Comments on fluid-intrusion measurements for determining the pore-size distribution in filter media

Commentson Fluid-Intrusion Measurementsfor Determiningthe Pore-SizeDistribution in Filter Media Peter R Johnston 302 Morningside Dr., Carrboro, NC 27510, USA Pore-size distributions in filter media are determined by fluid-intrusion measurements. But many measurements indicate distributions that differ from what is expected. This paper reviews what sort of distributions are expected in a medium containing a random array of fibers or particles, or in a microporous membrane produced by the solvent-cast method. The paper then goes on to show examples where one type of fluid-intrusion measurement fails to show the expected random distribution, while the second one does, as long as certain conditions are met.

o address the pore-size distribution in a filter medium, we consider a theoretical plane in which a single pore is an irregular-shaped opening, the “diameter” of which is the ratio of the cross-sectional area to the perimeter. The average diameter, p, the arithmetic mean is, of course, a function of the sizes of the solid materials making up the medium and how closely those solids are packed together. Packing density is expressed as 1 -E, where E is the porosity, the ratio of void volume to bulk volume. The spread of the pore-size distribution is expressed through the variance, c?, the square of the standard deviation, 0. More specifically, the spread is expressed as the ratio G/F., the coefficient of variation, cv. Which is to say, any distribution is characterized by p and 0, along with a statement, when possible, of the mathematical description of that distribution. Here we consider a random distribution expressed in three different ways. 1. On considering a plot of the frequencies of pores of different diameters, we refer to that plot as the number distribution of pore diameters. 2. On considering a plot of the cross-sectional areas of the pores (a function of the square of the diameters) versus diameter, we refer to that plot as the cross-sectional area distribution of pore diameters. Further, on considering that our theoretical plane has unit depth, we consider that plot as the volume distribution of pore diameters. 3. Finally, on considering fluid flow through our theoretical plane, we reason that the volumetric flowrate of a fluid through a pore (under a given driving pressure) is a function of the square of the cross-sectional area (from the Hagen-Poiseuille Law). We refer to that distribution as the flow distribution versus pore diameter. Thus, the flow-averaged pore diameter is larger than the volume-averaged diameter, which, in turn, is larger than the number-averaged diameter (as illustrated below). In a filter medium consisting of a random array of solids, we envision that medium as a stack of theoretical planes. When the porosity on one face of the medium is the same as on the other face, we reason that every plane has the same pore-size distribution, but a large pore in one plane does not necessarily occupy the same spot in an adjacent plane. Which is to say, a plane parallel to the path of fluid flow has the same pore-size distribution as a plane perpendicular to the flow path. (Below, we consider the case where the porosity on one face of the medium is smaller than on the other face.)

T

HISTORICAL

BACKGROUND

In 1966, Corte and Lloyd’, contemplated a theoretical plane with a random array of fibers. They considered the distribution of fiber lengths between which other fibers cross, reasoning that the probability of such a length in the x direction decreases exponentially as does such a length in the y direction. Hence, the cross-sectional area of a polygon defined by crossing fibers is the product of the two of these two separate probabilities. From their mathematical reasoning they concluded that the number distribution of pore radii

(or diameters) follows the plotted points in the present Figure 1, through which they drew Line A. That is, the cumulative numbers of pores versus diameter, plotted on log/probability paper comes close to defining a straight line to suggest that the distribution is log-normal, with a geometric standard deviation, gsd, of 2.5. In 1967, Piekaar and Clarenburg2 actually demonstrated the pore-size distribution via a mechanical model. They randomly tossed lines of different slopes onto a plane and measured the areas and the perimeters of the polygons. Reasoning that the diameter of a polygon is the ratio of area to perimeter, they presented data plotted in the form of Line B in Figure 1. They reported a gsd of 1.9. While the nine points through which Line B passes suggest that a curve is a better fit than a straight line, other plots in their paper, with many more points, make that suggestion more convincingly. That is, a true log-normal distribution is not the case. In 1 9833, and 1 98Ej4 the present writer attempted to offer a less complicated mathematical model (than Corte and Lloyd offered), to describe the pore-size distribution demonstrated by Piekaar and Clarenburg, and, more recently, 1 9985, showed that the model is, in fact, a form of the gamma distribution (reviewed below). That is, while Corte and Lloyd’, and later, Dodson and

Pore diameter

units

Figure 1: Data of Corte and Lloydl, Line A, and Piekaar and Clarenburg2, line B, for the random distribution of pore diameters in a plane of randomly arrayed fibres. The points around Line A were deduced from mathematical-probability reasoning. The points around Line B were demonstrated in a mechanical model wherein lines of different slopes were randomly tossed onto a plane. Both teams of investigators expressed the spread of their distributions via the geometric standard deviation, here 2.5 for Line A and 1.9 for line B.

Sampson [6] reasoned that “free-fiber” lengths in the x direction and in the y direction follow gamma distributions, and the pore (polygon) sizes are products of two independent gamma distributions, the present writers offered that pore diameters alone follow a single gamma distribution. From that single gamma distribution two correlations were shown51 1, In mats of random fibers, resistance to fluid flow increases with increased packing density, and a direct correlation was shown between actual measurements and theoretical considerations tied to the single gamma distribution. 2. In discs of sintered particles, formed from different-size particles and different forming pressures, where the areas of the pore walls were measured by microscopic means, a direct correlation was shown between actual measurements and theoretical considerations around the single gamma distribution. THE GAMMA The (single) gamma distribution cy function

DISTRIBUTION can be expressed

as the frequen-

When a = 5, the fluid-flow distribution

of pore diameters

is:

f(X) = +?-flx Plots of these density (frequency) functions are shown in Figure 2, when p = 1. To place the X scale on an actual scale, understand that the average X value of the a = 5 curve is ~$3 = 5, then relate that to the flow-averaged pore diameter, d, deduced from separate measurements of permeability, B, m*, and porosity, 1 [7]. That is

B=z where u = velocity of fluid approaching medium, m/s,

(5) the face of the filter

n = viscosity of the fluid, N s/m*, z = thickness of the medium, m, where X = pore diameter (or radius), arbitrary units; ~1 = the distribution shape factor (illustrated below); 6 = the scale factor, a function of the size of fibers or particles making up the filter medium, along with the packing density; aI6 = p, the average value of X; cJp* = cr*, the variance of X; thus, d/p = IY.-O.~ When 6 = 1, then p = IS*, and the gamma distribution becomes the continuous form of the discrete Poisson distribution, a random distribution with very many examples. But the expression for the Poisson distribution contains no shape factor (other than through understanding what is meant by the mean). Thus, to relate the data of Piekaar and Clarenburg to a gamma distribution, and to the meaning of p, we depend on their data to provide the shape factor. Which is say, when the shape factor, CL,in Equation 1 is 2.0, so that:

f(X)

= XP

then the frequency (number) distribution of pore sizes compares to what Piekaar and Clarenburg found in their mechanical model (as shown below). It thus follows: When M. = 3, the area (or volume) distribution of pore diameters is:

0.41

Figure 2: Plots of Equations 2,3, and 4 for different values of a, when p=1 .o

I

AP = pressure

drop across the two faces of the medium,

N/m*. Older writings 0.987 x 1O-l* m2.

express

permeability

Remembering through a cylinder,

the Hagen-Poiseuille

in darcies,

1 darcy =

Law for laminar

flow

d2 = 32uqz = 32B AP where u = velocity through the cylinder, for a pressure drop, AP along a length z, it follows7 that the flow-averaged pore diameter, d, in a porous medium (with opened pores) can be deduced from d2 = 32~~ &

(6)

where 6 = the tortuosity factor, equal to 11~ in a random array of fibers or particles5z7. For example, if d is found to be 12 micrometers, then, from Equations 1 and 4 p = alp = 12 = 5/p, and we deduce that 6, the scale factor, equals 0.417, while X is expressed in micrometers. Cumulative plots, F(X), of Equations 2, 3 and 4 are displayed in Figure 3, on log/probability paper, to show the approach to log-normal distributions. Specifically, see that the a = 2 curve corresponds to the slope and the curvature suggested by the points in Figure 1 through which Line B has been drawn. Which is to say, the a = 2 distribution corresponds to the number distribution demonstrated by Piekaar and Clarenburg. That is, if, in Figure 3, we draw a straight line through the ~1= 2 curve, we would deduce a gsd of near 1.9, as shown in Figure 1. On the other hand, an cc = 1.5 distribution (not shown in Figure 3) corresponds to the data of Corte and Lloyd, and their gsd of 2.5, in Figure 1. As mentioned above, the O/P ratios surrounding Equation 1 correspond to CL-O.~. Thus, the o/p ratio for the number distribution of pore diameters equals 2-“.5 = 0.707... For the volume distribution the G/P ratio is 3-“.5 = 0.577.., and, for the distribution of fluid flow, the ratio equals 5-“.5 = 0.447..., no matter the value of p in either distribution.

intr=ssure,

Fluid

where P = gas pressure, N/m*, y = surface tension of the liquid, N/m 6 = wetting angle of the liquid r = radius of the capillary, m. Since a pore wall is not a smooth-wall capillary we let cos 0 = 1 .O, so that

P -l/X

1

p=22/_=4r r d

1

2

Pore diameter,X Figure 3: Cumulative plots, F(X), of three distributions in Figure 2, indicating the mean, p, values of X, along the standard deviations, cr. The increasing air pressure, P, to force liquid from a theoretical filter medium, is expressed as I/X, following Equation 8.

r

Cumulative

air pressure,

arbitrary

units

,

,

I’

c1=3,* , , ,

TEST

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.

.

. .

.

. .

.

.

.

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*

..I ..a

.

.

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.

.

.

*::::: *::::.’ *::::. .::::. . . . . . . . . ..I

G5,” I

/

‘.‘.‘.‘.’

THE EXTENDED

/

Figure 4: Plots of drainage curves, wherein air pressures force liquid from porous materials, set against the backdrop of Figure 3. Curves lro and 28 are beds of glass beads; Curve 31°, silt loam; Curve 4 g, a mat of non-woven fibers. RESULTS TO BE EXPECTED FROM FLUID-INTRUSION STUDIES Figure 3 is the frame of reference for this paper. Specifically, consider the upper, horizontal, pressure scale. It corresponds to the reciprocal of the lower, pore-diameter scale. That is, when we consider air pressure forcing liquid from a pore, we envision the analogy of air forcing a liquid from a capillary tube.

p=

where d = pore diameter. That is, we view Figure 3 in terms of the increasing pressure required to reduce the liquid content of the filter medium (represented by the vertical scale), and focus our attention on the a = 5 (flow) curve, or perhaps the CL= 3 (pore-volume) curve. We consider two kinds of fluid-intrusion measurements, 1. In the drainage test increasing air pressure is applied to one face of a liquid-filled medium as the cumulative volume of expelled liquid is related to the increasing air pressure. The increasing air pressure stops short of actually forcing air out of the other face of the medium. 2. In the extended bubble-point fest, increasing air pressure is also applied to a liquid-filled medium, but here we look for that pressure to force the first perceptible flow of air out of the other face, calling that pressure the bubble point. We continue increasing the air pressure until the flow of air approaches the flow seen with the dry medium, as further explained below.

However, drainage-test data, shown in Figure 4, fail to follow the expected curves of Figure 3. It appears that the expelled liquid flows through the large pores by an even greater proportion we would expect in ordinary viscous, straight-through flow. Apparently, the expelled liquid flows in all directions while following the large pores. Then, with a measurable fraction of liquid remaining in the media, at the “elbow” of the curves, a sizeable increase in air pressure does not force out more liquid, obviously held in the small pores. In one case, Haring and Greenkort? offer that the drainage curve follows a beta distribution, a two- parameter distribution where X values are confined to values between 0 and 1. Yet, at the same time, they mention that the pore-size distribution follows a gamma distribution, but do not elaborate on that statement. However, Miller and Tyomking indicate (as Figure 4 shows with Curve 4 ) that they do not see the “elbow” in their drainage curve. Yet their curve does not indicate the shape of the distribution expected from Figure 3. Scheidegger” in discussing drainage curves (and showing examples with the “elbow”), implies that such data do not show the pore-size distribution.

/ /

(8)

THE DRAINAGE

;

I

, or

PerimeterForce = 2ry cose CrossSectionalArea my2

2y

case r

(7)

BUBBLE-POINTTEST

ASTM Method F316 I2 describes this test, performed in three steps: 1. Beginning with a dry medium, gather data to make the plot of airflow rate versus driving pressure (the pressure drop across the medium). 2. Soak the medium with a liquid and then apply a slowly increasing air pressure to one face and record the increasing airflow rate out the other face as the air expels liquid first from the large pores then from the small pores. 3. Compare the two airflow curves, so as to determine the increasing airflow rate with increasing air pressure, and thus learn the flow distribution. One example of such test results 3 is shown in Figure 5. Here, the first measurable airflow from a 0.8-micrometer-rated membrane, wetted with a silicon oil, occurs at a pressure near 2 atm. That airflow amounts to 0.004 (0.4%) of the flow through the dry membrane at the same pressure. In this example, that pressure is referred to as the bubble point.

n=: =l %. 2 P cl-l s i5 rC 2 . 2 2 z E . 2 0 5

_‘.

’

20 10 5

0.02 0.01 Gauge pressure on upstream face, atm Figure 5: Dry and Wet curves, via ASTM Method F 316, for a 0.8 micrometer rated microporous membrane3. A side comment about the bubble point: In the pharmaceutical industry, one looks for lower airflows in deducing the bubble point, typically, an airflow rate 10m6 (or less) of the flow through a dry membranei3. At the other extreme, as discussed below, some bubble points correspond to as much as 10% or more of the flow through a dry membrane. This brings a plea to writers who report bubble points: Please tell us where, on the wet-flow curve, you measured this bubble point. Back to Figure 5: As the pressure reaches 9 atm, the wet airflow rate reaches 0.70 (70%) of the rate through a dry membrane at the same pressure. This expulsion of liquid occurred over an increasing air pressure by a factor of 9/2 or 4.5. Now, look at Figure 3 to see that these Figure 5 data correspond to a curve with a slope laying between the slopes of the Figure 3 curves described by CL = 5 and CL = 3. Further, understand, in a sub-micrometer-rated membrane we expect viscous airflow to be diluted by Knudsen (diffusion, slip) flow “3 14, and such flow is proportional to X3, instead of X4 (corresponding, in Equation 1, to a=4 instead of a=5). Cole15 employed this method in examining an 8.0-micrometerrated felt. He came to the conclusion that the pore-size distribution follows a log-normal distribution with a geometric standard deviation, gsd, of 2.2, instead of an expected gsd near 1.5, illustrated by the a= 5 Curve of Figure 3. In a woven cloth he found a narrower distribution, as expected, reporting a gsd of 1.5. (While Cole, in these tests, employed a volatile alcohol as the wetting fluid, he maintained a pool of that liquid on the top, downstream, face.)

But more recent writers, examining non-woven media, or paper, report pore-size distributions narrower than expected. To illustrate those results, we present dry-versus-wet curves, on linear/linear coordinates, in the manner taught by ASTM Method F316, as illustrated in Figure 6. In Figure 6, the shape of the theoretical wet curve (comparable to the a= 5 curve of Figure 3) indicates that over a pressure rise (from 1 to 5 units) by a factor of 5/i, or 5, that 85% of the liquid is expelled, and thereafter, the wet curve approaches the dry curve asymptotically. However, the two experimental curves (of examples seen in the literature) indicate that such 85% expulsion occurred over a pressure-rise factor of only 4/2 = 2.0, or less. Wet Curve 1 shows the obvious symptom that much of the liquid expelled was the result of evaporation.Those investigators should have employed a liquid with a lower vapor pressure. Or, in employing a volatile liquid, should have presaturated the air with that liquid. Similarly, Wet Curve 2 shows evidence of evaporation, but to a less extent. Moreover, in some cases, writers, in measuring the bubble point, do not perceive a first airflow from the wet medium until the rate reaches a much higher proportion of the dry curve. For example, ASTM Method F 316 teaches us to report the bubble point, and also the pressure (and corresponding pore size) where the wet curve has reached half the airflow rate of the dry curve. In the example of the theoretical wet curve of Figure 6 that 50% flow is reached at a pressure of 3 units, corresponding to a middle-flow pore diameter of l/3 = 0.33 diameter units. If our meaning of bubble point is 1 .O pressure unit, then the “largest” pores have a diameter of l/l = 1 .OO diameter unit. Thus, the diameter ratio of the largest pores, L, to the middle pores, M, is l-/M =1/0.33 = 3.00. Yet, at least one writer (who did not display a Figure 6-kind of plot) has reported an L/M ratio as low as 1.08. This indicates his measurement of the bubble point was not precise enough to measure small airflows. He reported values of M and L for different “micron-rated” nonwoven fibrous media, and, with decreasing values of M, reported decreasing values of UM, instead of an expected constant ratio. Recall, the finer the filter medium, the smaller the airflow rate at a given pressure. That investigator’s airflow meter lacked the precision to measure the very low flow rates associated with fine media. In other situations, investigators start their analyses with the wet medium, then, after “all the liquid has been expelled,” reduce the air pressure so as to trace the flow/pressure curve for the dry medium. This writer has not seen any plots of dry and wet curves developed by this procedure and wonders how those writers knew when the liquid had been expelled. Understand, In the absence of evaporation, the wet curve approaches the dry curve asymptoti-

0.85

Wet medium

Experimental1..

Gauge pressure on upstream face, arbitrary units igure 6: Dry and Wet curves, via ASTM Method F316, plotted as taught by that procedure, showing two different kinds of wet curves against the backdrop of that expected from theory, that is, the a=5 curve of Figure 3.

tally, as illustrated in Figure 6, meaning that to reach the dry curve (How close is close?) the air pressure must be taken to extremely high levels. To what high pressures did those investigators reach before deciding that the medium was dry? In any event, they report finding narrow pore-size distributions. They do not report o/p ratios, but from plots of the frequency distribution (without values on the coordinates) it is apparent that the ~//CL ratios are smaller than expected. FILTER

MEDIA

WITH

GRADED

POROSITIES

Not all filter media have the same porosity on one face as on the other face. For example, in paper, the wood-pulp fibers are more closely packed on the lower “wire” face than on the top face. (And for that reason, as a filter medium, the wire face is employed as the downstream face, so as to gain more capacity at absorbing particles from a feed stream.) In examining such media via ASTM F 316, this writer suggests that the results would correspond to which face was the downstream face in that analysis. RANDOM

VERSUS

NON-RANDOM

FILTER

MEDIA

In the formation of fibrous mats we can conceive of non-random situations. In one case the fibers are almost parallel, and perhaps the pore-size distribution is narrower than in the random case. In another case, fibers are laid down in clumps, and perhaps the resulting mat consists of two separate pore-size distributions, one within the clumps, and one between the clumps. Of course, the description of a partially random distribution becomes complicated. Dodson and Sampson5 approached such a description by reviewing data around sheets of wood-pulp fibers. They considered the o/p ratios reported by previous investigators who examined such sheets via fluid-intrusion studies, and who reported o/p ratios varying from 0.286 to 1.305. As mentioned above, Dodson and Sampson, developed a double-gamma equation for defining pore-size distributions. Their equation (somewhat akin to the present Equation 1, but much more complicated) contains two parameters, one of which can be called the scale factor, and the other the shape factor. For each o/p ratio cited, they list values for the two factors defining such a ratio. Yet, interestingly enough, Dodson and Sampson mention that those previous investigators did not report their raw data. In light of the present paper, the present writer has obvious questions about how the fluid-intrusions studied were performed. And, of course, one must wonder about the randomness of the felt that Colei examined via ASTM Method F 316, where, as mentioned above, he found a pore-size distribution broader than expected from random considerations. CONCLUSIONS The drainage test does not show the pore-size distribution (Figure 4.) The extended bubble point test (ASTM Method F 316) does show the pore-size distribution, as long as: 1) the liquid employed to soak the medium has a very low vapour pressure; or 2) where the liquid is volatile, the air employed to force out the liquid has been presaturated with that liquid (Figures 5 and 6), and, of course; 3) the wetting liquid does not swell the solids. In reporting bubble points, that pressure to perceive the first airflow from the wet medium, investigators owe it to their readers to report how that flowrate compares to flow through a dry medi-

urn at that same pressure, to answer the question: How much of the distribution was embraced in determining the size of the “largest” pore? In performing the extended bubble point test on a medium with a suspected graded porosity, two separate tests should be performed, alternating between which face is placed downstream. NOMENCLATURE cv = coefficient if variation, o/p d = flow-averaged pore size, m, Equation 6 f(X) = the frequency (density) function of X F(X) = the cumulative function of X gsd = geometric standard deviation, illustrated in Figure 1. L = diameter of the “largest” pore, m. M = diameter of the middle (median), flow pore, m. P = air pressure to force liquid from a porous medium, N/m2 X = pore diameter, Equations 1 through 4, arbitrary units a = shape factor in Equation 1. p = scale factor in Equation 1. E= porosity, ratio of void volume to bulk volume p = averaged (arithmetic mean) pore diameter, m. h = surface tension liquid used to flood pores, N/m. o = standard deviation of pore diameters REFERENCES 1. Cone, H L., and E. H. Lloyd: ‘Fluid flow through paper and sheet structure,’ in Francis Bolan (ed), Consolidation of the paper web, pp. 981-1009 (British paper and board maker’s assoc., London, 1966). 2. Piekaar, H. W., and L. A. Clarenburg, ‘Aerosol filters-pore size distribution in fibrous filters,’ 1967, Chemical Engineering Science, 22, pp. 1399-1408. 3. Johnston, Peter R., ‘The most probable pore-size distribution in fluid filter media,’ J. Testing & Evaluation, 1983, 11(2), pp. 117-121. 4. Johnston, Peter R., ‘Fluid filter media: Measuring the average pore size and the pore-size distribution, and correlation with results of filtration tests,’ J. Testing & Evaluation, 1985, 13(4), pp. 308-315. 5. Johnston, Peter R., ‘Revisiting the most probable pore-size distribution in filter media. The gamma distribution,’ Filtration & Separation, 1998, 35, No. 3, pp287-292. 6. Dodson, C. T. J., and W. W. Sampson: ‘The effect of paper formation and grammage on its pore-size distribution,’ J. Pulp and Paper Science, 1966, 22(5), pp. J165-J169. 7. ASTM Method F 902, Average circular-capillary equivalent pore diameter in filter media. 8. Haring, R. E., and R. A. Greenkorn, ‘A statistical model of a porous medium with nonuniform pores,’ AlChE J., 1970, 16, pp.477-483. 9. Miller, Bernard, and liya Tyomkin, ‘An extended range liquid extrusion method for determining pore-size distributions’, Textile Research J.,1986, 56, pp. 35-40. 10. Bear, Jacob, Dynamics of fluids in porous media, p. 448. (1972, American Elsevier, New York) 11. Scheidegger, Adrain E., The physics of flow through porous media, p. 62. (1963, Univ. of Toronto Press). 12. ASTM Method F 316. Pore size characteristics of membrane filters by bubble point and mean [middle] flow pore test. 13. Johnston, Peter R., Fluidsterilization by filtration, 2nd ed., p. 22 (1997, lnterphatm Press, Buffalo Grove, Illinois) 14. Carman, P. C., Flow of gases through porous media, Chap. Ill, (1956, Butterworths, London) 15. Cole. Fred W., ‘Filter ratings-an alternative to “black art” ‘, Filtration & Separation, 1975, Jan/Feb, pp. 17-22, 83