Application of Rayleigh-Ritz method to forced convection turbulent heat transfer |
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Authors: | L. A. Diaz Prof. N. V. Suryanarayana |
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Affiliation: | (1) Department of Mechanical Engineering, Michigan Technological University, 49931 Houghton, Michigan, USA |
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Abstract: | An analytical model to predict heat transfer rates to an incompressible fluid in turbulent flow, with fully developed velocity profile, between a heated plate and a parallel, insulated plate is developed. The model employs van Driest's mixing length expression near the wall, a constant eddy diffusivitiy in the core and a constant turbulent Prandtl number. An approximate solution obtained by employing Rayleigh-Ritz method is shown to compare well with the exact solution obtained by numerical integration of the differential equations. The results are compared with the available experimental data and analytical solutions.
Anwendung der Rayleigh-Ritz-Methode auf die Wärmeübertragung bei erzwungener turbulenter Strömung Zusammenfassung Es wird ein analytisches Modell zur Berechnung der Wärmeübertragung an ein inkompressibles Fluid in turbulenter Strömung mit voll ausgebildetem Geschwindigkeitsprofil zwischen einer beheizten Platte und einer dazu parallelen isolierten Platte angegeben. Das Modell verwendet van Driest's Ausdruck für die wandnahe Mischungslänge, eine konstante Wirbeldiffusivität im Kern und eine konstante turbulente PrandtlZahl. Eine Näherungslösung nach der Rayleigh-Ritz-Methode läßt sich gut mit der exakten Lösung vergleichen, die durch numerische Integration der Differentialgleichungen erhalten wurde. Die Ergebnisse werden mit verfügbaren Versuchswerten und analytischen Lösungen verglichen. Nomenclature A+ dimensionless constant in van Driest formula - a+ dimensionless distance from the wall after which the eddy diffusivity of momentum is constant - b half-gap of passage - b+ dimensionless half-gap=bu*/ - Cf skin friction coefficient - Cp constant pressure specific heat - d hydraulic mean diameter defined as 4xarea/perimeter=4b - h convective heat transfer coefficient - K+ dimensionless constant in van Driest formula - k fluid thermal conductivity - m mass flow rate of fluid - Nu Nusselt number hd/k - P pressure - Pr Prandtl number=/ - Prt turbulent Prandtl number=m/ - qw heat flux at wall - Re Reynolds number=vmd/ - T Temperature - u+ dimensionless velocity=Vx/u* - u* friction velocity= - Vx axial velocity - x axial distance from the entrance - x+ dimensionless distance=x/d - y distance from the heated wall - y+ dimensionless distance=yu*/Greek Symbols thermal molecular diffusivity - function equal to (H+)/ - boundary layer thickness - H eddy diffusivity of heat - m eddy diffusivity of momentum - m0 uniform eddy diffusivity of momentum in the core - dimensionless temperature - T-Ti/qwd/k uniform heat flux - T-Tw/Ti-Tw uniform temperature - fluid kinematic viscosity - fluid density - fluid shearing stress - bulk mean temperature—fully developed region - fully developed transverse temperature profileSuffixes 1 fully developed - 2 in the entrance region - i at the inlet - m bulk mean value - w at the heated wall |
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