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Laminar free convection heat transfer from a horizontal ring

Wärmeübergang bei laminarer freier Konvektion von einem horizontalen Ring

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Abstract

The analytical solution of laminar free convective heat transfer in an unlimited space from an isothermal horizontal ring with an adiabatic plug is presented. The results of theoretical considerations are presented as relation of the Nusselt and Rayleigh numbers:

$$Nu_D = 1.151 \cdot (Ra_D )^{1/5} \cdot \Phi (\phi _0 )$$

\] where Φ(φ0) is a function of shape coefficient of the ring (φ0=d/D). The solution presented has been verified experimentally with rings of constant external diameter (D=0.06 [m]) and various internal diameters (d=0, 0.01, 0.02, 0.04 and 0.05 [m]). The fluid tested was glycerin. The theoretical predictions agree well with the experimental results.

Zusammenfassung

Es wird die analytische Lösung für den Wärmeübergang bei laminarer freier Konvektion von einem horizontalen Ring mit adiabater Innenscheibe an den unbegrenzten Außenraum angegeben. Die Ergebnisse der theoretischen Überlegungen sind als Funktion der Nusselt-Zahl und der Rayleigh-Zahl dargestellt

$$Nu_D = 1.151 \cdot (Ra_D )^{1/5} \cdot \Phi (\phi _0 ),$$

wobei Φ(φ0) eine Funktion des Formkoeffizienten der Ringgeometrie ist (φ0=d/D). Die angegebene Lösung wurde experimentell an Ringen konstanten Außendurchmessers (D=0,06 m) und verschiedenen Innendurchmessern (d=0; 0,01; 0,02; 0,04 und 0,5 m) verifiziert. Die Versuchsflüssigkeit war Glyzerin. Theoretische Vorausberechnungen und experimentelle Befunde stimmen gut überein.

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Abbreviations

a=λ/c pϱ:

thermal diffusivity

a=A/P=D/4:

characteristic length

A :

area of heated plate

b :

thickness of the tested fluid between rings

c p :

specific heat at constant pressure of the fluid

C D=1.151·Φ(φ 0):

coefficient

d :

internal diameter of the ring

D :

external diameter of the ring

F :

coefficient in Eq. (15)

g :

gravitational acceleration

H :

height of a fluid column above rings

h :

specific enthalpy

K :

constant (Eq. (15))

L=D/2:

characteristic length

m :

mass flux

Nu D=αD/λ, Nu R=αR/λ, Nu(D−d)=α(D−d)/λ, NuL=αL/λ:

Nusselt numbers

p :

pressure

P :

perimeter of the heated plate

q :

heat flux density

Q :

heat flux

r :

internal radius of the rings

R :

external radius of the rings

Ra D=gβΔTD 3/(νa),Ra R=gβΔTR 3/(νa),Ra (D−d)=gβΔT(D−d)3/(νa),Ra L=gβΔTL 3/(νa):

Rayleigh numbers

Ra cr=gβΔTb 3/(νa):

critical value of Rayleigh number

T :

temperature

u :

parameter in Eq. (18)

W :

velocity

x :

length of the boundary layer

x :

horizontal coordinate to the surface

y :

vertical coordinate to the surface

α :

heat transfer coefficient

β :

coefficient of volumetric expansion

δ :

thickness of boundary layer

θ :

dimensionless temperature

Δ :

difference

λ :

thermal conductivity

ν :

kinematic viscosity

ϱ :

fluid density

φ=r/R=d/D :

dimensionless radius or diameter of the rings

φ 0=d/D=r/R :

shape coefficient of the ring

Φ(φ 0):

coefficient defined by Eq. (34)

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Authors and Affiliations

  1. Dept. of Chemical Engineering, Technical University of Gdańsk, ul. Majakowskiego 11/12, 80-952, Gdańsk, Poland

    W. M. Lewandowski, P. Kubski & J. M. Khubeiz

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  1. W. M. Lewandowski
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  2. P. Kubski
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  3. J. M. Khubeiz
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Lewandowski, W.M., Kubski, P. & Khubeiz, J.M. Laminar free convection heat transfer from a horizontal ring. Wärme- und Stoffübertragung 29, 9–16 (1993). https://doi.org/10.1007/BF01577454

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