gravity-induced flocculation of binary particle suspensions

11
Available online at www.sciencedirect.com Separation and Purification Technology 61 (2008) 22–32 Gravity-induced flocculation of binary particle suspensions You-Im Chang , Tzu-Yen Hunag Department of Chemical Engineering, Tunghai University, Taichung 40704, Taiwan, ROC Received 11 April 2007; received in revised form 17 September 2007; accepted 20 September 2007 Abstract A theoretical investigation on the gravity-induced flocculation rates of non-Brownian/Brownian particles in quiescent media is provided in the present paper. Based on the method of trajectory analysis and incorporating gravitational and interparticle forces (as described by the DLVO theory), the effects of the particle size ratio and the reduced density ratio on the capture efficiencies of non-Brownian/Brownian particles in binary suspensions are systematically studied. We find that the capture efficiencies of non-Brownian/Brownian particles will always increase with the increase of either the particle size ratio or the reduced density ratio under the condition of high ionic strength where the electric double layer repulsive force diminishes. When the electric repulsive force is presented, we find that the dispersions will become unstable at some intermediate particle size values, and those non-Brownian/Brownian particles will not become flocculated until a critical ionic strength is reached for a fixed reduced size ratio. We also confirm that the Brownian diffusion behavior of particles can decrease their flocculation rates under the strong Brownian/weak gravity condition. © 2007 Elsevier B.V. All rights reserved. Keywords: Flocculation; Gravity; Colloidal particles; Size; Density; Microgravity 1. Introduction Most theoretical discussions on the gravity-induced floccu- lation behavior of colloidal particles start from the classic work of Von Smoluchowski [1], at which the rapid flocculation rate of colloidal particles was considered as a simple diffusion pro- cess. Later on, through the inclusion of the electrostatic repulsive forces of the DLVO theory [2], the slow flocculation rate of col- loidal particles was studied by Fuchs [3], where the concept of the stability ratio for a colloidal suspension was introduced; the higher the value of stability ratio, the more stable the col- loidal suspension (hence, the more difficult for those colloids to flocculate with each other). Since then, the flocculation behav- ior of binary un-equal sized colloidal suspension systems had been investigated by many scholars. For example, by apply- ing the trajectory analysis method, Spielman and Cukor [4], Melik and Fogler [5] and authors [6] had calculated the capture efficiencies of non-Brownian particles by incorporating gravi- tational and DLVO interparticle forces simultaneously in their models. By using the method of determining the limiting tra- jectory from the stagnation point at the rear of the collector Corresponding author. Fax: +886 4 23590009. E-mail address: [email protected] (Y.-I. Chang). [7], the stability diagram at various magnitudes of gravity were also presented in their works. Based on the stability diagram, the stability criteria for the different types of flocculation can be delineated, and also the effect of gravitational forces on the capture efficiencies of different sized non-Brownian particles under various ionic strengths of suspension can be investigated in details. On the other hand, when the Brownian motion behav- ior of colloidal particles was considered, by using a singular perturbation expansion formula to solve the governing steady state convective diffusion equation, Melik and Fogler [8] suc- cessfully obtained an analytical expression for calculating the coupled gravity-induced/Brownian particles flocculation rates (i.e. in case of strong gravity/weak Brownian or strong Brown- ian/weak gravity effects). Then, by using the same perturbation method, Qiao and Wen [9] and Qiao et al. [10] had established a contour map describing the probability density distribution between two approaching particles, and found that the colloidal suspension of certain ionic strength can be stabilized at some intermediate gravity forces, and the effect of Brownian diffu- sion can act to decrease the coupled gravity-induced/Brownian flocculation rates of colloidal particles. Based on an analytical continuation into the plane of complex Peclet number (where the Peclet number is a measure of the driving velocities relative to Brownian motion) and a special conformal mapping, Zinchenko and Davis [11] had devised a novel numerical technique for 1383-5866/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.seppur.2007.09.024

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Page 1: Gravity-induced flocculation of binary particle suspensions

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Available online at www.sciencedirect.com

Separation and Purification Technology 61 (2008) 22–32

Gravity-induced flocculation of binary particle suspensions

You-Im Chang ∗, Tzu-Yen HunagDepartment of Chemical Engineering, Tunghai University, Taichung 40704, Taiwan, ROC

Received 11 April 2007; received in revised form 17 September 2007; accepted 20 September 2007

bstract

A theoretical investigation on the gravity-induced flocculation rates of non-Brownian/Brownian particles in quiescent media is provided in theresent paper. Based on the method of trajectory analysis and incorporating gravitational and interparticle forces (as described by the DLVOheory), the effects of the particle size ratio and the reduced density ratio on the capture efficiencies of non-Brownian/Brownian particles in binaryuspensions are systematically studied. We find that the capture efficiencies of non-Brownian/Brownian particles will always increase with thencrease of either the particle size ratio or the reduced density ratio under the condition of high ionic strength where the electric double layer repulsive

orce diminishes. When the electric repulsive force is presented, we find that the dispersions will become unstable at some intermediate particleize values, and those non-Brownian/Brownian particles will not become flocculated until a critical ionic strength is reached for a fixed reducedize ratio. We also confirm that the Brownian diffusion behavior of particles can decrease their flocculation rates under the strong Brownian/weakravity condition. 2007 Elsevier B.V. All rights reserved.

ity

[atbcuiipscc(imabs

eywords: Flocculation; Gravity; Colloidal particles; Size; Density; Micrograv

. Introduction

Most theoretical discussions on the gravity-induced floccu-ation behavior of colloidal particles start from the classic workf Von Smoluchowski [1], at which the rapid flocculation ratef colloidal particles was considered as a simple diffusion pro-ess. Later on, through the inclusion of the electrostatic repulsiveorces of the DLVO theory [2], the slow flocculation rate of col-oidal particles was studied by Fuchs [3], where the conceptf the stability ratio for a colloidal suspension was introduced;he higher the value of stability ratio, the more stable the col-oidal suspension (hence, the more difficult for those colloids toocculate with each other). Since then, the flocculation behav-

or of binary un-equal sized colloidal suspension systems hadeen investigated by many scholars. For example, by apply-ng the trajectory analysis method, Spielman and Cukor [4],

elik and Fogler [5] and authors [6] had calculated the capturefficiencies of non-Brownian particles by incorporating gravi-

ational and DLVO interparticle forces simultaneously in their

odels. By using the method of determining the limiting tra-ectory from the stagnation point at the rear of the collector

∗ Corresponding author. Fax: +886 4 23590009.E-mail address: [email protected] (Y.-I. Chang).

isflcPBa

383-5866/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.seppur.2007.09.024

7], the stability diagram at various magnitudes of gravity werelso presented in their works. Based on the stability diagram,he stability criteria for the different types of flocculation cane delineated, and also the effect of gravitational forces on theapture efficiencies of different sized non-Brownian particlesnder various ionic strengths of suspension can be investigatedn details. On the other hand, when the Brownian motion behav-or of colloidal particles was considered, by using a singularerturbation expansion formula to solve the governing steadytate convective diffusion equation, Melik and Fogler [8] suc-essfully obtained an analytical expression for calculating theoupled gravity-induced/Brownian particles flocculation ratesi.e. in case of strong gravity/weak Brownian or strong Brown-an/weak gravity effects). Then, by using the same perturbation

ethod, Qiao and Wen [9] and Qiao et al. [10] had establishedcontour map describing the probability density distribution

etween two approaching particles, and found that the colloidaluspension of certain ionic strength can be stabilized at somentermediate gravity forces, and the effect of Brownian diffu-ion can act to decrease the coupled gravity-induced/Brownianocculation rates of colloidal particles. Based on an analytical

ontinuation into the plane of complex Peclet number (where theeclet number is a measure of the driving velocities relative torownian motion) and a special conformal mapping, Zinchenkond Davis [11] had devised a novel numerical technique for
Page 2: Gravity-induced flocculation of binary particle suspensions

d Purification Technology 61 (2008) 22–32 23

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Y.-I. Chang, T.-Y. Hunag / Separation an

tudying aggregation rates of drops at arbitrary Peclet num-ers and applied it to the gravity-induced coalescence of fluidrops in an emulsion. Recently, instead of using the perturbationethod to find that contour map, we had successfully solved the

oupled Langevin type equations on determining the Brownianarticles’ collision trajectories directly [12], and proved that theonclusion made by Qiao et al. is correct [10], e.g. the Brown-an diffusion behavior of particles can decrease their flocculationates under the weak gravity condition. However, all results citedbove are restricted to the case in which the particles have theame density.

Wacholder and Sather [13] were the pioneers devoted tonvestigate the effects of different particle density on the gravity-nduced flocculation rates of two rigid spheres by taking theydrodynamic interaction only into consideration. In their work,he concept of the reduced density ratio γ = (ρp2

− ρf)/(ρp1−

f) was introduced, where the reduced density is defined as theifference between the particle and fluid densities. Later on,ith the addition of the van der Waals force, Mazzolani et al.

14] had extended the work of Wacholder and Sather to the casef negligible Brownian diffusion effect. They found that theocculation rate of non-Brownian particles is strongly depen-ent on the reduced density ratio and generally increases as thisatio increases above unity. Also, larger particles with fixed radiiatio result in smaller flocculation rates as the reduced densityatio decreases. But, the electrostatic repulsion force was notonsidered in their work.

With the consideration of the London van der Waals attrac-ive forces and the electrostatic double layer repulsive forcesf the DLVO theory, the objective of the present paper iso systematically study the effects of the particle size rationd the reduced density ratio on the gravity-induced floc-ulation rates (i.e. in terms of the capture efficiencies) ofon-Brownian/Brownian particles in binary suspensions. Fromhe results of theoretical analyses, we found that the floccu-ation rates of non-Brownian/Brownian particles are stronglyependent on the ionic strength of suspensions, irrespectivef the magnitudes of the size ratio and the reduced densityatio considered in the present paper. The capture efficienciesf non-Brownian/Brownian particles will always increase withhe increase of either the particle size ratio or the reduced den-ity ratio when the ionic strength of the suspensions is strongnough. When the electric repulsive force is present, we findhat the dispersions will become unstable at some intermediatearticle size values (i.e., through the definition of the ν valueiven below). We also find that the particle’s Brownian motionehavior shall reduce the flocculation rates, and becomes moreronounced as the gravitational forces decrease.

. The interaction energy of DLVO theory

According to the DLVO theory [2], the total interactionnergy V between two approaching colloidal particles

DLVOhown in Fig. 1 is the sum of the van der Waals attractive andlectric double layer repulsive energies:

DLVO = VA + VR (1)

(nc

ig. 1. The schematic figure for binary encounter between two different sizedarticles in the gravity-induced flocculation system.

n the present paper, the retarded van der Waal equation estab-ished by Ho and Higachi [15] and Gregory [16] is adopted

A = − Aa1a2

3(a1 + a2)(R− 2)Z(P0) (2)

here A is Hamaker constant and Z(P0) is the correction factoro the retarded van der Waal attraction

(P0) = 1

1 + 1.7692P0for P0 ≤ 1.0

= 2.45

5P0− 2.17

15P20

+ 0.59

35P30

for P0 > 1.0 (3)

ith P0 = 2π(R − 2)/ν, ν = 2λL/(a1 + a2) (i.e. the magnitudearameter related to the sizes of two particles) and λL = 10−5 cmcharacteristic London wavelength of atoms).

For the electrostatic repulsion energy between two differ-nt sized spherical particles, the Hogg, Healy and Fuerstenau’squation is adopted [17]

R = εa1a2ψ20

a1 + a2ln{1 + exp[−κ(R− 2)]} (4)

here ε is the dielectric constant of suspending medium

0 = 1

2

[ψ2

1 + ψ22

2

](averaged surface potential)

=[

4πe2 ∑CiZ

2i

εk T

]1/2

× a1 + a2

2(5)

B

dimensionless reciprocal of Debye-Huckel double layer thick-ess), e is the charge of electron, 1.6 × 10−19 C, Ci theoncentration of ion i and Zi is the valence of ion i.

Page 3: Gravity-induced flocculation of binary particle suspensions

24 Y.-I. Chang, T.-Y. Hunag / Separation and

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ig. 2. The schematic diagram for the total interaction energy curve of the DLVOheory between two interacting particles.

Fig. 2 shows a typical interaction energy profile, at which therimary minimum and the secondary minimum are separatedy a primary maximum energy barrier. Prieve and Ruckenstein18] found that the deposition rate of particles is exponen-ially sensitive to the height of the primary maximum, if therownian diffusion of particles over this energy barrier is the

ate-determining step. The presence of the secondary minimumill raise the accumulation probability of particles near the

econdary minimum position of the energy profile. This accumu-ation will increase the convective transport of particles parallelo the collector surface if the particle’s tangent velocity is sig-ificant [19]. Usually, the height of the primary maximum cane increased by either increasing the surface potentials of parti-les or decreasing the ionic strength of the suspending medium,nd the depth of the secondary minimum can be increased byncreasing the sizes of interacting particles [2].

By using the dimensionless van der Waals force function

A = (a1 + a2)2

Aa1a2

∂VA

∂R> 0.0 (6)

ence

A = 1

3

[3.5384π

(1 + 1.7692P0)(R− 2)ν

+ 1

(1 + 1.7692P0)(R− 2)2

]for P0 ≤ 1.0 (7)

1{

2π 2.45 4.35 1.77 1

A = −

3 (R− 2)ν−

5P20

+15P3

0

−35P4

0

−(R− 2)2

×[

2.45

5P0− 2.17

15P20

+ 0.59

35P30

]}for P0 > 1.0 (8)

a[

V

Purification Technology 61 (2008) 22–32

nd the dimensionless electric double layer force function

R = − a1 + a2

εψ20a1a2

∂VR

∂R> 0.0 = + κe−κ(R−2)

1 + e−κ(R−2) (9)

hen, the dimensionless interaction force F∗int can be derived

rom Eq. (1)

∗int = − 1

kBT

∂VDLVO

∂R= −NA[fA −NRfR] (10)

here NA is the dimensionless van der Waals attractive number

A = Aa1a2

kBT (a1 + a2)2 = A

kBT

λ

(1 + λ)2 (11)

nd NR is the dimensionless electrostatic repulsive number

R = εψ20(a1 + a2)

A(12)

This repulsion number described the relative importance ofhe repulsive to the attractive interaction forces.

. The trajectory analysis

As shown in Fig. 1, since the relative movement betweenarticles a1 and a2 is in the low Reynolds number regime (i.e.ydrodynamic linear), hence its corresponding disturbance floweld can be separated into the normal and the tangential direc-

ions, separately, by using the method of superposition. In theresent paper, the relative velocity of the two particles shown inig. 1 is attributed to three sources, namely

= Vg + Vint + VBr (13)

here Vg is the motion due to the net gravitational forces, Vints the motion due to the interparticle forces (i.e. DLVO theory)nd VBr is the motion due to the Brownian diffusion forces. Theetails of deriving Vg and Vint can be found in the papers of Meliknd Fogler [5] and Chang and Ku [6], which are expressed as

g = v2 − v1

= (u02 − u01)

{A(R)

rr

r2 + B(R)

[I − rr

r2

]}�eg (14)

ith

0i = Fi

6πμfai= 2g(ρp − ρf)a2

i

9γμf(15)

int = − D0

kBT

∂VDLVO

∂r

{G(R)

rr

r2 +H(R)

[I − rr

r2

]}�er (16)

ith

0 = kBT

6πγμf

[1

a1+ 1

a2

](17)

nd VBr is represented by a random velocity deviate Rv(t) as

12,20]

Br = −Rv(t)

{G(R)

rr

r2 +H(R)

[I − rr

r2

]}�er (18)

Page 4: Gravity-induced flocculation of binary particle suspensions

d Pur

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Y.-I. Chang, T.-Y. Hunag / Separation an

ith

v(t) =∫ t

0eβ(ζ−1)A(ζ)dζ (19)

here ρp is the density of the particle, ρf the density of the fluid,DLVO the DLVO interaction energy between two approachingarticles, kB the Boltzmann constant, T the absolute tempera-ure and β is the friction coefficient per unit mass of the particle.he functions A(R) and B(R) shown in Eq. (14) represent theydrodynamic correction to Stokes relative velocity (u02 − u01)or particles moving parallel and perpendicular to their line ofenters, respectively. The functions G(R) and H(R) shown in Eq.16) represent the hydrodynamic corrections to Stokes-Einsteinelative diffusion coefficient D0. Those functions of A(R), B(R),(R) and H(R) are dependent on the magnitudes of the interparti-

le distance R = 2r/(a1 + a2) and the particle size ratio λ= a1/a2.heir detailed formula can be found in Wen’s book [20] and inelik’s dissertation [21].Applying with spherical coordinates shown in Fig. 1, Eqs.

14), (16) and (18) can be shown in the following forms

g = (u02 − u01)[−A(R) cos θ−→er + B(R) sin θ−→eθ ] (20)

int = −G(R)D0

kT

∂VDLVO

∂ r−→er (21)

nd

Br = − [G(R)Rvr(t)−→er +H(R)Rvθ(t)−→eθ

](22)

y adding the �er terms of Eqs. (20), (21) and (22) together, theelative velocity of particles at normal direction can be writtens

r = dr

dt= −(u02 − u01)A(R) cos θ

−G(R)D0

kBT

∂VDLVO

∂r−G(R)Rvr(t) (23)

y using the definition of R = 2r/(a1 + a2) and substitute F∗int

efined by Eqs. (10) and (23) becomes

a1 + a2

2(u02 − u01)

dR

dt

= − 2G(R)D0

(u02 − u01)(a1 + a2)NA(fA −NRfR) − A(R) cos θ

− G(R)

(u02 − u01)Rvr(t) (24)

fter introducing the dimensionless time τGr

Gr = t

[(a1 + a2)/2(u02 − u01)](25)

nd the gravity number N

G

G = Gr

NA= (u02 − u01)(a1 + a2)

2D0NA

= gravitational force

(Brownian diffusion force × attractive force)(26)

Bitrmc

ification Technology 61 (2008) 22–32 25

here

r = (u02 − u01)(a1 + a2)

2D0(27)

hen, Eq. (24) can be rearranged into the dimensionless form as

∗r = dR

dτGr= −A(R) cos θ − G(R)

NG[fA −NRfR]

− G(R)

u02 − u01Rvr(τGr) (28)

ith the same method, the relative velocity of particles at tan-ential direction can be written as

θ = rdθ

dt= (u02 − u01)B(R) sin θ −H(R)Rvθ(t) (29)

fter substituting R = 2r/(a1 + a2) and the definition of τGr, then

∗θ = dθ

dτGr= B(R) sin θ

R− H(R)

(u02 − u01)RRvθ(τGr) (30)

In Eqs. (28) and (30), Rvr(τGr) and Rv�(τGr) are the two ran-om deviates which are bivariate Gaussian distributed [22], andheir expressions can be found in the paper of Ramarao et al.23].

In the present paper, since we are interesting in studying theffects of the particle size ratio λ, the particle size magnitudearameter ν and the reduced density ratio γ on the gravity-nduced flocculation rates of Brownian particles, so we furtherxpress Gr shown in Eq. (27) in terms of λ, ν and γ as

r = 25πg(ρp2− ρf)λ4

Lλ(1 − λ2γ)

3kT (1 + λ)4ν4(31)

ith γ = (ρp2− ρf)/(ρp1

− ρf) (i.e. the reduced densityatio).Hence

G = Gr

NA= (u02 − u01)(a1 + a2)

2D0NA

= 25πg(ρp2− ρf)λ4

L

3A(1 + λ)2ν4(1 − λ2γ) (32)

By using the method suggested by Peters and Gupta [24], therajectory describing the relative movement between particles a1nd a2 can be solved by the coupled Langevin type equations ofqs. (28) and (30) integrated incrementally. In order to avoid thertificial tunneling error occurring in the numerical integrationf Eqs. (28) and (30), the value of the time step �t adoptedn the present paper remains as small as 10−6 s, which is inhe same order of the momentum relaxation time (∼1/β) of therownian particle [24]. From the test of accuracy analysis shown

n our previous paper [12], we found that the results obtained by

he method suggested by Peters and Gupta fits well with thoseesults of Melik and Fogler obtained by using the perturbationethod [9], especially under the strong Brownian/weak gravity

ondition.

Page 5: Gravity-induced flocculation of binary particle suspensions

26 Y.-I. Chang, T.-Y. Hunag / Separation and Purification Technology 61 (2008) 22–32

F ty numB on coi

4

tpGFλ

fitaodtstgwwotci0atfprbλ

oriγ

gw

mb

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wpimth

paths that terminate in capture from those escape. Consider thereis a plane normal to the flow and placed far upstream from thecollector (i.e. assigned as particle a1 in Fig. 1), then each limitingtrajectory will intersect this plane and a capture area of radius L

ig. 3. (a) The effect of the reduced density ratio γ on the magnitudes of the gravielow the dashed lines of Gr = 0.1, the strong Brownian/weak gravity flocculati

n Fig. 7(a).

. Features of the reduced density ratio γ

In order to study the effect of the reduced density ratio γ onhe flocculation rates of different sized particles considered in theresent paper, its effect on the magnitudes of the gravity numberr must be analyzed first. Through the definition of Eq. (31), inig. 3(a), the gravity number Gr is shown as a function of γ when= 0.2 and 0.5, and ν = 0.001, 0.01 and 0.1, respectively. We cannd that Gr will decrease with the increase of γ , and will drop

o zero abruptly beyond some critical values of γ for all those λnd ν considered in Fig. 3(a). When λ is fixed at either λ= 0.2r at λ= 0.5, Gr will increase with the decrease of ν. Also, theecreasing rates of Gr for those curves of λ= 0.5 are higher thanhose curves of λ= 0.2. As mentioned in our previous paper [12],ince the Brownian diffusion behavior of particles can decreaseheir flocculation rates only under the strong Brownian/weakravity condition when Gr < 0.1 (i.e. see Fig. 6 in ref. [12]), soe only adopt those results for the curves of λ= 0.2 and λ= 0.5hen ν = 0.1 shown in this figure to study the flocculation ratesf the non-Brownian/Brownian particles below. By enlarginghe region of the dashed line Gr < 0.1 shown in Fig. 3(a), theritical values of γ beyond which Gr drops to zero can be foundn Fig. 3(b). Those critical values of γ for Gr = 1.0, 0.1, 0.01 and.001 are γ = 16.881, 24.188, 24.919 and 24.991 for λ= 0.2,nd γ = 2.731, 3.873, 3.987 and 3.999 for λ= 0.5. Therefore,he smaller the value of λ, the larger the value of γ requiredor Gr < 0.1, beyond which the Brownian diffusion behavior ofarticles becomes important in determining their flocculationates. Also, through the definition of Eq. (31), the relationshipetween Gr and ν can be found for various γ . For example, at= 0.2, Fig. 4 shows the effect of increasing ν on the magnitudesf Gr when γ are fixed at 16.881, 24.188, 24.919 and 24.992,espectively. In this figure, we can find that Gr decreases with the

ncrease of ν when γ is fixed, and increases with the decrease of

when ν is fixed. Also, as shown in this figure, the case of weakravity field induced flocculation falls into the region of ν≥ 0.1here Gr is always smaller than 0.1 for all of those four γ values

Fa

ber Gr for ν = 0.001, ν = 0.01 and ν = 0.1 when λ= 0.2 and λ= 0.5, respectively.ndition is reached. (b) The enlarged parts of the region when Gr < 0.1 as shown

entioned above. On the contrary, the region of ν≤ 0.01 alwayselongs to the strong gravity field where Gr is greater than 1.0.

. Limiting trajectory and capture efficiency

As shown in Fig. 1, knowing the trajectory of particle a2hich will intersect particle a1, we can then determine whetherarticle a2 will be captured by particle a1. However, since theres a very large number of particles existing in the suspending

edium, it is impossible to determine the trajectories for all par-icles and the concept of limiting trajectories should be appliedere to calculate the particle’s capture efficiency as follows [9].

The limiting trajectory will distinguish approaching particle

ig. 4. The effect of increasing ν on the magnitudes of Gr when the values of γre fixed at 16.881, 24.188, 24.919 and 24.992, respectively.

Page 6: Gravity-induced flocculation of binary particle suspensions

d Purification Technology 61 (2008) 22–32 27

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j

waoii

j

T

α

ista

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Y.-I. Chang, T.-Y. Hunag / Separation an

nclosed by the foci of these intersection points will be thereforeormed. The net flux of particles passing through this captureross-section is

12 = πL2(u02 − u01)N02 (33)

here (u02 − u01) is the relative particle velocity for infinite sep-ration between two particles, and N02 is the bulk concentrationf particle a2. If j0

12 is defined as the net particle flux due to grav-ty in the absence of both interparticle forces and hydrodynamicnteractions

012 = π(a1 + a2)2(u02 − u01)N02 (34)

hen, the gravity-induced capture efficiency is given by

= j12

j012

=[

L

a1 + a2

]2

(35)

Using the spherical coordinates given in Fig. 1, since thencoming trajectory of particle a2 for virtual capture will inter-ect the line θ =π at the rear of collector (i.e. particle a1), hencehis intersect point R = R* can be determined by setting dR/dθ = 0t θ =π, which gives

∗r = −A(R∗) − G(R∗)

NG[fA(R∗) −NRfR(R∗)]

−G(R∗)Rvr(τGr)

u02 − u01= 0 (36)

Then, with the backward integration of this Eq. (36), theimiting trajectory of particles can be obtained. In the presentaper, this backward integration will not be terminated until= 1500 × R*, where the trajectory of particle will coincideith the fluid streamline, and its corresponding azimuth angle isesignated as θlim. From there, the radius of the capture cross-ection can be calculated, since L = R0(a1 + a2)sin θlim. Once Ls found, the capture efficiency is then evaluated from Eq. (35).ig. 5 illustrates a numerical path of the limiting trajectory for

he case of non-Brownian particles with λ= 0.5, γ = 1.0, κ = 600nd NR = 400.

Since we are interesting in studying the effect of the particleize on the flocculation rates of those Brownian particles whenheir Brownian diffusion behavior is included or excluded (i.e.ermed as the non-Brownian particles in the present paper), so itsffect on the stability of binary dispersions will be investigatedrst as follows.

. Effect of the particle size on the stability of binaryuspensions

Using the illustrative example of the limiting trajectoryf non-Brownian particles shown in Fig. 5, as particle a2pproaches particle a1, the azimuth angle θ increases with theecrease of the separation distance R. Finally, particle a2 will

est at the stagnation point R* at the rear of particle a1 (i.e.∗r = 0 at θ =π). Therefore, during this flocculation period, the

educed velocity V ∗r of particle a2 will be continuously reduced

ntil the position R* is arrived, i.e.V ∗r < 0 when 0 < θ <π and

N

r

ig. 5. The illustrative figure describes the limiting trajectory between twoifferent sized particles when λ= 0.5, κ = 450, ν = 0.1 and NR = 400.

∗r = 0 when θ =π. On the contrary, when those particles flowast outside their limiting trajectories and cannot be flocculatedhereafter, their radial velocities must be greater than zero eithert the front or stagnation point of particle a1 shown in Fig. 5,.e. V ∗

r > 0 at θ = 0 and θ =π. Substitute this condition into Eq.28), we can get the following “non-capture” equations at θ = 0nd θ =π, respectively.

G <G(R)

A(R)[NRfR − fA], θ = 0 (37a)

G >G(R)

A(R)[fA −NRfR], θ = π (37b)

here G(R)/A(R) is the hydrodynamic corrections for interpar-icle forces.

Hence, Eqs. (37a) and (37b) provides the criteria for judginghether the particle can be captured at θ = 0 or at θ =π. In otherords, at θ = 0, if these particles want to keep their stability and

gainst primary minimum flocculation, their gravity parameterG must be less than the maximum net repulsive force, which

s

G < MAX

{G(R)

A(R)[NRfR − fA]

}(38)

Under this condition, primary minimum flocculation will notccur but flocculation at the secondary minimum is possible.imilarly, at θ =π, these flocculated particles can be pulledpart (i.e. deflocculation) if the value of NG is greater than theaximum net attractive force, which is{ }

G > MAX

G(R)

A(R)[fA −NRfR] (39)

Note that the term in brackets of Eqs. (38) and (39) have twooots, which are corresponding to the positions of primary and

Page 7: Gravity-induced flocculation of binary particle suspensions

28 Y.-I. Chang, T.-Y. Hunag / Separation and Purification Technology 61 (2008) 22–32

F λ= 0r rce cuN

s[

gtirfaooetirfwaflttdmamtfliift

F

aad

cc�

gotrawrc−ttcfliFf

coasmtFnwt(o

ig. 6. (a) The κ vs. NG stability diagram of gravity-induced flocculation whenatio λ is also shown in this figure. (b) The dimensionless DLVO interaction fo

R = 400.

econdary minima in the interaction energy profile, respectively6].

By using the above criteria, we can construct a stability dia-ram of κ versus NG for a given set values of NR and λ describinghe flocculation behavior of non-Brownian particles [5,6]. A typ-cal stability diagram considering the effect of the particle sizeatio λ is shown in Fig. 6(a). In this stability diagram, there wereour distinct regions of flocculation delineated: (a) flocculationt the primary minimum of the total interaction energy curvef DLVO theory; (b) flocculation at the secondary minimumf the total interaction energy curve of DLVO theory; (c) anxtremely narrow region of simultaneously flocculation at bothhe primary minimum and the secondary minimum of the totalnteraction energy curve of DLVO theory (i.e. the narrow stripegion enclosed by curves ABC for λ= 0.2 and by curves AB′C′or λ= 0.5 as shown in Fig. 6(a)); (d) a region of deflocculationhere colloidal suspension remains stable. Note that both Melik

nd Fogler [5] and Qiao and Wen [10] ignored the third narrowocculation region in their analyses, but we proved the exis-

ence of this narrow region in our previous paper [25]. Since theerm of the hydrodynamic factor G(R)/A(R) increases with theecrease of λ at the same κ value (see Fig. 11 of Ref. [6]), whichakes the term in the brackets of Eqs. (38) and (39) increase,

nd consequently, both the values of NG required for the primaryinimum flocculation and for the secondary minimum floccula-

ion will increase as λ decreases. Therefore, both the regions ofocculation at the secondary minimum and deflocculation will

ncrease with the decrease of λ as shown in Fig. 6(a). This results proved in Fig. 6(b) where the dimensionless DLVO interactionorce curves with ν = 0.1, κ = 600 and NR = 400 is illustrated. Inhis figure

∗∗int = G(R) F∗

int = G(R)[NRfR − fA] (40)

A(R) NA A(R)

nd we can find that both the depth of the secondary minimumnd the height of the primary minimum increase as λ valueecreases.

Ntpf

.2 (and λ= 0.5), ν = 0.1 and NR = 400. The effect of decreasing the particle sizerves when λ= 0.2, λ= 0.5 and λ= 0.8, respectively, where ν = 0.1, κ = 600 and

The practical utility of the stability diagram shown in Fig. 6(a)an be properly described as follows: consider a gravity-inducedolloidal suspension with ionic strength of κ = 200, if λ= 0.5,ρ = 0.2 g/cm3, A = 10−14 erg and a1 = 1.0 �m, then, we can

et NG = 27.71, −(F∗∗int )sec = 138.90 (i.e. the depth of the sec-

ndary minimum) and (F∗∗int )max = 3.77 × 104 (i.e. the height of

he primary maximum) at ν = 0.1 and NR = 400 from Eq. (40),espectively. Assume that there is a pair of particles which islready flocculated at the secondary minimum and aligns itselfith the θ =π axis. Now, when moving horizontally from left to

ight in Fig. 6(a) at κ = 200 and NG = 27.71, these particles floc-ulated at the secondary minimum will deflocculate first when(F∗∗

int )sec < NG < (F∗∗int )max. Then, if the value of NG is fur-

her increased to NG > (F∗∗int )max (for example, by increasing

he centrifugal rate), these particles which are already defloc-ulated will overcome the height of primary energy barrier andocculate into the primary minimum. Note that when we exper-

mentally decrease NG value and move from right to left inig. 6(a), the reverse process of deflocculating these particlesrom the primary minimum to the stable region will not occur.

Then, by using the same construction method, the effect ofhanging the value of the particle size magnitude parameter νn the stability diagram is obtained in Fig. 7(a) where λ= 0.2nd NG = 400. It is found that, as the value of ν decreases at theame κ value, the value of NG required for the primary mini-um flocculation will increase and the value of NG required for

he secondary minimum flocculation will decrease. As shown inig. 7(b) with λ= 0.2, κ = 600 and NR = 400, because the mag-itude of fA (i.e. the van der Waals attractive force) decreasesith the decrease of ν (i.e. with the increase of the sizes of par-

icles a1 and a2), thereby resulting in a larger repulsive forceF∗∗

int )max, and which will require a larger gravitational force tovercome this enlarged energy barrier. On the other side, when

G in Fig. 7(a) is fixed (i.e. when NG < 104), since more elec-

rolytes are required to suppress this increased height of therimary energy barrier as ν decreases, hence a larger κ is requiredor the occurrence of the primary minimum flocculation in this

Page 8: Gravity-induced flocculation of binary particle suspensions

Y.-I. Chang, T.-Y. Hunag / Separation and Purification Technology 61 (2008) 22–32 29

F ameteN , ν = 0

fidccgdaeclefitwiκ

FfN

5toimtoν

hmmw

ig. 7. (a) The effect of decreasing the value of the particle size magnitude par

R = 400. (b) The dimensionless DLVO interaction force curves when ν = 0.001

gure. Same condition is required for these particles want toeflocculate from the region of the secondary minimum floc-ulation. Based on this stability diagram, the effect of ν on theapture efficiency α at different ionic strengths κ can be investi-ated. One example for the non-Brownian particles of the sameensity (i.e. γ = 1.0) is demonstrated in Fig. 8 where λ= 0.2nd NR = 400. As shown in this figure when κ≥ 4000, since thelectric double layer repulsive force diminishes because of theompression of the electric double layer, and also because aarger gravitational force is required to overcome the enlargednergy barrier with the decrease of ν as mentioned above, there-ore the capture efficiencies of particles will increase with thencrease of ν value (i.e. with the decrease of Gr and NG valueshrough the definitions given by Eqs. (31) and (32)). However,

hen κ < 4000, the suspensions will become unstable at some

ntermediate ranges of ν values. For example, for the curve of= 600, when ν < 5.0 × 10−3, α remains the same. But, when

ig. 8. The effect of ν on the capture efficiency α at different ionic strengths κor non-Brownian particles of the same density (i.e. γ = 1.0), when λ= 0.2 and

R = 400.

7n

tBioa

tsoftoiaWiufiλ

d

r ν (i.e. ν = 2λL/(a1 + a2)) on the κ vs. NG stability diagram where λ= 0.2 and.01 and ν = 0.1, respectively, where λ= 0.2, κ = 600 and NR = 400.

.0 × 10−3 ≤ ν < 3.6 × 10−2, because the corresponded gravita-ional forces are still not big enough to overcome the heightf the primary maximum energy barrier (i.e. (F∗∗

int )max shownn Fig. 7(b)), hence those particles flocculate at the secondaryinimum will start to deflocculate and the corresponded cap-

ure efficiency will drop to zero gradually with the increasef ν in this region. If the value of ν is further increased to= 3.6 × 10−2, then those already deflocculated particles willave enough kinetic energy to overcome the height of the pri-ary energy barrier and be able to flocculate at the primaryinimum as shown in Fig. 2, and the value of α will increaseith the value of ν thereafter.

. Comparison of the capture efficiencies betweenon-Brownian and Brownian particles

Since the present paper is concentrated on the effect of par-icle size and of particle density on the flocculation rates ofrownian particles whether the Brownian diffusion behavior is

ncluded or not, hence the effect of ν and γ on the magnitudesf α shown in Eq. (35) will be emphasized in the theoreticalnalyses as follows.

Fig. 9(a) shows the effect of γ on the capture efficienciesα forhe non-Brownian particles when the electric double layer repul-ive force is not considered, at which κ = 4000 and NR = 400. Asbserved in this figure, the αwill increase with the increase of νor the same λ, and the corresponded α at λ= 0.5 are higher thanhose α at λ= 0.2 with the same ν. When those critical γ valuesf λ= 0.2 and λ= 0.5 shown in Fig. 3(b) are approached, thencreasing rates of α become large. Same variation tendenciesre observed for those Brownian particles as shown in Fig. 9(b).

hen comparing with Fig. 9(a), we can find that the Brown-an diffusion behavior of particles will not affect their α values

nless the ν value is increased to ν = 0.1 (i.e. the weak gravityeld as shown in Fig. 4). Also, the α values for the curves of= 0.2 and λ= 0.5 at ν = 0.1 will drop to the finite values imme-iately when the corresponded critical γ values are reached. For
Page 9: Gravity-induced flocculation of binary particle suspensions

30 Y.-I. Chang, T.-Y. Hunag / Separation and Purification Technology 61 (2008) 22–32

F an paκ r Broww

ttd

FbsFnidadtdd

Favfi

ptidr

eieν

anounced at small Gr (i.e. Gr < 0.1, see the double dashed lines

ig. 9. (a) The effect of γ values on the capture efficiencies α for non-Browni= 4000 and NR = 400. (b) The effect of γ values on the capture efficiencies α fohere κ = 4000 and NR = 400.

he case of considering the electric double layer repulsive force,he similar results for the reduction of α caused by the Brownianiffusion behavior of particles are also obtained below.

Without considering the electric double layer repulsive force,ig. 10 shows the effect of ν on α for non-Brownian (showny the dashed lines) and Brownian particles (shown by theolid lines) at λ= 0.2 for those critical γ values obtained inig. 3(b), respectively, when κ = 4000 and NR = 400. For thoseon-Brownian particles, we can find that α increases with thencrease of ν at fixed γ . Because Gr will increase with theecrease of ν as shown in Fig. 4, then this increased in Gr willccelerate the sedimentation rates of particles and therefore can

ecrease their flocculation rates consequently. From our compu-ations, we found that because the interaction range of the vaner Waals forces decreases with the increase of Gr, hence theecrease of R* (see Eq. (28)) at the rear of the collector (i.e.

ig. 10. The effect of ν on α for non-Brownian (shown by the dashed lines)nd Brownian particles (shown by the solid lines) at λ= 0.2 for those critical γalues obtained in Fig. 3(b), respectively, when κ = 4000 and NR = 400. In thisgure, the electric double layer repulsive force is not considered.

scν

FaBrI

rticles when the electric double layer repulsive force is not considered, wherenian particles when the electric double layer repulsive force is not considered,

article a1 in Fig. 1) is the main reason to cause α′s decreasingendency. Also, as shown in Fig. 10, α will increase with thencrease of γ at fixed ν. This result indicates that the bigger theifferences in densities of particles, the larger the flocculationates of particles under the same acting gravitational forces.

When the Brownian diffusion behavior of particles is consid-red, as shown by the solid lines in Fig. 10, there is no differencen the capture efficiencies independent of the Brownian diffusionffect when ν < 0.1 for the four critical γ values. However, when≥ 0.1 which correspond to the region of the weak gravity fields shown in Fig. 4, the Brownian diffusion effect becomes pro-

hown in Fig. 10) which is unfavorable for those particles to floc-ulate with each other, hence α start to drop with the increase ofwhen ν≥ 0.1.

ig. 11. The effect of ν on α for those four critical γ values of λ= 0.2 obtainedt Fig. 3(b) when κ = 600 and NR = 400. Solid lines represent the case whenrownian diffusion behavior of particles is not considered, while dashed lines

epresent the case when Brownian diffusion behavior of particles is considered.n this figure, the electric double layer repulsive force is considered.

Page 10: Gravity-induced flocculation of binary particle suspensions

Y.-I. Chang, T.-Y. Hunag / Separation and Purification Technology 61 (2008) 22–32 31

Fig. 12. (a) The effect of the ionic strength κ on the capture efficiency α for those four critical γ values of ν = 0.1 and λ= 0.2 obtained at Fig. 3(b) when NR = 400.Dashed lines represent the case when Brownian diffusion behavior of particles is not considered, while solid lines represent the case when Brownian diffusionbehavior of particles is considered. (b) The effect of the ionic strength κ on the capture efficiency α for those four critical γ values of ν = 0.1 and λ= 0.5 obtained atF ion bB

nuuubldarttfls

γ

dafihflplitFnα

ss

8

tappctcstrpaficrtct

ptflfcrco

ig. 3(b) when NR = 400. Dashed lines represent the case when Brownian diffusrownian diffusion behavior of particles is considered.

When the electric double layer repulsive force is consid-red, for example, when κ = 600 and NR = 400, the effect ofon α for various γ at λ= 0.2 is shown in Fig. 11. For the

on-Brownian particles shown by the dashed lines in this fig-re, same as observed in Fig. 10, the dispersions will becomenstable at some intermediate ν values for fixed γ . In this fig-re, we can find that the smaller the γ , the later the dispersionsecomes unstable, which indicates that a smaller Gr force (i.e.,arger ν value) is required to pull apart those particles of smallerifference in density (i.e., smaller γ value) already flocculatedt the secondary minimum to become deflocculated. The sameesults are observed for those Brownian particles as shown byhe solid lines in this figure. Also, same as observed in Fig. 10,he Brownian diffusion behavior of particles will decrease theirocculation rates when ν≥ 0.1 for all of those four γ values ashown in Fig. 11.

Fig. 12(a) shows the effect of the ionic strength κ on theapture efficiency α for those critical γ values obtained at= 0.1, λ= 0.2 and NR = 400. In this figure, for example, when= 16.881 for the non-Brownian particles represented by the

ashed lines, we find that those particles will not flocculate untilcritical value of κ* = 1.2 × 102 is reached. Beyond which, at axed κ the higher the γ , the larger the α is observed. Also, theigher the γ , a smaller κ is required to let those particles becomeocculated. More important, the critical values of κ* are inde-endent of the Brownian motion effect (represented by the solidines), and the degree of the Brownian diffusion effect on reduc-ng α increases with the increase of γ . Since the Gr values forhe curve γ = 24.992 are the smallest when ν≥ 0.1 as shown inig. 4 (i.e., the weak gravity field), hence the dominated Brow-

ian diffusion effect will make the highest degree on reducingfor the curve of γ = 24.992 as shown in this Fig. 12(a). The

ame variation tendencies are observed for the case of λ= 0.5 ashown in Fig. 12(b).

fltg(

ehavior of particles is not considered, while solid lines represent the case when

. Conclusion

By using the coupled Langevin type equations on determininghe particles’ collision trajectories, the effects of the particle sizend the reduced density ratio on the non-Brownian/Brownianarticles are investigated in the present paper. In addition to thearticle’s Brownian diffusion behavior can decrease the floc-ulation rates under the weak gravity condition, we also findhat the capture efficiencies of non-Brownian/Brownian parti-les will always increase with the increase of either the particleize ratio or the reduced density ratio, when the ionic strength ofhe suspensions is strong enough that the electric double layerepulsive force diminishes. When the electric repulsive force isresented, we find that the suspensions will become unstablet some intermediate particle size values (i.e., ν values). For axed reduced size ratio, those non-Brownian/Brownian parti-les will not become flocculated until a critical ionic strength iseached. Beyond this critical ionic strength value, the effect ofhe Brownian diffusion behavior of particles on decreasing theirapture efficiencies becomes pronounced with the increase ofhe reduced density ratio.

One of the most important conclusions obtained in the presentaper is that, same as the conclusion made by Qiao et al. [10],he Brownian diffusion behavior of particles can decrease theirocculation rates under the weak gravity condition when ν≥ 0.1or all of those four γ values as shown in Figs. 10 and 11. Thisonclusion can be evidenced by the microgravity experimentalesults obtained by Okubo et al. [26]. In their experiments, theollision rates of colloidal silica spheres with a mean diameterf 110 nm were studied in microgravity produced by parabolic

ights of a Mitsubishi MU-300 type jet air-craft. They found

hat the collision rates of the particles can be decreased in micro-ravity (zero G) by about 25% compared with those in gravityone G) in the diluted suspensions. The main reason to cause

Page 11: Gravity-induced flocculation of binary particle suspensions

3 n and

tappoamtptflowomUnp

A

C9

R

[[[

[[

[[[

[[[

[

[[[[[

2 Y.-I. Chang, T.-Y. Hunag / Separatio

his decrease result in microgravity was attributed to the dis-ppearance of the downward diffusion of the particle a2 to thearticle a1 as shown in Fig. 1. However, as mentioned in theiraper, the effect of the vibration problem caused by the jet flightn the microgravity experiments was still unclear. In order tovoid the vibration problem, in our opinion, the proposal sub-itted by Ansari et al. [27] to NASA is a good method to prove

hose theoretical results obtained in the present paper. In theirroposal, by using an integrated fiber optic dynamic light scat-ering (DLS) system, the experiment of studying the colloidalocculation phenomena in microgravity can be well executednboard a space shuttle or the international space station. Thereas one related research topic: “The influence of microgravityn the yeast cell adhesion” already done on the space shuttleission of STS-107 (sponsored by European Space Agency).nfortunately, because of the tragedy happened during re-entry,o data was available. Therefore, more efforts are required tout in this research area in the future.

cknowledgement

The financial support received from the National Scienceouncil of the Republic of China, research Grand No. NSC-5-2221-E029-034, is greatly appreciated.

eferences

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[

Purification Technology 61 (2008) 22–32

[3] N. Fuchs, Z. Physik 89 (1934) 736.[4] L.A. Spielman, P.M. Cukor, J. Colloid Interface Sci. 43 (1973) 51.[5] D.H. Melik, H.S. Fogler, J. Colloid Interface Sci. 101 (1984) 72.[6] Y.I. Chang, M.H. Ku, Colloids Surf. A 178 (2001) 231.[7] L.A. Spielman, J.A. FitzPatrick, J. Colloid Interface Sci. 42 (1973)

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1638.18] D.C. Prieve, E. Ruckenstein, AIChE. J. 20 (1974) 1178.19] R. Rajagopalan, J.S. Kim, J. Colloid Interface Sci. 83 (1981) 428.20] C.S. Wen, The Fundamentals of Aerosol Dynamics, World Scientific Pub.,

Singapore, 1996, pp. 77–103 (chapter 4).21] D.H. Melik, Flocculation in quiescent media, PhD Thesis, University of

Michigan, Ann Arbor, Michigan, 1984, (chapters 2 and 3).22] S. Chandrasekhar, Rev. Mod. Phy. 15 (1943) 1.23] B.V. Ramarao, C. Tien, S. Mohan, J. Aerosol Sci. 25 (1994) 295.24] D. Gupta, M.H. Peters, J. Colloid Interface Sci. 104 (1985) 375.25] Y.I. Chang, M.H. Ku, J. Colloid Interface Sci. 271 (2004) 254.26] T. Okubo, A. Tsuchida, T. Okuda, K. Fujitsuna, M. Ishikawa, T. Morita, T.

Tada, Colloids Surf. A 153 (1999) 515.27] R.R. Ansari, H.S. Dhadwal, K.I. Suh, Flocculation and aggregation in a

microgravity environment (FAME), Proceedings of the 2nd MicrogravityFluid Physics Conference, Cleveland, Ohio, 1994, p.333 (also, see NTRSdocument no. 19950008158 of NASA).