Macroscopic modelling and parameter estimation of blueberries freeze-drying

Abstract

This work aimed to enhance comprehension of the processes involved in the primary stage of blueberry freeze-drying by applying and evaluating two mathematical models to whole fruits, assuming spherical geometry. This approach represents a novel contribution to the modeling of freeze-drying of whole solid plant foods. Blueberries were modeled as spheres with dried and icy layers delimited by the icy core radius. Two models were proposed, to predict the evolution of key state variables - interface temperature, internal temperature, interface pressure, surface pressure and total sublimation time during the process. Model 1 includes five parameters and a detailed representation of heat and mass transfer resistances, and assumes spherical symmetry, neglects shrinkage and desorption, and describes a one-dimensional heat and mass transfer in series. Model 2 is a simplified version of Model 1, represented with three parameters, by unifying mass and heat transfer coefficients. Experimental data from pressure rise tests (PRT) at 20 Pa chamber pressure ($P_{\mathrm{vc}}$) and shelf temperatures ($T_s$) of −20, 0, 10 and 20 °C were used for parameter estimation and validation. Particle swarm optimization was employed to determine heat and mass transfer parameters by minimizing mean square error (MSE) and mean absolute percentage error (MAPE). Performance metrics including MSE, MAPE and coefficient of determination ($R^2)$were considered to evaluate the models. The two models achieved errors below 10 % for PRT. The major contribution to overall mass transfer resistance lies in the blueberry skin layer. When models were used to predict sublimation rate, Model 2 significantly outperformed Model 1 in predicting the initial slope of PRT curves ($R^2$ = 0.96 vs. 0.73). Model 2 was used to simulate the effects of control variables ($P_{\mathrm{vc}}$ and $T_s$) and blueberry initial radius on sublimation time. This work enhances the understanding of heat and mass transfer mechanisms in blueberry freeze-drying, providing a valuable tool for process optimization.