CONVECTION IN MICROCHANNELS

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Noah Robbins
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1 CAPER. Intrductin CONVECION IN MICROCANNELS.. Cntinuu and hernaic ythei. Previu chater are baed n tw fundaental autin: () Cntinuu: Navier-Stke equatin, and the energy equatin are alicable () hernaic equilibriu: N-velcity li and n-teerature ju at bundarie. Validity criterin: he Knuden nuber: λ Kn (.) λ i the ean free ath. Cntinuu: valid fr: N-li, n-teerature ju: D e Kn < 0. (.3a) Kn < 0.00 (.3b).. Surface Frce Surface area t vlue rati increae a channel ize i decreae. Surface frce bece re irtant a channel ize i reduced... Chater Sce Claificatin Gae v. liquid Rarefactin Creibility Velcity li and teerature ju able. Analytic lutin: Cuette and Pieuille flw R ρ 7 μ 0 ga 3 J/kg K kg/. Baic Cnideratin kg/ Air Mean Free Path eliu Ideal ga: ydrgen μ π Nitrgen λ R (.) Oxygen Prertie if variu gae; able. λ μ

2 .. Why Micrchannel? he heat tranfer cefficient increae a channel ize i decreaed. Exaine fully develed flw thrugh tube and nte the effect f diaeter..3 Claificatin. Baed n Knuden nuber k h (.3) D Kn < < Kn < < Kn < 0 0 < Kn cntinuu, cntinuu, n li flw li flw tranitin flw free lecular flw (.).. Macr and Micrchannel Macrchannel Cntinuu and thernaic equilibriu del alie. N-velcity li and n-teerature ju. Micrchannel Failure f acrchannel thery and crrelatin. Ditinguihing factr: tw and three dieninal effect, axial cnductin, diiatin, teerature deendent rertie, velcity li and teerature ju at the bundarie and the increaing dinant rle f urface frce...5 Gae v. Liquid Mean free ath f liquid are uch aller than the f gae. Onet f failure f thernaic equilibriu and cntinuu i nt well defined fr liquid. Surface frce fr liquid bece re irtant. Liquid are alt increible while gae are creible..3 General Feature Rarefactin: Knuden nuber effect. Creibility: large channel reure dr, change in denity (creibility). Diiatin: Increaed vicu effect..3. Flw Rate N-velcity li: Fig..3a Velcity li: Fig..3b Flw rate Q : Macrchannel thery underetiate flw rate: (a) Fig..3 (b)

3 .3. Frictin Factr Frictin cefficientc f Frictin factr f C f f Q Q e t 3 > (.5) τ (.37a) w ( / ) ρ u D Δ (.6) L ρ u Fully develed lainar flw in acrchannel: f Re P (.7) P i knwn a the Pieuille nuber Macrchannel thery de nt redict P. he fllwing rati i a eaure f redictin errr ( P) ( P) P aear t deend n the Reynld nuber. Bth increae and a decreae in.3.3 ranitin t urbulent flw Macrchannel e C * (.8) t C are rerted. ud Re t 300 (6.) ν Micrchannel: rerted tranitin Reynld nuber ranged fr 300 t 6, Nuelt nuber Macrchannel: Fully develed lainar flw: cntant Nuelt nuber, indeendent f Reynld nuber. Micrchannel: Macrchannel thery de nt redict Nu. he fllwing rati i a eaure f rerted dearture fr acrchannel redictin. Gverning Equatin ( Nu) e 0. < < 00 ( Nu) In the li-flw dain, 0.00< Kn < 0., the cntinuity, Navier Stke equatin, and energy equatin are valid. Irtant effect: Creibility, axial cnductin, and diiatin. t (.9)

4 .. Creibility Creibility affect reure dr, Pieuille nuber and Nuelt nuber... Axial Cnductin Axial cnductin i neglected in acrchannel fr Peclet nuber greater than 00. Micrchannel tyically erate at lw Peclet nuber. Axial cnductin ay be irtant. Axial cnductin increae the Nuelt nuber in the velcity-li dain...3. Diiatin Diiatin bece irtant when the Mach nuber i cle t unity r larger..5 Velcity Sli and eerature Ju Bundary Cnditin In icrchannel fluid velcity i nt the ae a urface velcity. he velcity li cnditin i σ u u( x,0) u( x,0) u λ (.0) σ n u (x,0) fluid axial velcity at urface u urface axial velcity x axial crdinate n nral crdinate eaured fr the urface σ tangential entu accdating cefficient u Ga teerature at a urface differ fr urface teerature: σ γ λ ( x,0) ( x,0) σ + γ Pr n (x,0) fluid teerature at the bundary urface teerature γ c / c v, ecific heat rati σ energy accdating cefficient u σ andσ are aue equal t unity. u (.) (.0) and (.) are valid fr gae..6 Analytic Slutin: Sli Flw Cnider Cuette and Pieuille flw. Alicatin: MEMS. heral bundary cnditin: Unifr urface teerature and unifr urface heat flux. Exaine the effect f rarefactin and creibility.

5 .6. Autin () Stea tate () Lainar flw (3) w-dieninal () Sli flw regie (0.00 < Kn < 0.) (5) Ideal ga (6) Cntant vicity, cnductivity, and ecific heat (7) Negligible lateral variatin f denity and reure (8) Negligible diiatin (unle therwie tated) (9) Negligible gravity (0) he accdatin cefficient are aued equal t unity, σ σ Cuette Flw with Vicu Diiatin: Parallel Plate with Surface Cnvectin Statinary lwer late, ving uer late. Inulated lwer late, cnvectin at the uer late. y h u Deterine: () Velcity ditributin u x () Ma flw rate (3) Nuelt nuber Fig..6 Flw Field x-cnent f the Navier-Stke equatin fr creible, cntant vicity flw (.9), ilifie t d u 0 (.) u Bundary cnditin: aly (.0) du( x,0) u( x,0) λ (g) Slutin du( x, ) u( x, ) u λ (h) Ma Flw Rate. he flw rate,, fr a channel f width W i u u y ( + Kn) (.) + Kn W 0 ρ u (.5) 5

6 6 (.) int (.5) Macrchannel flw hu Nuelt Nuber. Defined a eat tranfer cefficient h: Subtitute int (l) u ρw (.6) u ρ W (.7) k theral cnductivity f fluid fluid teerature functin (variable) fluid ean teerature NOE: late teerature (.8) h Nu (l) k ( ) k h ( ) Nu (.9) Fluid teerature at the urface, ( x, ), i nt equal t urface teerature. Surface teerature i unknwn in thi exale Relatin between ( x, ) and i given by the teerature ju cnditin: γ λ ( x, ) ( x, ) + + γ Pr (.0) Mean teerature u 0 u eerature ditributin: Energy equatin ilifie t (.) k + μφ 0 (.3)

7 7 Diiatin functin Φ : (.) int (.3) d u Φ (.) μ k du (.5) Bundary cnditin Ue (.0) d (0) 0 d ( ) k h ( d ( ) γ λ ( x, ) k h ( x, ) + + γ Pr n Slutin: Ue (.) fr u, ubtitute int (.5), lve and ue bundary cnditin () and (n) ϕ kϕ ϕ γ Kn y ϕ + (.6) h γ + Pr NOE: Nuelt nuber: Ue (.6) t frulate, (.9) Nu + ) () (n) μ u ϕ ( ) () k + Kn 8 3 8( + Kn) 8γ ( + Kn) Kn Kn + γ + Pr he Nuelt nuber i indeendent f Bit nuber. d ( ) and, ubtitute int (.7) he Nuelt nuber i indeendent f the Reynld nuber. hi i al the cae with acrchannel flw. he Nuelt nuber deend n the fluid (Pr andγ ). Nuelt nuber fr acrchannel flw, Nu : et Kn 0 in (.7) hu Nu 8 (.8)

8 8 Nu Nu + Kn 8 8γ ( + Kn Kn + Kn + 3 γ + Pr ) (.9).6.3 Fully Develed Pieuille Channel Flw: Unifr Surface Flux Inlet and utlet reure are i and Surface heat flux: q Deterine: () Velcity ditributin () Preure ditributin (3) Ma flw rate () Nuelt nuber Pieuille flw in icrchannel differ fr acrchannel a fllw: Strealine are nt arallel. Lateral velcity cnent v de nt vanih. Axial velcity change with axial ditance. Axial reure gradient i nt linear. Creibility and rarefactin are irtant. Autin. See Sectin.6.. Additinal autin: () Itheral flw. () Negligible inertia frce. u (3) he dinant vicu frce i μ. Flw Field. Deterine the axial velcity ditributin. Axial cnent f the Navier-Stke equatin Bundary cnditin Slutin u + μ 0 (c) u( x,0) 0 (e) u( x, / ) u( x, / ) λ (f) d y u + Kn( ) 8μ dx (.30) Mut deterine reure ditributin and lateral velcity v. Cntinuity fr creible flw: / / y x Fig..7

9 ( ρ u) + ( ρv) 0 (h) Ue ideal ga law in (h) ( v) ( u) (i) (.30) int (i) d y ( v) ( + Kn( ) ) (j) 8μ dx Bundary cnditin n v v ( x,0) 0 (k) v ( x, / ) 0 (l) Multily (j) by, integrate y d y y d v ( ) ( + Kn( ) ) () 8μ dx 0 0 Evaluate the integral, lve fr v, and ue (l) 3 d y y [ + Kn( ) ] (n) dx y / Intrduce Knuden nuber λ μ π Kn R (.33) Evaluate (n) at y /, ubtitute (.33) int (n) and integrate μ + π R Cx + D () 6 where C and D are cntant f integratin. he lutin t thi quadratic equatin i 9 μ μ ( x) 3 π R + 8π R + 6Cx + 6D () Bundary cnditin n ( 0) i, ( L) (q) Ue (q) t find C and D, ubtitute int () and ue the definitin f Knuden nuber x ( ) i i i Kn Kn Kn ( ) + ( ) Ma Flw Rate. he flw rate fr a channel f width W i x L (.35)

10 Ue (.30), (.35) and the ideal ga law / 0 W ρ u () 3 W μ π d + 6 R (.38) μ R dx Uing (.35) t frulate the reure gradient, ubtituting int (.38), auing cntant teerature ( ), and rearranging, give 0 Fr acrchannel 3 W i i + Kn ( ) μ LR W i μ LR aking rati i + + Kn NOE: in icrchannel i very enitive t channel height. (.39) hw the effect f rarefactin and creibility. 3 (.39) (.0) (.) Nuelt Nuber. Fllw Sectin.6. Nu (v) k( ) i given by (.) i given by ( x, / ) γ λ ( x, + + γ Pr / / ) (.) 0 (.3) u 0 / u eerature ditributin. Slve the energy equatin. Additinal autin:

11 () Axial velcity ditributin i arxiated by the lutin t the itheral cae. (5) Negligible diiatin, Φ 0 (6) Negligible axial cnductin, / << / (7) Negligible effect f creibility n the energy equatin, u / + v / 0 (8) Nearly arallel flw, v 0 Energy equatin: (.5) ilifie t ρ c u k (.) Bundary cnditin ( x,0) 0 (w) and ( x, / ) k q (x) Aue: (9) Fully develed teerature. Define φ Fully develed teerature: hu Equatin (.5) and (.6) give ( x, / ) ( x, y) φ (.5) ( x, / ) ( x) φ φ(y) (.6) φ 0 (.7) d ( x, / ) d ( x, / ) d ( x) φ ( y) 0 (.8) dx dx dx he heat tranfer cefficient h, i given by Ue (.) and (.5) int (y) Newtn law f cling: Equating the abve with (.9) ( x, / ) k y h (y) ( x) ( x) k[ ( x, / ) ( x)] dφ( / ) h (.9) ( x) ( x) h ( x) ( x)

12 ( x, / ) ( x) cntant dφ( / ) Cbining thi with (.8), give d ( x, / ) d ( x) dx dx Cnervatin f energy fr the eleent in Fig..8 give d q Wdx + c c + dx dx Silify and eliinate d cntant (.5) dx ρc u (.5) int (.5) d ( x, / ) d ( x) dx dx ρc u (.53) int (.) where u i given by / q k u u (.50) (.5) (.5) u u (cc) 0 (.30) int (cc) d u [ + 6Kn] (.55) μ dx (.30) and (.55) u 6 y + Kn (.56) u + 6 Kn (.56) int (.5) q y + Kn (.57) + 6 y Kn k Integrating twice and ue (w) y ( x, y) ( + Kn) y + g( x) ( 6Kn) k (.58) + dx q Fig..8 + d dx dx

13 3 deterine g(x), find uing tw ethd. Firt ethd: Integrate (.5) Evaluating the integral d i ρc u i 0 x dx ( 0) (.59) ( x) x + i (.60) ρ c u Secnd ethd: ue definitin in (.3). (.30) and (.58) int (.3) ( x) ( Kn) Kn g( x) k( 6Kn) (.6) + (.60) and (.6) give g(x) g( x) i + ρc u 3 x k( + 6Kn) ( Kn) Kn + Surface teerature ( x, / ) : 3 5 γ ( x) Kn + Kn + g( x) k( 6Kn) γ + kpr Nuelt nuber: (.6) and (.63) int (v) NOE: Nu 3 Kn + ( + 6Kn) 5 8 ( Kn) ( + 6Kn) (i) he Nuelt nuber i an ilicit functin f x ince Kn i a functin which i a functin f x. (ii) Unlike acrchannel, the Nuelt nuber deend n the fluid, a indicated by Pr and γ in (.6). (iii) he effect f teerature ju n the Nuelt nuber i rereented by the lat ter in the deninatr f (.6). (iv) he Nuelt fr n-li, Nu, i deterined by etting Kn 0 in (.6) Nu Kn γ + γ + Pr Kn (.6) (.63) (.6) Kn Fig..9 Nuelt nuber fr air flw between arallel late at unifrr urface heat flux fr air, γ., Pr 0.7, σ u σ

14 0 Nu 8.35 (.65) 7 (v) Rarefactin and creibility have the effect f decreaing the Nuelt nuber..6. Fully Develed Pieuille Channel Flw: Unifr Surface eerature Autin: ae a the unifr flux cae. he velcity, reure, and a flw rate, are the ae a fr unifr flux. Surface bundary cnditin i different. Mut deterine teerature ditributin eerature Ditributin and Nuelt Nuber. Newtn law f cling the Nuelt nuber fr thi cae i given by h Nu k ( x) Energy equatin: Include axial cnductin Bundary cnditin: ( x, Axial velcity i given by (.56) Dieninle variable / ) u u ( x, y / ) ( + (.66a) ρ c u k ) (.67a) ( x,0) 0 γ γ + Pr i ( x, Kn / ) (.68a) (.69a) ( 0, y) (.70a) (, y) (.7a) 6 y + Kn + 6Kn (.56) x y ρu θ, ξ, η, Re, Pe RePr (.7) RePr μ i Ue (.56) and (.7), int (.66a)-(.7a) θ ( ξ, η / ) Nu (.66) θ η / / y x Fig..0

15 5 6 θ ( + Kn η ) + 6Kn ξ ( Pe) θ ( ξ,0) 0 η θ θ + ξ η (.67) (.68) γ θ ( ξ,/ ) θ ( ξ,/ ) Kn γ + Pr η (.69) θ ( 0, η) (.70) θ (, η) 0 (.7) Slutin: ethd f earatin f variable Reult: Fig... NOE: he Nuelt nuber decreae a the Knuden nuber i increaed. Axial cnductin increae the Nuelt nuber. N-li (Kn 0) and negligible axial cnductin ( Pe ) : Nu (.73) Nu Pe Kn Fig.. Nuelt nuber fr flw between arallel late at unifr urface teerature fr air, Pr 0.7, γ., σ u σ, [].6.5 Fully Develed Pieuille Flw in Micrtube: Unifr Surface Flux hi rble i identical t Pieuille flw between arallel late at unifr flux reented in Sectin.6.3. Deterine the fllwing: () Velcity ditributin () Nuelt nuber Autin. See Sectin.6.3. r z Fig.. r r Flw Field. Fllwing the analyi f Sectin.6.3. Ue cylindrical crdinate. Reult:

16 v z r μ d dz r + Kn r 6 (.7) v v z z + Kn ( r / r ) (.77) + 8Kn ( z) i i i Kn Kn Kn ( ) + 6 ( ) z L (.78) π r i i + 6 Kn ( ) 6 μ LR (.79a) Nuelt Nuber. Define Reult g( z) π 8 r μ LR i ( ) (.79b) r h Nu (d) k r Nu (e) k( ) r ( r, z) ( + Kn) r + g( z) ( 8Kn) kr (.9) + r r 7 6Kn Kn g( z) k( 8Kn) (.95) r 7 + z 6Kn + Kn + c r v k( + 8Kn) (.96) ρ 3 + i z Nu ( + 8Kn) r 3 γ r ( r, z) Kn + Kn + g( z) k( 8Kn) γ + kpr ( Kn ) ( + 8Kn) 6Kn + Kn γ + Kn γ + Pr (.97) (.98) Nuelt nuber variatin with Knuden nuber fr air, with γ. and Pr 0.7, i ltted in Fig... N-li Nuelt nuber, Nu, i btained by etting Kn 0 in (.98) 8 Nu.36 (.99)

17 8.6.6 Fully Develed Pieuille Flw in Micrtube: Unifr Surface eerature he unifr urface flux f Sectin.6.5 i reeated with the tube aintained at unifr urface teerature. r z r r eerature Ditributin and Nuelt Nuber Sae flw field a the unifr urface flux cae f Sectin.6.5 Fllw the analyi f Sectin.6.. and ue the flw field f Sectin.6.5. Dieninle variable z r ρ θ, ξ, R, u Re r, Pe RePr (.06) r RePr μ i Nuelt nuber, energy equatin, and bundary cnditin in dieninle fr r θ (, ξ ) Nu θ R Fig..5 (.00) + Kn R ( + 6Kn) θ ξ R R ( θ R ) + R (Pe) θ ξ (.0) θ (0. ξ ) 0 R (.0) γ Kn θ (, ξ ) θ (, ξ ) (.03) γ + Pr R θ ( R,0) (.0) θ ( R, ) 0 (.05) Nu 3.0 Pe 0 5 Slutin: By earatin f variable..5 Reult: Fig Kn Fig..6 Nuelt nuber fr flw thrugh tube at unifr urface teerature fr air, Pr 0.7, γ., σ u σ, []

Chapter 8 Sections 8.4 through 8.6 Internal Flow: Heat Transfer Correlations. In fully-developed region. Neglect axial conduction

Chapter 8 Sections 8.4 through 8.6 Internal Flow: Heat Transfer Correlations. In fully-developed region. Neglect axial conduction Chapter 8 Sectin 8.4 thrugh 8.6 Internal Flw: Heat Tranfer Crrelatin T v cu p cp ( rt) k r T T k x r r r r r x In fully-develped regin Neglect axial cnductin u ( rt) r x r r r r r x T v T T T T T u r x