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Heat and Mass Transfer

, Volume 30, Issue 5, pp 349–359 | Cite as

Combined free and forced convection flow in a cooled vertical duct with internal solidification

  • B. Weigand
  • O. Neumann
  • T. Strohmayer
  • H. Beer
Article

Abstract

The effect of mixed convection flow on the shape of the frozen crust in a cooled vertical channel was investigated numerically. For the prediction of the ice-layer thickness a simple numerical model which is based on the boundary layer equations was used. It can be seen that in case of assisting mixed convection flow the heat transfer at the solid crust increases because of inreasing velocity near the solid-liquid interface. On the other hand this increase of the velocity near the solid-liquid interface can lead to flow separation in the core region of the channel because of continuity of mass. By comparing the numerically obtained results for aiding mixed flow with measurements of Campbell and Incropera [10] good agreement can be observed.

In case of opposing mixed flow it can be shown that flow separation might occur near the solid-liquid interface. This can result in a wave-like structure of the ice-layer.

Keywords

Heat Transfer Flow Separation Forced Convection Mixed Convection Boundary Layer Equation 
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

a

thermal diffusivity

B

dimensionless freezing parameter (T W T F )/(T F T0)k s /k

F

modified stream function

Gr4h

Grashof number based on 4h, [∣ρ(T F )−ρ(T0)∣/ρ(T0)]g(4h)3/v2

Grh

Grashof number based onh, [∣ρ(T F )−ρ(T0)∣/ρ(T0)]gh3/v2

H

temperature function, defined by Eq. (14)

h

distance from centerline to wall

k

thermal conductivity

Nu

Nusselt number based on 4δ

P

pressure

Pr

prandtl number

Re4h

Reynolds number based on 4h, ū04h/v

Reh

Reynolds number based onh, ū0h/v

Ri4h

Richardson number based on 4h,Gr4th/Re 4h 2

Rih

Richardson number based onh,Gr h /Re th 2

T

temperature

Tb

bulk temperature,\(\int\limits_0^\delta {Tudy/} \int\limits_0^\delta {udy} \)

TF

freezing temperature of the liquid

T0

constant inlet temperature of the liquid

TW

wall temperature

u, v

velocity components

ū0

mean axial velocity at the inlet

x, y

coordinates

Greek symbols

βi

constants for the density-temperature relationship for water

δ

distance between centerline and solid-liquid interface

θ

dimensionless temperature, (T−T F )/(T0T F )

θs

dimensionless temperature in the solid,(T s T F )/(T W T F )

ν

kinematic viscosity

ρ

density

ψ

stream function

Subscripts

0

at the entrance

d

dynamic

s

solid

w

at the wall

Superscripts

*, ∼

dimensionless quantity

mean quantity

Gemischte Konvektion in einem gekühlten, vertikalen Kanal mit Eisschichtbildung

Zusammenfassung

Überlagerte freie und aufgezwungene Konvektionsströmung in einem gekühlten vertikalen Kanal mit Eisschichtbildung an den Wänden ist numerisch untersucht worden. Grundlage der Eisschichtberechnung ist das einfache numerische Modell der Grenzschichtgleichungen. Für den Fall der dem Gravitationsvektor gleichgerichteten, gemischten Konvektion wird eine Verstärkung des Wärmeübergangs beobachtet, da die Strömungsgeschwindigkeit in der Nähe der Phasengrenze zunimmt. Aufgrund der Massenerhaltung kann es bei einer ausgeprägten Geschwindigkeitszunahme in der Nähe der Phasengrenze zu Rückströmungen in der Kanalmitte kommen. Die numerischen Ergebnisse zeigen gute Übereinstimmung mit Messungen von Campbell und Incropera [10].

Für den Fall der dem Gravitationsvektor entgegengerichteten Strömung kann es zur Strömungsablösung nahe der Phasengrenze kommen, die eine wellenartige Eisschicht bewirkt.

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References

  1. 1.
    Zerkle, R. D.;Sunderland, J. E.: The effect of liquid solidification in a tube upon laminar-flow heat transfer and pressure drop. J. Heat Transfer 90 (1968) 183–190Google Scholar
  2. 2.
    Özisik, M. N.;Mulligan, J. C.: Transient freezing of liquids in forced flow inside circular tubes. J. Heat Transfer 91 (1969) 233–240Google Scholar
  3. 3.
    Bilenas, J. A.;Jiji, L. M.: Numerical solution of a non-linear free boundary problem of axisymmetric fluid flow in tubes with surface solidification. Proc. 4th Int. Heat Transfer Conf. 1, Paris, Amsterdam: Elsevier Publishing Company, Cu 2. 1 (1970) 1–11Google Scholar
  4. 4.
    Chida, K.: Heat Transfer in steady laminar pipe flow with liquid solidification. Heat Transfer: Japanese Research 81 (1983) 81–94Google Scholar
  5. 5.
    Lee, D. G.;Zerkle, R. D.: The effect of liquid solidification in a parallel plate channel upon laminar-flow heat transfer and pressure drop. J. Heat Transfer 91 (1969) 583–585Google Scholar
  6. 6.
    Kikuchi, Y.;Shigemasa, Y.;Oe, A.;Ogata, T.: Steady-state freezing of liquids in laminar flow between two parallel plates. J. Nucl. Sci. Technol. 23 (1986) 43–55Google Scholar
  7. 7.
    Weigand, B.;Beer, H.: Heat transfer and solidification of a laminar liquid flow in a cooled parallel plate channel. Wärme- und Stoffübertragung 26 (1991) 233–240Google Scholar
  8. 8.
    Weigand, B.;Höhn, W.;Beer, H.: A perturbation analysis of transient freezing of a laminar liquid flow in a cooled two-dimensional channel. J. Heat Transfer 115 (1993) 294–301Google Scholar
  9. 9.
    Bennon, W. D.;Incropera, F. P.: Developing laminar mixed convection with solidification in a vertical channel. J. Heat Transfer 110 (1988) 410–415Google Scholar
  10. 10.
    Campbell, J. S.;Incropera, F. P.: Mixed convection solidification in a vertical channel. proc. 9th Int. Heat Transfer Conf. 4, Jerusalem, 12PC17 (1990) 311–316Google Scholar
  11. 11.
    Cebeci, T.;Bradshaw, P.: Physical and Computational Aspects of Convective Heat Transfer, Springer, New York (1984)Google Scholar
  12. 12.
    Fujii, T.: Fundamentals of Free Convection Heat Transfer (in Japanese) Progress in Heat Transfer Engineering 3 (1974) 66Google Scholar
  13. 13.
    Cheng, K. C.;Takeuchi, M.: Transient natural convection of water in a horizontal pipe with constant cooling rate through 4°C. J. Heat Transfer 98 (1976) 581–587Google Scholar
  14. 14.
    Landolt-Börnstein, Zahlenwerte und Funktionen. Band II, Teil 1, Springer (1971) 36–37Google Scholar
  15. 15.
    Reyhner, T. A.;Flügge-Lotz, I.: The interaction of a shock wave with a laminar boundary layer. Int. J. Nonlinear Mech. 3 (1968) 173–199CrossRefGoogle Scholar
  16. 16.
    Cebeci, T.;Lee, K. H.;Wang, S.;Chang, K. C.: Heat Transfer in vertical duct flows. Math. Eng. Ind. 1 (1987) 67–81Google Scholar
  17. 17.
    Kwon, O. K.;Pletcher, R. H.;Lewis, J. P.: Prediction of sudden expansion flows using the boundary-layer equations. J. Fluid Engng. 106 (1984) 285–291Google Scholar
  18. 18.
    Aung, W.;Worku, G.: Developing flow and flow reversal in a vertical channel with asymmetric wall temperature. J. Heat Transfer 108 (1986) 299–304Google Scholar
  19. 19.
    Morton, B.;Ingham, D. B.;Keen, D. J.;Heggs, P. J.: Recirculating combined convection in laminar pipe flow. J. Heat Transfer 111 (1989) 106–113Google Scholar
  20. 20.
    Gilpin, R. R.: Ice formation in a pipe containing flows in the transition and turbulent regimes. J. Heat Transfer 103 (1981) 363–369Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • B. Weigand
    • 1
  • O. Neumann
    • 2
  • T. Strohmayer
    • 2
  • H. Beer
    • 2
  1. 1.ABB Power Generation LtdBadenSwitzerland
  2. 2.Institut für Technische ThermodynamikTechnische Hochschule DarmstadtDarmstadtGermany

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