Thermal resistance; composite wall with convection

Thermal resistance is the opposition a layer offers to heat flow. By analogy with Ohm's law (I = V/R), heat flow Q (like current) is driven by a temperature difference ΔT (like voltage) against a thermal resistance R_th:

Q = ΔT/R_th, so R_th = ΔT/Q (unit K/W or °C/W)

For a plane wall: R_cond = L/(kA). For a convective surface: R_conv = 1/(hA).

Composite wall with convection on both sides

A plane wall of area A has two layers: thicknesses L₁, L₂ and conductivities k₁, k₂. Hot fluid at T_∞1 with film coefficient h₁ touches layer 1; cold fluid at T_∞2 with coefficient h₂ touches layer 2. Steady, one-dimensional flow; the same Q passes through every resistance in series.

 T_inf1   T_s1      T_i      T_s2   T_inf2
   o--[1/h1A]--o--[L1/k1A]--o--[L2/k2A]--o--[1/h2A]--o
        Q ------------------------------------->

Rate of heat transfer

Each stage carries the same Q:

Q = h₁A (T_∞1 − T_s1) = k₁A (T_s1 − T_i)/L₁ = k₂A (T_i − T_s2)/L₂ = h₂A (T_s2 − T_∞2)

Write the temperature drops:

T_∞1 − T_s1 = Q/(h₁A) T_s1 − T_i = Q L₁/(k₁A) T_i − T_s2 = Q L₂/(k₂A) T_s2 − T_∞2 = Q/(h₂A)

Add all four equations:

T_∞1 − T_∞2 = Q [1/(h₁A) + L₁/(k₁A) + L₂/(k₂A) + 1/(h₂A)]

Q = (T_∞1 − T_∞2)/[1/(h₁A) + L₁/(k₁A) + L₂/(k₂A) + 1/(h₂A)] = ΔT_overall/R_total

Overall heat transfer coefficient

U is defined by Q = U A ΔT_overall, so U = 1/(A R_total):

U = 1/[1/h₁ + L₁/k₁ + L₂/k₂ + 1/h₂] (unit W/m²·K)