Clogging of Nonuniform Filter Media James A. Guin Department of Chemical Engineering, Auburn University, Auburn, Ala. 56850
The relationship of the increase in pressure drop owing to the retention of solids during deep filtration is studied using a capillaric model with nonunifcrm pores. The change in filter medium properties is examined using two limiting rates of solid deposition within the pore structure. The nonuniformity in the pore size distribution and the variations in the rate of particle deposition are found to b e significant factors. The Kozeny constant, previously assumed to b e unchanged during clogging, is found to depend upon certain moments of the evolving pore size distribution function and, therefore, to vary with retention.
T h e filtration of dilute suspensions of minute particles through granular beds has been for many years a principal step in the purification of water. This process is differentiated from cake filtration by the fact that, as the suspension seeps through the bed, the solid particles become attached to the surface of the filter medium a t varying depths throughout the bed; hence the name “depth filtration.’’ As the particles are removed from the suspension and deposited in the pores of the filter medium, the bed experiences a decrease in porosity with a consequent reduction in permeability, accompanied by an increase in pressure drop. The macroscopic bed properties continue to evolve in time and space until the filtration process is halted, usually when the pressure drop across the clogged bed becomes prohibitively large. At this time the filter bed is cleaned by backwashing and the cycle is allowed to begin again. The relationship between the increase in pressure drop (or equivalently the decrease in permeability), the bed and suspension properties, and the operating variables is a complex one and, for this reason, design of deep filters is usually based upon empiricism rather than upon fundamental theory. I n spite of the lack of unified theory for design, many transport and attachment mechanisms present in deep filtration have been identified and investigated both experimentally and theoretically (Herzig, et al., 1970; Irson and Ives, 1969; Ives, 1970; Maroudas and Eisenklam, 1964). Some progress in the design of deep filters has been made by Ives (1963, 1965), although his methods still require experimental determination of several coefficients. The transport phenomena of suspensions are complicated further by the fact that the irregularly shaped particles may wander uripredictively across stream lines due to the action of unbalanced shear forces, even if the flow is uniform and laminar (Bretherton, 1962). In spite of the complexity of accounting for the local attachment mechanisms on a microscopic level, several models have been proposed to describe the variation in macroscopic medium properties (e.g., permeability, porosity) during the course of the filtration process. All of these models have been obtained by a modification of the specific surface and porosity terms in the Carman-Kozeny equation (Carman, 1956). These modifications have been based upon an assumed form for the internal geometry of the porous medium together with the assumptions of uniform coating of deposited particles and constant tortuosity. Some agreement with experiment for the modified Kozeny models has been found (Ives and Pienvichitr,
1965; Sakthivadivel, 1966); however, agreement is not always obtained. The ultimate aim of calculations based upon the Kozeny model is an equation relating the increase in pressure drop to the accumulation of deposited solids. Relationships derived from the Kozeny equation do not indicate any dependence upon the nonuniformity of pore sizes present in the bed, nor do they provide any method for introducing the particular particle-surface interactions which cause deposition and which may vary from case to case. Furthermore, parameters appearing in equations derived from the Kozeny model are difficult to interpret in terms of medium and suspension properties. It thus seems useful to consider a more detailed model for bed clogging which will show the dependence of evolving bed properties upon bed pore structure and suspension properties. As pointed out in the recent review by Herzig, et al. (1970), a major shortcoming of previous models is that they do not account for the fact that porous media actually consist of a distribution of pore sizes rather than being completely uniform. Thus, in the Kozeny model, the local rate of particle deposition is the same in each pore. It is probable, however, that the rate a t which particles are deposited within a pore will depend in some manner upon local conditions such as the fluid velocity existing within each pore. Thus different sized pores will most likely be clogged a t different rates. If this is true, a medium having a distribution of pore sizes certainly will behave differently from one having uniform pores. In this paper a model is examined which considers the fact that the porous medium is nonuniform, having a distribution of pore sizes. It is shown that the evolution in bed properties during clogging is quite sensitive to the particular initial distribution of pore sizes, as well as to the rate of particle deposition, a conclusion which explains in part the wide variation which has been obtained in experimental results. Furthermore, these more general results reduce to those previously obtained from the Carman-Kozeny equation if the pore sizes are uniform. The Model Porous Medium
The idealized porous medium to be used in these calculations is of the capillary type wherein the actual porous medium is visualized as being replaced by a collection of short, nonuniform, cylindrical capillaries distributed randomly throughout a solid matrix, as shown in Figure 1. Such a model was used Ind. Eng. Cham. Fundam., Vol. 1 1 , No. 3, 1972
345
Here y is a geometric constant depending upon the crosssectional shape of the pores. Substitution of eq 4 into eq 3, together with a consideration of the number of pores intersecting the plane area S, gives the permeability for the idealized medium as
From eq 1 and eq 5 it should be noted that the porosity and the permeability of the porous medium are proportional to the first and second moments, respectively, of the pore size distribution function..In principle, all quantities of interest may be found from the evolution of the pore size distribution function o(A,zlt) as the filtration proceeds. The retention of deposited solids per unit volume of bed is related to the porosity by material balance as
DYZECTDN OF FLOW Figure 1.
Capillary model for fllter medium
by Schechter and Gidley (1969) in studying the effects of surface reactions upon porous media properties. Other capillaric models have been used to study properties of porous media such as permeability (Guin, et al., k971), dispersion (Haring and Greenkorn, 1970), capillary pressure (Purcell, 1949), etc., and a good review of such models is given by Scheidegger (1960). Although it is recognized that these geometrical models are highly idealized, it should be realized also that valuable information concerning the relationship of the observed macroscopic properties of the medium to the microscopic phenomena occurring locally within the pore structure can be gained from such model studies, at least in a qualitative sense. In t,he model pictured in Figure 1, the capillaries are interconnected to some extent, so that upon application of an external pressure gradient, fluid communication may be achieved. The cross-sectional areas of the capillaries are distributed according to some pore size distribution function s(A,z,t). Experimentally, the function q(A,z,t) may be determined using a capillary pressure curve, as shown by Pakula and Greenkorn (1971) and Leamer and Lutz (1940). In this manner, the capillary model proposed here may be used to represent unconsolidated filter media. The function s(A,z,t)may be normalized so that the effective porosity 4 of the matrix is given by F m
$ =
L
J
A?(A,z,t) dA
0
and in this way q(A,z,t)dA will represent the number of pores dA per unit matrix volume. of length L and of area A to A The flow through the filter is usually within the range of validity of Darcy’s law, Wiz.
+
q=--
-k dp w dx
where q is the filter velocity defined as the flow rate per unit matrix area and k is the permeability. The filter velocity in the capillary model is obtained as the spatial average over an element of area containing many .pores. By definition
Q =j h V d S
(3)
where V is the average velocity in an elemental pore which, for laminar flow, is given by (4) 346 lnd. Eng. Chem. Fundam., Vol. 11, No. 3, 1972
g=-
$0
-4 P
where /3 is an experimental parameter accounting for the fact that the deposited solid itself has some porosity and contains stagnant liquid. If the filtration is conducted a t a constant rate, the pressure drop over the bed may be obtained a t any time by integration of eq 2, using the local permeability given by eq 5. As the filtration proceeds] solid particles will deposit inside the pores, reducing their cross-sectional area. The exact mechanisms of solid deposition are many and complex and for the present it is enough to assume that the area of each individual pore is reduced a t some average rate denoted by
(7) Equation 7 implies that the surface deposit within any pore may, on the average, be considered as uniform. The assumption of uniform coating of the solid surface is made also in connection with the Kozeny models and has received some experimental verification when the suspended particles are very small (Herzig, et al., 1970). Fortunately, it is the usual case in depth filtration that the suspended particles are on the order of 100 to 1000 times smaller than the average pore size and the assumption of uniform deposition may be reasonable. Conversely, the model of uniform coating would not be expected to apply where the suspended particle size is of the same order its the pore size, in which case the filtration action of the bed would be primarily one of blocking of pores rather than of surf ace deposition inside pores. An equation describing the change in the pore size distribution may be obtained by making a population balance on any group of pores contained in an arbitrary matrix volume. Thus
if the pore areas Az and AI are constrained to evolve in time according to eq 7. Applying the Leibnitz rule for differentiation under the integral and noting that the limits of integration are arbitrary, we find (9) Equation 9 governs the evolution of the pore size distribution as the pores are clogged by depositing solids a t the rate given by eq 7.
In the next section it is suggested that a reasonable form for the clogging rate function is
$(A,z,t) = -f(z,t)Aa
(10)
That is, the rate of decrease in pore area is proportional to some power of the area. If eq 10 is substituted into eq 9, then application of the method of moments yields the infinitely coupled hierarchy
-
MRECWN OF R O W
CROSS-SECTIONAL AREA, A
Figure 2.
where the new independent variable r is defined as
[
tions is given by /we2
J0
Ai(l
+ .(a
- l)Aa
-
NORMAL
- - -
--- -------
Individual pore in the idealized matrix
=
f(z,t) dt. The general solution to this set of moment equa-
M i=
&A n
- X
where s(A,z,O) is the initial pore size distribution of the filter medium. Recalling that the porosity and the permeability of the porous medium are proportional to M I and M z , respectively, we see that eq 12 may be used to compute these properties of interest as the clogging process proceeds. From eq 12 one can see immediately that the evolution in medium properties is influenced in general by the initial pore size distribution s(A,z,O).Thus one must expect media with different pore size distributions to behave differently during filtration, even when filtering the same suspension.
periments. I n spite of this difficulty, it is useful to examine within the capillaric model two reasonable pore clogging rates to see the diverse effects these have on the outcome of the filtration process. We shall see that the difference in deposition rates coupled with a distribution of pore sizes is capable of producing the large variation in pressure-drop behavior during clogging which has been experimentally observed. This is a behavior which previous models have been unable to account for, even with adjustable parameters (Herzig, et al., 1970). To see the effect of deposition rate upon matrix properties, we shall consider two limiting rates corresponding to fast and slow deposition of particulate matter upon the filter substrate. Considering the elemental pore shown in Figure 2 into which flows a suspension containing an inlet concentration c(x,t)of suspended particles, a material balance yields the average rate of area decrease as
The Clogging Rate Function
The actual rate a t which an elemental pore’s area is decreased depends upon the rate of deposition of solid particles within that pore. There are two factors which are involved in determining the deposition rate. One involves a particle transport mechanism by which the particle reaches the pore surface; the other involves surface forces which cause the particle to adhere to the pore wall once contact has been achieved. Little information on the particle-substrate interaction can be gleaned from filtration experiments because local fluid velocities, particle concentrations, and substrate surface properties are not known a t each point in the filter medium. -4s an alternative procedure it should be possible to gain insight into the particle-surface interaction by studying the simple case of duct flow with particles adhering to the duct wall. The probability of a particle adhering to the duct wall after contact is related to the relative magnitude of the drag force on the particle as opposed to the surface force of particle adhesion. By performing experiments in ducts similar to those summarized by Corn (1966), it should be possible to relate the clogging rate 4 (A,z,t)to pertinent dimensionless parameters characterizing the deposition process (Irson and’Ives, 1969). At the present time, such experiments have not been performed in liquid media, the main emphasis having been on particle adhesion in gaseous media because of the importance to aerosol filtration. Thus it is suggested here that the clogging rate for a pore be determined by performing particle adhesion experiments in duct flow using the same fluid, particle, and substrate material present in the filtration process. The clogging rate so deduced can then be used in the capillary model to predict process performance of the filter. Because of lack of data from the particle adhesion experiments suggested above, we cannot apply with definiteness the capillary model to data obtained from previous filtration ex-
Introducing the average pore velocity V and defining the mean concentration as
we may write eq 13 as
In the analysis to follow it is assumed that the concentration distribution within a pore can be described by an effective diffusion model with a surface deposition rate R(c)so that a material balance results in
with e(zl,zz,O,t)= c(z,t)and D,(Vc.n) = R(c) on r. This equation has been used to describe diffusion deposition from aerosols (Pich, 1966). The effective diffusion coefficient D, is equal to the Brownian diffusivity in the limit as ~ ( X I , X Z ) approaches zero. Otherwise De may be considered as a mixing coefficient accounting for the mechanical dispersion of suspended particles due to unbalanced hydrodynamic forces and may be a function of the local velocity. h similar situation is encountered with the dispersion coefficient used in the continuum model for miscible displacement within a porous medium. I n the analysis to follow, the numerical value for De is not required. In writing eq 16 as a steady-state balance, use has been made of the fact that the average residence time for a suspended particle in a pore is much less than the time Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
347
for changes to occur in the macroscopic concentration a t the pore entrance. Turning our attention now to the form of the clogging rate function, as a first limiting case, we may consider an extremely rapid surface deposition rate. This situation would occur if the medium surface had an extremely high affinity for the suspended particles. Now, if the effective dzusivity permits, all the suspended particles entering a pore will be deposited along the wall and the effluent concentration from the pore will be zero. For this case, from eq 15 we find
--dA- - bVc(z,t)A
(17)
L
dt
The average velocity in a pore is directly proportional to the pore area; in addition if total flow is constant, all pore velocities will increase as the porous matrix clogs. These two facts may be accounted for using eq 4 and eq 5 so that eq 17 becomes -dA dt
- @(z,t)A2 L2Mz
= f(z,t)A2
Thus for the case of rapid particle deposition, the clogging rate of an individual pore is proportional to the square of its area. This mechanism implies that the area of the larger pores is decreased preferentially; however, i t may be pointed out that a pore which is initially smaller will always be smaller and larger pores cannot overtake smaller ones. Physically, the reason for the accelerated clogging rate of the larger pores is that the volume of suspension conducted into a pore is proportional to its area squared and hence the larger pores conduct a disproportionate number of suspended particles, leading to an accelerated clogging rate if all the particles are retained. On the other extreme, we may have a surface deposition rate which is extremely small. I n order to obtain a pore clogging rate for small R(c), we multiply eq 16 by dxl. dxz and integrate over the pore cross-sectional area. Applying Green's theorem to the resulting equation and making use of the boundary condition yields
5L
bF(x,t) -bz VA
R(c) ds
where ds is an element of arc along series and eq 15 we find for R
