Advanced Study to Heat and Mass Transfer in Arbitrary Shape Porous Materials: Foundations, Phenomenological Lumped Modeling and Applications

  • E. S. Lima
  • W. M. P. B. Lima
  • Antonio Gilson Barbosa de LimaEmail author
  • S. R. de Farias Neto
  • E. G. Silva
  • V. A. B. Oliveira
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 93)


This chapter provides information related to simultaneous heat and mass transfer in unsaturated porous bodies with particular reference to drying process of arbitrarily-shaped solids. Several important topics such as drying theory, moisture migration mechanisms, lumped and distributed modeling for homogeneous and heterogeneous bodies, and applications are presented and discussed. Herein, a new phenomenological and advanced lumped-parameter model written in any coordinate system is presented, and the analytical solutions of the governing equations, limitations of the modeling and general theoretical results are discussed. The proposed model includes different effects such as shape of the body (hollow or not hollow), heat and mass generation, and coupled heating, evaporation and convection phenomena.


Drying Heat Mass Theoretical Complex shape solid 



The authors thank to CNPq, FINEP and CAPES (Brazilian Research Agencies) for financial support and to the authors referred in this text that contributed for improvement of this work.


  1. 1.
    Park, K.J., Brod, F.P.R.: Comparative study of grated coconut (Cocos nucifera) drying using vertical and horizontal dryers. In: Inter-American Drying Conference (IADC), Itu, Brazil, B, pp. 469–475 (1997)Google Scholar
  2. 2.
    Steffe, J.F., Singh, R.P.: Liquid diffusivity of rough rice components. Trans. ASAE 23(3), 767–774 (1980)CrossRefGoogle Scholar
  3. 3.
    Fortes, M., Okos, M.R.: Drying theories: their bases and limitations as applied to foods and grains. In: Advances in Drying, vol. 1, pp. 119–154. Hemisphere Publishing Corporation, Washington (1980)Google Scholar
  4. 4.
    Alvarenga, L.C., Fortes, M., Pinheiro Filho, J.B., Hara, T.: Moisture transport inside the black bean grains under drying conditions. Rev. Bras. de Armazenamento 5(1), 5–18 (1980) (in Portuguese)Google Scholar
  5. 5.
    Mariz, T.F.: Drying of cotton seed shell in fixed bed. Master dissertation in Chemical Engineering, Federal University of Paraiba, Campina Grande, Brazil (1986) (in Portuguese)Google Scholar
  6. 6.
    Keey, R.B.: Drying of Loose and Particulate Materials. Hemisphere Publishing Corporation, New York (1992)Google Scholar
  7. 7.
    Lima, A.G.B.: Drying study and design of silkworm cocoon dryer. Master dissertation in Mechanical Engineering, Federal University of Paraiba, Campina Grande, Brazil (1995) (in Portuguese)Google Scholar
  8. 8.
    Ibrahim, M.H., Daud, W.R.W., Talib, M.Z.M.: Drying characteristics of oil palm kernels. Drying Technol. 15(3–4), 1103–1117 (1997)CrossRefGoogle Scholar
  9. 9.
    Lima, A.G.B.: Diffusion phenomena in prolate spheroidal solids: case studies: drying of bananas. Doctorate thesis in Mechanical Engineering, State University of Campinas, Campinas, Brazil (1999) (in Portuguese)Google Scholar
  10. 10.
    Strumillo, C., Kudra, T.: Drying: Principles, Science and Design. Gordon and Breach Science Publishers, New York (1986)Google Scholar
  11. 11.
    Brooker, D.B., Bakker-Arkema, F.W., Hall, C.W.: Drying and Storage of Grains and Oilseeds. AVI Book, New York (1992)Google Scholar
  12. 12.
    Erbay, Z., Icier, F.: A review of thin-layer drying of foods: theory, modeling, and experimental results. Crit. Rev. Food Sci. Nutr. 50(5), 441–464 (2010)CrossRefGoogle Scholar
  13. 13.
    Luikov, A.V.: Heat and Mass Transfer in Capillary Porous Bodies. Pergamon Press, New York (1966)CrossRefGoogle Scholar
  14. 14.
    Sarker, N.N., Kunze, O.R., Stroubolis, T.: Finite element simulation of rough rice drying. Drying Technol. 12(4), 761–775 (1994)CrossRefGoogle Scholar
  15. 15.
    Zogzas, N.P., Maroulis, Z.B.: Effective moisture diffusivity estimation from drying data: a comparison between various methods of analysis. Drying Technol. 14(7–8), 1543–1573 (1996)CrossRefGoogle Scholar
  16. 16.
    Liu, J.Y., Simpson, W.T.: Solutions of diffusion equation with constant diffusion and surface emission coefficients. In: Inter-American Drying Conference (IADC), Itu, Brazil, A, pp. 73–80 (1997)CrossRefGoogle Scholar
  17. 17.
    Freire, E.S., Chau, K.V.: Simulation of the drying process of fermented cacao beans. In: Inter-American Drying Conference (IADC), Itu, Brazil, B, pp. 356–363 (1997)Google Scholar
  18. 18.
    Baroni, A.F., Hubinger, M.D.: Drying of onion: effects of pre-treatment on moisture transport. In: Inter-American Drying Conference (IADC), Itu, Brazil, B, pp. 419–426 (1997)Google Scholar
  19. 19.
    Sabadini, E., Carvalho Jr., B.C., Sobral, P.J.A., Hubinger, M.D.: Mass transfer and diffusion coefficient determination in salted and dried meat pieces. In: Inter American Drying Conference (IADC), Itu, Brazil, B, pp. 441–447 (1997)Google Scholar
  20. 20.
    Quintana-Hernandez, P., Rodrigues-Ramirez, J., Mendes-Lagunas, L., Cornejo-Serrano, L.: Humidity diffusion within sugarcane fibers. In: Inter-American Drying Conference (IADC), Itu, Brazil, B, pp. 538–542 (1997)Google Scholar
  21. 21.
    Oliveira, V.A.B., Lima, A.G.B.: Unsteady state mass diffusion prolate spheroidal solids: an analytical solution. In: Inter-American Drying Conference (IADC), Boca del Rio, Mexico, pp. 163–172 (2001)Google Scholar
  22. 22.
    Carmo, J.E.F., Lima, A.G.B.: Modelling and simulation of mass transfer inside the oblate spheroidal solids. In: Inter-American Drying Conference (IADC), Boca del Rio, Mexico, pp. 173–183 (2001)Google Scholar
  23. 23.
    Nascimento, J.J.S., Belo, F.A., Lima, A.G.B.: Simultaneous moisture transport and shrinkage during drying of parallelepiped solids. In: Inter-American Drying Conference (IADC), Boca del Rio, Mexico, pp. 535–544 (2001)Google Scholar
  24. 24.
    Karathanos, V.T., Vagenas, G.K., Saravacos, G.D.: Water diffusivity in starches at high temperatures and pressures. Biotechnol. Prog. 7(2), 178–184 (1991)CrossRefGoogle Scholar
  25. 25.
    Fortes, M., Okos, M.R.: A non-equilibrium thermodynamics approach to transport phenomena in capillary porous media. Trans. ASAE 24, 756–760 (1981)CrossRefGoogle Scholar
  26. 26.
    Ozdemir, M., Devres, Y.O.: The thin-layer drying characteristics of hazelnuts during roasting. J. Food Eng. 42(4), 225–233 (2000)CrossRefGoogle Scholar
  27. 27.
    Panchariya, P.C., Popovic, D., Sharma, A.L.: Thin-layer modeling of black tea drying process. J. Food Eng. 52(4), 349–357 (2002)CrossRefGoogle Scholar
  28. 28.
    Akpinar, E.K.: Determination of suitable thin-layer drying curve model for some vegetables and fruits. J. Food Eng. 73(1), 75–84 (2006)CrossRefGoogle Scholar
  29. 29.
    Doymaz, I.: The kinetics of forced convective air-drying of pumpkin slices. J. Food Eng. 79(1), 243–248 (2007)CrossRefGoogle Scholar
  30. 30.
    Raquel, P.F., Susana, P., Maria, J.B.: Study of the convective drying of pumpkin (Cucurbita maxima). Food Bioprod. Process. 89(4), 422–428 (2011)CrossRefGoogle Scholar
  31. 31.
    Henderson, S.M.: Progress in developing the thin-layer drying equation. Trans. ASAE 17(6), 1167–1172 (1974)CrossRefGoogle Scholar
  32. 32.
    Bruce, D.M.: Exposed layer barley drying, three models fitted to new data up to 150°C. J. Agric. Eng. Res. 32(4), 337–348 (1985)CrossRefGoogle Scholar
  33. 33.
    Santos, G.M.: Study of thermal behavior of a tunnel kiln applied to red ceramic industry. Master dissertation in Mechanical Engineering, Federal University of Santa Catarina, Florianópolis, Brazil (2001) (in Portuguese)Google Scholar
  34. 34.
    Nishikawa, T., Gao, T., Hibi, M., Takatsu, M., Ogawa, M.: Heat transmission during thermal shock testing of ceramics. J. Mater. Sci. 29(1), 213–219 (1994)CrossRefGoogle Scholar
  35. 35.
    Janjai, S., Lamlert, N., Mahayothee, B., Bala, B.K., Precoppe, M., Muller, J.: Thin-layer drying of peeled longan (Dimocarpus longan Lour.). Food Sci. Technol. Res. 17(4), 279–288 (2011)CrossRefGoogle Scholar
  36. 36.
    Akpinar, E.K.: Mathematical modeling of thin-layer drying process under open sun of some aromatic plants. J. Food Eng. 77(4), 864–870 (2006)CrossRefGoogle Scholar
  37. 37.
    Babalis, S.J., Papanicolaou, E., Kyriakis, N., Belessiotis, V.G.: Evaluation of thin-layer drying models for describing drying kinetics of figs (Ficus carica). J. Food Eng. 75(2), 205–214 (2006)CrossRefGoogle Scholar
  38. 38.
    Menges, H.O., Ertekin, C.: Mathematical modeling of thin-layer drying of golden apples. J. Food Eng. 77(1), 119–125 (2006)CrossRefGoogle Scholar
  39. 39.
    Vega, A., Uribe, E., Lemus, R., Miranda, M.: Hot-air drying characteristics of aloe vera (Aloe barbadensis) and influence of temperature on kinetic parameters. LWT Food Sci. Technol. 40(10), 1698–1707 (2007)CrossRefGoogle Scholar
  40. 40.
    Saeed, I.E., Sopian, K., Abidin, Z.Z.: Drying characteristics of Roselle (1): mathematical modeling and drying experiments. Agric. Eng. Int. CIGR J. Manuscript FP 08 015. X, 1–25 (2008)Google Scholar
  41. 41.
    Fadhel, M.I., Abdo, R.A., Yousif, B.F., Zaharim, A., Sopian, K.: Thin-layer drying characteristics of banana slices in a force convection indirect solar drying. In: 6th IASME/WSEAS International Conference on Energy and Environment: Recent Researches in Energy and Environment, Cambridge, England, pp. 310–315 (2011)Google Scholar
  42. 42.
    Kadam, D.M., Goyal, R.K., Gupta, M.K.: Mathematical modeling of convective thin-layer drying of basil leaves. J. Med. Plants Res. 5(19), 4721–4730 (2011)Google Scholar
  43. 43.
    Rasouli, M., Seiiedlou, S., Ghasemzadeh, H.R., Nalbandi, H.: Convective drying of garlic (Allium sativum L.): part I: drying kinetics, mathematical modeling and change in color. Aust. J. Crop Sci. 5(13), 1707–1714 (2011)Google Scholar
  44. 44.
    Akoy, E.O.: Experimental characterization and modeling of thin-layer drying of mango slices. Int. Food Res. J. 21(5), 1911–1917 (2014)Google Scholar
  45. 45.
    Gan, P.L., Poh, P.E.: Investigation on the effect of shapes on the drying kinetics and sensory evaluation study of dried jackfruit. Int. J. Sci. Eng. 7(2), 193–198 (2014)CrossRefGoogle Scholar
  46. 46.
    Tzempelikos, D.A., Vouros, A.P., Bardakas, A.V., Filios, A.E., Margaris, D.P.: Case studies on the effect of the air drying conditions on the convective drying of quinces. Case Stud. Therm. Eng. 3, 79–85 (2014)CrossRefGoogle Scholar
  47. 47.
    Darıcı, S., Sen, S.: Experimental investigation of convective drying kinetics of kiwi under different conditions. Heat Mass Transf. 51(8), 1167–1176 (2015)CrossRefGoogle Scholar
  48. 48.
    Onwude, D.I., Hashim, N., Janius, R., Nawi, N., Abdan, K.: Computer simulation of convective hot air drying kinetics of pumpkin (Cucurbita moschata). In: 8th Asia-Pacific Drying Conference (ADC 2015) Kuala Lumpur, Malaysia, pp. 122–129 (2015)Google Scholar
  49. 49.
    Tzempelikos, D.A., Vouros, A.P., Bardakas, A.V., Filios, A.E., Margaris, D.P.: Experimental study on convective drying of quince slices and evaluation of thin-layer drying models. Eng. Agric. Environ. Food 8(3), 169–177 (2015)CrossRefGoogle Scholar
  50. 50.
    Kucuk, H., Midilli, A., Kilic, A., Dincer, I.: A review on thin-layer drying-curve equations. Drying Technol. 32(7), 757–773 (2014)CrossRefGoogle Scholar
  51. 51.
    Midilli, A., Kucuk, H., Yapar, Z.: A new model for single-layer drying. Drying Technol. 20(7), 1503–1513 (2002)CrossRefGoogle Scholar
  52. 52.
    Sacilik, K.: Effect of drying methods on thin-layer drying characteristics of hull-less seed pumpkin (Cucurbita pepo L.). J. Food Eng. 79(1), 23–30 (2007)CrossRefGoogle Scholar
  53. 53.
    Aghbashlo, M., Kianmehr, M.H., Khani, S., Ghasemi, M.: Mathematical modeling of thin-layer drying of carrot. Int. Agrophysics 23(4), 313–317 (2009)Google Scholar
  54. 54.
    Wang, C.Y., Singh, R.P.A.: single layer drying equation for rough rice. ASAE American Society of Agricultural and Biological Engineers, St. Joseph, MI, Paper No 78-3001 (1978)Google Scholar
  55. 55.
    Diamante, L., Durand, M., Savage, G., Vanhanen, L.: Effect of temperature on the drying characteristics, colour and ascorbic acid content of green and gold kiwifruits. Int. Food Res. J. 17(2), 441–451 (2010)Google Scholar
  56. 56.
    Pardeshi, I.L., Arora, S., Borker, P.A.: Thin-layer drying of green peas and selection of a suitable thin-layer drying model. Drying Technol. 27(2), 288–295 (2009)CrossRefGoogle Scholar
  57. 57.
    Silva, W.P., Silva, C.M.D.P.S., Gama, F.J.A.: Mathematical models to describe thin-layer drying and to determine drying rate of whole bananas. J. Saudi Soc. Agric. Sci. 13(1), 67–74 (2014)Google Scholar
  58. 58.
    Incropera, F.P., De Witt, D.P.: Fundamentals of Heat and Mass Transfer. Wiley, New York (2002)Google Scholar
  59. 59.
    Parti, M.: Selection of mathematical models for drying grain in thin-layers. J. Agric. Eng. Res. 54(4), 339–352 (1993)CrossRefGoogle Scholar
  60. 60.
    Lima, A.G.B., Farias Neto, S.R., Silva, W.P.: Heat and mass transfer in porous materials with complex geometry: fundamentals and applications. In: Delgado, J.M.P.Q. (Org.) Heat and Mass Transfer in Porous Media. Series: Advanced Structured Materials, vol. 13, 1st edn, pp. 161–185. Springer, Heidelberg (2011)Google Scholar
  61. 61.
    Silva, J.B.: Drying of solids in thin-layer via lumped analysis: modeling and simulation. Master dissertation in Mechanical Engineering, Federal University of Campina Grande. Campina Grande, Brazil (2002) (in Portuguese)Google Scholar
  62. 62.
    Lima, L.A., Silva, J.B., Lima, A.G.B.: Heat and mass transfer during drying of solids with arbitrary shape: a lumped analysis. J. Braz. Assoc. Agric. Eng. (Engenharia Agrícola, Jaboticabal) 23(1), 150–162 (2003) (in Portuguese)Google Scholar
  63. 63.
    Silva, V.S., Delgado, J.M.P.Q., Barbosa de Lima, W.M.P., Barbosa de Lima, A.G.: Heat and mass transfer in holed ceramic material using lumped model. Diffus. Found. 7, 30–52 (2016)CrossRefGoogle Scholar
  64. 64.
    Lima, W.M.P.B.: Heat and mass transfer in porous solids with complex shape via lumped analysis: modeling and simulation. Master dissertation in Mechanical Engineering, Federal University of Campina Grande. Campina Grande, Brazil (2017) (in Portuguese)Google Scholar
  65. 65.
    Almeida, G.S.: Heat and mass transfer in heterogeneous solids with arbitrary shape: a lumped analysis. Master dissertation in Mechanical Engineering, Federal University of Campina Grande, Brazil (2003) (in Portuguese)Google Scholar
  66. 66.
    Munem, M.A., Foulis, D.J.: Calculus, Guanabara Dois S.A., Rio de Janeiro, Brazil. 1 (1978) (in Portuguese)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • E. S. Lima
    • 1
  • W. M. P. B. Lima
    • 2
  • Antonio Gilson Barbosa de Lima
    • 2
    Email author
  • S. R. de Farias Neto
    • 3
  • E. G. Silva
    • 4
  • V. A. B. Oliveira
    • 5
  1. 1.Department of MathematicsState University of ParaibaCampina GrandeBrazil
  2. 2.Department of Mechanical EngineeringFederal University of Campina GrandeCampina GrandeBrazil
  3. 3.Department of Chemical EngineeringFederal University of Campina GrandeCampina GrandeBrazil
  4. 4.Department of PhysicsState University of ParaibaCampina GrandeBrazil
  5. 5.State University of ParaibaGuarabiraBrazil

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