Two rods in series: finding the junction temperature.

• Heat moves in three ways: conduction, convection and radiation. Conduction passes heat between neighbouring parts of a body through molecular collisions, with no flow of matter; it is the usual way heat moves in solids.
• A bar of length L and cross-section A with its ends held at T_C and T_D (T_C > T_D) reaches a steady state in which the temperature falls uniformly along it. The heat current is then H = KA(T_C − T_D)/L.
• K is the thermal conductivity, in W m⁻¹ K⁻¹ (J s⁻¹ m⁻¹ K⁻¹). A large K means a good conductor.
• K in W m⁻¹ K⁻¹. Metals, lowest first: mercury 8.3, lead 34.7, steel 50.2, brass 109, aluminium 205, copper 385, silver 406. Non-metals: felt and glass wool 0.04, wood 0.12, insulating brick 0.15, body fat 0.20, concrete, glass and water 0.8, ice 1.6. Gases: argon 0.016, air 0.024, hydrogen 0.14.
• Everyday uses: some cooking pots have copper-coated bottoms to spread heat evenly; plastic foam insulates because it traps pockets of air; a concrete roof gets very hot in summer, so a layer of earth or foam insulation keeps rooms cooler. In nuclear reactors elaborate heat-transfer systems carry away the heat that fission produces.
• Worked example: a steel rod 15.0 cm long (K = 50.2) and a copper rod 10.0 cm long (K = 385) are joined end to end, sides insulated, the steel end at 300 °C and the copper end at 0 °C; the steel rod has twice the copper's cross-section. In steady state the same heat current crosses both, which gives the junction temperature T = 44.4 °C.
• Working: 50.2 × 2A × (300 − T)/0.15 = 385 × A × (T − 0)/0.10, i.e. 669.3(300 − T) = 3850 T, so T = 200800/4519.3 ≈ 44.4 °C.
• Worked example: an iron rod (K = 79) and a brass rod (K = 109), each 0.1 m long and 0.02 m² in area, are joined end to end with the free ends at 373 K and 273 K. The junction settles at T₀ = (K₁T₁ + K₂T₂)/(K₁ + K₂) = 315 K.
• The pair behaves like a single rod of double length with K′ = 2K₁K₂/(K₁ + K₂) = 91.6 W m⁻¹ K⁻¹, and the heat current through it is H = 916.1 W.

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