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Research Papers: Conduction

Semi-Infinite Solid Solution Adjusted for Radial Systems Using Time-Dependent Participating Volume-to-Surface Ratio and Simplified Solutions for Finite Solids

[+] Author and Article Information
Aleksandar G. Ostrogorsky

Fellow ASME
Illinois Institute of Technology,
Room 246, MANE E1,
Chicago, IL 60616
e-mail: AOstrogo@IIT.edu

Borivoje B. Mikic

Fellow ASME
Massachusetts Institute of Technology,
Room 5-214, Massachusetts Avenue,
Cambridge, MA 02139
e-mail: Mikic@MIT.edu

1Corresponding author.

2For sphere, at Fo = 0.005 and Bi → ∞, error is > 3%.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received December 4, 2017; final manuscript received March 2, 2018; published online May 25, 2018. Assoc. Editor: Alan McGaughey.

J. Heat Transfer 140(10), 101301 (May 25, 2018) (10 pages) Paper No: HT-17-1732; doi: 10.1115/1.4039688 History: Received December 04, 2017; Revised March 02, 2018

Time-dependent participating volume-to-surface ratio, V(t)/A, is used to adjust the semi-infinite (SI) solid solutions to the radial systems. In cylinders and spheres, the present “radial” SI sold model extends the domain of the planar model from δ ≪ R to δ ≈ R (δ is transient penetration depth and R is radius). The corresponding increase in the time span is from 0 < Fo < 0.01 to 0 < Fo < 0.06). The erfc series solution for finite solids (FS), which converges rapidly at small values of time, is simplified, by truncating the first term of the solution. For cylinders and spheres, the resulting half-term approximations are far more precise than the planar SI solid solutions.

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References

Glicksman, L. R. , and Lienhard , and J. H., V , 2016, Modeling and Approximation in Heat Transfer, Cambridge University Press, New York.
Incropera, F. P. , and De Witt, D. P. , 1990, Introduction to Heat Transfer, 2nd ed., Willey, New York.
Rohsenow, W. M. , and Choi, H. Y. , 1961, Heat Mass and Momentum Transfer, Prentice Hall, Englewood Cliffs, NJ.
Arpaci, V. , 1966, Conduction Heat Transfer, Addison-Wesley, Reading, MA.
Yener, Y. , and Kakac, S. , 2008, Heat Conduction, 4th ed., Taylor & Fancis, London.
Carslaw, H. S. , and Jaeger, J. C. , 1959, Conduction of Heat in Solids, 2nd ed., Oxford University Press, Oxford, UK.
Ostrogorsky, A. G. , 2008, “ Transient Heat Conduction in Spheres for Fo < 0.3 and Finite Bi,” Heat Mass Transfer, 44(12), pp. 1557–1562. [CrossRef]
Ostrogorsky, A. G. , 2018, “ Transient Heat Conduction in Cylinders and Spheres for Small Values of Time,” J. Serbian Soc. Comput. (in press).
Ozisik, M. N. , 1993, Heat Conduction, 2nd ed., Wiley, Hoboken, NJ.

Figures

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Fig. 1

Participating volume-to-surface ratio, V(t)/A in (a) cylinders and spheres and (b) solids surrounding cylindrical or spherical cavities; (c) transient temperature of sphere surface for Bi = 10. Exact solution is Eq. (5), full black line. For Fo = 0.03, θcorrect = 0.24. The semi-infinite solid solution requires “extended” time, Fo = 0.05, to give correct temperature.

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Fig. 2

Temperature profiles in a large plate for Fo = 0.075 and Bi = 1, 10, 100, and ∞

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Fig. 3

Radial temperature profiles in long cylinders at (a) Fo = 0.04 and (b) Fo = 0.06. SI model, Eq. (1) and present approximation Eq. (21) for (FS_[8]) yield equal error at the surface. Equation (A4) [6] diverges for Bi > 1.

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Fig. 4

Radial temperature profiles in long cylinders for (a) flux and (b) temperature boundary condition. SI model, Eq. (2) and present approximation for (FS_½), Eq. (22) yield equal error at the surface.

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Fig. 5

Radial temperature profiles in spheres at (a) Fo = 0.01 and (b) Fo = 0.01. Sphere: temperature response θ/θi of the surface (r/R = 1) as a function of Fourier number. Bi = 1, 10, 100. Temperature profiles in spheres for Bi = 1, 10, 100 and (a) Fo = 0.01; (b) Fo = 0.04.

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Fig. 6

Radial temperature profiles in spheres for (a) known flux boundary condition at (b) known surface temperature

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Fig. 7

Temperature profiles (r > R) in solid regions surrounding (a) cylindrical and (b) spherical cavity for known surface temperature boundary condition

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Fig. 8

Maximum error as a function of Fo for cylinders: (a)specified temperature (Bi = ∞) and (b) Bi = 10. Fourier(1), Fourier_(1 + 2) and Fourier_(1 + 2 + 3) are one-, two-, and three-term Fourier series.

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Fig. 9

Maximum error as a function of Fo for spheres: (a) specified temperature (Bi = ∞) and (b) Bi = 10. Fourier(1), Fourier_(1 + 2), and Fourier_(1 + 2 + 3) are one-, two-, and three-term Fourier series, respectively.

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