I. The End Temperature Differences Differ—How to Average
In a heat exchanger, the hot fluid cools and the cold fluid heats up; the temperature difference at every point along the path changes. The inlet end has a large temperature difference and the outlet end a small one. Using (ΔT_in+ΔT_out)/2 directly overestimates the driving force; the correct approach is the logarithmic mean temperature difference:
ΔTm = (ΔT_large − ΔT_small) / ln(ΔT_large / ΔT_small)
When the two end temperature differences are ΔT1 and ΔT2, take the larger as ΔT_large. It is smaller than the arithmetic mean and is the real "average driving force" of the heat transfer equation.
II. Counter Flow vs. Parallel Flow
For the same hot and cold fluid inlet/outlet temperatures, counter flow gives a larger LMTD:
- Parallel flow: Hot and cold fluids travel the same direction; at the outlet end their temperatures are forced to approach, ΔT_small tends to 0, LMTD is small, and the cold end can never reach the hot end temperature;
- Counter flow: The cold fluid outlet faces the hot fluid inlet, the temperature difference distribution is more uniform, LMTD is large, and the cold end can be heated close to the hot inlet temperature.
Conclusion: To transfer the same heat Q, counter flow requires a smaller area A = Q/(K·ΔTm). Therefore plate and most high-efficiency air-to-air heat exchangers are built as counter flow or cross-counter flow.
III. Limit Case: Phase Change on One Side
When one side is condensation or evaporation (temperature essentially constant, e.g., steam releasing heat), the temperature difference is constant along the path, LMTD degenerates to a constant, and the difference between parallel and counter flow is small. This kind of "isothermal side" is common in waste heat boilers and condensers.
IV. Design Notes
- Determine the flow direction before calculating LMTD; mixed flow needs a correction factor F (<1);
- If ΔT_small is too small, it means the two ends are about to "catch up"; either add area or switch to counter flow;
- Be wary if a bid scheme only states "average temperature difference 50°C" without saying whether it is LMTD or arithmetic—it may be inflated.