Heat Exchanger LMTD Method

Overview

The Log Mean Temperature Difference (LMTD) method is a standard approach for evaluating heat transfer in heat exchangers. It provides an effective mean temperature difference that accounts for the variation in driving force along the exchanger length, assuming a constant overall heat transfer coefficient.

According to Newton's Law of Cooling, heat transfer rate is related to the instantaneous temperature difference between hot and cold media. In a heat transfer process the temperature difference varies with position and time.

The mean temperature difference in a heat transfer process depends on the direction of fluid flows involved in the process. The primary and secondary fluid in a heat exchanger may flow in the same direction as parallel or co-current flow, in opposite directions as counter-current flow, or perpendicular to each other as cross flow.

Key Formulas

The LMTD is defined by the equation:

$\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}$

where (\Delta T_1) and (\Delta T_2) represent the temperature differences at the two ends of the heat exchanger. The total heat transfer rate (Q) is then calculated as:

$Q = U \cdot A \cdot \Delta T_{lm}$

Variables

Arithmetic and Logarithmic Mean Temperature Difference Calculator

The original page includes an online calculator for arithmetic and logarithmic mean temperature difference in counter-flow and parallel-flow heat exchangers. Use counter-flow when the hot and cold fluids move in opposite directions, and parallel-flow when both fluids enter from the same end.

This calculator implements both original calculation scenarios. In counter-flow mode it uses $\Delta T_1 = T_{h,in} - T_{c,out}$ and $\Delta T_2 = T_{h,out} - T_{c,in}$. In parallel-flow mode it uses $\Delta T_1 = T_{h,in} - T_{c,in}$ and $\Delta T_2 = T_{h,out} - T_{c,out}$. It returns both AMTD and LMTD from the same input temperatures.

Unit Converter

Logarithmic Mean Temperature Difference Chart

The chart expresses LMTD as a fraction of the larger terminal temperature difference. The ratio on the x-axis is Delta T min / Delta T max; the y-axis multiplier is applied to Delta T max.

Example - Hot Water Heating Air

Hot water at 80 degC heats air from 0 degC to 20 degC in a parallel-flow heat exchanger. The water leaves the heat exchanger at 60 degC.

Arithmetic mean temperature difference:

$AMTD = \frac{80 + 60}{2} - \frac{0 + 20}{2} = 60 \text{ degC}$

Logarithmic mean temperature difference:

$LMTD = \frac{(60 - 20) - (80 - 0)}{\ln((60 - 20)/(80 - 0))} = 57.7 \text{ degC}$

Example - Steam Heating Water

Steam at 2 bar gauge heats water from 20 degC to 50 degC. The saturation temperature of steam at 2 bar gauge is 134 degC, so the steam-side temperature is treated as constant while steam condenses.

Arithmetic mean temperature difference:

$AMTD = \frac{134 + 134}{2} - \frac{20 + 50}{2} = 99 \text{ degC}$

Logarithmic mean temperature difference:

$LMTD = \frac{(134 - 20) - (134 - 50)}{\ln((134 - 20)/(134 - 50))} = 98.24 \text{ degC}$

Restored Original Source Tables

The following tables are restored from the original source page to preserve the complete reference data. The cached source tables for this page are shared search/layout artifacts with no heat-exchanger engineering rows; substantive source chart data and examples are represented in the calculator, InteractiveChart, and examples above.

Original Source Images

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Logarithmic mean temperature difference

Engineering Notes

• The LMTD method assumes a constant overall heat transfer coefficient (U), which may not hold if fluid properties change significantly with temperature.
• It is directly applicable to simple flow arrangements like pure counter-flow or parallel-flow. For shell-and-tube or cross-flow exchangers, correction factors are typically required.
• Ensure (\Delta T_1) and (\Delta T_2) are positive and not equal to avoid logarithmic errors. If they are equal, the LMTD simplifies to their common value.
• For design-critical applications, validate results with manufacturer data or more advanced simulation methods, as real-world effects like fouling and flow maldistribution are not captured.
• Input temperatures should be within practical ranges; extreme differences may necessitate checking material limits or additional safety margins.

References

• Original Source