The inward solidification of spheres and circular cylinders

Riley, D. S.; Smith, F. T.; Poots, G.

doi:10.1016/0017-9310(74)90061-1

Cited by 167 publications

(73 citation statements)

References 5 publications

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“…These expansions are regular when s = O(1), but become singular when s → 0, with singularities appearing in u 1 and higher order terms for the temperature distribution and in τ 2 for the time. The same type of singularity has been noticed by Riley et al (1974) also for the isothermal case. Therefore, solutions (3.6) and (3.7) are valid for τ e − τ ≥ O(1) and are referred to as outer solutions with respect to the time proximity to the dimensionless final freezing time τ e .…”

Section: (A) Outer Solutionsupporting

confidence: 80%

Freezing of a supercooled spherical droplet with mixed boundary conditions

Tabakova¹,

Feuillebois

Radev³

2009

Proc. R. Soc. A.

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The freezing of a supercooled droplet occurs in two steps: recalescence, that is, a rapid return to thermodynamic equilibrium at the freezing temperature leading to a liquid-solid mixture and a longer stage of complete freezing. The second freezing step can be modelled by the one-phase Stefan problem for an inward solidification of a sphere, assuming the droplet to be spherical. A convective heat transfer with the ambient immiscible fluid is modelled by a mixed boundary condition on the outer surface of the droplet. This condition depends on the Biot number (ratio of the heat transfer resistances inside the droplet and at its surface). A novel asymptotic solution is developed for a small Stefan number and an arbitrary Biot number. Applying the method of matched asymptotic expansions, uniformly valid solutions are obtained for the temperature profile and freezing front evolution in the whole stage of complete freezing. For an infinite Biot number, that is, for a fixed temperature at the droplet outer boundary, known solutions are recovered. In parallel, numerical results are obtained for an arbitrary Stefan number using a finitedifference scheme based on the enthalpy method. The asymptotic and numerical solutions are in good agreement.

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Section: (A) Outer Solutionsupporting

confidence: 80%

Freezing of a supercooled spherical droplet with mixed boundary conditions

Tabakova¹,

Feuillebois

Radev³

2009

Proc. R. Soc. A.

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“…Estimates and calculations of the phase change behavior occur as discussed in Jaeger (1964), Kreith and Romie (1955), Soward (1980), Stewartson and Waechter (1976), Riley et al (1974), and Pedroso and Domoto (1973). Drift wall temperature is assumed to be ~ 600°C.…”

Section: Solidification Of Magmamentioning

confidence: 99%

“…There is substantial literature describing solidification of a spherical and a cylindrical molten mass (e.g., Soward 1980;Stewartson and Waechter 1976;Riley et al 1974;Pedroso and Domoto 1973;Kreith and Romie 1955), which has provided series solutions for the temperature history and for the time of final solidification. The particular problem of a magma plug of finite length is not among the solutions mentioned or among the problems tried.…”

Section: Thermal Environmentmentioning

confidence: 99%

“…Several of the analyses (Kreith and Romie 1955;Riley et al 1974) provide simple expressions for the time for solidification of a cylinder filled with chilling magma, in terms of the diameter of the drift, the drift wall temperature, and the physical properties of the magma. Riley et al (1974Riley et al ( , p. 1514 (Hodgman et al 1955(Hodgman et al , p. 2251 Table Heat Table 1) T w = Drift wall temperature (~600 0 C) With these numbers, the time for solidification to reach the center of the magma filled drift is approximately 70 days. Estimates using other reference calculations suggest that to the order of accuracy of the expansions that the time is 70< t f < 82 days.…”

Section: Thermal Environmentmentioning

confidence: 99%

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Dike Propagation Near Drifts

NA¹

2002

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“…Since there is no known exact solution to this problem, various numerical and semi-analytical techniques have been devised. Numerical solutions for such problems are given by Tao [12] while a number of authors have presented perturbation and boundary layer approaches to the problem (for example, Pedroso and Domoto [3], [4], Riley, Smith and Poots [5], Soward [10] and Stewartson and Waechter [11].) Integral approaches are given, for example by Goodman [2], Savino and Siegel [6], Shih and Chou [7], Shih and Tsay [8], and Theofanous and Lim [13].…”

mentioning

confidence: 99%

On an integral formulation for moving boundary problems

Dewynne¹,

Hill²

1984

Quart. Appl. Math.

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Abstract. The classical moving boundary problems arising in the freezing or chemical reaction of spheres, cylinders and slabs are considered. An integral method is employed to formally effect the integration of the motion of the moving boundary. This formal integration permits upper and lower bounds to be deduced for the motion and in particular simple upper and lower bounds are established for the time to complete freezing or reaction (that is, when the moving boundary reaches the centre of the sphere or cylinder). In addition an improved second upper bound on the motion is achieved by demonstrating that the dimensionless temperature or concentration is bounded above by the standard pseudo steady state approximation. The use of the integral formulation as an iterative scheme and the generalisations for a time dependent surface condition and a non nlinear diffusivity are also briefly considered.

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The inward solidification of spheres and circular cylinders

Cited by 167 publications

References 5 publications

Freezing of a supercooled spherical droplet with mixed boundary conditions

Freezing of a supercooled spherical droplet with mixed boundary conditions

Dike Propagation Near Drifts

On an integral formulation for moving boundary problems

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