Novel Finite Element Method for Solving One-Dimensional Phase Change Problems

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A novel finite element method is developed for more accurate and physically representative tracking of the phase change boundary in a one-dimensional, two-phase, diffusion-dominated Stefan problem with a sharp phase discontinuity. The crux of the method is the derivation and application of a novel phase change element, which enforces energy conservation both locally at the phase change boundary and globally upon assemblage into the global matrix equation. The temperatures near the phase change boundary are modeled by custom, piecewise-linear shape functions within the phase change element. The exact location of the phase boundary is determined by equating two enthalpy balances for the boundary speed, one using heat flows in surrounding elements, the other using the temperature gradients inside the phase change element. The method conserves energy to the accuracy of a first-order backward difference in time.The optimal spatiotemporal discretization of the novel method was determined by benchmarking results against the closed-form solution for a Dirichlet boundary condition on a semi-infinite domain. The accuracy of the numerical method relative to the dimensionless closed-form solution was determined for the optimal spatiotemporal discretization: the L2 relative error norm for the phase change boundary location calculation was found to be within 1% of the closed-form solution. Additionally, the optimization process revealed that the fundamental tenets of Constructal Law can be seamlessly applied to the process of numerical optimization, further extending the scope of the theory. Dimensional studies were performed to obtain a rigorous performance and versatility assessment of the numerical method. The Dirichlet boundary condition was first modeled on a finite domain, where no closed-form analog exits. A closed-form solution for a specified time-dependent heat flux with instantaneous phase change was then used to benchmark the novel numerical method for a Neumann boundary condition. The time-dependent heat flux condition was then replaced with a constant heat flux on a finite domain with preheating, where again no closed-form analog exists. Finally, the numerical method was applied to a one-dimensional version of a real-world case study for a Latent Heat Thermal Energy Storage System designed to support human operations on a future manned mission to Mars.





Zajda, Alexander (2021). Novel Finite Element Method for Solving One-Dimensional Phase Change Problems. Dissertation, Duke University. Retrieved from


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