International Journal of Thermal Sciences 147 (2020) 106121 Available online 17 October 2019 1290-0729/© 2019 Published by Elsevier Masson SAS. Supercooling of phase change: A new modeling formulation using apparent specifc heat capacity Tanguy Davin * , Benoît Lefez, Alain Guillet Groupe de Physique des mat� eriaux, UMR CNRS 6634, Universit� e de Rouen Normandie, France A R T I C L E INFO Keywords: Numerical simulation Lumped systems (capacitances) models Phase change Supercooling (subcooling) Equivalent (apparent) specifc heat capacity ABSTRACT The thermal behavior of Phase Change Materials (PCMs) is a major issue for cooling, heat storage and thermal management of various systems in general. Unfortunately many PCMs present supercooling which is a major drawback regarding the effciency of cooling systems. Various solutions were proposed to model the thermal kinetics, in particular using the apparent specifc heat Cp(T) technique. But among them only few consider the supercooling effect. The present work considers this issue by focusing on the representation of the different steps of the supercooling phenomenon. This leads to different formulations of the apparent capacity Cp (T,f super ). The presented algorithm uses the lumped system analysis approach that is widely spread for frst-order and multi- scale resolutions. The apparent capacity laws are explicitly presented for the different melting-crystallization steps. In particular, the punctually negative Cp formulation, a helpful mathematical artifact, permits to repro- duce the thermal dynamics during the local crystallization. The formulations and the models are discussed. Eventually, the temperature evolutions of an experimental system are compared to calculated data and the crystallization rate is compared to the literature. 1. Introduction Enthalpy of the liquid to solid (and vice-versa) phase change has been widely studied for thermal storage or cooling purpose [1–4]. PCMs (for Phase Change Materials) work as a thermal buffer, which load (on heating) and unload (on cooling) energy at an assumed constant tem- perature, in agreement with the Gibbs’ phase rule. Diverse numerical representations of phase change on various materials are already available in the literature [5–7]. In particular, two types of modeling have been provided by the literature, whether an Eulerian or Lagrangian approach is considered. One is based on a moving grid, permitting the interface tracking [8] and the other on fxed grid, introducing enthalpy methods. This former type of methods can be very accurate but it is relatively complicated especially for the discretization of the moving interface and it is time-consuming. These multi-domain methods involve the resolution of several phases. The latter considers a fxed discretiza- tion grid and only one set of equations for the different phases of the material. These one-domain methods, have been widely developed, as described in a recent exhaustive review on fxed-grid methods [9]. The idea of these methods is to represent the latent heat parameter in the formulation of the chosen variable. Indeed, the released or absorbed energy during the phase change is not located on a moving interface, but is included in the enthalpy variable of the system, either enthalpy-based (Enthalpy Methods EM), or Temperature-based (Enthalpy-Temperature Methods ETM) [10,11]. The governing equations of enthalpy methods comprise the expression of the enthalpy variation by the mean of the specifc capacity Cp(T) or enthalpy h(T). Some authors directly use the enthalpy h(T) [12,13] while others prefer using the apparent specifc capacity Cp(T), leading to specifc heat capacity (ESHC) models [14], also called zero latent heat models [14,15]. Their implementation is hence simplifed as the general equa- tion remains unchanged. The actualization of the specifc heat for every iteration is the only constraint. The apparent specifc capacity variation permits to take into account the phase change transformation without modifying the general heat transfer equation and resolution. It is so convenient to consider this parameterization by using different formu- lation as Finite Element Method, Finite Difference or lumped systems, as it is proposed in the present study. Supercooling also referred to as subcooling occurs when a compound is still liquid below the thermodynamic equilibrium temperature. This metastability implying a delayed phase change results from slow ki- netics of crystallization. The nucleation/growth theory explains the formation of crystals [16–18]. It is possible to express the statistical * Corresponding author. E-mail address: tanguy.davin@gmail.com (T. Davin). Contents lists available at ScienceDirect International Journal of Thermal Sciences journal homepage: http://www.elsevier.com/locate/ijts https://doi.org/10.1016/j.ijthermalsci.2019.106121 Received 28 February 2018; Received in revised form 26 June 2019; Accepted 27 September 2019