Flow through porous media. Examination of the immobile fluid model

Powder Technology. li (1977) 235 - 252 @ Elsevier Sequoia S-A., Lausanne -Printed

Flow Through

Porous Media.

in the Netherlands

Examination

of the Immobile

Fluid Model

C. C. HARRIS Henry Kntmb (Received

School

November

of Mines. Columbia

University,

New

York. NY

IO027

(U.S.A.)

15, 1976)

SUMMARY A model is examined which postulates that during permeation a fraction of the interstitial fluid may not join in the general flow. If valid, this model could explain differences between data and flow equations in which it is usually assumed that all of the interstitial fluid is mobile. The Kozeny-Carman eq-uation is recast in a form enabling the effective particle volume and effective surface area to be obtained from measurements of flow rates at several different porosities. Both flow through porous media and sedimentation are considered. Good fits to data for aqueous systems are reported. A model in which fissures between protuberances on the particle surface restrain the movement of fluid films is suggested and found to be in reasonable agreement with the results. Related information from other topics involving the interaction of fluid with porous media including capillary potential and streaming potential is introduced. The model is discussed critically and suggestions are made for work designed to further test the model and its implications_

INTRODUCTION

Flow of fluids through porous media is a process of major importance in several areas of technology, including hydrology [ 1,2], soil physics (agriculture) [3], soil mechanics (civil engineering) [4 - 73, petroleum technology [S, 91, chemical engineering [lo - 131 and mineral processing [ld - 161 among others_ It is also the basis of an important method for estimating the surface area of powders [17 - 361. Various modes of flow,

different fluids, and the nature of the porous matrix [37,38] have been studied both esperimentally and theoretically, and the work is well documented [Z, 7,12,14, 39 - 52]_ Because of the wide ramifications of permeability studies and the extent of previous work, it is necessary to state at the outset what the limitations of the present study are. It is concerned with single-phase slow viscous flow (low Reynolds number flow) 1531 through unconsolidated media, specifically non-uniform packings, ideally random, and composed of relatively coarse particles. Some application to sedimenting systems will also be indicated. The “immobile interstitial fluid model” [17,19,22,23,25,28 - 30, 33, 36, 40, 541 argues that the porosity and specific surface area expressions in the widely used Kozeny flow equation should be interpreted in terms of the flow/non-flow interface, and not the liquid/solid interface_ A method will be given for estimating the effective porosity for liquid flow and effective specific surface area from a series of flow measurements; these equations fit data for liquid flow quite closely. X model postulating that liquid is retained at the particle surfaces in fissures between rough protuberances is suggested and equations derived from the model are found to be in fair conformity with the results. The immobile fluid model appears to provide a plausible account of the interstitial flow conditions. Nevertheless, it is infrequently used although the model is not new. It has appeared in a few publications, mainly between 1939 and 1947, which can be classified by application into two groups: (i) a procedure for estimating effective surface area and real density knowing apparent density [22,23,40] ; (ii) an immobile fluid model

236

purporting to account for deviations froln full agreement with flow [ 19, 281 and sedimentation 119, 541 equations, thereby allowing estimation of effective surface area by permeametry [I?, 25,28 - 30,33,353, and the volume of liquid carried with sedimenting particles [ 54]_ These two applications of the model have been treated independently, and one purpose of this paper is to demonstrate the unity of the procedures_ Criticism has been directed at the model 129, 30, 331, notably during a 1947 symposium 129.303, but without significant agreement among the critics. The model will be criticized in this paper too, but on balance the evidence suggests that the model may be appropriate for certain systems, and when properly applied to them it could yield useful information_ X few examples which illustrate the application of the model under a variety of circumstances will be given, and, as far as is possible with the limited data available, tentative conclusions will be drawn. The purpose of this paper is mainly to invite the model and the equations derived from it to general attention, hoping that the model will be examined and tested more widely than hitherto_ Outlines for further work will be provided: these are directed towards the investigation of a broad range of conditions in order to test the theory more stringently and to explore its implications_

Sub-sieve Sizer [Zlj j. and Blaine [ZS] designs of apparatus. The equations can be obtained in several ways. One method is to express a forcebalance relationship 1123 in terms of’the dimensionless groups, resistance coefficient, 0, and modified Reynolds or Blake 1571 number, B, which are then combined in the low Reynolds number flow form_ The equivalent hydraulic radius [12] concept is an important component in all of these treatments. The required dimensionless groups are t = A”e3P/Q’p

(1 -

E)SI

(3)

and B = Qp.!Ap(l - E)S These groups are correlated

(4) in the usual way as

+ = K/B”

(6)

in the procedure known as the Blake-Carman 1561 correlation_ In the low Blake number range (roughly, B < 1), n = 1, and rearranging the variables gives eqn (2). A_ treatment which illustrates the nature of the problems involved in obtaining flow relationships for porous media by analogy with capillary flow now folIows. Only the essential steps are given: more rigorous derivations can be found elsewhere [ 52]_ The Poiseuille [ 58]Hagen [59,61] equation 1621 for laminar tlow without inertial effects through a straight cylindricaI capillary can be written as Q = (r/2)‘(iir’)P/2pL

THE

ROZENY-C_lRbIAX

which is analogous to Darcy’s flow through porous media:

EQUATION

The best-known equation relating the rate of fluid flow through porous media with the parameters of the fluid, media, and pressure gradient under low Reynolds number conditions is the Kozeny [ 551 -Carman [ 563 equation [39 - 41,49 - 521 f Q =

AesP/Ci"p(l - E)~S~Z

Usually the product CT* is replaced referred to as Kozeny’s “constant”, the familiar equation Q = Ae3PfKp(1

-

e)*S=L

(I) by K, giving (2)

Applications of this equation are in the measurement of surface area by permeability 117 - 363 as, for example, with the Lea and Nurse [ 181, Gooden a?d Smith [ 201 (Fisher

equation

for

Q = kAP/pL The variables referring to a cylindrical capillary are now replaced with analogous terms for porous media. There are four steps in the procedure leading to eqn. (I)_ (i) The flow path length for a straight capillary, L, equals the mean tortuous path length through the packing of length 1. Thus, 7 = L/I, and L in eqn. (6) is replaced by Zr- (ii) Equivalent hydraulic radius, m, is defined as (wetted volume)/(wetted surface), or (wetted crosssectional area)/(wetted perimeter). For tubes

it is r/2, and for random packings, e/(1 Thus, (r/2)* in eqn. (6) is replaced by

E)S.

e*/(l- e)*S*. (iii) The mean void area contained in the cross-section

perpendicular

to

23-i

the axis of flow through a random packing is shown by Dupult’s [12,64] relationship to equal AE. It can be shown further that the mean void area perpendicular to the tortuous axis of flow in the interstices is AZ/~, which replaces in? in eqn. (6). Alternatively, the mean velocity tangential to the tortuous a_~is is Qr/ile, which replaces axial velocity, Q/X$, for a straight tube; this refinement of Kozeny’s model due to Carman 140,561 shows that the total contribution of tortuosity in the flow equation is T’, not r [52]_ (iv) The value 2 in the denominator of eqn. (6) is replaced by C. Several further matters concerning viscous flow in tubes can be disposed of now as of minor importance in the present application. These are: inertial effects; fluid compressibility; effect of curvature of the axis of the tube [65] _ In all instances the effects on flow rate are negligible at the low Reynolds number values to be considered here_ Car-man [40] observed the flow path in the interstices of a packing with the aid of coloured tracers. He estimated it to lie at an average angle of about 45 o to the axis of flow through the packing, giving 7 - J/2. Other estimates of r are in the literature [40. 52]_ The Kozeny “constant” takes values in the range about 4 - 6, and 5 is frequently assumed, especially for gas flow_ For K = 5 and T = 42, then C = 2.5; it will be recalled that for a straight circular capillary, C = 2. The factors that are held responsible for the increased value of C for a packing are the increased complesity of the tortuous directions of the flow path, its interconnections, large pore diameter to length ratio, and its non-uniformity of pore size. C assumes the role of a porous media shape factor referred to the circular capillary as the basis for comparison. More detailed treatment of flow relationships which cover their history 12,391 and the conditions for their applicability can be found elsewhere [40,4-i,44 - 52]_

IMMOBILE

FLUID

MODEL

Tke term “immobile” requires some qualification in the present context. It refers to fluid which does not join in the general flow through the porous matrix, although the fluid can circulate within the surface fissures.

Consequently, the term “retained” is perhaps preferable to “immobile”. Both terms win be used here: “immobile” referring to the model; “retained” referring to the state of the fluid. In his 1939 work, Carmau [17] considered the fluid to be “absorbed”. Orr and Dallavalle [33] refer to the fluid as “adhered”_ Discrepancies in the Kozeny equation are usually attributed to inadequacies in the porosity function, ~~(1 -E)“, and empirical corrections to it have been discussed in the literature [25,26, 29, 32, 35, 39]_ Non-uniform packing, especially of fibres 1231, could explain apparent discrepancies 124, 25, 301 in the flow equation when tested against data. Lea [25] first favoured this esplahation, but later [29] changed his viewsAk~o, Keyes 1281 pointed out that tests on his samples showed them to be sensibly free of non-uniformity and he accounted for his results by means of the immobile fluid model. By exercising proper care in esperimentation, non-uniformity can be escluded to a large estent; if discrepancies persist it is necessary to seek alternative esplanations. It is found that h’ depends upon particle shape, orientation, and the porosity of the packing [12,40,41,66] _ If the basic Kozeny model is correct, and if the pore size distribution is homogeneous, two possible reasons can be suggested to account for the variation in K: (i) tortuosity factor, 7, or the shape factor, C, or both, are functions of the parameters describing the porous matrix. especially the porosity, E; (ii) the values assumed for porosity. E. and specific surface, S, may not properly reflect the hydrodynamic conditions within the packing. Only the second possibility will be considered here. Tortuosity and constriction effects have been dealt with recently using packing models [36] _ Other methods of handling tortuosity effects are available, e.g. formation factor measurement [ 52] _ It is usually assumed that the boundaries containing the flow coincide with the solidliquid interface; to identify this case the porosity and specific surface area terms will be given subscripts, s, to refer to the solidliquid interface (E, and S, respectively). From the standpoint of the immobile fluid model this view is too restrictive, since it does not account for regions and pockets of fluid which hydrodynamically must be considered

238

as extensions of the solid surface; in the context of rhe derivation of the Kozeny-Carman equation this can be the only meaningful interpretation of the porosity and surface area terms. Accordingly, in eqn. (2), e and S will refer to the effective porosity and effective specific surface area respectively. Xodel equafrons The following definitions are needed: V is bulk volume of packing; o is effective solid volume; a is effective area; u, is actual solid voiume; and a, is actual surface area. Thus, E = 1 -U/V;E, = 1 - v,/V; S = a/v; and S, = a&,. From eqns. (2) and (‘i), A?= $/K(l

-

E )zsz

(8)

The following procedure is adapted from that of Fowler and Hertel [22] and Sullivar~ and Hertel [23] for dealing with materials of unknown density (see Carman [40] ). The procedure of Powers [19] and Keyes [28] leads to equivalent equations but with different algebraic arrangements; this will be shown later in connection with eqn. (10). Using the porosity and surface area definitions in eqn

[k/(1

-es)]

1’3 = [l/(1

-e,)

-

v/v,] jK113 (alv,)2i3 This equation is analogous to eqn. (9) derived from Fowler and Hertel’s model C22.23, do] _ For handling data expressed in terms of porosity, however. it is preferable to rearrange the equation into what is substantially the algebraic f-Drrn of Powers [ 191 and Keyes 1281 I p/3(1

-

E,)2/3 =

CE, -

(1 -

K=13 (a/~,)“~

us/v)1/ (u_ /II)

(10)

Powers 1191 assumed that the volume of immobile fluid, v - u,, is proportional to solid volume, us, which is equivalent to assuming that v/u, is constant; u/u, is the ratio of effective to real solid volume, and if it is constant it could be a useful parameter of the solid-fluid system_ Other relationships are: S = (a/u,)/(u/u,); the effective porosity is given by E=

1 - (1 - E,)V/U,

(II)

and the fraction of the interstitial containing retained fluid is

volume

(8), k = (1 = (V -

Q = (u/u, -

v/V)s/K(v/V)‘(a/u)’ u)3/KVa’

whence rearranging, (_+V)r’a = (V -

,)fK1’3a”‘3

Iz and V can be obtained

(9)

from a series of permeability tests with the same sample packed to several different bufk volumes. If the relationships are valid, a plot of (kV)lf3 uersus V should result in a straight line of slope 1/K113~213 having intercept -v/K1f3a2’3 at V = 0. The value of ;I can be obtained directly by division, and a can be obtained from a = 3C-“-5 (slope)-1-5 using an appropriate value of K. Equation (9) can be expressed in a more useful form for handling data referring to different quantities of the same particulate solid packed to different bulk volumes. The measurable quantities involved are only k and ls. and it is therefore especiaIIy useful for handling a considerable body of published data. Dividing eqn_ (9) throughout by us, eliminating V with the definition of porosity, and re arranging, gives

I)(1

-

%)/Es = (e, -

e)/e,

(12)

When v/v, = 1, eqns. (10) and (11) reduce to the conventional relationships. Some values of E and q calcuiated from eqns. (11) and (12) are given in Table 1 for ranges of v/v, corresponding to those found in this study. Data given in the literature are frequently quoted in one of three ways: (i) esperimental E, uersus k, from which S, is calculated using eqn. (2) and an assumed value of K, usually 5. Equation (10) can be used for plotting such data to obtain U/V, and S; (ii) experimental E, versus K,, KS having been calculated using eqn_ (2) (with appropriate s subscripts) with a previously measured value of S. Equation (10) can be adapted for handling such data by replacing the permeability-porosity function on the left-hand side, [k(l - es)21 ‘13, with e,IW&] lf3; (iii) experimental E, versns S,, S, having been calculated using eqn_ (2) with an assumed value of K_ usually 5_ Equation (10) can be adapted for handling such data by replacing the left-hand side with e,/[K&] 1’3,

239

TABLE1 Values ofeand

wk

q asa functionoft, and V/V, Es 03

0.3

0.1

0.5

O-6

0.7

OS

o-9

1.05

E 9

0.160 0.200

0.265 0.117

0.370 0.075

0.4'75 0.050

0.580 0.033

0.685 0.021

O-i90 0.013

0.595 0.006

1.10

E 4

0.120 o--zoo

0.230 0.233

0.340 0.150

0.450 0.100

0.560 0_06i

0.6'70 0.013

0.750 0.025

OS90 0.011

1.20

E 9

O.OdO OS00

O-160 0.467

0.280 0.300

0.-100 0.200

0.520 0.133

O-630 O.OS6

O-760 0.050

o_sso 0.032

1.30

E 9

0.090 0.700

0.220 0.150

0.350 0.300

0.180 0.200

0.610 0.1?9

O.i-.lO 0.075

0.033

0.5

0.6

O-7

0.8

0.9

0.7109 0.3627

1.2635 1_0234 ] 0.6191 0.3159

1.1664 0.9417 O.SilS 0.2916

1-10'71 0.8967 0.5425 O-2768

1.06il 0.8614 0.5229 0.2669

1.0355 OS-i12 0.5088 0.2596

1.0169 0.823'7 0.4983 0.25-rs

4.6296 3.7500 2-2685 1.1574

2.2191 l-7975 1 1.0874 j 0.5548

1.6283 1.3189 0.7979 0.4071

1.3717 1.1111 O-6722 0.3429

1.2300 0.9963 O-6027 0.3Oi5

1.1-101 0.9238 0.55ss 0.0551

1.07S9 OS539 0.5237 0.2697

1.0341 OS376 0.5067 0.2565

125.0000 101.2500 61-2500 31.2500

6.5918 5.3394 3.2300 l-6479

2.9155 2.3615 l-4286 [ O-7289

1.1663 0.9447 0.5715 0.2916

1.0697 0.8665 0.5242 0.2674

37.0370 30.0000 18.1481 9.2593

6.0105 4.8685 2.9452 1.5026

1 1.2635

l-1071 0.8967 O-5425 0.276s

C=

1 -(l

--E,)U/v, (eqn. 1l);q = (Es -_)/E,

OS70

(eqn. 12).

TABLE2 Value of h&lh'as a function L*l+

ah,

1.10

1.20

1.30

1.0 o-9 0.7 0.5 1.0 0.9 0.7 0.5 1.0 0.9 0.7 0.5

and u/us

es O-2

1.05

of ~,,a&

1

1

1.9531 l-5820 0.9570 0.4883

1.0 0.9 0.7 0.5

Enclosureindicatesrange

O-1

0.3 1.4509

1 l-1752

KS/K=

1

1.9531 1.5820 ] 0.95'70 [ 0.4883

r

examination

1 1.3084 1.059s 0.6111 0.3271

0.7525 0.3s40

1

1

1.9531 1.5820 0.9570 0.4883

1

1

l-5111 l-2240 0.7405 0.3'778

1.0234 1 0.6191 0.3159

1 2 0.3.

where KS is the assumed value mentioned above. Preliminary

2.9155 2.3615 1.4286 0.7289

1.5362

1 1_31a3

of model

Equation (10) with eS/[KSSs2] “a replacing the left-hand side can be rearranged to give (13) The ratio KS/K dculated from eqn- (13) for several values of the variable eS and of the parameters a/a, and u/u, are given in Table 2;

the ranges chosen for a/a, and u& correspond to those found in the present study. Compilations of data relating KS with E, for flow through particulate packings show that KS/K = 1% 0.1 accounts for much data, while KJK = 1 f 0.3 accounts for most of them: this range is indicated in Table 2 by the enclosures.

Because a variation in E, of 10% is large for any given sample, it is evident from Table 2 that modest changes in a/a, and U/U, can

account for substantial changes in KS/K_ Thus the present model can accommodate even large departures of KS/K from unity. The tabulated data show that as E, decreases, KS/K increases at a rate increasing with u/us. Increasing K, as E, decreases is a common esperimental finding, so that in this regard the model corresponds to reality. However, some of the data reported by Coulson [12,66] for flow through packings of particles of various shapes such as cubes, cylinders and plates show K, and eg decreasing together; for these cases regularities in the packings as the particles stack face against face are probably responsible for departure from the Kozeny model. Inspection of the region of Table 2 for KJK = 1 k O-3 suggests that as E, increases among samples it is likely that u/u, will increase and a/c, decrease; within limits an inverse relationship between a/a, and V/V, would be in conformity with the model. Equations (9) and (10) have been applied by the writer [ 671 to data in the literature which have been quoted [ 401 as supporting evidence for Kozeny’s relationship_ These data show that as E, varies. S,, calculated assuming K is constant, remains sensibly constant_ One typical case for a sample of flaky flint sand [ 403 of size 277 Mm had the following porosity, permeability, and specific surface areas: E, = 45.2 - 54.4%; k = 2.21 X 10-7 - 5.74 x 10-7 c.g.s. units; s, = 519 - 543 cm*/cms, assuming K = 5. For these data the following values were found using eqn. (10): and S = 512 cm2/cm3, assuming Q/V, = l-013, K = 5. A further set of data for spherical particles [ 281 gives u/u, zz l_ These results provide both strong support for Kozeny’s relationship, and evidence that E = es, and S = S,, within experimental error. Despite the fact that the above data, and some other data besides, indicate that K is sensibly constant, the overwhelming evidence is that K is variable. The several considerations presented until now are in accord with the view that if the Kozeny model is basically sound, then it requires some modifications in order to meet all contingencies: this wih be the point of departure in the following discussion. EXPERIMENTAL Permeability measurements performed on packings of crushed solids will be presented

and analysed in order to illustrate and test the immobile fluid model. Capillary potential measurements were also performed, and some details of this work as it relates to the model will be reported later. Choice of system The solid chosen was finely ground coal prepared in a laboratory hammer mill and divided into four size ranges [ 14,431 (Table 3). Owing to its inhomogeneous and complex nature, special problems are associated with the use of coal in these studies, but it was judged that the problems would be balanced by the availability of supplementary information on capillary potential [ 3,14,39,4X, 43, 47,6S - 741 for these particular samples, and the desire to subject the theory to a severe test. Coal substance consists of four lithotypes, each with different hardness. and each breaking with a characteristic shape. Consequently these components are present in different proportions in different size fractions; particle shape measurements [ 75 J showed that shape changed with size in an unsystematic way [ 76]_ Packing measurements showed that porosity increased as particle size decreased, which was attributed to the effects of particie surface roughness and particle shape [ 14,43]_ Ground coal was screened into close size ranges and washed free of adhering fine particles. Density measurements were performed on each samp!e, and to ensure proper values the samples contained in the pycnometer were wetted under vacuum_ The volume of the test sample was calculated from its weight and density. Tables and graphs showing density, porosity and specirlc surface area as functions of particle size have been given elsewhere [ 14, 431; some of this information will be repeated here, as necessary. Permeability measurements Water permeability measurements were carried out on vacuum-wetted samples (it is important to ensure that the state of saturation of the particles in the permeability test corresponds to that in the density determination) of about 100 cm3 volume (u,), which were packed to several different porosities in a liquid permeability apparatus similar to that described by Carman 1401. Details of the e-Yperimental procedure b -1e heen given else-

where [ 14,431. Standard soil mechanics procedures [ 51 were observed_ Large changes in flow rate as a function of time have been reported [ 14,41,43,77 - 791, and this effect was avoided in the present series of measurements by maintaining short flow times with frequent loosening and repacking of the sample between measurements. The ranges of the results obtained are given in Table 3_ Detailed plots of k uersus E,, K versus E,, and the Blake-Carman correlation (eqns_ (3), (4) and (5)) are given elsewhere [ 14, 431. Figure I shows the data plotted according to eqn_ (IO)_ The lines are leastsquare fits, and the calculated parameters are given in Table 3_ For purposes of ill-dstration in subsequent discussion the air permeability measurement of area will be assigned as S,. The straight lines in Fig. 1 show that the ratios v/u, and a/a, remain constant for a given sample packed to different porosities. Thus, the volume of retained liquid, LJ- us, is proportional to the solid volume, u,, irrespective of the interstitial volume_ This assump-

Key: -

- 30+36

BSS

-

-i-2+85

855

-

-120+150

BSS

-

Full swx range

cs

Porosity

(percent1

Fig. 1_ Water permeability data for coal samples. Permeability-porosity function us_ porosity (eqn. data from Harris and Smith [14], IGrris [43]_

10)

242

tion in the model is supported by the results, and it could imply that the retained fluid is an estension of the solid surface into the interstitial volume. A rough trend can be discerned in which as ufv, increases, S/S, or a/a, decreases, which is in accord with expectation_ Discussion wiU be presented nest attempting to account for the results by means of physical models and exploring its implications_

Mean hydraulic radius and surface-volume shape factor The results for the immobiIe fluid model can be compared with those from the conventional Kozeny model by means of mean hydraulic radius. By defmition, m = VeJa hence m,Jm = ae,JaSe and using eqn. (II), m,/m

DISCUSSION

Keyes’ [28] data reinterpreted in the light of eqn. (10) show that v/v, = 0 for spherical particIes, whiIe for sized fractions of pulverized quartz, u/u, - 1.1 over a tenfold change of specific surface area [66]_ Other materials of less simple shape, however, indicated higher and varying values of u/u,. The present resuhs for the three closely sized fractions of coal (Table 3) show small and unsystematic changes in L/V, which are probably due to the inhomogene ous nature of the substance_ In the following, attempts will be made to interpret the data ;~1 terms of the structure of the particle surface. TABLE

= ae,/a, [ 1 -

(1 - E,) vJu,]

The mean hydraulic radius ratios calculated for the data of Table 3 are given in Table 4. Many entries show m, < m, which may be explained as follows_ I he equivalent hydraulic radius term for rough-surfaced pores will be less than that for smooth-surfaced pores because the change in volume due to rough protuberances (numerator) wiiI be small compared with the increase in surface area (denominator) Particle surfaces covered by rough protuberances between which are fissures which retain fluid could therefore account for the resuIts shown in TabIe 4. Some fiIming of the entire particle surface is indicated for the full size range sample. The very low value of m,/m for sedimentation

4

Mean hydra=& Size range

(B.S.S.)

tadius ratio and surface-volume partitle size d (Pm)

shape factor m,/m*

LMean

Value of P calculated

from**

S,(K=5)

S(K=

-30

f 36

458

47.16 50.32

0.843 0.817

2-10

l-18

-72

-t 85

194

51.82 53.43

O-578 0.860

2.17

1.11

-I- 150

115

49-94 56.11

0.973 0.944

2.26

l-71

Full size range

29.60 34.97

l-051 1.029

Quartz sand (sedimentation)

67.0 64.4

0.513 0.463

45.2 54.4

0.999 1.007

2.43 2.40

2.36 2-36

-120

FIaky flint

sand 1401

277

*m&n = aes/as[l - (1 --E~)v/u~] **z = S (cm2/cm3) cl (~rn)l60.000

(eqn_ 14). (eqn lS)_

(14)

5)

243

(see later) could be due to the fine particles being enveloped in the retained film surrounding the coarse particles. The hypothesis that the effective particle diameter remains unchanged while the surface fissures retain fluid, thereby increasing the effective volume of the particle and decreasing its effective area, can be subjected to a further comparative test by means of the surface-volume shape factor, z_ This is defined by S (cm’/cm3)

= 6 X 10” z/d (pm)

(15)

d is mean particle size, usually geometric mean. For spheres, cubes and cylinders of size d, z = 1; this is the smallest value which z may assume. Values of z for some of the data examined here are given in Table 4. The hypothesis that the fissures between protuberances on the particle surface contain retained fluid is supported by the values of z calculated from S; that they are less than the corresponding values calculated from S, is espected, but in two cases they are just a little greater than the value for spheres, cubes and cylinders, suggesting that effective particle shape is tending towards an equidimensional fork. Surface roughness model It is postulated that fissures between rough protuberances on the particle surface isolate fluid films from the general flow, so that the effective particle volume is increased and the effective surface area is decreased from their real values (u 2 u, and a < as, hence S < S,). A simple appro_ximate model based on the concept of surface roughness will be presented nest, and equations derived from the model will be applied to the data contained in Table 3. Figure 2 illustrates the surface roughness model. The average height of the protuberances is h, and their total cross-section area ard perimeter measured in the flow/no-flow surface are c and p, respectively (0 < c < a). From these definitions, area and volume relationships are given by a, - a = ph, and u-u, = (a - c) h, respectively_ Assuming that the protuberances are formed during the comminution process and are not due to preexisting internal voidage, it follows that at the instant of breakage the peaks and valleys on the surface of any par-

Fig. 2. Surface roughness model; section of particle surfaceFlowing liquid and flowlno-flow layers ore shown cut away to reveal surface contours of undetIyin:: strata. Protuberance geometry: average height. h; tota cross-section area, c, perimeter. p. both

measured in fiow[no-flo\v layer. Total particle area, os; total particle area measured in norv:no-now layer, a.

title fit exactly into the contours of the neighbouring particles: consequently, as the protuberances are being formed, c = a - c, on average. But many protuberances will break away from the surface, so that c =Ga/2, and it will be useful to define the surface contour ratio, c/a = f. which is espected to assume values in the range 0 < f < O-5_ From the volume relationship an equation for h can be obtained thus: h = (1 -

vJv)fS(1-

fl

(16)

In order to evaluate h it is necessary to obtain a value for f, which will be considered nest. The protuberances will be assumed to be roughly equidimensional; hence, for cubic or cylindrical shapes, c/p = hf& Different shapes will require a shape factor in the denominator of the right-hand side, but in the present discussion only the simplest case will be considered. Eliminating c from the surface coverage and shape relationships gives ph/a = 4f. Eliminating p and h from the area relationship, an equation for f is obtained: f = (a&

-

= (S,vJSv

1)/4 -

1)/4

071

244 TABLE

5

Surface roughness model Size range (B_S_S_)

f*

-30 + 36 (420 - 500 .um)

0.1245

-72 -+ 85 (180 - 210 pm)

h** (Pm)

t***

11.5

6.7

0.1327

6.5

4.1

-120+150 (105 - 125 pm)

0.04382

1.0

1.1

Full range

0.01122

O-6

o-7

Orm)

*f = (S,c,lSr: - 1)/-Z (eqn. 17)_ **h = (1 - uJu)/S(l -f) (eqn. 16)_ ***t = (vlr; - 1 )/S, (eqn. 18).

This model tacitly assumes that all of the retained fluid is associated with the particle surface. Table 5 gives values off and h calculated from the data in Table 3 for the four coal samples. A less detailed model is given now for purposes of comparison. The volume of fluid occupied by a +&in film of average thickness t (t G h) coating the particle surface is approximately k-z,, and equating this to the difference between the effective and solid volumes gives t = (u -

v&z,

= (u&

-

1)/S*

(18)

Values of t calculated from the data in Table 3 are given in Table 5. Comparing values of h and t and the size ranges to which they refer, both sets appear to be consistent with their respective models and with each

Fig. 3. FiIm thickness vs. mean particle size; coal -pies.

other. Although several good relationships can be obtained among the variables (e.g. h versus lid, Fig. 3), nevertheless the amount of data examined so far is too restricted to allow conclusions to be drawn about the usefulness of the models, and more evaluation is necessary_ In judging between the relative merits of the two models, it is observed that while +-hesurface roughness model utilizes more data than the surface filming model (which does not involve S), and is therefore more comprehensive, it also includes more assumptions in its formulation_ An independent estimate of the value of the surface contour ratio, f, is an essential next stage in future investigations, and in this regard the scanning electron microscope (SE&I_) holds promise as a method of studying surface microtexture. Residual

saturation

in capillary potential

The fraction of the interstitial fluid volume which is retained is given by eqn (12)_ For the four samples studied here, the value of o is given in Table 6. Capillary potential measurements performed on these samples 114,431 showed that the residual saturation values, qO, were as given in Table 6; these values include pendular rings and entrapped pore water besides surface films [72 - 74]_ Using the thin film model (eqn. X3), which will be adequate for dealing with films of about micron thickness, the volume of fluid occupied by a film of average thickness t (t G h) coating the surfaces of the particles

Data from Tables 4 and 5_

TABLE

6

Surface film saturation Size range (B.S.S.)

Residual saturation

Retained interstitial volume fraction

90**

0.183 0.241 0.097 0.071

*q = (u/c.; -

1 )( 1

-

- Es)& by capillary

**Determined

0.207 0.257 0.124 O-090

0.139 0.155 OwliT O-160

-

O_l-11 0.159 O-186 0-16-I

-

(eqn. 13_)_ potential measurements

[ 11.

u,)

By definition, (V - ~,)/a, hydraulic radius, hence q. = t/m,

= m,,

mean (19)

A plot of the residual saturation data for the three narrow size range samples gives a fair straight line having the equation [43] q0 = 0.125

n=s

qo - 0.125

3-13 1.46 5.77 6.6

is approximately kz,_ Saturation, qO, is the ratio of residual fluid volume to interstitial tiolume, and is given by 90 = k/(V

Residual surface film saturation

(cm)

qf -30 + 36 -73 + s5 -120 + 150 Full range

Equivalent hydraulic radius

+ 4.9 X 10-“/m,

(20)

-

3.36 1.52 9-8 7.1

x 10-a x 10-a x lo+ x IO-+

0.011 0.03 0.047 0.035

-

0.016 0.031 O-061 O-039

43]_

Shear plane in streaming potential Classical streaming potential concepts [ 80, Sl] place the shear plane at a distance from the solid surface of up to tens of Angstrom units. In contrast, the present work suggests that liquid is retained to a depth of between 5 X lo3 X and 10” A, depending upon particle size and the degree of surface roughness_ X reconciliation between these estimates is necessary, and one possibility could be that the shear plane concept applies to the peaks of the surface protuberances which together with the boundary of the flow/no-flow surface constitute the effective “particle” surface- It

The constant additive term may be interpreted as indicating a saturation of 12.5% due

would be necessary for the charge density over the peaks to be higher than the mean

to pendular rings at points of contact and entrapped moisture, while the part of the equation corresponding to eqn_ (19) translates as film thickness, Z - 0.5 pm. Gray’s [71] coal data treated in the same way [ 43 ] gave pendular rings and entrapped moisture saturation of about 6X and film thickness t - 1.0 pm. For random packings of smooth equal spheres the residual saturation is about 670, while t + 0: the water is held as rings at the points of contact between the spheres and entrapped in unemptied pores. Comparing the volumes of fluid in the form of surface films given in Table 6, it is seen that Q,, O-125 < q_ This suggests that water which is retained under permeability conditions is not so immobile that it cannot be thinned by slow surf&e capillary flow such as occurs in the low saturation region of capillary potential measurement Mention was made earlier of the context in which the term %nmobiie” was used.

surface charge, and this is in accord with the direct relationship between charge and surface curvature_ Streaming potential measurements are usually performed without simultaneous permeability measurements, and vice versa, and unexpected or anomalous findings are not uncommon in both areas [14,41,43,77 79,82]_ It seems likely that potentially useful information on the nature of the particle surface is being lost by default. One design of apparatus for simultaneous measurement has been described [ 79]_ AppCication determination

to specific

surface

area

Relationships due to Powers [ 191, Keyes [28] and Carman [17.40] ~ derived from immobile fluid models, have been tested by Lea and Nurse [29] using their air permeability method for surface area measurement. They reported mostly curved rather than

246 TAELE

7

Lithophone

1 data (Lea

e

ss

0.5-I - 0.60 O-62 - 0.66 0.68

11,600 11.220 8,810

and Nurse [29] ) UIV,

(cm%) - ll,-rSO - 9,960

l-036 1.556

S, (nitrogen adsorption) 26,000 cm’/g. Lea and Nurse give S by the Keyes [ 3s 1 method

Effective

arealwt (cm%)

surface

Rematks

straight line plot straight tine plot curving

11,050 6,000

as 955 cmz/g_

linear plots, and several “improbably low” results for snecific surface [25, 29]- They also cor.*idered that Keyes’ procedure for de+,mg his relationship was fundamentally preferable to Carman’s. Applications of the model to surface area measurements have recently reappeared in the literature_ Schultz [35] has applied the Fowler and Hertel model to the determination of effective surface area of fine iron ore by a variable pressure gas permeability method (Blaine [ 26]), reporting satisfactory results. He identified aggregation of fine material which would result in nonuniformity of packing as a possible reason for departure from the linear equation_ Some of the data examined by Lea and Nurse [29] have been re-esamined by the writer [67] according to eqn. (lo)_ The largest set of data, designated Lithophone 1, gave results shown in Table 7. The curves reported by Lea and Nurse resolve into a pair of straight lines and a slight curve. If attention is confined to the lower porosity values, the results appear to be in confirmity with the model. Regarding the higher porosity values, according to the present model a substantial volume of retained fluid (air, in this case) is indicated, a possibility which is not immediately acceptable but cannot be dismissed_ In discussion on the Lea and Nurse paper, Carman 1303 criticized the application of the “immobile pore space” model to gas flow and urged caution in applying the model even to water flow systems. He suggested that pore space heterogeneity might account for the observations, a possibility which Keyes and Lea and Nurse had already considered_ Keyes, especially, gave good reasons for rejecting the heterogeneity explanation in favour of the immobile fluid model_ Carman also criticized the Keyes model as being contrary to Kozeny-

Carman derivation_ There is no similar criticism from Carman of the Fowler and Hertel model and related work which he deals with in his testbook 1401, even though the Keyes model and the Fowler and Hertel models are similar. In fact, the Carman and Keyes models are equivalent when account is taken of their different definitions of specific surface area. Orr and Dallavalle 1331 have also criticized the relationship based on the immobile fluid model due to Carman (1939 paper [17] ) and to Keyes [ 281 for calcuIating specific surface area. They comment: “Neither (relationship) gives satisfactory correction for the observed discrepancies between surface area measurements made by the permeametric method and the absolute gas adsorption method.” The immobile fluid model was not, however, proposed merely as a means for correcting these discrepancies, neither is it capable of doing so_ From the physical basis of this model, the effective surface is expected to lie towards the lower extreme of area estimates, in contrast to area determined by the adsorption method, which must lie towards the upper extreme. Because the present model gives S and not S, (usually S C S,), its only meaningful use for surface area measurement would be in parameter determination for application to similar flow systems. By the same reasoning, if the present model is proper, the surface area values obtained from the conventional use of Kozeny’s equation (i.e. with E, and S,) arise through improper interpretation, but they will be useful in similar flow systems which are similarly (mis)interpreted. The need for selecting a method of measurement appropriate to the purpose for which the results are to be used is to be stressed when dealing with non-absolute methods.

Other permeability relationships An empirical equation which fits all the coal permeability data reasonably well is Q = k’A&‘/&I

(21)

where k * = 3-3 and N = 4-95 [ 43]_ Similar equations have been in the literature for many years [39]_ The reason for the good fit probably lies in the two disposable parameters, k‘ and N, rather than in any physical content_ If eqns. (2) and (21) are compared. it is seen that &&(I

-

E,)2 = k-E:

(22)

Over the ranges covered by E, and K,, the right-hand side provides a good empirical fit with the values of k’ and N mentioned above. An equation describing sedimentation velocity [83, 841 involves E:, where N = 4.50 - 4.95, and it is possible that the above esplanation applies in this case too. The Blake-Carman correlation equations (3), (4) and (5) described the data quite well 114, 431, but with n = 1.062 and K = 3.93. By introducing the present interpretation of S and E in place of S, and es, the correlation provides the expected value, n = 1, while the value of K obtained equals the value used in calculating a/u, from the slope of the plot of eqn. (lo)_ Sedimentation and other flow systems In sedimenting systems, aggregates of particles may carry with them substantial volumes of liquid 1541, so that the effective particle size will be larger and the effective particle density less than actual. The present model can be applied to sedimentation of suspensions, provided that the solids concentration is high enough to produce a distinct interface; usually concentration should exceed about 5% A recent application to the determination of mean particle size is reported by Dollimore and 1McBride [SS] _ The resistance coefficient and Reynolds number relationships are, respectively [ 83, 841, 0 = E=(o -

p)g/U2Sp

(23)

and RP = Up /JL(~ -

E)S

(24)

where I$ = KRe-”

(25)

For low Reynolds n = 1, whence

number

U = e3(a -p)g/Kp(l

-

The effective density to the actual density,

of the solid, us, by

il(u -p)

= u,(u,

flow conditions,

l)S2

(26) CJ,is related

--p)

(27)

By the same procedure as before, and noting that the solid ccncentration C, is given by c,

= 1 -E,

(28)

it can be shown IuC,Pl(a,

[L?,/u -C,]

-

that

Pkl 1’3 = /K1’3(a/us)z’3(u,/u)

(29)

This equation has been applied [67] to a set of data for sedimentation in water of quartz sand over the concentration range C, = 0.15 - 0.33. It was found that u/u, = 1.30, and S (K = 5) = 644 cm’icm3, while S, (measured by air permeability) = 1132 cm2/cm3, which corresponds to Fisher Sub-Sieve Sizer particle size of 53 pm_ These figures give f = 0.088 from eqn. (17) and h = 3.9 pm from eqn_ (16)_ In comparison, eqn. (18) gives t = 2.7 pm. The fraction of interstitial water retained at the particle surface, 9, ranged from 5.3 to 14.8%. The same model can be applied to fluidization [12], and eqns- (10) and (29) are appropriate to this case. In topics involving the interaction between interstitial fluids and the particulate surface, as for esample in mass and heat transfer in packed beds [12,13], the thickness of a surface layer is an important rate-determining parameter. If the present model describes the flowing system properly, the conclusions regarding the thickness of the fluid film forming in surface fissures could have important implications in reacting systems involving particulates. Criticism “Immobile” is somewhat inappropriate as a description of the state of the fluid which does not participate in the general flow. Some discussion of this point was provided earlier, where the term “retained” was suggested. Comments by Lea [25] and Nurse [29], Carman [30] and Orr and Dallavalle [33] critical of the immobile fluid model applied

248

to permeametry have been already cited in connection with surface area measurement; these concern special forms of the model and particular applications more than its basic validity_ Curiously, Fowler and Hertel’s [ZZ, 231 form of the model proposed not only for surface area determination but density too appears to have fared better from the critics than Powers’ 1191 form. It is also better known, perhaps because Powers’ work was published in an industry association bulletin rather than in an accepted journal_ Non-uniformity of packing 123 - 25, 28 301 has been suggested as an alternative to the immobile fluid model. Keyes [ZS] examined this possibility and tested his samples for non-uniformity; he did not accept iE as a means of accounting for his data. Also, Lea, who supported the non-uniformity explanation in 1943 1253, had rejected it by 1947 [29] _ It appears that the major failing of the non-uniformity esplanation is that it does not account for systematic flow variation as a function of porosity. On the other hand, it may account for departures from the immobile fluid esplanation [35] _ Also, it is probably applicable to fibre packings [23]_ In the writer’s view, the strongest criticism which can be leveled at the immobile fluid model and the procedure resulting from it can be stated as follows. Equations (9) and (10) wouid fit data quite well whether the model is applicable or not_ By disregarding possible variations in k’ as a function of E, the equation is reduced to a straight-line relationship with two disposable p ammeters. Moreover, the range covered by porosity is necessarily small, so that reasonably good straight lines can be passed through the transformed data points (see Fig. 1). Because of this undiscriminating fitting procedure, the effects of C and 5 as functions of E are unrecognized while the present model is apparently substantiated_ A criticism in similar vein E--,:lst be ieveled at a theory which deals only with the effects of tortuosity and other geometrical factors_ Points favourable to the model are the following. The values of u/u, are in the vicinity of unity. and this intercept term has always been found to be constant and to have the proper negative sign as required by the equation. Also, the relationship between u/v, and a/a, can be given some plausible physical meaning, the results of which lead to reason-

able est’mates of the depth of the retained fluid. These results suggest that those terms are not merely curve-fitting parameters. On the other hand, because the equations derived from the immobile fluid model account so well for the variation in K, procedures must be evolved for describing the effects on the flow equation of tortuosity, geometrical factors and retained fluid, all together_ This survey has not discovered any compelling reason to warran t the comparative neglect of the immobile fluid model since about 1947. Much of the criticism leveled against the model is inconclusive, and some is confused_ Moreover, the unity of the several alternative forms of the model was not generally recognized_ Present evidence appears to support a position more in favour of limited accep’mce of the model than outright rejection_ It is suggested that the application of rhe procedures outlined here to the analysis of more data will lead to z means of resolving the situation_ Suggestions for further work This exploratory investigation has illustrated the application of the model equations to data, and possible interpretations of the results. Although the equations appear to describe data satisfactorily, they are in need of further testing with more data and with more varied data. It is necessary to esamine both the applicability of the equations and the implications of the model. The following is a brief list of variables and modes of testing whose investigations are espected to yield useful information: particle size, shape, and surface roughness; surface area measurement by several different methods for S,; studies on the same packing permeated by liquid and by gas (corrected for slip [29,33,36] if necessary); the same particulate sample tested in sedimentation, fluidization, and flow through stationary packings; simultaneous permeability and streaming potential measurements with electrolytes; use of SE-M. for independent determination of surface contour ratio, f_ Both the immobile fluid model and tortuosity considerations indicate that flow changes as porosity varies, and distinguishing between the two effects remains the outstanding theoretical problem. An experimental approach would be to use two sets of

particles, one with smooth and the other with rough surfaces, but having otherwise similar shape_ -4s a first choice, glass beads with flame polished surfaces on the one hand, and abraded surfaces on the other, appear to be suitable as an experimental material. CONcLUSION (2) The Kozeny-Car-man model is basically valid for low Reynolds number flow through coarse particle packings having simple homogeneous pore geometry. (2) A model assuming that a fraction of the interstitial fluid proportional to the solid volume does not join in the general flow through the packing is esamined as a means of accounting for deviations from the Kozeny equation_ (3) The basic model is in tne literature and it has been used for estimating (a) effective surface area and real density knowing apparent density, (b) effective surface area from a modified permeability equation, and (c) volume of liquid carried with sedimenting particles. Applications (b) and (c) were developed together and independently of (a). Criticism has been directed mainly at (b)_ (4) Linear equations from which effective particIe voIume and effective surface area can be obtained have been derived for both flow through porous media and sedimentation_ (5) Data in the literature are stated in one of several ways, and the model equations are cast into corresponding algebraic forms for convenient application to data. (6) Several sets of data have been examined and found to be closely described by the equations. (7) Equations derived from a model which

assumes that fluid is retained at the particle surface in fissures between rough protuberances are in reasonable conformity with the data. Because the model assumes that liquid occupies mainly re-entrant surfaces, therefore the average particle size remains unchanged while its effective volume is increased and effective surface area is decreased_ (8) Related information from capillary potential investigations on the residual liquid saturation is provided. (9) The film thickness estimated by the immobile fluid model can be several orders of magnitude more than the distance of the

shear plane from the solid as visualized in classical streaming potential concepts- A reconciliation between these estimates is necessary, and it is suggested that simultaneous streaming potential and permeability measurements would provide useful data. (10) The model is discussed critically and suggestions are given for further work- Taking into account and distinguishing between the effects of both retained fluid, and tortuosity and geometrical effects. is a necessary nest step. LIST

OF

SYMBOLS

cross-sectional area of packing perpendicular to direction of flow effective surface area solid surface area Blake, or modified Reynolds number shape factor for flow through packings, C = 2_5_ For a straight capillary of circular cross section, C = 2 concentration of solid in suspension (volume solid/volume suspension) = 1 -E, total cross-sectional area of protuberances averaged in flow/non-flow surface mean particle size (usually geometric) surface contour ratio: fraction of surface covered by rough protuberances (f = c/a) acceleration due to gravity mean depth of retained fluid film occupying volume u - u, between surface protuberances Kozeny’s constant (= CT*); parameter in Blake-Carman correlation Kozeny’s constant calculated using .sS and S, %rcy permeability permeability in an empirical flow equation length of circular capillary in Poiseuille’s equation length of packing in direction of flow equivalent hydra ulic radius (interstitial volume/p: rticle surface area) exponent of E in a permeability equation exponent in Blake-Carman correlation (laminar flow, R = 1; turbulent flow, 72 = 0)

pressure difference total perimeter of protuberances averaged in flow/non-flow surface volumetric flow rate (volume/time) fraction of interstitial volume occupied by retained fluid residual saturation in capillary potential esneriment (residual fluid volume/ interstitial volume) Reynolds number radius of circular capillary in Poisecille’s equation effective specific surface area (effective area/effective solid volume = a/u) specific surface area (area of solid/ solid volume = a,/~,) subscript denoting actual as distinct from effective i.e. actual solid-liquid interface, and also values of parameters calculated assuming usual values of es and S, thickness of retained fluid film coating particle surface descent velocity of suspension/supernatant Iiyuid interface bulk volume of packing effective solid volume (solid volume plus volume of retained fluid) solid volume sieve size (Iimiting) size mtidulus in Rosin-Rammler equation cumulative weight fraction undersize _x surface-volume shape factor (P = 1 for spheres, cubes, and equi-dimensional cylinders) distribution coefficient in RosinRammler equation effective porosity (interstitial flowing volume/bulk volume = (V- u)/V) porosity (interstitial volume/bulk volume = (V- u,)/V) fluid viscosity fluid density effective solid density solid density tortuosity factor (mean length of flow path/length of packing) resistance coefficient

J_ Bear, D_ Zaslavsky and 6. Irmay. Physical Principles of Water Percolation and Seepage, Series: Xrid Zone Research -XXI-X. UNESCO. Paris, 1968. L_ D. Baver. Soil Physics. Wiley. New York. 3rd edn.. 1956.. K. Teraaghi, Theoretical Soil Mechanics, Wiley. New York, 1943. T_ W. Lambe. Soi! Testing for Engineers, Wiley. New York, 1951. K. Teraaghi and R. B. Peck, Soil Mechanics in ‘Jew York, 2nd edn.. Engineering Practice, Wiley, _ 1963 ‘7 Am. Sot. Test. Mater. Spec. Tech. Pub). 163, 2

195-I_

s 9 10 11

12

13 11

15 16

1’7

1s

19

20

21

22

23 REFERENCES 1 0. E. Meinzer, Hydrology, Physics of the Earth

IX. Dover, New York, 1919.

24

AI_ Muscat. Physical Principles of Oil Production, McGraw-Hill. New York, 1919. P. D. Tmsk, Applied Sedimentation, Wiley, New York, 1950. G. D. Dickey, Filtration, Reinhold, New York, 1961. L. E. Brownell, Filtration, Kirk-Othmer Encyciopedia of Chemical Technology. Vol_ 9. Interscience, NewYork, 2nd edn., 1966, pp- 364 - 2S6. J. M. Coulson and J. F_ Richardson, Chemical Engineering, Vol. 2. Unit Operations, Pergamon Press, London, 2nd edn.. 196% J. H. Perry, Chemical Engineers’ Handbook, McGraw-Hill, New York, -4th edn., 1969. C_ C. Harris and H. G. Smith, The moisture retention properties of fine coal: a study by permeabiiity and suction potential methods, 2nd Symp. on Cozl Preparation, Leeds Univ., Ott_ 1957, pp_ 5’7 - ss, 191- 210. E. J. Pryor, Mineral Processing, Elsevier. Amsterdam, 3rd edn., 1965 D_ J. I_ Evans and R. S. Shoemaker, Int. Symp. on Hydrometallurgy. Chicago, Feb. 25 - Mar_ 1. 19’73. American Institute of Mining, Metallurgical and Petroleum Engineers, New York, 1973. F. C. Carman, Determination of specific surface of powders. J. Sot_ Chem. Ind.. 57 (1938) 225 336; 5s (1939) 1 - 7. F. M. Lea and R. W. Nurse, The specific surface of fine powders, J. Sot. Chem. Ind., 58 (1939) 277 - 263. T. C. Powers. The bleeding of Portland cement paste, mortar and concrete treated as a special case of sedimentation, Portland Cam_ Assoc. Bull. 2, 1939. E. L. Gooden and C. M. Smith, Measuring the average particle diameter of powders. Ind. Eng. Chem. Anal. Ed.. 12 (1940) 479 - 482. Fisher Scientific Company, Fisher Subsieve Sizer; Manufacturer’s literature, Fisher Scientific Co., 633 Greenwich Street. New York_ J_ L. Fowler and K. L. Hertel, Flow of gas through porous media, J. Appl. Fhys., 11 (1940) 496 - 502. R. R. Sullivan and K_ L. Hertel. The permeability method for determining specific surface of fibras and powders, Adv. Colloid Sci., 1 (1940) 38 - SO_ F_ J. Rigden, Discussion of report of air-permeability method for determining fineness of cement, Am. Sot. Test. Mater. Bull., 123 (1943) 49.

251 25

26

27

28

29

30

31 32

33

34 35

36

37

38 39

40 41

42 43 44

F- M. Lea, Discussion of report of air-permeability method for determining fineness of cement, Am_ Sot_ Test. Mater. Bull 123 (1943) 50. R. L. Blame, A simplified air permeability fineness ;xp_anE5$rs, Am. Sot. Test. Mater. Bull. 123 (1943)

45

W. Davies and W. J. Rees, The specific surface and grain shape of silica sands used for glass making, Trans. Sot. Glass Technol., 29 (19-15) 279 - 288. WT.F. Keyes, Determination of specific surface by permeability measurements. Ind. Eng. Chem. Anal. Ed., 18 (1946) 33 - 34_ F. M. Lea and R. W. Nurse, Permeability methods of fineness measurement, Symp. on Particle Size Analysis, Feb. 1947, pp_ 47 - 56, Discussion: pp_ il, i8,75.114,118,13_0: Authorsreplyrp. 115, Suppl. to Trans. Inst. Chem. Eng., 25 (1947)_ P. C. Carman. Discussion on Lea and Nurse (ref. [ 29 ] ). Symp. on Particle Size Analysis. Feb. 19-17, pp. 118 - 120, Suppl. to Trans. Inst. Chem. Eng.. 25 (191’7). H. E. Rose, The Measurement of Particle Size in Very Fine Powders, Constable, London. 1953. S. S. Ober and K. J_ Frederick, A study of the Blaine fineness tester and a determination of surface area from air permeability data, Symp. on Particle Size Measurement, 61st Annu. Meet. of ASTM, Boston, June 1958, American Society for Testing Materials, Philadelphia, pp_ 279 - 30L_ C. Orr, Jr. and J. M. Dallavalle, Fine Particle Measurement, Size, Surface and Pore Volume, Macmillan, New York, 1959. T. Allen. Particle Size Measurement, Chapman and Hall, London, 1968 N. F. Schultz, Measurement of surface area by permeametry, Int. J. Miner. Process., 1 (1974) 65 - 79. D. T. Wasan, W. Wnek, R. Davies, bI_ Jackson and B. H- Kaye. Analysis and evaluation of permeability techniques for characterizing fine particles_ Part I, Diffusion and flow through porous media. Part II, Measurement of specific surface area, Powder Technol., 1-I (1976) 209 - 225,229 - 214. E. H. Blum, Statistical geometric approach to the random packing of spheres, Ph.D. Thesis, Princeton Univ., 1961_ W. A. Gray, The Packing of Solid Particles, Chapman and Hall, London, 1968. J. BI. Dallavalle, Micromeritics - the Technology of Fine Particles, Pitmans, New York, 2nd edn., 1948. P. C. Carman, Flow of Gases through Porous Media, Butterwortbs, London, 1956. P. Eisenklam, Porous masses, in H. W. Cremer and T. Davies (Eds.), Chemical Engineering Practice, Vol. 2, Buttenvorths, London, 1956, pp_ 342 463. Crushing and Grinding, a Bibliography, HMSO, London, 1958. C_ C. Harris, Dewatering fine coal, Ph.D. Thesis, Leeds Univ., 1959. A. E- Scheidegger, The Physics of Flow through Porous Media, Macmillan. New York, 2nd edn., 1960.

4i

46

-I8

49 50 51

52 53

54

56

56 57

58

59

60

61 62

63

6-I

65

J_ G. Richardson, Flow through porous media, in V. L. Streeter (Ed.), Handbook of Fluid Dynamics, BIcGraw-Hill, New York, 1961, Sect. 16-1 to 16-112. R_ E. Collins, Flow of Fluids through Porous Xedia, Reinhold. New York. 1961. Am. Sot. Test. Mater. Spec. Tech. Publ. 417, 1966. J_ B. Poole and D. Doyle, Solid-Liquid Separation: -4 Review and Bibliography. HIISO. London. 1966. C. Orr, Particulate Techno!om, Xacmillan, New York, 1966. R. J. BI. Wiest (Ed.), Flow through Porous Media, Academic Press. New York, 1969. Flow through Porous Media, Sixth State of-theArt Symp.. American Chemical Society Publications. Washington, DC, 1950. J. Bear, Dynamics of Fluids in Porous Media. American Elsevier, New York, 1952. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media, Prentice-Hall, Englewood Cliffs, NJ, 1966. H_ H. Steinour, Rate of sedimentation: I. Nonflocculated suspensions of uniform spheres; II. Suspensions of uniform size, angular particles; III. Concentrated, flocculated suspensions of powders, Ind. Eng. Chem., 36 (1944) 618 - 624. S-IO - S-IT. 901 - 907 (also in Portland Cem. _A.ssoc. Res. Lab. Bull. 3, 1944) J. Kozeny, Uber kapillare Leitung des Wassers im Boden. Ber. Xkad. Wiss. Wien, IIa (136) (1927) 271 - 306. P. C. Carman, Fluid flow through granular beds, Trans. Inst. Chem. Eng., 15 (193’7) 150 - 166. F. C. Biake, The resistance of packing to fluid flow. Ttans_ Am. Inst. Chem_ Eng., 1-I (1932) 415. J. 31. L. Poiseuille, Recherches esp&%ment.ales sur le mouvement des liquides dans Ies tubes de t&s petit diametre, Mem. Savants Etrange (Paris), 9 (1816) 433 - 545. Engl. transl. Rheol. hIem_, 1 (Jan. 1940). G. Hagen, On the motion of water in narrow cylindrical tubes, Ann_ Phys. Chem.. -16 (1639) 123 - 142 (in German)_ G. Hagen, On the influence oi temperature on the movement of water through pipes, Abh. Xkad_ Wiss. Berlin, (185-Z) 17. L. Prandtl and 0. G. Tietjex, Applied Hydro and Aeromechanics, hIcGraw-H-11. New York, 1931. F. H. Newman and V_ H. L. Searle, The General Properties of Matter, E. Arnold, London, 5th edn., 1957. H_ P_ G. Darcy. Les Fontaines Publiques de la Ville de Dijon. Exposition et Application I Suivre et des Formules 2 Employer dans les Questions de Distribution d’Eau, Victor Dalamont, Paris.1856_ A-J. E-J. Dupuit.JhudesTh&oriquesetPratiques surIe MouvementdesEZauxdansIesCanaur Dbcouverts et B Travers les Terrains Permeables, Dunod, Paris, 2nd edn., 1863. J. Larrain and C_ F. Bonilla, Theoretical analysis

66

67 68

69

50

71

72

73

i-1

75

of pressure drop in the laminar flow of a fluid in a coiled pipe, Trans_ Sot. Rheol.. 14 (2) (1970) 135 - 14i_ J. 51. Coulsor., The flow of fluids through granular beds; effect of particle sha;.e and voids in streamline flow, Trans. Inst. Chem. Eng.. 27 (1949) 237 - 357. C- C. Harris. Particle technoiogy, lecture notes. Columbia Univ.. 1973_ D. Cronev. _ J. D_ Coleman and P. JI. Bridge. _ . The suction of moisture held in soil and other porous materials. D.S.I.R. Road Res. Tech. Pap. _ 21. . HlrISO, London, 1952_ D_ Payne, A method for the determination of the approximate surface areas of particulate solids, Sature . 172 (1953) 261 - 262. G. Ringqvist. Method for determination of specific surfaces applied to coarse-grained powders, Proc. Swed. Gem. Concr. Res_ Inst.. 38 (1955). V. R_ Gray, The dewatering of fine coal: J_ In&_ Fuel, 31 (19%) 96 - 108; Discussion. 32 (1959) '72 - SO. C. C_ Eiarris and N. R. Bforrow, Pendular moisture in packings of equal spheres, Nature. 203 (1961) ‘706 - 708. S. R_ Morrow and C_ C. Harris, Capillary equilibrium in porous materials, Sot. Pet. Eng. J.. 5 (1965) 15 - 24. i\;. R. Morrow, Phqsics and thermodynamics of capillary action in porous media, Flow Through Porous Media. American Chemical Society Publications, Washington. DC, 19’70, Chap. 6, pp_ 103 - 128. C. C. Harris and G. Stamboltzis, Relationship

76 77

78

i9

80 51 S2

s3

a4

85

between particle size distributions by number and weight. Powder Technol.. 2 (1968169) 58 - 60. C. C. Harris, A method for measuring particle shape factors, Nature, 187 (1960) 101 - 403. P. C. Carman, Some physical aspects of water flow in porous media, Interaction of Water and Porous Materials, Discuss. Faraday Sot., 3 (1918) 7s - 85. E. J. Lynch. Some uncertainties in the measurement of permeability. Am_ Pet. Inst. Prod. Mon., 25 (1961) 2 - 1. E. A. Thiers. C. C. Harris and P. Duby, Permeability and electrokinetic fluctuations in time, 1Olst Xnnu. Meet. Am. Inst. Mining Eng., Feb. 1972. San Francisco, reprinted in AIChE Symp. Ser.. 150 (1975) 118 - 123. J. T. Davies and E_ K. Rideal. Interfacial Phenomena, -4cademic Press. New York. 1961. A. W. -4damson. Physical Chemistry-of Surfaces, Interscience, Xew York, 2nd edn.. 1967. R. D. Kulkarni and P. Somasundaran, The effect of aging on the etectrokinetic properties of quartz in aqueous solutions, Proc. Symp. on OsideElectrolyte Interfaces, Miami. Oct. 8 - 13,1972, Electrochemical Society, 1973, pp_ 31 - 64. C. C. Harris, Correlation of sedimentation rates by dimensionless groups, Nature, 183 (1959) 530 - 531. C. C!. Harris, Sedimentation rates, fluidization, and fIow through porous media, Nature, 184 (1959) 716. D. Dollimore and G. B. McBride, The application of hindered settling methods to suspensions of osysalts, Powder Technol.. 6 (1973).