Wärme - und Stoffübertragung

, Volume 27, Issue 6, pp 377–384 | Cite as

Analytical and numerical solutions of inward spherical solidification of a superheated melt with radiative-convective heat transfer and density jump at freezing front

  • S. C. Gupta
  • P. R. Arora
Article

Abstract

Analytical and numerical solutions of a general problem related to the radially symmetric inward spherical solidification of a superheated melt have been studied in this paper. In the radiation-convection type boundary conditions, the heat transfer coefficient has been taken as time dependent which could be infinite, at time,t=0. This is necessary, for the initiation of instantaneous solidification of superheated melt, over its surface. The analytical solution consists of employing suitable fictitious initial temperatures and fictitious extensions of the original region occupied by the melt. The numerical solution consists of finite difference scheme in which the grid points move with the freezing front. The numerical scheme can handle with ease the density changes in the solid and liquid states and the shrinkage or expansions of volumes due to density changes. In the numerical results, obtained for the moving boundary and temperatures, the effects of several parameters such as latent heat, Boltzmann constant, density ratios, heat transfer coefficients, etc. have been shown. The correctness of numerical results has also been checked by satisfying the integral heat balance at every timestep.

Nomenclature

a2

constant,ktm/R02

An

coefficients in Eq. (16)

C

specific heat, J kg−1°C−1

d

constant in Eq. (6)

erf()

error function

erfc()

complementary error function

h0

heat transfer coefficient ·R0/KS, Eq. (6)

h1

heat transfer coefficient ·R0/KS, Eq. (6)

H1

term defined by Eq. (13)

k

thermal diffusivity, m2S−1

K

thermal conductivity, J m−1°C−1S−1

l

latent heat of fusion, J kg−1

P

ϱSL, Eq. (1)

q

dummy variable of integration in Eqs. (12) and (13)

R

radial coordinate/R0

R0

radius of the sphere, m

t

time, s

tm

time at which solidification starts atR=1, s

T(R, V)

temperature/Tm

\(\bar T\)(R, y)

T(R, V)

Tm

melting temperature, °C

T0

ambient temperature/Tm

V

dimensionless time, =2(asy)1/2

X(V)

dimensionless freezing front,R=X(V)

\(\bar X\)(y)

X(V)

y

dimensionless time, (t−tm)/tm

Greek symbols

α

dimensionless constant, (kS/kL)1/2

β

dimensionless constant,KL/KS

σ

Stefan-Boltzmann constant ·Tm3R0/KS

θL(1)(R)

melt temperature att=0/Tm

θL(2)(R)

dimensionless temperature, Eq. (11)

θS(1)(R)

dimensionless temperature, Eq. (12)

θS(2)(R)

dimensionless temperature, Eq. (12)

λ

dimensionless constant,l/CsTm

ϱ

density, kg m−3

Δy

time step in the numerical scheme

Subscripts

L

liquid

S

solid

Analytische und numerische Lösungen für die kugelförmige Erstarrung von einer ultrahocherhitzten Schmelze bei Wärmeabgabe durch Strahlung und Konvektion sowie Dichtesprüngen an der Erstarrungsfront

Zusammenfassung

Die analytische und numerische Lösung des Problems der symmetrischen radialen kugelförmigen Erstarrung einer ultrahocherhitzten Schmelze wurden in dieser Arbeit untersucht. Bei den Randbedingungen für Strahlung und Konvektion wurde der Wärmeübergangskoeffizient als zeitabhängig angenommen, welcher für die Zeitt=0 als unendlich betrachtet werden konnte. Dies ist notwendig für das Einsetzen des sofortigen Erstarrens der Schmelze über der gesamten Oberfläche. Die analytische Lösung besteht aus dem Verwenden geeigneter fiktiver Anfangstemperaturen und fiktiver Ausdehnungen des durch die Schmelze besetzten Anfangsbereiches. Die numerische Lösung besteht aus der Methode der finiten Differenzen, bei der die Gitterpunkte mit der Erstarrungsfront voranschreiten. Die numerische Methode kann ohne Probleme die Dichteänderungen in der flüssigen und festen Phase sowie das dadurch hervorgerufene Schrumpfen oder Expandieren des Volumens behandeln. Bei den für die sich ändernden Grenzen und Temperaturen erhaltenen numerischen Ergebnissen werden die Einflüsse der verschiedenen Parameter wie latente Wärme, Boltzmannkonstante, Dichteverhältnisse, Wärmeübergangskoeffizienten usw. gezeigt. Die Gültigkeit der numerischen Lösungen wurde zu jedem Zeitpunkt untersucht, indem die Erfüllung der Energiegleichung überprüft wurde.

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References

  1. 1.
    Crank, J.: Free and moving boundary problems. Oxford: Clarendon Press, 1984Google Scholar
  2. 2.
    Ruddle, R. W.: The solidification of Castings. Institute of Metals, London, 1957Google Scholar
  3. 3.
    Tiller, W. A.: Grain size control during ingot solidification. Part II: Columnar equiaxed transition. Trans. Metal. Soc. A.I.M.E. 224 (1962) 448–459Google Scholar
  4. 4.
    Sparrow, E. M.; Patankar, S. V.; Ramadhyni, S.: Analysis of melting in the presence of natural convection in the melt region. Transactions ASME, J. Heat Transfer 99 (1977) 520–526Google Scholar
  5. 5.
    Patel, P. D.; Boley, B. A.: Solidification problems with space and time varying boundary conditions and imperfect mold contact. Int. J. Engng. Sci. 7 (1969) 1041–1066Google Scholar
  6. 6.
    Gupta, S. C.: Two-dimensional solidification in a cylindrical mold with imperfect mold contact. Int. J. Engng. Sci. 23 (1985) 901–913Google Scholar
  7. 7.
    Gupta, S. C.: Axisymmetric melting of a long cylinder due to an infinite flux. Proc. Indian Acad. Sci. (Math. Sci.) 95 (1986) 1–12Google Scholar
  8. 8.
    Murray, W. D.; Landis, F.: Numerical and machine solutions of transient heat conduction problems involving melting or freezing. Part I: Method of analysis and sample solutions. Transactions ASME, J. Heat Transfer 81 (1959) 106–112Google Scholar
  9. 9.
    Wessling, F. C. Jr.: Radiation and convection heat transfer from an internally heated slab, cylinder, or sphere. Trans. ASME, J. Heat Transfer 92 (1972) 245–250Google Scholar
  10. 10.
    Goodling, J. S.; Khader, M. S.: Inward solidification with radiation-convection boundary condition. Transactions ASME, J. Heat Transfer 94 (1974) 114–115Google Scholar
  11. 11.
    Yeh, L. T.; Chung, B. T. F.: A variational analysis of freezing or melting in a finite medium subject to radiation and convection. Transactions ASME, J. Heat Transfer 99 (1979) 592–597Google Scholar
  12. 12.
    Seeniraj, R. V.; Bose, T. K.: Planar solidification of a warm flowing liquid under different boundary conditions. Wärme-Stoffübertrag. 16 (1982) 105–111Google Scholar
  13. 13.
    Yan, M. M.; Huang, P. N. S.: Perturbation solutions to phase change problems subjected to convection and radiation. Transactions ASME, J. Heat Transfer 96 (1974) 95–100Google Scholar
  14. 14.
    Carslaw, H. S.; Jaeger, J. C.: Conduction of heat in solids. 2nd edition. Oxford: Clarendon Press, 1959Google Scholar
  15. 15.
    Rubinstein, L. I.: Crystallization of a binary alloy. Section 3 of Chapter 2, pp. 52–60 in the Stefan Problem, English translation published by American mathematical society. Providence 1971Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • S. C. Gupta
    • 1
  • P. R. Arora
    • 2
  1. 1.Department of Applied MathematicsIndian Institute of ScienceBangaloreIndia
  2. 2.Department of Mechanical EngineeringIndian Institute of ScienceBangaloreIndia

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