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Wärme - und Stoffübertragung

, Volume 22, Issue 3–4, pp 159–168 | Cite as

Mixed convection on vertical plate for fluids of any prandtl number

  • H. -T. Lin
  • C. -C. Chen
Article

Abstract

A mixed convection parameterζ=(σRa)1/4/(ωRe)1/2, withσ=Pr/(1+Pr) andω=Pr/(1 +Pr)1/2, is proposed to replace the conventional Richardson number, Gr/Re2, for combined forced and free convection flow on an isothermal vertical plate. This parameter can readily be reduced to the controlling parameters for the relative importance of the forced and the free convection,Ra1/4/(Re1/2Pr1/3) forPr ≫ 1, and (RaPr)1/2/(RePr1/2 forPr ≪ 1. Furthermore, new coordinates and dependent variables are properly defined in terms ofζ, so that the transformed nonsimilar boundary-layer equations give numerical solutions that are uniformly valid over the entire range of mixed convection intensity from forced convection limit to free convection limit for fluids of any Prandtl number from 0.001 to 10,000. The effects of mixed convection intensity and the Prandtl number on the velocity profiles, the temperature profiles, the wall friction, and the heat transfer rate are illustrated for both cases of buoyancy assisting and opposing flow conditions.

Keywords

Convection Prandtl Number Heat Transfer Rate Free Convection Force Convection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

Cf

local friction coefficient

Cp

specific heat capacity

f

reduced stream function

g

gravitational acceleration

Gr

local Grashoff number,g βTw−T)x3/v2

Nu

local Nusselt number

Pr

Prandtl number,v/α

Ra

local Rayleigh number,g βTw−Tx3/(α v)

Re

local Reynolds number,ux/v

Ri

Richardson number,Gr/Re2

T

fluid temperature

Tw

wall temperature

T

free stream temperature

u

velocity component in thex direction

u

free stream velocity

v

velocity component in they direction

x

vertical coordinate measuring from the leading edge

y

horizontal coordinate

Greek symbols

α

thermal diffusivity

β

thermal expansion coefficient

ζ

mixed convection parameter (σRa)1/4/ωRe)1/2

η

pseudo-similarity variable,(y/x)λ

η0

conventional similarity variable,(y/x)Re1/2

θ

dimensionless temperature, (T−TTW−T

λ

unified mixed-flow parameter, [(ωRe)1/2 + (σRa)1/4]

μ

dynamic viscosity

ν

kinematic viscosity

ξ

stretched streamwise coordinate or mixed convection parameter, [1 + (ωRe)1/2/(σRa)1/4]−1=ζ/(1 +ζ)

ϱ

density

σ Pr/(1 + Pr)τw

wall shear stress

Ψ

stream function

ω

Pr/(l+Pr)1/3

Mischkonvektion an einer vertikalen Platte für Fluide beliebiger Prandtl-Zahl

Zusammenfassung

Für die kombinierte Zwangs- und freie Konvektion an einer isothermen senkrechten Platte wird ein Mischkonvektions-Parameterζ=(σ Ra)1/4 (ωRe)1/2, mitσ=Pr/(1 +Pr) undω=Pr/(1 +Pr)1/2 vorgeschlagen, den die gebräuchliche Richardson-Zahl, Gr/Re2, ersetzen soll. Dieser Parameter kann ohne weiteres auf die maßgebenden Kennzahlen für den relativen Einfluß der erzwungenen und der freien Konvektion reduziert werden,Ra1/4/(Re1/2Pr1/3) fürPr ≫ 1 und (RaPr)1/4/(RePr)1/2 fürPr ≪ 1. Weiterhin werden neue Koordinaten und abhängige Variablen als Funktion vonζ definiert, so daß für die transformierten Grenzschichtgleichungen numerische Lösungen erstellt werden können, die über den gesamten Bereich der Mischkonvektion, von der freien Konvektion bis zur Zwangskonvektion, für Fluide jeglicher Prandtl-Zahl von 0.001 bis 10.000 gleichmäßig gültig sind. Der Einfluß der Intensität der Mischkonvektion und der Prandtl-Zahl auf die Geschwindigkeitsprofile, die Temperaturprofile, die Wandreibung und den Wärmeübergangskoeffizienten werden für die beiden Fälle der Strömung in und entgegengesetzt zur Schwerkraftrichtung dargestellt.

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Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • H. -T. Lin
    • 1
  • C. -C. Chen
    • 1
  1. 1.Dept. of Chemical EngineeringNational Central UniversityChung-liTaiwan 32054, ROC

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