0
Technical Brief

Heat Transfer From a Wedge to Fluids at Any Prandtl Number Using the Asymptotic Model

[+] Author and Article Information
M. M. Awad

Mechanical Power Engineering Department,
Faculty of Engineering,
Mansoura University,
Mansoura 35516, Egypt
e-mail: m_m_awad@mans.edu.eg

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received May 4, 2013; final manuscript received May 26, 2014; published online June 24, 2014. Assoc. Editor: Jose L. Lage.

J. Heat Transfer 136(9), 094503 (Jun 24, 2014) (6 pages) Paper No: HT-13-1228; doi: 10.1115/1.4027769 History: Received May 04, 2013; Revised May 26, 2014

Heat transfer from a wedge to fluids at any Prandtl number can be predicted using the asymptotic model. In the asymptotic model, the dependent parameter Nux/Rex1/2 has two asymptotes. The first asymptote is Nux/Rex1/2Pr→0 that corresponds to very small value of the independent parameter Pr. The second asymptote is Nux/Rex1/2Pr→∞, that corresponds to very large value of the independent parameter Pr. The proposed model uses a concave downward asymptotic correlation method to develop a robust compact model. The solution has two general cases. The first case is β ≠ −0.198838. The second case is the special case of separated wedge flow (β = −0.198838) where the surface shear stress is zero, but the heat transfer rate is not zero. The reason for this division is Nux/Rex1/2 ∼ Pr1/3 for Pr ⪢ 1 in the first case while Nux/Rex1/2 ∼ Pr1/4 for Pr ⪢ 1 in the second case. In the first case, there are only two common examples of the wedge flow in practice. The first common example is the flow over a flat plate at zero incidence with constant external velocity, known as Blasius flow and corresponds to β = 0. The second common example is the two-dimensional stagnation flow, known as Hiemenez flow and corresponds to β = 1 (wedge half-angle 90 deg). Using the methods discussed by Churchill and Usagi (1972, “General Expression for the Correlation of Rates of Transfer and Other Phenomena,” AIChE J., 18(6), pp. 1121–1128), the fitting parameter in the proposed model for both isothermal wedges and uniform-flux wedges can be determined.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Pohlhausen, E., 1921, “Der Wärmeaustausch zwischen festen Körpern und Flüssigkeiten mit kleiner reibung und kleiner Wärmeleitung (The Exchange of Heat Between Solids and Liquids With Less Friction and Less Heat Conduction),” Z. Angew. Math. Mech., 1(2), pp. 115–121. [CrossRef]
Falkner, V. M., and Skan, S. W., 1931, “Some Approximate Solutions of the Boundary Layer Equations,” Philos. Mag., 12, pp. 865–896.
Fage, A., and Falkner, V. M., 1931,“ On the Relation Between Heat Transfer and Surface Friction for Laminar Flow,“ H. M. Stationary Office, London, Report No. 1408, pp. 172–201.
Eckert, E. R. G., 1942, “Die Berechnung des Wärmeübergangs in der laminaren Grenzschicht umströmter Körper (The Calculation of Heat Transfer in the Laminar Boundary Layer Immersed Body),” VDI-Forschungsh., 416, pp. 1–44.
Lighthill, M. J., 1950, “Contributions to the Theory of Heat Transfer Through a Laminar Boundary Layer,” Proc. R. Soc. A, 202, pp. 359–377. [CrossRef]
Spalding, D. B., 1958, “Heat Transfer From Surfaces of Non-Uniform Temperature,” J. Fluid Mech., 4(1), pp. 22–32. [CrossRef]
Spalding, D. B., and Evans, H. L., 1961, “Mass Transfer Through Laminar Boundary Layers—Similarity Solutions to the b-Equation,” Int. J. Heat Mass Transfer, 2(4), pp. 314–341. [CrossRef]
Evans, H. L., 1961, “Mass Transfer Through Laminar Boundary Layers—3a. Similar Solutions of the b-Equation when B = 0 and σ ≥ 0.5,” Int. J. Heat Mass Transfer, 3(1), pp. 26–41. [CrossRef]
Evans, H. L., 1961, “Mass Transfer Through Laminar Boundary Layers—6. Methods of Evaluating the Wall Gradient (b0'/B) for Similar Solutions; Some New Values for Zero Main-Stream Pressure Gradient,” Int. J. Heat Mass Transfer, 3(4), pp. 321–339. [CrossRef]
Evans, H. L., 1962, “Mass Transfer Through Laminar Boundary Layers—7. Further Similar Solutions to the b-Equation for the Case B = 0,” Int. J. Heat Mass Transfer, 5(1–2), pp. 35–57. [CrossRef]
Morgan, G. W., Pipkin, A. C., and Warner, V. H., 1958, “On Heat Transfer in Laminar Boundary Layer Flows of Liquids Having a Very Small Prandtl Number,” J. Aeronaut. Sci., 25(3), pp. 173–180. [CrossRef]
Adams, E. W., 1963, “Heat Transfer in Laminar Flows of Incompressible Fluids With Pr → 0 and Pr → ∞,” NASA, Washington, DC, Report No. NASA T ND 1527.
Stewartson, K., 1964, The Theory of Laminar Boundary Layer in Compressible Fluids, Oxford University, Oxford, UK.
Narasimha, R., and Vasantha, S., 1966, “Laminar Boundary Layer on a Flat Plate at High Prandtl Number,” Z. Angew. Math. Phys., 17(5), pp. 585–592. [CrossRef]
Goddard, J. D., and Acrivos, A., 1966, “Asymptotic Expansions for Laminar Forced Convection Heat and Mass Transfer. Part 2. Boundary Layer Flows,” J. Fluid Mech., 24(2), pp. 339–366. [CrossRef]
Acrivos, A., and Goddard, J. D., 1965, “Asymptotic Expansions for Laminar Forced Convection Heat and Mass Transfer. Part 1. Low Speed Flows,” J. Fluid Mech., 23(2), pp. 273–291. [CrossRef]
Narasimha, R., and Afzal, N., 1971, “Laminar Boundary Layer on a Flat Plate at Low Prandtl Number,” Int. J. Heat Mass Transfer, 14(2), pp. 279–292. [CrossRef]
Chao, B. T., and Cheema, L. S., 1971, “Forced Convection in Wedge Flow With Non-Isothermal Surfaces,” Int. J. Heat Mass Transfer, 14(9), pp. 1363–1375. [CrossRef]
Chao, B. T., 1972, “An Improved Lighthill's Analysis of Heat Transfer Through Boundary Layers,” Int. J. Heat Mass Transfer, 15(5), pp. 907–919. [CrossRef]
Bejan, A., 1984, Convection Heat Transfer, 1st ed., Wiley, New York.
Chen, Y. M., 1985, “Heat Transfer of a Laminar Flow Passing a Wedge at Small Prandtl Number: A New Approach,” Int. J. Heat Mass Transfer, 28(8), pp. 1517–1523. [CrossRef]
Chen, Y. M., 1986, “A Higher-Order Asymptotic Solution for Heat Transfer of a Laminar Flow Passing a Wedge at Small Prandtl number,” Int. J. Heat Mass Transfer, 29(3), pp. 490–492. [CrossRef]
Schlichting, H., 1979, Boundary-Layer Theory, 7th ed., McGraw-Hill, New York.
Lin, H. T., and Lin, L. K., 1987, “Similarity Solutions for Laminar Forced Convection Heat Transfer From Wedges to Fluids of Any Prandtl Number,” Int. J. Heat Mass Transfer, 30(6), pp. 1111–1118. [CrossRef]
Andersson, H. I., 1988, “On Approximate Formulas for Low Prandtl Number Heat Transfer in Laminar Wedge Flows,” Int. J. Heat Fluid Flow, 9(2), pp. 241–243. [CrossRef]
Cebeci, T., 2002, Convective Heat Transfer, Springer-Verlag, New York.
Cheng, W. T., and Lin, H. T., 2002, “Non-Similarity Solution and Correlation of Transient Heat Transfer in Laminar Boundary Layer Flow over a Wedge,” Int. J. Eng. Sci., 40(5), pp. 531–548. [CrossRef]
Kuo, B.-L., 2005, “Heat Transfer Analysis for the Falkner–Skan Wedge Flow by the Differential Transformation Method,” Int. J. Heat Mass Transfer, 48(23–24), pp. 5036–5046. [CrossRef]
Pantokratoras, A., 2006, “The Falkner-Skan Flow With Constant Wall Temperature and Variable Viscosity,” Int. J. Therm. Sci., 45(4), pp. 378–389. [CrossRef]
Bachiri, M., and Bouabdallah, A., 2012, “Analytical Study of the Convection Heat Transfer From an Isothermal Wedge Surface to Fluids,” ASME J. Heat Transfer, 134(6), p. 064502. [CrossRef]
Awad, M. M., 2008, “Heat Transfer From a Rotating Disk to Fluids for a Wide Range of Prandtl Numbers Using the Asymptotic Model,” ASME J. Heat Transfer, 130(1), p. 014505. [CrossRef]
Churchill, S. W., and Usagi, R., 1972, “A General Expression for the Correlation of Rates of Transfer and Other Phenomena,” AIChE J., 18(6), pp. 1121–1128. [CrossRef]
Churchill, S. W., 1988, Viscous Flows: The Practical Use of Theory, Butterworths, Boston, MA.
Kraus, A. D., and Bar-Cohen, A., 1983, Thermal Analysis and Control of Electronic Equipment, Hemisphere, New York.
Yovanovich, M. M., 2003, “Asymptotes and Asymptotic Analysis for Development of Compact Models for Microelectronic Cooling,” 19th Annual Semiconductor Thermal Measurement and Management Symposium and Exposition (SEMI-THERM), San Jose, CA, March 11–13.
Kays, W. M., and Crawford, M. E., 1993, Convective Heat and Mass Transfer, McGraw-Hill, New York.
Churchill, S. W., and Ozoe, H., 1973, “Correlations for Laminar Forced Convection in Flow Over an Isothermal Flat Plate and in Developing and Fully Developed Flow in an Isothermal Tube,” ASME J. Heat Transfer, 95(3), pp. 416–419. [CrossRef]
Incropera, F. P., and DeWitt, D. P., 1990, Fundamentals of Heat and Mass Transfer, Wiley, New York.
Bejan, A., 1993, Heat Transfer, Wiley, New York.

Figures

Grahic Jump Location
Fig. 1

Different principal configurations of the flow over a wedge surface

Grahic Jump Location
Fig. 2

Comparison of the asymptotic model with different sets of data for isothermal wedges (β = 0)

Grahic Jump Location
Fig. 3

Comparison of the asymptotic model with different sets of data for isothermal wedges (β = 1)

Grahic Jump Location
Fig. 4

Comparison of the asymptotic model with different sets of data for isothermal wedges (β = −0.198838)

Grahic Jump Location
Fig. 5

Comparison of the asymptotic model with different sets of data for isothermal wedges (β = 1)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In