Purpose of the LMTD

The log mean temperature difference (LMTD) quantifies the effective driving force for heat transfer in a heat exchanger. When a hot fluid transfers energy to a cold fluid, the temperature difference between them typically changes along the flow path. The LMTD condenses this varying difference into a single value representing the overall potential for heat exchange. Engineers multiply the LMTD by the heat-transfer coefficient and the exchanger area to estimate the heat rate.

Deriving the Formula

Consider a small element of a heat exchanger where the temperature difference between the hot and cold streams is $\mathrm{\backslash Delta\; T}$. The rate of heat transfer is proportional to this difference. When you integrate along the length of the exchanger, assuming constant specific heats and no phase change, you obtain

$Q=UA\times \mathrm{\backslash mathrm\{LMTD\}}$

where $U$ is the overall heat-transfer coefficient and $A$ is the surface area. The LMTD itself is defined by

$\mathrm{\backslash mathrm\{LMTD\}}=\frac{\mathrm{\backslash Delta\; T\_1}-\mathrm{\backslash Delta\; T\_2}}{\backslash ln\left(\frac{\mathrm{\backslash Delta\; T\_1}}{\mathrm{\backslash Delta\; T\_2}}\right)}$

where $\mathrm{\backslash Delta\; T\_1}$ is the temperature difference at one end of the exchanger and $\mathrm{\backslash Delta\; T\_2}$ is the difference at the other end. This expression arises from integrating the differential heat-transfer equation along the length.

Parallel vs. Counterflow

In a parallel-flow exchanger, both fluids enter at the same end and move in the same direction. The temperature difference decreases steadily along the flow. In a counterflow exchanger, the fluids move in opposite directions, usually yielding a larger average temperature difference. Our calculator assumes counterflow, which maximizes the LMTD for given inlet and outlet temperatures. The same formula applies to parallel flow if you adjust the temperature differences appropriately.

Using the Calculator

Enter the inlet and outlet temperatures of the hot and cold streams. The calculator computes $\mathrm{\backslash Delta\; T\_1}$ as $\mathrm{T\_\{h\_i\}}-\mathrm{T\_\{c\_o\}}$ and $\mathrm{\backslash Delta\; T\_2}$ as $\mathrm{T\_\{h\_o\}}-\mathrm{T\_\{c\_i\}}$. These values substitute into the LMTD equation. The result appears in degrees Celsius or Kelvin because a temperature difference has the same units in either scale.

Design Implications

A larger LMTD implies a higher potential heat-transfer rate for a given surface area and coefficient. If the LMTD is small, you may need a larger exchanger or a configuration that increases the temperature difference, such as counterflow rather than parallel flow. Engineers use the LMTD along with performance factors like fouling and phase changes to size equipment for chemical plants, refrigeration systems, and power generation.

Example Calculation

Suppose a hot fluid enters at 150 °C and leaves at 100 °C, while a cold fluid enters at 30 °C and exits at 80 °C. The temperature differences are $\mathrm{\backslash Delta\; T\_1}=70\mathrm{°C}$ and $\mathrm{\backslash Delta\; T\_2}=70\mathrm{°C}$. Because they are equal, the LMTD is simply 70 °C. Changing the outlet temperatures would produce a different result, illustrating how the approach temperatures influence exchanger effectiveness.

Assumptions and Limitations

The formula assumes steady-state operation, constant specific heats, and no heat losses to the environment. Phase changes such as boiling or condensation require more complex analysis because the temperature of one stream may remain nearly constant. Additionally, fouling on the heat exchanger surfaces can reduce the effective heat-transfer coefficient $U$, necessitating periodic cleaning or oversizing.

Broader Context

LMTD is only one method for analyzing heat exchangers. Another is the effectiveness-NTU approach, which directly relates exchanger performance to the number of transfer units. Nevertheless, LMTD remains popular for design because it provides an intuitive measure of the average temperature difference. It also connects directly to real-world measurements of inlet and outlet temperatures.

Conclusion

By calculating the log mean temperature difference, this tool helps engineers and students estimate heat-exchanger performance and understand how temperature profiles affect heat transfer. Adjust the input temperatures to explore different scenarios and gain intuition about how counterflow and parallel configurations influence efficiency.