pages224_259

224 8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

Growth of Alcaligenes eutrophus strain H16. Eur. J. Appl. Microbiol.BiotechnoL 11, 8.1

8.3 Fed Batch Reactors

8.3.1 Variable Volume Fermentation (VARVOL andVARVOLD)

System

Semi-continuous or fed batch cultivation of micro-organisms is common in thefermentation industries. The fed batch fermenter mode is shown in Fig. 1 andwas also presented in the example FEDBAT. In this procedure a substrate feedstream is added continuously to the reactor. After the tank is full or thebiomass concentration is too high, the medium can be partially emptied, andthe filling process repeated. Since the variables, volume, substrate and biomassconcentration change with time, simulation techniques are useful in analyzingthis operation. This example demonstrates the use of dimensionless equations.

Figure 1. Filling and emptying sequences in a fed batch fermenter.

Model

The balances are as follows:

Volume,dv

dT = FOSubstrate,

Biological Reaction Engineering, Second Edition. I. J. Dunn, E. Heinzle, J. Ingham, J. E. PfenosilCopyright 2003 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 3-527-30759-1

8.3 Fed Batch Reactors 225

- = F0S0Biomass,

d(VX)dt = rx

The kinetics arerx = MX

|LimS** - (Ks + S)

andrx

rs = -YThe dilution rate is defined as

In order to simplify the equations and to present the results more generally, themodel is written in dimensionless form. Defining the dimensionless variables:

Vv =

XX< =

ss =

F =

- _ JLMm

tf- F

t' = t


Expanding the derivatives gives

d(V S) = V dS + S dVand

d(V X) = V dX + X dVSubstituting, the dimensionless balances now become:

VolumedV'

BiomassdX'dt'

SubstratedS'dt = ( l - S ) D - j i X

The Monod equation is:

KS +S

In Fig. 2 a computer solution shows the approach to and attainment of thequasi-steady state of the dimensionless fed-batch model.


Quasi- steady

Figure 2. Dynamic simulation results for a fed batch culture.

Programs

The program VARVOL is based on the model equations with normaldimensions. The program VARVOLD is based on the dimensionless equationsas derived above. Both are on the CD-ROM.

Nomenclature

Symbols

DFKSr

SV

Dilution rateFlow rateSaturation constantReaction rateSubstrate concentrationReactor volume

1/hm3/hkg/m3

kg/m3 hkg/m3

m3


XY

Biomass concentrationYield coefficientSpecific growth rate

kg/m3kg/kg1/h

Indices

0fmSX

Refers to feed and initial valuesRefers to finalRefers to maximumRefers to substrateRefers to biomassRefers to dimensionless variables

Dimensionless Variables

S'VX1t'

Dimensionless flow rateDimensionless saturation constantDimensionless substrate concentrationDimensionless volumeDimensionless biomass concentrationDimensionless timeDimensionless specific growth rate

Exercises


Results

During the quasi-steady state, \l becomes equal to D, and this requires that Smust decrease steadily in order to maintain the quasi-steady state as the volumeincreases (Fig. 3). Increasing flow rates from 0.01 to 1.0 causes a delay in theonset of linear growth and causes the final biomass levels to be higher (Fig. 4).

Run 1:105 steps in 0 seconds4.5

Figure 3. Fed batch concentration and growth rate profiles, showing quasi-steady state.

Run 7:105 steps in 0 seconds5

2.5 C/>

10


Figure 4. Influence of flow rate on growth. Flow rate increase from 0.01 to 1.0.

References

Dunn, I.J., and Mor, J.R. (1975) Variable Volume Continuous Cultivation.Biotechnol. Bioeng. 17, 1805.

Keller, R., and Dunn, I.J. (1978) Computer Simulation of the BiomassProduction Rate of Cyclic Fed Batch Continuous Culture. J. AppL Chem.Biotechnol. 28, 784.

8.3.2 Penicillin Fermentation Using ElementalBalancing (PENFERM)

System

This example is based on the publication of Heijnen et al. (1979), andencompasses all the principles of elemental balancing, rate equationformulation, material balancing and computer simulation. A fed batch processfor the production of penicillin as shown in Fig. 1 is considered withcontinuous feeding of glucose. Ammonia, sulfuric acid and o-phosphoric acidare the sources of nitrogen, sulfur and phosphorous respectively. O-phosphoric acid is sufficiently present in the medium and is not fed. Oxygenand carbon dioxide are exchanged by the organism. The product of thehydrolysis of penicillin, penicilloic acid, is also considered, thus taking the slowhydrolysis of penicillin-G during the process into account.


Glucose Carbon dioxide

Oxygen

PrecursorPhenylacetic acid

Sulfuric acid

Ammonia

Figure 1. Streams in and out of the penicillin fed batch reactor.

Table 1. lists the components and their conversion rate designation.


Table 1. Component properties and rateCompound

GlucoseMycelium

PenicillinPenicilloic acidOxygenCarbon DioxideAmmoniaSulfuric AcidPhosphoric AcidPhenylacetic AcidWater

Chemical formula

C6H1206CHi.64Oo.52No. 16So.0046P


For hydrogen

12 RI + 1.64 R2 + 18 R3 + 20 R4 + 3 R8 + 2 R9 + 3 RIO + 8 Rn + 2 R12 = 0

For phosphorus

0.0054 R2 + RIO = 0

A steady state enthalpy balance gives the following

- 303 RI - 28.1 R2 - 115 R3 - 183 R4 - 94 R6 - 19 R8 - 194 R9 -

-319Rio-69Rn -68Ri2 + rH = 0

where TH is the rate of heat of production (kcal/h).A total of 12 unknowns (Ri through R6, Rg through Ri2 and TH) are involved

with a total of 7 equations (6 elemental balances and one heat balance). Thefive additional equations are provided by five reaction kinetic relationships.The remaining rates can be expressed in terms of these basic kinetic equations.

From the carbon balance

- R6 = 6 RI + R2 + 16 R3 + 16 R4 + 8 RH

From the nitrogen balance

- R 8 = 0.16R2 + 2R3 + 2R4

From the sulfur balance

- R9 = 0.0046 R2 + R3 + R4

From the phosphor balance

-Rio = 0.0054 R2

From the hydrogen and nitrogen balances

- R5 = -6 RI - 1.044 R2 - 18.5 R3 - 18.5 R4 - 9 Rn

From the enthalpy balance

rH = - 669 RI - 110.1 R2 - 1961 R3 - 1961 R4 - 955 RH


To complete the model, equations for glucose uptake rate (-Ri), biomassformation rate (R2), rate of penicillin formation (Rs), precursor consumptionrate (-Rn), and rate of penicillin hydrolysis (R4) must be known. Note that thereaction rates are defined with respect to total broth weight, since the process isthe fed-batch type and broth weight is variable with respect to time.

b) Formulation of the Kinetic Equations

Substrate (Glucose) Uptake Rate:A MONOD type equation for the uptake of sugar by P. Chrysogenum is used.

- Q l C i M 2

Biomass Formation Rate:A linear relationship between the glucose consumption rate and growth rate ofbiomass is assumed. Hence,

1- Rl = y^ ^2 + m M2

where Y2 is the maximum growth yield and m is the maintenance rate factor(mol glucose/mol mycelial biomass h).

Some sugar is used in the formation of the product. Hence,

- Rl = Yj R2 + m M2 + YJ (R3 + R*)where 3 is the conversion yield for glucose to penicillin (mol penicillin/molglucose).

The total rate of biomass formation equals the net rate of formation,corrected for the amount transformed to penicilloic acid. Therefore,

R2 = -Y 2Ri - Y 2 mM 2 - yf (Rs + 4)

Precursor Conversion RateIt is assumed that the precursor is only used for penicillin synthesis. Thus

-Rll = R3 + R4

where - RH is the precursor consumption rate.


Rate of Penicillin SynthesisThe specific rate of penicillin synthesis is assumed not to be a function ofspecific growth rate. So that

R3 = Q3 M2 - R4where Q3 is the maximum specific rate of penicillin synthesis (mol/mol h),

Equation for the Rate of Penicillin HydrolysisThe hydrolysis of penicillin takes place by a first-order reaction.

R4 = K3 M3

c) Balance Equations

Total Mass BalanceThe individual feed rates of glucose, sulfuric acid and ammonia are adjusted toequal their molar consumption rates. Water lost by evaporation is neglected.The change in mass due to gas uptake and production is neglected. The massflow rates are calculated from the molecular weights, the uptake rates and themass ratio compositions.

Feed rate of glucose stream (kg/h)180 F

Fl = F500 = 2.78where F = mol glucose /h.Feed rate of NH3 stream (kg/h)

F8 = R825Q = T4JT

Feed rate of [2804 stream (kg/h)18 R9

F9 - R95QO ~ 2.55

The total mass in reactor G (kg/h) changes with time according to

dG F R9 Rg"dT = Tn + 235" + TTTT

Component BalancesExpressed in mol/h the dynamic balances are,


Glucose

~3T = Rl + FBiomass

dM2~ar = R2

PenicillindM3 .

Penicilloic aciddM4

The concentrations in mol/kg are as follows:

M2TrM3

M4c4 =

where the masses MI, M2, M3 and M4 are in mol units.

d) Metabolism Relations

The various metabolic relationships are given from

Specific growth rate for cellsR2

* = MlRespiration quotient

ReRQ = R7

Oxygen uptake rateOUR = -R5

CO2 production rateCPR = R6

Fraction of N2 in mycelium


R2f2 = 0.16 R|N2 fraction in penicillin

f3 = 1-F2

Fraction of sulfur used for myceliumR2f4 = 0.046 R|

Sulfur fraction used for penicillin

f5 = 1 - F4

Fraction of glucose for cell growth=

R2

Fraction of glucose for penicillinR3 + R4S3 = - Y3 R!

Fraction of glucose for maintenanceM2g4 = -M R^-

Program

The Madonna program covers a fermentation time of 200 h starting from theinitial conditions of 5500 mol glucose, 4000 mol biomass, 0 mol penicillin and0.001 mol penicilloic acid in an initial broth weight of IxlO5 kg. The programis on the CD-ROM.

Nomenclature

Symbols

a, b Flow rate variables variousC Component concentration mol/kgCPR Carbon dioxide production rate mol/hF Feed rate kg/h


hflf5G82g3g4KlK3MmOURQRRQrqY

Fraction of nitrogen in myceliumNitrogen fraction in penicillin -Fraction of sulfur used for mycelium -Fraction of sulfur used for penicillin -Mass in reactor kgFraction of glucose for cell growthFraction of glucose for penicillinFraction glucose for maintenanceSaturation constant mol/kgHydrolysis rate constant 1/hMass of individual components molMaintenance rate factor mol/(mol h)Oxygen uptake rate mol/hMaximum specific rates mol/(mol h)Conversion mol/hRespiration quotient -Heat production rate kcal/hRespiratory quotient -Yield coefficient -Specific growth rate 1/h

Indices

012345689101112

initialglucosebiomasspenicillinpenicilloic acidoxygencarbon dioxideammoniasulfuric acidphosphoric acidphenylacetic acidwater

Exercises


Results

The results of Fig. 2 show the substrate MI to pass through a maximum, whilethe penicillin M2 develops linearly, for this constant feeding situation.Increasing the feeding linearly with time (F = 500 + 5* time) gave the results inFig. 3, where it is seen that maintenance accounts for about 70 % of glucoseconsumption at the end of the fermentation.

Run 1:215 steps in 0 seconds

0 20 40 60 80 100 120 140 160 180 200TIME

Figure 2. Penicillin fed batch fermentation with total masses of glucose (M]) and biomass(M2).


Run 1:215 steps in 0 seconds

8

0.9-,

0.8-

0.7-

0.6-

r-- 0.4-

0.3-

0.2-

0.1-

0-20 60 80 100

TIME120

Figure 3. Linear increase of feeding with time F = 500 + 5*T.

Reference

Heijnen, J., Roels, J. A., and Stouthamer, A.H. (1979). Application of BalancingMethods in Modeling the Penicillin Fermentation. Biotechnol. and Bioeng., 21,2175-2201.

8.3.3 Ethanol Fed Batch Diauxic Fermentation(ETHFERM)

System

Yeast exhibits diauxic behavior with respect to the glucose and ethanol in themedium as alternative substrates. In addition, the glucose effect, when glucoselevels are high, will cause fermentation, instead of respirative oxidation, to takeplace, such that the biomass yields are much reduced (Fig. 1). In this examplethe constant a designates the fraction of respiring biomass and (1 - a) thefraction of biomass that ferments. The rates of the process are controlled bythree enzymes.

8.3 Fed Batch Reactors 24 1

^^ C02 + X

Glucose^^ *^- Ethanol + X

Figure 1. Pathways of aerobic ethanol fermentation.

Model

The rates of the processes are as follows:

Respirative oxidation on glucose,

R, = Glu+KslFermentation to ethanol,

R2 = K2 (1 - a) XGlu + KS2Conversion of ethanol to biomass,

Enzyme activation for the transformation of ethanol to biomass is assumed toinvolve an initial concentration of starting enzyme EQ, which is converted to en-zyme 2 and which catalyzes growth on ethanol through an intermediateenzyme EI.

Thus, the production rate of enzyme EI is inhibited strongly by glucose,

R4 = - -rXEoKS4+Glu3

and the production rate of enzyme 2 controlling the conversion of biomass toethanol depends on EI,

R5 = K 5 XEi

The mass balances for the biomass, substrates and enzymes are those for a fedbatch with variable volume.


For the total mass balance with constant density,

dt

The component balances are written by separating the accumulation term,noting that

d(VC) _ VdC CdV _ VdC- - + - -

dt dt dt dt

Thus,

dt V

f

^o = _ E0Q

dt V

f -*-.-*

Program

Note that the program on the CD-ROM is formulated in terms of C-mol for thebiomass. This is defined as the formula weight written in terms of one C atom,thus for yeast CHL667Oo.5No.i67-


Nomenclature

Symbols

CEEtOUGluKQRVXYa

Component concentrationEnzyme concentrationEthanol concentrationSubstrate feed concentrationRate constantsFeed flow rateReaction rateReactor volumeBiomass concentrationYield coefficientFraction of respiring biomass

mol/m3mol/m3mol/m3mol/m3variousm3/hmol/m3 hm3

C-mol/m3mol/mol

Indices

012345

Refers to feedRefers to reaction 1Refers to reaction 2Refers to reaction 3Refers to reaction 4Refers to reaction 5

Exercises


Results

Seen in Fig. 3 are the simulation results giving the concentrations (glucose,ethanol and biomass) during the fed batch process. In Fig. 4 the maximum inethanol concentration as a function of feedrate is given from a Parameter Plot.

Run 1: 605 steps in 0.0167 seconds30

25

60

Figure 3. Batch yeast fermentation.


Run 2:12100 steps in 0.333 seconds30

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Figure 4. Influence of flowrate on the maximum ethanol concentration.

Reference

This example was contributed by C. Niklasson, Dept. of Chemical ReactionEngineering, Chalmers University of Technology, S - 41296 Goteborg,Sweden.

8.3.4 Repeated Fed Batch Culture (REPFED)System

A single cycle of a repeated fed batch fermentation is shown in Fig. 1. In thisoperation a substrate is added continuously to the reactor. After the tank is full,the culture is partially emptied, and the filling process is repeated to start thenext fed batch. The operating variables are initial volume, final volume,substrate feed concentration and flow rates of filling and emptying.


Figure 1. One cycle of a repeated fed batch.

Model

The equations are the same as given in the example FEDBAT (Section 8.1.3),where the balances for substrate and biomass are written in terms of masses,instead of concentrations. The only difference is that an outlet stream isconsidered here to empty the fermenter at the end of the production period.

Program

Since in a Madonna program, the initial conditions cannot be reset, an outletstream is added. The inlet and outlet streams are controlled by conditionalstatements as shown below. The full program is on the CD-ROM.

{Statements to switch the feed and emptying streams)Fin=if time> = 10 then Flin else 0 {batch start up}Flin= if time> = 33 then 0.5 else if time> = 32 then 0else if time> = 21 then 0.5 else if time> = 20 then 0else 0.5Fout= if time>=33 then 0 else if time>=32 then 5.39else if time> = 21 then 0 else if time> = 20 then 5.39else 0


Nomenclature

Symbols

D Dilution rateF Flow rateKl and K2 Product kinetic constantsKS Saturation constantP Product concentrationS Substrate concentrationX Biomass concentrationV Reactor volumeV0 Initial volume of liquidVX Biomass in reactorVS Substrate in reactorY Yield coefficient|i Specific growth rate

1/hm3/hvariouskg/m3

g/m3

kg/m3

kg/m3

m3

m3

kgkgkg/kg1/h

Indices

SX0 (zero)initialinout

Refers to substrateRefers to biomassRefers to initial and inlet valuesRefers to initial valuesRefers to inletRefers to exit

Exercises


Results

Shown below are results of a simulation with three filling cycles.

Run 1: 5004 steps in 0.15 seconds60 -|

50-

40-

30-

20-

10-

0-

A/ I

/ I/ I

f I/ I

/ I\l

0 5 10 15 20

.-80

pr^ Ti /-70| "vsi| , go

/"( / .50

' J -40

/ \

/ i /-V/ I / ^ \X V ,* \ -10

- - * ' " - ' L -Q25 30 35 40 45 50

TIME

Figure 2. Masses of substrate and biomass during filling and emptying cycles.


Run 1: 5004 steps in 0.35 seconds10

Figure 3. Concentrations of product, substrate and biomass during filling and emptyingcycles. The volume is also shown.

References

Dunn, I.J., Mor, J.R., (1975) Variable Volume Continuous CultivationBiotechnol. Bioeng. 17, 1805.

Keller, R., Dunn, I.J. (1978) Computer Simulation of the Biomass ProductionRate of Cyclic Fed Batch Continuous Culture J. AppL Chem. Biotechnol. 28,784.

8.3.5 Repeated Medium Replacement Culture(REPLCUL)

System

Slow-growing animal and plant cell cultures require certain growth factors andhormones which begin to limit growth after a period of time. To avoid this,part of the entire culture is replaced with fresh medium. A single cycle ofrepeated replacement culture is shown in Fig. 1. In this procedure part of themedium volume (with cells) is removed after a certain replacement time andreplaced with fresh medium. Each cycle operates as a constant volume batch in


which the concentration of substrate decreases, while that of biomass increases.The operating variables are replacement volume, replacement time, andsubstrate concentration in the replacement medium. The initial conditions foreach cycle are determined by the final values in the previous cycle and thereplacement volume and concentration.

Replacement

Final Conditions VX VS

Initial Conditions

Figure 1. One cycle for medium replacement culture.

Model

The equations are those of batch culture, where for convenience the totalmasses are used.

dVS"dT = r s Vdvx

Monod kinetics is used.The effective starting conditions for each batch can be calculated using the

final conditions of the previous cycle from the volume replaced, VR? and thetotal volume, V, by the equations,

* VR

f =

VX = (1 - ) VXF


VS= ( l - f )VS F +V R S 0where f is the volume fraction replaced.

Program

The program as shown on the CD-ROM makes use of the PULSE function tovary the biomass and substrate concentrations corresponding to thereplacement of a fraction F of the culture medium. The time for each batch isthe value of INTERVAL.

Nomenclature

Symbols

DfKSsXVvovxvsVRY

Dilution rateFraction of volume replacedSaturation constantSubstrate concentrationBiomass concentrationReactor volumeInitial volume of liquidBiomass in reactorSubstrate in reactorVolume replacedYield coefficientSpecific growth rate

1/h

g/m3

g/m3g/m3m3

m3

kgkgm3

1/h

Indices

FSX0

Refers to final values at end of the cycleRefers to substrateRefers to biomassRefers to initial and inlet values


Exercises

Results

Fig. 2 shows how the biomass increases, until after six cycles the time profilesbecome almost identical.

TIME= 19.29 X = 1.26

10 20 30 40 50 60 70TIME

90 100

Figure 2. Oscillations of biomass and substrate concentrations with replacement cycles forInterval 10 and F=0.8


8.3.6 Penicillin Production in a Fed BatchFermenter (PENOXY)

A fed batch process is considered for the production of penicillin, as describedby Muttzall (1), The original model was altered to include oxygen transfer andthe influence of oxygen on the growth kinetics.

Figure I . Fed batch reactor showing nomenclature.

Model

As explained in the example FEDBAT the balances are:

Total mass

Biomass:

Substrate:

Product:

dt

d(MassX)dt

d(MassS)dt

= Vr-X

= FS f+Vr s


d(MassP) _ .= V fpdt p

Dissolved oxygen, neglecting the content of the inlet stream is calculated from

d(MassO)dt = KLa*(Osat-0) + Vr0

The influence of biomass concentration on the oxygen transfer isapproximated here by

K X +X

The concentrations are calculated from

_MassX MassS MassP MassO1\. """""""""""""^ , J , L , U

V V V V

The growth kinetics take into account the oxygen influence

o

The substrate uptake kinetics includes that amount used for growth, for productand for maintenance

J*o ~ Jft o _/V^ V V" ^YXS YPS

Product production involves two terms whose constants are turned on and offaccording to the value of |ii, as seen in the program.

Oxygen uptake includes growth and maintenance

=-TTYxo


Program

The program is on the CD-ROM.

Nomenclature

Symbols

FKLaKoKSKXMassmoms

OsatSfV'maxYPSYXOYXSH-max

Feed flowrateOxygen transfer coeff.Monod constant for oxygenMonod constant for glucoseConstant for biomass effect onComponent massMaintenance coeff. for oxygenMaintenance coeff. for glucoseSaturation for oxygenFeed cone, of glucoseVolumeMaximum volumeYield product to substrateYield biomass to oxygenYield biomass to substratemax.specific growth rate

m3/h1/hkg/m3

kg/m3

kg/m3kgkg O/kg X hkg S/kg X hkg/m3

kg/m3

m3

m3

kg/kgkg/kgkg/kg1/h

Exercises


II!

References

K. Mutzall, "Modellierung von Bioprozesses", Behr's Verlag, 1994.

Program and model developed by Reto Mueller, ETH Zurich.

Results

Run 1: 2023 steps in 0.117 seconds0.008

20 40 60 80 100 120 140 160 180 200

Figure 2. Dynamics of the fed batch reactor.

8.4 Continuous Reactors 257

120

100

80

- 60

40

20

Run 3: 2021 steps in 0.15 seconds-0.008

! Ii I: I

L i II

0.007

0.006

0.005

0.004 O

-0.003

-0.002

-0.001

00 20 40 60 80 100 120 140 160 180 200

TIME

Figure 3. Influence of initial KLa value from 100 to 160 h"^ on the S and O profiles.

8.4 Continuous Reactors

8.4.1 Steady-State Chemostat (CHEMOSTA)

System

The steady state operation of a continuous fermentation having constantvolume, constant flow rate and sterile feed is considered here (Fig. 1).

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