Skip to main content
SpringerLink
Log in

A numerical study of the heat transfer through a fluid layer with recirculating flow between concentric and eccentric spheres

  • Published:
pure and applied geophysics Aims and scope Submit manuscript

Abstract

A numerical study has been made of the heat transfer through a fluid layer with recirculating flow. The outer fluid surface was assumed to be spherical, while the inner surface consisted of a sphere concentrically or eccentrically located with respect to the outer spherical surface. The recirculating flow was assumed to be driven by a gas flow creating stress on the fluid's outer surface so that creeping (low Reynolds number) flow developed in its interior. The present study solves the Stokes equation of motion and the convective diffusion equation in bispherical coordinates and presents the streamline and isotherm patterns.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

a i :

inner sphere radius

a d :

outer sphere radius

A 1 :

defined by equation (5)

A 2 :

defined by equation (6)

B 1 :

defined by equation (7)

B 2 :

defined by equation (8)

c :

dimensional factor for bispherical coordinates

C :

constant in equation (4)

d′ :

narrowest distance between the two eccentric spheres

E 2 :

operator defined by equation (1) in spherical coordinates and by equation (21) in bispherical coordinates

G :

modified vorticity, defined in equation (22)

G * :

non-dimensional modified vorticity, defined in equation (28)

h ϕ :

metric coefficient of bispherical coordinate system, defined in equation (18)

k w :

thermal conductivity of water

K 1 :

defined by equation (9)

K 2 :

defined by equation (10)

N Re :

Reynolds number=2a dU∞/gn

N Pe,h :

Peclet number=2a dU∞/κ

n :

integer counter

q :

heat flux

r :

radius

r * :

non-dimensional radius=r/a d

S :

surface area

t :

time

t * :

non-dimensional time=κt/a 2d

T :

temperature

T o :

temperature at inner sphere surface

T a :

temperature at outer sphere surface

T * :

non-dimensional temperature;=(T−T o)/(Ta−To)

u :

velocity

u r :

radial velocity in spherical coordinates

u θ :

angular velocity in spherical coordinates

u η :

radial velocity in bispherical coordinates

u ξ :

angular velocity in bispherical coordinates

U ∞ :

free stream velocity

u r * :

=u r/U∞

u θ * :

=u θ/U∞

u η * :

=u η/U∞

u ξ * :

=u ξ/U∞

δa 1 :

small displacement

ζ:

vorticity, defined in equation (17)

ζ* :

non-dimensional vorticity, defined in equation (27)

η:

radial bispherical coordinates

ηo :

bispherical coordinate of inner sphere

ηa :

bispherical coordinate of outer sphere

θ:

angular coordinate in spherical coordinates

κ:

thermal diffusivity

κw :

thermal diffusivity of water

ν:

kinematic viscosity

ξ:

angular bispherical coordinate

ϕ:

spherical coordinate

ψ:

streamfunction

ψ′ :

non-dimensional streamfunction for spherical coordinates, = ψ/(U ∞a 2d )

ψ* :

non-dimensional streamfunction for bispherical coordinates, defined in equation (26)

References

  • Astill, K. N. (1976),An analysis of laminar forced convection between concentric spheres, ASME J. Heat Transfer,98, 601–608.

    Google Scholar 

  • Carslaw, H. S., andJaeger, J. C.,Conduction of Heat in Solids, (Second Edition, Clarendon Press, Oxford 1959) 510 p.

    Google Scholar 

  • Douglass, R. W., Shaughnessy, E. J., andMunson, B. R. (1979)Small Reynolds number convection in rotating spherical annuli, ASME J. Heat Transfer101, 427–433.

    Google Scholar 

  • Happel, J., andBrenner, H.,Low Reynolds Number Hydrodynamics (Prentice-Hall, Inc., Englewood Cliffs, New Jersey 1965) 553 p.

    Google Scholar 

  • Kronig, R., andBrink, J. C.,On the theory of extraction from falling droplets, Appl. Sci. ResearchA2, 142–154.

  • Le Clair, B. P., Hamielec, A. E., Pruppacher, H. R., andHall, W. D. (1972),A theoretical and experimental study of the internal circulation in water drops falling at terminal velocity in air, J. Atmos. Sci.29, 728–740.

    Google Scholar 

  • Maolem, D. M., andMiloh, T. (1976),Theoretical analysis of heat and mass transfer through eccentric spherical fluid shells at large Peclet number, Appl. Sci. Research32, 395–414.

    Google Scholar 

  • Rasmussen, R. M. (1982),The Melting of Spherical Ice Particles Falling at Terminal Velocity in Air: An Experimental and Theoretical Study (Ph.D. Thesis, Public. No. 83/06, 106, University Microfilms, Ann Arbor, Michigan) 385 p.

    Google Scholar 

  • Rasmussen, R. M., andPruppacher, H. R. (1982),A wind tunnel and theoretical study of the melting behaviour of atmospheric ice particles: I. A. wind tunnel study of frozen drops of radius<500 μm, J. Atmos. Sci.39, 152–158.

    Google Scholar 

  • Rundell, H. A., Ward, E. G., andCox, J. E. (1968),Forced convention in concentric-sphere heat exchanges, ASME J. Heat Transfer90, 125–129.

    Google Scholar 

  • Weber, N., Powe, R. E., andBishop, E. H. (1973),Heat transfer by natural convection between vertically eccentric spheres, ASME J. Heat Transfer95, 47–52.

    Google Scholar 

  • Woo, S., andHamielec, A. E. (1971),A numerical method of determining the rate of evaporation of small drops falling at terminal velocity in air, J. Atmos. Sci.28, 1448–1454.

    Google Scholar 

  • Yao, L. S. (1980),Analysis of heat transfer in slightly eccentric annuli, ASME J. Heat Transfer102, 279–284.

    Google Scholar 

Download references

Author information

Author notes

  1. V. Levizzani

    Present address: Reparto Nubi e Precipitazioni, Istituto FISBAT-CNR, Bologna, Italy

Authors and Affiliations

  1. Department of Atmospheric Sciences, University of California, 90024, Los Angeles, CA

    R. M. Rasmussen, V. Levizzani & H. R. Pruppacher

Authors
  1. R. M. Rasmussen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Levizzani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. R. Pruppacher
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rasmussen, R.M., Levizzani, V. & Pruppacher, H.R. A numerical study of the heat transfer through a fluid layer with recirculating flow between concentric and eccentric spheres. PAGEOPH 120, 702–720 (1982). https://doi.org/10.1007/BF00876656

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00876656

Keywords

Search

Navigation