Log-Mean Temperature Difference

Find LMTD and heat duty from two positive terminal temperature differences and known UA.

How this calculation works

The logarithmic mean temperature difference represents the changing temperature driving force along a simple heat exchanger. Enter the hot-minus-cold temperature difference at each physical end, rather than a stream's inlet-to-outlet temperature change. Both differences must be positive. Supply the overall conductance UA in W/K, which already combines the heat-transfer coefficient and area. The calculator evaluates the logarithmic mean and multiplies it by UA to estimate duty in kilowatts. When the two differences are identical, their common value is the LMTD, avoiding division by zero in the usual expression. The inputs do not establish the flow arrangement or validate the four stream temperatures. Multipass and crossflow arrangements may need a correction factor that this calculation does not apply.

Inputs and units

Hot-minus-cold difference at end 1 (K)
Hot-minus-cold difference at end 2 (K)
Overall conductance UA (W/K)

Method and formula

LMTD = (ΔT1 − ΔT2)/ln(ΔT1/ΔT2); when equal, LMTD = ΔT1. Q = UA × LMTD/1000 kW.

Worked example

Example inputs

Hot-minus-cold difference at end 1: 40 K
Hot-minus-cold difference at end 2: 20 K
Overall conductance UA: 1000 W/K

Calculation steps

Use the two positive terminal differences, 40 K and 20 K: their difference is 40 − 20 = 20 K and their ratio is 40/20 = 2.
Calculate LMTD = (40 − 20)/ln(40/20) = 20/ln(2) = 28.853901 K.
Multiply by the supplied conductance and convert watts to kilowatts: Q = 1000 × 28.853901/1000 = 28.853901 kW.

Example results

LMTD: 28.85390082 K
Heat duty for supplied UA: 28.85390082 kW

Assumptions

Differences refer to the two physical exchanger ends and are both hot minus cold
Constant UA; simple co-current or counter-current exchanger

Limitations

Does not infer stream arrangement or verify the four stream temperatures
Multipass and crossflow exchangers may require an additional correction factor; zero or crossed local approach is unsupported

Sources

U.S. DOE Fundamentals Handbook: Heat Transfer, Volume 2: HT-02 p. 3 equation 2-2 and pp. 36–39, LMTD applications