~ --~

--1-< -.D -...... -.....:! --if] = .....-<

.....-< = <l) -~ f:Q = §

N § ....

N <l) '<!'

:r:: 10 0

~ .... z ......; 1-'0 Cll

:::i :::!

Journal Title: Journal of colloid and interface science

Volume: 54 Issue: 3 Month/Year: /1976 Pages: 375-

Article Author:

Article Title: Fine particle deposition in laminar flow through parallel-plate and cylindrical channels

Imprint: Elsevier:SD IU-Link

Call#: QD549 .J8 v.54 1976 (4/12 3:47)

Location: B-ALF

Item#:

CUSTOMER HAS REQUESTED: ARIEL

Christopher Volz(cjvolz)

Notice: This material may be protected by US copyright law (Title 17 U.S. Code)

:ARKER, R., Biochem-istty

D, G., AND BROWN, A., il, 184 (1947). (J1)KA 1 M. J., MORRISON:,

, ]. A mer. Chem. Soc, 69,

;;::ENDREW, J. C. (Eds.), n, 197, liulterworths,

!:AL, E. K., "Interfacial Press, New York, 1961.

ysical Chemistry," 2ncl ~ading, Mass., 1971. m CuATTORAJ, D. K. 4, 1053 (1969). ,

CnATTERJlm, A. K., .r. ' 159 (1966). termodynamics," p. 117. dam, 1957. ~oid Interface Sc-i. 28, 240

4, 10.18 (1937). OvERBEEK, J. Tn. G.,

' of Lyophobic Colloids," ~rdam/New York, 1948. :CHUMAKER1 J. B., Proc. l, 863 (19.12). ophobic Effect," p. 12t,

, Teor. Fiz. 19, 95 (1955).

~,B., AND WAlTJo:, F. A., 42, 262 (1973). ' 51 (1939). temical Engineers' Hand -78, McGraw-Hill, J\'ew

Fine Particle Deposition in Laminar Flow Through Parallel-Plate and Cylindrical Channels

B. D. HOWE:'<, S. LEVINE/ AND N. EPSTEIN

VejJitrlment of Chemical Enginem'ng, University of Bril-ish Columbia., Va.ncom•er 81 B.C., Ca.nad1~

Received Jww 1 t, 1975; accepted September 5, 1975

The deposition o{ colloidal particles (rom a suspension in steady fully developed laminal' 1low onto the walls of a channel is rationalized as equivalent to mass transfer in the bulk with a first-order reaction at the walls. The resulting extended Graetz problem is solved for both parallel-plate and cylindrical channels. Through the usc of confluent hypergeometric functions combined with asymp­totic techniques, an evaluation of the result-ing series solutions is made possible which is more accurate than all previous solutions, especially for the deposition of colloids and for cylindrical channels. Simple Leveque-typc asymptotic solutions are also obtained for the case of large Pcclct numbers, and when the reaction rate constant is infi.nite, these l'eclucc to the corresponding well-established results for convective diffusion.

I. L'lTRODUCTION

Ruckenstein and Pricve (1), Spielman and Friedlander (2), and Dahncke (3) have recently developed a thcoretictLl approximation for analyzing the role played by surface inter­actions in particle deposition by convective diffusion on .filtration collectors. Specifically, these authors have shown that under widely applicable conditions, particle collection in the presence of London-van der Waals, electrical double-layer, and hydrodynamic interaction forces can be treated by assuming convective mass transfer in the bulk of the fluid and a first-order reaction at the collector surface. They derive explicit expressions for the surface reaction coefficient in terms of these interactions.

In this paper, the approximation developed above is applied to the analogous problem of submicron particle deposition in channels under laminar flovv conditions. First, the arguments of the above authors are restated in order lo

1 Permanent address: Department of Mathematics, University of Manchester, Manchestcl', England.

375

eshtblish the range of applicability of the f1rst-ordcr-rcaction boundary condition. Exist­ing solutions to the convective mass transfer problem are then cited and new solutions, more suited to colloid systems, arc obtained for both cylindrical and parallel-plate channels. In addition, because colloidal particles are characterized by small Brownian diffusivities, approximate solutions for both channel types are developed for large Peclet numbers. The results and their implications are then discussed.

2. I' ARALLEL PLATE CHANNEL

First-Order-Reaction Approximation

Consider a parallel-plate channel of width 2b (Fig. 1) through which an incompressible fluid flows with a steady laminar motion. The velocity profile is fully established and there­fore parabolic. Suspended within the fluid are spherical particles of radius a which are assumed to move with the local fluid velocity; i.e., the particles are sufficiently fine that gravitational and lift forces can be neglected.

Copyright © 1976 by Academic Press, Inc. Journal of Colloid and lulajace Science, Vol. 54, No. 3, March 1976 All rights of reproduction in any form reserved.

376 BOWEN, LEVINE AND EPSTEIN I

'~ 0

FIG. 1. Parallel plate channel. Particle suspension with a fully developed laminar velocity profile flows from a region where the walls act as particle reflectors (x < 0) to a region where deposition can occur (x 50).

Let x be the distance in the direction of fluid flow parallel to the channel walls and y the distance normal to these walls, which are specifted by y =bandy= -b; y = 0 is the median plane. C = C(x, y) denotes the particle concentration at any point x, y; more pre­cisely, C(x, y)dV is the time average number of particle centers in a volume element dV. When interaction forces are included, the particle concentration C is given by the equation of convective diffusion in a potential energy field ¢ = ¢ (y) :

~(1 _ y')vm ac = !_[:o ac: + :oc a"'J. [2.1] 2 b' ax ay ay Ill' ay

Here Vm is the mean velocity of the fluid, k the Boltzmann constant, and T the absolute temperature. The potential energy 4> between . each particle and the channel wall is the sum of the electrical double-layer and the London­van der Waals interaction (1, 2). The hydro­dynamic interaction is accounted for by using a variable particle diffusivity, ~, which is a function of the separation between the particle and the wall (1, 3).

Equation [2.1 J is valid only if each particle maintains its terminal drift velocity in the interaction force field a¢/ ay. It is also assumed that diffusion in the x direction can be neg­lected. Equation [2.1] is subject to the boundary conditions

C(O, y) = Co, [2.2]

[aC(x, y)J = C'(x, 0) = O, [2.3]

ay ,_,

denoting differentiation with respect to y prime, and

C(x, b) = 0.

closest distance at which al ·int<omcti<Jns can be considered ncgli1

Thus, for the wall region (subscriJ . [2.1 J may be written

!_[:o ac, + :oc, a"'] = 0 ay ay kT ay

The initial condition, [2.2], signifies that the suspension is homogeneous at the constant concentration Co at x = 0. The second · ary condition, [2.3], is a symmetry condition about the median plane. The third bot@lii.F;,- j(j'w·ith boundary conditions

condition, [2.4], assumes that there exists an c,(x, b) = o, C,(x, b - o) = c,(, infinitely deep potential energy well (presmn-ably due to molecular attraction) at negligible C,(x) is the particle concenl particle-wall separations. This a priori assum.p- junction of the two regions. For tion that the wall behaves as a perfect particle (subscript 2), Eq. [2.1] becon sink is a characteristic of all deposition probM

~(1 - y')v .. ,ac, = 2 b' ax

lems regardless of the specific surface ·"----• action (if any) being considered. The la<:lt<)fit=::-jlr theoretical description of the potential energy

with boundary conditions field at extremely small separations necessitateS the use of this assumption here.

Because both </> and :D are complicated functions of the space variable y, Eq. [2.1].

C,(O, y) =Co, C,'(x, 0) = (

C,(x, b - o) = C,(x),

is intractable as it stands. Fortunately, the ::000

is the constant particle

nature of the physical problem allows sim<pp~li:?fy;:-:-:--1 ~:;'!~~~~·~~-:outside the region of hydr< ing assumptions to be made. Surface ii . To complete the descripti actions are restricted to a region close to the problem the additional conditiml channel wall. The double-layer thickness K-

1 fluxes at the junction of (De bye length) is a measure of the region of is required: double~layer interaction. Hydrodynamic inter­actions and in some cases, London-van det­Waals interactions, are significant over sepa~­rations of the order of the particle radius, a.: Thus, if it is assumed that both the double­layer thickness and the particle size are small compared with the channel dimension (i.e.,­Kb » 1, b/ a» 1), then the velocity and potential energy fields of the above problem

J,(x, b- o) = -:D.C,'(x, b - o)

:I!' where J 1 is the particle flux in the wa Solving the differential equation I

the wall region and applying the I conditi.om [2.6] yields

can be approximately uncoupled by dh·iding,"--.J the flow channel into two regions:

1. a wall region vvhere the convective te~ms can be neglected, ·

It = b - a - y, the closest bet•.ve<'n the particle surface and the ·

= ::000/ :D is the Stokes' law correcti1 ~=~~given by Brenner (4) to account

2. a core region where the potential energy and hydrodynamic interaction terms can­be neglected.

The junction of the two regions is located at a distance o from a channel wall, defined to be

friction encountered by l>lllovinu normal to a plane surface. Th!

Journal of Colloid and Jn(l:rjace Science, Vol. 54, No.3, March 1976

with respect to y by a

= 0. [2.4]

U], signifies that the eous at the constant 0. The second bound­

a symmetry condition '· The third boundary ~s that there exists an

energy well (presum­ttraction) at negligible ;, This a priori assump­·es as a perfect particle of all deposition prob­specific surface inter-

1sidered. The lack of a >f the potential energy ieparations necessitates ::m here.

i X> are complicated variable y, Eq. [2.1] .nds. Fortunately, the wblem allows simplify-

made. Surface inter­> a region close to the ble-layer thickness K-1

~asure of the region of . Hydrodynamic inter­ases, London-van der significant over sepa­the particle radius, a. that both the double­particle size are small

annel dimension (i.e., m the velocity and of the above problem uncoupled by dividing ro regions:

e the convective terms

·e the potential energy :: interaction terms can

> regions is located at a nel wall, defined to be

I' ARTICLE DEPOSITION 377

the closest distance at which all surface interactions can be considered negligible.

Thus, for the wall region (subscript 1), Eq. [Z.1] may be written

!_[:o ac, + :oc~ aq,J = 0 [ 2.5] ay ay l•T ay

with boundary conditions

C1(x, b) = 0, Ct(x, b - o) = C,(x), [2.6]

where C,(x) is the particle concentration at the junction of the two regions. For the core region (subscript 2), Eq. [2.1] becomes

~(1 - y')vmac, =

2 b' ax [2.7]

with boundary conditions

c,(o, y) = c,, C,'(x, O) = o, [2.8]

C2(x, b - o) = C5(x), [2.9]

where ::000 is the constant particle diffusion coefficient outside the region of hydrodynamic interaction. To complete the description of this new problem the additional condition of equal particle fluxes at the junction of the two regions is required :

J 1(x, b- o) = -X>~C,'(x, b- o), [2.10]

where J 1 is the particle flux in the wall region. Solving the differential equation [2.5] for

the wall region and applying the boundary conditions [2.6] yields

C,(x)c-<lk7'1" ac<lkTJ/t

C't(x, y) = , [2.11]

],' ae<i"''dh

where It = b - a - y, ti1e closest distance between the particle surface and the wall, and a= X>oo/X> is the Stokes' law correction factor given by Brenner (4) to account for the additional friction encountered by a sphere moving n?rmal to a plane surface. The particle

flux in the wall region is given by

[ ac1 :oc, aq,J

J.(;l',y) =- X>-+---. ay kT ay

[2.12]

By differentiating Eq. [2.11] and substituting into [2.12], it is found that the 1lux into the wall region at the junction is

X>~C,(x) J.(x, b - o) = , , [2.13]

!, ae<l "''dh

since X> = X>~ at y = b - o. Thus, by substi­tuting Eq. [2.13] into [2.10], the following explicit relationship for c,(x) may be obtained:

c,(x) = -[!,' ae•lk1'd!t]c,'(x, b- o). [2.14]

Hence, the third boundary condition for the core region, [2.9], combined with Eq. [2.14] becomes, in terms of fluxes,

-X>~ :o.c,' (x, b - o) = ----

' !, ae<fk2'dh

xc,(x, b - o). [2.15]

To eliminate the complication of specifying the distance 0 in the above expression, Spielman and Friedlander (2) suggest the use of an effective wall concentration, C2(x, b), for the core region. By expanding C,(x, y) and C2'(x, y) in a Taylor series about y = b and neglecting second and higher derivatives of C, with respect to y, Eq. [2.15] can be shown to be equivalent to

-:Doo X>~C,'(x, b)= ··-----

!,' [ae<lk2' - 1]<lh

xc,(x, b). [2.16]

Because ti1e viscous interaction between the particle and the wall is not accounted for by

Jma·mll of Colloid and Inler!ace Scimc~. Vol. 54, No.3, March 1976

378 BOWEN, LEVINE AND EPSTEIN

Spielman a.nd Friedlander, a = 1 in their analysis. This implies that at large particle~· wall separations (h :>: o) lor which </>--; 0, aert>lk'l' ~ 1 and the upper limit of integration o can be replaced by oc in Eq. [2.16]. How­ever, at large h, a-. 1 + (9a/8h) + O(h-2) (4), and so the integral becomes logarithmic­ally dependent on 0, diverging as 0-tw.2

Thus, when hydrodynamic interactions are accounted for, the limit of integration in Eq. [2.16] should not be extended to oc, and it becomes necessary to evaluate 0 for each new deposition situation. In fact the analysis can only be applied to those situations where it can be shown that the integral is insensitive to the value of 0 taken over a reasonable range. Preliminary evaluations of this integral for a feW practical cases indicate that the above conditions can only be met when the inter­action energy <P has a repulsive maximum. The applicability of the analysis to cases ·where only attractive forces exist appears doubtful.

The integral in Eq. [2.16] is a constant. Thus, when surface interactions are confined to a narrovi· zone near the channel walls, transfer within this thin layer appears to the core region as a first-order reaction at the walls, for which the surface rate constant, K1, is given by

:D. K, = .

!,' [ae>lkT- 1]dh

[2.17]

Hence, when lib» 1 and b/a» 1, the problem of particle deposition with surface interactions in a parallel-plate channel is reduced to a convective-diffusion problem with a first­order-reaction boundary condition at the walls.

M odijied Graetz Solution

In dimensionless terms, the convective­diffusion equation [2. 7] for the core region

2 This was brought to our attention by one of the referees, to whom grateful acknowledgment is hereby extended.

and its corresponding bounda.ry conditions cmi be rewritten

(1- ~2)(aojay) = awa~2 [2.18]

with boundary condilions

Y'(O) = 0,

Y(1) + (1/IC)Y'(1) = 0.

0 (0, n) = 1, [2.19] Because their Brownian diffusiviti, o' ( 0) o [ small, colloidal particles are charac y, = ' 2.20]

large Peclet numbers. Hence, com1 O(y, l) + (1/K)O'(y, 1) = 0, [2.21] countered values of the dimension

where 8 = O(y, n) = C2/Co, K = bK1/:D., ~·· tudinal distance y, which is inverse = y/b, and if Pe = 4vmb/:D. is the Pec!et tiona! to Pe, are correspondingly s number, y = (1/Pe)(8x/3b). In this section convergence of the series solution the prime denotes differentiation with respect slowest when 'Y ---t 0. Thus, for colloi< ton. ' many terms of Eq. [2.22] will be n

VVhcn the dimensionless reaction rate con- - provide an accurate solution. stant K =co, the infinite sink wall conditior~-- Traditionally, the eigenfunction results. Under these circumstances, the ~"'"~--icf2 has been solved by assuming ential equat.ion [2.18] and its accompanying encc of a simple power series solutio boundary conditions constitute the classical Y(~), e.g. (8, 9). Substitution of thi; "Graetz problem," named after the mathe- Eq. [2.23] yields a recurrence relati rnatician who originally examined the analog- coefficients. Although this series conv ous problem of heat transfer in a channel with absolute values of the successive tern a constant wall temperature. The Graetz , alternate in sign) go through an 1

problem has been extensively investigated in large maximum, making it very di the literature, and the results are vvell summar=-- -- 'evaluate eigenvalues An and coeffic ized by Drew (5), Jacob (6), and Sellars beyond n""' 5. Recently, Walker an et al. (7). (10) showed tl1at if a solution to Eq.

When ]( ~co, corresponding to the case expressed in terms of confluent hypeq where surface interactions affect the progress functions, it then becomes possible to of particle deposition, the "extended Graetz many more An and Dro. The use of problem" which ensues has also been treated · functions, proposed s1 in the literature. Sideman et al. (8) gave a ago by Lauwerier (11) for differential< solution to the analogous problem of heat of this type, will be adopted here. transfer with constant wall resistance, and By making the transformation, z =

Colton et al. (9) dealt with the correspond- Eq. [2.23] becomes Weber's equation

ing situation of mass transfer in a parallel d2Y ("' A) plate duct formed from two semipermeable dz' + -

4- - -

2 Y = 0.

membranes. By separation of variables, the solution of

Eq. [2.18] may be written as

• O(y, n) = L: D,Y.,(n)e-'"'> [2.22]

n=l

where Yn(?J) and An are the eigenfunctions and_ corresponding eigenvalues, respectively, ~<----'l-

(d2Y/dn2) +!.'(1 -n')Y = o [2.23]

general solution in terms of the v (12, p. 686)

(3- >, )

XM -4-, t, A112 ,

Jotmral of Colloid and Interface Science, Vol. 54, No.3, Murch 1976

mnduxy conditions can

r) ~ a'e;a~' [2.1SJ

0(0, ~) ~ 1, [2.19]

O'(y, 0) ~ 0, [2.20]

)O'(y, 1) ~ 0, [2.21]

,;c,, K ~ bK,;:n., n vmb/'J)<:/J is the Peclct c/3b). In this section :entiation with respect

less reaction rate con~ ite sink wall condition cumstances, the differ~ and its accompanying :mstitute the classical ned after the mathc~ · examined the analog~ nsfer in a channel with )erature. The Graetz 1sively investigated in ~suits are well sunmmr~ 1cob (6), and Sellars

·sponding to the case Jns affect the progress the "extended Graetz has also been treated

nan et al. (8) gave a ~ous problem of heat

wall resistance, and with the correspond~

transfer in a parallel m two semipermeable

1·iables, the solution of tten as

D.Y .(~)<'"'> [2.22]

the eigenfunctions and Jes, respectively, of

(1 - ~') y ~ 0 [2.23]

PARTICLE DEPOSITION :J79

with boundary conditions

Y'(O) ~ 0,

Y(1) + (1/K)Y'(1) ~ 0.

[2.24]

[2.25]

Because their Brownian diffusivities are very small, colloidal particles are characterized by large Peclet numbers. Hence, commonly en­countered values of the dimensionless longi­tudinal distance ')', which is inversely propor­tional to Pe, a..re correspondingly small. The convergence of the series solution [2.22] is slowest when 1' --t 0. Thus, for colloid systems, many terms of Eq. [2.22] will be required to provide an accurate solution.

Traditionally, the eigenfunction equation [2.23] has been solved by assuming the exist­ence of a simple power series solution in 't1 for Y(~), e.g. (8, 9). Substitution of this series in Eq. [2.23] yields a recurrence relation for its coefficients. Although this series converges, the absolute values of the successive terms (which alternate in sign) go through an extremely large maximum, making it very difficult to evaluate eigenvalues i\n and coefficients Dn

beyond n"" 5. Recently, Walker and Davies (10) showed that if a solution to Eq. [2.23] is expressed in terms of confluent hypergeometric functions, it then becomes possible to evaluate many more i\n and Dn. The use of confluent hypergeometric functions, proposed some time ago by Lauwerier (11) for differential equations of this type, will be adopted here.

By making the transformation, z ~ (21-)1 ''~, Eq. [2.23] becomes Weber's equation

d2

Y (z' ') dz2 + 4 - Z y = O. [2.26]

The general solution in terms of the variable 't1 is (12, p. 686)

Y(r,) ~ A1e-<11"'''MC: ~' t, ),~')

+A' (2),) (l/2J~e-ltlz»''

(3-), )

XM -4-, !, >v/]2 ' [2.27]

where A1 and A2 are arbitrary constants and M (a, b, x) is a confluent hypergeometric function of the first kind. Slater (13) provides an excellent treatise on this function and its properties. Using the relationships

(d/dx)M (a, b, x) ~ (a/b)M(a + 1, b + l, x), M(a, b, 0) ~ 1,

it is easily shown that the first boundary condition, Eq. [2.24], implies that A 2 ~ 0. Thus, if the confluent hypergeometric function is expanded in its infmitc series form, the solution of Eq. [2.2.1] may be written

• (1 - ),) · · · (4i - 3 - 1.) l +I: ),i~Zi ' [2.28]

i~l (2i)!

where the constant A1 is absorbed into the coefficients Dn in Eq. [2.22].

The eigenvalues are the roots of the second boundary condition [2.25], which reads

F(),) ~ Y(1) + (1/K)Y'(1)

( ),

= e-(1!2)}. 1 - J(

+I:' [(1 - 1-) .. · (4i- 3 - 1.)

• (2i)!

( 2i - ')] l X'' 1+---zz- ~o. [2.29]

Equation [2.29] is solved by means of a Newton-Raphson iteration technique. In practice, a good first approximation to any eigenvalue is obtained simply by adding 4 to the previous one. Using this scheme, Eq. [2.29] yields at least 50 eigenvalues before convergence problems begin leading to exces­sive computing time. This notable improve­ment in calculating the i\n and Dn, using Eq. [2.28] in place of the conventional power series solution referred to above, can be

Journal of ColWid aud Inler!acc Science, Vol. 54, No.3, Murch 1976

380 BOWEN, LEVINE AND EPSTEIN

attributed to the presence of the exponential term as a separate factor in Eq. [2.28]. When expanded in series form this exponential term displays oscillatory instabilities similar to those of the conventional series when A becomes large.

When n '> SO, the convergence of F(A) is unacceptably slow and higher eigenvalues can only be obtained if an asymptotic expansion is used. For the case where A -wo, Eq. [2.29] can be written as (Appendix I)

2'i'I'(~) (A" 5.-) F(A)""' ----'-sin ----- )\116

32/3 rm 4 12

--- _3'1"_. r(t~ sin()\" --- "-))\516 = 0. [2.30] . 34/3[( I'(~) 4 12

The coefficients, Dn, are found by substi­tuting Eq. [2.22] into the initial condition [2.19] and employing the usual orthogonality relationships. Thus,

f (1--- ~')Y,.(~)d~ Dn = [2.31]

Sideman et ai. (8) have shown that

f (1--- ~2)Y,.(~)d~ = -(1jA,.2)Y,.'(1) [2.32]

and

!' 1 [aY,.(1) (1--- ~2)Y,.2 (~)d~ = -Y,.'(1) --

o 2A,. aA

1 aY,.'(1)] + . [2.33] J( a>-. ,_,,.

Hence,

l [aY,.(1) 1 _av_._'(t_)J )-1

D,. = -2 A,.---+--a!< K al\ ,_,,

= -2J )\,.[aF(A)J l-1

, [2.34] \ a" ,_,,

where, from Eq. [2.29]

aF(A) = ,-(lfml ~(~ _ 1) _ ~ aA lz K K

~ ~ (1 --- A) .. · (4i --- 3 --- A) +I:

i-1 (2i)!

[(;1; 1)

XA' - --- - --- I; ---A 2 ;-1 4j --- 3 --- A

[2.35]

TABLE I

Number of Terms in Series .Expressir Om Required to Yield Answers Correct t cant Figures

N 5 1 o so 100 500

It is noteworthy that the appro here can theoretically yield me solutions than are currently avai literature for the special case } relevant expressions are easily ob

The corresponding asymptotic form easily the above formulas by dropping al follows from Eq. [2.30]. [( in the denominator.

The two quantities of practical interest wntcll---J•'-In the computations, the first may now be obtained from the solution are the Eqs. [2.36] and [2.39] can b< particle deposition rate and the mean particle · , exactly; additional terms, if re concentration as a function of the longitudinal ,- found approximately by using the distance clown the channel. In dimensionless · ; asymptotic forms. In general, the terms, the local particle deposition rate ] 0 is . and exact eigenvalues and coeffic given by ~ dose agreement in the region 11.

: example, for K = 1.0 the followin ; asymptotic) values arc obtained: )\ 5

~

Jo('Y) = -e'("!, 1) = I; D,.Y,.'(1) n=l

X exp( -)\,."'(), [2.36]

using Eq. [2.22], where Y,.'(1) is readily obtained from Eq. [2.28]. For large A., the asymptotic form is used (Appendix I), namely,

-2716 rm (An?!' ,. ) Y1/(1)~----sin --- An516

,

3'1' r(tl 4 12 [2.37]

The dimensionless mean concentration, Om, is found by integrating the concentration profile over the channel cross section. Hence,

Om('Y) =- (1 --- ~2)0("(, ~)d~. [2.38] 3 !1 2 0

By substituting Eqs. [2.22] and [2.32] into Eq. [2.38] it can be easily shown that

~

Om("!) = -! 2..; (D,./A.2)Y.'(1)

Xexp( -/,,."'(). [2.39]

(196.368), D50 = -0.00015233 ( -( ; and DooY:.o'(l) = -0.00035094 ( -(

Comparisons of Deposition 1

1( 7

Ref. (8) Ref. (9)

1.0 10° 0.2428 0.2428 tQ-1 0.6111 0.6111 tQ-2 0.7828 0.7831 w-a 0.8647 0.8885 J.Q-·1 0.8787 0.9450 lQ-6 0.8802 0.9733 J.Q-G 0.8803 0.9869 1Q-7 0.8803 0.9933

10' 0.1015 0.1015 J.Q-1 1.339 1.339 J.Q-2 3.033 3.041 1Q-3 4.954 6.680 1Q-4 5.381 14.51 w-o 5.428 31.38 w-a 5.433 67.73 J.Q-7 5.433 146.0

Joumal of Colloitl and Interface Scie11ce, Vol. 54, No. 3, March 1976

) t

- 1 - K

-3- A)

l ) '-3- A

ymptotic form easily

practical interest which om the solution are the and the mean particle

tion of the longitudinal nnel. In dimensionless e deposition rate 1 0 is

~

- I; D,.Y,.' (1) lt=l

< exp (-A,.2y), [2.36]

ere Y,.' (1) is readily 28]. For large An, the (Appendix I), namely,

n concentration, 8n,, is 1e concentration profile section. Hence,

. - n')O(y, n)dn. [2.38]

:2.22] and [2.32] into ily shown that

,')Y.'(l)

Xexp( -A.'y). [2.39]

PARTICLE DEPOSITION 381

TABLE I

Number of Terms in Series Expressions for In and 011, Required to Yield Answers Correct to Four Signift~ cant Figures

1 w-~

N 1 5 10 50 100 son 1000 5000

It is noteworthy that the approach adopted here can theoretically yield more accurate solutions than are currently available in the literature for the special case ]( = oo. The relevant expressions are easily obtained from the above formulas by dropping all terms witl1 K in the denominator.

In the computations, the first 50 terms of Eqs. [2.36] and [2.39] can be evaluated exactlyj additional terms, if required, arc found approximately by using the appropriate asymptotic forms. In general, the asymptotic and exact eigenvalues and coefficients are in close agreement in the region n'""' 50. For example, for K = 1.0 the following exact (vs asymptotic) values are obtained: Aoo = 196.365 (196.368), D, = -0.00015233 ( -0.00015207), tmd D,Y,'(1) = -0.00035094 ( -0.00035022).

As tt becomes larger, the agreement behvecn each asymptotic and exact term improves, while its contribution to the converged value of the series diminishes. On the basis of this information, it has been estimated that the maximum error which results from using asymptotic forms for n > 50 occurs only in the fifth significant figure of the computed quantity of interest. Thus, particle deposition rates and mean concentrations correct to four significant figures can be evaluated, provided a sufficient number of terms in Eqs. [2.36] and [2.39], respectively, are chosen. Table I indicates the approximate number of terms (N) required to yield a solution with four-figure accuracy for various values of 'Y and for all values of K.

Table II compares values of J D and Om obtained using the analysis presented here with those computed from other solutions available in the literature for the two cases K = 1.0 and K = oo. The table shows, for example, that the solution of Sideman et al. (8) involving five exact terms should only be used to predict values for 'Y ;(; 10-2• Table I indicates that the series solutions of Walker and Davies (10) extended to 50 exact terms

TABLE II

Comparisons of Deposition :Rates and Mean Concentrations in a ParallellJlate Channel

K ' Jn Om

Ref. (8) Ref. (9) Eq. [2.36] Eq, [2.45] Ref. (8) Ref. (9) Eq. [2.39] Eq. [2,46]

1.0 10° 0.2428 0.2428 0.2428 0.4042 0.3642 0.3642 0.3642 0.2720 lQ-1 0.6111 0.6111 0.6111 0.5937 0.8968 0.8968 0.8968 0.9000 lQ-2 0.7828 0.7831 0.7831 0.7589 0.9875 0.9875 0.9875 0.9879 1Q-3 0.8647 0.8885 0.8889 0.8715 0.9986 0.9989 0.9989 0.9987 10-4 0.8787 0.9450 0.9459 0.9360 0.9997 0.9999 0.9999 0.9999 to-ij 0.8802 0.9733 0.9743 0.9692 0.9999 1.0000 1.0000 1.0000 1Q-6 0.8803 0.9869 0.9879 0.9854 0.9999 1.0000 1.0000 1.0000 10-7 0.8803 0.9933 0.9943 0.9932 0.9999 1.0000 1.0000 1.0000

10° 0.101.1 0.101.1 0.1015 0.6783 0.05384 0.05384 0.05384 -0 . .1262 w-1 1.339 1.339 1.339 1.461 0.6883 0.6883 0.6883 0.6712 lQ-2 3.033 3.041 3.041 3.148 0.9307 0.9307 0.9307 0.9292 to-a 4.954 6.680 6.680 6.783 0.9815 0.9849 0.9849 0.9847 lQ-4 5.381 14.51 14.51 14.61 0.9885 0.9967 0.9967 0.9967 10-ij 5.428 31.38 31.38 31.48 0.9892 0.9993 0.9993 0.9993 lQ-6 5.433 67.73 67.73 67.83 0.9893 0.9999 0.9999 0.9999 lQ-1 5.433 146.0 146.0 146.1 0.9893 1.0000 1.0000 1.0000

Jourual of Colloid and Interface Sdcncc, Vol. 54, No.3, Mardi 1976

382 BOWEN, LEVINE AND EPSTEIN

analysis requires that the concentration ary layer thickness exceed the thickness the wall region and that only particles in core region be considered. For particles in the­core, h ;::: o, and thus Kh » 1 and h/ a» 1. This means that the approximation becomes valid when 1' is small but not zero.

conditions [2.43] is

f' c"'du + (1/ lC) (2/91'

o(X) = -~--~~~­

~,~ e-"'d,u + (1/ K)(2/9y)

can be used for y ;(; lQ-4• The analysis pre­sented by Colton et al. (9) provides for the evaluation of any number of terms of which the first five are exact, the remainder being supplied by the first-order asymptotic expan­sion first suggested by Sellars et a,/. (7) (see Appendix I). The close agreement shown in Table II between the computed results of Colton et al. and those obtained from the present analysis (i.e., Eqs. [2.36] and [2.39]) attests to the fact that the asymptotic forms provide an exceptionally good approximation to the exact forms, particularly for the special case [{ = oo. In fact, the present analysis leads to no significant improvement over that of Colton et al. for a parallel plate channel but, because the former has so many more exact terms, it does demonstrate the accuracy of Colton's results.

It is convenient to regard 0 = C?/C0 as a or function of y and of ~ = h/b. Using th~~~ ·. ·

Modified Leveque Approximation for Large Peclet Numbers

When the Peclet number is large, the dimensionless longitudinal distance /' is corre­spondingly small, and hence, many terms of Eqs. [2.36] and [2.39] are required. In this case, simpler approximate forms which are asymptotically correct for small y can be found.

Small -y corresponds to the entrance region of the channel where the particle concen­tration boundary layer (14, p. 601) is still developing. The simplest approximate form is obtained when this boundary layer lies entirely within the region near the wall where the parabolic velocity profile can be replaced by its tangent line at the wall. Thus, if virtually all of the changes in concentration in the direction normal to the wall take place in a region h « b, where h = b - a - y as before, then the two channel walls are effectively at infinite separation and the problem is reduced to one of particle d~position on a single wall in an infinite medium undergoing a simple shear flow. To take advantage of the first­order-reaction boundary condition, the present

conditions ajb « ~ « 1 appropriate to the.· above approximation, the convective diffusiOn problem described by Eqs. [2.18]-[2.21} becomes

with the single-wa.ll boundary conditions

0(0, I;) = 1, O(y, oc) = 1,

and

O(y, 0) - (1/K)O'(y, 0) = 0. [2.41]

In this section, the prime denotes differentia­tion with respect to 1;. The problem results when K = oo was first treated by Leveque (15), who found an entrance region solution to the original Graetz heat transfer problem. Colton et al. (9), using a perturbation technique, have given an approximate solution to the above problem which is valid only when the product K y113 « 1. Leveque's method is extended here to provide a general solution for all K. By employing the similarity variable, X = (2/9y) 1 i3~, Eq. [2.40] is transformed to the ordinary differential equation

d'O dO --+3X'- = 0, dX' dX

subject to boundary conditions

e( oo) = 1, [2.43]

0(0) - _!_ (!_)''' ('!!_) = 0. K 9y dX x-o

The solution of Eq. [2.42] satisfying the

Jottmal of Colloid and Iulerjace Science, VoL 54-, No. J, Mat"ch 1976

r(t) + (1/ K) (2/91')

since

I'(t) = !," e-"'du.

Tables of the function

are given by Abnunowitz (16). The dimensionless particle dcposi t

(2/9y )'1' J ( ) - 0'( 0) -D'Y- y, -I'(t)+(1/K)(2

In this case the dimensionless mea tration, Om(y), is obtained by eq1 total number of particles removed

' to the number deposited over th r. This yields

2I' (!) (2/9y)

the special case J( = oo, Eqs. [2.4 ·reduce to those obtained by Leve Also for this special case, Kennedy

he concentration bound~ xcecd the thickness of Lat only particles in the l·ed. For particles in the It» 1 and It/ a» 1. This ximation becomes valid >l zero. regard 0 = Cz/Cfj as a

,f ~ ~ It/b. Using the . 1 appropriate to the the convective diffusion 'Y Eqs. [2.18]-[2.21]

[2.40]

mndary conditions

O(y, x) ~ 1,

K)O'(-y, 0) ~ 0. [2.41]

·ime denotes differentia~ ~· The problem that was first treated by

,und an entrance region al Graetz heat transfer (9), using a perturbation an approximate solution m which is valid only "« 1. Leveque's method ·ovide a general solution g the similarity variable, :2.40] is transformed to ial equation

dO X'-= 0,

dX

onditions

8( 00) ~ 1,

a (dO) -- ~ 0 dX x-o

[2.42]

[2.43]

. [2.42] satisfying the

PARTICLE DEPOSIT!OX 383

conditions [2.~3] is

J,x e-•'du + (1/K)(2/9-y)' 13

O(X) ~

~,~ ,-•"du + (1/K)(2/9y)''"

or [2.44] , (2/~')')liO~

I c""du + (I/K)(2/9-y)'i3

• 0

O(r, tl ~ rm + (I/K)(2/9y)'i"

since

l'(f) ~ f c""du.

Tables of the function

X !, e-"'du

are given by Abramowitz (16). The dimensionless particle deposition rate is

(2/9y )'i"

lo(-r) ~ O'(-y, O) ~ r(') + (I/K)(2/9-y) 113 3

[2.45]

In this e<tse the dimensionless mean concen­tration, Om(y), is obtained by equating the total number of particles removed from the fluid to the number deposited over the distance y. This yields

+---Kl' Ct )2 (2/9-y )113

I

2r Ct l (2/9y )''"

For the special case K ~ oo, Eqs. [2.44]-[2.46] reduce to those obtained by Leveque (14). Also for this special case, Kennedy (17) has

given a higher-order approximation for hu·ge Peclet numbers which explicitly accounts for the parabolic nature of the velocity profile. Approximate values of Jn and Om obtained from Eqs. [2.45] and [2.46], respectively, are included in Table II for comparison with their exact counterparts. As expected, the exact and asymptotic solutions approach each other as 'Y --7 0.

Mercer and Mercer (18) have shown that, with an appropriately modified definition of 'Y, the core-region convective-diffusion equation [2.18 J also describes the problem of heat and mass transfer between parallel, circular discs. Thus, all of the results obtained here for a parallel-plate channel can also be applied to their situation.

3. CYLINDRICAL CHANNEL

Consider now the analogous problem of particle deposition in a cylindrical channel of radius R (Fig. 2). Using arguments similar to those for the parallel-plate channel, when !([( » I and R/ a» 1, the core-region .convec­tive-diffusion equation corresponding to [2. 7] is

2(1 - _:_)v,~c, R' ax

~ :ooo(a'C~ + ~ ac'), [J.!] ar2 r ar

where the effect of surface interactions is again approximated by a first-order reaction at the wall which now reads

In the integral [2.17] for the surface reaction rate constant K1, his replaced by fi~ R-a-r.

Modified Graetz Solution

In dimensionless terms, Eq. [3.1] may be written a.':l

ao a'o 1 ae (I - ~2)-- ~ -- +-~ [3.3]

a1 a~' ~ a~

J0/11'1/Gl of Colloid and lnlcljace Scimce, Vol. 54, No. J, March 1976

384 BOWEN, LEVINE AND EPSTEIN

x=O

l''IG. 2. Cylindrical channel. Suspension with fully developed laminar velocity profile. Particle deposition starts at x = 0.

with boundary conditions

0(0, ~) = 1, [3.4]

e'('i, O) = o, [3.5]

O('i, 1) + (1/K)O'('i, 1) = 0, [3.6]

where 0 = O('i, ~) = C,/Co, K = RK,fXJ~, ~ = r/R, and if Pe = 2vmR/XJ~ is the Peclet number, 'i = (1/Pe) (x/ R). In this section, the prime denotes differentiation with respect to ij. This "extended Graetz problem" has also received treatment in the literature. Sideman et al. (8) have given a solution to the constant wall resistance heat transfer analog, while Davis and Parkinson (19) have dealt with the semipermeable cylinder problem.

By separation of variables, the solution of Eq. [3.3] is

~

O('i, ~) = L: D,.Y,.(~)e-'"'", [3.7] n=l

where Y,.(~) and An are the eigenfunctions and corresponding eigenvalues, respectively, of

d'Y 1 dY

obtain a solution to Eq. [3.8]. Because of its superior convergence properties, the conflueht hypergeometric function will again be em~ played here.

If I = A~2 and ,P = ~ Y, Eq. [3.8] is trans. formed to Whittaker's equation

~"" + [~-+ ~- - ~],;, = 0. [3.11] dl' 412 41 4

The general solution is (12, p. 505)

(2- A + B,e-<112lti'''U --- 1 " 4 ) ) ~

which may be written

(2- A ) ·

+B2A'''r(li21><'U -4-, 1, A~' , [3.12]

where B1 and B2 arc arbitrary constants and U(a, b, x) is a confluent hypergeometric function of the second kind. Using tl1e appro­priate relationships (12, 13) it can be shown that the boundary condition [3.9] implies that B, = 0. Thus, if the confluent hypergeometric function is expanded in its series form, the solution of Eq. [3.8] becomes

-+---+A'(1-~2)Y=O [3.8] l d~' ~ d~ Y(~) = e-Om><' 1

subject to boundary conditions

Y'(O) = 0,

1 Y(1) + -Y'(1) = 0.

](

[3.9]

[3.10]

Both Sideman et al. (8) and Davis and Parkin­son (19) use the conventional power series to

~ (2 - A) ... (4i - 2 - A) l + L: A'~" , [3.13] i=l 4ii !i!

where the constants B1An112 are absorbed into ...

the coeflicients D. in Eq. [3.7]. The eigenvalues An in Eq. [3. 7] are the roots

of the second boundary condition [3.10],

Jmmwl of Colloid and Inter/au Science, Vol. 54, No.3, March 1976

·which can be wri ttcn

[1(/o.) = Y(1) + (1/K)Y'(1)

= e-(l/2)J.J 1 - ~ 1 ](

( 2i -A)] l XA' 1+~

; For large A, it becomes necessar : asymptotic form (Appendix I).

2''' r(1) (A"' 2"') • F(A)~- --sin ----. 3'1' r(t) 4 3

l 3 l 2''' xA-l,, --- 1 ---5K 3'''K

(A"' "') X sin ---- >._t/a 4 3

From the usual orthogonality ' the coefficients D, in Eq. [3. 7] ar•

f ~ (1 - ~') Y,.(~ Dn = ---·----

!,' ~(1- ii')Y ,.2 (~

It has been shown (8) that

··fl ~(1 - ~2)Y n(~)d~ : 0

:and

= - (1/A,.') Y,

1 ~(1- ~')Y,.2 (~)d~ = -Y,.'(l

2A,

[aY,.(1) 1 aY.'(1)]

X--+----'-a), K "' ,

. [3.8]. Because of its ·operties, the confluent n will again be em.

iY, Eq. [3.8] is traus. ~quation

~- ~} = 0. [3.11]

(12, p. 505)

2- I. ) ~4~, 1,!

(1/2ltt112lj ~~ 1 " (2- I. )

4 ) d '

r~~11.-• (2- A )

4 ' ' '11

2- A ) ~~, 1, A~' , [3.12]

4

.rbitrary constants and tluent hypergeometric kind. Using the appro-

2, 13) it can be shown lition [3.9] implies that nfluent hypergeometric in its series form, the :comes

- 2- 1.) } ~~~.!.'~" ' [3.13]

1A,~1 '2 are absorbed into

:q. [3.7]. 1 Eq. [3. 7] are the roots lary condition [3.10],

PARTICLE DEPOSITIOX 385

which can be written

F(l.) = Y(l) + (1/K)Y'(1)

( 2-i -·A)] l XI.' 1 + ----x = 0.

For large A, it becomes necessary to use the asymptotic form (Appendix I).

2''" r(1) (1-71' 271') F(l.) """~~-sin ----

3''' rm 4 3

13 \ 2"'" r(!)

Xl.-!13 --- 1 - --- -. -- [3.15] SK J 3''"K r(t)

(A7r 71')

X sin 4- J 1.113 = 0.

From the usual orthogonality arguments, the coeff1cients D, in Eq. [3. 7] are found to be

f ~(1 - fi')Y,.(~)d~ Dn = ~~~--~~~-

!,1 fi(l - ~')Y,.2(ij)d~ [3.16]

It has been shown (8) that

{ ~(1- ~')Y,(fi)dij = -(1/1.,2)Y,'(1) [3.17]

and

fl 1 ii(1 - n')Y ,.2(ij)dij = ~Y,' (1)

0 2An

[3.18]

Hence,

_ 1

[aF(I.)J ~-1 --2 An~~ , BA f..=Xn

[3.19]

where, from Eq. [3.14],

aF(I.) = e-<1''''1~(-~- 1) - _1_ a1. 2 K K

+ £: I (1 - 1.) ... (4-i _ 2 _ 1.)

i=l 4'1hl [3.20]

The asymptotic form of aF I a!. is easily ob­tained from Eq. [3.15].

Using Eq. [3. 7], the local dimensionless particle deposition rate, .Tn = .Tn('Y), is given by

In(?) = -8'(1, 1) 00

= - I: D,Y,'(1)e->n'i, [3.21] n=l

where Y,'(1) follows from Eq. [3.13]. For large A, the required asymptotic form (Ap­pendix I) is

-2''' r(1) (A,.7r ") Yn'(1)~~~~-sin- -- \~~.1 1 3

3''' r(t) 4 3

2'''3''' r(1) (1-,71' 271') + ---~~,-sin ~ - - A,-113, [3.22] o r(,) 4 3

For a cylindrical channel, the dimensionle.ss mean concentration, Bm = Om(Y), is

~ [3.23] -4 I: (D,./A,2) Y ,.' (1)e->,"r,

n-l

Journal of Colloid a11.d Interface Scimce, VoL 54, No. 3, March 1976

386 BOWEN, LEVINE AND EPSTEIN

TABLE III

Comparisons of Deposition Rates and Mean Concentrations in a Cylindrical Channel

K 1 Jo lim

Ref. {8) Ref. (19) Eq. [3.21] Eq. [2.45] Ref. (8) Ref. (19) Eq, [3.23] Eq, [3.24]

----------1.0 100 0.04466 0.04466 0.04466 0.4042 0.06632 0.06632 0.06632 0.2720 1Q-l 0.5104 0.5104 0.5104 0.5937 0.7498 0.7498 0.7498 0.9000 LQ-2 0.7580 0.7582 0.7582 0.7589 0.9674 0.9674 0.9674 0.9879 lQ-3 0.8598 0.8829 0.8830 0.8715 0.9962 0.9964 0.9964 0.9987 1Q-( 0.8772 0.9325 0.9445 0.9360 0.9993 0.9996 0.9996 0.9999 1Q-5 0.8791 0.9407 0.9740 0.9692 0.9997 1.0000 1.0000 1.0000 w-6 0.8792 0.9416 0.9879 0.9855 1.0000 1.0000 1.0000 1.0000 w-o 0.8793 0.9417 0.9940 0.9932 1.0000 1.0000 1.0000 1.0000

w 10° 0.0009980 0.0009980 0.0009980 0.6783 0.0005458 0.0005458 0.0005458 -3.070 lQ-1 0.7334 0.7334 0.7334 1.461 0.3953 0.3953 0.3953 0.1232 lQ-2 2.503 2.508 2.508 3.148 0.8363 0.8364 0.8364 0.8111 w-o 4.557 6.158 6.159 6.783 0.9540 0.9618 0.96\8 10-4 5.040 11.63 14.00 14.61 0.9713 0.9906 0.9914 1Q-6 5.094 13.34 30.87 3\.48 0.9731 0.9951 0.9981 0.9981 1()-0 5.099 13.54 67.21 67.83 0.9733 0.9956 0.9996 0.9996 10-7 5.100 13.56 145.5 146.1 0.9733 0.9956 1.0000 1.0000

using Eqs. [3. 7] and [3.17]. The relevant expressions for the special case ]{ = a;, are easily obtained from the above formulas by deleting those terms having K in the denominator.

As in the case of the parallel-plate channel, the asymptotic forms provide a very good estimate of the eigenvalues and coefficients for n <; 50. For example, when K ~ 1.0 the follow­ing exact (vs asymptotic) values are obtained: Aoo ~ 197.347 (197.347), Doo ~ -0.0026798 ( -0.0026797), and DsoYoo'(1) ~ -0.00035428 ( -0.00035430). Thus, it can also be sbown for the case of the cylindrical channel that the maximum error incurred by using asymptotic forms for n > 50 affects only the fifth signifi­cant figure in the result of interest. The total number of terms, N, of Eqs. [3.21] or [3.23] required to yield a converged answer of four­figure accuracy is again shown in Table I, with ')' replacing "f· Table III compares the deposition rates and mean concentrations obtained using the present analysis with results available in the literature for K ~ 1.0 and K ~ "'. As before, the table indicates that the five-term solution of Sideman et al. (8)

- . ·------------·-- ------

should not be used for predicting J n values where ')' ;S 10-2• Davis and Parkinson (19) obtained additional exact eigenvalues and coeiftcients for n > 5 by numerically inte-=--­grating the differential equation [3.3]. Their 15-term solution for K ~ 1.0 and 20-term solution for ]( = oo are accurate only when ')' <; 10-3, as shown in Table III.

Modified Leveque A pproximalion for Large l'eclet Numbers

The nature of this approximation for a cylindrical channel is very similar to that of a parallel-plate channel for which a detailed explanation has already been given. Since the flow region being considered here is so close-­to the wall that any curvature effects can be ignored, in both cases the channel can be treated as a single flat plate. The same set of approximate equations [2.40] and [2.41] are obtained for a cylindrical tube if 'Y is replaced by i', ~by~~ li/R, where a/R« ~« 1 and_ the prime in Eq. [2.41] now denotes entiation with respect to ~. The solutions for 0 ~ 0('?, ~) and Jo('?) can be written im·

Journal of Colloid awllulcrjace Science, Vol. 5~. No.3, Man.:h 1976

mediately and are identical t and [2.45], respectively, prov are again replaced by ')' and concentration Om, which depends geometry, is now given by

8 ( ~1- lo,1+ 3K'r(t)" g

8 +-----

3Kr(t)'(2/9i')1'

4

31'(t) (2/'

When K ~ oo, it can be easily the above solutions reduce to t by Leveque (14). For this special and Kennedy (20) and subseqm (21) have given a higher-order : which accounts for the para profile. Approximate values o predicted by Eq. [2.45] (with ' and Eq. [3.24], respectively, a Table III for comparison wit counterparts.

4. DISCUSSION AND CONC

Figure 3 shows the dimensic deposition rate J n plotted as a the dimensionless distance frorr entrance, at various values of t less reaction rate constant K, fc the parallel-plate channel. In t1 the subsequent one, the solid lin results obtained from the mo solution, which is accurate for : the dotted lines represent tl1ose , the modified Leveque approxin is asymptotically correct whet small. Figure 3 indicates the ra: ness of the approximate Eq. [2.L systems. This simple expression

lrical Channel

19) Eq, [:!.23]~ -----------i2 0.06632 0.2720 0.7498 0.9000 0.9614 0.9879 0.9964 0.9987 0.9996 0.9999 1. 0000 1. oooo 1. 0000 l. oooo 1.0000 1.0000

i458 0.0005458 -3.070 I 0.3953 0.1232 I 0.8364 0.8111

0.9618 0.9593 0.9914 0.9912 0.9981 0.9981 0.9996 0.9996 1.0000 1.0000

---

n· predicting In values is and Parkinson (19) exact eigenvalues and ; by numerically inte~

1 equation [3.3]. Their K = 1.0 and 20-term

.re accurate only when Table III.

1pro~imalion for Large Numbers

s approximation for a very similar to that of a

for which a detailed ly been given. Since the sidered here is so close curvature effects can be es the channel can be t plate. The same set of s [2.40] and [2.41] are ical tube if 7 is replaced where a/ R « ~ « 1 and 41] now denotes differ­. to ~- The solutions for i) can be written im-

PARTICLE DEPOSITION 387

mediately and are identical to Eqs. [2.44] and [2.45], respectively, provided 7 and ~ are again replaced by i' and ~- The mean concentration Om, which depends on the channel geometry, is now given by

o .. (i') = 1 - 4 J,f Jn(i')di'

8 ( Kr(t) ) = 1 - 3K'l'(!)' log, 1 + (2/91')'''

8 +-----

3Kr (t )'(2/91')' 13

4

31' (t) (2/91')'''

[3.24]

When K = oo, it can be easily verified that the above solutions reduce to those obtained by Leveque (14). For this special case, Gormley and Kennedy (20) and subsequently Newman (21) have given a higher-order approximation which accounts for the parabolic velocity profile. Approximate values of In and (J1n

predicted by Eq. [2.45] (with i' replacing 7) and Eq. [3.24], respectively, are included in Table III for comparison with their exact counterparts.

4. DISCUSSION AND CONCLUSIONS

Figure 3 shows the dimensionless particle deposition rate Jn plotted as a function of 7, the dimensionless distance from the channel entrance, at various values of the dimension­less reaction rate constant K, for the case of the parallel-plate channel. In this plot, as in the subsequent one, the solid lines refer to the results obtained from the modified Graetz solution, which is accurate for all ')', whereas the dotted lines represent those obtained from the modified Leveque approximation, which is asymptotically correct when 7 becomes small. Figure 3 indicates the range of useful­ness of the approximate Eq. [2.45] for colloid systems. This simple expression provides an

10',----, --~--~

Fro. 3. Dimensionless particle deposition rate Jn in a parallel plate channel versus dimensionless longi­tudinal distance -y for selected values of the dimension­less reaction rate constant K. The solid lines refer to the modified Graetz solution and the dotted lines to the modified Leveque approximation for small "Y·

adequate estimate of In for 'Y < 10-a when K = "' and for an even wider range of 7 as J(

decreases (see also Table II). In the entrance region of the channel, the

particle deposition rate decreases with de­creasing K as expected. However, because a high deposition rate corresponds to a high rate of depletion of particles in the suspension, at distances far down the channel, this relation between In and K is reversed, i.e., In increases as K decreases. When either 'Y or J( becomes very small, the dimensionless deposition rate approaches K as predicted by Eq. [2.45]. In other words, under conditions where the concentration boundary layer is sufficiently thin or the resistance to deposition sufficiently high, the particle deposition rate is surface reaction controlled. As either 'Y or K increases, diffusion from the core region becomes more prominent. In fact, if the ratio of deposition rate to mean concentration is plotted in place of In in Fig. 3, the range of surface reaction control for any K would correspond to the value of 7 beyond which the curve begins to deviate significantly from tl1e horizontal. When K = oo, surface interactions have no net effect on particle transport and deposition is com-

Jourual of Colloid aml hiler/ace Scieucc, Vol. 54, No, J, March 1976

388 BOWEN, LEVINE AND EPSTF.IN

plctely diffusion controlled. This case is equiv­alent to the classical situation where surface interactions (except for the infinitely deep potential well at zero separation between particle and wall) are not considered.

Figure 4 shows the dependence of the transverse mean particle concentration Om on 1' and J{ for the parallel plate channel. When JC = 0, there exists an infinite potential barrier to pa1ticle deposition at the channel -wall and thus the mean concentration remains constant at the inlet value for all 'Y· For any positive K, Om diminishes as y increases, and this decrease in Om becomes more pronounced with increase in K, which corresponds to a decrease in the surface resistance to particle deposition. This trend continues until the second extreme case K = oo is reached. Here, the modified Leveque approximation, Eq. [2.46], becomes valid when-y < 10-'. An even wider range of "' is covered by this approxi­mation for K < oo (see also Table II).

The overall collection efficiency ' ~ ' ( 1') is defmed as the ratio of the total number of particles deposited over a given length"' of a channel \-Yith reaction rate constant K to the number collected over the same distance in a diffusion-controlled channel (JC ~ oc ). Hence,

[4.1]

Figure 5 shows the dependence of the collection efficiency thus defmcd as a function of"' and K. When surface interactions are not important, K = oo and <: = 1. As J( decreases, the collection efficiency for any given length of channel also decreases. Eventually, provid­ing the channel is long enough, all of the particles in the suspension are deposited and the collection efficiency approaches unity.

12,------

0.81 8"' o5~·

' OAt-1

02L -E0[?.-391

- --~-~-----,

FlG. 4. Dimensionless mean particle concentration in a parallel plate channel versus dimensionless longi­tudinaL distance 'Y for various values of the dimension­less reaction rate constant .K. The solid lines refer lo

the modified Graetz solution and the dotted lines to the modified Leveque approximation for sma!l-y.

equations for the apparent activ and apparent frequency factor oft

relationship as functions of Hat stant, dielectric constant, ion surface potentials and partialrad

the simplest available expressi' interaction potential. It was founC tions of deposition rates using were orders of magnitude smalle

observed in the experimental colla study of Hull and Kitchener (22).

and Prieve (1) suggest that thi' agreement can be explained by associated with the theoretical <

·-·---1-- tl1e interaction potential. Anoth1 The behavior o£ J D, Om, and E as functions

of 1 and K for a cylindrical channel is very is that Eq. [2.1] does not adequat similar to that of the parallel-plate channel. In the system under consideration the case of the cylindrical channel, the modified fa.ilure to account for the effect Leveque approximations for J u (Eq. [2.45] inertia. Although colloidal part with "' and ~ replaced by y and ~) and (j111 assumed to achieve their tcrmir (Eq. [3.24]) may be applied at 1 :( to-• and almost instantaneously in mos 1 < 1.0-3, respectively, when K ='X;, and the present case is unusual inasn over an ever increasing range of i' asK becomes la.rge changes in the interactiot smaller (see Table III). occur over very small distances.

Ruckcnstein and Prievc (1) have shown In view of such uncertainties, ~ that when the interaction between the particle · Fnedlander (2) recommend the a and the dcposi Lion surface experiences a sufli- semi-empirical approach, mcasur: ciently high maximum, the temperature

single collector geometry and f dependence of the dimensional surface reaction rate constant (Eq. [2.17]) has the Arrhenius results to other geometries. B,

f F tl · th ] 'd d hydrodynamically unequivocal, i orm. 'or 11s case, · ey 1ave p.rov1 e

ld' 10'

FrG. 5. Particle collection efliciency (Eq. L4.lj).JJL!t____._J~ parallel plate channel versus dimensionless longitudinal distance 1' for selected values oi the dimensionless reaction rate constant J(,

and reproducibly defined, a c~

parallel plate channel would app< geometry most suited to this task.

APPENDIX I

Asymptotic A jJproximation ]< Eigenvalues

For the case of real A where 1J -t 0, the function

is RpproximRtecl by the Rsympto1

Jonnwl of Colloid amllulcr/ace Science, Vol. 54, No . .1, Murch 1976

=1.0

ld

1ean particle concentration versus dimensionless longi­

ms values of the dimension­K. The solid lines refer lo

n and the dotted lines to the mtion Ior small)'.

Om, ancl" € as functions ndrical channel is very arallel-plate channel. In :al channel, the modified ns for J n (Eq. [2.45] l by 1 and ~) and Om .pplied at 1 < w-• and r, when K = (;(;' and range of -y asK becomes

'rieve (1) have shown lon between the particle face experiences a suffi­um, the temperature ~nsional surface reaction 17]) has the Arrhenius

they have provided

PARTICLE DEPOSITION 389

equations for the apparent activation energy and apparent frequency factor of the Arrhenius relationship as functions of Hamaker's con­stant, dielectric constant, ionic strength, surface potentials and partial radius, based on the simplest available expressions for the interaction potential. It was found that predic­tions of deposition rates using this analysis were orders of magnitude smaller than those observed in the experimental colloid deposition study of Hull and Kitchener (22). Ruckenstein and Prieve (1) suggest that this radical dis­agreement can be explained by uncertain ties associated with the theoretical evaluation of the interaction potential. Another possibility is that Eq. [2.1] does not adequately represent the system under consideration due to its failure to account for the effects of particle inertia. Although colloidal particles can be assumed to achieve their terminal velocities almost instantaneously in most situations, the present case is unusual inasmuch as very large changes in the interaction force field occur over very small distances.

In view of such uncertainties, Spielman and Friedlander (2) recommend the adoption of a semi-empirical approach, measuring K.' for a single collector geometry and applying the results to other geometries. Because it is hydrodynamically unequivocal, i.e., uniquely

' and reproducibly defined, a cylindrical or parallel plate channel would appear to be the geometry most suited to this task.

APPENDIX I

Asymptotic Approximation for Large Eigenvalues

For the case of real A where A -tXJ and 1) -t 0, the function

Y(~) = e-(ltm,'M(a, b, i-~2) [I.l]

is approximated by the asymptotic expansion

(13)

Y(~) = r(bh1[Ai(t)

+ 3-113(it'Ai'(l) + a(t- f3)Ai(i) +3-112a{3Bi(i))(2k)-''3 + O(k-413)]

+ r(b)y2[Bi(i) + 3-113(!t2Bi'(t)

+ a(t + {3)Bi(l) -3112a{3Ai(i) )(2k )-2/3 + 0 (k-4/3) ],

where

'Yt = ---1--131'

3 cos(a1r) 31/3 (2k) H/3

[I.2]

201(3 sin(<V~r + "/6) I + + 0 (k--4/B) J1/2 (2/i )2/3 '

'Y2 = 1 131

'3 sin(a1r)

J1/3(2k)H/3

2a{3 sin(a, + 1r/6) ) - + 0 (/•--1/3) (2/, )2/3 '

"' = (5b - 2)/10, f3 = tcrm;r(~)J,

1 ( 3 )1/3 t = - - (4k - i-~2)

2 2k '

k = !b- a.

Here, Ai and Bi are Airy functions of the first and second kind, respectively (12, p. 446), and the prime denotes differentiation with respect to~·

For both the parallel-plate channel (Eq. [2.27]) and the cylindrical channel (Eq. [3.12]), '' = A/4. Thus, as ~-> 1, both t and i-> 0. Since

Ai(O) = 3-213/f(j), Bi(O) = 3-116jr(j),

it follows from Eq. [I.2] that

Substituting for 'Y1 and 'Y2 in Eq. [I.3] yields

2r (b) (')''3-' ( 11') Y(1)""' - sin ""'+- .

3'''r(t) 2 6 [I.4]

Journal of Colloid aud Iulc1jace Scirmce, VoL 54-, No. 3, March 1976

\

" ,_ ~-- \ .

' \;

I ~ ·: Ji 'I i ' \

390 BOWEN, LEVINE AND EPSTEIN

Furthermore, since

dl/d~

and

Ai'(O) = -3''"/r(t), Bi'(O) = 3'16/r(t),

then differentiating Eq. [L2] yields at~ = 1,

Y'(l)""" -3''"2"'[r(b)/r(t)J>-''"[3-'''"'- "']

4r(b) ( "If)(>-)''"-' = 34131' (j) sin a "If -- 6 Z

[T.S] 2 ·3''" r(b) . ( "If)(>-)'''-' - -----sm a1r +- -

5 rm 6 2

Equation [1.2] is a second-order asymptotic expansion to Eq. [I.l], whereas the asymptotic expansion first suggested by Sellars et al. (7) and later used by Colton et al. (9) for the parallel-plate Graetz problem is only first order, and in fact can be obtained from Eq. [1.2] by dropping the second-order terms in /1. The second-order term for Y(1) in Eq. [1.4] is identically zero. Thus, when K = oc, the eigenvalues, An, obtained as the roots of Y(1) = 0, are correct to the second rather than the first order as conservatively surmised by Sellars et al. The second-order term for Y' (1) in Eq. [I.S] is the one containing f.2/a-b. A comparison of the exact and asymp­totic eigenvalues and coefficients (n «:; SO) for a number of different values of K revealed that, in the case of the parallel plate channel, the first~order asymptotic expansion always gave a better approximation than the second~order, while in the case of the cylindrical channel, the reverse was true. Thus, in the analysis pre~ sen ted in the text, the second-order asymptotic expansion is used for the cylindrical channel while only the first is employed in the case of parallel plates.

ACKNOWLEDGMENTS

Continuing fmancial support from the National Research Council of Canada is gratefully acknowledged, S. L. was a Visiting Lecturer at the University of British Columbia under the sponsorship of the Nuffield Foundation of the United Kingdom and the National Research Council of Canada, during the winter o( 1975,

REFERENCES

[. RucrmNSTEIN, E., AND PRIEV.l!;, D. C., I. Cftem. Soc. Famday II69, 1522 (1973).

2. SPIELMAN, L. A., AND FRIEDLANDER, S. K., J, Caf.toid Jnte1jace Sci. 46, 22 (1974).

3. DAHNEim, B., J. Colloid Tnle1jt~ce Sci 48, S20 (1914).

4. BRENNER, H., Cltem., Eng. Sd. 16, 242 (1961), 5. DREW, T. B., 1'ra.1ts. AJC!tE 26, 26 (1931). 6. JAKon, M., "Heat Transfer," Vol. 1. \Vilcy, Nr,w----llc

York, 1949. 7. SELLARS, J. R., TRIBUs, M., AND KLEIN, J. 8.,

Tmns. ASME 78, 441 (1956). 8. SmEMAN, S., Luss, D., AND PEcK, R E., AjJpl. Sci.

Res. A 14, 151 (1965). 9. Cor:.roN, C. A., SMITH, K. A., SnWEVE, P., AND

MERRILL, E. w., AICI!E J. 17, 773 (1971). 10. WMxtm, G., AND DAVJI>~s, T., AICM,; J. 20, 881

(1914). t 1. LAUWERrER, H. i\. 1 AjJp/. Sci. Res. A 2, 184 (1950).-12. ABRAMOWITZ, M., AND Sl'KGUN, l. A. (Eds.),

"Handbook o£ Mathematical Functions." Dover, New York, 1965 .

.13. SLATER, L. J., ''Confluent Hypergeometric Func­tions.'' Cambridge University Press, Cambridge, 1960.

14, J~IRD, R. 13., STEWART, W. E., ANn LIGHTFOO'l', E. N., "Transport Vhcnomena." Wiley, New York, 1960.

15, LEVEQUE, M., Ann. Mines 13, 201. (1928). J6, AnRAMownz, M., J. Math. Phys. 30, 162 (1951). 17. KENNEDY, M., quoted by NOLAN, P. J., AND

KENNY, P., J. Atmos. Terr. Pltys. 3, 181 (1953). 18. MERCER, T. T., AND MERCER, R. L., AerosolSci.1,

219 (1910). . 19. DAVIS, H. R., AND PAl~KINsoN, G. V., Appt. Set.:

R". 22, 20 (1910). 20. GoRMLEY, P. G., AND KlmNEDY, M., Proc. Roy.

Irislr A cad. A 52, 163 (1949). 21. NEWMAN, J., J. Heat Tra1tsjer 91, 177 (1969). 22. HULL, M., AND KrTCimNER, J. A., Trans. Jlnra.da.y

Soc. 65, 3093 (1969).

Joamal of Colloid and Inlcljace Sciwcc, VoL 54, No.3, Murch 1976

Arsenate Adsorpt

MARC A. ANDERSO"f,

nepa.r!ment of GeogmjJhy

Received

Arsenate adsorption on amorph anion adsorption on oxide surfac sorbent particles were measured allowed tlte adsorption to be expl adsorption reaction. When used i allow calculation of levels o£ arset

ISTRODUCTIO"f

Anion adsorption on oxide ad~ for many years been of interest to~ concerned with nutrient retent systems. In highly weathered s< of phosphorus is related directly 1 dance of colloidal hydrated oxideE alumina (1).

Engineers have been concerned adsorption as a mechanism fo anions with hydrous metal oxid treatment processes. Phosphate p used in tertiary sewage treatmer may be enhanced by anion adson

Unfortunately, anion adsorptio: geneous oxide systems is not as stood and has not been as \.Yell cation adsorption in similar sy investigation of anion adsorpti should lead to a better understanc adsorption processes in aqueous general.

Several characteristics are adsorption of botl1 anions and

1 Present address: Water Chemist University of Wisconsin, Madison, Wis.

2 Present address: Department of Civi Ut1iversity of WaRhington, Seattle, Wasi-

Cop~Tlght © 1976 by Academic Press, Inc. All rights of reproduction in any form reserved