Convective Heat Transfer (6) Forced Convection (8) Martin Andersson

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1 Convecive Hea Tansfe (6) Foced Convecion (8) Main Andesson

2 Agenda Convecive hea ansfe Conini eq. Convecive dc flow (inodcion o ch. 8)

3 Convecive hea ansfe

4 Convecive hea ansfe

5 Convecive hea ansfe f flid vämeledande kopp Hea condcing bod w () (,) Q α( ) A f w

6 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

7 Convecive hea ansfe Objecive (of chape 6-11): Deemine α and he paamees inflencing i fo pescibed w () o q w () Q/A

8 Ode of magnide fo α Medim α W/m²K Ai (1ba); naal convecion -0 Ai (1ba); foced convecion Ai (50 ba); foced convecion Wae foced convecion Oganic liqids; foced convecion Condensaion (wae) Condensaion (oganic vapos) Evapoaion, boiling, (wae) Evapoaion, boiling (oganic liqids)

9 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

10 Conini eq. Epesses ha mass is consan and no desoable ρ ( ρ) ( ρv) ( ρw) τ z 0 (6 4) Especiall fo sead sae, incompessible flow, wo-dimensional case v 0 (6 5)

11 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 solve b hand, need o be simplified (chape 6, 7 and 8)

12 Tempeae Eqaion z c z w v p ρ λ Unpossible o solve b hand, need o be simplified (chape 6, 7 and 8)

13 Bonda lae appoimaions (lamina case) U (,) δ() Wh diffeen fields fo empeae and veloci??? (,) w δ T

14 Bonda lae heo developed b Pandl >> v If he bonda lae hickness is ve small If D >>, v, v >>

15 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

16 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

17 Bonda lae eqaions 0 v d du U v ρ µ P v ρ µ Mass consevaion Momenm consevaion Eneg consevaion

18 Bonda laes c U U fll blen lae bffe lae viscos sblae lamina bonda lae ansiion blen bonda lae Re c U c /ν N f (Re,P) 7

19 Conini eq. (epesses ha mass is consan and no desoable) m 1 ρ ddz : m ρ d dz ( ρ) d d dz 1 m m d 1 z d d dz Ne mass flow o in -diecion m ( ρ)d ddz Analogos in - and z-diecions m ( ρv) d d dz m z ( ρw) dz d d z Ne flow o : m m m z Ne mass flow o Redcion in mass wihin volme elemen

20 Con. conini eq. Redcion pe ime ni: ρ τ ρ τ d d dz ( ρ) ( 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)

21 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)

22 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, ni??? b. sesses σ ij ni???

23 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

24 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

25 Eamples of sesses σ p µ e p µ σ σ µ e µ v v σ p µ e p µ

26 Resling momenm eqaions : ˆ p F v µ ρ τ ρ : ˆ v v p F v v v v µ ρ τ ρ

27 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

28 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 6.4

29 . Analogos in - and z-diecions (6-7) )dz d d z ( z Q )dd dz ( Q z λ λ { } z Q Q Q Q d Q d signconvenionfo hea d ddz ) z ( z ) ( ) ( dq λ λ λ

30 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 d d dz 6.8

31 Enhalp changes - and z-diecions v h dh ρ h d d dz ρv w h dh z ρh d d dz ρ w z z d d dz d d dz

32 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

33 Enhalp vs empeae h ( p ) h, dh h p dp h p d 6.33

34 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

35 Tempeae Eqaion 6.36 Rewiing eqn 6.3 as a fcion of insead of h z c z w v p ρ λ

36 Smma Veloci pofile eqals he empeae pofile when P1 Re c 300 Re c

37 . Chape 8 - Convecive Dc Flow

38 . 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

39 Chape 8 Convecive Dc Flow. U 0 gänsskik, bonda lae käna, coe fll bildad sömning, fll developed flow b L D i Re D - gänsskik, bonda lae käna, coe Tpical enance egion flow pofile gänsskik, bonda lae

40 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

41 . Pesse dop fll developed flow p 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

42 . Pesse dop - enance egion (cicla pipe) 0.1 /D h Re Dh E p ρ / m Fige 8.4

43 Convecive hea ansfe fo an isohemal be d d w konsan; consan

44 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

45 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)

46 Inodce Inodce m (1 - ) o p w, c a R, R, ρ λ ϑ ϑ ϑ R R R R 1 R 1 a ϑ ϑ ) (1 1 a R m 0) (8 ) (1 1 P Re D ϑ ϑ Nssel appoach sill valid Copling beween veloci and aveage veloci

47 0) (8 ) (1 1 P Re D ϑ ϑ τ 1 a 1 a compaed wih nsead hea condcion: Sead hea condcion

48 Inspiaion fo one of he eoeical home assignmens Assme ϑ F( ) G( ). Afe some ( ) calclaions one finds ϑ i 0 C i G i ( )e β i / ReD P (8 9) β 0 < β 1 < β < β 3 < β 4

49 . Tempeae pofile in he hemal enace egion fo a cicla pipe wih nifom wall empeae and fll developed lamina flow β β β 10.67

50 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 " m" "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)

51 Local Nssel nmbe 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 /D Re P D

52 Aveage Hea Tansfe Coefficien 1. 0 w kons If he veloci field is fll developed N D αd λ Re ReD D P D / [ 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/ 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

53 q w kons. consan Fll developed flow and empeae fields N D (8-50) q w α ( w B ) kons kons w B kons. If B inceases, w ms incease as mch Aveage vale inclding effecs of he hemal enance lengh N D αd λ 1/ 3 1 / D / D om < 0.03 ReD P ReD P 0.07 / D ReD P om > 0.03 / D ReD P

54 q w kons. consan Fll developed flow and empeae fields N D (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!

Convective Heat Transfer (6) Forced Convection (8) Martin Andersson

Convective Heat Transfer (6) Forced Convection (8) Martin Andersson 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