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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*µ) / (dP*A)
where
K = permeability
Q = rate of volume discharge
l = specimen length
µ = 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 / µ
where
µ = 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
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*µ) / (dP*A)
where
K = permeability
Q = rate of volume discharge
l = specimen length
µ = 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 / µ
where
µ = 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