Heat transfer and Newton's law of cooling

Heat travels in three ways:

Conduction: when a bar's ends are kept at different temperatures, it soon reaches a steady state in which each point has a fixed temperature and the same heat current H flows through every cross-section: H = KA(T_C – T_D)/L.

• K, the thermal conductivity, is large for metals (copper 385 W m⁻¹ K⁻¹) and small for wood, glass, wool and air. Example: a copper rod 1 m long and 1 cm² across, with ends 100 °C apart, carries 385 × 10⁻⁴ × 100/1 = 3.85 W.
• Like electric current, H = ΔT/R with a thermal resistance R = L/(KA). For bars in series the same H flows through each, so K₁(T₁ – T₀) = K₂(T₀ – T₂) for equal A and L.

That is also why a metal spoon feels colder than a wooden one in the same room: the metal conducts heat away from your hand much faster.

Convection happens when warm fluid, being less dense, rises and cooler fluid sinks to take its place. It drives sea and land breezes (the land warms faster by day, so cool sea air blows in), the trade winds, and the monsoon. In forced convection a pump or a fan moves the fluid: a car's radiator, or the heart pumping warm blood round the body.

Radiation is how the Sun's heat reaches us through empty space. Some rules:

• A good absorber is a good emitter (Kirchhoff). A black body absorbs all the radiation falling on it; lamp black comes close.
• Hotter bodies radiate far more: the power per square metre is σT⁴ (Stefan's law, σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴), so doubling the absolute temperature gives 16 times as much.
• They also radiate at shorter wavelengths: λ_m T ≈ 2.9 × 10⁻³ m K (Wien). Iron in a fire glows dull red, then yellow, then white. The Sun's peak near 500 nm means a surface of about 5800 K.
• The greenhouse effect: the earth's warm surface radiates infrared, which gases like CO₂ absorb and send partly back, keeping the earth warmer.

Newton's law of cooling: for small temperature differences, the rate of cooling is proportional to the excess temperature, T – T_s. Over an interval, use the average temperature. Example: tea cools from 60 °C to 40 °C in 7 min in a 10 °C room. Then 20/7 = K(50 – 10), so K = 1/14 per min. For the next 7 min, (40 – T)/7 = (1/14)[(40 + T)/2 – 10], which gives T = 28 °C.