Transient behavior of deep-bed filtration of brownian particles

Chemical Engineering Science, Printed in Great Britain.

Vol.

42, No.

1 I, pp. 2729-2739.

1981.

0009%2509187 1987 Pergamon

0

TRANSIENT

BEHAVIOR OF DEEP-BED FILTRATION BROWNIAN PARTICLES

S. VIGNESWARAN? and CHI TIEN Department ofchemical Engineering and Materials Science, Syracuse University,

Syracuse,

$3.00+ Journals

0.00 Ltd.

OF

NY 13244-1240,

U.S.A. (Received

28 August 1986; accepted

11 March

1987)

Abstract-A new model is proposed for deep-bed filtration of Brownian particles. The model incorporates the unit-collector concept for estimating particle deposition rates and the application of the effective medium approximation (EMT) theory for predicting the instantaneous pressure drop across the filter bed. Case studies are used to show the efCects of various variables on filter performance.

INTRODUCTION

Accurate prediction of the dynamic behavior of deepbed filtration is required as the basis for the rational design, optimization and control of deep-bed filters. The main features of deep-bed filtration’s dynamic behavior are the histories of the filtrate quality (effluent concentration) and the pressure drop necessary to maintain a given flow rate. Because of the accumulation of deposited particles within the filter media, both the eflluent concentration and pressure drop vary with time. For any filtration process, when either the effluent concentration or pressure drop exceeds its threshold value, the operation must cease and the filter media must be replaced or regenerated. Numerous investigators in the past have attempted to model this transient behavior. Broadly speaking, their efforts can be grouped into two categories: macroscopic and microscopic. On a macroscopic level, as Tien and Payatakes (1979) have shown, transient behavior of deep-bed filtration can bedetermined from the solution of the conservation equation, the rate equation and the pressure gradient-flow rate relationship. These two latter relationships can, in principle, be obtained empirically from experimental data. Once they are available, one can readily solve these equations, to determine transient behavior. In utilizing this approach for estimating the transient behavior, specific experimental data are required. The microscopic approach is based on an understanding of the nature of the deposition process including the interactions between particles in suspensions and elements of the filter media and the consequences of these interactions for filter performance. To this end, a filter is often considered an assembly of collectors of specified geometry and size (and/or size distribution). The problem then becomes one concerning the interaction between a collector and particles flowing past it. A good example of applying this approach is the use of the trajectory analysis for

i Present address: Division of Environmental Engineering, Asian Institute

of Technology,

Bangkok,

Thailand. 2729

estimating the collection efficiencies of clean filters [for example, Yao et al. (1971); Spielman and Fitzpatrick (1973); Payatakes et al. (1974a, b); and Rajagopalan and Tien (1976, 1977)]. As the works of Pendse and Tien (1982) and Chiang and Tien (1985a, b) have demonstrated, the approach has also been found useful, although less so, in predicting the effect of deposition on filtrate quality. On the other hand, attempts to use this approach to predict pressure-drop increase as a result of the accum;lation of deposited particles in filter beds have not yielded satisfactory results (Chiang and Tien, 1985a, b). The difficulty in predicting pressure drop arises from the fact that the microscopic approach used in studying deep-bed filtration focuses exclusively on the interaction between the collector surface (which may also be viewed as the pore surface) and the suspension flowing framework

past

(or

through)

postulated

by

the

collector.

Payatakes

The et

al.

basic (1973)

assumes that the unit cells contained within a unit bed element are oriented along the axial direction and suspensions exiting from all the unit cells are thoroughly mixed before they enter the next unit bed element. At the same time, there is no lateral connection between unit cells present in the same unit bed element. This highly idealized version of the interconnectedness of the unit cells certainly cannot be expected to provide an accurate description of the manner in which the pore spaces within a filter bed are connected to one another. It is, therefore, not surprising, as Pendse and Tien (1982) show, that studies based on the microscopic approach have not been particularly successf-ul in estimating pressure-drop increases of clogged filters. In principle, as a result of particle deposition, filter media undergo two kinds of changes, restriction (or narrowing) of pore spaces and blocking of pore spaces. Both will change the flow field throughout the media, and the pressure drop across the filter bed will increase. Furthermore, because of the changes in flow field, collection efficiency will also vary with time, so that filtrate quality is not constant, but a function of time. Because a filter bed consists of many interconnected

S.

2730

VIGNESWARAN

pores, the local flow fields within each pore space are strongly interdependent. A logical way to include this interdependence in modeling is to view the flow through a filter bed as a flow through a network composed of various elements. The variables which characterize the local flow can be determined from the size and size distribution of these elements. Once the local flow variables are known, the degree of particle collection can be estimated. This particle collection information can, in turn, be used to determine the change in the structure of the elements of which the network is comprised. To the authors’ knowledge, such an approach has not previously been applied in modeling deep-bed filtration. The purpose of the present work is to formulate a model for deep-bed filtration of Brownian particles based on the above-mentioned principle. The method for formulating the model and the procedure for developing computation algorithms will be presented first, followed by case studies to demonstrate the potential use of this new modeling approach. MODEL

FORMULATION

The basic premise used in formulating the model is to consider a filter bed as a network of randomly connected capillaries of various sizes. The flow behavior throughout the network is estimated from both the capillary size and size distribution and from the mean flow behavior by using the effective medium approximation (EMA) theory (Koplik, 1982). Thus, the physical dimension of the network must by chosen such that the number of capillaries present in the network is sufficiently large to warrant using the EMA theory. At the same time, since the extent of particle deposition is known to depend strongly on the filter axial distalce, the physical dimensions of the network must be small compared to the filter height so that the assumption that deposition within the network is independent of the axial distance remains approximately valid. Since the capillary dimension is approximately of the same order as d,, the filter grain diameter, and the filter height is, in most cases, at least of the order of lo3 d,, the volume element containing the network may be assumed to have an axial dimension of the order of 10 d,. Because the model is constructed by using the EMA theory in conjunction with the deposition theory based on the constricted tube model these two aspects of the problem will be discussed separately. FLOW

8EHAVIOR

MEDIUM

ACCORDING

APPROXIMATION

TO THE (EMA)

EFFECTIVE

and CHI TIEN

particular type (the i-th type), the pressure drop, 6Pi, is equal to the sum of a mean pressure drop, 6P,. and a fluctuating pressure drop, aPA, or 6P, = 6Pm+6P,. Furthermore, given as

6Pfi =

(1)

pressure

drop,

6Ph,

6pm(gjm-gi)

is

(2)

(z-1 )

gi+

9m

where 2 is the coordination number, gi is the conductance of the capillary with radius ri (the i-th type), and gm is the mean conductance of the network satisfying the condition

( yi_g-g:)gm)G(g,= O (3)

as

where G(gi) is the conductance distribution function. The conductance, the applied field (namely pressure drop), and the volumetric flow rate are related by the expression Volumetric Flow Rate = (Applied

Field) (Conductance). (4)

If the Poiseuille equation is assumed to apply for the flow through the capillaries, then eq. (4) can be written as Qi = (dPinrf)/(8pii) (5) where p is the fluid viscosity and Ii is the length of the capillaries. If the applied field is taken to be (bPi/p), then conductance, gi, is given as

nrf

gi=81,According becomes

to

eq.

(3), the

mean

conductance,

g,,,,

nr;

s,-z ?

.fi‘=o

?rrf iL 81,-agm -

where fi is the number

(7)

i-1

fraction

radius ri in the flow network

of capillaries

with

and a = 5 - 1.

The local flow through each type of capillary can be characterized by the average velocity through the capillary, ui, defined as

THEORY

The results on the flow through porous media obtained by Koplik (1982) with the EMA theory can be summarized as follows: A network representing a porous medium is composed of segments of capillaries of various sizes. The pressure drop across the capillaries of different sizes which corresponds to a given superficial velocity, us, through the medium, constitutes a set of random parameters. For capillaries of a

the fluctuating

ui=Qi.

nri”

Combining

(8)

eqs (l), (5) and (8) yields

(9)

filtration of Brownian particles

Deep-bed

The total number of open capillaries volume element, n,, is given as

present in the

1 -1

nt = (NC-AL)

= (AL)&

xfi’(nrf).li

[i

(10)

where N, is the number of open capillaries per unit volume (i.e. capillary density) and E, the effective porosity of the volume element. One may assume that when the filter bed is clean, there is no distinction between the bed porosity and the effective porosity; in other words, all the capillaries are open. However, as filtration proceeds, some of the capillaries may be blocked because of particle deposition. Then the distinction between these two porosities must be noted. The superficial velocity, u,, of suspension flow through the medium is then given as uS= N;x:f;t&(m-f)

initial radius rio. According to this method of description, deposition alters over time the radii corresponding to various types of capillaries. The change in ri can be estimated by the following expression:

- (27Cri)(Zi)z

= (Qi)‘Tqi/(l

(14)

-Ed)

where vi is the collection efficiency of capillaries of the i-th type and sd is the porosity of the deposits f-ormed by the collected particles. The exact value of sd is not known; however, on physical grounds, one may argue that E,,should be comparable to the porosity of a bed packed with the same particles at the incipience of fluidization. Equation (14) is obtained on the basis that particIe deposition leads to the formation and presence of a uniform deposit layer along the surface of the capillaries. C is the particle concentration of the suspension entering the capillary. Since the capillaries are situated over an axial distance between z and z + AL, C is the arithmetic average of the concentration at z or z+AL. From eq. (14) together with eqs (8) and (9), the change in the radius of the i-type capillary, Ari, over a time interval At is

(11) Throughout a filtration cycle, U, is kept constant. Furthermore, the mean pressure gradient across a may be taken as the average mean capillary, bP,/I,, pressure gradient over the volume element, AP/AL. Thus, eq. (11) can be written as

2731

-Ari

“rli At

= -- Qi 27criZi 1 -Ed

I

.At

(15)

or

ri(t +At) The ratio of the pressure drop, AP. to its initial value, APO, that is, its value when the filter is clean, is given as

AP

(13)

where the subscript 0 denotes the initial state. Accordingly if the structure of the network is known (namely, N,, fi’, ri, u, li), then the flow distribution throughout the network can be determined from eqs (7) and (9). Furthermore, by knowing the change in the media structure, one can use eq. (13) to estimate the corresponding change in AP necessary to maintain a fixed flow rate. ESTIMATING

NETWORK

CHANGE

DUE

STRUCTURE

AND

ITS

TO DEPOSITION

The information necessary to define the structure of a flow network representing a filter medium are N,, the number of capillaries per unit filter volume, and A, the number fraction of capillaries of the i-th type (that is, capillaries with radius and length, ri and &). If one assumes that the extent of restriction of capillaries of the same initial size remains the same, one may characterize the i-th type capillary as that with an

z ri(r)+Ari.

The length of the capillary, change with time or li, = li,.

however,

(16) should

not (17)

According to the constricted model of porous media of Payatakes er al. (eq. 1 l), the initial capillary length, liO, and the initial radius rio, of the i-th type capillary can be related by the following expression:

(18) The evaluation of c, is given in the Appendix. With the assumption that particle deposition leads to the formation of a smooth deposit layer along the capillary surface and that capillaries are identified by their initial radii, both ff, the number fraction of the i-th type capillary, and its number density (N, .ji), remains unchanged until ri = 0 (in other words, until the entire capillary is filled with deposited particles). As shown in the sample calculations below, this condition occurs only after an exceedingly long period of operation. Moreover, the assumption of the formation of smooth deposit layers is not always valid. If the particles are sufficiently large (as compared with ri), then capillaries may become blocked even though they are not completely filled with deposited particles. For Brownian particles, because of their submicron size, one may assume that both N, and fi‘ remain constant.

S. V~GNESWARAN

2732 PARTICLE

COLLECTION PHASE

AND

PARTICLE

PROFILE

OF

FLUID

CONCENTRATION

In determining the rate at which capillary radius changes (i.e. eq. 16), one must know the capillary collector efficiency, vi. Based on the constricted-tube model for media characterization, and in the absence ofa repulsive force barrier between Brownian particles and filter grains, Chiang and Tien (1982) obtained a general expression for vi, which yields the following results for capillaries: (19) Knowing the extent of deposition allows direct calculation of the fluid-phase particle concentration throughout a filter bed. Based on the mass balance of particles extending from z to z+ AL over a filter element, one has u,’

[C(Z)-C(Z+

AL)]

= (AL).

NC CJ(nr?)Uicqi

(20)

where C may be taken to be the average value of c(z) and c(z + AL). The value of c(z + AL) is found to be

c(z + AL)

= (21)

We may obtain c vs z incrementally concentration.

TRANSIENT

BEHAVIOR

for a given influent

SIMULATION

The main features of the transient behavior of deepbed filtration are the variations of the etlluent concentration (or filtrate quality) and the pressure drop across the filter beds. The equations formulated and presented above allow direct calculation as described below of both the concentration and pressure drop histories. Method

of calculation

Assuming that at a given time, t, the structure of a filter bed in terms of the network representation is known (in other words, with the bed represented by M ( = L/AL) elements connected in series, for each element the value of ri, the radii of the capillaries constituting the network, the number fractions of the various types of capiilaries in the network,J and the capillary density N, are known), the calculation may proceed in the following manner:

(1) With (2)

N, known as well as J vs ri, the mean conductance of the network corresponding to each filter element, gm, can be determined from eq. (7). With g,,, known and the knowledge as stated earlier that I,,, remains the same as I,,, the mean capillary radius, rm, can be found from the definition of the conductance given by eq. (6).

and

CHI TIEN

(3) The pressure drop across each element can be calculated from eq. (13). The sum of these values gives the value of the pressure drop across the filter. (4) For each element the local flow velocity within the i-th type capillary, Uir can be found from eq. (9), with APJI, replaced by APIAL. (5) The knowledge of Ui and ri allows the calculation from eq. (19) of vi for each type of capillary in a given network for all the elements. (6) The concentration profile, c vs z, can be found from eq. (21) assuming that the influent concentration, cin[ = c(O)], is given. (7) The radius of the i-th type capillary for each element at time t + At can be determined from eqs (11) and (17). The above procedure can be repeated until some of the capillaries become enclosed. The enclosure will occur first for the smallest capillaries present in the first element, or when 0 < z < AL. To begin the calculation, the following procedure may be applied:

(1) Initially,

one assumes the capillary size distribution, A, vs riO. This information can be found from the saturation-capillary pressure measurements suggested by Paytakes et al. (1973). (2) The capillary density, N,, can be found from eq. (10). (3) The initial mean conductance, g,,,,, can be calculated from eq. (7). (4) From the relationship between the capillary radius and length (i.e. eq. 18), the value of r,,, and I,, (= I,,,) can be determined. (5) One also assumes the initial pressure drop across each element (for example, AP can be estimated from the Ergun equation for a given superficial velocity, u,). RESULTS

The model formulation presented above indicates that the transient behavior of granular filtration is controlled by a large number of variables. In order to obtain a better picture of the effects of these variables, a parametric study was conducted in which the transient behavior was simulated by systematically changing these variables. The conditions used for the simulations are presented in Table 1. Since the simulation was made incrementally in terms of both space and time, it was necessary to first specify the values of AL and Ar. As mentioned earlier, the EMA theory requires that the number of capillaries in the network be sufficiently large. At the same time, since the extent of deposition in a filter is strongly dependent upon the axial distance, the assumption that the extent of deposition is the same in capillaries of the same type in the network requires that AL cannot be too large. AL must, therefore, satisfy the condition r,

4 AL

-=gL.

Based on these general requirements, trial calculations were made using AL = 0.1, 0.2, 0.4, 0.8 and 1.6 cm. The results obtained from AL = 0.8 cm were

Deep-bed Table 1. Conditions Sample simulation

no. 1

filtration of Brownian particles

2733

used in sample simulations (ds = 0.05 cm; d, = 0.1 pm) Conditions

US

@m/s)

Z

0.15

6

Pore size and size distribution

Ed

.7 (cm )

r10 = 0.0035 cm, r~, = 0.0065 cm

0.6, 0.8

4-12

Bimodal distribution n1:nz = 1:l

2

0.15

6

r, = 0.0035 cm, r2, = 0.0065 cm Bi’modal distribution n, : n2 variable

0.6

4-12

3

0.15

4. 6. 8

rip = 0.0035 cm, ‘1, = 0.0065 cm Blmodal distribution n,:n, = 1:1

0.6

4-12

4

0.15, 0.30 0.60

6

rl.,, = 0.0035 cm, q0 = 0.0065 cm Blomodal distribution n,:nz = 1:l

0.6

4-12

5

0.15

6

Two arbitrary distribution functions

0.6

4-12

essentially the same as those from any smaller values. A similar procedure was used to select the time interval. The value of At chosen was 10 minutes. The following discussion describes the main features of the simulation for the various cases considered. (A) Simulation No. 1. The results were obtained by assuming that the capillaries are of two types with initial radii of 0.0035 cm and 0.0065 cm, respectively, and the ratio of the number of these two capillary types n, :n, = 1: 1. The primary purpose of this simulation was to determine the effect of the deposit porosity, cd, on the simulated transient behavior. The results of the simulation are shown in Fig. la-e, in which are shown the changes in the capillary radii (rl, r2 and r,); effluent concentration; filter coefficient; and pressure drop vs time. The deposit porosity, .Q, has negligible effect on the effluent concentration (see Fig. lc) but significant effect on the pressure drop and capillary radii (Fig. la, b and e). The filter coefficient shows a slight decline with time. The so-called ‘filterripening phenomenon,’ in which filtrate quality improves with time, was not observed. (B) Simulation No. 2. This simulation was made in order to determine the effect of capillary size distribution, with the results shown in Fig. 2. The capillaries were to be of two types (i.e. bimodal distribution) with the initial radii of 0.0035 cm and 0.0065 cm, respectively. The number ratios of the smaller to larger capillaries were assumed to vary from 1: 1, 1:2 and 1: 10. As the proportion of smaller capillaries increased, the filtration quality (i.e. lower effluent concentration) improved at the expense of a higher pressure drop. The higher pressure drop, of course, is a direct consequence of the more rapid decrease in the capillary ratio. The behavior of the effluent concentration’s remaining relatively constant over time was found not to be affected by the increase in the proportion of larger capillaries. (C) Simulation No. 3. The effect of the coordination number, Z, was examined and the results are shown in

Fig. 3. As the coordination number increases, both the pressure drop increases and filtrate quality improves. As Z is an indication of the extent of the interconnectedness of the capillaries, the distribution of the flow through the various segments of the flow network improves as Z increases. Accordingly, filter performance also improves as 2 increases. (D) Simulation No. 4. The effect of the superficial velocity was studied in this simulation. As shown in Fig. 4, at higher velocity, both the filtrate quality and pressure drop increase became less satisfactory. The observed relationship between pressure drop and flow rate is obvious and needs no comment. A lower flow rate means a greater residence time for each element in the suspension. Consequently, the probability of particle collection increases, which translates into better filtration. (E) Simulation No. 5. Unlike all the previous simulations which used bimodal distribution for the capillaries, this simulation assumed that capillary distributions are two additional types: an arbitrarily distribution and uniform distribution (as shown in Fig. 5). The results of the simulation are presented in Fig. 6a and b. A comparison of the predicted pressure drop histories using the three distribution function are shown in Fig. 7. From all the results presented above, it is clear that the type of distribution function plays perhaps the more important rate in determining pressure drop increase. After 900 minutes of operation. the pressure drop increase (expressed by AFlAP, - 1) may differ by 50%. The results obtained from the simulations indicate clearly the feasibility of applying the EMA theory to deep-bed filtration of Brownian particles. Predicted behavior is found to agree with physical reality, and the required computation is not excessive. Concerning the use of the EMA theory for modeling deep-bed filtration in general, perhaps the most important task yet to be accomplished is to devise a scheme which accurately relates the enclosure of capillaries to particle

S. VIGNESWARAN

2734

and

CHI

TIEN

b

t-

o 3.2-4.0cm.E,j=0.6 0 IZ0-12.6cm,~d= 6 32-4Dcm , Ed’ A 12.0-12.6cm.~ds

1.0

0 4.0cm ,t = 0.6 0 12.8cm.+0.6 6 4.Ocm,Ed=0.6 Al2.6Cm.6pO8

0.6 0.6 0.6

d t

3.0b

0 3.2 -4.0 l 3.2-4.0

’

’ 0.1

’

cm ,Ed cm&d=

’ 0.2

UX IO’

’

’

0.3

=0.6 0.6

*

e 0 0 6 P

0-4.0 Cm.E,j=0.6 O -12.0Cm&,,=O.6 O-4.OCm $d=0.8 0 -12.0~~1 ,6,,= 0.6

2 .s L?LD 2.0 /

hQ, I.5

.fA

/ I_

1.0 k 0

Fig.

1. Effect

300

900

of cd on filter performance

(bimodal distribution; U, = 0.15 cm/s; = 0.0065 cm; n, : n2 = 1: 1; d, = 0.05 cm).

of deep-bed filtration formulated on such a basis would accurately predict the pressure drop increase, a capability which, so far, has eluded all the investigators’ efforts.

deposition.

600 mid

t(

A model

NOTATION

c c Cl

Ci” ... Acknowledgemenr-This study was performed under Grant No. CPE 8309508, National Science Foundation.

2 = 6; rl,, = 0.0035 cm; ~2.

ds DBh4

particle average

concentration concentration in a capillary constant of eq. (18) influent particle concentration grain diameter Brownian diffusivity

pore

kp-bed

;

0.6

-

0.6

- -

‘0

5

rzo

--0

O-40.2

0

0

0

2735

particles

c

3.2-4.0em ---12.0~12.6cm

=QOO36~m =O.mScm

t-i, :ns n, :nz I 300

filtration of Brownian

nI :np =I:10.rmo~6.ZSx10~cm 0 n1 :nz =I:&‘m0 =s.s67~1o%fn 0

=i:to =t:t

0 n, : n 2

060

rmo

z~C#~&@C@I

O%7td+z+

I * 900

I 600

tImin)

dl,

1 t(m

n)

6.0

3.2-4.0cm

4.

0

0 0

XXIO’

“I

: n7 = I:10

n,

: nr

n,

: nz =

=

1:2

I:t

40

3

0

300

3.0@$zgY&

900

GX

oxlO’

e -

32-4.0cm l

nl

a

nI

0

nr

: nz = 130 : nz 8 I:2 : ne z I:1

600

t l mid

900

Fig. 2. Effect of n1/n2 on filter performance (bimodal distribution; u, = 0.15 cm/s; Z = 6; .Q = 0.6; r10 = 0.0035 cm; rzO= 0.0065 cm; d, = 0.05 cm).

x an a

gmo 9io G(gi) L Ii

number

fraction

of capillaries

with radius

mean conductance of capillary conductance of capillary with radius initial value of gm initial value of gi distribution function filter depth length of capillary

mean length of capillary initial value of ii

ri

initial value of I, defined as L/AL

ri n1,

of conductance

nt with radius

ri

NC

n2

number of capillaries with respectively total number of capillaries capillary density

radii

rl and

r2,

2736

S. VIGNESWARAN

and CHI TIEN

‘i ‘i.

0.6

_‘I* -_:

0.4

-

= 0.0035 ‘z. z= z=

= 00065 4 6

0.75

--

1.0

-

3.2 -4.Ocm

4.0cm 12.6~~1

I

I

0

1

600 t tmin)

3al

900)

c

d

4.5 xxld

--

a- -_++

4.0

-6

0.6 -

0.5

‘0

I 300

1 600 t(min)

I 900,

e -0-4 ---

cm O-12.6cm

rZ=4 02 =6

02-e

300

0

600 t (minl

900

Fig. 3. Effect of coordination number (Z) on filter performance (bimodal distribution; = 0.6; nI : n2 = 1: 1; r10 = 0.0035 cm; r2, = 0.0065 cm).

NC, ri

?-Ill

Qi

initial value of N, capillary radius mean capillary radius volumetric flow rate through radius

ri

ui

a capillary

with

US t z

average radius ri superficial time axial

velocity velocity

distance

uS = 0.15 cm/s; E,,

through

a capillary

with

-_ I-=-Deep-bed

filtration of Brownian

2737

particles

a

1.0

-_

__

----o --

0.8

li

ri8.6

0.90

_::,

-

= 0D03scm -_ 0.0065 cm US = 0.15cm/s

6

I

= 0.6

cm/s

--

0.60

3.2 - 4.0cm I2.0 -12.8cm

-

300

0

0

0

l

600 t (min)

900

“s =cUacm/s

u. =0.3Ocln/¶ “s=O.6Ocm~

-0-4.0cm --0 -12.8cm

0.8 c&S t

US = 0.3Ocmh = 0.60 cm/s t(min1 Fig.

4. Effect

of

uS on

filter

performance (bimodal distribution; Z = 6; = 0.0035 cm; ‘2, = 0.0065 cm; d, = 0.05 cm).

nl

: rz2= 1: 1; ed = 0.6; r,,,

5-

P-

0.: 3(i 0.: ?fi

3

An

I

40 5.0 6.0 Pore mdius. P x Idkm) Arbitrary Oistributkn Function 3.0

70

Fig. 5. Two arbitrary distribution

(bl

1 0.

functions

Pore

Brood

rodirrr.

Distribution

ri

x10’ km) Function

used in sample calculations.

.

T“0

rm

0.8

0.7

1

0.6

0.9t--,

1.0 ‘0

t 0.7t.

0.9i

1.0

2 44.0 cm - O-12.8cm

r (mln)

tlmin)

(b)

loL I[~

0.8 c/c,

Fig. 6. (a) Influence of different pore sizedistribution on filter performance;u, = 0.15 em/s; Z = 6; &d= 0.6; d = 0.05em; distribution function given by Fig. 5a. (b) Influence of different pore sizedistribution on filter phormance; u, = 0.15m/s; Z = 6; ed = 0.6; rlo = 0.0035cm; r& = 0.0065em; d,,= 0.05 em; distribution function given by Fig 5b.

Q- 3.2 - 4.0cm -a-12.0-IZ.Bcm

-_

I

Deep-bed

2739

filtration of Brownian particles

Rajagopalan, R. and Tien, C., 1977, Single collector analysis of collection mechanisms in water filtration. Cnn. J. Chem. Engng 55, 246255. Spielman, L. A. and Fitzpatrick, J. A., 1973, Theory of particle collection under London and gravity forces. J. Colloid Interface Sci. 42, 607-623. Tien, C. and Payatakes. A. C., 1979, Advances in deep bed filtration. A.1.Ch.E. J. 25, 737-759. Yao, K. M., Habibian, M. T. and O’Melia, C. R., 1971, Water and waste water filtration: concepts and applications. Environ. Sci. Technol. 5, 1105-l 112. APPENDIX.

EVALUATION

OF CONSTANT

e, OF

EQ. (18) For the constricted tube model proposed by Payatakes et al. (1972), the height of the constricted tube, h, and the maximum and the minimum diameter of the tube, dmin and d max are related by the following expressions. t(minl Fig.

for different distributions 7. Headloss comparison (2 = 6; us = 0.15 cm/s; Ed = 0.6).

d max = a1 dmin

(A-1)

h = a2dmin;

(A-2)

where aI and aZ are given as E(1 - SWJ w

a, =

l--E

Z Greek

L AP APO At SPi

6Plll % E ‘d rli

IJ

coordination

a2=

number

letters

equal to (Z/x) - 1 axial increment pressure drop initial value of AP time increment pressure drop across capillary with radius ri and length li mean pressure drop across a capillary fluctuating component of pressure drop porosity deposit porosity particle collection efficiency fluid viscosity

1

“3 (A-3)

(

(A-4)



where d, and d, are the grain and pore constriction > denotes the average value. E is diameters and the symbol < the bed porosity and swi the irreducible fraction obtained from the capillary pressure-saturation curve. If the height of the constricted tube is interpreted as the length of the capillary and the radius of the capillary, r is assumed to be (A-5) 2r = !Z(d,i” + d,,, ). Combining

eqs (A-l)

through (A-5), one has

h = a2.d,i, d

= a2 . dmin

+d,_

m’” dmin+ 4nax

a2

=

(dmin + drnax)

1 + (d,,,ld,in)

1=-r REFERENCES

Chiang, H. W. and Tien, C., 1982, Deposition of Brownian particles in packed beds. Chem. Engng Sci. 37, 1159-l 171. Chiang, H. W. and Tien, C., 1985a, Dynamics of deep bed filtration-Part I. Analysis of two limiting situations. A.I.Ch.E. J. 31, 1349-1359. Chiang, H. W. and Tien, C., 1985b, Dynamics of deep bed filtration-Part II. Experiment. A.1.Ch.E. J. 31, 13-1371. Koplik, J., 1982, Creeping flow in two dimensional networks. J. Fluid Mech. 119, 219-247. Payatakes, A. C., Tien, C. and Turian, R. M., 1973, A new model of granular porous media-Part I. Model formulation. A.1.Ch.E. J. 19, 5847. Payatakes, A. C., Tien, C. and Turian, R. M., 1974a. Trajectory calculation of particle deposition in deep bed filtration-Part I. A.1.Ch.E. J. 20, 889-899. Payatakes, A. C., Tien, C. and Turian, R. M., 1974b, Trajectory calculation of particle deposition deep bed filtration-Part II. A.1.Ch.E. J. 20, 9-905. Pendse, H. and Tien, C., 1982, A simulation model of aerosol collection in granular media. J. Colloid Interface Sci. 81, 225-241. Rajagopalan, R. and Tien, C. 1976, Trajectory analysis of deep bed filtration with sphere-in-cell porous media model. A.I.CA.E. J. 22, 523-533.

(d;).

(A-6)

46 1+LZ,



1

(A-7)

lf3”

namely 1 is linearly proportional eq. (18) is simply

to r. The constan! ,cl of

*
=

-

1+

E(l-Ss,,)~ 1-s

1

1’3.



(‘4-8)

Payatakes et al. found the S,i = 0.111 for a bed composed of exam spheres ofd, = 0.047 cm and E = 0.4. It-one assumes that
1’3 A!&)_ w [ 1 c, is readily found to be 3.28.