Calculation of the Thermal Properties (and Their Uncertainties) of Strawberry During Its Cooling Under Natural Convection

Many times, the thermal properties of a product are determined but their uncertainties (and, mainly, the covariance matrix) are not provided. Thus, in the simulations, it is not possible to establish a confidence band for a transient state described through the values obtained for these properties. In this article, a model was proposed to determine thermal diffusivity and convective heat transfer coefficient, providing the above-mentioned lack of information, for a product with spherical geometry during its cooling. The proposed model involved: 1) an experimental data set of the cooling kinetics in a point within the product; 2) a one-dimensional numerical solution of the heat conduction equation; 3) an optimizer based on the Levenberg-Marquardt algorithm to determine the thermal properties, their uncertainties, and the covariance between the parameters. Model was applied for determining thermal properties of strawberries, using an equivalent sphere to represent the geometry of the product, and the obtained results were compatible with the literature results.


Introduction
In the literature, several methods are found to determine the thermal properties of agricultural products during their cooling.One of these methods is based on the cooling kinetics of a point within the product, usually the central point (Dincer, 1995;Silva et al., 2011Silva et al., , 2012;;Erdogdu et al., 2014;Da Silva et al., 2018).In this method, a solution of the heat conduction equation is used to describe the experimental data set of the cooling kinetics, and the thermal properties are determined using some type optimization.As example, Dincer (1995) has used an analytical solution of the heat conduction equation, in which the series that represents the solution was truncated and only its first term was used in the calculation of the thermal parameters, using non-linear regression.In another example, Erdogdu et al. (2014) have also used with success a single term of the series for representing the analytical solution of one-dimensional heat conduction equation.The authors disregarded, by graphical observation of the logarithm applied on the dimensionless temperature versus time, the first experimental points through the identification of the linear portion of the graph.In this case in which logarithm is applied on the dimensionless temperatures, a simple linear regression enables to determine the thermal properties.Silva et al. (2012) have also used only the first term of the series for the non-linear regression, but these authors have repeated the procedure several times, eliminating the first experimental point; after that the authors eliminate two experimental points, etc. and, in each case, they determined the thermal properties by curve fitting.After that, the parameters determined in each case were used to calculate the corresponding chi-square, involving all the experimental points, in which more than 100 terms of the series were used.Finally, the authors compared all of the chi-squares, considering as optimal values for the parameters those corresponding to the lowest chi-square.To this algorithm the authors gave the name OREP (Optimal Removal of Experimental Points).purpose of this program was to determine parameters of differential equations and complex functions, including average values, their uncertainties and the covariance matrix (http://zeus.df.ufcg.edu.br/labfit/LS.htm).Da Silva et al. (2018) used this optimizer and an analytical solution of the diffusion equation (first 200 terms) for determining the thermal properties of banana with peel, their uncertainties and the covariance matrix.The authors have reported good results for the calculated parameters using this model involving LS Optimizer.
An observation in the literature (Sweat, 1986;ASHRAE, 1993;Fricke & Becker, 2001;Da Silva et al., 2010, 2011;Silva et al., 2011Silva et al., , 2012) ) makes it possible to realize that it is not so common to determine thermal properties of agricultural products, including their uncertainties and the covariance matrix, when only one experimental data set is available.However, LS Optimizer enables these calculations and, in this sense, the objectives of this article are defined in the following.
The main objective of this article was to propose a model to determine thermal properties of agricultural products, which may be considered as a sphere, during their cooling.The model have involved the following characteristics: 1) a solver to describe the direct problem, in order to simulate the kinetics cooling of a point within the sphere equivalent to the real geometry of the product; 2) an experimental data set of the cooling kinetics of a point within the equivalent sphere; 3) an optimization program for the inverse problem, which consists in determining of the thermal properties (their uncertainties and the covariance matrix) for the product.The model was used to study strawberry cooling, determining not only its thermal properties, but also simulating the process.

Diffusion Equation
In spherical coordinates, the one-dimensional heat conduction equation is written in the following way: where, T is the temperature, t is the cooling time, ρ is the density of the product, c p is the specific heat, r defines a position within the equivalent sphere (with respect to the center) and k is the thermal conductivity.On the other hand, in a cooling process, the temperature variation is usually equal or less than 25 °C and, in this case, ρ and c p can be assumed as constant properties.Thus, dividing Equation (1a) by the product ρc p , it is obtained: where, α = k/(ρc p ) is the thermal diffusivity.For the cooling process, the boundary condition is, usually, of the third kind, and it can be expressed by the following equations: ( ) where, h = h H /(ρc p ) is the convective heat transfer coefficient, T b is the temperature at the boundary, T ∞ is the value of temperature of the air in which the strawberry fruit is cooled, and h H is the heat transfer coefficient.

Direct Problem: Numerical Solution to Create a Solver
Equation (1b), subjected to the boundary condition defined by Equation (2b), was solved by using the finite volume method, with a fully implicit formulation (Patankar, 1980).Figure 1 presents a sphere and its uniform one-dimensional mesh, in which the control volumes have a common thickness given by ∆r; and the control volume number "i" has a nodal point "P".4) is obta he system of eq Algorithm) me

Inverse Problem: Optimization
In order to determine α and h, including their uncertainties, LS Optimizer Software was used.As above-mentioned, this software uses Levenberg-Marquardt algorithm (Levenberg, 1944;Marquardt, 1963) and executes the solver provided by the user for the direct problem, with the objective of determining the parameters α and h that make it possible a simulation as close as possible to the experimental data set (minimum chi-square).
According to the information available in the LS Optimizer manual, as long as the solver for the direct problem and the experimental data set are provided, this optimization software executes the solver in order to obtain the necessary information for the determination of parameters, using the statistical indicator chi-square as objective function: χ 2 = Σ(T exp -T sim ) 2 /σ 2 .If the uncertainties of the experimental points (σ) are not known at first, LS Optimizer determines the standard deviation associated with the optimization process using the expression σ' = [Σ(T exp -T sim ) 2 /(N p -n)] 1/2 in which (N p -n) is the number of degrees of freedom for the optimization process.Hence, at the end of the iterative process, one can impose σ = σ' and recalculate the covariance matrix, which provides the uncertainties of the parameters and the correlation between them.Thus, starting from initial values α 0 and h 0 , the optimizer corrects these parameters iteratively up to the convergence and, in the end, it delivers the optimal values for α and h, their uncertainties, and the covariance matrix.

Experimental Data
Silva et al. ( 2011) performed cooling of strawberry (X = 90%, wet basis) under natural convection, initially at 24.8 °C, using a home refrigerator Brastemp with internal temperature set about 0.6 ºC and relative humidity between 87 and 92%.A thermocouple was inserted within the strawberry, for measuring the temperature at regular time interval during 140 min, but its exact position was not determined during the experiment; and the group was placed inside the refrigerator.In the present article, the experimental data set was obtained by digitization, using the program xyExtract (V 5.1, 2011, http://zeus.df.ufcg.edu.br/labfit/index_xyExtract.htm).
The real geometry of the strawberry was considered as an equivalent sphere, with radius R = 0.0167 m, and this value was obtained by imposing same volume for the two geometries (real and sphere).For the equivalent sphere, the thermocouple position was estimated using optimization technique: r = 0.0091 m.
Due to the robust method of optimization used by Silva et al. (2011) to determine thermal properties, the uncertainties of the obtained values were not calculated.In cooling processes, the calculation of only the average values of the parameters is a common procedure found in the literature (Sweat, 1986;ASHRAE, 1993;Fricke & Becker, 2001;Silva et al., 2010Silva et al., , 2011)).In the present work, the data set obtained by Silva et al. (2011) in the cooling experiment was used to determine not only the thermal properties of strawberry, by optimization, but also the uncertainties and the covariance between these parameters.Although it can't be assumed a priori that the error distribution to be obtained is a Gaussian distribution, as the experimental data set contains 61 points, in practical terms, the uncertainties to be obtained will define intervals with confidence at least close to 68.3% (Bevington & Robinson, 1992;Taylor, 1997).

Results and Discussion
In this study, a fully implicit formulation was used for the numerical solution, due to the fact that this formulation ensures that such solution is unconditionally stable (Patankar, 1980).A preliminary study has shown that, in the simulations of a diffusion process in a small product with spherical geometry, 200 control volumes and 2000 steps of time are enough in order to disregard possible convergence errors for the iterative process (Da Silva et al., 2012).Thus, these refinements for the grid and time were used in all the calculations of this article.
Table 1.Thermal diffusivity and convective heat transfer coefficient for strawberry.The intervals were obtained with a confidence of 68.3% Thermal property Present study Silva et al. (2011) α (m 2 s -1 ) (1.42±0.11)× 10 -7 1.42 × 10 -7 h (ms -1 ) (3.196±0.039)× 10 -6 3.256 × 10 -6 It should be remembered that the LS program minimizes the chi-square obtained by the deviations calculated by the differences between the simulated values and the values of the corresponding experimental points, and the minimum chi-square value is used to determine the uncertainties of each calculated parameter.This procedure is Figure 1 Figur