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Analysis of Brownian particle deposition and reentrainment in granular beds
Analysis of Brownian Particle Deposition and Reentrainment in Granular Beds H I D E T O YOSHIDA 1 AND CHI TIEN
Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, New York 13210 Received April 18, 1985;acceptedSeptember3, 1985 An analysis is presented to describe the deposition of particles from suspensions flowing through granular beds, taking into account the possible reentrainment o f the deposited particles. The extent of reentrainment is found to be a monotonic function of the extent of deposition and declines drastically with the increase in particle size. © 1986AcademicPress.Inc.
INTRODUCTION
2(1 - p,5) As -
Granular filters have been used extensively to clarify suspensions containing small amounts of fine particles. The capacity of a filter bed can be described by its filter coefficient or by its efficiency of the unit collectors which comprise the bed. The traditional way of estimating the filter coefficient (or unit collector efficiency) of a filter bed is through the use of trajectory calculations. However, when the particles are sufficiently small and the Brownian diffusion force is the dominant force acting on these particles, the results of mass transfer analysis of packed beds can be readily applied. The unit collector efficiency, 7, can be estimated from ~1 = 4A~/3NP2/3 [ 1]
[2]
W
p ' = (1 - c)1/3 w = 2 - 3 p ' + 3p '5 - 2p '6.
[3] [4]
The derivation of Eq. [ 1] is based on the assumption that the surface interaction forces between the particles in the suspension and filter grains are favorable. On the other hand, Eq. [ 1] does not apply if the surface interactions are unfavorble, that is, the force is repulsive. Instead the relevant transport equations must be solved using the following boundary conditions at the collector surface:
D° l
= kclconector surface
[5]
collector surface
where Nee is the Peclet number, defined as (udp/D) with u, dp, and D denoting the superficial velocity through the bed, particle diameter, and the Brownian diffusivity, respectively. As is a parameter, dependent upon the bed porosity e and defined as
where c denotes the particle concentration and Oc/On denotes the concentration gradient along the normal from the collector. The pseudo first-order rate constant, k, is related to the surface interaction potential, q~, by (2) k =
J Present address: Department of Chemical Engineering, Hiroshima University, 724 Higashi-Hiroshima, Saijyo, Japan.
D
[6]
fo ~ (e ~ / ~ r - 1)dx 189
Journal of Colloid and Interface Science, Vol. 111,No. 1, May 1986
0021-9797/86 $3.00 Copyright© 1986by AcademicPress,Inc. All rightsof reproductionin any form reserved.
190
YOSHIDA AND TIEN
where kb is the Boltzman constant and T the absolute temperature. The effect of the surface interactions on Brownian particle deposition in granular beds has been examined by a number of investigators using fairly simple collector geometry (2-4). More recently, Chiang and Tien (5) considered the same problem but used a more complicated porous media model (the constricted tube model), which is known to provide a more realistic representation of granular beds (6). In all these analyses, the possibility that deposited particles may become reentrained was not considered. Particles in liquid media, even those bound to a collector, are subject to the Brownian diffusion force. As shown in the discussions by Rajagopalan and Chu (7) and Hirtzel and Rajagopalan (8), the so-called adsorption (deposition) and desorption (reentrainment) of colloidal particles result directly from the balance of between the Brownian force and the interaction force between the particle and the collector. A very thorough analysis of this problem has been made by Dahneke (9-11). His results indicate that the extent of reentrainment can be described by the term k~a, where a is the concentration of the deposited particles and k~, the repulsion constant. Dahneke also derived an expression for evaluating kx from the surface interaction potential, ~ (9). Thus, a more complete analysis of Brownian particle deposition in granular beds should consider both the retardation effect caused by
unfavorable surface interactions as well as the possible reentrainment of deposited particles. The boundary condition at the collector surface given by Eq. [5] should be modified to be
D ( Oc l = kclcol,ector surface - k~ a \On/collector surface [7] where a is the number of deposited particles per unit collector surface area. In the present work, an analysis was done concerning the deposition of Brownian particles in granular beds. The possibility of the reentrainment of deposited particles was included. In carrying out the analysis, the constricted tube model was used to characterize granular beds. The present work complements the earlier work of Chiang and Tien (5) and provides a more complete examination of the Brownian particle filtration in granular beds and, in particular, the effect of surface interactions on filter performance. REPRESENTATION OF GRANULAR BEDS A schematic diagram depicting the representation of a filter bed by the constricted tube model is shown in Fig. 1. The bed, composed of nearly monosized filter grains, is assumed to consist of a number of unit bed elements in series. Each unit bed element of axial length, l, in turn, consists of a number of unit cells in the form of sinusoidal constricted tubes. To simplify, one may assume these cells to be uniform in size and characterized by their height, h; their maximum diameter, dm; and
us
Granular Bed
Series ofunit bed • I ements
Constricted tube typeunit eei|s
FIG. 1. S c h e m a t i c r e p r e s e n t a t i o n o f a g r a n u l a r bed. Journal (fCollotd and Interface Science, Vol. 111, No. 1, May 1986
BROWNIAN PARTICLE DEPOSITION TABLE I
ANALYSIS OF BROWNIAN PARTICLE DEPOSITION
Determination of Porous Media Parameters (Cases of Uniform Unit Cell Size)
The pertinent equations and boundary conditions describing the deposition of the Brownian particles in dimensionless forms are
Number of constricted tubes per unit area, Nc N~o = Ncd~
= 6"~1/3"(1
-
Swi)l/3"(1
-
e)2/3
191
[1]
7r
Oc* Oc* D u* - + 1)* - - -- - Oy* h(Ul) Ox*
Length of unit bed element, l l
1
1113
y* =0,
I3]
0y .2
04 w hk klh Oy* =~Cw*--coD
Constricted diameter, dc = d~ = 2~ -~ 0.35
02c *
[4]
Maximum diameter of the unit cell, dm am
U~( 1 -- Sw/)'] 1/3
d*~=~-~-g= L
] - _ ~ - jI
=2~
[5]
the constriction diameter, dc. Expressions for determining these quantities as well as those for other model parameters are given in Table I. The geometry of the constricted tube is given in Fig. 2. With a filter represented in the m a n n e r described above, problems of particle deposition in granular beds can be studied by examining the deposition of particles from suspensions flowing through a constricted tube with surface interaction forces operative between the particles and the wall of the tube.
Co
:2 2 'g
[9]
y*--'~,
c*--~ 1
[10]
x*
C* =
[ll]
= 0,
1
Height of the unit cell, h h = dg
[8]
where c* = C/Co, x * = x / h , y * = y / h , u* = u~ ( u l ) , and v* = v / ( u ~ ) , c is the particle concentration of the suspension, and Co is the influent concentration, x and y are the coordinates along the tangential and normal directions of the tube wall (see Fig. 2), and u and v are velocity components along the x and y coordinates. (Ul) is the average velocity through the constricted tube. The boundary condition of Eq. [10] is based on the assumption that the Peclet number, ( u ~ ) d p / D , is large and, thus, the region over which significant concentration change occurs is thin. As a result, the boundary condition used in external flow can be applied here without significant error. A solution of the above equations can be found by using Lighthill's formula (5). The dimensionless wall concentration, c* = cw/co, is
a
FIG. 2. Representation of Brownian particle deposition with particle repulsion in constricted tube. Journal of Colloid and Interface Science, Vol. 111, No. 1, May 1986
192
YOSHIDA AND TIEN
D
Ou*
c~(~) = ffa* + 2 ' / 6 ( - - ~ V - - [
[ Ou* "11/2 1 r*/ \hk]l_Oy* ~ , 9'/3I'(4/3)~/3"
"] 1/2
M / = 2'/61--11 _
r*/
\hkJLOY*lw J
,
× 91/3i,(4/3)(1/3
[1-
[201 c*(O)
The unit collector efficiency, n, by definition, is given as
__~1/3£ ~ (iZ~'~/3dc*w/d~'dX,]
[121
Oc 21rrw.dx 1 f ~ D(--] n = qco \OY]w
where
£:
D
Ou* I
f2(rw*)3 / 2 -
= h"
dx
*
oy* [w
[13]
B + ~a* c~(o)
-
-
-
[14]
I+B
.=:¢)
,
9mP(4/3)
× lim
~{(°u*/°y*):*}m} [151 k,
a~ - Cokh2'
[16]
a* = h2a
[17]
and r* is the dimensionless tube wall radius, defined as rw/h. Using the numerical technique developed earlier (5), that is, by approximating the integral or the fight side with appropriate quadrature formula, the value of Cwat different values o f f (or x*, through Eq. [13]) can be calculated with C*w(~i) =
(1 + (1.5~ff3Mi)/(~i-
~i_1)1/3)
X {cba* + M i ( 1 - C*w(O)+ 1.5~ 13 ×
C*w(~,-3 1.5~/3A)} (~i- ~/-1)'/3
[ 18]
i-I
× [ % _ ~_,12/3 _ ( ( ~ _ ~j)2/31 Journal of Colloidand Interface Science, Vol. 111,No. 1, May 1986
[ 19]
[21]
where xj is the total arc length of the constricted tube and q is the volumetric flow rate through a given tube. By definition, the superficial velocity through the bed, Us, is equal to (N~). (q), where N¢ is the number of constricted tubes per unit cross-sectional area of the bed. The concentration gradient, (Oc/Oy)w, is related to the wall concentration, Cw,through Eqs. [7] and [9]. In terms of c*, ~ becomes
ox/
rl = 2rrX*ck* I " ( 4 - ~a*)r,~dx* do
[22]
where k* = k/us and N* = h 2° No. As an indication of the effect of reentrainmerit on filter performance, it is convenient to compare the value of n with and without reentrainment under otherwise identical conditions. No reentrainment will occur if a = 0. The ratio, n/(~):=0, is given as
n - -
(n)::o
£x~(~ _ ~ * ) . r * . d x * =
[23]
~x[, C'w[:=0r*wdx do
The extent of particle deposition of a filter bed is commonly expressed by the specific deposit, b, defined as the volume of particles retained in the bed per unit bed volume (volume per volume). On the other hand, the rate of reentrainment is related to the surface density of deposited particles, a, defined as the number of particles per unit collector surface area. The relationship between b and a can be seen as follows. Consider a bed of unit cross-sectional area and with an axial height of l. The total number of constricted tubes within the bed is
193
BROWNIAN PARTICLE DEPOSITION
Arc. The number of deposited particles in each constricted tube corresponding to a surface particle density ~ is
10-4
I
I
B Op = 0 . 2 p m
Xo=4A
.
Aqueous suspension
;J 27rrw.~. dx. 10" --
The volume of each deposited particle is (Tr/6)d 3. Accordingly, the total volume of all deposited particles is
(; d~)(Nc) f:J
2rrwa.
dx.
03 i 0 -~ _ Z 0 (_J
Since the volume of the bed is (l) (1), then, by definition, k is given as (r/6)d 3.N~ [~J 2rrwa.
gr=
dx
z 0
Dp = 2 . 0 p.m
I0"; --
l 2 ,T3 ~r,
71" ' I V R l * c
=
_¢
r'a*. dx*
3[(r/6)(1/(1 - e))] ~/3
5
[24] ~k-b
where NR, the relative size parameter or the interception parameter, is defined as dp/dg. NUMERICAL CALCULATION AND RESULTS The analysis presented above enables one to calculate the wall concentration, c*, for a given tube geometry and a given degree of particle deposition, ~*, as a function of three ~cx
_ Xo x ~(xl-a O-~-)+B(I~o)
z
(
ID x
Sepororion distonce
(--}
FIG. 4. Relation between k and
A4#kbT.
parameters: (~), (Npe = k(ul)/D), and (D/hk). Once the wall concentration is known, the unit collector efficiency, n, can be readily found from Eq. [22]. A parametric study of the effect of reentrainment on n can be done by combining the results of Eqs. [23] and [24]. A brief description of the method used in the calculations and some of the results obtained are given below. Tube geometry. In the original formulation of the constricted tube model (6), the wall geometry was assumed to be parabolic. Subsequent investigators have used both the sinusoidal and hyperbolic geometries. In the earlier work of Chiang and Tien (5), the effect of tube geometry was found to be negligible. In the present study, the tube geometry was assumed to be sinusoidal. In other words, r* = ( ~ ) I 1
FIG. 3. Surface interaction potential as a function of separation distance.
10
r~ - r~' . + (r-~-~r~) cOs(27rz )1"
[25] Journal of Colloid and Interface Science, Vol. 111, No. I, M a y 1986
194
YOSHIDA AND TIEN 102
'
#:,o,,o-,
XO= 4 A Aqueouss u s p e n s i o n
I0
where x denotes the separation distance, x0 is the minimum separation distance and x 1 the value at which 4~ is at its maximum. The exact expression of q~depends upon the nature of the surface interactions. In general, one may consider q~ to be composed of two parts corresponding to the attraction and repulsion forces, respectively. Dahneke argues that in proximity to a collector, the surface interaction potential can be approximated as
g ¢_)
0)I= l
z
_o
n~
td 2
l
I
0
5
I0
as shown in Fig. 3. a equals (Hap)/(6Xo), where H is the Hamaker constant and ap, the particle radius, assuming that the attraction surface interaction arises from the London-van der Waals flux./3 is the repulsion force acting on the particle. The separation distance which gives the maximum ~b, Xl, is
FIG, 5. Relation between kl and A4)/kbT.
and the maximum 4 is
The flow field within the tube was obtained by applying the perturbation solution of Chow and Soda (12) for flow through a circular tube with arbitrary deformation. Reaction rate constant and repulsion rate constant. The effect of the surface interactions between particles and filter gains is manifested in both the reaction rate constant, k (for deposition), and the repulsion rate constant, k~. The reaction rate constant, k, is related to the surface interaction potential, q~, through Eq. [6]. The repulsion rate constant, k~, according to Dahneke (9), can be expressed as
kz =
o[,_
[26]
fx:' Z=L exp '~(~x~)~(1
Parameter Values Used in Sample Calculations
Co
1 × 10" ~ 1 × 10-3 ( l / c m 3) 0,1685 0.402
a~
0.055 (cm) 0.413 0.1135 (cm/s) 6.10 × 10-6 ~ 2 × 10-4 (cm) 298 (K) 0.008937 (g/era, s) 4A 1.0 × 10-is ~ 1.0 X 10-4 1 0 - 5 ~ 10-1 1.0 (g/cm3) 1 ~ 10 9 . 3 × 1 0 5 ~ 4 . 6 X 106 0.64 ~ 2.66 X 106 5.53 X 1 0 - 1 ! ~ 5.53 × 1 0 - 7
Us T /z
#/a & Zx~/kbT
.
Journal of Colloid and Interface Science, Vol. 111, No. 1, May 1986
J "
TABLE II
Of
× exp~bT]av'ax
\cq
,/
)Co
D
14 ( .
[28]
xl = (a/B)UZxo
kbT
Npe "y
[291
195
BROWNIAN PARTICLE DEPOSITION I
~Cil~ = 11
+-.-+ \ \ \/ ~:~%~,.o.;,o
I
kbT ~) =5.53xlO'" y =6.23x10 "s Dp =O.061p.m
+=+
0.5 -
,,x,l,5
Npe = h )..= k--~--hD I
10.5
I I0 -2
I
i0 -4
10-3
I0
-
SPECIFIC DEPOSIT ~ (--)
FIG. 6. Effect of particle deposition on ~7for various values of A4Jlki,T.
The repulsion constant, k~, found from Eq. [27] is
DC-((~(Xl)/4/~T) kt
(1 +
ap/Xl)~Ax
[30]
where d2q~ 3" -
dx2 x, -
2tx(/~/oz) 3/2
out by Ruckenstein and Prieve (3), the evaluation of the indefinite integral can be greatly simplified if the total surface interaction potential, q$(x), in the neighborhood of the point where 4~ reaches its maximum, can be approximated by a parabolic expression. Under such an assumption, k is found to be
[31]
x~
De-(+(xl)/~o7~
k = (1 + ap/x,)2f2~b~kbT/~"
and Ax = Xl - x0.
[32]
The pseudo first-order rate constant, k, can be calculated according to Eq. [6]. As pointed
" q , ~ r/
=5.53;0-'
[331
The effect of the surface interactions on particle deposition and reentrainment can be seen from the change in values of k and kl as functions of ~ ( x l ) / k b T. Two sets of results are
(a~-~)=I.Ox I0" 15 ~ = 5.53x10-" 7= 6.23x10 z
~ × -~li0.89
0.5
(~)-'0 El
.... Co= I.~o'~/e~)
\\
\\
rl
Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, New York 13210 Received April 18, 1985;acceptedSeptember3, 1985 An analysis is presented to describe the deposition of particles from suspensions flowing through granular beds, taking into account the possible reentrainment o f the deposited particles. The extent of reentrainment is found to be a monotonic function of the extent of deposition and declines drastically with the increase in particle size. © 1986AcademicPress.Inc.
INTRODUCTION
2(1 - p,5) As -
Granular filters have been used extensively to clarify suspensions containing small amounts of fine particles. The capacity of a filter bed can be described by its filter coefficient or by its efficiency of the unit collectors which comprise the bed. The traditional way of estimating the filter coefficient (or unit collector efficiency) of a filter bed is through the use of trajectory calculations. However, when the particles are sufficiently small and the Brownian diffusion force is the dominant force acting on these particles, the results of mass transfer analysis of packed beds can be readily applied. The unit collector efficiency, 7, can be estimated from ~1 = 4A~/3NP2/3 [ 1]
[2]
W
p ' = (1 - c)1/3 w = 2 - 3 p ' + 3p '5 - 2p '6.
[3] [4]
The derivation of Eq. [ 1] is based on the assumption that the surface interaction forces between the particles in the suspension and filter grains are favorable. On the other hand, Eq. [ 1] does not apply if the surface interactions are unfavorble, that is, the force is repulsive. Instead the relevant transport equations must be solved using the following boundary conditions at the collector surface:
D° l
= kclconector surface
[5]
collector surface
where Nee is the Peclet number, defined as (udp/D) with u, dp, and D denoting the superficial velocity through the bed, particle diameter, and the Brownian diffusivity, respectively. As is a parameter, dependent upon the bed porosity e and defined as
where c denotes the particle concentration and Oc/On denotes the concentration gradient along the normal from the collector. The pseudo first-order rate constant, k, is related to the surface interaction potential, q~, by (2) k =
J Present address: Department of Chemical Engineering, Hiroshima University, 724 Higashi-Hiroshima, Saijyo, Japan.
D
[6]
fo ~ (e ~ / ~ r - 1)dx 189
Journal of Colloid and Interface Science, Vol. 111,No. 1, May 1986
0021-9797/86 $3.00 Copyright© 1986by AcademicPress,Inc. All rightsof reproductionin any form reserved.
190
YOSHIDA AND TIEN
where kb is the Boltzman constant and T the absolute temperature. The effect of the surface interactions on Brownian particle deposition in granular beds has been examined by a number of investigators using fairly simple collector geometry (2-4). More recently, Chiang and Tien (5) considered the same problem but used a more complicated porous media model (the constricted tube model), which is known to provide a more realistic representation of granular beds (6). In all these analyses, the possibility that deposited particles may become reentrained was not considered. Particles in liquid media, even those bound to a collector, are subject to the Brownian diffusion force. As shown in the discussions by Rajagopalan and Chu (7) and Hirtzel and Rajagopalan (8), the so-called adsorption (deposition) and desorption (reentrainment) of colloidal particles result directly from the balance of between the Brownian force and the interaction force between the particle and the collector. A very thorough analysis of this problem has been made by Dahneke (9-11). His results indicate that the extent of reentrainment can be described by the term k~a, where a is the concentration of the deposited particles and k~, the repulsion constant. Dahneke also derived an expression for evaluating kx from the surface interaction potential, ~ (9). Thus, a more complete analysis of Brownian particle deposition in granular beds should consider both the retardation effect caused by
unfavorable surface interactions as well as the possible reentrainment of deposited particles. The boundary condition at the collector surface given by Eq. [5] should be modified to be
D ( Oc l = kclcol,ector surface - k~ a \On/collector surface [7] where a is the number of deposited particles per unit collector surface area. In the present work, an analysis was done concerning the deposition of Brownian particles in granular beds. The possibility of the reentrainment of deposited particles was included. In carrying out the analysis, the constricted tube model was used to characterize granular beds. The present work complements the earlier work of Chiang and Tien (5) and provides a more complete examination of the Brownian particle filtration in granular beds and, in particular, the effect of surface interactions on filter performance. REPRESENTATION OF GRANULAR BEDS A schematic diagram depicting the representation of a filter bed by the constricted tube model is shown in Fig. 1. The bed, composed of nearly monosized filter grains, is assumed to consist of a number of unit bed elements in series. Each unit bed element of axial length, l, in turn, consists of a number of unit cells in the form of sinusoidal constricted tubes. To simplify, one may assume these cells to be uniform in size and characterized by their height, h; their maximum diameter, dm; and
us
Granular Bed
Series ofunit bed • I ements
Constricted tube typeunit eei|s
FIG. 1. S c h e m a t i c r e p r e s e n t a t i o n o f a g r a n u l a r bed. Journal (fCollotd and Interface Science, Vol. 111, No. 1, May 1986
BROWNIAN PARTICLE DEPOSITION TABLE I
ANALYSIS OF BROWNIAN PARTICLE DEPOSITION
Determination of Porous Media Parameters (Cases of Uniform Unit Cell Size)
The pertinent equations and boundary conditions describing the deposition of the Brownian particles in dimensionless forms are
Number of constricted tubes per unit area, Nc N~o = Ncd~
= 6"~1/3"(1
-
Swi)l/3"(1
-
e)2/3
191
[1]
7r
Oc* Oc* D u* - + 1)* - - -- - Oy* h(Ul) Ox*
Length of unit bed element, l l
1
1113
y* =0,
I3]
0y .2
04 w hk klh Oy* =~Cw*--coD
Constricted diameter, dc = d~ = 2~ -~ 0.35
02c *
[4]
Maximum diameter of the unit cell, dm am
U~( 1 -- Sw/)'] 1/3
d*~=~-~-g= L
] - _ ~ - jI
=2~
[5]
the constriction diameter, dc. Expressions for determining these quantities as well as those for other model parameters are given in Table I. The geometry of the constricted tube is given in Fig. 2. With a filter represented in the m a n n e r described above, problems of particle deposition in granular beds can be studied by examining the deposition of particles from suspensions flowing through a constricted tube with surface interaction forces operative between the particles and the wall of the tube.
Co
:2 2 'g
[9]
y*--'~,
c*--~ 1
[10]
x*
C* =
[ll]
= 0,
1
Height of the unit cell, h h = dg
[8]
where c* = C/Co, x * = x / h , y * = y / h , u* = u~ ( u l ) , and v* = v / ( u ~ ) , c is the particle concentration of the suspension, and Co is the influent concentration, x and y are the coordinates along the tangential and normal directions of the tube wall (see Fig. 2), and u and v are velocity components along the x and y coordinates. (Ul) is the average velocity through the constricted tube. The boundary condition of Eq. [10] is based on the assumption that the Peclet number, ( u ~ ) d p / D , is large and, thus, the region over which significant concentration change occurs is thin. As a result, the boundary condition used in external flow can be applied here without significant error. A solution of the above equations can be found by using Lighthill's formula (5). The dimensionless wall concentration, c* = cw/co, is
a
FIG. 2. Representation of Brownian particle deposition with particle repulsion in constricted tube. Journal of Colloid and Interface Science, Vol. 111, No. 1, May 1986
192
YOSHIDA AND TIEN
D
Ou*
c~(~) = ffa* + 2 ' / 6 ( - - ~ V - - [
[ Ou* "11/2 1 r*/ \hk]l_Oy* ~ , 9'/3I'(4/3)~/3"
"] 1/2
M / = 2'/61--11 _
r*/
\hkJLOY*lw J
,
× 91/3i,(4/3)(1/3
[1-
[201 c*(O)
The unit collector efficiency, n, by definition, is given as
__~1/3£ ~ (iZ~'~/3dc*w/d~'dX,]
[121
Oc 21rrw.dx 1 f ~ D(--] n = qco \OY]w
where
£:
D
Ou* I
f2(rw*)3 / 2 -
= h
dx
*
oy* [w
[13]
B + ~a* c~(o)
-
-
-
[14]
I+B
.=:¢)
,
9mP(4/3)
× lim
~{(°u*/°y*):*}m} [151 k,
a~ - Cokh2'
[16]
a* = h2a
[17]
and r* is the dimensionless tube wall radius, defined as rw/h. Using the numerical technique developed earlier (5), that is, by approximating the integral or the fight side with appropriate quadrature formula, the value of Cwat different values o f f (or x*, through Eq. [13]) can be calculated with C*w(~i) =
(1 + (1.5~ff3Mi)/(~i-
~i_1)1/3)
X {cba* + M i ( 1 - C*w(O)+ 1.5~ 13 ×
C*w(~,-3 1.5~/3A)} (~i- ~/-1)'/3
[ 18]
i-I
× [ % _ ~_,12/3 _ ( ( ~ _ ~j)2/31 Journal of Colloidand Interface Science, Vol. 111,No. 1, May 1986
[ 19]
[21]
where xj is the total arc length of the constricted tube and q is the volumetric flow rate through a given tube. By definition, the superficial velocity through the bed, Us, is equal to (N~). (q), where N¢ is the number of constricted tubes per unit cross-sectional area of the bed. The concentration gradient, (Oc/Oy)w, is related to the wall concentration, Cw,through Eqs. [7] and [9]. In terms of c*, ~ becomes
ox/
rl = 2rrX*ck* I " ( 4 - ~a*)r,~dx* do
[22]
where k* = k/us and N* = h 2° No. As an indication of the effect of reentrainmerit on filter performance, it is convenient to compare the value of n with and without reentrainment under otherwise identical conditions. No reentrainment will occur if a = 0. The ratio, n/(~):=0, is given as
n - -
(n)::o
£x~(~ _ ~ * ) . r * . d x * =
[23]
~x[, C'w[:=0r*wdx do
The extent of particle deposition of a filter bed is commonly expressed by the specific deposit, b, defined as the volume of particles retained in the bed per unit bed volume (volume per volume). On the other hand, the rate of reentrainment is related to the surface density of deposited particles, a, defined as the number of particles per unit collector surface area. The relationship between b and a can be seen as follows. Consider a bed of unit cross-sectional area and with an axial height of l. The total number of constricted tubes within the bed is
193
BROWNIAN PARTICLE DEPOSITION
Arc. The number of deposited particles in each constricted tube corresponding to a surface particle density ~ is
10-4
I
I
B Op = 0 . 2 p m
Xo=4A
.
Aqueous suspension
;J 27rrw.~. dx. 10" --
The volume of each deposited particle is (Tr/6)d 3. Accordingly, the total volume of all deposited particles is
(; d~)(Nc) f:J
2rrwa.
dx.
03 i 0 -~ _ Z 0 (_J
Since the volume of the bed is (l) (1), then, by definition, k is given as (r/6)d 3.N~ [~J 2rrwa.
gr=
dx
z 0
Dp = 2 . 0 p.m
I0"; --
l 2 ,T3 ~r,
71" ' I V R l * c
=
_¢
r'a*. dx*
3[(r/6)(1/(1 - e))] ~/3
5
[24] ~k-b
where NR, the relative size parameter or the interception parameter, is defined as dp/dg. NUMERICAL CALCULATION AND RESULTS The analysis presented above enables one to calculate the wall concentration, c*, for a given tube geometry and a given degree of particle deposition, ~*, as a function of three ~cx
_ Xo x ~(xl-a O-~-)+B(I~o)
z
(
ID x
Sepororion distonce
(--}
FIG. 4. Relation between k and
A4#kbT.
parameters: (~), (Npe = k(ul)/D), and (D/hk). Once the wall concentration is known, the unit collector efficiency, n, can be readily found from Eq. [22]. A parametric study of the effect of reentrainment on n can be done by combining the results of Eqs. [23] and [24]. A brief description of the method used in the calculations and some of the results obtained are given below. Tube geometry. In the original formulation of the constricted tube model (6), the wall geometry was assumed to be parabolic. Subsequent investigators have used both the sinusoidal and hyperbolic geometries. In the earlier work of Chiang and Tien (5), the effect of tube geometry was found to be negligible. In the present study, the tube geometry was assumed to be sinusoidal. In other words, r* = ( ~ ) I 1
FIG. 3. Surface interaction potential as a function of separation distance.
10
r~ - r~' . + (r-~-~r~) cOs(27rz )1"
[25] Journal of Colloid and Interface Science, Vol. 111, No. I, M a y 1986
194
YOSHIDA AND TIEN 102
'
#:,o,,o-,
XO= 4 A Aqueouss u s p e n s i o n
I0
where x denotes the separation distance, x0 is the minimum separation distance and x 1 the value at which 4~ is at its maximum. The exact expression of q~depends upon the nature of the surface interactions. In general, one may consider q~ to be composed of two parts corresponding to the attraction and repulsion forces, respectively. Dahneke argues that in proximity to a collector, the surface interaction potential can be approximated as
g ¢_)
0)I= l
z
_o
n~
td 2
l
I
0
5
I0
as shown in Fig. 3. a equals (Hap)/(6Xo), where H is the Hamaker constant and ap, the particle radius, assuming that the attraction surface interaction arises from the London-van der Waals flux./3 is the repulsion force acting on the particle. The separation distance which gives the maximum ~b, Xl, is
FIG, 5. Relation between kl and A4)/kbT.
and the maximum 4 is
The flow field within the tube was obtained by applying the perturbation solution of Chow and Soda (12) for flow through a circular tube with arbitrary deformation. Reaction rate constant and repulsion rate constant. The effect of the surface interactions between particles and filter gains is manifested in both the reaction rate constant, k (for deposition), and the repulsion rate constant, k~. The reaction rate constant, k, is related to the surface interaction potential, q~, through Eq. [6]. The repulsion rate constant, k~, according to Dahneke (9), can be expressed as
kz =
o[,_
[26]
fx:' Z=L exp '~(~x~)~(1
Parameter Values Used in Sample Calculations
Co
1 × 10" ~ 1 × 10-3 ( l / c m 3) 0,1685 0.402
a~
0.055 (cm) 0.413 0.1135 (cm/s) 6.10 × 10-6 ~ 2 × 10-4 (cm) 298 (K) 0.008937 (g/era, s) 4A 1.0 × 10-is ~ 1.0 X 10-4 1 0 - 5 ~ 10-1 1.0 (g/cm3) 1 ~ 10 9 . 3 × 1 0 5 ~ 4 . 6 X 106 0.64 ~ 2.66 X 106 5.53 X 1 0 - 1 ! ~ 5.53 × 1 0 - 7
Us T /z
#/a & Zx~/kbT
.
Journal of Colloid and Interface Science, Vol. 111, No. 1, May 1986
J "
TABLE II
Of
× exp~bT]av'ax
\cq
,/
)Co
D
14 ( .
[28]
xl = (a/B)UZxo
kbT
Npe "y
[291
195
BROWNIAN PARTICLE DEPOSITION I
~Cil~ = 11
+-.-+ \ \ \/ ~:~%~,.o.;,o
I
kbT ~) =5.53xlO'" y =6.23x10 "s Dp =O.061p.m
+=+
0.5 -
,,x,l,5
Npe = h
10.5
I I0 -2
I
i0 -4
10-3
I0
-
SPECIFIC DEPOSIT ~ (--)
FIG. 6. Effect of particle deposition on ~7for various values of A4Jlki,T.
The repulsion constant, k~, found from Eq. [27] is
DC-((~(Xl)/4/~T) kt
(1 +
ap/Xl)~Ax
[30]
where d2q~ 3" -
dx2 x, -
2tx(/~/oz) 3/2
out by Ruckenstein and Prieve (3), the evaluation of the indefinite integral can be greatly simplified if the total surface interaction potential, q$(x), in the neighborhood of the point where 4~ reaches its maximum, can be approximated by a parabolic expression. Under such an assumption, k is found to be
[31]
x~
De-(+(xl)/~o7~
k = (1 + ap/x,)2f2~b~kbT/~"
and Ax = Xl - x0.
[32]
The pseudo first-order rate constant, k, can be calculated according to Eq. [6]. As pointed
" q , ~ r/
=5.53;0-'
[331
The effect of the surface interactions on particle deposition and reentrainment can be seen from the change in values of k and kl as functions of ~ ( x l ) / k b T. Two sets of results are
(a~-~)=I.Ox I0" 15 ~ = 5.53x10-" 7= 6.23x10 z
~ × -~li0.89
0.5
(~)-'0 El
.... Co= I.~o'~/e~)
\\
\\
rl
Npe= ~
y= Npe :9.37x105 (--) i i0 -7
10. 6
~
I io -s SPECIFIC DEPOSIT ~ ( - )
5
hk
10. 4
FIG. 7. Relationship between 7t/~;=o and 3- at various values of
i0 -3
c~lfl.
Journal of Colloid and Interface Science, Vol. 111, No. 1, May 1986
196
YOSHIDA AND TIEN
\0;o,.o\ 0.5 - Co: I . o * i o '2 ~lzec)
\
y=6.23x10' (-) 0
\
\
\
-
\
~.
I
I
10-5
\
\
I0-'~
~. - - ) , :.~_
~
?
I0-3 SPECIFIC DEPOSIT ~"
h.
10-z
I0- I
FIG. 8. Relation between 01/~;=o) and Dp.
shown in Figs. 4 and 5. The conditions used to obtain these results are listed in Table II. The effect of 4~(Xl)/kbT on k is much stronger than that of kl. On the other hand, both k and kl depend strongly on particle sizes. An increase in particle size from 0.2 to 2.0 t~m causes a change in k (or kl) by more than two orders of magnitude. Effect of reentrainment on collection. The effect of particle deposition on particle reentrainment can be seen by computing ~/(n);=o as a function of b from Eqs. [23] and [24]. In Fig. 6, the values of n/(O);=0 are plotted against 8 for various values of 4a(xO/kbT. [Note that parameters • and 3' depend upon ¢(xl)/kbT.] It is clear that as the magnitude of the repulsive barrier increases, the extent of reentrainment increases correspondingly. The ratio of fl to a is a direct indication of the relative magnitude of the attractive to repulsive forces, which together make up the surface interaction. The results displayed in Fig. 7 demonstrate clearly that as the magnitude of the repulsive force decreases, the extent of reentrainment becomes significant. For 13/a = 1.0 × 10 -15, even with a value of ~ as high as 10-3, the decrease in ~ resulting from reentrainment is only 5%. The most striking feature of the results obtained in this study is the strong dependence of reentrainment on particle size. The results given in Fig. 8 illustrate this feature concluJournal of Colloid and Interface Science, Vol. t 11, No. 1, May 1986
sively. At b = 10-3, the value of ~ for 0.061~m particles is found to be approximately half of the value of n if there is no reentrainment. For particles of 0.3 um size, the decrease in ~7 is hardly perceptible. Figure 8 also suggests that as b = 10 -2, a filter bed becomes nonretentive for particles equal or to or tess than 0.1 /sm. ACKNOWLEDGMENT This study was performed under Grant CPE-8412705, National Science Foundation. REFERENCES 1. Cookson, J. T., Environ. Sei. Technol. 4, 128 (1970). 2. Spielman, L. A., and Friedlander, S. K., J. Colloid Interface Sci. 46, 22 (1974). 3. Ruckenstein, E., and Prieve, D. C., J. Chem. Soc. (London) Faraday Trans. 69, 1522 (1973). 4. Rajagopalan, R., and Karis, T. E., A1ChE Sym. Set. No. 190 75, 73 (1979). 5. Chiang, H. W., and Tien, C., Chem. Eng. Sei. 37, 1159 (1982). 6. Payatakes, A. C., Tien, C., and Turian, R. M.,AIChE J. 19, 58 (1973). 7. Rajagopalan, R., and Chu, R. Q., J. Colloidlnterfaee Sci. 86, 299 (1982). 8. Hirtzel, C. S., and Rajagopalan, R., "Colloidal Phenomena: Advanced Topics." Noyes, Park Ridge, N.J., 1985. 9. Dahneke, B., J. Colloid Interface Sci. 50, 89 (1975). 10. Dahneke, B., J. ColloidInterfaee Sci. 50, 194 (1975). 11. Zimmer, S. L., and Dahneke, B., J. Colloid Interface Sei. 54, 329 (1975). 12. Chow, J. C. F., and Soda, K., Phys. Fluids 15, 1700 (1972).