Viviana Cocco Mariani a,*, Antonio Gilson Barbosa de Lima b, Leandro dos Santos Coelho c
´ Mechanical Engineering Graduate Program, Pontiﬁcal Catholic University of Parana – PUCPR, Imaculada Conceicao, ß˜ 1155, Prado Velho, Zip code 81215-901, Curitiba, PR, Brazil b Department of Mechanical Engineering,Federal University of Campina Grande – UFCG, P.O. Box 10069, Zip code 58109-970, Campina Grande, PB, Brazil c ´ Industrial and Systems Engineering Graduate Program, Pontiﬁcal Catholic University of Parana – PUCPR, Imaculada Conceicao, 1155, Prado Velho, ß˜ Zip code 80215-901, Curitiba, PR, Brazil
Abstract A new approach for the estimation of apparent thermal diﬀusivity of foods at diﬀerentdrying temperatures was explored, analysed and discussed in this work. Temperature versus time was obtained numerically at the center of the food (banana, ‘‘nanicao” variety) ˜ using the 1D Fourier equation with drying temperatures in the range between approximately 17–65 °C and moisture content in the range between 0.01 and 3.43 (dry basis). The solution of the partial diﬀerential equation is madewith a ﬁnite diﬀerence method coupled to an optimization technique of Diﬀerential Evolution used in inverse method. The mathematical model proposed considered the eﬀects of shrinkage and convective heat transfer at surface of fruit. Parameters of two functions, the ﬁrst dependent of the moisture content and the second dependent also of the temperature were obtained by inverse method modelling theapparent thermal diﬀusivity. Such parameters that provide the best least square ﬁt between the experimental and predicted time-temperatures curves are presented in this work. This study demonstrated that a small change in the temperature and moisture content of banana cause an abrupt change in the apparent thermal diﬀusivity, which decrease with the decreasing of the moisture. Statistical analysisshows the excellent agreement between reported and estimated curves.
Keywords: Thermal diﬀusivity; Banana; Inverse problem; Diﬀerential Evolution; Food
1. Introduction Drying, wetting, heating, cooling and freezing are important parts of food processing operation. Interest in transport properties of foods (thermal conductivity, heat capacity, density, mass and thermal diﬀusivity and heat andmass transfer coeﬃcient) appears due to the importance to predict heat and mass transfer rates during pro-
Corresponding author. Fax: +55 041 3271 1345. E-mail addresses: firstname.lastname@example.org (V.C. Mariani), gilson@ dem.ufcg.edu.br (A.G. Barbosa de Lima), email@example.com (L. dos Santos Coelho).
cessing, preservation and optimal design of processing equipment. The mainobjective of present work is to estimate the apparent thermal diﬀusivity of bananas during the drying. There are numerous methods to measure the thermal diﬀusivity proposed in the specialized literature. Nevertheless, most of them need relatively complex instrumentation or experimental assemblies and demand an expertise of the thermal phenomena. Several papers present such methods and results of thermaldiﬀusivity of diﬀerent foods, some are cited as follow. Choi and Okos (1983a, 1983b) propose a line heat-source thermal conductivity probe with auxiliary thermocouple to determine thermal conductivity and thermal diﬀusivity, simultaneously. Sweat (1986) recommends
Nomenclature A1, A2, A3, A4 parameters used to obtain thermal diﬀusivity, Eqs. (20) and (21) (m2/s) CR crossover or recombinationrate cv speciﬁc heat of vapor of water (J/kg.K) Def diﬀusion coeﬃcient (m2/s) f objective function fm mutation factor kmax maximum number of generations h heat transfer convective coeﬃcient (W/m2 °C) hfg latent heat of vaporization of water (J/kg) hm mass transfer convective coeﬃcient (m/s) i index j index represents time interval k thermal conductivity of the banana or air (W/ m °C), counter of...