Natural convection within spherical annuli by symbolic algebra on workstations

Kevin G. TeBeest, Steven A. Trogdon, Jeffrey R. Dahl, Rod W. Douglass

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Presented is an asymptotic series approximate solution of steady, axisymmetric natural convection within arbitrary-gap spherical annulus enclosures. The boundaries are of uniform and constant temperature, the inner shell being hotter than the outer. The fluid within the annulus is Boussinesq of arbitrary Prandtl number. Both the dimensionless momentum and energy equations involve the parameter R, the square root of the Grashof number based on outer radius. Temperature and streamfunction are represented as truncated power series of R and are even and odd functions of R respectively. We show that a natural representation for the temperature and streamfunction bases are separable series involving Legendre polynomials and associated Legendre functions of order 1 respectively. By invocation of orthogonality, an uncoupled system of linear ordinary differential equations is obtained for the radially dependent functions and are solved exactly in sequence by the symbolic manipulator MACSYMA. We present results for air (Pr = 0.7) for various values of R for an annular gap-width of 0.5 and compare the surface average Nusselt numbers with previous results.

Original languageEnglish (US)
Title of host publicationComputational Techniques and Numerical Heat Transfer on PCs and Workstations
PublisherPubl by ASME
Pages101-106
Number of pages6
ISBN (Print)0791808424
StatePublished - Dec 1 1991
EventWinter Annual Meeting of the American Society of Mechanical Engineers - Atlanta, GA, USA
Duration: Dec 1 1991Dec 6 1991

Publication series

NameAmerican Society of Mechanical Engineers, Aerospace Division (Publication) AD
Volume23
ISSN (Print)0733-4230

Other

OtherWinter Annual Meeting of the American Society of Mechanical Engineers
CityAtlanta, GA, USA
Period12/1/9112/6/91

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    TeBeest, K. G., Trogdon, S. A., Dahl, J. R., & Douglass, R. W. (1991). Natural convection within spherical annuli by symbolic algebra on workstations. In Computational Techniques and Numerical Heat Transfer on PCs and Workstations (pp. 101-106). (American Society of Mechanical Engineers, Aerospace Division (Publication) AD; Vol. 23). Publ by ASME.