Thermal-Fluids I. Chapter 17 Steady heat conduction. Dr. Primal Fernando Ph: (850)

Transcription

1 emal-fluids I Capte 7 Steady eat conduction D. Pimal Fenando pimal@eng.fsu.edu P: (

2 Steady eat conduction Hee we conside one dimensional steady eat conduction. We conside eat tansfe in a plane wall, a cylinde, and a spee (multilaye. We ty to develop temal esistance elations to tis geometies. Also ty to develop temal esistance elations fo convection and adiation at te boundaies. Heat tansfe fom fin sufaces and complex sufaces.

3 One dimensional eat flow If te tempeatue of te wall vaies on one diection, te eat flow is one-dimensional. is will confim by measuing te tempeatues at vaious locations on te wall suface (isotemal sufaces. 3

4 e Fouie s law of eat conduction is given by cond, wall ka ( W x Constant eefoe tempeatue distibution in te wall is a staigt line 4

5 emal esistance Om s law povides electical esistance V I V I Electical esistance Diving potential Coesponding tansfe ate In eat tansfe Diving potential is tempeatue diffeence ( - Coesponding tansfe ate is eat tansfe ate (Q 5

6 emal esistance is define as te atio between te diving potential and tansfe ate temal Conduction Q cond ka ka t, cond x L cond ka L Convection conv A ( s conv s ( s conv conv As t, conv adiation ad A ( s ad s ( s ad ad As t, ad 6

7 Note: ad εσa s ( 4 s 4 ad εσa s ( s ( s ad εσas ( s ( s ( s ad { ( } A ( ad εσ ad ( s s s s εσas ( s ( s A ( ad s s Note: fo combined eat tansfe of adiation and convection combined con ad 7

8 emal esistance netwoks 8

9 Equivalent esistance (a In seies 3 V V V 3 V 4 V V 4 3 9

10 Equivalent esistance (b In paallel 3 V V V V 3 0

11 emal esistance netwoks (in seies convection conduction convection L conv, cond conv A ka, A Qconv, cond conv, Q,,

12 Oveall eat tansfe coefficient (U Oveall eat tansfe coefficient (U multiply by aea (A is epesent as UA And eat tansfe toug a medium is expessed in an analogous manne to Newton s law of cooling, UA

13 Multilaye plane walls L L conv, cond, cond, conv A k A k A, A 3

14 Single and double glass windows (Ex. 7- and 7-3 k glass 0.78 W / mk k ai 0.06 W / mk Lglass i glass 0 A k A i glass A o 0.7 C/W Q66 W Lglass, L L ai glass, i glass, ai glass, 0 A k, A k A k, A i glass ai glass A o C/W Q69. W 4

15 emal contact esistance Heat conduction toug multilaye solids, we assume pefect contact. is would be te case wen te sufaces ae pefectly smoot. But in eality te sufaces ae not pefectly smoot. 5

16 emal contact esistance Heat tansfe toug te intefacesolid contact spots gaps contact gap It can be expessed in an analogous manne to Newton s law of cooling as, emal contact conduction contact A int eface int / A eface c ( temal contact esis tan ce contact [ C W ] c entie suface / contact A A [ m. C / W ] 6

17 emal contact esistance Depends on suface ougness Mateial popeties empeatue and pessue of te inteface ype of te fluid tap Expeimental, c m C/W Expeimental, contact ,000 W/m C emal esistances of -cm tick insulating mateial and coppe ae 0.5 and m C/W emal esistance impotant emal esistance can disegad 7

18 Example 7-4, emal contact conductance at te inteface of two -cmtick aluminum plates is measued to be,000 W/m C. Detemine te tickness of te aluminum plate wose temal esistance is equal to te temal esistance of te inteface between te plates (k37w/m C. ( temal contact esis tan ce contact c 4 c m C / W, 000W / m C [ m. C / W ] L alu min um alu min um ( pe unit aea ka L k L alu 4 min um ( pe unit aea k m C / W 37W / m C. 5cm 8

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20 0 emal esistance netwoks (in paallel Q Q Q ( Q ( A k L A k L

21 Genealized temal esistance netwoks 3 con 3 con k L A k L A 3 k L 3 3 A 3 conv A 3

22 Genealized temal esistance netwoks

23 Heat conduction in cylindes cond d. cyl ka ( W d A L 3

24 Heat conduction fom to cond. cyl A cond d kd d. cyl ka ( W d cond. cyl L d kd cond. cyl L d kd cond. cyl L d kd cond. cyl L cond. cyl d k Lk ln( / d 4

25 5 Conduction esistance / ln(. cyl cond Lk Q Q temal Diving potential Coesponding tansfe ate cylinde cyl cond Q Lk. / ln(

26 6 Heat conduction in spees 4 A (. W d d ka Q sp cond kd d 4 Q kd d A Q sp cond sp cond.. sp cond d k d 4 Q. (. sp cond k Q (. sp cond k Q sp sp cond Q k. (

27 7 emal esistance netwoks total L Lk L ( / ln( ( total 4 k 4 4 ( ( Fo a cylinde Fo a spee

28 8 Multilaye cylindes total L Lk Lk Lk L ( / ln( / ln( / ln( (

29 Example 7-7: A 3m intenal diamete speical tank was made of cmtick stainless steel (k5w/m C is used to stoe iced wate at 0 C. e tank is located in a oom wose tempeatue is C. e walls of te oom ae also at C. e oute suface of te tank is black and eat tansfe between oute suface of te tank and te suounding is by natual convection and adiation. e convection eat tansfe coefficients at te inne and te oute sufaces of te tank ae 80W/m C and 0W/m C, espectively. Detemine te eat tansfe to te iced wate in te tank and te amount of ice at 0 C tat melts duing a 4 peiod. 9

30 Solution i i ad ad o A k ad o ad A 4 A ad o A A 4 εσa ( ( s s s s..? Assume a value between 0 - C close to 0 C, since eat tansfe coefficients in te tank muc lage. 30

31 Solution e-calculate te suface tempeatue s by, ad ad o o combine s Compae wit te assumed value. If tey ae not easonably close to eac ote, epeat te calculation wit a new assumed value otal eat tansfe duing 4 peiod Q Q m ice Q latent eat of melting ice 3

32 Example 7-8: Steam at 30 C flows in a cast ion pipe (k80w/m C wose inne and oute diametes ae d 5cm and d 5.5cm, espectively. e pipe is coveed wit 7cm-tick glass wool insulation wit k0.05w/m C. Heat is loss to te suoundings at 5 C by natual convection and adiation, wit a combine eat tansfe coefficient of 8W/m C. aking te eat tansfe coefficient inside te pipe to be 60W/m C, detemine te ate of eat loss fom te steam pe unit lengt of te pipe. Also detemine te tempeatue dops acoss te pipe sell and te insulation. 3

33 33 Solution com 3 3 total L Lk Lk L ( / ln( / ln( ( total total Q Q Q pipe Q insulation

34 Citical adius of a insulation Adding moe insulations to a wall, always educe te eat tansfe (incease te temal esistance. icke te insulation, lowe te eat tansfe. is is always tue since te suface aea A always constant. Fo cylindical o speical matte, it is diffeent adding insulations always incease te suface aea tat deceases convective esistance and inceasing conduction esistance te eat tansfe may incease o decease depending on wic effect is dominate conivection A ( L insulationcond ln( / Lk 34

35 Cylindical pipe: oute adius, suface tempeatue of te cylinde maintained at. It was insulated wit a mateial wose temal conductivity k and outside adius. emal esistance between te oute suface of te cylinde and te oute suface of te insulation; insulation cond conivection Q ln( / Lk ( L 35

36 36 Optimum insulation tickness ( / ln( L Lk ( ln( ln( ( / ln( L Lk Lk d d L Lk d d d d ( ( L Lk d d Fo optimum insulation tickness 0 d d 0 k 0 L Lk ( ( k Citical adius

37 37 k ( ( L Lk d d 3 3 Lk L L Lk d d ( ( ( ( 0 Lk k k k L k L d d 3 > / / ( eefoe, tis insulation tickness gives minimum temal esistance. eefoe, fo above tickness, te eat tansfe ate is maximum.

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39 Some notes e ate of eat tansfe fom a cylinde inceases wit addition of insulation fo < c, eac maximum value at citical adius and ten deceases fo > c,. Citical adius is lage fo ig temal conductive mateials (insulation mateials ave low temal conductivity and low wen convective eat tansfe coefficient ig (it sould ave been used combine eat tansfe coefficient instead of convective eat tansfe coefficient by taking into account of te adiation eat tansfe. eefoe te citical adius is even smalle tan te epoted value. k fo common insulating mateials 0.05 W/mK, and fo natual convection 5 W/m K wic gives citical adius appoximately cm. Hot wate pipes, steam pipes we do not need to conside tis because tey always, > c, adius of electical wies may be smalle tan citical adius Fo spee ci k/ 39

40 Heat tansfe fom finned sufaces A ( s s 40

41 Fin equation {ate of eat conduction into te element at x} {ate of eat of eat conduction fom te element at x x }{ate of eat convection fo te element} cond, x Qcond, x x conve Q conve ( p x(, ( p x( cond, x cond x x Element: coss aea A c, peimete p 4

42 Fin equation Qcond, x Qcond, x x ( p x( cond x x, cond, x ( p x( 0 cond, x x x cond, x ( p( 0 aking te limit as x 0 d cond ( p( 0 dx Fom Fouie s law d kac dx cond d dx ka c d dx ( p ( 0 4

43 Fin equation d kac dx d dx ( p ( 0 d kac dx d dx ( p ( In geneal A c and p of a fin vay wit x wic make tis diffeential equation difficult to solve. In te special case of constant k and constant Ac te diffeential equation educes to 0 d θ a dx θ 0 At te fin base θ θ b b a p ka c 43

44 Solutions to te diffeential equation d θ a dx θ is is a linea, omogeneous second ode diffeential equation wit constant coefficients. 0 Fundamental teoies of diffeential equations states tat suc an equation as two linealy independent solution functions, and its geneal solution is te linea combination of tose to solution functions. Geneal solution of te diffeential equation θ ( x C ax ax e Ce Bounday conditions at te fin base: You need one moe bounday condition θ ( 0 θ b b 44

45 Case :Infinitely long fin Fo sufficiently long fin, te tempeatue at te fin tip eac to envionmental tempeatue (to Bounday condition at te fin t θ ( L ( L 0 is will only satisfies e -ax of te geneal solution not e ax wen L e ax θ ( x C ax ax e Ce wen L e -ax 0 45

46 Note: unde steady conditions, eat tansfe fom te exposed sufaces of te fin is equal to eat conduction to te fin at te base. 46

47 Case :Infinitely long fin eefoe geneal solution in tis case θ ( x C e ax At te base x 0 and θ(0θ b b - a0 0 Ce C θb b θ ( Note: θ ( x θbe θ ax Fo a long fin ( x ax x p / ka e e c b ( x ( b e x p / ka c 47

48 Case :Infinitely long fin long e steady ate eat tansfe fom te fin can be detemine by Fouie s law fin Note: d kac pkac ( b dx x 0 ( x ( b e x p / ka c d dx p / ka c ( b e x p / ka c p / ka c ( b x 0 48

49 Case :Negligible eat loss at fin tip (Q fin-tip 0-insulated fin tip (adiabatic fin tip Bounday condition dθ dx x L 0 e tempeatue distibution ( x cos a( L cos al b x linsulated fin tip ka c d dx x 0 pka c ( tan b al 49

50 Case 3: Convection (o combined convection at fin tip e fin tips, in pactice ae exposed to suounding. eefoe te pope bounday conditions to fin tip is convection tat also includes te effects of adiation. adiation fom fin tip can neglect since te fin tip aea is muc smalle compaed to fin aea. In pactical way, fo accounting tis use coected fin lengt defined as Coected fin lengt Fo ectangula fin Fo cylindical fin Lc L Ac p c, L L ec c, L L cyl t D 4 50

51 Fin efficiency e eat tansfe fom a fin is maximum, if tempeatue of te fin is equal to te base tempeatue. en te maximum eat tansfe is given by A ( fin, max fin b η fin actual eat tanfe ate fom te fin ideal eat tanfe ate fom te fin η pkac ( b A ( long fin long, fin fin,max fin b fin fin,max al η pka ( tan al long fin c b insulated, fin fin,max A fin ( b tan al al 5

52 Fin efficiency elations ae developed fo vaious fin pofiles 5

53 Fin effectiveness Fins added to enance eat tansfe. In fact, tee is no assuance tat adding of fins enance te eat tansfe. e pefomances of fins is jugged on te bases on te enancement on te eat tansfe elative to no-fin case. ε no fin fin no fin fin b A b b ( fin η fina fin ( b ε fin A ( ε > fin fins b enance te fin eat A A fin b η tansfe fin ε < fin fins act as an insulation 53

54 54 ate of eat tansfe fom a finned suface [ ] ( ( ( A A A A Q Q Q b fin fin unfin b fin fin b unfin suface fin suface unfin suface finned total η η [ ] fin no fin fin unfin b fin no b fin fin unfin nofin total suface finned total oveall A A A A A A Q Q η η ε ( (

55 Pope lengt of a fin linsulated long fin fin tip pka c ( b pka c ( b tan al tan al 55

56 Heat tansfe in common configuations Many poblems encounteed in pactice ae two o tee dimensional and involve ate complicated geometies. e steady state eat tansfe of tese sufaces ae expessed by following simple equation: QSk( - S is te conduction sape facto wic as te units of lengt. Compeensive tables ae available to find S fo diffeent geometies in te liteatue. (able