N RECEST years t h e understunding of the bnaic mechanism
Of the numerous reasons that might be advanced for t h e failure of industry to exercise a better cont,rol over variable filtration behavior, perhaps the most important has been the lack of suitable methods of det’ecting and measuring variations in prefilt properties as these are influenced by changes in processing condit,ions. If adequate methods were made available, i t is possible t h a t a day-to-day correlation of prefilt properties with prefilt history might furnish information that would eventually lead to better control, and with it a narrowing of the present gap between theory and practice. This paper briefly reviews research and ideas t h a t appear to offer a clue t o t h e interpretation of variable filtration behavior: it also discusses means of quantitatively evaluating t h e factors responsible. The material presented has been gathering dust in the author’s files for nearly a decade; during t h a t time any claim to t h e importance vhich might have accompanied earlier publication has been lost. Nevertheless, i t is felt that certain phases are still capable of contributing t o t h e rapprochement of science and practice, and are &-orthy of being placed on record.
I
of separation by filtration and sedimentation operations has been considerably advanced and unified. l h e essential similarity of the two processes is well illustrated by comparing the Kozenyequation ( 7 ) , 21=
P3
5(l
pg
- p)S* 7&(1 - p )
for linear velocity of fluid flow through a stationary bed of granular solids. with t h e equation of Powers (IO),
for the velocity with ryhich a similar bed of solids Kill fall through a stationary column of fluid. Carman ( 5 ) discussed the application of Equation 1 t o filtration, and Steinour (16) showed that a modified form of Equation 2 is applicable to sedimentation. The application of Equation 1 to filtration is more apparent whenwrittenin the form,
SPECIFIC FILTRATION RESISTANCE
(3)
At a n early date (IS)attention was d r s y n t o the possibility of predicting filtration resistance from a knowledge of cake porosity and of average particle size and shape. This observation was based upon experiments in &-hich the permeabilities of beds of glass spheres and screened minerals viere correlated by means of:
where the term in brackets is the specific permeability of unit volume of solids, and a p is a specific filtration resistance per unit weight. If it is assumed that the porosity, p , is a function of pressure stress only, i t follows that the average resistance, a,of a filter cake is determined solely by t h e upper limit of pressure stress-that is, the filtration pressure. This provides a thcoretical basis for t h e experimentally observed fact that t,he timcvolume discharge curve of a constant pressure filtration is a portion of a more or less perfect parabola, regardless of the nature of t h e substance filtered. iictually, a short interval of time is required for a bed of solids in compression t o attain a new equilibrium void volume each time the pressure is increased. As a result, t h e average values of specific filtration resistance in various methods of testing diverge t o some extent as filtration pressure increases. This is fairly unimportant, however, since tests a t constant gradients of volume-time, pressure-time, or pressurevolume are easily carried out, if more complete data are desired. The failure of filtration equations and theory t o be more useful in industrial practice arises, not so much from inability t o correlate resistances in various methods of testing more precisely, as from inability of industry to control the properties that determine specific filtration resistance t o a degree sufficient t o make the application of mathematical analysis worth while. Without such control the design and selection of filters are necessarily based largely upon rule-of-thumb estimates intended t o provide for the least favorable conditions likely t o be encountered.
dV _ = - -~28fZR’AZPg de K”’ 7 W
(4)
where f v-as a shape or surface factor defined as the ratio of the surface area of unit volume of spheres passed and retained by a given pair of standard screens, t o t h e surface area of unit volume of irregularly shaped particles passed and retained by the same pair of screens, Average particle diameter was taken as the square root of t h e harmonic mean of the aperture areas of any two screens used t o prepare a given sample. For two successive perfect screens of t h e standard screen scale series, this is a diameter 1.155 times the opening width of the screen on which t h e sample is retained. Equation 4 is a simplified and approximate form of the more accurate expression dV v362j2R2 A2Pg (5) dB (U l/’~5)~ (18s) .
-_
+
obtained (8, fa)by expressing capillary radius and length in the Poiseuille equation in terms of t h e properties of a bed of uniformly sized spheres, in a manner similar t o that employed by Fair and Hatch ( 6 ) . I n this derivation E occurs as the ratio of the average t o the effective cross section of the pores, a quantity to be deter564
shape b y means of permeability tests on solids su6jected to increasing degrees of mechanical compression. Affording d separate quantitative measure of each factor, permeabilitycornpression testing appears to provide a means of correlating day-today and batch-to-bdtrb variations in filtration resist-
static pressure difference. The phenomenon i s somewhat dnd/ogous to the e/ectromotive force of polarization enpicture countered in the operation of electrolytic cells.--The shows a Kelly filter press used by Aluminum Company of America for sepsrating red mud from sodium aluminate liquor.
565
INDUSTRIAL AND ENGINEERING CHEMISTRY
566
mined by experimental tests. The essential similarity of Equation 5 and the Kozeny equation is easily shown by replacing the l / 6 ) z , with the equivalent group, p3,'l - p , notgroup, u 3 6 / ( u ing that the definition of surface factor permits 3/S to be substituted forfR, where Sis the surface of unit volume of solids. Equation 4 results from observing that the square of specific void volume is a n approximately linear function of v36 ( a 1/6)2 over a considerable range of porosity, a wubstitution equiv:ilent to replacing p3jK'(l - p ) in the Iiozeny equation with p2/K"(1- p)2. Using a value of K" = 4K' (exact at a porosity of 50YG),the simplified expression is in error by less than 10' between porosities of 35 and 657,. The author and C. I,. Neyette subjected Equation 5 to a n i'xhaustive series of tests in which the factor 18s was determind. using '*-inch ball bearings and screened glass spheres betn-wn t h e sizes of 28 and 65 mesh (8). The mean value of a large nuniher of such tests was 41.9 = 0.75, a n average deviation of 1.8(7. Utilizing the average value of 18e found in tests on sphcrc's, surface factors were determined for ten different crushed minei,:il. over the size range 28 to 200 mesh and between the maximum and minimum porosity limits obtainable without the application of mechanical pressure. While surface factor generally decreased several per cent as porosity decreased from maximum to minimum values (thereby appearing to confirm a belief that 18s was a variable depending upon porosity), the average values were extremely constant and characteristic of the mineral tested, ranging from 0.86 for galena to 0.47 for magnesite. Considerable interest was aroused by the fact that 41.9 ' I S gives a value for e corresponding closely to the ratio of n to minimum pore area, a result in agreement with the findings of Smith ( I ( , ) in his examination of the data of Green and Ampt. The hypothesis that effective pore area may be identified with minimum pore area permits the prediction of s as a function of porosity. Thus 1 8 e should have a value of 39.9 a t a poro.ity of 47.6% (corresponding t o spheres arranged in cubic packing), increasing to a value of 50 for spheres in rhombohedral packing (porosity 26Yc). For t>heporosity values which can be attnincd in the random packing of spheres (42 down to 38%). the predicted values of 18s are 41.2 and 42.7, respectively. Although no regular variation of this factor with porosity was detectable in tests on spheres, the agreement between the experimental average and the computed value was considered noteworthy. I n order to test the hypothesis and a t the same time verify the validity of Equation 5 at extreme ranges of porosity, permeability tests were carried out with some 8000 '/*-inch steel ball bearings in both cubic and rhombohedral packing arrangements (9). Two carefully machined cells were employed; one provided a channel 1.25 inches square and 8.00 inches long, and the other was 6.00 inches long n-ith a parallelogram-shaped cross section forming angles of 60' and 120'' between sides 1.43 inches in length. E a c h cell ivas tested in a n empty and a fully packed condition, after which corrections for flow through the side-\vall channels and t!orners were determined by tests carried out, upon singlr r o w , single l:q-crs, and multiple layers of spheres. The cell designed for cubic packing was also tested in rhombohedral packing aftt.1. blocking off the exceptionally large passages formed at the two vertical side ivalls by alternate layers of balls. It was because of some doubt as to the validity of the correction for wall flow in t,his ease that rhombohedral packing was M e r tested in the 60' cell. The results obtained were:
+
+
Packing
I Cubic
Rhombohedral in 90" ccll Rhombohedralin 60' ceil
P 0,476 0.26 0.26
1% 39.69
42.2 44.4
Although the accuracy of the experimental work leauinp to these results was unquestioned, some doubt existed as to whether the relatively simple method employed in correcting for side-mall flow was adequate. For this reason the results were considered inconclusive, neither verifying nor disproving the hypothesi3.
Vol. 38, No. 6
They did, however, furnish reason to believe that Equation 5 with a K constant of 41.9 in place of the variable quantity 1& ivas sufficiently valid over wide ranges of porosity. T o compare the preceding value of K with the constant in the ~ be substituted forf2R2in Equation Kozeny equation, ( 3 1 8 )may 5 . This yirlds a value for K' of (41.919) or 4.65, some 7% smiiller than the commonly accepted value of 5 , APPLICATION TO CHEMICAL PREClPlTpTES AND PARTICLES IN SUBSIEVE SIZE RANGE
\Then Equations 4 and 5 jvere applied to chemical precipitates iind finely ground mineral particles sized by sedimentation, the correlation of permeability \\-ith particle size as determined by microscopic measurement presented difficulties in that the value obtained for shape fact,or d s not only smaller than expected, hut in the case of ground minerals became smaller as particle size decreased. Suspicion vas aroused, moreover, by the fact that, n-hen permeabilit,y to the passage of air was compared with permeability to passage of liquid, the value obtained with air was ahvays considerably greater. Thus, a certain calcium carbonate precipitate employed in test filtrations n-as observed to be two to three times more permeable t o air than to water. Samples of silica and magnesite ground for more than 200 hours in a ball millwere found to be six or seven times more permeable to passage of air than to Tvater. Some doubt existed as t,o whether avcrage particle size as computed from microscopic measuremenrs corresponded very closely t o the size which might have b w n ascribed to particle beds had it been possible t o subject them to a standard screen analysis. Nevertheless, \-slues of j' computed from air permeabilities were usually in sufficiently good agreement with values o f f obtained from liquid f l o \ ~permenbilities in the screen size range of particle size t o indicate that fiow of liquids through small capillaries must be subject to a retarding force not encountered with gases, a factor for which Equations 4 and 5 make no allo\vance. Equations 4 and 5 can be made applicable to flow of liquids as well as gases if it is assumed that for liquids a portion of the void volume is not available for flow. Equation 4 then becomes
where vli represents the useless or "dead" volume. Such a hypot,hcsis was regarded as likely since it was corroborated by the behavior of precipitates tested under increasing pressure heads and by filter cakes analyzed for their moisture contents. The porosity variation from front to back of a single filter cake, :IS \vel1 as the variation of average porosity of entire cakes depoiited a t a series of constant pressures, was far too small t o explain the variation in specific resistance with filtration pressure < i n the basis of Equations 4 or 5. On the other hand, it was possihle to explain the observed variation by assuming that lla, rather than Y* alone. was proportional t o (V MEASUREMENT OF UNUSED VOID VOLUME
Thc problem as to R-hether the quantity constituted a r e d or apparent correction was investigated by observing the permeability of a constant weight of filter solids while varying the void volume. This was done by subjecting the solids t o pressure 5 mechanically applied in a small filter, one septum of which was a piston actuated by a load table resembling a large tlcad-weight gage tester. Clear fluid that had long been in cont a r t \\-\-iththe solids t o be tested was conducted to the face of the movahlc head or piston under a constant head of a few feet of fluid: at the same time the solids were subjected to uniform compression at pressures increasing from 0.25 to 70 pounds per square inch. The permeability of the compressed solids was determined by timing the discharge of fluid from the bottom septum into a rnicroburct over short, periods. d dial gage rc,ading t o O'.OOl inch
June, 1946
INDUSTRIAL AND ENGINEERING CHEMISTRY
prrmitted measurement of the descent of the piston and s u b w quent computation of the volume of the cell at each pressure load station. Thc m i g h t and density of the solids were determined h y weighing the compressed cake before and after drying. The data were customarily treated by plotting the square root of the rate of f l o against ~ specific void volume. Such plot!: were straight lines w-hich, when extrapolated to zero rate of flow, cut the void volume axis at values of y o varying with the nature and condition of the solids tested. Values ofjR andfcomputed from Equation 6 by utilizing the uo correction obtained in this manner gave much more reasonable surface factors n-hen correiatcd Jvith average particle size determined by microscopic measurements. PERMEABILITY-COMPRESSION TESTING
APPLICATION TO PREFILT COSTROL. Whatever the cause u i the dead void volume corrertion, it is clearly a factor capable of producing wide fluctuations in specific filtration resistance among prefilts identical as to average particle size and shapr. The simple expedient of decreasing void volume during permeability measurements provides a means of distinguishing the effect of dead volume from that of particle size and at the same time provides a quantitative measure of each. By witing Ikluation 4 in the form,
567
filtrate‘, ant1 thus any cliangc.. in ionic rquilibria are avoidrcl. l'he total volume uf fluid p d tlirougli the cake may be kept quit(' small, Irhich avoids the incrcase in resistance rhnt oftvn occurs whcn clisar fluid is p a s d through x membrane. Tlic following data were obtained in a test carried out on Xovc,mhcr 14, 1933:
Calcium carbonate slurry n-as taken from a large ~IATERIAL. stock supply used over a period of years in test filtrations; it was originally prepared by hot reaction of lime and soda ash, followed by leaching until free of sodium hydroxide: approximate age a t time of test, four years; average density of solids, 2.85 grams/cc.; particle form, mixture of aragonite and calcite; average size of most numerous particle, about 2.6 x 10-4 cm. In four years the specific filtration resistance of this material had decreased to about 75% of the original. This decrcase was believed due to the practice of returning accumulations of dried filt,er cake and slurry samples to the original supply a t the end of a series of tests. ADDITIOSAL DATA. Diameter of permeability-compresion cell, 5.21 cm.; area, 21.30 sq. em. Temperature of fluid passing through compressed solids, 24.2" C.; density assumed to be 1.00 gram/cc.; viscosity asFumed to be that of water, 0.915 centipoise. Weight of wet cake removed from cell, 81.600 grams. Weight of dried cake 45.003 grams; computed volume of solids, 15.79 cc. Compressed volume of wet cake computed from density of water and calcium carbonate, 52.39 cc. 1-olumedecreaseduringtest: 21.30 X 0.3078 X 2.54 = 1 6 . 6 5 ~ ~ . Initial volume of cake at start of test: 52.39 16.65 = 6 9 . 0 4 ~ ~ . Void volume per cc. of solid as a function of piston descent:
+
(69.04 - 15.79) - 21.30 X 2.540 = 3.376 15.79
where. K" has a value of about ( I X 41.9,'9) or 18.6, it provider a measure of the physical properties of prefilt solids in terms of hpecific surface. If it is assumed that y o depends upon some property of the prefilt fluid proportional to specific surface, Y O may IIe replaced by d'S, which affords a measure of this property in terms of an apparent film thickness, d'. AfETHOD O F SECURISG TEST FILTRATIOX DATA. From a theoretical point of view a t least, permeability tests under mcchanical compression provide a n ideal mean? of obtaining tc.t filtration data. This is because a single test carried out under easily controlled conditions supplies the complete equilibrium differential relation betn-een specific filtration resistance, q,, and pressure stress, from xhich the relation b e t w e n average reaistance, a, and filtration pressure can be obtained by integration. To obtain the same information by ordinary testing methods would require a multiple number of tests and the expenditure of much more time and labor. To .secure it with the same accuracy n-ould require a combination of technique, skill, and good fortune that is practically impossible to attain. The same data*might also be used to predict the timc-volumcpressure relations to be expected in other methods of testing (constant gradients of volumetime, pressure-time, and pressurevolume) if equilibrium void volume were attained instantaneously. Actually, increasing pressure during a filtration causes it to proceed as if somewhat less than the predicted resistance were being offered to flow of fluid. The reason becomes obvious when the behavior of irregular-shaped granular solids in compression is studied. Whether the solids are wet or dry, a considerable time is required before sufficient resistance to crushing develops to halt the downward movement of the piston. In the case of a wet chemical precipitate the viscous resistance offered by a fluid while being forced out of a cake by a large pressure load suddenly applied develops a hydrostatic pressure, the initial value of which approaches closely that of the load intensity. Since the sum of hydrostatic pressure and the mechanical stress on the solids must a t all times be equal to the load intensity, it is apparent that some time must elapse before the mechanical stress on the solids can rise t o full value. The difficulties usually encountered when attempts are made to perform constant-weight-of-solids testing in ordinary filter presses arc absent in this method, The cake cannot develop cracks or pull away from the side walls. The fluid used ma)- be
m here
D
- 3.4280
descent of the piston, inches
=
The fluid head producing flow decreased several centimeters during each test as a result of the rise of liquid level in the microburet. The arithmetic mean head effective during each test period is reported in Table I. T R E ~ T X EOFX D T a ~ . i . The square root of the flow rate, in (ec./sec.) per cm. of fluid head, is plotted against specific void volume in Figure 1. The elope of the plot is 0.00486, and the dead void volume intercept at zero rate of flow is 0.92 CC./CC. of solid. AppaPent specific surface is given by rearrangement of Equation 7 : (p
s
-
bgp - 21.30 4 - 2 . 5 5
A
yo)
= v
/
~
~
X 980.6 X 1.00 0.00486 ~ / 18.6 X d 0.00915 ~ X 45.00 ~
= 83,800 sq. cm./cc. of solid
Apparent thickness of film corresponding to dead void volume is: d'
=
VO/S= 0.92/83,800 = 1.098 X lod6cm
/
I
0
I
0.4
Figure 1 .
,
I
I
'
!
08 L2 /6 20 2.4 2.8 3.2 S P E C / F / C VO/D VOLUME, CC / C C
Method OF Determining Dead Void Volume in a Permeability-Compression Test
~
~
INDUSTRIAL AND ENGINEERING CHEMISTRY
S68
Average specific filtration resistance is then given by:
T h e shape factor is:
3 = j = -
SR
3 x 2 - 0.275 83,800 X 0.00026 -
SPECIFIC RESISTANCE.The specific filtration resistance of a mass of granular solids is defined by the equation:
The left-hand side of Equation 10 was integrated numerically in the present instance, employing Simpson’s rule and values of aP read from curve A of Figure 2. Values of a as given by Equation 11 are shonm as curve B in Figure 2, where they are plotted against a filtration pressure equal to the upper limit of pressure stress on the solids.
The resistance of either a differential amount of solids or of a larger amount in uniform compression is distinguished from average resistance by witing a pinstead of a . The nature of CY^ as a function of void volume and particle size .or specific surface becomes evident upon rearrangement of Equations 6 and 7 :
RELATIOXBETWEEN SPECIFIC RESIST.4SCE AND FILTRATIOS PRESSURE.A complete and accurate relation between void
volume and pressure stress cannot, in general, be secured in any single equation simple enough to permit formal integration of Equation 10. However, the relation between l / ( v - YO)* and the square r o o t of pressure stress is so nearly linear above some small critical pressure (usually less than 5 pounds/square inch) that no great error results from assuming that the relation is linear over the entire pressure range. This permits the expression of olP as
If time is ex ressed in hours, conveniently small numerical values of a an?a, result, either with metric or English units. If consistent English units are emplo ed, the dimensions of specific resistance are hr.? X Ib. force,’(&. mass)2, or if the distinction between mass pounds and force pounds is ignored, simply $r.2/lb. Values of a p in English units may be obtained from the data of Table I by introducing t,he proper conversion factors. Utilizing flow rates, (21.30)* X 1.00 = ( d V / d O A H ) X 0.00915 X 45.00 (
453.6 X g 3 m )
=
( 83,800 )* Y
- 0.92
18.6 2.85 X 980.6
( d H )
(m)4536 X g
=
(Y
1635 0.92)2
Values of mP computed from flow rates are listed in the last column of Table I ; as curve A of Figure 2, they are shown plotted against the mechanical pressure stress in pounds per square inch. PREDICTION OF AVERAGEFILTRATION RESISTANCE.During the separation of a finely divided solid from a fluid by filtration, the pressure stress P, responsible for void volume reduction arises from’$he cumulative drag of the fluid on the capillary walls. Since this stress increases in the direction of filtrate flow, a p also increases in the mme direction. The over-all or avera of CY can be obtained by writing Equation 8 for a dil%:z% amount of solids and integrating between the pressure stress limits at the faces of the.cake. I n the absence of any a preciable resistance to fluid flow through filtrate and slurry concfuits, the sum of hydrostatic pressure of the fluid and mechanical pressure stress on the solids is equal to the filtration pressure. For this reason dP, may be substituted for -dl”, giving for the expression to be integrated:
Piston Pressurea, Lb.:Sq. In. 0.41 1.13 1.51 1.89 2.27 3.01 4.92 6 81 8.71 10.60 12.49 16.38 20.17 23.9G 27.75 31.53 36.32 39.11 44,79 52.36
for a wide range of materials of low and medium compressibility. Integration of Equation 10 with alp expressed in this form yields for average filtration resistance
0.0386 =
Using void volumes, aP is given by ~~
Vol. 38, No. 6
If 6 is not too large, this rather cumbersome expression can be approximated with very good accuracy by another equation similar to 12 but with different coefficients. Application of the foregoing generalization to the data of Table 1 results in ap = 230(1
+ 0.02935 dE)
(14)
for the relation between specific resistance and uniform pressure stress. The integrated expression for average a as a function of filtration pressure becomes: f f =
3.377dF 1 - (78.4/d/F‘) log (1 0.02935.\/P’)
+
(15)
Values of a computed from Equation 15 yield a very slightly curved line when plotted against the square root of pressure. A straight line having the equrttion a = 233
(1
+ 0.01736dp)
(16)
gives values agreeingwith those of Equation 15, with a maximum error of about 17’. COMPARISON O F PREDICTED FILTRATION RESISTANCE WITH TE~T FILTRATION DATA. J’alues of CY, as found in a typical series of constant pressure filtrations w o n the same material at an earlier date, are plotted as curve C of Figure 2. Inasmuch as the time-volume discharge data utilized are easily PERJf EABILITY-COMPRESSION TEST TABLE I. TYPICAL accessible (t),they need not be .4v. Fluid Flow R a t e Sp. Void given here. I t will be noted Flaw Time of (Cc./Sec.) per 1 olume. Piston Head, that, although the test filtration Descent, Cm. of Volume, Flow. Cm. Head Cc./Cc. results are well fitted by an Inch Water cc. Seconds x 100 Solid Hr.?jLb. equation of the same form, they 271 14.27 3.35s 0.700 191.1 25.86 0.0052 are 10 to 15% higher. Since 317 3.214 224.2 12,20 0.700 0.0470 25.58 331 3.150 11.65 235,3 25,53 0.700 0.0658 other permeability-compression 354 248.8 3.096 ‘0 0818 25.41 11.06 0.700 tests made upon this same cal352 11.10 108.4 3.081 0.320 0.0860 26,58 cium carbonate slurry yielded 377 2.977 149,6 10.24 0.400 0.1160 26.10 426 2,877 9.06 192.2 0.450 0.1453 25.86 results agreeing well with those 450 2.823 lis,8 8 59 25 90 0.1610 0.400 of Table I, the writer is in474 190.0 8,l6 2.773 0.400 25,83 0.1752 clined to believe that the difer504 202.4 2.720 0,400 7.665 25,78 0.1910 518 2.693 208.7 7,45 0.1988 0,400 25.73 ence is primarily the result of a 556 225.5 6.94 2,630 0.2174 0.400 25.58 continued decrease in compressi597 2.580 242.0 6.47 0.400 0.2320 25.53 bility accompanying the aging 617 2.537 250.7 6.26 0.400 0.2443 25.48 652 5.92 2,500 265.7 0,400 0.2555 26.43 of the precipitate between the 682 278.5 2.468 0.400 5.665 0.2647 25.33 time of the filtration tests and 718 2.432 0.400 294.2 5.38 25.25 0,2750 the permeability-c o m p r es si0 n 736 0,400 303.1 5.245 2.410 0.2813 25.16 776 319.8 4.98 0.2938 0,400 2.368 25.11 tests. 0.3078
25.03
0.400
344.8
4.635
2.320
833
r 0 Half the hydrostatic pressure drop through the cake (0.18 pound/square inch) ha8 been added t o the mechanics1 pressure computed from the sum o f the known weights of the piston, load table, and gage testing weights.
*
COXPARATIYE PERMEABILITY FLOWOF LIQUID AND GAS.
TO
A dry powdered sample of the same carbonate precipitate
June, 1946
INDUSTRIAL AND ENGINEERING CHEMISTRY
( n e i g h t , 9 264 grams) was tamped tightly into a glass tube t o form a column 6 6 em. In length a n d 1.63 sq. cm. in area. The permeabiiity \$-as deteimined a t a constant void v o l u m e of 2 . 3 1 cc. /cc. of sohd n1th both air and water. The s p e c i f i c p e r meability x a s computed in each case, u>ina Eauation 3 . The- pe;mcability Figure 2. Comparison of Specific Filfraobtained n-ith air tion Resistance as Found in a Series of \v a s 3 0 i t i n i e s Constant Pressure Filtrations with the Bel a I g e r t h a 11 the havior Predicted from a Permeabilitvvalue found with rratrr, :I rtitio n-hich Compression Test can be explained if the permeability t o Now of miter is computed using Equation 6 and a “dead” void volumc of 0.993 c c . ‘cc. of solid. DEAD VOID VOLUME CONSIDERED AS ELECTROKINETIC PHENOMENON
I t il: p o G G i b l c to :itlr:uicc a number of explnnatioiis for the appe:w;incc i:i the vu qimntity i n the expresqion for spcrific permeability when particle size becomes miall: .I\mong these are the possibilities (1) it might represent void volume occluded in 1 agglomerates, and thus be physically incapable of sharing in the passage of fluid: (2) it might be due to decrease in particle size by crushing, followed by plugging of the voids with debris; ( 3 ) it might represent the volume of a strongly adsorbed layer of fluid; or (4) i t might be a phenomenon of electrokinetic origin. Although it is possible that all four of these factors contribute t o some estent, it seems unlikely t h a t the first three can be responsible for more than a minor share. Thus, although adsorption undoubtedly plays :I part, it is difficult t o believe that adsorption forces can act through the relatively great distances (up to several hundred molecular diameters) sometimes needed to explain the magnitude of vo. On the other hand, it is quite easy to explain both the magnitude and variable nature of the yo quantity from a n electrokinetic point of view. The idea that a considerable part of the resistance to fluid flow in filtration may be clertrokineric in origin i j not new, having been suggested by the author a number of years ago (If), According t o well established principles. the interface between a solid and a liquid is the seat of a double electrical layer in Tvhich ions of one kind (usually the negative) are adsorbed and tightly held a t tlie salid surface. The removal of oiie kind of ion leaves an excess of the other kind in solution. These arrange themselves i n a diffuse layer in n-hich the excess of concentration of the second kind oyer the first diminishes from a mnsimuni a t the ,mil to z13ro :it :in outrr boii~id:ii~y cstending t o a n indefinite distance, v i , imn tlie .urface. ‘l’lieory provides no measure of 711, hut it dlie.> nffcrd n good illen of the distance d between the writer of gravity of t h e chr: s in tlic cliffuse laycar nnd t h a t of thiJiC‘ :id,SorhJ 011 t!ir > ~ r f . l l i i - i- xivcn 1 ) ~ -the DebyeIIiickel o(1u:i t ion
569
strength increases as the square of ion valence, the value of d is much less for equimolar solutions of multivalent, ions. I n the discussion to follow it vi11 be convenient t o regard 711 a s a distance perhaps three or four times larger than d. I n a capillary tube of radius r (large in comparison to d j , T - m is tlie radius of a n electrically neutral cylinder containing equal numbers of positive and negative ions. It is surrounded by an annular , of two coilcentric cylinders of sheath of thickness ~ n composed thicknesses d and rn - d, each containing equal numbers of the exeesj kind of ion. Because the expression “thickness of the double electrical layer” is ordinarily understood t o rc’fw to distiincr d only, that por,tion of the double electrical 1:ryer lying beti\-ct,n d and v i n-ill be referred t o aq t h e inncsr or diffuse region. MECHANISM OF ELECTRO-OSMOSIS
The electric potcntial bc,tivivn t h e cvhw face and the double elwtricnl I ; \ ~ I , Iithe , zeta potential) gives rise to t h e n.ell known electrnkinc.tic phrnomenn of electro-osmosis and ;trcnminp potcntinl n-hm the w r i docs a c.apillnry t ~ l i e ) .:inti t o c:It:xpli pirtenli:il wlicn tlie sui,fnccb is t h t of more in R surrounding fluid If nn electric potential is applied t o fluid a t the ends of n system of capilhry tubes, ionic migration will take place as in a n electrolytic cell, each ion t h g g i n g Tvith it a n atmosphcre of fluid moleculcs. I n the central core of the tuhe thc effect of equal numbers of countermigrating ions is la tralized, but in the diffuse double layer the excess oi positive over negative ions (assuming it is the negative ion tlmt i.. adsorlxd) n-ill result in a11 unbalanced force on the fluid. This may be regarded a4 equivalent t o a hydroytatic pressure of variable intensity, decreasing from a maximum a t the \ d l t o zero at the inner boundary of the double layer. I n thtl absence of any ordinary hydrostatic pressure difference, the fluid movement initiated hy this electrically generated pressure will be resisted by the force:: of viscous shcar in the same manner as if the pressure w e r e applied mechanically. If a pressure distribution between the wall and the inner boundary m is assumed, the vrlocity profile of the resulting flox may be secured by a simple graphical integration of the basic equation relating preswre, viscosity, and rate of fluid shear. K h e n this is done, velocity incr through the double layer to reach a flat masimum extending from m to the asis of the tube. This makes apparent how the entire body of fluid can be pulled along by a force exerted Tvithin the confines of a n n r r m sheath or skin to produce the familiar phenomenon of electro-osmotic flon-. If electro-osmosis is oppoqed by hydro>t central portion of the velocity profile is depi 013. This impart.. to the profile a chnrncteristic, ~ine-\mvclike appearance, in which velocity in the direction of clertro-osmosis passes through maxima a t points near the i ~ a l h i,~ l i c r eelertroo,qmotic ani1 tiylroztatic pressure intrmities are ( ~ 4 u a l . Integration of iurh a profile yields f o i , the anniilnr reqirrn neiir thc \vall volumc of electro-osmotic flow tn n-liich is op1ioscd a hydloitntically created flon. in t h e core region. JT-liet1it.r tiischarpis of fluid from the capillary will take plncc, in onc direction or t h c other i q seen to tlq)enti upon the rclaiive vc~lurncsnf thca opi)o>inciion..;. ELECTRO-OSMOSIS AND STREAMING POTENTIAL
in tcrm. o f funil:immt:1l con;tniit:. tcmperaturc. nnd ionic ~trencrtlip. For :in>- p i ticu1:ir solvent m d temperature the distance tl tle!wnd. only upon ionic ;trenoth. Tlius for water rnoiar cona t 25‘ C . , (I is z i w n by 3.03 X 10-8 ’t/i. For a centr:itiiin of n univnlcnt electrolyte, it is a length of 1it~:irIy10-6 cm., clecrcasina to lo-‘ em. for R 0.1 .lI solution (4). Since ionic
JT-lien a fluid is mcchnnically Eorc.ixI through a capillnry, a small electromotive force knou-n as t h c >treaminp potrntkil can usu:illy be dctectcd. llntler the influc.nce of thi. wlf-generated potentinl, a smnll elcctro-or.motic:ilIy transported f l o ~ rof liquid \vi11 take place in the same ni:iiuier :IP il the c.m.f. W I P euternnlly applied. Although the retndution of flon- i. sni:ill until :i certnin rritirnl size is reached, its effect czn nevertheless be detectvd. Abr:inison ( I ) shoxved that in a certain case it amountcd t o 10%
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sine-wavelike profiie characteristic of electro-osmotic flox against hydrostatic pressure increases in amplitude; a t the shme time the null point a t which velocity again returns to zero moves out from the wall. Equilibrium is finally attained when sufficient potential difference i: built u p to transport ioiis electro-osmoticnl!y back t o the inlet :It 1111~.sime rate they arc swept from t h c inner portion of the ilecti.ic:tl double layer by hydrostatic flon-. spccrli of electro-osmotic and mechaiiicnl ion itli*rcdequal, equilibrium should occur n.hen t,he null point of a l ~ o l u t e\vlwity attains the tlistnncc d from the nail. This fo!!o\v? from t h e fact that n cylinder nt diitance (1 from the .sill1 divides thc tlouli!c Itlectricnl layrr into c.cincentric. sh(~11s containing equnl numbers of the excess ioni (Figure 3 ) . Therc. is no reaion to wppose that the tn-o spccds are vclii:ii, onable to assume that thc :ic.tual distance 111 wliivh 2 1 J W l U t f 2 velocity of flow changes dirclction must hccir a: close and c,)iiztaiit proportionality t o d as given by the Uc41yc~-FIiicltelq u . i t i o n , provided total thickness 771 of d o u t ~ l r eIrrtri(x1 h y c r is 10. th:in t lie radius of the capillary. APPLICATION TO EXPERIMENTAL OBSERVATIONS
,. 1 !le
Coregoing niecha ni provides a simple qualitntivi: 1’9p1nn:ition for n number well known phenonienn. Thus geuliiFigure 3. Effect of Streaming Potential Generation in Retarding gist5 h:iw lung tiecw awnre that seepage of water through mineral Flow of Liquids through Small Capillaries cleavage p1:ines trikes plnce much more slowly than ivould bc experted on the basis of Poiaeuille’s la^. Terzaghi (16) foulid tli:it T o p Lelt. Capillary wall with ions of one kind (assumed to be the negative) adsorbed in a stationary layer, leaving excess positive ions arranged the apparent viscosity oi m t e r increased abruptly tu many in a diffuse mobile layer extending to distance m from the wall: d , distance times its rioi,m,zlvalue n-hen made to flow between pnrailel planes to center of gravity of double electrical layer. Annular shells of thickness i .jpac.ed l e s tlilin IO-Scm. apart. and m-d thus contain equal numbers of e x c a s positive ions. continued passage of clenr lmter through almost TOP Right. abb”, initially ur.ifurr.1 hydrostatic pressure gradient P H producing downward fiow of fiuid: in the absence of streaming potential, any t y l x of fine-pored membrane is accompanied by a gradual velocity distribution would assume the parabolic profile oho. Curve cc‘, tnrice is 8 matter of common experience and has assumed equilibrium distribution of electro-osmotic pressure intensity, P E , been reported many times in the literature. I n making streaming opposing hydrostatic pressure: caused by upward streaming of fluid under potential measurements, i t is sometimes necessary to wait for the inEuence of potential generated by ejecting excess ions from the capillary outlet. Curve abb’, distribution of net pressure gradient acting after the periods of 15 to 24 hours before the ratio E,!P (developed potenratio E I P has become constant. tial t,o hydrostatic pressure head) becomes constant. Both of Boliom. oefgo, velocitv profile ot~taineil by numerical integration of these phenomena appear to have a rewonable explanation in the equation, fact that growth of the electro-osmotic flow region proceeds from a thickness of zero to an equilibrium value, a t a rate depending upon the extent to which the rate of mechanical transport exceeds taking values of P H - P B from curve abb’. The ratio of P E to P H has been that of elcctro-osmotic transport. If these rates are nearly equal, adjusted by trial and error to yield a profile in which absolute velocity it is conceivable that fairly long periods of time might elapse bepasses through zero a t distaxce d from the wall. Cpward streaming of fluid fore a n equilibrium potential would be reached. The gradual inin shell d then carries positive ions hack toward the inlet at the same rate t h a t ions in shell rn-d are sn-ept to the outlet: thus. electric neutrality of the crease in resistance that simultaneously takes place is explained system and a constant ratio of E I P are preserved. Volume rate of 00w by the fact that, as electro-osmotic velocity increases, the absolute corresponding t o profile of/Do is 37% of t h a t given by profile oho. Since velocity of flow throughout the central region of the capillary is the initial rate of flow will correspond closely tu diitriliution oho, a reduction correspondingly decreased. in flow rate upward of 60% may be exprctrd to take p1:ice duriiig the time E/P builds u p t o a niaximum value. If the foregoing hypothesis is correct, i t would be expected that, upon interrupting floiv of fluid through a capillary, the difference in ion concentration a t the two ends Tvould cause electro-osmosis to continue until the potential disappeared; or if flow were started in a capillary 1 X lo-‘ cm. in radius. increasing rapidly for capillaries of smaller size. again, that the resistance upon resuming flow rrould be less than 9 qualitative explanation uf the phenomenon follm-s: \\71ie~~ it had been nhen flow n’as stopped. Such behavior was actually fluid starts t o flow through a capillary, the velocity diFtrit)ution observed by Bishop, Urban, and TF’hite (Sj who found thnt, whcn the pressure producing floiv of a dilute potassium chloride soluprofile near the wall is at first substantially parabolic in the direction of flow. Since no potential differcnce exists to cause tion through a cellophane membrane was reduced to zero, a migration of ions, fluid movement in the doublr electrim1 layer hnckwartl flow of fluid took place for a short time. is without restraint. This rwults, howcvcr, in disch:irging il porIf the ionic strength of the fluid flon-ing through a blocked tion of the doublc layer f r i ~ mthe r r i d . 6 = density of granular solids, granis/ec. of solid e = ratio of average to effective cross section of pore formed by a unit cell of eight contacting spheres f = surface or shape factor, no units q = fluid viscosity in poises, grams mass/cm. X see. g = local value of gravitational acceleration, cm./sec.* or ftJsec.2 g. = dimensional constant in L-ewton’s law, 980.665 (gram mass X cm.)/(gram force X see.*) or 32.1740 (lb. mass X ft.)/(lb. force X spc.2) AH = fluid head, cm. K , K‘, K”. K”’ = constant. (41.9, 4.63. 18.6, and 168, respectively) L = depth of particle bed, cm. P = hydrostatic pressure difference, grams mass/sq. cm. P’ = h;drost’atic pressure difference, lb. force/sq. Et. p = packing porosity, cc. of void volumc/cc. of total volume p = fluid density, grams/cc. R = average radius of spheres or particles, em. S = specific surface, sq. cm./cc. of solid ti = linear velocity of fluid flow, cm./sec. dT-/dB = volume rate of flow, cc./sec. v = specific void volume, ce./cc. of solid Y O = dead void volume, cc./cc. of solid v = void volume per unit weight of solids, cc./gram IT = weight of particle bed, grams LITERATURE CITED
Abramson, H. A., J . Gen. Physiol., 15, 979 (19821. (2) Badger. W. L.. and McCabe, W. L., “Elements of Chemical Engineering”, 2nd ed., p. 509, New York. hf cGran;Hill Book Co., 1936. (3) Bishop, G. H., Urban, F., and White, H. E., J . P h y s . C‘hem.. 35, 137 (1931). (4) Bull, H. B., “Physical Biochemistry”, p, 1S1, S e w York, John Tiley &Sons. 19-13. ( 5 ) Carman, P. C., Trans. Inst. Chem. Engrs. (London), 16, 168 (1938). air G. M.. and Hatch, L. P., J . Am. F a t e r B‘orks .4Ssoc.. (6) 25, 1551 (1933). (7) Kozeny, J., Sitzber. Akad. Tt’iss. T i e n , .llnth. T L U ~ L L ~ Masse. ~JI. Abt. IIa, 136,271(1927). (8) Meyette, C. L., M. 5 . thesis, Univ. of Minn., 1933. (9) Meyette, C. L., unpublished report, 1935. (10) Powers, T. C . , Research Lab., Portland Cement L4ssoc., Biill. 2 (1939). (11) Ruth, B. F., ISD. E x . CHEM.,27, 808, 812 (1935). (12) Ruth, B. F., Ph.D. thesis, Univ. of Minn., 1931. (13) Ruth, B. F., Montonna, R. E., and.Montillon, G. H., IXD. ENQ. CHEH.,23,850 (1931). (1)
, K., 2. angeu. Math. d
