In order to solve certain heat exchanger problems, engineers often use a log mean temperature difference (LMTD), which is used to determine the temperature driving force for heat transfer in heat exchangers. LMTD is introduced due to the fact, the temperature change that takes place across the heat exchanger from the entrance to the exit is not linear.
The heat transfer through the wall of heat exchanger at a given location is given by the following equation:
Here the value of overall heat transfer coefficient can be assumed as a constant. On the other hand the temperature difference continuously varies with location (especially in counter-flow arrangement). In order to determine the total heat flow, either the heat flow should be summed up using elemental areas and the temperature difference at the location or more conveniently engineers can average the value of temperature difference. The heat exchanger equation can be solved much easier if we could define a “Mean Temperature Difference” (MTD). It can be seen from the figure that the temperature difference varies along the flow and the arithmetic average may not be the real average, therefore engineers use the logarithmic mean temperature difference. The “Logarithmic Mean Temperature Difference“ (LMTD) is a logarithmic average of the temperature difference between the hot and cold feeds at each end of the heat exchanger. The larger the LMTD, the more heat is transferred. It can be seen from the figure that the temperature difference varies along the flow and the arithmetic average may not be the real average.
For heat exchanger that has two ends (which we call “A” and “B”) at which the hot and cold streams enter or exit on either side, the LMTD is defined as:
The heat transfer is then given by:
This holds both for parallel-flow arrangement, where the streams enter from the same end, and for counter-flow arrangement, 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.