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Technical Brief

Cooling of a Hot Torus

[+] Author and Article Information
Rajai S. Alassar

Department of Mathematics and Statistics,
King Fahd University of Petroleum and Minerals,
Dhahran 31261, Saudi Arabia
e-mail: alassar@kfupm.edu.sa

Mohammed A. Abushoshah

Department of Mathematics and Statistics,
King Fahd University of Petroleum and Minerals,
Dhahran 31261, Saudi Arabia

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 22, 2015; final manuscript received November 26, 2015; published online December 29, 2015. Assoc. Editor: Wilson K. S. Chiu.

J. Heat Transfer 138(4), 044501 (Dec 29, 2015) (5 pages) Paper No: HT-15-1062; doi: 10.1115/1.4032149 History: Received January 22, 2015; Revised November 26, 2015

The problem of a hot torus left to cool in a medium of known temperature is studied. We write the governing equation in toroidal coordinates and expand the temperature in terms of a series in the angular direction. The resulting modes in the radial direction are numerically obtained. We consider both isothermal and convective boundary conditions and study the effect of Biot number and aspect ratio on the heat transfer rate.

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References

Wiesche, S. A. , 2001, “ Transient Heat Conduction in a Torus: Theory and Application,” HMT, 38(1–2), pp. 85–92.
Ozisik, M. N. , 1989, Boundary Value Problems of Heat Conduction, Dover, Dutchess County, New York.
Carslaw, H. S. , and Jaeger, J. C. , 1986, Conduction of Heat in Solids, 2nd ed., Clarendon Press, Oxford.
Kakac, S. , and Yener, Y. , 1985, Heat Conduction, Hemisphere Publishing, Washington, DC.
Andrews, M. , 2006, “ Alternative Separation of Laplace's Equation in Toroidal Coordinates and Its Application to Electrostatics,” J. Electrost., 64(10), pp. 664–672. [CrossRef]
Alassar, R. S. , 1999, “ Heat Conduction From Spheroids,” ASME J. Heat Transfer, 121(2), pp. 497–499. [CrossRef]
Alassar, R. S. , and Abushoshah, M. A. , 2013, “ Heat Conduction From Donuts,” Int. J. Mater. Mech. Manuf., 1(2), pp. 126–130.
Moon, P. , and Spencer, D. E. , 1961, Field Theory for Engineers, Van Nostrand, Princeton, NJ.
Morse, P. M. , and Feshbach, H. , 1953, Methods of Theoretical Physics, Part I, McGraw-Hill, New York.
Arfken, G. , 1985, Mathematical Methods for Physicists, 3rd ed., Academic Press, Orlando, FL.

Figures

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Fig. 1

Problem configuration

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Fig. 2

Variation of Nu¯ on the surface for different aspect ratios

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Fig. 3

Variation of φ at the center point for different aspect ratios

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Fig. 4

Variation of Nu¯ on the surface for tori with equal volume

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Fig. 5

Variation of φ at the center point for tori with equal volume

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Fig. 6

Time development (τ=0.0001, 0.01, 0.2, 0.5) of isotherms for R/r=1.5 and Bi=10

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Fig. 7

Time development of φ on the surface for the case R/r=1.5 and Bi=5

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Fig. 8

Variation of φ on the surface for the case R/r=1.5 and τ=10−6

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Fig. 9

Time variation of φ for the case R/r=1.5 in (a) the center and (b) the focal point

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