A new lattice Boltzmann model for solid–liquid phase change Rongzong Huang, Huiying Wu ⇑ , Ping Cheng Key Laboratory for Power Machinery and Engineering of Ministry of Education, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China article info Article history: Received 2 November 2012 Accepted 14 December 2012 Available online 16 January 2013 Keywords: Lattice Boltzmann method Solid–liquid phase change Latent-heat source term Total enthalpy Immersed moving boundary scheme Melting by conduction Melting by convection abstract The solid–liquid phase change problems were solved by the lattice Boltzmann method in this paper. By modifying the equilibrium distribution function for the temperature, a new approach was developed to treat the latent-heat source term. As compared with the previous work, the approach developed in this paper could avoid iteration steps or solving a group of linear equations, which guaranteed this approach’s high efficiency. The phase interface was traced by updating the total enthalpy, and the moving interface was treated by the immersed moving boundary scheme proposed by Noble and Torczynski for simulation of particulate suspensions. The approach was firstly validated by the problem of conduction-induced melting in a semi-infinite space, and good agreement with the analytical result was obtained. Then it was used to simulate melting problems coupled with natural convection, which demonstrated that the approach could produce consistent results as compared with other numerical method. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Phase change problems are of great importance in lots of engi- neering and natural systems. Related problems include the metal smelting and casting, thermal energy storage, crystal growth, and etc. In these processes, matter is subject to a solid–liquid phase change. Consequently, a moving interface separating two different phases develops and absorption or release of latent heat happens in the vicinity of this interface. Mathematical modeling of such problems is always a challenging task because of the complex boundary conditions, as well as varied thermophysical properties. A detailed introduction of various mathematical modeling for such problems could be found in the review article of Henry and Stavros [1]. Because of the complexity, analytical results cannot be ob- tained except for some extremely simple situations, and therefore, numerical methods have been developed [2–5]. The lattice Boltzmann method (LBM), developed over the last two decades, has been widely used to simulate series of challeng- ing problems owing to its some obvious advantages, such as the mesoscopic kinetic background, easy boundary treatment and inherently parallelizable computation property. The LBM proposed for modeling solid–liquid phase-change problems can generally be grouped into two major categories: (1) the phase-field method; (2) the enthalpy-based method. Wolf-Gladrow [6] was one of the first to derive a diffusion-based LBM for simulating heat conduction problem. De Fabritiis et al. [7] proposed a thermal LB model for solid–liquid phase change problems by using two types of quasi- particles for solid and liquid phases respectively. Subsequently, Miller [8] simplified and extended this model, using one type of quasi-particle and a phase-field method. They used a chemical term to represent phase change because of the fact that the melt- ing and solidification are just like exothermic and endothermic chemical reactions. Further work proceeded along similar lines [9–13], with the phase-field method to track the phase interface. To reproduce the dynamics of the sharp interface equation, the phase-field method requires extremely finer grid spacing to resolve the interfacial region in principle, while it is not required in the en- thalpy-based method. Jiaung et al. [14] firstly developed a lattice Boltzmann equation with an enthalpy formulation to simulate heat conduction problem. Subsequently, taking computational advan- tage of the enthalpy-based method, Chatterjee and Chakraborty [15–17], Chakraborty and Chatterjee [18], Chatterjee [19,20] pro- posed a series of lattice Boltzmann (LB) models for solid–liquid phase change problems coupling with the fluid flow. They success- fully adopted the enthalpy-based method to simulate the problem including the crystal growth [18]. Huber et al. [21] also developed a double distribution function model coupled with convection and phase change. In the enthalpy-based method, the phase interface is traced naturally by updating the total enthalpy. However, be- cause of the non-linear latent-heat source term included in energy equations, it needs iteration steps to solve the temperature evolu- tion equation. Recently, Eshraghi and Felicelli [22] proposed an im- plicit lattice Boltzmann model to treat the source term. It avoids iteration steps, while needs to solve a group of linear equations at each node. 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.12.027 ⇑ Corresponding author. E-mail address: whysrj@sjtu.edu.cn (H. Wu). International Journal of Heat and Mass Transfer 59 (2013) 295–301 Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt