Abstract
The problem of laminar forced convection heat transfer in short elliptical ducts with (i) uniform wall temperature and (ii) prescribed wall heat flux is examined in detail with the well known Lévêque theory of linear velocity profile near the wall. Moreover, consideration is given to the variation of the slope of the linear velocity profile with the position on the duct wall. A correction factor for the temperature dependent viscosity is included. Expressions for the local and average Nusselt numbers and wall temperatures are obtained. For the case of constant heat flux the Nusselt numbers are higher than for constant wall temperature.
The results corresponding to the classical Graetz and Purday problems are deduced as special cases.
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Abbreviations
- a, b :
-
semiaxes of ellipse, b<a
- A h :
-
area of heat transfer surface
- c=ae :
-
distance between focus and centre of the ellipse
- C :
-
heat capacity of the fluid
- D e :
-
equivalent diameter, (18)
- e :
-
eccentricity of the elliptical duct
- E(e):
-
complete elliptic integral
- g :
-
Laplace transform of T
- g w :
-
Laplace transform of T w
- G o z :
-
Graetz number (local), Re Pr D e/z
- \(\overline {Gz}\) :
-
Graetz number (average), Re Pr D e/Z
- h oiθ :
-
local heat transfer coefficient
- J n(x):
-
Bessel function of order n
- K :
-
thermal conductivity of the fluid
- ℒ [X]:
-
Laplace transform of X
- N o u :
-
local Nusselt number, h oiθ D e/K
- \(\overline {N_u^o }\) :
-
perimeter average Nusselt number
- \(\overline {Nu}\) :
-
overall average Nusselt number
- Nu w :
-
wall Nusselt number
- Nu ∞ :
-
Nusselt number at large distance from the inlet
- p :
-
Laplace transform parameter
- Pr :
-
Prandtl number, Cμ a/K
- Re :
-
Reynolds number, D e ūρ/μ a
- T :
-
temperature of the fluid
- T 1, T W :
-
inlet and wall temperatures, respectively
- u z :
-
local isothermal velocity along the axis of the duct
- ū :
-
average fluid velocity
- x, y, z :
-
Cartesian coordinates, z-axis parallel to the axis of the duct (z=0 at duct inlet)
- Z :
-
length of the duct
- α :
-
thermal diffusivity, K/ρC
- β*:
-
correction factor for the temperature dependent viscosity
- Γ(x):
-
gamma function
- η :
-
coordinate measured normal to the wall of the duct
- μ a, μ w :
-
viscosity of fluid at average and wall temperatures
- ξ, θ, z :
-
elliptic cylindrical coordinates
- ρ :
-
density of fluid
- φ(z):
-
heat flux
References
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Rao, S.S., Ramacharyulu, N.C.P. & Krishnamurty, V.V.G. Laminar forced convection in elliptic ducts. Appl. sci. Res. 21, 185–193 (1969). https://doi.org/10.1007/BF00411606
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DOI: https://doi.org/10.1007/BF00411606