• Korean Chemical Engineering Research
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• v.52 no.2
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• pp.199-204
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• 2014

In binary solidification compositional convection in a porous mushy layer influences the quality of the final products. We consider the mushy layer solidifying from below with a constant solidification velocity. The disturbance equations for the mushy layer are derived using linear stability theory. The basic-state temperature fields and the distribution of the porosity in the mushy layer are investigated numerically. When the superheat is large, the thickness of the mushy layer is relatively small compared to the thickness of the thermal boundary layer. With decreasing the superheat the critical Rayleigh number based on the thickness of the mushy layer increases and the mushy layer becomes stable to the compositional convection. The critical Rayleigh number obtained from the continuity conditions of temperature and heat flux at the mush-liquid interface is smaller than that from the isothermal condition at the upper boundary of the mushy layer.

• Korean Journal of Mathematics
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• v.16 no.2
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• pp.191-203
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• 2008

We present an axisymmetric model containing only one chimney to study how convection affects the flow in a steady state chimney. We find that the mass fraction of solid in a mush and the depth of a mush when the strength of convection is given. We use the knowledge of the variables in the mush to find the fluid flow in the chimney Our procedure employs the von $K{\acute{a}}rm{\acute{a}}n$-Pohlhausen technique for determining chimney flow and makes use of the fact that the radius of the chimney is much less than the thickness of the mush.

• Korean Chemical Engineering Research
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• v.50 no.1
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• pp.61-65
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• 2012

The onset of convection in a mushy layer is analyzed by using linear stability theory in time-dependent solidification of a binary melt. A simplified model of a near-eutectic mush, in which the mush is assumed to be a porous block, is used and the propagation theory is applied to determine the critical conditions for the onset of convection. The present critical Rayleigh number is higher than the existing experimental result and also theoretical results obtained by considering the mushy layer with an overlying liquid layer. The constant pressure (permeable) condition applied on the mush-liquid interface produces a lower critical Rayleigh number, which is closer to the experimental results of aqueous ammonium chloride solution, compared with the impermeable condition.

• Journal of applied mathematics & informatics
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• v.12 no.1_2
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• pp.419-427
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• 2003

We have developed and analyzed numerically a simple ordinary differential equation for a mush. Numerical solutions illustrate that the solid fraction at the bottom of the mush increases and the depth of the mush decreases when convection increases. The results are consistent to those observed in experiments with aqueous solutions of ammonium chloride（Loper ＆ Roberts［11］, Chen & Chen［2］, Tait ＆ Jaupart［11］）and in experiments with lead-tin alloys（Hellawell etc.［6］）.

• Bulletin of the Korean Mathematical Society
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• v.34 no.4
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• pp.583-593
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• 1997

We have derived a simple ODE system for the mush by assuming that the temperature T, the solid fraction $\phi$ and the vertical component $\omega$ of the velocity, depend on z only. Analytical solutions of the system have presentd in case of $\omega << 1 and \phi << 1$.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.21 no.3
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• pp.445-456
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• 1997

Continuous casting process is numerically analyzed using the continuum model in a non-orthogonal coordinate system. Flow damping in the mush is modeled by combining the viscosity dependence on liquid fraction in dilute mush and the permeability dependence on liquid fraction in concentrated mush. The effect of turbulence is indirectly considered by effective diffusivity determined elsewhere by experiment. The main objective is to investigate the effects of casting parameters such as casting speed and tundish superheat on the distribution of surface temperature, shell thickness, metallurgical length and centerline segregation. Some of the computed results are compared with available experiments, and reasonable agreements are obtained.

• Korean Chemical Engineering Research
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• v.47 no.2
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• pp.174-178
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• 2009

A mushy layer of dendritic crystals is often formed during solidification of a binary mixture. Natural convection in the mushy layer is analyzed by using the propagation theory we have developed. The critical Rayleigh numbers for the onset of convection are evaluated numerically using the self-similar stability equations based on Emms and Fowler's model. The present results approach those from quasi-static stability analysis in the limit of a large superheat or a small growth rate of the mushy layer.

• Bulletin of the Korean Mathematical Society
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• v.41 no.2
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• pp.283-297
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• 2004

We have obtained a simplified model for the mush under the assumption of the temperature, the solid fraction and the vertical component of the velocity, depend on upward coordinate z only. We have found solutions in the asymptotical limit and solved numerically for the model.

• Proceedings of the Korean Society of Crop Science Conference
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• 2006.04a
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• pp.186-187
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• 2006

• Korean Journal of Mathematics
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• v.7 no.1
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• pp.27-35
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• 1999

As seen from the ammonium chloride experiment (Chen & Chen [1], Roberts & Loper [11], the interface near chimneys has an up-rising shape and we observe thickening of mush next to chimney. We analyze the thermal boundary layer around chimney that forms as the mush is cooled locally by the fluid rising through the chimney. We obtain solutions of the temperature, the solid fraction, and the pressure in the chimney wall. Also, our expression of the pressure shows that the fluid flow can require a huge pressure in order to pass through the chimney wall if its permeability is very small. We present a simple analytic description of the up-rising shape near the exit of the chimney, due to the fact that the comparatively solute (i.e. $NH_4Cl$ in the case of the ammonium chloride experiment)-rich fluid near the chimney tends to crystallize as it is chilled by the rising jet of cold fluid in the chimney.