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The logarithmic mean temperature difference combines the varying local driving temperature difference along a heat exchanger.
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Formula
ΔT in K; Q̇ in W; U in W/(m2·K); A in m2
| Symbol | Meaning | Unit |
|---|---|---|
| Logarithmic mean driving temperature difference | K | |
| ΔT1, ΔT2 | Temperature differences at the two exchanger ends | K |
| Overall heat-transfer coefficient | W/(m2·K) | |
| Heat-transfer area | m2 |
\Delta T_{lm}=\frac{\Delta T_1-\Delta T_2}{\ln(\Delta T_1/\Delta T_2)},\quad \dot Q=UA\Delta T_{lm}The standard LMTD method assumes steady operation and suitable temperature profiles. ΔT1 and ΔT2 must match parallel- or counterflow arrangement; multipass or crossflow equipment often needs a correction factor. When the end differences approach one another, the logarithmic mean approaches their common value. If an end difference crosses zero, the simple relation is not applicable.
The standard LMTD method assumes steady operation and suitable temperature profiles. ΔT1 and ΔT2 must match parallel- or counterflow arrangement; multipass or crossflow equipment often needs a correction factor. When the end differences approach one another, the logarithmic mean approaches their common value. If an end difference crosses zero, the simple relation is not applicable.
The logarithmic mean temperature difference combines the varying local driving temperature difference along a heat exchanger.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Logarithmic mean temperature difference in a heat exchanger” and check its unit and magnitude.
For ΔT1=50 K and ΔT2=30 K, ΔTlm≈39.2 K; with U=500 W/(m2·K) and A=2 m2, Q̇≈39.2 kW.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: ΔT in K; Q̇ in W; U in W/(m2·K); A in m2.
Formula cluster
The logarithmic mean temperature difference combines the varying local driving temperature difference along a heat exchanger.
For ΔT1=50 K and ΔT2=30 K, ΔTlm≈39.2 K; with U=500 W/(m2·K) and A=2 m2, Q̇≈39.2 kW.
Estimate heat-exchanger area and heat rate from terminal temperatures.
Here is how to work through a typical Logarithmic mean temperature difference in a heat exchanger (ΔTlm = (ΔT₁−ΔT₂)/ln(ΔT₁/ΔT₂); Q̇=UAΔTlm) task step by step:
Task
Work through the numerical example for “Logarithmic mean temperature difference in a heat exchanger” and check its unit and magnitude.
Solution path
For ΔT1=50 K and ΔT2=30 K, ΔTlm≈39.2 K; with U=500 W/(m2·K) and A=2 m2, Q̇≈39.2 kW.
The local temperature difference changes along the exchanger as both fluid streams exchange heat and their temperatures shift. Integrating the local heat balance yields a logarithmic rather than arithmetic mean. ΔTlm therefore represents the effective driving temperature difference across the area under the model assumptions. LMTD uses the two terminal temperature differences for a specified flow arrangement and steady exchanger. Ensure both remain positive in the chosen direction. Multiplied by overall U and area, it gives heat rate; complex multipass or crossflow arrangements may need a correction factor.
For counterflow, pair hot- and cold-stream inlet/outlet temperatures at the same physical ends of the exchanger. One terminal difference is Th,in−Tc,out; the other is Th,out−Tc,in. Label both ends before assigning ΔT1 and ΔT2; mismatching pairs can produce plausible but wrong results. LMTD uses the two terminal temperature differences for a specified flow arrangement and steady exchanger. Ensure both remain positive in the chosen direction. Multiplied by overall U and area, it gives heat rate; complex multipass or crossflow arrangements may need a correction factor.
The expression appears to be 0/0, but its limit is exactly their common temperature difference. If both end differences are 40 K, ΔTlm=40 K. For nearly equal values, use the limiting form or a numerically stable logarithm to reduce rounding error. LMTD uses the two terminal temperature differences for a specified flow arrangement and steady exchanger. Ensure both remain positive in the chosen direction. Multiplied by overall U and area, it gives heat rate; complex multipass or crossflow arrangements may need a correction factor.
Write the quantity and unit from the variable list next to every value: ΔT in K; Q̇ in W; U in W/(m2·K); A in m2. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For ΔT1=50 K and ΔT2=30 K, ΔTlm≈39.2 K; with U=500 W/(m2·K) and A=2 m2, Q̇≈39.2 kW. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. LMTD uses the two terminal temperature differences for a specified flow arrangement and steady exchanger. Ensure both remain positive in the chosen direction. Multiplied by overall U and area, it gives heat rate; complex multipass or crossflow arrangements may need a correction factor.
The background states this starting condition: The standard LMTD method assumes steady operation and suitable temperature profiles. ΔT1 and ΔT2 must match parallel- or. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. LMTD uses the two terminal temperature differences for a specified flow arrangement and steady exchanger. Ensure both remain positive in the chosen direction. Multiplied by overall U and area, it gives heat rate; complex multipass or crossflow arrangements may need a correction factor.
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