Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications!

This is an AMP page - Open full page! for all features.

• the most efficient way to navigate the Engineering ToolBox!

Arithmetic and Logarithmic Mean Temperature Differences in Heat Exchangers

Arithmetic Mean Temperature Difference - AMTD - and Logarithmic Mean Temperature Difference - LMTD - formulas with examples - Online Mean Temperature Calculator

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 vary with position and time

Mean Temperature Difference

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 an heat exchanger process may

• flow in the same direction - parallel flow or co-current flow
• in the opposite direction - counter-current flow
• or perpendicular to each other - cross flow

With saturation steam as the primary fluid the primary temperature can be taken as a constant since the heat is transferred as a result of a change of phase only. The temperature profile in the primary fluid is not dependent on the direction of flow.

Logarithmic Mean Temperature Difference - LMTD

The rise in secondary temperature is non-linear and can best be represented by a logarithmic calculation. A logarithmic mean temperature difference is termed

• LMTD (or DTLM) - Logarithmic Mean Temperature Difference

LMTD can be expressed as

LMTD = (dto - dti) / ln(dto / dti)                                       (1)

where

LMTD = Logarithmic Mean Temperature Difference (oF, oC)

For parallel flow:

dti = tpi - tsi = inlet primary and secondary fluid temperature difference (oF, oC)

dto = tpo - tso = outlet primary and secondary fluid temperature difference (oF, oC)

For counter flow:

dti = tpi - tso = inlet primary and outlet secondary fluid temperature difference (oF, oC)

dto = tpo - tsi = outlet primary and inlet secondary fluid temperature difference (oF, oC)

The Logarithmic Mean Temperature Difference is always less than the Arithmetic Mean Temperature Difference.

Arithmetic Mean Temperature Difference - AMTD

An easier but less accurate way to calculate the mean temperature difference is the

• AMTD (or DTAM) - Arithmetic Mean Temperature Difference

AMTD can be expressed as:

AMTD = (tpi + tpo) / 2 - (tsi + tso) / 2                                  (2)

where

AMTD = Arithmetic Mean Temperature Difference (oF, oC)

tpi = primary inlet temperature (oF, oC)

tpo = primary outlet temperature (oF, oC)

tsi = secondary inlet temperature (oF, oC)

tso = secondary outlet temperature (oF, oC)

A linear increase in the secondary fluid temperature makes it more easy to do manual calculations. AMTD will in general give a satisfactory approximation for the mean temperature difference when the smallest of the inlet or outlet temperature differences is more than half the greatest of  the inlet or outlet temperature differences.

When heat is transferred as a result of a change of phase like condensation or evaporation the temperature of the primary or secondary fluid remains constant. The equations can then be simplified by setting

tp1 = tp2

or

ts1 = ts2

Arithmetic and Logarithmic Mean Temperature Difference Calculator

The calculator below can be used to calculate Arithmetic and Logarithmic Mean Temperature Difference of counter-flow an parallel-flow heat exchangers.

tpi - primary flow - inlet temperature (oF, oC)

tpo - primary flow - outlet temperature (oF, oC)

tsi - secondary flow - inlet temperature (oF, oC)

tso - secondary flow - outlet temperature (oF, oC)

Counter-flow Parallel-flow

Example - Arithmetic and Logarithmic Mean Temperature, Hot Water Heating Air

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

Arithmetic Mean Temperature Difference can be calculated as

AMTD = ((80 oC) + (60 oC)) / 2 - ((0 oC) + (20 oC)) / 2

= 60 oC

Logarithmic Mean Temperature Difference can be calculated as

LMTD = ((60 oC) - (20 oC)) - ((80 oC) - (0 oC))) / ln(((60 oC) - (20 oC)) / ((80 oC) - (0 oC)))

= 57.7 oC

Example - Arithmetic and Logarithmic Mean Temperature, Steam Heating Water

Steam at 2 bar gauge heats water from 20 oC to 50 oC. The saturation temperature of steam at 2 bar gauge is 134 oC.

Note! that steam condenses at a constant temperature. The temperature on the heat exchangers surface on the steam side is constant and determined by the steam pressure.

Arithmetic Mean Temperature Difference can be calculated like

AMTD = ((134 oC) + (134 oC)) / 2 - ((20 oC) + (50 oC)) / 2

= 99 oC

Log Mean Temperature Difference can be calculated like

LMTD = ((134 oC) - (20 oC) - ((134 oC) - (50 oC))) / ln(((134 oC) - (20 oC)) / ((134 oC) - (50 oC)))

= 98.24 oC

Related Topics

• Heat Loss and Insulation - Heat loss from pipes, tubes and tanks - with and without insulation - foam, fiberglass, rockwool and more
• Steam Thermodynamics - Thermodynamics of steam and condensate applications
• Insulation - Heat transfer and heat loss from buildings and technical applications - heat transfer coefficients and insulation methods and to reduce energy consumption
• Thermodynamics - Effects of work, heat and energy on systems

Search Engineering ToolBox

• the most efficient way to navigate the Engineering ToolBox!

SketchUp Extension - Online 3D modeling!

Add standard and customized parametric components - like flange beams, lumbers, piping, stairs and more - to your Sketchup model with the Engineering ToolBox - SketchUp Extension - enabled for use with the amazing, fun and free SketchUp Make and SketchUp Pro .Add the Engineering ToolBox extension to your SketchUp from the Sketchup Extension Warehouse!

Privacy

We don't collect information from our users. Only emails and answers are saved in our archive. Cookies are only used in the browser to improve user experience.

Some of our calculators and applications let you save application data to your local computer. These applications will - due to browser restrictions - send data between your browser and our server. We don't save this data.

Temperature

oC
oF

Length

m
km
in
ft
yards
miles
naut miles

Area

m2
km2
in2
ft2
miles2
acres

Volume

m3
liters
in3
ft3
us gal

Weight

kgf
N
lbf

Velocity

m/s
km/h
ft/min
ft/s
mph
knots

Pressure

Pa (N/m2)
bar
mm H2O
kg/cm2
psi
inches H2O

Flow

m3/s
m3/h
US gpm
cfm

6 17