Question

a.) Develop equations for X and S as a function of dilution rate for a continuous cell culture system. Assuming that the kinetics are governed by an uncompetetive inhibition model.

b.) Plot X, S and "yield of cell mass (g cells/l) per time" versus dilution rate and recommend a dilution rate to operate the reactor at if:

So = 20 g/l

Ks = 1 g/l

I = 0.08 g/l

Y x/s = 0.2 g cells/g substrate

Xo= 0

KI = .02 g/l

m = 0.75 h-1

Answer 1

Solution:

The specific growth rate is known to be inhibited and also exhibits a death rate which leads to the following expression for the net growth rate:

An unsteady-state material balance on the cells in the culture volume of a continuous reactor (CSTR):

Input + generation = Output + Accumulation

where,

F₀ = nutrient solution feed flow rate in dm³/h

X₀ = The cell feed concentration in g/dm³,

X = The cell concentration in the reactor and in the outlet stream in g/dm3

V = The liquid phase volume of the culture in dm3.

Rearranging above equation to derive the derivative of (VX) and take the limit delta t -> 0 for constant nutrient flow rate where F₀ = F and for no cells in the feed (X₀ = 0) gives the following differential equation:

Where, Dilution rate (D) = F/V

For steady state process (dX/dt = 0), So:

Similarly for substrate an unsteady-state material balance in the culture volume of the continuous reactor (CSTR) and no extra cellular product formation yields:

where,

S₀ = The substrate feed concentration (g/dm3)

S = The substrate concentration in the reactor and in the outlet stream (g/dm3)

Y_X/S = The yield coefficient for (g cell/g S).

Thus, the resulting differential equation becomes:

At steady state (dS/dt = 0) then above equation becomes:

By combining dilution rate (D) equation and above equation we will get:

X = Y_X/S (S₀-S)

(b)

After putting all the given values in above equation we will get:

Now drawing Plot of X,S and YX/S Vs dilution rate:

D	X	S	YX/s
0	4	0	0.2
0.1	3.846154	0.769231	0.2
0.2	3.636364	1.818182	0.2
0.3	3.333333	3.333333	0.2
0.4	2.857143	5.714286	0.2
0.5	2	10	0.2
0.6	0	20	0