Heat Transfer Handbook part 76 pptx

BOOKCOMP, Inc. — John Wiley & Sons / Page 745 / 2nd Proofs / Heat Transfer Handbook / Bejan CONDENSATION IN SMOOTH TUBES 745 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 [745], (27) Lines: 802 to 806 ——— -2.279pt PgVar ——— Normal Page PgEnds: T E X [745], (27) 0 0.2 0.4 0.6 0.8 1 0 100 200 300 400 600 700 800 500 Mass Flux (kg/m) Quality Annular Slug Stratified smooth Stratified wavy Figure 10.9 Taitel–Dukler (1976) predictions on G–x coordinates for R-134a at 35°C in a 7.04-mm-ID tube. (From Dobson and Chato, 1998.) regions of wavy, wavy–annular, then annular flow. The quality range over which the wavy and wavy–annular flow regimes occurred decreased as mass flux was increased. The predictions of the Taitel–Dukler map are translated onto mass flux–quality (G–x) coordinates in Fig. 10.9. At low mass fluxes, stratified flow is predicted across the entire range of quality. At slightly higher mass fluxes, wavy flow is predicted across most of the quality range with a small amount of stratified flow at low quality. At mass fluxes above 140 kg/s ·m 2 , slug flow is predicted for qualities below 11.8% and annular flow is predicted for all higher qualities. It is close to this boundary that the observed flow regimes deviated most significantly from the Taitel–Dukler predictions. The length of the slug flow region was underpredicted, and this was consistently followed by some wavy or wavy–annular flow that was not predicted by the Taitel–Dukler map. The apparent discrepancy between the observed and predicted flow regimes at mass fluxes slightly above the annular boundary of the Taitel–Dukler map is due largely to differences in terminology. In an early experimental verification of the Taitel–Dukler map, Barnea et al. (1980) used the term wavy–annular flow to refer to a hybrid pattern observed at the lowest gas rates where the slug-to-annular tran- sition occurred. A similar regime has been termed proto-slug flow by Nicholson et al. (1978), and pseudo-slug flow by Lin and Hanratty (1989). Because this pattern occurs after the wavy flow has become unstable, it is properly labeled as intermit- tent or annular flow in Taitel–Dukler terminology. From a heat transfer standpoint, however, the instability of the wavy flow near this boundary is less important than the significant stratification due to gravity. At higher mass fluxes, the range of quality BOOKCOMP, Inc. — John Wiley & Sons / Page 746 / 2nd Proofs / Heat Transfer Handbook / Bejan 746 CONDENSATION 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 [746], (28) Lines: 806 to 818 ——— 0.0pt PgVar ——— Long Page PgEnds: T E X [746], (28) occupied by this hybrid flow pattern becomes so small that proper classification is unimportant. Of the boundaries from the Taitel–Dukler map that were used, only that between wavy and intermittent or annular flow depends on the diameter. The parameter that is used for predicting this transition, F td , is proportional to D −0.5 at a fixed mass flux and quality. Thus, decreasing the diameter increases the Froude number and decreases the mass flux at which the annular transition is expected to occur. This is consistent with the observed trend of more annular flow in the smaller tubes. Although the trend is physically correct, the predictions themselves were incorrect when applied to the data in the 3.14-mm-inside-diameter tube. TheTaitel–Dukler method predicts annular flow across nearly the entire range of quality at a mass flux of 75 kg/s · m 2 , while wavy flow was observed exclusively at this mass flux. The predicted trend of the Taitel–Dukler map to an increase in the reduced pressure was consistent with the experimental observations. For example, the slug flow region was wider, as predicted by eq. (10.35). At higher reduced pressures one would also expect the stratified-to-wavy and wavy-to-annular transitions to be shifted to higher mass fluxes, due to the lower vapor velocity at a given mass flux and quality. This small shift is due to two opposing trends brought about by the changes in fluid properties. At constant mass flux and quality, the value of F td decreases with increasing pressures, moving the curve downward relative to the transition boundary on the Taitel–Dukler map. However, the value of X tt at constant quality increases with pressure, moving the curve to the right on the Taitel– Dukler map and therefore closer to the boundary. From a practical standpoint, this predicted shift in the transition boundary is insignificant. The apparent discrepancy between the predictions and the observations at mass fluxes slightly above the annular flow boundary was also present with all the refrigerants used. For example, at a mass flux of 150 kg/s · m 2 , wavy flow persisted at qualities up to 50%, while the Taitel– Dukler map predicted annular flow above 20% quality. Soliman Transitions Soliman (1982, 1986) developed criteria for two flow regime transitions for condensation: (1) wavy or slug flow to annular flow, and (2) annular flow to mist flow. His transition criteria are displayed on G–x coordinates in Fig. 10.10. Several interesting observations can be made from comparing the predictions of Soliman to those of Taitel and Dukler. First, at high qualities Soliman’s prediction of the wavy-to-annular transition agrees fairly well with that of Taitel and Dukler. Un- like the Taitel–Dukler map, though, Soliman predicts a wavy region at low qualities over the entire mass flux range of this study. This occurs partially because Soliman lumps the wavy and slug flow regions together. At high mass fluxes, the region pre- dicted to be wavy flow by Soliman corresponds almost exactly with the slug flow region on the Taitel–Dukler map. At lower mass fluxes, though, the region predicted to be occupied by wavy flow extends to higher qualities than the slug flow boundary on the Taitel–Dukler map. This is consistent with the experimental data in both mag- nitude and trend if the predicted transition line is considered to be that of wavy flow to wavy–annular flow. It was shown by Dobson (1994) and Dobson et al. (1994a,b) that the transition from wavy–annular flow to annular flow was well predicted by a value of Fr so = 18, as opposed to Fr so = 7 for the wavy-to-wavy–annular transition. BOOKCOMP, Inc. — John Wiley & Sons / Page 747 / 2nd Proofs / Heat Transfer Handbook / Bejan CONDENSATION IN SMOOTH TUBES 747 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 [747], (29) Lines: 818 to 827 ——— 0.25099pt PgVar ——— Long Page PgEnds: T E X [747], (29) Unlike the maps of Mandhane and Taitel–Dukler, Soliman’s map also includes a distinct mist flow region. According to Soliman, mist or spray flow is a regime with all the liquid flowing as entrained droplets in the core flow and no stable film on the wall. Annular mist flow would refer to a regime with a stable liquid film on the wall and significant entrainment in the core flow. According to the observations made by Dobson (1994), most of the region labeled as mist flow by Soliman’s map would more properly be called annular–mist flow. Although the amount of entrainment was very significant, a stable liquid film was always observed on the wall at qualities below 90%. Even when the flow entered the sight glass at the inlet of the test section as mist flow, the outlet sight glass always had annular–mist flow. This observation suggests that the net mass flux toward the wall during condensation always results in a stable liquid film, no matter what the observations might indicate in an adiabatic section. This finding is important for interpreting the annular–mist flow heat transfer data. If Soliman’s mist flow region is interpreted as annular–mist flow, the predictions seem quite reasonable. The diameter effects predicted by Soliman’s transition criteria are also shown in Fig. 10.10. The lower mass flux limit at which annular flow is predicted is relatively insensitive to the diameter change, much like the predictions of Taitel and Dukler. At mass fluxes slightly above this, however, the wavy-to-annular transition line was shifted to lower qualities with decreasing diameter. This was consistent with the 0 0.2 0.4 0.6 0.8 1 0 100 200 300 400 600 700 800 500 Mass Flux (kg/m) Quality Wavy to Annular Annular to Mist Flow Annular to Mist Flow Wavy to Annular Wavy Annular Mist flow (7.04) Mist flow (3.14) Figure 10.10 Soliman’s (1982, 1983) predicted flow regime on G–x coordinates for R-134a at 35°C. (From Dobson and Chato, 1998.) BOOKCOMP, Inc. — John Wiley & Sons / Page 748 / 2nd Proofs / Heat Transfer Handbook / Bejan 748 CONDENSATION 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 [748], (30) Lines: 827 to 836 ——— 2.25099pt PgVar ——— Normal Page PgEnds: T E X [748], (30) experimental observations in both direction and magnitude. The predicted effect on the mist flow regime was much more dramatic, with a significant stabilizing effect on the liquid film being predicted as the tube diameter was decreased. This was consistent with the trend of the observations, although the transition was very difficult to detect visually. The effect that increasing the reduced pressure has on the wavy-to-annular flow regime transition predicted by Soliman is shown in Fig. 10.11. This figure compares the predicted wavy-to-annular transition lines for R-134a at 35°C and R-32/R-125 at 45°C (low and high reduced pressures). The predicted trends are consistent with the experimental observations. Because reliable surface tension data were not yet available for R-32/R-125 and drawing any conclusions concerning mist flow from the visualization was difficult, only the lines for the wavy-to-annular transition were included in Fig. 10.11. As another way of assessing the effect of increasing reduced pressure, the magnitude of the Weber number was examined as the temperature was increased for both R- 134a and R-22. The Weber number increased as temperature increased due to the reduced liquid viscosity and surface tension, but only slightly (less than 10% as the temperature of both fluids was raised from 35°C to 55°C). This small change indicates that the decreased surface tension and liquid viscosity are nearly balanced by corresponding decreases in the density ratio. Based on these trends, one would 0 0.2 0.4 0.6 0.8 1 0 100 200 300 400 600 700 800 500 Mass Flux (kg/m) Quality R-134a, 35°C, 7.04 mm R-32/R-125, 45°C, 7.04 mm Wavy flow Annular flow Figure 10.11 Effect of reduced pressure on Soliman’s (1982) wavy-to-annular flow regime transition. (From Dobson and Chato, 1998.) BOOKCOMP, Inc. — John Wiley & Sons / Page 749 / 2nd Proofs / Heat Transfer Handbook / Bejan CONDENSATION IN SMOOTH TUBES 749 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 [749], (31) Lines: 836 to 850 ——— 0.927pt PgVar ——— Normal Page PgEnds: T E X [749], (31) expect slightly more entrainment to occur at higher reduced pressures at identical mass flux and quality. 10.6.3 Heat Transfer in Horizontal Tubes The following discussion is based on data of Dobson (1994) in the smooth–stratified, wavy–stratified, wavy–annular, annular, and annular–mist flow regimes. Effects of Mass Flux and Quality Figure 10.12 presents typical heat transfer data for R-32/R-125 (60%/40%) in a 3.14-mm-inside-diameter tube at a saturation temperature of 35°C. At the lowest mass flux of 75 kg/s · m 2 , the Nusselt number increases very modestly as the quality is increased. A similar quality dependence is exhibited as the mass flux is doubled to 150 kg/s · m 2 . The Nusselt numbers remain nearly identical as the mass flux is doubled. At a mass flux of 300 kg/s ·m 2 , a different trend emerges. At low qualities, the heat transfer coefficients remain nearly identical to the lower mass flux cases. As the quality increases to around 30%, the Nusselt number displays a much more pronounced effect of quality. At mass fluxes above 300 kg/s · m 2 , the dependence of the Nusselt number on quality remains similar. Even at low qualities, the Nusselt numbers are substantially higher than those for the low- mass-flux cases. If the same data were plotted as Nusselt number versus mass flux, the heat transfer coefficient would remain relatively constant at low mass fluxes. At a given mass flux, the slope of the heat transfer versus mass flux curve increases to a relatively constant value. The mass flux at which this change in slope occurs increases as the quality is decreased. For example, at 25% quality this shift occurs at a mass flux of 300 kg/s · m 2 . Figure 10.12 Variation of Nusselt number with quality for 60%/40% R-32/R-125 mxiture at 35°C in 3.14-mm-ID test section. Mass flux G is in kg/s · m. (From Dobson and Chato, 1998.) BOOKCOMP, Inc. — John Wiley & Sons / Page 750 / 2nd Proofs / Heat Transfer Handbook / Bejan 750 CONDENSATION 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 [750], (32) Lines: 850 to 870 ——— -1.80798pt PgVar ——— Normal Page PgEnds: T E X [750], (32) The change in heat transfer behavior exhibited in Fig. 10.12 is closely linked to changes in the two-phase flow regime. For the two lowest mass fluxes, 75 and 150 kg/s · m 2 , wavy or wavy–annular flow prevail over much of the quality range. The primary item affecting the heat transfer coefficient in this flow regime is the film thickness, which is insensitive to mass flux. Thus, wavy flow heat transfer coefficients are also relatively insensitive to mass flux. At the highest mass fluxes, annular flow prevails over most of the quality range. In the annular flow regime, correlations such as those of Soliman et al. (1968) and Traviss et al. (1973) clearly illustrate the interdependence between pressure drop and heat transfer (h is proportional to √ −∆P/∆Z). Because the pressure drop increases sharply as the quality is increased, the heat transfer coefficients in the annular flow regime show significant quality dependence. At the intermediate mass flux of 300 kg/s · m 2 , the flat Nusselt number versus quality behavior that is characteristic of wavy flow occurs at low qualities whereas annular flow behavior appears at higher qualities. The change in slope in Fig. 10.12 occurs around 30% quality, corresponding closely to the observed change from the wavy–annular to the annular flow regime. At this mass flux, it would be inappropriate to use a single heat transfer model over the entire quality range. Effects of Tube Diameter The relationships between h and D that were pre- dicted by annular and wavy flow heat transfer coefficients agreed well with the ex- perimental data. Although this was expected for the 7.04-mm-inside-diameter (ID) tube, a commonly used and tested size, some doubt existed about whether the heat transfer behavior in the 3.14-mm-ID tube would correspond with that predicted by “large tube” correlations. The most noticeable effect of the tube diameter has to do with the point at which the heat transfer mechanism changes from filmwise (wavy) to forced-convective (annular). The primary difference in the heat transfer behavior in the two tubes was observed at a mass flux of 300 kg/s · m 2 . In the 3.14-mm-ID tube, the heat transfer behavior showed a change in slope around 30% quality as the flow regime changed from wavy-annular to annular. In the larger 7.04-mm-ID tube, this transition was observed only at the highest quality point (89%), corresponding closely with observed transition to annular flow at around 80% quality. The heat trans- fer characteristics at the other mass fluxes were similar in both tubes. Effects of Fluid Properties In the stratified and wavy flow regimes, the property index from the Chato (1960, 1962) correlation,  ρ l (ρ l − ρ g )k 3 l λ/µ l  0.25 , can be used to compare the expected heat transfer behavior of different fluids. In the annular flow regime the single-phase liquid heat transfer index k 0.6 l c 0.4 pl /µ 0.4 l can be used at low qualities, but at higher qualities the two-phase multiplier, described below, becomes dominant. Effects of Temperature Difference The refrigerant-to-wall temperature differ- ence has an impact on the heat transfer coefficients in the wavy flow regime. This dependence occurs because for a falling film, a larger temperature difference results in a thicker film at a given location (hence, lower heat transfer coefficients). In the BOOKCOMP, Inc. — John Wiley & Sons / Page 751 / 2nd Proofs / Heat Transfer Handbook / Bejan CONDENSATION IN SMOOTH TUBES 751 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 [751], (33) Lines: 870 to 876 ——— 0.721pt PgVar ——— Normal Page * PgEnds: Eject [751], (33) 0 0.2 0.4 0.6 0.8 1 ⌬T =2°C ⌬T =3°C 0 50 100 150 200 250 300 Nu Average Quality Figure 10.13 Effect of ∆T on Nu for R-32/R-125 in a 7.04-mm test section, G = 75 (kg/ s · m), T sat = 35°C. (From Dobson and Chato, 1998.) annular flow regime, a significant amount of experimental and analytical evidence suggests a negligible impact of temperature difference. Studying the effect of tem- perature difference on the heat transfer coefficients then provides a nonvisual method of assessing the extent of film wise and/or forced-convective condensation. Precisely controlling the temperature difference during internal condensation ex- periments is very difficult. For this reason, few internal condensation data are avail- able for which the temperature differences were controlled deliberately. Figure 10.13 shows the variation of Nusselt number with quality for R-32/R-125 at a saturation temperature of 35°C and a mass flux of 75 kg/s·m 2 . The two sets of points correspond to temperature differences of approximately 2°C (1.88 to 2.12°C) and 3°C (2.87 to 3.11°C). As predicted by the Nusselt theory, the Nusselt numbers are lower for the higher temperature difference data across the full range of quality. Using the quantity Nu/(Ga · Pr l /Ja l ) 0.25 , based on liquid properties, on the vertical axis instead of Nu brings these data on a single line because at the very low mass flux almost all the heat transfer occurs by filmwise condensation on the top and very little occurs in the bottom of the tube. As the quality approaches unity and the liquid pool vanishes, the value of Nu/(Ga·Pr l /Ja l ) 0.25 properly approaches the value of 0.728 for condensation outside a horizontal cylinder. Gravity-Driven Condensation The gravity-driven flow regimes include the stratified, wavy, and slug flow regions. These regimes are lumped together primarily because the dominant heat transfer mechanism in each regime is conduction across the film at the top of the tube. This type of condensation is commonly referred to as film condensation. Nusselt’s (1916) heat transfer coefficient for gravity-driven condensation of a pure component on a vertical plate given by eq. (10.11) can be expressed as the mean Nusselt number at z = L: BOOKCOMP, Inc. — John Wiley & Sons / Page 752 / 2nd Proofs / Heat Transfer Handbook / Bejan 752 CONDENSATION 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 [752], (34) Lines: 876 to 921 ——— 0.81113pt PgVar ——— Normal Page PgEnds: T E X [752], (34) Nu L = hL k l = 0.943  ρ l g(ρ l − ρ g )L 3 λ k l µ l (T sat − T w )  1/4 (10.43) The bracketed term in eq. (10.43) can be expressed in dimensionless form as NuL = 0.943  Ga L · Pr l Ja l  1/4 (10.44) Dhir and Lienhard (1971) devised a simple way to extend the analysis for the vertical wall to arbitrary axisymmetric bodies. They showed that the local Nusselt number can be predicted by replacing g in eq. (10.43) with an effective acceleration of gravity: g eff = x(gr) 4/3  x 0 g 1/3 r 4/3 dx (10.45) In eq. (10.45), r(x) is the local radius of curvature and g(x) is the local gravity com- ponent in the x direction. For the horizontal cylinder, as was done for eq. (10.12), the effective gravity can be evaluated numerically and averaged over the circumference of the tube to obtain Nu = 0.729  Ga · Pr l Ja l  1/4 (10.46) The diameter is the length scale in Ga. Based on integral analyses, Bromley (1952) and Rohsenow (1956) corrected for the assumption of a linear temperature profile by replacing the latent heat in these equations by a modified latent heat given by λ  = λ(1 + 0.68Ja l ) (10.47) This correction shows that the assumption of a linear temperature profile in the original analysis is quite acceptable for Ja l much less than unity. During condensation inside horizontal, smooth tubes at low vapor velocities, grav- itational forces, which tend to pull condensate down the tube wall, are much stronger than vapor shear forces, which tend to pull the condensate in the direction of the mean flow. Thus, a condensate film forms on the top of the tube and grows in thickness as it flows around the circumference. The bottom portion of the tube is filled with a liquid pool that transports the condensed liquid along the tube in the direction of the mean flow. Chato (1960, 1962) concentrated on stratified flows with low vapor velocities, Re vo < 35,000. He developed a similarity solution for the condensate film, which was patterned after Chen’s (1961) analysis of falling-film condensation outside a horizontal cylinder. He applied this solution to the upper portion of the tube, where falling-film condensation existed (i.e., down to the liquid pool on the bottom). To BOOKCOMP, Inc. — John Wiley & Sons / Page 753 / 2nd Proofs / Heat Transfer Handbook / Bejan CONDENSATION IN SMOOTH TUBES 753 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 [753], (35) Lines: 921 to 952 ——— 0.18407pt PgVar ——— Normal Page * PgEnds: Eject [753], (35) predict the depth of the liquid pool, he developed a separate model based on open channel hydraulics. Both his analytical model and experimental results for R-113 showed that the depth of the liquid level was relatively constant. This allowed his heat transfer data to be approximated quite well by the following correlation for the average Nusselt number: Nu = hD k l = 0.555  ρ l g(ρ l − ρ g )D 3 λ k l µ l (T sat − T w )  1/4 (10.48) The constant 0.555 is 76% of the value of 0.728 for external condensation on a cylinder. This decrease in heat transfer is due to the thickness of the liquid pool on the bottom of the tube, which reduces the heat transfer to negligible amounts. Jaster and Kosky (1976) proposed a correlation similar to Chato’s for stratified flow condensation. To account for the variation of the liquid pool depth in a manner consistent with pressure-driven flow, they replaced the constant in the Chato correla- tion with a function of the void fraction α. This resulted in Nu = hD k l = 0.728α 3/4  ρ l g(ρ l − ρ g )D 3 λ k l µ l (T sat − T w )  1/4 (10.49) They recommend using Zivi’s (1964) correlation for the void fraction: α =  1 + 1 − x x  ρ g ρ l  2/3  −1 (10.50) Jaster and Kosky’s correlation overpredicts the Chato correlation for all qualities to greater than about 0.2. It had a mean deviation of 37% with their own data, which it appeared to overpredict consistently. The correlations of Chato and Jaster and Kosky both neglect the heat transfer that occurs in the liquid pool at the bottom of the tube. Chato showed that considering conduction only, this heat transfer was negligible compared to that through the upper part of the tube. This assumption is reasonable for low-speed stratified flows, but might not be so for higher-mass-flux low-quality situations where wavy or stratified flow could prevail, creating substantial convective heat transfer in the bottom of the tube. Rosson and Myers (1965) collected experimental data in what they called the intermittent flow regime, which included stratified, wavy, and slug flows. They measured the variation of heat transfer coefficient with angle around the tube and found, as expected, that the heat transfer coefficient decreased continuously from the top to the bottom of the tube. They proposed replacing the constant in the Nusselt’s solution with an empirically determined function of the vapor Reynolds number: Nu top = 0.31Re 0.12 g  ρ l g(ρ l − ρ g )D 3 λ k l µ l (T sat − T w )  1/4 (10.51) BOOKCOMP, Inc. — John Wiley & Sons / Page 754 / 2nd Proofs / Heat Transfer Handbook / Bejan 754 CONDENSATION 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 [754], (36) Lines: 952 to 994 ——— 0.98038pt PgVar ——— Normal Page PgEnds: T E X [754], (36) In the bottom of the tube, they postulated forced-convective heat transfer. Using a heat and momentum transfer analogy, they recommended the following correlation: Nu bot = φ l,lt √ 8Re l 5 [ 1 + ln(1 + 5Pr l )/Pr l ] (10.52) where φ l,lt =  1 + 1 X lt + 12 X 2 lt (10.53) Rosson and Myers defined a parameter β that represented the fraction of the tube perimeter over which filmwise condensation occurred. They recommended predicting the value of β as follows: β =          Re 0.1 g if Re 0.6 g · Re 0.5 l Ga < 6.4 × 10 −5 (10.54a) 1.74 × 10 −5 Ga  Re g · Re l if Re 0.6 g · Re 0.5 l Ga > 6.4 × 10 −5 (10.54b) Then the circumferentially averaged Nusselt number was given by Nu = β · Nu top + (1 −β)Nu bot (10.55) Rosson and Myers compared their predicted values to their own experimental data for acetone and methanol, and the agreement was reasonable. A large number of scatter was inherent due to inaccuracies in their experimental techniques, so it is difficult to discern whether the deviations were due to theoretical deficiencies or experimental scatter. Tien et al. (1988) presented an analysis for gravity-driven condensation that they proposed to be valid for stratified, wavy, and slug flow. Their analysis was similar to that of Rosson and Myers, although more deeply rooted in conservation equations than are empirically determined expressions. This analysis approaches the correct values in the asymptotic limits. That is, for a quality of zero it predicts a single-phase- liquid Nusselt number, and for situations where stratified flow exists rather than slug flow, it reduces to the form of Rosson and Myers. To use the Tien model, six simul- taneous nonlinear equations must be solved. Although novel and well structured, the technique is probably too involved for a practical design correlation. Dobson (1994), Dobson et al. (1994a,b), and Dobson and Chato (1998) devel- oped correlations for wavy flows based primarily on their own data obtained with refrigerants. As the vapor velocities increase from very low values, the vapor shear causes an increase in the convective heat transfer in the pool at the bottom of the tube and it generates an axial velocity component in the condensate film at the top of the tube. The development of the correlation was guided by a combination of careful . film at a given location (hence, lower heat transfer coefficients). In the BOOKCOMP, Inc. — John Wiley & Sons / Page 751 / 2nd Proofs / Heat Transfer Handbook / Bejan CONDENSATION IN SMOOTH. the heat transfer that occurs in the liquid pool at the bottom of the tube. Chato showed that considering conduction only, this heat transfer was negligible compared to that through the upper part. interdependence between pressure drop and heat transfer (h is proportional to √ −∆P/∆Z). Because the pressure drop increases sharply as the quality is increased, the heat transfer coefficients in the annular