Abstract
The steady free convection boundary layer flow of non-Darcy fluid along an isothermal vertical cylinder embedded in a saturated porous medium using the Ergun model has been studied. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme developed by Keller. It is found that the heat transfer is strongly affected by the modified Grashof number which characterizes the non-Darcy fluid, and the curvature parameter. Also the heat transfer is found to be more than that of the flat plate.
Zusammenfassung
Es wird die Grenzschichtströmung bei stationärer, freier Konvektion eines nicht-Darcyschen Fluids entlang eines isothermen, vertikalen Zylinders, der in ein gesättigtes, poröses Medium eingelagert ist, mit dem Ergun-Modell behandelt. Die partiellen Differentialgleichungen, welche die Strömung beschreiben, wurden numerisch unter Verwendung eines von Keller entwickelten, impliziten Finiten-Differenzen-Schemas gelöst. Es zeigte sich, daß der Wärmeübergang stark von der modifizierten Grashof-Zahl beeinflußt wird, welche das nicht-Darcysche Fluid und den Krümmungsparameter charakterisiert. Der Wärmeübergang am Zylinder ist höher als an einer ebenen Platte.
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Abbreviations
- f :
-
transformed stream function
- g :
-
acceleration due to gravity
- Gr * :
-
modified Grashof number
- k m :
-
thermal conductivity of the porous medium
- K :
-
permeability of the saturated porous medium
- K * :
-
inertial coefficient in the Ergun model
- L :
-
length of the cylinder
- p :
-
static pressure
- q :
-
local heat transfer rate per unit area
- Q :
-
total surface heat transfer rate
- r :
-
radi coordinate
- r0 :
-
radius of cylinder
- Ra x :
-
modified local Rayleigh number
- T :
-
temperature
- u, v :
-
velocity components in thex- andr-directions, respectively
- x :
-
axial coordinate
- α :
-
equivalent thermal diffusivity
- β :
-
coefficient of thermal expansion
- η :
-
pseudo-similarity variable
- θ :
-
dimensionless temperature
- μ :
-
viscosity of convective fluid
- v :
-
kinematic viscosity of convective fluid
- ϱ :
-
density of convective fluid
- ξ :
-
stretched streamwise coordinate
- ξ L :
-
constant
- ψ :
-
streamfunction
- r, x :
-
denote derivatives with respect tor andx, respectively
- w :
-
refers to wall conditions
- ∞:
-
refers to free stream conditions
- ':
-
denotes derivatives with respect toη
References
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Kumari, M.; Pop, I.; Nath, G.: Finite-difference and improved perturbation solutions for free convection on a vertical cylinder embedded in a porous medium. To be published
Cheng, P.: Recent advances in convective and boiling heat transfer in porous media. USA-China Binational Workshop on Heat Transfer, October 5–7, Peking 1983
Plumb, O. A.; Huenefeld, J. C.: Non-Darcy natural convection from heated surfaces in a saturated porous media. Int. J. Heat and Mass Transfer 24 (1981) 765–768
Ergun, S.: Fluid flow through packed columns. Chem. Engng. Prog. 48 (1952)89–94
Keller, H. B.: A new difference scheme for parabolic problems. In: Numerical Solution of Partial Differential Equations. Ed. J. Bramble, Vol. II, New York: Academic Press 1971
Keller, H. B.; Cebeci, T.: Accurate numerical methods for boundary layers. I. Two-dimensional laminar flows. Proceedings of the Second International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in Physics, Vol. 8, Berlin, Heidelberg, New York: Springer 1971
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Kumari, M., Pop, I. & Nath, G. Non-darcy natural convection on a vertical cylinder in a saturated porous medium. Wärme- und Stoffübertragung 20, 33–37 (1986). https://doi.org/10.1007/BF00999734
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DOI: https://doi.org/10.1007/BF00999734