Skip to main content
SpringerLink
Log in

Flow of a liquid metal coolant past a heat generating cylinder

  • Published:
Applied Scientific Research Aims and scope Submit manuscript

Abstract

The coupled problem whereby a solid heat generating cylinder is being cooled in steady state by a coolant in potential flow is investigated. An analytical technique for determining the temperature distributions in the solid and the fluid is presented. Numerical studies for six Péclet numbers (0.9<Pe<11.3) and three thermal conductivity ratios (0.31<K<3.1) were carried out.

The surface hot-spot temperature and center temperature are presented graphically as functions of the Péclet number with the thermal conductivity ratio as a parameter. The average Nusselt number is found to be proportional to the Péclet number to approximately the one-half power. For the special case of constant surface temperature (uncoupled problem), the variation of local Nusselt number with angle measured from the forward stagnation point is in excellent agreement with the result presented by Grosh and Cess [6].

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

ce m(Θ, −q):

Mathieu function, periodic

D n :

Fourier coefficient for solid temperature distribution

E n :

Fourier coefficient for fluid temperature distribution

E(γ):

a term defined by equation (12), degree

F(γ):

a term defined by equation (13), degree

Fek m(z, −q):

modified Mathieu function, non-periodic

Fek′ m(z, −q):

derivative of Fek m(z, −q)

h :

local heat transfer coefficient, energy/time area degree

\(\bar h\) :

average heat transfer coefficient, energy/time area degree

h m :

mean heat-transfer coefficient, energy/time area degree

k f :

thermal conductivity of fluid, energy/time length degree

K :

thermal conductivity ratio, k f/k s

k s :

thermal conductivity of solid, energy/time length degree

Nu :

local Nusselt number, 2Rh/k f

\(\overline {Nu}\) :

average Nusselt number defined by equation (55)

(Nu)m :

mean Nusselt number defined by equation (57)

Pe :

Péclet number, 2RU∞/α f

Q :

rate of heat generation per unit volume, energy/time volume

q :

parameter of Mathieu function, (Pe/4)2

q″ :

normal heat flux, energy/time area

R :

cylinder radius, length

Re :

Reynolds number, 2R∞/ν

r :

radial position variable, length

T :

temperature, degree

T 0 :

constant surface temperature, degree

T ∞ :

temperature of fluid at infinity, degree

T e :

temperature at center of cylinder, degree

T f :

temperature of fluid, degree

T s :

temperature of solid, degree

T w :

surface temperature, degree

\(T_{w_{max} }\) :

surface hot-spot temperature, degree

\(\bar T\) :

reduced temperature, (T−∞)/E(1)

U∞ :

approach velocity of flowing fluid, length/time

v Θ :

velocity component in the Θ direction, length/time

v r :

velocity component in the r direction, length/time

z :

logarithm of γ

α f :

thermal diffusivity of the fluid, (length)2/time

γ :

reduced radius, r/R

Θ :

angular position variable measured from the trailing stagnation point, radians

ν :

kinematic viscosity, (length)2/time

ψ :

angular position variable measured from the forward stagnation point, degree

References

  1. Hoe, R. J., D. Dropkin, and O. E. Dwyer, Trans. ASME 79 (1957) 899.

    Google Scholar 

  2. Andreevskii, A. A., Soviet J. of Atomic Energy 7 (1961) 745.

    Google Scholar 

  3. Borishanskii, V. M., A. A. Andreevskii, and V. B. Zhinkina, Soviet J. of Atomic Energy 13 (1963) 874.

    Google Scholar 

  4. Kalish, S. and O. E. Dwyer, Int. J. Heat Mass Transfer 10 (1967) 1533.

    Google Scholar 

  5. Richard, C. L., O. E. Dwyer, and D. Dropkin, Trans. ASME 80 (1958) 646.

    Google Scholar 

  6. Grosh, R. J. and R. Q. Cess, Trans. ASME 80 (1958) 667.

    Google Scholar 

  7. Cess, R. D. and R. J. Grosh, Trans. ASME 80 (1958) 677.

    Google Scholar 

  8. Hsu, C. J., Int. J. Heat Mass Transfer 7 (1964) 431.

    Google Scholar 

  9. Hsu, C. J., Int. J. Heat Mass Transfer 8 (1965) 303.

    Google Scholar 

  10. Fagan, W. and S. Leipziger, J. Heat Transfer (Trans. ASME) 88 (1966) 257.

    Google Scholar 

  11. Tomotika, S. and H. Yisinobu, J. Math. Physics 34 (1957) 112.

    Google Scholar 

  12. McLachlan, N. W., Theory and Application of Mathieu Functions, Dover Publications Inc., New York, 1964.

    Google Scholar 

  13. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications Inc., New York, 1965.

    Google Scholar 

  14. Knudsen, J. G. and D. L. Katz, Fluid Dynamics and Heat Transfer, McGraw-Hill Book Company, New York, 1958.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Inst. of Gas Techn., Chicago, Ill., U.S.A.

    N. Ishikawa & S. Leipziger

Authors
  1. N. Ishikawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Leipziger
    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

Ishikawa, N., Leipziger, S. Flow of a liquid metal coolant past a heat generating cylinder. Appl. Sci. Res. 23, 53–72 (1971). https://doi.org/10.1007/BF00413187

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00413187

Keywords

Search

Navigation