1. Your Biotech Company is interested in manufacturing catalyst particles to be used

Question

(suspended) in a stirred tank reactor. The manufacturing process will generate porous,

cylindrically shaped particles (i.e. with a characteristic height - h, and radius-R) - which will allow

for diffusion only through the end caps (i.e. axial, NOT radial diffusion). A local pharmaceutical

company requests that you immobilize an enzyme that they use in the production of an antibiotic

onto the internal surface (i.e. within the pores) of the cylindrical catalyst particles. When these

catalyst particles are created, it is determined that standard Michaelis. Mention kinetics are

observed, where:

V (mol/m² s) = Vm"[S] / Km + [S]

With

and

Vm" = 1 mol/m² min, defined per unit of catalyst surface area

Km = 10 mol/l.

The catalyst particle having a density of 1.4 g/ml and 2.0 m² of internal surface area per gram of

catalyst particle. The concentration of substrate in the antibiotic production process is 0.25 mol/l. The

effective diffusivity of the substrate in the interior of the catalysts is 1 x 10-⁹ m²/s. There is no enzyme

bound to the exterior of the particle. The radius of the particles is 8mm. The conditions in the stirred

tank are such that the bulk substrate concentration is equal to the substrate concentration at the

entrance to the pores (i.e. no external mass transfer resistance), and is constant over time (i.e. CSTR).

a.) Develop a differential equation that represents the conservation of substrate inside the

catalyst particle. List the boundary conditions.

b.) Make this differential equation dimensionless, and identify the Thiele modulus (and the

parameters, such as De, that make it up).

c.) Solve the dimensionless differential equation, obtaining the concentration profile of substrate

versus position inside the catalyst particle. Apply the boundary conditions to obtain the specific

solution.

d.) Determine the relationship between the effectiveness factor and the Thiele modulus for this

cylindrical catalyst particle, and plot this relationship.

e.) Recommend the maximum particle length to use for the antibiotic production process, that

ensures that the reaction is not significantly (i.e. less than 5% reduction from the max possible

reaction rate) reduced by diffusional limitations inside the particle.