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CHAPTER 3
DEVELOPMENT OF CORRELATION BASED
PREDICTION MODEL
3.1 INTRODUCTION
Though large number of experimental studies are available,
theoretical models proposed are less and are mainly due to the efforts of
Beatty and Katz (1948), Webb et al (1985), Honda and Nozu (1987), Adamek
and Webb (1990), Sreepathi (1994), Briggs and Rose (1994), Rose (1994),
Sreepathi et al (1996) and Kumar et al (2002).
The earlier Beatty and Katz (1948) model does not include the role
of surface tension forces and condensate retention aspects and hence found to
be inadequate. Webb et al (1985) model requires an assumption of a suitable
Adamek (1981)’s - profiles for the given fin geometry and no guidelines
exist for such selection. Hence, the use of Webb et al model is impractical.
Though the agreement between the predictions of the Adamek and Webb
(1990) model and Honda and Nozu (1987) model, and the experimental data,
is reported to be within 20%, the complexities involved in these models are
quiet high and hence are not readily usable in practice. The semi-empirical
model of Briggs and Rose (1994) appears to be relatively simpler. It is
reported to predict the available data for refrigerants within 20% band.
However, Rose (1994) himself reports that his model under predicts the water
data by more than 20%. The prediction of Sreepathi (1994) model is
successful to his R123 data but for other fluids, the deviations are
51
considerably larger in addition to its complexity. Moreover the model of
Sreepathi et al (1996) under predict R134a data at low condensate
temperature differences and slightly over predict at higher condensate
temperature differences. Kumar et al (2002) model predicts water data in the
range of 30% and for other refrigerants such as R11, R113, R12 and R22
the prediction is in the range of 35%. Hence, there exists a need for a still
simple and better model or correlation with a wider scope. This chapter presents
the development of such an empirical model with better prediction capabilities.
3.2 PROPOSED MODEL
When a pure saturated vapour condenses over a HIF tube, the
phenomenon of condensate retention divides the tube circumference into two
regions, viz., the un-flooded region and the flooded region. The mechanism of
condensation process in these regions is totally different. Hence, in the earlier
models (Webb et al (1985), Honda and Nozu (1987), Adamek and Webb
(1990), Rose (1994), Briggs and Rose (1994), Sreepathi (1996) and Kumar
et al (2002)), condensation in these two regions is separately calculated.
Individual contributions are then added to get the total condensation rate.
Similar approach has been adopted in the present work as well.
3.2.1 Calculation of Total Nusselt Number
The heat transfer in the flooded (Nuf) and un-flooded (Nuu) regions
of the HIF tube is accounted separately and then added to calculate the overall
performance (Nud) as given by equation (3.1). The Nusselt numbers (Nuf, Nuu
and Nud) used in equation (3.1) are defined using the root diameter of the HIF
tube as the characteristic length dimension.
Nud = (1-Cf) Nuu + Cf Nuf (3.1)
where Cf is the flooded fraction.
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3.2.2 Calculation of Flooded Fraction
The fraction (Cf) of the tube circumference which is flooded is
given by equation (3.2). The flooding angle ( f), measured from the tube
bottom, for the tubes having rectangular or trapezoidal integral-fins (Rudy
and Webb (1985), Honda et al (1983)) is given by equation (3.3). The validity
of equation (3.3) has been verified experimentally by several investigators.
For fins of other shapes, Rudy and Webb (1985) or Honda et al (1983) may be
referred for the calculation of f.
Cf = f / (3.2)
f = cos-1 {1 – [(4 cos ) / ( gbdo)]} (3.3)
In equation (3.3), If (4 cos ) > 0.5, then f = .
3.2.3 Calculation of Nusselt Number for the Flooded Region
Webb et al (1985) based on their R11 data had shown that the
contribution of flooded region for heat transfer is very less. Honda and Nozu
(1987) analysis has also indicated the same. However, Briggs et al (1992)
have shown that significant heat transfer occurs even for the tube whose
circumference is flooded in case of steam. Hence, it is considered worthy to
include the role of flooded zone in the present model in a simple way, without
much complication.
Since the condensate film is thick and gravity is the major
controlling force in the flooded zone, it is more appropriate to use the model
standard Nusselt (1916) equation for gravity controlled condensation over a
plain horizontal tube to calculate the contribution of the flooded zone. Masuda
and Rose (1985) and Webb et al (1985) have shown that due to condensate
retention, the fin flank and the fin root areas are covered by thick condensate
53
film and the heat transfer over these surfaces is negligible. Thus, in the
flooded zone, only the fin tip area is available for condensation. Hence, the
Nusselt number for the flooded part is calculated by using the Nusselt (1916)
type equation for horizontal tube with appropriate area ratio (t/p), which is
given by equation (3.4). The Gravity number (Gy) is defined as, Gy {= [( hfg)
/ ( k T)] . ( g p3)}, where the fin pitch (p) is used as the characteristic length
dimension. As Nuf (= h.dr / k) is defined with the root diameter (dr) of the HIF
tube as the characteristic dimension, the ratio dr* (= dr / p) is used in the
equation (3.4).
Nuf = 0.728 . (t*) . [Gy/dr*]1/4 (3.4)
3.2.4 Calculation of Nusselt Number for the Un-flooded Region
The numerical studies of Srinivasan (2001) covering various fluids
and fin geometry have shown that, the condensation process in the un-flooded
zone can be correlated by using the non-dimensional parameters – the Surface
tension number {Su = [( hfg) / ( k T)] . ( p)} and the normalized fin height
(e* = e / p).
For the given tube diameter and condensing fluid, the fin spacing
decides the flooding angle. Keeping this aspect in mind, ‘b*’ is included as
one of the parameters for correlating the condensation process in the un-
flooded zone. Srinivasan (2001) has also shown that by neglecting the effect
of gravity force, the numerical result for the average Nusselt number over fin
surface decreases only by 2%. Hence the role of gravity force could be
neglected while modeling the condensation over the un-flooded zone.
In all the parameters considered so for (i.e., Su, e*and b*), the fin
pitch is used as the characteristic length dimension. However, the average
Nusselt number for the un-flooded zone (Nuu) is defined using the root
54
diameter of the HIF tube as the characteristic length dimension. Thus the ratio
dr* (= dr / p) is included as another parameter.
Finally, the average Nusselt number for the un-flooded zone can be
expressed as a function of four independent, non-dimensional parameters as
given in equation (3.5).
Nuu = f [ Su, e*, b*, dr* ] (3.5)
The function ‘f [ ]’ in the equation (3.5) could be of any convenient
form. The only necessity is that it should correlate the experimental data with
a minimum possible deviation. The equation (3.5) can be expressed in the
power-law form as,
Nuu = A (Su)a1 (e*)a2 (b*)a3 (dr*)a4 (3.6)
where A, a1, a2, a3, and a4 are regression constants, which are to be
determined from the experimental data. If these constants are available, the
equations (3.1) to (3.4) and (3.6) can be easily used to estimate the heat
transfer coefficient during condensation over HIF tube in a simple manner.
The equation (3.6) in logarithmic form becomes equation (3.7), which is used
during regression analysis.
ln(Nuu) = ln(A) + (a1).ln(Su) + (a2).ln(e*) + (a3).ln(b*) + (a4).ln(dr*) (3.7)
3.2.5 Calculation of Nusselt Number for the Fully Flooded Tubes
The tubes used by Briggs et al (1992), Wanniarachchi et al (1985,
1986), Yau et al (1985, 1986) are fully flooded for the fin spacing of 0.5mm.
Hence the condensation process in the fully flooded tubes is correlated by a
separate empirical equation.
55
As the condensate film is thick and gravity is the only major
controlling force in the flooded zone, the Gravity number (Gy) is used along
with the common characteristic dr* for correlation. Hence, for fully flooded
tubes the heat transfer is calculated in the form as in equation (3.8).
Nuf = B. (t*) . [Gy/dr*]b1 (3.8)
where B and b1 are regression constants, which are to be determined from the
experimental data. For this case f = ; Cf = 1; and Nuu = 0. The equation
(3.8) in logarithmic form becomes equation (3.9), which is used during
regression analysis.
ln(Nuf / t*) = ln(B) + (b1).ln(Gy/dr*) (3.9)
3.3 SELECTION OF EXPERIMENTAL DATA
The use of input data, which is accurate and more reliable, decides
the success of the resultant correlation in predicting the experimental data.
Hence, a careful selection of experimental data is essential for better
correlation.
3.3.1 Difficulties in Selecting the Experimental Data for Correlation
Development
By providing a chronological listing of important references
pertaining to condensation over HIF tubes, over a period of 1945 ~ 1988,
Marto (1988) expressed his concern, with caution, as follows: “An important
observation pertains to the difficulty in interpreting and using the results. In
obtaining film condensation heat transfer data of pure vapours, there are
numerous operational difficulties that can alter the accuracy of the
experimental results. For example, the presence of non-condensing gases in
the vicinity of the test tube, or partial drop wise condensing conditions on the
56
surface of the tube, or a substantial vapour velocity near the tube can
influence the heat transfer significantly. Most of the investigations fail to
provide information on these operational problems in obtaining the data and
fail to quote the resultant uncertainties in the results. Another very important
issue pertains to the method of determining the average condensing-side heat
transfer coefficient. Certain investigations have used Wilson (1915) plot
technique or modified versions of it. Others have determined by direct
measurement of the tube wall temperature. A comparison of these techniques
was made by Wanniarachchi et al (1986). In general, depending upon which
technique is used, they found that there may be variations of 10~15% in the
quoted condensing-side heat transfer coefficients”.
Marto (1988) further adds, “More confusion exists with the various
results due to the choice of surface area used in calculating the results. For
example, some investigators have chosen the fin “envelop” surface area (i.e.,
the surface area of a smooth tube of diameter equal to the outer diameter of
the fins (do)). Others have used the area of a smooth tube of fin root diameter
(dr). Still others have used the total surface area (equal to the fin area plus the
tube surface area) or a total effective surface area which corrects the fin
surface area with fin efficiency. Finally, heat transfer enhancements with
relation to a smooth tube have been reported based either on constant heat
flux or on constant vapour to wall surface temperature difference. Masuda and
Rose (1985) and Yau et al (1986) had shown that these two definitions yield
very different numerical results [i.e., q = ( T)4/3 ]. As a result of these
inconsistencies, great care must be exercised while extracting the published
experimental data for practical use”.
Sukhatme (1990) also discussed the above said issues. In addition,
he adds: “Because of higher heat transfer coefficient and lower condensing
temperature differences, extreme care is needed while making the
measurements. In the same experimental set-up, if the measurements are
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made in such a way that the heat transfer coefficient can be determined by
two ways, it is not unusual to have differences of the order of 10 % among the
two calculated values. These reasons lead to a considerable scatter in the
experimental data and to some extent are responsible for the fact that the data
obtained for similar tube-fluid combinations by various investigations differ
widely”. Briggs et al (1992) have made accurate heat transfer measurements
on six tubes which differ in diameter and fin spacing during condensation of
Steam, Ethylene Glycol and R-113. They reiterated the statements of
Sukhatme (1990) and stressed the need for accurate measurements.
Regarding selection of experimental data, Rose (1994) quotes as:
“It is difficult to decide which data should be included in the determination of
empirical constants. Different investigators have used different methods to
determine the condensation heat transfer coefficient (direct measurement of
wall temperature or from overall temperature difference using the
predetermined coolant-side correlation or some form of ‘Wilson-plot’). In
some cases, significant vapour velocity may have been present. Moreover,
heat transfer enhancement ratio is not strictly independent of vapour to
surface temperature difference”.
The additional difficulty experienced in the present investigation is
that in some investigations, the results are given in the form: he = a ( T)-n, by
forcing n = 0.25. However, for HIF tubes, index (n) is found to vary in the
range 0.1 n 0.25 in most of the investigations. It would introduce
considerable error while extracting the experimental data.
3.3.2 Tube Details of Various Experimental Studies Referred in this
Thesis
Tables 3.1 to 3.6 list the dimensions of the tubes referred in this
thesis. A three-digit code number is assigned for each tube for easy reference.
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The first-digit is assigned based on the condensing fluid. The fluid and the
corresponding number assigned are water {1}, R11 {3}, R12 {4}, R407c {5},
R113 {6,7}, R134a {8} and R123 {9}. As the number of tube data available
for R113 is more, two numbers {6,7} are used to refer R113 itself. Second
and third digits are assigned based on the investigator(s) and tubes tested. The
code number of the tubes will not be continuous. It is because of the reason
that some investigator(s) have used the same tube against different fluids.
Under such circumstances, second and third digits are kept fixed and only the
first digit is changed according to the fluid used.
The tables also provide other related information, viz., the number
of experimental data points extracted for the present exercise, the flooding
angle obtained from the equation (3.3), condensing temperature of the vapour
(Tv), and the range of surface to condensing vapour temperature difference
(Tv - Tw) for which the experimental data are reported. Since the accuracy of
the reported experimental heat transfer coefficient data depends upon the
technique used for data reduction (Briggs et al (1992)), these tables also
provide the information on the technique used for obtaining the condensing-
side heat transfer coefficient. Wherever possible, the reported maximum error
in the estimation of the condensing-side heat transfer coefficient is also added
in these tables. Some of the investigators have written at great length about
the care exercised in obtaining the heat transfer coefficient, but they have
failed to provide the error estimates. Under such circumstances, they are
marked as NA (not available). Table 3.7 provides the overall picture. In total
626 experimental data points pertaining to 170 tubes during condensation of 7
fluids are used in the present exercise. These data are obtained from 32
investigations.
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3.3.3 Selection of Experimental Data for the Correlation Development
Considering the above said issues, enough care has been exercised
in data selection. Three specific correlations and a general correlation have
been developed. First, is the CFC data based correlation developed from the
reliable experimental data covering various CFC fluids viz., R11, R12 and
R113. In total 198 experimental data obtained from 85 tube geometries and 21
investigations are used.
The second specific correlation is based on the reliable
experimental data of water obtained from 8 investigations. In total, 94
experimental data obtained with 28 tube geometries are used for this model.
The third specific correlation is based on the experimental data of HCFC /
HFC refrigerants viz., fluids covering R407C, R134a and R123. For this
correlation 319 experimental data obtained from 54 tube geometries and 15
investigations are used.
A general model is developed using eco friendly water and
HCFC/HFC fluids, which are of current importance and in accordance with
Montreal protocol for industrial applications. In this, 57 water data from 13
tube geometries obtained from 4 investigations and 154 HCFC/HFC data
obtained from 21 geometries and 6 investigations are used. In total 211 data
points of water and HCFC/HFC fluids obtained from 34 tube geometries are
used for correlating the general model.
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Table 3.1 Details of the tubes used during condensation of steam
Reference TCTube
No.FC
No.of
data
Fin
half
angle,
deg.
dr
mm
p
mm
t
mm
b
mm
e
mmf
deg.
Tv
C
Tv
Range
K
Tech-
nique
Exp.
error
%
Srinivasan
(2001)11
111 1 7 0 19 2 1 1 1 101.1
100 10-24 W 10
112 1 7 0 19 2 1 1 2 95.12
113 1 7 0 19 2 1 1 2.5 92.52
114 1 7 0 19 2 1 1 3 90.12
115 1 7 0 19 2 1 1 3.5 87.91
Briggs et al
(1992)12a
121 1 5 0 19.1 1.5 1 0.5 1 FLD - - - -
122 1 5 0 19.1 2 1 1 1 100.8100 10-30 D 22
123 1 5 0 19.1 2.5 1 1.5 1 77.97
Briggs et al
(1992)12b
124 1 4 0 12.7 1 0.5 0.5 1.59 FLD - - - -
125 1 4 0 12.7 1.5 0.5 1.0 1.59 92.97100 10-30 D 22
126 1 4 0 12.7 2 0.5 1.5 1.59 92.97
Wanniarachchi
et al (1986)14a
141 1 3 0 19 1.5 1 0.5 1 FLD - - - -
142 1 3 0 19 2 1 1 1 101.1
100 14-30 W 4-20143 1 3 0 19 2.5 1 1.5 1 78.19
144 1 3 0 19 3 1 2 1 66.2
Wanniarachchi
et al (1985)14b
145 1 1 0 19 1 0.5 0.5 1 FLD - - - -
146 1 1 0 19 1.5 0.5 1 1 101.1
100 10 W 4-20147 1 1 0 19 2 0.5 1.5 1 78.19
148 1 1 0 19 2.5 0.5 2 1 66.2
Yau et al
(1985, 1986)15
151 1 1 0 12.7 1 0.5 0.5 1.59 FLD - - - -
152 1 1 0 12.7 1.5 0.5 1 1.59 125.3
100 15-30 W NA153 1 1 0 12.7 2 0.5 1.5 1.59 92.97
154 1 1 0 12.7 2.5 0.5 2 1.59 77.81
Wen (1990) 16
161 1 1 0 12.7 1.5 0.5 1 0.5 146
100 10 NA NA162 1 1 0 12.7 1.5 0.5 1 0.9 136.7
163 1 1 0 12.7 1.5 0.5 1 1.3 129.6
164 1 1 0 12.7 1.5 0.5 1 1.6 125.2
Kumar et al
(2002)17 171 1 8 0 22.7 2.57 1.11 1.46 1.1 71.77 100 4-22 NA 2-4
Total - Water 28 1 94
FC: 1- water, 3- R11, 4- R12, 5- R407c, 6,7- R113, 8- R134a, 9- R123.
Technique (used for the calculation of condensing-side heat transfer coefficient): D-direct tube-wall temperature measurements, W-Wilson plot technique, MW- Modified Wilson-plottechnique.
NA-Not Available, FLD- Fully Flooded, TC- Tube code, FC- Fluid code
All the tubes were made of copper.
61
Table 3.2 Details of the tubes used during condensation of R11 and R12
Reference TCTube
No.FC
No.of
data
Fin
half
angle,
deg.
dr
mm
p
mm
t
mm
b
mm
e
mmf
deg.
Tv
C
Tv
Range
K
Tech-
nique
Exp.
error
%
Sukhatme et al
(1990)31
311 3 1 30.0 22.7 1.06 0.53 0.53 0.69 35.07
40 5 D 5-9
312 3 1 30.0 23.1 0.71 0.35 0.35 0.46 43.36
313 3 1 30.0 23.4 0.53 0.27 0.27 0.34 50.74
314 3 1 30.0 23 0.45 0.22 0.22 0.29 56.17
315 3 1 10.0 23.4 0.71 0.17 0.54 0.46 46.09
316 3 1 10.0 25.6 0.71 0.22 0.49 0.71 43.66
317 3 1 10.0 22.8 0.71 0.25 0.46 0.92 45.85
318 3 1 10.0 22.6 0.71 0.31 0.41 1.22 45.46
Indulkar and
Sukhatme
(1992)
32
321 3 4 0 19 0.9 0.4 0.5 0.8 57.07
40 2-5 W 3.3
322 3 4 0 19 0.9 0.4 0.5 1.2 55.89
323 3 4 0 19 0.9 0.4 0.5 1.6 54.79
324 3 4 0 19 0.9 0.4 0.5 2 53.75
325 3 4 0 19 0.9 0.4 0.5 2.5 52.53
Carnavos
(1980)33
331 3 2 0 16.3 0.94 0.36 0.58 1.32 54.96
35 1-2 W NA332 3 2 0 17.2 0.62 0.25 0.37 0.91 71.09
333 3 2 0 17.6 0.82 0.25 0.57 0.78 55.43
Webb et al
(1982)34
341 3 3 2.9 17.2 0.73 0.25 0.49 0.89 57.73
37 3-8 D NA342 3 3 6.0 15.9 0.98 0.36 0.62 1.53 46.8
343 3 3 4.1 15.9 1.34 0.31 1.03 1.53 38.41
344 3 3 4.7 16 1.24 0.25 0.99 0.85 41.38
Sreepathi
(1994)35
351 3 5 1.3 22.2 0.48 0.22 0.26 0.88 48.33
35-45 1-8 W 8.8352 3 5 3.3 21.3 0.57 0.21 0.36 0.86 38.79
356 3 5 2.0 21.2 0.8 0.19 0.61 0.86 27.63
357 3 5 1.89 21.6 1 0.18 0.82 0.91 22.49
Total – R11 24 3 66
Cheng and
Tao (1994)41
411 4 3 16.1 14.3 1.27 0.65 0.62 0.78 38.5740 1- 4 MW 5-19
412 4 3 0 15.2 0.67 0.27 0.4 1.14 51.51
Kabov (1984) 42
421 4 4 0 17.7 1.5 0.5 1 1.09 29.84
40 1-11 W NA422 4 4 12.1 16.5 2 1.09 0.91 2.5 23.53
423 4 4 15.2 17.1 1.92 0.97 0.94 1.6 24.59
Huber et al
(1994c)43
431 4 4 0 15.9 0.98 0.33 0.65 1.45 38.3235 1-5 W
12
432 4 4 0 17.1 0.63 0.31 0.32 0.86 55.29 10
Total – R12 7 4 26
FC: 1- water, 3- R11, 4- R12, 5- R407c, 6,7- R113, 8- R134a, 9- R123.
Technique (used for the calculation of condensing-side heat transfer coefficient): D-direct tube-walltemperature measurements, W-Wilson plot technique, MW- Modified Wilson-plot technique.
NA-Not Available, FLD- Fully Flooded, TC- Tube code, FC- Fluid code
All the tubes were made of copper.
62
Table 3.3 Details of the tubes used during condensation of R113 (part-1)
Reference TCTube
No.FC
No.of
data
Fin
half
angle,
deg.
dr
mm
p
mm
t
mm
b
mm
e
mmf
deg.
Tv
C
Tv
Range
K
Tech-
nique
Exp.
error
%
Honda et al
(1983)61
611 6 4 4.5 15.8 0.98 0.39 0.59 1.46 48.77
50 2-10 D NA612 6 4 5.0 17.1 0.64 0.26 0.38 0.92 61.34
613 6 4 0 17.1 0.5 0.11 0.39 1.13 66.51
Briggs et al
(1992)62a
621 6 4 0 19.1 1.5 1 0.5 1 55.27
48 5-20 D 8622 6 4 0 19.1 2 1 1 1 38.29
623 6 4 0 19.1 2.5 1 1.5 1 31.06
Briggs et al
(1992)62b
624 6 4 0 12.7 1 0.5 0.5 1.59 64.64
48 7-15 D 8625 6 4 0 12.7 1.5 0.5 1 1.59 44.42
626 6 4 0 12.7 2 0.5 1.5 1.59 35.96
Masuda and
Rose (1985)65
651 6 1 0 12.7 0.85 0.6 0.25 1.6 98.16
48 10 MW NA
652 6 1 0 12.7 1.1 0.6 0.5 1.6 64.59
653 6 1 0 12.7 1.6 0.6 1 1.6 44.39
654 6 1 0 12.7 2.1 0.6 1.5 1.6 35.93
655 6 1 0 12.7 2.6 0.6 2 1.6 30.99
Wen (1990) 66
661 6 1 0 12.7 1.5 0.5 1 0.5 48.03
48 10 NA NA662 6 1 0 12.7 1.5 0.5 1 0.9 46.61
663 6 1 0 12.7 1.5 0.5 1 1.3 45.3
664 6 1 0 12.7 1.5 0.5 1 1.6 44.39
Michael et al
(1990)67
671 6 2 0 12.7 1.25 1 0.25 1 103.6
48 15-20 MW 12
672 6 2 0 12.7 1.5 1 0.5 1 67.51
673 6 2 0 12.7 2 1 1 1 46.27
674 6 2 0 12.7 2.5 1 1.5 1 37.42
675 6 2 0 12.7 3 1 2 1 32.26
Michael et al
(1990)68
681 6 2 0 19.1 1.25 1 0.25 1 82.1
48 15-20 MW 12
682 6 2 0 19.1 1.5 1 0.5 1 55.34
683 6 2 0 19.1 2 1 1 1 38.34
684 6 2 0 19.1 2.5 1 1.5 1 31.1
685 6 2 0 19.1 3 1 2 1 26.85
Michael et al
(1990)69
691 6 2 0 25 1.25 1 0.25 1 70.88
48 10-15 MW 12
692 6 2 0 25 1.5 1 0.5 1 48.41
693 6 2 0 25 2 1 1 1 33.71
694 6 2 0 25 2.5 1 1.5 1 27.39
695 6 2 0 25 3 1 2 1 23.66
Total – R113
(part-1)33 6 75
FC: 1- water, 3- R11, 4- R12, 5- R407c, 6,7- R113, 8- R134a, 9- R123.
Technique (used for the calculation of condensing-side heat transfer coefficient): D-direct tube-wall temperature measurements, W-Wilson plot technique, MW- Modified Wilson-plottechnique.
NA-Not Available, FLD- Fully Flooded, TC- Tube code, FC- Fluid code
All the tubes were made of copper.
63
Table 3.4 Details of the tubes used during condensation of R113 (part-2)
and R407c
Reference TCTubeNo.
FCNo.ofdata
Fin
halfangle,
deg.
dr
mmp
mmt
mmb
mme
mmf
deg.
Tv
C
Tv
RangeK
Tech-nique
Exp.
error
%
Marto et al
(1990)70
701 7 3 0 19.1 1.25 1 0.25 1 82.1
48 10-20 MW 7
702 7 3 0 19.1 1.5 1 0.5 1 55.34
703 7 3 0 19.1 2 1 1 1 38.34
704 7 3 0 19.1 2.5 1 1.5 1 31.1
705 7 3 0 19.1 3 1 2 1 26.85
Marto et al
(1990)71
712 7 1 0 19.1 1.25 0.75 0.5 1 55.34
48 15 MW 7713 7 1 0 19.1 1.75 0.75 1 1 38.34
714 7 1 0 19.1 2.25 0.75 1.5 1 31.1
715 7 1 0 19.1 2.75 0.75 2 1 26.85
Marto et al
(1990)72
721 7 1 0 19.1 0.75 0.5 0.25 1 82.1
48 15 MW 7
722 7 1 0 19.1 1 0.5 0.5 1 55.34
723 7 1 0 19.1 1.5 0.5 1 1 38.34
724 7 1 0 19.1 2 0.5 1.5 1 31.1
725 7 1 0 19.1 2.5 0.5 2 1 26.85
Marto et al
(1990)73 733 7 1 0 18.1 2 1 1 0.5 40.38 48 15 MW 7
Marto et al
(1990)74
743 7 1 0 20.1 2 1 1 1.5 36.57
48 15 MW 7744 7 1 0 20.1 2.5 1 1.5 1.5 29.69
745 7 1 0 20.1 3 1 2 1.5 25.64
Marto et al
(1990)75
753 7 1 0 21.1 2 1 1 2 35.03
48 15 MW 7754 7 1 0 21.1 2.5 1 1.5 2 28.45
755 7 1 0 21.1 3 1 2 2 24.58
Total – R113
(part-2)21 7 31
Honda et al
(2003)51
521 5 4 0 16.040.96 0.45 0.51 1.38 37.7650 2-12 NA 7
522 5 7 0 16.12 1.3 0.48 0.82 1.29 29.70
Total – R407c 2 5 11
FC: 1- water, 3- R11, 4- R12, 5- R407c, 6,7- R113, 8- R134a, 9- R123.
Technique (used for the calculation of condensing-side heat transfer coefficient): D-direct tube-
wall temperature measurements, W-Wilson plot technique, MW- Modified Wilson-plot
technique.
NA-Not Available, FLD- Fully Flooded, TC- Tube code, FC- Fluid code
All the tubes were made of copper.
64
Table 3.5 Details of the tubes used during condensation of R134a
Reference TCTube
No.FC
No.of
data
Fin
half
angle,
deg.
dr
mm
p
mm
t
mm
b
mm
e
mmf
deg.
Tv
C
Tv
Range
K
Tech-
nique
Exp.
error
%
Honda et al
(2002)81
811 8 10 0 16.04 0.96 0.45 0.51 1.38 40.8940 1.5-12 NA
7
812 8 5 0 16.12 1.30 0.48 0.82 1.29 32.12 7
Honda et al
(1999a)82 821 8 5 0 14.94 0.96 0.33 0.63 1.43 40.41 50 2-12 NA 7
Zhang et al
(2007)85
851 8 9 0 17.08 0.59 0.21 0.37 0.926 48.3142 2-20 NA
NA
852 8 10 2.5 16.93 0.6780.309 0.35 0.998 49.96 NA
Gstoehl and
Thome(2006)86 861 8 12 0 15.99 0.94 0.345 0.6 1.36 37.70 31 0.5 - 4 MW 8.3
Belghazi et al
(2002)87
871 8 10 0 16.00 2.31 0.38 1.93 1.45 20.69
40 2-12 W
9.8
872 8 12 0 16.00 1.34 0.33 1.01 1.45 28.74 8.3
873 8 11 0 15.80 0.97 0.25 0.72 1.50 34.29 4.2
874 8 10 0 16.20 0.82 0.20 0.62 1.30 37.04 13
875 8 9 0 16.3 0.635 0.16 0.48 1.30 42.44 3
Cheng and
Wang (1994)88
881 8 7 12.4 16.3 0.98 0.51 0.46 1.35 31.31
42 2-20 MW
8
882 8 7 8.72 16.2 0.79 0.31 0.49 1.01 34.67 8
883 8 7 5.83 16 0.62 0.31 0.31 1.42 41.27 12
Huber et al
(1994)89
891 8 4 0 15.9 0.98 0.33 0.65 1.45 36.1535 1-4 W
11
892 8 4 0 17.1 0.63 0.31 0.32 0.86 52.02 14
Total – R134a 16 8 132
FC: 1- water, 3- R11, 4- R12, 5- R407c, 6,7- R113, 8- R134a, 9- R123.
Technique (used for the calculation of condensing-side heat transfer coefficient): D-direct
tube-wall temperature measurements, W-Wilson plot technique, MW- Modified Wilson-plot
technique.
NA-Not Available, FLD- Fully Flooded, TC- Tube code, FC- Fluid code
All the tubes were made of copper.
65
Table 3.6 Details of the tubes used during condensation of R123
Refe-rence TCTube
No.FC
No.of
data
Fin half
angle,
deg.
dr
mm
p
mm
t
mm
b
mm
e
mmf
deg.
Tv
C
Tv
Range
K
Tech-
nique
Exp.
error
%
Honda et al(1996)
91
911* 9 5 4.7 12.74 0.96 0.45 0.51 1.43 59.44
501.5-13.3
D 5.0
912! 9 5 4.7 12.74 0.96 0.45 0.51 1.43 59.44
913$ 9 5 4.7 12.74 0.96 0.45 0.51 1.43 59.44
914^ 9 5 4.7 12.74 0.96 0.45 0.51 1.43 59.44
915~ 9 5 4.7 12.74 0.96 0.45 0.51 1.43 59.44
916` 9 5 4.7 12.74 0.96 0.45 0.51 1.43 59.44
921* 9 5 3.6 13.62 0.52 0.36 0.16 1.09 123.34
922! 9 5 3.6 13.62 0.52 0.36 0.16 1.09 123.34
923$ 9 5 3.6 13.62 0.52 0.36 0.16 1.09 123.34
924^ 9 5 3.6 13.62 0.52 0.36 0.16 1.09 123.34
925~ 9 5 3.6 13.62 0.52 0.36 0.16 1.09 123.34
926` 9 5 3.6 13.62 0.52 0.36 0.16 1.09 123.34
931* 9 5 0 12.78 0.5 0.17 0.33 1.41 76.25
932! 9 5 0 12.78 0.5 0.17 0.33 1.41 76.25
933$ 9 5 0 12.78 0.5 0.17 0.33 1.41 76.25
934^ 9 5 0 12.78 0.5 0.17 0.33 1.41 76.25
935~ 9 5 0 12.78 0.5 0.17 0.33 1.41 76.25
936` 9 5 0 12.78 0.5 0.17 0.33 1.41 76.25
941* 9 5 0 12.82 0.5 0.22 0.28 1.39 84.17
942! 9 5 0 12.82 0.5 0.22 0.28 1.39 84.17
943$ 9 5 0 12.82 0.5 0.22 0.28 1.39 84.17
944^ 9 5 0 12.82 0.5 0.22 0.28 1.39 84.17
945~ 9 5 0 12.82 0.5 0.22 0.28 1.39 84.17
946` 9 5 0 12.82 0.5 0.22 0.28 1.39 84.17
Sreepathi(1994)
95
951 9 5 1.3 22.2 0.48 0.22 0.26 0.88 68.07
35-45
1-8 W 8.8
952 9 5 3.3 21.3 0.57 0.21 0.36 0.86 58.25
953 9 5 4.94 21.7 0.62 0.21 0.41 0.81 53.74
954 9 5 3.11 21.7 0.64 0.25 0.39 0.81 55.32
955 9 5 3.11 21.5 0.75 0.26 0.49 1 48.71
956 9 5 2.0 21.2 0.8 0.19 0.61 0.86 43.94
957 9 5 1.89 21.6 1 0.18 0.82 0.91 37.29
958 9 5 5.1 22.4 0.58 0.21 0.37 0.58 56.55
959 9 5 2.88 21.1 0.58 0.2 0.38 1.29 55.61
Honda et al(1999b)
96
961 9 5 0 14.94 0.96 0.33 0.63 1.43 52.98
50 2-12 NA 7962 9 5 0 15.22 0.52 0.29 0.23 1.09 94.46
963 9 5 0 15.26 0.5 0.17 0.33 1.41 76.25
964 9 5 0 15.16 0.5 0.22 0.28 1.39 84.17
Rewerts etal (1996)
97971 9 3 0 15.9 0.98 0.33 0.65 1.45 47.24
35 1.0-1.5 W20
972 9 3 0 17.1 0.63 0.31 0.32 0.86 68.99 25
Total – R123 39 9 191
FC: 1- water, 3- R11, 4- R12, 5- R407c, 6,7- R113, 8- R134a, 9- R123. u, m/s 2.0 4.0 7.0 Staggered * ! $ In – line ̂ ~ ̀
Technique (used for the calculation of condensing-side heat transfer coefficient): D-direct tube-walltemperature measurements, W-Wilson plot technique, MW- Modified Wilson-plot technique.
NA-Not Available, FLD- Fully Flooded, TC- Tube code, FC- Fluid code
All the tubes were made of copper.
66
Table 3.7 Summary of tube details
Detail Fluid codeNo. of
tubesNo. of data points
Total - Water 1 28 94
Total – R11 3 24 66
Total – R12 4 07 26
Total – R407c 5 02 11
Total – R113 (part-1) 6 33 75
Total – R113 (part-2) 7 21 31
Total – R134a 8 16 132
Total – R123 9 39 191
Total - All 7-fluids 170 626
The ranges of various parameters covered during the development
of the correlations are listed in equation (3.10).
{ t = 0.4-1.0 mm, e = 0.5-3.0 mm, b = 0.25-2.0 mm, dr = 12-25 mm,
T = 2-30K, = 0.008-0.06 N/m, Vapour velocity < 1.0 m/s, Fin shape :
Rectangular or Trapezoidal ( 15 ), Tube material: Copper, Fluids covered:Refrigerants and Water } (3.10)
3.4 DETERMINATION OF REGRESSION CONSTANTS
For each tube, investigators have provided the experimental data in
the form of ‘he versus Tv’ or ‘he versus heat flux plots’. The experimental
data are directly extracted by enlarging these plots to the maximum possible
size. Few investigators provided the result plots in the form of enhancement
ratio either based on Nusselt value or based on their measured plain tube
condensation coefficients. Enlarged plots are used to obtain the experimental
data appropriately. In certain cases, both ‘he versus Tv’ plots as well as best
fit equations in the form he = a. Tv-n are available. Instead of equations,
enlarged graphs were used to extract the experimental data, as those equations
incorporate certain averaging while curve fitting. In few investigations, only
67
the equations with correlation constants (‘a’ and ‘n’) are available while in
few others, the value of regression constant ‘a’ is reported by forcing the
exponent ‘n’ equal to the Nusselt value of 0.25. Most of the investigations
have shown that the value of exponent ‘n’ is less than 0.25 for the HIF tubes.
This fact will introduce considerable error while correlating the experimental
data. Under these circumstances, less number of data points are used to reduce
such errors.
Rose (1994) points out that it is possible to employ appropriate
weighing factors according to the accuracy or reliability of the various
experimental data sets. But, no such weighing factors seem to have been
employed. In a similar line no weighing factors are applied here. However,
the number of experimental data points used (listed in Tables 3.1 to 3.6) for
correlation development from each investigation is varied depending upon
their accuracy and other factors as discussed earlier.
3.4.1 Data Reduction
The following procedure is adopted to reduce the available
experimental data to the form required for developing the correlation:
(i) Experimental data available are tube geometry (tube diameter
at fin root, fin shape, fin spacing, fin thickness, and fin
height), condensing fluid, condensing temperature and
condensing heat transfer data (he, T).
(ii) From the appropriate property tables, properties of liquid ( ,
, k, hfg, and ) at the condensing temperature are obtained.
(iii) Fin pitch is obtained from p = t + b for rectangular fins. In
case of trapezoidal fins, mean fin thickness and fin spacing
[i.e., ‘t’ and ‘b’ at the mean diameter (do + dr) / 2 ] is used to
find the fin pitch. Fin thickness, fin height and tube diameter
68
at fin root are normalized using fin pitch, as t* = t/p, e* = e/p
and dr* = dr / p respectively.
(iv) Using the equation (3.3), the flooding angle ( f) is estimated
and by using equation (3.2), the flooded fraction of tube
circumference Cf is calculated.
(v) Gravity number and Surface tension number are calculated as
Gy = {[( hfg) / ( k Tv)] . ( g p3)} and
Su = {[( hfg) / ( k Tv)]. ( .p)} respectively.
(vi) The average Nusselt number is calculated as Nud = he dr / k.
(vii) Using equation (3.4), Nusselt number for the flooded zone
(Nuf) is calculated.
(viii)Using equation (3.1), Nusselt number for the un-flooded zone
(Nuu) is obtained. Now, all the parameters in the equation (3.6),
Nuu, Su, t*, e*, b* and dr* are available.
(ix) If f = 180°, Cf = 1. Hence, the equation (3.1) becomes
Nud = Nuf. Now the parameters of equation (3.8) Nuf, Gy and
dr* are available.
3.4.2 The Correlation Technique
Taking the equations (3.6) and (3.8) in the logarithmic form,
equations (3.7) and (3.9) respectively are obtained. The equations (3.7) and
(3.9) are linear equations with a total of four independent parameters and
seven constants, which needs to be obtained. By using Log-linear least square
multiple regression analysis (Newbold (1984)), the constants (A, a1-a4, B, b1)
are obtained which are listed in Tables 3.8 and 3.9 along with the coefficient
of correlation (R2). To find out the regression constants, the “DataFit”
software has been used. By selecting number of independent variables, the
data feeding table is opened. The relevant experimental data are fed. Then the
equation type is entered and the values of constants and R2 values are
obtained. The R2 values lie in the range 0.80 - 0.91. It indicates that
69
the agreement between the calculated values and the experimental data are
better, as R2 = 1.0 for a perfect fit.
Table 3.8 Correlation constants for equation (3.6)
CategoryFluid
used
Experimental data used for
correlation
Correlation
constantsR
2
CFC data
based
R11
R12
R113
31, 32, 33, 34, 35, 41, 42, 43,61, 62, 65, 66, 67, 68, 69,70,71, 72, 73, 74, 75[ 85 tubes, 198 data points]
A = 39.2077
a1 = 0.1729
a2 = 0.2759
a3 = - 0.1486
a4 = 0.5141
0.82
Waterdata based
Water11,12,14,15,16,17
[ 23 tubes, 80 data points ]
A = 13.6515
a1 = 0.1377
a2 = 0.0916
a3 = - 0.0288
a4 = 1.2128
0.81
HCFC /HFC data
based
R407c
R134a
R123
51,81,82,85,86,87,88,89,91,92,93,94,95,96, 97
[ 53 tubes, 319 data points ]
A = 4.8847
a1 = 0.2282
a2 = 0.341
a3 = - 0.6199
a4 = 0.8247
0.91
General
Water
R407c
R134a
R123
11, 12, 16, 51, 81, 85, 86, 87,95 [ 34 tubes, 211 data points ]
A = 12.8661
a1 = 0.1814
a2 = 0.1629
a3 = - 0.3731
a4 = 0.8317
0.90
Table 3.9 Correlation constants for equation (3.8)
CategoryFluid
used
Experimental data used
for correlation
Correlation
constantsR
2
Fully Flooded
dataWater
12,14,15
[ 5 tubes, 14 data points ]
B = 39.822
b1 = 0.250.80
70
3.5 CONCLUDING REMARKS
The correlation based prediction models for predicting the
condensation of CFCs, water and HCFC/HFCs over HIF tube, which is
relatively simpler than the existing models has been developed by correlating
the reliable experimental data directly.
The present model is simple with only four non-dimensional
equations and seven empirical constants. It needs only a simple hand-
calculator for calculating the condensation heat transfer coefficient. The
model covers a wider range of experimental data and the agreement between
the calculated values and the experimental data are better. It is felt that this
correlation-based model is simplest of its kind reported until recently. It will
enable the researchers and the heat exchanger manufacturers to simplify the
design of condensers for various fluids. The next chapter provides the
prediction capability of the correlated models.