How this instrument works
Inside a heat exchanger, the gap between the hot and cold streams shrinks as they travel along it — steeply near the end where the gap is largest, then more slowly, an exponential decay rather than a straight-line drop. The log mean temperature difference is the single averaged gap that, multiplied by the surface area and the overall heat transfer coefficient, correctly reproduces the total heat duty: Q = U·A·LMTD. Because the underlying decay is exponential, the correct average is a logarithmic one, not the arithmetic mean of the two end gaps, and it is provably always the smaller of the two.
The formula falls out of integrating that decay along the exchanger's length: a gap that satisfies d(ΔT)/dx proportional to −ΔT has a length-averaged value between two endpoints ΔT1 and ΔT2 equal to exactly (ΔT1 − ΔT2) ⁄ ln(ΔT1 ⁄ ΔT2) — the same log-mean construction that turns up wherever a quantity decays exponentially between two known endpoints, such as pressure falling along a pipe run with friction. Swap ΔT1 and ΔT2 and the value is unchanged, since the numerator and the logarithm flip sign together; only how far apart the two gaps are, not which one gets called first, matters.
The formula breaks down at exactly one point: when ΔT1 equals ΔT2, the denominator becomes ln(1) = 0 and direct substitution is undefined, even though the exchanger itself is behaving perfectly reasonably — a constant gap the whole way through. Taking the limit as ΔT1 approaches ΔT2 recovers the arithmetic mean itself, so a process engineer facing near-identical end gaps can use either figure interchangeably; a genuinely undefined result, or a negative ΔT2, instead flags a temperature cross this formula was never built to handle.
- Enter the Temperature difference at end 1, ° — the gap between the hot and cold streams at whichever end of the exchanger you treat as the first.
- Enter the Temperature difference at end 2, ° — the same gap measured at the opposite end, in the same degree scale as the first.
- Both fields must be positive; a negative reading means the streams have crossed somewhere along the length and this formula no longer applies.
- If the two values come out identical, use their arithmetic mean directly — the log-mean formula is undefined exactly at that point.
- Read the Log mean temperature difference, ° — the ΔT to substitute into Q = U·A·LMTD when sizing or checking an exchanger.
Worked example — sizing a counter-flow oil cooler
A counter-flow oil cooler runs 40° hotter than its coolant at one end and 15° hotter at the other — Temperature difference at end 1, ° = 40 and Temperature difference at end 2, ° = 15. Averaging those two figures arithmetically gives 27.5°, but that is not the number this exchanger's design actually needs, because the gap between the streams does not fall in a straight line along the exchanger's length.
Plugging into the formula: LMTD = (40 − 15) ⁄ ln(40 ⁄ 15) = 25 ⁄ ln(2.6667) = 25 ⁄ 0.98083 = 25.4886°, which the instrument reports as 25.49°. That figure, not 27.5°, is the ΔT an engineer substitutes into Q = U·A·LMTD to size the exchanger's surface area — using the arithmetic mean here overstates the driving force by roughly 8 percent, which would leave the finished exchanger undersized.
Questions
Why isn't LMTD just the average of the two end temperature differences?
Because the gap between the two streams doesn't fall in a straight line along the exchanger — it decays exponentially, steep near the end where the gap is largest and shallow where it's small. Averaging that curve arithmetically overstates the true driving force; the logarithmic mean is the one average that, multiplied by area and the overall coefficient, reproduces the exchanger's actual heat duty exactly.
What happens if the two temperature differences are equal?
The formula's denominator becomes ln(1) = 0, undefined by direct substitution — but the exchanger itself isn't broken, it just has a constant gap along its whole length. The limit of the formula as ΔT1 approaches ΔT2 is the arithmetic mean of the two, so use that value directly rather than dividing by zero.
Does it matter which end I call ΔT1 and which I call ΔT2?
No. Swapping the two labels flips the sign of both the numerator and the logarithm in the denominator, so the two negatives cancel and the result is identical. Pick whichever end is more convenient to call first — inlet or outlet, hot end or cold end — the log mean temperature difference comes out the same.
Why does the formula need both end differences to be positive?
A negative or zero value means the two streams' temperatures have crossed somewhere along the exchanger, which the natural logarithm can't handle and which this sizing formula was never built to model. A temperature cross usually means the exchanger needs a counter-flow arrangement instead of parallel-flow, or multiple shell passes, before LMTD sizing applies again.
Is LMTD calculated the same way for parallel-flow and counter-flow exchangers?
The arithmetic is identical either way — only how ΔT1 and ΔT2 are defined changes. In parallel flow both streams enter at the same end, so ΔT1 is the gap at that shared inlet; in counter-flow they enter at opposite ends. For the same four terminal temperatures, counter-flow produces a higher LMTD, which is why it typically needs less surface area to move the same heat duty.
What is LMTD actually used for once I have it?
It's the ΔT term in Q = U·A·LMTD, the standard sizing equation for shell-and-tube and plate heat exchangers. Given a required heat duty and an estimated overall coefficient, engineers solve for the surface area a design needs; given an existing exchanger's area and coefficient, the same equation predicts the duty it can deliver. Process and HVAC engineers run this routinely when specifying or checking exchanger hardware.