In thermal engineering, the logarithmic mean temperature difference (LMTD) is used to determine the temperature driving force for heat transfer in flow systems, most notably in heat exchangers. The LMTD is a logarithmic average of the temperature difference between the hot and cold feeds at each end of the double pipe exchanger. For a given heat exchanger with constant area and heat transfer coefficient, the larger the LMTD, the more heat is transferred. The use of the LMTD arises straightforwardly from the analysis of a heat exchanger with constant flow rate and fluid thermal properties.
Definition We assume that a generic heat exchanger has two ends (which we call "A" and "B") at which the hot and cold streams enter or exit on either side; then, the LMTD is defined by the logarithmic mean as follows:
L M T D = Δ T A − Δ T B ln ( Δ T A Δ T B ) = Δ T A − Δ T B ln Δ T A − ln Δ T B {\displaystyle \mathrm {LMTD} ={\frac {\Delta T_{A}-\Delta T_{B}}{\ln \left({\frac {\Delta T_{A}}{\Delta T_{B}}}\right)}}={\frac {\Delta T_{A}-\Delta T_{B}}{\ln \Delta T_{A}-\ln \Delta T_{B}}}}
where ΔTA is the temperature difference between the two streams at end A, and ΔTB is the temperature difference between the two streams at end B. When the two temperature differences are equal, this formula does not directly resolve, so the LMTD is conventionally taken to equal its limit value, which is in this case trivially equal to the two differences. With this definition, the LMTD can be used to find the exchanged heat in a heat exchanger:
Q = α × A × L M T D {\displaystyle Q=\alpha \times A\times \mathrm {LMTD} }
where (in SI units):
Q is the exchanged heat duty (watts),
α {\displaystyle \alpha } is the heat transfer coefficient (watts per kelvin per square meter), A is the exchange area. Note that estimating the heat transfer coefficient may be quite complicated. This holds both for cocurrent flow, where the streams enter from the same end, and for countercurrent flow, where they enter from different ends. In a cross-flow, in which one system, usually the heat sink, has the same nominal temperature at all points on the heat transfer surface, a similar relation between exchanged heat and LMTD holds, but with a correction factor. A correction factor is also required for other more complex geometries, such as a shell and tube exchanger with baffles.
Derivation Assume heat transfer is occurring in a heat exchanger along an axis z, from generic coordinate A to B, between two fluids, identified as 1 and 2, whose temperatures along z are T1(z) and T2(z). The local exchanged heat flux at z is proportional to the temperature difference:
q ( z ) = α ( T 2 ( z ) − T 1 ( z ) ) = α Δ T ( z ) {\displaystyle q(z)=\alpha (T_{2}(z)-T_{1}(z))=\alpha \;\Delta T(z)}
The heat that leaves the fluids causes a temperature gradient according to Fourier's law:
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