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The finite element solutions of laminar flow and combined convection of air in a staggered or an in-line tube-bank

Die Finite-Elemente-Lösung von laminarer Strömung und kombinierter Konvektion von Luft in einer versetzten oder fluchtenden Rohranordnung

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Abstract

A finite element method is used to solve the full Navier-Stokes and energy equations for the problems of laminar combined convection from three isothermal heat horizontal cylinders in staggered tube-bank and four isothermal heat horizontal cylinders in in-line tube-bank. The variations of surface shear stress, pressure and Nusselt number are obtained over the entire cylinder surface including the zone beyond the separation point. The predicted values of total, pressure and friction drag coefficients, average Nusselt number and the plots of velocity flow fields and isotherms are also presented.

Zusammenfassung

Eine Methode der finiten Elemente wird zur Lösung der vollständigen Navier-Stokes- und der Energiegleichung für die Probleme der laminaren kombinierten Konvektion an drei isothermen geheizten horizontalen Zylindern in versetzter Rohranordnung sowie für vier isotherme geheizte horizontale Zylinder in fluchtender Anordnung verwendet.

Die Veränderung der Wandschubspannung, des Druckes und der Nusselt-Zahl werden für die gesamte Zylinderoberfläche, einschließlich des Bereiches nach dem Ablösepunkt, bestimmt. Die Werte des gesamten Widerstandsbeiwertes aufgrund von Druck und Reibung, die durchschnittliche Nusselt-Zahl und die Diagramme des Geschwindigkeitsfeldes und der Isothermen werden ebenfalls aufgezeigt.

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Abbreviations

C :

specifie heat

C D :

total drag coefficient

C f :

friction drag coefficient

C p :

pressure drag coefficient

D :

diameter of cylinder,L=2R 0

G, g :

gravitational acceleration

Gr :

Grashof number, gβ(Tw−T ∞)D 3/v 2

h :

local heat transfer coefficient

K :

thermal conductivity

L :

spacing between the centers of cylinder

M l :

shape function

N i :

shape function

Nu,\(\overline {Nu}\) :

local and average Nusselt numbers

P :

dimensionless pressure, p*/ϱu 2∞

p *,p ∞ :

pressure, free stream pressure

Pe :

Peclet number,RePr

Pr :

Prandtl number, μc/K

Ra :

Rayleigh number,Gr Pr

Re :

Reynolds number,Du ∞/v

R 0 :

radius of cylinder

T :

temperature

T w :

temperature on cylinder surface with fixed value

T ∞ :

free stream temperature

v :

dimensionless x-direction component of velocity,v */u ∞

u * :

x-direction component of velocity

u ∞ :

free stream velocity

v :

dimensionless Y-direction component of velocity,v */u ∞

v * :

Y-direction component of velocity

X :

x-direction axis

x :

dimensionless x-direction coordinate,x */D

x* :

x-direction coordinate

Y :

Y-direction axis

y :

dimensionless Y-direction coordinate,y */D

y * :

Y-direction coordinate

β :

coefficient of volumetric thermal expansion

θ :

plane angle

μ :

dynamic viscosity

ν :

kinematic viscosity, μ/ϱ

ϱ :

density of fluid

τ w :

dimensionless surface shear stress, τ w* /ϱu 2∞

τskw/* :

surface shear stress

φ :

dimensionless temperature,\(\frac{{T - T_\infty }}{{T_w - T_\infty }}\)

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Authors and Affiliations

  1. Dept. of Mechanical Engineering, National Cheng-Kung University, Tainan, Taiwan 700 Republic of China

    K. -L. Wong (Graduate Student) & C. -K. Chen (Professor)

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  1. K. -L. Wong
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  2. C. -K. Chen
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Wong, K.L., Chen, C.K. The finite element solutions of laminar flow and combined convection of air in a staggered or an in-line tube-bank. Wärme- und Stoffübertragung 23, 93–101 (1988). https://doi.org/10.1007/BF01637131

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  • DOI: https://doi.org/10.1007/BF01637131

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