of heat transport behavior on the wall region. The conven­ience of not having to introduce another parameter into the formulation is quite attractive.

Numerical Procedure The solution of equation (4) with the appropriate boundary

conditions was done numerically. In order to extend the range of y + to the order of 105, a variable step size was used. This step size was tailored to be a constant over each (approximate) decade in y+ and the same number of steps were used for each decade in y + . The eigenvalues for the resulting nonsymmetric matrices were numerically determined and an orthogonal set was constructed through the use of transposed matrix opera­tions (Carnahan et al. 1969).

With approximately 20 steps per decade, integrations out to y+ of 5x 10s are possible. This corresponds to x+ values of up to 107 (with the outer boundary conditions still being satisfied).

Four Prandtl (or Schmidt) numbers were chosen for the numerical calculations: 0.5, 0.7, 2.0, 7.0. A turbulent Prandtl number of 0.9 was used for all of the numerical calculations. A roughness Reynolds number Rek was chosen to be 70 for cases (c) and (d).

Results Values of St* for the four cases were calculated as functions

of x+ and Pr. For cases (a) and (c), values of St* were also evaluated. The rough surface cases were examined for Re* =70 only.

Adequate curve fits to these results are

Case (a).- St* = 0.155 Pr-°-79(x+r°-111Pr~°'36 (13)

Case (b) : St* = 0.124 Pr-°-77(.x+)-°-081Pr~°'42 (14)

Case (c) : St; = 0.227 Pr-°-40(x+ )-o.i44Pr~0093 ( 1 5 )

lx+ \ -0.122exp(-0.193 Pr)

These suggested curve fits are within ± 7 percent for the limits 0.5<Pr<7.0 and 10 4<x+<10 7 . The average Stanton numbers for cases (a) and (c) may be estimated from

Case ( a ) : St*/Stjf=1.13 Pr0040

Case (c) : St*/St*=1.17 pr-°-013

within ± 5 percent for 0.5 <Pr< 7.0 and 105<x+ <107, where the value of St* is found using equation (13) or equation (15), respectively.

The thermal boundary layer growth was also examined. An initial slow development was observed for x + ^10 3 , after which the boundary layer grew as

A+~0.3(x+)0-84

for case (a) and Pr = 0.7. The other cases were similar. This development length increases with increasing Prandtl number, while the thermal boundary layer thickness decreases slightly with increasing Pr. An exponent of 0.8 has been reported for most experiments on the growth of this "internal boundary layer" (Brutsaert, 1984).

References Bailey, R. T., Mitchell, J. W., and Beckman, W. A., 1975, "Convective Heat

Transfer From a Desert Surface," ASME JOURNAL OF HEAT TRANSFER, Vol. 97, pp. 104-109.

Brutsaert, W., 1984, Evaporation Into the Atmosphere, D. Reidel Publishing Co., Boston, corrected ed.

Carnahan, B., Luther, H. A., and Wilkes, J. O., 1969, Applied Numerical Methods, Wiley, New York.

Dipprey, D. F., and Sabersky, R. H., 1963, "Heat and Momentum Transfer in Smooth and Rough Tubes at Various Prandtl Numbers," Int. J. Heat Mass Transfer, Vol. 6, pp. 329-353.

Garg, H. P., 1982, Treatise on Solar Energy, Vol. 1 Fundamentals of Solar Energy, Wiley, New York.

Jaluria, Y., and Cha, C. K., 1985, "Heat Rejection to the Surface Layer of a Solar Pond," ASME JOURNAL OF HEAT TRANSFER, Vol. 107, pp. 99-106.

Kays, W. M., and M. E. Crawford, 1980, Convective Heat and Mass Transfer, 2nd ed., McGraw-Hill, New York.

Kind, R. J., Gladstone, D. H., and Moizer, A. D., 1983, "Convective Heat Losses From Flat-Plate Solar Collectors in Turbulent Winds," ASME J. Solar Energy Engineering, Vol. 105, pp. 80-85.

Nusselt, W., and Jurges, W., 1922, "Die Kuhlung Einer Ebenen Wand Durch Einen Luftstrom," Gesundheits Ingenieur, Vol. 52, No. 45, pp. 641-642.

Pimenta, M. M., Moffat, R. J„ and Kays, W. M., 1975, Report No. HMT-21, Dept. of Mechanical Engineering, Stanford University, Stanford, CA.

Schlichting, H., 1968, Boundary-Layer Theory, 6th ed., McGraw-Hill, New York, pp. 578-580.

Sutton, O. G., 1934, "Wind Structure and Evaporation in a Turbulent At­mosphere," Proc. Roy. Soc. London, A, Vol. 146, pp. 701-722.

Van Driest, E. R., 1956, "On Turbulent Flow Near a Wall," J. Aeronautical Sciences, Vol. 23, pp. 1007-1011.

Williams, G. P., 1976, "Design Heat Requirements for Embedded Snow-Melting Systems in Cold Climates," Transportation Research Record, Vol. 576, Transportation Research Board, National Research Council, Washington, DC, pp. 20-32.

Heat Transfer Measurements From a Surface With Uniform Heat Flux and an Impinging Jet

J. W. Baughn1 and S. Shimizu2

Introduction There are numerous studies, mostly experimental, on the

flow characteristics and heat transfer associated with jet im­pingement on surfaces. These studies have considered both single jets and multiple jets (i.e., arrays) and many different aspects of impinging jets including the effects of crossflow, jet orientation (oblique jets), jet temperature, rotating surfaces, and different surface shapes. The present study is concerned with the case of a single circular turbulent air jet at the am­bient air temperature impinging on a flat stationary surface. For even this simplest case, there is a very large body of literature and a full review is not possible in this note. Gaunt-ner et al. (1970) present an early survey of the literature on flow characteristics of such a jet, Donaldson et al. (1971) review heat transfer characteristics, and Martin (1977) reviews heat and mass transfer. Some of the earliest measurements of heat transfer were done by Gardon and Cobonpue (1962) and by Gardon and Akfirat (1965, 1966). More recent measurements have been made by Goldstein and Behbahani (1982), and Hrycak (1983). The effects of a jet temperature different from the ambient temperature have recently been studied by Hollworth and Gero (1985) and Striegl and Diller (1984).

There are also some recent attempts to do numerical studies of the heat transfer for an impinging jet on a surface. Amano and Brandt (1984) did a numerical study of the flow

Professor, Department of Mechanical Engineering, University of Califor­nia, Davis, CA 95616; Mem. ASME.

Research Engineer, Canon Corporation, Yokohama, Japan. Contributed by the Heat Transfer Division for publication in the JOURNAL OF

HEAT TRANSFER. Manuscript received by the Heat Transfer Division June 9, 1988. Keywords: Augmentation and Enhancement, Measurement Techniques, Turbulence.

1096/Vol. 111, NOVEMBER 1989 Transactions of the ASME Copyright © 1989 by ASME

Downloaded 15 Jun 2012 to 132.236.27.111. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Fig. 1 Diagram of impingement test section

characteristics of a turbulent jet and Polat et al. (1985) have done a numerical study of heat transfer.

One of the difficulties in comparing recent numerical work with previous experimental results is the lack of data on the jet characteristics and in some cases the mixed thermal boundary conditions at the surface (Launder, 1987). The present work provides some new experimental results that attempt to over­come this difficulty by using a fully developed jet and a well-controlled thermal boundary condition (i.e., a uniform heat flux). No other similar measurements were found in the literature.

Experimental Technique and Apparatus The experimental technique used is described by Simonich

and Moffat (1982) and Baughn et al. (1986). In this technique, a uniform heat flux is established by electrically heating a very thin gold coating on a plastic substrate. The surface temperature distribution is measured using liquid crystals. An isotherm on the surface represents a contour of constant heat transfer coefficient and is a line of a particular color (a light green was used for most of the data here). The liquid crystal used here had a narrow range of approximately 1°C over which the full color spectrum occurs. Temperature resolution of the green color was better than 0.1 °C. The position of the green line is shifted by changing the electrical heating of the gold coating and thus the surface heat flux. This allows a com­plete mapping of the heat transfer coefficient over the entire surface. Since the temperature differences are small (on the order of 10°C), the resulting heat transfer coefficients are in­dependent of the level of heat flux used.

The apparatus consisted essentially of a blower, a long pipe for development of the flow (2.5 cm i.d. and 72 diameters long), and a test section. The upstream development length of 72 diameters provides nearly fully developed flow at the jet ex­it. The turbulence level at the center of the jet at the exit was measured with a hot wire and was 4.1 percent at a Reynolds number of 21,000. This is consistent with the measurements of Hisida and Nagamo (1979). The velocity profile for a fully developed flow is available in Schlicting (1968) and the tur­bulence distribution is discussed by Hinze (1959). The test sec­tion, shown in Fig. 1, had a thin (0.64 cm thick) Plexiglas plate

Journal of Heat Transfer

O 2 4 6 8 i t r/D

Fig. 2 Nusselt number distribution along the surface (Re = 23,750)

Fig. 3 Effect of Jet distance on the heat transfer at the stagnation point (Re = 23,750)

on the front of which the plastic sheet containing the gold coating was glued and on the back of which there was Styrofoam for insulation. The liquid crystal was air brushed on the surface of the gold coating. A Mylar board surrounded the Plexiglas surface to ensure a flat smooth surface.

The data reduction was straightforward and consisted of computing the surface heat flux from the gold coating voltage, current (determined from a shunt resistor), and the area. A radiation correction, using a measured emissivity of 0.5, was made to determine the convective component of the surface heat flux. The radiation correction was usually less than 5 per­cent. Conduction losses with this technique have previously been shown to be less than 1 percent due to the low thermal conductivity of the plastic substrate and are neglected. Using the ambient temperature and liquid crystal temperature, the heat transfer coefficient and corresponding Nusselt number based on the jet diameter were then calculated.

A standard uncertainty analysis was performed using the method of Kline and McClintock. The uncertainty in the Nusselt number (based on 20:1 odds) was estimated at 2.4 per­cent. The uncertainty in the Reynolds number was estimated at 2.3 percent. The uncertainty in position r/D and Z/D was less than 1 percent. Details of the uncertainty analysis are given in Hechanova (1988).

Results The distribution of the Nusselt number along the surface

(for Re = 23,750) is shown in Fig. 2. The symmetry around the jet was quite good as evidenced by the fact that the color band was very close to a perfect circle. One of the most interesting

NOVEMBER 1989, Vol. 111 /1097

Downloaded 15 Jun 2012 to 132.236.27.111. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

distributions occurs when the jet is quite close to the surface (Z/D = 2). In this case, the maximum heat transfer is at the stagnation point; the heat transfer then has a minimum at r/D of approximately 1.3, and another maximum at approximate­ly 1.8. For certain runs this results in three concentric circles of color on the test section. The effect of jet distance from the surface on the stagnation point heat transfer is shown in Fig. 3. As found by other investigators, the maximum stagnation point heat transfer occurs at a Z/D of approximately 6. It is hoped that these results will be generally useful to those at­tempting to model turbulent jets impinging on a surface.

Acknowledgments The authors gratefully acknowledge the support of the

University of California UERG program. Professor Brian Launder made the original suggestion that these data were needed and made valuable suggestions during the course of work. The assistance of Anthony Hechanova, who has con­tinued this research, is also appreciated.

References Abuaf, N., Urbaetis, S. P., and Palmer, O. F., 1985, "Convection Ther­

mography," General Electric Corporation Research and Development Report No. 85CRD168.

Amano, R. S., and Brandt, H., 1984, "Numerical Study of Turbulent Ax-isymmetric Jets Impinging on a Flat Plate and Flowing Into an Axisymmetric Cavity," ASME Journal of Fluids Engineering, Vol. 106, pp. 410-417.

Baughn, J. W., Hoffman, M, A., andMakel, D. B., 1986, "Improvements in a New Technique for Measuring and Mapping Heat Transfer Coefficients," Review of Scientific Instruments, Vol. 57, pp. 650-654.

Donaldson, C. duP., Snedeker, R. S., and Margolis, D. P., 1971, "A Study of Free Jet Impingement. Part 2. Free Jet Turbulent Structure and Impingement Heat Transfer," Journal of Fluid Mechanics, Vol. 45, Part 3, pp. 477-512.

Gardon, R., and Cobonpue, J., 1962, "Heat Transfer Between a Flat Plate and Jets of Air Impinging on It," International Developments in Heat Transfer, ASME, pp. 454-460.

Gardon, R., and Akfirat, C. J., 1965, "The Role of Turbulence in Determin­ing the Heat-Transfer Characteristics of Impinging Jets," International Journal of Heat and Mass Transfer, Vol. 8, pp. 1261-1272.

Gardon, R., and Akfirat, C. J., 1966, "Heat Transfer Characteristics of Im­pinging Two-Dimensional Air Jets," ASME JOURNAL OF HEAT TRANSFER, Vol. 88, pp. 101-108.

Gauntner, J. W., Livingwood, J. N. B., and Hrycak, P., 1970, "Survey of Literature on Flow Characteristics of a Single Turbulent Jet Impinging on a Flat Plate," NASA TN D-5652.

Goldstein, R. J., and Behbahani, A. I., 1982, "Impingement of a Circular Jet With and Without Cross Flow," International Journal of Heat and Mass Transfer, Vol. 25, pp. 1377-1382.

Hechanova, T. E., 1988, "An Experimental Study of Entrainment Effects on Heat Transfer From a Surface With a Fully Developed Impinging Jet," M. S. Thesis, University of California, Davis.

Hinze, J. O., 1975, Turbulence, 2nd ed., McGraw-Hill, New York. Hippensteele, S. A., Russell, L. M., and Stepka, P. S., 1983, "Evaluation of

a Method for Heat Transfer Measurements and Thermal Visualization Using a Composite of a Heater Element and Liquid Crystals,'' ASME JOURNAL OF HEAT TRANSFER, Vol. 105, pp. 184-189.

Hishida, M., and Nagamo, Y., "Structure of Turbulent Velocity and Temperature Fluctuations in Fully Developed Pipe Flow," ASME JOURNAL OF HEAT TRANSFER, Vol. 101, pp. 15-22.

Hollworth, B. R., and Gero, L. R., 1985, "Entrainment Effects on Impinge­ment Heat Transfer: Part II—Local Heat Transfer Measurements," ASME JOURNAL OF HEAT TRANSFER, Vol. 107, pp. 910-915.

Hrycak, P., 1983, "Heat Transfer for Round Impinging Jets to a Flat Plate," International Journal of Heat and Mass Transfer, Vol. 26, pp. 1857-1865.

Launder, B. E., 1987, personal communication. Martin, H., 1977, "Heat and Mass Transfer Between Impinging Gas Jets and

Solid Surfaces," Advances in Heat Transfer, Vol. 13, pp. 1-60. Polat, S., Mujumdar, A. S., and Douglas, W. J. M., 1985, "Heat Transfer

Distribution Under a Turbulent Impinging Jet—A Numerical Study," Drying Technology, Vol. 3, pp. 15-37.

Schlicting, H., 1968, Boundary Layer Theory, 6th ed., McGraw-Hill, New York.

Simonich, J. C , and Moffat, R. J., 1982, "A New Technique for Mapping Heat-Transfer Coefficient Contours," Review of Scientific Instruments, Vol. 53, pp. 678-683.

Striegl, S. A., and Diller, T. E., 1984, "The Effect of Entrainment Temperature on Jet Impingement Heat Transfer," ASME JOURNAL OF HEAT TRANSFER, Vol. 106, pp. 27-33.

Convective Heat Transfer Measurement Involving Flow Past Stationary Circular Disks

G. L. Wedekind1

Introduction Considerable empirical data exist in the literature for forced

convection heat transfer involving external flow over a variety of geometries, and for various ranges of Reynolds number. In fact, many current heat transfer textbooks (Kreith and Bohn, 1986; Incropera and Dewitt, 1981; Holman, 1986) present em­pirical correlations for flow over a flat plate, a sphere, a spheroid, and tubes in cross flow, and for tubes of cylindrical, square, hexagonal, and other assorted cross sections. A geometry that appears to be missing from this list is that of a thin circular disk. Although there has been considerable theoretical and experimental research devoted to natural con­vection for stationary and rotating circular disks and disk systems, such as that reported by Zakerullah and Ackroyd (1979), Mochizuki and Yeng (1986), and Owens (1984), this author is not aware of any published empirical data for forced convection heat transfer involving flow past a simple sta­tionary circular disk, whose axis is perpendicular to the flow. Such is the purpose of this paper.

The disk geometry and its orientation to the external flow is schematically represented in Fig. 1. The disk has a diameter d, thickness t, is at a uniform temperature T, and is oriented as shown in a fluid having a uniform approach velocity ty and a free-stream temperature Tj. The symbol Q represents the rate of convective heat transfer from the disk to the moving fluid.

Experimental Apparatus and Measurement Techniques The circular disks that were used as heat transfer models for

the experimental data presented in this paper were commer­cially available disk-type thermistors.2 Five different models were tested. Thermistors were chosen as the heat transfer models because they provided a unique combination for in­directly measuring the surface temperature T, and the convec­tive heat transfer rate Q (Wedekind, 1977). The thermistor was self-heated by means of Joule heating. Losses through the thermistor lead wires (0.22 mm dia) were mini­mized (less than 3 percent) by using an alloyed steel (such as 20 percent nickel steel), which had a poor enough thermal con­ductivity to minimize any "fin effect," yet a sufficient elec­trical conductivity that a negligible amount of Joule heating would exist in the lead wires themselves.

Therefore, by setting up an electrical circuit such as that shown in Fig. 2, the thermistor current and resistance could be accurately and simultaneously measured during self-heating, thus making it possible indirectly to measure not only the con­vective heat transfer rate, but also the average temperature of the thermistor as well; the latter by having precalibrated the resistance/temperature characteristics of each thermistor heat transfer model. Thermistors have a high resistance coefficient; therefore, the heat transfer surface temperature T could be in-

Professor of Engineering, Oakland University, Rochester, MI 48309; Mem. ASME.

Thermistors are semiconductors of ceramic material made by sintering mix­tures of metalic oxides such as manganese, nickel, cobalt, copper, iron, and uranium. Disks are made by pressing thermistor material under high pressure in a round die to produce flat coinlike pieces. These pieces are sintered and then coated with silver on the two flat surfaces. Reference: Thermistor Manual; Fen-wal Electronics, Framingham, MA 01701.

Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received by the Heat Transfer Division June 1, 1987. Keywords: Forced Convection, Measurement Techniques.

1098/Vol. 111, NOVEMBER 1989 Transactions of the ASME

Downloaded 15 Jun 2012 to 132.236.27.111. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm