Mathematical particle model for microwave drying of leaves

  • V. H. Borda-Yepes
  • F. ChejneEmail author
  • D. A. Granados
  • B. Rojano
  • V. S. G. Raghavan


In this work, the model of particles for microwave drying by means of assisted force convection for blueberry leaves is described. A one-dimensional particle model is made in the direction of the thickness of the leaf. Only the phases of water during drying are considered. The mass and energy equation in the particle model develops. The effective diffusivity and the Arrhenius equation for the water phase (liquid and vapor) are considered in the mass equation. The energy equation considers the Lambert-Beer equation. The simulation is performed for different cases of microwave powers (100, 300, 400 W) and temperatures (50, 60 and 70 °C) The activation energy and the pre-exponential factor in the Arrhenius equation are taken from the kinetic analysis prior to this Work The temperature and mass profiles for some experimental and theoretical cases are compared, and it is observed that the model considered gives good results of adjustment between the experimental and the theoretical.



V.H. Borda-Yepes wish to thank the Colombian Administrative Department of Science, Technology and Innovation (COLCIENCIAS, #617) (Departamento Administrativo de Ciencia, Tecnología e Innovacion) for financial support awarded to the program Doctoral in Engineering - Energy System of the National University of Colombia, Sede Medellin, and to the stay of doctoral training at McGill University. F. Chejne wish to thank to the project “Strategy of transformation of the Colombian energy sector in the horizon 2030” funded by the call 788 of Colciencias: Scientific Ecosystem. Contract number FP44842-210-2018.

Supplementary material


  1. 1.
    Datta AK (2007) Porous media approaches to studying simultaneous heat and mass transfer in food processes. I: Problem formulations. J Food Eng 80:80–95. CrossRefGoogle Scholar
  2. 2.
    Warning AD, Arquiza JMR, Datta AK (2014) A multiphase porous medium transport model with distributed sublimation front to simulate vacuum freeze drying. Food Bioprod Process 94:637–648. CrossRefGoogle Scholar
  3. 3.
    Ho QT, Carmeliet J, Datta AK, Defraeye T, Delele MA, Herremans E et al (2013) Multiscale modeling in food engineering. J Food Eng 114:279–291. CrossRefGoogle Scholar
  4. 4.
    Defraeye T (2014) Advanced computational modelling for drying processes - A review. Appl Energy 131:323–344. CrossRefGoogle Scholar
  5. 5.
    Kapellos GE, Alexiou TS, Payatakes AC (2012) A multiscale theoretical model for fluid flow in cellular biological media. Int J Eng Sci 51:241–271. MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fanta SW, Abera MK, Ho QT, Verboven P, Carmeliet J, Nicolai BM (2013) Microscale modeling of water transport in fruit tissue. J Food Eng 118:229–237. CrossRefGoogle Scholar
  7. 7.
    Karunasena HCP, Senadeera W, Brown RJ, Gu YT (2014) A particle based model to simulate microscale morphological changes of plant tissues during drying. Soft Matter 10:5249–5268. CrossRefGoogle Scholar
  8. 8.
    Karunasena HCP, Senadeera W, Brown RJ, Gu YT (2014) A meshfree model for plant tissue deformations during drying. ANZIAM J 55:110–113MathSciNetzbMATHGoogle Scholar
  9. 9.
    Nam JH, Song CS (2007) Numerical simulation of conjugate heat and mass transfer during multi-dimensional freeze drying of slab-shaped food products. Int J Heat Mass Transf 50:4891–4900. CrossRefzbMATHGoogle Scholar
  10. 10.
    Verboven P, Kerckhofs G, Mebatsion HK, Ho QT, Temst K, Wevers M et al (2008) Three-dimensional gas exchange pathways in pome fruit characterized by synchrotron x-ray computed tomography. Plant Physiol 147:518–527. CrossRefGoogle Scholar
  11. 11.
    Luikov AV (1968) Two-Dimensional temperature field: Particular problems. Anal Heat Diffus Theory 460:245–248. Google Scholar
  12. 12.
    Budd CJ (2011) Hill a. DC. A comparison of models and methods for simulating the microwave heating of moist foodstuffs. Int J Heat Mass Transf 54:807–817. CrossRefzbMATHGoogle Scholar
  13. 13.
    Zhao J, Zhang T, Corless RM (2006) Convergence of the compact finite difference method for second-order elliptic equations. Appl Math Comput 182:1454–1469. MathSciNetzbMATHGoogle Scholar
  14. 14.
    Leonard BP (1995) Order of accuracy of QUICK and related convection-diffusion schemes. Appl Math Model 19:640–653. CrossRefzbMATHGoogle Scholar
  15. 15.
    Pettermann HE, Huber CO, Luxner MH, Nogales S, Böhm HJ (2010) An incremental mori-tanaka homogenization scheme for finite: Strain thermoelastoplasticity of mmcs. Materials (Basel) 3:434–451. CrossRefGoogle Scholar
  16. 16.
    Karunasena HCP, Brown RJ, Gu YT, Senadeera W (2015) Application of meshfree methods to numerically simulate microscale deformations of different plant food materials during drying. J Food Eng 146:209–226. CrossRefGoogle Scholar
  17. 17.
    Yao Z, Le Maguer M (1997) Mathematical Modelling and Simulation of Mass lkansfer in Osmotic Dehydration Processes. Part III: Parametric Study. J Food Eng 8774:33–46. CrossRefGoogle Scholar
  18. 18.
    Chen D-SD, Singh RK, Haghighi K, Nelson PE (1993) Finite element analysis of temperature distribution in microwaved cylindrical potato tissue. J Food Eng 18:351–368. CrossRefGoogle Scholar
  19. 19.
    Balaban M (1989) Effect of volume change in foods on the temperature and moisture content predictions of simultaneous heat and moisture transfer models. J Food Process Eng 12:67–88. CrossRefGoogle Scholar
  20. 20.
    Curcio S, Aversa M (2014) Influence of shrinkage on convective drying of fresh vegetables: A theoretical model. J Food Eng 123:36–49. CrossRefGoogle Scholar
  21. 21.
    Kowalski SJ (2002) Modelling of fracture phenomena in dried materials. Chem Eng J 86:145–151. CrossRefGoogle Scholar
  22. 22.
    Udell KS (1985) Heat transfer in porous media considering phase change and capillarity-the heat pipe effect. Int J Heat Mass Transf 28:485–495. CrossRefzbMATHGoogle Scholar
  23. 23.
    Campañone LA, Zaritzky NE (2005) Mathematical analysis of microwave heating process. J Food Eng 69:359–368. CrossRefGoogle Scholar
  24. 24.
    Doymaz I (2006) Thin-layer drying behaviour of mint leaves. J Food Eng 74:370–375. CrossRefGoogle Scholar
  25. 25.
    Doymaz I, Tugrul N, Pala M (2006) Drying characteristics of dill and parsley leaves. J Food Eng 77:559–565. CrossRefGoogle Scholar
  26. 26.
    Özbek B, Dadali G (2007) Thin-layer drying characteristics and modelling of mint leaves undergoing microwave treatment. J Food Eng 83:541–549. CrossRefGoogle Scholar
  27. 27.
    Onwude DI, Hashim N, Janius RB, Nawi NM, Abdan K (2016) Modeling the Thin-Layer Drying of Fruits and Vegetables: A Review. Compr Rev Food Sci Food Saf 15:599–618. CrossRefGoogle Scholar
  28. 28.
    De VD (1958) Simultaneous transfer of heat and moisture in porous media. Trans Am Geophys Union 39:909–916CrossRefGoogle Scholar
  29. 29.
    Datta AK (2007) Porous media approaches to studying simultaneous heat and mass transfer in food processes. II: Property data and representative results. J Food Eng 80:96–110. CrossRefGoogle Scholar
  30. 30.
    Batista LM, da Rosa CA, Pinto LAA (2007) Diffusive model with variable effective diffusivity considering shrinkage in thin layer drying of chitosan. J Food Eng 81:127–132. CrossRefGoogle Scholar
  31. 31.
    Chen XD, Sidhu H, Nelson M (2011) Theoretical probing of the phenomenon of the formation of the outermost surface layer of a multi-component particle, and the surface chemical composition after the rapid removal of water in spray drying. Chem Eng Sci 66:6375–6384. CrossRefGoogle Scholar
  32. 32.
    Aregawi W, Defraeye T, Saneinejad S, Vontobel P, Lehmann E, Carmeliet J et al (2013) Dehydration of apple tissue: Intercomparison of neutron tomography with numerical modelling. Int J Heat Mass Transf 67:173–182. CrossRefGoogle Scholar
  33. 33.
    Aregawi WA, Defraeye T, Verboven P, Herremans E, Roeck GD, Nicolai BM (2013) Modelling of coupled water transport and large deformation during dehydration of apple tissue. Food Bioprocess Technol 6:1963–1978. CrossRefGoogle Scholar
  34. 34.
    Erbay Z, Icier F (2010) A Review of Thin Layer Drying of Foods: Theory, Modeling, and Experimental Results. Crit Rev Food Sci Nutr 50:441–464. CrossRefGoogle Scholar
  35. 35.
    Rayaguru K, Routray W (2011) Microwave drying kinetics and quality characteristics of aromatic Pandanus amaryllifolius leaves. Int Food Res J 18:1035–1042Google Scholar
  36. 36.
    Wang R, Zhou W, R-AH W (2006) Kinetic study of the thermal stability of tea catechins in aqueous systems using a microwave reactor. J Agric Food Chem 54:5924–5932. CrossRefGoogle Scholar
  37. 37.
    Dadalı G, Kılıç Apar D, Özbek B (2007) Microwave Drying Kinetics of Okra. Dry Technol 25:917–924. CrossRefGoogle Scholar
  38. 38.
    Kasaoka S, Sakata Y, Shimada M, Matsutomi T (1985) New Kinetic Model for Temperature Programmed Thermogravimetry and Its Applications To the Gasification of Coal Chars With Steam and Carbon Dioxide. J Chem Eng Japan 18:426–432. CrossRefGoogle Scholar
  39. 39.
    Chizoba Ekezie FG, Sun DW, Han Z, Cheng JH (2017) Microwave-assisted food processing technologies for enhancing product quality and process efficiency: A review of recent developments. Trends Food Sci Technol 67:58–69. CrossRefGoogle Scholar
  40. 40.
    Marra F, De Bonis MV, Ruocco G (2010) Combined microwaves and convection heating: A conjugate approach. J Food Eng 97:31–39. CrossRefGoogle Scholar
  41. 41.
    Talens C, Castro-Giraldez M, Fito PJ (2016) A thermodynamic model for hot air microwave drying of orange peel. J Food Eng 175:33–42. CrossRefGoogle Scholar
  42. 42.
    Salagnac P, Glouannec P, Lecharpentier D (2004) Numerical modeling of heat and mass transfer in porous medium during combined hot air, infrared and microwaves drying. Int J Heat Mass Transf 47:4479–4489. CrossRefzbMATHGoogle Scholar
  43. 43.
    Venkatesh MS, Raghavan GSV (2004) An overview of microwave processing and dielectric properties of agri-food materials. Biosyst Eng 88:1–18. CrossRefGoogle Scholar
  44. 44.
    Whitaker S (1977) Simultaneous Heat, Mass, and Momentum Transfer in Porous Media: A Theory of Drying. Adv Heat Tran 13:119–203. CrossRefGoogle Scholar
  45. 45.
    Quintard M, Whitaker S (1993) Transport in ordered and disordered porous media: volume-averaged equations, closure problems, and comparison with experiment. Chem Eng Sci 48:2537–2564. CrossRefGoogle Scholar
  46. 46.
    Li N, Taylor LS, Ferruzzi MG, Mauer LJ (2012) Kinetic Study of Catechin Stability: Effects of pH, Concentration, and Temperature. J Agric Food Chem 60:12531–12539. CrossRefGoogle Scholar
  47. 47.
    Joardder MUH, Karim A, Kumar C, Brown RJ (2015) Porosity: Establishing the Relationship between Drying Parameters and Dried Food QualityGoogle Scholar
  48. 48.
    Barbosa-Canovas GV (1996) Dehydration of foods. Chapman & Hall, New YorkCrossRefGoogle Scholar
  49. 49.
    Liu JY, Shun C (1991) Solutions of Luikov equations of heat and mass transfer in capillary-porous bodies. Int J Heat Mass Transf 34:1747–1754. CrossRefGoogle Scholar
  50. 50.
    Halder A, Datta AK (2012) Surface heat and mass transfer coefficients for multiphase porous media transport models with rapid evaporation. Food Bioprod Process 90:475–490. CrossRefGoogle Scholar
  51. 51.
    Puangsuwan K, Chongcheawchamnan M, Tongurai C (2015) Effective Moisture Diffusivity, Activation Energy and Dielectric Model for Palm Fruit Using a Microwave Heating. J Microw Power Electromagn Energy 49:100–111. CrossRefGoogle Scholar
  52. 52.
    Nield DA, Bejan A (2017) Convection in Porous Media. 5 ed. Switzerland. Springer-Verlag, Berlin Heidelberg. CrossRefzbMATHGoogle Scholar
  53. 53.
    Khawam A, Flanagan DR (2006) Solid-State Kinetic Models: Basics and Mathematical Fundamentals. J Phys Chem 110:17315–17328. CrossRefGoogle Scholar
  54. 54.
    Chandrasekaran S, Ramanathan S, Basak T (2013) Microwave food processing-A review. Food Res Int 52:243–261. CrossRefGoogle Scholar
  55. 55.
    Yang J, Di Q, Zhao J, Wang L (2009) Mechanism on mass transfer in micro-scale during the microwave drying of plant porous materials. Proc ASME Summer Heat Transf Conf, HT2009 3:8–10. Google Scholar
  56. 56.
    Yang J, Chen JF, Zhao YY, Mao LC (2010) Effects of drying processes on the antioxidant properties in sweet potatoes. Agric Sci China 9:1522–1529. CrossRefGoogle Scholar
  57. 57.
    Balaban M, Piggot GM (1988) Mathematical model of simultaneous heat and mass transfer in food with dimensional changes and variable transport parameters. J Food Sci 53:935–939. CrossRefGoogle Scholar
  58. 58.
    Hasatani M, Itaya Y, Miura K (1988) Hybrid Drying of Granular Materials By Combined Radiative and Convective Heating. Dry Technol 6:43–68. CrossRefGoogle Scholar
  59. 59.
    Dhib R (2007) Infrared Drying: From Process Modeling to Advanced Process Control. Dry Technol 25:97–105. CrossRefGoogle Scholar
  60. 60.
    Shafiur Rahman M (2009) Food propertries handbook. Second edi. Boca RatonGoogle Scholar
  61. 61.
    Esveld DC, Van Der Sman RGM, Van Dalen G, Van Duynhoven JPM, Meinders MBJ (2012) Effect of morphology on water sorption in cellular solid foods. Part I: Pore scale network model. J Food Eng 109:301–310. CrossRefGoogle Scholar
  62. 62.
    Esveld DC, Van Der Sman RGM, Witek MM, Windt CW, Van AH, Van Duynhoven JPM et al (2012) Effect of morphology on water sorption in cellular solid foods. Part II: Sorption in cereal crackers. J Food Eng 109:311–320. CrossRefGoogle Scholar
  63. 63.
    Nguyen TA, Verboven P, Scheerlinck N, Veraverbeke E, Nicolai BM (2003) An estimation procedure of effective diffusivity in pear tissue by means of a numerical water diffusion model. Acta Hortic 599:541–548. CrossRefGoogle Scholar
  64. 64.
    Feng H, Yin Y, Tang J (2012) Microwave Drying of Food and Agricultural Materials: Basics and Heat and Mass Transfer Modeling. Food Eng Rev 4:89–106. CrossRefGoogle Scholar
  65. 65.
    Syamaladevi RM, Sablani SS, Tang J, Powers J, Swanson BG (2009) State diagram and water adsorption isotherm of raspberry (Rubus idaeus). J Food Eng 91:460–467. CrossRefGoogle Scholar
  66. 66.
    Botheju WS, Amarathunge KSP, Mohamed MTZ (2008) Modeling moisture desorption isotherms and thermodynamic properties of fermented tea Dhool (Camellia sinensis var. assamica). Dry Technol 26:1294–1299. CrossRefGoogle Scholar
  67. 67.
    Knani S, Aouaini F, Bahloul N, Khalfaoui M, Hachicha MA, Ben Lamine A et al (2014) Modeling of adsorption isotherms of water vapor on Tunisian olive leaves using statistical mechanical formulation. Phys A Stat Mech Its Appl 400:57–70. CrossRefGoogle Scholar
  68. 68.
    Santos MV, Vampa V, Califano A, Zaritzky N (2010) Numerical simulations of chilling and freezing processes applied to bakery products in irregularly 3D geometries. J Food Eng 100:32–42. CrossRefGoogle Scholar
  69. 69.
    Radhika GB, Shyma SM (2014) Estimation of Mass Transfer Parameters and Drying Characteristics of Black Pepper using Microwave Energy. Int J Innov Res Adv Eng 1:228–234Google Scholar
  70. 70.
    Zanoelo EF (2007) A theoretical and experimental study of simultaneous heat and mass transport resistances in a shallow fluidized bed dryer of mate leaves. Chem Eng Process Process Intensif 46:1365–1375. CrossRefGoogle Scholar
  71. 71.
    Bergman TL, Lavine A, Incropera FP (2017) Fundamentals of heat and mass transfer. John Wiley & Sons, Inc, HobokenGoogle Scholar
  72. 72.
    Krokida MK, Maroulis ZB (1997) Effect of drying method on shrinkage and porosity. Dry Technol 15:2441–2458. CrossRefGoogle Scholar
  73. 73.
    Rahman MS (2001) Toward Prediction of Porosity in Foods During Drying: a Brief Review. Dry Technol 19:1–13. CrossRefGoogle Scholar
  74. 74.
    Kajishima T, Taira K (2017) Computational Fluid Dynamics. Cambridge. Cambridge University Press, New York. CrossRefzbMATHGoogle Scholar
  75. 75.
    Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere Pub. Corp.; McGraw-Hill, Washington; New YorkzbMATHGoogle Scholar
  76. 76.
    Allanic N, Salagnac P, Glouannec P (2007) Convective and radiant drying of a polymer aqueous solution. Heat Mass Transf Und Stoffuebertragung 43:1087–1095. CrossRefGoogle Scholar
  77. 77.
    Rahman MS (2014) Mass–Volume–Area-Related Properties of Foods. In: Group T& F, editor. Eng. Prop. Foods. Fourth Edi, CRC Press, p. 1–36.Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • V. H. Borda-Yepes
    • 1
  • F. Chejne
    • 1
    Email author
  • D. A. Granados
    • 1
    • 2
  • B. Rojano
    • 3
  • V. S. G. Raghavan
    • 4
  1. 1.Facultad de MinasUniversidad Nacional de Colombia – MedellínMedellínColombia
  2. 2.Universidad Católica de OrienteRionegroColombia
  3. 3.Facultad de CienciasUniversidad Nacional de Colombia – MedellínMedellínColombia
  4. 4.Department of Bioresource Engineering, Faculty of Agricultural and Environmental SciencesMcGill UniversityQuebecCanada

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