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Particle transport in crossflow microfiltration – II. Effects of particle–particle interactions
Chemical Engineering Science 54 (1999) 281—289
Particle transport in crossflow microfiltration — II. Effects of particle—particle interactions Ingmar H. Huisman*, Gun Tra¨gas rdh, Christian Tra¨gas rdh Food Engineering, Lund University, P.O. Box 124, SE 221 00 Lund, Sweden Received 23 March 1998; received in revised form 15 June 1998; accepted 28 July 1998
Abstract A model previously developed for the calculation of limiting fluxes for crossflow microfiltration of non—interacting particles was extended to include the effect of physico-chemical particle—particle interactions. It was shown theoretically that the effect of particle—particle interactions on the microfiltration flux can be described by a diffusion-type equation with an effective interactioninduced diffusion coefficient. The microfiltration flux for a general situation, where particle transport is caused by convection, Brownian diffusion, shear-induced diffusion, and particle—particle interactions, was then calculated by adding the diffusion coefficients, and solving the governing convective-diffusion equation numerically. Results of these calculations agreed very well with experimental fluxes measured during crossflow microfiltration of model silica particle suspensions. The influence of wall shear stress, membrane length, particle size, and particle concentration on permeate flux was very well predicted. However, the effect of particle surface potential was quantitatively underpredicted. Shear-induced diffusion seems to be the main transport mechanism governing the flux in the microfiltration of suspensions of micron-sized particles, but charge effects can increase fluxes considerably. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Crossflow microfiltration; Concentration polarisation; Particle—particle interactions; Double-layer forces; Model; Silica particles
1. Introduction Physico-chemical particle—particle interactions have a substantial effect on the fouling behaviour in membrane filtration. Permeate fluxes have been reported to increase as the zeta potential of the feed suspension particles increases (Hoogland et al., 1990; McDonogh et al., 1989; Huisman et al., 1997), and as the salt concentration decreases (Chang et al., 1995; Elzo et al., 1998). Most authors explain the effect of zeta potential qualitatively by reasoning that high zeta potentials increase interparticle repulsion, thus causing less deposition (thinner cake layers) and more permeable cake layers. The effect of salt concentration is explained by noting that a decrease in salt concentration corresponds to an increase in the Debye length. This causes greater interparticle repulsion
*Corresponding author. Current address: Dpto. Termadina´mica, Fac. Ciencias, Real de Burgos, 4707, Valladolid, Spain. Tel.: #46 46 222 98 20; fax: #46 46 222 46 22; e-mail: [email protected].
and larger interparticle distances, resulting in less deposition and more permeable cake layers. A quantitative description of charge effects on the flux in ultrafiltration and microfiltration was first given by McDonogh et al. (1984, 1989). Their treatment was highly simplified, but explained observed fluxes qualitatively. In a later paper, these authors used a modified film theory to explain charge effects in crossflow filtration (Welsch et al., 1995). Although the theory in this later paper was more robust, it still failed to predict observed fluxes quantitatively. Bacchin et al. (1995) proposed a theoretical model, based on a parallel between cake layer build-up and particle aggregation. They assumed that a particle is deposited on a membrane (or on a cake layer) if the particle has enough energy to overcome the potential barrier caused by the repulsive interaction forces. This implies that fouling is always irreversible. However, it has been shown experimentally that cake-layer build-up can be either reversible or irreversible (Huisman et al., 1998). Palacek and Zydney (1994) proposed a very simple model based on a force balance on a single
0009-2509/99/$ — see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 2 2 3 - 1
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particle. This model predicts that the flux while filtering a suspension of charged particles is equal to the flux for uncharged particles plus a charge-induced term. This term is proportional to the square of the surface charge density of the particles. They also demonstrated experimentally that the flux increased with the square of the charge density in the filtration of protein solutions. It was realised by several authors that the effects of particle—particle interactions on the flux cannot be understood by considering only an isolated pair of particles. In general, a particle will be surrounded by many other particles, in a cake layer or a concentration polarisation layer, and the interactions between all these particles must be included. This reasoning led to the use of cell models. In cell models it is assumed that a concentration polarisation layer or cake layer can be approximated by an ordered lattice of spheres. Petsev et al. (1993) used a cell model to calculate the permeability of cake layers in ultrafiltration, and Huisman et al. (1998) used a cell model to calculate the reversibility of fouling in microfiltration. Bowen and Jenner (1995) used a cell model to calculate fluxes for dead-end ultrafiltration. This model included rigorous analysis of the particle—particle interaction forces, and included electroviscous effects which were neglected by other authors. The model was later adapted to crossflow filtration (Bowen et al., 1996); however, the handling of the hydrodynamics of the concentration polarisation layer was oversimplified. Jo¨nsson and Jo¨nsson (1996) used an approach similar to that of Bowen and Jenner, yet related the necessary parameters to experimentally easily obtainable system characteristics, such as the concentration dependence of the osmotic pressure and the sedimentation coefficient. Their model also dealt with the hydrodynamics of the concentration polarisation layer in a simplified manner. It has been shown by many authors that particle transport in crossflow microfiltration is not only the result of Brownian diffusion and physico-chemical interactions, but that other transport processes, such as shear-induced diffusion, are also of importance. Detailed descriptions of the effect of shear-induced diffusion on microfiltration flux are available (e.g. Huisman and Tra¨gas rdh, 1998). However, to the best of our knowledge, no theoretical description of the combined effect of shear-induced diffusion, Brownian diffusion, and particle—particle interactions has yet been presented. The aim of the present study was to calculate crossflow microfiltration fluxes for the general situation in which particle transport is caused by a combination of shearinduced diffusion, Brownian diffusion, and particle—particle interactions. It will be shown in this paper that the influence of particle—particle interactions can be described by an effective interaction-induced diffusion coefficient. The different transport mechanisms can be combined by adding this interaction-induced diffusion
coefficient to the shear-induced and the Brownian diffusion coefficients. The convective—diffusion equation can be solved using the approach described for non-interacting particles in an earlier paper (Huisman and Tra¨gas rdh, 1998).
2. Theory Although it has already been suggested that the influence of particle—particle interactions on particle transport can be modelled by an effective diffusion coefficient (Petsev et al., 1993; Bowen et al., 1996), this approach has not yet been widely applied to crossflow microfiltration modelling. In this section, an equation for the interaction-induced diffusion coefficient will be derived. It will also be shown that this diffusion coefficient can be easily combined with the Brownian and the shear-induced diffusion coefficients to give a full description of the combined effect of these transport mechanisms. Consider a particle P and its neighbours, as depicted in Fig. 1. A body-centred geometry has been assumed here, but any other geometry could be used. The particle P is subject to a hydrodynamic force F (caused by the F permeate flux) and interaction forces with other particles F (h ). At equilibrium, the net force on the particle G L vanishes, thus: F (y )#m cos h [F (h )!F (h )]"0 (1) F L G L> G L where m is the number of nearest neighbours in one plane (m"4, for body-centred packing), h is the angle between the interparticle force and the hydrodynamic force (h "54.74°, for body-centred packing), and h is the interparticle distance. The difference [F (h )! F (h )] G L> G L
Fig. 1. Schematic representation of a particle in a cake layer or a concentrated suspension.
I.H. Huisman et al. /Chemical Engineering Science 54 (1999) 281—289
in Eq. (1) can be approximated by *F /*h (h !h ), G L> L which can then be approximated by *F dh *F G (h !h ) + G (y !y ) L L *h L> *h dy L> *F dh " G [2a#h(y)] cos h *h dy
(2)
where a is the particle radius and y the transverse coordinate. h is related to the volume fraction of particles, , through
"1#h/2a
(3)
where is the closest packing in the chosen geometry
(0.6792 for body-centred packing). Combining Eqs. (1)—(3) results in
*F d
4m cos h a
G F (y)"! . F 3
*h dy
(4)
The hydrodynamic force is given by F "6nk a(v!v ) ) F N '( ), where k is the permeate viscosity, v is the local liquid velocity, v is the convective velocity of the particle, N and '( ) is a hydrodynamic friction factor, given by (Davis and Russel, 1989) '"(1! )\ . Substitution of F and ' in Eq. (4) gives F *F d
4m cos h a
G (v!v ) "! N 18nk ' ( )
*h dy d
"D ' dy
2ne e ati P F" G exp (ih)!1
(6)
where e e is the dielectric constant of the electrolyte P solution, t is the surface potential, and i is the inverse Debye length. Eq. (10) is only strictly valid for the interaction between two isolated particles. It is a good approximation for many-particle interactions if ia 1, which is usually the case in microfiltration. An equation for *F /*h G can then be obtained by differentiation, resulting in the following formula for the interaction-induced diffusion coefficient:
where
*F 4m cos ha
G. (7) D "! ' 18nk '( )
*h The convective—diffusion equation, governing particle transport in the concentration polarisation boundary layer over a membrane is given by (8)
where D is the effective diffusion coefficient caused M>1' by Brownian and shear-induced diffusion (Huisman and Tra¨ga rdh, 1998). Axial diffusion and axial slip are neglected, as other authors have done (e.g. Davis and Sherwood, 1990). Moreover, the particles are assumed to be sufficiently small, so that gravity and inertial-lift forces can be neglected. Combining Eqs. (6) and (8) leads to the relation v "(D
d
#D ) M>1' ' dy
where v is the local liquid velocity, not the convective particle velocity, and D is an effective interaction-induced ' diffusion coefficient. Comparing Eqs. (8) and (9) shows that there are two ways of describing the effects of particle—particle interactions on particle transport, which are equivalent: one can either use a convective velocity which differs from the liquid velocity, or an effective diffusion coefficient that differs from the Brownian and shear-induced diffusion coefficients. Note that the interaction-induced diffusion coefficient, D , is equal to the ' collective (generalised) diffusion coefficient of charged particles (Petsev et al., 1993; Bowen et al., 1996) minus the Brownian diffusion coefficient. D increases proportionally with *F /*h, which can be ' G obtained from the curve of interaction force versus interparticle distance. The interaction force F is the sum of G double-layer forces, van der Waals forces, and possibly hydrophobic and hydration forces. For the particle concentrations of interest in concentration polarisation layers, double-layer forces are often the dominating forces, and it will be assumed throughout this paper that the contribution of all other forces is negligible. The double-layer force for particles of oxides, such as silica, is well described by the constant charge approximation, Eq. (10) (Shaw, 1980; Huisman et al., 1998)
(5)
d
v ) "D N M>1' dy
283
(9)
(10)
m cos h
D" ' 9k '( )
e e [at i] P ;
sinh ia !1
.
(11)
It can be seen that the interaction-induced diffusion coefficient depends on properties of the electrolyte solution (e , k , i), properties of the particle ( , a, t ), and some P assumptions made regarding the packing geometry (m, h,
). Most of these properties can be easily measured or
calculated. However, the surface potential, t , of the feed suspension particles is generally unknown. Many authors report the zeta potential of the particles used. The zeta potential is the potential at the plane of shear, at about 0.55 nm from the surface. The effective surface potential can be calculated from the zeta potential using
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Gouy—Chapman theory (Israelachvili, 1991). This method has been described in more detail by Huisman et al. (1998). It has been shown, when solving the convective—diffusion equations for Brownian and shear-induced diffusion, that one can take advantage of the fact that the diffusion coefficients can be written in the form D"D (k¹,q ,a,k , 2 , ) D ( ) U
(12)
where D has the units m s\ and gives the scaling with all variables except for the volume fraction, , and D ( )is the dimensionless diffusion coefficient that depends only on (Huisman and Tra¨ga rdh, 1998). Unfortunately, it is not possible to factorise the interactioninduced diffusion coefficient exactly in the way expressed in Eq. (12). The concentration-dependent part is also a function of
and of i a, the dimensionless inverse
Debye length. Possible choices for D and D ( ) are ' ' m cosh D " e e t ' P 9k D " '
(13a)
(ia)(1! ) ( / )
.
sinh ia !1
(13b)
Solving the convective—diffusion equations using only interaction-induced diffusion results in numerical instability and is physically unrealistic. A collective diffusion coefficient was therefore used, which includes interaction-induced diffusion, Brownian diffusion, and shear-induced diffusion. Fluxes were then calculated numerically, using the approach outlined in detail by Huisman and Tra¨gas rdh (1998). This approach takes advantage of the factorisation given in Eq. (12). The flux in the case of two diffusional processes, with coefficients D and D , is then given by ? @ J"vN (D ( ), D ( ), k ( ), D /D ) U ? @ ? @
9q D U ? 8k ¸
3. Calculations for interaction-induced diffusion and Brownian diffusion Calculations were performed for the crossflow filtration of suspensions of particles with a radius of 100 nm at an ionic strength of 1;10\ M, and surface potentials (t ) of 50 and 100 mV, using a combination of Brownian diffusion and interaction-induced diffusion as the transport mechanism. The results are given in Fig. 2, which shows that the surface potential of the particles has a large impact on the permeate flux. The calculated flux is ten times higher for particles with a surface potential of 100 mV than for uncharged particles. For low concentrations, the flux increases as \ , behaviour typical for @ diffusional transport. Note that the parametric dependence of the flux, J, on wall shear stress, q , and membrane U length, ¸, is the same as in the case of pure Brownian diffusion, given by the non-dimensionalisation of the problem [Eq. (14)]. The microfiltration flux for dilute suspensions, under conditions in which Brownian diffusion and interaction-induced diffusion are of importance, is thus given by
k q U J" M>' k ¸
@
(16)
where k is a constant that depends on ia, D /D M>' ' M (where D is the Brownian diffusion coefficient for M dilute suspensions) and possibly slightly on .
The results of our numerical calculations were compared with the experimental results of Welsch et al. (1995), who performed crossflow microfiltration experiments with silica particles of varying zeta potential in laminar flow. The crossflow velocities were low, and particle sizes small, and it can be calculated that
(14)
where v is a dimensionless flux, which is obtained by U solving a differential equation [Eq. (16) in Huisman and Tra¨gas rdh, 1998], k is the dimensionless viscosity of the suspension, q is the wall shear stress, and ¸ is the U membrane length. It was assumed that the viscosity is given by (Leighton and Acrivos, 1987)
k"k k "k 1#1.5 . 1! /
(15)
The maximum volume fraction
was assumed to be
0.6792, because this is equal to the closest packing in a body-centred geometry, and because this value is close to experimental values obtained previously (Huisman et al., 1998).
Fig. 2. Numerical results showing dimensionless flux vN ("J ) (8k ¸/ U 9q D ) ) versus bulk concentration, for the combined effect of U M Brownian and interaction-induced diffusion for "0.6792. The
particles have a radius of 100 nm, and the salt concentration is 1;10\ M. The surface potentials (t ) modeled were 0, 50, and 100 mV.
I.H. Huisman et al. /Chemical Engineering Science 54 (1999) 281—289
Fig. 3. Flux versus the square of the surface potential for crossflow microfiltration of silica particle suspensions: particle radius"23 nm; salt concentration"0.2 M; wall shear stress"0.1632 and 0.6277 N/m\ (Re"78 and 300). The symbols represent experimental results of Welsch et al. (1995). The solid and dashed line represent calculated results incorporating Brownian diffusion and interaction-induced diffusion. The dotted lines represent Brownian theory.
shear-induced particle transport was negligible in their experiments. Results are given in Fig. 3, where the flux is plotted versus the square of the surface potential. The experiments of Welsch et al. were performed with particles with a radius of 23 nm, at a salt concentration of 0.2 M, and Reynolds numbers of 78 and 300 (corresponding to q "0.1632 and q "0.6277 N m\). For U U very low values of surface potential (below 20 mV) the particles coagulated and the flux increased with decreasing surface potential. As our model does not incorporate the effect of particle coagulation, only the data obtained for surface potentials above 20 mV are shown in Fig. 3. The correlation between experimental and calculated fluxes is fairly good. The trend of flux increase with both zeta potential and wall shear stress is predicted qualitatively correctly. The simulations predict the flux to increase almost proportionally to the square of the surface potential, in agreement with the simple theory and the experimental results of Palacek and Zydney (1994). The flux at vanishing surface potential is equal to the flux predicted by Brownian theory.
4. Materials and methods Crossflow microfiltration experiments were performed with model suspensions of spherical silica particles in electrolyte solutions. A detailed diagram and description of the experimental set-up used has been presented elsewhere (Huisman et al., 1997). Tubular ceramic a-alumina and titania membranes were used (SCT, France) with a diameter of 6.85 mm and a nominal pore size of 200 nm. The type of membrane material was found not to
285
influence the permeate flux. A uniform transmembrane pressure could be guaranteed, since a circuit on the permeate side created a pressure drop along the membrane equal to the pressure drop along the feed side. Since the permeate was regularly returned to the feed tank, the volume on the feed side of the membrane was constant. The steady-state permeate flux was measured at different transmembrane pressures and crossflow velocities. For each crossflow velocity a pressure-independent (or ‘limiting’) flux could be reached for high transmembrane pressures. These limiting fluxes are reported here. Both ends of the 250 mm long membranes were sealed by the manufacturer, leaving an effective membrane length of about 230 mm. For studying the effect of filter length, ¸, on permeate flux, the effective membrane length was changed by plugging the pores in part of the membrane with acetone-based glue. Plugging was achieved by partially inserting the membrane tube into the glue (Karlssons Klister, Sweden) further diluted with acetone. The membrane was left in the glue for about an hour followed by drying to remove the acetone. Water flux measurements confirmed that this was an effective and reproducible method of plugging the pores, rendering the part of the membrane that was inserted into the glue impermeable, but leaving the rest of the membrane unchanged. The layer of glue formed on the membrane surface was very thin compared with the tube diameter. The relation between axial pressure drop and crossflow velocity, which was earlier shown to follow Blasius’ law (Huisman et al., 1997), was not significantly altered, thus proving that the tube diameter was unaffected by the glue. Various silica particles were used for the filtration experiments. Particles with a radius of about 0.25 km were kindly supplied by Nissan Chemical Industries, Ltd, Japan (PST-5, MP-4540). These particles were either used as received (batches 1 and 2), or after narrowing the size distribution by allowing the particles to sediment for 24 h and removing the supernatant containing the smaller particles (batch 3). Other particles with various sizes (batches 4—6) were kindly supplied by Christian Kaiser, University of Mainz, Germany. Particle size distributions were determined by a laser light scattering method and by SEM. Batch between Batch between Batch between Batch between Batch between
1: Mean radius: 95 and 450 nm. 2: Mean radius: 85 and 470 nm. 3: Mean radius: 165 and 500 nm. 4: Mean radius: 205 and 370 nm. 5: Mean radius: 470 and 844 nm.
240 nm, 90% of all particles 266 nm, 90% of all particles 280 nm, 90% of all particles 260 nm, 90% of all particles 595 nm, 90% of all particles
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I.H. Huisman et al. /Chemical Engineering Science 54 (1999) 281—289
Batch 6: Mean radius: 911 nm, 90% of all particles between 749 and 1234 nm. Suspensions of these particles were prepared based on Milli-Q filtered deionised water. Reactant grade HCl or NaOH were added to adjust the pH, and NaCl was used to adjust the salinity. pH and salinity together determine the zeta potential (f) of the silica particles, as shown by Elzo et al. (1998). Experiments were performed under the following conditions: z Particle batch 1: I"1;10\ M, q "4.42 N/m, U
"0.69;10\ M, various values of f (Fig. 4). @ z Particle batch 2: f"!86 mV, I"1 ; 10\ M, various values of , various values of q (Figs. 5 @ U and 6).
z Particle batch 2: f "!26 mV, I"2 ; 10\ M,
@ "0.69;10\ M, various values of q (Fig. 5). U z Particle batch 2: f"!7 mV, I"0.1 M, various values of , various values of q (Figs. 5 and 6). @ U z Particle batch 4—6: f"!86 mV, I"1;10\ M,
"0.26 ; 10\, q "4.5 and 8.8 N/m (Fig. 7). @ U z Particle batch 3: f"!26 mV, I"2 ; 10\ M,
"0.69;10\ M, q "9 N m\, various values @ U of ¸ (Fig. 8). It can easily be shown that gravity and inertial-lift forces can be neglected for all combinations of experimental conditions considered here. ia values are at least 27, with higher values for the larger particles and for the higher salt concentrations. This warrants the use of Eq. (10), valid for ia 1. The experiments with particle batch 1 have been published by Elzo et al. (1998). The
Fig. 4. Flux versus the square of the surface potential for crossflow microfiltration of silica particle suspensions: particle radius"240 nm; salt concentration"10\ M; wall shear stress"4.42 N m\. Symbols represent experimental results. The solid line represents calculated results incorporating shear-induced diffusion, Brownian diffusion, and interaction-induced diffusion. The dashed line represents shear-induced diffusion theory and the dotted line Brownian diffusion theory.
Fig. 6. Flux versus particle volume fraction in the feed suspension for two values of zeta potential: particle radius"266 nm; wall shear stress "30 N m\. Symbols represent experimental values, lines represent the results of numerical calculations.
Fig. 5. Flux versus wall shear stress for various values of zeta potential; particle radius "266 nm; particle volume fraction"0.69;10\. Symbols represent experimental values, lines represent the results of numerical calculations.
Fig. 7. Flux versus particle radius at two values of wall shear stress; zeta potential"!86 mV; particle volume fraction"0.26;10\. Symbols represent experimental values, lines represent the results of numerical calculations.
I.H. Huisman et al. /Chemical Engineering Science 54 (1999) 281—289
Fig. 8. Flux versus membrane length; particle radius"280 nm; zeta potential"!26 mV; particle volume fraction"0.69;10\; wall shear stress"9 N m\. Symbols represent experimental values, lines represent the results of numerical calculations.
experiments using low zeta potentials (f"!7 mV) have been published by Huisman and Tra¨gas rdh (1998). 5. Results and discussion Measured permeate fluxes for various process conditions are given in Figs. 4—8 together with results of corresponding calculations. The value of wall shear stress, required for the flux calculations, was calculated from turbulent flow theory (Blasius’ law) and from measured pressure drops; both methods giving similar results. It can be seen that agreement between measured and calculated fluxes is in general good. Fig. 4 shows the crossflow microfiltration flux using particles with a radius of 240 nm, at a salt concentration of 10\ M, versus the square of the surface potential. The data point corresponding to the lowest value of surface potential (11 mV) in Fig. 4 was obtained at 0.1 M. It is clear from the graph that charge effects hardly influence the flux at surface potentials as low as 11 mV, so the actual value of the salt concentration is of very little importance. The correlation between experimental and calculated fluxes is reasonable. The trend of flux increase with increasing zeta potential is predicted qualitatively correctly. Again, the numerical calculations predict the flux to increase almost proportionally to the square of the surface potential, in agreement with the simple theory and the experimental results of Palacek and Zydney (1994). Flux predictions based on pure Brownian and pure shear-induced diffusion theory, according to Huisman and Tra¨gas rdh (1998), are also plotted in Fig. 4. The flux at vanishing surface potential is equal to the flux predicted by a combination of Brownian and shear-induced theory. Note that the data plotted in Fig. 4 were measured at very low crossflow velocity (1 m s\, q " U 4.42 N m\). Obtained fluxes are therefore very low.
287
The influence of wall shear stress on permeate flux is shown in Fig. 5 for various values of particle zeta potential. Crossflow velocities range from 1 to 5 m s\. Measured fluxes for low zeta potential and high ionic strength (f"!7 mV, I"0.1 M) are proportional to the wall shear stress, and are predicted very well by the model. The influence of interaction-induced diffusion is negligible for such low zeta potentials, and the numerical flux predictions are virtually identical to those presented by Huisman and Tra¨gas rdh (1998) for non-interacting particles. Increasing the repulsive particle—particle interactions, by increasing the zeta potential and decreasing the ionic strength, results in an increase in measured fluxes for all wall shear stresses. The calculated fluxes also increase, but not by as much as the experimental values. The influence of particle—particle interactions is thus underpredicted by the calculations. It can also be seen in Fig. 5 that the scaling of flux with the square of the surface potential, observed in Figs. 3 and 4, is lost at high wall shear stresses. Fig. 6 shows the flux versus particle volume fraction in the feed suspension. In the case of low zeta potential and high ionic strength (f"!7 mV, I"0.1 M) the measured fluxes decrease with volume fraction according to J) \ , and are predicted very well by the model. Again, @ for such low zeta potentials the numerical results are virtually identical to those presented earlier for noninteracting particles. Increasing the effect of particle—particle interactions, by increasing the zeta potential and decreasing the ionic strength, again results in an increase in measured fluxes. The calculated fluxes also increase, but not by as much as the experimental values. Experimental fluxes for the high zeta potential case scale as
\ for particle volume fractions greater than @ 0.4;10\, in agreement with model predictions. For lower particle volume fractions, fluxes deviate slightly from this trend. The influence of particle—particle interactions is again underpredicted by the calculations. Fluxes measured for suspensions of particles from batches 4—6 are shown in Fig. 7 versus the mean particle radius. Measurements were only performed for conditions with strong particle-particle repulsion (f"!86 mV, I"1;10\ M). It can be seen that the measured fluxes increase strongly with particle radius, and that measured fluxes are in very good agreement with the calculations. The agreement between model calculations and measured values is somewhat surprising, given that in Figs. 5 and 6, an underprediction of the flux was seen for such high zeta potentials. Particles used in the experiments depicted in Fig. 7 are larger than those used in the other experiments. It was pointed out earlier that the assumptions made about the interaction forces are more realistic for larger values of ia. Therefore, agreement between model calculations and experiments should increase with increasing particle size. Moreover, particles used in the experiments depicted in Fig. 7 are
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I.H. Huisman et al. /Chemical Engineering Science 54 (1999) 281—289
from a different source and have a narrower size distribution than the other particles, which could explain why the measured fluxes were closer to the predicted values. Fig. 8 shows the permeate flux for different values of filter length. It can clearly be seen that the measured flux decreases with increasing filter length. Moreover, these data indicate that flux scales with membrane length as J&¸\ , which is typical for diffusive particle transport. Fluxes are underpredicted again by the model calculations, probably because the effect of particle—particle interactions was underpredicted, in accordance with the results shown in Figs. 5 and 6. Overall, it can be concluded that crossflow microfiltration fluxes are predicted correctly within about 25% by the model. To improve the accuracy of the flux predictions, it appears to be necessary to describe the influence of particle—particle interactions in more detail. In the derivation of the interaction-induced diffusion coefficient the only assumptions made were that the particle size distribution was monodisperse, and that the concentration polarisation layer could be approximated by an ordered lattice of spheres. These assumptions are similar to those used by other researchers (e.g. Bowen and Jenner, 1995). The equation chosen for '( ) and the assumed packing geometry (body-centred, face-centred, etc.) influence the flux slightly, but are not expected to lead to large errors in flux prediction. More restrictive assumptions were made, however, when relating the interaction-induced diffusion coefficient to measurable quantities, such as the salt concentration and the zeta potential of the particles. It was assumed that the interaction force between two particles in the lattice could be approximated by the double-layer force between two isolated spheres, assuming constant surface charge, thus neglecting van der Waals forces and hydration forces. Moreover, the effect of the electrical double-layer on the suspension viscosity (Buscall, 1991) and on the water viscosity (electroviscous effect, Levine et al., 1975) was neglected. It can therefore be expected that some discrepancies exist between experimental fluxes and model predictions. The main conclusion that can be drawn from the present study is, that the numerical predictions show the same trends as the experimental results, and that the fluxes obtained are of the correct order of magnitude, thus supporting the approach employed. In future research, some of the effects neglected in the present model may be included, hopefully leading to even more accurate flux predictions.
dicted correctly within an error margin of $25%. If double-layer forces are negligible (low zeta potential, high ionic strength), the accuracy is even higher ($10%). The fact that accurate flux predictions were obtained especially for low zeta potentials (i.e. for small doublelayer forces) suggests that van der Waals forces and hydration forces have insignificant effects on the permeate flux. This justifies the a priori assumption that these forces are negligible. Trends observed for the measured fluxes are in agreement with the scaling predicted by the calculations. Thus, the experimental fluxes increase with particle size, increase with particle zeta potential, increase almost linearly with the wall shear stress, decrease with particle volume fraction according to J& \ , and decrease @ with membrane length as J&¸\ . The extended model gave good flux predictions both for situations in which Brownian diffusion and charge effects are dominant and for situations in which shear-induced diffusion and charge effects are dominant. From these observations, it can be concluded that the concept of interaction-induced diffusion is very useful for predicting fluxes in crossflow microfiltration. However, the quantitative effect of interaction forces on the flux is, in general, slightly underpredicted by the model. This is probably due to the assumptions made when relating the interaction-induced diffusion coefficient to measurable quantities. Especially the equation used for the particle—particle interaction force, Eq. (10), seems to be an oversimplification.
Acknowledgements The authors would like to thank Eert Vellenga and Iris Balestri for performing some of the experiments. We are grateful to Christian Kaiser, University of Mainz, Germany, and Nissan Chemicals, Ltd, Japan for providing the silica particles. This work was financially supported by the Swedish Foundation for Membrane Technology.
Notation a D D D
6. Conclusions
D
Calculating microfiltration fluxes by combining shear-induced diffusion, Brownian diffusion, and particle—particle interactions results in flux predictions that compare well with experimental values. Fluxes are pre-
D ' F F F G
M>1'
particle radius, m diffusion coefficient, m s\ concentration-dependent part of the diffusion coefficient, dimensionless concentration-independent part of the diffusion coefficient, m s\ sum of Brownian and shear-induced diffusion coefficients, m s\ interaction-induced diffusion coefficient, m s\ hydrodynamic drag force, N particle—particle interaction force, N
I.H. Huisman et al. /Chemical Engineering Science 54 (1999) 281—289
h I J k ¸ m Re ¹ v v N v U y
interparticle distance, m ionic strength, mol dm\ length-averaged permeate flux, m s\ Boltzmann’s constant membrane length, m number of nearest neighbours in one plane Reynolds number, dimensionless temperature, K liquid velocity in transverse direction, m s\ convective particle velocity in transverse direction, m s\ dimensionless flux, dimensionless transverse coordinate, m
Greek letters e permittivity of free space, C J\ m\ e relative dielectric constant, dimensionless P f zeta potential, V h angle between interparticle force and hydrodynamic force, rad i Debye constant, m\ k suspension viscosity, Pa s k dimensionless viscosity, dimensionless k viscosity of the permeate, Pa s q wall shear stress, N m\ U
particle volume fraction, dimensionless
particle volume fraction in the bulk, dimen@ sionless
maximum particle volume fraction (" close
packing), dimensionless ' ‘friction factor’ for concentrated suspensions, dimensionless t surface potential, V References Bacchin, P., Aimar, P., & Sanchez, V. (1995). Model for colloidal fouling of membranes. A.I.Ch.E. J., 41, 368—376. Bowen, W.R., & Jenner, F. (1995). Dynamic ultrafiltration model for charged colloidal dispersions: a Wigner—Seitz cell approach. Chem. Engng Sci., 50, 1707—1736. Bowen, W.R., Mongruel, A., & Williams, P.M. (1996). Prediction of the rate of crossflow membrane ultrafiltration: a colloidal interaction approach. Chem. Engng Sci., 51, 4321—4333. Buscall, R. (1991). Effect of long-range repulsive forces on the viscosity of concentrated latices: comparison of experimental data with an
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effective hard-sphere model. J. Chem. Soc. Faraday ¹rans., 87, 1365—1370. Chang, D.J., Hsu, F.C., & Hwang, S.J. (1995). Steady state permeate flux of crossflow microfiltration, J. Membrane Sci., 98, 97—106. Davis, K.E., & Russel, W.B. (1989). An asymptotic description of transient settling and ultrafiltration of colloidal dispersions. Phys. Fluids A, 1, 82—100. Davis, R.H., & Sherwood, J.D. (1990). A similarity solution for steadystate crossflow microfiltration. Chem. Engng Sci., 45, 3203—3209. Elzo, D., Huisman, I.H., Middelink, E., & Gekas, V. (1998). Charge effects on inorganic membrane performance in a crossflow microfiltration process. Colloids Surfaces A: Physicochem. Engng Aspects, 138, 145—159. Hoogland, M.R., Fane, A.G., & Fell, C.J.D. (1990). The effect of pH on the crossflow filtration of mineral slurries using ceramic membranes. In A.J. Burggraaf (Ed.). Proceedings of the Ist International Conference on Ceramic Membranes, Montpellier, (pp. 153—162). Huisman, I.H., Johansson, D., Tra¨ga rdh, G., & Tra¨ga rdh, C. (1997). Design of a crossflow microfiltration unit for studies of flux and particle transport. ¹rans. Inst. Chem. Engng, 75 A, 508—512. Huisman, I.H., Elzo, D., Middelink, E., & Tra¨ga rdh, C. (1998). Properties of the cake layer formed during crossflow microfiltration. Colloids Surfaces A: Physicochem. Engng Aspects, 138, 265—281. Huisman, I.H., & Tra¨ga rdh, C. (1998). Particle transport in crossflow microfiltration; effects of hydrodynamics and Brownian diffusion, 54, 269—278. Israelachvili, J.N. (1991). Intermolecular and surface forces (2nd Ed.). London: Academic Press. Jo¨nsson, A.S., & Jo¨nsson, B. (1996). Ultrafiltration of colloidal dispersions — a theoretical model of concentration polarization phenomena. J. Colloid Interface Sci., 180, 504—518. Leighton, D., & Acrivos, A. (1987). Measurement of shear-induced self-diffusion in concentrated suspensions of spheres. J. Fluid Mech., 177, 109—131. Levine, S., Marriott, J.R., Neale, G., & Epstein, N. (1975). Theory of electrokinetic flow in fine cylindrical capillaries at high zeta-potentials. J. Colloid Interface Sci., 52, 136—149. McDonogh, R.M., Fell, C.J.D., & Fane, A.G. (1984). Surface charge and permeability in the ultrafiltration of non flocculating colloids. J. Membrane Sci., 21, 285—294 McDonogh, R.M., Fane, A.G., & Fell, C.J.D. (1989). Charge effects in the crossflow filtration of colloids and particulates, J. Membrane Sci., 43, 69—85. Palecek, S.P., & Zydney, A.L. (1994). Intermolecular electrostatic interactions and their effect on flux and protein deposition during protein filtration. Biotechnol. Prog., 10, 207—213. Petsev, D.N., Starov, V.M., & Ivanov, I.B. (1993). Concentrated dispersions of charged colloidal particles. Colloids Surfaces A, 81, 65—81. Shaw, D.J. (1980). Introduction to Colloid and Surface Chemistry (3rd Edn.). London: Butterworths. Welsch, K., McDonogh, R.M., Fane, A.G., & Fell, C.J.D. (1995). Calculation of limiting fluxes in the ultrafiltration of colloids and fine particles. J. Membrane Sci., 99, 229—239.
Particle transport in crossflow microfiltration — II. Effects of particle—particle interactions Ingmar H. Huisman*, Gun Tra¨gas rdh, Christian Tra¨gas rdh Food Engineering, Lund University, P.O. Box 124, SE 221 00 Lund, Sweden Received 23 March 1998; received in revised form 15 June 1998; accepted 28 July 1998
Abstract A model previously developed for the calculation of limiting fluxes for crossflow microfiltration of non—interacting particles was extended to include the effect of physico-chemical particle—particle interactions. It was shown theoretically that the effect of particle—particle interactions on the microfiltration flux can be described by a diffusion-type equation with an effective interactioninduced diffusion coefficient. The microfiltration flux for a general situation, where particle transport is caused by convection, Brownian diffusion, shear-induced diffusion, and particle—particle interactions, was then calculated by adding the diffusion coefficients, and solving the governing convective-diffusion equation numerically. Results of these calculations agreed very well with experimental fluxes measured during crossflow microfiltration of model silica particle suspensions. The influence of wall shear stress, membrane length, particle size, and particle concentration on permeate flux was very well predicted. However, the effect of particle surface potential was quantitatively underpredicted. Shear-induced diffusion seems to be the main transport mechanism governing the flux in the microfiltration of suspensions of micron-sized particles, but charge effects can increase fluxes considerably. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Crossflow microfiltration; Concentration polarisation; Particle—particle interactions; Double-layer forces; Model; Silica particles
1. Introduction Physico-chemical particle—particle interactions have a substantial effect on the fouling behaviour in membrane filtration. Permeate fluxes have been reported to increase as the zeta potential of the feed suspension particles increases (Hoogland et al., 1990; McDonogh et al., 1989; Huisman et al., 1997), and as the salt concentration decreases (Chang et al., 1995; Elzo et al., 1998). Most authors explain the effect of zeta potential qualitatively by reasoning that high zeta potentials increase interparticle repulsion, thus causing less deposition (thinner cake layers) and more permeable cake layers. The effect of salt concentration is explained by noting that a decrease in salt concentration corresponds to an increase in the Debye length. This causes greater interparticle repulsion
*Corresponding author. Current address: Dpto. Termadina´mica, Fac. Ciencias, Real de Burgos, 4707, Valladolid, Spain. Tel.: #46 46 222 98 20; fax: #46 46 222 46 22; e-mail: [email protected].
and larger interparticle distances, resulting in less deposition and more permeable cake layers. A quantitative description of charge effects on the flux in ultrafiltration and microfiltration was first given by McDonogh et al. (1984, 1989). Their treatment was highly simplified, but explained observed fluxes qualitatively. In a later paper, these authors used a modified film theory to explain charge effects in crossflow filtration (Welsch et al., 1995). Although the theory in this later paper was more robust, it still failed to predict observed fluxes quantitatively. Bacchin et al. (1995) proposed a theoretical model, based on a parallel between cake layer build-up and particle aggregation. They assumed that a particle is deposited on a membrane (or on a cake layer) if the particle has enough energy to overcome the potential barrier caused by the repulsive interaction forces. This implies that fouling is always irreversible. However, it has been shown experimentally that cake-layer build-up can be either reversible or irreversible (Huisman et al., 1998). Palacek and Zydney (1994) proposed a very simple model based on a force balance on a single
0009-2509/99/$ — see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 2 2 3 - 1
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particle. This model predicts that the flux while filtering a suspension of charged particles is equal to the flux for uncharged particles plus a charge-induced term. This term is proportional to the square of the surface charge density of the particles. They also demonstrated experimentally that the flux increased with the square of the charge density in the filtration of protein solutions. It was realised by several authors that the effects of particle—particle interactions on the flux cannot be understood by considering only an isolated pair of particles. In general, a particle will be surrounded by many other particles, in a cake layer or a concentration polarisation layer, and the interactions between all these particles must be included. This reasoning led to the use of cell models. In cell models it is assumed that a concentration polarisation layer or cake layer can be approximated by an ordered lattice of spheres. Petsev et al. (1993) used a cell model to calculate the permeability of cake layers in ultrafiltration, and Huisman et al. (1998) used a cell model to calculate the reversibility of fouling in microfiltration. Bowen and Jenner (1995) used a cell model to calculate fluxes for dead-end ultrafiltration. This model included rigorous analysis of the particle—particle interaction forces, and included electroviscous effects which were neglected by other authors. The model was later adapted to crossflow filtration (Bowen et al., 1996); however, the handling of the hydrodynamics of the concentration polarisation layer was oversimplified. Jo¨nsson and Jo¨nsson (1996) used an approach similar to that of Bowen and Jenner, yet related the necessary parameters to experimentally easily obtainable system characteristics, such as the concentration dependence of the osmotic pressure and the sedimentation coefficient. Their model also dealt with the hydrodynamics of the concentration polarisation layer in a simplified manner. It has been shown by many authors that particle transport in crossflow microfiltration is not only the result of Brownian diffusion and physico-chemical interactions, but that other transport processes, such as shear-induced diffusion, are also of importance. Detailed descriptions of the effect of shear-induced diffusion on microfiltration flux are available (e.g. Huisman and Tra¨gas rdh, 1998). However, to the best of our knowledge, no theoretical description of the combined effect of shear-induced diffusion, Brownian diffusion, and particle—particle interactions has yet been presented. The aim of the present study was to calculate crossflow microfiltration fluxes for the general situation in which particle transport is caused by a combination of shearinduced diffusion, Brownian diffusion, and particle—particle interactions. It will be shown in this paper that the influence of particle—particle interactions can be described by an effective interaction-induced diffusion coefficient. The different transport mechanisms can be combined by adding this interaction-induced diffusion
coefficient to the shear-induced and the Brownian diffusion coefficients. The convective—diffusion equation can be solved using the approach described for non-interacting particles in an earlier paper (Huisman and Tra¨gas rdh, 1998).
2. Theory Although it has already been suggested that the influence of particle—particle interactions on particle transport can be modelled by an effective diffusion coefficient (Petsev et al., 1993; Bowen et al., 1996), this approach has not yet been widely applied to crossflow microfiltration modelling. In this section, an equation for the interaction-induced diffusion coefficient will be derived. It will also be shown that this diffusion coefficient can be easily combined with the Brownian and the shear-induced diffusion coefficients to give a full description of the combined effect of these transport mechanisms. Consider a particle P and its neighbours, as depicted in Fig. 1. A body-centred geometry has been assumed here, but any other geometry could be used. The particle P is subject to a hydrodynamic force F (caused by the F permeate flux) and interaction forces with other particles F (h ). At equilibrium, the net force on the particle G L vanishes, thus: F (y )#m cos h [F (h )!F (h )]"0 (1) F L G L> G L where m is the number of nearest neighbours in one plane (m"4, for body-centred packing), h is the angle between the interparticle force and the hydrodynamic force (h "54.74°, for body-centred packing), and h is the interparticle distance. The difference [F (h )! F (h )] G L> G L
Fig. 1. Schematic representation of a particle in a cake layer or a concentrated suspension.
I.H. Huisman et al. /Chemical Engineering Science 54 (1999) 281—289
in Eq. (1) can be approximated by *F /*h (h !h ), G L> L which can then be approximated by *F dh *F G (h !h ) + G (y !y ) L L *h L> *h dy L> *F dh " G [2a#h(y)] cos h *h dy
(2)
where a is the particle radius and y the transverse coordinate. h is related to the volume fraction of particles, , through
"1#h/2a
(3)
where is the closest packing in the chosen geometry
(0.6792 for body-centred packing). Combining Eqs. (1)—(3) results in
*F d
4m cos h a
G F (y)"! . F 3
*h dy
(4)
The hydrodynamic force is given by F "6nk a(v!v ) ) F N '( ), where k is the permeate viscosity, v is the local liquid velocity, v is the convective velocity of the particle, N and '( ) is a hydrodynamic friction factor, given by (Davis and Russel, 1989) '"(1! )\ . Substitution of F and ' in Eq. (4) gives F *F d
4m cos h a
G (v!v ) "! N 18nk ' ( )
*h dy d
"D ' dy
2ne e ati P F" G exp (ih)!1
(6)
where e e is the dielectric constant of the electrolyte P solution, t is the surface potential, and i is the inverse Debye length. Eq. (10) is only strictly valid for the interaction between two isolated particles. It is a good approximation for many-particle interactions if ia 1, which is usually the case in microfiltration. An equation for *F /*h G can then be obtained by differentiation, resulting in the following formula for the interaction-induced diffusion coefficient:
where
*F 4m cos ha
G. (7) D "! ' 18nk '( )
*h The convective—diffusion equation, governing particle transport in the concentration polarisation boundary layer over a membrane is given by (8)
where D is the effective diffusion coefficient caused M>1' by Brownian and shear-induced diffusion (Huisman and Tra¨ga rdh, 1998). Axial diffusion and axial slip are neglected, as other authors have done (e.g. Davis and Sherwood, 1990). Moreover, the particles are assumed to be sufficiently small, so that gravity and inertial-lift forces can be neglected. Combining Eqs. (6) and (8) leads to the relation v "(D
d
#D ) M>1' ' dy
where v is the local liquid velocity, not the convective particle velocity, and D is an effective interaction-induced ' diffusion coefficient. Comparing Eqs. (8) and (9) shows that there are two ways of describing the effects of particle—particle interactions on particle transport, which are equivalent: one can either use a convective velocity which differs from the liquid velocity, or an effective diffusion coefficient that differs from the Brownian and shear-induced diffusion coefficients. Note that the interaction-induced diffusion coefficient, D , is equal to the ' collective (generalised) diffusion coefficient of charged particles (Petsev et al., 1993; Bowen et al., 1996) minus the Brownian diffusion coefficient. D increases proportionally with *F /*h, which can be ' G obtained from the curve of interaction force versus interparticle distance. The interaction force F is the sum of G double-layer forces, van der Waals forces, and possibly hydrophobic and hydration forces. For the particle concentrations of interest in concentration polarisation layers, double-layer forces are often the dominating forces, and it will be assumed throughout this paper that the contribution of all other forces is negligible. The double-layer force for particles of oxides, such as silica, is well described by the constant charge approximation, Eq. (10) (Shaw, 1980; Huisman et al., 1998)
(5)
d
v ) "D N M>1' dy
283
(9)
(10)
m cos h
D" ' 9k '( )
e e [at i] P ;
sinh ia !1
.
(11)
It can be seen that the interaction-induced diffusion coefficient depends on properties of the electrolyte solution (e , k , i), properties of the particle ( , a, t ), and some P assumptions made regarding the packing geometry (m, h,
). Most of these properties can be easily measured or
calculated. However, the surface potential, t , of the feed suspension particles is generally unknown. Many authors report the zeta potential of the particles used. The zeta potential is the potential at the plane of shear, at about 0.55 nm from the surface. The effective surface potential can be calculated from the zeta potential using
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Gouy—Chapman theory (Israelachvili, 1991). This method has been described in more detail by Huisman et al. (1998). It has been shown, when solving the convective—diffusion equations for Brownian and shear-induced diffusion, that one can take advantage of the fact that the diffusion coefficients can be written in the form D"D (k¹,q ,a,k , 2 , ) D ( ) U
(12)
where D has the units m s\ and gives the scaling with all variables except for the volume fraction, , and D ( )is the dimensionless diffusion coefficient that depends only on (Huisman and Tra¨ga rdh, 1998). Unfortunately, it is not possible to factorise the interactioninduced diffusion coefficient exactly in the way expressed in Eq. (12). The concentration-dependent part is also a function of
and of i a, the dimensionless inverse
Debye length. Possible choices for D and D ( ) are ' ' m cosh D " e e t ' P 9k D " '
(13a)
(ia)(1! ) ( / )
.
sinh ia !1
(13b)
Solving the convective—diffusion equations using only interaction-induced diffusion results in numerical instability and is physically unrealistic. A collective diffusion coefficient was therefore used, which includes interaction-induced diffusion, Brownian diffusion, and shear-induced diffusion. Fluxes were then calculated numerically, using the approach outlined in detail by Huisman and Tra¨gas rdh (1998). This approach takes advantage of the factorisation given in Eq. (12). The flux in the case of two diffusional processes, with coefficients D and D , is then given by ? @ J"vN (D ( ), D ( ), k ( ), D /D ) U ? @ ? @
9q D U ? 8k ¸
3. Calculations for interaction-induced diffusion and Brownian diffusion Calculations were performed for the crossflow filtration of suspensions of particles with a radius of 100 nm at an ionic strength of 1;10\ M, and surface potentials (t ) of 50 and 100 mV, using a combination of Brownian diffusion and interaction-induced diffusion as the transport mechanism. The results are given in Fig. 2, which shows that the surface potential of the particles has a large impact on the permeate flux. The calculated flux is ten times higher for particles with a surface potential of 100 mV than for uncharged particles. For low concentrations, the flux increases as \ , behaviour typical for @ diffusional transport. Note that the parametric dependence of the flux, J, on wall shear stress, q , and membrane U length, ¸, is the same as in the case of pure Brownian diffusion, given by the non-dimensionalisation of the problem [Eq. (14)]. The microfiltration flux for dilute suspensions, under conditions in which Brownian diffusion and interaction-induced diffusion are of importance, is thus given by
k q U J" M>' k ¸
@
(16)
where k is a constant that depends on ia, D /D M>' ' M (where D is the Brownian diffusion coefficient for M dilute suspensions) and possibly slightly on .
The results of our numerical calculations were compared with the experimental results of Welsch et al. (1995), who performed crossflow microfiltration experiments with silica particles of varying zeta potential in laminar flow. The crossflow velocities were low, and particle sizes small, and it can be calculated that
(14)
where v is a dimensionless flux, which is obtained by U solving a differential equation [Eq. (16) in Huisman and Tra¨gas rdh, 1998], k is the dimensionless viscosity of the suspension, q is the wall shear stress, and ¸ is the U membrane length. It was assumed that the viscosity is given by (Leighton and Acrivos, 1987)
k"k k "k 1#1.5 . 1! /
(15)
The maximum volume fraction
was assumed to be
0.6792, because this is equal to the closest packing in a body-centred geometry, and because this value is close to experimental values obtained previously (Huisman et al., 1998).
Fig. 2. Numerical results showing dimensionless flux vN ("J ) (8k ¸/ U 9q D ) ) versus bulk concentration, for the combined effect of U M Brownian and interaction-induced diffusion for "0.6792. The
particles have a radius of 100 nm, and the salt concentration is 1;10\ M. The surface potentials (t ) modeled were 0, 50, and 100 mV.
I.H. Huisman et al. /Chemical Engineering Science 54 (1999) 281—289
Fig. 3. Flux versus the square of the surface potential for crossflow microfiltration of silica particle suspensions: particle radius"23 nm; salt concentration"0.2 M; wall shear stress"0.1632 and 0.6277 N/m\ (Re"78 and 300). The symbols represent experimental results of Welsch et al. (1995). The solid and dashed line represent calculated results incorporating Brownian diffusion and interaction-induced diffusion. The dotted lines represent Brownian theory.
shear-induced particle transport was negligible in their experiments. Results are given in Fig. 3, where the flux is plotted versus the square of the surface potential. The experiments of Welsch et al. were performed with particles with a radius of 23 nm, at a salt concentration of 0.2 M, and Reynolds numbers of 78 and 300 (corresponding to q "0.1632 and q "0.6277 N m\). For U U very low values of surface potential (below 20 mV) the particles coagulated and the flux increased with decreasing surface potential. As our model does not incorporate the effect of particle coagulation, only the data obtained for surface potentials above 20 mV are shown in Fig. 3. The correlation between experimental and calculated fluxes is fairly good. The trend of flux increase with both zeta potential and wall shear stress is predicted qualitatively correctly. The simulations predict the flux to increase almost proportionally to the square of the surface potential, in agreement with the simple theory and the experimental results of Palacek and Zydney (1994). The flux at vanishing surface potential is equal to the flux predicted by Brownian theory.
4. Materials and methods Crossflow microfiltration experiments were performed with model suspensions of spherical silica particles in electrolyte solutions. A detailed diagram and description of the experimental set-up used has been presented elsewhere (Huisman et al., 1997). Tubular ceramic a-alumina and titania membranes were used (SCT, France) with a diameter of 6.85 mm and a nominal pore size of 200 nm. The type of membrane material was found not to
285
influence the permeate flux. A uniform transmembrane pressure could be guaranteed, since a circuit on the permeate side created a pressure drop along the membrane equal to the pressure drop along the feed side. Since the permeate was regularly returned to the feed tank, the volume on the feed side of the membrane was constant. The steady-state permeate flux was measured at different transmembrane pressures and crossflow velocities. For each crossflow velocity a pressure-independent (or ‘limiting’) flux could be reached for high transmembrane pressures. These limiting fluxes are reported here. Both ends of the 250 mm long membranes were sealed by the manufacturer, leaving an effective membrane length of about 230 mm. For studying the effect of filter length, ¸, on permeate flux, the effective membrane length was changed by plugging the pores in part of the membrane with acetone-based glue. Plugging was achieved by partially inserting the membrane tube into the glue (Karlssons Klister, Sweden) further diluted with acetone. The membrane was left in the glue for about an hour followed by drying to remove the acetone. Water flux measurements confirmed that this was an effective and reproducible method of plugging the pores, rendering the part of the membrane that was inserted into the glue impermeable, but leaving the rest of the membrane unchanged. The layer of glue formed on the membrane surface was very thin compared with the tube diameter. The relation between axial pressure drop and crossflow velocity, which was earlier shown to follow Blasius’ law (Huisman et al., 1997), was not significantly altered, thus proving that the tube diameter was unaffected by the glue. Various silica particles were used for the filtration experiments. Particles with a radius of about 0.25 km were kindly supplied by Nissan Chemical Industries, Ltd, Japan (PST-5, MP-4540). These particles were either used as received (batches 1 and 2), or after narrowing the size distribution by allowing the particles to sediment for 24 h and removing the supernatant containing the smaller particles (batch 3). Other particles with various sizes (batches 4—6) were kindly supplied by Christian Kaiser, University of Mainz, Germany. Particle size distributions were determined by a laser light scattering method and by SEM. Batch between Batch between Batch between Batch between Batch between
1: Mean radius: 95 and 450 nm. 2: Mean radius: 85 and 470 nm. 3: Mean radius: 165 and 500 nm. 4: Mean radius: 205 and 370 nm. 5: Mean radius: 470 and 844 nm.
240 nm, 90% of all particles 266 nm, 90% of all particles 280 nm, 90% of all particles 260 nm, 90% of all particles 595 nm, 90% of all particles
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I.H. Huisman et al. /Chemical Engineering Science 54 (1999) 281—289
Batch 6: Mean radius: 911 nm, 90% of all particles between 749 and 1234 nm. Suspensions of these particles were prepared based on Milli-Q filtered deionised water. Reactant grade HCl or NaOH were added to adjust the pH, and NaCl was used to adjust the salinity. pH and salinity together determine the zeta potential (f) of the silica particles, as shown by Elzo et al. (1998). Experiments were performed under the following conditions: z Particle batch 1: I"1;10\ M, q "4.42 N/m, U
"0.69;10\ M, various values of f (Fig. 4). @ z Particle batch 2: f"!86 mV, I"1 ; 10\ M, various values of , various values of q (Figs. 5 @ U and 6).
z Particle batch 2: f "!26 mV, I"2 ; 10\ M,
@ "0.69;10\ M, various values of q (Fig. 5). U z Particle batch 2: f"!7 mV, I"0.1 M, various values of , various values of q (Figs. 5 and 6). @ U z Particle batch 4—6: f"!86 mV, I"1;10\ M,
"0.26 ; 10\, q "4.5 and 8.8 N/m (Fig. 7). @ U z Particle batch 3: f"!26 mV, I"2 ; 10\ M,
"0.69;10\ M, q "9 N m\, various values @ U of ¸ (Fig. 8). It can easily be shown that gravity and inertial-lift forces can be neglected for all combinations of experimental conditions considered here. ia values are at least 27, with higher values for the larger particles and for the higher salt concentrations. This warrants the use of Eq. (10), valid for ia 1. The experiments with particle batch 1 have been published by Elzo et al. (1998). The
Fig. 4. Flux versus the square of the surface potential for crossflow microfiltration of silica particle suspensions: particle radius"240 nm; salt concentration"10\ M; wall shear stress"4.42 N m\. Symbols represent experimental results. The solid line represents calculated results incorporating shear-induced diffusion, Brownian diffusion, and interaction-induced diffusion. The dashed line represents shear-induced diffusion theory and the dotted line Brownian diffusion theory.
Fig. 6. Flux versus particle volume fraction in the feed suspension for two values of zeta potential: particle radius"266 nm; wall shear stress "30 N m\. Symbols represent experimental values, lines represent the results of numerical calculations.
Fig. 5. Flux versus wall shear stress for various values of zeta potential; particle radius "266 nm; particle volume fraction"0.69;10\. Symbols represent experimental values, lines represent the results of numerical calculations.
Fig. 7. Flux versus particle radius at two values of wall shear stress; zeta potential"!86 mV; particle volume fraction"0.26;10\. Symbols represent experimental values, lines represent the results of numerical calculations.
I.H. Huisman et al. /Chemical Engineering Science 54 (1999) 281—289
Fig. 8. Flux versus membrane length; particle radius"280 nm; zeta potential"!26 mV; particle volume fraction"0.69;10\; wall shear stress"9 N m\. Symbols represent experimental values, lines represent the results of numerical calculations.
experiments using low zeta potentials (f"!7 mV) have been published by Huisman and Tra¨gas rdh (1998). 5. Results and discussion Measured permeate fluxes for various process conditions are given in Figs. 4—8 together with results of corresponding calculations. The value of wall shear stress, required for the flux calculations, was calculated from turbulent flow theory (Blasius’ law) and from measured pressure drops; both methods giving similar results. It can be seen that agreement between measured and calculated fluxes is in general good. Fig. 4 shows the crossflow microfiltration flux using particles with a radius of 240 nm, at a salt concentration of 10\ M, versus the square of the surface potential. The data point corresponding to the lowest value of surface potential (11 mV) in Fig. 4 was obtained at 0.1 M. It is clear from the graph that charge effects hardly influence the flux at surface potentials as low as 11 mV, so the actual value of the salt concentration is of very little importance. The correlation between experimental and calculated fluxes is reasonable. The trend of flux increase with increasing zeta potential is predicted qualitatively correctly. Again, the numerical calculations predict the flux to increase almost proportionally to the square of the surface potential, in agreement with the simple theory and the experimental results of Palacek and Zydney (1994). Flux predictions based on pure Brownian and pure shear-induced diffusion theory, according to Huisman and Tra¨gas rdh (1998), are also plotted in Fig. 4. The flux at vanishing surface potential is equal to the flux predicted by a combination of Brownian and shear-induced theory. Note that the data plotted in Fig. 4 were measured at very low crossflow velocity (1 m s\, q " U 4.42 N m\). Obtained fluxes are therefore very low.
287
The influence of wall shear stress on permeate flux is shown in Fig. 5 for various values of particle zeta potential. Crossflow velocities range from 1 to 5 m s\. Measured fluxes for low zeta potential and high ionic strength (f"!7 mV, I"0.1 M) are proportional to the wall shear stress, and are predicted very well by the model. The influence of interaction-induced diffusion is negligible for such low zeta potentials, and the numerical flux predictions are virtually identical to those presented by Huisman and Tra¨gas rdh (1998) for non-interacting particles. Increasing the repulsive particle—particle interactions, by increasing the zeta potential and decreasing the ionic strength, results in an increase in measured fluxes for all wall shear stresses. The calculated fluxes also increase, but not by as much as the experimental values. The influence of particle—particle interactions is thus underpredicted by the calculations. It can also be seen in Fig. 5 that the scaling of flux with the square of the surface potential, observed in Figs. 3 and 4, is lost at high wall shear stresses. Fig. 6 shows the flux versus particle volume fraction in the feed suspension. In the case of low zeta potential and high ionic strength (f"!7 mV, I"0.1 M) the measured fluxes decrease with volume fraction according to J) \ , and are predicted very well by the model. Again, @ for such low zeta potentials the numerical results are virtually identical to those presented earlier for noninteracting particles. Increasing the effect of particle—particle interactions, by increasing the zeta potential and decreasing the ionic strength, again results in an increase in measured fluxes. The calculated fluxes also increase, but not by as much as the experimental values. Experimental fluxes for the high zeta potential case scale as
\ for particle volume fractions greater than @ 0.4;10\, in agreement with model predictions. For lower particle volume fractions, fluxes deviate slightly from this trend. The influence of particle—particle interactions is again underpredicted by the calculations. Fluxes measured for suspensions of particles from batches 4—6 are shown in Fig. 7 versus the mean particle radius. Measurements were only performed for conditions with strong particle-particle repulsion (f"!86 mV, I"1;10\ M). It can be seen that the measured fluxes increase strongly with particle radius, and that measured fluxes are in very good agreement with the calculations. The agreement between model calculations and measured values is somewhat surprising, given that in Figs. 5 and 6, an underprediction of the flux was seen for such high zeta potentials. Particles used in the experiments depicted in Fig. 7 are larger than those used in the other experiments. It was pointed out earlier that the assumptions made about the interaction forces are more realistic for larger values of ia. Therefore, agreement between model calculations and experiments should increase with increasing particle size. Moreover, particles used in the experiments depicted in Fig. 7 are
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from a different source and have a narrower size distribution than the other particles, which could explain why the measured fluxes were closer to the predicted values. Fig. 8 shows the permeate flux for different values of filter length. It can clearly be seen that the measured flux decreases with increasing filter length. Moreover, these data indicate that flux scales with membrane length as J&¸\ , which is typical for diffusive particle transport. Fluxes are underpredicted again by the model calculations, probably because the effect of particle—particle interactions was underpredicted, in accordance with the results shown in Figs. 5 and 6. Overall, it can be concluded that crossflow microfiltration fluxes are predicted correctly within about 25% by the model. To improve the accuracy of the flux predictions, it appears to be necessary to describe the influence of particle—particle interactions in more detail. In the derivation of the interaction-induced diffusion coefficient the only assumptions made were that the particle size distribution was monodisperse, and that the concentration polarisation layer could be approximated by an ordered lattice of spheres. These assumptions are similar to those used by other researchers (e.g. Bowen and Jenner, 1995). The equation chosen for '( ) and the assumed packing geometry (body-centred, face-centred, etc.) influence the flux slightly, but are not expected to lead to large errors in flux prediction. More restrictive assumptions were made, however, when relating the interaction-induced diffusion coefficient to measurable quantities, such as the salt concentration and the zeta potential of the particles. It was assumed that the interaction force between two particles in the lattice could be approximated by the double-layer force between two isolated spheres, assuming constant surface charge, thus neglecting van der Waals forces and hydration forces. Moreover, the effect of the electrical double-layer on the suspension viscosity (Buscall, 1991) and on the water viscosity (electroviscous effect, Levine et al., 1975) was neglected. It can therefore be expected that some discrepancies exist between experimental fluxes and model predictions. The main conclusion that can be drawn from the present study is, that the numerical predictions show the same trends as the experimental results, and that the fluxes obtained are of the correct order of magnitude, thus supporting the approach employed. In future research, some of the effects neglected in the present model may be included, hopefully leading to even more accurate flux predictions.
dicted correctly within an error margin of $25%. If double-layer forces are negligible (low zeta potential, high ionic strength), the accuracy is even higher ($10%). The fact that accurate flux predictions were obtained especially for low zeta potentials (i.e. for small doublelayer forces) suggests that van der Waals forces and hydration forces have insignificant effects on the permeate flux. This justifies the a priori assumption that these forces are negligible. Trends observed for the measured fluxes are in agreement with the scaling predicted by the calculations. Thus, the experimental fluxes increase with particle size, increase with particle zeta potential, increase almost linearly with the wall shear stress, decrease with particle volume fraction according to J& \ , and decrease @ with membrane length as J&¸\ . The extended model gave good flux predictions both for situations in which Brownian diffusion and charge effects are dominant and for situations in which shear-induced diffusion and charge effects are dominant. From these observations, it can be concluded that the concept of interaction-induced diffusion is very useful for predicting fluxes in crossflow microfiltration. However, the quantitative effect of interaction forces on the flux is, in general, slightly underpredicted by the model. This is probably due to the assumptions made when relating the interaction-induced diffusion coefficient to measurable quantities. Especially the equation used for the particle—particle interaction force, Eq. (10), seems to be an oversimplification.
Acknowledgements The authors would like to thank Eert Vellenga and Iris Balestri for performing some of the experiments. We are grateful to Christian Kaiser, University of Mainz, Germany, and Nissan Chemicals, Ltd, Japan for providing the silica particles. This work was financially supported by the Swedish Foundation for Membrane Technology.
Notation a D D D
6. Conclusions
D
Calculating microfiltration fluxes by combining shear-induced diffusion, Brownian diffusion, and particle—particle interactions results in flux predictions that compare well with experimental values. Fluxes are pre-
D ' F F F G
M>1'
particle radius, m diffusion coefficient, m s\ concentration-dependent part of the diffusion coefficient, dimensionless concentration-independent part of the diffusion coefficient, m s\ sum of Brownian and shear-induced diffusion coefficients, m s\ interaction-induced diffusion coefficient, m s\ hydrodynamic drag force, N particle—particle interaction force, N
I.H. Huisman et al. /Chemical Engineering Science 54 (1999) 281—289
h I J k ¸ m Re ¹ v v N v U y
interparticle distance, m ionic strength, mol dm\ length-averaged permeate flux, m s\ Boltzmann’s constant membrane length, m number of nearest neighbours in one plane Reynolds number, dimensionless temperature, K liquid velocity in transverse direction, m s\ convective particle velocity in transverse direction, m s\ dimensionless flux, dimensionless transverse coordinate, m
Greek letters e permittivity of free space, C J\ m\ e relative dielectric constant, dimensionless P f zeta potential, V h angle between interparticle force and hydrodynamic force, rad i Debye constant, m\ k suspension viscosity, Pa s k dimensionless viscosity, dimensionless k viscosity of the permeate, Pa s q wall shear stress, N m\ U
particle volume fraction, dimensionless
particle volume fraction in the bulk, dimen@ sionless
maximum particle volume fraction (" close
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