Log mean temperature difference
Encyclopedia
The log mean temperature difference (also known by its acronym LMTD) is used to determine the temperature driving force for heat transfer
in flow systems, most notably in heat exchanger
s. The LMTD is a logarithmic average of the temperature difference between the hot and cold streams at each end of the exchanger. The larger the LMTD, the more heat is transferred. The use of the LMTD arises straightforwardly from the analysis of a heat exchanger with constant flow rate and fluid thermal properties.
as follows:
where ΔTA is the temperature difference between the two streams at end A, and ΔTB is the temperature difference between the two streams at end B.
This equation is valid both for parallel flow, where the streams enter from the same end, and for counter-current
flow, where they enter from different ends.
A third type of flow is cross-flow, 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.
Where Q is the exchanged heat duty (in watt
s), U is the heat transfer coefficient (in watts per kelvin
per square meter) and A is the exchange area. Note that estimating the heat transfer coefficient may be quite complicated.
The temperature differences are ΔT(A) at point A and ΔT(B) at point B, having defined ΔT(z)=T2(z)-T1(z).
Note that the direction of fluid flow does not need to be considered; it is also unimportant which stream is the hot and which is the cold one, as a change of role will be represented by negative numbers. Since LMTD is the average temperature difference of the two streams between A and B, it is defined by the following formula:
Assumption:The rate of change of the temperature of the two fluids is proportional to the temperature difference between them:
This gives:
where K=ka+kb.
We can now express dz as a function of ΔT:
Substituting this expression back into our formula for LMTD, we can remove dz from it:
K is constant and can be simplified. Integration is at this point trivial, and finally gives:
Heat transfer
Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...
in flow systems, most notably in heat exchanger
Heat exchanger
A heat exchanger is a piece of equipment built for efficient heat transfer from one medium to another. The media may be separated by a solid wall, so that they never mix, or they may be in direct contact...
s. The LMTD is a logarithmic average of the temperature difference between the hot and cold streams at each end of the exchanger. The larger the LMTD, the more heat is transferred. The use of the LMTD arises straightforwardly from the analysis of a heat exchanger with constant flow rate and fluid thermal properties.
Definition
We assume that a generic heat exchanger has two ends (which we call "A" and "B") at which the hot and cold streams enter or exit on either side; then, the LMTD is defined by the logarithmic meanLogarithmic mean
In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient...
as follows:
where ΔTA is the temperature difference between the two streams at end A, and ΔTB is the temperature difference between the two streams at end B.
This equation is valid both for parallel flow, where the streams enter from the same end, and for counter-current
Countercurrent exchange
Countercurrent exchange is a mechanism occurring in nature and mimicked in industry and engineering, in which there is a crossover of some property, usually heat or some component, between two flowing bodies flowing in opposite directions to each other. The flowing bodies can be liquids, gases, or...
flow, where they enter from different ends.
A third type of flow is cross-flow, 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
NTU method
The Number of Transfer Units Method is used to calculate the rate of heat transfer in heat exchangers when there is insufficient information to calculate the Log-Mean Temperature Difference...
may then be preferable.
Application
Once calculated, the LMTD is usually applied to calculate the heat transfer in an exchanger according to the simple equation:Where Q is the exchanged heat duty (in watt
Watt
The watt is a derived unit of power in the International System of Units , named after the Scottish engineer James Watt . The unit, defined as one joule per second, measures the rate of energy conversion.-Definition:...
s), U is the heat transfer coefficient (in watts per kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
per square meter) and A is the exchange area. Note that estimating the heat transfer coefficient may be quite complicated.
Derivation
Assume heat transfer is occurring in a heat exchanger along an axis z, from generic coordinate A to B, between two fluids, identified as 1 and 2, whose temperatures along z are T1(z) and T2(z).The temperature differences are ΔT(A) at point A and ΔT(B) at point B, having defined ΔT(z)=T2(z)-T1(z).
Note that the direction of fluid flow does not need to be considered; it is also unimportant which stream is the hot and which is the cold one, as a change of role will be represented by negative numbers. Since LMTD is the average temperature difference of the two streams between A and B, it is defined by the following formula:
Assumption:The rate of change of the temperature of the two fluids is proportional to the temperature difference between them:
This gives:
where K=ka+kb.
We can now express dz as a function of ΔT:
Substituting this expression back into our formula for LMTD, we can remove dz from it:
K is constant and can be simplified. Integration is at this point trivial, and finally gives:
Assumptions and Limitations
- 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 condensersCondenser (heat transfer)In systems involving heat transfer, a condenser is a device or unit used to condense a substance from its gaseous to its liquid state, typically by cooling it. In so doing, the latent heat is given up by the substance, and will transfer to the condenser coolant...
and reboilerReboilerReboilers are heat exchangers typically used to provide heat to the bottom of industrial distillation columns. They boil the liquid from the bottom of a distillation column to generate vapors which are returned to the column to drive the distillation separation....
s, where the latent heatLatent heatLatent heat is the heat released or absorbed by a chemical substance or a thermodynamic system during a process that occurs without a change in temperature. A typical example is a change of state of matter, meaning a phase transition such as the melting of ice or the boiling of water. The term was...
associated to phase change makes the hypothesis invalid.
- It has also been assumed that the heat transfer coefficient (U) is constant, and not a function of temperature. If this is not the case, the LMTD approach will again be less valid
- The LMTD is a steady-state concept, and cannot be used in dynamic analyses. In particular, if the LMTD were to be applied on a transient in which, for a brief time, the temperature differential had different signs on the two sides of the exchanger, the argument to the logarithm function would be negative, which is not allowable.