Skip to main content
SpringerLink
Log in

On the second fundamental problem of combined free and forced convection through vertical noncircular ducts

  • Published:
Applied Scientific Research Aims and scope Submit manuscript

Abstract

Analysis of combined free and forced convection through vertical noncircular ducts is carried out using a variational technique. Fully developed flow with uniform axial heat input and uniform peripheral heat flux is assumed. All fluid properties are considered invariant with temperature except the variation of density in the buoyancy term of the equation of motion. The condition of uniform peripheral heat flux is utilized in deriving the variational expression. This procedure releases the thermal boundary condition from satisfying exactly the condition at the wall. A finite-difference procedure is carried out. For pure forced convection case, a particularly simple variational expression is presented. Nusselt numbers for combined free and forced convection are computed for rectangular, rhombic and elliptical ducts. An exact solution is presented for laminar forced convection through elliptic ducts. Variational results are in agreement with this exact solution. The present results are compared with those in the published literature wherever possible, and good agreement is obtained.

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 :

area of cross-section

a, b :

semi axes

C p :

specific heat of the fluid at constant pressure

C 1 :

temperature gradient in flow direction,∂T/∂Z

C 2 :

heat flux normal to surface,∂φ/∂N

D h :

hydraulic diameter, (4× cross-sectional area)/(heat transfer perimeter)

F :

heat generation parameter, dimensionless,Q/ρC p C 1 U

g :

gravitational, acceleration

L :

pressure drop parameter, dimensionless

Nu :

hD h/κ, Nusselt number, dimensionless

P :

heat transfer perimeter

Q :

heat generation rate

Ra :

Rayleigh number, dimensionless, (ρ 2 gC p C 1 βD 4 h )/κμ

S :

duct circumference

T :

temperature

u :

axial velocity

U :

average axial velocity

V :

dimensionless axial velocity,u/U

x, y, X, Y :

rectangular co-ordinates,X=x/D h,Y=y/D h

z :

axial co-ordinate in flow direction

φ :

dimensionless temperature function,\({{(T - T_{ref} )} \mathord{\left/ {\vphantom {{(T - T_{ref} )} {\left[ {\frac{{\rho UC_p C_1 D_h^2 }}{\kappa }} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\frac{{\rho UC_p C_1 D_h^2 }}{\kappa }} \right]}}\)

β, ρ, κ, μ :

the fluid properties in standard notation

ψ :

aspect ratio,b/a

References

  1. Tao, L. N., Trans. ASME, Series E, J. Appl. Mechanics84 (1962) 415.

    Google Scholar 

  2. Tyagi, V. P., Proc. Camb. Phil. Soc.62 (3) (1966) 555.

    Google Scholar 

  3. Tyagi, V. P., Trans. ASME, Series E, J. Applied Mechanics88 (1966) 18.

    Google Scholar 

  4. Sparrow, E. M. andA. Haji-Sheikh, Trans. ASME, Series C, J. Heat Transfer88 (1966) 351.

    Google Scholar 

  5. Sparrow, E. M. andR. Siegel, Trans. ASME, Series C, J. Heat Transfer81 (1959) 157.

    Google Scholar 

  6. Han, L. S., Trans. ASME, Series C, Journal of Heat Transfer81 (1959) 121.

    Google Scholar 

  7. Lu, Pau-Chang, Trans. ASME, Series C, Journal of Heat Transfer82 (1960) 227.

    Google Scholar 

  8. Tao, L. N., Trans. ASME, Series C, Journal of Heat Transfer82 (1960) 233.

    Google Scholar 

  9. Tao, L. N., Applied Scientific Research9A (1960) 357.

    Google Scholar 

  10. Aggarwala, B. D., andM. Iqbal, Int. J. Heat Mass Transf.12 (1969) 737.

    Article  Google Scholar 

  11. Iqbal, M., B. D. Aggarwala andA. C. Fowler, Int. J. Heat Mass Transfer12 (1969) 1123.

    Article  Google Scholar 

  12. Iqbal, M., S. A. Ansari andB. D. Aggarwala, Trans. ASME, Series C, J. Heat Transfer92 (1970) 237.

    Google Scholar 

  13. Timoshenko, S., andJ. N. Goodier, Theory of Elasticity, McGraw-Hill Book Co., New York, N.Y., 2nd edition 1951.

    Google Scholar 

  14. Knudsen, J. G. andD. L. Katx, Fluid Dynamics and Heat Transfer, McGraw-Hill Book Co., New York, N.Y., 1958.

    Google Scholar 

  15. Tao, L. N., Trans. ASME, Series C, J. Heat Transfer83 (1961) 466.

    Google Scholar 

  16. Cheng, K. C., Trans. ASME, Series C, J. Heat Transfer91 (1969) 156.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mech. Engineering, The University of British Columbia, Vancouver, Canada

    M. Iqbal & A. K. Khatry

  2. Dept. of Mathematics, The University of Calgary, Calgary, Canada

    B. D. Aggarwala

Authors
  1. M. Iqbal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. K. Khatry
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. D. Aggarwala
    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

Iqbal, M., Khatry, A.K. & Aggarwala, B.D. On the second fundamental problem of combined free and forced convection through vertical noncircular ducts. Appl. Sci. Res. 26, 183–208 (1972). https://doi.org/10.1007/BF01897849

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01897849

Keywords

Search

Navigation