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Ind. Eng. Chem. Prod. Res. Dev. 1983, 22, 97-101
Physicochemical Aspects of the Filtration of Aqueous Suspensions of Fibers and Cement. 3. Influence of Morphological and Mechanical Properties of the Fibers on Filtration Efficiency Jacques Schultz, EugOne Paplrer, and Mlchel Nardln Centre de Recherches sur la Physlco-Chimie des Surfaces Solides, CNRS, 68200 Mulhouse. France, and Laboratoire de Recherches sur la Physico-Chimie des Interfaces de I'Ecole Nationale Sup6rleure de Chimie de Mulhouse, 68093 Mulhouse Cedex, France
In the third part of the general study of the filtration of aqueous suspensions of fibers and cement, the influence on filtration efficiency of the morphological and mechanical properties (length I , diameter d , and modulus E ) of fibers of different nature has been considered on a quantitative basis. It has been shown that the quantity (d5/3/1/3E2'3) describing this influence is explained by the permeability of the felt of fibers formed during the early stages of filtration.
Introduction In the previous parts (Schultz et al., 1983a,b) of this general study of the filtration of aqueous suspensions of fibers and cement, the principles and experimental conditions of filtration and the nature and characteristics of the fibers have been described. Firstly, a quantitative relationship between efficiency (e) and rate of filtration has been established. Secondly, the influence on filtration efficiency of the grid opening (y), the composition of the suspension, expressed as fiber content (7)and concentration by weight of the solids (Cs), has been studied. This study has enabled us to describe the role of these factors by the equation
e =
where bo is a constant when the characteristics of the fibers are kept constant. In this third part, this equation will be developed taking into account the influence of the dimensions of the fibers and their rigidity. Influence of Fiber Morphology The influence of the diameter and length of the fibers has been studied by using glass fibers a and b having the same Young's modulus. Table I of part 1 (Schultz et al., 1983a) gives the diameter ( d ) , length (I), and Young's modulus (E)of these fibers. Only fibers of 1 1 0.5 mm have been considered. For shorter fibers, one of the working conditions is not satisfied; i.e., the fibers have to be totally retained on the grid. This limiting value of 1 obviously depends on the grid opening. The value of 1 = 0.5 mm corresponds to an opening y = 64% which has been adopted for this study. Figure 1shows, as an example, the decrease of the filtration efficiency ( e ) with 1 for fibers b of diameters equal respectively to 10 and 14 pm, at different fiber content (7)in the suspension. This variation is probably connected with the relative matting ability of the fibers according to their length. The influence of d is shown in Figure 2 obtained by using glass fibers a and b of 1 N 0.6 mm at different values of 7. The variation of e with d may be explained in the same way as before. The effect of both parameters, length and diameter, may be quantitatively established through the variation of the initial filtration coefficient F. As shown in a previous part
of this work (Schultz et al., , 1983b), F is related to the filtration efficiency at low fiber content in the suspension and the derived equations thus apply to conditions where an almost perfect state of dispersion of the fibers can be assumed. It depends on the grid opening (y) and on the total volume (V) of the suspension
F = boyV
(2)
Experimentally, F varies as d5J3at constant 1 values and as l1I3 at constant d values. Figure 3 shows the linear relationship obtained by plotting F vs. d5/311/3for glass fibers a and b. Therefore, F = FOd5/311/3, and according to eq 2
F = goyVd5/311/3
(3)
where gois a constant equal to (3.75 f 0.2) X 1O'O SI units (kg m-5) in the case of glass fibers a and b. It must be noted that the quantity (d51311J3) has the same dimensions as a permeability coefficient ( K ) . Since the permeability of the felt formed by the fibers is obviously a determining parameter, the influence of the quantity (d4I3l1l3) will be justified by studying the permeability in the last part of the paper.
Influence of the Modulus of the Fibers Besides the morphological characteristics (1, d ) , the ability of the fibers to form a felt depends obviously also on the rigidity of the fibers. Clearly, more flexible fibers will entangle and form more easily a felt capable of retaining the cement. To a first approximation, the Young's modulus of the fibers has been chosen as being representative of their rigidity. As shown in the preceeding section, for glass fibers a and b having the same Young's modulus ( E ) , the ratio F/d5I31'l3 is constant. It is seen in Table I that this ratio varies with the nature of the fibers (glass fiber c, carbon fibers, polymeric fibers). The difference is hypothetically attributed to the difference in Young's modulus. Let us suppose, however, that this ratio is constant when Young's modulus is constant, whatever the type of fiber. This is justified by considering the constant ratio obtained in the case of rayon spun fibers (Table I). However, since different types of fibers are being considered, i.e., fibers of different specific gravities, the
0196-432118311222-0097$01.50/00 1983 American Chemical Society
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Ind. Eng. Chem. Prod. Res. Dev., Vol. 22, No. 1, 1983
Table I. Characteristics of Fibers of Different Nature F (kg)
nature of fibers class fibers a and b glass fibers c rayon spun fibers carbon fibers polyvinyl alcohol fibers
lo3
d, fim
I , mm
x
15 6 6.3 8 14.5 11
0.54
3 0.071 0.169 0.159 1.16 0.53
1 5 5
6 6
F/d5/3/1/3
E, (N m-2)
x
(kg m - 2 ) X
P,
lo-''
(kg m3-3)
x 10-
2.4 ? 0.1 4.04
0.79 1.07
0.02 0.02
2.66 2.8
0.37
0.06 i: 0.04
1.77
0.37 0.29
1.63 1.20
i:
0.09
0.74 0.54
f t
f
0.07
6
F ( kg).io3
I
4
0 .
2
0
i
Figure 3. Initial filtration coefficient (F)vs. (d5/311/3) (for glass fibers a and b).
I
0
Figure 1. Filtration efficiency ( e ) vs. fiber length (1) at different fiber content (7):(a) glass fibers b5, bo, and b7 (d = 10 pm); (b) glass fibers bl, b2, and b3 (d = 14 pm).
0
1
0.5
E( N.m-2),i0-'1 Figure 4. Plot of l l f v s . Young's modulus ( E ) for fibers of different nature.
quantitative study must take into account this fact. Therefore F , the initial filtration coefficient, defined as
( mbeing the q u a n t i t y of fibers in the suspension) will be replaced b y
f 0
I
I
10
n=O
where n is the number of fibers. The relationship between f and F is therefore
d(rim)
Figure 2. Filtration efficiency (e) vs. fiber diameter (d) at different fiber content (7)(for glass fibers a and b of constant length 1 = 0.6 mm).
(4) w h e r e p is the specific gravity of the fibers.
Ind. Eng. Chem. Prod. Res. Dev., Vol. 22, No. 1, 1983 9s
The values of f-' have been calculated for the different fibers at a constant and arbitrarily chosen value of d5/31'/3 = lo4 m2(i.e., d = 15.85 pm, 1 = 1mm). In Figure 4 these values off' are plotted vs. Young's modulus E. The curve can be described by
--'---kb I 0
In:I
or, according to (4)
Comparing eq 6 and 3 and remembering that the quantity go is independent of d , I , y, and V, gocan be expressed as go = kopE2J3
(7)
where ko is a constant independent of all the variables studied and equal to (0.77 f 0.07) SI units. Comparing eq 2, 3, and 7 and replacing the obtained value of bo in eq 1leads to the general relationship between the filtration efficiency e and all the parameters studied
(8)
This equation holds whatever the nature of the fibers if the following conditions are met: 1% 5 7 5 15%; 10 < C, < 500 gL-l; 12.5 < y I64%; 1 I d I 20 pm; 0.6 5 1 5 10 mm. As stated previously, the appearance of the quantity d5J3N3 in this general relationship could be attributed to the obvious role of the permeability of the fibrous felt formed during the early steps of the filtration. Therefore, in the second part of this study, the permeability coefficient of fibrous plugs will be studied independently of the filtration process.
Study of the Permeability of Fibrous Plugs Principle. The flow rate Q of a fluid, through a porous medium, due to a hydrostatic pressure difference (AP),is given by Darcy's law
where K is the permeability coefficient, B and L are respectively the cross section and the length of the porous plug, and 7 is the viscosity of the fluid. According to Kozeny-Carman's equation, K may be written €3 K = - -1 kSo2(1 - c
) ~
where is the porosity of the medium defined as the ratio of the void volume to total volume, k is a shape factor, and Sois the surface area of the fibers per unit volume of the fibers. This equation is valid for fibrous plugs if the porosity is less than a critical value e, of 0.85. Setting t = 1 - a c , where c is the concentration, by weight, of solids in the porous volume, a is the specific volume of the solid, and a = aso,the specific surface area of the solid, eq 10 becomes
Equation 11 shows that if Kozeny-Carman's theory is
Figure 5. Schematic representation of the apparatus for permeability determination of fibrous plugs.
valid, the quantity (Kc2)'I3is a linear function of c.
Experimental Section The permeability coefficient was measured using the equipment shown in Figure 5. A hydrostatic pressure difference is established because the flow level at the exit (V) is lower than the level of the liquid in the entrance tank (I). The maximum applicable pressure in this system is 80 cm of water. However, the pressures commonly used do not exceed 60 cm of water in order to avoid, during measurement, any important distortions of the fiber plug because of the pressure. Such distortions could hinder the reproducibility of the results. The liquid flows into the system, which is made of glass tubes of 5 mm inside diameter for the rigid part. I t then flows through the measurement cell (111). The stopcock bodies and the leakproof joints which are in contact with water are made of polytetrafluoroethylene. The cell is made of polymethyl methacrylate. The streaming direction of the liquid through the cell can be reversed without modification of the other experimental conditions (11). The fiber plug in the measurement cell (111) is in a cylinder with a length of 3 cm and a cross section of T cm2 in between two brass grids. A flowmeter (IV) is inserted in the streaming circuit. In order to obtain reproducible results, care should be taken in the preparation of the fibrous plugs. A good homogeneity of the plug is obtained by direct filtration in the cell of the fiber slurry in water. The cell is slowly fiied in a vertical position with water under a vacuum (20 mmHg) in order to eliminate air bubbles. The experiment being done at room temperature, the water viscosity will P. be considered as constant and equal to For each fibrous plug the flow rate Q of the liquid is measured as a function of the hydrostatic pressure difference AP. According to Darcy's law (eq 9), the slope of the linear relationship Q = f(AP)allows determination of the permeability coefficient to be made. However, a correction has to be made taking into account the flow through the empty cell. If p o and pmare respectively the slopes of the lines corresponding to the empty and filled cell, the corrected slope p is given by P=-
PO'Pm
Po - P m X p SI Numerically, K is expressed by K = 9.5493 X units (m2). Results Figure 6 gives an example of the variation of K with c for fibrous plugs constituted by fibers having approximately the same length (0.6 mm) and varying diameters. K decreases with increasing values of c and decreasing
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Ind. Eng. Chem. Prod. Res. Dev., Vol. 22, No. 1, 1983
Table 11. Concentration ( c ) and Number ( n )of Fibers in the Plug and Permeability Coefficient ( K ) (for glass Fibers a and b)
I
nature c , (kg m i 3 ) of fibers x 10a,
b,
hi
b3
I 02
0
04
06 C ( k g m-3)~10-3
08
ai
Figure 6. Permeability coefficient ( K ) vs. concentration of fibers per unit volume of the plug (c) (for glass fibers a and h). ab
ai
0.284 0.37 7 0.524 0.630 0.757 0.215 0.340 0.549 0.127 0.263 0.377 0.578 0.229 0.390 0.547 0.203 0.264 0.307 0.419 0.206 0.246 0.300 0.347 0.135 0.190 0.270 0.35 7
n x 10-7 0.494 0.655 0.911 1.094 1.313 8.905 14.070 22.690 0.988 2.041 2.923 4.483 0.788 1.342 1.883 0.339 0.441 0.512 0.700 0.098 0.118 0.144 0.166 0.032 0.045 0.064 0.085
K, ( m 2 )
x 10"
15.70 7.00 3.46 1.50 0.98 4.36 1.90 0.96 12.63 4.09 2.44 1.37 8.57 3.57 1.46 32.78 17.41 8.86 3.62 26.25 20.88 14.52 1.92 134.20 44.68 25.02 9.68
2
1
0
Figure 7. Verification of Kozeny-Carman's law (for glass fibers a and h).
values of the diameters of the fibers. However, Figure 7 shows that Kozeny-Carman's law is not followed even for values of t significantly lower than 0.85. Knowing that during the filtration process, the felt formed is even less compressed than the plugs used for the permeability measurements, it is clear that this law does not apply in the present study. Therefore another approach has to be taken in order to establish a relationship between K , the concentration c, and the morphological characteristics of the fibers. In order to simplify the parametric study, the concentration c will be replaced by the number ( n )of fibers in the plug. As previously, the variation of K was studied first as a function of n and d at constant 1, then as a function of n and 1 at constant d. From the values of Table 11, the following two equations have been established Kd2 = Aln-3/2; K1 = A2n-3/2 A , and A, are quantities depending respectively on 1 and d. Supposing that 1 and d intervene as multiplying factors, the following expression of K is derived
n-%
to be equal to A = (2.7 f 1.1) x m5. Introducing the concentration c enables eq 12 to be written as
with A' = A ( T ~ / ~ L Z ) ~ / ~ . Equation 12 implies that A is the only coefficient susceptible to dependence on E, the Young's modulus of the fibers, if this equation holds whatever the nature of the fiber. As shown in Figure 9, a good linear relationship is obtained between A and E for different types of fibers including the glass fibers a, b, c , a carbon fiber, rayon spun fibers, and a polymeric fiber A=A& with A. = (4.6 f 0.7) X in (12)
This relationship is experimentally verified in Figure 8 which relates Kd21 to n-3/2.The mean value of A is found
lo9
Figure 8. Experimental verification of eq 12.
SI units (m7N-l). Inserting
Ind. Eng. Chem. Prod. Res. Dev., Vol. 22, No. 1, 1983 101 L
-I
A (m5)x 1 0 ' ~
since from our definitions S = B and L = d . Relationship 19 proves that at the earlier stages of filtration, the filtration efficiency depends on the permeability of the felt of fibers. Moreover, the combination of (15), (4),and (19) with the general expression (8) for the filtration efficiency leads to (20)
0.5
0
1
E (N .m-*)
x10-l'
Figure 9. Variation of the quantity (A) with Young's modulus (E) (for fibers of different nature).
Having established a relationship between K and the main parameters affecting the filtration efficiency, the last step of this study will be devoted to showing the direct influence of K on the filtration efficiency. Let us consider the early stages of the filtration process which are described by the initial fitration coefficients (F or f ) defined by ( 3 ) and (4). In this region, the filtration efficiency (e),+, is given by
Replacing (3), (4), and (7) in (15) ~pd'J~P/~n (e)n4 = 4koyVpE2/3 Let us suppose now that the thickness of the felt formed at the beginning of the filtration is equal to the diameter of the fibers, or in other words, that less than a monolayer of fibers is deposited on the grid. The concentration of such a layer is given by c =
=P
-1dn 4s where S is the area of the grid. Combining with (13)
with D2 = A ' ( ~ S / T ~ Comparing )~/~. (16) and (18) leads to
This equation demonstrates that the influence of the morphological and mechanical parameters of the fibers through the quantity (d5J3l1I3E2J3) is explained by their effect on permeability.
Conclusion In this part of the general study of the filtration of aqueous suspensions of fibers and cement, the influence of the morphological and mechanical characteristics of the fibers (length, diameter, and modulus) on the filtration efficiency has been quantitatively established. It has been demonstrated that these parameters define the ability of the fibers to form a felt measured by its permeability coefficient. The variation of the filtration efficiency with these characteristics is therefore entirely explained by the variation of filtration efficiency with the permeability of the felt. The general study of filtration has enabled us to relate, on a quantitative basis, the efficiency and rate of filtration to all the determining factors, that is to say: (a) the geometrical characteristics of the grid (grid opening), (b) the composition of the fibers-cement suspension (fiber content and concentration of solids), and (c) the characteristics of the fibers (specificgravity, length, diameter, and modulus). This quantitative relationship holds, whatever the nature of the fibers, in the rather large range of filtration conditions studied. Acknowledgment We acknowledge with thanks the support of this project by both Saint-Gobain Recherche and Everitube and the cooperation and advice of Dn.J. J. Massol, F. Naudin, and A. Sabouraud. Registry No. Cellulose, 9004-34-6. Literature Cited
where
Schultz, J.; Papirer, E.; Nardin, M. Id. Eng. Chem. Prod. Res. D e v . I883a. Part 1 in this issue. Schuitz, J.; Papirer, E.; Nardin, M. I d . Eng. Chem. Rod. Res. D e v . I883b. Part 2 in this issue.
Received for review February 19, 1982 Accepted August 23, 1982
