Summary
The cooling of a hot fluid in laminar Newtonian flow through cooled elliptic tubes has been calculated theoretically. Numerical data have been computed for the two values 1.25 and 4 of the axial ratio of the elliptic cross-section ε. For ε=1.25 the influence of non-zero thermal resistance between outmost fluid layer and isothermal surroundings has also been investigated. Special attention has been given to the distribution of heat flux around the perimeter; when ε increases the flux varies more with the position at the circumference. This positional dependence becomes less pronounced, however, as the (position-independent) thermal resistance of the wall increases.
Flattening of the conduit, while maintaining its cross-sectional area constant, improves the cooling. Comparison with rectangular pipes shows that this improvement is not as marked with elliptic as with rectangular pipes.
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Abbreviations
- A k =A m, n :
-
coefficients of expansion (6)
- a, b :
-
half-axes of ellipse, b<a
- a p =a r, s :
-
coefficients of representation (V)
- D :
-
hydraulic diameter, = 4S/P; S = cross-sectional area, P = perimeter
- D e :
-
equivalent diameter, according to (13)
- n :
-
coordinate (outward) normal to the tube wall
- T :
-
temperature of fluid
- T i :
-
temperature of fluid at the inlet
- T s :
-
temperature of surroundings
- v 0 :
-
mean velocity of fluid
- v z :
-
longitudinal velocity of fluid
- x, y :
-
carthesian coordinates coinciding with axes of ellipse
- z :
-
coordinate in flow direction
- α, β :
-
dimensionless half-axes of ellipse, α=a/D and β=b/D
- α t :
-
heat transfer coefficient from fluid at bulk temperature to surroundings; equation (11)
- α w :
-
heat transfer coefficient at the wall; equation (3)
- ε :
-
axial ratio of ellipse, = a/b = α/β
- ξ, η, ζ, ν :
-
dimensionless coordinates; ξ=x/D, η=y/D, ζ=z/D, ν=n/D
- ϑ :
-
dimensionless temperature, = (T−T s)/(T i−T s)
- ϑ 0 :
-
cup-mixing mean value of ϑ; equation (10)
- λ :
-
thermal conductivity of fluid
- μ m,n =μ k :
-
eigenvalue
- ρc :
-
volumetric heat capacity of fluid
- φ m, n =φ k =θ k :
-
eigenfunction; equations (6) and (I)
- Nu :
-
total Nusselt number, = α t D/λ
- \(Nu_\infty \) :
-
Nusselt number at large distance from the inlet
- Nu w :
-
wall Nusselt number, = α w D/λ, based on α w
- Pé:
-
Péclet number, = ν 0 Dρc/λ
References
Brown, G. M., A. I. Ch. E. Journal 6 (1960) 179.
Van der Does de Bye, J. A. W. and J. Schenk, Appl. Sci. Res. A3 (1952) 308; Schenk, J. and J. M. Dumoré, Appl. Sci. Res. A4 (1953) 39.
Dennis, S. C. R., A. McD. Mercer and G. Poots, Quart. Appl. Mech. 17 (1959) 285.
Reynolds, W. C., R. E. Lundberg and P. A. McCuen, Int. J. Heat Mass Transfer 6 (1963) 483 and 495.
Dunwoody, N. T., J. Appl. Mech. 29 (1962) 165.
Bong Swij Han, Warmteoverdracht naar laminaire stroming in elliptische buizen. Afstudeerverslag, Technische Natuurkunde, Technische Hogeschool, Delft, 1965 (copies available on request at the authors).
Sparrow, E. M. and R. Siegel, Int. J. Heat Mass Transfer 1 (1960) 161.
Kramers, H. and P. J. Kreyger, Chem. Engng Sci. 6 (1956) 42. See also Appl. Sci. Res. A5 (1955) 243.
Schenk, J. and J. van Laar, Appl. Sci. Res. A7 (1958) 449.
Beek, W. J. and R. Eggink, De Ingenieur 74 (1962) Ch. 81.
Nieuwenhuijsen, H. and W. J. Beek, De Ingenieur 77 (1965) Ch. 70.
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Schenk, J., Han, B.S. Heat transfer from laminar flow in ducts with elliptic cross-section. Appl. sci. Res. 17, 96–114 (1967). https://doi.org/10.1007/BF00419779
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DOI: https://doi.org/10.1007/BF00419779