Simple formulas for the prediction of three important thermal quantities, the center temperature, the mean temperature, and the total heat transfer in regular configurations (large plane wall, long cylinder, and sphere) cooled/heated with prescribed uniform surface temperature during “small time,” are addressed in the present paper. Two immediate engineering applications deal with quenching of metals and sterilization of canned food. The simple formulas emanate from the truncated one term series of the supplementary infinite series at small time. The small time subregion has been traditionally characterized in the heat conduction literature by the dimensionless time or Fourier number τ< 0.24 in the large plane wall, τ< 0.21 in the long cylinder, and τ< 0.19 in the sphere. Excellent agreement between the obtained simple formulas and the traditional solutions (namely the exact, analytical infinite series for “all time”) are attained for the center temperature, mean temperature, and total heat transfer in the large plane wall, long cylinder, and sphere.

References

References
1.
Arpaci
,
V.
,
1966
,
Conduction Heat Transfer
,
Addison–Wesley
,
Reading, MA
.
2.
Luikov
,
A. V.
,
1968
,
Analytical Heat Diffusion Theory
,
Academic Press
,
London
.
3.
Grigull
,
U.
, and
Sandner
,
H.
,
1984
,
Heat Conduction
,
Hemisphere
,
New York
.
4.
Özişik
,
M. N.
,
1993
,
Heat Conduction
,
2nd ed.
,
Wiley
,
Hoboken, NJ
.
5.
Poulikakos
,
D.
,
1993
,
Conduction Heat Transfer
,
Prentice Hall
,
Englewood Cliffs, NJ
.
6.
Totten
,
G. E.
,
Bates
,
C. E.
, and
Clinton
,
N. A.
,
1993
,
Handbook of Quenchants and Quenching Technology
,
ASM International
,
Russell Township, OH
.
7.
Koutchma
,
T.
,
2011
, “
Pasteurization and Sterilization
,”
Handbook of Food Safety Engineering
,
D.-W.
Sun
, ed.,
Wiley–Blackwell
,
Oxford, UK
.
8.
Grigull
,
U.
,
Bach
,
J.
, and
Sandner
,
H.
,
1966
, “
Näherungslösungen Der Nichtstanionaren Wärmeleitung
,”
Forsch. Ing. Wes.
,
32
, pp.
11
18
.
9.
Lavine
,
A. S.
, and
Bergman
,
T. L.
,
2008
, “
Small and Large Time Solutions for Surface Temperature, Surface Heat Flux, and Energy Input in Transient, One-Dimensional Heat Conduction
,”
ASME J. Heat Transfer
,
130
(
10
), p. 101302.
10.
Milkhailov
,
M. D.
, and
Vulchanov
,
N. L.
,
1983
, “
Computational Procedure for Sturm-Liouville Problems
,”
J. Comput. Phys.
,
50
(
3
), pp.
323
336
.
11.
Haji–Sheikh
,
A.
, and
Beck
,
J. V.
,
2000
, “
An Efficient Method of Computing Eigenvalues in Heat Conduction
,”
Numer. Heat Transfer, Part B: Fundam.
,
38
(
2
), pp.
133
156
.
12.
Al–Gwaiz
,
M.
,
2008
,
Sturm–Liouville Theory and Its Applications
,
Springer–Verlag
,
London
.
13.
Mills
,
A. F.
, and
Coimbra
,
C. F. M.
,
2015
,
Heat Transfer
,
3rd ed.
,
Temporal Publishing LLC
,
San Diego, CA
.
14.
Mills
,
S.
,
1985
, “
The Independent Derivation of Leonhard Euler and Colin MacLaurin of the Euler–MacLaurin Summation Formula
,”
Arch. Hist. Exact Sci.
,
33
, pp.
1
13
.
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