Conduction and convection through hollow cylinder, two layers

Setup

A hollow cylinder (pipe) of length L has inner radius r₁, outer radius r₂ and conductivity k₁. It is covered by two insulation layers: layer 2 from r₂ to r₃ (conductivity k₂) and layer 3 from r₃ to r₄ (conductivity k₃). Hot fluid at T_i flows inside with film coefficient h_i. Cold fluid at T_o surrounds the outside with coefficient h_o. Steady, radial flow; the same Q crosses every layer.

 T_i --[conv]--[pipe]--[layer 2]--[layer 3]--[conv]-- T_o
     1/hi A1  ln(r2/r1) ln(r3/r2) ln(r4/r3)  1/ho A4
               2pi k1 L  2pi k2 L  2pi k3 L

Derivation

Inside convection (Newton): Q = h_i (2π r₁ L)(T_i − T₁), so T_i − T₁ = Q/(2π r₁ L h_i)

Conduction in each cylindrical layer, from Q = 2πkL (T_in − T_out)/ln(r_out/r_in):

T₁ − T₂ = Q ln(r₂/r₁)/(2π k₁ L) T₂ − T₃ = Q ln(r₃/r₂)/(2π k₂ L) T₃ − T₄ = Q ln(r₄/r₃)/(2π k₃ L)

Outside convection: T₄ − T_o = Q/(2π r₄ L h_o)

Add all five equations (the surface and interface temperatures cancel):

T_i − T_o = (Q/2πL) [1/(r₁h_i) + ln(r₂/r₁)/k₁ + ln(r₃/r₂)/k₂ + ln(r₄/r₃)/k₃ + 1/(r₄h_o)]

Q = 2πL (T_i − T_o)/[1/(r₁h_i) + ln(r₂/r₁)/k₁ + ln(r₃/r₂)/k₂ + ln(r₄/r₃)/k₃ + 1/(r₄h_o)]

In resistance form Q = (T_i − T_o)/ΣR, with R_conv,i = 1/(h_i 2π r₁ L), R_conv,o = 1/(h_o 2π r₄ L) and R_cond = ln(r_out/r_in)/(2πkL) for each layer.

The overall coefficient based on the outer area A₄ = 2π r₄ L is U₄ = 1/(A₄ ΣR).