# Log Mean Temperature Difference (LMTD) Discussion

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## Definition

Log Mean Temperature Difference (LMTD) is defined as follows:

- [math]\displaystyle{ \boldsymbol{\Delta}T_{LM} = \dfrac{\Delta T_2 -\Delta T_1}{\ln\left(\dfrac{\Delta T_2 }{\Delta T_1}\right)} }[/math]

- where:
- [math]\displaystyle{ \Delta T_1 = T_{Hi} - T_{Co} }[/math]
- [math]\displaystyle{ \Delta T_2 = T_{Ho} - T_{Ci} }[/math]

## Assumptions

The calculation of heat transfer based on LMTD makes a number of assumptions:

- The heat transfer coefficient, U, does not vary along the exchanger;
- The system is adiabatic - heat exchange occurs only between the two fluids;
- The temperatures of both fluids are constant over a given cross section and can be represented by the bulk temperatures; and
- The specific heats of the fluids are constant.

## Condensing Superheated Steam

In a number of applications, including Flash Trains, the steam coming off the liquor may be superheated due to Boiling Point Elevation. Therefore, the steam enters the condenser at a higher temperature than the final condensing temperature. This means that it must initially be desuperheated over some area of the exchanger, we refer to this as the desuperheating zone. See image below:

- The actual temperature profile follows the solid line.
- If the temperature of the superheated steam was used in the LMTD calculation, then the hot side profile would roughly follow the dotted line. This would obviously give an incorrect result.
- In the desuperheating zone, the heat transfer coefficient is much lower than in the condensing zone, though the temperature difference between the steam and the liquor may be higher. This tends to produce a small heat transfer value, or the order of 1 to 2%.
- Determining the true LMTD for a desuperheater-condenser is a complicated calculation. In practice it is adequate to calculate the LMTD based on the condensing temperature since the desuperheating zone is small and the effect is minimal.