Radial temperature gradients in an annular chemical reactor. Acatalytic reaction is being carried out at constant pressure in apacked bed between coaxial cylindrical walls with inner radius roand outer radius r1. Such a configuration occurs when temperaturesare measured with a centered thermowell, and is in addition usefulfor controlling temperature gradients if a thin annulus is used. The entire inner wall is at uniform temperature T0 and it can beassumed that there is no heat transfer through this surface. Thereaction releases heat at a uniform volumetric rate Sc throughoutthe reactor. The effective thermal conductivity of the reactorcontents is to be treated as a constant throughout.
A) Ba a shell energy balance, derive a second-order differentialequation that describes the temperature profiles, assuming that thetemperature gradients in the axial direction can be neglected. Whatboundary conditions must be used?
B) Rewrite the differential equation and boundary conditions interms of the dimensionless radial coordinate and dimensionlesstemperature defined as ξ =r/ro , θ=T-To/(Scro2/4keff) Explain why these are logical choices.
C) Integrate the dimensionless differential equation to get theradial temperature profile.
D) Develop expressions for the temperature at the outer wall andfor the volumetric average temperature of the catalyst bed.
E) Calculate the outer wall temperature when ro =0.45 in, r1=0.50in, keff=0.3 Btu/hr ft F, To=900 F and Sc = 4800 cal/hr cm3.
F) How would the results of part (e) be affected if the inner andouter radii were doubled?