Problem Statement: Consider a stirred batch bioreactor in which an enzyme is immobilized on the surface of a web of glass fibers through chemical covalent bonding. The enzymatic reaction is inhibited by the substrate and further modulated by mass transfer. Assuming the bulk phase is well mixed, find the time it takes for the substrate concentration to drop to one half of its initial value. Derive the applicable equation(s) to enable a solution. Identify the kinetic parameters and operating parameters.
Solution:
Note that had the enzyme been immobilized on the surface of
carrier beads (or any other shape) through chemical covalent
bonding, the treatment should remain completely unchanged.
The substrate concentration at the reacting surface satisfies the
following relationship:
rate of mass flux = rate of reaction at the surface
kL*(sb-s) = vm*s/(Km+s+Ki*s2)
Another way of expressing the above relationship is to say that the
substrate concentration at the surface is a function of that in the bulk.
s = s(sb)
Furthermore, in a batch reactor, the substrate concentration in the bulk
changes with time according to:
d(sb)/dt = - A/V*kL*(sb-s(sb))
Or, equivalently
d(sb)/dt = - A/V*vm*s/(Km+s+Ki*s2)
Since there is no analytical solution, integrate the above expression to
find how long it takes to react.
Kinetic parameters:
vm, Km, Ki
Operating parameters (those that an operator can change)
kL, A, V
Problem Statement: Numerical solution to the above problem. Find how long it takes for the substrate concentration to drop to one half of the initial value. How long does it take if there was no mass transfer resistance? Use the following parameters:
Bead surface area = A = 50000 cm2 Reactor volume = V = 1 liter Starting substrate concentration = sb0 = 20 g/liter Mass transfer coefficient = kL = 0.0015 cm/min Maximum rate constant = vm = 0.0001 g/min-cm2 Michaelis-Menten rate constant = Km = 1 g/liter Substrate inhibition constant = Ki = 1 liter/g(This is a case where mass transfer resistance can speed up rate of conversion. If it is not what you had expected, think hard what is happening.)
Solution:
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