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.
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:
| LMTD = | \DeltaTA-\DeltaTB | = | |||
|
| \DeltaTA-\DeltaTB | |
| ln\DeltaTA-ln\DeltaTB |
where is the temperature difference between the two streams at end, and is the temperature difference between the two streams at end . 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=U x A x LMTD
where (in SI units):
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.
Assume heat transfer [1] is occurring in a heat exchanger along an axis, from generic coordinate to, between two fluids, identified as and, whose temperatures along are and .
The local exchanged heat flux at is proportional to the temperature difference:
q(z)=U(T2(z)-T1(z))=U \DeltaT(z)
The heat that leaves the fluids causes a temperature gradient according to Fourier's law:
| \begin{align} | dT1 |
| dz |
&=ka(T1(z)-T2(z))=-ka\DeltaT(z)\\[4pt]
| dT2 | |
| dz |
&=kb(T2(z)-T1(z))=kb\DeltaT(z) \end{align}
where .
The total exchanged energy is found by integrating the local heat transfer from to :
Q=
| B | |
| D\int | |
| A |
q(z)dz=UD
| B | |
| \int | |
| A |
\DeltaT(z)dz=UD
| B | |
| \int | |
| A |
\DeltaTdz,
Notice that is clearly the pipe length, which is distance along, and is the circumference. Multiplying those gives the heat exchanger area of the pipe, and use this fact:
Q=
| UAr | |
| B-A |
| B | |
| \int | |
| A |
\DeltaTdz=
| |||||||||
|
In both integrals, make a change of variables from to :
Q=
| ||||||||||||
|
With the relation for (equation), this becomes
Q=
| ||||||||||||
|
Integration at this point is trivial, and finally gives:
Q=U x Ar x
| \DeltaT(B)-\DeltaT(A) | |||||
|
from which the definition of LMTD follows.
A related quantity, the logarithmic mean pressure difference or LMPD, is often used in mass transfer for stagnant solvents with dilute solutes to simplify the bulk flow problem.