Asymptotic analysis of Stefan problem


Droplet freezing schematic under the sub-zero environment

1. Asymptotic analysis of a two-phase Stefan problem in annulus: Application to outward solidification in phase change materials (Completed)

Stefan problems provide one of the most fundamental frameworks to capture phase change processes. The problem in cylindrical coordinates can model outward solidification, which ensures the thermal design and operation associated with phase change materials (PCMs). However, this moving boundary problem is highly nonlinear in most circumstances. Exact solutions are restricted to certain domains and boundary conditions. It is therefore vital to develop approximate analytical solutions based on physically tangible assumptions, e.g., a small Stefan number. A great amount of work has been done in one-phase Stefan problems, where the initial state is at its fusion temperature, yet very few literature has considered two-phase problems particularly in cylindrical coordinates. This project conducted an asymptotic analysis for a two-phase Stefan problem for outward solidification in a hollow cylinder, consisting of three temporal and four spatial scales. The results are compared with the enthalpy method that simulates a mushy region between two phases by numerical iterations, rather than a sharp interface in Stefan problems. After studying both mathematical models, the role of mushy-zone thickness in the enthalpy method is also unveiled. Moreover, a wide range of geometric ratios, thermophysical properties and Stefan numbers are selected from the literature to explore their effects on the developed model with regards to interface motion and temperature profile. It can be concluded that the asymptotic solution is capable of tracking the moving interface and evaluating the transient temperature for various geometric ratios and thermophysical properties in PCMs. The accuracy of this solution is found to be affected by Stefan number only, and the computational cost is much less compared with the enthalpy method and other numerical schemes.

2. An asymptotic solution for a two-phase Stefan problem in a droplet subjected to convective boundary condition (Completed)

Droplet solidification is governed by classical Stefan problems which have been commonly treated as a single-phase problem by the majority of the studies in the literature. This approach, however, is unable to capture the initial temperature and the start of freezing time correctly. The treatment of two-phase Stefan problem in spherical coordinates is limited. No known exact solution exists, albeit numerical solutions and asymptotics have proven to be useful. We present a singular perturbation solution in the limit of low Stefan number and arbitrary Biot number for the two-phase Stefan problem in a finite spherical domain. An asymptotic solution is developed for a droplet at a non-freezing initial temperature subjected to a convective boundary condition at the surface. The solution is developed for both long-time and short-time scales. The results from asymptotic expansion method are validated with the experimental results in the literature and are further verified by a numerical model of a freezing droplet using enthalpy–porosity method. The sensitivity of the asymptotic solution to the droplet initial temperature, Biot number, and Stefan number has also been studied. The results indicate that the solution from perturbation series and enthalpy–porosity method agrees to within 1%–10% for temperature profile and overall freezing times over a wide range of practical values for initial temperature, Stefan and Biot numbers for the application of spray freezing. Our perturbation series solution is also able to capture the effect of initial temperature on the overall freezing time of the droplet.

3. Two-Phase Stefan Problem in Artificial Ground Freezing Using Singular Perturbation Theory (Completed)

Artificial ground freezing (AGF) has historically been used to stabilize underground structure. Numerical methods generally require high computational power to be applicable in practice. Therefore, it is of interest to develop accurate and reliable analytical frameworks for minimizing computational cost. This paper proposes a singular perturbation solution for a two-phase Stefan problem that describes outward solidification in AGF. Specifically, the singular perturbation method separates two distinct temporal scales to capture the subcooling and freezing stages in the ground. The ground was considered as a porous medium with volume-averaged thermophysical properties. Further, Stefan number was assumed to be small, and effects of a few site-dependent parameters were investigated. The analytical solution was verified by numerical results and found to have similar conclusions yet with much lesser computational cost.


Publications


Verification and Validation of Droplet Freezing for Convective Boundary Condition Using Matched Asymptotic Perturbation Method and Computational Fluid Dynamics


Saad Akhtar, Minghan Xu, Agus Sasmito


ASME International Mechanical Engineering Congress and Exposition, 2019 Nov


Singular Perturbation Solution for a Two-Phase Stefan Problem in Outward Solidification


Minghan Xu, Saad Akhtar, Mahmoud A. Alzoubi, Agus P. Sasmito


ASME 2019 International Mechanical Engineering Congress and Exposition, 2019 Oct 10


Development of Analytical Solution for a Two-Phase Stefan Problem in Artificial Ground Freezing Using Singular Perturbation Theory


Minghan Xu, Saad Akhtar, Ahmad Zueter, Victor Auger, Mahmoud Alzoubi, Agus Pulung Sasmito


Journal of Heat Transfer-transactions of The Asme, vol. 142(12), 2020 Oct 30


Development and validation of an asymptotic solution for a two-phase Stefan problem in a droplet subjected to convective boundary condition


Saad Akhtar, Minghan Xu, Agus P. Sasmito


International Journal of Thermal Sciences, 2021 Apr 31