Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T12:58:07.836Z Has data issue: false hasContentIssue false
  • English
  • Français

On a heat flow problem in a hollow circular cylinder

Published online by Cambridge University Press:  24 October 2008

Nuretti̇n Y. Ölçer
Nuretti̇n Y. Ölçer
Affiliation:
Department of Mechanical and Aerospace Engineering, Illinois Institute of Technology, Chicago, Illinois, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

Recently, through a repeated application of one-dimensional finite integral transforms, Cinelli(1) gave a solution for the temperature distribution in a hollow circular cylinder of finite length. Since no new ideas or techniques are introduced, the extension claimed in (1) with regard to the finite Hankel transform technique employed in the transformation of the radial space variable in the hollow cylinder problem is trivial, in view of well-known works by Sneddon(2) and Tranter (3), to mention a few. The list of the finite Hankel transforms given in (1) for a variety of boundary conditions at r = a and r = b is the result of routine, algebraic manipulations well known from the general theory of eigenvalue problems specialized for the hollow cylinder. In this list a set of seemingly different series expansions is given for the inverse Hankel transform for each combination of boundary conditions at the two radial surfaces. In each case, the two expressions for inversion can readily be shown to be identical to each other when use is made of the frequency equation. One of the inversion forms is therefore unnecessary once the other is given. Furthermore, the general solution as given by equation (54) of Cinelli(1)does not satisfy his boundary conditions (27), (28), (29) and (30), unless these latter are homogeneous.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 1 , January 1968 , pp. 193 - 202
Copyright
Copyright © Cambridge Philosophical Society 1968

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Cinelli, G.An extension of the finite Hankel transform and applications. Int. J. Engng Sci. 3 (1965), 539–559.Google Scholar
(2)Sneddon, I. N.Handbuch der Physik II/II (Springer-Verlag; Berlin-Göttingen-Heidelberg, 1955), 314–319.Google Scholar
(3)Tranter, C. J.Integral transforms in mathematical physics (Methuen; London, 1956), 88–95.Google Scholar
(4)Ölçer, N. Y.On the theory of conductive heat transfer in finite regions. Int. J. Heat Mass Transfer 7 (1964), 307–314.CrossRefGoogle Scholar
(5)Ölçer, N. Y.On the theory of conductive heat transfer in finite regions with boundary conditions of the second kind. Int. J. Heat Mass Transfer 8 (1965), 529–556.CrossRefGoogle Scholar
(6)Ölçer, N. Y.On a class of boundary-initial value problems. Öst. Ing.-Arch. 18 (1964), 104–113.Google Scholar
(7)Elistratova, M. V.A finite integral Hankel transform (in Russian). Inzh.-Fiz. Zh. 3, n. 6 (1960), 120–125.Google Scholar
(8)Plyat, S. N.The solution of problems of nonstationary heat conduction of hollow cylinders by the Grinberg method (method of finite integral transformations) (in Russian). Inzh.-Fiz. Zh. 5, n. 6 (1962), 81–87.Google Scholar
(9)Plyat, S. N.The temperature distribution in abrasive workpieces in the process of thermal machining (in Russian). Inzh.-Fiz. Zh. 3, n. 7 (1960), 80–87.Google Scholar
(10)Nikitin, A. T.Temperature distribution in an unbounded hollow cylinder having heat exchange of the boundaries with variable temperature media (in Russian). Inzh.-Fiz. Zh. 3, n. 8 (1960), 74–77.Google Scholar
(11)Shimko, N. G.A finite integral Hankel transform for a hollow cylinder (in Russian). Inzh.-Fiz. Zh. 3, n. 10 (1960), 39–46.Google Scholar
(12)Trofimov, E. P.The nonstationary temperature field of an unlimited hollow cylinder (in Russian). Inzh.-Fiz. Zh. 3, n. 10 (1960), 47–53.Google Scholar
(13)Trofimov, A. S.Heat conduction of a hollow cylinder and plate with harmonic temperature oscillations (in Russian). Inzh.-Fiz. Zh. 4, n. 7 (1961), 79–83.Google Scholar
(14)Latyshev, L. A. and Kozhevnikov, N. I.Calculation of temperature fields in the wall of a finite hollow cylinder under non-uniform boundary conditions (in Russian). Inzh.-Fiz. Zh. 5, n. 9 (1962), 44–52.Google Scholar
(15)Ölçer, N. Y. and Miller, D. Radial heat conduction in hollow circular cylinders with time-dependent boundary conditions and heat sources (Argonne National Lab., Illinois, Oct. 1962, 21 pp., TID-17831, UAC-7018). Abstract No. 12425 in Nuclear Science Abstracts 17, n. 8 (April 1963), 1615.Google Scholar
(16)Ölçer, N. Y.Heat conduction in finite hollow circular cylinders with general time-dependent boundary conditions and heat generation (Argonne National Lab., Illinois, Nov. 1962, unpublished).Google Scholar
(17)Kim, V. Yu.Transient heat conduction in a hollow cylinder with the known boundary conditions (in Russian). Inzh.-Fiz. Zh. 6, n. 7 (1963), 66–69.Google Scholar
(18)Khristichenko, P. I.Transient heat conduction and thermoelastic stresses in a hollow cylinder (in Russian). Inzh.-Fiz. Zh. 6, n. 7 (1963), 76–82.Google Scholar
(19)Marchi, E. and Zgrablich, G.Heat conduction in hollow cylinders with radiation. Proc. Edinb. Math. Soc. 14 (II) (1964), 159–164.Google Scholar
(20)Kleiner, M. K.A temperature field of infinite hollow cylinders heated asymmetrically with respect to perimeter under general non-uniform boundary conditions (in Russian). Inzh.-Fiz. Zh. 11, n. 3 (1966), 307–314.Google Scholar
(21)Kirkham, D.Graphs and formulas for zeros of cross product Bessel functions. J. Math. Phys. 36 (1958), 371–377.Google Scholar
(22)Bridge, J. F. and Angrist, S. W.An extended table of roots of J′n(x) Y′n (βx)-J′n(x)=0. Math. Comp. 16 (1962), 198–204.Google Scholar
(23)Laslett, L. J. and Lewish, W.Evaluation of the zeros of cross-product Bessel functions. Math. Comp. 16 (1962), 226–232.Google Scholar
(24)Carslaw, H. S. and Jaeger, J. C.Conduction of heat in solids (Oxford University Press, 1959), 332–334.Google Scholar