Question

A hollow cylinder is made of material of thermal conductivity k. The dimensions of the cylinder are as follows inside diameter D_i, outside diameter D_o, and length L. The inside and outside surfaces are at constant temperatures T_i and T_o, respectively, whereas the top and bottom surfaces are both perfectly insulated. Find the expression for the lumped conductive thermal resistance of the cylinder for heat transfer in the radial direction only.

Answer 1

R_i = D_i/2 r_0 = D_0/2 let insode Radius = r_i outside Radius = r_0 For steady state condition in Radial direction, with no Heat Losses, the form of Heat conduction equation is 1/r middot partial differential/ partial differential r (kr middot partial differential T/ partial differential r) + 1/r^2 partial differential/ partial differential straightphi (k partial differential T/ partial differential straightphi) + partial differential/ partial differential z (k partial differential T/ partial differential z) + q = rho c partial differential T/ partial differential t Now because d/dr (r middot dT/dr) = 0 (only Radial flow) & Boundary condition, are T(r_i = D_i/2) = T_i T(r_0 = -D_0/2) = T_0 doublerightarrow temperature distribution (T) T = T_i + T_0 - T_i/ln(D_i/D_0) ln (r/r_1) and Heat flow q = 2 pi KL (T_i - T_0)/ln (D_0/D_i) For thermal Resistance R_conv = L/K.A = ln(D_0/D_i)/2 pi KL X convection Resistance by Newton's law