Unsteady-State Heating and Cooling of Solid Objects

  • Octave Levenspiel
Part of the The Plenum Chemical Engineering Series book series (PCES)


If a hot object is plunged into cold water it cools, but not instantaneously. Two factors govern the cooling rate of the object:
  • the film resistance at the surface of the object, characterized by the h value for that situation; and

  • The rate of heat flow out of the interior of the object. The governing differential equation for this conduction process is
    $$\frac{\partial T_{s}}{\partial t} = \alpha \left ( \frac{\partial^{2}T_{s}}{\partial x^{2}} + \frac{\partial^{2}T_{s}}{\partial y^{2}} + \frac{\partial^{2}T_{s}}{\partial z^{2}}\right )$$
    $$\alpha = \frac{k_{s}}{\rho_{s}C_{s}},\;\;\;\;\textup{thermal\;diffusivity}\;[\textup{m}^{2}/\textup{s}]$$
    $$T_{s} = \textup{temperature\;at\;any\;point\;in\;the\;object}\;\;[\textup{K}]$$


Surface Resistance Biot Number Solid Object Fourier Number Roasted Peanut 
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References and Notes

  1. L. M. K. Boelter V. H. Cherry, H. A. Johnson, and R. C. Martinelli, Heat Transfer Notes, McGraw-Hill, New York (1956). Gives detailed derivations of the many equations for unsteady state conduction used in this chapter.Google Scholar
  2. M. Colakyan, R. Turton, and O. Levenspiel, Unsteady-state transfer to various shaped objects, Heat Transfer Engineering 5, 82 (1984).CrossRefGoogle Scholar
  3. H. Gröber, S. Erk, and U. Gringull, Fundamentals of Heat Transfer (translated from the German by J. R. Moszynski), McGraw-Hill, New York (1961). Also a good source book for many of the underlying equations of this chapter.Google Scholar
  4. H. P. Gurney and J. Lurie, Charts for estimating temperature distributions in heating and cooling solid shapes, Ind. Eng. Chem. 15, 1170 (1923).CrossRefGoogle Scholar
  5. J. Mator, M. S. Project, Chemical Engineering Department, Oregon State University, Corvallis (1982).Google Scholar
  6. A. Schack, Industrial Heat Transfer (translated from the 6th German ed., by I. Gutman), Wiley, New York (1965).Google Scholar
  7. J. Sucec, Heat Transfer, Simon and Schuster, New York (1975).Google Scholar
  8. J. R. Welty, Engineering Heat Transfer, p. 135, Wiley, New York (1974).Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Octave Levenspiel
    • 1
  1. 1.Oregon State UniversityCorvallisUSA

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