Glossary term

Log Mean Temperature Difference

Heat exchanger temperature-driving-force metric used with UA to estimate duty, area, fouling effect and thermal performance when terminal temperature differences vary.

Branch: Energy Engineering
Glossary type: quantity
Content: Glossary term
Updated: Jun 26, 2026
Revision: v1.0.0 · reviewed

Definition

quantity

Log mean temperature difference is the effective temperature driving force for heat transfer when the hot-to-cold temperature difference changes along a heat exchanger.

Log mean temperature difference, abbreviated LMTD, is used in heat exchanger sizing, performance testing, UA estimation and fouling diagnosis. It is valid when the terminal temperature differences are defined consistently and remain positive for the selected flow arrangement. Multipass, crossflow and nonideal exchangers often require an LMTD correction factor.

Log mean temperature difference is the effective temperature driving force for heat transfer when the hot-to-cold temperature difference changes along a heat exchanger. It is abbreviated LMTD and usually written as (\Delta T_{lm}).

LMTD is not a measured temperature. It is a calculated driving force based on terminal temperature differences. If those terminal differences are wrong, inconsistent or invalid for the flow arrangement, the resulting heat-transfer area or (UA) estimate can be misleading.

Engineering Meaning

For a simple counterflow or parallel-flow exchanger, define two positive terminal temperature differences, (\Delta T_1) and (\Delta T_2), between hot and cold streams at the two ends of the exchanger. The log mean temperature difference is:

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

The heat-transfer relation is:

\dot Q=UA\Delta T_{lm}

where (\dot Q) is heat duty, (U) is overall heat-transfer coefficient and (A) is area.

Terminal Difference Setup

For a counterflow exchanger, a common terminal-difference setup is:

\Delta T_1=T_{h,in}-T_{c,out}

\Delta T_2=T_{h,out}-T_{c,in}

For a parallel-flow exchanger, the terminal pair is usually:

\Delta T_1=T_{h,in}-T_{c,in}

\Delta T_2=T_{h,out}-T_{c,out}

The labels are less important than consistency. The two differences must represent the two ends of the same exchanger and the selected flow arrangement. Mixing a counterflow temperature pair with a parallel-flow assumption can produce an area that looks precise but has no physical basis.

Worked LMTD Example

If the terminal differences are:

\Delta T_1=80\ \text{K},\quad \Delta T_2=30\ \text{K}

then:

\displaystyle \Delta T_{lm}=\frac{80-30}{\ln(80/30)}=51.0\ \text{K}

The LMTD is between the two terminal differences, but it is not their arithmetic mean. The logarithmic form comes from integrating a changing temperature difference along the heat-transfer area.

Nearly Equal Terminal Differences

When the two terminal differences are nearly equal, the formula can suffer numerical cancellation. The limiting value is the common temperature difference:

\Delta T_1\approx \Delta T_2\Rightarrow \Delta T_{lm}\approx \Delta T_1

For (\Delta T_1=42\ \text{K}) and (\Delta T_2=40\ \text{K}):

\displaystyle \Delta T_{lm}=\frac{42-40}{\ln(42/40)}=41.0\ \text{K}

Using the limiting idea avoids treating a harmless numerical condition as a physical problem.

Area Sizing with Correction Factor

Multipass shell-and-tube, crossflow and other nonideal arrangements usually need a correction factor (F):

\dot Q=UA F\Delta T_{lm}

The required area is:

\displaystyle A=\frac{\dot Q}{UF\Delta T_{lm}}

For (\dot Q=1.80\ \text{MW}), (U=650\ \text{W/(m}^2\text{K)}), (F=0.92) and (\Delta T_{lm}=51.0\ \text{K}):

\displaystyle A=\frac{1.80\times10^6}{650\cdot0.92\cdot51.0}=59.0\ \text{m}^2

A low correction factor can signal an inefficient temperature program or unsuitable exchanger arrangement.

UA Estimation

For performance testing, LMTD is often used to estimate effective (UA):

\displaystyle UA_{eff}=\frac{\dot Q}{\Delta T_{lm}}

This should only be done after checking heat-balance closure. If hot-side and cold-side duties disagree, (UA_{eff}) may be dominated by measurement error rather than exchanger performance.

Invalid Cases

The simple LMTD formula requires positive terminal differences with a consistent heat-transfer direction. If temperature profiles cross, if one terminal difference is zero or negative, or if the exchanger has strong phase-change, bypassing, maldistribution or nonlinear property behavior, the simple calculation may not represent the real driving force.

In those cases, use the correct exchanger method, correction factor, segmented model, effectiveness-NTU method or process simulation basis.

Validation Evidence

Useful LMTD evidence includes hot and cold inlet/outlet temperatures, flow arrangement, correction factor, heat-balance closure, sensor calibration, flow rates, fluid properties, phase state, bypass position, fouling condition, uncertainty and whether the same operating window was used for duty and temperature data.

For a digital twin, LMTD should be recomputed from validated sensor data and flagged when terminal differences are too small, inconsistent or outside the approved model range.

Limits and Common Mistakes

Common mistakes include swapping terminal differences, using arithmetic mean temperature difference, ignoring correction factor (F), applying counterflow LMTD to a multipass exchanger without review, accepting a temperature-crossing case, estimating (UA) from an unclosed heat balance and comparing clean and fouled (UA) at different flows or utility conditions.

A strong LMTD review states terminal temperatures, terminal differences, flow arrangement, correction factor, calculated (\Delta T_{lm}), duty basis, uncertainty and the design or operating decision tied to the result.

REF

Disciplines