Heat Exchanger Effectiveness: Method

The ε–NTU Method

Two streams exchange heat across a wall. Each carries a capacity rate C = ṁ c_p, the heat needed to change its temperature by one degree per unit time, and the smaller of the two, C_min, limits what can be exchanged: the most that can ever pass is q_max = C_min (T_h,in − T_c,in), the duty of an infinitely long counterflow exchanger in which the C_min stream leaves at the other stream's inlet temperature. The effectiveness ε is the actual duty as a fraction of that maximum.

The effectiveness depends on only two numbers once the arrangement is chosen: the number of transfer units NTU = UA/C_min, the size of the exchanger measured in units of the limiting stream, and the capacity ratio C_r = C_min/C_max. The relations are closed forms for counterflow, parallel flow, the mixed crossflow cases and shell-and-tube exchangers, and a close correlation for crossflow with both streams unmixed. At C_r = 0, a condensing or boiling stream, every arrangement gives ε = 1 − e^−NTU. Counterflow is best at every NTU; parallel flow is capped at 1/(1 + C_r) because the streams can only approach a common temperature; the others fall between.

Two problems are posed on an exchanger. Rating: UA is known (from the manufacturer, or from U and the area), so NTU and then ε, the duty and both outlets follow directly. Sizing: an outlet temperature is the target, which fixes the duty and the required ε; the NTU that gives it is solved from the arrangement's relation (the tool bisects) and UA = NTU C_min is what to buy. A target that needs more than the arrangement's limiting ε cannot be met by any size; the tool says so and suggests counterflow.

The older LMTD method writes the same duty as q = F UA ΔT_lm with the log-mean difference on the counterflow basis and a correction factor F for other arrangements. The tool reports both as a cross-check; F is 1 for counterflow and falls as the arrangement departs from it. For a design, keep F above about 0.75: below it the exchanger is near a temperature pinch and small changes in the flows move the duty a lot.