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Journal ArticleDOI

The permeability of porous materials

TL;DR: In this article, a simple statistical theory, based upon the calculation of the probability of occurrence of sequences of pairs of pores of all the possible sizes, and of the contribution to the permeability made by each such pair, leads to an expression of the porosity as the sum of a series of terms.
Abstract: The permeability of a porous material to water is a function of the geometry of the boundary between the solid component and the pore space. Expressions of the Kozeny type purporting to represent this function are based upon the particle size or specific surface of the solids, and whilst, for engineering practice, they have given satisfaction for saturated sands, they may fail badly in other cases. By developing a Kozeny type of expression for the particular structure of a bundle of capillary tubes of assorted radii, we demonstrate the cause of the failure. Such failure may be avoided by relating permeability to pore-size distribution, which is the factor of prime concern and which may be measured directly by even simpler means than are used to determine particle-size distribution. The pore-size distribution is arrived at by an interpretation of the moisture characteristic of the material, i.e. of the curve of moisture content plotted against pressure deficiency. A simple statistical theory, based upon the calculation of the probability of occurrence of sequences of pairs of pores of all the possible sizes, and of the contribution to the permeability made by each such pair, leads to an expression of the permeability as the sum of a series of terms. By stopping the summation at a selected upper limit of pore size one may calculate the permeability at any chosen moisture content and plot it as a function of that content. An example is presented, using a coarse graded sand specified by its moisture characteristic. To check these calculations, experimental determinations of the permeabilities of unsaturated materials are presented, using two different grades of sand and a sample of slate dust, the results being compared with computed values. The agreement seems good, and is certainly better than that provided by the Kozeny formula as developed, with difficulty, for the purpose. The limitations and possible improvements of our concept are very briefly discussed, and finally it is shown how a combined use of the moisture characteristic and the permeability (which is itself derivable from the moisture characteristic) leads to an expression for the coefficient of diffusion of water in the material as a function of moisture content. From this it should be possible, in principle, to calculate in suitable cases the course of water movement down a gradient of moisture content. Such a calculation awaits a satisfactory solution of the problem of non-linear diffusion.

Summary (3 min read)

INTRODUCTION

  • Accurate and comprehensive information on the permeability characteristics of porous materials is essential to the proper design of areasuction installations on wings, flaps, inlets, and wind tunnels.
  • It was found in the course of designing a wing using suction through a porous material that available calibration information was not sufficient for predicting permeability characteristics for the range of operating conditions (absolute pressure levels) anticipated.
  • If a calibration test is made with the.
  • In addition, previous investigations were found to include no information regarding choking of the flow following the occurrence of sonic velocity within the pores, bending the material, and, in the case of wire cloth, rolling the material.

NACA TN 3596

  • The need for information on the aforementioned factors is emphasized by the following considerations.
  • Consideration of the limitation on inflow velocity imposed by flow choking is important in some installations.
  • The investigation reported herein was undertaken to determine the influence on the permeability of porous material of the aforementioned factors.
  • The analysis includes comparison of the wire-cloth calibrations with a calibration for the same material bent to form the,leading edge of a wing model which was tested in the Langley 19-foot pressure tunnel.
  • An occurrence of this sort shows that correlation based on anything but the fundamental variables, Mach number, and Reynolds number, should be used only with great caution.

APPARATUS

  • The apparatus used in the tests with constant upstream pressure is shown schematically in figure 1 .
  • This apparatus was the same as that used for constant upstream pressure except that an additional length of 2-inch pipe was attached upstream of the sample, and the downstream pipe and valve were moved upstream of the entire setup.
  • Fine wire screens were placed 2 diameters upstream of the sample to help maintain flow uniformity.
  • High and low pressure sources used were the Langley 19-foot pressure tunnel and the suction side of a centrifugal compressor.
  • Downstream flow velocities were determined by applying onedimensional-flow relations between stations d and t (figs. 1 and 2) as follows: Continuity and isdthermal conditions were used to relate stations u and d.

MATERIAL

  • Two flat samples of 30-by 250-mesh Dutch weave Monel metal wire cloth were calibrated.
  • The wire diameters for these samples were O.OOSO inch for the wires that were 30 per inch and 0.00^0 inch for the wires that were 250 per inch.
  • Results presented herein for the wire cloth are, therefore, considered to be typical.
  • A 6-percent difference occurred between the flow velocities for the two samples at a given value of Ap.
  • A 16 theoretical and experimental investigation of the permeability of various sintered metals is reported in reference k.

PROCEDURE

  • In the calibrations made for this investigation, three methods of pressure variation were used.
  • The first method employed a pressure p upstream of the sample of approximately 1 atmosphere and a variable suction pressure p-, downstream; the second method employed p u of approximately 2.-atmospheres with a variable p^; and the third method employed a variable p with p^ of 1 atmosphere.
  • Before each calibration the sample was thoroughly cleaned with acetone.
  • The pressures indicated by the manometers shown in figures 1 and 2 were recorded for various values of £p.
  • For each value of £p, the flow velocities were obtained from the calibration equation (eq. l).

ACCURACY

  • The repeatability of the'wire-cloth calibrations was determined by making several duplicate calibrations.
  • The results indicated that the calibrations were repeatable within about 1 percent.
  • No repeatability check was made for the sintered materials.
  • The venturi calibration was based on the assumption of incompressible flow in the venturi.
  • This assumption resulted in a maximum error of about 1 percent at the highest velocities obtained.

RESULTS AND DISCUSSION

  • The basic permeability data are presented as plots of V against tip, inasmuch as the quantity desired for boundary-layer-suction applications is inflow velocity for a given pressure drop.
  • Also, since the porous-material flow phenomena depend only on Mach number and Reynolds number, any dimensionless flow characteristic (for example, ^P/Pu or £gp/q u ) plotted as a function of M and R will completely describe the flow.
  • For purposes of analysis, plots of this type are used herein.
  • The quotient Ru/DMu was chosen because this quantity is constant for calibrations with constant p u and T u .
  • Since the length which appears in the expression for R is significant only in comparisons between different materials, and since only a single wire cloth is considered, this length has been eliminated from the Reynolds number by using the quantity RU/D instead of R u .

Effect of Rolling the Material

  • The effect of rolling on the permeability of metal filter cloth is shown in figure 9 as obtained from data taken in the Langley cascade aerodynamics laboratory.
  • Rolling or hammering the material apparently has little effect on the form of the calibration curve but changes only the magnitude of the velocities.
  • Figure 10 is a cross plot of the data of figure 9 -This plot and the other data examined indicate approximately linear variation of downstream velocity V^ with thickness, at least over the range of thickness presented.
  • The scatter of some of the points of figure 10 is attributed to the fact that the thicknesses were determined with a precision no better than ±0.0001 inch.
  • Differences between the permeabilities of two samples of the same thickness were mentioned previously in the section entitled "Material".

Effect of Bending the Material

  • Figure 8 presents a comparison of V u and V^ curves for flat and curved specimens of the JO-by 250-mesh filter cloth.
  • The V u curve for the 0.017-inch thickness is that of the present investigation.
  • Inch, measured perpendicular to the leading edge.
  • A comparison of these curves for V u indicates that the amount of tending present in the model leading edge caused no noticeable change in the form of the .curve, although more extreme bending might possibly produce a change.

SINTERED METAL

  • Calibration curves are shown in figure 11 for two samples (designated A and B) of sintered bronze similar to that used in references 6, 11, and 12.
  • The curves for samples A and B (fig. 11 ) show a variation with absolute pressure level similar to that for the wire-cloth, calibrations.
  • In view of this difference of results each set of data, that is, for sample A, sample B, and the sample of reference 13, was examined for consistency within itself.
  • At higher velocities, n becomes greater, and the effect of absolute pressure level should be more noticeable.
  • Equation (3) indicates that for the Darcy flow, £p will vary linearly with V u .

CONCLUSIONS

  • The permeability characteristics of representative samples of wire cloth and sintered metal have been investigated by means of flow calibration tests made over an independent range of Mach number and Reynolds number.
  • The results and comparison with other data revealed the following information: 1. Since the permeability characteristics of a given porous material depend only on Mach number and Reynolds number, values of absolute pressure and temperature affect the usual calibration curve of upstream velocity as a function of pressure drop by changing the Mach number and Reynolds number of the flow.
  • The results may be expected to be reasonably accurate up to conditions where appreciable choking occurs in the pores.
  • For a given wire cloth, the spread between calibrations obtained by using, different absolute pressure levels can be greatly reduced by plotting upstream velocity against the ratio of pressure drop to upstream pressure.
  • When the thickness of wire cloth is reduced by rolling or hammering, the velocity downstream of the material varies almost linearly with thickness for a given pressure drop, at least within the range of this study.

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CASE
FILE
NATIONAL ADVISORY COMMITTEE
FOR
AERONAUTICS
TECHNICAL
NOTE 3596
ON
THE
PERMEABILITY
OF
POROUS
MATERIALS
By
E.
Carson Yates,
Jr.
Langley
Aeronautical Laboratory
Langley
Field,
Va.
Washington
January
1956
(NASA-TH-79825)
ON THE
PEBHEABHITY
OF
IPOBOUS
HA.TEBIAIS
(National
Advisory
(Committee
for; Aeronaoitics.)
32 p
N78-78604
Dnclas
00/24
32228

NATIONAL ADVISORY COMMITTEE
FOR
AERONAUTICS
TECHNICAL
NOTE
3596
ON
THE
PERMEABILITY
OF
POROUS MATERIALS
By
E.
Carson Yates,
Jr.
SUMMARY
The
effects
on
porous-material permeability characteristics
of the
absolute pressure level
(and
associated scale effects), choking
of the
flow, bending
the
material^
and
other factors have been investigated.
Samples
of
rolled
30-by
250-mesh Dutch weave Monel metal cloth
and
l/l6-inch-thick sintered bronze were calibrated with constant upstream
pressures
of 1
atmosphere
and 2
atmospheres (varying downstream pressure)
and
with constant downstream pressure
of 1
atmosphere (varying upstream
pressure). Experiments showed permeability characteristics
to be
appre-
ciably
affected
by
absolute pressure level, flow choking,
and
thickness
of
the
material. Moderate bending
of the
material caused
no
noticeable
change
in the
permeability. Simple calculation
and
correlation proce-
dures
are
presented
for
determining permeability characteristics with
reasonable accuracy when experimental data
are
limited.
INTRODUCTION
Accurate
and
comprehensive information
on the
permeability charac-
teristics
of
porous materials
is
essential
to the
proper design
of
area-
suction
installations
on
wings, flaps,
inlets,
and
wind tunnels.
It
was
found
in the
course
of
designing
a
wing using suction through
a
porous
material that available calibration information
was not
sufficient
for
predicting permeability characteristics
for the
range
of
operating
conditions
(absolute pressure levels) anticipated. Since most previous
calibrations (for example, refs.
1 and 2)
were found
to
have been made
with
a
single fixed relation existing between
the
Mach number
and
Reynolds
number
of the
flow,
the
effect
on
permeability
of
independent variation
of
these quantities (caused,
for
example,
by
differences
of
absolute
pressure
level)
could
not be
evaluated.
For
example,
if a
calibration
test
is
made with the. upstream pressure held constant
and the
downstream
pressure
varied,
it can
easily
be
shown that
the
ratio
of
Reynolds num-
ber
to
Mach number (both based
on
upstream conditions)
is a
constant.
In
addition, previous investigations were found
to
include
no
information
regarding choking
of the
flow following
the
occurrence
of
sonic
velocity
within
the
pores, bending
the
material, and,
in the
case
of
wire cloth,
rolling
the
material.

NACA
TN
3596
The
need
for
information
on the
aforementioned factors
is
emphasized
by
the
following considerations.
For a
given pressure difference
across
a
porous material, scale effects associated with absolute pressure
level
were believed
to
influence flow velocities.
If
this influence
is
appre-
ciable,
it
should
be
considered
in the
selection
of a
porous material.
Consideration
of the
limitation
on
inflow velocity
imposed
by
flow
choking
is
important
in
some
installations.
For
example,
in
wing-suction
applications choking
may be
used advantageously
to
limit inflow
over
areas
where large pressure differences exist,
so
that chordwise grading
of
permeability
may be
dispensed with. Bending
the
porous material could
affect permeability
by
changing
the
sizes
of
openings
in the
material
and by
changing
the
flow from
one
dimensional
to two
dimensional.
These
changes could have significant effectSj
for
example,
in
flow near
the
leading
edge
of a
wing.
The
investigation reported herein
was
undertaken
to
determine
the
influence
on the
permeability
of
porous material
of the
aforementioned
factors. Representative samples
of
wire cloth
and
sintered bronze (two
samples
of
each) were calibrated
by
holding
the
upstream pressure con-
stant
at 1
atmosphere
and
varying
the
downstream pressure,
by
holding
the
upstream pressure constant
at 2
atmospheres
and
varying
the
downstream
pressure,
and by
holding
the
downstream pressure constant
at one
atmos-
phere
and
varying
the
upstream pressure. Testing
in
this manner yielded
independent variation
of
Mach number
and
Reynolds number.
The
analysis
includes comparison
of the
wire-cloth calibrations with
a
calibration
for
the
same material bent
to
form the,leading
edge
of a
wing model which
was
tested
in the
Langley
19-foot
pressure tunnel. Data obtained
in the
Langley
cascade aerodynamics laboratory
for the
wire cloth
rolled
to
var-
ious thicknesses
are
also shown.
The
analysis indicates methods
for
cal-
culating
or
estimating permeability calibration curves,
and
these methods
are
presented herein.
The
possible large effect
on
permeability
of the
presence
of a
com-
ponent
of the
inlet velocity which
is
parallel
to the
porous surface
(as
discussed
in
ref.
3) is not
considered herein. However, since
the
pores
in the
materials investigated
are
very
much
smaller than those
in
the
perforated materials
of
reference
3, it is
believed that
the
influ-
ence
of a
parallel
flow component
in the
present case would
be
consid-
erably
less than that reported
in
reference
3-
SYMBOLS
A .
cross-sectional area
of
venturi throat
D
effective diameter
of
pores,
ft
g
gravitational
acceleration,
ft/sec^

NACA
TN
3596
K
porous-material flow coefficient
M
Mach number
p
static pressure, Ib/sq
in. or
lb/sq
ft
£sp
pressure drop across
the
porous material,
p - p.,
Ib/sq.
in. or in. H
2
0
q
dynamic pressure,
ipV ,
Ib/sq.
ft
R
Reynolds number,
£^S
V-
T
temperature,
°F or °R
t
thickness
of
porous material,
in.
V
velocity
in
test apparatus
(flow
considered
to be
one-
dimensional), ft/sec
a
Darcy flow coefficient
7
ratio
of
specific heat
at
constant pressure
to
specific
heat
at
constant
volume
(i
viscosity coefficient, slugs/ft-sec
p
mass density, slugs/cu
ft
Subscripts:
d
station just downstream
of the
porous material
t
station
at the
venturi throat
u
station just upstream
of the
porous material
DEFINITIONS
;
The
term "porosity"
as
used herein
is
defined
as the
percentage
of
void
present
in the
porous material.
Permeability
is a
qualitative term related
to the
resistance
of the
material
to
fluid flow.
The
greater
the
resistance,
the
less,the
permeability,
and
vice
versa.

NACA
TN
3596
APPARATUS
AMD
TECHNIQUE
APPARATUS
The
apparatus used
in the
tests with constant upstream pressure
is
shown
schematically
in
figure
1. It
consisted
of a
2-inch inside-
diameter pipe with
a
flange, clamp plate,
and
gaskets
at one end to
hold
the
sample
and a
wooden venturi
at the
other end. Downstream
of the
i-inch-diameter-venturi throat
was an
exhaust pipe containing
a
gate
valve
for
regulating
the
downstream pressure
p,.
The
apparatus used
in the
tests with varying upstream pressure
is
shown
in
figure
2.
This
apparatus
was the
same
as
that used
for
con-
stant upstream pressure except that
an
additional length
of
2-inch pipe
was
attached upstream
of the
sample,
and the
downstream pipe
and
valve
were moved upstream
of the
entire setup. Fine wire screens were placed
2
diameters upstream
of the
sample
to
help maintain flow
uniformity.
High
and low
pressure sources used were
the
Langley
19-foot
pressure
tunnel
and the
suction side
of a
centrifugal compressor.
Downstream
flow velocities were determined
by
applying one-
dimensional-flow
relations between stations
d and t
(figs.
1 and 2)
as
follows:
Continuity
and
isdthermal conditions were used
to
relate stations
u
and d.
MATERIAL
Two
flat samples
of
30-by 250-mesh Dutch weave Monel metal wire
cloth were calibrated.
The
wire diameters
for
these samples were
O.OOSO inch
for the
wires that were
30 per
inch
and
0.00^0
inch
for the
wires that were
250 per
inch.
The
samples were rolled
to a
thickness
of
0.017 inch from
an
original thickness
of
0.026
inch.
For
purposes
of
this investigation,
tests
of
wire cloth
of
other mesh dimensions were
not
made because examination
of
data available from
the
Langley
cascade
aerodynamics
laboratory
indicated that other meshes would have generally

Citations
More filters
Journal ArticleDOI
TL;DR: In this article, a simple analytic model is proposed which predicts the unsaturated hydraulic conductivity curves by using the moisture content-capillary head curve and the measured value of the hydraulic conductivities at saturation.
Abstract: A simple analytic model is proposed which predicts the unsaturated hydraulic conductivity curves by using the moisture content-capillary head curve and the measured value of the hydraulic conductivity at saturation. It is similar to the Childs and Collis-George (1950) model but uses a modified assumption concerning the hydraulic conductivity of the pore sequence in order to take into account the effect of the larger pore section. A computational method is derived for the determination of the residual water content and for the extrapolation of the water content-capillary head curve as measured in a limited range. The proposed model is compared with the existing practical models of Averjanov (1950), Wyllie and Gardner (1958), and Millington and Quirk (1961) on the basis of the measured data of 45 soils. It seems that the new model is in better agreement with observations.

6,529 citations

Journal ArticleDOI
TL;DR: In this paper, a theory of moisture movement in porous materials under temperature gradients is developed which explains apparently discordant experimental information, including (a) the large value of the apparent vapor transfer, (b) effect of moisture content on net moisture transfer, and (c) the transfer of latent heat by distillation.
Abstract: A theory of moisture movement in porous, materials under temperature gradients is developed which explains apparently discordant experimental information, including (a) the large value of the apparent vapor transfer, (b) effect of moisture content on net moisture transfer, and (c) the transfer of latent heat by distillation. The previous simple theory of water vapor diffusion in porous media under temperature gradients neglected the interaction of vapor, liquid and solid phases, and the difference between average temperature gradient in the air-filled pores and in the soil as a whole. With these factors taken into account, an (admittedly approximate) analysis is developed which predicts orders of magnitude and general behavior in satisfactory agreement with the experimental facts. An important implication of the present approach is that experimental methods used to distinguish between liquid and vapor transfer have not done so, since what has been supposed to be vapor transfer has actually been series-parallel flow through liquid ‘islands’ located in a vapor continuum. Equations describing moisture and heat transfer in porous materials under combined moisture and temperature gradients are developed. Four moisture-dependent diffusivities arising in this connection are discussed briefly.

2,179 citations

01 Jan 1992
TL;DR: The RETC computer code as mentioned in this paper uses the parametric models of Brooks-Corey and van Genuchten to represent the soil water retention curve, and the theoretical pore-size distribution models of Mualem and Burdine to predict the unsaturated hydraulic conductivity function from observed water retention data.
Abstract: This report describes the RETC computer code for analyzing the soil water retention and hydraulic conductivity functions of unsaturated soils. These hydraulic properties are key parameters in any quantitative description of water flow into and through the unsaturated zone of soils. The program uses the parametric models of Brooks-Corey and van Genuchten to represent the soil water retention curve, and the theoretical pore-size distribution models of Mualem and Burdine to predict the unsaturated hydraulic conductivity function from observed soil water retention data. The report gives a detailed discussion of the different analytical expressions used for quantifying the soil water retention and hydraulic conductivity functions. A brief review is also given of the nonlinear least-squares parameter optimization method used for estimating the unknown coefficients in the hydraulic models. Several examples are presented to illustrate a variety of program options. The program may be used to predict the hydraulic conductivity from observed soil water retention data assuming that one observed conductivity value (not necessarily at saturation) is available. The program also permits one to fit analytical functions simultaneously to observed water retention and hydraulic conductivity data. The report serves as both a user manual and reference document. Detailed information is given on the computer program along with instructions for data input preparation and sample input and output files. A listing of the source code is also provided.

1,553 citations

Book ChapterDOI
01 Jan 1969

1,477 citations

References
More filters
Book
01 Jan 1937

1,591 citations

01 Jan 1937

1,390 citations


"The permeability of porous material..." refers background in this paper

  • ...The fact seems to be that it has been impossible to secure a sufficiently wide range of variation of p and of pore-size distribution to test the formulas severely; and such tests as have been reported indicate a wide range of error (Muskat 1937)....

    [...]

Journal ArticleDOI
TL;DR: In this paper, it was shown that the magnetic energy of the disturbance will increase provided the conductivity is greater than a critical value determined by the viscosity of the fluid, and the rate of growth of magnetic energy is approximately exponential, with a doubling time which can be simply related to the properties of the turbulence.
Abstract: Several recent investigations in geophysics and astrophysics have involved a consideration of the hydrodynamics of a fluid which is a good electrical conductor. In this paper one of the problems which seem likely to arise in such investigations is discussed. The fluid is assumed to be incompressible and in homogeneous turbulent motion, and externally imposed electric and magnetic fields are assumed to be absent. The equations governing the interaction of the electromagnetic field and the turbulent motion are set up with the same assumptions as are used to obtain the Maxwell and current flow equations for a metallic conductor. It is shown that the equation for the magnetic field is identical in form with that for the vorticity in a non-conducting fluid; immediate deductions are that lines of magnetic force move with the fluid when the conductivity is infinite, and that the small-scale components of the turbulence have the more powerful effect on the magnetic field. The first question considered is the stability of a purely hydrodynamical system to small disturbing magnetic fields, and it is shown that the magnetic energy of the disturbance will increase provided the conductivity is greater than a critical value determined by the viscosity of the fluid. The rate of growth of magnetic energy is approximately exponential, with a doubling time which can be simply related to the properties of the turbulence. General mechanical considerations suggest that a steady state is reached when the magnetic field has as much energy as is contained in the small-scale components of the turbulence. Estimates of this amount of energy and of the region of the spectrum in which it will lie are given in terms of observable properties of the turbulence.

409 citations

Journal ArticleDOI

145 citations


"The permeability of porous material..." refers background in this paper

  • ...For a non-shrinking material such as sand, the distribution function/(r) may be derived satisfactorily from the moisture characteristic, i.e. from the curve relating moisture content to the hydrostatic pressure a t th a t content (Childs 1940, 1942)....

    [...]

  • ...We have sought to relate permeability more rationally with pore-size distri bution, which, in recent years, has become almost as common a determination as mechanical composition (Donat 1937; Childs 1940, 1942; Feng & Browning 1946)....

    [...]

Journal ArticleDOI

113 citations


"The permeability of porous material..." refers background in this paper

  • ...Some of these findings are summarized by Christiansen (1944)....

    [...]