Question

Write a cell balance for a batch bioreactor where microbes are being cultivated. Assuming a constant specific growth rate for cells, derive an equation for time required in the batch bioreactor for the cells to reach as a concentration of X. Assume an initial cell inoculum concentration of X0.

Answer 1

For a general reactor, the cell balance equation is given by:

Input + Cell generation= Output+ Cell accumulation

By definition, there is no input or output during the process in a batch reactor.

Therefore,

Cell generation = Cell accumulation

This is given by the formula,

r(V) = d(X)/dt

where,

V is the Volume  of the reaction mixture,

r is the rate of growth in time t,

and X is the concentration at a given time t.

Derivation for the time required to reach concentration X from X0.

To derive the time taken, the formula mentioned above is used:

r(V) = d (X)/ dt

dX/dt=rV

rearranging,

dt=dX/rV

and integrating with limits that at t = 0, then X = X0, and at t = t₁, then X = X, we obtain

This equation is the integral form of the balanced reaction on a batch reactor. The formula is used to calculate the time, t₁, necessary for the cells to reach a concentration of X from initial inoculum volume Xo.