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1 Theoretical Foundations of Chemical Engineering, Vol. 39, No. 4, 5, pp Translated from Teoreticheskie Osnovy Khimicheskoi Tekhnologii, Vol. 39, No. 4, 5, pp Original Russian Text Copyright 5 by Polyakov, Kazenin. Membrane Filtration with Reversible Adsorption: The Effect of Transmembrane Pressure, Feed Flow Rate, and the Geometry of Hollow Fiber Filters on Their Performance Yu. S. Polyakov* and D. A. Kazenin** * New Jersey Institute of Technology, Newark, NJ, USA ** Moscow State University of Environmental Engineering, Staraya Basmannaya ul. /4, Moscow, 7884 Russia yuriypolyakov@lycos.com Received April 6, 4 Abstract The effect of transmembrane pressure, feed flow rate, and the geometry of hollow-fiber filters on their performance is studied for depth membrane filtration with reversible adsorption. Mathematical models for rectangular and radial filters are used to find the maximal operation times with a filter retention of not less than.9 at constant product (permeate plus filtrate flow rates for the treatment of a latex suspension in continuousflow and batch operations. It is shown that the maximal operation time and, hence, product volume is achieved when the feed flow rate approaches the initial permeate flow rate. It is also found that several filters with a lower initial permeate flow rate allow one to achieve a much longer operation time than one filter with the same total initial permeate flow rate. Depth membrane filtration (DMF with reversible adsorption is a novel pressure-driven membrane separation process in which the feed suspension is treated in a hollow fiber (HF filter so that the clarified liquid produced by the semipermeable membranes is made up of a mixture of permeate and filtrate []. The permeate is a liquid that passed through the semipermeable membranes, which reject almost all suspended particles. The filtrate is a liquid in which the concentration of suspended particles was considerably reduced due to their adsorption on the membrane surface as the suspension flowed around the hollow fibers. DMF can be implemented in rectangular or radial hollow fiber devices. Generally speaking, it can be implemented in any membrane device with a high membrane packing density in which the suspension moves around the external surface of tubular, capillary, or hollow fiber membranes in a direction normal to their axes. DMF differs from crossflow membrane filtration in that it produces permeate and filtrate instead of permeate and retentate, which are the conventional outlet streams in crossflow filtration [, 3]. As compared to dead-end membrane filtration, DMF produces one more outlet stream, filtrate. By contrast with conventional membrane separation processes, DMF plants are designed to operate in a single-pass treatment (continuous-flow or batch mode, rather than in a feed-andbleed or a multistage recycle operation, because these plants do not produce any retentate stream []. In [], we developed a mathematical model describing the process of depth membrane filtration with reversible adsorption in a rectangular hollow-fiber filter. The differential equations that account for the adsorption and peptization mass fluxes of particles in the boundary layer of surface interaction forces at the membrane surface and for the local and overall material and volume balance in the filter were solved using an approximate method based on the averaging of permeate flow rate. Example calculations with a latex suspension demonstrated the feasibility of DMF with reversible adsorption and its advantages over conventional ultrafiltration (UF and microfiltration (MF operations. The reason for the latter is that DMF beneficially uses the adsorption of suspended particles on the membrane surface instead of wasting power or other expensive resources on reducing the rate of particle deposition, which is now typical for UF and MF plants [, 3]. The purpose of this paper is to derive analytical expressions for evaluating the performance of a radial DMF filter and study the effect of transmembrane pressure, feed flow rate, and the geometry of DMF filters on their performance. As in [], we assume that the suspension under treatment in a radial DMF filter (Fig. is a dilute incompressible liquid with constant viscosity, which contains suspended quasi-lyophilic (quasi-stable particles. The feed concentration, along with the process temperature, remains constant. The porous HF membranes completely reject the suspended particles. The electrokinetic properties of the membrane surface and suspended particles, along with other assumptions, are adopted from []. The equation of mass conservation and the equation for the rate of reversible adsorption with appropriate initial and boundary conditions that govern DMF in a radial filter can be written as in []: /5/ MAIK Nauka /Interperiodica

2 MEMBRANE FILTRATION WITH REVERSIBLE ADSORPTION Fig.. Radial hollow fiber DMF filter: ( feed, ( hollow fiber, (3 filtrate, and (4 permeate. ( ( c = c when r =, t >, (3 c =, Γ = when t =, > r. (4 Here, w c ( rwc = s É, r r É/ = βc αé, = r r G r p rdr, (5 (6 (7 where χ, χ, and s are adopted from []. The permeate velocity averaged over the filter depth is written as V p ' (8 The filtrate concentration can be found from the expression (9 The product (permeate plus filtrate concentration and filter retention for a continuous-flow operation, in which the product withdrawn from the filter is continu- r G p = χ V p, V V p = χ É, = V r d r. p c f = c s É rdr c r r d / G p rdr. ously supplied for use in another process, can be determined from the equations É c c pf = c s rdr rdr / ( w r, ( ( For a batch operation, in which the product is accumulated in a separate product tank until the treatment of the entire slurry batch is completed, one can write c pf ( (3 The expressions for the adsorption coefficient β and peptization coefficient α are adopted from []. As in [], we will find analytical expressions for Γ and c for the case where V p = V av = const and by solving problem ( (4 and substitute the solutions into Eqs. (6 (3 to determine the desired performance parameters of the filter. We rewrite Eqs. ( (4 in the same form as we already studied in [] for a rectangular HF filter: where (4 (5 q c = c when x =, t >, (6 q c =, q γ = when t =, x >, (7 R = c pf /c. ' = c t s Érdr crdr / ( w t, w = R' = c pf ' /c r G av ( r q c q w c s q γ = , x q γ / = β av q c α av q γ, THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 39 No. 4 5

3 44 POLYAKOV, KAZENIN x V av = T( T (8 Problem (4 (7 has an analytical solution similar to that derived in []. With a new variable it takes the following form: c = at t < /, (9 ( ( ( The expressions for the time derivatives of the concentration of suspended particles and the specific deposit can be written in the following form: c/ = at t < /, (3 V χ É( rt, r d r dt, G q c c av ( r = , G q γ É av ( r = , ( r ξ ξ r = ln [ ( /( ], ξ G av ( = , G w av = χ V av ( r ξ ξ r = ln[ ( /( ], c c ξ( r = [ /( ] exp α av t ( α av sβ av I m α avsβ av t / m = α av w t sβ av m/ at t > ϑ ( r /, É = at t < /, É β av c α av ξ( r = [ ( /( ] exp α av t ( α av sβ av I m α avsβ av t / m = α av w t sβ av m/ at t > ϑ ( r /. c α av c ξ r = [ ( /( ] exp α av t ( α w av sβ av (4 sβ av / α av ( t / I α avsβ av w t / at t > ϑ ( r /, É/ = at t < /, (5 É β av c ξ r = ( ( /( exp α av t ( α av sβ av (6 I α avsβ av w t / at t > ϑ ( r /. The above expressions become the same as those for a rectangular DMF filter [] if we take z r = ( /, d = ( /, (7 where z is the axial coordinate of the rectangular filter and d is its depth. This means that all our calculations for a radial DMF filter will be applicable to a rectangular DMF filter with z and d determined from Eq. (7. The iterative procedure for determining V av will be the same as in the case of a rectangular DMF filter []. As a rule, the performance of industrially manufactured hollow fiber filters is maximized by determining the optimal values of feed flow rate and transmembrane pressure P. The value of pressure dictates the value of V and, hence, β and α. The value of V, together with the values of and d, determines the value of ξ, which is equal to χ V d/. Equation ( implies that the parameters and d are involved in the expression for calculating the filtrate concentration c f, or c(t, d, only as their ratio. It can easily be shown that the product concentrations c pf and c pf ' and the permeate velocity V p ' are also functions of this ratio, rather than separately and d. This fact can be helpful in maximizing the performance of a DMF filter at its design stage. In our calculations, we used the same values of electrokinetic and process parameters as in [] along with =. m, =.65 m, P = (5 5 4 Pa [], V = ( 3 5 m/s [], and = ( m/s. The relation between and V accounts for the proportions of filtrate and permeate in the product. Clearly, the least value of is dictated by the practical requirement that the feed flow rate should be greater than the initial permeate flow rate. In our calculations, THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 39 No. 4 5

4 MEMBRANE FILTRATION WITH REVERSIBLE ADSORPTION 45 Effect of feed flow velocity and transmembrane pressure on the performance of a DMF filter with a retention not lower than.9 Continuous flow Batch 3, m/s P 5, Pa V 5, m/s ξ t op, s V av /V t op, s V av /V the operation was terminated as the filter retention dropped to.9. The effect of and P, the latter determining the value of V, on the filter performance can be seen from the data summarized in the table. The data show that the maximal operation time and, hence, the greatest product volume for a given value of P are attained when the feed flow rate approaches the initial permeate flow rate (ξ for either continuous-flow or batch operation. The first reason for this is that the product in this case contains the highest proportion of permeate, which according to our assumption does not contain suspended solids at all. Second, the carry-away of suspended particles is the lowest due to the lowest liquid velocity in the interfiber space. Third, the lag time of the concentration front is the highest, which facilitates the adsorption of particles on the membrane surface [4]. This implies that DMF filters should be operated in a ξ mode. Evaluation of the effect of pressure on the performance of a DMF filter with ξ shows that the operation time decreases with increasing P. With a double increase in the pressure, which corresponds to a similar double increase in the feed flow rate, the operation time decreases by about.5 times. The reasons for this are the higher rate of particle carry-away and the lower lag time of the concentration front. It is interesting to compare the data for the continuous-flow and batch operations. The batch operation time is seen to be approximately 5% higher than the continuous-flow time, all other conditions being the same. It should be noted that all conclusions about the effect of feed flow rate on filter performance can easily be translated to the effect of filter depth on performance. As we already noticed above, the effect of increasing d is mathematically equivalent to the effect of decreasing and vice versa. As for the geometry of a rectangular DMF filter, it follows from the above that it should have a square cross section to provide the lowest feed flow rate and, hence, the best performance. The above relations can be very helpful in designing commercial DMF plants. For example, the fact that two DMF filters allow us to achieve an operation time.5 times longer than one filter with a double transmembrane pressure can become the determining factor in designing the plant flow diagram and cost analysis, especially when the filters are used as expendables, without flushing or backflushing. The data in the table enable us to demonstrate the advantage of DMF filters over conventional dead-end and cross-flow HF devices. Taking a dead-end HF device the same as the above DMF filter, we see that it could provide only about half the product volume for the same operation time. It should be noted that the product (permeate alone flow rate of the dead-end filter sharply declines in the first small portion of operation time (Fig.. This practically excludes its use in V ' p /V t 4, s Fig.. Variation of permeate velocity with time for a deadend HF filter ( P = Pa. THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 39 No. 4 5

5 46 POLYAKOV, KAZENIN continuous-flow operation. On the other hand, taking a cross-flow HF device the same as the above DMF filter, we see that the power consumption for the cross-flow device would be about ten times higher than that for the DMF filter because cross-flow inside-out UF and MF devices are operated at recoveries (ratios of permeate flow rate to feed flow rate lower than.. Thus, we showed that the performance of rectangular and radial DMF filters can be easily maximized by choosing a feed flow rate close to the initial permeate flow rate. It was also shown that the lower the initial permeate velocity, the longer the operation time with a retention equal to or greater than.9, and that two DMF filters yield an operation time two and a half times longer than one with a double transmembrane pressure. The results of this study imply that DMF filters could be much more efficient than existing dead-end and cross-flow UF and MF devices. ACKNOWLEDGMENTS We are grateful to S.V. Polyakov for his valuable remarks and comments, which were taken into account in preparing this paper. This study was supported in part by the Russian Foundation for Basic Research, project no NOTATION c concentration of particles in suspension, kg/m 3 ; c concentration of suspended particles in feed suspension, kg/m 3 ; c f filtrate concentration, kg/m 3 ; c pf product (permeate plus filtrate concentration for continuous-flow operation, kg/m 3 ; c pf ' product (permeate plus filtrate concentration for batch operation, kg/m 3 ; d equivalent filter depth for a rectangular HF filter, m; G p permeate flux per unit of slurry volume, /s; I m modified Bessel function of order m; q c radial concentration flux, kg/(m s; q γ radial specific deposit flux, kg/(m s; P pressure drop across the membrane, Pa; R filter retention for continuous operation; R' filter retention for batch operation; r radial coordinate, m; external radius of a hollow-fiber bundle, m; internal radius of a hollow-fiber bundle, m; s specific surface area of a DMF filter, m ; t time, s; t op operation time, s; V initial permeate velocity, m/s; V p permeate velocity, m/s; V p ' permeate velocity averaged over the filter depth, m/s; w filtration velocity, m/s; filtration velocity at filtelet, m/s; x corrected coordinate, m; z equivalent axial coordinate for a rectangular HF filter, m; α peptization (desorption coefficient, /s; β adsorption coefficient, m/s; Γ specific deposit of particles, kg/m ; ϑ corrected distance, m; ξ ratio of permeate flow rate to feed flow rate. SUBSCRIPTS av average value; f filtrate; p permeate; initial value, or value at filtelet. REFERENCES. Polyakov, Yu.S. and Kazenin, D.A., Membrane Filtration with Reversible Adsorption: Hollow Fiber Membranes as Collectors of Colloidal Particles, Teor. Osn. Khim. Tekhnol., 5, vol. 39, no., pp [Theor. Found. Chem. Eng. (Engl. Transl., vol. 39, no., pp. 8 8].. Cheryan, M., Ultrafiltration and Microfiltration Handbook, Lancaster: Technomic, Zeman, L.J. and Zydney, A.L., Microfiltration and Ultrafiltration: Principles and Applications, New York: Marcel Dekker, Tien, C., Granular Filtration of Aerosols and Hydrosols, Stoneham: Butterworth, 989. THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 39 No. 4 5

Beneficial Effect of Particle Adsorption in UF/MF Outside-In Hollow Fiber Filters. Yuriy Polyakov USPolyResearch, New Jersey Institute of Technology

Beneficial Effect of Particle Adsorption in UF/MF Outside-In Hollow Fiber Filters Yuriy Polyakov USPolyResearch, New Jersey Institute of Technology NAMS 2005 BENEFICIAL EFFECT OF PARTICLE ADSORPTION Slide