Kinetic models of suspension flocculation by polymers

COLLOIDS AND ELSEVIER

Colloids and Surfaces A: Physicochemical and Engineering Aspects 104 (1995) 67 72

A

SURFACES

Kinetic models of suspension flocculation by polymers Ad.A. Berlin *, V.N. Kislenko Department of General Chemistry, Lviv Polytechnic State University, 12 S. Bandera St., Lviv 290646, Ukraine

Received 23 November 1994; accepted 21 March 1995

Abstract

A comparison of mathematical models that describe the kinetics of flocculation of a suspension is presented. The models enable us to describe the process over a wide time interval, including late stages, and to find values of the kinetic constants. This provides the possibility of calculating the relationships of the floc number in the system and of the number of primary particles in a floc with time and with the initial coverage of the particle surface with flocculant. The agreement between the kinetic models and experimental data for the flocculation of polystyrene latex by poly(methyl vinyl pyridine) is shown. The model that takes both the changing floc surface coverage with polymer during flocculation and the floc break-up into account, has the highest accuracy. Keywords: Polymers; Suspension flocculation

1. Introduction

Investigations of the flocculation kinetics of suspensions are important for understanding the mechanism and for process control of solidqiquid system separation. The creation of physically grounded, adequate mathematical models of flocculation kinetics promotes such understanding. A system of nonlinear differential equations may be used to describe the process kinetics [ 1-4]. The system considers the interaction of two aggregates with different numbers of primary particles and with different surface coverages by the flocculant, floc break-up under hydrodynamic action, and the adsorption of flocculant macromolecules by particles from solution. However, the absence of information on the distribution of flocs according to ~ Paper presented at the 10th International Symposium on Surfactants in Solution, held in Caracas, Venezuela, 26-30 June 1994. * Corresponding author. 0927-7757/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0927-7757(95)03219-3

size and surface coverage with polymer, and on values of the kinetic constants for different aggregate types hinders the solving of this system in an analytical form. However, numerical calculations are frequently connected with computing limitations, i.e. insufficient memory and inadequate performance [3]. Therefore, it is of interest to consider kinetic models using certain simplifications and averaged values of the above-mentioned parameters. For example, the problem can be simplified when we assume that flocculation proceeds in the kinetic field, and that equilibrium is quickly reached at flocculant adsorption. In this case, particle aggregation controls the rate of the whole process [-5]. The limiting stage of flocculation is the biparticle collision when a surface area covered by a flocculant of one particle interacts with the uncovered surface area of another particle. In this case, the following equation has been suggested to describe flocculation kinetics [5]: va = - d N / d t = O( 1 - O)Ka N2,

( 1)

Ad.A. Berlitz, V.N. Kislenko/Colloids SurJaces A: Physicochem. Eng. Aspects 104 (1995) 67-72

68

where va is the rate of aggregation of particles, N is the number of flocs and unaggregated primary particles per unit volume, t is time, 0 is the fraction of the particle surface covered by polymer flocculant, and Ka is the rate constant of floc formation. Here, as a first approach, the factor 0 ( 1 - 0) is applied to take the collision efficiency of flocs into account. Integration of Eq. (1) under the initial condition N ( t = 0 ) = N o for 0 = c o n s t leads to a linear relationship between the average number of primary particles in an aggregate and time: No~N= 1 + N00(1 - O)Kat.

(2)

However, the experimental data correspond to this relationship only in the early stages of flocculation (Fig. 1, curves 1 5). Then they show a marked reduction in flocculation rate, and finally the kinetic curves reach plateaus. This may be caused by floc break-up [6] or (and) by the changing aggregate surface coverage with polymer flocculant during the process [-7,8]. In our opinion, it is interesting to compare such mathematical models that describe flocculation kinetics with various degrees of approximation.

2. Model A

It has been assumed that the reduction in flocculation rate, as expected from simple Smoluchowski kinetics, can be explained by the changing coverage of the aggregate surface with polymer during the process /-7,8]. As a result of contacts between polymer loops and tails on one particle with the bare adsorption sites of another particle, the coverage of the surface with flocculant decreases (Fig. 2a) or increases (Fig. 2b). Then 0 of the aggregate differs from 0o of initial particles [7]. Such an approach may be transferred to the process of interaction of already formed aggregates with each other or with initial particles. Therefore, each floc may look like a branched chain of the initial particles (Fig. 3) [4]; this is consistent with a floc model based on cluster-cluster interactions [-2]. The model leads to a fractal dimensionality of about 1.75, indicating a "stringy" floc. As shown in Fig. 3, two cases of floc formation leading to aggregates covered or uncovered by polymer are possible. These two cases were combined in Refs. [4,7,8] by introducing the concept of the fraction of the prevailing surface S. Provided that 0 < 1/2, S corresponds to the fraction of the particle surface not covered by flocculant, S = 1 - 0,

e

22

(3)

240

6

I60 f8

O0

~

0

~0

Fig. 2. Scheme of aggregate formation by interaction between two initial particles when 0o< 1/2 (a) and 0o> 1/2 (b).

6 2

~-

I

t

40

I

I

80

f

I

120

T, m e (mL'~) Fig. 1. Changes in the particle numbers (curves 1 5) and Y (curve 6) with time during the flocculation of polystyrene latex by poly(methyl vinyl pyridine) for 0o=0.091 (1), 0.91 (2), 0.18 (3), 0.68 (4) and 0.36 (5). The experimental data (points) on curves 1 5 were taken from Ref. [13]; curves 1 5 were calculated according to Eq. (5).

Fig. 3. Scheme of floc structure in the form of a branched chain of primary particles when 0o< 1/2 (a) and 0o> I/2 (b).

AriA. Berlin, V.N. Kislenko/Colloids Surfaces A: Physicochem. Eng. Aspects 104 (19953 67 72

and provided that 0 > 1/2, S corresponds to the fraction of the particle surface covered by flocculant: S = 0.

(4)

To simplify the following considerations we use Eqs. (1) and (3) or (4) to obtain P'a =

- d N / d t = K a S ( 1 - S ) N 2.

(5)

For the prevailing surface fraction of the aggregate consisting of i primary particles (see Fig. 3), from a simple geometric consideration [8] we obtain

S~ = [iSo - ( i - 1 ) c ] / [ i - 2 ( i - 1)c],

(6)

where So is the prevailing surface fraction of initial particles, c is the average fraction of the particle surface corresponding to the segment area of a given primary particle in contact with a neighboring one in the floc. The number of primary particles in a floc,

n = No~N,

(7)

found experimentally is an average value. The particle size distribution can be described mathematically by the Smoluchowski theory [9] or by its modifications [ 10], by the Poisson distribution [8,11] or by a self-preserving distribution based on the maximum entropy principle [ 12], etc. Only the Poisson distribution fits our discrete model because it allows us to obtain S and n in the form of an explicit function of the particle number per unit volume of suspension. According to Ref. [8] the value of S can be described by a Poisson distribution with the parameter 2 = n - 1 . The probabilities of its separate values are represented by

p(i-l,2)=exp(-2)2~-l/(i-1)! S = ~ p ( i - 1,23S i.

(i=1,2 .... ),

(8)

average number n in the plateau section and equating expression (9) to 1. Integration of Eq. (5) at the initial condition N ( t = 0 ) = No leads to the relationship No/N

Y=

~ S - I ( 1 - S ) - l d ( N o / N ) = N o K a t.

(11)

l

For the experimentally found values of n = No/N for each kinetic curve of flocculation, it is possible to calculate S according to formulas (6), (8) and (9), and then numerical integration of the left side of Eq. (11) can be accomplished. This allows the rate constant of floc formation K a (Table l) be determined from the tangent of the slope angle of the straight line (Fig. 1). As seen in Fig. 1 (curve 6), the experimental data for the flocculation kinetics of polystyrene latex by poly(methyl vinyl pyridine) lie on a straight line passing through the origin when plotted according to Eq. ( l l ) . Using the found value of K s we can determine the dependence of No/N on time by Eq. (5) for the flocculation of polystyrene latex at different values of surface coverage with poly(methyl vinyl pyridine). Fig. 1 (curves 1-5) illustrates the agreement of the calculated kinetic curves with the experimental data.

3. Model B

On the other hand, the deceleration of flocculation and the levelling off of the kinetic curves can Table 1 The rate constants of floc formation and break-up for the flocculation of polystyrene latex by poly(methyl vinyl pyridine), as well as the sums of squares of deviations (SSD) between the results calculated using the kinetic models and the experimental data

(9)

i

Parameter

In the final stage of the flocculation of a suspension, the kinetic curves level off where function (5) becomes equal to zero. This is possible under the condition S--.I

69

(10)

Thus with the help of a computer, we can define the value of c for a given kinetic curve, using the

Kinetic model

A [8] (IKb/Ka} x 10 8 cm 3 K a x 1011 cm 3 min -1 K b x 103 min 1 SSD for 0o=0.18 00=0.36 00=0.68

5.6+0.5 2.63 15.41 35.30

B [6]

C

4.6 +_0.3 2.6_+0.8 6.1 _+ 0.3 4.82 31.20 4.61

3.6 + 0.5 2.7+0.6 4.9 + 0.9 0.65 13.90 3.47

70

Ad.A. Berlin, V.N. Kislenko/Colloids Surjaces A: Physicochem. Eng. Aspects 104 (1995) 67-72

be explained by the break-up of flocs formed, under the action of hydrodynamic forces. Laminar flow can even lead to the break-up of some aggregates of particles, such as polystyrene [14]. For simple shear flow, the destruction of an aggregate by the action of normal, tangential and centrifugal forces on a segment of its sphere can be stipulated [- 15]. The shear rate and strength of flocs influence a limiting size of aggregate formed [ 16]. Such a size of flocs consisting of an average N o / N of primary particles is reached in the plateau section of the kinetic curves. The deformation and rupture of flocs, the erosion of primary particles from the surface, or the fragmentation of the aggregates can occur, depending on their size, under a turbulent regime [-1,2]. The critical modified Weber number defines floc break-up under the flocculation of a suspension [ 17]. If the flocculation rate decreases with time owing to floc break-up, the process kinetics can be described by Eq. (12) [-6], which takes into account the rate of floc formation Va according to Eq. (5), and the rate of change of the particle number per unit volume of suspension by floc break-up, Ub. In the authors' opinion [6], as a first approximation the latter must be proportional to the aggregate number in the system; see Eq. (13). v = -dN/dt Vb

=

= ua

--

(12)

1)b

I K b ( N - N1)

(13)

where K b is the rate constant of floc break-up, N1 is the number of unaggregated primary particles per unit volume of suspension, and 1 is the number of particles forming during floc break-up under the action of hydrodynamic forces. According to Ref. [-8] the experimentally found average number of primary particles in a floc n given by Eq. (7) can be described by a Poisson distribution (see Eq. (8)). It can be supposed that

N 1 = N exp( 1 - N o / N ).

where Ni is the number of flocs consisting o f / primary particles per unit volume. From Eqs. (7), (8) and (14) it follows that 1)~ 1 / ( i - 1)!, (15)

(16)

In the plateau section of the kinetic curve, the number of particles in the system does not practically change, i.e. Eq. (12) becomes zero. In this case, from Eqs. (12), (5), (13) and (16) for S = So = const [6], it follows that So( 1 - So) = ( I K b / K a ) [ 1 - exp( 1 - N o / N p ) ] / N p ,

(17) where Np is the particle number per unit volume of suspension in the equilibrium state. Fig. 4 illustrates the linear relationship between So and Np when plotted according to Eq. (17). The ratio of the constants l K b / K a is found from the tangent of the slope angle of the straight line for the flocculation of polystyrene latex by poly(methyl vinyl pyridine) (Table 1). For N o / N > 3, exp(1 -No~N)<< 1; this allows us to simplify Eq. (13) taking formula (16) into account. Then, by integrating Eqs. (12), (5) and (13) under the initial condition N ( t = 0 ) = No, the following relationship is obtained [6]: (18)

ln(F/Fo) = lKbt,

where F = ] N / [ I K b / K a - So(1 - So)N]I,

(19)

Fo = I N o / [ l K b / K a -- So(1 -- So)No]l.

(20)

Fig. 5 shows a plot of the experimental data on the flocculation of polystyrene latex by poly(methyl vinyl pyridine) in the coordinates of Eq. (18) with the straight line passing through the origin. Thus

(14)

p(i-1,).) = N ~ / N ,

N, = N exp(1 - N o / N ) ( N o / N -

which for i = 1 therefore yields

o

o

I

o

2

4,

6

~~,_ e ' - ~ o / ~ ) . A l l ¢ ¢0 ¢° [cm ~)

Fig. 4. Relationshipbetween So and Np, plotted according to Eq. (17).

Ad.A. Berlin, V.N. Kislenko/Colloids Surfaces A: Physicochem. Eng. Aspects 104 (1995) 67-72 -., 2

O

¢D

#o

8o

t2o

T~'t"ne (rni~z )

Fig. 5. Change in the particle number of polystyrene latex over time during flocculation by poly(methyl vinyl pyridine), plotted according to Eq. (18).

we can find the value of I K b from the tangent of the slope angle of the straight line (Table 1). As a first approach, it can be supposed that floc breakup into two particles prevails; then /=2. This allows us to calculate the values of Kb and K~ from the ratio of constants l K b / K a found above (Table 1 ). Using the found values of the rate constants of K, and K b we can calculate the relationship between N o / N and time, obtained from Eq. (18): N o / N = [ N o K ~ / ( I K b ) ] exp( - I K b t ) / F o - So( 1 - So)l.

4. Model C

This model considers both factors that decelerate flocculation: floc break-up and the changing cover-

I00

age of the aggregate surface with polymer during the process. In this case, the dynamics of the change in the particle number in the system can be described by Eqs, (12), (5) and (13). Here one has to take into account changes in S during the process, using Eqs. (6) and (9). If investigations are made in the sections of the kinetic curves where i>> 1 then, from Eqs. (6) and (9), it follows Si = (So - c)/( 1 - 2c).

(22)

By designating D = S(1 - S), one can obtain from Eqs. (9) and (22) the expression D = l-So( 1 - So) - c(1 - c)]/(1-2c) 2.

(23)

In the late stages of flocculation when the kinetic curves reach plateaus, function (12) can be equal to zero. Then, from Eqs. (12), (16) and (23), it follows that So(1 - S o ) N p / [ 1 - e x p ( 1 - N o / N p ) ] = ( I K b / K a ) ×

( 1 - 2 c ) 2 + c(1 - c)Np/[1 - e x p ( 1 - No/Np)]. (24) (21)

The calculated kinetic curves in Fig. 6 are in good agreement with the experimental data points. At the same time, one can note that Eq. (17) describes the experimental data markedly less well for low values of S o ( 1 - So) (see Fig. 4), perhaps indicating the necessity to take into account changes in S during the process.

0

71

The linear relationship between So and Np, when plotted according to Eq. (24), is shown in Fig. 7. The ratio of the rate constants I K b / K . is found from the intercept of the straight line on the ordinate axis (Table 1). For N o / N > 3, exp( 1 - N o ~ N ) << 1; this allows us to simplify Eq. (13) in the function (12) taking (16) into account. After this simplification at the initial condition N ( t = O ) = N o integration of Eq. (12) yields the relationship: (25)

l n ( E / E o ) = IKbt

200

Tdme ( m , ' n )

0

1

2

s

(¢- e t'~'°/a~P) - ~"4o-'° (cn, - b Fig. 6. Kinetic curves of polystyrene latex flocculation by poly(methyl vinyl pyridine) for 0o=0.36 (1), 0.68 (2) and 0.18 (3). The experimental data (points) were taken from Re[ [ 13]; the curves were calculated according to Eq. (21).

Fig. 7. Relationship between So and Np, plotted according to Eq. (24). The experimental data (points) were taken from Ref. [13].

72

Ad A. Berlin. V.N. Kislenko/Colloids SuiTJitcesA: Physicochem. Eng. Aspects 104 (1995) 67-72

5. Conclusions

where E = I N / ( I K b / K , -- D N ) I,

(26)

Eo = [No/(IKb/Ka - DNo)I.

(27)

The experimental data on polystyrene latex flocculation by poly(methyl vinyl pyridine) converted according to Eq. (25) fall on a straight line (Fig. 8). The tangent of the slope angle of the straight line gives the value of K b shown in Table 1 (when I= 2). From this we can calculate the value of Ka from the earlier found ratio of constants I K b / K a (Table l ). With these values of the rate constants we can calculate N o / N versus time using Eq. (25): N o / N = N o K j ( l K b ) l e x p ( -- IKbt)/Eo + DI.

(28)

Fig. 9 shows the agreement between the experimental data (points) and the kinetic curves calculated in this way.

It is interesting to compare the three data sets for the flocculation of polystyrene latex by poly(methyl vinyl pyridine) obtained using the kinetic models A, B and C. As can be seen from the data in Table 1, the values of the rate constants found are quite close to each other. However, the sums of squares of errors are the lowest for kinetic model C taking both the changing coverage of the aggregate surface with polymer and floc break-up into account.

Acknowledgement This research is a part of the Program 2.2.2. Clear Water of State Committee of Science and Technology of Ukraine (Project N2.02.02/045).

References

£ o

4o

8o

¢20

7~me (m<'tz) Fig. 8. Change in the particle number of polystyrene latex over time during flocculation by poly(methyl vinyl pyridine), plotted according to Eq. (25).

¢6

8 I

o

i

t

/2o

i

2~o

7-gr~ e ( m i ~ )

Fig. 9. Kinetic curves of flocculation of polystyrene latex by poly(methyl vinyl pyridine) for the initial fractions of the particle surface covered by flocculant: 0.18 (1), 0.68 (2), 0.36 (3). The experimental data (points) were taken from Ref. [ 13]; the curves were calculated according to Eq. (28).

[1] T. Neesse, A. Ivanauskas and K. Mtihle, in Zur Modellirung des Flockungprocesses, Freiberger Forschungshefte, Leipzig, A 720 (1985) 63. [2] J. Gregory, Progr. Filtr. Separ., 4 (1986) 55. [3] Z.M. Yaremko, V.D. Gavriliv and M.I. Soltys, Colloid J. (USSR), 51 (1989) 1164. [4] Ad.A. Berlin, V.N. Kislenko and M.A. Moldovanov, Colloid Polym. Sci., 270 (1992) 1042. [-5] Th. Healy and V.K. La Mer, J. Colloid Interface Sci., 19 (1964) 323. [6] Ad.A. Berlin, V.N. Kislenko and M.A. Moldovanov, Water Chem. Technol. (Ukraine), 15 (1993) 19. [7] V.N. Kislenko, Ad.A. Berlin and M.A. Moldovanov, Water Chem. Technol. (Ukraine), 13 (1991) 486. [8] V.N. Kislenko, Ad.A. Berlin and M.A. Moldovanov, Colloid. J. (USSR), 53 (1991) 499. [9] D.A. Fridrichsberg, Course of Colloid Chemistry, Chemistry, Leningrad, 1984. [10] Y. Tambour and J.H. Seinfeld, J. Colloid Interface Sci., 74 (1980) 260. [11] A. Molski and W. Nowicki, Colloid Polym. Sci., 267 ( 19891 506. [12] J.M. Rosen, J. Colloid Interface Sci., 99 (19841 9. [13] I.M. Solomentseva, I.K. Tusunbayev, A.A. Baran and K.B. Musabekov, Ukr. Chem. J., 46 (1980) 928. [14] R.L. Powell and S.G. Mason, AIChE J., 28 (1982) 286. [15] U.A. Buevich and A.A. Rivkin, Colloid J, (USSR), 42 (1980) 19. [16] J. Gregory, Colloids Surfaces, 31 (1988) 231. [17] A. Ivanauskas and K. M0hle, in: Zur Modellirung des Flockungprocesses, Freiberger Forschungshefte, Leipzig, A 720 (19851 82.