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Heat and Mass Transfer

, 46:119 | Cite as

On the Stefan problem with volumetric energy generation

  • John C. CrepeauEmail author
  • Ali Siahpush
  • Blaine Spotten
Original

Abstract

This paper presents results of solid–liquid phase change, driven by volumetric energy generation (VEG), in a vertical cylinder. We show excellent agreement between a quasi-static, approximate analytical solution valid for Stefan numbers less than one, and a computational model solved using the computational fluid dynamics code FLUENT®. A computational study also shows the effect that the VEG has on both the mushy zone thickness and convection in the melt during phase change.

Keywords

Phase Change Rayleigh Number Mushy Zone Recirculation Zone Stefan Problem 
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.

List of symbols

cp

Specific heat, J/kg K

D

Diameter, m

fLF

Liquid fraction

g

Gravitational constant, m/s2

H

Height of cylinder, m

Δhf

Latent heat of fusion, J/kg

k

Thermal conductivity, W/m K

lz

Unit vector, z-direction

p

Pressure, Pa

\( \dot{q} \)

Volumetric energy generation, W/m3

Q

Nondimensional volumetric energy generation, constant surface temperature

\( \dot{Q} \)

Nondimensional volumetric energy generation, constant surface heat flux

\( Q^{\prime\prime} \)

Nondimensional heat flux

r

Radius, m

r0

Radius of cylinder, m

Ra

Rayleigh number

s

Distance from centerline to the phase change front, m

St

Stefan number

t

Time, s

T

Temperature, K

T0

Surface temperature, K

TL

Liquidus temperature, K

Tm

Melt, or fusion, temperature, K

TS

Solidus temperature, K

ΔT

T L − T S

u

Velocity, m/s

Greek symbols

α

Thermal diffusivity, m2/s

β

Thermal expansion coefficient, 1/K

θ

Nondimensional temperature

μ

Dynamic viscosity, kg/m s

ν

Kinematic viscosity, m2/s

ρ

Density, kg/m3

τ

Nondimensional time

ξ

Nondimensional distance to phase change front

Subscripts

liq

Liquid

sol

Solid

ss

Steady-state

Notes

Acknowledgments

The authors would like to thank Mr. Michael Kennedy for his assistance with the figures.

References

  1. 1.
    Stefan J (1889) Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere. Sitzungsberichte der k.k. Akademie der Wissenschaften in Wien, Mathematische-Naturwissenschaften, Abteilung II, 965–983Google Scholar
  2. 2.
    Rubinstein LI (1971) The Stefan problem. AMS Publications, ProvidenceGoogle Scholar
  3. 3.
    Viskanta R (1988) Heat transfer during melting and solidification of metals. J Heat Transfer 110:1205–1219CrossRefGoogle Scholar
  4. 4.
    Yao LS, Prusa J (1989) Melting and freezing. Adv Heat Transfer 19:1–95zbMATHGoogle Scholar
  5. 5.
    Huppert HE (1990) The fluid mechanics of solidification. J Fluid Mech 212:209–240CrossRefMathSciNetGoogle Scholar
  6. 6.
    Tien CH, Yen YC (1966) Approximate solution of a melting problem with natural convection. AICHE J 62:166–172Google Scholar
  7. 7.
    Mbaye M, Bilgen E (2001) Phase change process by natural convection-diffusion in rectangular enclosures. Heat Mass Transfer 37:35–42CrossRefGoogle Scholar
  8. 8.
    Tritton DJ, Zarraga MN (1967) Convection in horizontal layers with internal heat generation. Experiments. J Fluid Mech 30:21–31CrossRefGoogle Scholar
  9. 9.
    Roberts PH (1967) Convection in horizontal layers with internal heat generation. Theory. J Fluid Mech 30:33–49CrossRefGoogle Scholar
  10. 10.
    Martin BW (1967) Free convection in a vertical cylinder with internal heat generation. Proc R Soc A 301:327–341CrossRefGoogle Scholar
  11. 11.
    Dhir VK (1997) Heat transfer from heat generating pools and particulate beds. Adv Heat Transfer 29:1–57Google Scholar
  12. 12.
    Worster MG (1992) Instabilities of the liquid and mushy regions during solidification of alloys. J Fluid Mech 237:649–669zbMATHCrossRefGoogle Scholar
  13. 13.
    Worster MG (1997) Convection in mushy layers. Ann Rev Fluid Mech 29:91–122CrossRefMathSciNetGoogle Scholar
  14. 14.
    Al-Rawahi N, Tryggvason G (2004) Numerical simulation of dendritic solidification with convection: three-dimensional flow. J Comput Phys 194:677–696zbMATHCrossRefGoogle Scholar
  15. 15.
    Chen WL, Ishii M, Grolmes MA (1976) Simple heat conduction model with phase change for reactor fuel pin. Nucl Sci Eng 60:452–460Google Scholar
  16. 16.
    El-Genk M, Cronenburg AW (1978) An assessment of fuel freezing and drainage phenomena in a reactor shield plug following a core disruptive accident. Nucl Eng Des 47:195–225CrossRefGoogle Scholar
  17. 17.
    Kikuchi Y, Shigemasa Y (1982) Liquid solidification in laminar tube flow with internal heat sources. Nucl Eng Des 75:73–80CrossRefGoogle Scholar
  18. 18.
    Cheung FB, Chawla TC, Pedersen DR (1984) The effects of heat generation and wall interaction on freezing and melting in a finite slab. Int J Heat Mass Transfer 27:29–37CrossRefGoogle Scholar
  19. 19.
    Chan SH, Hsu KY (1987) Generalized phase change model for melting and solidification with internal heat generation. J Thermophysics 1:171–174CrossRefGoogle Scholar
  20. 20.
    Crepeau J, Siahpush A (2008) Approximate solutions to the Stefan problem with internal heat generation. Heat Mass Transfer 44:787–794CrossRefGoogle Scholar
  21. 21.
    Incropera F, DeWitt D (2002) Fundamentals of heat and mass transfer, 4th edn. Wiley, New York, pp 100–110Google Scholar
  22. 22.
    Poulikakos D (1994) Conduction heat transfer. Prentice Hall, Englewood CliffsGoogle Scholar
  23. 23.
    Voller R, Prakash C (1987) A fixed-grid numerical modeling methodology for convection-diffusion mushy region phase-change problems. Int J Heat Mass Transfer 30:1709–1720CrossRefGoogle Scholar
  24. 24.
    Spotten B (2008) Computational fluid dynamic simulations of the Stefan problem with internal heat generation. Thesis, University of IdahoGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • John C. Crepeau
    • 1
    Email author
  • Ali Siahpush
    • 2
  • Blaine Spotten
    • 1
  1. 1.Department of Mechanical EngineeringUniversity of IdahoMoscowUSA
  2. 2.Idaho National LaboratoryIdaho FallsUSA

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