Nonequilibrium Planar Interface Model for Solidification of Semitransparent Radiating Materials

Abstract

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

Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium

“…Design of boilers, furnaces, internal combustion engines and insulations are some of the systems which require a correct analysis of thermal radiation (Marakis et al , 2000; Spinnler et al , 2004). Analysis of phase change process of semitransparent materials such as glass and semiconductor materials requires knowledge of the volumetric radiation Yao et al (2000); Shu et al (2004); Sadooghi (2005) and Raj et al (2006). Correct estimates of volumetric radiation is also important in weather forecasting which relies on atmospheric radiation budget Grabowski (2003) and medium characterization of an optically participating medium like human tissue and laser surgery of a human organ Kim and Guo (2004) and Pal et al (2005).…”

A Lattice Boltzmann Formulation for the Analysis of Radiative Heat Transfer Problems in a Participating Medium

“…Design of boilers, furnaces, internal combustion engines and insulations are some of the systems which require a correct analysis of thermal radiation [1,2]. Analysis of phase change process of semitransparent materials such as glass and semiconductor materials requires knowledge of the volumetric radiation [3][4][5][6]. Correct estimates of volumetric radiation is also important in weather forecasting which relies on atmospheric radiation budget [7] and medium characterization of an optically participating medium like human tissue and laser surgery of a human organ [8][9].…”