1. Multi-stage Chemostat System
In some fermentations, the growth and product-formation steps need to be separated.
e.g. secondary metabolite,
culture of genetically engineered cells.
P1≈0 P2
Growth stage
Product formation stage
At steady state, Vn, Xn,Sn,,Pn in the reactor of each stage don’t change with time.

2. Multi-stage Chemostat System
X0=0, Vi, i=1,2….n constant.
Stage 1: cell growth condition, Kd=0, qp=0
Cell mass:
At steady state
Limiting substrate:
At steady state

3. Multi-stage Chemostat System
Stage 2 – product formation conditions, Kd=0, F’=0
Cell mass: ‘
At steady state
P1≈0 P2 F
V2 are constant.

4. Multi-stage Chemostat System
Stage 2 – product formation conditions, Kd=0, F’=0
Limiting substrate:
At steady state
Product:
At steady state

5. Multi-stage Chemostat System
Stage n – product formation conditions, Kd=0, F’=0
Similarly, equations could be obtained for nth stage.
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.

6. Multi-stage Chemostat System
To 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
rx,n: cell growth rate at nth stage (g/l-h);
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)

7. Multi-stage Chemostat System
Similarly at steady state
rp,n: product formation at nth stage (g/l-h);
rs,n: substrate consumption at nth stage (g/l-h);

8. Multi-stage Chemostat System
To determine the parameters (X, P) graphically.
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.
Get dx/dt to determine X1 first, vice versa.
originating from (Pn-1, 0).

9. 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.

10. dX/dt (g/L-h)
X (g/L)

11. To determine the parameters (X, P) graphically in a 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.

12. Multi-stage Chemostat-Graphical Solution
- Get data of dX/dt~X from the kinetics data in the batch culture and plot.
dX/dt (g/L-h)
X (g/L)
y=0.167X1
x1 is 7.1g/l
Determine the x1 in the first stage of chemostat.
On the graph of dX/dt~X, draw a line of
originating from (X 0 , 0). x1 can be found from the intersection.

13. To determine the parameters (X, P) graphically in a 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=F/V1 (X1 -X0)
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=F/V2(P2-P1), P2 is determined from the intersection.
On the graph of X,P~t, X2 is determined by the P2 at the respect time.

14. Multi-stage Chemostat-Graphical Solution
- From the graph of kinetics data get the P1 in respect with the X1.
X1=7.1 g/l
P1=0.09 g/l

15. To determine the parameters (X, P) graphically in a 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=F/V1 (X1 -X0)
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=F/V2(P2-P1), P2 is determined from the intersection.
On the graph of X,P~t, X2 is determined by the P2 at the respect time.

16. Multi-stage Chemostat-Graphical Solution
Determine the X2 and P2 For the second stage.
X1=7.1 g/l
As X>X1, the growth rate is small and product formation is increased, it is better to use the kinetic data of P to determine P2 first in order to get more accurate results.
P1=0.09 g/l

17. To determine the parameters (X, P) graphically in a 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=F/V1 (X1 -X0)
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~t,
draw a line of y=F/V2(P2-P1), P2 is determined from the intersection.
On the graph of X,P~t, X2 is determined by the P2 at the respect time.

18. Multi-stage Chemostat-Graphical Solution
Determine the X2 and P2 For the second stage.
-get dP/dt ~ P from the kinetics data in the batch culture and plot.
dP/dt (g/L-h)
P (g/L)
on the graph of dP/dt ~ P, draw a line of
P2=0.53 g/l
P1=0.09 g/l
y=0.111(P2-0.09)
P2 can be determined from the intersection.

19. To determine the parameters (X, P) graphically in a 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=F/V1 (X1 -X0)
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=F/V2(P2-P1), P2 is determined from the intersection.
On the graph of X,P~t, X2 is determined by the P2 at the respect time.

20. Multi-stage Chemostat-Graphical Solution
- From the graph of kinetics data get the X2 in respect with the P2
X1=7.1 g/l
P1=0.09 g/l
X (g/L)
Time (hr)
X2=7.95 g/l
P2=0.53 g/l
- Similarly, Xn and Pn can be determined for the nth stage of reactor.

21. Multi-stage Chemostat-Graphical Solution
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 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)
X (g/L)
Time (hr)
P (g/L)
X=7.5g/l
P=0.14g/l
dX/dt (g/L-h)
X (g/L)
y=0.067X
X=7.5g/l
Product concentration can be increased by using multistage chemostats
instead of single stage with the same total reactor working volume. Ex.9.2

22. Problems for Practice Textbook (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: 9.4 Ex