Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Transient temperature response of spherical bodies

Author: Bejoy K. Choudhury
Journal: Quart. Appl. Math. 69 (2011), 205-225
MSC (2000): Primary 80A20, 44A10
DOI: https://doi.org/10.1090/S0033-569X-2011-01193-7
Published electronically: March 3, 2011
MathSciNet review: 2814525
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Fundamental temperature solutions in closed form of composite spherical bodies are given for mixed and time-dependent boundary conditions. Solid and hollow spherical bodies are included as further examples. The solution requires calculating the roots of certain transcendental equations. A method is developed to find the roots rapidly. As a practical application, the two-layer composite solution is used to determine the available fuel mass of an orbiting spacecraft.

References [Enhancements On Off] (What's this?)

  • 1. N. I. Ahiezer and I. M. Glazman, \cyr Teoriya lineĭnykh operatorov v Gil′bertovom prostranstve. Tom II, “Vishcha Shkola”, Kharkov, 1978 (Russian). Third edition, corrected and augmented. MR 509335
  • 2. J. V. Beck, B. Blackwell, and C. R. St. Clair, Jr., Inverse Heat Conduction: Ill-Posed Problems, Wiley-Interscience, New York, 1985.
  • 3. B. A. Boley and J. H. Weiner, Theory of Thermal Stresses, Dover reprint, New York, 1977.
  • 4. T. J. I'a. Bromwich, Symbolic methods in the theory of conduction of heat, Proc. Cambridge Phil. Soc., 20, 1921, 411-427. Here Bromwich devises a contour and method that now bears his name. Bromwich's and Carslaw's [5] papers were read the same day (2 May 1921) at Society's meeting.
  • 5. H. S. Carslaw, The cooling of a solid sphere with a concentric core of a different material, Proc. Cambridge Phil. Soc., 20, 1921, 399-410. The Tripos problem mentioned in the Introduction is reproduced in this paper.
  • 6. H. S. Carslaw and J. C. Jaeger, A problem in conduction of heat, Proc. Cambridge Philos. Soc. 35 (1939), 394–404. MR 0000456
  • 7. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford, at the Clarendon Press, 1947. MR 0022294
  • 8. Bejoy K. Choudhury, Transient temperature and elastic response of a space-based mirror in the radiation-conduction environment, Quart. Appl. Math. 64 (2006), no. 2, 201–228. MR 2243860, https://doi.org/10.1090/S0033-569X-06-00970-7
  • 9. J. Crank, The mathematics of diffusion, Oxford, at the Clarendon Press, 1956. MR 0082827
  • 10. William Alan Day, Heat conduction within linear thermoelasticity, Springer Tracts in Natural Philosophy, vol. 30, Springer-Verlag, New York, 1985. MR 804043
  • 11. G. M. L. Gladwell and J. R. Barber, Thermoelastic contact problems with radiation boundary conditions, Quart. J. Mech. Appl. Math. 36 (1983), no. 3, 403–417. MR 714309, https://doi.org/10.1093/qjmam/36.3.403
  • 12. A. Haji-Sheikh and J. V. Beck, An efficient method of computing eigenvalues in heat conduction, Numerical Heat Transfer B 38, 2000, 133-156.
  • 13. G. H. Hardy, A Course of Pure Mathematics (8th edition), Cambridge University Press, London, 1941.
  • 14. T. Hata, Transient thermal and residual stresses caused by creep in a sphere, in Theoretical and Applied Mechanics, vol. 22, Proc. $ 22nd$ Japan National Congress, University of Tokyo Press, Tokyo, 1972, 235-244.
  • 15. P. K. Jain, S. Singh, and Rizwan-uddin, Analytical solution to transient asymmetric heat conduction in a multilayer annulus, J. Heat Transfer, 131, 2009, 011304 (7 pages).
  • 16. N. W. McLachlan, Complex variable theory and transform calculus with technical applications, Cambridge, at the University Press, 1953. 2d ed. MR 0059397
  • 17. N. Noda, R. B. Hetnarski, and Y. Tanigawa, Thermal Stresses (2nd edition), Taylor and Francis, London, 2003.
  • 18. M. N. Özişik, Heat Conduction (2nd edition), Wiley-Interscience, New York, 1993.
  • 19. M. N. Özişik and H. R. B. Orlande, Inverse Heat Transfer: Fundamentals and Applications, Taylor and Francis, London, 2000.
  • 20. William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, Numerical recipes, 3rd ed., Cambridge University Press, Cambridge, 2007. The art of scientific computing. MR 2371990
  • 21. S. Singh, P. K. Jain, and Rizwan-uddin, Analytical solution to transient heat conduction in polar coordinates with multiple layers in radial direction, Int. J. Therm. Sciences, 47, 2008, 261-273.
  • 22. J. W. Stevens and R. Luck, Explicit approximations for all eigenvalues of the $ 1$-D transient heat conduction equations, Heat Transfer Engineering, 20, 1999, 35-41.
  • 23. Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 80A20, 44A10

Retrieve articles in all journals with MSC (2000): 80A20, 44A10

Additional Information

Bejoy K. Choudhury
Affiliation: Lockheed Martin Space Systems, Sunnyvale, California
Email: bejoy.choudhury@lmco.com

DOI: https://doi.org/10.1090/S0033-569X-2011-01193-7
Keywords: Heat conduction, diffusion, radiation, Laplace transform
Received by editor(s): May 1, 2009
Published electronically: March 3, 2011
Dedicated: In memory of late Professor I. M. Cohen [1937–2007], whose lifelong dedication to teaching and research has inspired, and continues to inspire
Article copyright: © Copyright 2011 Brown University
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society