Theory and Simulation of Selfcycling Fermentation a Population Balance Approach

7/24/2019 Theory and Simulation of Selfcycling Fermentation a Population Balance Approach

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Theory and simulation self cycling fermentation

a population balance approach

by

Francois Godin

Department

Chemical

Engineering

McGill

University

Montreal

A thesis submitted

ta

the Faculty

Graduate Studies

and

Research

in

partial

fulfillment

the requirements

the

Degree ofMaster

Engineering

McGill

University

Francois Godin 997

August

997


4/136

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0 612 44012 5

Cana


5/136

STR CT

Self cycling fermentation SCf

is a technique in

which

sequential batch

fermentations

are performed using a

computerized

feedback control scheme. In

this

method half

the reactor

volume is

periodically removed and replaced by

fresh

medium.

This results in very stable and repeatable growth cycles and synchronized

cell

cultures.

In

this wode two

segregated

microbial

population

balance models

are

developed

and

used

to simulate

SCF.

Cell

age

and

cell mass distnDution models

are bath

used

to

study the behavior microorganisms in various systems. One example is the study

autonomous oscillations and partial synchronization in cultures of

accharomyces

cerevisiae However when the cell age

and

cell mass models are compared for the

modeDg

cell synchrony

in

SCF two contrasting

population profiles

arise from the

simulations.

The SCF

technique with

its existing data can be

used

as a powerful tool

to

test and

validate

models

ofmicrobial systems. When

used

ta simulate SCF the

cell

age model was

able

to predict cell

synchrony however

the ceU number profile obtained was remarkably

different than

that observed in experiments. The

cell mass model as

proposed

by

Eakman

al was

able to capture the dissolved oxygen concentration

the

limiting substrate

concentration and the biomass concentration

but

was not able ta describe the cell number

profile or the feature cell synchrony. By

introducing

a feedback mechanism

between

the critical division mass and the limiting substrate

ceU

synchrony wu achieved and the

experimental

cell profile

was

captured.


6/136

RSUM

Le procd de fermentation auto-cyclique

AC

est une mthode de fermentation

squentielle discontinue qui est obtenue par

l entremise d un

contrle raction

informatis.

Pour cette

mthode,

la

moiti

du

volume

du racteur est priodiquement

remplac par

du

bouillon de culture

frais.

On obtient des cycles de croissances stables

et

rptitifs ainsi que ds cultures cellulaires

synchronises.

Pour ce mmoire, deux modles cellulaires de population sgrgue sont

dveloppes et utilises pour simuler le F

AC.

Les modles de distribution cellulaires

d ge

et de masse sont tous deux utiliss pour tudier le comportement des microorganismes

dans diffrents

systmes. Par

exemple, les

oscillations autonomes spontanes et

les

synchronisations panielles dans les cultures

de

accharomycescerev siae ont dj t

tudies. Cependant quand ces modles sont

utiliss

pour

simuler

le FAC, ils produisent

des rsultats diffrents.

La

technique FAC

avec

les donnes existantes peut tre utilise comme un outil

puissant pour tester et valider les modles de

systme

microbiologique. Lorsqututilis

pour simuler

le

F

AC, le

modle dtge cellulaire pouvait prdire

la

synchronisation

cellulaire. Par contre le nombre de cellules en fonction du temps qui est obtenu est trs

diffrent de celui observ dans les

exprimentations. Le modle

de

masse

cellulaire

comme

propos par

Eakman

pouvait captur

la

concentration d oxygne dissoute la

concentration de substrat limit, et la concentration de biomasse

sans

pouvoir dcrire le

profile numrique ou la synchronisation cellulaire. En introduisant une relation entre la

masse critique de division

et

la concentration de substrat

limit,

la synchronisation et

le

profil cellulaire ont

obtenus.


7/136

ACKNOWLEDGMENTS

This work was made possible through gr nts from the

Natural

Sciences and

Engineering

Research

Council

Canada NSERC and the

Fonds

pour

la

formation

de

chercheurs

et

l aide

la

recherche FeAR .

Additional thanks are extended

to

the

entire

Falcon

research

group, for

their

ftequent encouragement and good cheer,

and

to Michael Silverberg

for

his

help.

Of

course, ny thanks go ta my parents for their unconditionallove and for putting up with

ail my erratic moods. Extreme gratitude is also

felt

towards

Isabelle

Joubert,

for

her

immense

support and patience throughout the duration this research.

Finally, l wou

Id

especially like to

thank

Dr. D.G. Cooper and Dr. AD. Rey for

their guidance, support

and fiiendship

throughout the

duration

of my

studies.

ili


8/136

TABLE

OF

CONTENTS

INftODUgION M

1 1

SELF eYCLlNG

F ERMEN rATlON

1

1 2 CLASSIFICATIONOF

CELL POPUL A TION MODELS

Il

1 3 SEOREGATED

SnuCJlJRED

MODELS 12

1 4

PREVIOUS

MODELmOWou 13

1 S

VALIOATION CRITERIA FOR

n

SCF MaDEL 5

1 6 OBJECI1VES 16

1 7 THEsIS

OROANIZATION

17

H PnR CELL AGE

MO

DEL

18

2 1

lNTRODuC110N 18

2 2

FOWULATION OF

nIE

CEU AOE MODEL

18

2 3 A CELL AO E

Mo

DEL FOR lNDucnON SYNCHRONY 21

2 4

SOLUTION SCHEME FOR THE CELL

AO E

DlSTRlBurION MODEL:

MEnlOD OF

CHARACTERISTICS

27

2 5 REsULTS AND

DISCUSSION 33

2 6

CONCLUSIONS

38

CllA PTER 3 CELL

MASS

MODEL 40

3 1 NTRoouCI10N 40

3 2 FORMULATION

OF

THE

CEIJ

MAss MODEL 4

3 3 S LlTl1 N

SCHEME 57

3 3 1 Galerkin Fin te ElementMethod 57

3 3 2

Prediclor Coweclor

Euler

Scheme 61

3 4

REsULTS

AND DISCUSSION

63

3 4 1 Verification the Solution 63

IV


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3 4 2 Application a the Ce Mass Population Balance Madel

ta

the SCFProblem

3 4 3 Dispersion Effects within the Cell

Moss

Population Ba ance Model 84

3 S MODIFIC TION

TO

CELL

MAss

POPUL TION

B L NCE MaDEL 87

3 6

R sutTS

NDDISCUSSION fm MODIFD CELL

MAss MaDEL

9

3 7 CONCLUSIONS 97

CH PTER CONCLUSIONS 99

R nUN S

2

v


10/136

LIST

OF T LES

TABLE

PARAMETER VALUES

USED

FOR

THE SIMULATIONS OF

lIE

CHEMOSTAT

AND

T

OnaS

TABLE 2

PARAMETER

VALUES ESTIMATED

FROM

THE WORK OF WINCURE 79

vi


11/136

LIST O FIGURES

FIGURE 1 CONCENTRATION

PROFILES

FORCINEI OBACTER CALCOACEI1CUS RAG l

OROWN

usmo

IllE

SCF lEClINIQlJE 2

FIGURE 2 1NTRAcYCLE

CELL

COUNT

PROFILE

FOR

ONE

CYCLE OF PSEUDOMONAS PUTITA

GROWN USING

nm

SCF TECIOOQtJE

5

FIGURE 3

1NrERCYCLE PROFll E

OF DO AND

CELL

NUMBER

FOR

ONE CYCLE OF

ACl \ ETOBACTER

CALCOAC 77cusRAG l

GROWN USINO

1H E

SCF

TECHNIQUE

6

FIGURE

4 1NTRACYCLE DO AND CELL NUMBER PROFILES

FOR

ONE CYCLE OF

PSEUOO \fO \ AS PUT1TA

GROWN

USING

THE SCF TECHNIQUE 7

FIGURE S 1NrERCYCLE PROFILE

OF

DO

AND

CELL NUMBER

OF

CANDIDA UPOLYTI

GROWN

USINO n

SCF l ECfOOQUE

FIGURE 6 1NTERCYCLE PROFILE

OF

DO AND CELL NUMBER OF CANDIDA UPOLrrI GROWN

UsrnGnm SCF lECflNIQUE

9

FIGURE 7

1NTERcYCLE

PROFILE OF DO AND CELL NUMBER OF CANDIDA UPOLYTI

GROWN USING nI

SCF TECHNIQUE

10

FIGURE

8

CHANGE

IN

DISTANCE

BETWEEN

IWO

CELL

UNES FOR A VARVlNG DMSION AGE .

4

FIGURE

9

CELL

UNES REPRESENTING SlEADY CYCLE SOLurION

Ta

THE

POPULATION

BALANCE EQUATION

WHEN A PERIODIC SHIFT

IN

nmDMSION AGE

IS

IMPOSED ..... 26

vii


12/136

FIGURE 10. CHARACTERISTIC

CURVES

SPANNINOTIIE

AGE TIME

PLANE 30

FIGURE

Il

SIMULATION OF

EVOLurION OF

A

SYNCHRONIZED CULTIJRE IN

SCF

3S

FIGURE 2 NITIALRECfANGULAR CELL AGE DISTRIBunON AND nIE

FINAL

DISTRIBtmON

OF

SYNCHRONIZED

CELL POPULATION 36

FIGURE 13. TRANsIENT PROBABILITY OF CELL

DMsrON

ft

m

I VERSUS CELL MASS

M.

EFFECT OF Cs 50

FIGURE 14. TRANSIENTPROBABILITYOFCEllDMSION f m,C

s

) VERSUS CELL MASS M.

EFFECT OF 51

FIGURE

15.

DISTRIBunON OF DAUOHTER.

CELL MASS FOR TWO DIFFERENT

g

VALUES 53

FIGURE

16.

DISTRIBUTION OF DAUOHrER.

CELL MASS AS ONEN

BY SUBRAMANIAN ET

AL 5

FIGURE 17. DO LIMlTING SUBSTRATE BIOMASS AND CELL

NUMBER

PROFILES FOR A

CIMOSTAT

USINO nm INITIAL CONDmONS GIVEN BY EQUATION

77 66

FIGURE

8

CELL

MASS DISTRIBunONS FOR A CflEMOSTAT USING

mE

INITIAL CONDmONS

GIVENSY

EQUATION

77 u

67

FIOURE

19. DO, LIMITING SUBSTRATE BIOMASS AND CEIL

NUMBER PROFILES

FOR A

CHEMOSTAT USINO

1BE

INITIAL

CONDmONS

GIVEN

DY

EQUATION 78 68

viii


13/136

FIGURE 20

CELL

MASS

DIS1RIBUTIONS

FOR A

CHEMOSTAT USINO

mE INITIAL CONDmONS

GIvm.r

BY EQUATION 78) 69

FIGURE 21 DO LIMlTING SUBSTRATE

BIOMASS

AND CELL

NUMBER

PROFILES FOR A

CHEMOSTAT USING

m INITIAL

CONDmONS

GIVEN

BY EQUATION

79) 70

FIGURE 22 CELL MASS DISTRIBUTIONS FOR A CHEMOSTAT USING mE INITIAL CONDmONS

GIVEN BY EQUATION 79)

71

FIGURE 23 SOLUTION OF

mE CELL MASS

POPULATION MODEL

AS

GIVEN BY

n

EQUATIONS

l olREFERmeE

[15]

73

FlOURE

24. RESULTS FROM nm MODIFIED

POPULATION

BALANCE EQUATION FOR THE DO,

LIMlTING

SUBSTRATE

BIOMASS

AND

CELL

NUMBERPROFILES

OF A BATCH REACTOR

USINO

nIE

INITIAL CONDmONS ONEN BY EQUATION 77) 76

FIGURE

25. RESULTS FOR nm MODIFIED POPULATION BALANCE

EQUATION FOR

nm CELL

MASS

DIS1 RIBunONS OF

A

BATCH REACTOR

77

FIGURE

26

BIOMASS, UMlTlNG

SOBSTRATE AND DO CONCENTRATION PROFILES FOR nIE

SCF SYSTEM 80

FIGURE 27 CELL

NUMBER

AND

DO

PROFILES FOR mE

SCF

SYSTEM....................... 81

FIGURE

28

CELL

MASS DISTRIBUllONFOR mE SCF SYSTEM 82

FIGURE 29.

CELL

NUMBER BIOMASS

AND aLMASS

DISTRIBtmON

PROFILES

FOR A

CHEMOSTAT REACTOR USING

A

NARROW INITIAL

CELL

MASS

DIS1RIBtmON

86

ix


14/136

FIGURE 30 CRrnCAL DMSIONMASS McAS AFUNeTION OF

LIMlTING

SUBSmATE Cs

91

FIGURE 31.

SIMULATION

RESULTS

OF CELL

NUMBER

AS A

FUNCTION

OF

TIME

FOR

mE SCF

SYSTEM WlTIIEllMASS MSIONCON IROL

9

FIGURE 32 SIMULATION RESULTS

FOR

1HECELL NUMBER PROFILE OF

mE S F

SYSTEM

COMPARED Ta nm PROFILE

OBTAINED WHENnIE DMSIONMASS

IS fJ L

CONSTANT 94

FIGURE

33

EFFEcr

OF

nIE

V

ARYINGCRITICAL

DMSIONMASS Mc ON

nm

CELL MASS

DIS I RIB1ITION 96

x


15/136

hapter

Introduction

Self ycling Fermentation

The self-cycling fermentation SCF) process was first developed

by

Sheppard and

Cooper [41,42] as an

improvement

of continuous phasing [11,12,13]. SCF

is

also

described

in

later references [3,4,26,31,40,50,52,54]. This fermentation process

is

based

on monitoring a growth associated parameter

while

growing microorganisms in a

bioreactor. As the limiting nutrient approaches depletion, a decrease in the

metabolic

activity of an organism

is

reflected

in

the

measured

parameter. At this time, half

of

the

broth is

removed

from the reactor harvesting) and

is

replaced with fresh medium

dosing . The action of harvesting

and dosing is

known as cycling, while the period

between successive dosing steps

is

known as a

cycle.

After a few transient cycles,

the

system settles into a stable periodic state. Figure 1

demonstrates typical biomass,

limiting

substrate and

dissolved

oxygen

DO)

concentration

profiles

as

a

function of

time

for

the SCF process

for severa

cycles

of

cinetobacter

ca coaceticus

RAG-I

[53].

In this example, ethanol was the limiting nutrient.

The

biomass concentration profile top graph

is

seen ta increase exponentially throughout the

cycle while the limiting substrate concentration

profile

middle graph decreases

exponentially ta a value below

detectable

Iimits The

growth associated parameter

monitored du ring

this

fermentation was the

DO

concentration bottom graph . Air was

supplied to the system at a constant

rate.

For a

given cycle,

the

concentration initiaUy

decreased exponentially. As the limiting nutrient approached depletion

leveI,

a decrease

1


16/136

7r

8

3

2

1

J

J

0 . 5 . .

o

0 3

1

o

1

1

JL

_

1

~ . . . . .

8

4

2

o o

1

TIIIIC

2

es

9

_M

JI:

85

8

55

S O . f ~ . . . . . _ f

o

igure

1 Concentration profdes for

AC;lIsD6aeter clI1cDtlCSclIS RAG 11I 0wn usinl

the SCF technique [53]

2


17/136

the

metabolic aetivity caused the oconcentration to increase Thus

a

minimum

in

the

growth associated parameter

was observed

toward the

end of

the

cycle and once

detected the system

was allowed

to

cycle

The

biomass

profile

also

demonstrates

another

important feature

of

SCF

The

biomass concentration

prior

to harvesting was found to

be the

same

for

all cycles Snce

half the

biomass was removed during

harvesting

and was

recovered

by the end

of the

cycle the length of the

cycle the

cycle

time

must

be

equal to the

doubling

time of the

microorganism Thus the

cells

double exaetly once during

cach

cycle

One

advantage

that

this

type

of

fermentation

has

over

conventional batch

fermentations is that no

lag

phase or stationary growth

phase

are observed

during

a

cycle

These

periods of slow

growth are

common in batch fermentations

[1]

SCF

aIso

has

the

advantage

of

not having long down tintes

for

cleaning

sterilization

etc which are

inevitable between batch

fermentations Thus the microorganisms

can

grow at

the

maximum

growth rate for

prolonged

periods of

time Maximum

growth

rates

can

also

be

achieved in chemostats However in these

systems

the limiting substrate is ooly

completely consumed

at

very low dilution

rates

for which the

growth

rate is Iow [1] SCF

has the advantage of supporting

i

growth rates for extended periods of time

with

the

complete

utilization

of the limiting substrate This

fermentation method

has

been

used

for

both the

biodegradation of various industrial pollutants [4 26 40] and for the enhanced

production ofvarious biological products

[31 41 42 52 54]

Another important feature of

SCF is

the

synchronization

of the microorganisms

in

the system Figure 2 depiets a

typical cell

number profile in s for one cycle

[40] while

3


18/136

Figures 3 and 4 show the

DO

profile

and

the

cen

number profile

as

a function time for

two

difFerent

mieroorganisms [3,26]. These

figures

show that the total

cell

number

in

the

reactor

do

es not increase

exponentially

throughout the cycles, but rather increases

a

step-like

fasbion

towards

the end

the

cycle.

This

synchranization

in

cellular

division

suggests a temporal

alignment

the microorganisms cellular cycle. The location

the

lep increase during

the

cycle also corresponds to

where the

minimum in

the

DO profile

occurs. This is evident in

Figure

3, and

in

Figures

S,

6 and 7 where the

SCF

technique

was

used

to grow

Candida lipolyti [5 ]

During these runs, the fermentation

was

allowed

ta continue

beyond the

minimum

DO which

corresponded to the exhaustion

the

limiting

nutrient

CNHthSO... The

system

was allowed

to continue without

cycling,

until

a second nutnent (glucose) was

cornpletely

cansumed, at which point the DO

concentration

was

seen ta

rise rapidly. The system was only

allowed to cycle upon

this

sharp increase in

DO.

AlIowing the cycle to continue after

CNHt)2S0

had being

exhausted

was

termed

extended nutrient starvation. Cen synchrony

was

still rnaintained

using tbis mode

cycling,

with the step increase in

cell

number

corresponding

to

the

depletion

the

limiting

nutrient.

No

other data exists on the

cell

number profile

during

extended nutnent starvation

using SCF. However

data obtained

by

Dawson

[11 12 13]

doing work on the synchronization organisms using continuous phasing, showed that

cellular

division also

occurred

upon exhaustion

the

limiting

substrate.

The ability

this

method

to

generate

and maintain

synchronized

cell

populations

is very useful for the

study

cell cycle

events. The synchronization the

microorganisms

t

result

an amplification

cellular

events. Since

a large fraction

4


19/136

~ .

2

1 2

End o Cycle

0

1 ~

o

Figure

2

Intracyde

ceU

count

prorale

for one

cyde

o

Pse domolltB

p tittlgrown

us nl

the SCF tecbnique [40]


20/136

to

10

End

of

Cycle

70

c S

l

o t

5

J

J

40

Q

3

2

tO

u

2 3

2.t

1

1 1

t.7

a

1 5

ri

1.3

.a

:1

t t

Z

o

0.7

5

tO

2

3

40

eo

10

70

1 90

n. .IIIUI.

Figure 3 Intercycle prorlle of DO ADd ceU Dumber

fo r

one cycle ofAcinetobacter

c lco ceticl s RAG l growa UliDI the SCF technique

[3]

6


21/136

Sl

ao

70

8

JI 5

J 40

I

l

Q

10

0

U

1 5

_ 1 4

E

i 1 3

t 2

1

1 t

1

t

10 9

i

o a

U

0 7

11

0

1

2

30

40 50

10

70

10

ThIl

Figure

4

Intracyde DO and ceU number pro il for one

cycle of

selldolll ftllS

pllt t grown usinl the SCF technique [26]

7


22/136

85 r

8

65

60

400

300

200

nme mlnutal

1

~ .

17

f

16

i

15

u

;:; 14

.

c

13

i

12

5

11

u

i

10

u 9

8..10.----------------

o

Figure

5

Intercyde prorde

o

n

ceU

Dumber

o

c lldidt

lipolyti

growD UliDI

the SCF technique [52

8


23/136

1

SIS

c

80

1

15

la

1

~

Q

70

20

18

1e

12

u

10

Il

w

T w

1

2

3

4

lOO eoo

700

TI

Cftlln

Figure 6 Intercycle profde of

DO

and ceU number of ndid

l polyt tllrown

usinl


9


24/136

95

90

c

85

JI 80

j

f

75

7

13

2

1

a

10

9

1

1

8

7

8

.

.

.

.

.

0

2

3

00

5

lm

(1IIlnutal

Figure 7 ntercyde profile of DO nd eeU

Bumber

of lldid

lipolyti l

grown usinl



25/136

the

cell

population

go

through cell

cycle

events simultaneously, the entice system can be

studied as a representation single cell aetivity. The faet that the SCF cycles are very

repeatable,

and

that the

duration

these

cycles correspond

exactly ta the

micro

rgani

sm

s

doubling

time, aIso

facilitate these

types

of studies.

2 Classification Cell Population Modela

8iological systems, and their interactions w t their environment,

are

very

complex

and the models used ta

study

them generally do not attempt to capture all minor details.

For

tms

reason,

scientists

and engineers

have

developed

models

that

usually

deal with

specifie and fundamental

aspects

of

biological

systems.

Engineers

have derived a host of

mathematieal

models

with the objective

controlling

and optimizing biological processes.

This

section deals with

the

classification ofthese

classical models

as proposed

by

Tsuchiya

l [45]

and

discussed

by

Ramkrishna

[36,37] and Balley and

Dllis [1].

A mathematical

model

of a biological system can be c1assified as

segregated

or

non-segregated. Segregated models

recognize

the faet that a population

is

composed of

distinct

individuals.

Non-segregated distributed)

microbial models,

such as Monod s

models

[1], do

not recognize

individuals

ceUs

but lump

them

into an

average

biophase

such

as

dry biomass

concentration.

Microbial

models

can

also

be charaeterized as

struetured or non-struetured.

Struetured models

take into account the state of the

microorganisms.

In the

case of segregated struetured

models,

the population

is treated as

individual

cells

which

can

be

dift erentiated from

one

another. This

is

accomplished by

specifying the state of the microorganism. For

example

the

chemical composition

the

ceU,

the

cell

age, the eeU mass, the morphology or

ceU size,

or a combination

indices


26/136

can he used to indicate the physiologica1 state

the

microorganism. In the

case

a

distributed model this would

imply

the specification the

state

the

lumped biophase.

Non structured

models cannot

differentiate

between individual ceUs,

or between

different

states

the

lumped biophase. Finally, microbial models can be classified as

detenninistic

or stochastic. Although cellular

division

and birth processes are thought ta

be

probabilistic [45], for large

cell

populations, these processes can be readily

described

in

terms deterministic funetions [1,36].

The

population

balance models

dealt with tbis

study are

segregated structured

microbial models.

1 3 Segregated. Structured Modela

Segregated, struetured

microbial

models treat

cellular

populations as

distinct

individuals which can be

differentiated

trom one another. This differentiation between

organisms can be charaeterized

by a

number different indices physiological state.

Ramkrishna [36,37] and Fredrickson

l [18] bath

discuss

the mathematica1 framework

for a

general

population

balance

model. They

discuss

the case when

an

arbitrary number

variables are

used

ta describe the state

the organisms, and

the state

the growth

environment.

However, from

an experimental

point

of view,

monitoring

a large number of

physiological

indices

at once

can

praye ta

be impraetical.

Rey

and

Mackey [8 38 39]

have worked

with a population balance model

where

age

and cell maturation were considered. The proliferation

stage was considered to

be

composed

four distinct major phases

Go,

G

I

S and

Ch . This

description cellular

proliferation lead

ta

the formulation

a

differential, delay equation. Rey and Mackey

12


27/136

studied the rich

dynamic

behavior exhibited by the equation. This model

cell replieation

is weil suited for the eucaryotic cell cycle which

can

be

described

terms discrete

sequential

events.

In

the

case

the

microbial cell cycle

there

is

much less

differentiation between

the

various

parts

the

cycle. For

example DNA synthesis can

oecur throughout

most

the cycle and proceeds in parallel

wit

other growth processes such

as

protein

and

RNA

synthesis [1]. This is in contrast with the eucaryotic ceU cycle were

DNA

synthesis occurs

only

the S phase the

cycle.

Microbial

cultures are often

mode1ed

using

a

single index

physiological state.

In

this worle a cell age

model and

a ceU

mass

model

SCF will be

developed

and used.

1.4 revious odeling Work

The

proposed population balance models presented this thesis are not the tirst

model

SCF

to

be

developed. Wmcure

t l [53] bas

developed

and

solved a non

segregated model the system. The constitutive equations

used

tms Madel were the

Monod constitutive equations which

were modified

to account for the instantaneous

cycling

the system. The model predieted the behavior

the biomass concentration, the

limiting substrate concentration and the DO concentration. The equations were:

dX

=

PIUXC. X_ ~ j X 6 t t .

.

C

l IuaOt

6 l

1

2


28/136

where X

=

biomass concentration gIL ,

CI, Co =lirniting substrate, and DO concentrations gIL ,

C

SF

, a

=limiting

substrate,

and DO

concentrations the

fresh medium gIL ,

s

f

o

=

limiting

substrate, and

DO

yield coefficients,

- -

t =

time

br ,

lmax=

maximum specifie

growth rate br-I ,

l

=saturation constant

gIL ,

kLa liquid side mass transfer coefficient he-

Co

saturation concentration

dissolved oxygen in

the

medium,

f

=emptyinglfill

fraction,

S t =deltaDirac

function,

tmin 02J =time

at the

DO minimum br ,

j =cycle

number.

These differential equations, along with the appropriate irtial conditions, were integrated

numerically.

The model was able

to prediet the

major macro

scopie features

SeF. It captured

the stable periodicity

the system, the

biomass production, the limiting substrate

consumption,

and

the DO concentration profiles.

When

comparing these results with

14


29/136

experimental data the simulation results

are

seen t capture the

major

trends along with

the end o cycle values.

However

the model output was

seen

to

become

out

o

phase

when compared to experimental data. W mcure attnouted this to the faet that

instantaneous

cycling was assumed

in

the

model

construction.

This

explanation

seems

unlikely since taking account

o

the finite cycling time

would

prolong the cycles

o

the

model. The experimental data

suggests

that the simulation cycles were longer than those

o the

actual data.

A possible

explanation

could

be

that the

kinetic

parameters

o

the

system were poorly

estimated

or might have

changed

~

time due

ta adaptation o the

organisms to

the

growth

conditions

o

the system. In

faet a1though

the predicted end of-

cycle values correspond ta that

o

the experimentai values the simulated values took

longer ta reach these

end of cycle

values.

The

model was

also

able

to predict

the

stability

o

the system when

the

emptying/filling fraction

was

other than 1/2.

However

this model does not reveal

any

information

on

the total

cell

number profile l

nor

does it

provide

any

insight

iota the

synchronization

o

the

organisms. To study

the

phenomenon

o cell

synchrony

a

different

model had to be

developed.

1 5 Validation Criteria or the S Mode

Any new model o

SCF should

be

able to capture at least the main macroscopie

features accounted

for by

the

previous model. n addition

the

cell number profile and

the

feature

o

synchrony should be explained

by

tbis model. OveraU the experimental features

to be captured

by

the proposed new model were:

The stable periodicity o the system

including

the

cycle length


30/136

The macroscopie profiles

biomass

limiting substrate and

DO

concentration

The total cell number profile

and

eell synchrony

1 6

Objective

This work

deals

with the modeling and simulation

the SCF process. The

approach taken

is

to develop and solve a segregated structured

model

ta study

the

system. Two

different

models were considered: a cell

age

model and a cell

mass model.

Bath these models have been

used

to study various microbial

systems and

can give rise ta

very different cell number

profiles.

The specifie objectives this study were:

1.

Develop

cell

age

and cell mass microbial

population

balance

models for

SCF.

2.

Develop numerical

methods and algorithms to solve the population

balance models for

SCf

3. Select the MOst appropriate population model

using

criteria

based on available

experimental

data.

4. Validate the model and

select the

model

parameters using available data.

S. Provide a fundamental understanding

the various physieal and biological processes

operating in

the

SCF process.

6. Establish the process

conditions

and mechanism that y lead to the

convergence

cell synchrony.

16


31/136

7 Thesis Organization

This thesis

is organized into four

chapters

Two different

population

balance

models

were applied ta simulate the SeF process Each these models is dealt with

in

separate chapters Chapter 2 deals

w t

the

development application

and discussion

the cell age population balance applied ta SCF Chapter 3 discusses the development

application and

discussion

the

cell

mass

population

balance

applied to

SCF

In both

these chapters the simulation

results are

compared with

experimental

data

n

order ta

determine their suitability to

model the SCF

process Chapter 4 is an

overall

summary

the work

Finally an

Appendix is

also

included

and

contains the computer program that

was written to numerically solve

the cell

mass model for different fennentation systems

17


32/136

~ t 2 The Cell Age Madel

2 Introduction

This chapter deals with the fonnulation

of

a segregated, structured microbial

population balance

model

for the

SCF

process, which used

ceU

age

as

the single

index

of

physiological state. eeu

age

distribution models have the advantage

that no

assumptions

have ta

be made

about

the single eell

growth

kinetics. In

addition, the population balance

equation is

simpler for

the

age

distribution

when

compared ta other

types

of distribution

models. However cell age

cannat

be measured experimentally

unless a

cell

has been

followed since

binh,

and therefore its predictions can not

be

validated without further

assumptions.

The derivation that fol1ows is based

on

work by

Trocco

[46] and the resultant

population balance equation

is

known as the Von

Foerster equation.

The cell

age

model

is

also discussed in [45].

2 2 Formulation

the Cell Age Model

Given a cell population, let

N. t)

be the

number

of ceUs, l

time 1,

that have ages

between

a

and

a

a.

Assuming

that

lim.-.o[AN. t)

a]

exist, we can define the cell age

density funetion

o 1,a)

=

lim oN. t)

l

Integrating

0 1,1)

over

all

ages

a)

gives:

GD

N t

=f

n t,a da

o

18

4)


33/136

where N t is the total cell

number in

the culture cells/volume . Therefore, the number o

eells

which are in the age interval

a,

a

Aa at time t is equal to o 1,a

a.

In a

small

time

interval dt,

the age

o

each

cell

increases

by

dt.

t

is

worth ooting here

that

the

units

o

time and eell age

should

be

the

same and

that

cell

age can ooly be positive.

addition, a

eell o age zero

is defined

as a eell that was just ereated

from

cellular division. For the

time interval dt, the

following

expression

can

be written:

net dt, a dt

Aa

cell death =n 1,a

Aa.

5

Cell

death is

assumed

to

he

proportional to the number o cells

in

a given

cell

age group

n t,a a,

and

to

dt.

It

can

be written as:

ce// death -. t,a, ...)n t,a) ldt

6

where is the

loss

function l/time

and

could depend on

1,

and

other parameters

o

the

system.

Equation 5 can

he

rewritten

as:

n t

dt,

a dt n t,a -

t,a,...

n t,a

Aadt.

19

7


34/136

Expanding net dt, a dt in

powers

o dt yields:

Il

D

net

dt, a

dt = n t, a dt -d t+ O dt

2

.

a

Dividing equation

7

by

and substituting

in

equation

8

gives:

on t ,a n t ,a

~ ~

= - l t , a , ... n t ,a

l

o

which

is

the

Von

Foerster equation

[46].

8

9

For the integral in equation 4 to converge, n 1,8 s 8 o must go to

zero.

Integrating the Von

F

oerster equation trom

a

=

0 to a

= CI results in

the total cell

balance:

dN t

o

d = n t,O - l t ,a , ... n t,a da.

t 0

10

To solve equation

9

the boundary and initial conditions must

be specified.

The

boundary condition

is

expressed for

a

=

0

as:

n t ,O

=

2 I r a n t , a da

o

11

where

r a

is the

division modulus

defined

50ch

that the probability that a

cell

with

age

8

20


35/136

will divide

between t and t + dt

is

equal to r a dt. Equation 11

is

the renewal equation

which accounts for the number newbom ce1Is as a funetion time. Here, binary

division

is

assumed.

Division is expressed in terrns a probability funetion which bas to be integrated

over all ages. For simplification, and without lost generality for the upcoming

discussion,

it

will

he

assumed that

all cells

divide at the

sarne

age

e

The renewal equation

can then

be

rewritten without

the

division modulus and the integral as:

n t,O =

20 t,

S

12

The initial condition for the population balance equation

is

the initial age

distribution:

n O,a = Ilo a .

13

2 3 A

eU

Age Model rInduction Synchrony

Two different methods are generally

used

to obtain il synchrony

in

pure cultures,

sele tion

te hniques

and in u tion te hniques [5,12,33]. Selection techniques usually

involve the

physical

isolation ceUs that are close together with respect their

progression through the cell

cycle.

These ceUs

are

often differentiated trom the rest the

population

based

on marphological differences. For example, centrifugation is often

21


36/136

employed to segregate cells of ditrerent size and mass

in

a gradient Cens having similar

charaeteristics can then be isolated and used as an

inoculum into

a sterile medium

These

cens

will

generally

produce

a synchronous

culture

which can exhtbit sorne

cenular

synchrony

for a

few

generations The

synchrony

is

eventually

lost

due to randomizing

factors.

Induction

techniques usually involve imposing sorne shift

in

the growth

environment

of

the organisms to bring about cen synchrony This can be

accomplished

through singleshock treatments were a single disturbance is introduced which

causes the

cells

to

align

themselves

with

respect to

their

cell cycle

or through

periodic shocks where

a disturbance is applied to the system at

fixed time

intervals This later

method has the

advantage of

providing an

environmental pressure

to rnaintain cell synchrony

for

prolonged

periods of

time

Hjortso has

proposed

a cell

age

model for induction

synchrony

[23] Cell

synchrony

was

studied

using

a

cell

age distribution

model

in

which

the age

at

division

was subjected to periodic forcing The population

balance model

for

the cell

age

distribution

assuming no

cell

death

=0, cao

be

written as:

n t

l1 t,

=

t

a

with

the

renewal equation

0 1,0

=

20 1,0 .

22

14

15


37/136

Under

certain conditions, periodic shifting in the age at division resulted

in

synchronized il populations. This change in division age

could

result ftam a change in

the growth environment.

This

idea

was

exploited

by

Hjortso

and

Nielsen

[25] who

modeled

oscillations

and

partial synchrony

in

continuous cultures accharomyces

cerevisiae

using

an age distribution

model.

They reasoned that,

as

the limiting substrate

concentration

increased,

the duration the

cell cycle

length should decrease.

Figure

8 shows the

growth and division

of ceUs

along

two ditrerent

cell

lines.

Cell

lines

represent the growth

curve,

in

the age-time plane,

ceUs

having

the

same cell age.

The graph depicts the behavior

two

celllines before and after

division when the age

at

division, e t , decreases with time.

The age

difference between these two ceIllines prier

to

division

is

.180,

while their difference after division is Aal_

Since

the

cell

lines have a

slope 1, the age difference between two given cell lines is equal to the distance

separating them in time, At. Assuming

binary

fission,

as

At, becomes differentially smalt,

the following number balance over

dividing

ceUs can be written:

n t,O dtl

=2n t,0 dte,

A relationship between

the

two time intervals can be expressed as:

23

16


38/136

CellAge

Time

Figure 8 Change in distance between two ceillines for a varying division age

8

[ 3]


39/136

where 8

is the change

in

the division age during the time interval 4th and is

positive

when

0 increases over

Atl Thus

it can

be

observed that

i f

the

division

oftwo celllines

occurs while the

division

age

is

decreasing A8

0


40/136

Cell Age

t

Time

Figure

9

Ceillines representiDI the steady

cyde

lolution to the population balance

equation when a periodie sbift in tbe division age il impoled The solid lines

represents the stable ttr etonwhile the dashed lines represent the unstable

repellen

6


41/136

where t is the period of 8 t ,

and al

and a2

are

the lower and upper mits of 8 t ,

respectively.

Assuming

that 0 t

bas

no local extrema between al and a2,

and

that the

condition

represented

by

equation

19

is fulfilled, then for

each

cycle

of0 t) there will be

two celllines

which divide

when

0 t)

=

t .

One ofthese celllines will interseet the

division

age when 0 t) is increasing and one will intersect

when

0 t is decreasing. These cell

lines

represent the

steady

state

cycle

solutions to the population

balance equation.

The

cells in

these

cell

lines will

always divide

at the same relative position in

each of

the

dividing age cycles. Hjortso

abserved

that amang these cell lines, the ones which

divided when

8 t

was

decreasing

were

attracting neighboring cell

lines, while

those

which divided

when

e t)

was increasing

repeUed their

neighboring

celllines. These cell

Unes were tenned attraetors and repellers, respeetively. The

cell

Hnes between two

repellers

will

therefore converge onto

the

attraetor cell line in

this

region.

Hjortso

also

demonstrated

that a

rich

array

of

dynamic

behavior could

he

achieved

when

the periadic

forcing

did not

meet condition 19 . He

described

examples where

bifurcations gave rise

ta

muitimodai synchrony, and

he discussed

cases

exhibiting behavior

similar to period

doubling,

halving,

and

chaos. A brief discussion of how 0 t could be

modeled was also given.

2 4 Solution Scheme

for

the Ce Age Distribution Model: Method

of

Characteristics

An analyticai solution of

the

cell age distnoution model can be

obtained

using

the

method

of charaeteristics [24]. In

this

method, partial

differential

equations are changed

27


42/136

into sets

of

ordinary differential equations ODEts . These ODEts are then

solved

along

characteristics curves in

the plane spanned by

the two

independent variables.

The population balance equation to be

solved

was:

n t, a n t, a =

t

a

with boundary condition:

and initial

condition:

n O,a

=

l1o a .

Equation 20

can be

written as an

ODE such

that:

dn t,

a

=

n t,

a

da n t, a = 0

t t t

20

21

22

23


43/136

where the

cell age growth rate, d

= l

is the

differential

fonn of

the charaeteristic curves,

which can be integrated to

give:

a=t e

24)

where 8

is

a parameter. Figure

10 shows

the family of charaeteristic curves straight Unes)

which span

the age-time

plane,

along which equation 23) cm be

directly

integrated. For

>

0,

equation

23)

is

integrated from the initial condition described

by

equation 22),

over the independent

variable time, while

for

t,

substituting

equation

19)

and

integrating equation 23)

over

time

yields:

I I t . t + ~

f n fo

0. ) 0

fora>t

26)


44/136

CellAge

> 1,

27)

28)

where n O,a-t) is the initial condition l1o a-t). For

< 0, a

15

)

u

0

10

-

.a

5

a.

c

0

0

.

-

0

c

l

1

Figure 13. Transient probability of cela division

f i m

s venus cell mass m. T he

gr ph

illustrates the

efTect

of

the limitinl

lubstr te

concentration. The values uled

in this plot were: me

=

3 X 10.

12

1, E

=

4.242 X 10.

13

1, J.1

=

6 X 10.

5

gI cm

1

hr), K.

=

0.02 IlL

nd De

=1

br

[44].

so


65/136

.

.

.

.

3.5 4

.

=1x 10.

13

.

.

.

.

.

c:

90

...

c

80

0

70

>

a

-

60

-

D

u

50

.2

40

.

-

30

a

ca

.a

e

2

a.

c

1

0

0

i

c:

0

0.5 1

1.5

2

2.5 3

t

Cell mass x 1

12

g)

Figure 14. Transient probability orcell division

r m,C

venus

cell mass

m.

The

graph illustrates the etrect of the

varyinl tbe

spread 8 about the division masse The

values

used in tbis plot were: me 3 S 10

12

1,

Cs=0.034 gIL J,l S 10

5

gI(cm

2

hr),

Ka

= 0.02

gIL

.ad De = 1

l

[44].

SI


66/136

During binary ceU division, assuming

no

loss

of cell

mass during

the

division

process, the mass of the parent cell must

be

divided between the two daughter

cells.

Eakman et l

[15]

proposed an expression

for

the

density

of daughter cell m ass

distribution

p m,m ).

They

assumed

that

randomness exists

in the partition

of

mass

between the daughter

cells and

that this randomness follows a Gaussian-type distribution.

They proposed:

57)

m- .1II. 1

r

p m,m )

m )

s lier

-

2

where

m is the mass

of

the

parent

cell

and

e Ji

is the standard

deviation of

this

distribution.

This

expression

is

plotted

as

a

function

of

daughter

ceU

mass

in

Figure

15.

Again the distribution cannat be

Gaussian since

the daughter

ceUs

cannot have a

mass

less

than

zero or greater

than

that

of

the parent cell.

The

distribution

of

daughter

cell mass

bas

to

be symmetrical about .m

since

2

p m,m )

=pern -m,m ). 58)

The

efFect of

e

can alse be seen, where

the

smaller

the

spread in

the distribution of

mass

the narrower the distnoution.

52


67/136

0

el =5

x

10

13

ft

El

=1

x 10 13

i :

.

.

.

.

.

.

.

:

.

.

.

..

:

Cen

mass X 10

12

(g)

o

M

C

o

:s

.a

.

i

i

i

E

1

~ 6 ~

1

Filure

15. Distribution

of daulhter ceU lalS

for two difTerent

e

values.

m = 4

x

10.

g.

53


68/136

Subramanian

al. [44] have

solved

the

cell mass

population

balance

equation for

various

systems and conditions.

Later, a

different solution scheme

will be

derived

ta

solve

the model. Ta verify the solution, the

simulation

results

will

be compared

with

those

obtained

by

Subramanian

al.

[44].

However

in

their

simulations,

Subramanian

al.

used a simpler relation for the distribution

ofmass

between the daughter

cells:

_ 30

m

m -m)2

p m m -

m

This expression has no adjustable parameters

and

its graph

is

shown

in

Figure 16.

59 .

To simulate

the SCF

process, a substrate

balance on the

system must

be

considered. For any arbitrary i

th

substrate or produet which enters and/or leaves

the

reaetor through the

feed and

effluent

streams,

the

following mass

balance

can be

written:

60

where COli

is the concentration of

the i

dl

substrate

in

the

feed stream, Y

m is the fraction

of

component i

the

mass

taken

up

by the ceU [g of the i

lh

substrate 1g of

cell

mass] and

r ; m is the fraction ofcompanent i in the

mass

released

by the

cell

g of the i

th

substrate

1

g

of cell

mass]. Bath

Y, m

and ri

m

depend on the physiologieal state

of

the

eell

-

54


69/136

m

0 50

N

0 45

c

0 40

c

0 35

0

0 30

.a

0 25

i

0 20

i

i

c

0 15

E

-

0 10

D

U

r

0 05

1

0

=

0

0 5

1 0

1 5 2

2 5 3

3 5

4 0

ca

c

Cell mass

X

10

12

1/g

Figure 16 Distribution

of

ulhter

eeD

mus a liven by Subramanian

[ ]

55


70/136

and are therefore funetions o the cell

mass.

The

SCF

simulations also require that a

mass

balance

for

oxygen be performed on the reador. This equation is the control equation

since the DO concentration is the parameter

monitored

for Y Dg For any system,

assuming

that the

oxygen

is

sparged into the reaetor

and

that

no

oxygen

is

released

by

the

eell, the oxygen balance m be written

as:

61

where Co is the

DO

concentration in the reactor

[gIL],

o is the

DO

concentration o the

inIet

stream

[gIL],

Co is the saturated

DO

concentration

[gIL],

kLa is the volumetrie

oxygen transfer

coefficient

[br-il and Y0 is the fraction o

oxygen in

the mass taken up

y

x

the

cell [g oxygen

/ g

cell

mass].

Equations 39 , 60 and 61 , along

with

the

boundary

condition 49 and the

initial conditions W O,m , Ca O and Co O

constitute

the fully

defined

cell mass model

for

the continuous 0

e X

and

batch 9

=

oc reaetor

problems.

Later

it

will be seen how

these

equations

are modified

to simulate

the SCF

process.

Eakman

al

[1 ]

also

presented a

discussion on

the relation

o

the cell

mass

model to

the segregated unstNetured model total

il density

and the distributed

model

viable biomass concentration , addition to the relation

between

the

cell mass model

and

the eell age model. It is

aIso

worth noting that the

viable

biomass concentration C

[gIL]

I be

obtain ftom the cell mass model by taking

the first moment

o the cell mass

56


71/136

distribution:

C=mW t,m dm.

o

3 3 Solution Scheme

62

The

system equations which have

to

be

solved

consist

one

non-linear,

partial

integro differential equation population balance equation , coupled to two non-linear

differential equations limiting

substrate

and

oxygen balances . The

cell mass

population

balance

model

has

been

solved

by

Subramanian and Ramkrishna

using

the

method

moment

equations along with the Laguerre function

expansion [35,43].

Other techniques

used to solved the general population equation

for

particles undergoing a cambination

growth, comminution, and collection are reviewed

by

Ramkrishna [37]. More

recently,

Liou et

al [29] has

obtained the solution to the ceU

mass

population

balance

equation

using a successive generations approaeh.

This

work uses the Galerkin

Finite

Element Method [17,28] along

with

the implicit

predictor-correctorEuler scheme [17,19] to solve the microbial population model.

3 3 1 Galerki Fillite Element

Met1lod

To

solve the

eeU

mass

population

balance

equation

39 for

the

ceU

mass

distribution, W t,m , the foUowing trial solution is defined:

W t,m = w J t 8 j m

j

57

63


72/136

where

W

t,m

is

the trial solution for the

cell mass

distribution, Wj t are unknown

funetions oftime, Sj m are known nearly orthogonal basis functons, and N

is

the

number

of

Dodes

the

mesh spanning

the

cell

mass domain 0

S

m

S 1JJmq,

where x

is

the upper

cell

mass limit

above which, for ail practical purposes,

no

cells exist.

By

substituting this

trial solution

into

equation

39 ,

the

residual

R was defined as:

R=W t,m blr m,C.)W t,m)]

f IJ

r

M ,C.)W t,m )p m,1d)dnt

G

[

m C.>

0 m

W t m>

0

64

The

residual is

a

measure

of the error occurred when the trial solution

is

substituted

into

the

cell

mass population balance

equation.

The problem lies in obtaining the funetions Wj

that

minimize

the residual. This

is done

by setting

the inner

product

of

the residual and of

a set ofweighing

funetions equal

to zero:

65

where are the weighing

funetions,

and

l x

is the

upper cell mass mit over

which the

finite

mesh

is defined. To find a numerical solution to equation 39 , the mesh had to

be

defined such

as

to cover the entire domain over which the cell mass

distnDution

has a

non-

zero solution. Applying the

Galerkin

method, the weighing

funetions

were set equal to the

58


73/136

basis

funetion such

that:

66

In

addition

the population

balance

equation is

also

coupled

to

the miting

substrate

balance

In

the foUowing simulations

a

single limiting

substrate

will

be

assumed

This assumption was also

foUowed by

Wincure tal

[53]. Similarly, the oxygen

balance

equation

is

coupled to both

the

population balance

equation

and

the limiting

substrate

equation

Therefore

a total

N

2

unknowns must solved in

N

2

equations

Rewriting this

system

equations

in veetor notation yields:

F ~ O

67

f Cl

w

1

f l ~

w

2

f3


74/136

= ras

R. 9 dm =m..

f

W t,m) /J

dm+

m

I

bll(m,C

s

)W t,m)]

/J.dm

Jo

t n

o 0

T

21r m C.)W t.ml)p m.m ) in)e,cbn i =

1.2

...N. 69)

T

+r m,c.)+9 m,c.)W t.m)8,cbn=o

o

dC 1

IO

[ ]

S=

--cc;

-C -I

ys(m) (m}-Ys(m) (m,C,) W(t,m)dm=O,

o x x

and

7D

71)

The

solution vector

o

equation 67)

was

solved using

the

Newton Raphson

iteration

scheme

[17,19]. For the vector equation

this

scheme

may

be written

as:

72)

where k is the iteration index and

is the

Jacobian

matrix:

73)

60


75/136

This iterative scheme can be rewritten

as:

y l

yt

_

J

F yt .

_

74

With

each i t e r t i o ~ the veetor converges quadratically towards its true value. The

iteration is carried out until the difference between successive solutions reaches a

value

below a user specified tolerance.

Chapeau basis

funetions

(Le. the function 9

j

in

equation

(63

were used ta solve

the

cell mass

population balance

equation. These

funetions

are linear and nearly

orthogonal in that they hardly overlap.

Therefore

when

evaluating the integrals the

integration limits may be reduced to values covering

the

range

where

is non

zero.

In

addition,

over

each element,

there are

only two contributions

trom the

basis

funetions.

This reduces

the

likelihood having to

solve ill-conditioned

matrices. The integrals were

solved using a 3-point

Gaussian

quadrature method [19].

3 3 2

Predictor orrector

E ler Scheme

The solution to the N 2 equations discussed above must be found

as

a funetion

time. This was accompli

shed

using

the

implicit

Predietor-Correetor Euler

scheme

[17,19]. This numerical method

consists

two step. The first is an explicit predictor

step in which a solution is approximated tram previous known solutions.

Using

the notation

developed

the previous

section, the predictor

lep can be

61


76/136

written as:

y

y J

Y

=

Y At :..11-1

~ 1 : 1 1 M

75)

where l Pn+1 is

the

predieted value

o

the dependent variable

veetor

l and l n-I

are

the

known

values

o the

dependent variable at time

ln and t

l respeetively, Atll 1

and

n

are

the user

specified time

step from

t

n

ta tn+l and to-I ta ln respectively, and n is the solution

index.

The

term in

the

bracket

is

the

first

arder difference approximation o the tirst

derivative o l

at time

t

n

The

second

step consists o

an implicit

procedure

which

corrects the predicted

value l

n

1

to yield

a

more

accurate solution for

ln+l.

This

procedure

uses

the

predieted

solution l

n

1

to estimate the

tinte derivatives

at tn l

ylf l

_

= 1

i

l i tlf l

76)

This approximation,

along with

the predieted value P

a+1

is

then re-substituted

back into

equation

67 and the Newton-Raphson

iteration

scheme is used to

find the correeted

solution ll:n+l at time

tn+l.

the absolute ditrerence between the predieted

and

correeted

values is

greater than a user-speclfed

tolerance, in this

case I

l

cn l

-

lelll

1 x 10 , the

solution

is rejected and

the

process is repeated with

a

smaller time step.

this di1ference

62


77/136

is

within the specified

tolerance,

the

solution

is accepted

and the

process is continued to

find the next solution Ytt+2. Ta

veritY for

mesh independence, the number ofnodes

where

increased

unt no

change

in the

solution

could

be

noticed.

The number

of nodes

used

for

the SCF simulations was 61. The distribution of nodes were accommodated as to be

concentrated

in the space spanned by the cell

mass

distribution W m,t .

4

Results and Discussion

3 4 Verification oll Solution

In arder to verny the validity of the

solution

scheme, the cell mass population

balance was solved,

using

the new solution scheme, for the chemostat 9 Xl and

batch

8

= J problems. The model for

the

chemostat

and

batch fermentation is defined

by

equations 39 , 60

and

61 , with boundary

conditions

49 . The simulation results were

then compared

to

the results

obtained by Subramanian et

l

[44].

Table

1 shows the

parameter vaiues, taken tram reference [44], that were used for these simulations. The

organisms

were

assumed

to

be bacilli with

cylindrica1

radius R No

cell death was

assumed. The distribution ofdaughter cell mass W S given by equation 59 .

In arder

to

study the

behavior

of the cell population balance model, Subramanian

et l [44] used

three

different sets of initial conditions to simulate

the

chemostat reactar.

These

initial

conditions are given by the foUowing equations:

63


78/136

arameters

alues

e

3 x 10

g

R

5

X

10.

5

cm

K.

0.02

WL

J

1br

1

J

6 X 10.

5

gI cm

2

.hr

0.75

..

0

z

e

3 12

x 10.

13

g

P

1.01 g/cm

3

C.o

2.5 gIL

kLa

300 br

1

1

6.7

z

Co

0.2624

g L

Table Parameter values used for the simulations

the chemostat

and

batch

modell


79/136

m

W O,m) 10

w(o,m)=o.ot(:re

:10

W O,m)

=

o {: e :10

(77)

(78)

79

Equation (77) represents an

initial cell mass distribution composed

oforganisms of

relatively

small cell

mass,

while

that of equation (78) represents a

broader, more

uniformly

distributed

cell mass

distribution. The

initial

condition

given

by equation (79)

has

the

same

distribution

as

equation (77)

except that the total cell number has been reduced by

a

factor

of 10. These tbree

sets

of

initial conditions lead

to three very

different solutions

to

the chemostat case, as is observed from the

simulation

results Figures 17 to

22.

These

graphs plot the

DO

concentration,

the

limiting substrate concentration, the biomass

concentration, and the

cell number

concentration

as

a function of time. The evolution of

the cell mass distribution is also

shawn.

The

cell

number

concentration profiles

presented

here are

normalized

with respect

to

the

parameter E.

This was

done

to facilitate

the

comparison of the results obtained

this

thesis wit those published

by Subramanian

[44]. The solutions

obtained

by

the

new solution scheme developed the present

thesis reproduce

those

published

the

Iiterature [44].

A

discussion

ofthese results

is

also

given

tbis

reference.

65


80/136

Dissolved Oxygen

Lim

iting Su bs trate

0.6

~

S

~

0.4

l

c

= 0.3

ri

DQ

=

0.2

E

0.1

7 ~ r 265

0 260

0 255

0 250

~

~

0 245 =

~

0 240

0 235

1

0.230 j

0 225 Q

0 220

0.0 ~ _ . . . . 0 215

1.8 1.8

6

Cali

Number 1.6

c

o 1.4 1.4

0

=

w

2 Biomass 1.2 ;

x ~

c

i

1.0 1.0

fi

~

~

~

E u 0.8 0.8 u .

:::s--

0.6 0.6

:1

E

u 0.4 0.4 0

2 2

0.0 1.0 2.0 3.0 4.0 5.0

Tlme hours)

Figure 17 DO, limitiDllubstnte, biomul Ind ceU Damber prordes for a

chemoltat, UliDI the initial conditions

liven by

equatioD 77 .

66


81/136

0.40

. . . . . . .

t=O hr

1.8

6

4

Cell

Mass

1 12

1

0.00

-II _ _ _ _ _ 4 _ : : ~ ~

_ _ _ _ _ ~

o

0.35

en

0.30

1

1

0.25 i

o

1

; 0.20

1

~

0.15

CIl

0.10

LI. 0.05 ,

Figure 8 Cell mass distributions

for

a chemostat,

us nl

the initial conditions given

by equation 77 . The dasbed

li

ne represents the

stu y

state distribution.

67


82/136

0.6

0.270

s

0.260

Limiting Substrate

:y

0.4

-

.

Dissolved Oxygen

jO

\

ri

0.2

-

e

0.1

\

0.250

~

0.240

1:

1

0.230

0

l

0.220

-;

0.210

ell

Number

0.0 0.200

2.5 2.5

2.3

C

2 1 0

1.9

I

c

1.7

I

U....

C J

1.5 0

Cft

u

1.3 II )

ft

ftS

1 1 E

o

0.9 iii

0.7

0.0 . . . . . . ~ I ~ 0.5

o

1

4

5

l m

hours)

-

)

U 0.5

~ 2 0

Q

-

~

)

:i1.5

.. en

CD

a

e

cu

::s ~ 1

z

Figure 19. no, limiting substrate, bioDlIIS, aad ceu

aumber

profiles for a

cbeDlostat, usinl the initial conditioDs

liven

by

equatioD

78 .

68


83/136

6

4

Cell Mass 10.

g)

1

5 r

~ O 4 5

..J

0.40

-

~

35

. ;

~

0.30

o

0.25

~ 2

c

0.15

cr

0.10

iL

0.05

0.00

~ ~ t _ _ _ _ t ~ _ _ _ _ _ _ . : : : : = = ~

o

Figure 20. CeU mass distributions for a cbemostlt, us nl the initia. conditions given

by

equation 78 . The dashed line represents the steady state distributioD.

69


84/136

1.6

27

1.4

Limiting

Substrate

26

. 1.2

25

1.0

Dissolved Oxygen

24

ii

-

i

0.8

u

23

=

0.6

1

c:

1 0.4

22

0.21

0.2

0.0

2

2.5 2.5

2.0 2.0

c

N

0

e

-

l

-

Cell

Number

C

~ ~ 1 5

1.5 D

u

.....

en

C I

=

a

:a

D

u ....

E

~ 1

1.0

::1

1

Z

-

E

-

D

0.5

0

u

0.5

ii

0.0 0.0

0 2

4

6

8

10

lime hours)

Figure 21. DO, limiting substrate, biomass, Ind ceU Bumber pronles for a

chemostat, using the initial conditions gjven by equatioD 79).

70


85/136

4 8

0.50 . .

0.45

0.40

-

8

0.35

~

0.30

c

... 0.25

~ 2

i

0.15

=

~ 1

~ 0.05

0.00 ~ c ~ ~ ~ ~ ~

_ _ t _ _ ~

a

1

4

Cell Mass 10.

2

g

5

6

Figure 22. Cell mass distributions for a chemostat, us n the initial conditions given

by equation 79 . Tbe dasbed line represents the steady state distribution.

7


86/136

The case of the batch fennentation revealed a mathematical inconsistency between

the

population

balance, as

described by Eakman

[15], and the

simulation

results

obtained

by

Subramanian

et l [44].

The simulation results, in reference [44],

for

the

batch

problem show

the fermentation proceeding

up

to

the exhaustion

the

limiting

substrate. At

low

limiting substrate concentrations,

the

single cell growth rate is allowed

to

become

negative, indicating that

the

ceUs are experiencing a

net mass

loss.

faet,

the

single cell growth rate attains a

negative

value

when

80

This is

also

true for the transition probability funetion for

cell

division

[ o1,C.), which

leads

ta the

inconsistencies

between the published simulations and the results obtained

here.

These results for

the cell number profile

and

the

biomass

concentration

are

seen

in

Figure 23.

This

profile is

very difTerent from that

obtained

by

Subramanian

l

[44]

for

the

same

conditions.

Figure 23

shows the cell

number

profile decreasing

in time once

the

miting

substrate

reaches a

concentration

below that specified

by

equation 80 . Since no

cell

death

was assumed, this

decrease

should

not

be

predieted

by

the

model.

The

negative

[ 01,C.) function eventually causes

eeUs

to be

lost

ftom the

eeU mass

mesh and ceeates a

sharp profile

in the ceU

mass distnDution. This

causes the

steep

deerease

in

the

cell

density, which is allowed ta attained negative values.

72


87/136

2.5

. .

2.0

-

i

u 1.5

Theory and Simulation of Selfcycling Fermentation a Population Balance Approach

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Transcript of Theory and Simulation of Selfcycling Fermentation a Population Balance Approach