Steady-state catalytic chemical reactor (described by a single ODE). Your task as a design engineer.

Steady-state catalytic chemical reactor (described by a single ODE). Your task as a design engineer in a chemical company is to model a fixed bed reactor packed with the company proprietary catalyst of spherical shape. The catalyst is specific for the removal of a toxic gas at very low concentration in air, and the information provided from the catalytic division is that the reaction is first-order with respect to the toxic gas concentration. The reaction rate has units of moles of toxic gas removed per mass of catalyst per time. Your first attempt, therefore, is to model the reactor in the

»Steady-state catalytic chemical reactor (described by a single ODE). Your task as a design engineer in a chemical company is to model a fixed bed reactor packed with the company proprietary catalyst of spherical shape. The catalyst is specific for the removal of a toxic gas at very low concentration in air, and the information provided from the catalytic division is that the reaction is first-order with respect to the toxic gas concentration. The reaction rate has units of moles of toxic gas removed per mass of catalyst per time. Your first attempt, therefore, is to model the reactor in the simplest possible way so that you can develop some intuition about the system before any further modeling attempts are made to describe it exactly.a)      For simplicity, assume that there is no significant diffusion inside the catalyst and that diffusion along the axial direction is negligible. An isothermal condition is also assumed.(This assumption is known to be invalid when the reaction is very fast and the heat of reaction is high). The coordinate z is measured from entrance of the packed bed. Perform a steady-state mass balance around a thin shell at the position z with the shell thickness of _z and show that when the shell thickness approaches zero, the following relation is obtained:Where uo is the superficial velocity, C is the toxic gas concentration, _ is the bed porosity, _p is the catalyst density, and k is the rate constant for a heterogeneous reaction, which has units of  for such a case (i.e., first-order reaction). 

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