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  • hydraulic permeability coefficient

    Dear BIOMCH-L,

    Does anyone have a reference that would explain the definition of
    the hydraulic permeability coefficient used to express fluid flow through
    cortical or trabecular bone?

    I read in several articles different units for this permeability
    coefficient and some of them don't seem to be right.

    1) In [Mow 1991] the rate of volume discharge Q across an area A is
    related to the
    hydraulic permeability coefficient k by Darcy's law:

    Q = (k*A*dP) / h

    where h is the thickness of the specimen and dP the pressure gradient.

    As results form permeation experiments for normal cartilage and meniscus, k
    ranges
    form 10^-15 to 10^-16 m^4/(N*s).

    [Mow 1991]
    Mow V.C., Hayes W.C.: Basic Orthopaedic Biomechanics. Library of Congress
    Cataloging-in-Publication Data. Raven Press New York, pp. 158-159.



    2) On the other hand [Grimm 1997] explains Darcy's law by using following
    equation:

    K = (Q*l*) / (dP*A)

    where
    K = permeability
    Q = rate of volume discharge
    l = specimen length
    = kinematic viscosity
    dP = pressure gradient
    A = specimen cross sectional area

    Out of his model the permeability coefficient ranges from 0.4 to 11*10^-9
    with the unit [m^2]. In this case the hydraulic permeability coefficient of
    trabecular bone is proportional to the area and a function of the porosity
    factor.


    [Grimm 1997]
    Grimm M. J., Williams J.L.: Measurements of permeability in human calcaneal
    trabecular bone. Journal of Biomechanics, Vol. 30, No. 7, pp. 743-745,
    1997.



    3) [Keanini 1995] used the time averaged form of Darcy's law with:

    u = -k*w*P /

    where
    = fluid viscosity
    k = permeability
    P = time averaged fluid pressure
    w = gradient operator
    u = time averaged velocity component

    To do their calculations they used a permeability for cortical bone ranging
    form 10^-14 to 10^-11 m^2. And here again the permeability coefficient has
    the unit [m^2].


    [Keanini 1995]
    Keanini R.G., Roer R.D., Dillaman R.M.: A theoretical model of circulatory
    interstitial fluid flow and species transport within porous cortical bone.
    Journal of Biomechanics, Vol. 18, No. 8, pp. 901-914, 1995.


    And that's where I have a problem, how can it be that three different
    references have used the same law but with different units in some
    parameters?

    Isn't it true that the kinematic viscosity is already taken into account by
    explaining the hydraulic permeability coefficient, so that the variable for
    the kinematic viscosity is not part of Darcy's law?

    Does anyone has any further information on Darcy's law and the definition
    of it's parameters?


    Thanks in advance.

    Best regards,
    Michael L.


    ************************************************** **********
    Dipl.-Ing. Michael Liebschner E-mail: mliebsch@emba.uvm.edu
    University of Vermont
    Mechanical Engineering Phone: (802) 656-1432
    119 Votey Building
    Burlington, VT 05405 Fax: (802) 656-4441
    ************************************************** **********
    You can not solve problems with the same level of thinking
    that existed when the problems were created.
    --Dr. Albert Einstein
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