Convection and heat exchangers

When a fluid flows past a surface it carries heat away (or brings it): convection. Newton's law of cooling sums it up as q = h × A × ΔT, where h is the heat transfer coefficient in W/(m²·K). Rough values: still air 5 to 25, flowing air 25 to 250, flowing water 500 to 10,000, and boiling or condensing water 2,500 to 100,000. Moving faster and turbulence raise h.

A heat exchanger passes heat from a hot fluid to a cold one through a wall. Going across, the heat meets three resistances in series: the hot film, the wall and the cold film. They're combined into one overall coefficient U, with 1/U = 1/h_hot + (wall) + 1/h_cold, and q = U × A × ΔT. Dirt and scale (fouling) add more resistance over time, so designs allow for it.

The temperature difference changes along the exchanger, so the right average is the log mean temperature difference: LMTD = (ΔT₁ − ΔT₂) ÷ ln(ΔT₁ ÷ ΔT₂), using the differences at the two ends. Oil cooled from 150 to 90 °C by water warming from 30 to 70 °C, flowing in opposite directions (counter-current): the ends have 150 − 70 = 80 and 90 − 30 = 60 K, and LMTD = 20 ÷ ln(80/60) = 69.5 K.

To size it: q = U × A × LMTD, so A = q ÷ (U × LMTD). For 100 kW with U = 500 W/(m²·K): A = 100,000 ÷ (500 × 69.5) = 2.9 m² of surface.

Counter-current flow gives a bigger LMTD than co-current (both flowing the same way), so it needs less area, and it lets the cold stream leave hotter than the hot stream leaves. That's why most exchangers are built counter-current, or close to it. The common shell-and-tube design runs one fluid inside a bundle of tubes and the other around them in a shell.