The Cooling and Freezing of Parallelepiped-Shaped Solid: Foundations and Application to Food Product


Article Preview

Freezing is a physical treatment commonly used in operations such as drying, conservation and lyophilization of foods. In the processing and potato industries, parameters like dimension and initial moisture content of the product has a great effect on the cooling, freezing and post-freezing kinetics. Therefore, this work presents a transient three-dimensional mathematical modeling including phase change to describe the heat transfer during the process, of cooling and freezing parallelepiped foods. The governing equation was solved numerically using the finite-volume technique and a full implicit formulation. As an application, this methodology was used to describe the freezing process of potato (french-fry). Numerical results of the temperature in the center of the product were compared to the experimental data reported in the literature and a good agreement was obtained. Results of the temperature distribution inside the solid and cooling, freezing and post-freezing kinetics are presented and analyzed. It was verified that, the smaller the dimensions and lower the initial moisture content of the product, the solidification of water inside the solid occurs even faster. The largest temperature gradients were identified in the surface, close to the regions of the borders of the solid.



Diffusion Foundations (Volume 20)

Edited by:

João Delgado and A.G. Barbosa de Lima




M. Mesquita da Silva et al., "The Cooling and Freezing of Parallelepiped-Shaped Solid: Foundations and Application to Food Product", Diffusion Foundations, Vol. 20, pp. 78-91, 2019

Online since:

December 2018




* - Corresponding Author

[1] A.E. Delgado, D. W. Sun, Heat and mass transfer models for predicting freezing process – A review, J. Food Eng., 47(3) (2001) 157-174.

[2] I. Dincer, O.F. Genceli, Cooling process and heat transfer parameters of cylindrical products cooled both in water and air, Int. J. Heat Mass Transf., 37(4) (1994) 625-633.


[3] B.J. Teruel Mederos, Theoretical-experimental study of cooling of orange and banana with forced air", Doctoral Thesis in Mechanical Engineering, State University of Campinas, Campinas, Brazil, 2000. (In Portuguese).

[4] G.A. Montague, J. Glassey, M.J. Willis, French fry quality improvement using advanced control techniques, J. Food Eng., 57(4) (2003) 357-365.


[5] F.A. Ansari, V. Charan, V.K. Varma, Heat and mass transfer in fruits and vegetables and measurement of thermal diffusivity, Int. Commun. Heat Mass Transf., 11(6) (1984) 583-590.


[6] Z. Lin, A.C. Cleland, D.J. Cleland, G.F. Serrallach, A simple method for prediction of chilling times: extension to three-dimensional irregular shapes, Int. J. Refrig., 19(2) (1996) 107-114.


[7] F.A. Ansari, A. Afaq, New method to measuring thermal diffusivity of spherical produce, Int. J. Refrig., 9(3) (1986) 158-160.

[8] K.V. Chau, J. J. Gaffney, A finite-difference model for heat and mass transfer in products with internal generation and transpiration, J. Food Sci., 55(2) (1990) 484-487.


[9] D.I. LeBlanc, R. Kok, G. E. Timbres, Freezing of a parallelepiped food product. part 2. comparison of experimental and calculated results, Int. J. Refrig., 13(6) (1990) 379-392.


[10] Z. Lin, A.C. Cleland, D.J. Cleland, G.F. Serrallach, A simple method for prediction of chilling times: extension to three-dimensional irregular shapes, Int. J. Refrig., 19(2) (1996) 107-114.


[11] S. Chuntranuluck, C.M. Wells, A.C. Cleland, Prediction of chilling times of foods in situations where evaporative cooling is significant – Part 1. Method development, J. Food Eng., 37(2) (1998) 111-125.


[12] K. McDonald, D. Sun, Vacuum cooling technology for the food processing industry: a review, J. Food Eng., 45(2) (2000) 55-65.

[13] W.P. Silva, C.M.D.P.S. Silva, V.S.O. Farias, D.D.P.S. Silva, Calculation of the convective heat transfer coefficient and cooling kinetics of an individual fig fruit. Heat Mass Transf., 46(3) (2010a) 371-380.


[14] C.R. Maliska, Computational Heat Transfer and Fluid Mechanics, 2.ed., LTC Editora S.A., Rio de Janeiro, Brazil, 2004 (In Portuguese).

[15] W.P. Silva, J.W. Precker, D.D.P.S. Silva, C.D.P.S. Silva, A.G.B. Lima, Numerical simulation of diffusive processes in solids of revolution via the finite volume method and generalized coordinates, Int. J. Heat Mass Transf., 52(21-22) (2009).


[16] W.P. Silva, C.M.D.P.S. Silva, D.D.P.S. Silva, G.A. Neves, A.G.B. Lima, Mass and heat transfer study in solids of revolution via numerical simulationsusing finite volume method and generalized coordinates for the Cauchyboundary condition. Int. J. Heat and Mass Transf., 53(5-6) (2010b) 1183–1194.


[17] M.M. Silva, Modeling and simulation of heat transfer in foods with parallelepiped shape. studied case: cooling and freezing of potatoes. Master´s Dissertation in Agricultural Engineering, Federal University of Campina Grande, Campina Grande, Paraíba, Brazil. 2005. (In Portuguese).


[18] S.V. Patankar,Numerical Heat Transfer and Fluid Flow, Ed. Hemisphere Publishing Corporation, New York, (1980).

[19] H.K. Versteeg, W. Malalasekera,An Introduction to Computational Fluid Dynamics – The Finite Volume Method, Prentice Hall, London, (1995).

[20] A.O. Fortuna,Computational Techniques for Fluid Dynamics - Basic Concepts and applications, Edusp, São Paulo, 2000. (In Portuguese).

[21] J.J.S. Nascimento,Transient diffusion phenomena in parallelepiped solids. Case studies: Drying of ceramic materials", Doctoral Thesis in Mechanical Engineering, Federal Universityof Paraíba, João Pessoa, Paraíba, Brazil, 2002. (In Portuguese).

[22] N. Wang, J.G. Brennan,The Influence of moisture content and temperature on the specific heat of potato measured by differential scanning calorimetry, J. Food Eng., 19(3) (1993) 303-310.


[23] N. Wang, J.G. Brennan,Changes in structure, density and porosity of potato during dehydratation, J. Food Eng., 24(1) (1995) 61-76.

[24] N. Wang, J.G. Brennan,Thermal conductivity of potato as a function of moisture content", J. Food Eng., 17(2) (1992) 153-160.

[25] N.N. Mohsenin, Thermal Properties of Foods and Agricultural Materials. Gordon and Breach Science Publishers, New York, (1980).

[26] ASHRAE Handbook. Thermal properties of foods. American Society of Heating, Refrigeration and air-Conditioning Engineers, Inc., Atlanta, (1993).

[27] A. López-Ramos, E. Palmisano, A. Dombey, J.A. Pimentel, D. Fayés, D. González-Mendizável, Propriedades térmicas de frutas y hortalizas tropicales. Ciência e Tecnologia de Alimentos, 33(3) (1993) 271-283.

[28] A.B. Buhri, R.P. Sigh, Measurement of food thermal conductivity using differential scanning calorimetry. J. Food Sci., 58(5) (1993) 1145-1147.