chapter 3 development of correlation based...

50

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.

52

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

57

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.

58

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.

59

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.

60

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


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.



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


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.



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


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.



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


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.



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.



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


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


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.

chapter 3 development of correlation based...

Documents