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

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