Conduction within relatively complex geometries can sometimes be evaluated using the finite-difference methods of this text that are applied to subdomains and patched together. Consider the two-dimensional domain formed by rectangular and cylindrical subdomains patched at the common, dashed control surface.

Calculate the heat transfer per unit depth into the page, q , using  Δx = Δy = r = 10 mm and Φ  =π/8. The base of the rectangular subdomain is held at T_h  = 20C, while the vertical surface of the cylindrical subdomain and the surface at outer radius r_o are at T_c = 0 C. The remaining surfaces are adiabatic, and the thermal conductivity is k = 10 W/m · K.

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Adiabatic surfaces H = 30 mm y r₁ = 30 mm r = 50 mm - W = 20 mm X T₁ = 20°C T₂ = 0°C