Influence of the geometry on the numerical simulation of the cooling kinetics of cucumbers
Abstract
In this paper, the effect of the geometric representation of cucumbers on the numerical simulation of its cooling kinetics is studied. It is supposed that the diffusion model with boundary condition of the third kind satisfactorily describes the cooling, and that the thermo-physical parameters are constant during the process. The geometries used to represent the cucumber are: infinite cylinder, finite cylinder, and ellipsoid. The diffusion equation was solved through the finite volume method, with a fully implicit formulation, using cylindrical and generalized coordinates. The convective heat transfer coefficient and the thermal diffusivity were determined through optimization, using the inverse method. The best model in the representation of the cucumber’s shape was the ellipsoid, but the time demanded in its optimization was about 66 times greater than the time for the infinite cylinder.
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References
Amendola M., Queiroz M.R., 2007. Mathematical methodologies for calculating the mass diffusion coefficient of bananas during drying. Agriambi 11, 623-627.
ASHRAE, 1993. Handbook of fundamentals. American Society of Heating, Refrigerating and Air Conditioning Engineers, Atlanta, GA, USA.
Crank J., 1992. The mathematics of diffusion. Clarendon Press, Oxford, UK. PMCid:464097
Cuesta F.J., Lamúa M., 2009. Fourier series solution to the heat conduction equation with an internal heat source linearly dependent on temperature: application to chilling of fruit and vegetables. J Food Eng 90, 291-299. http://dx.doi.org/10.1016/j.jfoodeng.2008.06.036
Dincer I., 1996. Determination of thermal diffusivities of cylindrical bodies being cooled. Int Commun Heat Mass 23, 713-720. http://dx.doi.org/10.1016/0735-1933(96)00054-1
Doymaz I., 2005. Drying behaviour of green beans. J Food Eng 65, 161-165. http://dx.doi.org/10.1016/j.jfoodeng.2004.08.009
Erdogdu F., Turhan M., 2006. Analysis of dimensional ratios of regular geometries for infinite geometry assumptions in conduction heat transfer problems. J Food Eng 77, 818-824. http://dx.doi.org/10.1016/j.jfoodeng.2005.08.008
Fasina O.O., Fleming H.P., 2001. Heat transfer characteristics of cucumbers during blanching. J Food Eng 47, 203-210. http://dx.doi.org/10.1016/S0260-8774(00)00117-5
Gastón A.L., Abalone R.M., Giner S.A., 2002. Wheat drying kinetics. Diffusivities for sphere and ellipsoid by finite elements. J Food Eng 52, 313-322. http://dx.doi.org/10.1016/S0260-8774(01)00121-2
Gastón A.L., Abalone R.M., Giner S.A., Bruce D.M., 2003. Geometry effect on water diffusivity estimation in PROINTA-Isla Verde and broom wheat cultivars. Latin Am Appl Res 33, 327-331.
Hacihafizoglu O., Cihan A., Kahveci K., Lima A.G.B., 2008. A liquid diffusion model for thin-layer drying of rough rice. Eur Food Res Technol 226, 787-793. http://dx.doi.org/10.1007/s00217-007-0593-0
Le Niliot C., Lefèvre F., 2004. A parameter estimation approach to solve the inverse problem of point heat sources identification. Int J Heat Mass Tran 47, 827-841. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2003.08.011
Luikov A.V., 1968. Analytical heat diffusion theory. Academic Press, NY.
Maliska C.R., 2004. Transferência de calor e mecânica dos fluidos computacional. LTC Editora SA, Rio de Janeiro. [In Portuguese].
Mariani V.C., Amarante A.C.C., Coelho L.S., 2009. Estimation of apparent thermal conductivity of carrot purée during freezing using inverse problem. Int J Food Sci Tech 44, 1292-1303. http://dx.doi.org/10.1111/j.1365-2621.2009.01958.x
Patankar S.V., 1980. Numerical heat transfer and fluid flow. Hemisphere Publ Co, NY. PMCid:3013606
Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P., 1996. Numerical recipes in Fortran 77, the art of scientific computing. Cambridge Univ Press, NY.
Resende O., Corr A.P.C., Jarén C., Moure A.J., 2007. Bean moisture diffusivity and drying kinetics: a comparison of the liquid diffusion model when taking into account and neglecting grain shrinkage. Span J Agric Res 5, 51-58.
Ruiz-López I.I., García-Alvarado M.A., 2007. Analytical solution for food-drying kinetics considering shrinkage and variable diffusivity. J Food Eng 79, 208-216. http://dx.doi.org/10.1016/j.jfoodeng.2006.01.051
Silva W.P., Silva C.M.D.P.S., Silva D.D.P.S., Silva C.D.P.S., Lima A.G.B., 2006. Determinação de funções aproximadas para a solução numérica de uma equação diferencial ordinária. Revista de la Facultad de Ingeniería de la UCV 21, 29-37.
Silva W.P., Silva C.M.D.P.S., Silva D.D.P.S., Silva C.D.P.S., 2008. Numerical simulation of the water diffusion in cylindrical solids. Int J Food Eng 4, article 6.
Silva W.P., Precker J.W., Silva D.D.P.S., Silva C.D.P.S., Lima A.G.B., 2009a. Numerical simulation of diffusive processes in solids of revolution via the finite volume method and generalized coordinates. Int J Heat Mass Tran 52, 4976-4985. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2009.05.008
Silva W.P., Silva C.M.D.P.S., Nascimento P.L., Silva D.D.P.S., Silva C.D.P.S., 2009b. Influence of the geometry on the numerical simulation of isothermal drying kinetics of bananas. World Appl Sci J 7, 846-855.
Silva W.P., Silva C.M.D.P.S., Silva D.D.P.S., Neves G.A., Lima A.G.B., 2010a. Mass and heat transfer study in solids of revolution via numerical simulations using finite volume method and generalized coordinates for the Cauchy boundary condition. Int J Heat Mass Tran 53, 1183-1194. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2009.10.028
Silva W.P., Silva C.M.D.P.S., Farias V.S.O., Silva D.D.P.S., 2010b. Calculation of the convective heat transfer coefficient and cooling kinetics of an individual fig fruit. Heat Mass Transfer 46, 371-380. http://dx.doi.org/10.1007/s00231-010-0577-7
Sweat V.E., 1986. Thermal properties of foods. In: Engineering properties of foods (Rao M.A., Rizvi S.S.H., eds). Marcel Dekker, NY. PMid:2875716
Tannehill J.C., Anderson D.A., Pletcher R.H., 1997. Computational fluid mechanics and heat transfer, 2nd ed. Taylor & Francis, Philadelphia. PMid:9256128
Taylor J.R., 1997. An introduction to error analysis. University Science Books, 2nd ed. Sausalito, CA, USA.
Thompson J.F., Warsi Z.U.A., Mastin C.W., 1985. Numerical grid generation. Elsevier Sci Publ Co, NY.
Wu B., Yang W., Jia C., 2004. A three-dimensional numerical simulation of transient heat and mass transfer inside a single rice kernel during the drying process. Biosyst Eng 87, 191-200. http://dx.doi.org/10.1016/j.biosystemseng.2003.09.004
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