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Effect of particle loading on granular bed filtration—The cluster enhanced filter model
J. Aerosol Sci., Vol. 19, No. 4, pp. 425~141, 1988. Printed in Great Britain.
0021-8502/88 $3.00+0.00 © 1988 Pergamon Press plc
EFFECT OF PARTICLE LOADING ON GRANULAR BED FILTRATION--THE CLUSTER E N H A N C E D FILTER MODEL M . FICHMAN, C. GUTFINGER a n d D. PNUELI Faculty of Mechanical Engineering, Technion, Israel Institute of Technology, Haifa, 32000, Israel (Received 11 March 1987; and in final form 4 July 1987)
Abstract--A cluster enhanced filter (CEF) model is proposed for the filtration process in granular bed filters. The formulation is made in terms of particle deposition on a single spherical collector. The model accounts for the effect of particle loading on the dynamic behavior of the filter. It predicts the formation of a high porosity layer, which consists of clusters of particles, on the surface of the collector, with a very high increase in the calculated filtration efficiency. The theory is presented in terms of filtration efficiency data as a function of the mass of the loaded material. The results compare favorably with experimental data.
NOMENCLATURE a ao c
C ea elp en el
E
En h H k kt, k2, k3 L M n nin ni
N Nc P P r Yp ri, r2
R. Re
Rt sc Si Sn So t
T Ul, U2, 1,13
Uo Ux, Uy, Uz V
V x, y, z
radius of filter element (granule) radius of clean filter element length of cell side dimensionless group mean capture efficiency of unit clean surface specific mean capture efficiency of a unit surface with a particle specific capture efficiency of a 'node' specific efficiency in the neighborhood of point i single element (sphere) efficiency capture efficiency of k-particle cluster capture efficiency of a 'node' correction factor height of porous layer filter length number of particles in a cluster permeability radius of unit cell deposited material mass number particle concentration density in the flow particle number concentration density at filter entrance number of particles per unit area captured on site i number particle concentration number of particles captured by the whole collector pressure probability of a given cell containing a single particle radius dust particle radius bounds to particle radius radius of a 'node' interception parameter non-dimensional bound to particle radius change area of a cell area of neighborhood of point i area of a 'node' area of a 'node' not occupied by particle time dimensionless time flow velocity in three different parts of flow field model superficial velocity flow velocity vector near collector surface volume fraction of solids in gas volume of filter bed cartesian coordinates 425
426
M. FICHMAN et al.
Greek symbols
effective void fraction, i.e. void fraction available for flow through bed clean bed void fraction ea porosity of deposits £p porous layer void fraction tl overall filtration efficiency ~'o
= r / k 1/2
viscosity P density of solids ff volume fraction of deposited solids t;r mean volume fraction of deposited solids in filter bed fluid stress n/nin
capture probability of the surface si q,n capture probability of a 'node' fluid stress ~ck capture probability of k-particle cluster stream function in different regions
INTRODUCTION Granular bed filters are used for the separation of particulate matter from gas streams in a wide range of particle sizes. The smaller submicron particles are removed by the mechanism of diffusion, while the larger particles are separated by interception and inertial mechanisms. In certain ranges of particle sizes, gravitation is also important. In the presence of electric fields, electric forces may be the most important. Several theoretical models which predict filtration efficiencies in various processes have been suggested (Paretsky et al., 1971; Gutfinger and Tardos, 1979). Each of these models starts with the analysis of a single spherical collector. The granular bed is assumed to consist of many such spheres of equal size, homogeneously distributed. To obtain the filter efficiency, one must first calculate the efficiency of a single sphere collector and then integrate over the whole filter bed. The single sphere efficiency depends on the trajectories of the particles flowing toward the spherical collector. These trajectories depend, of course, on the flow field around the sphere, and on other effects such as inertia, diffusion, interception, etc. The flow field must include the mutual influences of the neighboring spheres in the bed. The literature presents several theoretical models for the flow field around a single sphere in porous media (Happel, 1958; Kuwabara, 1959; Neale and Nader, 1974). These models assume creeping flows, for which analytical solutions are obtained. In addition to the spherical collector models, there are others which assume capillary tubes and constricted tube geometries (Payatakes et al., 1973a, b). Fichman et al. (1981) suggested modifications of the single sphere model, which took into account the direct interaction of neighboring spheres with the particle capture process, and not just their indirect influence through the modification of the flow field. Two new mechanisms were proposed: a 'sieve' mechanism, which affects particles in the inertial size range and a 'starvation' mechanism, important in the diffusional range. All these existing models are related to the performance of clean filters. THE TRANSIENT PROCESS The present work investigates the time dependent behavior of granular bed filters during particle deposition. Several studies concerned with the transient behavior of deep bed filters have been reported; these, however, considered only the filtration of particles from liquids. An extensive survey of these investigations appears in a review by Tien and Payatakes (1979). In filtration from liquids the ratio between the filtered particle size and the size of the collector is usually much larger than in gas filtration. In filtration from liquids this high ratio causes a very quick blocking of the filter by straining. To explain this deposition process Irmay (1969) suggests the formation of 'bridges' between adjacent collectors, which cause the narrow passage to be clogged, and the fluid to flow with higher velocity through the still open passages. Tien et al. (1979) use a combination of the spherical and the constricted tube
Particle loading on granular bed filtration
427
configuration to describe the transient behavior of filters. They assume the existence of two major stages in the deposition process. In the first stage the deposition occurs primarily by the adhesion of individual particles to the filter grains. The deposits form a smooth coating layer on the filter grains. This causes an increase in the collector size and a decrease in the bed porosity. Thus the models of clean filters can be extended to transient filtration, using modified filter parameters. In the second stage of the deposition process particle aggregates are formed. Some of these aggregates are reentrained and redeposited into the constricted pores. In this stage some of the pores become closed up. The pressure drop in the filter increases and the filter efficiency decreases. Experiments of aerosol filtration in granular beds, performed in the Technion laboratories (Fichman, 1983) have shown an increase in filter efficiencies and in the pressure drops, the extent of which cannot be explained by the formation of smooth coatings during the first stage of the filtration process. The fast increase in the filter efficiency in the first stage of filtration also appears in fibrous filters in which the deposits form so-called 'trees'. These 'trees' increase the filter efficiency due to a 'shadow effect' as shown by Wang et al. (1977), who calculated trajectories of particles, randomly injected into the fluid. The major reason for the formation of these 'trees' in fibrous filters is that in these cases the dust particles are just one order of magnitude smaller than the collector fibers, and the particles can very easily overshadow the fibers. On the other hand, in filtration of gas in granular beds the dust particles are much smaller than the collectors and no 'shadow effects' appear. Indeed, photographs of dirty collectors (Fichman, 1983) show no 'trees', but rather particle clusters with no preferred direction. All this leads to the conclusion that a different model is needed in order to explain the dynamics of aerosol filtration in granular bed filters. In this model the effect of the clusters on the enhancement of filtration should play a major role.
THE TIME D E P E N D E N T EQUATIONS The transport equations which govern monodisperse filtration were obtained by Herzig et al. (1970). The general form of the one-dimensional equations for filtration of discrete polydisperse particles was obtained by Tien and Payatakes (1979). Their equations do not rely on any particular model, and are therefore quite general. Several basic concepts which are necessary in the following exposition, are now introduced: The single collector efficiency, E. This is defined as the ratio between the amount of dust particles captured by the spherical filter element, whose diameter is 2a, and the total amount of particles flowing toward it in a cylindrical region of the same diameter, 2a, oriented along the flow direction. The number of particle concentration density function, n(r). This is defined as the number of particles in the gas per unit bed volume between the radii r and r + dr. The number particle concentration, N, is the total number of particles per unit bed volume and is related to n through: N =
;oon(r)dr.
(1)
Mass balance performed on a single layer of filter elements leads to the rate equation for particle deposition (Gutfinger and Tardos, 1979): IOn 1-e E n c~-~= 1.5 - - e --2a"
(2)
Aerosol mass conservation during the deposition is now considered. The volume fraction of solids in the dust laden gas, v, is given by: v =
rcr n(rp)drp,
(3)
drl
where rt, and r 2 are the limits of the particles radii. Considering a small control volume
M. FICHMANet al.
428
around a given point, the equation of aerosol mass conservation is obtained (Herzig et al., 1970) as:
Ov ~a Uo ~x + - ~ = O,
(4)
where a is the solidity of the deposit, i.e. the volume of solid deposit per unit volume of bed. Uo is the superficial gas velocity, i.e. the volumetric flowrate per unit cross-section area of the filter bed. Gas flow through the bed depends on its void fraction e. The space occupied by the deposit itself is not available for the gas flow, and the total volume of the porous deposit (including the volume of the pores) should be subtracted from that of the voids available for gas flow. Thus the 'effective' void fraction of the bed is 1 -a ee '
e = eo
(5)
where/30 is the bed void fraction before deposition started, ee is the porosity of the deposited layer and a/(1 - ed) is the volume fraction of the porous deposit. Substitution of equations (3) and (5) into equation (4) yields
4
("2
Uo ~ ~
Jr
On
t)/3
rp3 Ux dFp = (1 -- ed)--. at
(6)
1
Substitution of dn/Ox from equation (2) yields
uo l as_~,2
r3vnE dr v = (1 - ee) ~ '
(7)
1
The following dimensionless variables are defined:
n t# = - - ,
rp x R =--, X =
nin
r2
t Wo H '
T-
H'
(8)
where ni,(r) is the number of particles per unit volume at the filter entrance, between r and r+dr. Equations (2) and (7) can now be recast in a dimensionless form as ~q~ Ox
3H 1 -e Eth = 0 4 a /3
(9)
de C 1 - ~- fR R3Et#dR, O~-t-
(10)
where
C=
nr ~ noH (1 - ed) a"
The initial and boundary conditions for q~ and/3 are: th=l
at
X=0
/3 =/3o, ck = dpo(X,R)
(at all T) at
T = 0,
(11) (12)
where ~o is the dimensionless concentration, in a clean filter, which can be obtained from the rate equation (3) as: t~o = exp
(
31-e°EoX 4 ao/3~
)
"
(13)
Equations (9) and (10) with the initial and boundary conditions, equations (11) and (12), describe the dynamic operation of a granular bed filter. They can be solved analytically for monodisperse filtration. For the general case these equations have to be solved numerically.
Particle loading on granular bed filtration
429
N U M E R I C A L S O L U T I O N OF THE D Y N A M I C E Q U A T I O N S Equation (9) may be integrated, and together with boundary condition (11) it yields: 4 = exp
-4 a
e
A predictor-corrector method is used to calculate the void fraction from equation (10), while the relation between q~and e is obtained from equation (14) by a numerical integration. The grid for the numerical calculation is presented in Fig. 1. The discrete form of equation (14) is:
(f~,-f~)+Ax,~_2f~n
qS~j= exp
,
(15)
where f ik =
3 H 1 - e0 Ek ' 4 a eij
and the subscripts i, j, k denote the appropriate steps in time, position and particle radius, respectively. The trapezoidal rule of numerical integration is used in replacing the integral in equation (14) by the summation in equation (15). Now the summation in equation (15) may be rewritten in terms of a recursive relationship k FAx ] = @,j expLT (f/k S-, +f/kj) ,
(16)
The advance in the T-direction is done by rewriting equation (10) in discrete form, using the predictor-corrector method. The predictor-equation for e is: ei+½,j- ei~ _ AT
C
Eijq~ijR 3 dR.
(17)
JR, e~j
The equation for ~bmay be deduced from equation (17) in terms of the half steps required in the predictor-corrector method:
= O,+½J"-, expLyrAx(fJ+½j-l'k
+fi+½,j)l,'k
T
I
X
~(jl
i x
Fig. 1. Grid for numerical calculations.
(18)
430
M. FICHMAN et al.
The corrector-equations are: ?'i+ 1, j -- gij = -- C
AT
f l 1 - ~ i +½"J ~i+½,j ~ i +½j R 3 dR; I~i_l.½,j
(19)
,
4~+1,~ = 4~+1,j-1 exp ~-(f~+,,~_l ~f~+l,j) •
(20)
Trapezoidal numerical integration is used in all integrals. Equations (17)-(20) together with the initial and boundary conditions given by equations (11) and (12) set the numerical scheme for the solution of the dynamic filtration equations as functions of time, T. In practice this solution can be performed once the single collector filtration efficiency, E, is known as a function of position and time. The solution of the problem is sought in terms of the total deposited material, rather than of time. The total mass, M, of solids of density p deposited in a filter of volume V is: M
~=pv
(1 - e , )
Io
leo
e(T)]dx.
(21)
Equation (21) now constitutes an expression for the total mass of material deposited in the filter as a function of filter porosity, e, which itself is a function of time and position in the filter. The function e(X, T) can be calculated from the integration of equations (17)-(20), which can be performed once the single collector efficiency, E, is known. In what follows we propose the cluster enhanced filter (CEF) model for the filtration process. The model predicts the single collector efficiency, E, and thus makes the solution of the dynamic filtration problem possible. DYNAMIC FILTRATION MODELS The single collector efficiency, E, in a dirty filter must be known before the dynamic equations for the filter can be solved. The simplest way to proceed seems to use the clean filter model for description of the dynamic process. In this model the deposited material is assumed to form a smooth layer, resulting in a monotonous increase in the size of the collectors and a decrease in bed void fraction. The change in the single collector efficiency with time is expressed here through the change of the bed void fraction and of the collector size. Such a smooth layer deposit model was reasonably successful in describing liquid filtration (Tien et al., 1979), and it was hoped to be applicable to gas filtration also. Figure 2 shows typical filtration data obtained at Technion (Fichman, 1983) and some filter efficiencies
!
•
H=gcm 2a=l.Smm U = 7.5cm • E x p e r i m e n t a l data
0,~
N
I !
Smooth coating
theory
I 2
M(g)
Fig. 2. Overall filtration efficiency as a function of total mass of loaded material---comparison between experimental data and theoretical calculations using smooth coating theory.
Particle loading on granular bed filtration
431
calculated from equation (17) + (21) under the assumption of a dynamic smooth coating filter model. As seen, this smooth coating model cannot be applied to aerosol filtration, as it grossly underpredicts filtration efficiencies in comparison to these, obtained by experiments. Moreover, observations under a microscope show that the deposited particles do not form a smooth layer on the collector. The deposition is in the form of clusters, where each cluster acts as a secondary collector, thus increasing the collection efficiency of the primary collector. In the description of the deposition model it is very difficult to relate to each and every secondary collector and thus calculate the combined collection efficiency. A more effective approach is to consider the whole porous layer which these secondary collectors form. Thus, to obtain the single collector efficiency in a dirty filter, a model of a deposited high porosity layer on the collector surface is employed. This layer grows as the deposition process proceeds. It describes the dynamic behavior of granular bed filter at the first phase of the filtration process, before reentrainment takes place. This first phase is divided into three stages, as described presently. THE L E N G T H SCALES OF THE M O D E L The filtration of particles from a flow in porous media presents two different length scales. The scale connected with the filter efficiency and the particle trajectory is the large scale. Its characteristic length is the collector diameter. On this scale, the deviation of the collector from a perfect spherical shape can be neglected. The effect of a dust particle attached to the collector surface on the overall flow field can also be disregarded. Many physical parameters which characterize the filtration effects are obtained as averages on this large scale. The small scale is of the order of a dust particle. To describe the dust particle arrangement on the collector surface this smaller scale must be used. The general form of the mean flow that brings dust particles towards the collector is not important on this scale. What is important is the local surface shape, the local flow near the collector surface and the shape of the particle. All these determine the arrangement of the particles on the collector surface. A number of assumptions must now be made to successfully describe what happens on the smaller scale. The irregularities in the collector surface structure are assumed to be random, resulting in the local flow directions also being of random nature. The particles are assumed to be randomly collected on the collector surface, and to be randomly arranged, and therefore, on the average, arranged uniformly. The model assumes that the particles are distributed on the surface of the collector in such a way that equal cells with particles in their centers are formed (Fig. 3). The transient filtration process, before reentrainment takes place, is now considered. This process is divided into three different stages.
•
L
I. /
•
",,,---'--(
/-~
/
i
S •
l -L
cell boundary
\
~
~ "--/
"
~--
/
,
\ e
I
/
particle
Fig. 3. Cell structure formed by depositedparticles on the collectorsurface.
432
M. FXCHMANet al.
The first staoe---cell formation At the beginning of the filtration process the particles are deposited on the collector, forming an arrangement of equal cells. The particles which come next are collected, on the average, between those already deposited, in such a way that each cell size decreases with time. Through this stage the capture of particles by those captured earlier can be neglected. This approximation is valid as long as the total number of collected particles is small. Once this approximation is no longer valid, the second stage of the collection begins.
The second stage--cluster formation In this stage aerosol particles are collected in both new, clean areas, and on top of the earlier deposited particles. Clusters that consist of particles captured one on top of the other start to grow. The manner of capture and the position of the particles in the cluster are assumed to be random. The location where a cluster is formed is called a node. When the number of clusters increases sufficiently the combined capture efficiency of the clusters becomes greater than that of the clean areas and the third stage of collection commences.
The third staoe--cluster enhanced filtration (CEF ) In this stage particles are captured mostly by clusters and the capture by clean areas can be practically neglected. The present model states the following sequence of events: in the first stage the quantity of the deposited material is small and the single collector efficiency increases slowly. The second stage covers the formation of cell nodes, i.e. the locations of the clusters, and the final characteristic cell size. In the third stage the single collector efficiency increases at a high rate. At this stage the particles in the clusters penetrate into the flow field where the gas velocity is high and, therefore, the aerosol particle flux is also high. This leads to an enhancement in single collector efficiency. In the calculations that follow, the size of the cell is taken as that corresponding to the end of the second stage, when the cell size stops changing. This size is kept constant through the deposition process during the third stage.
THE C H A R A C T E R I S T I C L E N G T H OF AN I N D I V I D U A L CELL The characteristic length of an individual cell is assumed fixed at the end of the second stage. The second stage is completed when the probability of capture of a particle by a node or a cluster is equal to the probability of its capture by a clean area and the formation of a new cell. Thus, to obtain this characteristic cell length, these probabilities must be found. The local specific capture efficiency of a certain site on a collector surface is defined as the ratio between the number of particles captured per unit area there and the total number of particles captured by the whole collector: ni
ei = ~ .
(22)
Here ni is the number of particles per unit area captured on site i and Nc is the number of particles captured by the whole collector. To obtain the local specific capture efficiency at a certain point, a small area around this point must be chosen and the number of particles captured by this area must be calculated. A local cartesian coordinate system is now introduced such that the local flow velocity is in the x-direction (Fig. 4). Because the flow region considered is near the surface, u~ ~ 0. The xcomponent of the flow can be written as:
ux = bz + 0(z2).
(23)
The number of particles captured by the small area Asl is equal to the number of particles in
Particle loading on granular bed filtration
(Q)
433
(b)
Fig. 4. Capture of particles (a) by clean area and (b) by previously deposited particle.
the layer extending from the collector surface up to the height rp (Fig. 4a):
~rpnAyi UxA t dz 0
nl Asi = i~
= nAyibAt
f ~pzdz
= nAyibAtr~/2,
(24)
d
where n is the number concentration density of particles in the flow. Hence, the local specific capture efficiency at the point i is: bat ei = ~ c
2 Ayi
(25)
rlrp Asi "
To calculate the number of particles captured on a site already occupied by a particle, the integration must proceed up to the height of 3rp (see Fig. 4b). The particles already deposited on the collector have a negligible influence on the flow field. Therefore, to obtain the capture efficiency at a point which already has a captured particle, eip, it suffices to replace rp in equation (25) by 3r~, hence, eip ~ ~
n(3rp) 2
•
(26)
Denote the area-average capture efficiency by eo. Then division of equation (26) by equation (25) yields: eo~ ~ 9. ep
(27)
This means that on the average the local specific capture efficiency of a site with a captured particle is 9 times higher than that of a clean area. Moreover, nodes for the formation of particle clusters are not just at a point where a particle is found, but also in the neighboring domain. The definition of the size of such a neighboring domain is based on the following considerations: if a second particle is captured within a distance less than 4rp from the first particle, no additional particle can squeeze between these two particles and form a node. Thus the radius, Rn, of the domain in the neighborhood of a captured particle, i.e. where the additional particle causes the formation of a cluster, is defined as (see Fig. 5): Rn = 4rp
(28)
and the whole neighborhood domain is denoted as a 'node'. The capture efficiency of the empty area in a 'node' (the shadowed area in Fig. 5) is: eoSo = eo nl-(4rp)2 -- r 2] = eo" 15 nr 2 .
(29)
The capture efficiency of the whole 'node' comes out as: E~ = eopS, + eoSo = 9eonr 2 + 15eonr 2 = 24nr2eo.
(30)
Now the probability of the capture of one particle on the site surface s~and its incorporation into the 'node' must be defined. This probability function is normalized such that the probability of capture of one particle in the cell is 1. If No is the number of particles captured
434
M. FICHMAN et al.
/
\
/
/
\ "node"
\
/ \
I
/
Fig. 5. Structure of a single cell.
in the cell and Ni is the number of particles captured on the surface si, the capture probability o f the surface si, ~b~,is defined as: N i
d/i
Ilisi
noeiSi
. . . . . Nc
eisi -
ncsi
noecSc
.
(31)
ecSc
The area of a cell of characteristic length c is: s~ = c 2 .
(32)
The probability of capture on the 'node' and forming a cluster can be calculated by using the definition o f equation (31): e.s.
~b.
(33)
(c 2 - s.)eo + e.s.
where e.s. = E . is substituted from equation (30) and s. is calculated using equation (28): s. = rcR2 = 16~r2
(34)
resulting in 3 ~k, = 1 + 0.0398(c/rp) 2"
(35)
Now let the probability o f the capture of a particle by a cluster be calculated. First consider a cluster consisting of two particles. The probability of capture by such a cluster is designated by q&2. The two particles in the cluster can be located either one on top of the other or one beside the other. For a two-particle cluster, structured as one particle on top of the other, the capture efficiency can be obtained by replacing 3r~ by 5r~ in equation (26). This leads to Ec2 = 25eonr 2 + 15eonr 2 = 40nr~eo.
(36)
When the two particles in the cluster are located one beside the other the capture efficiency is enhanced to about twice that o f a node, equation (30). E~2 ~ 2E. = 48nr2eo.
(37)
Comparison of equations (36) and (37) yields that in both cases the capture efficiency is roughly the same. Hence, the probability of capture by a two-particle cluster, qsc2, is approximated by twice that of a node.
0~2 = 20..
(38)
Similarly, it can be shown that every additional particle contributes 4. to the probability of capture; hence, for a cluster of k particles @ok = k~0..
(39)
Particle loading on granular bed filtration
435
To obtain the probability of capture by an 'average' cluster, the distribution of the particles between the clusters is needed. This distribution can be obtained by reconsidering the problem from the point of view of the large scale. On the macroscale, the probability of capture by a cell does not depend on the cell surface structure and, hence, it is independent of the particles already occupying the cell. On the other hand, on the microscale, the particles located inside a given cell do influence the incoming particles in how they arrange themselves in the cell. Returning to the large scale picture, it is noted that because particles are distributed homogeneously in the main flow, the capture probability of each cell should be the same. This probability is equal to the ratio between the cell area and the total collector area available for capture. Let the probability of a given cell to contain a single particle be denoted by p. Now the probability o f capturing a particle is equal for all the cells, and does not depend on the number of particles already in it. Therefore, the probability of a given cell to contain two particles is p2, and generally, the probability of containing k particles is pk. By definition, each cell is occupied by some particles, one or more, therefore p + p2 + p3 + . . . . pk = 1.
(40)
The solution of equation (40) for k ~ ~ yields the probability for a given cell to contain a single particle 1
P
2
(41)
The probability of a particle that approaches the collector to be captured by a node, i.e. by 'single-particle cluster' is @,. The probability of a particle to be captured by a cluster containing any number of particles, including one, is @c = @,,P + ¢c2P 2 + I/.Ic3P3 + . . . .
~b
+ 2" + 3"~ + .
----2¢,.
(42)
The second stage of filtration is defined to end when the probability of capture by a cluster or a node is equal to the probability of capture by a clean surface, i.e. 1
¢c = ~.
(43)
Or, from equation (42): 1
¢n
(44)
4
Substitution of this ft, into equation (35) yields 3
1
1 + 0.0398(c/rp) 2 = -4'
(45)
which results in the characteristic length of an individual cell c/rp = 16.6.
(46)
THE FILTER PARAMETERS IN THE THIRD STAGE The cell size has been obtained from the second stage. The single sphere collection efficiency is now sought, as a function of time. For this the void fraction of the deposited layer must first be obtained. To do this the individual clusters are regarded as single elements of the deposited porous layer. Let the characteristic length of a cluster be denoted by h, and that of a cell containing the duster by c. The void fraction of the porous layer, ep, protruding from the collector surface at
436
M. FICI-tMANet al.
the height h, is the cluster-free volume divided by the total volume of the porous layer
Now, the height of the porous layer, which equals that of a cluster, is calculated as a function of the solidity of the deposited material, a. To do this, the 'unit cell' model of a single collector (Happel, 1958; Kuwabara 1959; Neal and Nader, 1974) is used. The model consists of a test sphere surrounded by a concentric cell of radius L, as shown in Fig. 6. The volume of each unit cell is equal to the volume of the filter divided by the number of the spherical collectors in the filter. As the number of collectors does not change during the filtration process the cell radius, L, remains constant while the effective collector radius increases from ao to a. Thus L = ao
( )lj3 / )13 = a
(48)
The fraction of the volume of the unit cell occupied by the material deposited on the collector is the solidity, a, of the deposited material which forms a porous layer of thickness h and void fraction ep on the collector surface. The solidity is calculated as a --
4na2h(1 - ~p) (I - ed) 4/3rrL 3 ,
(49)
where ~d is the porosity of each individual cluster. Substitution ofep from equation (47) yields an expression for h .r
O'(C/a) 2 11/3
h = LL-3~d) j
,
(50)
and, finally, using equation (48) to eliminate L, results in h=
3 ( 1 - ~ o ) ( 1 - ~.)
"
(51)
Equations (47) and (51) connect the large scale parameters of the problem: the initial bed void fraction, ~o, the deposit solidity, e, the porous layer void fraction, ep, and the size, c, of the cell containing the cluster. These relations are now used in the determination of the flow field around a single collector in a dirty filter.
Fig. 6. Model of a dirty collector inside a porous matrix.
Particle loading on granular bed filtration
437
M O D E L FOR THE F L O W F I E L D IN A DIRTY F I L T E R The flow field inside the filter is assumed to be creeping flow. The unit cell model of a single collector for a clean filter consists of a sphere surrounded by a concentric cell (Happel, 1958), with this cell embedded inside an infinite region of porous media (Neal and Nader, 1974). The difference between the clean filter model and the one proposed here for a dirty filter is that in the model for the dirty filter there is a porous layer on the collector surface also. Thus the suggested model of a dirty filter contains three different regions: the first region (region 1, Fig. 6) is the porous layer on the collector surface; the second region (region 2, Fig. 6) is the empty space inside the cell; and the third region (region 3, Fig. 6), is the infinite porous medium outside the cell. The governing flow equations in regions 1 and 3 are the Brinkman equations (Brinkman, 1947, 1949) and those in region 2 are the Stokes equations for creeping flow. Let these equations now be written out: For a <~ r <~ a + h , -~Tu~ +~V2ul = AP~
VUl =
0
(52)
for a + h <~ r <~ L 2u2 =
f]IV
AP2
(53)
Vu2 = 0,
and for r I> L (
- ~ 3 u 3 +/.tV2ua = VPa Vua = 0
(54)
As was first suggested by Brinkman (Brinkman, 1947), the conditions on the surfaces between the regions are continuity of velocities and equality of stresses, i.e. ul, = Ulo = 0
{ulr
U2r; UlO
U20
P1
= P2
"ElrO ~- T2rO;
u2,
U3r; 1120
"E2rO "~" "~3rO U3r ~
U30
at r = a
(55)
at r = a + h
(56)
at r = L
(57)
P2 = P3 U o COS0
u3o --* Uo sin0
r --} ~ .
(58)
Define non-dimensional coordinate ~ and ~ for regions 1 and 3, respectively r
rl
--- k~/2 " ~ - k~/2 •
(59)
The solutions for the stream functions, assuming spherical symmetry (Happel and Brenner, 1965; Neal and Nader, 1974) everywhere are for~ ~< ~ < / ~ qJl = k ~ ° [ E ~ - l + f ~ - 2 + G l e ~ ( l + ( - ~ ) + H ~ e ~ ( 1 - ( - ~ ) ] s i n 2 0
(60)
forfl~<~
(61)
438
M. FICHMAN et al.
fore > ~ ~b3 = ~
[E3~ - 1 + F3~2 + G3e-~(1 + ~- 1) + H3eg(1 _ ~- 1)]sin20,
(62)
where
a &-k~/2" The boundary conditions constants. This system can dirty single filter collector. analysis may be continued
a+h a+h L [3= k~/2 ; f l = k~/2 " y = k ~ / 2 .
(54)-(58) give a system of linear algebraic equations for the easily be solved numerically, resulting in the flow field around a Now that the flow field in region 2 is known, equation (60), the to calculate the single element collection efficiency E.
F I L T R A T I O N E F F I C I E N C Y OF A DIRTY F I L T E R With the flow field just developed the single collector efficiency can be obtained from the equations of particle motion which yield the particles' trajectories (Gutfinger and Tardos, 1979). When inertia can be neglected, particles move along streamlines and are collected by interception. In this case the single collector efficiency is (Gutfinger and Tardos, 1979)
E =
a2
(63)
Inertia causes a particle to transfer from one streamline to another, thus enhancing filtration efficiency. Far away from the collector, where the curvature of the streamlines is small, a particle moves along some initial streamline; when getting close to the collector, where the streamlines curve appreciably, inertia makes the particle leave its initial streamline and move closer to the collector surface. The overall difference between the locations of the initial and the final streamlines at 0 = ~t/2, At, can be calculated from the equations of particle motion. When the effect of inertia is small, the difference Ar can be approximated by the expression (Fichman and Pnueli, 1982): Ar = 3.37. St. Rp(St 2 + 16Rp2) 1/4
(64)
The resulting Ar, exact or approximated, is added to a + h in equation (64) to account for inertia effects in the calculation of the single collector filtration efficiency. This efficiency is calculated using the stream function in region 2, Oz, given by equation (61), and this stream function depends on the porous layer thickness, h v, and on the void fraction ev, which, in turn, are functions of the deposit solidity, equations (47) and (51). Thus, using equations (64), (47), (51) and (61), the single filter element collection efficiency, E, can be calculated as a function of the deposit solidity. Once E is known it is used to calculate the overall filtration efficiency. The overall filtration efficiency, ~/, is defined as: r/= 1
n°ut- 1-~b(1),
(65)
F/in
where ~(1) is calculated by integrating numerically the time dependent dynamic filter equations, equations (9) and (10), using the algorithms given by equations (16)-(20). In these calculations, the time is used as a parameter, which by means of equation (21) connects ~band e (or a) with the mean volume fraction of the deposited material tr. Thus, the overall filtration efficiency is a function of several parameters including the mean volume fraction of the deposited material, a,
n/no = FEE(St, Rp, eo, 6), H/a],
(66)
where r/o is the overall efficiency of a clean filter. Figure 7 presents the overall filtration efficiency as a function of 6 for several values of the different parameters in equation (67). These filter efficiencies are calculated for interceptional
Particle loading on granular bed filtration
'I3
,0-3
439
2-~=60
Rp=3.75.10-~
o
?2
IF [ i0-4
I
I
5.10-4
I
[
10-4
10-3
I
5.10-4 o"
iO-S
Fig. 7. Normalized filtration efficiency as a 'function of mean volume fraction of the deposited material: (a) H/2a = 30; (b) H/2a = 60.
=120 2•0 St =0.O4
Rp=5"lO-3 o
O( ~
°
'
=
~_
0.2 I
I
O
~ . _ ~ I
02
I
4
~"='=='~" O"= 1.8' 10 .4 I °'=3"10-4
06 x/H 1"t=120
~-O.78'10"
1
St=O.Q3
~.~ 0.1" ,-
°"=0"85"10-4 or 1.56-10-4 o- =7.7.10-'~
O-2~ ~
I
I
I
0.2
l
1
11=120 2a St=O.007 Rp=3.75.10_s
I c~ 1 "~ c 0.,
I
0-6 x/H
6
~
°'=0 O"=1.5.10-4 G'=4.~" IO-4
0"2 1
0'2
I
I
0"6 x/H
I
I
1
Fig. 8. Number particle concentration density distribution in the gas phase along the filter for different filtration parameters.
440
M. FICHMANet al.
R
H=9cm 2a=l-5mm U=7-5 cm • Experimental --
0.5
dato
Smooth coating theory Present theory
I o
I
2
I
M(g)
Fig. 9. Overall filtration efficiency as a function of total mass of loaded material---comparison between experimental data and theoretical calculations, using smooth coating theory and the CEF model.
and inertial depositions only. It can be seen that for Rp > 10-3 the filtration efficiency increases very quickly with 5. Figure 8 shows the change in the particle concentration in the gas as it flows along the filter bed, for various values of d. Figure 9 reproduces the experimental data of Fig. 2, together with the curves corresponding to the 'smooth coating theory' presented there. The theoretical curve obtained from the present theory is added to the figure. The plot is given in terms of the overall filter efficiency as a function of particle loading. It seems that the agreement between the present theoretical results and the experimental measurements is quite good. SUMMARY This work treats the time dependent behavior of a granular bed filter while it filters small particles from a gas, starting in a clean condition, and then being loaded with particles. A cluster enhanced filter (CEF) model has been proposed to describe the process as it develops in time. The first stage is deposition on clean surfaces. It is followed by the formation of clusters of particles, which constitutes the second stage. Then, in the third stage, deposition takes place on the clusters with a very high enhancement in efficiency. The transitions between the stages are well defined and clearly formulated. The results, which predict a very high increase in efficiency, are quite different from those of the 'smooth coating' theory developed for liquid filtration, which predicts only moderate increase in filtration efficiency, and which could not explain the very high deposition rates in the case of gas filtration. The C E F model provides filtration efficiency data as a function of the mass of the deposited material in the filter. The theory compares favorably with experimental data. REFERENCES Brinkman, H. C. (1947) A calculation of the viscous force exerted by a flowingfluid on a dense swarm of particles. Appl. Scient. Res. AI, 27. Brinkman, H. C. (1949)Problems of fluid flowthrough swarmsof particlesand through macromoleculesin solution. Research (Lond.) 2, 190.
Fichman, M. (1983) D.Sc. thesis, Technion, Haifa. Fichman, M., Gutfinger, C. and Pnueli, D. (1981) A modified model for deposition of dust in a granular bed filter. Atmos. Envir. 15, 1669. Fichman, M. and Pnueli, D. (1982) More detailed analysis of the effect of inertia on the filtration efficiencyof granular bed filters. Israel J. Tech. 20, 60.
Particle loading on granular bed filtration
441
G utfinger, C. and Tardos, G. (1979) Theoretical and experimental studies on granular bed filtration. A tmos. Envir. 13, 853-867. Happel, J. (1958) Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles. A.I.Ch.E.J. 4, 197. Happel, J. and Brenner, H. (1965) Low Reynolds Number Hydrodanamics, Prentice Hall, New York, Herzig, J. P., Leclerc, D. M. and Le Goff, P. (1970) Flow of suspensions through porous media--application to deep bed filtration. Ind. Engng. Chem. 62, 8. lrmay, S. (1969) The mechanism of filtration of non-colloidal fines in porous media. Symposium of the Fundamentals of Transport Phenomena in Porous Media, Haifa, 5. Kuwabara, S. (1959) The forces experienced by randomly distributed parallel circular cylinder or spheres in viscous fluid at small Reynolds numbers. J. Phys. Soc. Japan 14, 527. Neal, H. N. and Nader, W. K. (1974) Prediction of transport processes within porous media. Creeping flow relative to a fixed swarm of spherical particles. A.I.Ch.E.J. 20, 530. Paretsky, L., Theodore, L., Pfeffer, R. and Squires, A. M. (1971 ) Panel bed filters for simultaneous removal of fly ash and sulfur dioxide, li. J. Air Pollut. Control Ass. 21,204. Payatakes, A. C., Tien, C. and Turian, R. M. (1973a) A new model for granular porous media: Part 1. Model formulation. A.I.Ch.E.J. 19, 58. Payatakes, A. C., Tien, C. and Turian, R. M. (1973b) A new model for granular porous media: Part II. Numerical solution of steady state incompressible Newtonian flow through periodically constricted tubes. A.I.Ch.E.J. 19, 67. Tien, C. and Payatakes, A. C. (1979) Advances in deep bed filtration. A.I.Ch.E.J. 25, 737. Tien, C., Turian, R. M. and Pendse, H. (1979) Simulation of the dynamic behavior of deep bed filters. A.I.Ch.E.J. 25, 385. Wang, C. S., Beizaie, M. and Tien, C. (1977) Deposition of solid particles on a collector: formulation of a new theory. A.I.Ch.E.J. 23, 879.
0021-8502/88 $3.00+0.00 © 1988 Pergamon Press plc
EFFECT OF PARTICLE LOADING ON GRANULAR BED FILTRATION--THE CLUSTER E N H A N C E D FILTER MODEL M . FICHMAN, C. GUTFINGER a n d D. PNUELI Faculty of Mechanical Engineering, Technion, Israel Institute of Technology, Haifa, 32000, Israel (Received 11 March 1987; and in final form 4 July 1987)
Abstract--A cluster enhanced filter (CEF) model is proposed for the filtration process in granular bed filters. The formulation is made in terms of particle deposition on a single spherical collector. The model accounts for the effect of particle loading on the dynamic behavior of the filter. It predicts the formation of a high porosity layer, which consists of clusters of particles, on the surface of the collector, with a very high increase in the calculated filtration efficiency. The theory is presented in terms of filtration efficiency data as a function of the mass of the loaded material. The results compare favorably with experimental data.
NOMENCLATURE a ao c
C ea elp en el
E
En h H k kt, k2, k3 L M n nin ni
N Nc P P r Yp ri, r2
R. Re
Rt sc Si Sn So t
T Ul, U2, 1,13
Uo Ux, Uy, Uz V
V x, y, z
radius of filter element (granule) radius of clean filter element length of cell side dimensionless group mean capture efficiency of unit clean surface specific mean capture efficiency of a unit surface with a particle specific capture efficiency of a 'node' specific efficiency in the neighborhood of point i single element (sphere) efficiency capture efficiency of k-particle cluster capture efficiency of a 'node' correction factor height of porous layer filter length number of particles in a cluster permeability radius of unit cell deposited material mass number particle concentration density in the flow particle number concentration density at filter entrance number of particles per unit area captured on site i number particle concentration number of particles captured by the whole collector pressure probability of a given cell containing a single particle radius dust particle radius bounds to particle radius radius of a 'node' interception parameter non-dimensional bound to particle radius change area of a cell area of neighborhood of point i area of a 'node' area of a 'node' not occupied by particle time dimensionless time flow velocity in three different parts of flow field model superficial velocity flow velocity vector near collector surface volume fraction of solids in gas volume of filter bed cartesian coordinates 425
426
M. FICHMAN et al.
Greek symbols
effective void fraction, i.e. void fraction available for flow through bed clean bed void fraction ea porosity of deposits £p porous layer void fraction tl overall filtration efficiency ~'o
= r / k 1/2
viscosity P density of solids ff volume fraction of deposited solids t;r mean volume fraction of deposited solids in filter bed fluid stress n/nin
capture probability of the surface si q,n capture probability of a 'node' fluid stress ~ck capture probability of k-particle cluster stream function in different regions
INTRODUCTION Granular bed filters are used for the separation of particulate matter from gas streams in a wide range of particle sizes. The smaller submicron particles are removed by the mechanism of diffusion, while the larger particles are separated by interception and inertial mechanisms. In certain ranges of particle sizes, gravitation is also important. In the presence of electric fields, electric forces may be the most important. Several theoretical models which predict filtration efficiencies in various processes have been suggested (Paretsky et al., 1971; Gutfinger and Tardos, 1979). Each of these models starts with the analysis of a single spherical collector. The granular bed is assumed to consist of many such spheres of equal size, homogeneously distributed. To obtain the filter efficiency, one must first calculate the efficiency of a single sphere collector and then integrate over the whole filter bed. The single sphere efficiency depends on the trajectories of the particles flowing toward the spherical collector. These trajectories depend, of course, on the flow field around the sphere, and on other effects such as inertia, diffusion, interception, etc. The flow field must include the mutual influences of the neighboring spheres in the bed. The literature presents several theoretical models for the flow field around a single sphere in porous media (Happel, 1958; Kuwabara, 1959; Neale and Nader, 1974). These models assume creeping flows, for which analytical solutions are obtained. In addition to the spherical collector models, there are others which assume capillary tubes and constricted tube geometries (Payatakes et al., 1973a, b). Fichman et al. (1981) suggested modifications of the single sphere model, which took into account the direct interaction of neighboring spheres with the particle capture process, and not just their indirect influence through the modification of the flow field. Two new mechanisms were proposed: a 'sieve' mechanism, which affects particles in the inertial size range and a 'starvation' mechanism, important in the diffusional range. All these existing models are related to the performance of clean filters. THE TRANSIENT PROCESS The present work investigates the time dependent behavior of granular bed filters during particle deposition. Several studies concerned with the transient behavior of deep bed filters have been reported; these, however, considered only the filtration of particles from liquids. An extensive survey of these investigations appears in a review by Tien and Payatakes (1979). In filtration from liquids the ratio between the filtered particle size and the size of the collector is usually much larger than in gas filtration. In filtration from liquids this high ratio causes a very quick blocking of the filter by straining. To explain this deposition process Irmay (1969) suggests the formation of 'bridges' between adjacent collectors, which cause the narrow passage to be clogged, and the fluid to flow with higher velocity through the still open passages. Tien et al. (1979) use a combination of the spherical and the constricted tube
Particle loading on granular bed filtration
427
configuration to describe the transient behavior of filters. They assume the existence of two major stages in the deposition process. In the first stage the deposition occurs primarily by the adhesion of individual particles to the filter grains. The deposits form a smooth coating layer on the filter grains. This causes an increase in the collector size and a decrease in the bed porosity. Thus the models of clean filters can be extended to transient filtration, using modified filter parameters. In the second stage of the deposition process particle aggregates are formed. Some of these aggregates are reentrained and redeposited into the constricted pores. In this stage some of the pores become closed up. The pressure drop in the filter increases and the filter efficiency decreases. Experiments of aerosol filtration in granular beds, performed in the Technion laboratories (Fichman, 1983) have shown an increase in filter efficiencies and in the pressure drops, the extent of which cannot be explained by the formation of smooth coatings during the first stage of the filtration process. The fast increase in the filter efficiency in the first stage of filtration also appears in fibrous filters in which the deposits form so-called 'trees'. These 'trees' increase the filter efficiency due to a 'shadow effect' as shown by Wang et al. (1977), who calculated trajectories of particles, randomly injected into the fluid. The major reason for the formation of these 'trees' in fibrous filters is that in these cases the dust particles are just one order of magnitude smaller than the collector fibers, and the particles can very easily overshadow the fibers. On the other hand, in filtration of gas in granular beds the dust particles are much smaller than the collectors and no 'shadow effects' appear. Indeed, photographs of dirty collectors (Fichman, 1983) show no 'trees', but rather particle clusters with no preferred direction. All this leads to the conclusion that a different model is needed in order to explain the dynamics of aerosol filtration in granular bed filters. In this model the effect of the clusters on the enhancement of filtration should play a major role.
THE TIME D E P E N D E N T EQUATIONS The transport equations which govern monodisperse filtration were obtained by Herzig et al. (1970). The general form of the one-dimensional equations for filtration of discrete polydisperse particles was obtained by Tien and Payatakes (1979). Their equations do not rely on any particular model, and are therefore quite general. Several basic concepts which are necessary in the following exposition, are now introduced: The single collector efficiency, E. This is defined as the ratio between the amount of dust particles captured by the spherical filter element, whose diameter is 2a, and the total amount of particles flowing toward it in a cylindrical region of the same diameter, 2a, oriented along the flow direction. The number of particle concentration density function, n(r). This is defined as the number of particles in the gas per unit bed volume between the radii r and r + dr. The number particle concentration, N, is the total number of particles per unit bed volume and is related to n through: N =
;oon(r)dr.
(1)
Mass balance performed on a single layer of filter elements leads to the rate equation for particle deposition (Gutfinger and Tardos, 1979): IOn 1-e E n c~-~= 1.5 - - e --2a"
(2)
Aerosol mass conservation during the deposition is now considered. The volume fraction of solids in the dust laden gas, v, is given by: v =
rcr n(rp)drp,
(3)
drl
where rt, and r 2 are the limits of the particles radii. Considering a small control volume
M. FICHMANet al.
428
around a given point, the equation of aerosol mass conservation is obtained (Herzig et al., 1970) as:
Ov ~a Uo ~x + - ~ = O,
(4)
where a is the solidity of the deposit, i.e. the volume of solid deposit per unit volume of bed. Uo is the superficial gas velocity, i.e. the volumetric flowrate per unit cross-section area of the filter bed. Gas flow through the bed depends on its void fraction e. The space occupied by the deposit itself is not available for the gas flow, and the total volume of the porous deposit (including the volume of the pores) should be subtracted from that of the voids available for gas flow. Thus the 'effective' void fraction of the bed is 1 -a ee '
e = eo
(5)
where/30 is the bed void fraction before deposition started, ee is the porosity of the deposited layer and a/(1 - ed) is the volume fraction of the porous deposit. Substitution of equations (3) and (5) into equation (4) yields
4
("2
Uo ~ ~
Jr
On
t)/3
rp3 Ux dFp = (1 -- ed)--. at
(6)
1
Substitution of dn/Ox from equation (2) yields
uo l as_~,2
r3vnE dr v = (1 - ee) ~ '
(7)
1
The following dimensionless variables are defined:
n t# = - - ,
rp x R =--, X =
nin
r2
t Wo H '
T-
H'
(8)
where ni,(r) is the number of particles per unit volume at the filter entrance, between r and r+dr. Equations (2) and (7) can now be recast in a dimensionless form as ~q~ Ox
3H 1 -e Eth = 0 4 a /3
(9)
de C 1 - ~- fR R3Et#dR, O~-t-
(10)
where
C=
nr ~ noH (1 - ed) a"
The initial and boundary conditions for q~ and/3 are: th=l
at
X=0
/3 =/3o, ck = dpo(X,R)
(at all T) at
T = 0,
(11) (12)
where ~o is the dimensionless concentration, in a clean filter, which can be obtained from the rate equation (3) as: t~o = exp
(
31-e°EoX 4 ao/3~
)
"
(13)
Equations (9) and (10) with the initial and boundary conditions, equations (11) and (12), describe the dynamic operation of a granular bed filter. They can be solved analytically for monodisperse filtration. For the general case these equations have to be solved numerically.
Particle loading on granular bed filtration
429
N U M E R I C A L S O L U T I O N OF THE D Y N A M I C E Q U A T I O N S Equation (9) may be integrated, and together with boundary condition (11) it yields: 4 = exp
-4 a
e
A predictor-corrector method is used to calculate the void fraction from equation (10), while the relation between q~and e is obtained from equation (14) by a numerical integration. The grid for the numerical calculation is presented in Fig. 1. The discrete form of equation (14) is:
(f~,-f~)+Ax,~_2f~n
qS~j= exp
,
(15)
where f ik =
3 H 1 - e0 Ek ' 4 a eij
and the subscripts i, j, k denote the appropriate steps in time, position and particle radius, respectively. The trapezoidal rule of numerical integration is used in replacing the integral in equation (14) by the summation in equation (15). Now the summation in equation (15) may be rewritten in terms of a recursive relationship k FAx ] = @,j expLT (f/k S-, +f/kj) ,
(16)
The advance in the T-direction is done by rewriting equation (10) in discrete form, using the predictor-corrector method. The predictor-equation for e is: ei+½,j- ei~ _ AT
C
Eijq~ijR 3 dR.
(17)
JR, e~j
The equation for ~bmay be deduced from equation (17) in terms of the half steps required in the predictor-corrector method:
= O,+½J"-, expLyrAx(fJ+½j-l'k
+fi+½,j)l,'k
T
I
X
~(jl
i x
Fig. 1. Grid for numerical calculations.
(18)
430
M. FICHMAN et al.
The corrector-equations are: ?'i+ 1, j -- gij = -- C
AT
f l 1 - ~ i +½"J ~i+½,j ~ i +½j R 3 dR; I~i_l.½,j
(19)
,
4~+1,~ = 4~+1,j-1 exp ~-(f~+,,~_l ~f~+l,j) •
(20)
Trapezoidal numerical integration is used in all integrals. Equations (17)-(20) together with the initial and boundary conditions given by equations (11) and (12) set the numerical scheme for the solution of the dynamic filtration equations as functions of time, T. In practice this solution can be performed once the single collector filtration efficiency, E, is known as a function of position and time. The solution of the problem is sought in terms of the total deposited material, rather than of time. The total mass, M, of solids of density p deposited in a filter of volume V is: M
~=pv
(1 - e , )
Io
leo
e(T)]dx.
(21)
Equation (21) now constitutes an expression for the total mass of material deposited in the filter as a function of filter porosity, e, which itself is a function of time and position in the filter. The function e(X, T) can be calculated from the integration of equations (17)-(20), which can be performed once the single collector efficiency, E, is known. In what follows we propose the cluster enhanced filter (CEF) model for the filtration process. The model predicts the single collector efficiency, E, and thus makes the solution of the dynamic filtration problem possible. DYNAMIC FILTRATION MODELS The single collector efficiency, E, in a dirty filter must be known before the dynamic equations for the filter can be solved. The simplest way to proceed seems to use the clean filter model for description of the dynamic process. In this model the deposited material is assumed to form a smooth layer, resulting in a monotonous increase in the size of the collectors and a decrease in bed void fraction. The change in the single collector efficiency with time is expressed here through the change of the bed void fraction and of the collector size. Such a smooth layer deposit model was reasonably successful in describing liquid filtration (Tien et al., 1979), and it was hoped to be applicable to gas filtration also. Figure 2 shows typical filtration data obtained at Technion (Fichman, 1983) and some filter efficiencies
!
•
H=gcm 2a=l.Smm U = 7.5cm • E x p e r i m e n t a l data
0,~
N
I !
Smooth coating
theory
I 2
M(g)
Fig. 2. Overall filtration efficiency as a function of total mass of loaded material---comparison between experimental data and theoretical calculations using smooth coating theory.
Particle loading on granular bed filtration
431
calculated from equation (17) + (21) under the assumption of a dynamic smooth coating filter model. As seen, this smooth coating model cannot be applied to aerosol filtration, as it grossly underpredicts filtration efficiencies in comparison to these, obtained by experiments. Moreover, observations under a microscope show that the deposited particles do not form a smooth layer on the collector. The deposition is in the form of clusters, where each cluster acts as a secondary collector, thus increasing the collection efficiency of the primary collector. In the description of the deposition model it is very difficult to relate to each and every secondary collector and thus calculate the combined collection efficiency. A more effective approach is to consider the whole porous layer which these secondary collectors form. Thus, to obtain the single collector efficiency in a dirty filter, a model of a deposited high porosity layer on the collector surface is employed. This layer grows as the deposition process proceeds. It describes the dynamic behavior of granular bed filter at the first phase of the filtration process, before reentrainment takes place. This first phase is divided into three stages, as described presently. THE L E N G T H SCALES OF THE M O D E L The filtration of particles from a flow in porous media presents two different length scales. The scale connected with the filter efficiency and the particle trajectory is the large scale. Its characteristic length is the collector diameter. On this scale, the deviation of the collector from a perfect spherical shape can be neglected. The effect of a dust particle attached to the collector surface on the overall flow field can also be disregarded. Many physical parameters which characterize the filtration effects are obtained as averages on this large scale. The small scale is of the order of a dust particle. To describe the dust particle arrangement on the collector surface this smaller scale must be used. The general form of the mean flow that brings dust particles towards the collector is not important on this scale. What is important is the local surface shape, the local flow near the collector surface and the shape of the particle. All these determine the arrangement of the particles on the collector surface. A number of assumptions must now be made to successfully describe what happens on the smaller scale. The irregularities in the collector surface structure are assumed to be random, resulting in the local flow directions also being of random nature. The particles are assumed to be randomly collected on the collector surface, and to be randomly arranged, and therefore, on the average, arranged uniformly. The model assumes that the particles are distributed on the surface of the collector in such a way that equal cells with particles in their centers are formed (Fig. 3). The transient filtration process, before reentrainment takes place, is now considered. This process is divided into three different stages.
•
L
I. /
•
",,,---'--(
/-~
/
i
S •
l -L
cell boundary
\
~
~ "--/
"
~--
/
,
\ e
I
/
particle
Fig. 3. Cell structure formed by depositedparticles on the collectorsurface.
432
M. FXCHMANet al.
The first staoe---cell formation At the beginning of the filtration process the particles are deposited on the collector, forming an arrangement of equal cells. The particles which come next are collected, on the average, between those already deposited, in such a way that each cell size decreases with time. Through this stage the capture of particles by those captured earlier can be neglected. This approximation is valid as long as the total number of collected particles is small. Once this approximation is no longer valid, the second stage of the collection begins.
The second stage--cluster formation In this stage aerosol particles are collected in both new, clean areas, and on top of the earlier deposited particles. Clusters that consist of particles captured one on top of the other start to grow. The manner of capture and the position of the particles in the cluster are assumed to be random. The location where a cluster is formed is called a node. When the number of clusters increases sufficiently the combined capture efficiency of the clusters becomes greater than that of the clean areas and the third stage of collection commences.
The third staoe--cluster enhanced filtration (CEF ) In this stage particles are captured mostly by clusters and the capture by clean areas can be practically neglected. The present model states the following sequence of events: in the first stage the quantity of the deposited material is small and the single collector efficiency increases slowly. The second stage covers the formation of cell nodes, i.e. the locations of the clusters, and the final characteristic cell size. In the third stage the single collector efficiency increases at a high rate. At this stage the particles in the clusters penetrate into the flow field where the gas velocity is high and, therefore, the aerosol particle flux is also high. This leads to an enhancement in single collector efficiency. In the calculations that follow, the size of the cell is taken as that corresponding to the end of the second stage, when the cell size stops changing. This size is kept constant through the deposition process during the third stage.
THE C H A R A C T E R I S T I C L E N G T H OF AN I N D I V I D U A L CELL The characteristic length of an individual cell is assumed fixed at the end of the second stage. The second stage is completed when the probability of capture of a particle by a node or a cluster is equal to the probability of its capture by a clean area and the formation of a new cell. Thus, to obtain this characteristic cell length, these probabilities must be found. The local specific capture efficiency of a certain site on a collector surface is defined as the ratio between the number of particles captured per unit area there and the total number of particles captured by the whole collector: ni
ei = ~ .
(22)
Here ni is the number of particles per unit area captured on site i and Nc is the number of particles captured by the whole collector. To obtain the local specific capture efficiency at a certain point, a small area around this point must be chosen and the number of particles captured by this area must be calculated. A local cartesian coordinate system is now introduced such that the local flow velocity is in the x-direction (Fig. 4). Because the flow region considered is near the surface, u~ ~ 0. The xcomponent of the flow can be written as:
ux = bz + 0(z2).
(23)
The number of particles captured by the small area Asl is equal to the number of particles in
Particle loading on granular bed filtration
(Q)
433
(b)
Fig. 4. Capture of particles (a) by clean area and (b) by previously deposited particle.
the layer extending from the collector surface up to the height rp (Fig. 4a):
~rpnAyi UxA t dz 0
nl Asi = i~
= nAyibAt
f ~pzdz
= nAyibAtr~/2,
(24)
d
where n is the number concentration density of particles in the flow. Hence, the local specific capture efficiency at the point i is: bat ei = ~ c
2 Ayi
(25)
rlrp Asi "
To calculate the number of particles captured on a site already occupied by a particle, the integration must proceed up to the height of 3rp (see Fig. 4b). The particles already deposited on the collector have a negligible influence on the flow field. Therefore, to obtain the capture efficiency at a point which already has a captured particle, eip, it suffices to replace rp in equation (25) by 3r~, hence, eip ~ ~
n(3rp) 2
•
(26)
Denote the area-average capture efficiency by eo. Then division of equation (26) by equation (25) yields: eo~ ~ 9. ep
(27)
This means that on the average the local specific capture efficiency of a site with a captured particle is 9 times higher than that of a clean area. Moreover, nodes for the formation of particle clusters are not just at a point where a particle is found, but also in the neighboring domain. The definition of the size of such a neighboring domain is based on the following considerations: if a second particle is captured within a distance less than 4rp from the first particle, no additional particle can squeeze between these two particles and form a node. Thus the radius, Rn, of the domain in the neighborhood of a captured particle, i.e. where the additional particle causes the formation of a cluster, is defined as (see Fig. 5): Rn = 4rp
(28)
and the whole neighborhood domain is denoted as a 'node'. The capture efficiency of the empty area in a 'node' (the shadowed area in Fig. 5) is: eoSo = eo nl-(4rp)2 -- r 2] = eo" 15 nr 2 .
(29)
The capture efficiency of the whole 'node' comes out as: E~ = eopS, + eoSo = 9eonr 2 + 15eonr 2 = 24nr2eo.
(30)
Now the probability of the capture of one particle on the site surface s~and its incorporation into the 'node' must be defined. This probability function is normalized such that the probability of capture of one particle in the cell is 1. If No is the number of particles captured
434
M. FICHMAN et al.
/
\
/
/
\ "node"
\
/ \
I
/
Fig. 5. Structure of a single cell.
in the cell and Ni is the number of particles captured on the surface si, the capture probability o f the surface si, ~b~,is defined as: N i
d/i
Ilisi
noeiSi
. . . . . Nc
eisi -
ncsi
noecSc
.
(31)
ecSc
The area of a cell of characteristic length c is: s~ = c 2 .
(32)
The probability of capture on the 'node' and forming a cluster can be calculated by using the definition o f equation (31): e.s.
~b.
(33)
(c 2 - s.)eo + e.s.
where e.s. = E . is substituted from equation (30) and s. is calculated using equation (28): s. = rcR2 = 16~r2
(34)
resulting in 3 ~k, = 1 + 0.0398(c/rp) 2"
(35)
Now let the probability o f the capture of a particle by a cluster be calculated. First consider a cluster consisting of two particles. The probability of capture by such a cluster is designated by q&2. The two particles in the cluster can be located either one on top of the other or one beside the other. For a two-particle cluster, structured as one particle on top of the other, the capture efficiency can be obtained by replacing 3r~ by 5r~ in equation (26). This leads to Ec2 = 25eonr 2 + 15eonr 2 = 40nr~eo.
(36)
When the two particles in the cluster are located one beside the other the capture efficiency is enhanced to about twice that o f a node, equation (30). E~2 ~ 2E. = 48nr2eo.
(37)
Comparison of equations (36) and (37) yields that in both cases the capture efficiency is roughly the same. Hence, the probability of capture by a two-particle cluster, qsc2, is approximated by twice that of a node.
0~2 = 20..
(38)
Similarly, it can be shown that every additional particle contributes 4. to the probability of capture; hence, for a cluster of k particles @ok = k~0..
(39)
Particle loading on granular bed filtration
435
To obtain the probability of capture by an 'average' cluster, the distribution of the particles between the clusters is needed. This distribution can be obtained by reconsidering the problem from the point of view of the large scale. On the macroscale, the probability of capture by a cell does not depend on the cell surface structure and, hence, it is independent of the particles already occupying the cell. On the other hand, on the microscale, the particles located inside a given cell do influence the incoming particles in how they arrange themselves in the cell. Returning to the large scale picture, it is noted that because particles are distributed homogeneously in the main flow, the capture probability of each cell should be the same. This probability is equal to the ratio between the cell area and the total collector area available for capture. Let the probability of a given cell to contain a single particle be denoted by p. Now the probability o f capturing a particle is equal for all the cells, and does not depend on the number of particles already in it. Therefore, the probability of a given cell to contain two particles is p2, and generally, the probability of containing k particles is pk. By definition, each cell is occupied by some particles, one or more, therefore p + p2 + p3 + . . . . pk = 1.
(40)
The solution of equation (40) for k ~ ~ yields the probability for a given cell to contain a single particle 1
P
2
(41)
The probability of a particle that approaches the collector to be captured by a node, i.e. by 'single-particle cluster' is @,. The probability of a particle to be captured by a cluster containing any number of particles, including one, is @c = @,,P + ¢c2P 2 + I/.Ic3P3 + . . . .
~b
+ 2" + 3"~ + .
----2¢,.
(42)
The second stage of filtration is defined to end when the probability of capture by a cluster or a node is equal to the probability of capture by a clean surface, i.e. 1
¢c = ~.
(43)
Or, from equation (42): 1
¢n
(44)
4
Substitution of this ft, into equation (35) yields 3
1
1 + 0.0398(c/rp) 2 = -4'
(45)
which results in the characteristic length of an individual cell c/rp = 16.6.
(46)
THE FILTER PARAMETERS IN THE THIRD STAGE The cell size has been obtained from the second stage. The single sphere collection efficiency is now sought, as a function of time. For this the void fraction of the deposited layer must first be obtained. To do this the individual clusters are regarded as single elements of the deposited porous layer. Let the characteristic length of a cluster be denoted by h, and that of a cell containing the duster by c. The void fraction of the porous layer, ep, protruding from the collector surface at
436
M. FICI-tMANet al.
the height h, is the cluster-free volume divided by the total volume of the porous layer
Now, the height of the porous layer, which equals that of a cluster, is calculated as a function of the solidity of the deposited material, a. To do this, the 'unit cell' model of a single collector (Happel, 1958; Kuwabara 1959; Neal and Nader, 1974) is used. The model consists of a test sphere surrounded by a concentric cell of radius L, as shown in Fig. 6. The volume of each unit cell is equal to the volume of the filter divided by the number of the spherical collectors in the filter. As the number of collectors does not change during the filtration process the cell radius, L, remains constant while the effective collector radius increases from ao to a. Thus L = ao
( )lj3 / )13 = a
(48)
The fraction of the volume of the unit cell occupied by the material deposited on the collector is the solidity, a, of the deposited material which forms a porous layer of thickness h and void fraction ep on the collector surface. The solidity is calculated as a --
4na2h(1 - ~p) (I - ed) 4/3rrL 3 ,
(49)
where ~d is the porosity of each individual cluster. Substitution ofep from equation (47) yields an expression for h .r
O'(C/a) 2 11/3
h = LL-3~d) j
,
(50)
and, finally, using equation (48) to eliminate L, results in h=
3 ( 1 - ~ o ) ( 1 - ~.)
"
(51)
Equations (47) and (51) connect the large scale parameters of the problem: the initial bed void fraction, ~o, the deposit solidity, e, the porous layer void fraction, ep, and the size, c, of the cell containing the cluster. These relations are now used in the determination of the flow field around a single collector in a dirty filter.
Fig. 6. Model of a dirty collector inside a porous matrix.
Particle loading on granular bed filtration
437
M O D E L FOR THE F L O W F I E L D IN A DIRTY F I L T E R The flow field inside the filter is assumed to be creeping flow. The unit cell model of a single collector for a clean filter consists of a sphere surrounded by a concentric cell (Happel, 1958), with this cell embedded inside an infinite region of porous media (Neal and Nader, 1974). The difference between the clean filter model and the one proposed here for a dirty filter is that in the model for the dirty filter there is a porous layer on the collector surface also. Thus the suggested model of a dirty filter contains three different regions: the first region (region 1, Fig. 6) is the porous layer on the collector surface; the second region (region 2, Fig. 6) is the empty space inside the cell; and the third region (region 3, Fig. 6), is the infinite porous medium outside the cell. The governing flow equations in regions 1 and 3 are the Brinkman equations (Brinkman, 1947, 1949) and those in region 2 are the Stokes equations for creeping flow. Let these equations now be written out: For a <~ r <~ a + h , -~Tu~ +~V2ul = AP~
VUl =
0
(52)
for a + h <~ r <~ L 2u2 =
f]IV
AP2
(53)
Vu2 = 0,
and for r I> L (
- ~ 3 u 3 +/.tV2ua = VPa Vua = 0
(54)
As was first suggested by Brinkman (Brinkman, 1947), the conditions on the surfaces between the regions are continuity of velocities and equality of stresses, i.e. ul, = Ulo = 0
{ulr
U2r; UlO
U20
P1
= P2
"ElrO ~- T2rO;
u2,
U3r; 1120
"E2rO "~" "~3rO U3r ~
U30
at r = a
(55)
at r = a + h
(56)
at r = L
(57)
P2 = P3 U o COS0
u3o --* Uo sin0
r --} ~ .
(58)
Define non-dimensional coordinate ~ and ~ for regions 1 and 3, respectively r
rl
--- k~/2 " ~ - k~/2 •
(59)
The solutions for the stream functions, assuming spherical symmetry (Happel and Brenner, 1965; Neal and Nader, 1974) everywhere are for~ ~< ~ < / ~ qJl = k ~ ° [ E ~ - l + f ~ - 2 + G l e ~ ( l + ( - ~ ) + H ~ e ~ ( 1 - ( - ~ ) ] s i n 2 0
(60)
forfl~<~
(61)
438
M. FICHMAN et al.
fore > ~ ~b3 = ~
[E3~ - 1 + F3~2 + G3e-~(1 + ~- 1) + H3eg(1 _ ~- 1)]sin20,
(62)
where
a &-k~/2" The boundary conditions constants. This system can dirty single filter collector. analysis may be continued
a+h a+h L [3= k~/2 ; f l = k~/2 " y = k ~ / 2 .
(54)-(58) give a system of linear algebraic equations for the easily be solved numerically, resulting in the flow field around a Now that the flow field in region 2 is known, equation (60), the to calculate the single element collection efficiency E.
F I L T R A T I O N E F F I C I E N C Y OF A DIRTY F I L T E R With the flow field just developed the single collector efficiency can be obtained from the equations of particle motion which yield the particles' trajectories (Gutfinger and Tardos, 1979). When inertia can be neglected, particles move along streamlines and are collected by interception. In this case the single collector efficiency is (Gutfinger and Tardos, 1979)
E =
a2
(63)
Inertia causes a particle to transfer from one streamline to another, thus enhancing filtration efficiency. Far away from the collector, where the curvature of the streamlines is small, a particle moves along some initial streamline; when getting close to the collector, where the streamlines curve appreciably, inertia makes the particle leave its initial streamline and move closer to the collector surface. The overall difference between the locations of the initial and the final streamlines at 0 = ~t/2, At, can be calculated from the equations of particle motion. When the effect of inertia is small, the difference Ar can be approximated by the expression (Fichman and Pnueli, 1982): Ar = 3.37. St. Rp(St 2 + 16Rp2) 1/4
(64)
The resulting Ar, exact or approximated, is added to a + h in equation (64) to account for inertia effects in the calculation of the single collector filtration efficiency. This efficiency is calculated using the stream function in region 2, Oz, given by equation (61), and this stream function depends on the porous layer thickness, h v, and on the void fraction ev, which, in turn, are functions of the deposit solidity, equations (47) and (51). Thus, using equations (64), (47), (51) and (61), the single filter element collection efficiency, E, can be calculated as a function of the deposit solidity. Once E is known it is used to calculate the overall filtration efficiency. The overall filtration efficiency, ~/, is defined as: r/= 1
n°ut- 1-~b(1),
(65)
F/in
where ~(1) is calculated by integrating numerically the time dependent dynamic filter equations, equations (9) and (10), using the algorithms given by equations (16)-(20). In these calculations, the time is used as a parameter, which by means of equation (21) connects ~band e (or a) with the mean volume fraction of the deposited material tr. Thus, the overall filtration efficiency is a function of several parameters including the mean volume fraction of the deposited material, a,
n/no = FEE(St, Rp, eo, 6), H/a],
(66)
where r/o is the overall efficiency of a clean filter. Figure 7 presents the overall filtration efficiency as a function of 6 for several values of the different parameters in equation (67). These filter efficiencies are calculated for interceptional
Particle loading on granular bed filtration
'I3
,0-3
439
2-~=60
Rp=3.75.10-~
o
?2
IF [ i0-4
I
I
5.10-4
I
[
10-4
10-3
I
5.10-4 o"
iO-S
Fig. 7. Normalized filtration efficiency as a 'function of mean volume fraction of the deposited material: (a) H/2a = 30; (b) H/2a = 60.
=120 2•0 St =0.O4
Rp=5"lO-3 o
O( ~
°
'
=
~_
0.2 I
I
O
~ . _ ~ I
02
I
4
~"='=='~" O"= 1.8' 10 .4 I °'=3"10-4
06 x/H 1"t=120
~-O.78'10"
1
St=O.Q3
~.~ 0.1" ,-
°"=0"85"10-4 or 1.56-10-4 o- =7.7.10-'~
O-2~ ~
I
I
I
0.2
l
1
11=120 2a St=O.007 Rp=3.75.10_s
I c~ 1 "~ c 0.,
I
0-6 x/H
6
~
°'=0 O"=1.5.10-4 G'=4.~" IO-4
0"2 1
0'2
I
I
0"6 x/H
I
I
1
Fig. 8. Number particle concentration density distribution in the gas phase along the filter for different filtration parameters.
440
M. FICHMANet al.
R
H=9cm 2a=l-5mm U=7-5 cm • Experimental --
0.5
dato
Smooth coating theory Present theory
I o
I
2
I
M(g)
Fig. 9. Overall filtration efficiency as a function of total mass of loaded material---comparison between experimental data and theoretical calculations, using smooth coating theory and the CEF model.
and inertial depositions only. It can be seen that for Rp > 10-3 the filtration efficiency increases very quickly with 5. Figure 8 shows the change in the particle concentration in the gas as it flows along the filter bed, for various values of d. Figure 9 reproduces the experimental data of Fig. 2, together with the curves corresponding to the 'smooth coating theory' presented there. The theoretical curve obtained from the present theory is added to the figure. The plot is given in terms of the overall filter efficiency as a function of particle loading. It seems that the agreement between the present theoretical results and the experimental measurements is quite good. SUMMARY This work treats the time dependent behavior of a granular bed filter while it filters small particles from a gas, starting in a clean condition, and then being loaded with particles. A cluster enhanced filter (CEF) model has been proposed to describe the process as it develops in time. The first stage is deposition on clean surfaces. It is followed by the formation of clusters of particles, which constitutes the second stage. Then, in the third stage, deposition takes place on the clusters with a very high enhancement in efficiency. The transitions between the stages are well defined and clearly formulated. The results, which predict a very high increase in efficiency, are quite different from those of the 'smooth coating' theory developed for liquid filtration, which predicts only moderate increase in filtration efficiency, and which could not explain the very high deposition rates in the case of gas filtration. The C E F model provides filtration efficiency data as a function of the mass of the deposited material in the filter. The theory compares favorably with experimental data. REFERENCES Brinkman, H. C. (1947) A calculation of the viscous force exerted by a flowingfluid on a dense swarm of particles. Appl. Scient. Res. AI, 27. Brinkman, H. C. (1949)Problems of fluid flowthrough swarmsof particlesand through macromoleculesin solution. Research (Lond.) 2, 190.
Fichman, M. (1983) D.Sc. thesis, Technion, Haifa. Fichman, M., Gutfinger, C. and Pnueli, D. (1981) A modified model for deposition of dust in a granular bed filter. Atmos. Envir. 15, 1669. Fichman, M. and Pnueli, D. (1982) More detailed analysis of the effect of inertia on the filtration efficiencyof granular bed filters. Israel J. Tech. 20, 60.
Particle loading on granular bed filtration
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G utfinger, C. and Tardos, G. (1979) Theoretical and experimental studies on granular bed filtration. A tmos. Envir. 13, 853-867. Happel, J. (1958) Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles. A.I.Ch.E.J. 4, 197. Happel, J. and Brenner, H. (1965) Low Reynolds Number Hydrodanamics, Prentice Hall, New York, Herzig, J. P., Leclerc, D. M. and Le Goff, P. (1970) Flow of suspensions through porous media--application to deep bed filtration. Ind. Engng. Chem. 62, 8. lrmay, S. (1969) The mechanism of filtration of non-colloidal fines in porous media. Symposium of the Fundamentals of Transport Phenomena in Porous Media, Haifa, 5. Kuwabara, S. (1959) The forces experienced by randomly distributed parallel circular cylinder or spheres in viscous fluid at small Reynolds numbers. J. Phys. Soc. Japan 14, 527. Neal, H. N. and Nader, W. K. (1974) Prediction of transport processes within porous media. Creeping flow relative to a fixed swarm of spherical particles. A.I.Ch.E.J. 20, 530. Paretsky, L., Theodore, L., Pfeffer, R. and Squires, A. M. (1971 ) Panel bed filters for simultaneous removal of fly ash and sulfur dioxide, li. J. Air Pollut. Control Ass. 21,204. Payatakes, A. C., Tien, C. and Turian, R. M. (1973a) A new model for granular porous media: Part 1. Model formulation. A.I.Ch.E.J. 19, 58. Payatakes, A. C., Tien, C. and Turian, R. M. (1973b) A new model for granular porous media: Part II. Numerical solution of steady state incompressible Newtonian flow through periodically constricted tubes. A.I.Ch.E.J. 19, 67. Tien, C. and Payatakes, A. C. (1979) Advances in deep bed filtration. A.I.Ch.E.J. 25, 737. Tien, C., Turian, R. M. and Pendse, H. (1979) Simulation of the dynamic behavior of deep bed filters. A.I.Ch.E.J. 25, 385. Wang, C. S., Beizaie, M. and Tien, C. (1977) Deposition of solid particles on a collector: formulation of a new theory. A.I.Ch.E.J. 23, 879.