Newton's Law of Cooling (Q = hAΔT)

Newton published this in 1701, anonymously and in Latin, as a throwaway note on how a red-hot iron bar cools: the heat leaving a surface is proportional to how far that surface is from the fluid around it. Everything hard about convection is hidden in h, the film coefficient, which is not a material property at all but a shorthand for the whole boundary layer — geometry, velocity, viscosity, whether the fluid is boiling. Still air gives h ≈ 5–25 W/(m²·K); a fan raises it to 25–250; water in a tube runs 500–10,000; and boiling or condensing water can exceed 50,000.

Worked example: a 2.5 m² transformer tank sitting 40 K above ambient with h = 25 W/(m²·K) sheds 25 × 2.5 × 40 = 2500 W. Run it backwards and it becomes the field diagnostic every service technician uses — measure the duty and the surface ΔT, and the h you compute tells you whether the airflow is what the nameplate assumed. The classic trap is using the mean fluid temperature where the correlation wanted the film temperature, or forgetting that a fouled, painted or dusty surface has quietly halved its h since commissioning day.