# Log mean temperature difference

Log mean temperature difference

The log mean temperature difference (LMTD) is used to determine the temperature driving force for heat transfer in flow systems (most notably in heat exchangers). The LMTD is a logarithmic average of the temperature difference between the hot and cold streams at each end of the exchanger. The use of the LMTD arises straightforwardly from the analysis of a heat exchanger with constant flow rate and fluid thermal properties.

For Countercurrent flow (i.e. where the hot stream, liquid or gas, goes from say left to right, and the cold stream, again liquid or gas goes from right to left), is given by the following equation:

:$LMTD=frac\left\{\left(T_1-t_2\right)-\left(T_2-t_1\right)\right\}\left\{lnfrac\left\{\left(T_1-t_2\right)\right\}\left\{\left(T_2-t_1\right)$

And for Parallel flow (i.e. where the hot stream,nliquid or gas, goes from say left to right, and so does the cold stream), is given by the following equation:

:$LMTD=frac\left\{\left(T_1-t_1\right)-\left(T_2-t_2\right)\right\}\left\{lnfrac\left\{\left(T_1-t_1\right)\right\}\left\{\left(T_2-t_2\right)$

:$T_1=$Hot Stream Inlet Temp. :$T_2=$Hot Stream Outlet Temp.

:$t_1=$Cold Stream Inlet Temp. :$t_2=$Cold Stream Outlet Temp.

It makes no difference which temperature differential is 1 or 2 as long as the nomenclature is consistent.The larger the LMTD, the more heat is transferred.

Yet a third type of unit is a cross-flow exchanger, in which one system (usually the heat sink) has the same nominal temperature at all points on the heat transfer surface. This follows similar mathematics, in its dependence on the LMTD, except that a correction factor F often needs to be included in the heat transfer relationship.

There are times when the four temperatures used to calculate the LMTD are not available, and the NTU method may then be preferable.

Derivation

Assume heat transfer is occurring between two fluids ($T_1$ and $T_2$) with a temperature "difference" of $Delta T\left(A\right)$ at point A and $Delta T\left(B\right)$ at point B (where $Delta T\left(z\right)=T_2\left(z\right)-T_1\left(z\right)$). The direction of fluid flow does not need to be considered. Since LMTD is the "average" temperature difference of the two streams between points A and B the following formula defines LMTD:

:$LMTD=frac\left\{int^\left\{B\right\}_\left\{A\right\} Delta T\left(z\right),dz\right\}\left\{int^\left\{B\right\}_\left\{A\right\},dz\right\}$ where $z$ is the distance parallel to the motion of the two fluids.

The rate of change of the temperature of the two fluids, $T_1$ and $T_2$ respectively, is proportional to the temperature difference between the two fluids:

:$frac\left\{dT_1\right\}\left\{dz\right\}=k_a \left(T_1\left(z\right)-T_2\left(z\right)\right)=-k_aDelta T\left(z\right)$

:$frac\left\{dT_2\right\}\left\{dz\right\}=k_b \left(T_2\left(z\right)-T_1\left(z\right)\right)=k_bDelta T\left(z\right)$

Therefore:

:$frac\left\{d\left(Delta T\right)\right\}\left\{dz\right\}=frac\left\{d\left(T_2-T_1\right)\right\}\left\{dz\right\}=frac\left\{dT_2\right\}\left\{dz\right\}-frac\left\{dT_1\right\}\left\{dz\right\}=KDelta T\left(z\right)$ where $K=k_a+k_b$

This gives us a value for $dz$:

:$dz=frac\left\{1\right\}\left\{K\right\},frac\left\{d\left(Delta T\right)\right\}\left\{Delta T\right\}$

Substituting back into our formula for LMTD:

:$LMTD=frac\left\{int^\left\{B\right\}_\left\{A\right\} Delta T\left(z\right),dz\right\}\left\{int^\left\{B\right\}_\left\{A\right\},dz\right\}=frac\left\{int^\left\{Delta T\left(B\right)\right\}_\left\{Delta T\left(A\right)\right\} frac\left\{1\right\}\left\{K\right\},d\left(Delta T\right)\right\}\left\{int^\left\{Delta T\left(B\right)\right\}_\left\{Delta T\left(A\right)\right\} frac\left\{1\right\}\left\{K\right\},frac\left\{d\left(Delta T\right)\right\}\left\{Delta T$

Integrating gives:

:$LMTD=frac\left\{int^\left\{Delta T\left(B\right)\right\}_\left\{Delta T\left(A\right)\right\} frac\left\{1\right\}\left\{K\right\},d\left(Delta T\right)\right\}\left\{int^\left\{Delta T\left(B\right)\right\}_\left\{Delta T\left(A\right)\right\} frac\left\{1\right\}\left\{K\right\},frac\left\{d\left(Delta T\right)\right\}\left\{Delta T=frac\left\{Delta T\right] ^\left\{Delta T\left(B\right)\right\}_\left\{Delta T\left(A\right)\left\{ln\left\{Delta T\right\}\right] ^\left\{Delta T\left(B\right)\right\}_\left\{Delta T\left(A\right)=frac\left\{Delta T\left(B\right)-Delta T\left(A\right)\right\}\left\{ln\left(frac\left\{Delta T\left(B\right)\right\}\left\{Delta T\left(A\right)\right\}\right)\right\}$

Trivial case:

:$\left\{Delta T\left(z\right) = C\right\}$ for all $z$

:$LMTD=frac\left\{int^\left\{B\right\}_\left\{A\right\} Delta T\left(z\right),dz\right\}\left\{int^\left\{B\right\}_\left\{A\right\},dz\right\}=frac\left\{Cint^\left\{B\right\}_\left\{A\right\},dz\right\}\left\{int^\left\{B\right\}_\left\{A\right\},dz\right\}=C$

Assumptions

It has been assumed that the rate of change for the temperature of both fluids is proportional to the temperature difference; this assumption is valid for fluids with a constant specific heat, which is a good description of fluids changing temperature over a relatively small range. However, if the specific heat changes, the LMTD approach will no longer be accurate.

A particular case where the LMTD is not applicable are condensers and reboilers, where the latent heat associated to phase change makes the hypothesis invalid.

References

Kay J M & Nedderman R M (1985) "Fluid Mechanics and Transfer Processes", Cambridge University Press

Simple LMTD calculator: http://www.fridgetech.com/calculators/lmtd2.html

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