Log mean temperature difference

Remarks

The value at $\Delta T_1 = \Delta T_2$ is the limit of the formula, so $\Delta T_{\mathrm{lm}}$ is a continuous function of $(\Delta T_1, \Delta T_2)$. It lies between the geometric and the arithmetic mean: $\sqrt{\Delta T_1 \Delta T_2} \le \Delta T_{\mathrm{lm}} \le (\Delta T_1 + \Delta T_2)/2$.

Example: $\Delta T_1 = 40$ K and $\Delta T_2 = 10$ K give $\Delta T_{\mathrm{lm}} = 30 / \ln 4 \approx 21.6$ K, which lies between $20$ K and $25$ K.

Non-example: suppose the cold stream leaves at the inlet temperature of the hot stream. Then $\Delta T_2 = 0$ at that end, the positivity condition fails, and $\Delta T_{\mathrm{lm}}$ is not defined. Such an exchanger would need infinite area.

For parallel flow or counterflow with constant overall heat transfer coefficient $U$ and area $A$, the heat rate is $q = U A \, \Delta T_{\mathrm{lm}}$. For crossflow and multipass exchangers one multiplies the counterflow value by a correction factor $F \le 1$.