The LMTD calculator above computes the log mean temperature difference for a heat exchanger from the four terminal temperatures, applies the published correction factor for a one-shell-pass, two-tube-pass shell-and-tube geometry, and returns the heat duty when you give it an overall heat transfer coefficient and an area. It reports both terminal temperature differences separately, because those are what tell you whether a design is comfortable or marginal.
Arb Digital publishes free engineering calculators. This one sits next to the live heat transfer calculator, and the boundary between them is worth stating: that tool solves one mode of heat transfer at a time — conduction, convection or radiation — across a single surface using one temperature difference. This page is about an exchanger with two flowing streams whose temperature difference changes continuously along the length, which is precisely why a log mean is needed rather than a simple subtraction.
What This LMTD Calculator Does
In a heat exchanger, the difference between the hot and cold streams is not constant. It is largest at one end and smallest at the other, and it varies exponentially in between. The driving force that governs the whole exchanger is therefore not the arithmetic average of the two ends but their logarithmic mean, which is the correct average for an exponentially varying quantity.
The tool works out the two terminal differences for your chosen arrangement, computes their logarithmic mean, and if you have selected a shell-and-tube geometry it multiplies by a correction factor F that accounts for the fact that such an exchanger is neither purely counter-flow nor purely parallel. Multiply the result by the overall coefficient and the area and you have the heat duty.
It does not size an exchanger, select a geometry, estimate a coefficient or provide any fouling allowance. Those depend on fluids, velocities, materials and service, and they belong in a design calculation with a thermal designer or the manufacturer's software.
How to Use It
- Choose the flow arrangement. Counter-flow and parallel flow are the two ideal cases; the shell-and-tube option applies the correction factor for the most common real geometry.
- Enter the four terminal temperatures — hot in, hot out, cold in, cold out — in Celsius, Fahrenheit or kelvin.
- Read the LMTD and both terminal differences. The smaller of the two, the approach temperature, is the number that usually governs the design.
- Optionally enter U and A to get the heat duty. Both must be referenced to the same surface.
- Check the correction factor. If it drops below about 0.8, the published guidance is to reconsider the arrangement rather than accept the penalty.
The Formula: How LMTD Is Calculated
With ΔT₁ and ΔT₂ the temperature differences at the two ends of the exchanger, the log mean is
LMTD = (ΔT₁ − ΔT₂) / ln(ΔT₁ / ΔT₂)
Which physical temperatures those are depends on the arrangement. In counter-flow the streams travel in opposite directions, so ΔT₁ = Th,in − Tc,out and ΔT₂ = Th,out − Tc,in. In parallel flow they enter at the same end, so ΔT₁ = Th,in − Tc,in and ΔT₂ = Th,out − Tc,out.
The relation comes from integrating the local energy balance along the exchanger. Because the temperature difference decays exponentially with area, the correct mean of that decay is logarithmic. When the two terminal differences happen to be equal the expression is indeterminate, and the limiting value is simply that common difference; the tool handles that case directly. The derivation and the correction-factor charts are standard material in any graduate heat transfer course — the syllabus of MIT OpenCourseWare 2.51 Intermediate Heat and Mass Transfer is a freely available example.
For a shell-and-tube exchanger with one shell pass and two tube passes, some of the tube length runs counter to the shell flow and some runs with it, so the effective driving force is lower than the counter-flow value. The published correction factor is written in terms of two dimensionless groups, the thermal effectiveness P and the capacity ratio R:
P = (Tc,out − Tc,in) / (Th,in − Tc,in) · R = (Th,in − Th,out) / (Tc,out − Tc,in)
and with S = √(R² + 1), the Bowman expression is
F = [S / (R − 1)] · ln[(1 − P) / (1 − PR)] ÷ ln[(2/P − 1 − R + S) / (2/P − 1 − R − S)]
The corrected mean temperature difference is then ΔTm = F × LMTD, and the duty is Q = U A ΔTm.
Work the defaults through by hand. Hot stream 150 °C in and 90 °C out, cold stream 30 °C in and 80 °C out, counter-flow. The terminal differences are 150 − 80 = 70 K and 90 − 30 = 60 K. The log mean is (70 − 60) / ln(70/60) = 10 / 0.15415 = 64.87 K, which is below the arithmetic mean of 65 K as it always must be. Switch to the shell-and-tube option: P = 50/120 = 0.4167, R = 60/50 = 1.2, S = √2.44 = 1.5621. The numerator term is (1.5621/0.2) × ln(0.5833/0.5) = 7.8103 × 0.15415 = 1.2040. The denominator is ln(4.1621/1.0380) = ln(4.0101) = 1.3888. So F = 0.8669, and the corrected mean difference is 64.87 × 0.8669 = 56.24 K. With U = 500 W/m²·K and A = 20 m², the duty is 500 × 20 × 56.24 = 562.4 kW.
Why Counter-Flow Beats Parallel Flow, Every Time
Run the same four temperatures through both arrangements and the difference is stark. The defaults give an LMTD of 64.87 K in counter-flow and 44.27 K in parallel flow — a third less driving force for exactly the same duty, which means roughly half as much again in surface area to do the same job.
The reason is structural. In parallel flow both streams enter at the same end, so the difference starts very large and collapses towards a common outlet temperature that neither stream can pass. In counter-flow the hot outlet meets the cold inlet, so the difference stays comparatively uniform along the whole length and the exchanger works evenly.
The decisive advantage is that counter-flow permits a temperature cross: the cold stream can leave hotter than the hot stream leaves. In the defaults the cold outlet is 80 °C while the hot outlet is 90 °C, which is not a cross, but push the cold outlet to 95 °C and counter-flow still works while parallel flow becomes thermodynamically impossible. Parallel flow has only one genuine advantage: it produces a lower peak wall temperature at the inlet, which matters for a fluid that degrades when overheated.
The Approach Temperature Is the Number That Governs the Design
The smaller of the two terminal differences is the approach temperature, and it sets both the cost and the risk of the exchanger. As the approach shrinks, the LMTD falls faster than the duty does, so the area needed climbs steeply. Halving the approach from 20 K to 10 K on an otherwise fixed duty can add far more than half again to the surface.
Below roughly 5 K the design becomes sensitive in an uncomfortable way. A small error in the estimated overall coefficient, or a modest amount of fouling in service, eats a large fraction of a small driving force, and the exchanger misses its outlet temperature. This is why designers treat a close approach as a commercial decision rather than a purely thermal one: it trades capital cost against energy recovery, and it narrows the margin for everything that is uncertain.
The LMTD approaches zero as the two terminal differences do, so an exchanger with a truly zero approach would need infinite area. That limit is the practical statement of the second law in this context.
When LMTD Is the Wrong Method
The LMTD method assumes a constant overall coefficient, constant specific heats, no phase change partway through, negligible heat loss to surroundings, and steady state. Those hold well enough for most single-phase liquid-to-liquid duties, and the method is exact for pure counter-flow and parallel flow under them.
Two situations break it. The first is a condenser or evaporator where one stream changes phase: within the two-phase region that stream sits at constant temperature, which the relations handle fine, but a unit that desuperheats, condenses and subcools has to be split into zones and each zone computed separately. Averaging across all three gives a meaningfully wrong answer, and each zone needs its own saturation temperature and enthalpy data — the reference fluid property data published by the National Institute of Standards and Technology is the usual source for that.
The second is when the outlet temperatures are unknown. LMTD needs all four terminal temperatures, so if you are rating an existing exchanger and only know the inlets and the flow rates, you would have to iterate. The effectiveness-NTU method exists for exactly that case and solves it directly, which is what the NTU effectiveness calculator handles. The two methods are equivalent when both apply; they simply take different inputs.
A third caution concerns the correction factor. When F falls below about 0.8 the curve becomes steep, so a small error in a temperature produces a large error in F and the design is poorly conditioned. Standard practice is to add shell passes or units in series rather than accept a low F.
Where This Sits Next to the Other Thermal Tools
This page gives the driving force. The heat transfer coefficient calculator gives the U that multiplies it, and the NTU effectiveness calculator solves the same exchanger from inlet conditions alone. The live heat transfer calculator covers single-mode conduction, convection and radiation across one surface, which is a different problem from a two-stream exchanger.
For material and stream properties, the thermal conductivity calculator and specific heat calculator supply the inputs a duty calculation needs, the temperature converter handles the units, and the heat flux converter moves between flux units. The thermal efficiency calculator covers the cycle-level question of what the recovered heat is worth.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using the arithmetic mean instead of the log mean — it is always larger, so it always undersizes the exchanger, and the error grows as the two terminal differences diverge.
- Pairing the wrong terminal temperatures — counter-flow pairs hot inlet with cold outlet; parallel flow pairs both inlets. Using the counter-flow pairing on a parallel-flow unit overstates the driving force badly.
- Applying a correction factor to a true counter-flow exchanger — F is exactly one there, and multiplying by anything else is double-counting.
- Averaging across a phase change — a unit that desuperheats, condenses and subcools must be split into zones and each zone computed on its own.
- Accepting a correction factor below about 0.8 — the curve is steep there, so the design is badly conditioned and a small temperature error moves F a long way.
Related Free Tools From Arb Digital
Pair this with the heat transfer coefficient calculator for U and the NTU effectiveness calculator when the outlet temperatures are unknown. The live heat transfer calculator covers single-surface conduction, convection and radiation, while the thermal conductivity calculator and specific heat calculator supply stream and material properties. The temperature converter, heat flux converter and thermal efficiency calculator complete the set. Everything Arb Digital publishes sits on the free online tools hub.
Frequently Asked Questions
It is the correct average driving force across a heat exchanger whose temperature difference varies exponentially from one end to the other. It equals the difference between the two terminal temperature differences divided by the natural logarithm of their ratio, and it is always smaller than their arithmetic mean.
Because the difference does not vary linearly along the exchanger, it decays exponentially with area. The arithmetic mean is always larger than the log mean, so using it overstates the driving force and undersizes the surface. The two agree closely when the terminal differences are similar and diverge sharply when they are not.
For pure counter-flow and pure parallel flow, both of which are genuinely one-dimensional cases where the log mean is exact. Any real geometry that mixes counter-current and co-current sections, such as a shell-and-tube exchanger with multiple tube passes or a cross-flow unit, has an F below one because part of its surface works against a smaller driving force.
The formula becomes indeterminate, because it reduces to zero divided by zero. The limiting value is simply the common terminal difference itself, and this calculator returns that directly. Physically it corresponds to a balanced counter-flow exchanger where both streams have the same heat capacity rate.
Yes, in counter-flow, and this is called a temperature cross. It is possible because the cold outlet meets the hot inlet rather than the hot outlet, so no thermodynamic limit is violated. Parallel flow cannot do it: both streams converge on a common intermediate temperature that neither can pass.
Use LMTD when you know all four terminal temperatures and want the required area. Use effectiveness-NTU when you know only the inlet temperatures, the flow rates and the geometry, and want to predict the outlets. The two are mathematically equivalent for the same exchanger; they simply suit different sets of known quantities.
Within the condensing region it works well, because a pure fluid condenses at essentially constant temperature and one terminal difference is simply the saturation temperature minus the coolant temperature. A unit that also desuperheats the vapour or subcools the liquid must be divided into zones, with the log mean computed separately in each, because the coefficient and the profile change between them.
Because the required area is the duty divided by the product of the overall coefficient and the mean temperature difference. As the approach shrinks the mean difference shrinks with it, so the area rises steeply and tends to infinity as the approach tends to zero. Below about five kelvin the design also becomes very sensitive to fouling and to errors in the estimated coefficient.
This tool is provided for educational and preliminary engineering use only. It evaluates a published thermal relation from figures you supply and does not size, rate or select a heat exchanger. Overall heat transfer coefficients, fouling allowances, mechanical design, pressure containment and material selection are outside its scope and are governed by the applicable design codes and the manufacturer's data. Confirm any exchanger design with a qualified thermal engineer.
