190:
36:
1793:
397:
996:
1992:
537:
In a cross-flow, in which one system, usually the heat sink, has the same nominal temperature at all points on the heat transfer surface, a similar relation between exchanged heat and LMTD holds, but with a correction factor. A correction factor is also required for other more complex geometries,
2173:
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 difference had different signs on the two sides of the exchanger, the argument to the logarithm function would be
1572:
1594:
1197:
2129:
202:
727:
1817:
1395:
1428:
430:. When the two temperature differences are equal, this formula does not directly resolve, so the LMTD is conventionally taken to equal its limit value, which is in this case trivially equal to the two differences.
1788:{\displaystyle Q={\frac {UAr\displaystyle \int _{\Delta T(A)}^{\Delta T(B)}\Delta T{\frac {dz}{d\Delta T}}\,d(\Delta T)}{\displaystyle \int _{\Delta T(A)}^{\Delta T(B)}{\frac {dz}{d\Delta T}}\,d(\Delta T)}}}
732:
710:
1030:
392:{\displaystyle \mathrm {LMTD} ={\frac {\Delta T_{A}-\Delta T_{B}}{\ln \left({\frac {\Delta T_{A}}{\Delta T_{B}}}\right)}}={\frac {\Delta T_{A}-\Delta T_{B}}{\ln \Delta T_{A}-\ln \Delta T_{B}}}}
483:
2003:
991:{\displaystyle {\begin{aligned}{\frac {d\,T_{1}}{dz}}&=k_{a}(T_{1}(z)-T_{2}(z))=-k_{a}\,\Delta T(z)\\{\frac {d\,T_{2}}{dz}}&=k_{b}(T_{2}(z)-T_{1}(z))=k_{b}\,\Delta T(z)\end{aligned}}}
1987:{\displaystyle Q={\frac {UAr\displaystyle \int _{\Delta T(A)}^{\Delta T(B)}{\frac {1}{K}}\,d(\Delta T)}{\displaystyle \int _{\Delta T(A)}^{\Delta T(B)}{\frac {1}{K\Delta T}}\,d(\Delta T)}}}
53:
2144:
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
180:
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
2148:, 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.
2163:
associated to phase change is a special case of the hypothesis. For a condenser, the hot fluid inlet temperature is then equivalent to the hot fluid exit temperature.
1256:
168:, 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
164:
of the temperature difference between the hot and cold feeds at each end of the double pipe exchanger. For a given heat exchanger with constant area and
100:
72:
1567:{\displaystyle Q={\frac {UAr}{B-A}}\int _{A}^{B}\Delta T\,dz={\frac {UAr\displaystyle \int _{A}^{B}\Delta T\,dz}{\displaystyle \int _{A}^{B}\,dz}}}
79:
86:
119:
68:
1192:{\displaystyle {\frac {d\Delta T}{dz}}={\frac {d(T_{2}-T_{1})}{dz}}={\frac {d\,T_{2}}{dz}}-{\frac {d\,T_{1}}{dz}}=K\Delta T(z)}
611:
57:
2124:{\displaystyle Q=U\times Ar\times {\frac {\Delta T(B)-\Delta T(A)}{\ln \left({\frac {\Delta T(B)}{\Delta T(A)}}\right)}}}
439:
2170:) is constant, and not a function of temperature. If this is not the case, the LMTD approach will again be less valid
93:
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2152:
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2218:
1390:{\displaystyle Q=D\int _{A}^{B}q(z)dz=UD\int _{A}^{B}\Delta T(z)dz=UD\int _{A}^{B}\Delta T\,dz,}
17:
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181:
433:
With this definition, the LMTD can be used to find the exchanged heat in a heat exchanger:
530:
This holds both for cocurrent flow, where the streams enter from the same end, and for
169:
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for stagnant solvents with dilute solutes to simplify the bulk flow problem.
527:
Note that estimating the heat transfer coefficient may be quite complicated.
1238:
The total exchanged energy is found by integrating the local heat transfer
715:
The heat that leaves the fluids causes a temperature gradient according to
2156:
489:
546:
Assume heat transfer is occurring in a heat exchanger along an axis
1014:
are the thermal conductivities of the intervening material at points
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29:
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It has also been assumed that the heat transfer coefficient (
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is the temperature difference between the two streams at end
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is the temperature difference between the two streams at end
2180:
Changes in kinetic energy and potential energy are neglected
193:
The LMTD illustrated in a countercurrent temperature profile
1997:
Integration at this point is trivial, and finally gives:
1422:
the heat exchanger area of the pipe, and use this fact:
705:{\displaystyle q(z)=U(T_{2}(z)-T_{1}(z))=U\;\Delta T(z)}
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is clearly the pipe length, which is distance along
1577:In both integrals, make a change of variables from
60:. Unsourced material may be challenged and removed.
27:
Method of calculating heat transfer in flow systems
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1787:
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477:
391:
538:such as a shell and tube exchanger with baffles.
478:{\displaystyle Q=U\times A\times \mathrm {LMTD} }
605:is proportional to the temperature difference:
1418:is the circumference. Multiplying those gives
8:
1022:respectively. Summed together, this becomes
534:flow, where they enter from different ends.
2135:from which the definition of LMTD follows.
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120:Learn how and when to remove this message
69:"Logarithmic mean temperature difference"
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138:logarithmic mean temperature difference
2256:Fluid Mechanics and Transfer Processes
7:
2191:logarithmic mean pressure difference
2185:Logarithmic Mean Pressure Difference
2177:No phase change during heat transfer
1024:
558:, between two fluids, identified as
58:adding citations to reliable sources
2254:Kay J M & Nedderman R M (1985)
2243:"MIT web course on Heat Exchangers"
2151:A particular case for the LMTD are
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2174:negative, which is not allowable.
601:The local exchanged heat flux at
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45:needs additional citations for
18:Log mean temperature difference
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2258:, Cambridge University Press
499:is the exchanged heat duty (
2139:Assumptions and limitations
1807:
566:, whose temperatures along
144:) is used to determine the
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550:, from generic coordinate
511:heat transfer coefficient
166:heat transfer coefficient
146:temperature driving force
2189:A related quantity, the
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1798:With the relation for
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2219:"Basic Heat Transfer"
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523:is the exchange area.
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54:improve this article
2197:, is often used in
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162:logarithmic average
134:thermal engineering
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517:per square meter),
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182:logarithmic mean
160:. The LMTD is a
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158:heat exchangers
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2199:mass transfer
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2223:www.swep.net
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184:as follows:
179:
172:properties.
154:flow systems
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52:Please help
47:verification
44:
2161:latent heat
513:(watts per
2228:2020-05-12
2205:References
2153:condensers
1805:(equation
542:Derivation
488:where (in
176:Definition
80:newspapers
2157:reboilers
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110:May 2020
509:is the
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101:JSTOR
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