Convecive Hea Tansfe (6) Foced Convecion (8) Main Andesson
Agenda Convecive hea ansfe Conini eq. Convecive dc flow (inodcion o ch. 8)
Convecive hea ansfe
Convecive hea ansfe
Convecive hea ansfe f flid vämeledande kopp Hea condcing bod w () (,) Q! α( ) A f w
Convecive hea ansfe f vämeledande kopp hea condcing bod w () Q! α( f w) A flid (,)! 0, v, w 0 hea condcion in he flid Inodcion of hea ansfe coeficien: λ & (6 3) f Q/ A 0 α w f w f
Convecive hea ansfe Objecive (of chape 6-11): Deemine α and he paamees inflencing i fo pescibed w () o q w () Q/A
Ode of magnide fo α Medim α W/m²K Ai (1ba); naal convecion -0 Ai (1ba); foced convecion 10-00 Ai (50 ba); foced convecion 00-1000 Wae foced convecion 500-5000 Oganic liqids; foced convecion 100-1000 Condensaion (wae) 000-50000 Condensaion (oganic vapos) 500-10000 Evapoaion, boiling, (wae) 000-100000 Evapoaion, boiling (oganic liqids) 500-50000
How o do i? (o descibe convecive HT) Wha ae he ools? Flid moion: Mass consevaion eqaion (Conini eqn) Momenm eqaions (Newon s second law) Eneg balance in he flid Fis law of hemodnamics fo an open ssem
Conini eq. Epesses ha mass is consan and no desoable ρ τ ( ρ) ( ρv) ( ρw) z 0 (6 4) Especiall fo sead sae, incompessible flow, wo-dimensional case v 0 (6 5)
Resling momenm eqaions dim. : ˆ p F v µ ρ τ ρ : ˆ v v p F v v v v µ ρ τ ρ Inpossible o solveb hand, need o be simplified (chape 6, 7 and 8)
Tempeae Eqaion z c z w v p ρ λ Unpossible o solve b hand, need o be simplified (chape 6, 7 and 8)
Bonda lae appoimaions (lamina case) U (,) δ () Wh diffeen fields fo empeae and veloci??? (,) w δ T
Bonda lae heo developed b Pandl >> v If he bonda lae hickness is ve small If D >>, v, v >>
Bonda lae appoimaions Pandl s heo p() p d dp F v µ ρ ρ c v p ρ λ Fom Navie-Sokes eqn in he -diecion i is fond ha he pesse is independen of Then Navie Sokes in he -diecion is simplified o: Also he empeae field is simplified
Bonda lae appoimaions Pandl s heo p 1 ρu konsan Benollis eqn descibes he flow oside he bonda lae dp d ρu du d P ν ρc p µ c p λ λ The dimensionless Pandl nmbe is inodced
Bonda lae eqaions 0 v d du U v ρ µ P v ρ µ Mass consevaion Momenm consevaion Eneg consevaion
Bonda laes c U U fll blen lae bffe lae viscos sblae lamina bonda lae ansiion blen bonda lae 5 5 10 Re c U c /ν N f (Re,P) 7
Conini eq. (epesses ha mass is consan and no desoable)! m 1 ρ ddz : & m& & ρddz ( ρ) dddz 1 m m d 1 z d d dz Ne mass flow o in -diecion Δ! m ( ρ)d ddz Analogos in - and z-diecions Δ m! ( ρv) d d dz Δm! z ( ρw) dz d d z Neo sömma, ne flow o : Δm! Δm! Δm! z Ne mass flow o Redcion in mass wihin volme elemen
Con. conini eq.! Redcion pe ime ni: ρ τ ρ τ ( ρ) d ddz ( ρv) z ( ρw) Conside a mass balance fo he volme elemen a he pevios page ρ τ ( ρ) ( ρv) ( ρw) z 0 (6 4) Especiall fo sead sae, incompessible flow, wo-dimensional case v 0 (6 5)
Navie Sokes ekvaione (eqs.) Deived fom Newon s second law!! m a F m ρ d d dz a! d dτ, dv dτ, dw dτ b (,, z, τ), v v(,, z, τ), w w(,, z, τ)
Foces F! The sface foces ac on he bonda sfaces of he flid elemen and ae acing as eihe nomal foces o shea foces a. volme foces (F, F, Fz) ae callaed pe ni mass, N/kg b. sesses σ ij N/m
Foces The sface foces ae callaed pe ni aea and ae called sesses F! a. volme foces (F, F, F z ) ae callaed pe ni mass, N/kg b. sesses σ ij N/m σ ij σ σ σ z σ σ σ z σ σ σ z z zz!!! "" "" "z" Sesses fo an elemen dddz
Signs fo he sesses d σ σ σ σ σ σ
. Signs fo he sesses d σ σ σ σ σ σ Resling sesses σ σ ( σ )d σ σ ( σ )d ( σ )d σ σ σ σ ( σ )d
Ne foce in -diecion (fo he 3D case) ( σ )dddz ( σ )dddz ( σ z z )dzdd j ( σ ji )dddz
Eamples of sesses σ p µ e p µ σ σ µ e µ v v σ p µ e p µ
Resling momenm eqaions : ˆ p F v µ ρ τ ρ : ˆ v v p F v v v v µ ρ τ ρ 6.0 6.1
Eneg eq. (Fis law of hemodnamics of an open ssem), Tempeae field eq. d dz z d d Q! dh Neglecing kineic and poenial eneg Ne hea o elemen Change of enhalp flow
. Hea condcion in he flid (callaing he hea flow as in chape 1) Q d! )dddz ( ddz d Q Q Q ddz A Q d λ λ λ λ!!!! )dddz ( Q Q Q d λ Δ!!! 6.4
. Analogos in - and z-diecions (6-7) )dz d d z ( z Q )dd dz ( Q z λ Δ λ Δ!! { } z Q Q Q Q d!!!! Δ Δ Δ Q d! sign convenion fo hea d ddz ) z ( z ) ( ) ( dq λ λ λ!! 6.5 6.6
Enhalp flows and changes Flow of enhalph in he -diecion H! m! h ρ d dz h h dh ρ h d d dz ρ! 6.8 d d dz
Enhalp changes - and z-diecions v h dh! ρ h d d dz ρv w h dh! z ρh d d dz ρw z z d d dz d d dz 6.9 6.30
Toal change in enhalp dh& dh& dh& dh& z v w ρh d d dz z h h h ρ v w d d dz z
Enhalp vs empeae h ( p ) h, dh h p dp h p d 6.33
Enhalp vs empeae c p h p B definiion Fo ideal gases he enhalp is independen of pesse, i.e., ( h / p) 0.Fo liqids, one commonl assmes ha he deivaive ( h / p) is small and/o ha he pesse vaiaion dp is small compaed o he change in empeae. Then geneall one saes dh c p d i.e., enhalph is copled o empeae via he hea capaci
Tempeae Eqaion z c z w v p ρ λ 6.36 Rewiing eqn 6.3 as a fcion of T insead of h
. Chape 8 - Convecive Dc Flow
. Chape 8 - Convecive Dc Flow U 0 gänsskik, bonda lae käna, coe fll bildad sömning, fll developed flow b m! ρa m Re D md ν ma 1 b Paallel plae dc m 1 R Cicla pipe, be lamina if pipe o be Re D < 300
Chape 8 Convecive Dc Flow. U 0 gänsskik, bonda lae käna, coe fll bildad sömning, fll developed flow b L D i 0.0575Re D - gänsskik, bonda lae käna, coe Tpical enance egion flow pofile gänsskik, bonda lae
Con. dc flow If Re D > 300 omslag, ansiion U 0 fll bildad blen sömning Fll developed blen flow laminä gänsskik blen gänsskik Lamina bonda lae Tblen bonda lae
. Pesse dop fll developed flow Δp f L D h ρ m f C Re Re m D h ν D h hdalic diamee 4 väsnisaean 4 coss secion aea mediebeöd omkes peimee m m! ρ A
. Pesse dop - enance egion (cicla pipe) 0.1 /D h Re Dh 0.01 0.001 1E-4 0 1 3 4 5 6 Δp ρ / m Fige 8.4
Convecive hea ansfe fo an isohemal be d d w konsan; consan
Convecive hea ansfe fo an isohemal be d d w konsan; consan Veloci field fll developed m 1 R Hea balance fo an elemen ddπ Hea condcion in adial diecion Enhalp anspo in -diecion
Remembe he fige fom pevios page Eneg eqn fo sead sae ρ c p λ 1 "(8 1)" Bonda condiions: 0 : 0 Commonl called Gaez poblem R : w 0 : 0 (Smme)
Inodce Inodce m (1 - ʹ) o p w, c a R, R, ρ λ ϑ ʹ ʹ ʹ ϑ ʹ ʹ ʹ ʹ ϑ R R R R 1 R 1 a ʹ ϑ ʹ ʹ ʹ ʹ ʹ ϑ ) (1 1 a R m 0) (8 ) (1 1 P Re D ʹ ϑ ʹ ʹ ʹ ʹ ʹ ϑ Nssel appoach sill valid Copling beween velociand aveage veloci
0) (8 ) (1 1 P Re D ʹ ϑ ʹ ʹ ʹ ʹ ʹ ϑ τ 1 a 1 a compaed wih nsead hea condcion: Sead hea condcion
Inspiaion fo one of he eoeical home assignmens Assme ϑ F(ʹ) G(ʹ). Afe some (8.4-8.8) calclaions one finds ϑ i 0 C i G i (ʹ)e β ʹ i / ReD P (8 9) β 0 < β 1 < β < β 3 < β 4
. Tempeae pofile in he hemal enace egion fo a cicla pipe wih nifom wall empeae and fll developed lamina flow β 0.705 β 1 6.667 β 10.67
Blkemp. B m d he enhalp flow of he mie: m! c p B he enhalp flow can be wien R ρ 0 πd cp %"$"#! " Δm" & "h"! B m! 1 ρπ m! R 0 R 0 ρπd d πd ρ 4 m B R 0 R 0 d d (8 34)
Local Nssel nmbe 100 50 N D konsan vämeflöde, nifom hea fl Hasighesfäl ej fll bilda; veloci field no fll developed 10 konsan vämeflöde, nifom hea fl 5 konsan empea, consan wall empeae 4.364 3.656 1 0.001 0.01 0.1 1 Re /D D P
Aveage Hea Tansfe Coefficien 1. 0 w kons If he veloci field is fll developed N D αd λ 0.0668Re 3.656 1 0.04 D D P D / [ Re P D / ] / 3 µ B µ w 0.14 N.B.! Highe vales if veloci field no fll develped. Eq. (8-38) gives he aveage vale. 1/ 3 0.14 αd Re P N 1.86 D µ B D (8 38) λ L / D µ w L / D Re P D < 0.1 w consan
... 0.... q w kons. consan Fll developed flow and empeae fields N D 4.364 (8-50) q w α ( w B ) kons $!#!" kons! w B kons. If B inceases, w ms incease as mch Aveagevale inclding effecs of he hemal enance lengh N D αd λ 1/ 3 1 1.953 / D ReD P 0.07 4.364 ReD P / D / D om < 0.03 Re P / D om > 0.03 Re P D D
.... 0.... q w kons. consan Fll developed flow and empeae fields N D 4.364 (8-50) q w α ( w B ) cons cons $!!! #!!! " w B kons. If B inceases, w ms incease as mch w highes a he ei!