Comparison between moving-boundary and distributed models for predicting the time evolution of the solidification front in ice trays
Introduction
Initiated in ancient civilizations and boosted by the popularization of refrigerators in the first half of the 20th century, household consumption of edible ice has increased steadily over the years. Nowadays, automatic ice makers feature most modern refrigerators. For instance, the number of international patents filed yearly increased from ~100 in 1980 to ~1400 in 2018, revealing a growing interest in the ice making market, whose value is expected to exceed US$ 5.9 billion by 2023 [8].
An automatic (or cyclic) ice maker is a device installed inside a low-temperature compartment of a household refrigerator and directly connected to a water source to continuously produce ice to be stored in a container placed inside the freezer compartment. In general, cyclic ice making relies on filling the tray with water, cooling and freezing water into ice, and harvesting the ice from the tray [19]. A typical ice production rate for a household appliance is around one kilogram per day, although a substantial amount of energy is demanded if compared to the total amount consumed by the refrigerator itself [13].
This study is part of a major project aimed at increasing the ice production rates in household refrigerators with a minimal penalty in terms of energy consumption. This paper is particularly focused on modelling the processes of cooling and freezing water into ice, as they respond for half of the total energy consumption of an automatic ice maker, whereas the other half is required for ice harvesting and storage [7]. Albeit solidification involves complex phenomena that make its mathematical modelling a nontrivial effort, simplifications were made possible due to the pioneering works of Stefan and Plank[3][18].
The present work brings about a simplified formulation for (two-dimensional) rectangular cavities, whose predictions for the time evolution of the solidification front have been compared against numerical data obtained from a more sophisticated model developed by means of an enthalpic formulation based on the first-principles of mass, momentum and energy conservation.
Section snippets
Theoretical background
The time evolution of the water temperature according to a theoretical solidification curve is illustrated in Fig. 1. The process starts from a temperature T0 > Tm (a), where Tm is the datum freezing/melting temperature of water(0 °C), and it goes on until the solidification is complete and then the ice is cooled down below Tm. The real behavior is quite different.
In most cases, metastable water is cooled down until Ts < Tm, where the onset of solidification occurs (b). The temperature
Moving-boundary approach
Basically, the moving-boundary model relies on a lumped energy balance applied to each of the k walls of the ice tray, which follows closely the Stefan formulation [4], i.e. the latent heat released during freezing is conducted through the solid phase and then rejected by convection from the tray walls to the cooling medium. Since the volume of the liquid region changes with time, the freezing front has a different heat transfer area (Ai,k) than the external solid wall (Ao,k), as can be seen in
Results
Before comparing the moving-boundary and distributed models, the accuracy of the latter was checked against benchmark solutions. For this purpose, the work of Michalek and Kowaleski[14] was considered, where a square cavity with a 38 mm side filled with liquid water at 278 K was exposed to a wall initially at 283 K (left) and another wall initially at 273 K (right), whereas the top and bottom boundaries were considered adiabatic. No phase change takes place, so that the solution (velocity and
Conclusions
The process of ice making was assessed in this work. Two approaches for predicting the solidification boundary over time were developed and implemented, namely the moving-boundary and the distributed models. The predictions of the latter for the time evolution of the ice front were verified against a benchmark solution. Both approaches were then compared for different geometries and cooling conditions. Albeit the moving-boundary model was not able to identify precisely the onset of ice
Declaration of Competing Interest
This article has not any actual or potential conflict of interest.
Acknowledgments
Financial support from the Brazilian Government funding agencies CAPES, CNPq and Embrapii is duly acknowledged. The authors are thankful to Whirlpool for technical and financial support, particularly to Messrs. Alisson C. Silva and Arthur A. Marcon who championed this research project.
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