Comprehensive Mathematical Model for Freezing Time Prediction of Finite Object

Abstract

Frozen food sees a continuous increase in consumption thanks to the capability to preserve the organoleptic properties and, at the same time, to increase the shelf-life of the food itself. Therefore, a proper design of the freezing process is crucial and is strictly related to an accurate evaluation of freezing time because this establishes the minimum residence time of the product in a continuous freezer. For this reason, several mathematical models have been proposed and investigated for predicting the freezing time, starting from empirical models (lower accuracy and computational time) up to computational fluid dynamics simulation (higher accuracy and computational time). An excellent compromise between accuracy and computational effort seems to be a model that combines empirical laws for property evaluation and heat diffusion equation solved in one dimension. This model can numerically be solved using the method of lines in which spatial derivatives are discretized by the finite difference method and the resulting system of ordinary differential equations is integrated using an appropriate solver. This work aimed to fill some gaps to develop a comprehensive and more accurate model for freezing time prediction. Indeed, the key idea is to validate a model that could be used to optimize the refrigeration process for energy-saving and be the base for a design of experiment in case of lack of experimental data. The newly developed model includes the evaluation of freezing time for finite food shapes because, in some cases, it is not possible to assume a characteristic direction for heat flux. The property of unfrozen and frozen food (e.g. density, thermal conductivity, apparent specific heat, etc.) are evaluated basing on the principal constituent of food (e.g. water, fiber, protein, etc.), while heat transfer coefficient is evaluated using empirical equations, depending on adimensional numbers. In this way, it is possible to be flexible and not strictly related to the evaluation of the properties of food for each different case. The proposed model was validated with different sets of experimental data related to beef and strawberry pulp freezing. For these cases, the R2 index is equal to 0.982 and 0.996 respectively, showing a good agreement between experimental and predicted data. Finally, a case study of spinach cubes freezing was provided to show the potentiality of the novel model.