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Analysis of fully developed opposing mixed convection between inclined parallel plates

Berechnung der voll entwickelten entgegengesetzt gerichteten Misch-Konvektion zwischen geneigten parallelen Platten

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Abstract

An exact solution is presented of fully developed, laminar flow between inclined parallel plates with a uniform wall heat flux boundary condition. The flow is downward and the heat flux is into the channel, so that natural convection opposes the forced flow. The solution depends on the two parametersP 1=Gr sinθ/Re andP 2=Gr cosθ/Re 2 Pr. Four different flow reversal regimes are observed: 1) no reversal, 2) top reversal, 3) bottom reversal, and 4) top and bottom reversal. Velocity profiles, temperature profiles, wall friction, and Nusselt numbers are presented. Despite the simplicity of the problem which has been analyzed, it does display some features which have been observed in real mixed convection flows, such as flow reversal and nonmonotonic dependence on tilt angle.

Zusammenfassung

Es wird eine exakte Lösung für voll entwickelte laminare Strömung zwischen geneigten parallelen Platten mit einheitlichem Wand-Wärmestrom als Randbedingung dargestellt. Die Strömung ist abwärts gerichtet und der Wärmestrom führt in den Kanal, so daß die freie Konvektion der erzwungenen entgegengesetzt gerichtet ist. Die Lösung hängt von den beiden Parametern P1=Gr sinθ/Re und P2=Gr cosθ/Re 2 Pr ab. Vier verschiedene Bereiche der Strömungsumkehr wurden betrachtet: 1) keine Richtungsumkehr, 2) Umkehr an der Oberseite, 3) Umkehr an der Unterseite und 4) Umkehr an Ober- und Unterseite. Es wurden Geschwindigkeits- und Temperaturprofile, Wandreibung und Nusselt-Zahlen dargestellt. Trotz der Einfachheit des analysierten Problems werden einige Dinge dargestellt, welche in realer gemischter Konvektion untersucht wurden, so z.B. Strömungsumkehr und die nicht-monotone Abhängigkeit vom Schrägungswinkel.

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Abbreviations

f :

friction factor=τw/(1/2ϱou−2)

g :

gravitational acceleration constant=9.8 m/s2

Gr :

Grashof number=βqL4/kv2

k :

fluid thermal conductivity

L:

channel width

Nu :

Nusselt number=2qL/k(T w −T h )

p :

fluid thermodynamic pressure

P :

nondimensional pressure=[p−ϱ o g(x sinθ−y cosθ)]/Pr ϱ o u −

Pr :

Prandtl number=v/α

P 1 :

Gr sinθ/Re

P2 :

Gr cosθ/Re 2 Pr

q :

wall heat flux

Re :

Reynolds number=−u L/v

T :

fluid temperature

T b :

fluid bulk temperature

T o :

constant reference temperature

T w :

fluid temperature

u :

axial velocity

ū :

average velocity

U :

nondimensional velocity=u/ū

x :

axial coordinate

X :

nondimensional axial coordinate=xα/−uL 2

y :

transverse coordinate

Y :

nondimensional transverse coordinate=y/L

α :

fluid thermal diffusivity

β :

fluid thermal expansion coefficient

θ :

tilt angle, measured counterclockwise from horizontal in Fig. 1

v:

fluid kinematic viscosity

ϱ 0 :

fluid density evaluated atT 0

τw :

wall shear stress

φ :

nondimensional temperature=(T−T o )/(q L/k)

φ b :

nondimensional bulk temperature=f U φdY

φw :

nondimensional wall temperature=(T w −T 0)/(qL/k)

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Authors and Affiliations

  1. Mechanical, Aerospace and Nuclear Engineering Department, University of California, Los Angeles, 90024, Los Angeles, CA, USA

    A. S. Lavine (Assistant Professor)

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  1. A. S. Lavine
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Lavine, A.S. Analysis of fully developed opposing mixed convection between inclined parallel plates. Wärme- und Stoffübertragung 23, 249–257 (1988). https://doi.org/10.1007/BF01807328

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  • DOI: https://doi.org/10.1007/BF01807328

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