Log mean temperature differenceFrom

Log mean temperature difference
From Wikipedia, the free encyclopedia
The log mean temperature difference (also known by its initialism 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 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.

Contents [hide]
1 Definition
2 Derivation
3 Assumptions and Limitations
4 References
5 External links
Definition[edit]
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 mean as follows:

LMTD=frac{Delta T_A - Delta T_B}{ln left( frac{Delta T_A}{Delta T_B}
ight ) }
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. With this definition, the LMTD can be used to find the exchanged heat in a heat exchanger:

Q = U imes Ar imes LMTD
Where Q is the exchanged heat duty (in watts), U is the heat transfer coefficient (in watts per kelvin per square meter) and Ar is the exchange area. Note that estimating the heat transfer coefficient may be quite complicated.

This holds both for cocurrent flow, where the streams enter from the same end, and for counter-current flow, where they enter from different ends.

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, such as a shell and tube exchanger with baffles.

Derivation[edit]
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 local exchanged heat at z is proportional to the temperature difference:

q(z) = U (T_2(z)-T_1(z))/D = U (Delta;T(z))/D,
where D is the distance between the two fluids.

The heat that leaves the fluids causes a temperature gradient according to Fourier's law:

frac{mathrm{d},T_1}{mathrm{d},z}=k_a (T_1(z)-T_2(z))=-k_a,Delta T(z)
frac{mathrm{d},T_2}{mathrm{d},z}=k_b (T_2(z)-T_1(z))=k_b,Delta T(z)
Summed together, this becomes

frac{mathrm{d},Delta T}{mathrm{d},z}=frac{mathrm{d},(T_2-T_1)}{mathrm{d},z}=frac{mathrm{d},T_2}{mathrm{d},z}-frac{mathrm{d},T_1}{mathrm{d},z}=KDelta T(z)
where K=ka+kb.

The total exchanged energy is found by integrating the local heat transfer q from A to B:

Q = int^{B}_{A} q(z) dz = frac{U}{D} int^{B}_{A} Delta T(z) dz = frac{U}{D} int^{B}_{A} Delta T ,dz
Use the fact that the heat exchanger area Ar is the pipe length A-B multiplied by the interpipe distance D:

Q = frac{U Ar}{(B-A)} int^{B}_{A} Delta T ,dz = frac{U Ar int^{B}_{A} Delta T ,dz}{int^{B}_{A} ,dz}
In both integrals, make a change of variables from z to Δ T:

Q = frac{U Ar int^{Delta T(B)}_{Delta T(A)} Delta T frac{mathrm{d},z}{mathrm{d},Delta T},d(Delta T)}{int^{Delta T(B)}_{Delta T(A)} frac{mathrm{d},z}{mathrm{d},Delta T},d(Delta T)}
With the relation for Δ T found above, this becomes

Q = frac{U Ar int^{Delta T(B)}_{Delta T(A)} frac{1}{K},d(Delta T)}{int^{Delta T(B)}_{Delta T(A)} frac{1}{K Delta T},d(Delta T)}
Integration is at this point trivial, and finally gives:

Q = U imes Ar imes frac{Delta T(B)-Delta T(A)}{ln [ Delta T(B) / Delta T(A) ]} ,
from which the definition of LMTD follows.

Assumptions and Limitations[edit]
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.
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.
References[edit]
Kay J M & Nedderman R M (1985) Fluid Mechanics and Transfer Processes, Cambridge University Press
External links[edit]
Categories: Heat transfer
Navigation menu
Create accountLog inArticleTalkReadEditView history

Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Wikimedia Shop
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact page
Tools
What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Data item
Cite this page
Print/export
Create a book
Download as PDF
Printable version
Languages
Afrikaans
Español
فارسی
日本語
Polski
Português
Edit links
This page was last modified on 12 April 2014 at 07:49.
Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policyAbout WikipediaDisclaimersContact WikipediaDevelopersMobile viewWikimedia Foundation Powered by MediaWiki

Hasil (Bahasa Indonesia) 1:

Perbedaan suhu rata-rata logDari Wikipedia bahasa Indonesia, ensiklopedia bebasLog berarti perbedaan suhu (juga dikenal oleh para pengembangnya LMTD) digunakan untuk menentukan suhu kekuatan untuk proses transfer panas pendorong dalam sistem aliran, terutama di penukar panas. LMTD adalah rata-rata perbedaan suhu panas dan dingin sungai di setiap akhir penukar logaritma. Semakin besar LMTD, semakin banyak panas yang ditransfer. Penggunaan LMTD muncul secara langsung dari analisis penukar panas dengan laju aliran konstan dan cairan termal properti.Isi [hide] 1 definisiTerbitan 23 asumsi dan keterbatasan4 referensi5 Pranala luarDefinisi [sunting]Kita berasumsi bahwa penukar panas generik memiliki kedua ujung (yang kita sebut "" dan "B") yang berarus panas dan dingin masuk atau keluar di kedua sisi; kemudian, LMTD didefinisikan oleh mean logaritma sebagai berikut:LMTD = frac {T_A Delta - Delta T_B} {ln kiri (frac {T_A Delta} {Delta T_B} ight)}mana ΔTA adalah perbedaan suhu antara dua aliran pada akhir A, dan ΔTB adalah perbedaan suhu antara dua aliran pada akhir B. Dengan definisi ini, LMTD dapat digunakan untuk menemukan bertukar panas di penukar panas: Q = U imes Ar imes LMTDMana Q tugas bertukar panas (dalam watt), U adalah koefisien transfer panas (dalam watt per kelvin per meter persegi) dan Ar adalah daerah asing. Perhatikan bahwa memperkirakan koefisien transfer panas mungkin cukup rumit.Ini memegang baik untuk aliran cocurrent, dimana aliran masuk dari ujung yang sama, dan kontra arus, di mana mereka masuk dari ujung yang berbeda.Dalam salib-aliran, di mana satu sistem, biasanya heat sink, memiliki suhu nominal sama di semua poin pada permukaan transfer panas, hubungan yang sama antara bertukar panas dan memegang LMTD, tetapi dengan faktor koreksi. Faktor koreksi yang diperlukan untuk geometri lain lebih kompleks, seperti shell dan tube exchanger dengan baffle.Derivasi [sunting]Menganggap perpindahan panas ini terjadi di penukar panas sepanjang sumbu z, dari generik koordinat A ke B, antara dua cairan, diidentifikasi sebagai 1 dan 2, suhu yang sepanjang z adalah T1(z) dan T2(z).Panas bertukar lokal di z sebanding dengan perbedaan suhu: q(z) = waktu bumi U (T_2(z)-T_1(z)) = U (Delta; T(z)) / D,dimana D adalah jarak antara dua cairan.Panas yang meninggalkan cairan menyebabkan gradien suhu menurut undang-undang Fourier:Frac {mathrm {d}, T_1} {mathrm {d}, z} = k_a (T_1(z)-T_2(z)) =-k_a, Delta T(z)Frac {mathrm {d}, T_2} {mathrm {d}, z} = k_b (T_2(z)-T_1(z)) = k_b, Delta T(z)Disimpulkan bersama-sama, ini menjadiFrac {mathrm {d}, Delta T}{mathrm{d},z}=frac{mathrm{d},(T_2-T_1)} {mathrm {d}, z} = frac {mathrm {d}, T_2} {mathrm {d}, z}-frac {mathrm {d}, T_1} {mathrm {d}, z} = KDelta T(z)mana K = ka + kb.Total energi bertukar ditemukan dengan mengintegrasikan q transfer panas lokal dari A untuk B: Q = int ^ _ {A} {B} q(z) dz = frac {U} {D} int ^ _ {A} {B} Delta T(z) dz = frac {U} {D} int ^ _ {B} {} Delta T, dzMenggunakan fakta bahwa daerah penukar panas Ar panjang pipa dikalikan dengan jarak interpipe D: A-B Q = int frac {U Ar}{(B-A)} ^ _ {B} {} Delta T, dz = frac {int U Ar ^ _ {B} {} Delta T, dz} {int ^ _ {B} {A}, dz} Dalam integral kedua, membuat perubahan variabel dari z Δ T: Q = frac {int U Ar ^ _ {Delta T(B)} {Delta T(A)} Delta T frac {{d} mathrm, z} {mathrm {d}, Delta T}, d (Delta T)} {int ^ frac _ {Delta T(A)} {Delta T(B)} {{d} mathrm, z} {mathrm {d}, Delta T}, d (Delta T)} Dengan hubungan untuk Δ T ditemukan di atas, ini menjadi Q = frac {int U Ar ^ frac _ {Delta T(A)} {Delta T(B)} {1} {K}, d (Delta T)} {int ^ frac _ {Delta T(A)} {Delta T(B)} {1} {K Delta T}, d (Delta T)} Integrasi saat ini sepele, dan akhirnya memberikan: Q = U imes Ar imes frac {Delta T (B)-Delta T(A)} {ln [Delta T(B) / Delta T(A)]},yang definisi LMTD berikut.Asumsi-asumsi dan keterbatasan [sunting]Telah diasumsikan bahwa laju perubahan untuk suhu cairan kedua sebanding dengan perbedaan suhu; asumsi ini ini berlaku untuk cairan dengan panas spesifik konstan, yang merupakan penjelasan yang baik cairan mengubah suhu rentang relatif kecil. Namun, jika perubahan panas spesifik, pendekatan LMTD tidak lagi akan akurat.Kasus tertentu mana LMTD ini tidak berlaku di condensers dan reboilers, mana panas laten yang terkait dengan perubahan fase membuat hipotesis tidak valid.Juga telah diasumsikan bahwa koefisien transfer panas (U) adalah konstan, dan tidak fungsi dari suhu. Jika hal ini tidak terjadi, pendekatan LMTD lagi akan kurang berlakuThe 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.References[edit]Kay J M & Nedderman R M (1985) Fluid Mechanics and Transfer Processes, Cambridge University PressExternal links[edit]Categories: Heat transferNavigation menuCreate accountLog inArticleTalkReadEditView historyMain pageContentsFeatured contentCurrent eventsRandom articleDonate to WikipediaWikimedia ShopInteractionHelpAbout WikipediaCommunity portalRecent changesContact pageToolsWhat links hereRelated changesUpload fileSpecial pagesPermanent linkPage informationData itemCite this pagePrint/exportCreate a bookDownload as PDFPrintable versionLanguagesAfrikaansEspañolفارسی日本語PolskiPortuguêsEdit linksThis page was last modified on 12 April 2014 at 07:49.Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.Privacy policyAbout WikipediaDisclaimersContact WikipediaDevelopersMobile viewWikimedia Foundation Powered by MediaWiki