The role of colloid chemistry in modeling deep bed liquid filtration

Chemical Engineering Science, 1975, Vol. 30. pp. 103%lW7.

Pergamon Press.

Printed in Great Britain

THE ROLE OF COLLOID CHEMISTRY IN MODELING DEEP BED LIQUID FILTRATION W. J. WNEKt and D. GIDASPOW Instituteof Gas Technology, IIT Center, Chicago, IL 60616,U.S.A.

and

D. T. WASAN Illinois Institute of Technology, Chicago, IL 60616,U.S.A. (Received 22 April 1974;accepted 5 December 1974) Absbnct-From mass balance on the suspension and lilter and a charge balance on the filter, a set of hyperbolic partial differential equations is derived which describes how the suspension concentration, surface charges of the filter and particles, porosity of the filter, and the pressure drop vary with filter depth and time. These equations include a local deposition term which is evaluated by considering transport of the suspension particles to the filter particles by Brownian diffusion, interception, and sedimentation. The effect of the surface forces due to electrical double layer and van der Waals interactions was taken into account by treating the surface of the filter particles as possessing lkst order intermediate reaction kinetics, for which the rate constant is a function of the stability ratio of colloid chemistry. The governing equations were solved numerically, and the results compared with experimental data for unflocculated particles. The proposed tIltration model is an advance over present models in that it contains no empirical factors which must be evaluatedfrom filter runs and the effects of surfactants, pH, and ionic strength are accounted for.

Although deep bed liquid filtration has been extensively studied for years, it is still not presently possible to predict the removal efficiency of such filters from the characteristics of the suspension and lilter particles. Several mathematical models for liquid filtration have been proposed[l]. Unfortunately, all of them contain a number of empirical constants which have little physical significance. Further, the effect of surface forces due to van der Waals and electrical double layer interactions has been neglected. In this paper, it is shown that consideration of these surface forces is the missing factor in current liquid filtration theory and that their inclusion allows the efficiency of a filter to be predicted from information on the surface characteristics of the suspension and filter particles. The qualitative basis of the proposed filtration model is as follows. From a mass balance on the colloidal suspension, a partial differential equation describing how the concentration of the suspension varies with time and bed depth is obtained. This equation includes the local rate of deposition of the colloid onto the filter which depends on the charge characteristics of the colloid and filter. In order to complete the system of equations, this local rate and the manner in which the charge and porosity of the lilter vary with time and bed depth are needed. The latter are obtained by performing a charge and mass balance on the tilter which results in two additional partial differential equations. The three equations are coupled through the expressions for the local deposition rate. For a specific deposition mechanism, an expression for the deposition rate may be derived, and then the system is

completely defined and may be solved to give the breakthrough curves and the pressure drop variations with time. Iit order to calculate the deposition rate, the work of Yao et al. [2] is used. They considered transport in a filter due to interception, sedimentation, and Brownian ditfusion in the absence of surface forces. It has been shown by other investigators that it is justified to neglect the effect of inertial impingement in liquid tiltration[l, 3,4]. The results of their study showed that numerical solutions of the convective-diffusion equation were in agreement with calculations based on adding the contributions of each mechanism considered separately of the others. It was found that for sand beds diffusion was the controlling mechanism for particles below 1 m in diameter while above 6 pm sedimentation and interception were controlling. The result of interest which simplifies the inclusion of surface forces is that the various mechanisms can be treated separately and then combined to give the total rate of deposition. In order to develop design procedures for the removal of colloidal particles from water by deep bed lilters it is necessary to describe mathematically the mechanism by which deposition of these particles onto much larger particles occurs. However, due to the complicated flow patterns in a filter bed, this is a very diacult problem. Thus, in order to gain some insight to the basic mechanisms involved, the deposition of charged colloidal particles onto the charged surface of a rotating disk was analyzed by Wnek[fl and Wnek et al. [6]. The analysis is similar to that of Spielman and Friedlander[‘ll. In this system, diffusion was the controlling transport tPresent address: Aerojet Nuclear Co., Idaho Falls, ID 83401, step, and due to the fact that the surface of a rotating disk is U.S.A. uniformly accessible, the only independent variable was 1035

1036

W. J.

WNEK,

D. Gm~w

time. Thus, this problem is similar to considering a fixed point in a tilter and following how the various properties vary with time there. The important result that emerged from this analysis was the deposition of charged particles to a charged surface can be treated as a convectivediffusion problem with first-order surface kinetics accounting for the surface force interactions. Using this result, it is possible to extend the heat and mass transfer model of Pfeffer[8] for packed beds. He combined the free surface model of Happel[9] which implies that the concentration boundary layer starts anew on each particle with the analysis of mass transfer from an isolated moving sphere[lO]. Efeffer’s solution is for the case of a zero concentration boundary condition at the surface of the sphere and is extended to the case of a first-order surface reaction. Justification for using this model is that it is in agreement with data on heat and mass transfer in packed and fluidized beds. Also, Yao et al. [2] and Cookson[ll] have been reasonably successful in applying it to their filtration data. In order to calculate the deposition rate due to interception and sedimentation under the inAuence of surface forces, the work of Spielman[l2] and Fitzpatrick1131 is used. The basis of the method is to perform a force balance on a colloidal particle approaching tilter particle which results in an ordinary differential equation for the particle’s trajectory. MASS BALANCE ON SUSPENSION

A mass balance may be performed on the colloidal suspension within a differential element of volume of the void space of the filter to yield

fl)

and D. T.

WASAN

W, = stability ratio &r =l for i#j l/2 for i = j I = velocity gradient in liquid from which

z i(mi, mi) Bi = x (J(l?lj,mi -

Di=

I

mj).

I

If Eq. (1) is local volume averaged over the cross-sectional area of the filter as in Slattery[l4], the result is

where the bar denotes an average taken over the liquid volume, V = superficial velocity z = axial coordinate Jd”=.f. (--D. (ac&n)tu.c,)dS = rate of particle deposition/unit filter volume S = specific surface area of filter (area/unit filter volume) n = outward normal to the surface of the fllter particles. .I, is also subject to the constraint that whenever the shear stress at the surface of the deposit due to the fluid exceeds the shear strength of the deposit, its value is zero. If one is willing to associate the shear strength with the tensile strength, estimates of the latter can be made for various .bonding mechanisms[lS, 161.The shear stress of the fluid may be estimated from du, r=CLz

where cl = number of suspension particles of the ith size/unit liquid volume t=time u) = velocity in the jth direction xi = jth coordinate 9, = diffusivity in the jth direction rX = rate of disappearance of the ith particles J3,= rate of appearance of the ith particles. The birth and death terms are a result of agglomeration of the suspension particles due to Brownian movement, turbulence, and/or velocity gradients in the liquids. The number of collisions between particles of mass rnr and mi can be expressed as [ lo], J(rni, ml) = %CiCl

(2)

where

where r.~= fluid viscosity u, = velocity tangent to the surface of the deposit. Since the accumulation and diffusion terms are small in comparison with the convection term[lTJ, they may be dropped. Also the average concentration is essentially equal to that in the bulk. Further, from this point on, the birth and death terms will be omitted and a single particle size considered. As a result, Eq. (3) becomes

pB_

XT-

_J

”

where CS = the bulk concentration. The deposition term may be expressed as

4?r&,(ai t aj)91jWij’ for Brownian movement Kij =

J,=NFJ

or turbulence 4 ? IQ, + Uj)’

for gradient coagulation

g,, = Brownian or eddy ditfusivity

(5)

where J = number of suspension particles deposited on a fllter particle unit time

The role of colloidchemistryin modelingdeepbed liquidfiltration

1037

NF = number of filter particles/unit titer volume. Since (lla)

I _ E= Solid Volume Total Volume subject to the initial condition

= Number of filter particles x Volume of a filter particle unit filter volume

qF = qo(Z)

then 1-r N’” (4/3)7ra2

(6)

(7)

t=0

at

Ulb)

The reason that an effective particle charge $ is used in Eq. (11)rather than the actual value is to account for the fact that when a surface is completely covered by colloidal particles, it tends to accept their surface properties. One way of representing this effect is to consider that the charge over an area 4ap2due to the filter particle at y = 2a, and the actual particle charge are additive. This results in the expression 151

Further, defining the removal efficiency as & = qp -4a’q;jl J 17= ?r&’VCB

to the linearized Poisson-Boltzmann equation for a flat plate has been used[lSl.

cw and is subject to the boundary conditions

C,=O

for

(12)

(8) where q$is the filter charge per unit area and the solution

Equation (7) may be expressed as

CB= C, at Z=O,

-e-““p)

allt

ZzyI

EFIXCI’OF DEFCMTION ON !JIZE OF FILTER PARTKL.ES, POROSITY, AND PRWWRJZ DROP

In order to determine how the size of the filter particles, porosity, and pressure drop vary due to deposition, the number of suspension particles that are deposited on a CW filter particle is needed and is given by

w

The flux and thus 77 depend on the mechanism of deposition, the flow pattern in the filter, and the surface charges of the bed and suspension particles. Since the quantities which may change during filtration are the charges of the filter, and suspension particles, porosity, and the size of the filter particles, J and 7 will be of the form

Flux of particles Rate of particle deposition/unit filter = onto the filter/unit filter volume volume M$!XL&J aN+ atsubject to the initial condition N=O

J= J(qF, qP,4 ap) = 7 (qF9qP.4 OF)

(lOa) (lob)

where qF = surface charge on a filter particle qp = surface charge on a suspension particle

aF = radius of a filter particle. In order to determine J, a specific deposition mechanism must be chosen and an analysis carried out. This will be discussed later. CRARGE BALANCE ON FILTJZR

In order to complete the set of governing equations, a charge balance is performed on a differential element of volume of the filter Accumulation of = Flux of charge charge on fdter to the filter /unit filter volume /unit filter volume

Wa)

at

t=O

W)

where N is the number of suspension particles deposited on a filter particle. If the charge of the suspension particles is constant, then N may be obtained from qF according to N,&k tip where AqF is the change in the surface charge of a filter particle. The increase in the volume of a filter particle due to deposition is AV~=i.gap’N so that the total volume of a tilter particle is VF =irra&,+AV, =:lra&[l+(z)lN].

1038

W. J.

WNEK,

D. GTDASPOW and

Defining an effective radius according to V, = (4/3)7x1:,

D. T. WASAN

subject to c = 0 at the particle’s surface

..=aFo[l+(~)1N]“3

(14)

pp = density of suspended particles m = mass of suspended particle

where aF0= initial radius of filter particle.

dp = 2ap.

From Eq. (6), 1-eo=;7r&N~ and 1-•=+Np so that their ratio gives 1-E = (1 - EC!) (z)3

where

(15)

where e. = initial porosity. The pressure drop across the lilter is given by

Equation (18) takes into account transport due to interception, sedimentation, and Brownian diffusion in the absence of surface forces. Inertial impingement was considered insignificant which has been shown to be valid by other investigatorsI 1,3,4]. The results of their study showed that numerical solutions of Eq. (18) were in agreement with calculations based on adding the contributions of each mechanism considered acting separately of the others. It was found that for sand beds diffusion was the controlling mechanism for particles below 1 pm. The result of interest which will simplify the inclusion of surface forces is that the various mechanisms can be treated separately and then combined to give the total rate of deposition. DEF’MTION RATE DUE TO BROWNIANDIFFUSION

=dP AP= -dZ I o dZ

(16)

where the pressure gradient may be calculated from the Ergun Equation [ 111 _&dP=q+bk VdZ

(17)

where L = length of filter a = 150(1 -E)2 CL 7z 6 = *.757$ rir=pv g = gravitational constant p = viscosity of liquid

In order to determine the deposition rate due to Brownian diffusion, the free surface model of Happel[9] will be used which implies that the concentration boundary layer starts anew on each particle. Pfeffer[8] has shown that when this model is applied to heat and mass transfer in a packed bed, the theoretical results are in agreement with data. The assumptions for this model are a Reynolds number less than one and a Schmidt number much greater than one. In the case of liquid filtration these conditions are fulfilled. Further, Yao et al. [2] and Cookson[ 111have been reasonably successful in applying this model to their filtration data. The derivation of the deposition rate including surface forces will be only sketched. It follows that given by Levich[lO]. Details may be found in [5]. Following Levich[lO], the convective diffusion equation in spherical coordinates and the stream function in the vicinity of the filter particle surface as given by Pfeffer 181 which is Levich’s Eq. 14.2 with the free stream velocity U defined as

dF = 2a.n p = density of liquid.

(19)

It should be noted that in the above analysis it has been implicitly assumed that the deposit is not significantly flocculated. Otherwise, the density of the floe would have to be taken into consideration in the analysis as discussed by Fox and Cleasbyll91.

where W=2-3y+3ys--2y6 y=(l-c)‘” are combined to yield

THECALCULATlON OF THE RATE OF DEPOSITION Yao et nl.[2] have analyzed the transport step of the filtration process by considering the convective-diffusion equation for a single filter particle $+“vc=Av%+

(

1-e

mg ilc -PP> 3rpdp ar

(18)

$-$(+-g-J

(20)

The associated boundary conditions may be formulated as

(214

1039

The roleof colloidchemistryin modelingdeepbed liquidfiltration

Equation (26) satisfies the governing partial differential equation, and the constants C,, and C, are to be chosen so c = CD,0 = 0, + = 0 (2lc) as to satisfy the boundary conditions. Unfortunately, such a procedure fails due to the surface reaction condition. where the rate constant is given by[5,7] However, a solution may be possible by allowing CZand CJ to be functions of a variable other than Z[21] which %J turns out to be 0. Substitution of Eq. (26) into the (22) =(aF+aF.)WFF’ boundary conditions 21 allows Cz and C, to be evaluated as It should be noted that the effect of the surface forces is accounted for in k. The expression used for the stability CE ratio Wm between the filter and suspension particles is c*(e) = (27) that obtained by Wnek[S]. F(8) Although the first boundary condition does not permit a similarity transformation to go through, let us consider c C,(e) = i F(B)&(B) (28) as a function of n and t where as

c=cB

4+--m

(21b)

k

where

q=-+. Then E@.(20) becomes ac 2

dc

d (23)

Since a solution is desired for small t, the first term in Eq. (23) is small in comparison with the others and can be dropped. A related problem has been treated by Solbrig[20] for convective diffusion in the entrance regions of a parallel plate duct and was solved as a Leveque type problem which is mathematically equivalent to Row over a single plate. Exact numerical solutions showed that this term could be dropped provided

The local deposition rate can then be calculated as jD=kC

IA=

jD” 1 tjDo/jsd

where jDois the deposition rate for diffusion controlling as given by Levich’s Eq. (14.16) and jsOthat for surface forces controlling given by js0= kCB. The total deposition rate is expressed as (30)

where Pe

=

2aFu

where S is the surface of the filter particle. Noting that in(e) is not extremely sensitive to 8 except around f3= ?T where the deposition is very small, Equation (30) may be approximated by

-z-

L = length of plate. For flow across a sphere, L can be associated with 2?raF so that an approximate criterion for neglecting this term is Pe > l&r

(31)

which is more than satisfied for particulate suspensions. Now, letting

where JD’ is the total deposition rate for diffusion controlling as given by Levi&s Eq. (14.19) and Jso that for surface forces controlling given by

Equation (23) becomes d2c 4 2 dc @f?Z ;iz=O

(25)

whose general solution is c(Z)=&l’exp(-!Z’)dZ+G.

(26)

JSO = 4?raFzkG. It should be pointed out that Spielman and Friedlander 171 have also considered this problem and obtained a solution by solving an integral equation numerically. In the preceding analysis, k has been taken as a constant. However, since the deposition rate varies over

W. J.

1040

WNEK, D.

Gmurow and D. T.

the surface of a filter particle, the charge and thus k also vary. If the variation in charge is not too extreme as would be expected, the first term in Eq. (23) can still be neglected. This again leads to Eq. (29) but with k as a function of 13.When Eq. (30) is now evaluated, f is also a function of 0, and its average value may be approximated at the average charge. Thus, Eq. (31) is again obtained where .I,”is evaluated in terms of the average charge of a filter particle. DEPOSITIONRATE DUE TO INTERCEPTIONAND SEDIMEZNTATION

In order to determine the deposition rate due to interception and sedimentation, the single particle efficiency for classical interception will be used with a correction due to gravitational, van der Waals, and viscous interaction as given by Spielmen and FitzPatrick[22]. The effect of repulsion due to double layer interaction will be taken into account by considering that the limiting deposition rates in the presence and absence of double layer repulsion may be combined as accomplished previously to yield the actual deposition rate. In classical interception, it is assumed that the suspension particles follow the streamlines of the fluid and are deposited if they pass within a distance a, of the surface of the filter particle at the angle 8 = 90”. Under these assumptions, the single efficiency by classical interception is 27rr dr (32) and yields [30] (33)

WASAN

applied to the classical interception efficiency due to hydrodynamic, gravitational, and van der Waals interactions (34) where &I, = adhesion group = Aail9vpA, Va,’ NG,,= gravity group = 2(p, - p)ga:/9pA.V As=2(1-y’)/W

which may be obtained from Fig. 1. Alternately, one could also use the similar work of Payatakes et al.[23]. Since the attractive force due to double layer interaction acts over a distance of hundreds of angstroms and the viscous and van der Waals interaction on the order of a,, double layer attraction will not significantly increase aid. However, Spielman and Cukor[24] have also considered the effect of double layer repulsion in the absence of sedimentation in similar calculations and presented their results in graphical form in terms of &d, and a zeta potential group. They showed that double layer repulsion decreases nya. Unfortunately, these results are not sufficiently complete due to the many variables involved. Further, the calculation of the efficiency according to their method in conjunction with the numerical solution of the set of partial differential equations describing the overall filter efficiency would be impractical. Thus, an alternate approach is needed which is to combine the deposition rate J!G in the absence of double layer repulsion and that .I: for when the surface forces are controlling as done previously. Then, the deposition rate due to interception and sedimentation is

For an isolated filter particle (e = l), JIG

=-&

(35)

JJ in agreement with Yao et ai. (1971). Due to viscous interaction, the motion of a suspension particle approaching a filter particle is retarded. If there were no attractive forces acting such as the van der Waals force, contact could not be made because the rate of drainage of fluid between the particles becomes inlinitesimally small as the gap narrows. However, since the van der Waals force increases very rapidly as the separation decreases, the slow drainage is eventually overcome, and contact is made. Spielman and FitzPatrick[22] have analyzed this problem and obtained the equations of motion which describe the trajectory of a suspension particle in the vicinity of a filter particle. By determining the limiting particle trajectory which just misses hitting the filter particle, the single particle efficiency can be determined. Spielman and Fitzpatrick 1221have numerically integrated the governing differential equations and presented their results in terms of a correction factor

Since the double layer repulsion acts over a smaller distance than the van der Waals and viscous interactions, flu and not J% is the appropriate term to use in conjunction with A“. Although one would have liked to correct the classical result by absorbing the effect of both surface forces and the viscous interaction into Jp as was done for the diffusion case, this was not convenient because in contrast to the diffusion case the distance S where the surface forces are insignificant would have had to be determined[S]. Then the flux at S would have been given by Eq. (33) with ap replaced by ap + S from which it is seen that the result is very dependent on 8. Just the opposite was true for the diffusion where the result was insensitive to S [5]. It should also be recalled that the assumption underlying Eq. (35) is that the velocity component due to the surface forces is much greater than that due to fluid flow and gravity in the close vicinity of the surface of a filter particle.

The role of colloid chemistry in modeling deep bed liquidfiltration

,0-c

,0-s

10-4

10-3

10-2

10-1

100

IO’

Id

I03

IO4

1041

lo5

IO6

IO’

NA4=oq%p&JJ

Fig. 1. F(N,.,,, NO,,) vs NM, and No,.

The filter coefficient is an empirical expression which may be written in a general form as[l]

Df?KSITION RATE DUE TO BROWNIAN DIFF’USION, INTwCXl’ITON, AND SEDIMENTATION The deposition rate due to Brownian diffusion, interception, and sedimentation is obtained by using the result of Yao et a/.[21 that in the absence of surface forces the separate rates are additive and then combining this total rate with that for surface forces controlling as shown previously.

A = ho(1+ @r/a)’ (1 - o/e)’ (1 - U/U”)”

(38)

where ho = initial value A ultimate value of u p = a geometric constant related to the packing of the filter particles x, y, 2 = empirical constants. U" =

J=*

(36)

I

where P=J,“+& COMPARISON OF PROPOSED FILTRATION MODEL WITH OTHER MODELS

Present mathematical models of liquid filtration may be expressed as -$ = -h(u)c

(37a)

au -=-v-

(37b)

at

ac az

subject to c = co at Z = 0 and u = 0 at t = 0 where A = the filter coefficient u = volume of deposit/unit filter volume c = suspension concentration on a volume basis. The basis of Eq. (37a) is that experimental observations show the removal of suspension particles to be proportional to their concentration. Equation (37b) is simply a mass balance on the suspension.

Equation (38) has little theoretical basis and is merely a composite of experimental observations. Further, even the qualitative reasoning behind this expression is questionable. It is proposed that break-through occurs as a result of decreasing porosity due to deposition which in turn decreases the filter coefficient. However, the theoretical expressions presented indicate that a decrease in porosity from Oe4to 0.3 does not sufficiently diminish the filter coefficient to cause breakthrough. On the other hand, the pressure drop is significantly increased which means that the shear stress acting on the deposit is also increased. This suggests that in the absence of repulsive surface forces or for high ionic strength breakthrough occurs as a result of scouring of the deposit. It is apparent from Eq. (38) that the filter coefficient is not a very well defined quantity and is a function of several empirical constants which have no physical meaning. As a result, it is not feasible to use these models for design work since the information to be generated by the model must first be obtained experimentally in order to determine the empirical constants. Such a procedure is no more than a sophisticated version of curve fitting data to an equation.

W. J.

1042

WNEK, D. GIDASPOWand D. T. WASAN

It is of interest to compare these models with that proposed in this paper. A comparison of the analogous Eqs. (9a) and (37a) shows that the filter coefficient can be associated with the single particle efficiency according to

c,m’= c0 Cs"'=O if iAZ+ qF(‘)= qo(‘),t =

31-e A =iapn

(39)

Also, Eqs. (12a) and (37b) can be associated with each other. However, no equation has been presented for taking into account the effect of surface forces as does Eq. (11). Further, it should be noted that the difference between the models is not simply the inclusion of a charge balance but that the filter coefficient can now be calculated a priori on the basis of the deposition mechanisms occurring in a filter and does not contain any empirical factors. In addition, the effect of surfactants pH, and ionic strength in liquid filtration has been taken into account for the first time. SOLUTIONOFTHEFlLTRATION EQUATIONS Due to the complexity of the filtration equations, it is

not possible to obtain an analytical solution so that a numerical solution is required. Since the governing equations constitute a set of hyperbolic partial differential equations, the method of characteristics may be used. For convenience, let us restate the problem to be solved. The set of equations is given by Eqs. (9), (11) and (13) which may be expressed as

$=

subject to

-h(qF, N)CB

N”’ =

for i = 12 , ,.a., M where C,“-‘) = value of CB entering the ith segment

qF(‘),N”’ = values of sp and N within the ith segment Since C,“’ can be eliminated in Eqs. (40b) and (4Oc) by means of Eq. (4Oa),it is seen that Eqs. (40) define a set of ordinary differential equations which may be solved by any standard numerical technique such as Runge Kutta. An asymptotic solution to the filtration equations is derived in the Appendix of this paper. NUMERlCALCALCULATIONSANDCOMF%RlSONOF FILTRATION MODEZLWITHDATA

Numerical calculations were performed based on the experimental data of Heertjes and Lerk[25] for the filtration of iron hydroxide particles and that of Bums, Bauman, and Oulman[26] for the filtration of clay particles by diatomite particles[5]. The values used for iron hydroxide particles are given in Table 1. The effective charge of the suspension particles was taken as equal to their actual charge. The mass concentration of the suspension was converted to a particle concentration according to

c, = 1 x 1o-6G/m,

(114

g

Wa)

= J(Cs, q1;, N)

subject to ca,z=o,

all

t

=

1 o,z+ qF

=qO,t=O

N

=O,t=O

differential equations are also coupled to the algebraic equations (14), (15), (36) and (39). The pressure drop is obtained by integrating Eq. (16) using Eq. (17). If the length of the filter is divided into M segments of length AZ, the governing equations can be written as These

C,‘” = Ci’-l) exp [ - A (qi”, N”‘)AZ] dq;” 7 = 4J( cp, -dN”’ = J(@“, At Y.

q/j, N”‘) qp, N”‘)

(aa)

W-W

(41)

Table 1. Valuesof the parametersused for the data of Heertjes and Lerk[27lfor the tiltrationof iron hydroxideparticlesby glass particles 7.7 ms Felt Mass Conccntralian of Suspension Radius of Suspension Particle8

CB

0

0, t = 0

200

II

Density of Suspension Material (Fe (OH), )

3.61

Particle Concentration of Suspension

Equations (41) and (42) (Number of particles / cm’)

Radius of Filter Particles

250 Y

Superficial

0.07

Velocity

g/cm’

cm /a&c

Length of Filter

12 and 24 cm

Porosity

0.36

Temperature

250 C

Conductivity

5.6 x IO-$ ohm-’

Ionic Strength

4.43

Hamaker

0.5

Constant

Y. lo-’

mole/L

to 1.5 x lo-”

Initial Zeta Potential of Filter Particle6

-55 rn”

Initial Surface Charge of a Filter Particle

-2.5

zeta Potential of Suspension Particles

+ 34.5 (43)

cm-’

erg

x 1O-9 coul mv or Equation

Theroleof colloidchemistryin modelingdeepbed liquid filtration where C, = number of particles/cm” C, = mg/l m, = mass of a suspension particle = (4/3)Ta,)pb pb = density suspension material (g/cm’). In the case of the Heertjes and Lerk data, the mass concentration was given in terms of the amount of iron in the suspension. Assuming the particles to be Fe(OH),, m, was obtained from

where mFc = mg Fell. Also, the conductivity K was given as 5.6 x 10m5 ohm-’ cm-’ from which the ionic strength C may be estimated according to

CJf)OoK A

where A = equivalent conductivity. Assuming the solution to be a dilute sodium chloride solution, the value of A is 126.45 mho cm’lequiv. Thus, c

=

looo(5.6x 10-P =

4.43

x

1o-4

& 1

126.45

Heertjes and Lerk[27] have further shown that the zeta potential of iron hydroxide particles is a function of the suspension concentration. It was possible to fit their data with the equation &(mv) = 13.2 In G(mg/Fe/l) - 4.0. Since this equation gives t = 23 mv for 7.7 mg Fe/l while Heertjes and Lerk (1967)give { = 38 mv, this equation was modified to read [13’21u cFe-4’0]

(43)

where the factor 1.5 approximately converts the Smoluchowski value to the Wiersema one. Equation (43) was used for describing an iron hydroxide suspension whose zeta potential varies with suspension concentration in the filter. For the clay particles, a zeta potential of - 100mv was selected based on the data of Lorenz[28] and a Hamaker constant of 1 x lo-” erg. The zeta potential of diatomite particles coated with Purifloc 601 was taken from Burns et al. [26]. Unfortunately, the zeta potential measurements were carried out for an ionic strength of 4~ 10m4mole/l while the tilter runs were made for 1 x 10m6mole/l. The breakthrough curves based on the Heertjes and Lerk data for a constant and varying suspension zeta potential are presented in Figs. 2 and 3 respectively. The zeta potential of the effluent particles is given in Fig. 4. It is seen that a constant zeta potential for the suspension better represents the data than a varying one. The latter predicts that the initial etlluent concentration is not greatly effected by the filter length as a result of a charge reversal around 1.5 mg Fe/l, while the former does, which is in agreement with the data. The explanation for this is that as the suspension particles are being removed, equilibrium between the potential determining ferric ions on the particles and in solution is being disturbed. The residence time of the particles in the filter is on the order of a minute, and the time between taking a sample and making a measurement is no doubt much greater. Thus, equilibrium is probably far from reached within the filter while for the measurements equilibrium has been attained. The effect of increasing the value of the Hamaker constant was studied by Wnek[5] and it was shown that the breakthrough curve is only slightly altered toward a higher removal. For the case of a constant suspension zeta potential, the surface charge profiles for the filter are shown in Figs. 5 and 6 while the corresponding zeta potential profile of the filter is given in Fig. 7. Although Heertjes and Lerk measured the zeta potential of the filter, it was not possible to make a quantitative comparison because they did not make a non-flow or background correction on the streaming potential. However, the predicted curve does give the same trends as their data.

TIME iminl

Fig.2. Breakthroughcurves for &, = 34.5 mv, A = 0.5 x IO-‘* erg, L = 12,24cmand data of Heertjes and Lerk[27]. CES Vol. 30, No. 9-C

1043

W. J. WNEK,G. GIDASPOW and D. T. WASAN

1044

3-

0-i 0

1 20

10

11 ?a

11 5-0

40

I To

60

I 80

I 90

I loo

I 110

I 120

11 I3o

11 I5o

140

1 110

160

I

TIME (min)

Fig. 3. Breakthrough curves for & as a function of suspension concentration. 40

30

-

20

-

0

IO

I

I

I

I

I

I

I

I

I

20

30

40

M

60

70

60

90

100

TIME

I 110

I

I

I

I20

130

140

I

I

I

I60

160

170

I60

(mm)

Fig. 4. Zeta potential of efftuentparticles vs time for avarying suspension zetapotential (L = 12cm).

0

I

2

3

4

6

6

7

8

9

IO

II

0

BED DEPTH km1

Fig. 5. Surface charge of a filter particle vs depth at various times for constant & = 34.5mv (L = 12cm).

10

PO

3”

4”

50

60

70

TIME (mini

Fii.

80

90

IO0

110

120

6. Surface charge of a filter particle vs time at various depths for constant t = 34.5 mv (L = 12cm).

1045

The role of colloid chemistry in modelingdeep bed liquid liltration

J 90 TIME

(min)

Fig. 7. Zeta potential of a filter particle vs time at various depths for constant & = 34.5 mv (L = 12cm). It was found that for the Heertjes and Lerk data the controlling deposition mechanism was Brownian diffusion and that the increase in pressure drop was negligible as observed experimentally. Breakthrough curves based on the data of Burns et a1.[26] were also computed by Wnek[S]. However, agreement between the predicted and experimental curves was not as good, in contrast to the previous case. However, the information on the zeta potentials of the suspension and tilter was somewhat limited. In addition, some stmining was taking place. The reason for performing these calculations is that for the Burns et al. data both Brownian diffusion and interception were the controlling mechanisms and pressure drop increased with time. It is felt that this discrepancy is not a deficiency of the model but rather that of the data because by judiciously selecting the zeta potentials or using effective rather than actual particle charge a better fit probably could have been obtained. The increase in the pressure drop with time is shown in Fig. 8 where it is seen that the pressure drop increases linearly with time which is in accord with experimental

g

0.7

i% 4

0.6

observations[l]. Although it was not possible to make a quantitative comparison because Burns et al. [26], plotted the pressure drop against the amount of clay removed, the increase in the pressure drop and its absolute magnitude are in agreement with their data. Acknowledgmm&One of the authors (WJW) would like to express his gratitude to the IIT ResearchInstitute for supporting this study. NOTATION

A‘ 2(1-73/w a 150((1- E)*/c’)(p/&)orradiusof aparticle ai radius of sphere i aF radius of a filter particle aF0 initial radius of a filter particle up radius of a suspension particle b 1.75 (1- e/c3) (l/d,) El rate of appearance of the ith particles ce concentration of the colloidal suspension in the bulk ci number of suspension particles of the ith size/unit liquid volume

a. g

0.5

*z

0.4

aw z

0.3

0

5

IO

I5

20 TlMf

25

2:)

:; *

(mini

Fig. 8. Increase in pressure drop vs time for data of Burns et al. [26](initial pressuredrop = 71.9 cm H20).

W. J.

WNEK, D. GIDASPOWand

concentration of the suspension entering the filter 2aF

diameter of a suspension particle rate of disappearance of the ith particles Brownian diffusivity diffusivity in the jth direction Brownian or eddy dzusivity gravitational constant flux or deposition rate per filter particle flux or deposition rate due to Brownian diffusion deposition rate due to Brownian diffusion in the absence of surface forces deposition rate due to interception and sedimentation deposition rate due to interception and sedimentation in the absence of electrical double layer interaction collision rate between particles of type i and j J;+.& flux or deposition rate for surface forces controlling deposition rate/unit filter volume %/(qF+qP)%P

coagulation constant for the ith and jth particles length of filter mass of jth particles mass flux = pV outward normal to the filter particle surface number of colloidal particles deposited onto a filter particle adhesion group, AaF’19~~A,Va~4 number of filter particles/unit filter volume gravity group = 2(pp- p)gap2/9&V pressure drop across filter Peclet number, dJJl% surface charge of a filter particle change in the surface charge of a filter particle initial surface charge of a filter particle actual surface charge of a suspension particle effective surface charge of a suspension particle radial distance from the center of a particle shortest distance between two spheres or specific surface area of filter time 2V(l- $)/W velocity in jth direction velocity in radial direction velocity in O-direction ttigential velocity along the deposit superficial velocity volume of a filter particle 2-3yt3y’-2y6

stability ratio for particles of type i and j stability ratio for filter and suspension particles jth coordinate distance away from the surface of a filter particle axial coordinate along the length of the filter

D. T. WASAN

Greek symbols

geometric constant related to the packing of the filter particles (1 - E)“~ or kinematic viscosity velocity gradient in liquid 1 for i# j, l/2 for i = j porosity of filter initial porosity of filter zeta potential removal efficiency of a single filter particle removal efficiency of a single filter particle due to classical interception removal efficiency of a single filter particle due to interception and sedimentation in the absence of electrical double layer interaction angular coordinate filter coefficient = $(I - e)/aF initial valut Of filter coefficient viscosity density of liquid density of a suspension particle volume of depositelunit filter volume ultimate value of u shear stress stream function denotes an average value

REFERENCES

HI Ives K. .I., Review Paper: Rapid Filtration, Water&search, 19704 201.

PI Yao K. M., Habibian M. T. and O’Melia C. R., Water and

Waste Water Filtration: Concepts and Applications, Environ. Sci. and Tech., 19715 1105. [31O’MeliaC. R. and Crapps D. K., Some Chemical Aspects of Ranid Sand Filtration. Jour. AWWA. 196456 1326. [41O’Ikelia C. R. and St&m W., Theoi of Water Filtration, I. Am. Water Works Assn., 1%7 59 1393. 151Wnek W. J., Ph. D. Thesis, Illinois Institute of Technology, Chicago, Ill. 1973. t61Wnek W. J., Gidaspow D. and Wasan D. T., The Deposition of Colloidal Particles onto the Surface of a Rotating-&k, to be published in J. Colloid. Interface Sci. 1975. [71Spielman L. A. and Friedlander S. K. The Role of the Electrical Double Layer in Particle Deposition by Convective Diffusion, J. Colloid Interface Sci., 197446 22. WI Pfeffer R., Heat and Mass’ Transport in Multiparticle Systems, I & EC Fund., 1964 3 380. 191Happel J., Viscous Flow in Multi-particle Systems: Slow Motion of Fluids Relative to Beds of Spherical Particles, AICHE I., 1958 4 197.

[lo] Levich V. G., Physicochemical Hydrodynamics, Prentice Hall, New York, 1%2. [ll] Cookson J. T. Jr., Removal of Submicron Particles in Packed Beds, Environ. Sci. and Tech., 19704 128. [12] Spielman L. A. and Goren S. L., Capture of Small Particles by London Forces From Low-Speed Liquid Flows, Environ. Sci. and Tech., 19704 135. [13] Fitzpatrick J. A., Ph.D. Thesis, Harvard University, Cambridge, Mass., 1972. 1141Slatt&y J. C., Momentum, Energy, and Mass Transfer in Continua. McGraw-Hi. New York 1972. [15] Rumpf, I% in Knepper’ W. A., Agglomeration, Int. Symp. Philadelphia, Pa., p. 379, 1%1. [la] Pietsch W., The Agglomerative Behavior of Fine Particles, Staub, 1%7 27 24. [17] Roy D. and Gidaspow D., Prediction of Nusselt Numbers for Regenerators with Fully Developed Velocity Profiles,

1047

The role of colloid chemistry in modeling deep bed liquidfiltration TwelfthNational Heat Transfer Conference, AZCHE-AWE, Tulsa, Oklahoma, August 15-18, [IS] Kruyt, H. R., Colloid Science, Vol. I, Elsevier Publ. Co., Amsterdam, 1952. [19] Fox D. M. and Cleasby J. L. Experimental Evaluation of Sand Filtration Theory, J. SanitaryEng. Div.,ASCE, 92, No. SA5, 196661. [20] Solbrig C. W., Ph.D. Thesis, Illinois Institute of Technology, Chicago, Jan. 1966. [21] Lyczkowski R. W., MS. Thesis, Illinois Institute of Technology, Chicago, 1%6. [22] Spielman L. A. and Fitzpatrick J. A., Theory for Particle Collection under London and Gravity Forces, J. Colloid Interface Sci., 197342 607. [23] Payatakes A. C., Tien C. and Turian R. M., Trajectory Calculation of Particle Deuosition in Deeu Bed Filtration: Part I. Model Formulation; AZCHEJ., 19j4 20 889. [24] Spielman L. A. and Cukor P. M., Deposition of NonBrownian Particles under Colloidal Forces. J. Colloid. Interface Sci., 197343 51. [25] Heerijes P. M. and Lerk C. F., The Functioning of Deep-Bed Filters: Part I: The Filtration of Colloidal Solutions. 1%7 45 Tl29. [26] Bums D. E., Baumann E. R. and Oulman C. S., Particulate Removal of Coated Filter Media. J. AWWA. 197062 121. [27] Heertjes P. M. and Lerk C. F., So&e Aspects bf the Removal of Iron From Groundwater, Interaction Between Fluids and Particles (London: Instn. Chem. Engrs.) p. 269 (1%2). [28] Lorenz P. B., Surface Conductance and Electrokinetic Properties of Kaolinite Beds, Clays and CIay Minerals, I%9 17 223. [29] Paretsky L., et al., Panel Bed Filters for Simultaneous Removal of Fly Ash and Sulfur Dioxide: I. Filtration of Dilute Aerosols by Sand Beds, JAPCA, 197121 204.

Asymptotic solution of the filtration equations Suppose that there exists a concentration front which moves down the length of the filter at a constant velocity Us such that above it the titer is completely saturated and no longer removes any suspension particles. Such a situation usually develops after the filtration has proceeded for a certain time, especially in long beds. This time period would be that required to saturate the top of the filter. The governing filtration equations are a mass balance on the suspension

and a mass balance on the deposit

aN_J

at-

where J, is the rate of deposition per unit filter volume. It should be noted that J, is a function of C, and N These equations can be transformed into a set of ordinary differential equations via the change of variables I=Z-v&

(2)

The resulting set of equations is

dN I --Z---J” dt VF

(3b)

In order to determine the value of V~the ratio of Eqs. (3a)and (3b) may be taken to give

dG

VP

dN

V

-=-

(4)

Since the fdter is saturated above the front, C,=C,,,

N=Nst

forlc0

where C, is the feed concentration while far down the iilter C,=O,

N=N,

forl=m

Thus, integrating Eq. 4 from 4 = 0 to .$= m and solving for vF we obtain a simple expression for the propagation velocity

vF=q&)v The value of N,, is obtained by solving the filtration equations for the top of the filter which is accomplished by integrating Eq. (lb) with Cg set equal to C,. This yields an ordinary diierential equation. The value of qslt is that when 4 + m or alternately when N does not vary significantly.

qsat=qo+qP The initial conditions for Eq. (3) are

“5%-J az

”

C.=C,,

N=N,,

at,$=O.