal environment withinthebioreactor. The greaterthe complexity of the model, however, the greater is then the difficulty inidentifying the numerical values for the increased number of model parameters,and one of the skills of modelling is to derive the simplest possible model thatis capable of a realistic representation of the process.A basic use of a process model is thus to analyze experimental data and touse this to characterize the process, by assigning numerical values to theimportant process variables. The model can then also be solved withappropriate numerical data values and the model predictions compared withactual practical results. This procedu re is kn ow n as simulation and may beused to confirm that the model and the appropriate parameter values are"correct". Simu lations, how ever, can also be used in a predictive m an ne r to testprobable behavior under varying conditions; this leads on to the use of modelsfor process optimization an d their use in advanced control strategies.The application of a combined modelling and simulation approach leads tothe following advantages:

1. Modelling improvesunderstanding, and it is through understanding thatprogress is made. In formulating a mathematical model, the modeller isforced to consider the complex cause-and-effect sequences of theprocess in detail, together with all the complex inter-relationships thatmay be involved in the process. The com parison of a model predictionwith actual behavior usually leads to an increased understanding of theprocess, simplyby havingto consider the ways in which the model mightbe inerror. The resultsof a simulation can also often suggest reasons asto why certain observed, and app arently inexplicable, phen om ena occu rin practice.

2. M odels help inexperimentaldesign. It is important that experiments bedesigned in such a way that the model can be properly tested. Often themodel itself will suggest the need for data for certain parameters, whichmight otherwise be neglected, and hence the need for a particular type ofexperim ent to prov ide the required data. Con versely, sensitivity tests onthe model may indicate that certain parameters may have a negligible

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1.1FundamentalsofModelling 13

Physical Model

LJtRevise ideasandequations

Mathematical Model

Newexperiments

NO Experimental DataSolution: C =f(t) ComparisonOK?

YES

Use fordesign,optimization andcontrol

Figure 1.1. Information flow diagram fo rmodel building.and perhaps alternative physical models for the process need to be developedandexamined. A tthis stage, it is often helpful tostart withthesimplest possibleconception of the process and to introduce complexities as the developmentproceeds, rather than trying to formulate the full model with all its complexitiesat the beginning of the modelling procedure.(ii) The available theory must then be formulated in mathematical terms.Most bioreactor operations involve quite a large number of variables (cell,substrate and product concentrations, rates of growth, consumption andproduction) and many of these vary as functions of time (batch, fed-batchoperation). For these reasons the resulting mathematical relationships oftenconsist of quite large sets of differential equations. The thick arrow in Fig. 1.1designates both the importance and the difficulty of this mathematicalformulat ion.(iii) Having developed a model, the model equations must then be solved.Mathematical models of biological systems are usually quite complex andhighly non-linear and are such that th e mathematical complexity of the

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equations is usually sufficient to prohibit the use of an analytical means ofsolution. N um erical methods of solution must therefore be em ployed, with themethod preferred in this text being that of digital simulation. W ith thismethod, the solution of very complex models is accomplished with relative ease,since digital simulation provides a very easy and a very direct method ofsolution.Digital simulation languages are designed specially for the solution of setsof simultaneous differential equations using num erical integration. M any fastan d efficient numerical integration routines are now available and areimplemented within the structure of the languages, such that m any digitalsimulation languages are able to offer a choice of integration routin e. So rtingalgorithms within the structureof the language enable very simple programs tobe written, havingan almost one-to-one correspondence with the way in whichthe basic model equations were originally formulated. The resulting simulationprograms are therefore very easy to understand and also to write. A furthermajor a dvan tage is a conven ient outpu t of results, in both tabulated andgraphical form, that can be obtained via very simple program commands.(iv) The validity of the computer prediction must be checked an d steps (i) to(iii) will often need to be revised at frequent intervals during the modellingprocedure. The validity of the model depends on the correct choice of theavailable theory (physical and mathematical model), the ability to identify themodel parameters correctly and the accuracy of the numerical solution method.In many cases, owing to the complexity and very interactive nature ofbiological processes,the system willnot be fully understood, thus leaving largeareas of un certainty in the mod el. A lso the relevant theory m ay be verydifficult to apply. In such cases, it is then often very necessary to make rathergross simplifying assumptions, which may subsequently be eliminated orimproved as a better understanding is subsequen tly obtained. Care andjudgement must also be used such that the model does not become overcomplex and so that it is not defined in terms of too m an y imm easurab leparameters. Often a lack of agreement between the model and practice can becaused by an incorrect choice of parameter values. This can even lead to qu itedifferent trends being observed in the variation of particular parameters duringthe simulation.It should be noted, however, that often the results of a simulation model donot haveto give an exact fit to the experimental data, an d often it is sufficient tosimply have a qualitative agreement. Thus a very useful qualitativeunderstanding of the process and its natural cause-and-effect relationships isobtained.

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1.1 Fundamentals of Modelling 15

1.1.4 Sim ulation ToolsMany different digital simulation software packages are available on the marketfor PC and Mac application. Modern tools are numerically powerful, h ighlyinteractive an d allow sophisticated types of graphical and numerical output.M ost packages also allow optimisation an d parameter estimation. BE RK EL EYMADONNA is very user-friendly an d very fast. W ehave chosen it for use inthis book, and details can be found in the Appendix. With it data fitting andoptimisation can be done very easily. M ODE LM A KER is also a more recent,powerful and easy to use program , wh ich also allows optimisation andparameter estimation. ACSL-OPTIMIZE has quite a long history ofapplication in the control field, and also for chemical reaction engineering.MATLAB-SIMULINK is a popular an d powerful software fo r dynamicsimulation and includes manypowerful algorithms for non-linear optimisation,which can also be applied for parameter estimation.

1.1.5 Teaching A pplicationsFor effective teaching, the introduction of computer simulation methods intomodelling courses can be achieved in various ways, and the method chosen willdepend largely on how much time can be devoted, both inside and outside theclassroom. The most t ime-consuming method for the student is to assignmodelling problems to be solved outside the classroom on any availablecomputer. If schedu ling time allows, com puter laboratory sessions areeffective, withthe student working either alone or in groups of up to three oneach m onitor or compu ter. This requires the availability of m any com puters,but has the advantage that pre-programmed examples, as found in this text, canbe used to emphasize particular points related to a previous theoreticalpresen tation. This method has been found to be particularly effective whenused fo r short, continuing-education, professional courses. By use of thecomputer examples the student may vary parameters interactively and makeprogram alterations, as well as working through the suggested exercises at his orher ownpace. Dem onstration of a particular simu lation problem via a singlepersonal computer an d video projector is also an effective way of conveyingthe basic ideas in a short period of time, since students can still be very active insuggesting parametric chang es and in anticipating the results. The bestapproach is probably to combine all three methods.

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1.2 DevelopmentandMeaningof DynamicDifferentialBalances

A s indicated in Section 1.1, many models for biological systems are expressedin terms of sets of differential equations, which arise mainly as a result of thepredominantly time-dependent nature of the process phenomena concerned.For many people and especially for many students in the life sciences, themention of differential equations can cause substantial difficulty. This sectionis therefore intended, hopefully, to bring the question of differential equationsinto perspective. The differential equations arise in the model formulation,simply by havingto express rates of change of material, due to flow effects orchem ical and biological reaction e ffects. The method for solution of thedifferential equations will be handled automatically by the computer. It ishoped that much of the difficulty can be overcome by considering thefollowingcase. In this section a simple examp le, based on the filling of a tan kof water, is used to develop the derivation of a mass balance equation from thebasic p hysical model and thereby to give meaning to the terms in the equ ations.Following thedetailed derivation, a short-cut method based on rates is given toderive the dynamic balance equations.Consider a tank into which water is flowing at a constant rate F (m3/s), asshown in Fig. 1.2. A t any time t, the volum e of water in the tank is V (m 3) andthedensity ofwateris p (kg/m3).

Figure 1.2. Tank of water being filled by stream with flow rateF.During the time intervalA t (s), a mass of water p F At(kg) flows into the t ank.A s longas no water leaves the tank, the mass of water in the tank will increaseby aqu antityp F At,causinga corresponding increase involume,A V .Equating the accumulation of mass in the tank to the mass that entered the tankduring the time intervalA tgives,

p A V = p F A tSince p iscon stant, -FA t - h

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1.2 Development and Meaning of Dynamic Differential Equations 17

Applying this to very small differential time intervals (A t > dt) and replacingthe A signs by the differential operator "d", gives the following simple firstorder differential equation, to describe the tank filling operation,

dVdT = FWhat do weknow about the solution of this equa tion? That is, how does thevolume change with time or in model terms, how does the dependent variable,V , change with respect to the independent variable, t? To answer this, we canrearrange the equation and integrate it between appropriate limits to give,

or for constantF,to

= F f l l d t = F ( t i - t o )JtoIntegration is equivalent to sum m ing all the contributions, such that the totalchangeof volume is equal to the total volumeofwater added to the tank,

IV =IFAtFor the case of constant F, it is clear that the analytical solution to thedifferential equation is, V = F t + constantIn this case, asshownin Fig. 1.3, theconstantofintegration is the initial volumeof water in the tank, V Q , attime t = 0.

V o dt

Figure 1.3. V olume change with t ime fo r constant flow rate.

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18 1 Modelling PrinciplesNote thatthe slope in thevariation of Vwith respect to t,dV/dt, is constant, an dthat from the differential equation it can be seen that the slope is equal to F.Suppose F is not constant but varies linearly with time.

F = F o - k tThe above model equation applies also to this situation.Solving the model equation to obtain the functional dependence of V withrespectto t,

JdV = |F dt = J(F0- kt)dt = F O Jdt - k Jt dtIntegrating analytically,

The solution is,V = FOt - k t" + constant

kt'v = F O t - + V 0

. tFigure 1.4. Variation of F and V for the tank-filling problem.Note that the dependent variable startsat the initial condition, (Vo), and that theslope is alwa ys F. W hen F becomes zero, the slope of the curve relating V and talso becomes zero. In other words, the volume in the tank remains constantan d does not change an y further as long as the valueof F remains zero.Derivationof aBalance EquationUsingRatesA differential balance can best be derived directly in terms of rates of change .For the above example, the balance can then be expressed as:

/The rate ofaccumu lation^ /Theflow rateofmass^V ofmass withinthe tank ) enteringthe tank )

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1.2Developmentand M eaning ofDynamic Differential Equations 19

Thu s, the rate of accumulationof mass withinthe tankcan be written directly asdM /dt wherethemassM is equal to pV . The rate of mass entering the tank isgivenby p F,where both sides of the equation have unitso fkg/h.

= PFand d ( o V ) = pFThus this approach leads directly to a differential equation model, w hich is thedesired form for dyna mic simulation. N ote that both terms in the aboverelationship are expressed in mass quantities per unit time or kg/h.A t constant density, the equation again reduces to,

dVdT = Fwhich is to be solved for the initial condition, that attime t = 0, V = V o and for avariation in flow rate, givenby,

F = F0 - k twhichis valid un til F = 0.These two equations, plus the initial condition, form the mathematicalrepresentation or the mathematical model of thephysical model, representedby the tank filling with an entering flow of water. Thus this approach leadsdirectly to a differential equation model, which is the desired form forsimulation. This approach can be applied not only to the total massbut also tothe mass of any component.We have seen an analytical solution to this model, but it is also interesting toconsider how a computer solution can be obtained by a numerical integrationof the model equations. This is imp ortant since analytical integration is seldompossible in the case of real complex problems.Computer SolutionThe numerical integration can in principle be performed using the relations:

dV

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wheret - torepresents a very small time interval and V - V Q is the resultingchan ge in volum e of the w ater in the tank. As before, the flow is assumed todecrease with time accordingto F =F Q - k t.This integration procedure is equivalent to the following steps:1) Setting the integration time interval.2) Assigning a value to the inlet water flow rate at the initial value, timet = to-3) The term involving the water flow rate, F-kt, is equal to the derivativevalue,(dV /dt), attime t = to.4) Kn ow ing the initial value of V and the slope dV /dt, enables a new value ofV to be calculated over the small interval of time, equivalent to theintegration time interval or integration step length.

5) At the end of the integration time interval, the value of V will havechanged to a new value, representing the changeof Vwithrespect to timefrom its original value.The new valueof V can thusbe calculated.6) Using the new value of V, a new value for the rate of change of V withrespect to time, (dV/dt), at the end of the integration time interval cannow be calculated.7) Knowing the value of V and the value of dV/dt at the end of theintegration time interval, a new value of V can be estimated over a furtherstep forwardintime or integration time interval.8) The entire procedure, asrepresented by steps (2) to (7) in Fig. 1.5 below,is then repeated with the calculation moving forward withrespect to time,until the value of F reaches zero. At this point the volume no longerincreases, and the resulting steady-state value of V is obtained, includingall the intermediate values of V and F, which were determined during thecourse of the calculation.

Figure 1.5. Graphical portrayal of numerical integration, showing slopes and approximatedvaluesof V ateach time interval.

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1.3FormulationofBalance Equations 21

Using such a numerical integration procedure, th e computer can thus be usedto generate data concerning the time variations of both F and V. In practice,more complex numerical procedures are employed in digital simulationlanguagestogive improved accuracy and speed of solution than illustrated byth e above simplified integration technique.

1.3 FormulationofBalance Equations

1.3.1 Types of Mass Balance EquationsSteady State BalancesOne of the basic principles of modelling is that of the conservation of mass,which for asteady-state flowprocess can beexpressedby the statement,

( Rate ofmass flow^ fRateofmass flow^jintothesystem J ^out of thesystemJDynamic TotalMass BalancesMany bioreactor applications are, however, such that conditions are in factchanging with respect to time. Under these circumstances, a steady-state massbalance is inappropriate and must be replaced by a dynamic or unsteady-statemassbalance, whichcan beexpressed as:

( Rate ofaccumulationof ( Rateof ^ ( Rateof ^massin thesystem J ^massflowinj ^massflowoutJHere the rate of accumulation term represents the rate of change in the totalmassof the system,withrespect totime,and atsteady-state this isequal to zero.Thus the steady-state mass balance represented earlier is seen to be asimplification of the more general dynamic balance, involving the rate ofaccumulation.A tsteady-state:

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( Rateof ^I accumulationof mass . = 0 = (Massflow in) - (Massflowout)

hence, when a steady-state is reached:(Mass flow in) = (Mass flow out)

Component BalancesThe previous discussion hasbeen in terms of the total mass of the system, butmost fluid streams, encountered in practice, contain more than one chemical orbiological species. Provided no chemical change occurs, the generalizeddynamic equation for the conservation of mass can also be applied to eachcompon ent. Thus for any particular compo nent:

Rate ofaccumulation of massof componentin the system ( Mass flow of Athe compon ent _intothesystemJ ( Mass flow of >th e component ou tofthesystem ,Component Balances with ReactionWhere chemical or biological reactions occur, this can be taken into account bythe addition of a further reaction rate term into the generalized componentbalance. Th us in the case of material produced by the reaction:

Rateofproduction

of thecomponent

by the reaction>

Rateofconsumption

of thecomponent

by thereaction

' Rateof ^accumulation

of massof component

^ in the system,

=Massflow

of thecomponent

into^thesystem,

-^ Mass f l o w ^

of thecomponent

out of^thesystem,

+

an d in the case of material consumed by the reaction:Rateof

accumulationof mass

of component^ inthe system,

=Massflow

of thecomponent

intov th e system,

-Massflow

of thecomponent

out of^thesystem^

-

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1.3Form ulation of Balance Equations 23

Elemental BalancesThe principle of the mass balance can also be extended to the atomic level andapplied to particular elements. Thus in the case of bioreactor operation, thegeneral mass balance equation can also be applied to the four main elements,carbon, hydrogen, oxygen and nitrogen and also to other elements if relevantto the particular problem. Thu s for the case ofcarbon:

Rateofaccumulation^ Massflowrateof \ (Massflowrateof\of carbonin = carbon into - thecarbonoutthe system J ^ the system J V of the system )Note the elemental balances do not involve reaction terms since the elements donot change by reaction.The computer example PENFERM, is based on the use of elemental massbalance equations for C, H, O and N which, when combined with otherempirical rate data, provide a working model for a penicillin productionprocess.While the principle of the mass balance is very simple, its application canoften be quite difficult. It is important therefore to have a clear un de rstan din gof both the nature of the system (physical model), which is to be modelled bymeans of the mass balance equations, an d also of the methodology ofmodelling.

1.3.2 Balancing ProcedureThe methodology described below outlines six steps, I through VI, to establishthe model balances. The first task is to define the system by choosing thebalance or control region. This is done using the following procedure:I. Choose the balance region such that the variables areconstant or change little within the system. Drawboundaries around the balance regionThe balance region may be a reactor, a reactor region, a single phase within areactor, a single cell, or a region within a cell, but will always be based on aregion of assumed constant composition. Generally the modelling exercises willinvolve some prior simplification. Often the system being modelled is usuallyconsidered to be composed of either systems of tanks (stagewise or lumped

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1.3 Formulation of Balance Equations 25that of the time of passage of material as it flows along the reactor and isequivalent to the time available for reaction to occur. Un der steady-statecon ditions the concen tration at any position along the reactor w ill be con stantwithrespect to time, though not withposition. This type of behavior can beapproximated by choosing the balance regions sufficiently small so that theconcentration of any compo nent within a region can be assumed to beapproximately un iform. Thus in this case, m an y unifo rm property subsystems(well-stirred tanks or increments of different volume but of uni formconcentration) comprise the total reactor volume.

13.2.3 Case C. R iver with Eddy CurrentFor this example, the combined principles of both the stirred tank anddifferential tu bu lar modelling approaches need to be applied. A s show n in F ig.1.8 the main flow along the river is very analogous to that of a column ortubular process, whereas the eddy region can be approximated by the behaviorof a well-mixed tank . The interaction between the main flow of the river andthe eddy, with flow into the eddy from the river and flow out from the eddyback into theriver'smain flow,mustbe included in any realistic model.The real-life and rather complex behavior of the eddying flow of the river,might thus be represented, by a series of many well-mixed subsystems (ortanks) representing themain flow of the river. This interacts at some particularstage of the river with a single well-mixed tank, representing the turbulent eddy.In modelling this system by means of mass balance equations, it would benecessary to draw bou nd ary regions aroun d each of the individual subsystemsrepresenting the mainriver flow, sections 1 to 8 in Fig. 1.9, and also around thetank system representing the eddy. This wo uld lead to a very minim um of nine

i v e r

Eddy

Figure 1.8. A complex river flow system.

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26 1Modelling Principlescomponent balance equations being required. The resulting model could beused, for example, to describe the flow of a pollutant down the river in rathersimple terms.

>I I

1 2 3 1 4 1 5 1 6 1 7 1 8 2 3 4 5 6 7 8i i i i i i i

V

>

fk Flow interaction

fitSli ^:iiK ^ ffK< y^

>fcCr

Figure 1.9. A multi-tank model for the complexriver flow system.//. Identify the transport streams which flow across thesystem boundariesHaving defined the balance regions, th e next task is to identify all the relevantinputs and outputs to the system. These may be well-defined physical flowrates (convective streams), diffusive fluxes, and also interphase transfer rates.It is important to assume adirection oftransferand to specify this by means ofan arrow. This direction might reverse itself, but will be accomodated by areversal in sign.

Out by d iffusion

ConvectiveflowinConvectiveflow out

n by diffusionFigure 1.10. Balance region showing convectiveand diffusive flows in andout.

///. Write the mass balance in word formThis is an important step because it helps to ensure that the resultingmathem atical equation will have an und erstandable physical mea ning . Just

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1.3Formulationof BalanceEquations 27

starting off by writing down equations is often liable to lead to fundamenta lerrors, at least on the part of the begin ner. A ll balance equations have a basiclogic as expressed by the generalized statemen t of the com pon en t balan cegiven below, and it is very important that the mathematical equations shouldretain this. Thus: Rateofaccumulationof massof com pon en tthe system )

f Mass flow ^ \of th ecom pon en tinto\thesystem^

f Massflowof th ecom pon en tout of^thesystem^

/ Rateofproduct ionof th ecom pon en t by\ the reaction /This can be abbreviated as,

(Accumulation) = (In) - (Out) + (Production)

IV. Exp ress each balance term in mathematical form withmeasurable variablesA. RateofAccum ulation TermThis is given by the derivative of the mass of the system, or the mass of somecomponent within the system,with respect totime. Hence:

dM i(Rate of accumulation ofmass ofcomponent iwithin the system) =where M is in kg or mol an d time is in h, min or s.Volume, concentration and, in the case of gaseous systems, partial pressureare usua lly the measured variables. Thus for any compo nent i

d M j _ d (CjV)dt = dt

where,Q is the concentration of i (kmol/m3 or kg/m3), and pi is the partialpressure of i within the gas phase system. In the case of gases, the Ideal GasLaw can be used to relate concentration to partial pressure and mol fraction.Thus, p i V = n i R Twhere R is in units compatible with p, V, n and T.In terms of concentration,

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28 1Modelling Principles

_ ni Pi y i Pci=T= RT = "RTwhere yi is the mol fraction of the component in the gasphase and p is the totalpressure.The accumulation term for the gas phase can be written as,/piVdM j _ d (CjV) _ d(QV) _ . d _

For the total massof thesystem:dM _ d (p V )dt = dtwithunits

B. Convective Flow Termsm3 s

Total mass flow rates are given by the product of volumetric flow multiplied b ydensity. Component mass flows are given by the product of volumetric flowrates times concentration.

( Mass\Volu meJk g m3 kgs -s m3

Total mass flow = F pComponent massflow MI = FQ

A stream leaving a well-mixed region, such as a well stirred tank, has the sameproperties as the system volume as a whole, since for perfect mixing thecontents of the tank will have un iform properties, identical to the properties ofthe fluid leaving at the outlet. Thu s, the concentrations of component i bothwithin the tank and in the tank effluent areequal toQi, as showninFig. 1.11.

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1.3 Formulation of Balance Equations 29

lilf

Figure 1.11. Convective flow terms for a well-mixed tank bioreactor.

C.Diffusion of ComponentsA s shown in Fig. 1.12, diffusional flow contributions can be expressed byanalogy to Pick's Law for molecular diffusion

Ji =-idZwherejt is the flux of any component i flowing across an interface (kmol/m2hor kg/m2 h) and dQ/dZ (kmol/m) is the concentration gradient as shown inFig. 1.12.

Figure 1.12.surfacearea A.

Diffusion flux jj driven by concen tration gradient (Qo - Cji) / AZt h rough

In accordance withPick's Law,diffusive flow always occurs in the direction ofdecreasing concentration and at a rate proportional to the concentrationgradient. Un der true cond itions of molecular diffusion, the constant ofproportionality is equal to the molecular diffusivity for the system, D j (m2/h).For other cases, such as diffusion inporous matrices an d turbulent diffusion, an

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effective diffusivity value is used, which must be determined experimentally.The concentration gradient may have to be approximated in finite differenceterms (Finite differencin g techniques are described in more detail in Sec. 6.2).Calculating the mass rate requires the area, through which diffusive transferoccurs.

( Massrateof(^componenti( Diffusivity Yof(^component ij\^

ConcentrationY Area ^ (gradient perpendicular =-DJo f i J^ totransport J

kg 2 _ m2 kg 2 _ kgsm 2 m s m4m T

D. Interphase TransportInterphase mass transport also represents a possible flow into or out of thesystem. In bioreactor modelling applications, this is most frequentlyrepresented by the case of oxygen transfer from air to the liquid medium,followed by oxygen taken up by the cells during respiration. In this case, thetransfer of oxygen occurs across the gas liquid interface, which exists betweenthe surface of the airbubbles and the surrounding liquid medium, as shown inFig. 1.13.

Figure 1.13. Transfer of oxygen across agas-liquid interface of specific area"a" into a liquidphase of volumeV .Other applications may involve th e supply of oxygen to the bioreactor bytransfer from the air, across a membrane and then into the bulk liquid. W herethere is interfacial transfer from one phase to another, the component balanceequations will need appropriate m odification to take this into account. Thus, anoxygen balance for the well-mixed gas phase, with transfer from the gas to theliquid, can bewrittenas ,

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1.3FormulationofBalanceEquations 31

/ Rateof \accumulationof themass ofoxygenin the gasV phase system J

=/Mass flow\of theoxygen

into th eVgasphase )

/Massflow\of theoxygenfrom the^gasphase>

-/ Rate \ofinterfacialmass transfer

fromthe gasphase intoV theliquid sThis form of transfer rate equation will be exam ined in m uch more detail inChapter 5. Suffice it to sayhere that the rate of transfer can be expressed in theform shown below:f RateOf \ ( Mass ^Uasst r a n s f e r J = trasPrtfV coefficient.

^AreapeA/Concentration /SystemA^ volume) ^driving force ) VvolumeJ= K a A C V

where, a is a specific area fo r mass transfer, A/V (m2/m3), A is the totalinterfacial area for mass transfer(m2), V is the liquid phase volume (m3), AC isthe concentration driving force (kmol/m3or kg/m3) and, K is the overall masstransfer coefficient (1/s). M ass transfer rate expressions are usually expressedin terms ofkmol/s, and can beconverted tomass flows (kg/s), if desired.

Theunitsof the terms in the equation (with appropriate mass quan tity units)are: kg 1 kg mjProductionRate

This term in the com pon ent b alance equation allows for the production orof material bv reaction and is incorporated into the componentThis term in the com pon ent b alance equation allows for the production orconsumption of material by reaction and is incorporated into the componentbalance equation. Th us,Rateof >\accumulationof massof componentth esystem )

/Massflow\of thecomponen tinto\thesystem//Massflow\of thecomponen tout ofthe system/

/ Rateofproductionofthecomponentby thereaction /

Chemical production rates are often expressed on a molar basis and, as in thecase of the interfacial mass transfer rate expressions, can be easily converted tomass flow quantities (kg/s). The production rate can then be expressed as

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f M ass rate \production of^component A y/^Reactionrate\= rA V = pervo lume ) (Volume of system)

kg _ _ kgsm3 m 3

Equivalent molar quantities may also beused. Thequantityr ispositivewhenA is formed asproduct, an d T A is negative whena reactant A is consumed.The grow th rate for cells can be expressed in the same m anner, u sing thesymbol rx- Thus,/ M ass rateof ^ /^Growthrate^Vbiomassproduction^ = rxV = pervolume) (Volume ofsystem)

kg _ kgs m3 m jThe consumption rate of substrate, r$ , is often directly related to the cell growthrate by means of a constant yield coefficient YX/S, which has the units of kgbiomass produced per kg substrate consum ed. Thu s,

( MansumptionJ = M>ervolumeABiomass-substrateyield)( V o l u m e )

kg ~ kg biomass kg substratesm3 m = sm3 kgbiomassV . Introduce other relationships and balances such that th enumber of equations equals the number of dependentvariablesThe system mass balance equations are often the most important elements ofan y modelling exercise, but are themselves rarely sufficient to completelyformulate the model. Other relationships are therefore needed to supplementthe material balance relations, both to complete the model in terms of otherimportant aspects of behavior and to satisfy the mathematical rigor of themodelling, such that the number of unknown variables must be equal to thenumber of defining equations.

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1.3 Formu lation ofBalance Equations 33

Examples of this type of relationships which are not based on balances, butwhich nevertheless form an important part of any model are:Reaction rates asfunctions of concentration, temperature, pHStoichiometric or yield relationships fo r reaction ratesIdeal gas lawbehaviorPhysical property correlations as functions of concentrationPressure variations as afunction of flow rate- Dynamicsof measurement instruments as a function of the instrumentresponse time- Equ ilibrium relationships (e.g.,Henry's law)- Con troller equations- Correlations of mass transfer coefficient, gas holdup volume, andinterfacial area, as functions of system physical properties an d degree ofagitation or flowvelocity

How these and other relationships are incorporated within the development ofparticular modelling instances are shown later in the cases given throughout thetext and in the simulation examples.V I . For additional insight with com plex problems drawan information flow diagramInformation flow diagrams can be useful in understanding complexinteractions (Franks, 1966). They help to identify missing relationships an dprovide a graphical aid to a full understanding of the interactive nature ofsystem. Anexample isgiven in the simulation example BATFERM.

1 . 3 . 3 Total Mass BalancesIn this section the application of the total mass balance principles will bepresented. Consider some arbitrary balance region, as shown in Fig. 1.14 bythe shaded area. Mass accum ulates within the system at a rate dM/dt, owing tothe competing effects of a convective flow input (mass flow rate in) and anoutput stream (mass flow rate out).

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Mass flow rate o utMass flow rate In

Figure 1.14. Balancing the total mass of an arbitrary system.The total mass balance is expressed by,

( Mass flow\ fMassflowout\=U.hesystem hesystem JdM = Massratein - M ass rateout

or in terms of volumetric flow rates, F, densities,(p),and volume, Vd(p V ) system3t = F o P o - F i P i

When densities areequal, as in the case of water flowing in and out of atank,dVdT = FO - F I

The steady-state condition of constant volume in the tank (dV/dt = 0) occurswhen the volumetric flow in, F Q , isexactly balanced by the volumetricflowout,F I . Total mass balances therefore are mostly important fo r those bioreactormodelling situations in whichvolumes are subject to change.

1.3 .4 Component Balances for Reacting SystemsEach chemical species can be described with a com pon ent balance aroun d anarbitrary, well-mixed, balance region, as shown inFig. 1.15.

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1.3Form ulation ofBalance Equations 35

Species iinflowSpecies ioutflow

Figure 1.15. Com ponent balancing for species i.Thus for any species i,involved in the system, the component mass balance isgiven by:

/ ^Massf lowoA f Rateof

component i production of/ Rateof \

accumulationof massof component ii in tnp cvct

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FQ CAO C BO

> .; . ? ? : 5: :; f.< XI :?-'?; ':-MI

|

F1 CA1 CB1

Figure 1.16. Continuou s stirred tank reactor with reaction A > 2B .Component A is converted to component B in a 1 to 2 molar ratio.The component balances for A and B are:

d(VCA1)dtd(VCBi)dt

= F0 C A O -= F0CBo-

Hereit is convenient to use molar masses, such that each term has the units ofkmol/h.Under constant volume conditions:d(VCA) = VdCAd(VCB) = VdCB

and in addition F Q =FI. Thus the two model equations, then simplify to give:dCAi F=Van d - CAI) +dCBi F~dT~ = V (c B O ~ C B i )

In these tw o balances there are four unknowns CAI, Q*i, rA l an( rBl8kinetics are assumed to be first order, as often found in biological systems atlow concentration. Then: rA l =-According to the molar stoichiometry,

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1.3Formulationof Balance Equations 37

rfi l =-2rM = + 2k CATogether withthekinetic relations there are 4 equations and 4 unknowns, thussatisfying the conditions necessary for the model solution. With the initialconditions, CAI an d CBI at time t = 0, specified, the solution to these tw osimultaneous equations, combined with the two kinetic relations, will give theresulting changes of concentrations CAI and CBIas functions of time. Thesimulation example ENZCON,is similarto the situationofCase A .

1.3.4.2 Case B. Semi-continuous Reactor with VolumeChange

The chemical reaction and reaction rate data are the same as in the precedingexample, but now the reactor has no effluent stream. The operation of thereactor is therefore semi-continuous.AO

* 2BFigure 1.17. A semi-continuous reactor example.The kinetics are as before:

t^rA = -kCAIn terms of moles the stoichiometry gives,

molesT m s

rB = - 2rA = + 2 kCAThe component balances withn o flow ofmaterial leavingthe reactor are now:

= F CA O + rA Vd(VCB) a t = IB v

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38 1 Modelling PrinciplesThe number of u nknowns is now five and the number of equations is four, sothat an additional defining relationship is required fo r solution. Note that Vmust remain withinthe differential, because the volume of the reactor contentsis now also a variable and must be determined by a total mass balance.Assuming constant density p, this gives the defining equation as:

Fdt =FWith initial conditions for the initial molar quantities of A and B,(VGA,an d the initial volume of the contents, V, at time t = 0 specified, the resultingsystem of equations can be solved to obtain the time varying quantities VCA(t),VCs(t),V(t)and hence also concentrations CA and CB as fun ction s of time.Similar variable volume situations are found in examples FEDBAT, an dV A R V O L .

1.3.4.3 Case C. Steady-State Oxygen Balancing inFermentationCalculation of the oxygen uptake rate, OUR,by meansof a steady-state oxygenbalance is an important application of component balancing for fermentation.In the reactor of Fig. 1.1.8, the entering air stream flow rate, oxygenconcentration, temperature and pressure conditions are shown by the subscript0 and the exit conditions by the subscript 1. F i > y i > T i > P Ias

F0 y O T0 P 0Air

Figure 1.18. Entering air and exit ga s during the continuou s aeration of a bioreactor.Writing a balance aroun d the combin ed gas and liquid phases in the reactorgives,

fRateof accum -^j_ fFlowrate^i (Flowrate"\ /^RateofO2uptake A( u l a t iono fO 2 J~ [ofO2in J~(ofO2outJ~I bythecells J

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1.3Formulation of Balance Equations 39

A t steady-state, the accumulation terms for both phases are zero andFlow of O2 in - Flow of C > 2 out = Rateof C > 2 uptake.

For gaseous systems, the quantities are often expressed in terms of molarquantities.Often onlythe inlet air flow rate F Q and the mol fraction of 62 in the outletgas, yi9 are measured. It is often assumed that the total molar flow rate of gas isconstant. This is a valid assumption as long as the number of carbon dioxidemols produced is nearly equal to the number of oxygen mols consumed or ifthe amounts of oxygen consumed are very small, relative to the total flow ofgas.Converting to molar quantities, usingthe Ideal GasLaw,p V = n R Tor in flow terms: pF = N R Twhere N is the m olar flow rate, R is the gas con stant and F is the volumetric flowrate. Th us, for the inlet gas flow:

Pwh ere NO is molar flow rate of the oxyge n entering. N ote that the pressure, po ,an d temperature, TO , are measured at the point of flow measurement.Assuming NO = N I , then measurement of NOgives enough inform ation tocalculate oxygen u ptake rate, OUR , from the steady-state balance. Thus,

0 = yoNO- y i NI - ro2VLOUR = ro2 VL = yoNO- y i N I

If NO is not equal to N I , then this equation will give large errors in oxygenuptake rate, and N I must be measured, or determined indirectly by an inertbalance. This isexplained in the Sec. 1.3.4.4below.

1.3.4.4 Case D. Inert Gas Balance to Calculate Flow RatesDifferences in the inlet an d outlet gas flow rates of a tank fermenter can becalculated by measuring one gas flow rate and the mol fraction of an inert gasin the gas stream. Since inert gases, such as nitrogen or argon, are n otconsumed or produced within the system (rinert= 0), their mass rates must

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therefore be equal at the inlet and outlet streams of the reactor, assumingsteady-state conditions apply. Then for nitrogen

/Molar flow of\ /Molarflowof\V nitrogen in ) = nitrogen out/and in terms ofmol fractions,

N O Y O inert = NI yi inertFrom this balance, calculation of N I can be made on the basis of a combinationof measurements of N Oand the inert gas partial pressures (yinertXat both inletand outlet conditions.

N, = yi inertSince theinlet mol fraction fo rnitrogen in air isknown,the outlet mol fraction,yi inert'must be measured. This is often done by difference, having measuredthe mol fraction ofoxygen and carbon dioxide concentration in the exit gas.

1.3.5 Stoichiometry, Elemental Balancing and theYield Coefficient ConceptStoichiometry is the basis for any quantitative treatment of chemical andbiochemical reactions. In biochemical processes it is a necessary basis forbuilding kinetic models.

1.3.5.1 Simple StoichiometryTheStoichiometryof chemical reactions isused to relate the relative quantitiesof the different materials which react with one another and also the relativequantities ofproduct thatare formed. Most chemical and biochemical reactionsare relatively simple in terms of their molar relationship or Stoichiometry. Forsingle reactionsstoichiometric coefficients are clearly defined and may usuallyeasilybedetermined. Some examples aregiven below:

C3H4O3+ NADH +H+

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1.3 Formulation of Balance Equations 43

< 1 > O O O < E > 1 (biomass), O2(substrate) and < J > 3 (product) are known, the unknown fluxes 04, 05, Og and< I > 7 can be obtained by methods of linear algebra and which are detailed byRoels (1983).

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In more com plex cases with grow th and product formation, m oreinformation is needed. The introdu ction of the concept of the degree ofreduction is use ful. For organic c om pou nd s this is defined as the nu m ber ofequivalent available electrons per gram atom C , that wou ld be transferred toC C > 2 ,H2O and NH3upon oxidation. Taking charge nu mb ers:C = 4,H =1 , O =-2, and N = -3, reductance degrees (y) can be defined fo r

substrate (S) ys = 4 + m - 2 1biomass (X ) yx = 4 +p - 2 n - 3 qproduct (P ) yp= 4 + r - 2 s - 3 t

The reductances fo r NHs,f^O an d C C > 2 are of course zero.Often the elemental composition of the substrate is not known and then thereductance method may be supplemented by the following regularities, whichapply to a wide variety of organic molecules.Qo2 = 27 J per g equivalent of available electrons transferred to oxygenYx =4.29 g equivalent of available electrons per equivalent 1 g atom C inbiomassGX =0.462 gcarbon / g dry biomass

1.3.5.3 Mass Yield CoefficientsYield coefficients are biological variables, which are used to relate the ratiobetween various consum ption and produ ction rates of mass and energy. Th eyare typically assumed to be time -indepen dent and are calculated on an overallbasis. This concept should n ot be con fused with the overall yield of a reactionor a process. The biomass yield coefficient on substrate(Yx/s) is defined as:

vx/S =rsIn batch systems, reaction rates are equal to accumulation rates, and therefore

/dX\IdTj dXY X /S = - TdST = - dS"IdTjAfter integration from time 0 to time t the integral value is obtained:

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1.3 Formu lationof Balance Equations 45

.. amountofbiomass produced* x/s = totalamountofsubstrate consumedX ( t ) - X ( t = 0 )S(t=0)-S(t)

For a steady-state continuous system the mass balances givers S Q - S

where index 0 and 1indicate feed andeffluent values, respectively.In the literature, yield coefficients for biomass with respect to nutrients aremost often used (e.g. Dekkers, 1983; Mou and Cooney, 1983; Roels, 1983;Moser, 1988). In many cases this is very useful because the biomasscomposition is quite uniform , and often product selectivity does not changevery much during an experiment involving exponential growth and associatedproduction. Some useful typical values are given in Table 1.1.

1.3.5.4 Energy Yield CoefficientsEnergy yield coefficients may be defined similarly to mass yield coefficients.In terms of oxygen uptake,

T O amountofheat releasedYq/02 == 7T Q 2 amountoroxygen consumedIn terms of carbon substrate consumed,

v - rQ- amountofheat releasedrs amountofcarbon source consumed

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Table 1.1. Typical mass and energy yield values (Roels, 1983; Atkinson an dMavituna, 1991).Type of yield coefficient Dim ension V alueYX/S,aerYx/S,anaerYx/02 (Glucose)YX/ATPYQ/O2YQ/C02YQ/x,aer(Glucose)Yq/x,anaer

c-mol / c-molc-mol / c-molc-mol / molc-mol/ molkJ/ molkJ/ molkJ /c-molkJ /c-mol

0.4-0.70.1-0.21-20.35380-490460325-500120-190

Note: The molecular weight of biomass is taken here as 24.6 g/C-molThe yield coefficien ts are usu ally determined as a result of a large num ber ofelementary biochemical reactions and it can easily be understood that theirvalues might vary depending on environmental and operating conditions.A detailed description of some of these dependencies is given in the literature.Despite this inconsistency, measured yield coefficients are often very useful fo rpractical purposes of process description an d modelling.

1.3 .6 Equilibrium Relationships

1.3.6.1 General ConsiderationsIn many biological systems, processes with large ranges of time constants haveto be described. Usu ally it is im portant to start with a simp lification of a system,focusing on the most important time constant or rate. For example, if thegrowth of an organ ism is to be modelled with a time constan t of the order ofhours, it is very useful to ignore all aspects of biological evolution with timeconstants of years. Also fast equilibrium reactions or conformational changesof proteins havin g time constants below milliseconds should be ignored. Fastreactions can, however,be very impo rtant wh en considering allosteric activationor deactivation of proteins or simp ly pH-changes during biochemical reactions.

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1.3 Form ulation ofBalance E quations 47

pH changes can have dramatic effects on the enzymeand microbial activity butcan also strongly influence absorption and desorption of carbon dioxide.A typical equ ilibrium reactions is the dissociation of a receptor-protein ligan dcomplex, RL, into the freeprotein,L , and thereceptorprotein,P

This reaction is characterized by the corresponding dissociation equ ilibriumconstantKDCL P k _ j

In most cases such relationships can be used to express the concentration of allconcentrations in explicit form using, e.g. a protein balance.C r1 _ i _ cPtot~ P~ r^LP

^ K D-^ptot Cr + KDThe total concentration, Cptot,is then included in a material balance equationand the concentrations of the free receptor and the receptor-ligand complex aredetermined by the equilibrium relationship. This is also true for a simple acid-base equilibrium relationship.In more complex cases with interactions ofv arious receptors or with a buffersystem containing several components, it is not possible to express theconcentrations in explicit forms and a non-linear algebraic equation has to besolved during the simulation. The implementation of such problems intoBerkeleyMadonna is shown below with the example of pH calculation

1.3.6.2 Case A. Calculation of pH with an Ion ChargeBalance.Modelling systemswith variable pH requires modelling of acid-base equilibria,whose reactions are almost instantaneous. Production of acids or bases causes avariation of pH, which depends on the buffer capacity of the system. pH alsoinfluences the biological kinetics. It has been shown that only the undissociatedacid forms are kinetically important substrates in anaerobic systems. The

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concentration of these species is a function of the pH as can be seen in theequilibrium equationAcid Base' + H+

with dissociation constantCBase-H+

CAcidwhere CAcid is the concentration of the undissociated acid an d CBase" is theconcentration of the corresponding base (salt).An ion charge balance can be written

(cations*charge) =(anions*charge)In the pH range of interest (usually around pH = 7) all strong acids and strongbases are completely dissociated. Moderately strong acids and bases exist inboth the dissociated and non-dissociated forms,In the usual pH range the sum of the cations are much larger than the H+ions. ICK+CH+where]CK+ is the total cation concentration.

Negative ions originate mainly from strong acids (e.g.C l % SO42') but alsoarise from weak acids (Ac", Pr, B u ~ , HCO3'). The concentration of CO32' isalwaysmuch smaller than that ofThe ion balance reduces to

K B j V1 KAiKW CBtot,i+CK+ =

where KAI are the acid dissociation constants (e.g. KAC); KBI are the basedissociation constants (e.g.KNHS); KW is the dissociation constant of water;Cfitot,iare the total concentrationsofbase i;CAtot,iare the total concentrationsof acid i andECA n " is the sum of the anions.The pH can be estimated from the above equation for any situation bysolving the resulting non-linear implicit algebraic equation, provided the totalconcentrations of the weak acids, C A t o t, i > weak bases, C B t o t ,i cations of strongbases, CK+,an d anionsof strong acids, C A I T > areknown.

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1.3F ormulationofB alance Equations 49

It isconvenient to use onlythe difference between cations an d anions

After neglecting any ammonia buffering effect, it is useful to rearrange theabove equations in the form,

The example A NA M EA S, Sec. 8.8.6 includes this ion balance for pHcalculation. This equation represents an algebraic loop in a dynamic simulationwhich is solved by iteration at each time interval until 8 approaches zero. This isaccomplished with the root-finding feature of Berkeley Madonna.If there is pHcontrol, then strong base or acid wouldbe usually added. Theaddition of strong alkali for pH control would cause an increase in CK+ which in accordance with the above equation would result in adecrease ofCH+.An alternative approach, which avoids an algebraic loop, is to treat theinstantaneous equilibrium reactions as reactions with finite forward andbackward rates. These rates must be adjusted with their kinetic constants tomaintain the equilibrium for the particular system ; that is, these rates m ust bevery fast compared withthe other rates of the model. This approach replacesthe algebraic loop iteration witha stiffer an dlarger set of differential equations.This could be an advantage in some cases.

1 .3 .7 Energy Balancing for BioreactorsEnergy balances are needed wh enev er temperature chan ges are imp ortant, ascaused by reaction heating effects or by cooling and heating fo r temperaturecontrol. For example, such a balance isneeded when the heat of fermentationcauses a variation in bioreactor temperature. Energy balances are writtenfollowing the same set of rules as given above for mass balances in Sec. 1.3.Thus the general form is asfollow s:

Accumu-^lationrateof

Ênergy ,

Rate ofênergyin by

^flow ,

'Rateof ênergyout byflow

'Rateof"energyout by

< transfer>

'Rateof >energygenerated

^by reactiony

'Rateof >energyadded by

âgitation

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50 1 Modelling PrinciplesThe above balance in word form is now applied to the measurable energyquantities of the continuous reactor shown inFig.1.20.

A HagitM) PO

U , A , T S

P

Figure 1.20. A continuous tank fermenter showing only the energy-related variables.Anexact derivationof theenergy balancewasgivenbyAris (1989) as,

S"dT = - hn) )+ U A(Ta- T I)+rQV +A Hagigit

where ni is the number of moles of component i,cpi are the partial molar heatcapacities an d hi are the partial molar en thalpies. In this equation the rate ofheat production, T Q , takes place at temperature T I .If the heat capacities, cpi, areindependent of temperature, the enthalpies at T I can be expressed in terms ofheat capacities as h n = h io + C p i ( T I - T O )andwith S2> i01 = 1

Thus w ith these simplifications,

S- 2X1 = 1

= v p

Vpcp L = F0pcp(T0- T I) + U A(Ta-T I)+rQV + A HagiThe unitsof each term of the equation areenergy per time (kJ/h orkcal/h).

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1.3Formulationof Balance Equations 5 1

Accumulation TermDensities and heat capacities of liquids can be taken as essentially constant.dTV p c P d Fhasun its: m 3 (kg/m3) (J/kg K) K _ kJs ~ s

Here (p cp T) is an energy "concentration" term and has the units,/nass\/ energy \ _ /energy\Vvolumey Vmassdegree V

aeg

reeJ - Vyolume,/

Thus the accumulation term has theunitsof energy/time (e.g. J/s)FlowTermsTheflowtermis F pCP(T0- T I)

/energy\/volume\ /energy\w i t h th e u n i t s , i^nn^J\ctisr) =This term actually describes heating of the stream entering the system withTOto the reaction temperature TI. It is important to note here that this term isexactly the same for a continuous reactor as for a fed-batch system.Heat Transfer TermThe important quantities in this term are the heat transfer area A, thetemperature driving force or difference(Ta-Ti), whereTa is the temperature ofthe heating or cooling source, and the overall heat transfer coefficient, U. Theheat transfer coefficient,U, hasunitsof energy/time area degree, e.g. J/sm2 C .The units for U A A T arethus,(heat transfer rate) = U A(Ta- T I)

energy energy~B55~ = area time degree (area) (deg ree)The sign of the temperature difference determines the direction of heat flow.Here ifTa>T I heat flows into th e reactor.Reaction Heat TermThe term rqV gives the rate of heat released by the bioreaction and has theunits of

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energy _ ene rgyvolumetime(volume) - timeThe rate term T Q can alternativelybe writteninvariousways as follows:In terms of substrate uptake and a substrate-related heat yield,

rq = rsYQ/SIn terms of oxygen uptake and an oxygen-related heat yield,

rq = ro2YQ/Q2In terms of aheat of reaction per mol of substrate and asubstrate uptake rate,

rQ = AHr,srsHere rs is the substrate uptake rate and A Hr? s is the heat of reaction for thesubstrate, for example J/mol or kcal/kg. The rs A Hr> s term therefore hasdimensions of (energy/time volume) and is equal to T Q .Other Heat TermsThe heat of agitation may be the most important heat effect for slowgrowingcultures, particularly with viscous cultures. Other terms, such as heat lossesfrom thereactor due to evaporation, can also be important.

1.3.6.3 Case B. Determining Heat Transfer Area or CoolingWaterTemperatureFor aerobic fermentation, the heats of reaction per unit volume of reactor areusually directly related to theoxygen uptake rate, ro2-Thus for a constant-volume batch reaction with no agitation heat effects, thegeneral energy balance is

/Accumulationrate /Energy o u t ^ /Energygenerated\V ofenergy ) ~ ~bytransferJ + V byreaction )

where YQ/Q2often has avalue near460 kJ/mol 2, asgiven inTable 1.1.

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1.3Formu lation ofBalance Equations 53

IfT is con stant (dT/dt =0):UA(Ti-Ta) = r02YQ/o2V

(heat tran sfer rate) = (rate of heat release)Using this steady-state energy balance, it is possible to calculate the coolingwater temperature (Ta) for agiven o xygen u ptake rate andcooling device.Thus, _-ro2

U AAlternatively this same relation can be used in other ways:

1) To calculate the additional heat transfer area required for a kn ow nincrease in cooling water temperature.2) To calculate the biomass co ncen tration allowable for a given coo lingsystem, knowing the specific oxygen uptake rate (kgO2 / kg biomass h).3) To calculate the cooling area required for a con tinuo us fermenter withknown volume inlet, temperature, flow rate and biomass production rate.

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