 Fig. 1
The log 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 streams at each end of the exchanger. 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.

A generic heat exchanger has two sides ("A" and "B") at which the hot and cold streams enter or exit. The LMTD is defined by the logarithmic mean as in Fig. 1. Fig. 2
This equation is valid both for parallel flow, where the streams enter from the same side, and for counter-current flow, where they enter from different sides. A third type of flow is cross-flow, in which one system – usually the heat sink – has the same nominal temperature at all points on the heat-transfer surface. This follows similar mathematics in its dependence on the LMTD, except that a correction factor “F" often needs to be included in the heat-transfer relationship. There are times when the four temperatures used to calculate the LMTD are not available.

Once determined, the LMTD is usually applied to calculate the heat transfer in an exchanger according to the simple equation:

(2) Q = U°-°A°-°LMTD

where:
Q = exchanged heat duty (in watts)
U = is the heat-transfer coefficient (in watts per Kelvin per square meter)
A = the exchange area

It has been assumed that the rate of change for the temperature of both fluids is proportional to the temperature difference. This assumption is valid for fluids with a constant specific heat, which is a good description of fluids changing temperature over a relatively small range. If the specific heat changes, however, the LMTD approach will no longer be accurate.

A particular case where the LMTD is not applicable is condensers and reboilers, where the latent heat associated to phase change makes the hypothesis invalid. It has also been assumed that the heat-transfer coefficient (U) is constant and not a function of temperature. If this is not the case, the LMTD approach will again be less valid.

The LMTD is a steady-state concept and cannot be used in dynamic analyses. In particular, if the LMTD were to be applied on a transient in which, for a brief time, the temperature differential had different signs on the two sides of the exchanger, the argument to the logarithm function would be negative, which is not allowable.

Example
A liquid-to-liquid counterflow heat exchanger is used as part of an auxiliary system at a nuclear facility. The heat exchanger is used to heat a cold fluid from 120°F to 310°F. Assuming that the hot fluid enters at 500°F and leaves at 400°F, calculate the LMTD for the exchanger.

Solution
Calculate the LMTD using Fig. 2.