LMTD: How to Calculate the Log Mean Temperature Difference
The temperature difference that drives heat from the hot fluid to the cold one in a heat exchanger is not constant: it is largest at one end and shrinks along the flow path. The design equation needs a single effective value, and the right one is the log mean temperature difference (LMTD), not the inlet difference, the outlet difference or their simple average. The LMTD Heat Exchanger Calculator computes it from the four terminal temperatures; this guide shows the formula, a worked example and how to use the result to size an exchanger.
What the LMTD Is and Why It Matters
The LMTD is the effective average temperature difference between the hot and cold streams in a heat exchanger, weighted to account for the way that difference varies exponentially along the flow path. It is the term that turns the basic heat-transfer design equation into something usable:
Q = U · A · LMTD
Here Q is the heat duty (how much energy you need to move per unit time), U is the overall heat-transfer coefficient (how readily heat crosses the wall between fluids), and A is the surface area available for transfer. The LMTD ties them together. Rearranged as A = Q / (U · LMTD), it tells you exactly how much area — and therefore how big and expensive — an exchanger must be to hit a target duty.
This matters because using the wrong temperature difference quietly wrecks a design. Substitute a simple arithmetic average and you will systematically overstate the driving force, undersize the area, and build an exchanger that cannot meet its duty in the field. The LMTD exists precisely because the temperature profile is curved, not linear, and getting it right is the difference between an exchanger that works and one that disappoints.
Why a Logarithmic Average and Not a Simple One
Picture a counterflow exchanger. The hot fluid enters at one end while the cold fluid enters at the other, and they flow in opposite directions. At each end you can measure a temperature difference:
- ΔT₁ = (hot inlet) − (cold outlet), the gap at the hot end
- ΔT₂ = (hot outlet) − (cold inlet), the gap at the cold end
Heat transfer at any point is proportional to the local temperature difference, and that difference decays exponentially as the fluids exchange energy. Averaging an exponential curve with a plain arithmetic mean overweights the high end. The logarithmic mean corrects for the curvature, giving the true average driving force across the whole surface. The result is always less than or equal to the arithmetic average, which is why skipping it makes an exchanger look more capable than it is.
How to Calculate the LMTD
The formula is:
LMTD = (ΔT₁ − ΔT₂) ÷ ln(ΔT₁ ÷ ΔT₂)
where ΔT₁ and ΔT₂ are the temperature differences at the two ends and ln is the natural logarithm. There is one special case: if ΔT₁ and ΔT₂ happen to be equal, the formula divides zero by zero, and the LMTD is simply that common value. Physically, equal end differences mean the gap never changes, so no averaging is needed.
Worked example. Suppose you are cooling a hot process stream with water in a counterflow exchanger:
- Hot fluid in: 120 °C
- Hot fluid out: 70 °C
- Cold water in: 25 °C
- Cold water out: 55 °C
First, find the two end differences:
1. ΔT₁ = 120 − 55 = 65 °C (hot end)
2. ΔT₂ = 70 − 25 = 45 °C (cold end)
Then apply the formula:
3. Numerator: 65 − 45 = 20
4. Ratio: 65 ÷ 45 = 1.444
5. ln(1.444) = 0.3677
6. LMTD = 20 ÷ 0.3677 ≈ 54.4 °C
The simple arithmetic average would have been (65 + 45) ÷ 2 = 55 °C, close here because the end differences are similar; the gap widens when one end difference is much larger than the other (for ΔT₁ = 80 and ΔT₂ = 20 the arithmetic mean is 50 but the LMTD is 43.3). The LMTD Heat Exchanger Calculator takes the four terminal temperatures and a flow-arrangement setting: in counterflow it pairs hot in with cold out and hot out with cold in, as above; in parallel flow it pairs the two inlets and the two outlets. It returns the common value when the two differences are equal and refuses temperatures where the cold stream is hotter than the hot stream at either end.
Using the LMTD in Real Designs and Avoiding Mistakes
Once you have the LMTD, sizing follows directly. If your duty is 200 kW and your overall coefficient U is 500 W/m²·K, then A = Q / (U · LMTD) = 200,000 / (500 × 54.4) ≈ 7.4 m² of heat-transfer area. Change any temperature and the required area moves with it.
Match your ΔT pairing to the flow arrangement. In counterflow, pair hot-in with cold-out and hot-out with cold-in, as above. In parallel (co-current) flow, both fluids enter the same end, so you pair hot-in with cold-in and hot-out with cold-out. Mixing up the pairing is the most common LMTD error and produces a meaningless number.
Apply a correction factor for real shell-and-tube units. Pure counterflow is an idealization. Multi-pass shell-and-tube and cross-flow exchangers mix flow directions, so the design equation becomes Q = U · A · F · LMTD, with the correction factor F (always ≤ 1, often 0.8–0.9 for common configurations) read from standard charts for the exchanger geometry. Forgetting F undersizes multi-pass designs. When the outlet temperatures are unknown and you want to rate an existing exchanger instead, the effectiveness-NTU method is more direct: it gives the heat transfer rate from the flow rates, the inlet temperatures and the exchanger effectiveness, with no logarithm to evaluate.
Watch for a temperature cross. If the cold outlet rises above the hot outlet, a single counterflow pass cannot achieve it, and a naive calculation may even produce a negative argument inside the logarithm. That is a signal to rethink the configuration, not to ignore the math.
Keep your units consistent. Because LMTD is a difference of temperatures, a difference in °C equals the same difference in kelvin, so either works — but never mix Fahrenheit differences into an SI calculation.
Key takeaways
• Know the formula: LMTD = (ΔT₁ − ΔT₂) ÷ ln(ΔT₁ ÷ ΔT₂), where ΔT₁ and ΔT₂ are the temperature differences at the two ends of the exchanger
• Pair ends by flow type: In counterflow pair hot-in with cold-out; in parallel flow pair the two inlets — getting the pairing wrong invalidates the result
• Use it to size: Feed the LMTD from the LMTD Heat Exchanger Calculator into A = Q ÷ (U · LMTD) to find the heat-transfer area you need; 200 kW at U = 500 W/m²·K and 54.4 °C needs about 7.4 m²
• Correct for reality: Multi-pass shell-and-tube exchangers need an F correction factor below 1, and a temperature cross signals the configuration must change