Pressure drop coefficients for elliptic and circular sections in one, two and three-row arrangements of plate fin and tube heat exchangers

Pressure Drop Coefficients for Elliptic and Circular Sections in One, Two and Three-Row Arrangements of Plate Fin and Tube Heat Exchangers

Sérgio Nascimento Bordalo

Universidade Estadual de Campinas

Faculdade de Engenharia Mecânica

13083-970 Campinas, SP Brazil

[e-mail]

Francisco Eduardo Mourão Saboya

Universidade Federal Fluminense

Departamento de Engenharia Mecânica

24210-240 Niterói, RJ Brazil

[e-mail]

Abstract

The objective of the present work is the experimental determination of pressure drop coefficients (loss coefficients) for elliptic and circular sections in one, two and three-row arrangements of plate fin and tube heat exchangers. The experiments permitted to correlate the dimensionless loss coefficient with the flow Reynolds number in the rectangular channel formed by the plate fins. The experimental technique consisted of the measurement of the longitudinal pressure distribution along the flow channel, for several values of air mass flow rate. The total number of data runs, each one characterized by the flow Reynolds number, was 216. The present geometry is used in compact heat exchangers for air conditioning systems, heaters, radiators, and others. Also, it is verified the influence of the utilization of elliptic tubes, instead of circular ones, in the pressure drop. The measurements were performed for Reynolds numbers ranging from 200 to 1900.

Keywords: pressure drop, loss coefficient, elliptic tube, circular tube, rectangular channel.

Introduction

Plate fin and tube heat exchangers are widely employed in such applications as air conditioning systems, heaters and radiators. Of fundamental importance for design is the pressure drop that occurs in these devices due to fluid viscosity. The subject investigated in this research deals with the experimental determination of the pressure drop that occurs in plate fin and tube compact heat exchangers. Tubes with elliptic and circular cross sections are analyzed. Configurations with one, two and three-row arrangements are compared, as far as pressure drop is concerned. The geometric characteristics of the exchangers under study in this work are shown in Fig. 1. It is seen that for the cases of two and three-row arrangements the tubes are disposed in staggered fashion.

Fig. 1 Geometric Characteristics of Heat Exchangers

In a pioneering study, Schulenberg (1966) analyzed the potential of the application of elliptic tubes in industrial heat exchangers. In this work, it is reported that Schulenberg's company produced in 1966 more 70 miles of elliptic tubes per week. This suggests that there is no commercial disadvantage when cost comparison is made with conventional heat exchanger tubes.

The elliptic tube geometry has a better aerodynamic shape than the circular one; therefore, it is reasonable to expect a reduction in total drag force when comparing the former to the latter, both submitted to a cross flow free stream. According to Webb (1980), the performance advantage of the elliptical tubes results from their lower pressure drop due to the smaller wake region on the fin behind the tube.

Heat transfer coefficients for plate fin and elliptical tube compact exchangers were experimentally determined by Saboya and Saboya (1981). These coefficients are applicable to one and two-row tube arrangements and were compared to circular tube results, found in the pertinent literature. The most important finding of this research is that the substitution of circular tubes by elliptical ones does not affect the rate of heat transfer adversely. It is shown that there are no major differences between the two configurations, as far as heat transfer coefficients are concerned.

With regard to the influence of the replacement of circular tubes by elliptic tubes, on the plate fin efficiency, Rocha, Saboya and Vargas (1997) have shown that the elliptical tube configuration is more efficient than the circular one. Among the cases studied in this paper, the maximum efficiency gain was 18%. The last two just mentioned results, found in the pertinent literature, are very positive outcomes since the use of elliptical tubes may reduce substantially the pressure drop without affecting the heat transfer coefficients and, at same time, increasing the fin efficiency.

The increasing interest in the elliptic tube geometry is evidenced by the recent work of Jang and Yang (1998). In this research experimental investigation and numerical analysis for elliptical finned-tube heat exchangers are reported.

The objective of the present experimental work is to produce curve-fitting equations providing the pressure drop coefficients (loss coefficients) as functions of the flow Reynolds number in the rectangular channel formed by the plate fins (see Fig. 1). The experimental method consisted of the measurement of the longitudinal pressure distribution along the flow channel, for several values of the mass flow rate (or Reynolds number). The total number of data runs, each one characterized by the flow Reynolds number was 216. The measurements were performed for 18 Reynolds number values in the range 200-1900, which is relevant for air conditioning applications. In the tests, the working fluid was air. Figures 2 and 3, for one-row circular and elliptic tube heat exchangers, respectively, represent schematic diagrams of the physical problem under study. The heat exchangers with two and three tube rows are obtained by repetition, side by side, of the modules shown in Figs. 2 and 3. For heat exchangers with more than one tube row the centers of the tubes are disposed in equilateral triangular arrangements.

Fig. 2

Fig. 3

Figures 2 and 3 also contain dimensional nomenclature designated by S, L, d, 2a, 2b and D. In the case of circular tube heat exchangers it is clear that 2b = D and b / a = 1.0.

Nomenclature

A = frontal air flow area, m²

a = bigger semi-axis of the elliptical tube section, m

b = smaller semi-axis of the elliptical tube section, m

D = diameter of the circular tube section, m

f = friction factor, dimensionless

K = loss coefficient per tube row, dimensionless

L = length of the one-row arrangement, m

= air mass flow rate (discharge), kg / s

N = number of tube rows, dimensionless

Re = Reynolds number, dimensionless

S = tube-to-tube center distance, m

V = average air frontal velocity, m/s

Greek Symbols

d = fin-to-fin distance, m

DP_d = pressure drop due only to the presence of the tubes, Pa

DP_tot = total pressure drop, Pa

DP_u = pressure drop in the unobstructed (free from the tubes) rectangular channel, Pa

m = air dynamic viscosity, kg / (m.s)

r = air density, kg / m³

The dimensionless parameters which govern the results may be taken as: b / a; S / L; S / 2b; d / L and the flow Reynolds number, Re. The dimension ratios of the present test apparatus were: S / 2b = 2.50 and 3.54; S / D = 2.50; b / a = 0.5, 0.65 and 1.0; S / L = 1.1541; d / L = 0.0919. These values are typical of heat exchangers encountered in air conditioning machines. The actual values of the apparatus dimensions were: L = 18.5 mm; S = 21.35 mm; D = 8.53 mm; d = 1.70 mm; 2a = 17.06 mm, 13.12 mm and 12.06 mm; 2b = 8.53 mm and 6.03 mm. The experiments were performed employing twelve heat exchanger configurations: four configurations with one tube row (three configurations were elliptic and one circular), another four configurations with two tube rows (three elliptic configurations and one circular) and the final four configurations with three tube rows (three elliptic configurations and one circular).

Test Apparatus and Experimental Procedure

Figure 4 shows a schematic side view of the test section arrangement used in the pressure drop experiments of the present research. There in one has:

a – buffer

b – rectangular channel with several pressure taps; inside the channel are the tubes

c – plenum chamber

d – valve to control the air flow

e – calibrated orifice to measure air mass flow rate

f – cut-off valve to stop the air flow

g – pressure taps to measure the pressure drop across the calibrated orifice

h – by-pass valve

i – blower

Fig. 4 View of the Test Section Used in the Experiments

As shown in Fig. 4, air from the laboratory room is suctioned through the rectangular channel in an open-loop flow circuit. The rectangular channel possess additional lengths before and after the heat exchanger section to permit the development and redevelopment of the flow. As it will be seen later, the development and redevelopment of the flow are necessary for the pressure drop determination across the tube rows. Upon leaving the rectangular duct formed by the plates, the air exits to a plenum chamber from which it passes successively to a control valve, a calibrated flow meter, a cut-off valve and a blower, and then to an exhaust system. The by-pass valve, h, in Fig. 4 is used to avoid heating of the blower at low air mass flow rates.

To obtain the loss coefficients it was necessary to determine the pressure distribution before (7 pressure taps) and after (6 pressure taps) the heat exchanger section. The pressure was sensed by a Baratron solid-state capacitance-type meter capable of resolving 10 ^{– 3} mm Hg. Once the air mass flow rate (or Reynolds number) was fixed, all 13 static pressure measurements were taken. This procedure was repeated for 18 values of Reynolds number and 12 heat exchanger configurations, totaling 216 data runs. Typically, the experimental uncertainty of the air mass flow rate measurements was 2%. The atmospheric pressure was monitored by a barometer of column of mercury with a smallest scale division of 0.1 mm. The ambient temperature was sensed by a precision thermometer that could be read up to 0.1 ^oC.

Data Reduction Procedure

The main objective of the data reduction procedure is to obtain the loss coefficient as function of the Reynolds number. The calculation of the loss coefficient, K, is facilitated by reference to Fig. 5, which presents a typical gage pressure distribution for Reynolds number, Re, equal to 1900 and a plate fin and tube heat exchanger configuration with three circular tube rows. The two additional lengths, before and after the circular tubes, are long enough to permit the development and redevelopment of the flow. For this reason, the two curves in Fig. 5 are parallel straight lines. They represent curve fits of experimental data, by the method of least squares. The low degree of scattering attests to the quality of the data.

Fig. 5 Typical gage pressure distribution

To determine the value of the loss coefficient for each run, (fixed air flow Reynolds number) the following definition is employed:

(1)

where DP_d is the pressure drop due only to the presence of the tubes (vertical distance between the two parallel straight lines in Fig. 5); N is the number of tube rows; r is the air density and V is the average air frontal velocity. The average air frontal velocity is given by the following equation:

(2)

where is the air mass flow rate and A (=181.3 mm²) is the frontal area of the air flow.

Although the determination of K, given by Eq. (1), does not require the knowledge of the total pressure drop, DP_tot, that occurs in the test section, it may be obtained from the following expression:

(3)

where DP_u is the pressure drop that would occur if the rectangular channel were free from the obstacle caused by the heat exchanger tubes (smooth channel). Such an additional pressure drop is determined from the usual definition of the friction factor, f, expressed as:

(4)

Jones (1976) reported the following expression for the friction factor for laminar flow in rectangular channels:

(5)

The flow Reynolds number can be calculated from:

(6)

where m is the air dynamic viscosity and 2d is the cross-section hydraulic diameter. Once K and Re are obtained from Eqs. (1) and (6), respectively, it is possible to produce curve-fitting equations giving K = K (Re). Of course this is the main objective of the present experimental research.

Results and Discussion

To correlate K with Re, 216 data runs were performed. Such a number of data runs was divided among the 12 heat exchanger configurations constructed to make the experimental tests. Thus, for each one of the 12 configurations, 18 experimental runs characterized by the Reynolds number, were carried out. As mentioned before, one, two and three-row arrangements, possessing circular and elliptic tubes, were employed. The problem has 4 dimensionless geometric parameters (see the last paragraph of the Introduction section) from which S / L and d / L were kept constant, during the runs.

Figures 6 and 7 present K as function of Re for one-row plate fin and tube heat exchangers. In Fig. 6 there are two cases: one is for circular tubes and the other for elliptic tubes. In Fig. 7 there are also two cases: both for elliptic tubes. The values of the pertinent dimensionless geometric parameters are shown in the figures. The points in Figs. 6 and 7 are experimental results and the solid lines are fits obtained by the method of least squares.

Fig. 6 K for one-row circular and elliptic tubes

Fig. 7 K for One-Row Elliptic Tubes

The curve-fitting equations, associated to Figs. 6 and 7 (one-row) are given by:

a) circular tube; one-row; b / a = 1.0; S / 2b = 2.5

(7)

The mean deviation of Eq. (7) in relation to the data points is 16.42%.

b) elliptic tube; one-row; b / a = 0.65; S / 2b = 2.5

(8)

The mean deviation of Eq. (8) in relation to the data points is 11.60%.

c) elliptic tube; one-row; b / a = 0.5; S / 2b = 2.5

(9)

The mean deviation of Eq. (9) in relation to the data points is 13.76%.

d) elliptic tube; one-row; b / a = 0.5; S / 2b = 3.54

(10)

The mean deviation of Eq. (10) in relation to the data points is 13.74%.

Equations (7) – (10) are now employed to construct the following numerical table:

Inspection of Figs. 6 and 7 and Table 1 reveals that, for Re > 1000, the elliptic tubes provoke less pressure drop than the circular tubes. The inverse occurs when Re < 1000. The reason for this behavior is that, for higher Reynolds numbers, the pressure drag is predominant. In this case, due to the better aerodynamic shape of the ellipse, the elliptic tubes present better performance. For lower Reynolds, the viscous drag is predominant. Since the elliptic tubes possess greater lateral area, they present poorer performance. It may also be observed (last column in Table 1) that the increase of the separation between the tubes (S / 2b = 3.54) causes a decrease in the pressure drop. However, this also will give rise to a prejudicial decrease in the heat transfer, as pointed out by Saboya and Saboya (1981).

Table 1 K as Function of Re. One-Row
Circular and Elliptic tubes

It should be mentioned that the relatively high scattering of the experimental points, shown in Figs. 6 and 7, in the low Reynolds number range, is due to the very small pressure drop values that occurred in this range. Values of DP_d as low as 0.002 mm Hg, were determined. Also one must consider that the situations displayed in Figs. 6 and 7 refer to N = 1, which is the case that presents the lowest pressure drop.

The values of the loss coefficient, K, as function of Re for two-row configurations are shown in Figs. 8 and 9. Figure 8 is concerned with circular and elliptic tubes, while Fig. 9 displays results for elliptic tubes. The values of the dimensionless geometric parameters that characterize the experimental situation are shown in both figures. In Figs. 8 and 9, the points are experimental data while the solid lines are fits determined by the method of least squares.

Fig. 8 K for two-row circular and elliptic tubes

Fig. 9 K for two-row elliptic tubes

The curve-fitting expressions, which are associated to Figs. 8 and 9, (two-row) are given by:

a) circular tube; two-row; b / a = 1.0; S / 2b = 2.5

(11)

The mean deviation of Eq. (11) in relation to the experimental points is 8.34%.

b) elliptic tube; two-row; b / a = 0.65; S / 2b = 2.5

(12)

The mean deviation of Eq. (12) in relation to the experimental points is 7.62%.

c) elliptic tube; two-row; b / a = 0.5; S / 2b = 2.5

(13)

The mean deviation of Eq. (13) in relation to the data points is 7.06%.

d) elliptic tube; two-row; b / a = 0.5; S / 2b = 3.54

(14)

The mean discrepancy of Eq. (14) in relation to the experimental points is 13.83%.

Equations (11) – (14) can now be used to construct the following numerical table:

It may be observed from Figs. 8 and 9 and Table 2 that the elliptic tube with b / a = 0.65 has a better performance than the circular tube, for Re > 200, and also than the elliptic tube with b / a = 0.5, for Re £ 800. For Re ³ 1000, the elliptic tube with b /a = 0.5 is the one that possess the better performance. For Re > 1000, the pressure drag seems to predominate. In this case, the better aerodynamic shape of the ellipse, with b / a = 0.5, gives rise to a better performance. For lower Reynolds number, the viscous drag becomes more important and the greater lateral area of the elliptic tubes with b / a = 0.5 causes a poorer performance.

Table 2 K as Function of Re. Two-Row Circular and Elliptic Tubes

Inspection of Table 2 (last column) shows that the increase of the separation between the tubes ( S / 2b = 3.54) causes a decrease in the pressure drop. As mentioned before, this also causes a prejudicial decrease in the heat transfer of the heat exchanger.

It should be noted that the mean deviations of Eqs. (11) – (14), for N = 2, decreased in comparison with Eqs. (7) – (10), for N = 1. In addition, the scattering in Figs. 8 and 9 is lesser than in Figs. 6 and 7.

Figures 10 and 11 display K as function of Re for three-row plate fin and tube exchangers. Figure 10 refers to circular and elliptic tubes, while Fig. 11 is concerned with elliptic tubes. As in the preceding situations, (Figs. 6 _ 9) the dimensionless geometric parameters that govern the problem are shown in Figs. 10 and 11 and the points and solid lines are experimental data and fits (method of least squares), respectively.

Fig. 10 K for Three-row circular and elliptic tubes

Fig. 11 K for three-row elliptic tubes

The equations of the fitted curves shown in Figs. 10 and 11 (three-row) are given by:

a) circular tube; three-row; b / a = 1.0; S / 2b = 2.5

(15)

The mean discrepancy of Eq. (15) in relation to the experimental points is only 4.78%.

b) elliptic tube; three-row; b / a = 0.65; S / 2b = 2.5

(16)

The mean deviation of Eq. (16) in relation to the data points is only 5.49%.

c) elliptic tube; three-row; b / a = 0.5; S / 2b = 2.5

(17)

The mean discrepancy of Eq. (17) in relation to the experimental points is only 6.43%.

d) elliptic tube; three-row; b / a = 0.5; S / 2b = 3.54

(18)

The mean discrepancy of Eq. (18) in relation to the data points is only 9.94%.

Based on Eqs. (15) – (18), the following numerical table can now be constructed:

Inspection of Figs. 10 and 11, as well as of Table 3, reveals that the elliptic tube with b / a = 0.65 presents a better performance than the circular tube, for Re ³ 200, and also than the elliptic tube with b / a = 0.5, for Re £ 800. For Re ³ 1000, the elliptic tube with b / a = 0.5 is more efficient, as long as pressure drop is concerned. As mentioned before, for Re > 1000 the pressure drag seems to predominate giving rise to a better performance of the elliptic tube with b / a = 0.5, due to its better aerodynamic shape. In the lower Reynolds number range, the viscous drag predominates and the greater lateral area of the elliptic tube with b / a = 0.5 causes a poorer performance. The last column in Table 3 (S / 2b = 3.54) shows, that the increase of the separation between the tubes decreases the pressure drop. As mentioned before, this provokes a reduction of the heat transfer of the heat exchanger.

Table 3 K as function of Re. three-row circular and elliptic tubes

One should observe that the mean discrepancy associated to Eqs. (15) – (18), for N = 3, reduced in comparison with the cases for N = 1 and N = 2. In addition, the scattering displayed in Figs. 10 and 11 is lesser than in Figs. 6 and 9.

Uncertainty Analysis

An uncertainty analysis for the Reynolds number was performed using the well-known Kline and McClintock (1953) methodology. Moffat (1988) also presented this methodology of describing uncertainties in experimental results. Typically, the experimental uncertainty associated to the Reynolds number was 2.5%, with maximum values of less than 5.0%. The uncertainties were evaluated by the responses of the Reynolds numbers to changes in each of the variables used in the data reduction procedure. (see Eqs. (2) and (6)). For the Reynolds number, the most relevant parameter is the mass flow rate given by the calibrated flow meter.

The uncertainty associated to the experimental determination of the loss coefficient, K, was obtained by the same method. (see Eqs. (1) and (2)). For the loss coefficient, the most relevant parameter is the pressure drop, DP_d. Typically, the experimental uncertainties of the loss coefficients are: 3.0% for N = 3; 5.0% for N = 2 and 8.5% for N = 1. In the low Reynolds number range and for one tube row the uncertainty in K reached maximum values as high as 40%. In these situations the pressure drop, DP_d, is very small (0.004 – 0.002 mm Hg) causing very large experimental errors. It is seen that the experimental uncertainty decreases as the Reynolds number and the number of tube rows increase.

In addition to the uncertainties that are present in all experimental work, a careful observation of Figs. 6-11, in the Reynolds number range from 500 to 800, reveals a systematic behavior of the loss coefficient, K, as far as deviations from the curve-fitting equations (solid lines in Figs. 6-11) are concerned. Such a behavior is most probably due to the generation of horse-shoe vortices around the tubes of the heat exchanger. Saboya and Sparrow (1974) have shown the presence of these vortices in the flow of air in plate fin and tube heat exchangers. In this work the authors have shown that in the Reynolds number range 500-800 an additional horse-shoe vortex is generated. This may cause a flow perturbation making the measurements a difficult task.

Concluding Remarks

Within the knowledge of the authors, the results reported in the present research are original. They are applicable to plate fin and tube compact heat exchangers. For Reynolds numbers in the range 1000 ^_ 1800, the replacement of the circular tubes by elliptic tubes, with b / a = 0.5, yields a reduction on the loss coefficient of about 30%. A greater reduction on the pressure drop could be obtained by increasing the separation between the tubes. However, such a procedure is not recommendable since the heat transfer of the heat exchanger is also reduced, as pointed out by Saboya and Saboya (1981).

Before closing, it should be mentioned that, in engineering applications, the loss coefficient results for three-row plate fin and tube heat exchangers can be applied, without great error, to multi-row configurations. Of course this is an approximation based on the assumption that for N >3 a pattern was established. Nevertheless, the values of the geometric dimensionless parameters that govern the problem can not be different from those of the present work.

Acknowledgments

The authors gratefully acknowledge the Brazilian agency, National Council of Scientific and Technological Development-CNPq, for the financial support provided during the course of the present research through grant 302130 / 84-5(RN).

Manuscript received: May 1999; Technical Editor: Angela Ourívio Nieckele

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Publication Dates

• Publication in this collection
  11 Oct 2001
• Date of issue
  Dec 1999

History

• Received
  May 1999