Particulate Membrane Fouling and Recent ... - ACS Publications

28 Particulate Membrane Fouling and Recent Developments in Fluid Mechanics of Dilute Suspensions GEORGES BELFORT, ROGER J. WEIGAND, and JEFFREY T. MAHAR

Downloaded by YORK UNIV on July 2, 2012 | http://pubs.acs.org Publication Date: January 1, 1985 | doi: 10.1021/bk-1985-0281.ch028

Department of Chemical Engineering and Environmental Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180

Fluid dynamic analysis is used to better understand the mechanism of particulate membrane fouling from dilute suspensions. The fluid mechanics of a neutrally buoyant particle moving in laminar flow in a slit with two porous walls is compared with that in a porous tube of similar dimensions. Trajectory calculations are used for several commercial membrane modules to estimate the propensity of one and ten micron radius particles to foul. Under the assumptions of the theoretical models, and all things being equal, tubular systems capture more particles from dilute suspensions than slits with two porous walls. Also, higher flux membranes lead to particle capture and fouling at shorter axial lengths for both tube and slit configurations. Although pressure-driven membrane processes are gaining wide acceptance in many new laboratory and industrial applications, and are even replacing traditional separation techniques such as distillation, ion exchange and centrifugation, they all suffer from a potentially serious and limiting phenomenon called concentration polarization (cp). Complications such as membrane fouling (mf), thought to be a direct result of cp, have proved to be extremely difficult to model theoretically or even predict experimentally. Well-defined and simple systems have thus been actively studied in order to gain a better understanding of mf. By arbitrarily defining mf as a twostep process: transport and solute association, Belfort and Altena (I) were able to separate the fluid mechanics of foulant materials in a membrane duct with the complicated physicochemical and electrokinetic phenomena that occur when these materials enter the solution- membrane interfacial (or near-field) region. They and others (2) have argued that a detailed understanding of each step will eventually lead to a better understanding of mf. Recently, several researchers have reported on solute association studies in which the effects of a model solute such as 0097-6156/85/0281-0383$06.00/0 © 1985 American Chemical Society

In Reverse Osmosis and Ultrafiltration; Sourirajan, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1985.


384

REVERSE OSMOSIS AND ULTRAFILTRATION

bovine serum albumin (bsa) on membrane performance has been evaluated (2"Z)* R api d irreversible adsorption of the solute onto and probably into the membrane surface has been reported. Also, studies on how suspended particles such as monodispersed polystyrene colloids behave in the presence and absence of dissolved salt or in the absence of a moving fluid have appeared in the literature (8^9). Transport processes or cp near the membrane-solution interface have been modeled using mass balances. Michaels (10) and later Blatt, et al. (11) were the first to present and analyze the gelpolarizatior (gpj model. Belfort and Altena (1_) give a detailed analyses of the limitations and refinements of the gp model. Modifications of the theory in the near-field region include osmotic effects (5,6,9,12) concentration-dependent viscosity and diffusivity effects (13,IT), and lateral migration effects (15,16). Only recently has some work been reported in the far-field region in which hydrodynamic forces dominate (17-19). Their results were used to explain the inability of the gp model to predict the correct permeation flux and power relationship between flux and axial velocity for suspensions. Analysis in a slit with one porous wall and in a porous tube have been reported. The work we report in this paper compares the fluid mechanics of a particle moving in laminar flow in a slit with two porous walls with that in a porous tube of similar dimension. Several commercial membrane modules are available with both these configurations and it is of practical as well as theoretical interest to compare these systems. After reviewing in the next section the Altena and Belfort (17) and the Weigand, et al. (18) theories for particle motion in a slit and tube, respectively, the results are presented. The significance of these results are then discussed especially in terms of currently available commercial modules. The paper closes with the conclusion that under the assumptions of the theoretical models, and all things being equal, tubular systems will capture more particles from dilute suspensions than slits with two porous walls. Theory Flow in a Porous Duct. Lateral migration of spherical rigid neutrally buoyant particles moving in a laminar flow f i e l d in a porous channel is induced by an inertial l i f t force (tubular pinch effect) and by a permeation drag force due to convection into the porous walls. Altena and Belfort (17) extended the analysis of Cox and Brenner (20) for particle motion in a non-porous duct to include the effect of wall porosity. They considered non-interacting particles in dilute solutions at some distance from the duct walls. Criteria were established under which the inertial and permeation drag force in the lateral direction can be vectorially added. Thus, when (i) (ii)

X ~ RepK2,

superposition is feasible since these terms are of the same order, A « Repic2, permeation is negligible and l i f t drag dominates in essentially a non-porous duct.


28.

BELFORT ET AL.

(iii)

A»

Fluid Mechanics of Dilute Suspensions

385

RepK2, inertial l i f t is negligible and permeation drag dominates in a very porous duct.

Solution of the Navier-Stokes equations impose additional restrictions on the system,

and


where Equation 1 . implies

For the case of a spherical particle of radius, a, moving in a laminar fluid flow f i e l d in a s l i t of half-height, £, or a porous tube of radius, R, where the wall flux is constant and independent of the axial coordinate z, (Figure 1) the convective flow term of order Rew i s , in dimensional form, given by (21,22)

where

and

Cox and Brenner (20) derived a general form for the lateral velocity due to inertial effects. Vasseur and Cox (23) and Ishii and Hasimoto (24) obtained the detailed coefficients of the general form given by "Cox and Brenner for the case of a plane and tube Poiseuille flow using intergal and Fourier-Bessel transform techniques, respectively. Since the boundary conditions for the disturbance flow at the porous wall is effectively zero at low relative wall permeabilities [Rew « 1], the Green's function for the porous wall problem was taken idential to the Green's function for the nonporous case (17). Consequently, for the inertial particle velocity the expression derived by Vasseur and Cox (23) and Ishii and Hasimoto (24) were used. For a neutrally buoyant particle (which is allowed to rotate) the lift velocity in dimensional form is given by:



386

where f ( 3 ) is the result of a numerical integration involving the undisturbed velocity profile and the Green's function. The function f ( 3 ) has been calculated using the expressions of Vasseur and Cox for a s l i t [ f s ( 3 ) ] and by Ishii and Hasimoto for a tube [ ^ ( 3 ) ] , and presented graphically in Altena and Belfort (J7) and in Weigand, et a l . (18) and in tabular form in Otis (25). The tabular data was used by Otis in a regression analysis to yield the following sixth order polynomial f i t for a s l i t :


f S (3) = 1.532139 - 12.182786 a + 21.652283 a2 + 4.495068 a3 28.176666 a* + 10.950694 a5 + 0.198042 a6

(8a)

where a = ( l + 3 ) / 2 , and a 5th order polynomial f i t for a tube (26):

^ ( 3 ) = 1.601512 3 - 0.860212 3 2 - 0.70634 3 3 - 2.734119 31* + 1.382202 3 5

(8b)

The functions g(3) and f(3) from Equations 5.-8. are shown in Figure 2. The maximum shown in g^(3) arises from neglecting terms of order Re w and higher order terms in Re w in Equation 4. Only the leading order term is used as a good approximation for the total permeation velocity. Two major differences between the twodimensional slit and the three dimensional tube are: (1) the tube shows higher wall suction for all values of 3, i.e., g*(3) > g s (3), and (2) the tube shows higher inertial lift away from the centerline but lower inertial lift away from the porous wall, w. Because of this the tube and slit have different equilibrium positions of 3* = 0.71 and 3* = 0.62, respectively. Thus, for 3 < 3* lift and permeation suction drag forces will enhance each other, while for 3 > 3* they will oppose each other. Particle Trajectories. Under the condition of superposition, when A ~ Rep* 2 , the permeation and inertial drag forces are added vectorial ly to give the net motion in the 3-direction. Terms of order Re w and higher (small wall velocities) and the pressure drop in the z-direction are ignored. Then, from Berman (21) for a slit and Yuan and Finkelstein (22) for a tube, the particle moves in the zdirection with a velocity:

In the 3-direction, adding Equations 4. and 7.



28.

BELFORTETAL.


Figure 1. Spherical p a r t i c l e , B, suspended in Poiseuille flow in a porous tube or a s l i t with two porous w a l l s .

Figure 2. Permeation functions g($) and l i f t functions f ( 3 ) f o r a porous tube ( t ) and a s l i t with two porous walls (s).


387

388


where Re p is the local Reynolds number Re p = atlm/v, i is replaced by R for a tube, and the appropriate functions for g(3) and f(3) are used. Dividing Equation 9. by Equation 10. we obtain:


* Substituting for U m from a material balance:

where n = 4 (tube) and n = 3/2 (slit) and % is replaced by R for a tube, we obtain after rearrangement:

which obtains the desired particle trajectory in a tube {% = R, n = 4) or in a slit (n = 3/2). Equation (13) gives the derived particle trajectory from the coordinates of the initial position of the particle entering the duct (z 0 , 3 0 ) to some later position (z,$). The well-behaved trajectories were solved numerically using an IBM 3033 computer with Euler's method of integration. In the above equations Y is a measure of the ratio of permeation drag to initial lift drag, and is given by

and 0 is a characteristic dimensional distance:

Results and Discussion Particle Trajectories. In Figure 3 trajectories are presented for laminar flow in a slit (a) and in a tube (b) for initial positions 3 0 = 0.3 and 3 0 = 0.9 (with z 0 = 0 in all cases) with Y as a parameter. It takes a particle about 6.0 and 8.0 dimensionless distance units (z/0) to reach the equilibrium positions of 3* = 0.71 and 0.62 in a non-porous tube and slit, respectively. It is important to note that at the low values of Y shown in Figure 3 the particles are captured at the permeable wall at the same axial position, (z/0) g a p o



28.

BELFORT ET AL.


Figure 3. P a r t i c l e t r a j e c t o r i e s in a s l i t (a) and tube (b) f o r 1. y = 0.0 and 2. y = 0.03.


389

390



independent of entrance radial position for 3 0 < 3 0 < 30-a where 3 0 is some critical position near the centerline. For this to occur the trajectories from each entrance position must converge to a unified pseudo-equilibrium trajectory prior to capture. For high values of y however, the particles move toward the wall with separate trajectories and are captured at different axial positions dependent on entrance radial position. This is shown in Figure 4 for 3 0 = 0.3 and 0.9. For 0.1 < 3 0 < 0.9 a critical flux value, Y C , is defined where for any value of y > yc the particles follow separate trajectories for different values of 3 0 . Figure 5 shows the critical capture distance (z/o) ap as a function of y for a porous tube and a slit with two porous°walls with 0.1 < 3 0 < 0.9. Figure 5 shows that Y c = 0.02 for a tube and Y C = 0.05 for a slit. It is also apparent that for any y for a given 3 0 the particle will reach the wall in a shorter axial distance (z/0) ap , in a tube than in a slit.

This

p

arises from the fact that the permeation function g(3) is higher for a tube than a slit in all cases. Also the parameter n from the material balance in Equation 12. is higher for a tube than a slit indicating a greater area for permeation. At starting positions near the wall (3 0 > 0.7) the curves for a slit and tube seem to approach one another as y increases. This is due to the fact that at high values of Y the particle reaches the wall in \/ery short axial distances for both the tube and slit. In summary Figure 5 shows that, under the constraints of the theory, a membrane in the porous tube configuration will foul at shorter axial lengths than a membrane in the slit configuration. Also higher fluxes lead to fouling at shorter axial lengths for both the tube and slit. Commercial Relevance. The above analysis can be used to test the design of commercial membrane modules for the treatment of dilute suspensions of neutrally buoyant particles in laminar flow. Design data were chosen for the membrane filtration modules of seven manufacturers including Abcor (USA), Berghof (Germany), DDS (Denmark), Dorr-Oliver (USA), Enka/Membrana (Germany), Romicon/ Amicon (USA), and Schleicher and Schull (Germany). Table I gives the product name, design configuration and membrane type for ultra-, hyper- and microfiltration units studied (27-33). Also in Table I are the physical parameters needed for a trajectory analysis of each unit. In many cases the necessary data were incomplete in the commercial literature, therefore estimates were made for some of the values

(3±935).

Figure 6 is a two dimensional velocity plot of maximum axial entrance cross-flow velocity, U m , versus membrane permeation velocity, v w . For the above trajectory theory to hold the stipulations Re p < 1 and Re w < 1 must both be met. These are both shown graphically for particles of 1 ym and 10 pm radii in Figure 6. Also shown are the lines of 100% recovery for the tube and slit configurations. Along these lines all of the fluid entering the duct permeates the membrane. Practical operating conditions with sufficient flow exiting the unit requires an operating region to the



28.

BELFORT ET AL.


Figure 4. P a r t i c l e t r a j e c t o r i e s in a s l i t (a) and tube (b) f o r 3. y = 0.05 and 4. y = 0.10.

Figure 5. C r i t i c a l capture distance, ( z / e ) g a p , as a function of y f o r a tube and a s l i t . o


391



392

Figure 6. Two dimensional velocity plot of maximum entrance crossflow velocity, U m , versus permeation velocity, v w , showning the operating ranges of commercial membrane filtration units by code (see Table I.).



28.

BELFORT ET AL.


393

left of this line. Also shown are lines of constant y which apply for the tube and slit. The operating ranges of the various commercial units studied are shown (see Table I for product code) and for the most part lie within or slightly outside the area in which the theory applies for model particles of 1 and 10 urn radii. Tables II and III list parameters K, y, 0 and L/0 calculated for each unit based on a 1 ym and 10 ym particle size, respectively. Also given are the critical capture distances, (z/0) g a p , for starting positions 3 0 = 0.1 and 0.9. By comparing th8 capture distances with the membrane unit length, L/0, it may be determined whether the system will capture particles and foul and to what degree fouling will take place. If (z/0) ap < L/0 for both the case of 3 0 = 0.9 and 0.1 the unit will be foule9 by most of the particles entering the duct. If, however (z/0)g P < L/0 for 3 0 = 0.9 but not for 3 0 = 0.1 only some of the particle? will be captured and cause fouling. If (z/0) g a p > L/0 for all 3 0 then the unit will not be fouled by particles°of the particular size considered. In Figure 6, the different regions of operation of hyper-, ultra- and micro-filtration are shown. They show that, in general, as the membrane permeability increases, axial velocity is increased. Some question remains regarding the applicability of the theories presented above to these results. Measurements of single particle trajectories (25) have shown that the theory applies for laminar flow above channel Reynolds numbers of one. How far above a value of one is not clear at present. What is clear is that the flow must be laminar. For porous tubes and slits, the onset of transition flow from laminar to turbulent occurs at a Reynolds number of about 4000 as compared to 2100 for flow in a smooth non-porous tube (36). Thus, the theory is probably not applicable to several of the entries shown in Table I (i.e., US1, US3, US5, UT1, UT2, and MT1). The entrance Reynolds numbers listed in Table I are probably on the high side since the entrance velocity used is a maximum and decreases along the flow path in the module. Also, the viscosity value used (water) is lower than that usually encountered and increases along the flow path. For the case with the 1 micron particle in laminar flow (Table II) capture and fouling occurs for the modules HT1, UT3-5, and MT2, while for the modules HS1 and US4 all the particles escape capture and ex-it the module downstream. When the y-value was yery large (> 10 ) , the critical capture distance was extremely short and could not be estimated with the current computer program. For this case the symbol NC - not calculable, is used in Table II. For the case with the 10 micron particle in laminar flow (Table III) capture and fouling occur for only the HT1 and UT5 modules while for all the other modules with laminar flow the particles escape capture and exit the module downstream. In summary, the data shown in Tables II and III suggest that for the standard operating conditions most commercial modules will not capture 10 micron diameter particles. Several modules will


394


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Particulate Membrane Fouling and Recent ... - ACS Publications

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