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Research Papers: Conduction

Generalized Solution for Two-Dimensional Transient Heat Conduction Problems With Partial Heating Near a Corner

[+] Author and Article Information
Robert L. McMasters

Department of Mechanical Engineering,
Virginia Military Institute,
Lexington, VA 24450
e-mail: mcmastersrl@vmi.edu

Filippo de Monte

Department of Industrial and Information
Engineering and Economics,
University of L'Aquila,
Via G. Gronchi No. 18,
L'Aquila 67100, Italy

James V. Beck

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received July 31, 2018; final manuscript received April 9, 2019; published online May 14, 2019. Assoc. Editor: Antonio Barletta.

J. Heat Transfer 141(7), 071301 (May 14, 2019) (8 pages) Paper No: HT-18-1491; doi: 10.1115/1.4043568 History: Received July 31, 2018; Revised April 09, 2019

A generalized solution for a two-dimensional (2D) transient heat conduction problem with a partial-heating boundary condition in rectangular coordinates is developed. The solution accommodates three kinds of boundary conditions: prescribed temperature, prescribed heat flux and convective. Also, the possibility of combining prescribed heat flux and convective heating/cooling on the same boundary is addressed. The means of dealing with these conditions involves adjusting the convection coefficient. Large convective coefficients such as 1010 effectively produce a prescribed-temperature boundary condition and small ones such as 10−10 produce an insulated boundary condition. This paper also presents three different methods to develop the computationally difficult steady-state component of the solution, as separation of variables (SOV) can be less efficient at the heated surface and another method (non-SOV) is more efficient there. Then, the use of the complementary transient part of the solution at early times is presented as a unique way to compute the steady-state solution. The solution method builds upon previous work done in generating analytical solutions in 2D problems with partial heating. But the generalized solution proposed here contains the possibility of hundreds or even thousands of individual solutions. An indexed numbering system is used in order to highlight these individual solutions. Heating along a variable length on the nonhomogeneous boundary is featured as part of the geometry and examples of the solution output are included in the results.

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References

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Cole, K. , Beck, J. , Haji-Sheikh, A. , and Litkouhi, B. , 2011, Heat Conduction Using Green's Functions, Taylor and Francis, Boca Raton, FL.
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Yen, D. H. Y. , Beck, J. V. , McMasters, R. , and Amos, D. E. , 2002, “ Solution of an Initial Boundary Value Problem for Heat Conduction in a Parallelepiped by Time Partitioning,” Int. J. Heat Mass Transfer, 45(21), pp. 4267–4279. [CrossRef]
de Monte, F. , Beck, J. V. , and Amos, D. E. , 2012, “ Solving Two-Dimensional Cartesian Unsteady Heat Conduction Problems for Small Values of the Time,” Int. J. Therm. Sci., 60, pp. 106–113. [CrossRef]
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McMasters, R. , de Monte, F. , Beck, J. , and Amos, D. , 2018, “ Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners,” ASME J. Heat Transfer, 140(2), p. 021301. [CrossRef]
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Figures

Grahic Jump Location
Fig. 1

Schematic diagram of rectangular body with boundary conditions

Grahic Jump Location
Fig. 2

Temperature solution as a function of ỹ at various values of x̃ with W̃=4 and W̃1=0.25

Grahic Jump Location
Fig. 3

Temperature solution as a function of ỹ at various values of x̃ with W̃=1 and W̃1=1

Grahic Jump Location
Fig. 4

Temperature solution as a function of ỹ at various values of x̃ with W̃=2 and W̃1=0.5

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