Advertisement

Heat and Mass Transfer

, Volume 30, Issue 1, pp 9–16 | Cite as

Ice-water convection in an inclined rectangular cavity filled with a porous medium

  • X. Zhang
  • R. Kahawita
Originals

Abstract

This paper reports on the results of a numerical study on the equilibrium state of the convection of water in the presence of ice in an inclined rectangular cavity filled with a porous medium. One side of the cavity is maintained at a temperature higher than the fusion temperature while the opposite side is cooled to a temperature lower than the fusion temperature. The two remaining sides are insulated. Results are analysed in terms of the density inversion parameter, the tilt angle, and the cooling temperature. It appears that the phenomenon of density inversion plays an important role in the equilibrium of an ice-water system when the heating temperature is below 20°. In a vertical cavity, the density inversion causes the formation of two counterrotating vortices leading to a water volume which is wider at the bottom than at the top. When the cavity is inclined, there exist two branches of solutions which exhibit the bottom heating and the side heating characteristics, respectively (the Bénard and side heating branches). Due to the inversion of density, the solution on the Bénard branch may fail to converge to a steady state at small tilt angles and exhibits an oscillating behavior. On the side heating branch, a maximum heat transfer rate is obtained at a tilt angle of about 70° but the water volume was found to depend very weakly on the inclination of the cavity. Under the effect of subcooling, the interplay between conduction in the solid phase and convection in the liquid leads to an equilibrium ice-water interface which is most distorted at some intermediate cooling temperature.

Keywords

Porous Medium Tilt Angle Heat Transfer Rate Starke Fusion Temperature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

cp

heat capacity, J/kg°C

g

gravitational acceleration, m/s2

H

height of the cavity, m

k

conductivity, W/m °C

K

permeability, m2

L

length of the cavity, m

n

unit vector normal to the solid-liquid interface

Nu

average temperature gradient at the heated surface

P

pressure, N/m2

q

1.894816, constant in the water density expression

R

k e s /k e l

Ra

λgΔT q KH/(να e l ), Rayleigh number

S

S*/H, dimensionless interface position

Tc*

cooling temperature, °C

Tf*

fusion temperature, °C

Th*

superheating temperature, °C

Tl

(Tl*T f * )/ΔT, dimensionless temperature in the liquid

Tm

4.029325 °C, maximum density temperature

TR

T c * /T h *

Ts

(Ts*T f * )/ΔT, dimensionless temperature in the solid

V

velocity in the liquid region, m/s

x, y

Cartesian coordinates

XL

H/L, aspect ratio of the cavity

Greek symbols

αel

k e l /(ρc)l, thermal diffusivity, m2/s

β

(T m T f * )/(T h * T f * ), inversion parameter

ΔT

T h * T f * , temperature scale, °C

θ

tilt angle of the cavity

λ

9.297173·10−6°Cq, constant in the water density expression

μ

viscosity, N s/m2

v

kinematic viscosity, m2/s

ρ

density, kg/m3

ρm

999.972 kg/m3, maximum density of water at 4°C

ϕ

ϕ e l , dimensionless stream function

Superscripts

l

refers to the liquid

m

refers to the porous matrix

s

refers to the solid

*

dimensional variables

Subscripts

e

effective property of the saturated porous medium

f

quantity at fusion point

m

refers to maximum density point

max

maximum value

min

minimum value

Eis/Wasser-Konvektion in einem mit porösem Medium befüllten, geneigten, rechteckigen Behälter

Zusammenfassung

Es wird über Ergebnisse einer numerischen Studie berichtet, welche die Ermittlung des Gleichgewichtszustandes bei Konvektion von Wasser in Gegenwart von Eis in einem geneigten, rechteckigen, mit porösem Medium befüllten Behälter zum Ziele hatte. Eine Seite des Behälters wird über Schmelztemperatur, die andere darunter gehalten, während die übrigen zwei Seiten wärmedicht sind. Die Auswertung der Resultate erfolgt hinsichtlich des Dichteinversionsparameters, des Neigungswinkels und der Kühlplattentemperatur. Unterhalb einer Heiztemperatur von 25°C wird der Gleichgewichtszustand des Eis/Wasser-Systems entscheidend durch das Phänomen der Dichteanomalie beeinflußt. In einem Vertikalbehälter verursacht letztere die Ausbildung von zwei gegensinnig drehenden Walzen, was zu einer Verbreiterung der Wasserzone am Boden gegenüber der im oberen Bereich führt. Bei geneigtem Behälter existieren zwei Lösungszweige, welche die Boden-bzw. die Seitenheizungscharakteristik repräsentieren (Bénard- und Seitenheizungszweig). Aufgrund der Dichteinversion kann die Lösung auf dem Bénardzweig bei kleinen Neigungswinkeln oszillatorisches Verhalten anstatt Konvergenz liefern. Auf dem anderen Zweig ergibt sich bei 70° Neigungswinkel ein Maximum des Wärmestroms. Das Volumen der Wasserzone zeigt nur schwache Abhängigkeit vom Neigungswinkel. Bei Berücksichtigung des Unterkühlungseffektes führt die Wechselwirkung zwischen Leitung in der festen und Konvektion in der flüssigen Phase zu einer Gleichgewichtsstruktur der Eis/Wasser-Grenzfläche, welche bei einigen intermediären Kühlungsplattentemperaturen äußerst starke Unregelmäßigkeiten aufweist.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Jany, P.;Bejan, A.: The scales of melting in the presence of natural convection in a rectangular cavity filled with porous medium. J. Heat Transfer 110 (1988) 526–529Google Scholar
  2. 2.
    Bejan, A.: Theory of melting with natural convection in an enclosed porous medium. J. Heat Transfer 111 (1989) 407–415Google Scholar
  3. 3.
    Kazmierczak, M.;Poulikakos, D.;Pop, I.: Melting from flat plate embedded in a porous medium in the presence of steady natural convection. Numerical Heat Transfer 10 (1986) 571–581Google Scholar
  4. 4.
    Okada, M.;Fukumoto, R.: Melting around a horizontal pipe embedded in a frozen porous medium. Trans. Japan Soc. Mech. Engrs. 48B (1982) 2041–2049Google Scholar
  5. 5.
    Okada, M.: Analysis of heat transfer during melting from a vertical wall. Int. J. Heat Transfer 27 (1984) 2057–2066CrossRefMATHGoogle Scholar
  6. 6.
    Kazmierczak, M.;Poulikakos, D.;Sadowski, D.: Melting of a vertical plate in porous medium controlled by forced convection of a dissimilar fluid. Int. Comm. Heat Mass Transfer 14 (1987) 507–518Google Scholar
  7. 7.
    Beckman, C.;Viskanta, R.: Natural convection solid/liquid phase change in porous media. Int. J. Heat Mass Transfer 31(1) (1988) 35–46Google Scholar
  8. 8.
    Prud'homme, M.;Nguyen, T. H.;Nguyen, D. L.: A heat transfer analysis for solidification of slabs, cylinders and spheres. J. Heat Transfer 111 (1989) 690–705Google Scholar
  9. 9.
    Wu, Y. K.;Prud'homme, M.;Nguyen, T. H.: Etude de la fusion autour d'un cylindre vertical soumis à deux types de conditions limites. Int. J. Heat Mass Transfer 32 (1989) 1927–1938CrossRefGoogle Scholar
  10. 10.
    Prud'homme, M.;Nguyen, T. H.;Wu, Y. K.: Simulation numérique de la fusion à l'intérieur d'un cylindre adiabatique chauffé par le bas. Int. J. Heat Mass Transfer 34 (1991) 2275–2286CrossRefGoogle Scholar
  11. 11.
    Nguyen, T. H.; Zhang, X.: Onset and evolution of penetrative convection during the melting process in a porous medium. In Heat and Mass Transfer in Porous Media, Elsevier Science Publisher (1992) 381–392Google Scholar
  12. 12.
    Caltagirone, J. P.;Bories, S.: Solutions and stability criteria of natural convective flow in an inclined porous layer. J. Fluid mech. 155 (1985) 267–287Google Scholar
  13. 13.
    Moya, S. L.;Sen, M.: Numerical study of natural convection in a tilted rectangular porous material. Int. J. Heat Mass Transfer 30(4) (1987) 741–756CrossRefGoogle Scholar
  14. 14.
    Gebhart, B.;Mollendorf, J.: A new density relation for pure and saline water. Deep-Sea Research 124 (1977) 831–848Google Scholar
  15. 15.
    Zhang, X.;Nguyen, T. H.;Kahawita, R.: Melting of ice in a porous medium heated from below. Int. J. Heat Mass Transfer 34(2) (1991) 389–405CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • X. Zhang
    • 1
  • R. Kahawita
    • 2
  1. 1.Department of Mechanical EngineeringEcole Polytechnique de MontréalMontréalCanada
  2. 2.Department of Civil EngineeringEcole Polytechnique de MontréalMontréalCanada

Personalised recommendations