A Population Balance Model for Flocculation of Colloidal Suspensions Incorporating the Influence of Surface Forces Venkataramana Runkanaa,b, P.Somasundarana,* and P.C.Kapurb NSF Industry/University Cooperative Research Center for Advanced Studies in Novel Surfactants, Columbia University, New York, NY, U.S.A. 10027 b Tata Research Development and Design Centre, 54-B, Hadapsar Industrial Estate, Pune, India 411013 a
ABSTRACT Flocculation is a key unit operation in domestic and industrial water treatment, beneficiation of minerals, dewatering of sludges and in many industrial solid-liquid separation processes. Coagulants and flocculants are commonly employed as additives to induce aggregation and improve subsequent filtration or dewatering operations. The characteristics of these additives have a strong influence on suspension stability and hence on rate of flocculation. When collisions between particles take place, the probability of aggregation depends mainly on the surface forces acting between colloidal particles. Population balance models for flocculation generally include kinetics of colloidal aggregation but mostly ignore surface and colloid chemistry of the suspension. Here, a population balance model is presented which incorporates fundamental theories of surface forces involved in flocculation and importantly in fragmentation. The collision efficiency is calculated as a function of total interaction energy between particles, which is estimated using the classical DLVO theory. The Hamaker constant of solids in solvent, an important model parameter, is computed by applying the approximate Lifshitz theory. The irregular and open structure of flocs is taken into account by estimating the collision frequency factor as a function of mass fractal dimension and permeability of aggregates. The model is tested and validated with published experimental data for flocculation of colloidal polystyrene latex and alumina suspensions in different devices. Simulation results on the effect of shear rate and solution pH are in close agreement with experimental data. Since the model incorporates the influence of variables such as pH and electrolyte concentration, it is more amenable for real-time optimization and control of industrial coagulation and flocculation units than existing flocculation models.
KEYWORDS: Flocculation, Population Balance, Surface Forces, Fractal Aggregates, Aggregation, Fragmentation
* Corresponding Author: P.Somasundaran, Fax: 1-212-854 8362; Email:
[email protected]
I
INTRODUCTION
Recovery or removal of colloidal particles is a serious problem in many processes involving particulate solids. Coagulation or flocculation, using inorganic electrolytes or coagulants, polymers or polyelectrolytes and surfactants, is normally employed for this purpose. Industrial examples include water treatment [1], mineral beneficiation [2], pulp and papermaking [3], fabrication of ceramic components [4], processing of paints [5] etc. Flocculation is essentially a process of aggregation of small particles into larger aggregates or flocs. Several phenomena occur simultaneously during flocculation: adsorption of dissolved ions and polymer chains on particles, collisions between particles and/or aggregates and colloid aggregation. Generally shear is applied to enhance frequency of collisions between particles and hence the rate of aggregation. This leads to desorption and/or rupture of polymer chains, fragmentation, restructuring and consolidation of flocs and reflocculation of fractured aggregates. The characteristics of coagulants/flocculants have a strong influence on suspension stability and hence on rate of flocculation [6]. When collisions between particles take place, the probability of aggregation depends mainly on the surface forces acting between colloidal particles. Many experimental studies have shown that aggregation does not occur, even at enhanced shear rates, if the net interaction force between particles is repulsive in nature. The surface forces include van der Waals attraction, electrical double layer repulsion, steric repulsion or bridging attraction in the presence of polymers and structural forces such as hydration repulsion [7]. The nature and magnitude of these forces, in turn, depend on the surface chemistry of the suspension, usually represented in terms of variables like pH, type and concentration of electrolyte species, type and concentration of polymer used and its properties such as molecular weight distribution, charge density and functional groups present [8]. The surface forces affect not only the probability of particle attachment but also floc fragmentation because rate of floc breakage depends on the forces holding the particles together [9]. Experimental studies on flocculation in shear flow [10-13] have clearly demonstrated that the rate of flocculation is strongly dependent on surface and colloid chemistry of the system. Shear enhances the rate of coagulation when the repulsive forces are absent or weak. It does not appear to influence the rate of flocculation when electrical double layer forces are strong. Delichatsios and Probstein [11] concluded from experimental studies on coagulation of latex suspensions that the coagulation efficiency is independent of the mode of particle transport for partially destabilized suspensions. Warren [10] did not observe any aggregation of scheelite (CaWO4), a common ore of tungsten, at low salt (NaCl) concentrations, even when the speed of agitation was increased considerably. This is because scheelite particles are highly negatively charged (-40 mV) in the pH range of 4-12. Coagulation was found to occur only when the repulsive forces were suppressed at higher salt concentrations. Rattanakawin and Hogg [13] found that median size of alumina aggregates increased with pH and reached the maximum at the isoelectric point (IEP) of alumina, and then it decreased as pH was increased further. This is because the surface potential of alumina is very high at low pH and it decreases as pH increases until the IEP of alumina is reached. Above the IEP,
2
charge reversal takes place and as pH is increased further, the particles become highly negatively charged. This behavior is reflected in the manner in which commonly measured variables such as solids settling rate and transmittance of alumina suspensions vary as a function of pH [12, 14]. It is evident from these studies that surface and colloid chemistry of the suspension plays a dominant role in determining the effectiveness of flocculation. Experimental and modeling studies on coagulation usually concentrate on flow of spherical particles in shear flow or on the non-uniform distribution of shear rate in stirred tanks and its effect on coagulation [15-18]. These studies are of great utility for process design and optimization but not for real-time process control because a process operator does not usually change the shear rate while the process is in operation. It is important to optimize the dosage of additives used for flocculation in order to minimize cost of solidliquid separation and to reduce potential environmental pollution by these additives as well as suspended particles which do not settle fast due to ineffective flocculation. It is variables such as pH, coagulant concentration, electrolyte concentration and temperature that are amenable for control and coagulation models need to incorporate the influence of these variables in order to be useful for on-line optimization and control. Accordingly, the objective of our work was to develop a model that rigorously captures the dynamics of aggregation and fragmentation in a population balance framework which incorporates microscopic interaction forces between particles. Previously, we have presented models for flocculation of colloidal suspensions by inorganic electrolytes and polymers in the absence of applied shear [19-21]. In the current work, the model is extended for flocculation in shear environments by incorporating appropriate kernels for shear-induced collisions between particles/aggregates and by including additional terms for floc breakage in the population balance. The model for flocculation by inorganic electrolytes is presented here while modeling of polymer-induced flocculation in shear flow is discussed elsewhere [22]. II
FLOCCULATION MODEL
The mathematical model for flocculation developed in the current work is based on a time-continuous and discrete-size population balance equation with geometrically lumped size classes. It is assumed here that the kinetics of colloidal aggregation and floc fragmentation are additive in nature. The open, irregular and porous structure of flocs is taken into account by the mass fractal dimension and permeability of aggregates. The classical DLVO theory [23, 24] is used to incorporate the influence of surface forces in the presence of inorganic electrolytes. II.A Population Balance Equation The rate of change of particle number concentration during simultaneous aggregation and fragmentation is given by the following discretized population balance equation [25-27]:
3
i−2 dN i = N i −1 ∑ 2 j −i +1 α i −1, j β i −1, j N j + 21 α i −1,i −1 β i −1,i −1 N i2−1 dt j =1 i −1
−Ni ∑ 2 j =1
j −i
max1
max2
j =i
j =i
(1)
α i, j βi, j N j − N i ∑ α i, j βi, j N j − S i N i + ∑ γi, j S j N j
where Ni is number concentration of particles in section i, t is flocculation time, β is collision frequency factor, α is collision efficiency factor, S is specific rate constant of floc fragmentation, γ is breakage distribution function, max1 is maximum number of sections used to represent the complete size spectrum and max2 corresponds to the largest section from which flocs in the current section are produced by fragmentation. The first and second terms on right hand side account for floc growth while the third and fourth terms represent floc loss by aggregation, respectively. The fifth and sixth terms represent loss and growth of the flocs by fragmentation, respectively. II.B Collision Frequency Factor The frequency of collisions between particles and/or aggregates depends on the structure as well as porosity of the aggregates as the hydrodynamic drag on porous objects is significantly different from that on impermeable objects. In order to account for aggregate porosity, the modified collision kernels, proposed by Veerapaneni and Wiesner [28], are used in the present work. These authors modified the kernels by introducing two factors, the drag force correction factor and the fluid collection efficiency. Both these parameters depend on aggregate permeability. The total frequency of collisions is assumed to be a sum of the frequencies due to Brownian motion, applied shear and differential sedimentation. The collision frequency factor for perikinetic aggregation (Brownian motion) β i,Brj is given by [28, 29]:
βi,Brj =
2k BT ⎛⎜ 1 1 + ⎜ 3μ ⎝ Ωi rci Ω j rc j
⎞ ⎟ r +r ⎟ ci c j ⎠
(
)
(2)
where kB is Boltzmann constant, T is suspension temperature, μ is dynamic viscosity of the suspending fluid, rci and rc j are collision radii of particles or aggregates belonging to B
sections i and j, respectively, and Ωi is drag force correction factor, defined as the ratio of the force exerted by the fluid on a permeable aggregate to that on an impermeable aggregate [30]. The collision radius of an aggregate rci containing n0 monodisperse primary particles of radius r0 is defined as [31, 32]: 1/ d F
⎛n ⎞ rci = r0 ⎜⎜ 0 i ⎟⎟ ⎝ CL ⎠
(3)
where dF is mass fractal dimension of the aggregate and CL is the structure prefactor, which is related to the lacunarity or packing structure of the aggregate. Experimental
4
measurements by de Boer [33] indicated that it is of the order of one. The collision frequency factor for orthokinetic aggregation (due to applied shear) β i,Shj is given by [28, 34, 35]:
(
βi,Shj =CSh G E fi rci + E f j rc j
)
3
(4)
where G is volume-averaged fluid velocity gradient or shear rate and CSh is 1.3333 for laminar shear [34] and 1.294 for turbulent shear [35]. E f i is fluid collection efficiency of an aggregate, defined as the ratio of the fluid flow moving through the aggregate to that approaching it [36]. The collision frequency factor for differential sedimentation β iDS , j is given by [28, 37]:
β
DS i, j
2πg = 9μ
(
E f i rci + E f j rc j
)
rc2i
2
Ωi
( ρi − ρl ) −
rc2j Ωj
(ρ
j
− ρl )
(5)
where g is acceleration due to gravity, and ρi and ρl are densities of aggregate and liquid, respectively. The equations for the drag force correction factor and the fluid collection efficiency can be found in the cited references [30, 36]. Both the drag force correction factor and the fluid collection of an aggregate depend on aggregate permeability which in turn is a function of solid volume fraction of the aggregate. There are a number of empirical expressions for estimating aggregate permeability [28, 38]. In general, they show similar trends and the predicted permeability values are reasonably close to each other. We have incorporated the cell model of Happel as it was found to be valid over a wider range of aggregate porosities [28]. The aggregate permeability κ H i is given by [39]:
κH
i
⎡ 9φ 1i / 3 9φi5 / 3 ⎤ + − 3φi2 ⎥ 2 ⎢3 − 2r 2 2 ⎥ = o ⎢ 9φ i ⎢ 3 + 2φi5 / 3 ⎥ ⎢ ⎥ ⎣ ⎦
(6)
where solids fraction of an aggregate, φi is given by [40, 41]:
φi = C
dF / 3 L
⎛ rci ⎜ ⎜r ⎝ o
⎞ ⎟ ⎟ ⎠
d F −3
(7)
II.C Collision Efficiency Factor The collision efficiency factor for aggregates is computed as reciprocal of the modified Fuchs’ stability ratio Wk,l for two primary particles k and l [42-45]:
5
exp(VT /k BT) ds 2 r0k + r0l s Wk ,l = ∞ exp(VvdW /k BT) ds ∫r0k + r0l Dk ,l s2 ∞
∫
Dk ,l
(8)
where Dk,l is hydrodynamic correction factor [46], VT is total energy of interaction , VvdW is van der Waals energy of attraction between two primary particles of radii r0k and
r0l (assumed spherical), s = r0 + r0 + h0 is distance between particle centers, h0 is k
l
distance of closest approach between particle surfaces. The total interaction energy is assumed to be a sum of the surface forces present between particles under given conditions of pH, temperature, type and concentration of electrolyte species and type and concentration of coagulant or flocculant employed and their properties. These forces include van der Waals attraction, electrical double layer attraction/repulsion, hydration repulsion and hydrophobic attraction. When polymers are present, the electrochemical nature of particle surfaces also gets modified, which leads to a change in the nature and magnitude of van der Waals attraction and electrical double layer force. In addition, steric repulsion or bridging attraction has to be included while computing the total interaction energy. As mentioned earlier, the focus of the present paper is on forces in the presence of inorganic electrolytes only. Hence the classical DLVO theory was used to compute the total interaction energy. The van der Waals energy of attraction between particles may be computed using either the Hamaker microscopic theory [47] or the Lifshitz macroscopic theory [48]. The relative merits of these theories are discussed elsewhere [49]. The latter is more rigorous than the former but requires accurate dielectric data of the materials involved. Because it is easier to implement, we have employed the Hamaker theory to compute the van der Waals energy of attraction VvdW but an approximate form of the Lifshitz theory [50] was used for calculating the Hamaker constant. The van der Waals energy of attraction between a pair of spherical primary particles is given by [47]: VvdW
A = − 131 6
⎧⎪ ⎡ s 2 − (r0k + r0l ) 2 ⎤ ⎫⎪ 2r0k r0l 2r0k r0l + 2 + ln ⎢ 2 ⎨ 2 2 2 ⎥⎬ s − (r0k − r0l ) 2 ⎪⎩ s − (r0k + r0l ) ⎢⎣ s − (r0k − r0l ) ⎦⎥ ⎪⎭
(9)
where A131 is Hamaker constant of solids (1) across the solvent medium (3). The Hamaker constant is not actually a constant and is influenced by electrolyte concentration and retardation. Here, it is computed as a sum of two contributions: zero-frequency energy and dispersion energy. The former is affected by the electrolyte while the latter is influenced by retardation. The screened and retarded Hamaker constant is given by [51]: A131 (κ , h0 ) = Aξ o (1 + 2κ h0 )e −2κh0 + Aξ ≥1 FR (hR )
(10)
6
where FR is retardation function, κ is Debye-Huckel parameter, Aξ 0 is zero-frequency energy contribution and Aξ ≥1 is dispersion energy contribution. The Debye-Huckel parameter κ is a function of temperature T, ionic strength I and dielectric constant εr of solution. The zero-frequency energy contribution is a function of dielectric constants of solids and the solvent while the dispersion energy contribution depends on characteristic frequencies of electromagnetic radiation and refractive indexes of solids and solvent [50]. Retardation sets in at about 5-10 nm from the particle surface [7] and its effect is automatically taken care of in the macroscopic Lifshitz theory, as seen above. The retardation function proposed by Russel et al. [52] is employed in the present work. The attraction or repulsion due to electrical double layers around the particles is calculated using the analytical expression proposed by Bell et al. [53] for two spherical particles of radii, r0k and r0 l and surface potentials, ψ 0k and ψ 0l : ⎛k T ⎞ Vedl = 64πε r ε 0 ⎜⎜ B ⎟⎟ ⎝ zc e ⎠
2
⎛ r0k r0l ⎜ ⎜ r0 + r0 l ⎝ k
⎞ z eψ ⎟ tanh ⎛⎜ c 0k ⎜ 4k T ⎟ B ⎝ ⎠
⎞ ⎛ z eψ ⎟⎟ tanh ⎜⎜ c 0l ⎠ ⎝ 4 k BT
⎞ ⎟⎟ exp (− κh0 ) ⎠
(11)
where e is elementary charge, zc is valence of the counterion, ε0 permittivity of free space and εr is relative permittivity or dielectric constant of solvent. II.D Fragmentation of Fractal Aggregates Application of shear results in enhanced collisions between particles but it also leads to fragmentation of aggregates concurrently. It was postulated that floc fragmentation occurs either by splitting or by surface erosion [54]. The mode of breakage is truly influenced by the aggregate structure. The fragmentation of aggregates with low fractal dimensions can occur by splitting as they have an open structure while surface erosion could be the primary mechanism of breakage of aggregates with high fractal dimensions. Floc fragmentation is stochastic in nature and it has been hypothesized in the literature that it resembles either breakage of particulate solids, as in particle size reduction by grinding [9] or splitting of droplets, as in liquid-liquid extraction [55]. The specific rate constant of fragmentation, Si assuming aggregate fragmentation similar to the droplet breakage in liquid-liquid dispersions, is given by [56]:
⎛ − ε b,i ⎞ ⎛ 4 ⎞ ⎛ εΕ ⎞ ⎟⎟ Si = ⎜ ⎟ ⎜ ⎟ exp⎜⎜ ⎝ 15π ⎠ ⎝ ν ⎠ ⎝ εΕ ⎠ 1/ 2
1/2
(12)
where εb,i and εE are respectively critical and average turbulent energy dissipation rates at which aggregate breakage takes place and ν is kinematic viscosity of the suspending fluid. The mean aggregate size decreases as fluid shear increases. Hence, it has been assumed that εb,i is inversely proportional to the aggregate collision radius.
7
ε b,i =
Bf
(13)
rci
where Bf is a proportionality constant. An aggregate can dissociate into two or three or more smaller fragments, but binary breakage appears to be more likely than the others [9, 57]. Although we have incorporated binary, ternary and normal breakage distribution functions in our model, only binary breakage was invoked in the simulation results presented next. B
III RESULTS AND DISCUSSION
The population balance model was tested with published experimental data for the flocculation of polystyrene latex and alumina in different devices and under conditions in which repulsive forces are comparable in magnitude to attractive forces. It was, however, initially tested to study the effect of shear rate in the absence of repulsive forces using experimental data for flocculation of polystyrene latex particles in a stirred tank. Next, it was used to simulate the effect of shear in a couette flow apparatus under conditions in which repulsive forces are not negligible. Later, the effect of pH on aggregation of alumina in stirred tanks was simulated and the model was validated with experimental data at different pH. The population balance in Eq. (1) was divided into 30 geometric sections and solved numerically by Gear’s predictor-corrector technique [58] for nonlinear ordinary differential equations. The stability ratio in Eq. (8) was evaluated by numerical integration using the Romberg algorithm [59]. The total interaction energy was computed as a sum of van der Waals attraction (Eq. 9) and electrical double layer repulsion (Eq. 11). The Hamaker constant was calculated using Eq. (10). The collision frequency factor was calculated as a sum of contributions due to perikinetic aggregation (Eq. 2), orthokinetic aggregation (Eq. 4) and differential sedimentation (Eq. 5). During integration of population balance equations, the computed floc size distribution was tested at each time step for conservation of solid volume. The loss of total solid volume was less than 1% for the results presented here. III.A Flocculation of Polystyrene Latex in a Stirred Tank The experimental data for flocculation of polystyrene latex published by Flesch et al. [32] was used for initial model testing. These authors studied the effect of shear rate on coagulation of monodisperse and spherical polystyrene latex particles in a 2.8 lit baffled stirred tank. The mean diameter of primary particles was 0.87 μm and the volume fraction of solids was 1.4×10-5, corresponding to an initial number concentration of 4×1013 m-3. The flocculant was aluminum sulfate hydrate (Al2(SO4)3.16H2O). The experiments were conducted using a constant Al2(SO4)3.16H2O concentration of 10 mg/lit. Sodium hydrogen carbonate (1 mM) was used to buffer the solution and pH was 7.2 ± 0.05. The suspension was mixed using a six-bladed Rushton impeller and floc size distribution (FSD) data are reported for three different shear rates, 50, 100 and 150 sec-1. Complete details of the materials and the experimental procedure can be found in the cited reference [32].
8
The population balance in Eq. (1) was solved to simulate the evolution of FSD with time and at different shear rates. The collision efficiency was assumed to be 1 as there were no repulsive forces during coagulation. The mass fractal dimension of aggregates was taken as 2.05, as reported by Flesch et al. [32]. The proportionality constant, Bf in the equation for fragmentation rate constant (Eq.13), was used as the fitting parameter. Its value was obtained as 1.0, 2.95 and 5.35 μm at shear rates of 50, 100 and 150 sec-1 respectively. These values are slightly different from those obtained by Flesch et al. [32] because of the difference in computing the collision frequency factor. It will be noticed that Bf increases as G increases. Similar trends for the constant in the breakage rate coefficient were also observed previously [9, 27]. The predicted and experimental steady state aggregate size distributions at a shear rate of 100 sec-1 are shown in Figure 1. Results obtained with other shear rates were presented elsewhere [60, 61]. It will be seen that the predicted size distribution is reasonably close to the measured distribution, though the former is relatively narrow. This is possibly because a binary breakage distribution function was used. Flesch et al. [32] have used both the binary and the normal distribution functions and found that the predicted aggregate size distributions become broader if a normal breakage distribution function with a larger variance is used. The normal distribution function was not tested in this work simply because it introduces two more model fitting parameters. The other reason for the discrepancy between predictions and experimental results is that we have assumed a monodisperse feed while in reality it could have been polydisperse. The total volume of solids of monodisperse feed will be different from that of a polydisperse feed and hence it may not be possible to achieve a very close agreement between experimental aggregate size distributions and predicted results. B
Normalized Volume Fraction, μm
-1
B
0.010 expt model
0.008 0.006 0.004 0.002 0.000 0
100
200
300
400
500
600
Aggregate Collision Diameter, μm
Figure 1. Comparison of simulated and experimental size distributions of polystyrene latex flocs obtained after 30 min of flocculation at a global average shear rate of 100 sec1 . Mean primary particle diameter: 0.87 μm; Solids volume fraction: 1.4×10-5; pH: 7.2; Aggregate fractal dimension: 2.05; NaHCO3: 1 mM; (Symbols – Experimental data from [32]; Line – computed using Eq. 1).
9
III.B Flocculation of Polystyrene Latex in a Couette Flow Apparatus Selomulya et al. [62] studied coagulation of polystyrene latex in a Couette flow apparatus in the presence of 0.05 M MgCl2, and at pH 9.15 ± 0.05. These experiments were conducted at different shear rates, ranging from 32 sec-1 to 330 sec-1, using primary particles of three different sizes, 60 nm, 380 nm and 810 nm. The corresponding solid volume fractions were 3.83×10-6, 3.74×10-5, and 3.76×10-5, and the number densities were 3.4×1016, 1.3×1015, and 1.4×1014 m-3, respectively. They reported data for time evolution of mean aggregate size and the scattering exponent, measured using light scattering. The results obtained with primary particles of diameter 810 nm and at shear rates of 32, 100 and 246 sec-1 are used in the present work for testing the model.
The collision efficiency was taken as reciprocal of the stability ratio in Eq. (8), which was obtained by numerical integration. The zeta potential of 810 nm particles was taken as –19 mV [62]. The dielectric constant, refractive index and UV absorption frequency data for water and polystyrene were taken from the literature [49]. The time evolution of computed normalized mass mean aggregate diameter, obtained by solving Eq. (1), is compared with experimental data in Figure 2. A constant value of 2.3 for floc fractal dimension was used at all three shear rates. The proportionality constant, Bf in the expression for breakage rate constant was adjusted by matching the predicted equilibrium mean aggregate diameter with experimental data at different shear rates. The fitted Bf values were, 0.27, 1.55 and 5.2 μm at shear rates of 32, 100 and 246 sec-1, respectively. The agreement between computed and actual time evolution of mean aggregate size could be improved by adjusting the fractal dimension as a function of time, but was not attempted here. The aggregate size distributions corresponding to the three shear rates are plotted in Figure 3. It can be observed that the size distribution becomes narrow as shear rate increases, which is in agreement with trends observed experimentally [32]. B
B
Normalized Floc Diameter (dfloc/dfeed)
200 expt model
G = 32 sec
-1
150
100
G = 100 sec
50
G = 246 sec
-1
-1
0 0
1
2
3
4
5
6
Normalized Flocculation Time
Figure 2. Comparison of predicted and experimental time evolution of the mass mean aggregate diameter in a couette flow apparatus as a function of global average shear rate. Mean primary particle diameter: 0.81 μm; Solids volume fraction: 3.76×10-5; pH: 9.15; MgCl2: 0.05 M (Symbols – Experimental data from [62]; Lines – computed using Eq. 1).
10
Volume Percent of Aggregates, %
50 G=246s-1 100 32
40
30
20
10
0 0
100
200
300
400
Aggregate Diameter, μm
Figure 3. Aggregate size distributions at different shear rates during flocculation of polystyrene latex in a couette flow apparatus, corresponding to experimental mean size data shown in Figure 2. III.C Flocculation of Alumina in a Stirred Tank – Effect of pH Rattanakawin and Hogg [13] studied coagulation of alumina in a stirred tank. The experiments were conducted at a constant shear rate of 1000 sec-1 and the pH was varied from 4 to 12. The median primary particle size was 0.43 μm. The isoelectric point (IEP) of alumina was 8.6. The median aggregate size was found to increase as the pH increased, up to the IEP and then it decreased as the pH increased. The main reason for this behavior was the change in the surface potential with pH. It is well known that the surface potential of alumina decreases as pH increases and charge reversal takes place above the IEP. As pH is increased further, the surface becomes more negatively charged. This is actually reflected in the interaction energy diagram. The stability ratio, or in other words, the collision efficiency, has a direct relationship with the maximum value of the interaction energy [63]. In order to show this relationship, the total interaction energy was computed as a function of pH. The zeta potential-pH data were taken from Rattanakawin [64]. The data required to compute the Hamaker constant of alumina using the approximate Lifshitz theory were taken from Israelachvili [7]. The computed maximum interaction energy is shown in Figure 4 along with the measured median aggregate size. It can be seen that as the interaction energy maximum decreases, the median size increases. This clearly reflects the influence of pH on interaction forces and as a result, on coagulation of colloidal suspensions.
Das [65] studied coagulation of alumina at different pH. The primary particle diameter was 700 nm and the solids concentration was 0.53 %wt. This corresponds to a number density of approximately 7.4×1015 m-3. The experiments were carried out in a 250 ml beaker fitted with four baffles and the suspension was mixed using a three bladed propeller of 2.5 cm diameter. The experiments were conducted in the presence of 0.03 M NaCl and at 300 rpm. The aggregate size distribution obtained at pH 9 was used to test the model. The shear rate was computed to be approximately 200 sec-1. The zeta potential-pH data were taken from Das [65]. The stability ratio was computed using Eq. (8). The Hamaker constant of alumina in water was calculated using Eq. (10). The
11
population balance in Eq. (1) was solved for these experimental conditions and the computed and measured aggregate size distributions after 6 min of flocculation are shown in Figures 5–7 at different pH. The floc breakage parameter, Bf and the fractal dimension of aggregates, dF were used as fitting parameters. The fractal dimension was obtained as 2.4 at all three pH values. The breakage rate parameter had to be adjusted and it was found to be 0.115 μm at pH 3.5 and 10.5 while at pH 9 it was 0.15 μm. It can be seen from Figures 5–7 that the agreement between computed and experimental size distributions is close. 8
1.50
-17
Total Interaction Energy - Maximum (x10 )(J)
B
(a)
6
1.00
5 4 3
0.50 (b)
2
Median Floc Size (μm)
7
1 0.00 2
4
6
8
10
12
0 14
pH
Normalized Volume Fraction, μm
-1
Figure 4. Effect of pH on flocculation of alumina (a) total interaction energy maximum – computed using DLVO theory (b) median aggregate size – experimental data from [13]. 0.5
pH = 3.5
expt model
0.4
0.3
0.2
0.1
0.0 1
10
Aggregate Diameter, μm
Figure 5. Comparison of simulated and experimental size distributions of alumina flocs after 6 min of flocculation. Mean primary particle diameter: 700 nm; pH: 3.5; 0.03 M NaCl (Symbols – experimental data from [65]; Line – computed using Eq. 1). The shear rate in a baffled stirred tank is normally non-uniform because of the complex flow patterns generated by the impeller. Characterization of fluid flow patterns is a difficult task and generally a volume-averaged or time-averaged shear rate is used for designing coagulation and dispersion tanks. Since we are simulating laboratory scale
12
Normalized Volume Fraction, μm
-1
devices in the current work, it was assumed that the shear rate is uniform throughout the suspension. It, is, however, acknowledged this assumption may be incorrect for largescale flocculation tanks and the current model can be readily extended by combining it with a suitable fluid/suspension hydrodynamics model. On the other hand, it is also possible to combine it with a residence time distribution model of the large-scale unit, determined from tracer experiments. 0.35
pH = 9
expt model
0.30 0.25 0.20 0.15 0.10 0.05 0.00 1
10
Aggregate Diameter, μm
Normalized Volume Fraction, μm
-1
Figure 6. Comparison of simulated and experimental size distributions of alumina flocs after 6 min of flocculation. Mean primary particle diameter: 700 nm; pH: 9; 0.03 M NaCl (Symbols – experimental data from [65]; Line – computed using Eq. 1). 0.5
pH = 10.5 expt model
0.4
0.3
0.2
0.1
0.0 1
10
Aggregate Diameter, μm
Figure 7. Comparison of simulated and experimental size distributions of alumina flocs after 6 min of flocculation. Mean primary particle diameter: 700 nm; pH: 10.5; 0.03 M NaCl (Symbols – experimental data from [65]; Line – computed using Eq. 1).
13
IV CONCLUSIONS
Surface and colloid chemistry of the suspension plays a critical role in coagulation and flocculation operations encountered in many industrial processes. A population balance model is presented in this paper which incorporates this by including fundamental theories of surface forces involved in flocculation including fragmentation. Unlike previous flocculation models, the current model can be employed to simulate the effect of important variables such as pH and electrolyte concentration besides the global average shear rate. The model was validated with published experimental data for flocculation of polystyrene and alumina suspensions in stirred tanks and a couette flow apparatus. The model can be utilized gainfully for determining optimum settings for shear rate, pH and electrolyte concentration for any given initial feed condition. V
ACKNOWLEDGMENTS
This work was supported by the National Science Foundation (NSF Grants # INT-9605197 and INT-01-17622) and the NSF Industry/University Cooperative Research Center (IUCRC) for Advanced Studies in Novel Surfactants at Columbia University (NSF Grant # EEC-98-04618). The authors thank the management of Tata Research Development and Design Centre for the permission to publish this paper. VR thanks Prof. E.C.Subbarao, Prof. Mathai Joseph and Dr. Pradip for their advice and encouragement. VI REFERENCES
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