Arithmetic and Logarithmic Mean Temperature Difference

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

According to Newton's Law of Cooling heat transfer rate is related to the instantaneous temperature difference between a hot and a cold media

• in a heat transfer process, the temperature difference vary with position and time

Mean Temperature Difference

The determination of the mean temperature difference in a heat transfer process depends upon the direction of fluid flow involved in the process. The primary and secondary fluid in an heat exchanger process may

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

With saturation of steam the primary fluid temperature can be taken as a constant because 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.

When the secondary fluid passes over the heat transfer surface, the highest rate of heat transfer occurs at the inlet and progressively decays with higher secondary fluid temperature along its way to the outlet.

Logarithmic Mean Temperature Difference - LMTD

The rise in secondary temperature is non-linear and is best represented by a logarithmic calculation. For this purpose the mean temperature difference chosen is termed the

• Logarithmic Mean Temperature Difference or LMTD or DT_LM

LMTD can be expressed as:

LMTD = (TD₂ - TD₁) / ln(TD₂ / TD₁) (1)

where

LMTD = Logarithmic Mean Temperature Difference ^oF (^oC)

TD₁ = T_p1 - T_s2 - Entering primary fluid and leaving secondary fluid temperature difference ^oF (^oC)

TD₂ = T_p2 - T_s1 - Leaving primary fluid and entering secondary fluid temperature difference ^oF (^oC)

Arithmetic Mean Temperature Difference - AMTD

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

• Arithmetic Mean Temperature Difference or AMTD or DT_AM

AMTD can be expressed as:

AMTD = (T_p1 + T_p2) / 2 - (T_s1 + T_s2) / 2 (2)

where

AMTD = Arithmetic Mean Temperature Difference ^oF (^oC)

T_p1 = primary inlet temperature ^oF (^oC)

T_p2 = primary outlet temperature ^oF (^oC)

T_s1 = secondary inlet temperature ^oF (^oC)

T_s2 = 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 heat is transferred as a result of a change of phase in condensation or evaporation heat exchangers, the temperature of the primary or secondary fluid remains constant. The equation can then be simplified by setting

T_p1 = T_p2 or T_s1 = T_s2 ^oF (^oC)

Example - Arithmetic and Log Mean Temperature

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.

• SteamTable

Arithmetic Mean Temperature Difference can be calculated as:

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

= 99 ^oC

Log Mean Temperature Difference can be calculated as:

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

= 98.24 ^oC

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