An inventor has the bright idea of running a gas-solid catalytic reaction with an ultrasonic assist. The concept is to induce acoustic waves inside the pores of the solid catalyst and therefore increase the reaction rate and effectiveness of the catalyst. As a patent examiner you must evaluate the idea shown in Figure P8.38. Consider a cylindrical catalyst pore of diameter, \(d\), and length, \(L\).

a. Assuming that diffusion is negligible compared to the speed of the acoustic wave, show that the differential equation describing the concentration/density profile within the pore for an \(\mathrm{n}^{\text {th }}\)-order reaction is given by:

\[\left[\left(\frac{\partial P_{a}}{\partial ho_{a}}\right)_{\Delta S=0}\right] \frac{\partial^{2} ho_{a}}{\partial x^{2}}=\frac{\partial^{2} ho_{a}}{\partial t^{2}}+k^{\prime \prime} ho_{a}^{n-1} \frac{\partial ho_{a}}{\partial t}\]

b. Considering only a first-order reaction and periodic solutions of the form \(ho=A(x)\) \(\exp (i \omega t)\), solve the differential equation for the concentration/density profile. Assume the pore has an impermeable tip.

c. What is the pore effectiveness for this scheme for a first-order reaction? For a zeroorder reaction?

Transcribed Image Text:

FIGURE P8.38 Pao(t) L -Tak"Pa