# Three dimensional freezing around a coolant-carrying tube

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## Abstract

The 3-D freezing of water around a coolant carrying horizontal tube placed in an adiabatic rectangular cavity is investigated mainly by means of a numerical analysis. The results are not sensitive to the coolant flow velocity and the tube length, but are very responsive to the coolant inlet temperature and the initial water temperature. The numerical analysis predicts fairly experimental results without introducing a heat transfer coefficient.

## Keywords

Heat Transfer Coefficient Natural Convection Phase Change Material Thermal Boundary Layer Thickness Stanton Number## List of symbols

- Bi
Biot number,

*α*(*d*_{ i }/2)/*k*_{ s }*c*Specific heat

*C*Coefficient in the flow resistance (=−10

^{−6}, Eq. 4)*d*_{i}Inner diameter of the tube (=17.05 mm)

*d*_{0}Outer diameter of the tube (=19.05 mm)

*DTF*Temperature range (0.2°C) in the mushy zone (0 <

*f*_{s}< 1)*f*Mass fraction

*F*Flow resistance (Eq. 4)

- FS
Solid mass fraction (=

*f*_{ s })*h*_{f}Latent heat of freezing

*H*Height/width of the rectangular cavity

*k*(Effective) thermal conductivity

*L*Tube length

*r*_{i}Inner radius of the tube (=

*d*_{i}/2)- St
Stanton number,

*α*/*w*_{ m }(*cρ*)_{ c }- Ste
Stefan number,

*c*_{ s }(*T*_{ ph }−*T*_{inlt})/*h*_{ f }*t*Time

*T*Temperature

*T*_{ph}Freezing temperature (=0°C)

*u**X*-component velocity*v**Y*-component velocity**V**Velocity vector composed of

*u, v*and*w**w**Z*-component velocity*w*_{m}Mean velocity of the coolant in the tube

*x*Horizontal coordinate

*y*Vertical coordinate

*z*Axial coordinate

## Greek symbols

- α
Heat transfer coefficient of the coolant

*γ*Volume fraction

- δ
Local frozen layer thickness from the tube outer surface

*δ*_{m}Mean frozen layer thickness on

*x*–*y*cross section*δ*_{mz}Total mean frozen layer thickness (=(

*δ*_{m,z=0}+*δ*_{m,z=0.5L}+*δ*_{m,z=L})/3, corresponding to the frozen layer thickness measured by experiment)*ρ*Density

## Subscripts

- c
Coolant

- ini
Initial

- inlt
Inlet

- ℓ
Liquid (water)

- s
Solid (ice)

*x**x*-direction*y**y*-direction*z**z*-direction

## Notes

### Acknowledgments

The authors wish to acknowledge support for this study by the technical official T. Fujita.

## References

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