Chapter 11: Two-Phase Flow and Heat Transfer Forced Convective Boiling in Tubes

Transcription

1 11.5 Forced Convective Regimes in Horizonta and Vertica Tubes The typica sequence of fow regimes for upward fow forced convective boiing in a uniformy-heated vertica tube (q =const) is shown in Fig Since gravity acts parae to the fow direction, the upward fow boiing in the tube is axisymmetric. The gravitationa force pays a more dominant roe whie the iquid-vapor interfacia shear force is ess important. If the fuid enters the tube as subcooed iquid and eaves the tube as superheated vapor, a of the fow regimes for horizonta tubes discussed in Section 11.2 wi be encountered in the interim. Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 1

2 Figure Regimes for convective boiing in a vertica tube Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 2

3 The void fraction increases from zero at the inet of the tube to one at the outet of the tube. Since the density of the vapor is significanty ower than that of the iquid, the density of the two-phase mixture significanty decreases aong the fow direction. To maintain constant mass fux aong the fow direction, the mean veocity must increase substantiay to match the significant increase in vapor phase veocity. The growing disparity between vapor and iquid veocities aong the fow direction wi resut in changing fow patterns aong the fow direction. Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 3

4 Figure Fow regimes for convective boiing in a horizonta tube Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 4

5 Figure Kattan-Thome-Favrat fow pattern map Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 5

6 An eectric fied in two-phase fow adds additiona body forces in the fuid as we as at the iquid-vapor interface. The eectric force, F e, appied to the fied is given as Fe = ρ ee E ε + ρ E (11.135) Some other eectrodynamic equations needed in the probem are ( ε E) div = ρ E = gradϕ divj = 0 (11.136) (11.137) (11.138) where ρ e is the charge density, φ is the eectric potentia, and J is the current density given by J = ρ V + σ E + e E e ( ε E) t ε ρ T (11.139) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 6

7 The two dimensioness EHD numbers that wi resut from the anaysis of the eectric body force are the EHD number or conductive Rayeigh number E hd = o J L o 2 ρ v µ (11.140) The Masuda number or dieectric Rayeigh number is given as ε 2 ( ) 2 oeoto ε s / T L ρ M d = 2 2ρ v (11.141) 3 o c A Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 7

8 Figure Proposed reconstructed fow pattern from surface temperature measurements and inet and outet fow observations for increasing DC votage eves: (a) 0 appied votage, (b) moderate votage, and (c) high votage Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 8

9 Bubbe Lift-Off Size in Forced Convective Boiing The force baance of the bubbe at the nuceate site is shown in Fig (b). The momentum equations aong the x- and y-directions are duv Fsx + Fdux + Fs = ρ vvb dt (11.142) dvv sy + duy + p + g + qs = ρ v b (11.143) F F F F F V dt The surface tension force in the x- and y- directions are Fsx = π Dwσ (cosθ r cos θ a ) θ θ F a π ( θ θ ) = 1.25 D σ (sinθ + sin θ ) r a sy w 2 2 a r π ( θ r θ a ) r (11.144) (11.145) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 9

10 (a) bubbe nuceation phenomena (b) force baance at a nuceation site Figure Physica mode of bubbe ift-off (Situ et a., 2005). Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 10

11 The unsteady drag force (aso referred to as growth force) is 2 d( ρ V u ) b d H dh dv Fdu = = ρ V + 2 dt dt dt dt (11.146) where is the voume of the virtua added mass (Chen, 2003) 11 3 V (11.147) = π R b 12 The unsteady drag force for spherica bubbe can be expressed as F (11.148) du = ρ π Rb R& b + Rb R&& b 12 6 The components of the unsteady drag force in the x- and y- directions are Fdux = Fdu cosθ i (11.149) F = F sinθ duy du i (11.150) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 11

12 The transient radius of the vapor bubbe can be obtained from Zuber s (1961) bubbe growth mode (Chapter 10) ( ) 2b R Ja b t = α t π (11.151) where (Zeng et a., 1993) and the Jakob number is defined as ρ c ( T T ) (11.152) Whie eq. (11.151) can provide a good estimate for saturated boiing, the bubbe radius wi depend on effective Jakob number, Reynods number, and Prandt number where w sat Ja = ρ v h v R ( t) = f (Ja,Re,Pr, t) b Ja e v e ρ c T = ρ h v e (11.153) (11.154) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 12

13 where S is the suppression factor. The shear ift force is The shear ift coefficient is C = G (Re G ) where and is the bubbe Reynods number. T = S( T T ) e w sat 1 F C R v s = ρ π b r 1/ / 4 s b s G s dv = R dx v 2R v b r Reb = ν b r (11.155) (11.156) (11.157) (11.158) (11.159) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 13

14 The pressure and gravity forces are F ρ gv The quasi-steady drag force is Fqs = 6π ρ vvr Rb Re b F p = = ρ (11.160) (11.161) (11.162) The unsteady drag force (growth force) is aso negigibe since the incination ange is zero at ift-off. The force baance in the x- direction is Fdu + Fs = 0 (11.163) Substituting eqs. (11.148) and (11.156) into eq. (11.163) yieds ρ π Rb R& b + Rb R&& b + C ρ π Rb vr = (11.164) b gv g v b 1.54 Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 14

15 Figure Force baance of a vapor bubbe at ift-off (Situ et a., 2005). Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 15

16 Substituting eq. (11.151) into eq. (11.164), one obtains α t o = 2 r 2 2 3π C v 22b Ja (11.165) where t o is the time at which a bubbe ifts-off, which can be obtained from eq. (11.151), i.e., 2 π ro to = 2 2 4b Ja α (11.166) Combining eqs. (11.165) and (11.166), a dimensioness ift-diameter is obtained 2 * 4 22 / 3b 2 1 Do = Ja Pr π (11.167) where * vr Do Do = C Reb = C ν (11.168) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 16

17 Heat Transfer Predictions The overa heat transfer coefficient for convective boiing in an upward vertica tube h = Fh + Sh b (11.169) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 17

18 Singe-phase heat transfer coefficient for iquid aone can be obtained using the Dittus- Boeter/McAdams equation h k = D Re Pr The contribution of nuceate boiing is determined by using the correation for poo boiing h T T p p T p k cp, ρ b = [ w sat ( )] [ sat ( w) ] σ µ hfg ρ v (11.170) (11.171) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 18

19 The convective factor F can be obtained by a regression of experimenta data The nuceate boiing suppression factor S is where F = 1 tt 1 X tt X tt 2.35( X ) > 0.1 S = Re TP Re TP = Re F 1.25 (11.172) (11.173) (11.174) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 19

20 Exampe 11.5 Cacuate the fow boiing two-phase heat transfer coefficient for 300 kg/(sm 2 ) of water at 1atm and 20% quaity in an upward vertica tube with a diameter of 2 cm and a wa temperature of 140 C. Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 20

21 Soution: The saturation temperature at o p = 1 atm = Pa is T = 100 C. The properties of water at this temperature are σ sat 3 = N/m, h = kg/m, o kj/kg C, The mass fux of µ v kj/kg, 7 2 = N-s/m, µ v kg/m, v ρ = ρ = = N-s/m, c p = o 0.68W/m- C, and Pr = µ cp / k = k = 2 yieds a iquid Reynods number of m = 300kg/s-m m& (1 x) D 300 (1 0.2) 0.02 Re = = = µ Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 21

22 The singe-phase heat transfer coefficient for iquid aone can be obtained from eq. (11.170), i.e., k h = Re Pr D = ( ) 1.73 = W/m -K 0.02 The saturation pressure corresponding to the wa temperature is. The contribution of nuceate boiing is determined from eq. (11.171): k cp, ρ hb = [ Tw Tsat ( p )] [ psat ( Tw ) p ] σ µ hfg ρ v ( ) = ( ) ( ) ( ) [ ] = W/m -K 0.75 Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 22

23 The Martinei parameter for turbuent fow in both iquid and vapor phases, X tt, is obtained from eq. (11.126): x ρ v µ X tt = x ρ µ v = = The convective boiing factor F is obtained from eq. (11.172), i.e., F = 2.35( X ) = 2.35 ( ) = The oca two-phase Reynods number is tt Re = Re F = = TP The nuceate boiing suppression factor S is obtained from eq. (11.173), i.e., 1 1 S = = = Re ( ) TP 0.1 Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 23

24 Therefore, the heat transfer coefficient is obtained from eq. (11.169) as h = Fh + Sh b = = W/m -K Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 24

25 Generaized heat transfer coefficient for convective boiing in both vertica and hoizonta tubes where h h h = max NBD, CBD h = Co f ( Fr ) Bo F (1 x) h NBD 2 o f o (11.175) (11.176) h = Co f ( Fr ) Bo F (1 x) h CBD 2 o f o (11.177) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 25

26 Co is the convective number, Bo is the boiing number, and Fr o is the Froude number ρ v 1 x Co = ρ x (11.178) Bo = q w AGh fg (11.179) Fr Froude number mutipier is f 2 o 0.3 Fr o o = 2 G ρ gd 1 vertica tubes or horizonta tubes ( Fr o >0.04) ( Fr ) = (25 ) horizonta tubes ( Fr o >0.04) (11.180) (11.181) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 26

27 For Prandt numbers between 0.5 and 2000 the singe-phase a-iquid heta transfer coefficient is h o k (Reo 1000)Pr ( f / 2) Re 10 2/3 0.5 o D (Pr 1)( f / 2) = k Reo Pr ( f / 2) 10 Re / D (Pr 1)( f / 2) Friction factor f ( 1.58n Re 3.28) = o 4 6 o 2 (11.182) (11.183) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 27

28 Tabe 11.4 Ff for copper tube (Kandikar, 1990; 1991) Fuid Fuid Ff Ff Water R-11 R-12 R-13B1 R-22 R R-114 R-134a R-152a Nitrogen Neon Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 28

29 The most generaized mode to predict CHF for forced convective boiing is recommended by Katto and Ohno For q max = q co 1 + K γ = ρ / ρ < 0.15 v q co = K ( ) h, sat h v h q q < q in co1 co1 co2 min q, q q > q co2 co3 co1 co2 (11.184) (11.185) ( ) K = max K, K K K1 K 2 (11.186) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 29

30 For where γ = ρ / ρ > 0.15 v q q < q q co = min ( q, q ) q > q K K co1 co1 co5 co4 co5 co1 co5 K K > K = min ( K, K ) K K K1 K1 K 2 K 2 K 3 K1 K L q co1 = CKGh vwek D / 3 1 q co2 = 0.10Gh v γ We k ( L / D) ( L / D) k q co3 = 0.098Gh vγ We ( L / D) (11.187) (11.188) (11.189) (11.190) (11.191) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 30

31 q co4 = Ghv γ Wek We k ( L / D) (11.192) ( L / D) co Gh vγ Wek q = ( L / D) K K K = K CK Wek D / L = K / 3 6 γ Wek We k + / K 3 = γ Wek D L (11.193) (11.194) (11.195) (11.196) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 31

32 γ = ρ ρ v (11.197) We k = 2 G L ρ σ (11.198) CK 0.25 L/D < 50 = [ L / D 50 ] 50 L/D L/D > 150 (11.199) Transport Phenomena in Mutiphase Systems by A. Faghri & Y. Zhang 32