Log-Mean Temperature Difference

The engineering

The correct average driving force for Q = U·A·LMTD when U is constant along the exchanger — a plain arithmetic mean overstates it whenever the two end ΔTs differ much. Counterflow pairs hot-in with cold-out; parallel pairs the two inlets. Same four temperatures, different ΔTs, and counterflow always wins or ties — it is the only arrangement that lets the cold outlet climb above the hot outlet.

Assumptions: steady state, constant specific heats, no phase change part-way, and a single pass. Shell-and-tube exchangers with multiple passes need the F correction factor on top of this number (F below about 0.8 means buy a different exchanger). Equal ΔTs at both ends are handled exactly — the log-mean limit is just ΔT.

Where this math comes from

Fourier gave conduction its law in 1822, but exchangers needed the convective side: Wilhelm Nusselt's papers around 1910–1916 built the modern theory of heat transfer between flowing streams, and the log-mean average of the terminal differences emerged from that era's integration of dQ = U·ΔT·dA along the exchanger.

The 1940s industrialized it: Bowman, Mueller, and Nagle published the F-factor charts for multi-pass shells in 1940, and TEMA's first standards (1941) made shell-and-tube design a catalog exercise. LMTD remains the rating engineer's first number, with effectiveness-NTU as the modern alternative when outlet temperatures are the unknowns.

1. 1822Joseph FourierThéorie analytique de la chaleur — heat flow proportional to ΔT.
2. 1915Wilhelm NusseltConvective heat-transfer theory; LMTD framework matures (circa 1910s).
3. 1940Bowman, Mueller & NagleF-correction charts for multi-pass exchangers.
4. 1941TEMAFirst shell-and-tube exchanger standards.

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