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Transient laminar free convection in horizontal cylinders

Laminare freie Konvektion in horizontalen Zylindern

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Abstract

A numerical study of laminar natural convection inside uniformly heated, partially or fully filled horizontal cylinders is made. A coordinate transformation which simplifies the discretization of the equations of motion and energy is utilized. The resulting system of partial differential equations with their boundary conditions is solved using central differences for various Prandtl and Grashof numbers for two different grid sizes. The flow in completely filled cylinders for which experimental data are available is predicted. Close agreement between steady-state predictions and experiments is obtained for temperature and velocity profiles as well as for the streamline contours and isotherms. The technique is further demonstrated by solving the transient natural convection flow inside a partially filled horizontal cylinder with an adiabatic free surface and subjected to uniform wall heating.

Zusammenfassung

Es wurde eine numerische Berechnung der laminaren, freien Konvektion in gleichmäßig beheizten, teilweise oder ganz gefüllten, horizontalen Zylindern durchgeführt. Dabei wird eine Koordinatentransformation benützt, welche die Diskretisierung der Bewegungs- und der Energiegleichung vereinfacht. Das so resultierende System von partiellen Differentialgleichungen wird, zusammen mit seinen Randbedingungen, unter Verwendung einer Differenzenmethode für verschiedene Prandtl und Grashof-Zahlen sowie für zwei verschiedene Gittergrößen gelöst. Für den vollständig gefüllten Zylinder, für den experimentelle Daten verfügbar sind, wird die Strömung vorhergesagt. Dabei wird für stationäre Zustände gute Übereinstimmung zwischen Rechnung und Experiment erzielt. Dies gilt sowohl für den Verlauf der Stromlinien als auch für den der Isothermen. Das Verfahren wird weiterhin am Beispiel der Berechnung instationärer, freier Konvektion in einem partiell gefüllten, horizontalen Zylinder demonstriert, wobei eine adiabate, freie Oberfläche und gleichmäßige Beheizung der Wand angenommen sind.

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Abbreviations

g :

acceleration due to gravity, m/s2

Gr *R :

modified Grashof number =gβqR4/kv2

Gr R :

Grashof number =gβΔTR3/v2

H :

heat function vector, dimensionless

k :

thermal conductivity, W/mK

L(Y) :

cord length associated with coordinateY, dimensionless

Pr :

Prandtl number=v/α

q :

wall heat flux, W/m2

R :

radius, m

r(X, Y,Z) :

distance of a boundary point from the reference axis, dimensionless

S :

vector derived from the flow field solution, dimensionless

T :

temperature, K

T w :

wall temperature, K

T ∞ :

reference temperature, K

t :

time, s

u, v :

velocity components inx, y directions, m/s

U, V :

dimensionless velocity components inX- and Y-direction normalized withU ∞

U∞ :

reference velocity=√gβqR2/k or √gβΔTR, m/s

V :

velocity vector, dimensionless

W :

vorticity vector, dimensionless

W :

vorticity, dimensionless

x, y, z :

cartesian coordinates, m

X, Y, Z :

cartesian coordinates normalized with a reference length, dimensionless

α :

thermal diffusivity, m2/s

β :

coefficient of thermal expansion, K−1

ζ,η′,ζ′,η′ :

non-dimensional coordinates in the transformed domain

θ :

non-dimensional temperature =(T−T∞)k/qR or T−T∞/Tw−T ∞

vτ :

kinematic viscosity, m2/s

τ :

non-dimensional time=v/R2 √GrRt or v/R2 √G *R t

ϕ :

angle measured from the bottom of the cylinder, rads

ϕ * :

angle measured from the ζ axis on (ζ− η) plane, rads

Φ :

heat potential, dimensionless

Φ :

angle of incidence of the heat flux vector, rads

ψ :

non-dimensional stream function

Ψ :

vector potential, dimensionless

Δ ζ:

grid size, dimensionless

▽2 :

Laplacian operator

▽:

gradient vector

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of New Brunswick, E3B 5A3, Fredericton, N.B., Canada

    N. U. Aydemir (Research Assistant), A. C. M. Sousa (Associate Professor) & J. E. S. Venart (Professor)

Authors
  1. N. U. Aydemir
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  2. A. C. M. Sousa
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  3. J. E. S. Venart
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Aydemir, N.U., Sousa, A.C.M. & Venart, J.E.S. Transient laminar free convection in horizontal cylinders. Wärme- und Stoffübertragung 20, 59–67 (1986). https://doi.org/10.1007/BF00999738

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  • DOI: https://doi.org/10.1007/BF00999738

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