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Transient natural convection in horizontal cylinders with constant cooling rate

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Abstract

Transient natural convection in horizontal cylinders with wall temperature decreasing at a constant rate is approached by a numerical method. Numerical solutions are obtained for Pr=0.7, 10, 100 and a range of modified Grashof numbers Gra=103 ∼ 109. The time-dependent flow and temperature fields, local and overall heat transfer rates are presented. For quasi-steady state, a generalized correlation equation for Nusselt number valid for Pr ⩾0.7 and GrPr < 107 is developed.

Zusammenfassung

Nichtstationäre natürliche Konvektion in horizontalen Zylindern mit stetig fallender Wand-temperatur wurde numerisch berechnet und zwar für Pr=0,7, 10 und 100 im Bereich der modifizierten Grashof-Zahl Gra von 103 bis 109. Mitgeteilt werden die zeitabhängigen Stromlinien- und Temperaturf eider so wie der örtliche und der totale Wärmeübergang. Für den quasistationären Fall wird eine verallgemeinerte Nusselt-Funktion für Pr ⩾0,7 und GrPr < 107 angegeben.

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Abbreviations

a:

inside radius of cylinder [m]

cp :

specific heat at constant pressure [J/kg°C]

Gr:

Grashof number, Eq(l3)

Gra :

modified Grashof number, gβa3 ΔT/ν2

g:

gravitational acceleration [m/h2]

H:

wall cooling rate [°C/h]

k:

thermal conductivity of fluid [J/mh°C]

M, N:

number of divisions in r and Φ directions

Nu:

Nusselt number, Eq(6)

Nua :

dimensionless average wall heat flux, Eq(5)

(Nua)Φ :

dimensionless local wall heat flux

Pr:

Prandtl number

q:

average wall heat flux [J/m2h]

R:

radial coordinate [m]

r:

dimensionless radial coordinate, R/a

T:

temperature [°C]

Tb :

mixed mean temperature [°C]

Tw :

wall temperature [°C]

T0 :

uniform initial fluid temperature [°C]

t:

time [h]

U:

velocity in R direction [m/h]

u:

dimensionless velocity in r direction, aU/χ

V:

velocity in Φ direction [m/h]

v:

dimensionless velocity in Φ direction, aV/χ

β:

coefficient of thermal expansion [1/°C]

θ:

dimensionless temperature, (T-Tw)/ΔT

θb :

dimensionless mixed mean temperature, Eq.(6)

χ:

thermal diffusivity [m2/h]

ν:

Kinematic viscosity [m2/h]

ξ:

dimensionless vorticity, a2Ω/χ

ρ:

fluid density [kg/m3]

τ:

dimensionless time (Fourier number), χt/a2

Φ:

polar coordinate [radian]

Ψ:

stream function [m2/h]

ψ:

dimensionless stream function, ψ/χ

Ω:

vorticity [1/h]

ΔT:

characteristic temperature difference, a2h/χ [°C]

i,j:

indices of a grid point in r and Φ directions

k:

kth time step

References

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Authors and Affiliations

  1. Department of Mechnical Engineering, University of Alberta, Edmonton, Alberta, Canada

    M. Takeuchi &  K. C. Cheng

Authors
  1. M. Takeuchi
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  2. K. C. Cheng
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Takeuchi, M., Cheng, K.C. Transient natural convection in horizontal cylinders with constant cooling rate. Wärme- und Stoffübertragung 9, 215–225 (1976). https://doi.org/10.1007/BF01003574

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  • DOI: https://doi.org/10.1007/BF01003574

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