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Particle aggregation at the membrane surface in crossflow microfiltration
171
Journalof Membmne Science, 84 (1993) 171-183 Elsevier Science Publishers B.V., Amsterdam
Particle aggregation at the membrane surface in crossflow microfiltration P. Schmitz*, B. Wandelt, D. Houi and M. Hildenbrand Znstitut de Mkanique des E&ides, URA DO005, Auenuc Camille SOL& 31400 Toulouse (France) (Received November 11,1992; accepted in revised form May 27,1993 )
Abstract We present a theoretical model which describes particle accumulation at a membrane surface during a crossflow microfiltration operation. We define empirical but realistic mechanisms for the motion and adhesion of suspended particles. These mechanisms take into account the complex balance of forces to which all particles are subjected in the proximity of the membrane. The model proposed here allows us to generate particle aggregates by a statistical method. We first characterize the influence of the model parameters on the morphology of the aggregates. Then we assign the appropriate values to simulate aggregates analogous to those occurring in crossflow microfltration. Furthermore we perform a comparison with experimental measurements. Key words: microfiltration; crossflow microfiltration; particle-aggregation;
Introduction Pressure driven membrane processes have become very promising techniques in many industrial fields. However several questions remain concerning their optimal working conditions. In particular, we need to better understand the complex problem of membrane fouling which still remains the limiting phenomenon in many applications using crossflow microfiltration. Many investigations have been carried out on this subject. They generally show that particle deposition at the membrane is not only the first step but also one of the major reasons for filtrate flux decline with time and subsequently for membrane fouling [ 11. The membrane user observes the phenom%
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statistical model
ena resulting from fouling at a macroscopic scale, i.e. filtrate flux decline and reduction of the porous channel gap. However, the mechanisms responsible occur at a microscopic scale involving, for instance, particle attractive or repulsive forces. So first we have to characterize the aggregation mechanisms of particles at the membrane surface on a local scale, in order to determine the impact of these mechanisms on the structure of the aggregates inducing negative consequences on a larger scale. F’urthermore we need to analyze the structure variations for a wide range of hydrodynamic conditions and suspension types. At a further stage it will be possible to predict and control the industrial operation of crossflow microfiltration processes. We have previously discussed the lack of local approaches concerning the behaviour of a
0 1993 Elsevier Science Publishers B.V. All rights reserved.
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suspended particle moving in the proximity of a membrane surface [ 21. In this earlier paper we determined and analyzed the hydrodynamic field near a few pores of the membrane surface under various filtration conditions. This study at pore level confirmed the existence of two flow structures which determine the possibility of particle deposition. Finally we proposed a simplified model of particle trajectories in this nearfield region, showing particle deposition on the membrane surface. The model took into account only hydrodynamic and Van der Waals forces. The phenomena active near the membrane include not only particle-fluid and particlemembrane but also particle-particle interactions. On the one hand, some studies have been undertaken to determine the hydrodynamic force on a particle close to a wall under creeping flow. Sherwood [ 31 and Payatakes and Dassios [4] were interested in the case of frontal flow. Spielman and Goren [ 51, Hubbe [ 61 and Lea1 [ 71 considered shear flow without suction. These works were restricted to the analysis of particular mechanisms in specific situations. The derived hydrodynamic force was generally proportional to the classical Stokes force. On the other hand, other studies have described and modelled surface forces due to physico-chemical effects related to particle-obstacle-solution interactions [ 5,8]. These works show the difficulty of modelling effects other than Van der Waals forces. We note the absence of a particle trajectory calculation taking into account all forces in the more complex case of shear flow in the proximity of a porous wall with suction. To avoid the complexity of surface forces, Kleinstreur and Chin [9] proposed a particle trajectory model in a porous tube, in which particles follow the streamlines of fluid-flow. The approach was restricted to particle behavior in the far-field region, since a porous wall with suction was taken as a boundary condition. The authors assumed that surface forces were so
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strong that deposition occurred if a particle trajectory intersected the wall or a particle within the cake surface. They employed the model to investigate the influence of particle layer growth on permeation flux decline. Hung and Tien [ 11 presented a similar analysis for the case of reverse osmosis. They considered surface forces, and also assumed that particle deposition was uniform over the membrane in determining the variation of the thickness of the deposited layer. These studies neglect particle-particle interactions at the cake surface level, which can induce the formation of aggregates having very different morphologies. If the aim is to understand cake formation, which is responsible for the fouling phenomenon, we must consider not only the complete balance of forces to which suspended particles are submitted near the porous wall, but also the possible motion of particles after they made contact with the wall or cake surface. Such a deterministic problem requires knowledge of the time-dependent boundary conditions on a variable domain bounded by complex surfaces. Studies have been undertaken by Tousi et al. [lo], Brochard [ 111 and Durlofsky et al. [ 121, on the behaviour of a few particles interfering in a flow, but a deterministic model for the growth of an aggregate of a great number of particles still does not exist. In this paper we propose a statistical model, able to simulate the aggregation of a thousand particles on a porous membrane, during crossflow microfiltration. Such a method has already been used by Houi and Lenormand [ 131 in frontal microfiltration. They obtained aggregates in good agreement with experimental visualisations and characterized their morphology. Witten and Sander [14] had previously employed this method to study the aggregation of particles in Brownian motion. They proposed that the aggregation process could be understood without reference to the details of forces holding them together, since they had
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neglected long-range electrostatic and magnetic forces. These statistical approaches are in keeping with the recent scientific studies which use fractal concepts to simulate and characterize irreversible growth phenomena. Here we define empirical but realistic mechanisms for the motion and adhesion of each suspended particle. These mechanisms take into account the complex balance of forces to which all particles are submitted in the proximity of the membrane, before and after contact with an obstacle, that is either the membrane or another particle. The model simulates and allows the visualization of the deposition of micrometer-sized particles on the membrane. We first describe the two-dimensional model, i.e. the membrane, the particles, and the rules governing their motion and adhesion. Then the model parameters are related to the geometric, hydrodynamic and physico-chemical conditions of the crossflow microfiltration problem. After that we characterize the influence of the main model parameters on the morphology of the aggregates. Finally we determine the appropriate values to simulate aggregates analogous to those visualized experimentally in crossflow microfiltration. Model The theoretical analysis is aimed at simulating the aggregation of a great number of particles at the membrane surface, i.e. the formation of a deposition layer during crossflow microfiltration. The domain is two dimensional. The model membrane is a flat plate with rectangular pores, i.e. slots, uniformly distributed depending on the surface porosity, E, of a given membrane. We assume that suspended particles are circular, non-deformable, non-brownian, monodispersed and micrometer-sized. We define particle size by the diameter ratio, d, equal to the pore size over the particle diameter. Of course, particles are not always cap-
tured, since they can pass through the membrane if their diameter is smaller than the pore diameter (d 2 1).We also assume that their concentration in the fluid is low, in order to justify that they enter one by one into the nearfield region of the filtration membrane. We perform the simulation of the deposit formation mechanisms by releasing each particle individually from a randomly selected initial location upstream of the porous wall, and following a linear trajectory towards the membrane (Fig. 1). Specific motion and adhesion rules are proposed to define particle behaviour, from its initial starting point to its final aggregation. These empirical rules are used to model what happens when the suspended particle makes contact with the membrane or with another particle already belonging to the aggregate. We believe that this method describes the physical reality when numerous phenomena such as hydrodynamics, Van der Waals and electrostatic forces, and concentration, density and shape effects, are involved. Adhesion rules We assume that a suspended particle with center A, launched in the near-field region, is immediately captured and stopped when it makes contact with the membrane. However when it touches a particle with center B, belonging to the growing aggregate, the particle in motion is only captured provided that the centerline (AB) is inside the circular sticking
Fig. 1. Principle of the aggregation model.
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Incident Trajkctory
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Veriical
Fig. 2. Adhesion rules when a suspended particle A makes contact with a particle B belonging to the growing aggregate.
how many times the particle is allowed to recover its original incidence angle (Y, instead of assuming a vertical trajectory after contact and rotation around a previously aggregated particle. With this model we can simulate aggregates of particles of various materials in very different physical domains. The empirical parameters introduced must be relevant to the context of crossflow microfiltration. Physical aspects
Fig. 3. Motion rules of particle A depending on the value of N.
sector defined by the sticking angle 8. We thus define two parameters 0, and 0, which characterize adhesion in the incident and vertical directions, respectively, as shown in Fig. 2.
Motion rules Each particle of center A, launched from its random initial location, follows a linear incident trajectory controlled by the incidence angle (Y. After making contact with a particle of center B already aggregated, and provided it is outside the &zone, the incident particle can turn right or left around the second particle and attain a vertical or a new incident trajectory (see Fig. 3 ) . The type of release depends on another parameter, N,, called the number of reentrainments. This integer number determines
We defined above the motion and adhesion rules, introducing four parameters to be chosen for the simulation of aggregate formation on a membrane. The choice of these parameters is based on experimental observations of aggregate formation. On the one hand, Houi and Lenormand [ 131 visualized particle behaviour in the case of frontal filtration of glass beads and rilsan particles, analogous to that which we have assumed to be valid in crossflow microfiltration. Furthermore, they demonstrated successful modelling with empirical rules utilizing a capture angle. On the other hand, Ritter and Houi [ 151 identified the effect of shear flow on the growth of the aggregate in the case of crossflow microfiltration of well mud using a flat ceramic membrane. In the dynamic simulation presented herein, the incidence angle (Y is a measure of the ratio of the radial suction flow to the tangential shear flow, in the vicinity of the filtering surface, i.e. the membrane. The number of reentrainments, N,, is used to quantify the influence of the tangential shear flow on the suspended particle motion at the interface, after the first and subsequent contacts of the particle at and within the growing aggregate. We have defined such an empirical parameter from experimental observations [151 showing the motion of particles at the surface of the deposit under tangential shear flow. We saw clearly that a particle
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had a succession of contacts and reentrainments before reaching its final location in a free space of the deposit capture interface. This ability to be removed after contact seemed to be directly related to the shear intensity. Thus, the hydrodynamic conditions are taken into account in the modelling by the aforementioned two parameters cy and N,. Two other parameters determine the particle capture mechanics. We assume that the balance of forces acting on a particle, when it makes contact, determines whether the particle sticks. An approach similar to that in Ref. [13] is adopted here. The magnitudes of the two sticking angles, 0, and t9,,measure the importance of the attractive and repulsive physice-chemical forces respectively, relative to the hydrodynamic force. More precisely, a low value of the sticking angle characterizes a case where the flow entrainment effect is the most prevalent among particle-particle interactions. In the other extreme, a high value of the sticking angle assigns the dominant effect to attractive forces. The incident sticking angle, 0i, is applicable when contact occurs in the upper part of the aggregate, i.e. where the shear flow effect is active, while the vertical sticking angle, & is used deeper within the aggregate, where the particles are sensitive to vertical suction flow. It is necessary to study the influence of the aforementioned parameters on the calculated morphology of the aggregates. The aim is to identify the best set of parameters for the description of aggregates encountered in crossflow microfiltration. We must also verify that the same type of structure cannot also be obtained with a different set of parameters, corresponding to a different physical situation. Results Study of aggregate morphology The morphology of the simulated aggregates
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has been investigated over wide ranges of the most important model parameters, i.e. the incident sticking angle and the number of reentrainments. Indeed these parameters depict not only the effect of the complex force balance acting on a particle when captured but also the action of the tangential shear flow on particle mobility. We treat the aggregates by analysis of the variations of three statistical parameters. The first one is the vertical concentration, C ( Y/ I&>, expressed as the relative number of particle centers located inside parallel slices which discretixe the deposit above the membrane. The variable Y is the vertical coordinate while DP means the particle size. Secondly, we compute the mean porosity of the deposits, c& Finally, we take the root mean square of the particle gap to express the mean thickness of the deposit as a function of the number of particles. It is defined as follows: ri i=AL. +/2
where Yi is the distance between the center of particle i and the membrane surface. In all the simulations, 5000 particles are injected. The model membrane has a porosity of 15%. The incident angle equals 20 O. No significant differences were obtained in varying 0,. This result implies that it is the mechanic blocking instead of particle-particle forces which controls particle capture within the aggregate. The vertical sticking angle was therefore held constant at lo since it is unnecessary. Influence of i3i Recall that this parameter represents the magnitude of the particle-particle adhesion force compared with the hydrodynamic release force. We vary (3,from 5 ’ to 60 O.We present in Figs. 4(a) and 4 (b) two aggregates obtained with 0, is 10 and 30”, respectively. The small
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II
Fig. 4. (a) Aggregates of 5000 particles a=20”, &=l”, N,=O.
for &=lOO, a=20”,
angles lead to very compact structures. However, the mean porosity increases quickly when the sticking angle increases (as shown in Fig. 5). Furthermore, the curves in Fig. 6 characterize the significant mean thickness difference in
II
S,=l”,
II
N,=O.
II
II
II
II
II
(b) Aggregates of 5000 particles
II
for 8,=30”,
the different cases. We also notice the existence of de&its directed in the incidence direction of the suspension in this case. This formation is caused by the mechanism of contactcapture expected with such assumptions. The vertical concentration diagram shown in Fig. 7
degrees 70
4
20 / 0
30
50
90
120
150
30
180
Fig. 5. Variations of the mean porosity cd with 0, for LY= 20 ‘, &=l”, N,=O.
Fig. 6. Evolution of the mean thickness Td as a function of the number of particles N, for different values of the incident sticking angle Bi and for LY= 20 ‘, 0, = 1 O,N, = 0.
P. Schmitz et al. / J. Membrane Sci. 84 (1993) 171-183 degrees 1
c (%) Fig. 7. Vertical variations of the concentration C with 0, for cY=20”, e,=io, N,=O.
provides an adequate classification of the solid particle deposit simulations obtained. Let us now analyze more precisely the shape of each concentration profile. We determine three specific regions in which the aggregate morphology is different. The closest region to the membrane, corresponding to a rapid increase of concentration, can be interpreted as the germination region, i.e. the region of contact between the aggregate and the membrane. It represents the aggregate foundation. The region in which the particle concentration decreases slowly is considered as the second region. This is the depth of the aggregate because only a small surface is available to capture particles in this region of high concentration. The third region is named the capture region because it is the place where the particles are intercepted by the deposit surface. Its thickness represents the surface rugosity of the aggregate. Influence of N, This parameter determines the sensitivity of a suspended particle to the entrainment effect due to crossflow when it makes contact with a previously aggregated particle. We can seen in
177
Figs. 8 (a) and 8 (b) the simulations of two different deposits corresponding to two different values of the number of reentrainments, from a branching out structure (N, equals zero) to a regular structure (N, high). We find that high N, gives optimal arrangement of the particles resulting in more complete space filling. The concentration profiles (Fig. 9) indicate the same three regions of deposit structure as depicted previously. Furthermore the increase of deposit uniformity corresponding to the increase of N, is not only quantified by the concentration variations but also by the mean thickness variations for different N, values (see Fig. 10). In this figure we notice the superposition of mean thickness profiles for N, greater or equal to 3 certainly due to the final capture of most particles before the third reentrainment. This is also confirmed by the asymptotic limit of the mean porosity curve plotted in Fig.
Homogeneity Some authors [16,17] have shown that the fractal dimension of an aggregate can be deducted from the following relation:
T,=BN;
(2)
in which the power 3,is expressed as a function of the fractal dimension Df and the euclidian dimension D,, as follows: A=
l-D,
l +I&
(3)
Of course these statistical parameters have only a physical meaning for a great number NP of aggregated particles. So the heterogeneity degree of the aggregate can be evaluated with the arithmetic difference D,-& A zero value ensures perfect homogeneity of the aggregate instead of a non zero value giving an heterogeneous nature. The variations of Td, plotted in Figs. 6 and
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P. Schmitz et al. /J. Membrane Sci. 84 (1993) 171-183
Fig. 8. (a) Aggregates of 5000 particles for N. = 1, (Y= 20 ‘, tl,= lo, 0,= 1 O.(b ) Aggregates of 5000 particles for N. = 3, CY= 20 O, 0,=1°, 19,=1°. ::
Y
Nr 0
Nr
0
P
0” \ P P-
lo
100
0.00
0.02
0.04 c
0.06
I
I
I
2000
3000
I
4000
5
10
NP
(%I
Fig. 9. Vertical variations of the concentration C with N, for a=20”, 0,=1’, ei=lO.
I
1000
0
Fig. 10. Evolution of the mean thickness Td as a function of the number of particles N, for different values of the number of reentrainments N, and for a = 20”, 0, = lo,
e,=io.
10 show the linear dependance of Td on N,, i.e. J equals 1 and D, -Dr equals 0. This result is true for all the simulated aggregates with this model. So we confirm that the aggregates made from particles which follow motion and adhesion rules defined here are statistically homogeneous. As a major consequence it is valid to describe mass transfer through these aggregates by Darcy’s law.
Comparisons with experimental
measurements
Numerical simulations of deposits are carried out for a great number of particles entering individually the filtering region specified by the model. It means physically that such aggregates can be compared to the real deposits we can observe after a long crossflow microfiltra-
P. Schmitz et al. /J. Membrane Sci. 84 (1993) 171-183
.-ri;i 28
+I
27 28
:
;.-
::I‘--0
1
2
3
4
6
N, Fig. 11. Variations of the mean porosity Ed with N. for CY=20°, e,=1o, &=lO.
tion period. We have shown previously [ 181 that such simulations were qualitatively in agreement with experimental visualizations from video recording of an optical microscope image of well mud filtration on a flat mineral membrane. Here our concern is to use our theoretical model to justify the existence of specific deposit structures obtained experimentally [ 19,201. Indeed- Wandelt et al. have developed an experimental method to study crossflow microfiltration of be&mite suspensions in hollow fibers. They have used NMR imaging allowing for in situ observations of deposit growth without any perturbations of the dynamic process. The measurements of deposit thickness are combined with the estimation of deposited mass from a turbidimetry device. These two complementary measurements enable calculation of cake porosities. We can see in Figs. 12(a), 12(b) and 12(c) cross sections of a hollow fiber module related to two types of hydrodynamic conditions, i.e. medium and low tangential shear flows. The structure of the particle aggregate at the membrane surface looks compact in the case of a Reynolds num-
Fig. 12. NMR image of cross-section of a module of 3 hollow fibers with a spatial resolution of 20 micrometer: (a) at t=O, (b) at t=17 min with Re=220, (c) at t=94 min with Re = 60.
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Fig. 13. Aggregates of 5000 particles for N. = 5, CY = 20 ‘, 0, = 1O,6,= 1’.
Fig. 14. Aggregates of 5000 particles for N,= 0, (Y= 20”) f3,= lo, 8, = 90 ‘.
ber Re equal to 220. The effect of tangential shear flow seems to be significant on interface erosion, reducing vertical increase of aggregates before complete space filling, i.e. dendrites development. This result is confirmed by porosity variations as a function of time. Wandelt et al. give low final porosities of about 40% and low cake thickness for this case. On the contrary, the deposit appears highly porous on the NMR image of Fig. 12(c) at a lower Reynolds number (60). The corresponding measurement indicates here a thick cake and a mean porosity of the deposit equal to 80% at the end of the operation. We should also mention that the local cake porosity was observed to be inhomogeneous. Of course these are only estimations of deposit thickness and porosities for several reasons. Firstly the NMR image consists of an integration over a cross slice of 3 mm thickness. Secondly the resolution, i.e. the pixel size is equal to 20 pm giving in fact the representation of more than a particle. But never-
theless these experimental results show the influence of the hydrodynamic regime on deposit morphology. That is why it is relevant to consider two parameters such as the incident sticking angle 0, and the number of reentrainments N, in our theoretical model to quantify on the one hand the complex balance of forces including the release force and on the other hand the competition between filtration flow and entrainment flow. We present two different types of deposits of 5000 solid particles performed with our simulation model. These simulations use two different sets of parameters adapted to the physical situations of interest cited above. In each case the suspension is assumed to flow from the left to the right. The first case in Fig. 13 models a fouling layer due to a suspension flow where particle-particle and particle-wall attractive forces are much greater than the hydrodynamic forces carrying the particles away. Here a suspended particle
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making contact with the filter surface of with a particle of the aggregate is immediately captured. Thus N, equals 0 and the given value for 0, is 90”. The deposit structure is dendritic, the porosity has a mean value of 61%, and the deposit is locally quite heterogeneous (see figure) but stays statistically homogeneous as already mentioned previously. This kind of aggregate is analogous to the experimental case of Fig. 12 (c) giving also a high porous deposit in the hollow fibers. The second case shown in Fig. 14 presents a very compact accumulation of particles which has been observed experimentally (figure) when the Reynolds number was relatively high. So we can understand that the effect or parallel shear flow is efficient even inside the deposit. It facilitates the optimal arrangement of particle deposition giving a very low porosity. To take into account the importance of tangential flow, N, is chosen equal to 4 while 0, has a very small value. The mean porosity is found to be 24% resulting in a mean thickness lower than the one obtained in the first case for the same number of particles. These two simulations allow us to give a tentative explanation of the very different deposit structures observed experimentally by Wandelt et al. Furthermore, despite the fact that the parameters of the model are not physical a priori, we believe that they result in realistic behaviour of particles during deposit formation at the membrane surface. Discussion The model presented in this paper aims at a simplified description of particle aggregation at the membrane surface in crossflow microfiltration. This description should be viewed as rather qualitative since the assumptions adopted here are to a certain extend restrictive: l perfectly flat membrane with uniformly distributed pores,
monodispersed, circular, non-deformable particles, 0 suspension at low concentration, l constant hydrodynamic conditions. Nevertheless, the model is able to show the influence of experimental conditions such as hydrodynamics and physico-chemistry on the morphology of the deposits, which yield significant effects on the evolution of the transmembrane flux in experiments. To exemplify, in the comparison presented above, our model allowed us to explain why specific aggregates having drastically different porosities could be various identified under experimental conditions. We believe that this phenomenological model is applicable to the crossflow microfiltration of a suspension at low concentration of rigid, nonbrownian particles, which are supposed to be of the same nature. In this case we can neglect particle-particle interactions in the flow. Ultrapure water is considered as the perfect fluid to ensure particle transport. Admittedly the model duplicates laboratory operating conditions rather than complex industrial conditions. It is common knowledge that any existing model is able to predict the behavior of a crossflow microfiltration process under certain conditions. Nevertheless, we have tried to give some indications to experimentalists that will guide them in selecting conditions for membrane microfiltration processes. Thus we have determined the corresponding values of the two key parameters, which are the incident sticking angle, 0i, and the number of reentrainments N,, for different hydrodynamic and physico-chemical conditions. D. Houi [21] studied numerically the variations of 0, as a function of the intensity of the resulting adhesion force, F,, and the length of the contact zone, r,, between the collector particle and the moving particle. The hydrodynamic force, Fh, which acted on the particle was obtained from the complete calculation of the velocity fields in the geometrical l
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182
domain surrounding the two particles in contact. The particle was supposed to be captured whenever the resulting force, R, from the total balance cut the contact zone as shown in Fig. 15. Values of 0, are reported in Table 1. Concerning particle reentrainment, we recognize that it is impossible to give a relationship between IV, and fundamental laws. The values proposed in Table 2 are the estimations of the authors from experimental results [ 15,19,20]. Additional experiments will be necessary to further validate the present model especially to improve quantitatively the model parameters. In the future, a complete three dimensional model will be uonsidered to investigate the permeability variations under a host of experimental conditions.
Conclusion We have presented a theoretical approach to the prediction of deposition and aggregate formation of a large number of particles at a membrane surface under crossflow microfiltration conditions. Real particle behaviour is translated into specific empirical mechanisms of motion and adhesion. The analysis of the influence of model parameters on the morphology of the aggregates has shown the ability to generate aggregates of different structures corresponding to very different physical situations. Finally the model has been successfully applied in the description of deposit formation in two experimental cases related to medium or low magnitude of tangential entrainment flow over the membrane. In a further study we will try to validate by microscope observations the motion and adhesion mechanisms adopted in our model. Acknowledgements
TABLE 1
We thank Doctor Pradere from C.H.U. Toulouse Rangueil for his constant help and advice in obtaining NMR images and G.D.R. ‘Filtration et Fibres Creuses’ for financial support.
Variations of 0, in degrees as a function of adhesion force and contact length for Re = 1
References
Fig. 15. Balance of forces on a particle in contact with a collector [ 211.
2 r,lD,
F, (N)
lo-‘0 10-a 10-E
0.5
0.25
0.05
3 25 180
1 14 180
1 6 27
TABLE 2 Variations of N, as a function of Reynolds number Re N,
<50 0
100 1 to 2
150 2to3
> 200 5
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Journalof Membmne Science, 84 (1993) 171-183 Elsevier Science Publishers B.V., Amsterdam
Particle aggregation at the membrane surface in crossflow microfiltration P. Schmitz*, B. Wandelt, D. Houi and M. Hildenbrand Znstitut de Mkanique des E&ides, URA DO005, Auenuc Camille SOL& 31400 Toulouse (France) (Received November 11,1992; accepted in revised form May 27,1993 )
Abstract We present a theoretical model which describes particle accumulation at a membrane surface during a crossflow microfiltration operation. We define empirical but realistic mechanisms for the motion and adhesion of suspended particles. These mechanisms take into account the complex balance of forces to which all particles are subjected in the proximity of the membrane. The model proposed here allows us to generate particle aggregates by a statistical method. We first characterize the influence of the model parameters on the morphology of the aggregates. Then we assign the appropriate values to simulate aggregates analogous to those occurring in crossflow microfltration. Furthermore we perform a comparison with experimental measurements. Key words: microfiltration; crossflow microfiltration; particle-aggregation;
Introduction Pressure driven membrane processes have become very promising techniques in many industrial fields. However several questions remain concerning their optimal working conditions. In particular, we need to better understand the complex problem of membrane fouling which still remains the limiting phenomenon in many applications using crossflow microfiltration. Many investigations have been carried out on this subject. They generally show that particle deposition at the membrane is not only the first step but also one of the major reasons for filtrate flux decline with time and subsequently for membrane fouling [ 11. The membrane user observes the phenom%
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statistical model
ena resulting from fouling at a macroscopic scale, i.e. filtrate flux decline and reduction of the porous channel gap. However, the mechanisms responsible occur at a microscopic scale involving, for instance, particle attractive or repulsive forces. So first we have to characterize the aggregation mechanisms of particles at the membrane surface on a local scale, in order to determine the impact of these mechanisms on the structure of the aggregates inducing negative consequences on a larger scale. F’urthermore we need to analyze the structure variations for a wide range of hydrodynamic conditions and suspension types. At a further stage it will be possible to predict and control the industrial operation of crossflow microfiltration processes. We have previously discussed the lack of local approaches concerning the behaviour of a
0 1993 Elsevier Science Publishers B.V. All rights reserved.
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suspended particle moving in the proximity of a membrane surface [ 21. In this earlier paper we determined and analyzed the hydrodynamic field near a few pores of the membrane surface under various filtration conditions. This study at pore level confirmed the existence of two flow structures which determine the possibility of particle deposition. Finally we proposed a simplified model of particle trajectories in this nearfield region, showing particle deposition on the membrane surface. The model took into account only hydrodynamic and Van der Waals forces. The phenomena active near the membrane include not only particle-fluid and particlemembrane but also particle-particle interactions. On the one hand, some studies have been undertaken to determine the hydrodynamic force on a particle close to a wall under creeping flow. Sherwood [ 31 and Payatakes and Dassios [4] were interested in the case of frontal flow. Spielman and Goren [ 51, Hubbe [ 61 and Lea1 [ 71 considered shear flow without suction. These works were restricted to the analysis of particular mechanisms in specific situations. The derived hydrodynamic force was generally proportional to the classical Stokes force. On the other hand, other studies have described and modelled surface forces due to physico-chemical effects related to particle-obstacle-solution interactions [ 5,8]. These works show the difficulty of modelling effects other than Van der Waals forces. We note the absence of a particle trajectory calculation taking into account all forces in the more complex case of shear flow in the proximity of a porous wall with suction. To avoid the complexity of surface forces, Kleinstreur and Chin [9] proposed a particle trajectory model in a porous tube, in which particles follow the streamlines of fluid-flow. The approach was restricted to particle behavior in the far-field region, since a porous wall with suction was taken as a boundary condition. The authors assumed that surface forces were so
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strong that deposition occurred if a particle trajectory intersected the wall or a particle within the cake surface. They employed the model to investigate the influence of particle layer growth on permeation flux decline. Hung and Tien [ 11 presented a similar analysis for the case of reverse osmosis. They considered surface forces, and also assumed that particle deposition was uniform over the membrane in determining the variation of the thickness of the deposited layer. These studies neglect particle-particle interactions at the cake surface level, which can induce the formation of aggregates having very different morphologies. If the aim is to understand cake formation, which is responsible for the fouling phenomenon, we must consider not only the complete balance of forces to which suspended particles are submitted near the porous wall, but also the possible motion of particles after they made contact with the wall or cake surface. Such a deterministic problem requires knowledge of the time-dependent boundary conditions on a variable domain bounded by complex surfaces. Studies have been undertaken by Tousi et al. [lo], Brochard [ 111 and Durlofsky et al. [ 121, on the behaviour of a few particles interfering in a flow, but a deterministic model for the growth of an aggregate of a great number of particles still does not exist. In this paper we propose a statistical model, able to simulate the aggregation of a thousand particles on a porous membrane, during crossflow microfiltration. Such a method has already been used by Houi and Lenormand [ 131 in frontal microfiltration. They obtained aggregates in good agreement with experimental visualisations and characterized their morphology. Witten and Sander [14] had previously employed this method to study the aggregation of particles in Brownian motion. They proposed that the aggregation process could be understood without reference to the details of forces holding them together, since they had
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neglected long-range electrostatic and magnetic forces. These statistical approaches are in keeping with the recent scientific studies which use fractal concepts to simulate and characterize irreversible growth phenomena. Here we define empirical but realistic mechanisms for the motion and adhesion of each suspended particle. These mechanisms take into account the complex balance of forces to which all particles are submitted in the proximity of the membrane, before and after contact with an obstacle, that is either the membrane or another particle. The model simulates and allows the visualization of the deposition of micrometer-sized particles on the membrane. We first describe the two-dimensional model, i.e. the membrane, the particles, and the rules governing their motion and adhesion. Then the model parameters are related to the geometric, hydrodynamic and physico-chemical conditions of the crossflow microfiltration problem. After that we characterize the influence of the main model parameters on the morphology of the aggregates. Finally we determine the appropriate values to simulate aggregates analogous to those visualized experimentally in crossflow microfiltration. Model The theoretical analysis is aimed at simulating the aggregation of a great number of particles at the membrane surface, i.e. the formation of a deposition layer during crossflow microfiltration. The domain is two dimensional. The model membrane is a flat plate with rectangular pores, i.e. slots, uniformly distributed depending on the surface porosity, E, of a given membrane. We assume that suspended particles are circular, non-deformable, non-brownian, monodispersed and micrometer-sized. We define particle size by the diameter ratio, d, equal to the pore size over the particle diameter. Of course, particles are not always cap-
tured, since they can pass through the membrane if their diameter is smaller than the pore diameter (d 2 1).We also assume that their concentration in the fluid is low, in order to justify that they enter one by one into the nearfield region of the filtration membrane. We perform the simulation of the deposit formation mechanisms by releasing each particle individually from a randomly selected initial location upstream of the porous wall, and following a linear trajectory towards the membrane (Fig. 1). Specific motion and adhesion rules are proposed to define particle behaviour, from its initial starting point to its final aggregation. These empirical rules are used to model what happens when the suspended particle makes contact with the membrane or with another particle already belonging to the aggregate. We believe that this method describes the physical reality when numerous phenomena such as hydrodynamics, Van der Waals and electrostatic forces, and concentration, density and shape effects, are involved. Adhesion rules We assume that a suspended particle with center A, launched in the near-field region, is immediately captured and stopped when it makes contact with the membrane. However when it touches a particle with center B, belonging to the growing aggregate, the particle in motion is only captured provided that the centerline (AB) is inside the circular sticking
Fig. 1. Principle of the aggregation model.
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Incident Trajkctory
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Veriical
Fig. 2. Adhesion rules when a suspended particle A makes contact with a particle B belonging to the growing aggregate.
how many times the particle is allowed to recover its original incidence angle (Y, instead of assuming a vertical trajectory after contact and rotation around a previously aggregated particle. With this model we can simulate aggregates of particles of various materials in very different physical domains. The empirical parameters introduced must be relevant to the context of crossflow microfiltration. Physical aspects
Fig. 3. Motion rules of particle A depending on the value of N.
sector defined by the sticking angle 8. We thus define two parameters 0, and 0, which characterize adhesion in the incident and vertical directions, respectively, as shown in Fig. 2.
Motion rules Each particle of center A, launched from its random initial location, follows a linear incident trajectory controlled by the incidence angle (Y. After making contact with a particle of center B already aggregated, and provided it is outside the &zone, the incident particle can turn right or left around the second particle and attain a vertical or a new incident trajectory (see Fig. 3 ) . The type of release depends on another parameter, N,, called the number of reentrainments. This integer number determines
We defined above the motion and adhesion rules, introducing four parameters to be chosen for the simulation of aggregate formation on a membrane. The choice of these parameters is based on experimental observations of aggregate formation. On the one hand, Houi and Lenormand [ 131 visualized particle behaviour in the case of frontal filtration of glass beads and rilsan particles, analogous to that which we have assumed to be valid in crossflow microfiltration. Furthermore, they demonstrated successful modelling with empirical rules utilizing a capture angle. On the other hand, Ritter and Houi [ 151 identified the effect of shear flow on the growth of the aggregate in the case of crossflow microfiltration of well mud using a flat ceramic membrane. In the dynamic simulation presented herein, the incidence angle (Y is a measure of the ratio of the radial suction flow to the tangential shear flow, in the vicinity of the filtering surface, i.e. the membrane. The number of reentrainments, N,, is used to quantify the influence of the tangential shear flow on the suspended particle motion at the interface, after the first and subsequent contacts of the particle at and within the growing aggregate. We have defined such an empirical parameter from experimental observations [151 showing the motion of particles at the surface of the deposit under tangential shear flow. We saw clearly that a particle
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had a succession of contacts and reentrainments before reaching its final location in a free space of the deposit capture interface. This ability to be removed after contact seemed to be directly related to the shear intensity. Thus, the hydrodynamic conditions are taken into account in the modelling by the aforementioned two parameters cy and N,. Two other parameters determine the particle capture mechanics. We assume that the balance of forces acting on a particle, when it makes contact, determines whether the particle sticks. An approach similar to that in Ref. [13] is adopted here. The magnitudes of the two sticking angles, 0, and t9,,measure the importance of the attractive and repulsive physice-chemical forces respectively, relative to the hydrodynamic force. More precisely, a low value of the sticking angle characterizes a case where the flow entrainment effect is the most prevalent among particle-particle interactions. In the other extreme, a high value of the sticking angle assigns the dominant effect to attractive forces. The incident sticking angle, 0i, is applicable when contact occurs in the upper part of the aggregate, i.e. where the shear flow effect is active, while the vertical sticking angle, & is used deeper within the aggregate, where the particles are sensitive to vertical suction flow. It is necessary to study the influence of the aforementioned parameters on the calculated morphology of the aggregates. The aim is to identify the best set of parameters for the description of aggregates encountered in crossflow microfiltration. We must also verify that the same type of structure cannot also be obtained with a different set of parameters, corresponding to a different physical situation. Results Study of aggregate morphology The morphology of the simulated aggregates
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has been investigated over wide ranges of the most important model parameters, i.e. the incident sticking angle and the number of reentrainments. Indeed these parameters depict not only the effect of the complex force balance acting on a particle when captured but also the action of the tangential shear flow on particle mobility. We treat the aggregates by analysis of the variations of three statistical parameters. The first one is the vertical concentration, C ( Y/ I&>, expressed as the relative number of particle centers located inside parallel slices which discretixe the deposit above the membrane. The variable Y is the vertical coordinate while DP means the particle size. Secondly, we compute the mean porosity of the deposits, c& Finally, we take the root mean square of the particle gap to express the mean thickness of the deposit as a function of the number of particles. It is defined as follows: ri i=AL. +/2
where Yi is the distance between the center of particle i and the membrane surface. In all the simulations, 5000 particles are injected. The model membrane has a porosity of 15%. The incident angle equals 20 O. No significant differences were obtained in varying 0,. This result implies that it is the mechanic blocking instead of particle-particle forces which controls particle capture within the aggregate. The vertical sticking angle was therefore held constant at lo since it is unnecessary. Influence of i3i Recall that this parameter represents the magnitude of the particle-particle adhesion force compared with the hydrodynamic release force. We vary (3,from 5 ’ to 60 O.We present in Figs. 4(a) and 4 (b) two aggregates obtained with 0, is 10 and 30”, respectively. The small
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II
Fig. 4. (a) Aggregates of 5000 particles a=20”, &=l”, N,=O.
for &=lOO, a=20”,
angles lead to very compact structures. However, the mean porosity increases quickly when the sticking angle increases (as shown in Fig. 5). Furthermore, the curves in Fig. 6 characterize the significant mean thickness difference in
II
S,=l”,
II
N,=O.
II
II
II
II
II
(b) Aggregates of 5000 particles
II
for 8,=30”,
the different cases. We also notice the existence of de&its directed in the incidence direction of the suspension in this case. This formation is caused by the mechanism of contactcapture expected with such assumptions. The vertical concentration diagram shown in Fig. 7
degrees 70
4
20 / 0
30
50
90
120
150
30
180
Fig. 5. Variations of the mean porosity cd with 0, for LY= 20 ‘, &=l”, N,=O.
Fig. 6. Evolution of the mean thickness Td as a function of the number of particles N, for different values of the incident sticking angle Bi and for LY= 20 ‘, 0, = 1 O,N, = 0.
P. Schmitz et al. / J. Membrane Sci. 84 (1993) 171-183 degrees 1
c (%) Fig. 7. Vertical variations of the concentration C with 0, for cY=20”, e,=io, N,=O.
provides an adequate classification of the solid particle deposit simulations obtained. Let us now analyze more precisely the shape of each concentration profile. We determine three specific regions in which the aggregate morphology is different. The closest region to the membrane, corresponding to a rapid increase of concentration, can be interpreted as the germination region, i.e. the region of contact between the aggregate and the membrane. It represents the aggregate foundation. The region in which the particle concentration decreases slowly is considered as the second region. This is the depth of the aggregate because only a small surface is available to capture particles in this region of high concentration. The third region is named the capture region because it is the place where the particles are intercepted by the deposit surface. Its thickness represents the surface rugosity of the aggregate. Influence of N, This parameter determines the sensitivity of a suspended particle to the entrainment effect due to crossflow when it makes contact with a previously aggregated particle. We can seen in
177
Figs. 8 (a) and 8 (b) the simulations of two different deposits corresponding to two different values of the number of reentrainments, from a branching out structure (N, equals zero) to a regular structure (N, high). We find that high N, gives optimal arrangement of the particles resulting in more complete space filling. The concentration profiles (Fig. 9) indicate the same three regions of deposit structure as depicted previously. Furthermore the increase of deposit uniformity corresponding to the increase of N, is not only quantified by the concentration variations but also by the mean thickness variations for different N, values (see Fig. 10). In this figure we notice the superposition of mean thickness profiles for N, greater or equal to 3 certainly due to the final capture of most particles before the third reentrainment. This is also confirmed by the asymptotic limit of the mean porosity curve plotted in Fig.
Homogeneity Some authors [16,17] have shown that the fractal dimension of an aggregate can be deducted from the following relation:
T,=BN;
(2)
in which the power 3,is expressed as a function of the fractal dimension Df and the euclidian dimension D,, as follows: A=
l-D,
l +I&
(3)
Of course these statistical parameters have only a physical meaning for a great number NP of aggregated particles. So the heterogeneity degree of the aggregate can be evaluated with the arithmetic difference D,-& A zero value ensures perfect homogeneity of the aggregate instead of a non zero value giving an heterogeneous nature. The variations of Td, plotted in Figs. 6 and
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Fig. 8. (a) Aggregates of 5000 particles for N. = 1, (Y= 20 ‘, tl,= lo, 0,= 1 O.(b ) Aggregates of 5000 particles for N. = 3, CY= 20 O, 0,=1°, 19,=1°. ::
Y
Nr 0
Nr
0
P
0” \ P P-
lo
100
0.00
0.02
0.04 c
0.06
I
I
I
2000
3000
I
4000
5
10
NP
(%I
Fig. 9. Vertical variations of the concentration C with N, for a=20”, 0,=1’, ei=lO.
I
1000
0
Fig. 10. Evolution of the mean thickness Td as a function of the number of particles N, for different values of the number of reentrainments N, and for a = 20”, 0, = lo,
e,=io.
10 show the linear dependance of Td on N,, i.e. J equals 1 and D, -Dr equals 0. This result is true for all the simulated aggregates with this model. So we confirm that the aggregates made from particles which follow motion and adhesion rules defined here are statistically homogeneous. As a major consequence it is valid to describe mass transfer through these aggregates by Darcy’s law.
Comparisons with experimental
measurements
Numerical simulations of deposits are carried out for a great number of particles entering individually the filtering region specified by the model. It means physically that such aggregates can be compared to the real deposits we can observe after a long crossflow microfiltra-
P. Schmitz et al. /J. Membrane Sci. 84 (1993) 171-183
.-ri;i 28
+I
27 28
:
;.-
::I‘--0
1
2
3
4
6
N, Fig. 11. Variations of the mean porosity Ed with N. for CY=20°, e,=1o, &=lO.
tion period. We have shown previously [ 181 that such simulations were qualitatively in agreement with experimental visualizations from video recording of an optical microscope image of well mud filtration on a flat mineral membrane. Here our concern is to use our theoretical model to justify the existence of specific deposit structures obtained experimentally [ 19,201. Indeed- Wandelt et al. have developed an experimental method to study crossflow microfiltration of be&mite suspensions in hollow fibers. They have used NMR imaging allowing for in situ observations of deposit growth without any perturbations of the dynamic process. The measurements of deposit thickness are combined with the estimation of deposited mass from a turbidimetry device. These two complementary measurements enable calculation of cake porosities. We can see in Figs. 12(a), 12(b) and 12(c) cross sections of a hollow fiber module related to two types of hydrodynamic conditions, i.e. medium and low tangential shear flows. The structure of the particle aggregate at the membrane surface looks compact in the case of a Reynolds num-
Fig. 12. NMR image of cross-section of a module of 3 hollow fibers with a spatial resolution of 20 micrometer: (a) at t=O, (b) at t=17 min with Re=220, (c) at t=94 min with Re = 60.
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Fig. 13. Aggregates of 5000 particles for N. = 5, CY = 20 ‘, 0, = 1O,6,= 1’.
Fig. 14. Aggregates of 5000 particles for N,= 0, (Y= 20”) f3,= lo, 8, = 90 ‘.
ber Re equal to 220. The effect of tangential shear flow seems to be significant on interface erosion, reducing vertical increase of aggregates before complete space filling, i.e. dendrites development. This result is confirmed by porosity variations as a function of time. Wandelt et al. give low final porosities of about 40% and low cake thickness for this case. On the contrary, the deposit appears highly porous on the NMR image of Fig. 12(c) at a lower Reynolds number (60). The corresponding measurement indicates here a thick cake and a mean porosity of the deposit equal to 80% at the end of the operation. We should also mention that the local cake porosity was observed to be inhomogeneous. Of course these are only estimations of deposit thickness and porosities for several reasons. Firstly the NMR image consists of an integration over a cross slice of 3 mm thickness. Secondly the resolution, i.e. the pixel size is equal to 20 pm giving in fact the representation of more than a particle. But never-
theless these experimental results show the influence of the hydrodynamic regime on deposit morphology. That is why it is relevant to consider two parameters such as the incident sticking angle 0, and the number of reentrainments N, in our theoretical model to quantify on the one hand the complex balance of forces including the release force and on the other hand the competition between filtration flow and entrainment flow. We present two different types of deposits of 5000 solid particles performed with our simulation model. These simulations use two different sets of parameters adapted to the physical situations of interest cited above. In each case the suspension is assumed to flow from the left to the right. The first case in Fig. 13 models a fouling layer due to a suspension flow where particle-particle and particle-wall attractive forces are much greater than the hydrodynamic forces carrying the particles away. Here a suspended particle
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making contact with the filter surface of with a particle of the aggregate is immediately captured. Thus N, equals 0 and the given value for 0, is 90”. The deposit structure is dendritic, the porosity has a mean value of 61%, and the deposit is locally quite heterogeneous (see figure) but stays statistically homogeneous as already mentioned previously. This kind of aggregate is analogous to the experimental case of Fig. 12 (c) giving also a high porous deposit in the hollow fibers. The second case shown in Fig. 14 presents a very compact accumulation of particles which has been observed experimentally (figure) when the Reynolds number was relatively high. So we can understand that the effect or parallel shear flow is efficient even inside the deposit. It facilitates the optimal arrangement of particle deposition giving a very low porosity. To take into account the importance of tangential flow, N, is chosen equal to 4 while 0, has a very small value. The mean porosity is found to be 24% resulting in a mean thickness lower than the one obtained in the first case for the same number of particles. These two simulations allow us to give a tentative explanation of the very different deposit structures observed experimentally by Wandelt et al. Furthermore, despite the fact that the parameters of the model are not physical a priori, we believe that they result in realistic behaviour of particles during deposit formation at the membrane surface. Discussion The model presented in this paper aims at a simplified description of particle aggregation at the membrane surface in crossflow microfiltration. This description should be viewed as rather qualitative since the assumptions adopted here are to a certain extend restrictive: l perfectly flat membrane with uniformly distributed pores,
monodispersed, circular, non-deformable particles, 0 suspension at low concentration, l constant hydrodynamic conditions. Nevertheless, the model is able to show the influence of experimental conditions such as hydrodynamics and physico-chemistry on the morphology of the deposits, which yield significant effects on the evolution of the transmembrane flux in experiments. To exemplify, in the comparison presented above, our model allowed us to explain why specific aggregates having drastically different porosities could be various identified under experimental conditions. We believe that this phenomenological model is applicable to the crossflow microfiltration of a suspension at low concentration of rigid, nonbrownian particles, which are supposed to be of the same nature. In this case we can neglect particle-particle interactions in the flow. Ultrapure water is considered as the perfect fluid to ensure particle transport. Admittedly the model duplicates laboratory operating conditions rather than complex industrial conditions. It is common knowledge that any existing model is able to predict the behavior of a crossflow microfiltration process under certain conditions. Nevertheless, we have tried to give some indications to experimentalists that will guide them in selecting conditions for membrane microfiltration processes. Thus we have determined the corresponding values of the two key parameters, which are the incident sticking angle, 0i, and the number of reentrainments N,, for different hydrodynamic and physico-chemical conditions. D. Houi [21] studied numerically the variations of 0, as a function of the intensity of the resulting adhesion force, F,, and the length of the contact zone, r,, between the collector particle and the moving particle. The hydrodynamic force, Fh, which acted on the particle was obtained from the complete calculation of the velocity fields in the geometrical l
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domain surrounding the two particles in contact. The particle was supposed to be captured whenever the resulting force, R, from the total balance cut the contact zone as shown in Fig. 15. Values of 0, are reported in Table 1. Concerning particle reentrainment, we recognize that it is impossible to give a relationship between IV, and fundamental laws. The values proposed in Table 2 are the estimations of the authors from experimental results [ 15,19,20]. Additional experiments will be necessary to further validate the present model especially to improve quantitatively the model parameters. In the future, a complete three dimensional model will be uonsidered to investigate the permeability variations under a host of experimental conditions.
Conclusion We have presented a theoretical approach to the prediction of deposition and aggregate formation of a large number of particles at a membrane surface under crossflow microfiltration conditions. Real particle behaviour is translated into specific empirical mechanisms of motion and adhesion. The analysis of the influence of model parameters on the morphology of the aggregates has shown the ability to generate aggregates of different structures corresponding to very different physical situations. Finally the model has been successfully applied in the description of deposit formation in two experimental cases related to medium or low magnitude of tangential entrainment flow over the membrane. In a further study we will try to validate by microscope observations the motion and adhesion mechanisms adopted in our model. Acknowledgements
TABLE 1
We thank Doctor Pradere from C.H.U. Toulouse Rangueil for his constant help and advice in obtaining NMR images and G.D.R. ‘Filtration et Fibres Creuses’ for financial support.
Variations of 0, in degrees as a function of adhesion force and contact length for Re = 1
References
Fig. 15. Balance of forces on a particle in contact with a collector [ 211.
2 r,lD,
F, (N)
lo-‘0 10-a 10-E
0.5
0.25
0.05
3 25 180
1 14 180
1 6 27
TABLE 2 Variations of N, as a function of Reynolds number Re N,
<50 0
100 1 to 2
150 2to3
> 200 5
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