- 2070 Magh · 6 marks
Overall heat transfer coefficient of hollow cylinder
Short answer
The inside overall coefficient is U_i = 1/[1/h_i + (r_i/k) ln(r_o/r_i) + (r_i/r_o)/h_o] and the outside overall coefficient is U_o = 1/[(r_o/r_i)/h_i + (r_o/k) ln(r_o/r_i) + 1/h_o]. They satisfy U_i A_i = U_o A_o = 1/R_total.
Derive expressions for inside and outside overall heat transfer co-efficient for a hollow cylinder subjected to convection medium on both sides.
Setup
A hollow cylinder of length L has inner radius r_i, outer radius r_o and conductivity k. Fluid at T_i with film coefficient h_i is inside; fluid at T_o with coefficient h_o is outside (T_i > T_o). Steady, radial flow with no generation. A_i = 2π r_i L and A_o = 2π r_o L are the inside and outside surface areas.
T_i --[ 1/(hi Ai) ]--[ ln(ro/ri)/(2 pi k L) ]--[ 1/(ho Ao) ]-- T_o
Heat flow
The three resistances are in series:
R_total = 1/(h_i A_i) + ln(r_o/r_i)/(2πkL) + 1/(h_o A_o)
Q = (T_i − T_o)/R_total
Overall coefficient based on the inside area
Define Q = U_i A_i (T_i − T_o). Then U_i = 1/(A_i R_total):
U_i = 1/[A_i/(h_i A_i) + A_i ln(r_o/r_i)/(2πkL) + A_i/(h_o A_o)]
Using A_i = 2π r_i L and A_i/A_o = r_i/r_o:
U_i = 1/[1/h_i + (r_i/k) ln(r_o/r_i) + (r_i/r_o)(1/h_o)]
Overall coefficient based on the outside area
Define Q = U_o A_o (T_i − T_o). Then U_o = 1/(A_o R_total), and with A_o/A_i = r_o/r_i:
U_o = 1/[(r_o/r_i)(1/h_i) + (r_o/k) ln(r_o/r_i) + 1/h_o]
Relation
U_i A_i = U_o A_o = 1/R_total, so U_o = U_i (r_i/r_o). The overall coefficient depends on the surface chosen, because the area of a cylinder changes with radius (unlike a plane wall).
✓ Answer checked by Claude Opus 5.5 (high effort)
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