Problem Statement: Consider a continuous bioreactor in which a product-inhibited enzyme is entrapped inside gel beads of radius R. The gel beads are completely fluidized and have a total volume of Vb. Substrate of concentration sf is continuously fed into the reactor of void volume V at a flow rate of F, and the reactor fluid content is also continuously removed at the same rate. List all the applicable equations to be solved to find the steady-state rate of conversion in the reactor.
Solution:
The number of beads (n) is calculated from: n*4/3*p*R3 = Vb
The total surface area (A) is: n*4*p*R2 = 3*Vb/R
The substrate and product flux on the bead surface satisfy the following
relationship:
rate of mass flux = rate of reaction in the gel bead
d2s 2 ds 1
--- + - -- = --*v(s,p) B.C. s(R)=sb ds(0)/dr=0
dr2 r dr De
Given sb and pb, solve the above two-point boundary value (TPBV) problem.
When this is done, we have the substrate and product flux, which in turn,
tells us the rate of conversion achieved.
The rate of conversion achieved is equal to A*Js:
ds
Js(sb,pb) = De*--(R)
dr
Another way of expressing the above relationship is to say that the
substrate and product flux at the bead surface are functions of
that in the bulk.
Js = Js(sb,pb)
Jp = Jp(sb,pb)
Furthermore, in a continuous reactor operated at steady-state, the
substrate and product concentrations in the bulk are described by the
following coupled set of algebraic equations:
V*d(sb)/dt = 0 = F*(sf-sb) - A*Js(sb,pb)
V*d(pb)/dt = 0 = -F*pb + A*Jp(sb,pb)
The solution specific to this steady-state problem is given below:
A series of Mathcad files contain the solution to this problem.
Substrate Profile in a Spherical Gel-Bead with Substrate Inhibition
Substrate/Product Profile in a Spherical Gel-Bead with Substrate/Product Inhibition
Plot of Effectiveness Factor with Substrate Inhibition
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