BOOKCOMP, Inc. — John Wiley & Sons / Page 674 / 2nd Proofs / Heat Transfer Handbook / Bejan 674 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [674], (40) Lines: 1350 to 1394 ——— 5.10521pt PgVar ——— Normal Page * PgEnds: Eject [674], (40) is determined from the liquid fraction of the flow, ˙m(1 − χ), using eq. (9.91). The effect of heat flux on nucleate boiling is introduced using the boiling number, which is defined as Bo = q ˙mh LG (9.98) This dimensionless group represents the ratio of the actual heat flux to the maximum heat flux achievable by complete evaporation of the liquid. For Bo > 0.0003, α nb α L = 230Bo 0.5 (9.99) and for Bo < 0.0003, α nb α L = 1 + 46Bo 0.5 (9.100) α cb α L = 1.8 N 0.8 (9.101) For 1.0 >N>0.1, the value α cb is determined from eq. (9.101) and the value of α nb in the bubble suppression regime is calculated from α nb α L = F ·Bo 0.5 exp(2.74N − 0.1) (9.102) Then the larger value is chosen for α tp .ForN<0.1, the value of α cb is calculated with eq. (9.101) while α nb in the bubble suppression regime is determined as α nb α L = F ·Bo 0.5 exp(2.74N − 0.15) (9.103) Again one chooses the larger value for α tp . In the equations above, the constant F is 14.7 when Bo > 0.0011 and is 15.43 when Bo < 0.0011. This method is also applicable to vertical annuli. For annular gaps between the inner and outer tubes greater than 4 mm, the equivalent diameter is the difference between the two diameters. For annuli less than 4 mm, the equivalent diameter is obtained by evaluating the hydraulic diameter using only the heated perimeter. Kan- dlikar (1990) has also proposed a fluid-specific correlation similar to that of Shah using an empirical constant specific to the particular fluid. 9.8.3 Gungor–Winterton Correlation Gungor and Winterton (1986) proposed a Chen type of correlation based on a large database (3693 points) covering water, refrigerants (R-11, R-12, R-22, R-113, and R-114), and ethylene glycol for mostly vertical up flows and some vertical down flows as BOOKCOMP, Inc. — John Wiley & Sons / Page 675 / 2nd Proofs / Heat Transfer Handbook / Bejan FLOW BOILING IN VERTICAL TUBES 675 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [675], (41) Lines: 1394 to 1445 ——— 4.92807pt PgVar ——— Normal Page * PgEnds: Eject [675], (41) α tp = Eα L + Sα nb (9.104) with α L calculated from eq. (9.91) using the local liquid fraction of the flow, ˙m(1−χ). Their convection enhancement factor E is E = 1 +24,000Bo 1.16 + 1.37 1 X tt 0.86 (9.105) where X tt is the Martinelli parameter defined in eq. (9.94). Their boiling suppression factor S is S = 1 +0.00000115E 2 · Re 1.17 L −1 (9.106) with Re L based on ˙m(1 −χ). The nucleate pool boiling coefficient is calculated with the Cooper (1984) reduced pressure correlation with R p set to 1 µm α nb = 55p 0.12 r −log 10 p r −0.55 M −0.5 q 0.67 (9.107) This dimensional correlation gives the heat transfer coefficient in W/m 2 · K and the heat flux q must be input in W/m 2 . M is the molecular weight and p r is the reduced pressure. Gungor and Winterton (1987) have also proposed a newer variation of this correlation. 9.8.4 Steiner–Taborek Method Steiner and Taborek (1992) proposed a model for vertical tubes using a large database containing 10,262 data points for water plus 2345 data points for four refrigerants (R-11, R-12, R-22, and R-113), seven hydrocarbons (benzene, n-pentane, n-heptane, cyclohexane, methanol, ethanol, and n-butanol), three cryogens (nitrogen, hydrogen, and helium), and ammonia. It is considered to be the most accurate vertical tube boil- ing correlation currently available. As opposed to the methods above, they assumed an asymptotic equation for the local two-phase flow boiling coefficient α tp : α tp = (α nb ) n + (α cb ) n 1/n (9.108) where n = 3 and the local flow boiling heat transfer coefficient is defined as α tp = q T wall − T sat (9.109) Here q,T wall , and T sat are local values along the evaporator tube and T sat corresponds to the local saturation pressure. Expression (9.108) is rewritten introducing the pa- rameters that affect heat transfer as α tp = (α nb,0 F nb ) 3 + (α L F tp ) 3 1/3 (9.110) BOOKCOMP, Inc. — John Wiley & Sons / Page 676 / 2nd Proofs / Heat Transfer Handbook / Bejan 676 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [676], (42) Lines: 1445 to 1493 ——— 0.85814pt PgVar ——— Normal Page * PgEnds: Eject [676], (42) where: • α nb,0 is the local nucleate pool boiling coefficient at a reference heat flux q 0 and reference reduced pressure of 0.1. • F nb is the nucleate boiling correction factor that accounts for the differences be- tween pool and flow boiling, and the effects of pressure, heat flux, tube diameter, surface roughness, and so on, but not boiling suppression (which is not required in an asymptotic model). • α L is the local liquid-phase forced-convection coefficient based on the total flow as liquid obtained with the Gnielinski (1976) correlation. • F tp is the two-phase multiplier that accounts empirically for the enhancement of convection in a two-phase flow. The term α nb,0 F nb may be obtained from experimental data or from the nucleate pool boiling correlation of the user’s choice (or from the method recommended below). The Gnielinski correlation is Nu L = α L d i λ L = (f/8)(Re L − 1000)Pr L 1 +12.7(f/8) 1/2 (Pr 2/3 L − 1) (9.111) where the Fanning friction factor f is f = [ 0.7904 ln(Re L ) −1.64 ] −2 (9.112) The total mass velocity of liquid plus vapor is used for evaluating Re L in eqs. (9.111) and (9.112). Only liquid physical properties are used. The two-phase multiplier F tp is for convective evaporation that will occur if χ < χ crit and q>q ONB or over the entire range of χ if q<q ONB . For applications where χ < χ crit at the tube exit, the following equation is used: F tp = (1 −χ) 1.5 + 1.9χ 0.6 ρ L ρ G 0.35 1.1 (9.113) This expression is valid for values of ρ L /ρ G from 3.75 to 5000 and converges to unity as χ goes to zero. At low heat fluxes where only pure convective evaporation is present over the entire range from χ = 0.0toχ = 1.0 (i.e., where the heat flux is too low for nucleation to occur), the following expression is used: F tp = (1 −χ) 1.5 + 1.9χ 0.6 (1 −χ) 0.01 ρ L ρ G 0.35 −2.2 + α G α L χ 0.01 1 +8(1 −χ) 0.7 ρ L ρ G 0.67 −2 −0.5 (9.114) BOOKCOMP, Inc. — John Wiley & Sons / Page 677 / 2nd Proofs / Heat Transfer Handbook / Bejan FLOW BOILING IN VERTICAL TUBES 677 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [677], (43) Lines: 1493 to 1613 ——— 5.75116pt PgVar ——— Normal Page PgEnds: T E X [677], (43) where χ 0.01 and (1 − χ) 0.01 take the expression to its proper limits at χ = 0 and χ = 1. This expression is valid for values of ρ L /ρ G from 3.75 to 1017. At χ = 1.0, the value of α tp corresponds to α G , which is the forced-convection coefficient with the total flow as all vapor. The nucleate boiling coefficient α nb is determined similar to the Gorenflo (1993) nucleate pool boiling method. The standard nucleate flow boiling coefficients α nb,0 to use are listed in Table 9.5, where the standard conditions are a reduced pressure of 0.1, a mean surface roughness of 1 µm, and a heat flux q 0 equal to the value listed for each fluid. The nucleate boiling correction factor F nb applied to the value of α nb,0 is F nb = F pf q q 0 nf d i d i,0 −0.4 R p R p,0 0.133 F(M) (9.115) where the pressure correction factor F pf , valid for p r < 0.95, is F pf = 2.816p 0.45 r + 3.4 + 1.7 1 −p 7 r p 3.7 r (9.116) The nucleate boiling exponent nf for all fluids is nf = 0.8 −0.1exp(1.75p r ) (9.117) except for cryogens, where it is nf = 0.7 −0.13 exp(1.105p r ) (9.118) The standard tube reference diameter d i,0 is equal to 0.01 m and the surface roughness term covers values of R p from 0.1 to 18 µm. The standard value of R p,0 is 1 µm and should be used if the surface roughness is not known. The Steiner–Taborek catch-all residual correction factor in terms of liquid molecular weight M is F(M) = 0.377 +0.199 ln M + 0.000028427M 2 (9.119) This expression is valid for 10 <M<187, with the maximum value of F(M) of 2.5, even for cases when the expression gives a larger value. For cryogenic liquids H 2 and He, the designated values of F(M)are 0.35 and 0.86, respectively. The minimum heat flux q ONB for determining the threshold for the onset of nucle- ate boiling is q ONB = 2σT sat α L r nuc ρ G h LG (9.120) where r nuc = 0.3 × 10 −6 m is used for the critical nucleation radius. For q>q ONB , nucleate boiling occurs, whereas below this value it does not. Notably, their method is applicable to pure fluids, but there is no simple way to determine the values of α nb,0 for mixtures. BOOKCOMP, Inc. — John Wiley & Sons / Page 678 / 2nd Proofs / Heat Transfer Handbook / Bejan 678 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [678], (44) Lines: 1613 to 1613 ——— 0.9385pt PgVar ——— Normal Page PgEnds: T E X [678], (44) TABLE 9.5 Standard Nucleate Flow Boiling Coefficients α nb,0 in W/m 2 · K at p r = 0.1forq 0 in W/m 2 and R p,0 = 1 µm with p crit in Bar Fluid p crit Mq 0 α nb,0 Methane 46.0 16.04 20,000 8,060 Ethane 48.8 30.07 20,000 5,210 Propane 42.4 44.10 20,000 4,000 n-Butane 38.0 58.12 20,000 3,300 n-Pentane 33.7 72.15 20,000 3,070 Isopentane 33.3 72.15 20,000 2,940 n-Hexane 29.7 86.18 20,000 2,840 n-Heptane 27.3 100.2 20,000 2,420 Cyclohexane 40.8 84.16 20,000 2,420 Benzene 48.9 78.11 20,000 2,730 Toluene 41.1 92.14 20,000 2,910 Diphenyl 38.5 154.2 20,000 2,030 Methanol 81.0 32.04 20,000 2,770 Ethanol 63.8 46.07 20,000 3,690 n-Propanol 51.7 60.10 20,000 3,170 Isopropanol 47.6 60.10 20,000 2,920 n-Butanol 49.6 74.12 20,000 2,750 Isobutanol 43.0 74.12 20,000 2,940 Acetone 47.0 58.08 20,000 3,270 R-11 44.0 137.4 20,000 2,690 R-12 41.6 120.9 20,000 3,290 R-13 38.6 104.5 20,000 3,910 R-13B1 39.8 148.9 20,000 3,380 R-22 49.9 86.47 20,000 3,930 R-23 48.7 70.02 20,000 4,870 R-113 34.1 187.4 20,000 2,180 R-114 32.6 170.9 20,000 2,460 R-115 31.3 154.5 20,000 2,890 R-123 36.7 152.9 20,000 2,600 R-134a 40.6 102.0 20,000 3,500 R-152a 45.2 66.05 20,000 4,000 R-226 30.6 186.5 20,000 3,700 R-227 29.3 170.0 20,000 3,800 RC318 28.0 200.0 20,000 2,710 R-502 40.8 111.6 20,000 2,900 Chloromethane 66.8 50.49 20,000 4,790 Tetrachloromethane 45.6 153.8 20,000 2,320 Tetrafluoromethane 37.4 88.00 20,000 4,500 Helium a 2.275 4.0 1,000 1,990 Hydrogen (para) 12.97 2.02 10,000 12,220 Neon 26.5 20.18 10,000 8,920 Nitrogen 34.0 28.02 10,000 4,380 Argon 49.0 39.95 10,000 3,870 Oxygen 50.8 32.00 10,000 4,120 Water 220.6 18.02 150,000 25,580 Ammonia 113.0 17.03 150,000 36,640 Carbon dioxide b 73.8 44.01 150,000 18,890 Sulfur hexafluoride 37.6 146.1 150,000 12,230 a Physical properties at p r = 0.3 rather than 0.1. b Calculated with properties at T crit . BOOKCOMP, Inc. — John Wiley & Sons / Page 679 / 2nd Proofs / Heat Transfer Handbook / Bejan FLOW BOILING IN HORIZONTAL TUBES 679 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [679], (45) Lines: 1613 to 1640 ——— 0.59909pt PgVar ——— Normal Page PgEnds: T E X [679], (45) 9.9 FLOW BOILING IN HORIZONTAL TUBES A composite schematic of convective evaporation in a horizontal tube is depicted in Fig. 9.14. At the inlet, the liquid enters subcooled. Farther along the tube, the liquid reaches its saturation temperature and the convective boiling process passes through various possible flow regimes, and flow can be either stratified or unstratified. In the latter case, dryout occurs at the top of the tube where the film thickness is thinnest, and dryout progresses around the perimeter from top to bottom along the tube. 9.9.1 Horizontal Tube Correlations Based on Vertical Tube Methods Some of the better known methods proposed over the years for predicting local flow boiling heat transfer coefficients inside vertical plain tubes have been adapted to horizontal tubes: for example, those of Shah (1982), Gungor and Winterton (1986, 1987), Klimenko (1988), and Kandlikar (1990). The first two are described below. The Shah (1982) vertical tube method described in Section 9.8 is applied to hor- izontal tubes by making an adjustment to N when the flow is stratified: that is, for Fr L < 0.04, N = 0.38Fr −0.3 L C 0 (9.121) where the liquid Froude number Fr L is defined as Fr L = ˙m 2 ρ 2 L gd i (9.122) Otherwise, for horizontal flow without stratification (i.e., Fr L ≥ 0.04), the Shah vertical tube method is used without change with N = C 0 . Figure 9.14 Flow patterns during evaporation in a horizontal tube with a uniform heat flux. (From Collier and Thome, 1994.) BOOKCOMP, Inc. — John Wiley & Sons / Page 680 / 2nd Proofs / Heat Transfer Handbook / Bejan 680 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [680], (46) Lines: 1640 to 1676 ——— 8.0pt PgVar ——— Normal Page PgEnds: T E X [680], (46) Gungor and Winterton (1986) also used the liquid Froude number as the threshold criterion for flow stratification, but they set the threshold value a little higher: at Fr L < 0.05. Their stratified flow correction factor E 2 is E 2 = Fr 0.1−2Fr L L (9.123) and is applied to their convection enhancement factor E in eq. (9.104). In addition, when Fr L < 0.05, their boiling suppression factor S must be multiplied by the correction factor S 2 = (Fr L ) 1/2 (9.124) Kattan et al. (1998a) have pointed out that the liquid Froude number is not reliable for predicting the stratification threshold, identifying it correctly only about 50% of the time. Therefore, the methods above can be used for annular and intermittent flows but do not predict heat transfer in stratified types of flow very accurately. Thome (1995a) summarized the pitfalls of applying vertical methods to horizontal flow boiling as follows: • Vertical tube methods do not recognize the different flow patterns occurring in horizontal flow boiling, and their stratification criteria are not reliable. • These methods are incapable of predicting the sharp peak in α tp versus χ found in many experimental data sets, nor can they predict the subsequent sharp decline in α tp after the onset of dryout at the top of the tube in annular flows at high vapor qualities. • They do not attempt to model the two-phase flow structure itself and utilize a tubular flow correlation (Dittus–Boelter) as their starting point to predict annular film flow heat transfer. Still, these methods are simple to implement and they predict flow boiling coeffi- cients in the annular flow regime with reasonable accuracy as long as they are limited to vapor qualities below the peak in α tp versus χ, which occurs at about 80 to 90% vapor quality. 9.9.2 Horizontal Flow Boiling Model Based on Local Flow Regime Kattan et al. (1998a,b,c) proposed a flow boiling model that takes a more fundamental approach to predicting local flow boiling heat transfer coefficients by incorporating a simplified local two-phase flow structure into the prediction method as a function of the local flow pattern. So far, their model covers annular flows, annular flows with partial dryout, intermittent flows, stratified-wavy flows, and fully stratified flows, and methods for predicting heat transfer in bubbly flows and mist flows are under development. Their method identifies the local two-phase flow pattern using their flow pattern map described in Section 9.7. BOOKCOMP, Inc. — John Wiley & Sons / Page 681 / 2nd Proofs / Heat Transfer Handbook / Bejan FLOW BOILING IN HORIZONTAL TUBES 681 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [681], (47) Lines: 1676 to 1698 ——— 1.8231pt PgVar ——— Normal Page * PgEnds: Eject [681], (47) Figure 9.15 Geometric illustration of liquid and vapor areas, stratified and dry angles, and liquid film thickness used in the flow boiling model. Their simplified two-phase flow structures assumed for the fully stratified, strat- ified-wavy, and annular flow regimes are depicted in Fig. 9.15. In annular flow, the liquid is assumed to form a uniform liquid ring on the tube wall, while for stratified- wavy flow a truncated annular ring is assumed to represent the flow structure. For fully stratified flow, the liquid is also assumed to form a truncated annular ring, with the same cross-sectional area of liquid as in its true stratified shape. Annular flow with partial dryout around the top of the tube is similar to stratified-wavy flow, and the heat transfer model treats it so. Currently, the complex intermittent flow structure is represented by the annular flow structure, an assumption that provides good representation of that heat transfer data. Their general equation for the local flow boiling coefficient α tp for evaporation in a horizontal, plain tube is α tp = r i θ dry α vapor + r i (2π −θ dry )α wet 2πr i (9.125) where the internal tube radius is r i . The dry perimeter of the tube, if any, is given by r i θ dry , where θ dry is the dry angle around the top of the tube, and the wetted perimeter of the tube is r i (2π −θ dry ). The vapor-phase heat transfer coefficient α vapor is applied to the dry perimeter, and the wet wall heat transfer coefficient α wet is applied to the wet perimeter. The wet wall heat transfer coefficient is obtained from an asymptotic model that combines the nucleate boiling α nb and convective boiling α cb contributions to heat transfer by the third power as α wet = α 3 nb + α 3 cb 1/3 (9.126) The nucleate boiling heat transfer coefficient α nb is determined with the dimensional reduced pressure correlation of Cooper (1984): α nb = 55p 0.12 r (−log 10 p r ) −0.55 M −0.5 q 0.67 (9.127) BOOKCOMP, Inc. — John Wiley & Sons / Page 682 / 2nd Proofs / Heat Transfer Handbook / Bejan 682 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [682], (48) Lines: 1698 to 1745 ——— 0.25806pt PgVar ——— Long Page PgEnds: T E X [682], (48) where a roughness equal to his standard surface roughness factor is assumed, α nb is in W/m 2 · K,p r is the reduced pressure, M is the liquid molecular weight, and q is the heat flux at the tube wall in W/m 2 . The convective boiling heat transfer coefficient α cb for annular liquid film flow was formulated as a film flow, not as a tubular flow, as follows: α cb = 0.0133 4 ˙m(1 −χ)δ (1 −ε)µ L 0.69 c pL µ L λ L 0.4 λ L δ (9.128) The term in brackets is the liquid film Reynolds number Re L and the following term is the liquid Prandtl number Pr L . The mean liquid velocity in the annular film is utilized in determining the liquid Reynolds number, which changes as a local function of the vapor quality χ, annular liquid film thickness δ, and vapor void fraction ε. Note that in this formulation, neither an empirical boiling suppression factor nor two-phase convection multiplier is required. The vapor-phase heat transfer coefficient α vapor is calculated with the Dittus– Boelter turbulent flow heat transfer correlation assuming tubular flow on the dry perimeter of the tube using the vapor properties and the mass velocity of the vapor: α vapor = 0.023 ˙mχd i εµ G 0.8 c pG µ G λ G 0.4 λ G d i (9.129) The vapor Reynolds number Re G in the first term in parentheses above includes the vapor void fraction ε such that the mean vapor velocity in the cross section of the tube occupied by the vapor is used in its determination. The internal tube diameter is d i , λ L and λ G are the liquid and vapor thermal conductivities, c pL and c pG are the liquid and vapor specific heats, and µ L and µ G are the liquid and vapor dynamic viscosities. The total mass velocity of the liquid plus vapor through the tube is ˙m, and χ is the local vapor quality. The dry angle θ dry in eq. (9.125) is the circumferential angle of the tube wall, which is assumed to be constantly dry for stratified types of flow and for annular flows with partial dryout. On the other hand, for annular and intermittent flows, the entire tube perimeter is always wetted and hence θ dry is equal to zero, in which case α tp is equal to α wet . Methods for determining θ dry , ε, and δ are described below. The Rouhani–Axelsson (1970) drift flux type of void fraction model is used to calculate the vapor void fraction ε: ε = χ ρ G [ 1 +0.12(1 − χ) ] χ ρ G + 1 −χ ρ L + 1.18 ˙m gσ(ρ L − ρ G ) ρ 2 L 1/4 (1 −χ) −1 (9.130) where ρ L and ρ G are the liquid and vapor densities and σ is the surface tension (all in SI units). The cross-sectional area of the tube occupied by the liquid-phase A L is then obtainable as A L = A(1 −ε) (9.131) BOOKCOMP, Inc. — John Wiley & Sons / Page 683 / 2nd Proofs / Heat Transfer Handbook / Bejan FLOW BOILING IN HORIZONTAL TUBES 683 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [683], (49) Lines: 1745 to 1797 ——— 1.1032pt PgVar ——— Long Page PgEnds: T E X [683], (49) and A is the total cross-sectional area of the tube. For the fully stratified flow regime illustrated on the right in Fig. 9.15, the stratified angle θ strat of the liquid is defined geometrically as A L = 0.5r 2 i [ (2π −θ strat ) −sin(2π −θ strat ) ] (9.132) where θ strat is in radians. Equation (9.132) is an implicit expression and has to be solved iteratively using A L to find the value of the stratified angle θ strat . A simplified version of this expression that avoids an iterative solution is θ strat 2 π −cos −1 (2ε −1) (9.133) The concept of a dry angle θ dry is introduced for the dry perimeter of the stratified- wavy flow regime. The dry angle θ dry at a particular vapor quality varies from its lower limit of θ dry = 0 for annular or intermittent flow on the ˙m wavy transition curve with complete wall wetting to its maximum value of θ dry = θ strat for fully stratified flow on the ˙m strat transition curve. These mass velocities, called ˙m high and ˙m low , respectively, themselves are functions of vapor quality, and hence θ dry changes as a function of vapor quality and mass velocity. The simple linear expression for θ dry between ˙m high and ˙m low for any value of χ < χ max is θ dry = θ strat ˙m high −˙m ˙m high −˙m low (9.134) At vapor qualities where χ > χ max , an additional step is required to determine θ dry since ˙m high runs into the intersection of the ˙m wavy and ˙m mist curves, and for χ > χ max there is no ˙m wavy curve for determining ˙m high . Thus when χ > χ max , the dry angle θ dry is prorated horizontally as a linear function of vapor quality between θ max and 2π, the latter being the upper limit since the tube wall is completely dry at χ = 1, so that θ dry = (2π −θ max ) χ −χ max 1 −χ max + θ max (9.135) Note that θ max is determined from eq. (9.134) at χ = χ max . The annular liquid film thickness δ is determined by equating the cross-sectional area occupied by the liquid phase for this void fraction and dry angle to that of a truncated annular liquid ring, assuming that the thickness δ is small compared to the tube radius r i , obtaining δ = A L r i (2π −θ dry ) = A(1 −ε) r i (2π −θ dry ) = πd i (1 −ε) 2(2π − θ dry ) (9.136) Figure 9.16 presents the heat transfer coefficients predicted by the Kattan–Thome– Favrat flow boiling model using their flow pattern map with the updated corrections to their wavy flow and stratified flow transitions made by Z ¨ urcher et al. (1999) to the flow pattern map. A 19.86-mm (0.01986-m)-internal-diameter tube is simulated for n-butane evaporating at saturation conditions of 60°C and 0.6394 MPa and a heat flux of 15,000 W/m 2 . The heat transfer curves for α tp cover: . The vapor-phase heat transfer coefficient α vapor is applied to the dry perimeter, and the wet wall heat transfer coefficient α wet is applied to the wet perimeter. The wet wall heat transfer coefficient. pa- rameters that affect heat transfer as α tp = (α nb,0 F nb ) 3 + (α L F tp ) 3 1/3 (9.110) BOOKCOMP, Inc. — John Wiley & Sons / Page 676 / 2nd Proofs / Heat Transfer Handbook / Bejan 676. in a horizontal tube with a uniform heat flux. (From Collier and Thome, 1994.) BOOKCOMP, Inc. — John Wiley & Sons / Page 680 / 2nd Proofs / Heat Transfer Handbook / Bejan 680 BOILING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [680],