Multi-stage Chemostat System

Multi-stage Chemostat SystemIn some fermentations, the growth and product-

formation steps need to be separated.e.g. secondary metabolite,

culture of genetically engineered cells.

Growth stage Product formation stageAt steady state, Vn, Xn,Sn,,Pn in the reactor of each stage don’t change with time.

P1≈0P2

Multi-stage Chemostat SystemX0=0, Vi, i=1,2….n constant.

stage.each in ratesgrowth gross specific respective the toequal are they 0,KdWhen

ely.recspectiv n, and 2,..., 1, Stagein ratesgrowth specificnet are and ... , 21

n

At steady state

Limiting substrate:dt

dSV

Y

XVFSFS

MSX

11

/

11110

MSXYD

XSS

/1

1121

Stage 1: cell growth condition, Kd=0, qp=0

dt

dXVVXFXFX 1

111110 Cell mass:

)1(1

011 X

XD

At steady state

Multi-stage Chemostat System

)1(2

122 X

XD

dt

dXVVXFXFX 2

222221

22 V

FD where

‘

Stage 2 – product formation conditions, Kd=0, F’=0

At steady state

Cell mass:

P1≈0P2

F

V2 are constant.

Multi-stage Chemostat SystemStage 2 – product formation conditions, Kd=0, F’=0

At steady state

Limiting substrate:

dt

dSV

Y

XqV

Y

XVFSFS

SP

p

MSX

22

/

2

/

22221

SP

p

MSX YD

Xq

YD

XSS

/2

2

/2

2212

dt

dPVXqVFPFP p

222221

At steady state

Product:

2

212 D

XqPP p


Stage n – product formation conditions, Kd=0, F’=0

If μn (e.g. Monod model) and qp are known functions, Xn , Pn, and Sn at nth stage could be determined by the above equations.

)1( 1

n

nnn X

XD

SPn

np

MSXn

nnnn YD

Xq

YD

XSS

//1

n

npnn D

XqPP 1

Similarly, equations could be obtained for nth stage.

Multi-stage Chemostat SystemTo determine the parameters (X, P) graphically when growth

kinetics cannot be expressed analytically.

With no additional streams added to the second or subsequent units, mass balance around the nth stage on cell, substrate and product yields

dt

dXVVXFXFX nnnnnnn 1

F: the volumetric flowrate (l/h); Dn=F/Vn: the dilution rate at the nth stage (1/h); Vn: the liquid volume of the nth chemostat (l)

,)( Since ,, nnnxnx XSXr )()( 1,, nnnnxnx XXDSXr

)( 1 nnnnn XXDX ,0 statesteady at dt

dX n

rx,n: cell growth rate at nth stage (g/l-h);


dt

dPVVrFPFP nnnnpnn ,1

dt

dSVVrFSFS nnnnsnn ,1

Similarly at steady state

)()( 1,, nnnnxnp PPDSXr

)()(1

)( 1,,,,

/

nnnnxnxnxns SSDSXrY

SXrSX

rp,n: product formation at nth stage (g/l-h);rs,n: substrate consumption at nth stage (g/l-h);

Multi-stage Chemostat SystemTo determine the parameters (X, P) graphically.

originating from (Pn-1, 0).

For an example, in a two-stage chemostat system, the first tank is 600 l and the second one is 900 l. The flowrate is 100l/h, determine the concentrations of cells and product in each of the tank.

Solution: - Get kinetics data (X~t, P~t) from the batch culture.

To determine the parameters (X, P) graphically in multi-stage chemostat system

Get the kinetics data of X~t and P~t from batch culture and plot

Get dX/dt~X from the kinetics data and plot

Determine X1 and P1 in the first stage:

On the graph of dX/dt~X, draw a line of y=D1 (X1 -X0)D1=F/V1, X0=0 in the feed, X1 is determined from the intersection.

On the graph of X,P~t, P1 is determined by the X1 at the respect time

Determine X2 and P2 in the second stage:

Plot the graph of dP/dt~P from the kinetics data of P, draw a line of y=D2(P2-P1), D2=F/V2 , P2 is determined from the intersection.

On the graph of X,P~t, X2 is determined by the P2 at the respect time.

dX

/dt

(g/L

-h)

X (g/L)

To determine the parameters (X, P) graphically in a multi-stage chemostat system



Determine X1 and P1 in the first stage:

On the graph of dX/dt~X, draw a line of y=D1 (X1 -X0)D1=F/V1, X0=0 in the feed, X1 is determined from the intersection.


Determine X2 and P2 in the second stage:

Plot the graph of dP/dt~P from the kinetics data of P, draw a line of y=D2(P2-P1), D2=F/V2 , P2 is determined from the intersection.


dX

/dt

(g/L

-h)

X (g/L)

Multi-stage Chemostat-Graphical Solution

10.167

)01(l 600

l/h 100

)01(1V

F

)01(1

X

X

XX

XXDy

)(

)(

1

,,

nnn

nxnx

XXD

SXr

- Get data of dX/dt~X from the kinetics data in the batch culture and plot.

-Determine the x1 in the first stage of chemostat.

On the graph of dX/dt~X, draw a line oforiginating from (X 0 , 0). x1 can be found from the intersection.

)01(1 XXDy

x1 is 7.1g/l

y=0.167X 1




Determine X1 and P1 in the first stage

On the graph of dX/dt~X, draw a line of y=F/V1 (X1 -X0)X0=0 in the feed, X1 is determined from the intersection.


Determine X2 and P2 in the second stage

Plot the graph of dP/dt~P from the kinetics data of P, draw a line of y=F/V2(P2-P1), P2 is determined from the intersection.



- From the graph of kinetics data get the P1 in respect with the X1.

X1=7.1 g/l

P1=0.09 g/l










-Determine the X2 and P2 For the second stage.

X1=7.1 g/l

P1=0.09 g/l









Plot the graph of dP/dt~P from the kinetics data of P~t, draw a line of y=F/V2(P2-P1), P2 is determined from the intersection.


dP

/dt

(g/L

-h)

P (g/L)

-Determine the X2 and P2 For the second stage.

P2 can be determined from the intersection.

-get dP/dt ~ P from the kinetics data in the batch culture and plot.

)(...)( 1,, nnnnxnp PPDSXr

)12(2 PPDy -on the graph of dP/dt ~ P, draw a line of

)/09.0(0.111

)/09.0(900

/100

)(

)(

2

2

122

122

lgP

lgPl

hl

PPV

F

PPDy

P2=0.53 g/lP1=0.09 g/l

y=0.111(P2-0.09)






On the graph of dX/dt ~X, draw a line of y=F/V1 (X1 -X0)X0=0 in the feed, X1 is determined from the intersection.





X1=7.1 g/l

P1=0.09 g/l

X (

g/L

)

Time (hr)

- From the graph of kinetics data get the X2 in respect with the P2

X2=7.95 g/l

P2=0.53 g/l

- Similarly, Xn and Pn can be determined for the nth stage of reactor.


In summary, the first tank is 600 l and the second one is 900 l. The flowrate is 100l/h, X1=7.1 g/l, P1=0.09 g/l; X2=7.95 g/l, P2=0.53 g/l

If one stage chemostat is used with the same total working volume i.e. 1500 l as that in the above example. The flowrate is also 100l/h. What is the achieved concentrations of the cells and the product? D=F/V=100/1500=0.067 (h-1)

dX

/dt

(g/L

-h)

X (g/L)

y=0.067X

X=7.5g/l

X (

g/L

)

Time (hr)

P (

g/L

)

X=7.5g/l

P=0.14g/l

Product concentration can be increased by using multistage chemostats instead of single stage with the same total reactor working volume. Ex.9.2


Problems for PracticeTextbook (M.Shular)Q: 9.3: • You are not required to determine the configuration of

tanks that would maximize product formation. Instead, determine the concentration of the product in each tank for the following configurations using the graphical method introduced in the lecture:

• Reactor 1 (600 L), reactor 2 (900L) and reactor 3 (300 L).

• Reactor 1 (900 L), reactor 2 (300L) and reactor 3 (600 L).

• Which of the above configurations gives higher product concentration?

• If 300 L reactor is the first reactor in series, what would happen?

Q: 9.4Ex.9.1-9.2

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Multi-stage Chemostat System