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Ind. Eng. Chem. Prod. Res. Dev. 1986, 2 5 , 11 1-1 18
possible tank mix be prepared before proceeding to full-size batches. Field testing occurred primarily in the Shenandoah Valley of western Virginia. Figures 8 and 9 illustrate the results obtained on standing ryegrass before and after application of fertilizer containing paraquat. Desiccation of the grass occurred rapidly, and the grass was brown and dying after only a few days. These results are typical of all trials. In this case the fertilizers contained paraquat applied at the rate of 1 q per acre in a 16-1.33-16.4 formulation stabilized with- dispersible xanthan gum a t a concentration of only 2.5 lb/ton. Normal use levels of the two polysaccharide suspending agents range from 2-3 lb/ton vs. 20-60 lb/ton for attapulgite clay. All trials proved the effectiveness of Paraquat when polysaccharides are used as suspending agents in suspension fertilizers.
When used within their compatibility limits, both xanthan gum and S-194 offer viable alternatives to multiple applications of fertilizer and herbicide. Registry No. Xanthan g u m , 11138-66-2;paraquat, 4685-14-7. Literature Cited Burdon, J. et ai. J . Environ. Sci. Health 1977, 812. Sandford, P. A. "Solution Properties of Polysaccharides"; Brant, D. A,, Ed.; American Chemical Society: Washington, DC, 1981; ACS Symp. Ser. No.
150. Summers, L. A. "The Bipyridinium Herbicides"; Academic Press: New York, ,.-,On
IYOU.
Received f o r review September 9, 985 Accepted November 18, 985
This paper was presented at the 188th National Meeting c the American Chemical Society, Philadelphia, PA, 1984.
Filtration on a Sieve of Aqueous Suspensions of Fibers and Solid Particles. 1= Relationship between the Retention of Particles by Fibers and Their Morphological Parameters Mlchel Nardin, Jacques Schultz, and Eugine Paplrer * Centre de Recherches sur la Physico-Chimie des Surfaces Solides, 68200 Mulhouse, France, and Laboratoire de Recherches sur la Physico-Chimie des Interfaces de I'Ecole Nationale Supdrieure de Chimie de Mulhouse, 68093 Mulhouse Cedex, France
A general model of the filtration of aqueous suspensions of glass fibers and solid particles on a sieve is proposed. Unit retention of particles by fibers, Le., the mean number of particles retained by one fiber in the filtration cake, is simply related to the morphological parameters of the constituents of the aqueous suspension. This relationship is applicable in a large domain of particle diameter (15 to about 400 pm) and also for small particles (1-15 pm) as long as the electrical repulsion between particles remains efficient. Finally, the mathematical form of the relationship is based on the random stacking of fibers and particles described as spheres.
Introduction As a rule, filtration can be defined as the process of separating dispersed particles from a dispersing fluid, generally on a well-defined filter medium. For the filtration on a grid of aqueous suspensions of fibers and solid particles, the grid is not exactly a filter medium, because all the particles pass over it and no cake is formed when the suspensions do not contain fibers. In the presence of fibers, the sieve is only a geometrical means of allowing the retention of both fibers and particles at the first stage of filtration. Of course, retention of fibers on the grid is the predominant phenomenon during this first step of filtration. After that, a cake builds up and progresses with time. In previous works (Nardin, 1980; Schultz et al., 1983a-d), the study of filtration on a metallic grid of aqueous suspensions of fibers and cement was described in detail. The aim of these works was the substitution of toxic asbestos fibers by glass fibers for the preparation of asbestos-cement composites, keeping the same manufacturing procedure. A restricted model was proposed relating filtration characteristics, in particular the filtration efficiency (defined as the ratio of the weights of solids in the cake at the end of the filtration and in the initial suspension), to all other parameters of the system, except to cement particles characteristics that were supposed to remain constant. 0196-432118611225-0111$01.50/0
As described in these previous papers, the value of filtration efficiency completely depends on the constitution of the first layers of both fibers and particles on the grid. Thus, the object of the present paper is the study of the formation of these first layers and not the growth of cakes. In order to test and to extend the validity of our previously proposed model to solid particles differing by their chemical nature, their surface properties, and their morphology, Le., glass spheres, alumina, kaolin, and latex particles are now used instead of cement. A different, more fundamental quantity than filtration efficiency is introduced, i.e., the unit retention of particles by fibers or the number of particles retained by an individual fiber in the filtration cake. In particular, this new quantity is fundamental for the study of the first step of the filtration. Two points have to be stressed: First, the fibers are glass fibers of the same density and modulus of elasticity. They differ only by their morphological characteristics (diameter and length). Second, experimental conditions were chosen in order to favor strong electrorepulsive forces between the particles. In a subsequent paper, this alternative aspect will be described (Nardin et al., following paper in this issue). Theory Filtration Efficiency. (See Nardin, 1980; Schultz et al., 1983a-c). If mc and mF are respectively the weights 0 1986 American Chemical Society
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Ind. Eng. Chem. Prod. Res. Dev., Vol. 25, No. 1, 1986
of solids in the cake and in the filtrate at the end of the filtration, the filtration efficiency, e, is expressed by (1) e = mc/(mc + md Further, if m and m: are the weights of fibers and particles in the initial suspension and if M is the total amount of solids, M is given by
M =m
+ mpo= mc + mF
(2)
In practice, almost all the fibers are retained by the grid and in the absence of fibers, no cake is formed. Under these conditions, the filtration efficiency is only a function of the weight percentage of fibers ( T = m / M : e = 7/[(1 - a ) T + a ] (3) In this equation, a is a dimensionless coefficient that is dependent on all other parameters, except T , and is called “initial filtration coefficient”. In fact ( d e / d ~ ) , ==~l / a (4) and hence, in the linear domain of variation, it follows that e =T/a (5) For the filtration of aqueous suspensions of various fibers (glass, carbon, and polymeric fibers) and cement, the following relationship was established
with k’ = coefficient, pf = density of the fibers, y = grid opening (the ratio between the void and the total surface area of the grid), C, = the total concentration of solids in the volume, V, of suspension (C, = M / V ) , E = modulus of elasticity of the fibers, d = diameter of the fibers, and 1 = mean length of the fibers. For the sake of simplicity, when fibers differ only by their morphological properties (d and I ) and when all other experimental conditions are kept constant, which holds for the present study, eq 6 can be written a = a’,,dj/31’/3
(7)
where a b is a coefficient. The domain of validity of eq 3, 6, and 7 is given by 12.5 < y 5 64%, 1 5 T 5 15%, 10 5 C, I 500 kgm-3, 10+ I I 1 I m, and lo9 I EI m, 0.2 X d I20 X 10l1 N.mW2. Unit Retention of Particles by the Fibers. The main mechanisms (Orr, 1977) that have been outlined to account for the retention of solid particles by a fibrous medium are diffusion deposition, direct interception, inertial deposition, gravitational deposition, and electrostatic deposition. Each kind of deposition is described by one or several dimensionless parameters whose numerical value is a measure of the magnitude of the individual mechanisms. For the particular type of filtration considered in this study, due to the morphological characteristics of both fibers and particles, the cause of particle retention is the direct interception mechanism. In this instance, a particle is caught as soon as its distance to the fiber surface is equal to its radius. Hence, it is possible to define a dimensionless parameter (NR)as the ratio of the particle diameter (@) to the fiber diameter ( d ) , whereas in the classical filtration process, it is the ratio of the particle diameter to the mean diameter of the pores of the filter. When is high, the capture coefficient in the direct interception mode is equal to NR.This is also the case for a sieving process.
Table I. Denomination and Morphological Characteristics of Glass Fibers diameter, mean length, denomination d X lo6, m 1 x io3, m a1 85
a6
as b3
b,
20 18 18 10 14 5
0.67 6.0 3.0 0.60 0.69 0.45
So far, for the filtration of aqueous suspensions of fibers and solid particles, the filtration efficiency was measured. Presently, a more fundamental quantity is considered, i.e., the unit retention, n, or the number of particles retained by an individual fiber in the filtration cake
n = np/nf
(8)
where np is the number of particles retained by a number nEof fibers. Taking into account eq 1 and 2, it follows that
Me = m
+ mp
where mpis the weight of solid particles in the filtration cake. Hence -m~ = - - Ie m~
Finally, a relationship between filtration efficiency and the unit retention is achieved
where pp is the density of the particles. Experimentally n is calculated from the relation
In the linear domain of variation of e vs. 7, eq 5 and 9 allow the determination of nu, the so-called “initial unit retention”:
-(
3 pf d21 1 nu = - - 1) 2 P p @3 a
Experimental Section Glass Fibers. Fibers (SociBtB Saint Gobain) were already described in the previous papers (Nardin, 1980; Schultz et al., 1983a; Nardin et al., 1982). Their general characteristics, useful for this study, are recalled in Table I. Glass fibers “b” are of “E”type. Fibers “a” are alkali-resistant fibers. Fibers “a5”,and “a6”are obtained by cutting monofilaments. Other fibers, in the form of waddings, were ground for 1 min in an aqueous medium, and their mean length was determined on macrophotographs. The density of the fibers is of the order of (2.66 f 0.07) X lo3 kg.m-3 and their elasticity modulus close to (0.79 f 0.02) x 10” N.m-2. Cement. An artificial Portland cement (Origny, CPA 55) with a density :f (3.1 f 0.1) X lo3 kgm-3 was used. Its m (Nardin, 1984). mean particle diameter is 39.5 X Glass Beads. The characteristics of glass beads (SocieG Verre Industrie) are shown on Table 11. These beads are made of spherical particles having a distribution of their diameters that is satisfactorily monodisperse. Their density is equal to (2.6 f 0.1) X lo3 kg-m-3.
Ind. Eng. Chem. Prod. Res. Dev., Vol. 25, No. 1, 1986 113 Table 11. Denomination and Characteristics of Glass mean weight mean number diameter, 9, X diameter, Q X denomination lo6, m lo6, m B1 51.0 30.7 B2 88.1 64.8 B3 125.1 115.8 B4 190.5 143.2 B5 248.2 221.7 B6 288.4 277.0 B7 369.2 331.1
Beads
2,
1.66 1.36 1.08 1.33 1.12 1.04 1.11
a
0,2
'iP = polydispersity index = @/a, Table 111. Denomination and Diameter of Kaolin Particles denomination diameter. Q X lo6. m K1 0.75 K2 2.0 K3 2.5
Alumina. Alumina particles (SociBtB Produits Chimiques Ugine Kuhlmann) appear as agglomerates of small elementary particles (diameter N 0.25 X lo4 m) having a mean size of 15 X lo* m. Their structure is essentially, at 95 % , of the a! alumina type. Their specific surface area is equal to (6 f 1) mZ@. Before use the particles were treated for 24 h in an alkaline medium a t pH 12. This treatment modifies in a known manner (Robinson et al., 1964) the surface properties of alumina. Kaolins. Table I11 indicates the references of the clay particles and their mean diameter. Kaolins K1 and K3 originate from the French Alps (Vercors), whereas kaolin K2 is from another source. Their density is equal to about 2.55 X lo3 kg.m-3. A water dispersion of kaolins K1 or K3 has a pH of 7.5. The corresponding value for K2 is 5.5. Latex. A latex, based on a vinyl acetatefethylenefvinyl chloride terpolymer, was used. The mean diameter of the spherical particles is 0.7 X lo4 m. The pH of the suspension is 4.7, and the actual weight content is 50 f 1%. Filtration Conditions. Aqueous (demineralized water) suspensions of fibers and solid particles are filtered, at 20 f 3 "C, on a circular sieve having a surface of 2.47 X m2, under conditions described earlier (Schultz et al., 1983a). Only one metallic sieve, with a constant opening of 64%, was used. The diameter of the steel wire is 0.2 X m. m, and their spacing is equal to 0.8 X The weight percentage of fibers, 7,is thus the only experimental parameter that is varied from 0 to 15%. The pH of the Alz03suspension is adjusted to 1.5 with nitric acid in order to create strong electrorepulsive forces between the particles (Nardin et al., following paper in this issue). In all cases, the cake and filtrate are recovered, drained on a Buchner, and dried in an oven for 3 h at 110 OC before determining their weight. Results First, extension of the validity of the restricted model of filtration of aqueous suspensions of fibers and cement to the filtration of other solid particles (glass beads) and fibers will be examined. Second, if the restricted model remains applicable, a modification will be proposed to take into account the dimension of the particles. Third, the model will be tested with particles of various nature and different size. Finally, a possible explanation of all the results will be proposed on the basis of a random stacking of fibers and spheres. Validity of the Restricted Model. In this part, only the filtration of aqueous suspensions of glass fibers and
0.1
0
20
Figure 1. Initial filtration coefficient a vs. (d5/31'/3)for different glass beads: (1)B1; (2) B2; (3) B3; (4) B4. Table IV. Values of the Term ( d 5 / 3 1 1 / 3for ) Glass Fibers fibers ( d 5 / 3 P / 3 ) x 1010, m2 b!3 1.12 a8 3.91 b3 7.18 a1 12.89 a6 17.83 a5 22.46
glass beads will be described. Whatever the fibers and the beads, the general eq 3, which relates efficiency, e, to the weight percentage, T , of fibers, is verified. In its linear form, this equation is l / e = (1 - a)
+U/T
Since the experimental 7 interval is limited ( T < 15%), the initial filtration coefficient, a, is determined from the slope of the straight line e-l vs. 7-l rather than the intercept. For each determination of a, five experiments were conducted: their results entirely support the general equation. Yet, to confirm the validity of the model, it is essential to check eq 7, which relates linearly the initial filtration coefficient to the morphology of the fibers: a =
~'&5/311/3
(7)
The values of the product (d5/311/3) for the glass fibers are given in Table IV. Figure 1shows the linear variations of the initial filtration coefficient, a, with this product for four types of glass beads. I t must be noted that only the coefficient a b , which is the slope of these straight lines, varies with the mean diameter, +, of the glass beads. Modification of the Filtration Model When Q Is Large. Table V gives the experimental values of ab, which are obtained by a linear fitting procedure of the variations and also some earlier results concerning of a with d5/311/3 the filtration of fibers and cement particles (Nardin, 1980; Schultz et al., 1983~).
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Ind. Eng. Chem. Prod. Res. Dev., Vol. 25, No. 1, 1986 1
Table V. Relationship between the Coefficient ab (Equation 7) and the Mean Weight Diameter of the Particles particles @ x lo6, m ab x m-2 cement 39.5 1.20 f 0.05 B1 51.0 1.50 f 0.02 B2 1.09 f 0.03 88.1 0.155 f 0.001 B3 125.1 B4 0.110 f 0.005 190.5 0.085 f 0.005 B5 248.2 0.037 f 0.001 B6 288.4 €37 0.015 f 0.001 369.2
. aA.10-8 (m-21
Table VI. Relationship between the Coefficient ab (Equation 7 ) and the Mean Weight Diameter of the Mixtures of Glass Beads @ x 106, m a b x IO+, m-2 69.5 1.27 f 0.09 88.1 0.43 f 0.06 120.7 0.17 f 0.02 106.6 0.51 f 0.09 157.8 0.16 f 0.02 0.43 f 0.05 116.7 69.5 0.99 f 0.09 0.25 f 0.05 100.3
~
0
400
Figure 2. Plot of coefficient ab vs. mean diameter Q of particles: ( 0 )glass beads; (0) mixture of glass beads; (m) cement.
of fibers. Therefore, the coefficient a is significant lower than 1. Thus, eq 11 becomes
The relationship between a b and the diameter, @, of the particles takes the following form
a b = p@6
3 Pf d21 1 nu = - - - 2 Pp @3 a
Or, considering eq 13, nu is equal to
where 6 and p are respectively equal to (-2.0 f 0.2) and (0.32 f 0.15). Therefore, a b is inversely proportional to @. In order to verify the generality of this relation, filtration experiments were performed with b3 type of glass fibers and glass beads mixed in 50/50,75/25, and 67/33 proportions. The mean diameter of each mixture is taken as the arithmetic mean value of the two populations. Table VI contains the values of coefficient a'o and the mean diameter of the mixed glass beads. Further, Figure 2 shows a satisfactory agreement between the results recorded either with the mixtures or with individual glass beads. When all experimental data of Tables V and VI are taken into account, the variation of a b with @ takes the form
a b = (0.25 f 0.09)@(-2~09*0~19) with an adjustment coefficient equal to 0.89. In short a'o =
200
(12)
where a. is a dimensionless coefficient, independent of all the parameters of filtration considered so far. In this instance, a,, is equal to 0.25 f 0.09. The combination of eq 7, 12, and 6 leads to (13) with
where k is a coefficient equal to (2.0 f 0.7) X m4/3.N-2/3 (Schultz et al., 1983~). The filtration may also be described in terms of previously defined unit retention values of particles by the fibers. In practice, the amount, mp, of particles retained in the filtration cake is much larger than the amount, m,
3 pf nu=--2ao P p
( 1 2 ~ 3
@
Since the values of pp of the glass beads and the cement are comparable and since pf is constant, eq 16 may be rewritten as
By definition, the quantity nu is a capture coefficient. However, as seen in the theoretical section, the basic theory of filtration on a porous medium states that this kind of capture coefficient (NR)must be equal to the ratio of the particle diameter, a, to the fiber diameter, d, or to the ratio of @ to the mean diameter of the pores of the filter. Of course, eq 17 proves that the quantity nu does not follow the same law. The term (12d)1/3 can be interpreted as the mean diameter of the pores of a particular arrangement; whether or not this is the case, nu will be inversely proportional to NR. Consequently, the basic theory of filtration on fibrous medium is not applicable to the present case, which concerns the formation of the first layers of the filtration cakes. Indeed, a well-defined and stable porous medium (fibers and particles) is not formed at this first step of filtration. Filtration Model When @ Is Small. To further verify and extend the model, the rather large glass particles were replaced by very small kaolins, alumina, and latex particles. Since these particles are indeed very tiny, the quantity of solids in the filtration cake is low. The filtration efficiency, e, varies linearly with the weight percentage, T, of the fibers when T is inferior to 5%. As previously, at least five experiments are made to obtain a mean value of e. From these data, the unit retention values are calculated according to eq 9 or 10. Since e is a linear function of 7, the unit retention, n, is in fact equal to the initial unit retention, nu. Equation 17, which is valid for large particles, is no longer applicable. The experimental data with small particles lead to values higher than those expected from eq 17. T o account for this new situation, it is interesting
Ind. Eng. Chem. Prod. Res. Dev., Vol. 25, No. 1, 1986
Table VII. Relationship between the Initial of Particles by Fibers, nu,and 12d/a3 fibers particles n" b9 latex 9.72 X lo4 K1 1.11 x 105 K2 5.19 x 103 K3 3.95 x 103 1.34 x 1 0 2 a1 K1 2.15 X lo5 K2 1.94 x 104 K3 8.37 x 103 a5 K1 3.96 X 10' 2.08 x 106 K2 1.04 X lo6 K3 a6 K1 1.42 X lo7 K2 8.77 x 105 K3 3.51 x 105 a8 K1 2.58 X lo5 K2 1.84 x 104 K3 7.95 x 103
Unit Retention 1+q(~3 2.95 X lo6 2.40 x 106 1.27 x 105 6.48 x 104 2.10 x 103 2.13 X lo7 1.12 x 106 5.75 x 105 1.54 X lo9 8.10 x 107 4.15 X lo7 3.84 X lo8 2.02 x 107 1.04 x 107 8.53 X lo6 4.50 x 105 2.30 x 105
Table VIII. Values of the Virtual Diameter, Glass Fibers fibers 9, x lo6, m 55.7 85.1 104.4 115.3 302.5 480.2
115
for the
The question that arises is to find the fundamental reason for the difference when going from small to large particles. In previous works (Nardin, 1984; Nardin et al., 1985) about the random packing of fibers and spheres, it was shown that in the fibers felt, the individual fiber can be described by a virtual sphere. The diameter of this sphere, aV,is calculated by considering that the volume of the fiber felt is equal to the volume of a random packing of spheres constituted by a number of spheres equal to the number of fibers:
aV= (0.555 f 0.005)(12d)1/3
(21)
Table VI11 gathers the values of the diameter of the virtual spheres relative to the studied fibers. This new parameter is already implicitly contained in eq 17 and 19 which give the variations of nu with the filtration conditions. Combining eq 21, 17, and 19, it follows that
nu = (0.55 f 0.07)(@v/@)3
j
,
, ,'O/
, Iog[GJ],
5
0
1
when
10
to try to find a correlation between nu and 12d/a3, a quantity that is already contained in eq 17. Table VI1 presents the values of nu for the different cases. For the system based on A1203,the mean diameter of glass fibers was corrected by taking into account the adsorption of A1203on the glass fibers, as will be demonstrated later (Nardin et al., following paper in this issue). Figure 3 clearly indicates that log nu varies linearly with log [ ( 1 2 d ) / @ 3independently ], of the filtration conditions. Hence
o(
5)'
(19)
Discussion The two most important results presented so far (eq 17 and 19) can easily enter into only one relationship between the initial unit retention, nu, and a function, f , of the morphological parameters of both fibers and particles: nu = f(Z2d/a3)
In the following part, the two different situations described by eq 22 and 23 will be discussed. When @,/a 2 4.4. The definition of nu is given by eq 22. Since the dimensions of the particles are very small, it seems reasonable to assume that the spatial arrangement of the fibers will be the predominant factor for filtration and that the small particles will essentially fill up the interstitial volume between the fibers. One fiber will define a virtual volume, ( ~ / 6 ) @ , 3containing , a given number, nu, of particles. The volume of this number of particles is equal to (r/6)a3nU.Thus, the volume fraction a p of particles is ap=
where 0and 6 are respectively equal to 0.095 f 0.009 and 0.938 f 0.015. Therefore, because 6 is close to 1, eq 18 becomes
nu = (0.095 f 0.009)(L2d/a3)
(23)
when is large. When plotted, the abscissa x , of the intersection of the previous functions corresponds to x , = (a"/@),= 4.4 f 1.2 (24)
Figure 3. Plot of log nu vs. log [(12d)/03].
nu =
is small and nu = (10.7 f 4.3)aV/@
(22)
(20)
In this expression f(L2d/a3)= 0.095(Z2d/a3)when the dim ameter of the particle is inferior or equal to 15 X (eq 19) or f(L2d/@3)= 6(12d)1/3/@3) (eq 17) when the particle m. diameter is superior to about 15 x
(@3nu)/av3
Numerically, aP is obtained through eq 22 and is equal to 0.55 f 0.07. Taking into account the experimental errors, this calculated value is very close to the measured one considering loose or dense packings of spheres for which aP is respectively equal to 0.601 and 0.637 (Scott, 1960). Consequently, eq 19 or 22 may be simply explained by a particular type of spatial arrangement of fibers forming a network filled with particles. This network is not disturbed by the presence of the particles, as illustrated by Figure 4a or by Figure 5. The volume fraction, aic, of solids in the cake a t the beginning of the filtration is thus easily calculated, since the volume of the cake is equal to one of the fibers network a, =
A A -a3nP + -d21nf 6 4 I,
Vf
116
Ind. Eng. Chem. Prod. Res. Dev.. VoI. 25. No. 1. 1986
a
I
b
a
C
Figure 6. Spatial arrangement of fibers and particles when < 4.4: (a) top view; (b) side view; (e) definition of the parameters.
will essentially favor the formation of a particles network. Thus a special arrangement of spheres of diameter @ and of virtual spheres of diameter Qv will he established (Figure 4b). Figure 6a illustrates the case of a compact first layer of particles. The number of particles (Schultz et al., 1982) per unit surface area in this layer is equal to 2/(3'1*Q2). If the fibers themselves create an independent network above the initial layer, the number of fibers per unit surface area will be proportional to the product (**"I, the proportionality coefficient being dependent on the value of Q, In this case, the initial unit retention value, n,, will vary in the same way as QJQ, as predicted by eq 23. However, theroretically it is impossible to compute the proportionality coefficient between n, and QJQ. Erperimentally, its mean value is 10.7 4.3 (eq 23). The large scatter of the experimental results stems from the complex dependence of this coefficient and According to Figure 6c,the half-thickness,h, ofthe first layer of fibers and particles is
b
*".
h = (3'12/3)* tan 8 where 8 = cos-'
Figure 4. Photographs of random packing of fibers and spheres: (a) polymeric fibers and glass beads (av/+= 4.8); (b) iron fibers and steel beads (av/* = 1.1).
Particle*
Virtual sphere
Hence, the total volume fraction, a,,of the system is given by
(mV)
0.004)12dnf
d21 + 3(3l/3 2(3)'/* X 10.7 WQV 7
T
ac=
where V ,defined in previous papers (Nardm, 19M Nardm et al., 1985). is
(27)
or in its numerical form
(25)
Or, according to eq 17 and 25
+ 5.307(d/l)
Q"
where %Oand np are respectively the numbers of particles and fibers per unit surface area. Therefore
Figure 5. Spatial arrangement of fibers and particles when 5 4.4.
a, = 0.334
2(3'12)
(a,)
Fiber
V , = (0.148
[3-*1+
0.605 a. =
d21 + O0.085 .( @*"
(28)
(26)
< 4.4. Contrary to the previous case, the When particle diameters are now important when compared to the morphology of the fibers or to their virtual diameter (QJ. Hence, at the s t a r t of the filtration experiment, fibers
*.
This calculation holds when (2h) is superior to If not, the loose spatial arrangement of particles will not be disturbed by the presence of fibers, since the fibers will be
Ind. Eng. Chem. Prod. Res. Dev., Vol. 25, No. 1, 1986 117
Table IX. Comparison between Experimental (a,B.P*')and Theoretical (CY,)"Values of the Volume Fraction of Filtration Cakes glass fibers glass beads m, X lo3, kg (V - V,) X lo6, m3 a,/@ (y,exptl acteq 28 ac,eq 30 1.09 0.47 f 0.05 0.350 13.3 b9 B1 16.6 0.606 0.45 0.59 f 0.08 10.0 15.6 B3 12.9 0.385 0.49 f 0.06 0.97 16.8 88 B2 0.51 f 0.06 0.499 12.4 0.68 16.7 B3 0.229 0.16 f 0.01 24.2 2.05 b3 B1 10.1 0.45 f 0.07 0.429 7.9 0.83 9.4 B3 0.608 0.71 f 0.11 0.47 8.4 15.9 a1 B5 0.149 3.43 0.18 f 0.02 a, B2 17.0 8.0 0.321 1.22 0.35 f 0.03 18.6 17.4 B5 Equations 28 a n d 30.
located in the voids between the particles. When (2h) is inferior to 9
the virtual spheres associated with the fibers in a random packing 9, 1
y=--
xc
where x , is defined by eq 24, and P is an exponent. When y 1 1, P = 3, and when y C 1, P = 1. Hence, the initial
and numerically
unit retention, nu,of the particles by the fibers is directly related to the morphological characteristics of the solids to be filtered. Finally, model arrangements of fibers and spheres, like those shown schematically on Figures 5 and 6, explain quite well the experimental results and also the mathematical form of eq 31. It should be also noted that when the value of nu is known, eq 3 and 11easily allow the calculation of the final filtration efficiency, e.
@V
- I0.528
9 Hence, the total volume fraction, a,,is
a, =
3'12 2 When the value of nu (given by eq 23) is inserted -93
a, = 0.605
d21 + 0.085-a+@.,
The last term in this equation is usually small, and a, is thus equal to 0.605, i.e., the volume fraction of a loose packing of spheres (Scott, 1960). To verify eq 28 and 30, the volume fraction, a,, was determined by filtrating various fibers-glass beads associations. Further, in these experiments the volume of filtrate, VF,as well as the weight of solids, mc, contained in the cake were noted. Considering that pp N pf, the experimental volume fraction, aYPt1,is given by
The error on VF is of the order of lo+ m3. Table IX shows the measuring conditions, the experimental results, and the calculated values of a,. A satisfactory agreement between the experimental and the theoretical values of a, is observed. The experimental results are relative to finished cakes containing high percentages of fibers, between 1.25 and 7.5%, whereas theoretical expressions describe solely the buildup of the first layers of the filtration cake. This fact explains some of the discrepancies between theory and experiment. Nevertheless, these results largely confirm the proposed model. In the literature such low values of acare scarcely discussed (Ben Aim, 1968,1969). The main results of this study can be summarized by a simple expression. The combination of eq 20, 22, and 23 leads to nu = QYP
(31)
In this expression q is equal to 47 f 6 and y is a function of the diameter of the particles and of the diameter ip, of
Conclusion This study leads to a general representation of the first steps of the filtration of aqueous suspensions of particles and fibers. The knowledge of the initial unit retention, nu, of particles by fibers allows a fundamental approach of the filtration process. In the first part of this study, nu is related simply to the relevant parameters of filtration, i.e., the morphological characteristics of both fibers and particles. This relationship holds in a large domain of particle dimensions, between 15 and 400 pm. For smaller particles having diameters between 1 and 15 hm, electrostatic repulsions between them play an additional role during the filtration. Nevertheless, the model remains applicable as long 8s the electrostatic repulsions are efficient, a condition which is experimentally realized by adjusting the pH of the suspension. This point will be further detailed in a subsequent paper. Finally, the mathematical expression relating nu and the morphology of fibers and particles is evidenced by considerations about the random packing of fibers and spheres. When the particle diameter is small, the volume of the first layers of the filtration cake is determined by the random packing of fibers, whereas when the particle diameter becomes important, it is the random packing of the particles that takes over the predominant role. Literature Cited Ben Aim, R.; Le Goff, P. Powder Techno/. 1968/89, 2 , 169. Nardin, M. "Etude physico-chimique de la filtration de suspensions aqueuses de fibres et de ciment", Master Degree, UniversitB de Haute-Alsace, Mulhouse, France, Nov 1980. Nardin, M. "Filtration de suspensions aqueuses de fibres et de particuldes solides", Ph.D. Thesis, UniversitB de Haute-Alsace, Muihouse, France, April 1984. Nardin, M.; Papirer, E.; Schultz, J. J . Colloid Interface Sci. 1982, 88, 204. Nardin, M.; Papirer, E.; Schuitz, J. Powder Techno/. 1985, 4 4 , 131. Nardin, M.; Schultz, J.; Papirer, E. Ind. Eng. Chem. Prod. Res. D e v . , following paper in this issue. Orr, C. "Filtration-Principles and Practices"; Marcel Dekker: New York, 1977; Vol. I.
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Ind. Eng. Chem. Prod. Res. Dev. 1986, 25, 118-123
Robinson, M. D.;Pask, J. A,; Fuerstenau. D. W. J . Am. Ceram. SOC.1964, 47(10), 516. Schultz. J.; Papirer, E.: Nardin, M. Verres Refract. 1982. 36(4),709. Schultz, J.; Papirer. E.: Nardin. M. Ind. Eng. Chem. Prod. Res. Dev. I983a,
Schultz, J.; Papirer, E.: Nardin, M. Ind. Eng. Chem. Prod. Res. Dev. 1983d,
22, 102. Scott, G.
D. Nature (London) 1980, 788,908.
22, 90. Schultz, J.; Papirer. E.; Nardin, M. Ind. Eng. Chem. Prod. Res. Dev. 1983b, 22,94. Schultz, J.: Papirer, E.; Nardin. M . Ind. Eng. Chem. Prod. Res. Dev. 1983c,
Received for review February 19, 1985 Revised manuscript received July 8, 1985 Accepted September 27, 1985
22, 97
Filtration on a Sieve of Aqueous Suspensions of Fibers and Solid Particles. 2. Influence of Interparticle Electrostatic Interactions on the Retention of Particles by Fibers Mlchel Nardln, Jacques Schultz, and Eug6ne Paplrer’ Centre de Recherches sur la Physico-Chimie des Surfaces Solides. 68200 Mulhouse, France, and Laboratoire de Recherches la Physico-Chimre des Interfaces de I’Ecole Nationale Supgrieure de Chimie de Mulhouse, 68093 Mulhouse Cedex, France
sur
The filtration model proposed earlier is modified to take into account interparticle electrostatic interactions in the aqueous media. These interactions are indeed responsible for the flocculation of the particles or their adsorption on the glass fibers, essentially when the particle diameters are inferior to about 15 hm, and also for the retention of the particles in the filtration cake. The absolute value of the electrokinetic surface potential is introduced into the general expression of filtration to account for the interparticle interactions and thus to extend even further the validity of the filtration model.
Introduction In the first part of this study (Nardin et al., preceding paper in this issue), a general model of the filtration of aqueous suspensions of fibers and solid particles on a sieve was proposed. In its mathematical representation, it relates the initial unit retention of particles by fibers, in the filtration cake, to the morphological characteristics of both fibers and particles, on the assumption that the level of electrical interactions between the solids is negligible. However, it is known (Maher and Irvin, 1979; Orr, 1977; Wishart and Gregory, 1979) that when the diameter of the particles is small, the electrostatic phenomena can no longer be ignored. This study is devoted to this particular aspect of filtration. In an aqueous medium, the measurement of electrokinetic potential, or {-potential, of the particles and the fibers allows an estimation (Kruyt, 1952) of the electrical interactions between solids. To highlight the role of electrical interactions, the filtration of small particles and glass fibers will be examined. Moreover, to change the intensity of the interactions, the {-potential is varied by performing experiments at different pH’s of the suspension, since it is established that the {-potential varies largely with pH. Experimental Section Only two types, al and b9, of glass fibers (Nardin et al., preceding paper in this issue) of quite different morphological properties are used. Alumina (A1203)and kaolins (Kl, K2, and K3) of small particle diameter are employed in conjunction with the glass fibers. In fact, the alumina appears as agglomerates built of small elementary particles. The mean diameter, cp, of these agglomerates was measured by a sedimentation method in an aqueous medium at different pH’s (Nardin, 1984; 0196-4321/86/1225-0118$01.50/0
Table I. Mean Diameter of Alumina Particles (Agglomerates) at Various pH’s PH p x lo6, m 3.5 16.7 5.5 15.2 5.75 14.6 6.5 14.6 8.25 15.7 17.6 8.5 12.0 15.0 Table 11. Mean Diameters of Individual Particle (po)and Agglomerates (CP)of Kaolins at Various pH’s kaolins 90 x lo6, m pH c x lo6, m K1 0.75 1.5 17.6 4.0 4.2 2.0 3.25 9.3 K2 K3 2.5 1.5 16.3 4.0 15.9
Nardin et al., 1985). Table I shows that cp is independent of pH. The mean value of cp is (15 zt 1 ) X m. The density of the agglomerates is equal to about 1.65 X lo3 kgm-3. At the pH of a kaolin suspension, close to neutrality, the particles are well dispersed, but a slight variation of pH will bring about an agglomeration of the individual particles. Table I1 contains the experimental values of the diameter of individual particles, cpo, and agglomerates, (0, determined also by sedimentation a t various pH. The density of the agglomerates is 1.3 x lo3 k ~ m - ~ . In order to check the adsorption capacity of the particles on glass fibers, the following procedure (Hull and Kitchener, 1969) was used: Flat and clean glass surfaces are dipped in a stirred aqueous suspension containing 2 X lo2 0 1986
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