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

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

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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.

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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Abbreviations

a 2 :

constant,kt m/R 20

A n :

coefficients in Eq. (16)

C :

specific heat, J kg−1°C−1

d :

constant in Eq. (6)

erf():

error function

erfc():

complementary error function

h 0 :

heat transfer coefficient ·R 0/K S, Eq. (6)

h 1 :

heat transfer coefficient ·R 0/K S, Eq. (6)

H 1 :

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 :

ϱ S/ϱL, Eq. (1)

q :

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

R :

radial coordinate/R 0

R 0 :

radius of the sphere, m

t :

time, s

t m :

time at which solidification starts atR=1, s

T(R, V):

temperature/T m

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

T(R, V)

T m :

melting temperature, °C

T 0 :

ambient temperature/T m

V :

dimensionless time, =2(a sy)1/2

X(V):

dimensionless freezing front,R=X(V)

\(\bar X\)(y):

X(V)

y :

dimensionless time, (t−t m)/t m

α :

dimensionless constant, (k S/kL)1/2

β :

dimensionless constant,K L/KS

σ :

Stefan-Boltzmann constant ·T 3 m R 0/K S

θ (1) L (R):

melt temperature att=0/T m

θ (2) L (R):

dimensionless temperature, Eq. (11)

θ (1) S (R):

dimensionless temperature, Eq. (12)

θ (2) S (R):

dimensionless temperature, Eq. (12)

λ :

dimensionless constant,l/C sTm

ϱ :

density, kg m−3

Δy :

time step in the numerical scheme

L :

liquid

S :

solid

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Authors and Affiliations

  1. Department of Applied Mathematics, Indian Institute of Science, 560 012, Bangalore, India

    S. C. Gupta

  2. Department of Mechanical Engineering, Indian Institute of Science, 560 012, Bangalore, India

    P. R. Arora

Authors
  1. S. C. Gupta
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  2. P. R. Arora
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Gupta, S.C., Arora, P.R. Analytical and numerical solutions of inward spherical solidification of a superheated melt with radiative-convective heat transfer and density jump at freezing front. Wärme- und Stoffübertragung 27, 377–384 (1992). https://doi.org/10.1007/BF01600027

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  • DOI: https://doi.org/10.1007/BF01600027

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