Log Mean Temperature Difference (Counterflow)

The driving ΔT in an exchanger is not constant along its length, so you cannot use the arithmetic mean: integrating Q̇ = UAΔT along the tube produces the logarithmic mean of the two terminal differences instead. In counterflow the streams run opposite ways, so the hot end pairs the hot inlet with the cold outlet: ΔT₁ = Th,in − Tc,out and ΔT₂ = Th,out − Tc,in. Cool 150 °C oil to 90 °C against water warming from 30 °C to 70 °C and the terminals are 80 K and 60 K, giving $\Delta T_{\mathrm{lm}} = (80 - 60)/\ln(80/60) = 69.5$ K, not the 70 K an average would suggest. The log mean is always the smaller of the two, and the gap widens fast as the terminals diverge.

Counterflow is the reason so many exchangers are plumbed the way they are: it permits a temperature cross, where the cold stream leaves hotter than the hot stream leaves, which parallel flow can never do. Two traps. First, a genuine cross at either terminal makes the logarithm undefined: the calculator refuses, because the arrangement you described cannot exist. Second, the four terminal temperatures sit inside a logarithm and a difference at once, so recovering an inlet temperature from a known $\Delta T_{\mathrm{lm}}$ is transcendental: it takes the Lambert W function, which this page uses when you leave a temperature blank.