The permeability of fibrous porous media

The Permeability of Fibrous Porous Media

GRAHAM W . JACKSON* and DAVID F . JAMES

Department of Mechanical Engineering, University of Toronto, Toronto, Ontario M5S lA4

The literature was searched for experiments and theories related to low-Reynolds-number flow through fibrous porous media, particularly highly porous structures. Experimental data wcrc found for a wide range of materials, from polymer chains to fiberglass, and the results collapse reasonably well when the appropriate dimensionless co-ordinates are employed. Of the theories, accurate solutions of Stokes equation are available for regular arrays of parallel rods, either aligned with or normal to the flow. For irregular arrays and three-dimensional media, approximate permeabilities can be calculated from several flow models.

On a fait une ttude bibliographique des travaux exptrimentaux et theoriqucs relatifs aux ecoulements faible nombre de Reynolds travers les milieux poreux fibreux et particulikrement a travers les structures hautement poreuses. Des donnCes expkrimentales ont kt6 recueillies pour dc nombreux materiaux, depuis les chaines de polymkres jusqu’a la fibre de verre; les rksultats sont raisonnablement bien corr6lCs lorsqu’on utilise des coordonnCes adimensionnelles appro- prikes. Parmi les diverses thCories propodes, il existe des solutions prkcises de I’equation de Stokes pour des disposi- tions rkgulikres de fibres parallkles, alignees ou normales a I’Ccoulement. Pour des empilements irrkguliers et des milieux a trois dimensions, il est possible de calculer des permeabilitts approximatives 5 partir de plusieurs moddes d’kcou- lement.

ost porous media are granular, but some are composed M of very long particles and therefore may be described as fibrous. Some obvious examples of fibrous porous media are steel wool and cotton batting, but we will also include in this category some structures not ordinarily thought of as porous media, like banks of heat exchanger tubes and entangled polymer chains in solution. The corrstituent fibers, then, may be straight or curved, man-made or natural, randomly oriented or in regular arrays, but whatever the nature of the fibers, our interest here is confined to those which are long enough that aspect ratio is not a parameter. Fiber size, then, is given by a single cross dimension.

A significant feature of fibrous porous media is that the fraction of solid material is often much lower than that for a granular material. The latter media are inherently compact and so the grains generally occupy about 60% to 70% of the total volume. With fibers, however, much less material is needed to form a structure and so the solid volume fraction can easily be as low as 0.01. In fact, we are most interested in high-porosity materials because the fibrous character of the medium becomes most important in this regime.

Since fibrous media are not as common as granular materials, they have been studied less extensively and much less is understood about the way resistance depends on the structure of the medium. From an initial survey of the literature, it became apparent that few comparisons had been made, particularly between theory and experiment, but also between experiments and between theories. To remedy this situation, we collected all the data and theories that we could find related to the resistance of fibrous porous media, crit- ically examined them, and then made as many appropriate comparisons as possible.

The comparisons were made on the basis of hydrodynamic permeability. This property is usually denoted by k and is defined by Darcy’s law:

- Q - k b A P L

*Now at Johnson Wire Industries Ltd., Kanata. Ontario.

where Q is the volumetric flow rate through a cross-section of area A , is the fluid dynamic viscosity, and A p is the pressure drop over a length L . This equation, widely used in engineering applications, is valid for Newtonian fluids at low flow rates, i.e., at low Reynolds numbers where fluid inertial effects are negligible. While permeability is a natural choice to express the flow resistance of a porous medium, in much of the literature resistance was expressed in other forms and these were converted to equivalent values of k.

The permeability of a fibrous medium depends on the size, concentration and arrangement of fibers. It is being presumed here that the fibers are circular, or approximately so, and uniform in diameter. For size, then, the logical dimension is diameter or radius a. To characterize fiber concentration, we selected the volume fraction of solid material, 4. The other obvious candidate to represent density is the porosity E, or ( I - +), but since 4 varies over several orders of magnitude in the present work while E

stays around unity, 4 is the more useful parameter. The permeability k is then a function of a, 4, and fiber arrangement. In non-dimensional terms, the relation is:

k - = f(4) a*

for a particular fiber arrangement. This format will be used throughout the paper to express the theoretical and experimental results found in the literature.

We first present all the experimental results on a single graph, ignoring variations in fiber arrangement. Fiber configurations in real porous materials are generally known only approximately and so no attempt was made to separate the data according to fiber arrangement. A truncated version of the master graph is then used as the backdrop for subsequent plots of theoretical predictions. Since fiber arrangement is specified in the theoretical work, and since the arrangements investigated fall into several natural categories, the various predictions were grouped and plotted accordingly. The graphs provide interesting comparisons and indicate the reliability of predictions, but we also hope

364 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 64. J U N E 19x6

they will be useful in design work where one needs some estimate of the flow resistance in a fibrous material.

Experimental

Only a fraction of the experimental data we found turned out to be suitable for inclusion in this project. For reasons that will be explained shortly, many results had to be rejected.

The fundamental criterion for accepting a set of results was that the data followed Darcy's law. In cases where proportionality between pressure drop and flow rate could not be verified, the criterion was that the Reynolds number, based on fiber diameter and superficial velocity, was less than 10. This upper limit is based on data'by Bergelin et al. (19501, the details of which are given later in this section. A second criterion was applied when the experimental fluid was a gas. To ensure that the fluid did not slip at the fiber surface, the Knudsen number was required to be less than 0.01. A few sets of data were accepted when Reynolds and Knudsen numbers were not known exactly but when best estimates of these numbers were well inside the acceptable range. A considerable quantity of data had to be discarded because the Reynolds number and Knudsen number criteria were not met. Still other data could not be used because the diameter of the fibers or the solidity of the medium was not given. In all, useful data were obtained for about 25 fibrous media.

These data are given in Table I and most are plotted in Figure I ; the remainder were omitted from the plot because they coincided with other points. The table also provides some relevant details not given elsewhere in the paper; for example,ofiber diameters are given and are shown to range from 10 A to 10 mm. The values of permeability generally had to be converted from parameters such as fiber drag, drag coefficient, or friction factor. In some cases, critical information was not provided and had to be found indirectly; for example, fiber diameter was sometimes determined from a separate source. In effect, each set of data required individual treatment and the important details of each case are described below.

The earliest work on porous media comprising long particles was that by Carman (1938) for packed beds of steel wire crimps. Carman presented his results in terms of permeability divided by kinematic viscosity, k / u , from which k was found from his measurements of tr.

In the following year, Wiggins et al. (1939) determined the permeability of a wide variety of fibrous media: glass rods, glass wool, fiberglass, copper wire and Celanese yarn. Each medium was a random packing of fibers in a tube, and several test liquids were used. Fiber diameters ranged from 3.5 p,m to 200 pm, and length-to-diameter ratios varied from 20 for the glass rods to infinity for a continuous strand of fiberglass. The copper wires were uniform in diameter but the glass rod diameters varied by as much as ?30%. Since the fiberglass, glass wool and plastic yarn were not circular, Wiggins et al. determined the specific surface of each fiber. This quantity, which is commonly used to characterize granular media, is the perimeter divided by the cross-sectional area (in the case of fibers). Since specific surface equals 2 / a for circular fibers, this relation was used to calculate the equivalent radius a for these non-circular strands.

Sullivan (1942) supplied abundant data on the permeability of various wools and hairs. The most important detail

A

A

A

A A A

LEGEND L%11111s are given m Table 1

0 stainless steel wire crimps (Carman) 6 glass rods. copper wire. glass wool.

fiberglass and Celanese yarn (Wiggins) m goat wool and various hairs (Sullivan) 0 glass wool (Brown) 0 heat exchanger tubes (Bergelin el al) + Davies correlation for filter pads 0 filter pads (Chen) + nylon fibers (Ingmanson el al) A acrylamide polymer gel (White) I glass fiber filters (Wheat) v nylon fibers (Labrecque) m Kapron fibers (Kysch 8 Fuchs) a collagen membranes (Stenzel e l all m Cr-Ni metal liber filters

(Kostornov 8 Shevchuk) 0 collagen membranes

(Viswanadhan el al) A hyaluronic acid gel (Jackson 8 James)

5 0.001 0.01 0.1 1 0.0001

Solid Volume Fraction (@)

Figure 1 - Non-dimensional permeability as a function of fiber volume fraction, for various fibrous porous media.

of his work is that the fibers were packed in a tube such that they were aligned with the flow.

Brown (1950) measured the flow resistance of dry air through glass wool and paper fibers, and reported the data in terms of Kozeny constants. Since diameter measurements were made only for the glass wool, only the results for this medium are presented in Figure 1.

The media described so far have generally consisted of small flexible fibers more or less randomly packed. In con- trast, the porous media tested by Bergelin et al. (1950) were large rods in ordered arrangements; to be specific, the media were banks of heat exchanger tubes oriented normal to the flow. Because the tubes had diameters of 9.53 and 10.05 mm, the Reynolds numbers were generally much larger than 1. However, for Reynolds numbers up to 10, the relation between pressure drop and flow rate was virtually linear, thus establishing the allowable upper limit specified earlier. The work of Bergelin is particularly valuable because it is one of the few in which the fiber arrangement is known precisely.

Davies ( 1952) presented a considerable quantity of experimental data for the permeability of glass wool, glass rods, cotton wool, camel hair, kapok, merino wool, rayon,

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 64, JUNE 1986 365

TABLE I Permeability of Fibrous Porous Media, Experimental Results

Dimensionless data Radius a Re Kn

Author( s) 4 k /a2 Porous material Test fluid (mm) (max) (max)

W iggins

Sullivan

Brown

C a n a n 0.319 0.312 0.278 0.269 0.243 0.235 0.315 0.170 0.090 0.154 0. I43 0. I05 0.070 0.115 0.081 0.072 0.102 0.096 0.362 0.360 0. I95 0.156 0.139 0. I17 0.112 0.098 0.0935 0.0779 0.0595 0.0468 0.0390 0.0334 0.0292 0.0234 0.0213 0.0195 0.0180 0.0167 0.0156 0.550 0.344 0.654 0.645 0. I34 0.262 0.258 0.221 0.218 0.172 0.172 0.131 0.129 0.089 0.088

Bergelin et al. 0.580 0.503 0.503 0.403 0.349 0.503 0.503

0. I59 0.234 0.245 0.322 0.379 0. I18 0.846

1.60 3.64

4.42

0.317

6.10

19.5 25.6 31.0 36.6 49.3 56.5 64.5 72.3 80.8 90.3

0.414

0.0233 2.99

I .37 1.6 2.8 3.4

0.0076 0.035 0.046 0.0095 0.0074

Stainless steel wire crimps

Glass rods Copper wire

Glass wool

Fiberglass

Celanese yarn

Goat wool

Blond hair

Chinese hair

Glass wool

Glass wool

Flow normal to various arrangements of tubes

Glycerol 0.164

Water 0.204 Water 0.0508 Benzene

Water 9.09XlO ' Water

Benzene 3.5x lo--'

Benzene 6.7X lo-'

0.5 NA

8 NA 8 NA

8 NA

8 NA

8 NA

Air 0.0195 2 0.002

Air 0.0327

Air 0.0367

Air 0.0380

0.2 0.001

0.03 0.001

0.001 0.01

Air Implied

4.6X lo-' < I No slip NQ

4.76 Oil

9.53

10 NA

366 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 64. JUNE 1986

TABLE I . Continued

Dimensionless data Radius a Re Kn

Author(s) 4 k/a2 Porous material Test fluid (mm) (max) (ma)

Davies

Chen

Ingmanson et al

White

Wheat

Labrecque

Kirsch and Fuchs

k / 0 2 = I l6+'.'( I + 56+')]- I

Best fit curve through data, 0.006 < 4 < 0.3

0.195 0. I76 0. I69 0.115 0.0645 0.048 0.0329 0.0194 0.0153 0.0151 0.00897 0.0083 0.0064 0.258 0.232 0.21 0.20 0. I7 0. I5 0.14 0.13 0.10 0.32 0.29 0. I9 0.20 0. I7 0.15 0.13 0.10 0.09 0.08 0.07 0.06 0.045 0.245 0.210 0. I75 0.140 0. I05 0.070 0.054 0.035 0.253 0.231 0.275 0.235 0.20 0. I6 0.3 0.2 0.15 0.11 0.065 0.045 0.036 0.018 0.01 0.0055

0.721 0.669

18. I 16.2 51.8 34.2

55.2 104

128

0.80 0.97 1 . 1 1 1.33 1.92

1.9 2.5 3.0 3.8 5.2 8.0

0.410 0.334 0.838 I .33 I .74 2.57 3.62

0.412

0.57 0.95 "'i 4.4 5.8

17.4 39 82

Merino cotton and glass wools, rayon, kapok, down

Filter pads

Nylon fibers

Glass fibers

Acrylamide polymer gel

Glass fibers

Nylon fibers

Kapron fibers

Air

NQ

Air

Air

Water

Air

Water

95% glycerc

NQ

8.5X lo-' 6.0x lo-' I .5x lo--' 5 . o ~ 4 . 7 ~

7 . o ~ I O - ~ 4 . 7 ~ 7 . o ~ I O - ~ 4 . 7 ~ 7 . o ~ I O - ~ 7 . o ~

1.8X10-' 4.7x

0.0965

NQ

1

< I

0.082 < I

3 . 5 ~ lo-'

3.9x lo+ 7.2X

0.0012

0. I5 0.225 and 0.4

10-10

I .5

NQ

<O.W

NQ

NQ

N o slip

N o slip

N A

0.008

N A

N A


TABLE I . Concluded

Dimensionless data Radius (I Re K n

Author( s) $ klo’ Porous material Test fluid (mm) (max) (max)

Stenzel et al.

Kostornov and Shevchuck

Viswanadham et al.

Jackson and James

0.239 0.107

0.7 I 0.64 0.5 I 0.4 I 0.31 0.71 0.64 0.51 0.41 0.3 I 0.270 0.215 0.120 0. I17 0.101 0.0875

0.0104 0.0062 I 0.00414 0.00276 0.00207 0.00138 0.000690 0.000345

Collagen 2.70

Ni-Cr alloy metal fibers

0.000143 0.00141 0.00600 0.0215 0.057 I 0.000766 0.00214 0.0084 0.0282 0.077

0.495

I .80 2.08

308 500

3000 I330 0 1

Collagen

Hyaluronic acid polymer

Water 1.5x 10 - h

Water

2.5X10 ’

Alcohol

1.55x10 1.45XlO 1.15x10

1.0x10

Water

Water 4.9x10 ’

10 * NA

10

NA

8

NQ N A

10 - ” NA

NA = not applicable. NQ = not quoted

down, and a mixture of glass wool and copper wire. His data are plotted in a form equivalent to the present dimensionless format, and a notable aspect of the graph i s the small scatter-small at least when compared with Figure I . I t is difficult to extract accurate values for each material from his graph, but Davies also presented an empirical correlation which fit the data, namely, k/a’ = 16 +”?( I + 56 +9-’. In Figure 1, we plotted eight points from his correlation covering the solidity range of the media. Davies does not specifically state what the Reynolds numbers were and they cannot be calculated, but fiber diameters were very small (0.8 to 40 pm) and he warns the reader against employing his correlation when the Reynolds number is greater than I . Hence, it is most likely that his results are well within the acceptable Reynolds number range.

A few years later, Chen ( 1955) gathered data from various experiments on filter mats (in which the fibers were randomly oriented in planes normal to the flow) and presented the collected results as drag coefficient versus Reynolds number. Chen’s graphs are small and logarithmic and thus some accuracy has been lost in transferring his results to Figure I .

lngmanson et al. (1959) measured the flow of air through glass, nylon and paper fibers which were compressed as the pressure drop across the medium increased. They determined Kozeny constants for all materials but diameters were given only for the nylon and glass fibers.

A polymer gel or concentrated polymer solution can also be considered a fibrous medium because the polymer chains form a matrix through which the solvent flows. Since the

Reynolds number is very low, the pressure drop should be proportional to flow rate, and White ( 1960) found this to bc the case for flows of distilled water through various concen- trations of cross-linked acrylamide polymer gels. White estimated that cross-linking affected the permeability, but the effect was small and within the reported repeatability of the data. We converted his data to values of k/u’ and + using a density of 1.43 gm/cm’ for acrylamide (Weiss and Sil- berberg, 1976) and a polymer diameter of 6.5 A estimated from the atomic structure of the chain (Weiss and Sil- berberg, 1977). Inclusion of these data (and subsequent data for hyaluronic acid solutions) assumes that continuum equations apply at this scale. It is not known when the continuum model becomes invalid, but since the chain diameter is five times the size of the solvent molecule, it is not unreasonable to use continuum relations in this case.

Some further data on filter mats were provided by Wheat (1963) and Labrecque (1968). In Wheat’s experiments, slip flow was generally prevalent, but for the two runs where it was negligible, we used the fiber diameters, flow rates and pressure drops quoted by Wheat to calculate non-dimensional permeability. Labrecque was concerned with the effect of fiber cross-sectional shape and by experiments showed that this factor was not important when the aspect ratio of the cross-section was less than 3: I . For larger ratios, however, with the flat side facing the flow, the permeability was significantly lower. His data for circular fibers are included in Figure I .

Kirsch and Fuchs (1966) appear to be the only in- vestigators other than Bergelin et a]. to provide data on


well-defined media. Kirsch and Fuchs tested different arrangements of rows of parallel fibers normal to the flow, specifically: rectangular arrays, ‘staggered’ arrays in which alternate rows where shifted to make a diamond pattern, and ‘fan’ arrays in which each row was turned through an angle in its own plane. Their results indicate that, if the distance between rows is equal to the distance between fibers, then the permeabilities of the different arrays are the same. If the distance is less, then the fan array has a lower permeability than the comparable rectangular array, and the staggered array has a lower permeability still. This pattern confirms the intuition that the more fully a cylinder is exposed to the flow, the higher is the drag on it. The data of Kirsch and Fuchs selected for Figure 1 are those for square arrays and corresponding staggered arrays. Kirsch and Fuchs also investigated cases of non-uniformity in parallel fiber arrangements, and measured permeabilities up to 100% higher than those for corresponding uniform arrays.

Collagen is a protein which forms fibrous structures found throughout the human body. The earliest measurements of collagen permeability with sufficient information to determine fiber volume fraction were those reported by Stenzel et al. (1971); older data are available but c$ unfortunately cannot be calculated. To determine the permeability, Sten- zel et al. measured the flow rate through the wall of a collagen tube at a single driving pressure. Two tubes were tested with very different porosities. We calculated + of each wall using the given values of collagen density and weight concentration, and determined the permeability using the cylindrical co-ordinate version of oDarcy’s law. The collagen fiber radius was taken to be 15 A, as reported by Viswanadham et al. (1978).

So far, solid volume fractions have not been greater than approximately 0.3, except for some of the data by Sullivan and by Bergelin et al. The materials tested by Kostornov and Shevchuck ( 1977), however, were decidely more dense, as 4 varied from 0.31 to 0.71. Their materials were discs of compressed 20% Cr - 80% Ni alloy fibers, 50 pm in diameter and 3 mm long. Measurements obtained with water, alcohol, glycerin and gases all followed Darcy’s law. Since static pressures were not quoted for the gases, Knudsen numbers could not be determined and it is suspected that slip occurred because the calculated permeabilities are exces- sively high. Hence the data for gases have been excluded. For the glycerin data, discrepancies were found between our calculated values of permeability and those given by the authors. Since neither set agrees with the other data in Figure 1, the glycerin data too have been rejected (Kostor- nov and Shevchuck themselves expressed some concern over the glycerin data because of the significant deviation from expected results). Only their water and alcohol data, therefore, have been included in the table and figure.

Visvanadham et al. (1978) were mentioned earlier re- garding the radius of collagen fibers. Like Stenzel et al., they measured the permeability of collagen tubes, but they tested more tubes and tubes with a much wider variation in solidity. The variation was achieved by controlling exposure of the tubes to ultraviolet radiation during manufacture and by varying the pH of the test fluid. Their measurements of fiber radii were made by X-ray techniques.

Our earlier work on tissue permeability (Jackson and James, 1982) was actually a study of the permeability of entangled polymer chains in solution. The polymer was hyaluronic acid, a linear flexible long-chain biopolymer, and the amounts of polymer in solution were such that the

volume fraction was of order 0.001. When solvent was forced through the chains, the pressure loss was proportional to flow rate, indicating that the entangled chains were not deformed by the flow and acted like a rigid structure.

The data in Figure 1 represent a wide range of materials and the large scatter is probably mostly due to variation in fiber arrangement. Fiber arrangement is actually a generic term, for it encompasses a number of factors. In the present case, the two most important factors and their effects on permeability are thought to be: (i) the fiber alignment. If fibers are orientated across the

flow, the resistance is double that when they are aligned with the flow, as theory in the next section will show. Moreover, even for fibers all normal to the flow, but not parallel as in the fan arrays of Kirsch and Fuchs, the permeability can change significantly. Hence orientation alone can easily lead to differences of 2: 1 in permeability.

(ii) homogeneity. The more uniform the distribution of fibers in a medium, the lower the permeability. It is difficult to make a truly homogeneous fibrous material or even to assess the degree of inhomogeneity in a fab- ricated material. Kirsch and Fuchs’ data provide some insight into this factor and an analysis in the next section shows that non-uniformities can increase the permeability by as much as 50%. Another aspect of this factor is channelling, or inhomogeneity caused by the flow. When an excessive pressure drop is applied to loose fibers or particularly to flow-aligned fibers, the structure can open in spots and create channels of relatively low resistance.

Since each of these factors has a significance influence on permeability, the scatter of data in Figure 1 is under- standable.

A portion of Figure 1 will be used as the background for comparing theories in the next section. This abridged version is restricted to solid volume fractions below 0.3, in keeping with our intention to focus on media which are fibrous in character. We have also removed Sullivan’s data because channelling most likely occurred in his tests; his media were loose bundles of fibers aligned with the flow and the permeabilities are significantly higher than surrounding data. The abridged version, then, contains the most reliable experimental data to which theories on fibrous porous media should be compared.

Theoretical predictions

The body of literature on theoretical work is smaller than that for experiments, but more comparisons will be neces- sary because fiber arrangement is now an identifiable parameter. The general approach taken by theoreticians has been to idealize the porous medium as a matrix of rods and then to solve Stokes equation, or a similar equation, for a particular configuration. The theories can be divided into three categories which depend on the structure of the matrix and on the flow direction: (i) flow parallel to an array of parallel rods, (ii) flow normal to an array of parallel rods, and (iii) flow through three-dimensional arrays. The theories and their predictions are presented in the text and figures according to these groups.

FLOW PARALLEL TO AN ARRAY OF RODS

In much of the theoretical work, the rods are parallel and


130001 100 -

10

1

0.1 I A

- Langmuir-Happel prediction. equalion 1 Drummond B Tahir square array. equation 2 and triangular array equation 3

........ _ _ _ _

Drumrnond B Tahir rectangular army equalmn 5

0.0001 0.001 0.01 0.1 1

Solid Volume Fraction (9)

Figure 2 - A comparison of theories for arrays of parallel rods aligned with the flow.

arranged in a periodic pattern like a square or triangular array. For each array, it is possible to define a ‘unit cell’-a polygon with a rod at the center-such that flow in the cell is the same as that in the array. The problem then reduces to solving Stokes equation in the unit cell, with zero velocity at the rod surface and usually with zero velocity gradient at the edge of the cell. This cell approach can be applied whether the flow is parallel or transverse to the array. There is only one velocity component in the parallel flow case and consequently this situation is treated first.

The earliest solution we found for flow parallel to an array was by Langmuir (1942). Instead of choosing a regular polygon for the shape of the unit cell, he chose a circle; this greatly simplifies the mathematics but dissociates the solution from an actual array. Langmuir specified a condition of zero shear stress at the boundary of the circle, and the non-dimensional form of his solution is:

k I 3 42 a’ 4 4 2 2 - =-(-In 4 .. + 24 . -1 . . . . . . . . .

Happel (1959) found the same result and his work is the better known today, probably because Langmuir’s work appears in a government report.

The first attempt at a non-circular cell was made by Sparrow and Loeffler (1959). They tackled square and triangular arrays and solved Stokes equation by a power series technique. The series constants were determined by re- quiring zero shear stress at discrete points on the boundary. They presented numerical results for a pressure drop parameter in graphical form. From their graphs, we determined values of kla’ and 4.

The only other work in this category has been done re- cently. Drummond and Tahir (1984) used a method of distributed singularities to find the flow field in square, trian-

gular, hexagonal and rectangular arrays. They solved for cylinder drag, which we converted to non-dimensional permeability by applying a force balance on the fiber. For the various polygons, their predictions are:

- k l = - (-In 4 - 1.476 + 24 - T 4? + O(4‘)) . . (2) u2 4 4

for the square array,

k 1 - = - (-In 4 - 1.498 + 24 - u2 4 4

for the equilateral triangular array,

for the hexagonal array, and

_ - - I (-In 4 - 1.130 + 24 - 1.197+* a’ 4 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 5 )

for a two-by-one rectangular array. The similarity of Equations (2) to ( 5 ) prompted Drum-

mond and Tahir to propose the following general equation for flow parallel to an array of rods:

(6) k I 4? a’ 4 4 - = - (-In + + K + 24 - T ) . . . . . . . . . . . .

where K depends on array geometry. The Langmuir-Happel equation, the Sparrow and

Loeffler results, and Equations (21, (3) and (5) of Drum- mond and Tahir have been plotted in Figure 2. The predictions agree well with each other, the only exception being Sparrow and Loeffler’s results for the triangular array. The best fit of a theory to the experimental data, particularly for low volume fractions (4 < 0. I ) , is the Langmuir-Happel equation or equation 6 with K = I . 5 . In general, the theories overpredict experimental permeability, which is to be expected since fibers in a real medium are not aligned with the flow but are randomly oriented or across the flow and therefore offer more resistance.

FLOW PERPENDICULAR TO A N A R R A Y OF RODS

We now consider flow perpendicular to an array. We first present solutions based on the ‘unit cell’ approach intro- duced in the previous section and then follow with ‘swarm’ theories. Happel (1959) provided the first solution for flow normal to an array, again using the circular unit cell and applying zero shear stress at the perimeter. His result, in terms of our parameters, is:

k I 4* - I ) - = - (-In 4 + - . . . . . . . . . . . . . . . . u2 84 @ + I (7)

In the same year, Kuwabara (1959) solved the problem with the same unit cell but specified zero vorticity at the boundary instead of zero shear. He solved for the drag coefficient of the rod, from which the following relation was found:

k 1 - = - (-In 4 - + 24) . . . . . . . . . . . . . . . . . . (8) a? 84 A third solution to this problem appeared in the same

year. Hasimoto ( 1959) used a Fourier series method to solve


\\ -1

1 Fibers normal to the flow

Happel prediclion. equation 7 - Kuwabara. equation 8

Hasmolo. equation 10

_ _ _ _

0.0001 0.001 0.01 0.1 1


Figure 3 - A comparison of theories for arrays of parallel rods perpendicular to the flow.

Stokes equation for a square array, and he too developed an equation for the drag coefficient of the rod. The equivalent dimensionless permeability is:

1.476) . . . . . . . . . . . . . . . . . . . . (9)

Hasimoto also reported he had solved the same problem with elliptic functions and found an almost identical equation:

I _ - . - (-ln 4 . 1.476 + 24 + O(4’)) . . . . . . a* 84 The unit cell approach was also the basis of recent work

by Sangani and Acrivos (1982). They used a least-squares technique similar to that of Sparrow and Loeffler to solve the problem of normal flow through square and hexagonal arrays. They presented numerical results for the drag coefficient and checked the accuracy by showing that the results agreed with values from asymptotic solutions. That is, at high solidities the results were asymptotic to predictions based on lubrication theory, and at low solidities the results were asymptotic to an analytical solution which they developed. The equations from their analytical solution are:

I -- - -(-In 4 - 1.476 + 24 - 1.7744* a* 84

+ 4.0764’ + 0(44)) . . . . . . . . . . . . . . . . . . . ( 1 I) for the square array, and

k 1 - = -(-In 4 - 1.490 + 24 - 4’/2 a2 8 4

+ 0(4~)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 12)

for the hexagonal array.

Drummond and Tahir, referred to in the preceding section, used their method of distributed singularities to treat the problem of flow normal to a square array. Their solution is:

k 1 - = - (-ln + - 1.476 + 24 - 1.7744’ a’ 8 4

The results of Kuwabara, Hasimoto, Sangani & Acrivos and Drummond & Tahir agree so closely that they are presented as a single curve in Figure 3. It is not surprising that Happel’s solution does not agree with the exact solutions because it is based on the inexact circular unit cell; at the same time, one would not expect Kuwabara’s solution to be so accurate, because it is based on the same cell. It should also be pointed out that the permeability of these arrays in cross flow is almost exactly half that when the arrays are parallel to the flow. This is seen most clearly when Equations (8) to (13) of this section are compared with Equations (1) to (5) of the previous section. This ratio of one half is the same as the well known ratio for parallel and perpendicular drag on a long rod in uniform flow (Cox, 1970).

The effect of fiber cross-section was studied theoretically by Epstein and Masliyah (1972). They placed elliptical fibers in the circular unit cell and numerically solved the problem of normal flow. For fibers having an axis ratio of 5 : 1, they demonstrated that (i) with the major axis aligned with the flow, the permeability is about 80% higher than for circular fibers, and (ii) when the minor axis is aligned with the flow, the permeability is roughly 75% below circular fiber values. The above results are for 4 > 0.1. For lower values of 4, further results indicate that shape becomes less important. In mats made of non-circular fibers, packing would cause the major axis to be across the flow and hence the results of Epstein and Masliyah are consistant with Labrecque’s experimental findings, described in the previous section.

The theories described so far are for uniform media, and it is apparent that non-uniformity will cause a medium to be more permeable. Yu and Soong (1975) specifically ad- dressed this issue of inhomogeneity and devised the following way of estimating the effect. Considering the medium to consist of parallel fibers normal to the flow, they divded the medium into equal-sized compartments. The fibers were randomly distributed to the compartments, with the conditions that each compartment contain at least one fiber and that the arrangement in each compartment be homogeneous. The permeability of each compartment was calculated by Kuwabara’s or Happel’s formula, and the overall resistance was found by summing the constituent resistances, weighted according to probability theory. The result depends on the number of compartments, and so this number was varied to maximize the permeability, i.e. to find the maximum effect of non-homogeneity according to this model. The results presented in Figure 3 are based on Kuwabara’s equation and show that inhomogeneity can increase the permeability by as much as 50%. This finding is consistent with the experiments on non-uniform media by Kirsch and Fuchs, described earlier.

A completely different analytical approach was put forward by Spielman and Goren (1968). Their idea was that flow through a fibrous porous medium can be modelled by the flow around a single circular cylinder surrounded by an infinite homogenous porous medium. To solve the

+ 0(+3)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 64, JUNE 1986 37 I

10,000

1,000 - N m , Y, x c 0 - - g 100 E a, a In u) a, C

In c

- 0 10

._ E n

1

0.1

1 Three-dimensional fiber networks

a - A

spielman a ~ o r m equat~on 15 Jackson 8 James equation 16

0.0001 0.001 0.01 0.1 1


Figure 4 - A comparison of three-dimensional models for fibrous porous media.

single cylinder problem, they used the Brinkman-Debye- Bueche (BDB) equation:

v p = p i 7 - 14V k

which is the superposition of Stokes equation and Darcy’s law. The velocity was required to vanish at the cylinder and to match the superficial velocity at infinity. Having avoided Stokes’ paradox because the Darcy term is dominant at large distances, they found the flow field and drag on an infinitely long cylinder. From the drag, the equivalent permeability was determined and this was taken to be the permeability k in the governing equation. Hence an implicit relation was found for k:

where K,, and Ki are modified Bessel functions of the second kind. This equation is plotted in Figure 3 and the curve lies about 25% above the accurate collective curve.

This novel approach of Spielman and Goren-now generally referred to as ‘swarm’ theory-led to a number of works which attempted to improve the concept. Howells (1974) extended the analysis by including a second fiber at random locations and accounting for its influence. His results agree with those of Spielman and Goren for + 5 0.3, and give lower permeabilities at higher +. Neale and Masliyah (1975) and Guzy et al. (1983) supposed that there is a fluid annulus around the single cylinder, inside of which Stokes’ equation applies and outside of which either Darcy’s equation or the BDB equation applies. This approach is, in effect, a combination of the circular cell model and the

two-zone concept of Spielman and Goren. Matching conditions at the edge of the envelope depend on which equation is specified for the outer region, but the resulting two predictions differ little for 4 < 0. I , as Guzy et al. show. If the two predictions were plotted in Figure 3, the curves would fall between the Happel curve and the collective curve (Kuwabara, etc.). The main point is that these refinements to a model flow do lead to results which are closer to the accurate solutions, i.e., to those of Hasimoto, Sangani & Acrivos and Drummond & Tahir.

FLOW THROUGH THREE-DIMENSIONAL ARRAYS

Spielman and Goren used the same technique to estimate the permeability of rods randomly oriented in all three directions. They solved the problem of a rod oblique to the superficial velocity by allowing the permeability to have different values in the directions parallel and perpendicular to the flow. Then, by averaging over all rod directions, they determined the permeability of this three-dimensional medium. Using the same notation as before, this result is

The only other work on three-dimensional arrays appears to be our own development of a cubical lattice model (Jack- son and James, 1982). In that work, it was argued that the permeability of a random medium like steel wool is equivalent to the permeability of a cubical lattice formed of the same material. The flow resistance of the lattice was estimated by adding the resistance of rods across the flow to that of rods aligned with the flow. The individual resistances were calculated from Happel’s relations and an overall permeability was predicted. Now that Drummond and Tahir’s more accurate equations are available, we now present a revised prediction for k/a’ , based on their Equations (2) and ( 1 I ) for square arrays:

3 - = - (-In + - 0.931 + O(ln + ) - I ) . . . . . . . a* 204

This equation and Spielman & Goren’s result (Equation 15) have been plotted in Figure 4. It is apparent that the two models agree reasonably well with each other and with the experimental data.

Conclusions

The conclusions of this work are best stated with refer- ence to the figures: (i) Figure 1. The graph shows that permeability data for

diverse fibrous materials are correlated reasonably well when the co-ordinates k /a2 and 4 are used. The scatter is caused by inhomogeneity and by variations in fiber shape and arrangement. For example, if the graph could be restricted to media with uniformly distributed fibers, or to media where the fibers had a single orientation, then the scatter would be reduced considerably. Even with the scatter, though, the graph is useful for order of magnitude estimates of flow resistance when flow data are not available for a porous material but when fiber size and volume fraction are known.

(ii) Figure 2. For arrays of parallel fibers aligned with the flow, predictions by all analytical techniques agree

372 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 64, JUNE 1986

when the array configuration is fixed. Furthermore, different configurations produce permeabilities which vary by no more than 20%. Hence, one can predict the resistance of a flow-aligned array with some accuracy, but such predictions have limited usefulness because actual media rarely consist of fibers aligned in the direction of flow.

(iii) Figure 3. For arrays normal to the flow, the various predictions are virtually identical except for those based on model flows, viz., the unit cell model of Happel and the swam models of Spielman & Goren and others. The weak dependence on array configuration in the exact solutions means that the collective curve represents the minimum permeability for uniform arrays of parallel, circular fibers. Lower permeabilities can be produced only by non-parallel arrangements or by fibers which are highly non-circular. Higher permeabilities are produced when non-circular fibers are on edge to the flow or when the medium is not uniform. While all these factors contribute to the scatter in the experimental data, the main factor is likely non-uniformity.

(iv) Figure 4. Predictions by the pair of three-dimensional models agree reasonably well with the data, but a satis- factory prediction would be expected of any model which realistically combines the flow resistances of rods parallel and perpendicular to the flow.

Acknowledgement

The support of this work by the Natural Sciences and Engi- neering Research Council of Canada is gratefully acknowledged.

Nomenclature

A = cross-sectional area of porous medium a = radius of fiber k = Darcy permeability, as defined by Darcy’s law KO = modified Bessel function of the second kind, of order zero K I = modified Bessel function of the second kind, of order one Kn = Knudsen number L = length over which pressure drop occurs Ap = pressure drop in porous medium V p = pressure gradient Q = volumetric flow rate Re = Reynolds number V = velocity vector

= absolute viscosity y = kinematic viscosity E = porosity, void fraction I$ = solid volume fraction (volume of fibers/total volume)

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