Advertisement

Transport Phenomena in Porous Structures

  • Frank A. CoutelierisEmail author
  • J. M. P. Q. Delgado
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 20)

Abstract

The phenomenon of diffusion and dispersion (axial and radial) in packed beds is summarized and reviewed for literature data. Several variables should be considered, such as the length of the packed column, viscosity and density of the fluid, ratio of column diameter to particle diameter, ratio of column length to particle diameter, particle size distribution, particle shape, effect of fluid velocity, and temperature effect (or Schmidt number). Empirical correlations are presented for the prediction of dispersion coefficients (axial and radial) over the entire range of practical values of Schmidt number and Peclet numbers. Engineering aspects of radial and axial dispersion on non-Newtonian fluids in packed beds are also considered.

Keywords

Porous Medium Molecular Diffusion Dispersion Coefficient Peclet Number Supercritical Carbon Dioxide 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Adler, P.M., Jacquin, C.G., Thovert, J.F.: The formation factor of reconstructed porous media. Water Resour. Res. 28, 1571–1576 (1992)CrossRefGoogle Scholar
  2. 2.
    Ahn, B.J., Zoulalian, A., Smith, J.M.: Axial dispersion in packed beds with large wall effect. AIChE J. 32, 170–174 (1986)CrossRefGoogle Scholar
  3. 3.
    Akehata, T., Sato, K.: Flow distribution in packed beds. Chem. Eng. Jpn. 22, 430–436 (1958)Google Scholar
  4. 4.
    Archie, G.E.: The electrical resistivity log as an aid in determining some reservoir characteristics. Trans. AIME 146, 54–61 (1942)Google Scholar
  5. 5.
    Aris, R.: On the dispersion of a solute in a fluid flowing through a tube. Proc. Roy. Soc. A, Math. Phy. 235, 67–77 (1956)CrossRefGoogle Scholar
  6. 6.
    Aris, R.: On the dispersion of a solute by diffusion, convection and exchange between phases. Proc. Roy. Soc. A, Math. Phy. 252, 538–550 (1959)CrossRefGoogle Scholar
  7. 7.
    Baron, T.: Generalized graphical method for the design of fixed bed catalytic reactors. Chem. Eng. Prog. 48, 118–124 (1952)Google Scholar
  8. 8.
    Baumeister, E., Klose, U., Albert, K., Bayer, E.: Determination of the apparent transverse and axial dispersion coefficients in a chromatographic column by pulsed field gradient nuclear magnetic resonance. J. Chromatogr. A 694, 321–331 (1995)CrossRefGoogle Scholar
  9. 9.
    Bear, J.: Dynamics of Fluids in Porous Media. Dover Publications, New York (1972)Google Scholar
  10. 10.
    Bear, J., Verruijt, A.: Modelling Groundwater Flow and Pollution, 1st edn. Springer, Berlin (1987)Google Scholar
  11. 11.
    Beck, R.E., Schultz, J.S.: Hindered diffusion in microporous membranes with known pore geometry. Sci 170, 1302–1305 (1970)Google Scholar
  12. 12.
    Benneker, A.H., Kronberg, A.E., Post, J.W., Van der Ham, A.G.J., Westerterp, K.R.: Axial dispersion in gases flowing though a packed bed at elevated pressures. Chem. Eng. Sci. 51, 2099–2108 (1996)CrossRefGoogle Scholar
  13. 13.
    Beran, M.J.: Dispersion of soluble matter in slowly moving fluids. Dissertation, Harvard University, Cambridge (1955) Google Scholar
  14. 14.
    Bernard, R.A., Wilhelm, R.H.: Turbulent diffusion in fixed beds of packed solids. Chem. Eng. Prog. 46, 233–244 (1950)Google Scholar
  15. 15.
    Bischoff, K.B.: A note on gas dispersion in packed beds. Chem. Eng. Sci. 24, 607–608 (1969)CrossRefGoogle Scholar
  16. 16.
    Blackwell, R.J.: Laboratory studies of microscopic dispersion phenomena. Soc. Petrol. Eng. J. 225, 1–8 (1962)Google Scholar
  17. 17.
    Blackwell, R.J., Rayne, J.R., Terry, W.M.: Factors influencing the efficiency of miscible displacement. Petrol. Trans. AIME 216, 1–8 (1959)Google Scholar
  18. 18.
    Boudreau, B.P.: The diffusive tortuosity of fine-grained unlithified sediments. Geochim. Cosmochim. Acta 60, 3139–3142 (1996)CrossRefGoogle Scholar
  19. 19.
    Brenner, H.: The diffusion model of longitudinal mixing in beds of finite length. Numerical values. Chem. Eng. Sci. 17, 229–243 (1962)CrossRefGoogle Scholar
  20. 20.
    Brenner, H.: Dispersion resulting from flow through spatially periodic porous media. Philos. Trans. R. Soc. A 297, 81–133 (1980)CrossRefGoogle Scholar
  21. 21.
    Bruggeman, D.A.G.: Calculation of different physical constants of heterogeneous substances i, dielectric constant and conductivity of media of isotropic substances. Ann. Phys. Ser.-5 24, 636–664 (1935)CrossRefGoogle Scholar
  22. 22.
    Bruinzeel, C., Reman, G.H., van der Laan, E.T.H.: Axial dispersion. Proceedings of the 3rd Congress of the European Federation of Chemical Engineering, Olympia, London, UK (1962)Google Scholar
  23. 23.
    Cairns, E.J., Prausnitz, J.M.: Longitudinal mixing in packed beds. Chem. Eng. Sci. 12, 20–34 (1960)CrossRefGoogle Scholar
  24. 24.
    Carberry, J.J.: Chemical and Catalytic Reaction Engineering. McGraw-Hill Chemical Engineering Series, USA (1976)Google Scholar
  25. 25.
    Carberry, J.J., Bretton, R.H.: Axial dispersion of mass in flow through fixed beds. AIChE J. 4, 367–375 (1958)CrossRefGoogle Scholar
  26. 26.
    Carbonell, R.G., Whitaker, S.: Dispersion in pulsed systems. Part II: Theoretical developments for passive dispersion in porous media. Chem. Eng. Sci. 38, 1795–1801 (1983)CrossRefGoogle Scholar
  27. 27.
    Catchpole, O.J., Berning, R., King, M.B.: Measurement and correlations of packed bed axial dispersion coefficients in supercritical carbon dioxide. Ind. Eng. Chem. Res. 35, 824–828 (1996)CrossRefGoogle Scholar
  28. 28.
    Chantong, A., Massoth, F.E.: Restrictive diffusion in aluminas. AIChE J. 29, 725–731 (1983)CrossRefGoogle Scholar
  29. 29.
    Chao, R., Hoelscher, H.E.: Simultaneous axial dispersion and adsorption in packed beds. AIChE J. 12, 271–278 (1966)CrossRefGoogle Scholar
  30. 30.
    Choudhary, M., Szekely, J., Weller, S.W.: The effect of flow maldistribution on conversion in a catalytic packed bed reactor. AIChE J. 22, 1021–1032 (1976)CrossRefGoogle Scholar
  31. 31.
    Chung, S.F., Wen, C.Y.: Longitudinal dispersion of liquid flowing through fixed and fluidized beds. AIChE J. 14, 857–866 (1968)CrossRefGoogle Scholar
  32. 32.
    Coelho, D., Thovert, J.F., Adler, P.M.: Geometrical and transport properties of random packings of spheres and aspherical particles. Phys. Rev. E 55, 1959–1978 (1997)CrossRefGoogle Scholar
  33. 33.
    Coelho, M.A.N., Guedes de Carvalho, J.R.F.: Transverse dispersion in granular beds. Part I: Mass transfer from a wall and the dispersion coefficient in packed beds. Chem. Eng. Res. Des. 66, 165–177 (1988)Google Scholar
  34. 34.
    Collins, M.: Velocity distribution in packed beds. Dissertation, University of Delaware (1958)Google Scholar
  35. 35.
    Comiti, J., Renaud, M.: A new model for determining mean structure parameters of fixed beds from pressure drops measurements, application to beds packed with parallelepipedal particles. Chem. Eng. Sci. 44, 1539–1545 (1989)CrossRefGoogle Scholar
  36. 36.
    Crank, J.: The Mathematics of Diffusion, 2nd edn. Oxford University Press, Oxford (1975)Google Scholar
  37. 37.
    Danckwerts, P.V.: Continuous flow systems. Chem. Eng. Sci. 2, 1–13 (1953)CrossRefGoogle Scholar
  38. 38.
    De Jong, G.J.: Longitudinal and transverse diffusion in granular deposits. Trans. AGU 39, 67–75 (1958)Google Scholar
  39. 39.
    Deisler, P.F., Wilhelm, R.H.: Diffusion in beds of porous solids - measurement by frequency response techniques. Ind. Eng. Chem. 45, 1219–1227 (1953)CrossRefGoogle Scholar
  40. 40.
    Delgado, J.M.P.Q., Guedes de Carvalho, J.R.F.: Measurement of the coefficient of transverse dispersion in packed beds over a range of values of Schmidt number 50–1000. Transp. Porous Med. 44, 165–180 (2001)CrossRefGoogle Scholar
  41. 41.
    Delmas, H., Froment, G.F.: Simulation model accounting for structural radial nonuniformities in fixed bed reactors. Chem. Eng. Sci. 43, 2281–2287 (1988)CrossRefGoogle Scholar
  42. 42.
    Dorweiler, V.P., Fahien, R.W.: Mass transfer at low flow rates in a packed column. AIChE J. 5, 139–144 (1959)CrossRefGoogle Scholar
  43. 43.
    Dullien, F.A.L.: Single phase flow through porous media and pore structure. Chem. Eng. J. 10, 1–34 (1975)Google Scholar
  44. 44.
    Dullien, F.A.L.: Porous media, fluid transport and pore structure. Academic, San Diego (1979)Google Scholar
  45. 45.
    Ebach, E.A., White, R.R.: Mixing of fluids flowing through beds of oacked solids. AIChE J. 4, 161–169 (1958)CrossRefGoogle Scholar
  46. 46.
    Edwards, M.F., Helail, T.R.: Axial dispersion in porous media. Proceedings of the 2nd European Conference of Mixing, BHRA Fluids Engineering, Cranfield, UK (1977)Google Scholar
  47. 47.
    Edwards, M.F., Richardson, J.F.: Gas dispersion in packed beds. Chem. Eng. Sci. 23, 109–123 (1968)CrossRefGoogle Scholar
  48. 48.
    Eidsath, A., Carbonell, R.G., Whitaker, S., Herrmann, L.R.: Dispersion in pulsed systems. Part III: Comparison between theory and experiments for packed beds. Chem. Eng. Sci. 38, 1803–1816 (1983)CrossRefGoogle Scholar
  49. 49.
    England, R., Gunn, D.J.: Dispersion, pressure drop, and chemical reaction in packed beds of cylindrical particles. Trans. IChemE. 48, T265–T275 (1970)Google Scholar
  50. 50.
    Evans, E.V., Kenney, C.N.: Gaseous dispersion in packed beds at low Reynolds numbers. Trans. IChemE. 44, T189–T197 (1966)Google Scholar
  51. 51.
    Fahien, R.W., Smith, J.M.: Mass transfer in packed beds. AIChE J. 1, 28–37 (1955)CrossRefGoogle Scholar
  52. 52.
    Fetter, C.W.: Contaminant hydrogeology, 2nd edn. Prentice-Hall, New Jersey (1999)Google Scholar
  53. 53.
    Freeze, R.A., Cherry, J.A.: Groundwater. Prentice-Hall Inc, Englewood Cliffs (1979)Google Scholar
  54. 54.
    Fried, J.J., Combarnous, M.A.: Dispersion in porous media. Advances in Hydroscience nº 7. Academic, New York (1971)Google Scholar
  55. 55.
    Froment, G.F., Bischoff, K.B.: Chemical reactor analysis and design, 2nd edn. Wiley, New York (1990)Google Scholar
  56. 56.
    Ghoreishi S.M., Akgermanb, A.: Dispersion coefficients of supercritical fluid in fixed beds. Sep. Purif. Technol. 39, 39–50 (2004)Google Scholar
  57. 57.
    Gibbs, S.J., Lightfoot, E.N., Root, T.W.: Protein diffusion in porous gel filtration chromatography media studied by pulsed field gradient NMR spectroscopy. J. Phys. Chem. 96, 7458–7462 (1992)CrossRefGoogle Scholar
  58. 58.
    Grane, F.E., Gardner, G.H.F.: Measurements of transverse dispersion in granular media. J. Chem. Eng. Data 6, 283–287 (1961)CrossRefGoogle Scholar
  59. 59.
    Gray, W.G.: A derivation of the equations for multi-phase transport. Chem. Eng. Sci. 30, 229–233 (1975)CrossRefGoogle Scholar
  60. 60.
    Greenkorn, R.A., Kessler, D.P.: Dispersion in heterogeneous nonuniform anisotropic porous media. Ind. Eng. Chem. 61, 8–15 (1969)CrossRefGoogle Scholar
  61. 61.
    Guedes de Carvalho, J.R.F., Delgado, J.M.P.Q.: The effect of fluid properties on dispersion in flow through packed. AIChE J. 49, 1980–1985 (2003)CrossRefGoogle Scholar
  62. 62.
    Guedes de Carvalho, J.R.F., Delgado, J.M.P.Q.: Radial dispersion in liquid flow through packed beds for 50 < Sc < 750 and 103 < Pem < 105. Proceedings of the 5th World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, Thessaloniki, Greece (2001)Google Scholar
  63. 63.
    Guedes de Carvalho, J.R.F., Delgado, J.M.P.Q.: Lateral dispersion in liquid flow through packed beds at Pem < 1,400. AIChE J. 46, 1089–1095 (2000)CrossRefGoogle Scholar
  64. 64.
    Gunn, D.J.: Axial and radial dispersion in fixed beds. Chem. Eng. Sci. 42, 363–373 (1987)CrossRefGoogle Scholar
  65. 65.
    Gunn, D.J.: Theory of axial and radial dispersion in packed beds. Trans. IChemE. 47, T351–T359 (1969)Google Scholar
  66. 66.
    Gunn, D.J.: Mixing in packed and fluidised beds. Chem. Eng. J. 5, E153–CE172 (1968)Google Scholar
  67. 67.
    Gunn, D.J., Malik, A.A.: Flow through expanded beds of solids. Trans. IChemE. 44, T371–T379 (1966)Google Scholar
  68. 68.
    Gunn, D.J., Pryce, C.: Dispersion in packed beds. Trans. IChemE. 47, T341–T350 (1969)Google Scholar
  69. 69.
    Han, N.W., Bhakta, J., Carbonell, R.G.: Longitudinal and lateral dispersion in packed beds, effect of column length and particle size distribution. AIChE J. 31, 277–288 (1985)CrossRefGoogle Scholar
  70. 70.
    Haring, R.E., Greenkorn, R.A.: Statistical model of a porous medium with nonuniform pores. AIChE J. 16, 477–483 (1970)CrossRefGoogle Scholar
  71. 71.
    Harleman, D.R.F., Rumer, R.: Longitudinal and lateral dispersion in an isotropic porous medium. J. Fluid Mech. 16, 1–12 (1963)CrossRefGoogle Scholar
  72. 72.
    Hartman, M.E., Wevers, C.J.H., Kramers, H.: Lateral diffusion with liquid flow through a packed bed of ion-exchange particles. Chem. Eng. Sci. 9, 80–82 (1958)CrossRefGoogle Scholar
  73. 73.
    Hassel, H.L., Bondi, A.: Mixing of viscous non-newtonian fluids in packed beds. AIChE J. 11, 217–221 (1965)CrossRefGoogle Scholar
  74. 74.
    Hennico, A., Jacques, G., Vermeulen, T.: Longitudinal dispersion in single-phase liquid flow through ordered and random packings. Lawrence Rad. Laboratory Report UCRL, 10696 (1963)Google Scholar
  75. 75.
    Hiby, J.W.: Longitudinal and transverse mixing during single-phase flow through granular beds. In: Interaction between fluid and particles, London IChemE, pp. 312–325, (1962)Google Scholar
  76. 76.
    Hiby, J.W., Schummer, P.: Zur messung der transversalen effektiven diffusion in durchstromten fullkorpersaulen. Chem. Eng. Sci. 13, 69–74 (1960)CrossRefGoogle Scholar
  77. 77.
    Hilal, M., Brunjail, D., Combi, J.: Electrodiffusion characterization of non-newtonian flow through packed-beds. J. Appl. Electrochem. 21, 1087–1090 (1991)CrossRefGoogle Scholar
  78. 78.
    Hoopes, J.A., Harleman, D.R.F.: Waste water recharge and dispersion in porous media. MIT Hydrodynamics Laboratory Report 75, 55–60 (1965)Google Scholar
  79. 79.
    Jacques, G.L., Vermeulen, T.: Longitudinal dispersion in solvent-extraction columns, peclet numbers for random and ordered packings. University of California Rad. Laboratory Report Nº 8029, US Atomic Energy Commission, Washington DC, USA (1958)Google Scholar
  80. 80.
    Johnson, G.W., Kapner, R.S.: The dependence of axial-dispersion on non-uniform flows in beds of uniform packing. Chem. Eng. Sci. 45, 329–3339 (1990)Google Scholar
  81. 81.
    Klinkenberg, A., Krajenbrink, H.J., Lauwerier, H.A.: Diffusion in a fluid moving at uniform velocity in a tube. Ind. Eng. Chem. 45, 1202–1208 (1953)CrossRefGoogle Scholar
  82. 82.
    Klotz, D.: Untersuchungen zur dispersion in porosen medien. Z. Deutsch. Geol. Ges. 124, 523–533 (1973)Google Scholar
  83. 83.
    Koch, D.C., Brady, J.F.: Dispersion in fixed beds. J. Fluid Mech. 154, 399–427 (1985)CrossRefGoogle Scholar
  84. 84.
    Koplik, J., Redner, S., Wilkinson, D.: Transport and dispersion in random networks with percolation disorder. Phys. Rev. A 37, 2619–2636 (1988)CrossRefGoogle Scholar
  85. 85.
    Kramers, H., Alberda, G.: Frequency response analysis of continuous flow systems. Chem. Eng. Sci. 2, 173–181 (1953)CrossRefGoogle Scholar
  86. 86.
    Langer, G., Roethe, A., Roethe, K.P., Gelbin, D.: Heat and mass-transfer in packed beds. Part III: Axial mass dispersion. Int. J. Heat Mass Tran. 21, 751–759 (1978)CrossRefGoogle Scholar
  87. 87.
    Latinen, G.A.: Mechanism of fluid-phase mixing in fixed and fluidised beds of uniformly sized spherical particles. Dissertation, Princeton University (1951)Google Scholar
  88. 88.
    Lerou, J.J., Froment, G.F.: Velocity, temperature and conversion profiles in fixed bed catalytic reactors. Chem. Eng. Sci. 32, 853–861 (1977)CrossRefGoogle Scholar
  89. 89.
    Levenspiel, O., Smith, W.K.: Notes on the diffusion-type model for the longitudinal mixing of fluids in flow. Chem. Eng. Sci. 6, 227–233 (1957)CrossRefGoogle Scholar
  90. 90.
    Liles, A.W., Geankopolis, C.J.: Axial diffusion of liquids in packed beds and end effects. AIChE J. 6, 591–595 (1960)CrossRefGoogle Scholar
  91. 91.
    List, E.J.: The stability and mixing of a density-stratified horizontal flow in a saturated porous medium. WM Keck Lab Hydraulics Water Resources Reporter KH-R-11, USA (1965)Google Scholar
  92. 92.
    Marsily, G.: Quantitative Hydrogeology, 1st edn. Academic, Orlando (1986)Google Scholar
  93. 93.
    Maxwell, J.C.: Treatise on Electricity and Magnetism. Oxford University Press, London (1873)Google Scholar
  94. 94.
    McHenry, J.R., Wilhelm, R.H.: Axial mixing of binary gas mixtures flowing in a random bed of spheres. AIChE J. 3, 83–91 (1957)CrossRefGoogle Scholar
  95. 95.
    Miller, S.T., King, C.J.: Axial dispersion in liquid flow through packed beds. AIChE J. 12, 767–773 (1966)CrossRefGoogle Scholar
  96. 96.
    Millington, R.J., Quirk, J.P.: Transport in porous media. Proceedings of the 7th International Congress of Soil Science, Madison, Wisc., USA (1960)Google Scholar
  97. 97.
    Miyauchi, T., Kikuchi, T.: Axial dispersion in packed beds. Chem. Eng. Sci. 30, 343–348 (1975)CrossRefGoogle Scholar
  98. 98.
    Niemann, E.H.; Dispersion during flow nonuniform heterogeneous porous media. Dissertation, Purdue University (1969)Google Scholar
  99. 99.
    Ogata, A.: The spread of a dye stream in an isotropic granular medium. US Geol. Surv. Prof. Pap. 411-G, pp. 1–11. USA (1964)Google Scholar
  100. 100.
    Otake, T., Kunugita, E.: Axial dispersion of the gas phase in countercurrent packed bed columns. Chem. Eng. Jpn. 22, 144–150 (1958)Google Scholar
  101. 101.
    Payne, L.W., Parker, H.W.: Axial dispersion of non-newtonian fluids in porous media. AIChE J. 19, 202–204 (1973)CrossRefGoogle Scholar
  102. 102.
    Perkins, T.K., Johnston, O.C.: A review of diffusion and dispersion in porous media. Soc. Petrol. Eng. J. 2, 70–84 (1963)Google Scholar
  103. 103.
    Pfannkuch, H.O.: Contribution a l’etude des déplacements de fluids miscibles dans un milieu poreux. Rev. Inst. Franc. Pétrol. 18, 215–219 (1963)Google Scholar
  104. 104.
    Pickens, J.F., Grisak, G.E.: Scale-dependent dispersion in a stratified granular aquifer. Water Resour. Res. 17, 1191–1211 (1981)CrossRefGoogle Scholar
  105. 105.
    Plautz, D.A., Johnstone, H.F.: Heat and mass transfer in packed beds. AIChE J. 1, 193–199 (1955)CrossRefGoogle Scholar
  106. 106.
    Pozzi, A.L., Blackwell, R.J.: Design of laboratory models for study of miscible displacement. Soc. Petrol. Eng. J. 3, 28–40 (1963)Google Scholar
  107. 107.
    Probst, K., Wohlfahrt, K.: Empirische abschätzung effektiver diffusionskoeffizienten in porösen systemen. Chem. Eng. Tech. 1, 737–739 (1979)Google Scholar
  108. 108.
    Raimondi, P., Gardner, G.H.F., Petrick, C.B.: Effect of pore structure and molecular diffusion on the mixing of miscible liquids flowing in porous media. In: AIChE-SPE joint symposium, San Francisco (1959)Google Scholar
  109. 109.
    Rifai, M.N.E., Kaufman, W.J., Todd, D.K.: Dispersion phenomena in laminar flow through porous media. University of California, Sanitary Engineering Report 3. Inst. Eng. Res. Ser. 90, 1–157 (1956)Google Scholar
  110. 110.
    Roblee, L.H.S., Baird, R.M., Tierney, J.W.: Radial porosity variations in packed beds. AIChE J. 4, 460–468 (1958)CrossRefGoogle Scholar
  111. 111.
    Roemer, G., Dranoff, J.S., Smith, J.M.: Diffusion in packed beds at low flow rates. Ind. Eng. Chem. Fund. 1, 284–287 (1962)CrossRefGoogle Scholar
  112. 112.
    Rosenberg, D.V.: Axial dispersion in porous media. J. Am. Inst. Chem. Engrs. 2, 55–64 (1956)Google Scholar
  113. 113.
    Saffman, P.G.: A theory of dispersion in a porous medium. J. Fluid Mech. 6, 321–325 (1959)CrossRefGoogle Scholar
  114. 114.
    Saffman, P.C.: Dispersion in flow through a network of capillaries. J. Fluid Mech. 7, 194–207 (1960)CrossRefGoogle Scholar
  115. 115.
    Sahimi, M.: Flow and transport in porous media and fractured rock. VCH Velagsgesellschaft mbH, Weinheim (1995)Google Scholar
  116. 116.
    Scheidegger, A.E.: The physics of flow through porous media, 3rd edn. University of Toronto Press, Toronto (1974)Google Scholar
  117. 117.
    Schuster, J., Vortmeyer, D.: Ein einfaches verfahren zur näherungsweisen bestimmung der porosität in schttungen als funktion des wasdabstandes. Chem. Eng. Tech. 52, 848–855 (1980)Google Scholar
  118. 118.
    Schwartz, C.E., Smith, J.M.: Flow distribution in packed beds. Ind. Eng. Chem. 45, 1209–1218 (1953)CrossRefGoogle Scholar
  119. 119.
    Scott, D.S., Lee, W., Papa, J.: The measurement of transport coefficients in gas-solid heterogeneous reactions. Chem. Eng. Sci. 29, 2155–2167 (1974)CrossRefGoogle Scholar
  120. 120.
    Sen, P.N., Scala, C., Cohen, M.H.: A self-similar model for sedimentary-rocks with application to the dielectric-constant of fused glass-beads. Geophysics 46, 781–795 (1981)CrossRefGoogle Scholar
  121. 121.
    Sherwood, T.K., Pigford, R.L., Wilke, C.R.: Mass transfer. McGraw-Hill, Tokyo (1975)Google Scholar
  122. 122.
    Shimamura, K.: Gas diffusion through compacted sands. Soil Sci. 153, 274–279 (1992)CrossRefGoogle Scholar
  123. 123.
    Simpson, E.S.: Transverse dispersion in liquid flow through porous media. US Geol. Surv. Prof. Pap. 411-C, pp. 1–30. USA (1962)Google Scholar
  124. 124.
    Sinclair, R.J., Potter, O.E.: The dispersion of gas in flow through a bed of packed solids. Trans. IChemE. 43, T3–T9 (1965)Google Scholar
  125. 125.
    Singer, E., Wilhelm, R.H.: Heat transfer in packed beds, analytical solution and design method. Fluid flow, solids flow and chemical reaction. Chem. Eng. Prog. 46, 343–352 (1950)Google Scholar
  126. 126.
    Slattery, J.C.: Momentum, energy and non transfer in continua. McGraw-Hill, New York (1972)Google Scholar
  127. 127.
    Slichter, C.S.: Field measurement of the rate of movement of underground waters. US Geol. Surv. Water Supply Pap. 140, USA (1905)Google Scholar
  128. 128.
    Smith, W.D., Bretton, R.H.: Paper presented at AIChE Houston meeting, Texas, USA (1967)Google Scholar
  129. 129.
    Stephenson, J.L., Stewart, W.E.: Optical measurements of porosity and fluid motion in packed beds. Chem. Eng. Sci. 41, 2161–2170 (1986)CrossRefGoogle Scholar
  130. 130.
    Strang, D.A., Geankopolis, C.J.: Longitudinal diffusivity of liquids in packed beds. Ind. Eng. Chem. 50, 1305–1308 (1958)CrossRefGoogle Scholar
  131. 131.
    Suzuki, M., Smith, J.M.: Dynamics of diffusion and adsorption in a single catalyst pellet. AIChE J. 18, 326–333 (1972)CrossRefGoogle Scholar
  132. 132.
    Tan, C.S., Liou, D.C.: Axial dispersion of supercritical carbon dioxide in packed beds. Ind. Eng. Chem. Res. 28, 1246–1250 (1989)CrossRefGoogle Scholar
  133. 133.
    Taylor, G.: Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. Roy. Soc. A, Math. Phy. 219, 186–203 (1953)CrossRefGoogle Scholar
  134. 134.
    Towle, W.L., Sherwood, T.K.: Studies in eddy diffusion. Ind. Eng. Chem. 31, 457–467 (1939)CrossRefGoogle Scholar
  135. 135.
    Tsotsas, E., Schlunder, E.U.: On axial dispersion in packed beds with fluid flow. Chem. Eng. Proc. 24, 15–31 (1988)Google Scholar
  136. 136.
    Urban, J.C., Gomezplata, A.: Axial dispersion coefficients in packed beds at low Reynolds numbers. Can. J. Chem. Eng. 47, 353–363 (1969)CrossRefGoogle Scholar
  137. 137.
    Vortmeyer, D., Schuster, J.: Evaluation of steady flow profiles in rectangular and circular packed beds by a variational method. Chem. Eng. Sci. 38, 1691–1699 (1983)CrossRefGoogle Scholar
  138. 138.
    Vortmeyer, D., Winter, R.P.: Impact of porosity and velocity distribution of the theoretical prediction of fixed-bed chemical reactor performance, comparison with experimental data. ACS symposium series, pp. 49–61. USA (1982)Google Scholar
  139. 139.
    Weissberg, H.L.: Effective diffusion coefficient in porous media. J. Appl. Phys. 34, 2636–2639 (1963)CrossRefGoogle Scholar
  140. 140.
    Wen, C.Y., Fan, L.S.: Axial dispersion of non-Newtonian liquids in fluidized beds. Chem. Eng. Sci. 28, 1768–1773 (1973)Google Scholar
  141. 141.
    Wen, C.Y., Fan, L.T.: Models for systems and chemical reactors. Marcel Dekker, New York (1975)Google Scholar
  142. 142.
    Wen, C.Y., Yin, J.: Axial dispersion of a non-newtonian liquid in a packed bed. AIChE J. 17, 1503–1504 (1971)CrossRefGoogle Scholar
  143. 143.
    Whitaker, S.: Diffusion and dispersion in porous media. AIChE J. 13, 420–432 (1967)CrossRefGoogle Scholar
  144. 144.
    Wilhelm, R.H.: Progress towards the a priori design of chemical reactors. Pure Appl. Chem. 5, 403–421 (1962)CrossRefGoogle Scholar
  145. 145.
    Wilson, H.A.: On convection of heat. P. Camb. Philos. Soc. 12, 406–423 (1904)Google Scholar
  146. 146.
    Wronski, S., Molga, E.: Axial dispersion in packed beds: the effect of particle size non-uniformities. Chem. Eng. Prog. 22, 123–135 (1987)Google Scholar
  147. 147.
    Yu, D., Jackson, K., Harmon, T.C.: Dispersion and diffusion in porous media under supercritical conditions. Chem. Eng. Sci. 54, 357–367 (1999)CrossRefGoogle Scholar
  148. 148.
    Yun, M.J., Yu, B.M., Zhang, B., Huang, M.Y.: A geometry model for tortuosity of streamtubes in porous media with spherical particles. Chin. Phys. Lett. 22, 1464–1467 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg  2012

Authors and Affiliations

  1. 1.Department of Environmental and Natural Resources ManagementUniversity of IoanninaAgrinioGreece
  2. 2.Laboratorio de Fisica das ConstruccoesFaculdade de Engenharia Universidade do PortoPortoPortugal

Personalised recommendations