Thermal Radiation Heat Transfer Effects on Solidification of Finite Concentric Cylindrical Medium-Enthalpy Model and P-1 Approximation

Kim, Kwang S.; Yimer, B.

doi:10.1080/10407788808913657

Numerical Heat Transfer

1988

DOI: 10.1080/10407788808913657

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Thermal Radiation Heat Transfer Effects on Solidification of Finite Concentric Cylindrical Medium-Enthalpy Model and P-1 Approximation

Kwang S. Kim

B. Yimer

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Cited by 4 publications

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“…Effects of internal radiation on phase change were also investigated by Seki et al, 11 Cho and Ozisik, 12 Oruma et al, 13 and Kim and Yimer. 14 All of these phase-change models of semitransparent materials assume a local equilibrium condition at the solidi cation front. This assumption is valid only for the situation when the material has a large dimension and heat-transfer rate is relatively small.…”

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Nonequilibrium Planar Interface Model for Solidification of Semitransparent Radiating Materials

Yao

Wang

Chung

2000

Journal of Thermophysics and Heat Transfer

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A nonequilibrium solidi cation model for semitransparent materials is presented. Consideration is given to a planar layer of emitting, absorbing, and scattering medium subject to radiative and convective cooling. The enthalpy method is used to formulate the phase-change problem together with radiative transfer equation taking into account internal emitting, absorbing, and scattering. A planar interface nonequilibrium solidi cation is assumed with crystalline phase nucleated on the surface at a given nucleation temperature, which may be signi cantly lower than the equilibrium melting temperature of the material. A linear kinetics relationship is introduced to correlate the unknown solidi cation temperature to the interface velocity. A fully implicit nite volume scheme is used to solve the problem with the solidi cation interface tracked by a modi ed interface tracking method. The radiative transfer equation is solved using the discrete ordinates method. Internal radiation enhances the latent heat removal and thus leads to a higher interface velocity and a larger melt undercooling. Optical thickness and the conduction-radiation parameter are two important parameters that affect the solidi cation process. In the presence of external convective cooling, effect of internal radiation is small in the early stage of solidi cation. Nomenclaturec = speci c heat, J/kg K D = thickness of the layer, m H = dimensionless enthalpy, (h-cT m )/ cT m H R = convection-radiationparameter, h c / (r T 3 m ) h = enthalpy, J/kg h c = convective heat-transfer coef cient, W/m 2 K I = intensity of radiation, W/m 2 sr I = normalized intensity of radiation, I / (4r T 4 m ) k = thermal conductivity, W/m K N = conduction-radiationparameter, k / (4r T 3 m D) n = refractive index q r = radiative heat ux, W/m 2 q r = dimensionless radiative heat ux, q r / (4r T 4 m ) S = dimensionless interface position, s / D St = Stefan number, cT m / k s = interface position, m T = absolute temperature, K T e = temperature of the environment, K T m = equilibrium freezing temperature, K T N = nucleation temperature, K T s = interface temperature, K T 0 = initial temperature, K t = time, s V s = dimensionless interface velocity, dS/ ds X = dimensionless coordinate, x / D x = coordinate in the direction across the slab, m b = extinction coef cient of the medium, m ¡ 1 h = dimensionless temperature, T / T m h e = dimensionless environment temperature, T e / T m h s = dimensionless interface temperature, T s / T m j D = optical thickness of the layer, b D k= latent heat of fusion, J/kg . l = dimensionless linear kinetics coef cient, (q cl k / 4r T 2 m ) l k = linear kinetics coef cient, m/s K n = direction cosine of a radiation ray q = density, kg/m 3 q 0 , q i = re ectivities to external and internal incidence r = Stefan-Boltzmann constant, 5.6705 £ 10 ¡ 8 W/m 2 K 4 s = dimensionless time, (4r T 3 m / q c D)t X = solid angle, sr x = single scattering albedo

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Nonequilibrium Planar Interface Model for Solidification of Semitransparent Radiating Materials

Yao

Wang

Chung

2000

Journal of Thermophysics and Heat Transfer

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Internal radiative transport in the vertical Bridgman growth of semitransparent crystals

Brandon

Derby

1991

Journal of Crystal Growth

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Solidification of a 2-D semitransparent medium using the lattice Boltzmann method and the finite volume method

Mishra

Behera

Garg

et al. 2008

International Journal of Heat and Mass Transfer

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Thermal Radiation Heat Transfer Effects on Solidification of Finite Concentric Cylindrical Medium-Enthalpy Model and P-1 Approximation

Cited by 4 publications

References 8 publications

Nonequilibrium Planar Interface Model for Solidification of Semitransparent Radiating Materials

Nonequilibrium Planar Interface Model for Solidification of Semitransparent Radiating Materials

Internal radiative transport in the vertical Bridgman growth of semitransparent crystals

Solidification of a 2-D semitransparent medium using the lattice Boltzmann method and the finite volume method

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