Abstract
An integral treatment was proposed for analysis of non-Darcy free convection over a vertical flat plate and cone within a fluid-saturated porous medium. A flexible one-parameter family of third order polynomials was employed to cope with vast changes in the velocity and temperature profiles encountered in the Darcy flow limit through to the Forchheimer flow limit. Zero curvature requirement for the temperature profile at the wall was exploited as an auxiliary relation to determine the shape parameter. Comparison of the approximate results with the exact solution reveals a high performance of the present integral procedure for heat transfer rat prediction.
Zusammenfassung
Eine Integral-Methode wurde zur Analyse von freier „Non-Darcy“-Konvektion entlang einer vertikalen flachen Platte und einem Kegel in einem gesättigten porösen Medium herangezogen. Eine flexible einparametrige Schar von Polynomen dritter Ordnung wurde verwendet, um große Änderungen in den Geschwindigkeits- und Temperaturprofilen im „Darcy“- bis hin zum „Forchheimer“-Bereich erfassen zu können. Die Forderung, daß das Temperaturprofil an der Wand keine Krümmung aufweisen darf, wurde als Hilfsbeziehung benutzt, um den Formfaktor zu bestimmen. Ein Vergleich der angenäherten Ergebnisse mit der exakten Lösung offenbart, daß die dargestellte Integral-Methode sich sehr gut eignet, um Wärmeübertragungswerte berechnen zu können.
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Abbreviations
- A, B, D :
-
shape factors
- C:
-
empirical constant associated with the Forchheimer term
- g :
-
tangential component of the acceleration due to gravity
- Gr :
-
modified Grashof number
- i :
-
1 for a flat plate and 3 for a cone
- K :
-
permeability
- r :
-
1 for a flat plate andx for a cone
- Rax :
-
local Rayleigh number
- T :
-
temperature
- u, υ :
-
Darcian velocity components
- x, y :
-
boundary layer coordinates
- α:
-
effective thermal diffusivity
- δ :
-
boundary layer thickness
- ν :
-
kinematic viscosity of the fluid
- Λ :
-
shape parameter
- e:
-
external
- w:
-
wall
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Nakayama, A., Kokudai, T. & Koyama, H. An integral treatment for non-Darcy free convection over a vertical flat plate and cone embedded in a fluid-saturated porous medium. Warme- und Stoffubertragung 23, 337–341 (1988). https://doi.org/10.1007/BF01650469
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DOI: https://doi.org/10.1007/BF01650469