The quality range over which the wavy and wavy–annular flow regimes occurred decreased as mass flux was increased.. At low mass fluxes, stratified flow is predicted across the entire range of
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0 100 200 300 400
600 700 800
500
Quality
Annular Slug
Stratified smooth Stratified wavy
Figure 10.9 Taitel–Dukler(1976) predictions onG–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 · m2, slug flow is predicted for qualities below 11.8%
and annularflow is predicted forall higherqualities 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 orwavy–annularflow that was not predicted
by the Taitel–Duklermap
The apparent discrepancy between the observed and predicted flow regimes at mass fluxes slightly above the annularboundary of the Taitel–Duklermap 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
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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 orannularflow depends on the diameter The parameterthat
is used for predicting this transition,F td, is proportional toD −0.5at a fixed mass flux
and quality Thus, decreasing the diameter increases the Froude number and decreases the mass flux at which the annulartransition 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-diametertube The Taitel–Duklermethod predicts annular flow across nearly the entire range of quality at a mass flux of 75 kg/s · m2, 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-annulartransitions to be shifted to highermass fluxes, due to the lowervapor 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 ofF tddecreases with increasing pressures, moving the curve downward relative
to the transition boundary on the Taitel–Dukler map However, the value ofX 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 · m2, wavy flow persisted at qualities up to 50%, while the Taitel–
Duklermap predicted annularflow above 20% quality
Soliman Transitions Soliman (1982, 1986) developed criteria for two flow regime transitions forcondensation: (1) wavy orslug flow to annularflow, 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 higherqualities than the slug flow boundary
on the Taitel–Duklermap 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–annularflow 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 Frso= 18, as opposed to Frso = 7 forthe wavy-to-wavy–annulartransition
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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 Annularmist flow would referto 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 matterwhat 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 lowermass flux limit at which annularflow is predicted is relatively insensitive to the diameterchange, much like the predictions of Taitel and Dukler
At mass fluxes slightly above this, however, the wavy-to-annular transition line was shifted to lowerqualities with decreasing diameter This was consistent with the
0 100 200 300 400
600 700 800
500
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.)
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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 forthe wavy-to-annulartransition 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
0 100 200 300 400
600 700 800
500
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.)
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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 forR-32/R-125 (60%/40%) in a 3.14-mm-inside-diametertube at a saturation temperature of 35°C At the lowest mass flux of 75 kg/s· m2, 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 · m2 The Nusselt numbers remain nearly identical as the mass flux is doubled At a mass flux of 300 kg/s· m2, 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 · m2, the dependence of the Nusselt numberon 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 · m2
Figure 10.12 Variation of Nusselt numberwith quality for60%/40% R-32/R-125 mxiture at 35°C in 3.14-mm-ID test section Mass fluxG is in kg/s · m (From Dobson and Chato, 1998.)
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The change in heat transferbehaviorexhibited 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 · m2, wavy orwavy–annularflow prevail overmuch 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, annularflow prevails overmost 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 · m2, the flat Nusselt numberversus quality behavior that is characteristic of wavy flow occurs at low qualities whereas annularflow behaviorappears at higherqualities 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 betweenh and D that were
pre-dicted by annularand wavy flow heat transfercoefficients agreed well with the ex-perimental data Although this was expected forthe 7.04-mm-inside-diameter(ID) tube, a commonly used and tested size, some doubt existed about whetherthe 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 · m2 In the 3.14-mm-ID tube, the heat transferbehaviorshowed 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, ρl (ρ l− ρg )k3
, can be used
to compare the expected heat transfer behavior of different fluids In the annular flow regime the single-phase liquid heat transfer indexk0.6
l can be used at low
qualities, but at higherqualities 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 thickerfilm at a given location (hence, lowerheat transfercoefficients) In the
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⌬T = 2°C
⌬T = 3°C
0 50 100 150 200 250 300
Average Quality
Figure 10.13 Effect of∆T on Nu forR-32/R-125 in a 7.04-mm test section, G = 75 (kg/
s· m), Tsat= 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/orforced-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 numberwith quality forR-32/R-125 at a saturation temperature of 35°C and a mass flux of 75 kg/s·m2 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 )0.25properly 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 atz = L:
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NuL =hL k
l = 0.943
ρl g(ρ l− ρg )L3λ
k lµl (Tsat− T w )
1/4
(10.43)
The bracketed term in eq (10.43) can be expressed in dimensionless form as
NuL= 0.943
GaL· Prl
Jal
1/4
(10.44)
Dhirand Lienhard (1971) devised a simple way to extend the analysis forthe vertical wall to arbitrary axisymmetric bodies They showed that the local Nusselt number can be predicted by replacingg in eq (10.43) with an effective acceleration
of gravity:
geff = x x(gr)4/3
In eq (10.45),r(x) is the local radius of curvature and g(x) is the local gravity
com-ponent in thex 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· Prl
Jal
1/4
(10.46)
The diameteris 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
This correction shows that the assumption of a linear temperature profile in the original analysis is quite acceptable for Jalmuch 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 vaporshearforces, 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,
Revo < 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
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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
ρl g(ρ l− ρg )D3λ
k lµl (Tsat− T w )
1/4
(10.48)
The constant 0.555 is 76% of the value of 0.728 forexternal 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 forthe 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 funccorrela-tion of the void fraccorrela-tionα This resulted in
Nu=hD k
l = 0.728α3/4
ρl g(ρ l− ρg )D3λ
k lµl (Tsat− T w )
1/4
(10.49)
They recommend using Zivi’s (1964) correlation for the void fraction:
α =
1+1− x x ρg
ρl
2/3 −1
(10.50)
Jaster and Kosky’s correlation overpredicts the Chato correlation for all qualities to greaterthan about 0.2 It had a mean deviation of 37% with theirown 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 forhigher-mass-flux low-quality situations where wavy orstratified 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:
Nutop= 0.31Re0.12
g
ρl g(ρ l− ρg )D3λ
k lµl (Tsat− T w )
1/4
(10.51)
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In the bottom of the tube, they postulated forced-convective heat transfer Using a heat and momentum transfer analogy, they recommended the following correlation:
Nubot= φl,lt
√ 8Rel
5 [1+ ln(1 + 5 Pr l )/Pr l] (10.52) where
φl,lt =
1+X1
X2
lt
(10.53)
Rosson and Myers defined a parameterβ that represented the fraction of the tube perimeter over which filmwise condensation occurred They recommended predicting the value ofβ as follows:
β =
Re0.1
Re0g .6· Re0.5
l
Ga < 6.4 × 10−5 (10.54a)
1.74 × 10−5Ga
Reg· Rel if
Re0g .6· Re0.5
l
Ga > 6.4 × 10−5 (10.54b) Then the circumferentially averaged Nusselt number was given by
Nu= β · Nutop+ (1 − β)Nubot (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