A variety of approaches for predicting heat transfer during annular flow condensation have been developed.. Two-Phase Multiplier Correlations The simplest method of heat transfer predic-t
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data analysis and physical guidance from analytical solutions The final correlation separates the heat transfer by film condensation in the upper part of the horizontal tube from the forced-convective heat transfer in the bottom pool:
Nu= 0.23Re0vo .12
1+ 1.11X0.58
tt
Ga· Prl
Jal
0.25
+
1−θπl
Nuf (10.56) whereθlis the angle subtended from the top of tube to the liquid level and
Nuf = 0.0195Re0.8
l · Pr0.4 l
1.376 + C1
X C2
tt
(10.57)
For0< Fr l · 0.7,
C1 = 4.172 + 5.48Fr l − 1.564Fr2
C2 = 1.773 − 0.169Fr l (10.58b) ForFrl > 0.7,
where Frl is the liquid Froude number Due to the 1.376 inside the radical of eq
(10.57), the correlation above matches the Dittus–Boelter single-phase correlation whenx = 0.
If the area occupied by the thin condensate film is neglected,θlis geometrically
related to the void fraction by
α = θπl −sin 2θl
If a void fraction model is assumed, this transcendental equation must be solved
to obtain the desired quantity,θl Jasterand Kosky (1976) deduced an approximate relationship which is much easier to use In the context of the present topic, their simplification can be stated as
1−θl
π
arccos(2α − 1)
The simplicity achieved by this assumption is well worth the modest errors as-sociated with it These errors are themselves mitigated by the fact that the forced-convective Nusselt number, by which the quantity in eq (10.61) is multiplied, is normally considerably smaller than the filmwise Nusselt number The void frac-tion correlafrac-tion of Zivi (1964), eq (10.50), was used with this correlafrac-tion Equafrac-tion (10.56) is to be used whenG < 500 kg/s · m2and Frso< 20 Even though the
Dob-son correlations yield reaDob-sonable results, available data suggest the need for further
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development of heat transfermodels in the wavy orwavy-annularregions at higher mass fluxes
Shear-Driven Annular Flow Condensation The annular flow regime repre-sents the situation where the interfacial shear stresses dominate and create a nearly symmetric annular film with a high-speed vapor core A variety of approaches for predicting heat transfer during annular flow condensation have been developed Al-though these approaches can be divided into many different categories, they can be reduced to three for the purposes of this review: two-phase multiplier approaches, shear-based approaches, and boundary layer approaches
Two-Phase Multiplier Correlations The simplest method of heat transfer predic-tion in the annular flow regime is the two-phase multiplier approach This approach was pioneered for predicting convective evaporation data by Denglor and Addoms (1956) and was adapted forcondensation by Shah (1979) The theoretical hypothesis
is that the heat transferprocess in annulartwo-phase flow is similarto that in single-phase flow of the liquid (through which all the heat is transferred), and thus their ratio may be characterized by a two-phase multiplier This reasoning is in fact very similar
to that of Lockhart and Martinelli (1947), who pioneered the two-phase multiplier approach for predicting two-phase pressure drop The single-phase heat transfer co-efficients are typically predicted by modifications of the Dittus and Boelter (1930) correlation, which results in the form
Nu= 0.023Re0.8
l · Prm
l · F
x,ρρl
g ,µµl
g , Fr l
(10.62)
wherem is a constant between 0.3 and 0.4 and F is the two-phase multiplier Although
the two-phase multiplier can depend on more dimensionless groups than those indi-cated in eq (10.62), the groups shown are the most relevant The type of single-phase correlation shown is valid for turbulent flow and is based primarily on an analogy between heat and momentum transfer
One of the most widely cited correlations of the two-phase multiplier type is that of Shah (1979) It was developed from his observation that the mechanisms of condensation and evaporation were very similar in the absence of nucleate boiling
With this idea, he set out to modify the convective component of his flow boiling correlation for use during condensation The form of his correlation is
Nu= 0.023Re0.8
l · Pr0.4 l
1+ 3.8
p0.38 r
x
1− x
0.76
(10.63)
The bracketed term is the two-phase multiplier It properly approaches unity asx
ap-proaches 0, indicating that it predicts the single-phase liquid heat transfer coefficient when only liquid is present As the reduced pressure is increased, the properties of the liquid and vaporbecome more alike and the two-phase multiplierdecreases, as expected
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Cavallini and Zecchin (1974) used the results of a theoretical annular flow analysis
to deduce the dimensionless groups that should be present in an annular flow correla-tion They then used regression analysis to justify neglecting many of the groups that did not appear in their empirically developed correlation, which can be shown to be
of the two-phase multiplier form by writing it in the following way:
Nu= 0.023Re0.8
l · Pr0.33 l
2.64
1+
ρ
l
ρg
0.5 x
1− x
0.8
(10.64)
Here the bracketed term represents the two-phase multiplier
The two-phase multiplier approach was selected for correlating the annular flow heat transfer data by Dobson (1994), Dobson et al (1994a,b), and Dobson and Chato (1998) To make sure that the correlation was not biased by data outside the annular flow regime, only data with Frso > 20 were used to develop the correlation This
value was reported by Dobson (1994) to provide a good indicator of the transition from wavy-annular to annular flow and agreed well with the data from his study The correlation developed was
Nu= 0.023Re0.8
l · Pr0.4 l
1+ 2.22
X0.89 tt
(10.65)
This form utilizes the single-phase heat transfer correlation of Dittus–Boelter (1930) with a Prandtl exponent of 0.4 At a quality of zero, the Lockhart–Martinelli parameter approaches infinity and eq (10.65) becomes the single-phase liquid Nus-selt number This correlation is to be used forG ≥ 500 kg/s·m2orforall mass fluxes
if Frso> 20.
Shear-Based Correlations The use of shear-based correlations for annular flow condensation dates back to the early work of Carpenter and Colburn (1951) They argued that the resistance to heat transfer in the turbulent liquid flow was entirely inside the laminarsublayerand that the wall shearstress was composed of additive components due to friction, acceleration, and gravity Although it was later pointed out by Soliman et al (1968) that theirequation forthe accelerational shearcomponent was incorrect, the framework that they established at a relatively early point in the history of forced-convective condensation remains useful
Soliman et al (1968) utilized the framework established by Carpenter and Col-burn to develop their own semiempirical heat transfer correlation for annular flow
Neglecting the gravitational term (which is appropriate for horizontal flow), the Soli-man correlation can be written as
Nu= 0.036Re lo· Pr0.65
l
ρl
ρg
0.5
!2(0.046)x2
Re0g .2 φ2
g+ Bo
5
"
n=1
a nρg
ρl
n/3
(10.66a)
where
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a3 = 2(γ − 1)(x − 1) (10.66d)
a4= 1
a5= γ
2−1
x − x
(10.66f)
γ = interface velocity
mean film velocity = 1.25 forturbulent liquid (10.66g) Soliman et al (1968) compared the predictions of their correlation to data for steam, R-113, R-22, ethanol, methanol, toluene, and tricholoroethylene The agree-ment was correct in trend, although even on log-log axes the deviations appear quite large No statistical information regarding deviations was given
Chen et al (1987) developed a generalized correlation for vertical flow condensa-tion, which included several effects combined with an asymptotic model They stated,
as did Carey (1992), that their correlation for the shear-dominated regime was also appropriate for horizontal flow but made no comparison with horizontal flow data
Their correlation use the general form of Soliman et al (1968), but the acceleration terms were neglected and the pressure drop model was replaced by one from Dukler (1960) The final result for the average modified Nusselt number is given in what follows in the discussion of flows in vertical channels, eq (10.79)
Boundary Layer Correlation The most theoretical correlation, based on boundary layer considerations, is that of Traviss et al (1973) Under its rather stringent assump-tions, this method provides an analytical prediction of the Nusselt number Before a pressure drop model is assumed, the correlation can be written as
Nu= D+· Prl
The termD+is the tube diameterscaled by the turbulent length scale,µl /√τ wρl
A simple force balance indicates the proportionality between the wall shear and the pressure drop, establishing the fact that the annular flow Nusselt number is propor-tional to the square root of the pressure drop per unit length
The denominatorof eq (10.67),F2, can be thought of as a dimensionless heat
transfer resistance Guidance as to its evaluation is given in eq (10.69) Physically, this resistance increases as the dimensionless film thickness increases, as would be expected from conduction arguments A plot ofF2versus Relforvarious values of Prl shows that as Relincreases from 0 to 1125 (the value where the fully turbulent region begins),F2 increases very rapidly As Rel is increased further,F2 increases much more slowly Physically, this occurs because the primary resistance to heat transfer is contained in the laminarsublayerand bufferregions
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Although the Traviss et al (1973) analysis was performed after the advent of the simple shear-based correlations, it provides a useful method for understanding them Forrelatively small changes in Rel when Rel > 1125, one could reasonably
assume a constant value of F2 at a fixed Prandtl number If the Prandtl number dependence could be expressed as a powerlaw function, the Nusselt numbercould then be expressed as
Nu= aD+· Prm
wherea is a constant This is exactly the form of the original shear-based correlation
of Carpenter and Colburn (1951) Thus, these correlations are justified for a narrow range of conditions by the more theoretically sound analysis of Traviss et al (1973)
Only a few manipulations are required to show the equivalence between the Traviss analysis and the two-phase-multiplier approach The first important observation is that annularflow is seldom encountered forliquid Reynolds numbers less than 1125
Using the criterion for annular flow that Frso= 18 [eqs (10.40a,b)], the
correspond-ing equation was solved forthe quality above which annularflow could exist with
Rel = 1125 The results indicated that the liquid film is seldom so thin that the fully
turbulent region is not reached; thus the piecewise definition ofF2is seldom necessary and its value can be well approximated by the function
F2 10.25Re0.0605
l · Pr0.592
If this approximation is used in eq (10.68) andD+ is evaluated with a
pres-sure drop correlation using Lockhart and Martinelli’s two-phase liquid multiplier approach, the following equation is obtained for the Nusselt number:
Nu= 0.0194Re0.815
l · Pr0.408
l φ2
l (X tt ) (10.70) This is identical in form, and close in value, to the commonly used two-phase multi-plier correlations, such as eq (10.63)
Comparison of Heat Transfer Correlations The following comparisons were developed by Dobson (1994), Dobson et al (1994a,b), and Dobson and Chato (1998)
Details may be obtained from these publications
Gravity-Dominated Correlations Chato’s correlation was developed for stratified flow and was recommended foruse at vaporReynolds numbers of less than 35,000, that is, at low mass fluxes in the stratified flow regime It was compared with 210 experimental data points that met this criterion and had a mean deviation of 12.8%
The range of applicability of the Jaster and Kosky (1976) correlation is specified
by an upperlimit of a dimensionless wall shear The mean deviation between the
213 experimental points that met this criterion and the values predicted was 14.5%, slightly higherthan forthe simplerChato (1962) correlation Although the deviations between the Jaster and Kosky correlation and the present data were sometimes large,
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the mean deviation of 14.5% was substantially betterthan the 37% standard deviation
of theirown data
The correlations of Chato and Jaster and Kosky were both able to predict most
of the experimental data for the wavy flow regime within a range of±25% Chato’s
analysis implies an essentially constant void fraction which is independent of quality
Jaster and Kosky’s correlation does not predict the variation with quality accurately
Neitheraccounted forheat transferin the bottom of the liquid pool Forlow mass fluxes, this approach is reasonable
Because no guidelines were given for use of the Rosson and Myers (1965) cor-relation, it was compared against the full database of points that were later used to develop the wavy flow correlation Although Rosson and Myers attempted to account forforced-convective condensation in the liquid pool at the bottom of the tube, their correlation was actually a poor predictor of the experimental data The correlation had
a mean deviation of 21.3% from the experimental data, almost 10% worse than the simpler Chato or Jaster and Kosky correlations The most problematic part of the cor-relation seemed to be the prediction of the parameterβ, which represents the fraction
of the tube circumference occupied by filmwise condensation At low mass fluxes, this parameter should clearly be related to the void fraction and approach unity as the quality approaches unity However, the empirical expressions developed by Rosson and Myers do not behave in this manner The trends were very erratic, particularly formass fluxes over25 kg/s · m2, where the relationship was not even monotonic
Annular Flow Correlations The annular flow correlations that were selected for comparison with the experimental data encompass at least one member of each of the three broad classes: two-phase multiplier correlations (Shah, 1979; Cavallini and Zecchin, 1974; Dobson, 1994), shear-based correlations (Chen et al., 1987) and boundary layer analyses (Traviss et al., 1973)
Of the five correlations, Shah’s and Dobson’s came with specific guidelines for a lower limit of applicability The Shah correlation should not be used at mass fluxes where the vapor velocity withx = 1 was less than 3 m/s In all cases this represents
a vaporvelocity well above the wavy-to-annulartransition line on the Taitel–Dukler map This criterion was selected for each of the correlations, so that they would be compared on an equal basis
The predictions of the Shah correlation agree fairly well with the data, with a mean deviation of 9.1% Nearly all the data were predicted within±25% The most
significant deviations occurred for some low Nusselt number data that were in the wavy-annularflow regime, and forsome very high-mass-flux, high-quality data In general, the Shah correlation underpredicted the experimental data
The mean deviation of the Cavallini and Zecchin correlation from the experimental data was 11.6%, slightly higher than that of the Shah correlation Despite the slightly higher mean deviation, though, the predictions of the Cavallini and Zecchin correla-tion were more correct in trend than those of the Shah correlacorrela-tion When the Cavallini and Zecchin correlation was in error, it tended to overpredict the experimental data
The largest errors occurred at low qualities because this correlation approaches a value of 2.18 times the single-phase Nusselt numberat a quality of zero
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The mean deviation of the Traviss correlation was 11.8%, slightly higher than the Cavallini and Zecchin correlation The Traviss correlation tended to overpredict the experimental data, particularly at high qualities, where their empirical correction was used Without this correction, their correlation would have underpredicted the high-quality data
The Chen correlation was the worst predictor of the annular flow data, with a mean deviation of 23.3% This correlation significantly underpredicted nearly all the data
The correlation of Soliman (1968) generally predicts lower Nusselt numbers than the Chen correlation Thus, it would have performed even worse against the data of Dobson
The mean deviation of Dobson’s wavy flow correlation, eq (10.56), from his experimental data was 6.6% The mean deviation of his annular flow correlation, eq
(10.65), was 4.5% The maximum mean deviation of both of these correlations from other experimental data in the literature was 13.7%
One problem with the annular flow correlations that is not apparent in a plot of experimental versus predicted Nusselt numbers concerns their range of applicability
The Nusselt numbers were well above the annular flow predictions at low qualities As the quality reached about 70% and the flow pattern became fully annular, the predic-tions of the Cavallini and Zecchin correlation agreed very well with the experimental data These data suggest the need forfurtherdevelopment of heat transfermodels
in the wavy orwavy-annularregions at highermass fluxes Recently, Cavallini et
al (2002) suggested a set of correlations for annular, stratified, and slug flow which provide better results for high pressure refrigerants
10.6.4 Pressure Drop
The discussion of pressure drop could take up a chapter of its own Here only one representative method is given The pressure drop∆P in a tube is caused by friction
and by the acceleration due to phase change Souza et al (1992, 1993) developed an expression for the total pressure drop in a short section∆z in which the quality can
be considered constant at a mean value:
− ∆P = 2f lo G2
ρl D φ2lo ∆z + G2
x2
o
ρgαo + (1 − x o )2
ρl (1 − α o )
−
x2
l
ρgαi + (1 − x i )2
ρl (1 − α i )
(10.71) where
f lo = 0.0791Re −0.25 lo (10.72)
φ2
lo=
1.376 + C1
X C2
tt
(1 − x)1.75 (10.73a)
C1andC2are given in eqs (10.58) and (10.59)
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Souza and Pimenta (1995) developed another correlation:
φ2
lo = 1 + (Γ − 1)x1.75
1+ 0.952ΓX0.4126
tt
(10.73b)
Γ =
ρl
ρg
0.5µg
µl
0.125
(10.73c)
Other pressure drop correlations were suggested by Friedel (1979) and Jung and Rademacher (1989), but neither one produced more accurate predictions than Souza’s
Cavallini et al (2002) proposed a correlation that better predicted the pressure drops for high pressure refrigerants
10.6.5 Effects of Oil
Oil in the refrigerant decreases the heat transfer and increases the pressure drop
Gaibel et al (1994) discussed these effects He found that foroil mass fractions
ωo < 0.05, the correction factor developed by Schlager et al (1990) gave acceptable
values forthe Nusselt numberwith oil (Nuo) when applied to the Dobson correlations (Nu) [eqs (10.56) and (10.65)]:
Nuo = Nu · e −3.2ω o (10.74) For the pressure drop with oil (∆P o), the Souza et al (1992, 1993) correction factor applied to the pure refrigerant pressure drop (∆P p) was found acceptable in the same oil concentration range:
∆P o = ∆P p
1+ 12.4ω o − 110.8ω2
o
(10.75)
10.6.6 Condensation of Zeotropes
Sweeney (1996) and Sweeney and Chato (1996) correlated the data of Kenney et
al (1994) obtained with Refrigerant 407c, a zeotropic mixture of R-32, R-125, and R-134a (23, 25, and 52% by mass), in a smooth tube They found that the Nusselt numbers for zeotropic mixtures (Num) could be predicted by the following simple modification of the Dobson correlation (Nu) [eq (10.65)] for the annular flow regime:
Num = 0.7
G
300
0.3
Forthe wavy regime, the mixture Nusselt numberis obtained by a similarmodifica-tion of the Dobson correlasimilarmodifica-tion (Nu) [eq (10.56)]:
Num=
G
300
0.3
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where the mass fluxG is in kg/s · m2 These correlations cannot be assumed to be generally applicable to zeotropes without additional data on other zeotropes, but they
do indicate that the equations represent the underlying physical phenomena well
10.6.7 Inclined and Vertical Tubes
Inclining the tube from the horizontal will affect only the gravity-dominated flows, (i.e., stratified and wavy patterns) significantly For the shear-dominated annular flows, the correlations are essentially independent of tube orientation Chato (1960) studied both analytically and experimentally the effects of a downward inclination of
a condenser tube with stratified flows He found improvement in the heat transfer up
to about a 15° inclination, caused by the reduction in depth of the bottom conden-sate pool However, this improvement decreases at increasing angles as the vertical orientation is approached because of the thickening of the condensate layer on the wall as it traverses longer distances before reaching the bottom pool It can be shown from eqs (10.43) and (10.48) that if theL/D ratio of the tube is greater than 8.3, the
horizontal tube will have better heat transfer It is obvious that an upward inclination
of the tube is counterproductive because gravity will retard the liquid flow, reducing the heat transfer
Chen et al (1987) analyzed annularfilm condensation in a vertical tube and proposed the following approximate correlation for the average Nusselt number for complete condensation in the tube:
Nud =
Re−0.44 T + Re0T .8· Pr1.3
l
1.718 × 105 +C · Pr1l .3· Re1.8
T
2075.3
0.5
(10.78) where
Nud = h T
k l
ν2
l g
1/3
ReT = 4m
µl πD
C = 0.252µ
1.177
l µ0.156 g
D2g0.667ρ0.553
l ρ0.78 g
(dimensionless)
Chen et al., contended that similar derivations can be applied to horizontal annular flows with the following result:
Nud = 0.022√C · Pr0.65
l · Re0.9
However, they did not verify this correlation with comparisons to experimental data
By the same argument, it can be suggested that the Dobson (1994) correlation of eq
(10.65) can be used for vertical, downward flow tubes at high mass fluxes Although condensation in upward flow is worse than in horizontal or downward flow, it does occur in reflux condensers Chen et al (1987) treated this case in some detail
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10.7.1 Microfin Tubes
Most evaporators and condensers of new unitary refrigeration and air-conditioning equipment are manufactured with microfin tubes The microfin tube dominates uni-tary equipment design because it provides the highest heat transfer with the lowest pressure drop of the commercially available internal enhancements (Webb, 1994)
Cavallini et al (2000) quotes an 80 to 180% heat transferenhancement overan equiv-alent smooth tube with a relatively modest 20 to 80% increase in pressure drop To-gether, R-134a, R-22, and replacements for R-22 replacements constitute by mass nearly all the refrigerants used in unitary products (Muir, 1989) As a result, much of the predictive development for convective condensation in microfin tubes has been focused on R-134a, R-22, and replacements for R-22 replacements
Figure 10.14 shows the cross section and characteristic dimensions of a microfin tube The outside diameter(Do) of commercially available microfin tubes ranges from 4 to 15 mm The root diameter (Dr) is depicted in Fig 10.14 Microfin tubes
Figure 10.14 Cross section of microfin tube
... will have better heat transfer It is obvious that an upward inclinationof the tube is counterproductive because gravity will retard the liquid flow, reducing the heat transfer
Chen... provides the highest heat transfer with the lowest pressure drop of the commercially available internal enhancements (Webb, 1994)
Cavallini et al (2000) quotes an 80 to 180% heat transferenhancement... refrigerants
10.6.5 Effects of Oil
Oil in the refrigerant decreases the heat transfer and increases the pressure drop
Gaibel et al (1994) discussed these effects He