Skip to main content
SpringerLink
Log in
Book cover

Principles of Heat Transfer in Porous Media pp 15–113Cite as

Fluid Mechanics

  • Chapter

Part of the book series: Mechanical Engineering Series ((MES))

Abstract

In this chapter we examine the isothermal single-phase flow through porous media by starting from the Darcy law. We then examine the permeability tensor and its relation with the matrix structure. The deviation from the Darcy law observed at high velocities and the pore-level fluid dynamics are then examined. Attempts at arriving at the Darcy law (the macroscopic momentum equation) from the point description of the flow field (Navier-Stokes equation) by the local volume-averaging and homogenization techniques are reviewed. Then, a semiempirical momentum equation, which includes the bulk and boundary viscous effects, the flow development in porous media, and the high-velocity effects, is given. The significance of these terms is assessed by the order-of-magnitude analyses and some estimations. When the porous media are bounded by the fluid occupying them, the hydrodynamic boundary conditions on these interfaces must be specified. The available slip velocity model and the Brinkman no-slip (uniform and variable effective viscosity) model for this interface are examined. The dependence of the slip coefficient (and the interfacial effective viscosity) on the bulk and surface structures of the matrix are studied. The velocity nonuniformities observed near the bounding impermeable surfaces and the various theoretical treatments of them are investigated. The chapter ends with an examination of the analogy between porous media- and magneto-hydrodynamics.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Armour, J. C. and Cannon, J. N., 1968, “Fluid Flow Through Woven Screens,” AIChE J., 14, 415–420.

    Article  Google Scholar 

  • Bachmat, Y. and Bear, J., 1986, “Macroscopic Modeling of Transport Phenomena in Porous Media,” Transport in Porous Media, 1, 241–269.

    Article  Google Scholar 

  • Bear, J., 1988, Dynamics of Fluids in Porous Media, Dover.

    MATH  Google Scholar 

  • Beasley, D. E. and Clark, J. A., 1984, “Transient Response of Packed Beds for Thermal Energy Storage,” Int. J. Heat Mass Transfer, 27, 1659–1669.

    Article  Google Scholar 

  • Beavers, G. S. and Joseph, D. D., 1967, “Boundary Condition at a Naturally Permeable Wall,” J. Fluid Mech., 30, 197–207.

    Article  ADS  Google Scholar 

  • Beavers, G. S., Sparrow, E. M., and Magnuson, R. A., 1970, “Experiments on Coupled Parallel Flow in a Channel and a Bounding Porous Medium,” J. Basic Eng., 92 D, 843–848.

    Article  Google Scholar 

  • Beavers, G. S., Sparrow, E. M., and Masha, B. A., 1974, “Boundary Condition at a Porous Surface Which Bounds a Fluid Flow,” AIChE J., 20, 596–597.

    Article  Google Scholar 

  • Beavers, G. S., Sparrow, E. M., and Rodenz, D. E., 1973, “Influence of Bed Size on the Flow Characteristics and Porosity of Randomly Packed Beds of Spheres,” J. Appl. Mech., 40, 655–660.

    Article  ADS  Google Scholar 

  • Benenati, R. F. and Brosilow, C. B., 1962, “Void Fraction Distribution in Packed Beds,” AIChE J., 8, 359–361.

    Article  Google Scholar 

  • Bensoussan, A., Lions, J. L., and Papanicolaou, G., 1978, Asymptotic Analysis for Periodic Structures, North-Holland.

    MATH  Google Scholar 

  • Brinkman, H. C., 1947, “A Calculation of the Viscous Force Exerted by a Flowing Fluid on a Dense Swarm of Particles,” Appl. Sci. Res., A1, 27–34.

    Google Scholar 

  • Brinkman, H. C., 1948, “On the Permeability of Media Consisting of Closely Packed Porous Particles,” Appl. Sci. Res., A1, 81–86.

    Google Scholar 

  • Carbonell, R. G. and Whitaker, S., 1984, “Heat and Mass Transfer in Porous Media,” in Fundamentals of Transport Phenomena in Porous Media, Bear and Corapcioglu, eds., Martinus Nijhoff publishers, 121–198.

    Google Scholar 

  • Carman, P. C., 1937, “The Determination of the Specific Surface Area of Powder I,” J. Soc. Chem. Ind., 57, 225–234.

    Google Scholar 

  • Chandrasekhar, B. C. and Vortmeyer, D., 1979, “Flow Model for Velocity Distribution in Fixed Porous Beds under Isothermal Conditions,” Warme-und Stoffubertrigung, 12, 105–111.

    Article  ADS  Google Scholar 

  • Cheng, P. and Vortmeyer, D., 1988, “Transverse Thermal Dispersion and Wall Channelling in a Packed Bed with Forced Convective Flow,” Chem. Eng. Sci., 43, 2523–2532.

    Article  Google Scholar 

  • Chu, C. F. and Ng, K. M., 1989, “Flow in Packed Tubes with a Small Tube to Particle Diameter Ratio,” AIChE J., 35, 148–158.

    Article  Google Scholar 

  • Cramer, K. R., 1963, “Several Magneto-Hydrodynamic Free-Convection Solutions,” ASME J. Heat Transfer, 85, 35–39.

    Google Scholar 

  • Cramer, K. R. and Pai, S.-I., 1973, Magneto Fluid Dynamics for Engineers and Applied Physicists, McGraw-Hill.

    Google Scholar 

  • Darcy, H., 1856, Les Fontaines Publiques de la ville de Dijon, Dalmont, Paris.

    Google Scholar 

  • Dullien, F. A. L., 1979, Porous Media: Fluid Transport and Pore Structure, Academic.

    Google Scholar 

  • Durlofsky, L. and Brady, J. F., 1987, “Analysis of the Brinkman Equation as a Model for Flow in Porous Media,” Phys. Fluids, 11, 3329–3341.

    Article  ADS  Google Scholar 

  • Dybbs, A. and Edwards, R. V., 1984, “A New Look at Porous Media Fluid Mechanics—Darcy to Turbulent,” in Fundamentals of Transport Phenomena in Porous Media, Bear and Corapcioglu, eds., Martinus Nijhoff Publishers, 199–254.

    Google Scholar 

  • Eidsath, A., Carbonell, R. G., Whitaker, S., and Herrmann, L. R., 1983, “Dispersion in Pulsed Systems—III, Comparison between Theory and Experiments in Packed Beds,” Chem. Engng. Sci., 38, 1803–1816.

    Article  Google Scholar 

  • Ene, H. I. and Polisevski, D., 1987, Thermal Flow in Porous Media, D. Reidel Publishing Company.

    Book  Google Scholar 

  • Ene, H. I. and Sanchez-Palencia, E., 1975, “Equations et Phenomenes de Surface Pour L’ecoulement dans un Modelle de Milieu Poreux,” J. de Meca., 14, 73–107.

    MathSciNet  MATH  Google Scholar 

  • Ergun, S., 1952, “Fluid Flow Through Packed Column,” Chem. Eng. Prog., 48, 89–94.

    Google Scholar 

  • Fatehi, M. and Kaviany, M., 1990, “Analysis of Levitation of Saturated Liquid Droplets on Permeable Surfaces,” Int. J. Heat Mass Transfer, 33, 983–994.

    Article  Google Scholar 

  • Happel, J. and Brenner, H., 1986, Low Reynolds Number Hydrodynamics, Martinus Nijhoff publishers.

    MATH  Google Scholar 

  • Jolls, K. R. and Hanratty, T. J., 1966, “Transition to Turbulence for Flow Through a Dumped Bed of Spheres,” Chem. Engng. Sci., 21, 1185–1190.

    Article  Google Scholar 

  • Jones, I. P., 1973, “Low Reynolds Number Flow Past a Porous Spherical Shell,” Proc. Camb. Phil. Soc., 73, 231–238.

    Article  ADS  MATH  Google Scholar 

  • Kaviany, M., 1987, “Boundary-Layer Treatment of Forced Convection Heat Transfer from a Semi-Infinite Flat Plate Embedded in Porous Media,” ASME J. Heat Transfer, 109, 345–349.

    Article  Google Scholar 

  • Koplik, J., Levine, H., and Zee, A., 1983, “Viscosity Renormalization in the Brinkman Equation,” Phys. Fluids, 26, 2864–2870.

    Article  ADS  MATH  Google Scholar 

  • Kyan, C. P., Wasan, D. T., and Kintner, R. C., 1970, “Flow of Single-Phase Fluids Through Fibrous Beds,” Ind. Eng. Chem. Fund., 9, 596–603.

    Article  Google Scholar 

  • Larson, R. E. and Higdon, J. J. L., 1986, “Microscopic Flow Near the Surface of Two-Dimensional Porous Media, Part 1: Axial Flow,” J. Fluid. Mech., 166, 449–472.

    Article  ADS  MATH  Google Scholar 

  • Larson, R. E. and Higdon, J. J. L., 1987, “Microscopic Flow Near the Surface of Two-Dimensional Porous Media, Part 2: Transverse Flow,” J. Fluid Mech., 178, 119–136.

    Article  ADS  MATH  Google Scholar 

  • Larson, R. E. and Higdon, J. J. L., 1989, “A Periodic Grain Consolidation Model of Porous Media,” Phys. Fluids, A1, 38–46.

    ADS  Google Scholar 

  • Levy, Th. and Sanchez-Palencia, E., 1975, “On Boundary Conditions for Fluid Flow in Porous Media,” Int. J. Eng. Sci., 13, 923–940.

    Article  MathSciNet  MATH  Google Scholar 

  • Levy, Th. and Sanchez-Palencia, E., 1977, “Equation and Interface Conditions for Acoustic Phenomena in Porous Media,” J. Math. Anal. Appl., 61, 813–834.

    Article  MathSciNet  MATH  Google Scholar 

  • Ling, J.-X., 1988, Physical Model for Heat Transfer in Porous Media, Ph.D. thesis, Case Western Reserve University, 74–78.

    Google Scholar 

  • Lundgren, T. S., 1972, “Slow Flow Through Stationary Random Beds and Suspensions of Spheres,” J. Fluid Mech., 51, 273–299.

    Article  ADS  MATH  Google Scholar 

  • Lykoudis, P. S., 1962, “Natural Convection of an Electrically Conducting Fluid in the Presence of a Magnetic Field,” Int. J. Heat Mass Transfer, 4, 23–34.

    Article  Google Scholar 

  • Macdonald, I. F., El-Sayed, M. S., Mow, K., and Dullien, F. A. L., 1979, “Flow Through Porous Media—Ergun Equation Revisited,” Ind. Eng. Chem. Fund., 18, 199–208.

    Article  Google Scholar 

  • Martin, H., 1978, “Low Peclet Number Particle-To-Fluid Heat and Mass Transfer in Packed Beds,” Chem. Engng. Sci., 33, 913–919.

    Article  Google Scholar 

  • Mei, C. C. and Auriault, J.-L., 1991, “The Effect of Weak Inertia on Flow Through a Porous Medium,” J. Fluid Mech., 222, 647–663.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Mickley, H. S., Smith, K. A., and Korchak, E. I., 1965, “Fluid Flow in Packed Beds,” Chem. Engng. Sci., 20, 237–246.

    Article  Google Scholar 

  • Neale, G. H. and Nader, W. K., 1974a, “Prediction of Transport Process within Porous Media: Creeping Flow Relative to a Fixed Swarm of Spherical Particles,” AIChE J., 20, 530–538.

    Article  Google Scholar 

  • Neale, G. H. and Nader, W. K., 1974b, “Practical Significance of Brinkman Extension of Darcy’s Law: Coupled Parallel Flows within a Channel and a Bounding Porous Medium,” Can. I. Chem. Eng., 52, 475–478.

    Article  Google Scholar 

  • Newell, R. and Standish, N., 1973, “Velocity Distribution in Rectangular Packed Beds and Non-Ferrous Blast Furnaces,” Metall. Trans., 4, 1851–1857.

    Article  Google Scholar 

  • Poots, G., 1961, “Laminar Natural Convection Flow in Magneto-Hydrodynamics,” Int. J. Heat Mass Transfer, 3, 1–25.

    Article  Google Scholar 

  • Richardson, S., 1971, “A Model for the Boundary Condition of a Porous Material: Part 2,” J. Fluid Mech., 49, 327–336.

    Article  ADS  MATH  Google Scholar 

  • Romig, M. F., 1964, “The Influence of Electric and Magnetic Fields on Heat Transfer to Electrically Conducting Fluids,” Adv. Heat Transfer, 1, 267–354, Academic.

    Article  Google Scholar 

  • Rossow, V. J., 1957, “On Flow of Electrically Conducting Fluids of a Transverse Magnetic Field,” NASA Technical Note, 3971.

    Google Scholar 

  • Rubenstein, J. and Torquato, S., 1989, “Flow in Random Porous Media: Mathematical Formulation, Variational Principles, and Rigorous Bounds,” J. Fluid Mech., 206, 25–46.

    Article  MathSciNet  ADS  Google Scholar 

  • Rumpf, H. and Gupte, A. R., 1971, “Einfüsse und Korngrössenverteilung in Widerstandsdesetz der Porenströmung,” Chemie-Ing. Techn., 43, 367–375.

    Article  Google Scholar 

  • Saffman, P. G., 1971, “On the Boundary Condition at the Surface of a Porous Medium,” Stud. Appl. Math., 50, 93–101.

    MATH  Google Scholar 

  • Sahraoui, M. and Kaviany, M., 1992, “Slip and No-Slip Boundary Condition at Interface of Porous, Plain Media,” Int. J. Heat Mass Transfer, to appear.

    Google Scholar 

  • Sanchez-Palencia, E., 1980, Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, 127, Springer-Verlag.

    MATH  Google Scholar 

  • Sangani, A. S. and Behl, S., 1989, “The Planar Singular Solutions of Stokes and Laplace Equations and Their Application to Transport Processes near Porous Surfaces,” Phys. Fluids, A1, 21–37.

    ADS  Google Scholar 

  • Scheidegger, A. E., 1974, The Physics of Flow Through Porous Media, Third Edition, University of Toronto Press.

    Google Scholar 

  • Schertz, W. W. and Bischoff, K. B., 1969, “Thermal and Material Transport in Nonisothermal Packed Beds,” AIChE J., 15, 597–604.

    Article  Google Scholar 

  • Schwartz, C. E. and Smith, J. M., 1953, “Flow Distribution in Packed Beds,” Ind. Engng. Chem., 45, 1209–1218.

    Article  Google Scholar 

  • Slattery, J. C., 1969, “Single-phase Flow Through Porous Media,” AIChE J., 15, 866–872.

    Article  Google Scholar 

  • Slattery, J. C., 1981, Momentum, Energy, and Mass Transfer in Continua, Second Edition, R. F. Krieger Publishing Co.

    Google Scholar 

  • Sparrow, E. M. and Cess, R. D., 1961, “The Effect of a Magnetic Field on Free Convection Heat Transfer,” Int. J. Heat Mass Transfer, 3, 267–274.

    Article  Google Scholar 

  • Sparrow, E. M. and Loemer, A. L. Jr., 1959, “Longitudinal Laminar Flow Between Cylinders Arranged in Regular Array,” AIChE J., 5, 325–330.

    Article  Google Scholar 

  • Stanek, V. and Szekely, J., 1974, “Three-Dimensional Flow of Fluids Through Nonuniform Packed Beds,” AIChE J., 20, 974–980.

    Article  Google Scholar 

  • Sullivan, R. R., 1942, “Specific Surface Measurement on Compact Bundles of ParaUel Fibers,” J. Appl. Phys., 13, 725–730.

    Article  ADS  Google Scholar 

  • Taylor, G. I., 1971, “A Model for the Boundary Condition of a Porous Material: Part 1,” 7. Fluid Mech., 49, 319–326.

    MATH  Google Scholar 

  • Vafai, K., 1984, “Convective Flow and Heat Transfer in Variable-Porosity Media,” J. Fluid Mech., 147, 233–259.

    Article  ADS  MATH  Google Scholar 

  • Vafai, K. and Thiyagaraja, R., 1987, “Analysis of Flow and Heat Transfer at the Interface Region of a Porous Medium,” Int. J. Heat Mass Transfer, 30, 1391–1405.

    Article  MATH  Google Scholar 

  • Vafai, K. and Tien, C.-L., 1981, “Boundary and Inertia Effects on Flow and Heat Transfer in Porous Media,” Int. J. Heat Mass Transfer, 24, 195–203.

    Article  MATH  Google Scholar 

  • van der Merwe, D. F. and Gauvin, W. H., 1971, “Velocity and Turbulence Measurement of Air Flow Through a Packed Bed,” AIChE J., 17, 519–528.

    Article  Google Scholar 

  • Ward, J. C., 1964, “Turbulent Flow in Porous Media,” J. Hyd. Div. ASCE, 90, HY5, 1–12.

    Google Scholar 

  • Whitaker, S., 1969, “Advances in Theory of Fluid Motion in Porous Media,” Ind. Engng. Chem., 61, 14–28.

    Article  Google Scholar 

  • Whitaker, S., 1986, “Flow in Porous Media I: A Theoretical Derivation of Darcy’s Law,” Transp. Porous Media, 1, 3–25.

    Article  Google Scholar 

  • Wooding, R. A., 1957, “Steady State Free Thermal Convection of Liquid in a Saturated Permeable Medium,” Phys. Fluids, 12, 273–285.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering and Applied Mechanics, The University of Michigan, Ann Arbor, MI, 48109-2125, USA

    M. Kaviany

Authors
  1. M. Kaviany
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag New York Inc.

About this chapter

Cite this chapter

Kaviany, M. (1991). Fluid Mechanics. In: Principles of Heat Transfer in Porous Media. Mechanical Engineering Series. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-0412-8_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-4684-0412-8_2

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4684-0414-2

  • Online ISBN: 978-1-4684-0412-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation