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07 Oct 2011 Prof. R. Shanthini 1

Cellular kinetics and associated reactor design:

Reactor Design for Cell Growth

CP504 ecture 7

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07 Oct 2011 Prof. R. Shanthini 2

Cell Growth Kinetics

r! " # C! $41%

&here

# : s'ecific gro&th rate $'er ti(e%

C! : cell concentration $dr) cell &eight 'er unit *olu(e%

+sing the 'o'ulation gro&th (odel, &e could &rite the cell

gro&th rate $r!% as

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07 Oct 2011 Prof. R. Shanthini -

for *olu(e of the reacting (i/ture at ti(e t

C!for concentration of the cells in at ti(e t

$r!% for cell gro&th rate in at ti(e t

ass alance for the cell:

0 $r!% " 0 d$C!% 3 dt

&hich for a atch reactor &ith constant *olu(e

reacting (i/ture gi*es

$42%

Batch Fermenter

dC! 3 dt " r!

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07 Oct 2011 Prof. R. Shanthini 4

Co(ining $41% and $42%, &e get

dC!" # C!

dt$4-%

Batch Fermenter

f # is a constant then integrating $4-% gi*es,

C! " C!0e/'#$t6t0% $44%

&here C!" C!0&hen t " t0.

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07 Oct 2011 Prof. R. Shanthini 5

Cell Growth Kinetics

&here #(a/and 8Sare kno&n as the onod kinetic 'ara(eters.

ostl), ho&e*er, # is not a constant &ith ti(e. t de'ends on

CS

, the sustrate concentration.

9he (ost co((onl) used (odel for # is gi*en ) the onod

(odel:

Monod Modelis an o*er si('lification of the co('licated

(echanis( of cell gro&th.

o&e*er, it ade;uatel) descries the kinetics &hen the

concentrations of inhiitors to cell gro&th are lo&.

# "8

S C

S

#(CS $45%

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07 Oct 2011 Prof. R. Shanthini ut &e don?t kno& $6rS%.

9herefore otain an e/'ression for $6rS% as sho&n in the ne/t

slide.

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07 Oct 2011 Prof. R. Shanthini B

for *olu(e of the reacting (i/ture at ti(e t

CSfor concentration of the Cells in at ti(e t

$rS% for sustrate utiliation rate in at ti(e t

ass alance for sustrate:

0 " 0 $6rS% d$CS% 3 dt

&hich for a atch reactor &ith constant *olu(e

reacting (i/ture gi*es

$4B%dCS 3 dt " 6$6rS%

Batch Fermenter

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07 Oct 2011 Prof. R. Shanthini D

$4B%dCS 3 dt " 6$6rS%

@!3S" 6 rS

r!$47%

$42%dC! 3 dt " r!

Co(ining the ao*e e;uations, &e get

dC! 3 dCS" 6@!3S

&hich u'on integration gi*es

(CX CX!" # $X%& (C&! C&" $4D%

Batch Fermenter

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07 Oct 2011 Prof. R. Shanthini 10

Sustituting CSfro( $4D% in $47% and integrating, &e get

#( $t 6 t0% "8S@!3S

C!0 CS0@!3S

1 lnC!0

C!

8S@!3S

C!0 CS0@!3S

lnCS

CS0$50%

&here

$C! C!0% " @!3S $CS0 CS% $4D%

Batch Fermenter

( )( )

))( (

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07 Oct 2011 Prof. R. Shanthini 11

'ercise ):

9he gro&th rate of E. colie e/'ressed ) onod kinetics

&ith #(" 0.D-5 hr61and 8S" 0.71 g3.

Essu(e that @!3Sis 0.< g dr) cells 'er g sustrate.

C!0is 1 g3 and CS0" 10 g3 &hen the cells start to gro&

e/'onentiall) $i.e., at t " 0%.

sho& ho& C!, CS, and dC!3dt change &ith res'ect to ti(e.

Batch Fermenter

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07 Oct 2011 Prof. R. Shanthini 12

CSis *aried fro( 10 g3 to 0.

( )( )0.D-5t "

0.71 / 0.CX= C&

Continuous Stirred 9ank Ler(enter $CS9L% at stead)6state

ass alance for cells o*er :

LC!i r! " LC! $5-%

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07 Oct 2011 Prof. R. Shanthini 21

=;uation $5-% gi*es

L"

C!6 C!i

r!$54%

Continuous Stirred 9ank Ler(enter $CS9L% at stead)6state

ntroducing Nilution Rate Nas

" $55%L

N "

1

M

in $54%, &e get

1

N"

C!6 C!i

r!$5

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07 Oct 2011 Prof. R. Shanthini 22

Since r!" # C!, $5

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07 Oct 2011 Prof. R. Shanthini 2-

CS " $

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07 Oct 2011 Prof. R. Shanthini 24

FCXi= C&i

F

CX= C&

>CX= C&

ass alance for sustrate o*er :

LCSi" LCS $6rS%

Continuous Stirred 9ank Ler(enter $CS9L% at stead)6state

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07 Oct 2011 Prof. R. Shanthini 25

&hich is rearranged to gi*e

$6rS% " N $CSi6 CS% $

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07 Oct 2011 Prof. R. Shanthini 2