Transient temperature response of spherical bodies

Author:
Bejoy K. Choudhury

Journal:
Quart. Appl. Math. **69** (2011), 205-225

MSC (2000):
Primary 80A20, 44A10

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.

**1.**N. I. Ahiezer and I. M. Glazman,*Teoriya lineinykh operatorov v Gilbertovom 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**, 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**, 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. 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 -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**

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.