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Properties of the cake layer formed during crossflow microfiltration
Colloids and Surfaces A: Physicochemical and Engineering Aspects 138 (1998) 265–281
Properties of the cake layer formed during crossflow microfiltration Ingmar H. Huisman *, Dominique Elzo, Erik Middelink, A. Christian Tra¨ga˚rdh Food Engineering Department, Lund University, P.O. Box 124, 221 00 Lund, Sweden Received 17 June 1996; accepted 23 November 1996
Abstract The conditions necessary for the formation of a reversible cake layer during crossflow microfiltration were studied both experimentally and theoretically. Crossflow microfiltration experiments were performed with suspensions of silica particles with a narrow size distribution. The steady-state flux was first measured at a low transmembrane pressure (TMP), then at increased TMP, and again at the original low TMP. The cake-layer thickness was measured indirectly using a light absorbance technique. The thickness of the cake layer increased with increased TMP. Upon decreasing the TMP, the cake-layer thickness either decreased (reversible cake), or stayed constant (irreversible cake). It was shown that irreversible cakes are formed when the silica particles have a relatively low charge, whereas reversible cakes are formed when the silica particles have a relatively high charge. The occurrence of irreversible cakes is unexpected, since approaching silica particles are reported to always repel each other. The irreversibility of the cakes was explained by the assumption that bridging between the particles can occur, causing the interparticle interaction to be attractive when the particles retreat. To explain the reversibility results quantitatively, a model was developed which links the physicochemical interaction forces of the silica particles to the permeate flux through the cake layer. A detailed description of the interaction forces of silica particles was given in order to feed this model with accurate parameters. A reversibility index was introduced which quantifies the amount of reversibility. Model calculations of the reversibility index were in excellent agreement with measurements. © 1998 Elsevier Science B.V. Keywords: Cake layer; Microfiltration; Pore plugging; Silica particles; Zeta potential
1. Introduction Crossflow microfiltration, a separation technique for removing colloidal particles from suspensions, has been successfully applied in industry. However, the potentially high capacity of crossflow microfiltration is still limited by fouling of the membrane [1]. The term ‘‘fouling’’ includes * Corresponding author. Tel.: +46 46 2229820; Fax: +46 46 2224622. 0927-7757/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0 9 2 7 -7 7 5 7 ( 9 6 ) 0 3 97 6 - 3
different processes, such as adsorption of macromolecules on the membrane surface or within the pores, and the formation of a filter cake-layer. Cake-layer formation is an important cause of flux decline in the microfiltration of colloidal suspensions, for example in the clarification of fruit juices or the recovery of pigment from paint. To limit the build-up of a cake layer, regular backflushing or membrane cleaning are commonly applied. Both methods will be relatively effective if the cake layer is reversible (i.e. removing the
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transmembrane pressure ( TMP) resolves the cake). These techniques will, however, be more time- and energy-consuming if the cake is irreversible (i.e. upon removing the TMP, the cake stays on the membrane). Information on the reversibility of cake-layer build-up is therefore of great practical importance in microfiltration applications. Benkahla et al. [2] and Ziani and Ben Aim [3] measured the reversibility of cakes formed during crossflow microfiltration of CaCO suspensions. 3 They increased the TMP stepwise up to a maximum value and then decreased it stepwise to zero. They found that a cake layer was formed on the membrane, and that the resistance of this cake layer increased upon increasing the TMP. However, this resistance stayed at a constant level upon decreasing the TMP. ( The plot of flux vs. TMP was a straight line.) In other words, cakelayer formation was found to be irreversible. This phenomenon was qualitatively explained by assuming that ‘‘some cohesive forces’’ hold the particles together in the cake. Ziani and Ben Aim [3] found that upon increasing the wall shear stress, this hysteresis disappeared: the plot of flux vs. TMP followed more or less the same curve with decreasing TMP as it had done with increasing TMP. A quantitative description of cake-layer properties generally starts off with Darcy’s law ( Eq. (1), which links the permeate flux J to the TMP: J=
TMP R m tot
(1)
where R is the total hydraulic resistance and m tot is the viscosity of the permeate. Resistance in series models are often used to describe the total resistance as the sum of two contributions, one caused by the membrane and one caused by the cake layer (R ). The resistance caused by the memcake brane during microfiltration of a suspension can be expected to be higher than the membrane resistance for pure water (R ), since some of the mem pores might be plugged, reducing the effective porosity and thus increasing the hydraulic resistance. This effect can be described by adding an extra term (R ) to the membrane resistance. This pp
leads to: R =R +R +R (2) tot mem pp cake The cake resistance can be calculated by the Kozeny–Carman equation [16 ] if the cake’s void fraction e, and the cake-layer thickness d are c known: =
180(1−e)2
d (3) c d2 e3 p where d is the particle diameter. p For compressible particles, e can be as low as zero. For the laboratory experiments described in this paper, hard silica particles were used, which were assumed to be practically incompressible. For incompressible monodisperse particles e can theoretically vary between 0.2595 and 0.97 [4]. Benkahla et al. [2] observed that e depends slightly on the TMP (e decreases as TMP increases), and that e could be as low as 0.18 for polydisperse systems. Solutions flowing through narrow capillaries show an increased apparent viscosity [17]. This phenomenon, called the electroviscous effect, is caused by the interaction between solutes and fixed wall charges. In microfiltration applications, both the membrane and the cake layer form capillary channels which give rise to different electroviscous effects. These effects have been reported [11,18] to cause an additional decrease in permeate flux. Increasing the charge of the suspended particles in a microfiltration process might thus give two different effects. The interparticle repulsion increases, thus increasing the cake’s void fraction and increasing the permeate flux. At the same time, electroviscous effects increase the effective viscosity of the permeate and thus decrease the flux. Both effects have been described theoretically and observed experimentally [19,20]. It can thus be seen that many different phenomena play a role in the characterization of cakelayer properties, such as void fraction, thickness, and reversibility. A complete understanding of the microfiltration cake layer has therefore not yet been obtained. Particularly little is known about the reversibility properties of the cake layer. In this paper we present results from microR
cake
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filtration experiments performed with silica particles with a narrow size distribution. The irreversibility observed by other authors [2,3] for CaCO suspensions was also found for silica sus3 pensions, but only if the silica particles had a low surface charge. The main aim of this paper is to explain this irreversibility quantitatively, and to predict under which circumstances it can be expected. In order to do this, an equation is derived which links the permeate flux through the cake layer to the repulsive interaction forces between particles in the cake. In addition to this equation, some specific information on the interparticle forces is necessary. Therefore, a description of the interparticle forces for silica is presented in which special attention is paid to the coagulation phenomenon, since this appears to be the cause of irreversible cake-layer build-up. Since the equation derived assumes a uniform cake with a constant void fraction, the void fraction e of the cake layers obtained was determined in order to prove the validity of this assumption.
2. Theory
2.1. Interaction forces between silica particles Grabbe and Horn [7] and Ducker et al. [8] measured the interaction forces between silica surfaces, and showed clearly that silica surfaces repel each other, upon approaching, for all distances, even if surface charges are low and salt concentrations are high (see Fig. 2). For many other materials there is a region where the net interaction force is attractive (primary minimum [5]). The interaction forces between silica surfaces do not show a primary minimum because of two effects [9]. The van der Waals attraction for silica is low, and an additional repulsive force is present, i.e. the hydration force. Silica surfaces are hydrated, and in order to move them very close together (distances of less than 1 nm), these hydration shells have to be removed. To do this, a large force is necessary. It is, however, a well known fact [9] that silica
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particles coagulate under certain circumstances ( low pH (=low surface potential ) and high salinity), even though the observed interaction forces are always repulsive. To solve this apparent paradox, it is assumed that chemical bridges can be formed upon contact (silica bridges [10] or ionic bridges (for example Na+, Ca2+) [9]), which cannot be observed when the interparticle distance decreases (particles approaching each other), but which give an attractive force when the interparticle distance is increased again (particles retreating). The plot of interparticle force vs. distance should therefore show hysteresis if the particles are pressed sufficiently close together (see Fig. 1). This is observed experimentally for the force between two glass particles [10]. To be able to do quantitative work on cake reversibility for silica particles, a description of the force–distance relation for approaching and retreating particles would be helpful. The present authors have not seen any quantitative description for the case of retreating particles. For the case of approaching particles, the situation is slightly better. Combining standard DLVO theory [6 ] with a charge regulation model [12] gives a good description of the interparticle forces [7,8]. For distances less than about 4 nm, hydration forces have to be taken into account. Grabbe and Horn [7] show that the hydration force depends only slightly or not at all on physicochemical parameters like surface potential and ionic strength. They give a description of the hydration force obtained by fitting experimental data by: F =F exp(−D/D )+F exp(−D/D ) (4) hydr 1 1 2 2 where F is the hydration force, D is the hydr interparticle distance, and F , F , D and D are 1 2 1 2 empirical parameters. Grabbe and Horn obtained the parameters F /2pa=0.14 J m−2, D =0.057 nm, 1 1 F /2pa=5.4×10−3 J m−2 and D =0.48 nm (a is 2 2 the particle radius). Eq. (4) is somewhat arbitrary, and the accuracy is certainly not better than ±50%. Since the description of the hydration forces has only limited accuracy, there is no use trying to describe the other forces with a higher accuracy. The total interaction force (F ) is described as i the sum of the van der Waals force (F ), the vdW
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Fig. 1. Schematic plot of interparticle force for silica particles which are approaching each other and then retreating again.
double-layer force (F ) and the hydration force dl (F ): hydr F =F +F +F (5) i vdW dl hydr The van der Waals force is defined as: F = vdW
−∂E
vdW ∂D
(6)
where the van der Waals energy, E , can be vdW given by: A(D) E (D)=− vdW 12pD2
(7)
An empirical expression [7] for the Hamaker function A(D) is given by: A(D)=A (1+2kD) exp(−2kD) 0 +A /[1+(D/l)p]1/p (8) v where k=the Debye constant, and A , A , l and 0 v p are empirical or semi-empirical parameters. For silica surfaces interacting across water, the values A =0.288×10−20 J, A =0.726×10−20, l= 0 v 5.82 nm and p=1.433 give a good approximation [7]. It is observed for both mica and silica surfaces [12] that the standard ‘‘constant charge’’ approximation describes experimental data well for interparticle distances down to a few nm. Healy and
co-workers [5] have derived the following expressions for the constant charge double-layer interaction between two spheres of radius a: ∂E F =− dl dl ∂D
(9)
with the double-layer energy E given by: dl E =−2pe e ay2 ln[1−exp(−kD)] (10) dl r 0 0 where e e is the electromagnetic permittivity of r 0 the solution, a is the particle radius, and y is the 0 surface potential. For the silica particles used in this study, the value of the zeta potential is known. The zeta potential is the potential at the hydrodynamic shear plane. This plane is situated very close the Stern plane. In this study, following Israelachvili [12], we assume that both planes coincide and that they are at a distance of 0.55 nm from the surface (0.55 nm=one sodium-ion diameter+one hydrated sodium-ion radius). The effective surface potential can then be calculated from the zeta potential using the Gouy–Chapman theory [12], which states that:
C
2kT 1+c exp(−kx) y = log x e 1−c exp(−kx)
D
(11)
where y is the potential at distance x from the x
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surface, k=Boltzmann’s constant, T is the absolute temperature, e is the electron charge, and c is defined by: c=tanh(ey /4kT ) (12) 0 y is then calculated by solving Eqs (11) and (12) 0 for x=0.55 nm and y =zeta potential. x The use of Eqs (10) and (11) is questionable at close distances. This means that potentials calculated by Eq. (11) will not be quantitatively correct. However the objective of using Eq. (11) is not to predict the ‘‘real’’ surface potential, but rather to predict an effective surface potential which can be used in Eq. (10) to calculate interaction energies for larger distances (outside of the Stern planes). Combining Eqs. (4)–(10) relates the interparticle force to the surface potential. Since some approximations are made, the accuracy will be limited. To obtain an idea about the accuracy, we calculated force–distance curves for four test cases found in the literature ( Fig. 2 gives an example). From the results of these test cases, it can be concluded that our calculations give a reasonable approximation of the interaction between two silica surfaces. Errors made are not more then a factor of two, which could be expected based on
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the ±50% error in the expression for the hydration force. Bowen and Jenner [11] argue that errors are made if a two-body force is generalized to a concentrated suspension ( like a cake layer). Their arguments only hold if the average distance between particle ‘‘contact’’ is of the same order as the double-layer thickness. In our case the doublelayer thickness is typically 10 nm, whereas the particles have a much larger radius (about 240 nm), so that the interaction between one particle and all its neighbors is simply the sum of all two-body forces. The above discussion has resulted in an equation for the interparticle force between approaching spheres in concentrated suspensions. For the case of retreating spheres, a theoretical description is not available. For the current study, however, it is only necessary to know whether or not the forces upon retreating will differ from those upon approaching, in other words, whether or not bridging occurs between the particles. It is now assumed that if the particles approach each other closer than a certain distance D*, then bridging can occur and aggregates can be formed. If the interparticle repulsion is so strong that the particles cannot get
Fig. 2. Interaction force between two spheres of silica. Experimental values are replotted from Ref. [5], and lines are calculated by combining Eqs. (5)–(10).
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as close as D*, no bridging can occur and the system is reversible. The distance D* can be estimated to be of the order of 1 nm (a silica atom has a diameter of about 0.23 nm, a hydrated Na+ ion about 0.7 nm). 2.2. Electroviscous effects It has been reported [18] that changes in salt concentration of the feed water cause changes in the clear-water flux of ultrafiltration membranes. These changes are ascribed to electroviscous effects: the viscosity of a salt solution increases if flowing through a thin capillary, because of interaction between solutes and fixed wall charges. These effects were described quantitatively by Levine et al. [17]:
C
m
a = 1− m
D
8b(ef/kT )2(1−G)F −1 (kr)2
(13)
where m is the apparent viscosity, b=37/L a (where L denotes the molar conductivity in V−1 cm2 mol−1), f is the zeta potential of the capillary surface, k is the Debye constant, and r is the capillary radius. G#0 if kr>10. F#1 if both kr>20 and |f|<100 mV. These effects are expected not only when a solution flows through the membrane pores, but also when a solution flows through a cake layer of colloidal particles. Since the channels in the cake are not cylindrical capillaries, Eq. (13) will be only an approximation. Since the cake layer and the membrane each have their own zeta potential and effective capillary radius, they will give rise to different apparent viscosities. Darcy’s law should therefore be rewritten to give: J=
TMP
(R +R )+m R a,mem mem pp a,cake cake This equation can then be rewritten as: m
(14)
TMP
J= m
A
m
m m a,mem R + a,mem R + a,cake R mem pp cake m m m
B
(15)
or TMP
(16) +R ) a,mem a,pp a,cake where the apparent resistance R is defined by: a m R ¬ a R (17) a m J=
m(R
+R
Eq. (16) looks exactly like Darcy’s law, but note that the apparent resistances are functions of the solution’s ionic strength (through the Debye constant) and of the surface’s zeta potential, unlike the resistances used in Darcy’s law. This has an important consequence for the relation between the observed cake resistance and the void fraction of the cake. The Kozeny–Carman equation ( Eq. (3)) only applies to R , not to R . cake a,cake Eqs. (13) and (17) then have to be used to transform the observed R into R , which a,cake cake can be used in the Kozeny–Carman equation. 2.3. Relation between interparticle forces within the filter cake and permeate flux It is intuitively clear that the particles within a filter cake will be pressed more strongly together if the pressure drop over the cake increases, and that the particles which are deeper down in the cake layer (i.e. closer to the membrane) are pressed more strongly together than the particles which lie at the surface of the cake layer. Particles which are pressed strongly together will show a mutual repulsion, according to Newton’s third law (action=−reaction). We now derive an equation which relates the repulsion between neighboring particles in the cake to the location of these particles within the cake, and to the pressure drop over the cake. The analysis is in parallel with the ‘‘disjoining pressure’’ approach, first developed by Derjaguin et al. [13] and used for silica suspensions by Bowen and Jenner [11]. In all practical microfiltration applications the pressure gradient (in bar m−1) across the cake in the z-direction (perpendicular to the membrane) is many orders of magnitude higher than the pressure gradient along the filter in the tangential direction. The pressure gradients in the x and
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y directions are therefore neglected in the following analysis. It is assumed in this section that the filter cake is constituted of identical spherical particles with a diameter of d (see Fig. 3). The membrane lies p below these spheres, and the bulk suspension lies above these spheres. The particles in the cake are in a hexagonal close packing. The particle under consideration is called S (sphere) and is situated at the origin of the coordinate system. Three particles lie on top of S and three below S. Six particles lie around S, but do not exert forces on S since it is assumed that there is no pressure gradient in the x or y directions. A standard mathematical derivation renders that the point where one of the neighbors touches S makes an angle w* with the vertical, and that sin w*=1/E3 (or w*=35.26°). Since the particle S does not move, the sum of all forces exerted on it must be zero. Two different forces are exerted on S: the forces of interaction with neighboring particles according to the theory described in Section 2.1, and the drag force caused by the permeate flux. Because of the symmetry of the system, the magnitude of these forces is a function of the z-coordinate only, not of x and y. Adding the z-components of the forces then gives:
by:
∑ interaction forces=−3F (z*) cos w* (18) i above where z*=d /2 cos w*, the z coordinate of the p position where the particles touch. In a similar manner, the interaction forces caused by the three particles below S are obtained. The total sum of interaction forces is then given
ˆ is the specific hydrodynamic resistance of where R the packed bed. Different equations are found ˆ (for example the to describe the value of R Kozeny–Carman equation and the Blake–Kozeny equation [1]), which assume that the resistance is a function of particle size and of the void fraction of the cake. The particle size is constant in the experiments described in this paper, and it will be shown in Section 3 that the void fraction is independent of the pressure drop over the cake. ˆ can be taken as a Therefore the value of R constant. Since in Eq. (22) the permeate flux J and the viscosity m cannot be functions of z, the product ˆ is a constant K. Eq. (22) can be integrated to JmR give:
Fig. 3. Schematic picture of a particle S and its neighbors in a cake layer.
∑ interaction forces=3[F (−z*)−F (z*)] cos w* i i (19) From sin w*=1/E3, it is found that cos w*= E(2/3), so that
C A B A BD
d ∑ interaction forces=E6 F z=− p i E6 −F
d z= p i E6
(20)
The drag force caused by the permeate flux (=the hydrodynamic force F ) can be calculated by the hyd well-known equation: F = hyd
P
P dA
(21)
surface where P is the absolute pressure in the fluid, and A is the particle surface area. In general, the pressure drop over a cake layer or any other packed bed is given by a variant on Darcy’s law: dP dz
ˆ =JmR
(22)
P(z)=P +Kz (23) 0 where P is the pressure at z=0. 0 As stated before, the hydrodynamic force has only a component in the z direction. Integrating only the z-component of the forces, using
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dA=2pa dz and cos w=z/a, gives: F = hyd
P
p
neighbors. Integrating Eq. (29) gives the solution
P(z)(−cos w)2pa2 sin w dw
(24)
0 Substituting Eq. (23) in Eq. (24) gives:
F =2pa2 hyd
P
p 0
(P +Ka cos w)(−cos w) sin w dw 0 (25)
Solving this equation results in F =−4/3pK(d /2)3=−1pKd3 hyd p p 6
(26)
The sum of the hydrodynamic forces ( Eq. (21)) and interaction forces ( Eq. (15)) acting on particle S must vanish, so that:
C A
B A
d d E6 F z=− p −F z= p i i E6 E6
BD
−1pKd3 =0 p 6
The last term on the left-hand side does not depend on z, so that F has to be linearly dependent on z, i and Eq. (27) can be rewritten as: dF 2d i p =−1pd3 K 6 p dz E6
(28)
where ˆ. K=JmR Eq. (28) can thus be rewritten to give: −p i= ˆ Jm d2 R dz 12 p
dF
(30)
Eq. (30) gives the repulsive interaction force as a function of the location within the cake (z) and the permeate flux (J ). This equation clearly shows that the particles which are deeper down in the cake layer are subject to larger repulsive interactions. To obtain Eq. (30) a hexagonal close packing was assumed for mathematical convenience. Similar analyses can be done assuming other packings; these render equations similar to Eq. (30), but with leading constants which differ slightly from p/12. 2.4. Reversibility index
(27)
E6
p ˆ Jm(d −z) F = d2 R i 12 p c
The experiments described in this study are so-called ‘‘reversibility’’ experiments. The flux (J ) is first measured at a low TMP ( TMP1), then measured at a high TMP ( TMP2), and finally measured again at TMP1. In the total reversible case the flux finally measured at TMP1 will be the same as the flux originally measured at TMP1, i.e. J( TMP1,2nd)=J( TMP1,1st). In the total irreversible case, the cake formed at TMP2 will stay on the membrane after decreasing the TMP to TMP1. The total resistance R will therefore stay constant, or tot R ( TMP1,2nd)=R ( TMP2). Using Darcy’s law, tot tot this results in: J( TMP1,2nd)/TMP1=J( TMP2)/TMP2.
(29)
Up to now, it was assumed that the origin of the coordinate system was at the center of particle S. Since Eq. (29) is independent of the actual place of this origin, we can move the origin to the surface of the membrane, where the cake formation starts. If the cake has a thickness of d , then c F (d )=0, since the particles situated at the upper i c surface of the cake layer are not submitted to repulsive interaction forces from higher-lying
A reversibility index (RI ) is now introduced by: RI=
J( TMP1,2nd)/TMP1−J( TMP2)/TMP2 J( TMP1,1st)/TMP1−J( TMP2)/TMP2 (31)
The use of this equation can be seen in Fig. 4, where a typical reversibility experiment is shown. The RI as defined above is given by distance A/distance B. If the flux J( TMP1,2nd) is equal to J( TMP1,1st), then A=B, and RI=1 (total reversibility). If the flux J( TMP1,2nd) lies on a straight
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Fig. 4. A typical reversibility experiment. Reversibility index (RI )=A/B.
line going through J( TMP2) and 0, then A=0 and RI=0 (total irreversible case). The reversibility index RI, defined as in Eq. (31) is thus a parameter which shows the amount of reversibility in cake-layer formation. In standard cases, its value will be between 0 (irreversible cake layer formation) and 1 (reversible cake formation). The reason for irreversibility of cake layers in general might be that strong attractive van der Waals forces keep particles together, even if they are no longer pressed together by the pressure drop over the cake. In the case of silica particles, irreversibility is assumed to be caused by the formation of chemical interparticle bridges. These bridges can only be built if the particles are pressed closer together than D*, the critical distance for bridging. Particles pressed so close together show a repulsive interaction of more than F (D*). F (D*) is calculated by the analysis given i i in Section 2.1. The repulsive interaction between particles in a cake is a function of the location within the cake (z), and of the flux of water through the cake (J ), as given in Eq. (30). At a given flux J, there will be a position within the cake (z*) where the interaction force will be F (D*). This means that i all particles lying below z* will be pressed closer
together than D*, and they can therefore build interparticle bridges. The part of the cake which will stay on the membrane after decreasing the TMP will therefore be the part below z*. If z* is equal to the total thickness of the cake, obtained at 820 mbar, then the whole cake will be irreversible and RI=0. If z* is equal to or smaller than the thickness of the cake originally obtained at 420 mbar, then the part of the cake which was formed extra at 820 mbar will be totally washed away, and cake-layer formation is reversible, RI=1. If z* is somewhere between these extremes, the cake will be partly reversible, and 0
C
1 TMP2 d = −R −R 2 R mem pp ˆ mJ(TMP2)
D
(33)
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2.4.1. Case 1: reversible cake The thickness of the cake, originally obtained at TMP1 (d ) is calculated by Eq. (33) where the 1 index 2 is replaced by index 1. If z*
−
12F (D*) TMP1 i < pd2 J( TMP2) J( TMP1) p
(34)
2.4.2. Case 2: irreversible cake If F (D*) is very small, then Eq. (32) predicts i that z*$d . In that case the whole cake will be 2 irreversible, or RI=0. A criterion for an irreversible cake is therefore TMP2 J( TMP2)
−
12F (D*) i %m(R +R ) mem pp pd2 J( TMP2) p (35)
2.4.3. Case 3: partly reversible cake If d
TMP1
(36)
ˆ +R m(z*R
+R ) mem pp Inserting Eq. (32) and Eq. (33) gives: TMP1 J( TMP2)
(37) 12F (D*) i TMP2− pd2 p Combining this with the definition for the reversibility index (Eq. (31)) finally yields: J( TMP1,2nd)=
RI=
C
J( TMP2)
−
J(TMP2)
TMP2−12F (D*)/(pd2 ) TMP2 i p J( TMP1,1st) J( TMP2) − TMP1 TMP2
C
D
D (38)
3. Materials and methods The experiments discussed in this paper are crossflow microfiltration experiments of a model suspension of silica particles. The permeate flux was measured as a function of time at different transmembrane pressures ( TMP) and crossflow velocities. The experimental set-up used is depicted in Fig. 5, and described in detail elsewhere [15]. It used a tubular ceramic a-alumina membrane, purchased from SCT, France, with a nominal pore size of 200 nm. A uniform TMP 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 flux was regularly returned to the feed tank, the volume at the feed side of the membrane was constant. The crossflow velocity was 1 m s−1 (Re=6800, t =4.42 N m−2) for all experiments w described here. The silica particles used were kindly supplied by Nissan Chemical Industries Ltd. (Japan). They had a mean diameter of 480 nm; 90% of the particles are within the range 190–900 nm. Their characteristics are described in detail elsewhere [14]. The feed suspension was prepared based on double-distilled and Milli-Q filtrated 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 of the silica particles, as shown by Elzo et al. [14]. The particle concentration in the feed was continuously determined by measuring the turbidity ( light absorbance at 700 nm) of the feed suspension. If particles were deposited on the membrane, the particle concentration in the feed decreased. The deposited amount was then calculated from these particle concentration measurements. All experiments started off at a particle concentration of 1.7 g l−1. During an experiment, the particle concentration typically dropped to about 1.55 g l−1. In order to obtain information on the reversibility of cake-layer formation, experiments were performed according to the following scheme. The TMP was set at 420 mbar, and was kept at 420 mbar until a steady-state flux was reached
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Fig. 5. The microfiltration set-up.
(1–2 h). The TMP was then increased to 820 mbar and was kept at 820 mbar until a steady-state flux was attained (2–3 h). The TMP was reduced again to 420 mbar, and kept at 420 mbar until a steadystate flux was reached (this usually took less than half an hour). In addition to the experiments based on 420–820–420 mbar cycles, experiments were performed for 110–420–110 mbar cycles.
4. Results and discussion Typical results are given in Figs. 6 and 7, which depict two similar experiments, performed at the same crossflow velocity, and using the same series of TMPs. By adjusting the pH, the zeta potential of these particles has been changed. In Fig. 6 the zeta potential of the particles is slightly negative; in Fig. 7 this potential is more negative. The fluxes at 420 and 820 mbar are similar for both cases (a small difference in flux is observed, in accordance with earlier measurements [14]). However, upon reducing the TMP back to 420 (or 415) mbar, the
flux is reduced more for the case of the slightly negative zeta potential than for the case of the strongly negative zeta potential. The results of Figs. 6 and 7 can be replotted as flux vs. TMP, as shown in Figs. 8 and 9. In such plots, the irreversibility (or hysteresis) is shown more clearly: for low zeta potential (Fig. 8) it is seen that the flux decreases with a straight line back to zero if the TMP is decreased, as also observed by Benkahla et al. [2]. A straight line in such plots shows that the cake-layer resistance is constant, and that the cake must be irreversible. For strongly negative zeta potentials, a reversible flux vs. TMP curve is observed (Fig. 9). Figs. 6 and 7 also show the amount of deposition as a function of time. The deposition results agree with the reasoning given above. In the case of a low zeta potential ( Fig. 6), the amount of deposition hardly changes when the TMP is decreased from 820 to 415 mbar: the cake stays on the membrane. For the high zeta potential case, the amount of deposition decreases upon decreasing the TMP, and finally reaches a value close to the
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Fig. 6. Flux and amount of deposition vs. time. f=−26 mV, Crossflow velocity=1 m s−1. Both the flux and the deposited amount show irreversibility.
value obtained originally at 420 mbar: cake formation is totally reversible. From the measured flux, a total apparent resis-
tance (R ) can be calculated, using Darcy’s law a,tot ( Eq. (16)). The total apparent resistance is plotted vs. the amount of deposition in Fig. 10. It can be
Fig. 7. Flux and amount of deposition vs. time. f=−80 mV, crossflow velocity=1 m s−1. Both the flux and the deposited amount show reversibility.
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277
Fig. 8. Flux vs. TMP for a reversibility experiment. f=−26 mV, crossflow velocity=1 m s−1. An irreversible cake is formed.
seen that the points follow a straight line, which indicates that a uniform cake was formed. The specific resistance does not seem to be a function of time, nor a function of the cake thickness. It can be seen in Fig. 10 that changing the TMP changes the slope of the plot slightly. However, the change is very small. According to Eq. (16), R is the sum of three a,tot different contributions: R ,R and R . a,mem a,cake a,pp
Note that only R depends on the amount of a,cake deposition. Extrapolating the curve of Fig. 10 to zero deposition gives an intercept, which represents R +R . Since R is known from a,mem a,pp a,mem clear water flux measurements, a value for the pore plugging resistance can be obtained. It can be seen that the pore plugging resistance is a large effect (compare R #150×109 m−1). a,mem From Fig. 10 a value of the cake’s void fraction
Fig. 9. Flux vs. TMP for a reversibility experiment. f=−80 mV, crossflow velocity=1 m s−1. A reversible cake is formed.
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Fig. 10. Cake resistance vs. amount of deposition. f=−26 mV, crossflow velocity=1 m s−1.
e can be determined by the Kozeny–Carman equation ( Eq. (3)), while taking into account electroviscous effects, as described in Section 2.2. It is found for Fig. 10 that e=0.3060. e and R were detera,pp mined for cakes formed under different conditions. The pH of the feed suspension (zeta potential of the particles) and the TMP were varied. In Table 1 it can be seen that the scattering in values of R is so large that no relation can be shown a,pp between the pore plugging resistance and either TMP or zeta potential. The cake’s void fraction is not significantly influenced by the TMP, but seems to increase slightly when increasing the absolute value of the zeta potential, as shown more clearly Table 1 Void fraction e and apparent pore plugging resistance R f potential (mV ) −26 −26 −44 −45 −58 −61 −69 −80 −84
e ( TMP=420 mbar) 0.2916 0.3060 0.2992 0.2815 0.2833
a,pp
in Fig. 11. The mean void fraction is 0.30, which is reasonable compared to the value for a monodisperse hexagonal close packing (0.2595). To obtain values for the amount of reversibility, the reversibility index (RI ) was calculated according to its definition ( Eq. (31)). Besides this, a theoretical calculation of the RI was performed using Eq. (38). Adjusting D* to obtain the best fit resulted in a value of D*=2.0 nm. However, reasonable fits were obtained for all values of D* within the interval between 1.5 and 2.5 nm. The results for the two different sets of TMPs are shown in Figs. 12 and 13. It can be seen in both graphs that RI=0 for very low zeta potentials
of filter cakes formed under different conditions
e ( TMP=820 mbar)
0.2994
R (109 m−1) a,pp ( TMP=420 mbar) 886 485 600 181 213
0.3018 0.2887 0.3123 0.3104
0.3040 0.3192
R (109 m−1) a,pp ( TMP=820 mbar) 453
250 420 359 417
365 717
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279
Fig. 11. Void fraction vs. zeta potential for cakes formed at two different TMPs.
(irreversible case), and RI=1 for fairly high zeta potentials (reversible case). There is a transition zone in between. For the experiments at 420–820–420 mbar this transition zone is broader than for the experiments at 110–420–110 mbar. This is reflected in the calculated values. The main results obtained are that the void
fraction of the filter cake increases slightly, and that reversibility increases dramatically upon increasing the absolute value of the particles’ zeta potential. The latter phenomenon was explained by assuming that the interparticle distance is less than 2.0 nm for slightly negative zeta potentials and more than 2.0 nm for stronger negative zeta
Fig. 12. Calculated and experimental values of the reversibility index vs. zeta potential of the particles for TMP=420–820–420 mbar.
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Fig. 13. Calculated and experimental values of the reversibility index vs. zeta potential for TMP=110–420–110 mbar.
potentials. In other words, the interparticle distance increases with the absolute value of the zeta potential, which is in agreement with the observed increase in void fraction. If one assumes that the particles are in a monodisperse hexagonal close packing, it can be calculated that an increase in interparticle distance of 2 nm would result in an increase in e of about 0.008. This is about the order of magnitude which is observed in Fig. 9.
5. Conclusions During the crossflow microfiltration of suspensions of spherical silica particles, the flux is decreased by two effects: pore plugging and cakelayer formation. It was found that the filter cakes formed in the case of a feed suspension with a narrow size distribution had a void fraction e of 0.30±0.012. This value was independent of TMP, but increased slightly with the absolute value of the particles’ zeta potential. The cake-layer formation can be totally reversible (the cake resolves again after removing the transmembrane pressure ( TMP)), totally irreversible (the cake stays on the membrane after removing the TMP), or partly irreversible (a part of the
cake stays on the membrane after removing the TMP). It was shown that the reversibility of the filter cake depends strongly on the zeta potential of the silica particles, and besides this on the TMP. Irreversible cakes were observed if the silica particles had a relatively small charge (slightly negative zeta potential ), whereas reversible cakes were observed if the particles had a higher charge (stronger negative zeta potential ). A reversibility index was defined which ranged from RI=0 (totally irreversible case) to RI=1 (totally reversible case). The reason for irreversibility of cake layers of silica particles might be the formation of chemical bridges. In this paper it is assumed that these bridges are formed if the particles are closer together than a critical distance D*. Within the cake the particles are pressed together by the drag force caused by the permeate flux; the deeper the particles are in the cake, the closer they are pressed together. There will therefore be a position within the cake below which the particles are closer together than D*. The part of the cake below this position will be aggregated, and thus irreversible. To be able to predict which part of a filter cake will be reversible, it is necessary to find equations
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which link the interparticle distance to the flux and the location within the cake. This was done in two steps. First, an equation was derived which links the interparticle distance to the particle– particle interaction force (F ). This equation used i DLVO theory, to which hydration forces were added. The equation obtained was tested against experimental results found in the literature, and it was concluded that it was sufficiently accurate. Besides this, an equation was derived which links the interparticle forces to the flux and the location within the cake. The amount of reversibility was calculated by combining these two equations. The calculated values were in good agreement with the experimentally obtained values. Based on the model formulated in this paper, it could also be concluded that the cake’s void fraction e increases slightly if the zeta potential becomes more negative. This is in agreement with experimental observations.
Acknowledgment This work was financially supported by the Swedish Foundation for Membrane Technology (I.H.H.), and by an EC Human Capital and Mobility program (D.E.).
References [1] G. Belfort, R.H. Davis and A.L. Zydney, J. Membrane Sci., 96 (1994) 1–58.
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[2] Y.K. Benkahla, A. Ould-Dris, M.Y. Jaffrin and D. Si-Hassen, J. Membrane Sci., 98 (1995) 107–117. [3] Y. Ziani and R. Ben Aim, in Proceedings of the Euromembrane ’95 Conference, Bath, 1995. [4] T. Kawakatsu, M. Nakajima, S. Nakao and S. Kimura, Desalination, 101 (1995) 203–209. [5] D.J. Shaw, Introduction to Colloid and Surface Chemistry, 3rd ed., Butterworths, London, 1980. [6 ] E.J.W. Verwey and J.T.G. Overbeek, Theory of the Stability of Lyophobic Colloids, Elsevier, 1948. [7] A. Grabbe and R.G. Horn, J. Colloid Interface Sci., 157 (1993) 375–383. [8] W.A. Ducker, T.J. Senden and R.M. Pashley, Nature, 353 (1991) 239–241. [9] R.K. Iler, The Chemistry of Silica, Wiley, 1979. [10] F. Tiberg, personal communication. [11] W.R. Bowen and F. Jenner, Chem. Eng. Sci., 50 (11) (1995) 1707–1736. [12] J.N. Israelachvili, Intermolecular and Surface Forces, 2nd ed., Academic Press, London, 1991. [13] B.V. Derjaguin, T.N. Voropayeva, B.N. Kabanov and A.S. Titiyevskaja, J. Colloid Interface Sci., 19 (1964) 113–135. [14] D. Elzo, I. Huisman, E. Middelink and V. Gekas, Colloids Surfaces A: Physico Chem. Eng. Aspects, accepted for publication. [15] I.H. Huisman, D. Johansson, G. Tra¨ga˚rdh and A.Ch. Tra¨ga˚rdh, J. Membrane Sci., submitted for publication. [16 ] F.A.L. Dullien, Porous Media, Fluid Transport and Pore Structure, 1979. [17] J. Levine, R. Marriott, G. Neale and N. Epstein, J. Colloid Interface Sci., 52 (1) (1975) 136–149. [18] K.M. Persson, Ph.D. thesis, Lund University, Sweden, 1994. [19] F.F. Nazzal and M.R. Wiesner, J. Membrane Sci., 93 (1994) 91–103. [20] R.M. McDonogh, C.J.D. Fell and A.G. Fane, J. Membrane Sci., 21 (1984) 285–294.
Properties of the cake layer formed during crossflow microfiltration Ingmar H. Huisman *, Dominique Elzo, Erik Middelink, A. Christian Tra¨ga˚rdh Food Engineering Department, Lund University, P.O. Box 124, 221 00 Lund, Sweden Received 17 June 1996; accepted 23 November 1996
Abstract The conditions necessary for the formation of a reversible cake layer during crossflow microfiltration were studied both experimentally and theoretically. Crossflow microfiltration experiments were performed with suspensions of silica particles with a narrow size distribution. The steady-state flux was first measured at a low transmembrane pressure (TMP), then at increased TMP, and again at the original low TMP. The cake-layer thickness was measured indirectly using a light absorbance technique. The thickness of the cake layer increased with increased TMP. Upon decreasing the TMP, the cake-layer thickness either decreased (reversible cake), or stayed constant (irreversible cake). It was shown that irreversible cakes are formed when the silica particles have a relatively low charge, whereas reversible cakes are formed when the silica particles have a relatively high charge. The occurrence of irreversible cakes is unexpected, since approaching silica particles are reported to always repel each other. The irreversibility of the cakes was explained by the assumption that bridging between the particles can occur, causing the interparticle interaction to be attractive when the particles retreat. To explain the reversibility results quantitatively, a model was developed which links the physicochemical interaction forces of the silica particles to the permeate flux through the cake layer. A detailed description of the interaction forces of silica particles was given in order to feed this model with accurate parameters. A reversibility index was introduced which quantifies the amount of reversibility. Model calculations of the reversibility index were in excellent agreement with measurements. © 1998 Elsevier Science B.V. Keywords: Cake layer; Microfiltration; Pore plugging; Silica particles; Zeta potential
1. Introduction Crossflow microfiltration, a separation technique for removing colloidal particles from suspensions, has been successfully applied in industry. However, the potentially high capacity of crossflow microfiltration is still limited by fouling of the membrane [1]. The term ‘‘fouling’’ includes * Corresponding author. Tel.: +46 46 2229820; Fax: +46 46 2224622. 0927-7757/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0 9 2 7 -7 7 5 7 ( 9 6 ) 0 3 97 6 - 3
different processes, such as adsorption of macromolecules on the membrane surface or within the pores, and the formation of a filter cake-layer. Cake-layer formation is an important cause of flux decline in the microfiltration of colloidal suspensions, for example in the clarification of fruit juices or the recovery of pigment from paint. To limit the build-up of a cake layer, regular backflushing or membrane cleaning are commonly applied. Both methods will be relatively effective if the cake layer is reversible (i.e. removing the
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transmembrane pressure ( TMP) resolves the cake). These techniques will, however, be more time- and energy-consuming if the cake is irreversible (i.e. upon removing the TMP, the cake stays on the membrane). Information on the reversibility of cake-layer build-up is therefore of great practical importance in microfiltration applications. Benkahla et al. [2] and Ziani and Ben Aim [3] measured the reversibility of cakes formed during crossflow microfiltration of CaCO suspensions. 3 They increased the TMP stepwise up to a maximum value and then decreased it stepwise to zero. They found that a cake layer was formed on the membrane, and that the resistance of this cake layer increased upon increasing the TMP. However, this resistance stayed at a constant level upon decreasing the TMP. ( The plot of flux vs. TMP was a straight line.) In other words, cakelayer formation was found to be irreversible. This phenomenon was qualitatively explained by assuming that ‘‘some cohesive forces’’ hold the particles together in the cake. Ziani and Ben Aim [3] found that upon increasing the wall shear stress, this hysteresis disappeared: the plot of flux vs. TMP followed more or less the same curve with decreasing TMP as it had done with increasing TMP. A quantitative description of cake-layer properties generally starts off with Darcy’s law ( Eq. (1), which links the permeate flux J to the TMP: J=
TMP R m tot
(1)
where R is the total hydraulic resistance and m tot is the viscosity of the permeate. Resistance in series models are often used to describe the total resistance as the sum of two contributions, one caused by the membrane and one caused by the cake layer (R ). The resistance caused by the memcake brane during microfiltration of a suspension can be expected to be higher than the membrane resistance for pure water (R ), since some of the mem pores might be plugged, reducing the effective porosity and thus increasing the hydraulic resistance. This effect can be described by adding an extra term (R ) to the membrane resistance. This pp
leads to: R =R +R +R (2) tot mem pp cake The cake resistance can be calculated by the Kozeny–Carman equation [16 ] if the cake’s void fraction e, and the cake-layer thickness d are c known: =
180(1−e)2
d (3) c d2 e3 p where d is the particle diameter. p For compressible particles, e can be as low as zero. For the laboratory experiments described in this paper, hard silica particles were used, which were assumed to be practically incompressible. For incompressible monodisperse particles e can theoretically vary between 0.2595 and 0.97 [4]. Benkahla et al. [2] observed that e depends slightly on the TMP (e decreases as TMP increases), and that e could be as low as 0.18 for polydisperse systems. Solutions flowing through narrow capillaries show an increased apparent viscosity [17]. This phenomenon, called the electroviscous effect, is caused by the interaction between solutes and fixed wall charges. In microfiltration applications, both the membrane and the cake layer form capillary channels which give rise to different electroviscous effects. These effects have been reported [11,18] to cause an additional decrease in permeate flux. Increasing the charge of the suspended particles in a microfiltration process might thus give two different effects. The interparticle repulsion increases, thus increasing the cake’s void fraction and increasing the permeate flux. At the same time, electroviscous effects increase the effective viscosity of the permeate and thus decrease the flux. Both effects have been described theoretically and observed experimentally [19,20]. It can thus be seen that many different phenomena play a role in the characterization of cakelayer properties, such as void fraction, thickness, and reversibility. A complete understanding of the microfiltration cake layer has therefore not yet been obtained. Particularly little is known about the reversibility properties of the cake layer. In this paper we present results from microR
cake
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filtration experiments performed with silica particles with a narrow size distribution. The irreversibility observed by other authors [2,3] for CaCO suspensions was also found for silica sus3 pensions, but only if the silica particles had a low surface charge. The main aim of this paper is to explain this irreversibility quantitatively, and to predict under which circumstances it can be expected. In order to do this, an equation is derived which links the permeate flux through the cake layer to the repulsive interaction forces between particles in the cake. In addition to this equation, some specific information on the interparticle forces is necessary. Therefore, a description of the interparticle forces for silica is presented in which special attention is paid to the coagulation phenomenon, since this appears to be the cause of irreversible cake-layer build-up. Since the equation derived assumes a uniform cake with a constant void fraction, the void fraction e of the cake layers obtained was determined in order to prove the validity of this assumption.
2. Theory
2.1. Interaction forces between silica particles Grabbe and Horn [7] and Ducker et al. [8] measured the interaction forces between silica surfaces, and showed clearly that silica surfaces repel each other, upon approaching, for all distances, even if surface charges are low and salt concentrations are high (see Fig. 2). For many other materials there is a region where the net interaction force is attractive (primary minimum [5]). The interaction forces between silica surfaces do not show a primary minimum because of two effects [9]. The van der Waals attraction for silica is low, and an additional repulsive force is present, i.e. the hydration force. Silica surfaces are hydrated, and in order to move them very close together (distances of less than 1 nm), these hydration shells have to be removed. To do this, a large force is necessary. It is, however, a well known fact [9] that silica
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particles coagulate under certain circumstances ( low pH (=low surface potential ) and high salinity), even though the observed interaction forces are always repulsive. To solve this apparent paradox, it is assumed that chemical bridges can be formed upon contact (silica bridges [10] or ionic bridges (for example Na+, Ca2+) [9]), which cannot be observed when the interparticle distance decreases (particles approaching each other), but which give an attractive force when the interparticle distance is increased again (particles retreating). The plot of interparticle force vs. distance should therefore show hysteresis if the particles are pressed sufficiently close together (see Fig. 1). This is observed experimentally for the force between two glass particles [10]. To be able to do quantitative work on cake reversibility for silica particles, a description of the force–distance relation for approaching and retreating particles would be helpful. The present authors have not seen any quantitative description for the case of retreating particles. For the case of approaching particles, the situation is slightly better. Combining standard DLVO theory [6 ] with a charge regulation model [12] gives a good description of the interparticle forces [7,8]. For distances less than about 4 nm, hydration forces have to be taken into account. Grabbe and Horn [7] show that the hydration force depends only slightly or not at all on physicochemical parameters like surface potential and ionic strength. They give a description of the hydration force obtained by fitting experimental data by: F =F exp(−D/D )+F exp(−D/D ) (4) hydr 1 1 2 2 where F is the hydration force, D is the hydr interparticle distance, and F , F , D and D are 1 2 1 2 empirical parameters. Grabbe and Horn obtained the parameters F /2pa=0.14 J m−2, D =0.057 nm, 1 1 F /2pa=5.4×10−3 J m−2 and D =0.48 nm (a is 2 2 the particle radius). Eq. (4) is somewhat arbitrary, and the accuracy is certainly not better than ±50%. Since the description of the hydration forces has only limited accuracy, there is no use trying to describe the other forces with a higher accuracy. The total interaction force (F ) is described as i the sum of the van der Waals force (F ), the vdW
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Fig. 1. Schematic plot of interparticle force for silica particles which are approaching each other and then retreating again.
double-layer force (F ) and the hydration force dl (F ): hydr F =F +F +F (5) i vdW dl hydr The van der Waals force is defined as: F = vdW
−∂E
vdW ∂D
(6)
where the van der Waals energy, E , can be vdW given by: A(D) E (D)=− vdW 12pD2
(7)
An empirical expression [7] for the Hamaker function A(D) is given by: A(D)=A (1+2kD) exp(−2kD) 0 +A /[1+(D/l)p]1/p (8) v where k=the Debye constant, and A , A , l and 0 v p are empirical or semi-empirical parameters. For silica surfaces interacting across water, the values A =0.288×10−20 J, A =0.726×10−20, l= 0 v 5.82 nm and p=1.433 give a good approximation [7]. It is observed for both mica and silica surfaces [12] that the standard ‘‘constant charge’’ approximation describes experimental data well for interparticle distances down to a few nm. Healy and
co-workers [5] have derived the following expressions for the constant charge double-layer interaction between two spheres of radius a: ∂E F =− dl dl ∂D
(9)
with the double-layer energy E given by: dl E =−2pe e ay2 ln[1−exp(−kD)] (10) dl r 0 0 where e e is the electromagnetic permittivity of r 0 the solution, a is the particle radius, and y is the 0 surface potential. For the silica particles used in this study, the value of the zeta potential is known. The zeta potential is the potential at the hydrodynamic shear plane. This plane is situated very close the Stern plane. In this study, following Israelachvili [12], we assume that both planes coincide and that they are at a distance of 0.55 nm from the surface (0.55 nm=one sodium-ion diameter+one hydrated sodium-ion radius). The effective surface potential can then be calculated from the zeta potential using the Gouy–Chapman theory [12], which states that:
C
2kT 1+c exp(−kx) y = log x e 1−c exp(−kx)
D
(11)
where y is the potential at distance x from the x
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surface, k=Boltzmann’s constant, T is the absolute temperature, e is the electron charge, and c is defined by: c=tanh(ey /4kT ) (12) 0 y is then calculated by solving Eqs (11) and (12) 0 for x=0.55 nm and y =zeta potential. x The use of Eqs (10) and (11) is questionable at close distances. This means that potentials calculated by Eq. (11) will not be quantitatively correct. However the objective of using Eq. (11) is not to predict the ‘‘real’’ surface potential, but rather to predict an effective surface potential which can be used in Eq. (10) to calculate interaction energies for larger distances (outside of the Stern planes). Combining Eqs. (4)–(10) relates the interparticle force to the surface potential. Since some approximations are made, the accuracy will be limited. To obtain an idea about the accuracy, we calculated force–distance curves for four test cases found in the literature ( Fig. 2 gives an example). From the results of these test cases, it can be concluded that our calculations give a reasonable approximation of the interaction between two silica surfaces. Errors made are not more then a factor of two, which could be expected based on
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the ±50% error in the expression for the hydration force. Bowen and Jenner [11] argue that errors are made if a two-body force is generalized to a concentrated suspension ( like a cake layer). Their arguments only hold if the average distance between particle ‘‘contact’’ is of the same order as the double-layer thickness. In our case the doublelayer thickness is typically 10 nm, whereas the particles have a much larger radius (about 240 nm), so that the interaction between one particle and all its neighbors is simply the sum of all two-body forces. The above discussion has resulted in an equation for the interparticle force between approaching spheres in concentrated suspensions. For the case of retreating spheres, a theoretical description is not available. For the current study, however, it is only necessary to know whether or not the forces upon retreating will differ from those upon approaching, in other words, whether or not bridging occurs between the particles. It is now assumed that if the particles approach each other closer than a certain distance D*, then bridging can occur and aggregates can be formed. If the interparticle repulsion is so strong that the particles cannot get
Fig. 2. Interaction force between two spheres of silica. Experimental values are replotted from Ref. [5], and lines are calculated by combining Eqs. (5)–(10).
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as close as D*, no bridging can occur and the system is reversible. The distance D* can be estimated to be of the order of 1 nm (a silica atom has a diameter of about 0.23 nm, a hydrated Na+ ion about 0.7 nm). 2.2. Electroviscous effects It has been reported [18] that changes in salt concentration of the feed water cause changes in the clear-water flux of ultrafiltration membranes. These changes are ascribed to electroviscous effects: the viscosity of a salt solution increases if flowing through a thin capillary, because of interaction between solutes and fixed wall charges. These effects were described quantitatively by Levine et al. [17]:
C
m
a = 1− m
D
8b(ef/kT )2(1−G)F −1 (kr)2
(13)
where m is the apparent viscosity, b=37/L a (where L denotes the molar conductivity in V−1 cm2 mol−1), f is the zeta potential of the capillary surface, k is the Debye constant, and r is the capillary radius. G#0 if kr>10. F#1 if both kr>20 and |f|<100 mV. These effects are expected not only when a solution flows through the membrane pores, but also when a solution flows through a cake layer of colloidal particles. Since the channels in the cake are not cylindrical capillaries, Eq. (13) will be only an approximation. Since the cake layer and the membrane each have their own zeta potential and effective capillary radius, they will give rise to different apparent viscosities. Darcy’s law should therefore be rewritten to give: J=
TMP
(R +R )+m R a,mem mem pp a,cake cake This equation can then be rewritten as: m
(14)
TMP
J= m
A
m
m m a,mem R + a,mem R + a,cake R mem pp cake m m m
B
(15)
or TMP
(16) +R ) a,mem a,pp a,cake where the apparent resistance R is defined by: a m R ¬ a R (17) a m J=
m(R
+R
Eq. (16) looks exactly like Darcy’s law, but note that the apparent resistances are functions of the solution’s ionic strength (through the Debye constant) and of the surface’s zeta potential, unlike the resistances used in Darcy’s law. This has an important consequence for the relation between the observed cake resistance and the void fraction of the cake. The Kozeny–Carman equation ( Eq. (3)) only applies to R , not to R . cake a,cake Eqs. (13) and (17) then have to be used to transform the observed R into R , which a,cake cake can be used in the Kozeny–Carman equation. 2.3. Relation between interparticle forces within the filter cake and permeate flux It is intuitively clear that the particles within a filter cake will be pressed more strongly together if the pressure drop over the cake increases, and that the particles which are deeper down in the cake layer (i.e. closer to the membrane) are pressed more strongly together than the particles which lie at the surface of the cake layer. Particles which are pressed strongly together will show a mutual repulsion, according to Newton’s third law (action=−reaction). We now derive an equation which relates the repulsion between neighboring particles in the cake to the location of these particles within the cake, and to the pressure drop over the cake. The analysis is in parallel with the ‘‘disjoining pressure’’ approach, first developed by Derjaguin et al. [13] and used for silica suspensions by Bowen and Jenner [11]. In all practical microfiltration applications the pressure gradient (in bar m−1) across the cake in the z-direction (perpendicular to the membrane) is many orders of magnitude higher than the pressure gradient along the filter in the tangential direction. The pressure gradients in the x and
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y directions are therefore neglected in the following analysis. It is assumed in this section that the filter cake is constituted of identical spherical particles with a diameter of d (see Fig. 3). The membrane lies p below these spheres, and the bulk suspension lies above these spheres. The particles in the cake are in a hexagonal close packing. The particle under consideration is called S (sphere) and is situated at the origin of the coordinate system. Three particles lie on top of S and three below S. Six particles lie around S, but do not exert forces on S since it is assumed that there is no pressure gradient in the x or y directions. A standard mathematical derivation renders that the point where one of the neighbors touches S makes an angle w* with the vertical, and that sin w*=1/E3 (or w*=35.26°). Since the particle S does not move, the sum of all forces exerted on it must be zero. Two different forces are exerted on S: the forces of interaction with neighboring particles according to the theory described in Section 2.1, and the drag force caused by the permeate flux. Because of the symmetry of the system, the magnitude of these forces is a function of the z-coordinate only, not of x and y. Adding the z-components of the forces then gives:
by:
∑ interaction forces=−3F (z*) cos w* (18) i above where z*=d /2 cos w*, the z coordinate of the p position where the particles touch. In a similar manner, the interaction forces caused by the three particles below S are obtained. The total sum of interaction forces is then given
ˆ is the specific hydrodynamic resistance of where R the packed bed. Different equations are found ˆ (for example the to describe the value of R Kozeny–Carman equation and the Blake–Kozeny equation [1]), which assume that the resistance is a function of particle size and of the void fraction of the cake. The particle size is constant in the experiments described in this paper, and it will be shown in Section 3 that the void fraction is independent of the pressure drop over the cake. ˆ can be taken as a Therefore the value of R constant. Since in Eq. (22) the permeate flux J and the viscosity m cannot be functions of z, the product ˆ is a constant K. Eq. (22) can be integrated to JmR give:
Fig. 3. Schematic picture of a particle S and its neighbors in a cake layer.
∑ interaction forces=3[F (−z*)−F (z*)] cos w* i i (19) From sin w*=1/E3, it is found that cos w*= E(2/3), so that
C A B A BD
d ∑ interaction forces=E6 F z=− p i E6 −F
d z= p i E6
(20)
The drag force caused by the permeate flux (=the hydrodynamic force F ) can be calculated by the hyd well-known equation: F = hyd
P
P dA
(21)
surface where P is the absolute pressure in the fluid, and A is the particle surface area. In general, the pressure drop over a cake layer or any other packed bed is given by a variant on Darcy’s law: dP dz
ˆ =JmR
(22)
P(z)=P +Kz (23) 0 where P is the pressure at z=0. 0 As stated before, the hydrodynamic force has only a component in the z direction. Integrating only the z-component of the forces, using
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dA=2pa dz and cos w=z/a, gives: F = hyd
P
p
neighbors. Integrating Eq. (29) gives the solution
P(z)(−cos w)2pa2 sin w dw
(24)
0 Substituting Eq. (23) in Eq. (24) gives:
F =2pa2 hyd
P
p 0
(P +Ka cos w)(−cos w) sin w dw 0 (25)
Solving this equation results in F =−4/3pK(d /2)3=−1pKd3 hyd p p 6
(26)
The sum of the hydrodynamic forces ( Eq. (21)) and interaction forces ( Eq. (15)) acting on particle S must vanish, so that:
C A
B A
d d E6 F z=− p −F z= p i i E6 E6
BD
−1pKd3 =0 p 6
The last term on the left-hand side does not depend on z, so that F has to be linearly dependent on z, i and Eq. (27) can be rewritten as: dF 2d i p =−1pd3 K 6 p dz E6
(28)
where ˆ. K=JmR Eq. (28) can thus be rewritten to give: −p i= ˆ Jm d2 R dz 12 p
dF
(30)
Eq. (30) gives the repulsive interaction force as a function of the location within the cake (z) and the permeate flux (J ). This equation clearly shows that the particles which are deeper down in the cake layer are subject to larger repulsive interactions. To obtain Eq. (30) a hexagonal close packing was assumed for mathematical convenience. Similar analyses can be done assuming other packings; these render equations similar to Eq. (30), but with leading constants which differ slightly from p/12. 2.4. Reversibility index
(27)
E6
p ˆ Jm(d −z) F = d2 R i 12 p c
The experiments described in this study are so-called ‘‘reversibility’’ experiments. The flux (J ) is first measured at a low TMP ( TMP1), then measured at a high TMP ( TMP2), and finally measured again at TMP1. In the total reversible case the flux finally measured at TMP1 will be the same as the flux originally measured at TMP1, i.e. J( TMP1,2nd)=J( TMP1,1st). In the total irreversible case, the cake formed at TMP2 will stay on the membrane after decreasing the TMP to TMP1. The total resistance R will therefore stay constant, or tot R ( TMP1,2nd)=R ( TMP2). Using Darcy’s law, tot tot this results in: J( TMP1,2nd)/TMP1=J( TMP2)/TMP2.
(29)
Up to now, it was assumed that the origin of the coordinate system was at the center of particle S. Since Eq. (29) is independent of the actual place of this origin, we can move the origin to the surface of the membrane, where the cake formation starts. If the cake has a thickness of d , then c F (d )=0, since the particles situated at the upper i c surface of the cake layer are not submitted to repulsive interaction forces from higher-lying
A reversibility index (RI ) is now introduced by: RI=
J( TMP1,2nd)/TMP1−J( TMP2)/TMP2 J( TMP1,1st)/TMP1−J( TMP2)/TMP2 (31)
The use of this equation can be seen in Fig. 4, where a typical reversibility experiment is shown. The RI as defined above is given by distance A/distance B. If the flux J( TMP1,2nd) is equal to J( TMP1,1st), then A=B, and RI=1 (total reversibility). If the flux J( TMP1,2nd) lies on a straight
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Fig. 4. A typical reversibility experiment. Reversibility index (RI )=A/B.
line going through J( TMP2) and 0, then A=0 and RI=0 (total irreversible case). The reversibility index RI, defined as in Eq. (31) is thus a parameter which shows the amount of reversibility in cake-layer formation. In standard cases, its value will be between 0 (irreversible cake layer formation) and 1 (reversible cake formation). The reason for irreversibility of cake layers in general might be that strong attractive van der Waals forces keep particles together, even if they are no longer pressed together by the pressure drop over the cake. In the case of silica particles, irreversibility is assumed to be caused by the formation of chemical interparticle bridges. These bridges can only be built if the particles are pressed closer together than D*, the critical distance for bridging. Particles pressed so close together show a repulsive interaction of more than F (D*). F (D*) is calculated by the analysis given i i in Section 2.1. The repulsive interaction between particles in a cake is a function of the location within the cake (z), and of the flux of water through the cake (J ), as given in Eq. (30). At a given flux J, there will be a position within the cake (z*) where the interaction force will be F (D*). This means that i all particles lying below z* will be pressed closer
together than D*, and they can therefore build interparticle bridges. The part of the cake which will stay on the membrane after decreasing the TMP will therefore be the part below z*. If z* is equal to the total thickness of the cake, obtained at 820 mbar, then the whole cake will be irreversible and RI=0. If z* is equal to or smaller than the thickness of the cake originally obtained at 420 mbar, then the part of the cake which was formed extra at 820 mbar will be totally washed away, and cake-layer formation is reversible, RI=1. If z* is somewhere between these extremes, the cake will be partly reversible, and 0
C
1 TMP2 d = −R −R 2 R mem pp ˆ mJ(TMP2)
D
(33)
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2.4.1. Case 1: reversible cake The thickness of the cake, originally obtained at TMP1 (d ) is calculated by Eq. (33) where the 1 index 2 is replaced by index 1. If z*
−
12F (D*) TMP1 i < pd2 J( TMP2) J( TMP1) p
(34)
2.4.2. Case 2: irreversible cake If F (D*) is very small, then Eq. (32) predicts i that z*$d . In that case the whole cake will be 2 irreversible, or RI=0. A criterion for an irreversible cake is therefore TMP2 J( TMP2)
−
12F (D*) i %m(R +R ) mem pp pd2 J( TMP2) p (35)
2.4.3. Case 3: partly reversible cake If d
TMP1
(36)
ˆ +R m(z*R
+R ) mem pp Inserting Eq. (32) and Eq. (33) gives: TMP1 J( TMP2)
(37) 12F (D*) i TMP2− pd2 p Combining this with the definition for the reversibility index (Eq. (31)) finally yields: J( TMP1,2nd)=
RI=
C
J( TMP2)
−
J(TMP2)
TMP2−12F (D*)/(pd2 ) TMP2 i p J( TMP1,1st) J( TMP2) − TMP1 TMP2
C
D
D (38)
3. Materials and methods The experiments discussed in this paper are crossflow microfiltration experiments of a model suspension of silica particles. The permeate flux was measured as a function of time at different transmembrane pressures ( TMP) and crossflow velocities. The experimental set-up used is depicted in Fig. 5, and described in detail elsewhere [15]. It used a tubular ceramic a-alumina membrane, purchased from SCT, France, with a nominal pore size of 200 nm. A uniform TMP 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 flux was regularly returned to the feed tank, the volume at the feed side of the membrane was constant. The crossflow velocity was 1 m s−1 (Re=6800, t =4.42 N m−2) for all experiments w described here. The silica particles used were kindly supplied by Nissan Chemical Industries Ltd. (Japan). They had a mean diameter of 480 nm; 90% of the particles are within the range 190–900 nm. Their characteristics are described in detail elsewhere [14]. The feed suspension was prepared based on double-distilled and Milli-Q filtrated 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 of the silica particles, as shown by Elzo et al. [14]. The particle concentration in the feed was continuously determined by measuring the turbidity ( light absorbance at 700 nm) of the feed suspension. If particles were deposited on the membrane, the particle concentration in the feed decreased. The deposited amount was then calculated from these particle concentration measurements. All experiments started off at a particle concentration of 1.7 g l−1. During an experiment, the particle concentration typically dropped to about 1.55 g l−1. In order to obtain information on the reversibility of cake-layer formation, experiments were performed according to the following scheme. The TMP was set at 420 mbar, and was kept at 420 mbar until a steady-state flux was reached
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Fig. 5. The microfiltration set-up.
(1–2 h). The TMP was then increased to 820 mbar and was kept at 820 mbar until a steady-state flux was attained (2–3 h). The TMP was reduced again to 420 mbar, and kept at 420 mbar until a steadystate flux was reached (this usually took less than half an hour). In addition to the experiments based on 420–820–420 mbar cycles, experiments were performed for 110–420–110 mbar cycles.
4. Results and discussion Typical results are given in Figs. 6 and 7, which depict two similar experiments, performed at the same crossflow velocity, and using the same series of TMPs. By adjusting the pH, the zeta potential of these particles has been changed. In Fig. 6 the zeta potential of the particles is slightly negative; in Fig. 7 this potential is more negative. The fluxes at 420 and 820 mbar are similar for both cases (a small difference in flux is observed, in accordance with earlier measurements [14]). However, upon reducing the TMP back to 420 (or 415) mbar, the
flux is reduced more for the case of the slightly negative zeta potential than for the case of the strongly negative zeta potential. The results of Figs. 6 and 7 can be replotted as flux vs. TMP, as shown in Figs. 8 and 9. In such plots, the irreversibility (or hysteresis) is shown more clearly: for low zeta potential (Fig. 8) it is seen that the flux decreases with a straight line back to zero if the TMP is decreased, as also observed by Benkahla et al. [2]. A straight line in such plots shows that the cake-layer resistance is constant, and that the cake must be irreversible. For strongly negative zeta potentials, a reversible flux vs. TMP curve is observed (Fig. 9). Figs. 6 and 7 also show the amount of deposition as a function of time. The deposition results agree with the reasoning given above. In the case of a low zeta potential ( Fig. 6), the amount of deposition hardly changes when the TMP is decreased from 820 to 415 mbar: the cake stays on the membrane. For the high zeta potential case, the amount of deposition decreases upon decreasing the TMP, and finally reaches a value close to the
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Fig. 6. Flux and amount of deposition vs. time. f=−26 mV, Crossflow velocity=1 m s−1. Both the flux and the deposited amount show irreversibility.
value obtained originally at 420 mbar: cake formation is totally reversible. From the measured flux, a total apparent resis-
tance (R ) can be calculated, using Darcy’s law a,tot ( Eq. (16)). The total apparent resistance is plotted vs. the amount of deposition in Fig. 10. It can be
Fig. 7. Flux and amount of deposition vs. time. f=−80 mV, crossflow velocity=1 m s−1. Both the flux and the deposited amount show reversibility.
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277
Fig. 8. Flux vs. TMP for a reversibility experiment. f=−26 mV, crossflow velocity=1 m s−1. An irreversible cake is formed.
seen that the points follow a straight line, which indicates that a uniform cake was formed. The specific resistance does not seem to be a function of time, nor a function of the cake thickness. It can be seen in Fig. 10 that changing the TMP changes the slope of the plot slightly. However, the change is very small. According to Eq. (16), R is the sum of three a,tot different contributions: R ,R and R . a,mem a,cake a,pp
Note that only R depends on the amount of a,cake deposition. Extrapolating the curve of Fig. 10 to zero deposition gives an intercept, which represents R +R . Since R is known from a,mem a,pp a,mem clear water flux measurements, a value for the pore plugging resistance can be obtained. It can be seen that the pore plugging resistance is a large effect (compare R #150×109 m−1). a,mem From Fig. 10 a value of the cake’s void fraction
Fig. 9. Flux vs. TMP for a reversibility experiment. f=−80 mV, crossflow velocity=1 m s−1. A reversible cake is formed.
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Fig. 10. Cake resistance vs. amount of deposition. f=−26 mV, crossflow velocity=1 m s−1.
e can be determined by the Kozeny–Carman equation ( Eq. (3)), while taking into account electroviscous effects, as described in Section 2.2. It is found for Fig. 10 that e=0.3060. e and R were detera,pp mined for cakes formed under different conditions. The pH of the feed suspension (zeta potential of the particles) and the TMP were varied. In Table 1 it can be seen that the scattering in values of R is so large that no relation can be shown a,pp between the pore plugging resistance and either TMP or zeta potential. The cake’s void fraction is not significantly influenced by the TMP, but seems to increase slightly when increasing the absolute value of the zeta potential, as shown more clearly Table 1 Void fraction e and apparent pore plugging resistance R f potential (mV ) −26 −26 −44 −45 −58 −61 −69 −80 −84
e ( TMP=420 mbar) 0.2916 0.3060 0.2992 0.2815 0.2833
a,pp
in Fig. 11. The mean void fraction is 0.30, which is reasonable compared to the value for a monodisperse hexagonal close packing (0.2595). To obtain values for the amount of reversibility, the reversibility index (RI ) was calculated according to its definition ( Eq. (31)). Besides this, a theoretical calculation of the RI was performed using Eq. (38). Adjusting D* to obtain the best fit resulted in a value of D*=2.0 nm. However, reasonable fits were obtained for all values of D* within the interval between 1.5 and 2.5 nm. The results for the two different sets of TMPs are shown in Figs. 12 and 13. It can be seen in both graphs that RI=0 for very low zeta potentials
of filter cakes formed under different conditions
e ( TMP=820 mbar)
0.2994
R (109 m−1) a,pp ( TMP=420 mbar) 886 485 600 181 213
0.3018 0.2887 0.3123 0.3104
0.3040 0.3192
R (109 m−1) a,pp ( TMP=820 mbar) 453
250 420 359 417
365 717
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279
Fig. 11. Void fraction vs. zeta potential for cakes formed at two different TMPs.
(irreversible case), and RI=1 for fairly high zeta potentials (reversible case). There is a transition zone in between. For the experiments at 420–820–420 mbar this transition zone is broader than for the experiments at 110–420–110 mbar. This is reflected in the calculated values. The main results obtained are that the void
fraction of the filter cake increases slightly, and that reversibility increases dramatically upon increasing the absolute value of the particles’ zeta potential. The latter phenomenon was explained by assuming that the interparticle distance is less than 2.0 nm for slightly negative zeta potentials and more than 2.0 nm for stronger negative zeta
Fig. 12. Calculated and experimental values of the reversibility index vs. zeta potential of the particles for TMP=420–820–420 mbar.
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Fig. 13. Calculated and experimental values of the reversibility index vs. zeta potential for TMP=110–420–110 mbar.
potentials. In other words, the interparticle distance increases with the absolute value of the zeta potential, which is in agreement with the observed increase in void fraction. If one assumes that the particles are in a monodisperse hexagonal close packing, it can be calculated that an increase in interparticle distance of 2 nm would result in an increase in e of about 0.008. This is about the order of magnitude which is observed in Fig. 9.
5. Conclusions During the crossflow microfiltration of suspensions of spherical silica particles, the flux is decreased by two effects: pore plugging and cakelayer formation. It was found that the filter cakes formed in the case of a feed suspension with a narrow size distribution had a void fraction e of 0.30±0.012. This value was independent of TMP, but increased slightly with the absolute value of the particles’ zeta potential. The cake-layer formation can be totally reversible (the cake resolves again after removing the transmembrane pressure ( TMP)), totally irreversible (the cake stays on the membrane after removing the TMP), or partly irreversible (a part of the
cake stays on the membrane after removing the TMP). It was shown that the reversibility of the filter cake depends strongly on the zeta potential of the silica particles, and besides this on the TMP. Irreversible cakes were observed if the silica particles had a relatively small charge (slightly negative zeta potential ), whereas reversible cakes were observed if the particles had a higher charge (stronger negative zeta potential ). A reversibility index was defined which ranged from RI=0 (totally irreversible case) to RI=1 (totally reversible case). The reason for irreversibility of cake layers of silica particles might be the formation of chemical bridges. In this paper it is assumed that these bridges are formed if the particles are closer together than a critical distance D*. Within the cake the particles are pressed together by the drag force caused by the permeate flux; the deeper the particles are in the cake, the closer they are pressed together. There will therefore be a position within the cake below which the particles are closer together than D*. The part of the cake below this position will be aggregated, and thus irreversible. To be able to predict which part of a filter cake will be reversible, it is necessary to find equations
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which link the interparticle distance to the flux and the location within the cake. This was done in two steps. First, an equation was derived which links the interparticle distance to the particle– particle interaction force (F ). This equation used i DLVO theory, to which hydration forces were added. The equation obtained was tested against experimental results found in the literature, and it was concluded that it was sufficiently accurate. Besides this, an equation was derived which links the interparticle forces to the flux and the location within the cake. The amount of reversibility was calculated by combining these two equations. The calculated values were in good agreement with the experimentally obtained values. Based on the model formulated in this paper, it could also be concluded that the cake’s void fraction e increases slightly if the zeta potential becomes more negative. This is in agreement with experimental observations.
Acknowledgment This work was financially supported by the Swedish Foundation for Membrane Technology (I.H.H.), and by an EC Human Capital and Mobility program (D.E.).
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