Proceedings of the Asian Conference on Theral Sciences 2017, 1st ACTS March 26-30, 2017, Jeu Island, Korea ACTS-P00337 REPRODUCING KERNEL PARTICLE METHOD FOR CONDUCTION RADIATION INTERACTION IN 3D PARTICIPATING MEDIA TANG Jia-dong, HE Zhi-hong, WANG Zhen-hua, DONG Shi-kui * Harbin Institute of Technology, 92 Xidazhi Street, Nangang District, Harbin 150001, China Presenting Author: td863@126.co * Corresponding Author: dongsk@hit.edu.cn ABSTRACT The reproducing kernel particle ethod, which is a Lagrangian eshless ethod, is eployed for the calculation of conduction radiation proble in participating edia. Interaction of conduction and radiation in 3D enclosure is carried out with a participating ediu. Effects of various paraeters such as the extinction coefficient, the scattering albedo, the conduction radiation paraeter and the boundary eissivities are studied on teperature distributions in the ediu. These paraeters are found to have significant effect on results. Solidification of a 2D seitransparent absorbing, eitting and scattering ediu is siulated. The equivalent heat capacity ethod has been studied to solve the probles of latent heat absorption and release with transient heat transfer. Solidification is assued to occur over a range of teperatures, and accordingly distinct liquid-, ushy- and solid-zones are considered. Two test cases are exained and copared with other published works to verify this reproducing kernel particle ethod. Coparisons show that the reproducing kernel particle ethod is stable and has a good accuracy. KEYWORDS: Conduction radiation, Meshless ethod, Reproducing kernel particle ethod, Participating edia, Phase change 1. INTRODUCTION The phase change process within seitransparent edia is an iportant research field due to its relevancy in various kinds of engineering applications like fils adopted in solar energy, alloy processing, fiber preparation, and nuclear engineering. The analytic solutions are liited to uncoplicated exaples. Nueric ethods are used to analyze the phase change process ore frequently. A lot of nueric ethods have been developed to deal with the coupled conduction and radiation heat transfer probles in seitransparent edia, such as the finite eleent ethod, the finite difference ethod, the finite volue ethod, and the discrete ordinates ethod. These traditional ethods, however, depend on the pre-defined esh quality severely for coplex geoetry and reeshing is required when large deforation occurs. Therefore, to avoid the drawback, soe so-called eshless ethods are propounded. Meshless ethods have been adopted to solve cobined radiation and conduction heat transfer probles successfully, such as the eshless local Petrov-Galerkin ethod, the least squares collocation eshless ethod, and the natural eleent (NEM) ethod, etc. The reproducing kernel particle (RKPM) ethod is a Lagrangian eshless ethod. Copared to other eshless approaches, where the spacial node is siply the interpolated node, the Lagrangian eshless approach has got itself advantages that the discrete particles can carries density, volue, and ass of the aterial, while oving under the effect of internal interaction and external force. RKPM has wide application in plenty of engineering fields, such as large deforation of nonlinear elastic and inelastic structures [1-2], etal foring processes [3], elastic-plastic deforation probles[4], and convection diffusion proble [5]. 1
It has been discovered that none of studies that were reported by now has analyzed for solidification of 2D seitransparent edia considering the effect of voluetric radiation by eploying the reproducing kernel particle approach. Thus, the present works focus on the analysis of solidification porcess in 2D and conduction radiation proble in 3D participating ediu. 2. MATHEMATICAL FORMULATION 2.1 REPRODUCING KERNEL PARTICLE METHOD Constructing an approxiation uˆ( x, y, z ) to u( x, y, z ) by applying a corrected kernel is the core idea in the reproducing kernel particle ethod [6]. A corrected kernel approxiation in 3D proble can be given by uˆ( x, y, z) w( x - s, y - t, z - p) u( s, t, p) dsdtdp (1) where, uˆ( x, y, z) is the physical field function at spatial coordinates ( x, y, z ), u( s, t, p) is the physical field function at spatial coordinates ( s, t, p ) ; is the supported doain of the spatial node; w( x - s, y - t, z - p ) is the corrected kernel function, which is written as the product of kernel function w( x - s, y - t, z - p) and correction function C( x, y, z, s, t, p ). The corrected kernel function can be given by w( x - s, y - t, z - p) C( x, y, z, s, t, p) w( x - s, y - t, z - p) (2) The RKPM constructs the interpolation function on the discrete particles in the whole supported doain, so the discretization of equation (1) could be given by uˆ( x, y, z) w( x - s, y - t, z - p ) u( s, t, p ) V ( x - s, y - t, z - p ) u( s, t, p ) i i i i i i i i i i i i i i i1 i1 (3) Where, ( si, ti, pi ) is spatial coordinate of the i th discrete particle in the supported doain; is the nuber of the discrete particles in the supported doain; Vi is expressed as the volue; i is the shape function. 2.2 THE ENERGY EQUATION The enthalpy-based energy equation that describes the proble of coupled radiation and phase-change conduction heat transfer can be written as ( H) ( kt ) qr (4) t where, is the density, H cpt fll is the total enthalpy, k is the theral conductivity, T is the teperature, and qr is the radiative heat flux. cp is the specific heat at constant pressure, L is the latent heat, and f l is the liquid fraction. 2.3 THE RADIATIVE TRANSFER EQUATION The radiation intensity inforation is norally acquired through solving the radiative transfer equation. In the for of discrete ordinates approach, the RTE is discrete as a series of differential equations in a liited nuber of directions M I I I 2 Ks ( Ka Ks ) I n KaIb I 1,2,, M (5) x y z Where,, are direction cosine coponents in x, y, z direction, respectively. Ks is the scattering coefficient, n is the refractive index, and is the scattering phase function. is one discrete direction of radiation and I represents the radiative intensity in the th direction. 2.4 DISCRETIZATION OF RADIATIVE TRANSFER EQUATION 1 2
According to (3), the approxiation of the radiative intensity was constructed on the the i th discrete particle, shown as follows i i i 1 th discrete direction at I ( x, y, z ) ( x, y, z ) I ( x, y, z ) (6) where I ( x, y, z ) is the radiative intensity of the th discrete particle in the th discrete direction. The radiative transfer equation with RKPM discretization schees under the interpolation constrains is written as s,, x( x, y, z ), y( x, y, z ), z ( x, y, z ) ( a, s, w ) ( x, y, z) I 1 (7) M 2 s ' ' ' n a, Ib, ( xi, yi, zi) I ( xi, yi, zi) i 1,2, L, N ' 1, ' 3. RESULTS AND DISCUSSION Fig. 1 and 2 show the validation of the present 2-D RKPM solidification code without considering the effect of radiation. Solidification of a liquid in an internal corner with two surfaces aintained at equal teperatures lower than the freezing teperature was considered. This validation is done with the analytic results of H. Yang [7] for a 2-D planar ediu. In Figure 1, the teperatures along the diagonal, as predicted by the RKPM and the analytical solution, are presented. Figure 2 shows the predicted position of the solidification front. There is good agreeent between the nuerical results and the analytical solutions. Fig.1 The contrast of teperature distribution on the Fig.2 The contrast of solidification front distribution diagonal of the corner region at 0.02s at the different oents While solidification results involving radiation in Fig. 3 4, the aterial properties were kept the sae as considered in [7]. The effect of the extinction coefficient and the scattering albedo on the teperature T distributions and the liquid fraction f l on the diagonal are shown in Fig. 3-4, respectively. Fig.3 Teperature distribution on the diagonal of the corner region with the effect of β and ω at 0.02s 3
Fig.4 Liquid fraction on the diagonal of the corner region with the effect of β and ω at 0.02s Variation of teperature distributions is presented for different conduction-radiation paraeters, extinction coefficients, scatting albedos and boundary eissivities in Fig. 5a 5d. The botto wall is at a high teperature T h = 1000K, and other walls are at T c and Tc/ Th 0.5. Teperature results along z direction at x = 0.5 and y = 0.5 are presented. It is clearly shown that a good agreeent between results by the RKPM and the FVM [8] is observed in Figure 5a 5d. In Figure 5a, the teperature profile is near to a pure conduction profile for N 1 as it represents a conduction doinated situation. As seen fro Fig. 5b, the teperature profile for lower (=0.1) approaches the pure conduction profile. As expected, scattering has little influence over the ediu teperature distribution in Figure 5c. As seen fro Figure 5d, the ediu teperatures increase ore and ore as the hot boundary eissivity ε increases. (a) (b) (c) (d) Fig.5 Centerline teperature along z-direction at x = 0.5 and y = 0.5 in the cubical enclosure by RKPM and FVM (dot) for various paraeters. (a) N, (b) β, (c) ω, and (d) ε 4
4. CONCLUSIONS Cobined radiation and conduction heat transfer in 2D and 3D enclosures with participating edia were analyzed using the RKPM, the two test cases were exained and copared with other published works to verify the RKPM, coparisons show that the reproducing kernel particle ethod is stable and has a good accuracy. To extend the application areas of the RKPM, the solidification porcess of a 2D participating ediu was analyzed, an enthalpy based forulation was used to siulate the solidification process, the teperature profiles and liquid fraction in the aterial were investigated for the effect of the scattering albedo and the extinction coefficient, these paraeters are found to have significant effect on results. In the 3D enclosure, the effects of paraeters such as the scattering albedo, the extinction coefficient, the radiationconduction paraeter, and the boundary eissivity were investigated on teperature distributions within the coputational doain. ACKNOWLEDGMENT This study was supported by the National Natural Science Foundation of China (No.51576054) NOMENCLATURE c p specific heat capacity (J/kg K) q r radiative heat flux (W/ 2 ) f l liquid fraction ( - ) T teperature ( ) G incident radiation (W/ 2 ) x, y, z coordinate directions ( - ) I radiation intensity (W/ 2 sr) phase function ( - ) H total enthalpy (kj/kg),, direction cosine ( - ) k theral conductivity (W/K) supported doain ( - ) k absorption coefficient ( -1 ) density (kg/ 3 ) a k scattering coefficient ( -1 ) Subscripts s L latent heat (kj/kg) b black body ( - ) k N conduction-radiation paraeter ( - ) Superscript 3 T n refractive index ( - ) discrete direction ( - ) REFERENCE [1] J.S. Chen, C. Pan, and C.T. Wu, "Large deforation analysis of rubber based on a reproducing kernel particle ethod," Coputational Mechanics, vol. 19, pp. 211 227, 1997. [2] W.K. Liu and S. Jun, "Multiple-scale reproducing kernel particle ethods for large deforation probles," International Journal for Nuerical Methods in Engineering, vol.41, pp. 1339 1362, 1998. [3] H.Wang, G. Li, X. Han, and Z.H. Zhong, "Developentof parallel 3D RKPM eshless bulk foring siulation syste," Advances in Engineering Software, vol. 38, pp. 87 101, 2007. [4] K.M. Liew, Y.C. Wu, G.P. Zou, and T.Y. Ng, "Elasto-plasticity revisited: nuerical analysis via reproducing kernel particle ethod and paraetric quadratic prograing," International Journal for Nuerical Methods in Engineering, vol. 55, pp. 669 683, 2002. [5] F. Colin, R. Egli, and A. S. Mouni, "A stabilized eshfree reproducing kernel-based ethod for convection-diffusion probles, " International Journal for Nuerical Methods in Engineering, vol. 87, pp. 869 888, 2011. [6] N. R. and Aluru, "A point collocation ethod based on reproducing kernel approxiations, " International Journal for Nuerical Methods in Engineering, vol. 47, pp. 1083 1121, 2000. [7] H. Yang and Y. Q. He. "Solving Heat Transfer Probles with Phase Change Via Soothed Effective Heat Capacity and Eleent-free Galerkin Methods". International Counications in Heat and Mass Transfer. vol. 37, pp. 385-392, 2010. [8] P. Talukdar, F. V. Issendorff, D. Triis, and C. J. Sionson, "Conduction Radiation Interaction in 3-D Irregular Enclosures using the Finite Volue Method", Heat and Mass Transfer, vol. 44, pp. 695 704, 2008. 5