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Effect of wall heat conduction on laminar mixed convection in the thermal entrance region of horizontal rectangular channels

Einfluß der Wandwärmeleitung auf die laminare Mischkonvektion in der thermischen Eintrittszone von horizontalen, rechteckigen Kanälen

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Abstract

This paper presents a numerical study of the effect of peripheral wall heat conduction on laminar mixed convection in the thermal entrance region of horizontal rectangular channels. In addition to the Rayleigh numberRa and the channel aspect ratio (width-to-height) γ, the wall heat conduction parameterK=k w t/(k f D e ) plays an important role in heat transfer. A numerical method of solution utilizing the vorticity-velocity formulation is developed to solve the system of governing partial differential equations coupled with the boundary condition equation considering peripheral wall heat conduction. Local friction factor ratio and Nusselt number variations are shown forPr=0.7,Ra=104, 3×104, and 105, γ=0.2, 0.5, 1, 2 and 5,K=0.01, 0.1 and 1. The effect of peripheral wall heat conduction on Nusselt number is found to be significant when the value of γ is less than 1. The asymptotic solutions forz→∞ are compared with the existing numerical results and good agreement is indicated.

Zusammenfassung

Diese Arbeit zeigt eine numerische Studie über den Einfluß der Wärmeleitung an der Wandoberfläche auf die laminare Mischkonvektion in der thermischen Eintrittszone von horizontalen, rechteckigen Kanälen. Neben der RayleighzahlRa und dem Kanalverhältnis γ von Breite zu Höhe, spielt der WärmeleitungsparameterK=k w t/(k f D e ) eine wichtige Rolle im Wärmetransport. Ein numerisches Verfahren der Lösungsauswertung der Wirbel-Geschwindigkeits-Gleichung ist entwickelt worden, um bestehende Differenzial-Gleichungssysteme zu lösen, die als Randbedingung gekoppelt sind die Wärmeleitung der Wandoberfläche berücksichtigen. Das lokale Reibungsfaktorverhältnis und Nusseltzahlvariationen sind fürPr=0,7,Ra=104, 3×104 und 105 und γ=0,2, 0,5, 1, 2 und 5 undK=0,01, 0,1 und 1 aufgezeigt. Der Einfluß der Wärmeleitung an der Wandoberfläche auf die Nusseltzahl ist als sehr bedeutend angesehen worden, wenn der Wert von γ kleiner als 1 ist. Die asymptotischen Lösungen fürz → ∞ sind mit den vorhandenen numerischen Ergebnissen verglichen worden und dabei wurde eine gute Übereinstimmung festgestellt.

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Abbreviations

A :

cross-sectional area of a channel

a, b :

width and height of a rectangular channel, respectively

C :

a constant, (D 2 e /μ \(\bar W\) f ) ∂P f /∂Z

D e :

equivalent hydraulic diameter, 4A/S

f :

friction factor, 2\(\bar \tau\) w /(ϱ \(\overline {w\prime }\) 2)

f 1 :

a function, −(∂p/∂z)/Pe 2

g :

gravitational acceleration

Gr :

Grashof number,g β ϑ c D 3 e /ν 2

\(\bar h\) :

average heat transfer coefficient

K :

wall heat conduction parameter,k w t/(k f D e )

k f :

thermal conductivity of fluid

k w :

thermal conductivity of channel wall material

M, N :

number of divisions inX andY directions, respectively

n :

dimensionless inward normal direction to the wall

Nu :

local Nusselt number,\(\bar h\) D e /k f

P :

pressure deviation

P f :

pressure for fully developed laminar flow before thermal entrance

p :

dimensionless quantity forP

Pe :

Peclet number,Pr Re

Pr :

Prandtl number,ν/α

Ra :

Rayleigh number,Pr Gr

Re :

Reynolds number,\(\overline {W\prime }\) D e /ν

S :

circumference of cross-section

s :

dimensionless tangential coordinate on the wall inside periphery

T :

temperature

T 0 :

uniform fluid temperature at entrance

U, V, W :

velocity components inX, Y, Z directions due to buoyancy effect

u, v, w :

dimensionless quantities forU, V, W

W f :

fully developed axial velocity before thermal entrance

w f :

dimensionless quantity forW f

W′ :

axial velocity in the thermal entrance region,W f +W

w′ :

dimensionless axial velocity,w f +Ra·w

X, Y, Z :

rectangular coordinates

x, y, z :

dimensionless rectangular coordinates

α :

thermal diffusivity

β :

coefficient of thermal expansion

γ :

aspect ratio of a rectangular channel,a/b

ϑ :

dimensionless temperature difference, (T−T 0)/ϑ c

ϑ c :

characteristic temperature,q w D e /k f

μ :

viscosity

τ :

shear stress

ν :

kinematic viscosity

ξ :

axial-direction vorticity, ∂u/∂y−∂v/∂x

ϱ :

density

c :

characteristic quantity

f :

fully developed quantity before thermal entrance

w :

value at wall

0:

condition for purely forced convection

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  1. Department of Mechanical Engineering, National Central University, Chungli, Taiwan 32054 Republic of China

    F. -C. Chou & W. -Y. Lien

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Chou, F.C., Lien, W.Y. Effect of wall heat conduction on laminar mixed convection in the thermal entrance region of horizontal rectangular channels. Wärme - Und Stoffübertragung 26, 121–127 (1991). https://doi.org/10.1007/BF01590110

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