Local free convection around inclined cylinders in air: An interferometric study

Local Free Convection Around Inclined Cylinders in Air: An Interferometric Study Jisheng Li J. D. Tarasuk Department of Mechanical Engineering, The University of Western Ontario, London, Ontario, Canada

• Free convective heat transfer from horizontal, inclined, and vertical cylinders in air was experimentally studied by means of a Mach-Zehnder interferometer. The effect of the Rayleigh number on both local Nusselt numbers along the perimeter of the cylinder and the average Nusselt number over the surface of the cylinder was investigated by varying the experimental Rayleigh numbers in a range of 1.1 x 104 to 3.4 x 105. The effect of orientation of the cylinder on heat transfer rates was also examined by varying the angle of inclination of the cylinder with respect to the horizontal. Angles of 0", 45", 60", 75", and 90* were studied. The experimental results in terms of the Nusselt number were correlated as a function of the Rayleigh number for different angles. A universal equation is given that correlates the experimental results for all angles studied. The present results are in good agreement with those of previous researchers.

Keywords: free convection, cylinders

INTRODUCTION Numerous heat transfer applications arise in which a heated cylinder is accepting or rejecting energy. Typical examples are heat exchangers, solar concentrator absorbers, currentcarrying conductors, and heating and process supply pipes. Prediction of free convection heating or cooling of the cylinder is often important because such heating or cooling may be the governing heat transfer mechanism.This paper focuses on research on heat transfer from a cylinder heated or cooled by free convection in air and in inclined orientations. Several studies [1-8] have been reported in the literature for the cylinder in several orientations and are summarized in Table 1. Generally, vertical and horizontal cylinders have been studied extensively. However, inclined cylinders, possibly because of the three-dimensional complex flow patterns, have not been studied to the same extent. This research reports on an experimental study of local coefficients for free convection about isothermal cylinders of three different diameters for the horizontal, 45", 60", 75", and vertical orientations. This research employed the Mach-Zehnder interferometer in the study of three-dimensional flow and heat transfer patterns. Local beam-averaged coefficients were obtained and were compared with those obtained with the energy balance method used by other researchers. EXPERIMENTAL

SETUP

The arrangement of the interferometer and model is shown in Fig. 1. The interferometer was oriented in a vertical plane, and the optical bench was isolated from vibrations at three

points by air sacs. The diameter of the major optics was 20 cm, and a continuous 5 mW linearly polarized, single-mode helium-neon laser was used as a light source. It was possible to have a variety of inclined test beams by suitably locating the optics on the supporting frame. The output of the interferometer was recorded by an S&K view SP-45 camera that used Type 55 Polaroid film. The infinite fringe alignment was used in all the experiments. An optical glass window was used to minimize the end effect for vertical cylinder experiments. It was placed at the top of the cylinder and perpendicular to the test beam. The gap between the optical window and the cylinder end was about 4 mm.

DESCRIPTION OF THE THREE ISOTHERMAL MODELS Three models, with three different diameters and length-todiameter ( L / D ) ratios, were constructed for this investigation. The diameters were 19.1, 31.7, and 50.8 mm, with lengths of 408, 680, and 700 mm, respectively. The L I D ratios were 14 and 21. All models were constructed with an electrically heated center core. The heat was transferred to an outer brass cylinder through an annulus of copper powder. Figure 2 shows the details of the cylinder. Twelve thermocouples were used to monitor the cylinder's exterior temperatures longitudinally and circumferentially. To offset end loss effects on the cylinder, additional electrical windings were added at the ends of the heater core as shown in Fig. 2. The maximum temperature variation was found to

Add~sscor~spondenceto Pro~ssorJ. D. Tarasuk, Depa~mentofMechanical Engineering, Universi~ of Wes~m Ontario, London, Onmfio, Canada N6A 5B9.

Experimental Thermal and Fluid Science 1992; 5:235-242 © 1992 by Elsevier Science Publishing Co., Inc., 655 Avenue of the Americas, New York, NY 10010

0894-1777/92/$5.00

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Free Convection Around Cylinders in Air

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Figure 1. Diagram of the MachZehnder interferometer and model arrangement.

analyze the three-dimensional fringe shift field, the following procedure was used. The Gladstone-Dale equation is given by

ANALYSIS 2 ( n o - 1)/3p = Kgd

Fringes on the interferogram were located with respect to the model edge by means of a traveling microscope, and the data were analyzed by the following method. The flow patterns along the three models studied were three-dimensional for all orientations except the horizontal position. For a three-dimensional flow, the fringe patterns on a interferogram with infinite fringe background could only represent fringe shifts. (For two-dimensional flow the fringe shifts on the interferograms also represent lines of constant temperature around a heated surface.) The technique developed by Papple and Tarasuk [9] was used to establish circumferential Nusselt numbers, averaged along the test beam. To

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Figure 2. Construction details of the heated isothermal cylinders. 1, nut; 2, end section; 3, brass tube; 4, copper powder; 5, resistance (nichrome) wire; 6, wooden rod; 7, thermocouple wire; 8, brass tube; 9, steel plate; 10, steel tube.

238 Jisheng Li and J. D. Tarasuk index by the equation

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horizontal (~ = 0 °) and inclined (0 ° < ~o < 90 °) cylinder, the local heat transfer rate is the highest at the lowest point of the cylinder, and decreases circumferentially toward the top. For the vertical (¢ = 90 °) cylinder, uniform local heat transfer rates are expected, as shown in Fig. 3. Figures 3 and 4 also present the effect of the angle of inclination on the heat transfer rates. The local heat transfer rates are the highest for the horizontal cylinder and decrease gradually as the angle of inclination increases. As shown in Fig. 3, the local heat transfer rates for the horizontal and inclined cylinders are lower than the value for the vertical cylinder in the region close to the top of the cylinder. Figure 5 shows plots of the average heat transfer results obtained in this study versus the Rayleigh number with the angle of inclination as a flee parameter. As shown in Fig. 5, there is a good degree of linearity of the data points for each angle of inclination studied. This linearity of data points suggests a correlation of the form Nu o = m Ra~

(9)

for each angle of inclination. The coefficients m and n in Eq. (9) are determined by carrying out a best-fit least squares straight-line approximation of the logarithm of the average Nusselt number Nu o and the Rayleigh number Ra D. The values of m, n, and the error of correlation as well the range of the Rayleigh number are given in Table 2 for each angle of inclination studied. As indicated in Table 2, the correlation errors are larger for the horizontal and vertical cylinders. These errors may be caused by the end effects. For the horizontal cylinder, the end-effect errors result from the deviation of the actual density field from the ideal two-dimensional fields at the ends of the test cylinder. An optical glass window was used to minimize the end effects for the vertical cylinders. However, the end-effect errors become more significant for the shorter test cylinder as well as for increasing angle of inclination of the cylinder. The correlations are also plotted in Fig. 5 (solid lines) for different angles of inclination. They show that the Nusselt numbers increase with increasing Rayleigh number and decrease as the angle of inclination with the horizontal increases.

where 10.0

The Nusselt number averaged along the test beam is directly proportional to the derivative of the fringe shift evaluated at the surface of the model. We used Eq. (7) to analyze interferograms for the inclined and vertical cylinders. The uncertainty estimated for the Nusselt number is +_4%, and for Rayleigh number, +2 %.

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Typical experimental results are shown in terms of the local Nusselt number distributions along the perimeter of the cylinders in Fig. 3. These distributions are evaluated from the interferograms o f R A o = 1.4 × 10 4 given in Fig. 4. For the

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Figure 3. Effect of the angle of inclination on the free convection heat transfer rate, Ra = 1.4 × 104.

Free Convection Around Cylinders in Air

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Figure 4. Interferograms showing qo =

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A fourth-order polynomial equation was used to fit the correlation values o f m and n with a least squares fit to produce a universal equation to estimate the average heat transfer rate in terms o f the Nusselt n u m b e r for inclination angles from 0* to 90*. A single correlation is given by Nu D = m ( , p ) R a ~ ~) 1 02

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240

Jisheng Li and J. D. Tarasuk

Table 2. Correlations of the Average Nusselt Numbers Angle

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n

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COMPARISON OF PRESENT RESULTS WITH PREVIOUS WORK

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It is difficult to compare our results with those of other researchers on a single figure because the constraints and limitations are frequently different from one research project to another. However, Fig. 7 shows the comparison of the present results for the horizontal cylinder with analytical results presented by Morgan [1], Churchill and Chu [3], Brdlik [5], and the experimental correlations of Fand et al [4] and Eckert [2]. As shown in Fig. 7, good agreement exists between our results and previous work. In particular, our results are in close agreement with those of Morgan and Fand et al. For the inclined cylinders, the present results are compared with the experimental correlations of Oosthuizen [6] and A1-Arabi and Salman [7] for angles of inclination of 45*, 60*, and 75*. There is good agreement between our results and Oosthuizen's as shown in Figs. 8-10. The differences are small at low Rayleigh numbers for larger angles of inclination. This can be attributed to the end-effect error. The differences between A1-Arabi and Salman's results and ours decrease as the angle of inclination increases. Figure 11 is a comparison of our results for the vertical cylinder with those of A1-Arabi and Salman [7] and Morgan [1]. The data obtained in the present study lie between the results of those researchers and show very good agreement for Rayleigh numbers from 4 × 104 to 3.5 × l05. For relatively small Rayleigh numbers on the order of 1.1 × 10 4 tO 3.1 × 104, the present experimental data are higher than those of AI-Arabi and Salman and Morgan. The difference may be due in part to model end effects. Only two aspect ratios were used in this study. This is justified in view of the close comparison to the results of

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Figure 9. Comparison of average Nusselt numbers for the in-

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In this study the maximum error occurred with the shortest cylinder (408 mm), which had an estimated end-effect error of 5 % in the vertical position. PRACTICAL SIGNIFICANCE

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Oosthuizen [6], AI-Arabi and Salman [7], and Stewart and Buck [8], whose research had an aspect ratio range from 8 to 25.

EXPERIMENTAL UNCERTAINTY The errors associated with the construction of the model, the interferometer alignment, and edge effects were minimized as much as possible. The techniques employed are described in Refs. 10-12. The error associated with fringe measurement was estimated to be less than ___2%. The major source of error is model end effects. This error is inversely proportional to the length-to-diameter ratio (L/D) of the cylinder. An end-effect error of less than 3 % was estimated for the cylinder in the horizontal position and is found to be consistent with the estimated error for L/D = 13 in Ref. 2. The L / D ratios in this study are all greater than 13. The models used in this study were polished to minimize radiation. For the worst condition the estimate of radiant energy lost was estimated to be 29% of convection. This estimate assumes that the air absorbed all of the radiation loss. Since air is transparent to long-wave radiation at 330 K, the radiant contribution to fringe shifts was considered to be negligible.

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Accurate knowledge of the overall free convection heat transfer, particularly from cylindrical systems, is important for many engineering designs. This study originated from the need to predict heat losses from an inclined cylindrical solar energy absorber. In addition, numerous devices depend on this mode of heat transfer. For example, radiators used for space heating transfer heat by free convection. Also, current-carrying conductors and pipes conveying heated gases or liquids are often cooled by free convection. Even though a great number of heat transfer studies for this fundamental geometry have been reported, there still is a lack of data for such engineering applications. CONCLUSIONS This work involved an experimental study of the free convection heat transfer from cylinders. The local heat transfer rates for both horizontal and inclined cylinders exhibit similar distributions along the perimeter of the cylinder. That is, the local heat transfer rate is the highest at the bottom and decreases gradually toward the top. For the vertical cylinder, the heat transfer rates are uniform along the perimeter of the cylinder. The average heat transfer rate is the highest for the horizontal cylinder and decreases as the angle of inclination increases. The results of the present work can be correlated, for each angle of inclination studied, by the correlation given in Table 2. Equation (10) predicts all the experiment data to within _+5% in the Rayleigh number range of 1.4 x 104 to 2.9 × 105 for inclination angles of 0-90*. The results of the present work agree well with the work of Brdlik [5], Churchill and Chu [3], Morgan [1], Eckert [2], and Oosthuizen [6]. This research was sponsored in part by the National Science and Engineering Research Council of Canada.

NOMENCLATURE D Gro GrL g h k Kgd

i

PRESENT STUDY AL-ARABI and 8ALMAN IJ~IGAN

2

L m n

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100

~ 104

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6 S

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ii

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,

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|

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6

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10s

Figure 11. Comparison of average Nusselt numbers for the vertical cylinder.

n o

Nu D Nu D Nuav NUL

cylinder diameter, m Grashof number (= gila TD3/ua), dimensionless Grashof number (= g~ATLa/u2), dimensionless gravitational acceleration, m/s 2 convective heat transfer coefficient, W/(m 2 • K) thermal conductivity, W/(m • K) Gladstone-Dale constant, 1.504 × 10 -4 m3/kg, dry air length of cylinder, m constant in Eq. (9) exponent in Eq. (9) index of refraction, dimensionless Nusselt number (= hD/k), dimensionless average Nusselt number, dimensionless average Nusselt number, dimensionless Nussel number (= hL/k), dimensionless

242

Jisheng Li and J. D. Tarasuk

p Pr q r R0 R Ra o T AT z

pressure, Pa Prandtl n u m b e r ( = v / a ) , heat flux, W / m 2

dimensionless

radial coordinate in a cylindrical coordinate system radius of the cylinder, m ideal gas constant, 287.0987 N • m / ( k g • K) Rayleigh n u m b e r ( = G r D Pr), dimensionless temperature, oC temperature difference between the model surface and ambient, °C axial coordinate in a cylindrical coordinate system in the direction o f test b e a m direction o f the cylinder

Greek Symbols /3 e 19 19

p

volume coefficient o f expansion ( = 1 / T ) , °C 1 fringe shift, dimensionless angular position, deg angular coordinate in a cylindrical coordinate system, deg laser wavelength, 6.328 × 1 0 - 7 m density, k g / m 3 angle of inclination of the cylinder, deg

Subscripts D f L s 0 oo

diameter o f a cylinder as a characteristic length evaluated at film t e m p e r a t u r e length of cylinder as a characteristic length surface angular direction or a local position on the surface of the cylinder ambient

2. Eckert, E. R. G., Studies on Heat Transfer in Laminar Free Convection with the Zehnder-Mach Interferometer, USAF Tech. Rep. 5747, 1948. 3. Churchill, S. W., and Chu, H. H. S., Correlating Equations for Laminar and Turbulent Free Convection from a Horizontal Cylinder, Int. J. Heat Mass Transfer, 18, 1049-1053, 1975. 4. Fand, R. M., Morris, E. W., and Lum, M., Natural Convection Heat Transfer from Horizontal Cylinders to Air, Water and Silicone Oils for Rayleigh Numbers Between 3 x 102 and 2 x 10 7, Int. J. Heat Mass Transfer, 20, 1173-1184, 1977. 5. Brdlik, P. M., Heat Transfer of Isothermal Cylinder with Natural Convection, Moscow Wood-Technology Institute, translated from Teplofiz. Vys. Temp., 21(4), 701-706, 1983. 6. Oosthuizen, P. H., Experimental Study of Free Convective Heat Transfer from Inclined Cylinders, J. Heat Transfer, 98, 673-674, 1976. 7, AI-Arabi, M., and Salman, Y. K., Laminar Natural Convection Heat Transfer from an Inclined Cylinder, Int. J. Heat Mass Transfer, 23, 45-51, 1980. 8. Stewart, W. E., Jr., and Buck, S. L., Experimental Free Convection from an Inclined Cylinder, presented at Heat Transfer Division of ASME Winter Annual Meeting, Nov. 16-21, 1980, Chicago, 111. 9. Papple, M. L. C., and Tarasuk, J. D., Developing Natural Convective Flow in Vertical and Inclined Isothermal Ducts, Proc. First World Conf. on Experimental Heat Transfer Fluid Mechanics and Thermodynamics, Sept. 4-9, 1988, Dubrovnik, Yugoslavia, pp. 433 -440. 10. Hauf, W., and Grigull, U., Optical Methods in Heat Transfer, Adv. Heat Transfer, 6, 283-293, 1970. 11. Li, J., An Interferometric Study of Free Convection Heat Transfer from Horizontal, Vertical and Inclined Isothermal Cylinder, M.E.Sc. Thesis, Univ. Western Ontario, London, Ont., 1989. 12. Krause, J. R., An lnterferometric Study of Mixed Convection from a Horizontal Cylinder to a Cross Flow of Air, M.E.Sc. Thesis, Univ. Western Ontario, London, Ont., 1985.

REFERENCES 1. Morgan, V. T., The Overall Convective Heat Transfer from Smooth Circular Cylinders, Adv. Heat Transfer, 11, 199-212, 1975.

Received November 12, 1990; revised December 18, 1991