External Forced Convection. The Empirical Method. Chapter 7. The empirical correlation
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1 Chapter 7 Eternal Forced Convection N f ( *,Re,Pr) N f (Re,Pr) he Empirical Method he empirical correlation N C Re he vale of C, m, n are often independent of natre of the flid m Pr n he vale of C, m, n var with the natre of the rface geometr and the tpe of the flow. Film temperatre f
2 Propert Ratio Evalate all propertie at, Pr Pr r or r
3 he Flat Plate in Parallel Flow.A with all eternal flow, the bondar laer develop freel withot contraint..bondar laer condition ma be entirel laminar, laminar and trblent, or entirel trblent..o determine the condition, compte Re Re Re, c aminar flow throghot Re Re, c tranition to trblent flow at c / Re, c / Re - Vale of depend on free tream trblence and rface roghne. Re,c - Nominall, Re, c 10
4 - If bondar laer i tripped at the leading edge Re, c 0 - Srface thermal condition are commonl idealized a being of niform temperatre or niform heat fl q. I it poible for a rface to be concrrentl characterized b niform temperatre and niform heat fl? - hermal bondar laer development ma be delaed b an nheated tarting length. Eqivalent rface and free tream temperatre for and niform (or q ) for.
5 Similarit Soltion for aminar, Contant-Propert Flow over an Iothermal Plate Conervation of Ma: 0 v Conervation of Momentm: X d dp v Conervation of Energ: c v p For flat plate with contant propertie and negligible vico diipation, the above bondar laer eqation redce to Continit 0 v Momentm v Energ v Stream fnction v, the continit eqation i atomaticall atified. -Baed on premie that the dimenionle -velocit component,, / and temperatre, / *, can be repreented eclivel in term of a dimenionle imilarit parameter.
6 () ( / ) ( ) -Similarit permit tranformation of the partial differential eqation aociated with the tranfer of -momentm and thermal energ to ordinar differential eqation of the form. 1/ df d df d v 1 df f d f f d f d d f d 3 d f 3 d d f d f d d 3 f 3 0 where ( / ) df / d, and d d Pr d f 0 d * * Bondar condition 0 at 0, f ' 0 at 0, v0 at 0, f 0 at 0, at f ' 1 at 1,
7 * at 0, 0 at t 0, * at 1 at t 1 t -Sbject to precribed bondar condition, nmerical oltion to the momentm and energ eqation ield the following relt for important local bondar laer parameter: -with / at. 0 (.0 / ) 1/ 1/ Re -with 0 d f / d 0 and d f / d 0. 33, 0 C f, 0.664Re /, 1/
8 -with h q h ( k( k q /( k * 0 / ) ) 1/ ) 0 k * * d d 0 0 * and d / d 0.33Pr for Pr 0.6, 0 he local Nelt nmber i N h k 1/ 0.33 Re Pr and 1/3 Pr t - How wold o characterize relative laminar velocit and thermal bondar laer growth for a ga? An oil? A liqid metal? - How do the local hear tre and convection coefficient var with ditance from the leading edge? - Average Bondar aer Parameter: 1 d, C f 0, 1.38Re 1/ h 1 0 h d
9 N 1/ Re - he effect of variable propertie ma be conidered b evalating all propertie at the film temperatre. For Uniform rface Heat Fl f N 0.43Re Pr Pr 1/ 1/3 For iqid metal (Pr<<1) N 0.6 Re Pr 0.6Pe 1/ 1/ 1/ where Pe Re Pr i the Peclet nmber.
10 Integral Method (See Appendi F on page A-6-8.)
11 rblent Flow Pat a Flat Plate C f, 0.09 Re 1/, 10 Re / 錯誤! 尚未定義書籤 錯誤! 尚未定義書籤 0.37 Re 錯誤! 4 / 尚未定義書籤, 錯誤! 尚未定義書籤 varie a t (independent of Pr ) C f St Pr / Re 1/ N St, N Re PrSt Re 4 / Pr Re Pr N 4 / Re PrSt Re Pr, 0.6 Pr 60 ranition Renold nmber Re, 10 c 0 laminar c c trblent h 1 c ( hlamd 0 k / Re Pr htrbd) c 1/ 4 / c d d / 1/ c 0.037(Re 4/ Re 4/,c )Pr 1/3 Pr
12 If Re 10,, c N Re 4 / Pr 1/, 0.6 Pr Re 10 8 Re 10 c, C f 10 Re , 1/ Re Re Re, c If c N Re 4 / Pr For Uniform rface Heat Fl N Re Pr, 0.6 Pr 60 4/ 1/3
13 Special Cae: Unheated Starting ength (US) and/or Uniform Heat Fl For both niform rface temperatre (US) and niform rface heat fl (USF), the effect of the US on the local Nelt nmber ma be repreented a follow: N N 1 ( 0 a / ) b N C Re m Pr 0 aminar rblent US USF US USF a 3/4 3/4 9/10 9/10 b 1/3 1/3 1/9 1/9 C m 1/ 1/ 4/ 4/ Sketch the variation of h ver ( ) for two condition: 0 and 0. What effect doe an US have on the local convection coefficient?
14 - US: - USF: q h ( ) q h A ( ) ( P /( ) /( 1) 1 ( / ) 1) P P P N N 0 ( ) P for laminar throghot P 8 for trblent throghot nmerical integration for laminar/trblent flow 1 c h hlamd htrbd c h q * q q A h -reatment of Non-Contant Propert Effect: Evalate propertie at the film temperatre. f
15 Methodolog for a Convection Calclation 1. Flow geometr. Evalate the flid propertie at the appropriate reference temperatre 3. Calclate the Renold nmber laminar or trblent 4. Determine local or average coefficient. Select the appropriate correlation.
16 Eample 7.1 Known : Air flow over a flat plate Find : Etimate the cooling rate per nit length of the plate needed to maintain it at a rface temperatre of 7 C. Amption : (1) Stead tate () Negligible radiation effect Film temperatre f C 437 K From able A.4 (on p. A-1), air m /, k W/mK, Pr Uall, k, Pr, ma be amed to be independent of prere P 1 P,, R P1 he Renold nmber 10 m/ 0. m Re, c m / Hence the flow i laminar over the entire plate. N Re 1/ Pr 0.664(997) 7.4 he average convection coefficient i 1/ (0.687) N k W/mK h 4.18 W/m K 0. m he reqired cooling rate per nit width of plate i
17 q h( ) 4.18 W/m 70 W/m K 0. m (300 7) K
18 Eample 7. Known : Air flow over a flat plate with egmented heater. Find : At which heater i the electrical inpt a maimm Amption : (1) Stead tate () Negligible radiation effect (3) Bottom rface of the plate i adiabatic 30 Film temperatre f 17. C 400 K 6 From able A.4, air m /, k W/m K, Pr c 60 m/ c Re, c m / c 0. m he heater reqiring the maimm electrical power i that for which the average convection coefficient i larget. From knowledge of how the local convection coefficient varie with ditance from the leading edge, we conclde that there are three poibilitie: 1. Heater 1, ince it correpond to the larget local, laminar convection coefficient.. Heater, ince it correpond to the larget local, trblent convection coefficient. 3. Heater 6, ince trblent condition eit over the entire heater. For heater 1, qelec q conv q h w( conv, )
19 From Eqation 7., 1/ 1/ N Re1 Pr 0.664( ) (0.69) 198 h 1 N k W/m K 134 W/m K 0.0 m q conv, W/m K(0.01) m (30 ) C 1340 W For heater, q ( h1 h14 4) w( conv, ) From Eqation 7., with Re 4Re / 1/ N Re4 Pr 0.664(4.610 ) (0.69) 396 h N k W/m K 67 W/m 0. m From Eqation 7.33, with Re Re K N ( Re 871) Pr h 4 / 4 / 0.037(.7010 ) 871 (0.69) 46 N 1 q conv, 100 W For heater 6, q k W/m K 74 W/m 0. m (74 W/m K 0. m 67 W/m ( h16 h1 ) w( conv, 6 6 ) K K 0.0 m)1 m(30 ) C From Eqation 7.33, with Re6 6Re N h 4 / 0.037( ) 871 (0.69) 73 6 N k W/m K 8 W/m 0.30 m q conv,6 (8 W/m 1440 W K 0.30 m 74 W/m K K 0. m)1 m(30 ) C
20 Hence q q q conv, 6 conv,1 conv,, and the ith plate ha the larget power reqirement.
21 Problem7.1 SCHEMAIC: Amption: (1) Srface B i fficient rogh to trip the bondar laer when in the ptream poition (Configration ); () Re for flow over A in Configration 1., c 10 6 Propertie: able A-4, Air ( 333 K, 1atm): m /, 3 k W/m K, Pr 0. 7 Anali: With 0m / 1m Re m / f tranition will occr jt before the rogh rface ( 0. 48m) for Configration 1. Hence, N / , / 3 ince N, (0.7) 1/ 139 N, 1, lowet heat tranfer i aociated with Configration 1. h,1 For Configration 1: N, k 6 c it follow that the
22 Hence h W/m K/1m 39. W/m K, q h 1,1 A( 39. W/m ) K(0. m 1m)(100 0) K 168 W Comment: For a ver hort plate, a lower heat lo ma be aociated with Configration. In fact, parametric calclation reveal that for <30mm, thi configration provide the preferred orientation.
23 Problem 7.4 Known: Plate dimenion and initial temperatre. Velocit and temperatre of air in parallel flow over plate. Find: Initial rate of heat tranfer from plate. Rate of change of plate temperatre. SCHEMAIC: Air = 0o oo C oo = 10 m/ = 6 mm q = 1 m q = 300 o C i = 1 m Amption: (1) Negligible radiation () Negligible effect of conveor velocit on bondar laer development (3) Plate are iothermal (4) Negligible heat tranfer from edge of plate () Re c, 10 (6) Contant Propertie Propertie: able A-1, AISI 1010 teel (73 K) : k 49. W/m K, p
24 3 c 49 J/kg K, 783 kg/m. able A-4, Air ( p 1atm, 433 K ): m /, k W/m K, Pr Anali: he initial rate of heart tranfer from a plate i f q h ( A i ) h ( i ) With Re / 10 m/ 1m / flow i laminar over the entire rface. Hence, 6 m / N 0.664Re 1/ Pr / ( ) 1/ h ( k / ) N ( W/m K /1m) W/m K q 1.1W/m K(1m) (300 0) C 6780 W Performing an energ balance at an intant of time for a control rface abot the plate, E, d dt Comment: i d c dt i ot E t h ( i ) K (300 ) C C/ 1.1W/m kg/m m 49 J/kg K 4 (1)With Bi h( / ) / 7.410, e of the lmped capacitance method k p i appropriate. () Depite the large plate temperatre and the mall convection coefficient, if adjoining plate are in cloe proimit, radiation echange with the rronding will be mall and the amption of negligible radiation i jtifiable.
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