Theoretical understanding of phase change heat transfer problems is of much interest for multiple engineering applications. Exact solutions for phase change heat transfer problems are often not available, and approximate analytical methods are needed to be used. This paper presents a solution for a one-dimensional (1D) phase change problem with time-dependent heat flux boundary condition using the perturbation method. Two different expressions for propagation of the phase change front are derived. For the special case of constant heat flux, the present solution is shown to offer key advantages over past papers. Specifically, the present solution results in greater accuracy and does not diverge at large times unlike past results. The theoretical result is used for understanding the nature of phase change propagation for linear and periodic heat flux boundary conditions. In addition to improving the theoretical understanding of phase change heat transfer problems, these results may contribute toward design of phase change based thermal management for a variety of engineering applications, such as cooling of Li-ion batteries.
Mathematical modeling of species diffusion in Li-ion cell electrodes is critical for improving performance and efficiency of electrochemical energy storage. There is a relative lack of literature in this direction for time-dependent flux boundary conditions. In this work, the method of Green’s functions is used to solve the solid-phase diffusion problem in electrodes with a time-dependent flux boundary condition. While Green’s functions have been used extensively for thermal transport problems, there is limited past work on application of Green’s functions for solving species transport problems in electrochemical systems. The concentration distribution is first determined for a thin film electrode and spherical electrode particle. The method is then extended to determine the concentration profile in two-layer composite electrodes. The mathematical models presented in this work are validated by comparison with past studies and numerical simulations. Concentration profiles for a variety of time-dependent boundary conditions are presented. It is expected that improved understanding of diffusion under time-dependent flux boundary conditions may help improve the performance and efficiency of Li-ion based electrochemical energy storage devices and systems.
Real-time State of Charge (SoC) estimation of a Li-ion cell is necessary for an accurate estimation of the state of the cell, and to ensure safety and efficient performance by avoiding overcharge or overdischarge. While past papers have presented analytical models for predicting voltage and SoC for constant-current conditions, there is a need for analytical models that account for timevarying charge/discharge currents representative of realistic conditions. This paper presents an analytical SPM-based model to predict the terminal voltage and SoC of a Li-ion cell operating under a general time-dependent current profile. Concentration distributions in the positive and negative electrodes are determined analytically using Green's function approach, followed by determination of the electrode voltages as functions of time using the Butler-Volmer kinetic equation. The analytical model is validated through good agreement with numerical simulations and past experimental data for a number of different operating conditions. Cell voltage and SoC are predicted for a variety of time-varying currents, including drive cycles representative of realistic driving conditions. It is expected that the analytical model developed here will help improve the performance of battery management systems by obtaining more accurate information about the internal state of the cell in realistic charge/discharge conditions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.