# Exchanged Heat in a Heat Exchanger

## Description

The logarithmic mean temperature difference (also known as log mean temperature difference or simply by its initialism 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. 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.

The LMTD can be used to find the exchanged heat in a heat exchanger as shown here.

Assumptions and Limitations

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. However, if the specific heat changes, the LMTD approach will no longer be accurate.

A particular case where the LMTD is not applicable are 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.

## Variables

Q | exchanged heat duty (W) (dimensionless) |

U | heat transfer coefficient (W/(K*m^2) (dimensionless) |

A | exchange area (m^2) (dimensionless) |

LMTD | logarithmic mean temperature difference (dimensionless) |