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A novel approach to analyzing concentration polarization of polysaccharide solutions S. Chang, S. Jamal, H. Chen, H. Zhou, Y. Hong and N. Adams
ABSTRACT This paper describes a novel approach to analyzing the concentration polarization (CP) in the shorttime constant flux filtration of polysaccharide solutions. With application of the pearl-necklace model based polymer solution permeability correlation, a transmembrane pressure (TMP) increase rate model was developed to relate the linear TMP increase rate measured in the short-time constant flux filtration to the solid concentration at the membrane surface, the polymer critical overlapping concentration, and the radius of gyration of the polymer molecules. Establishment of such a theoretical model allows analysis of the gelation propensity of the CP layer formed by polysaccharides based on the experimentally determined linear TMP increase rate. Key words
| concentration polarization, membrane bioreactor (MBR), membrane filtration, membrane fouling, polysaccharides
S. Chang (corresponding author) S. Jamal H. Chen H. Zhou School of Engineering, University of Guelph, 50 Stone Rd E. Guelph, Ontario, Canada N1G 2W1 E-mail: [email protected] Y. Hong N. Adams GE Water and Process Technologies, 3239, Dundas St. W, Oakville, Ontario, Canada L6J 4Z3
INTRODUCTION Gaining further understanding of membrane fouling is critical for the optimization and advancement of membrane bioreactors (MBRs). The short-time constant flux filtration is a widely used method to determine the critical flux and the filterability of MBR mixed liquor. Many researchers showed that the transmembrane pressure (TMP) rise in the short-time constant flux filtration of MBR mixed liquor can be characterized by a non-linear initial TMP increase followed by a linear TMP rise stage (Le Clech et al. ; Koseoglu et al. ; Wu et al. ; Navaratna & Jegatheesan ; Li et al. ). The linear TMP increase rate measured in the short-time constant flux filtration is usually used as an indicator to determine the critical flux and the fouling propensity of MBR mixed liquor. For the filtration of solutions containing polysaccharides, determination of the concentration polarization (CP) profile at the membrane surface is essential to develop deep understanding of the gelation propensity of the CP layer, the effect of water chemistry, and the interactions between the CP layer and the membrane. Dynamic modeling is usually used to study the CP at the membrane surface by solving a group of time-dependent equations, including the mass transport equation, the filtration equation, the permeability–concentration correlation, and the pressurespecific resistance correlations (Petsev et al. ; Bowen & Jenner ; Chen et al. ; Wang et al. ). doi: 10.2166/wst.2014.038
In this study, a model was developed to establish the relationship between the linear TMP increase rate measured in the short-time constant flux filtration, the solid concentration at the membrane surface, and the critical polymer properties affecting gelling behavior of the CP layer. The current common approach to analyzing the CP or cake properties is to link the filtration resistance to the specific resistance and the solid fraction in the CP or cake layer by using the Carmen–Kozeny or Happel correlation (Belfort et al. ; Foley ). Since the Carmen–Kozeny or Happel correlation treats the retained species as solid particles, the current filtration theory does not reflect the effect of molecular configuration and physical properties of polymer molecules in the CP layer. Such an inadequacy of the current filtration theory limits its efficiency in analyzing the CP layer formed by polysaccharides where the physical properties of molecules could play a critical role in determining the resistance characteristics of the CP layer. The model presented in this paper demonstrates the effect of the radius gyration and the critical overlapping concentration of macromolecules on the TMP increase rate and provides a practical approach to analyzing the gelling propensity of macromolecules in the CP layer based on the linear TMP increase rate measured in the short-time constant flux filtration. By using alginate as a model polysaccharide, we illustrated the methodology to analyse the CP of polysaccharide solution
S. Chang et al.
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based on the experimentally measured TMP increase rate. Establishment of such an approach is of significance to understand the resistance mechanisms, the gelation behavior, and the interactions between the macromolecules and the membrane surface. The developed model also provides a theoretical basis to determine the critical flux based on the gelling propensity of the CP layer.
MATERIALS AND METHODS The filtration system used in this study, as shown in Figure 1, consisted of a one-litre feed tank (1), a single hollow-fiber loop module (2), a pressure transducer with a pressure range of 14.5 to 15 psi (Cole Parmer, 68075-32) (3), a peristaltic permeate pump (Minipuls 3, Gilson) (4), a balance (CPA6235, Sartorius) for the permeate flow rate measurement (5), and a computer data logging system (6). The hollow fibers used in this study, which had a normalized pore size 0.04 μm, inner diameter 0.9 mm, and outer diameter 1.8 mm, were provided by GE Water Process Technologies (Oakville, Ontario, Canada). The fiber loop was 22 cm long with a total filtration area of 12.4 cm2. The filtration experiments were carried out in a dead-end constant flux mode with the permeate recycled back to the membrane tank. Filtration experiments were conducted at 10, 15, 20, 30, 40, and 50 L/(m2·h) (LMH) for 10 and 20 ppm alginate solution with a fresh hollow-fiber loop used for each of the filtration experiments. Sodium alginate used in this study was produced by Acros Organics (Fisher Scientific) and the alginate solutions used in the filtration experiments were made by adding 5 g/L concentrated alginate stock solution into 0.1 M sodium nitrate buffer solution to achieve a desired concentration. The pH of alginate solution was 8.33. An advanced tetra-detector size exclusion chromatograph (SEC) (Viscotek GPCmax, Malvern) was used to
Figure 1
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determine the molecular weight and intrinsic viscosity of the alginate molecules tested. The tetra-detector SEC used in this study was equipped with a multiple detector system that included a refractive index (RI), a photo-diode UV detector, a right and lower angle light scattering detector, and a viscometric detector. The tetra-detector SEC can directly determine the molecular weight by using the light scattering detectors without relying on the conventional SEC calibration, and measure the intrinsic viscosity of polymer using the equipped viscometric detector. The SEC columns used in this study were PolyAnalytik aqueous columns and the main SEC conditions included: mobile phase: 0.1 N KNO3; mobile phase flow rate: 1 mL/min; sample concentration: 5 mg/mL; injection volume: 100 μL; and temperature: 30 C. W
RESULTS AND DISCUSSION Filtration behavior of alginate Figure 2 shows an example of linear TMP increase profile a obtained in the dead-end filtration of 10 ppm alginate solution at different fluxes. From this figure, it can be seen that the TMP time-profiles obtained in the dead-end constant flux filtration of the alginate solution were characterized by a nonlinear initial TMP rise, followed by a linear TMP increase phase. Figure 3 shows the linear TMP increase rates measured under different flux conditions for the 10 and 20 ppm alginate solutions. Other experiments conducted in our group also showed that such a linear TMP increase stage occurred in the filtration of alginate solutions with addition of calcium and protein (data are not presented in this paper).
Schematics of the experimental set-up. (1) feed tank; (2) hollow-fiber membrane; (3) pressure transducer; (4) permeate pump; (5) balance; (6) computer data log system.
Figure 2
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TMP–time profiles of filtration of sodium alginate solutions of 10 ppm sodium alginate solution.
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TMP ¼ ηJRm þ ηJRCt þ ηJRC0 þ Δπ pb
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(1)
where J is the permeate flux (m/s); RCt is the hydraulic resistance caused by the bottom ‘dense’ CP layer, which changes with time (1/m); RC0 is the resistance caused by the concentration distribution beyond the bottom ‘dense layer’, which is time-independent at the linear TMP increase stage (1/m); Rm is the time–independent membrane resistance (1/m); Δπpb is the osmotic pressure difference between the permeate and the feed (Pa), and η is the viscosity of water (Pa s). The Δπbp can be formulated as Figure 3
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Linear TMP increase rate measured in the filtration of the 10 and 20 ppm
Δπ pb ¼ Δπ pc Δπ bc
(2)
sodium alginate solutions.
Linear TMP increase rate model Dynamic simulations of CP of charged colloidal solutions (Petsev et al. ; Bowen & Jenner ) showed that the solid fraction of a CP layer at the membrane surface reaches its maximum value when the long-range inter-molecule forces are no longer able to sustain the solid compression caused by the hydraulic drag induced by the permeate flow. Hence, the linear TMP increase phase observed in the short-time constant flux filtration can be assumed to be associated with the concentration profile described in Figure 4, which consists of a bottom CP layer with an equilibrium solid concentration and, beyond that, a concentration distribution zone with the concentration changing from the equilibrium concentration to the bulk concentration. The concentration distribution profile beyond the bottom ‘dense’ layer does not change with time for the constant flux filtration and the filtration resistance increase is caused by the increase of the thickness of the ‘condensed’ CP layer (Petsev et al. ; Bowen & Jenner ). For the concentration profiles described in Figure 4, the overall TMP at the linear TMP increase stage can be formulated as
where Δπbc is the osmotic pressure difference between the bulk solution and the CP layer and Δπpc is the osmotic pressure difference between the permeate and the CP layer. For a dilute polymer solution, Δπpb ≈ 0, so the TMP can be formulated as TMP ¼ ηJRm þ ηJRCt þ ηJRC0
(3)
Based on Equation (3), the TMP change rate under the constant flux condition can be expressed as dTMP dRCt ¼ ηJ dt dt
(4)
Introducing the thickness-based specific resistance, Equation (4) becomes dTMP dδ ¼ ηJαL dt dt
(5)
where δ is the thickness of the bottom CP layer (m), αL is the thickness-based specific resistance (1/m2). The growth of the equilibrium CP layer can be related to the mass deposition rate based on the mass balance ϕc dδ ¼ J(ϕb ϕp )dt
(6)
where φb is the bulk solid volume fraction; φp is the solid volume fraction in the permeate; φc is the solid volume fraction of the bottom CP layer. Combining Equations (5) and (6), the TMP increase rate is formulated as
Figure 4
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Schematics of the CP profile in the linear TMP increase phase.
dTMP ϕ ¼ ηJ 2 αL b ϕp dt ϕc
(7)
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A novel approach to analyzing CP of polysaccharide solutions
where φp is the membrane rejection to the species retained at the membrane surface. In this study, the membrane rejection to the alginate tested was determined to be around 98%. TMP increase rate model based on the permeability correlation The concentration of polymer solutions can be characterized by three different regimes: the dilute solution, the semi-dilute solution, and the concentrated regimes. In the dilute concentration regime, polymer molecules are separated from each other and there is a negligible overlapping between the polymer molecule chains; in the semi-dilute regime, polymer molecular chains are overlapped and entangled; and in the concentrated regime, polymer chains are closely packed so that no segment of a molecule chain can claim its own territory (Teraoka ). The semi-dilute concentration regime covers a wide range of concentration with its upper limit around 0.2 to 0.3 g/mL (Teraoka ). Johnson et al. () derived a model based on the pearl-necklace model (Flory ), as shown below, to calculate the permeability of polymer solution in the semi-dilute concentration regime 3=2 1 ϕ K ¼ R2G c 9 ϕ
(8)
where K is the permeability that is the reciprocal of the specific resistance αL; RG is the radius of gyration of the polymer; φ* is the critical overlapping concentration of the polymer, which is the low end boundary of the semidilution concentration regime. Then, combining Equations (8) and (7), the following equation can be derived 1=2
dTMP 9ηJ 2 ϕb ϕp ϕc ¼ dt R2G ϕ(3=2)
(9)
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flux filtration. To implement such an application, it is necessary to determine the critical overlapping concentration and the radius of gyration of the polysaccharides retained in the CP layer. In this study, such properties of the alginate molecules tested were measured by using the tetra-detector SEC, which can directly measure the molecule weight distribution by using the light-scattering detector, and the intrinsic viscosity of the molecules by using the equipped viscometer. Based on the determined molecular weight and intrinsic viscosity, the equipped tetra-detector SEC software can determine the radius of gyration using the theory of polymer physics. Thus, the average number-based molecular weight, the intrinsic viscosity, and the radius of gyration of the tested alginate molecules were determined by the tetra-detector measurement as 130.8 kDa, 9.7 dL/g, and 34.5 nm, respectively. The critical overlapping concentration of polymer solution can be calculated by (Teraoka ) ϕ
4π 3 Mn RG ¼ 3 NA ρp
(11)
where NA is the Avogadro number, ρp is the specific gravity of polymer, and Mn is the number averaged molecular weight. Based on Equation (11) and the SEC determined molecular properties, the critical overlapping volume fraction for the alginate tested in this study was determined as 0.00079. Figure 5 shows φc values estimated using Equation (10) based on the TMP increase rates shown in Figure 3. For the alginate solution tested in this study, the calculated solid volume fraction of the CP layer formed at the flux range of 15 to 50 LMH is higher than its critical overlapping concentration (0.00079), so it is in the semi-dilute concentration range, while the values calculated for the filtration at 10 LMH is still in the dilute concentration regime. It is also interesting to see that the effect of the flux on the CP varies
Then, φc, the solid concentration at the membrane surface, can be expressed as
ϕc ¼
!2 R2G ϕ(3=2) dTMP 9ηJ 2 ϕb ϕp dt
(10)
Analysis of CP of alginate solution One of the main applications of Equation (10) is to analyse the CP of polysaccharides at the membrane surface based on the linear TMP increase rate measured in the short-time constant
Figure 5
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Solid volume fraction of the CP layer calculated based on Equation (10) for the filtration of 10 and 20 ppm alginate solutions.
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in different flux ranges. For the flux range of 10 to 30 LMH, the estimated solid fraction of the CP layer evidently increases with the flux, while a less sensitive and an opposite trend is shown in the flux range of 30 to 50 LMH. It is easy to understand that the solid fraction of the CP layer increases with the flux because of the increased hydraulic compression at the higher fluxes. The slightly reduced solid fraction with the increased flux in the flux range of 30 to 50 LMH, however, cannot be elucidated by the existing filtration theory without considering the effect of the molecular configurations of alginates. Taking account of such a factor, the slightly reduced concentration of the CP layer at the higher fluxes in the flux range of 30 to 50 LMH could be attributed to the change in the packing randomness and molecular configuration under the effect of the increased permeate flow. Figure 5 also shows that there is an insignificant difference in the solid fraction of the CP layer for the filtration with the 10 and 20 ppm alginate solutions under the same flux condition, indicating that the evident higher TMP increase rate observed in the filtration of the 20 ppm solution (Figure 3) was mainly attributed to the increased growth rate of the CP layer at the higher feed concentration rather than to a more compressed CP layer.
CONCLUSIONS With application of the polymer solution permeability correlation based on the pearl-necklace model, a TMP increase rate model was developed to relate the linear TMP increase rate measured in the short-time constant flux filtration to the solid concentration at the membrane surface, the polymer critical overlapping concentration, and the radius of gyration of the polymer molecules. Establishment of such a theoretical model allows analysis of the gelation propensity of the concentration polarization layer formed by polysaccharides based on the experimentally determined linear TMP increase rate.
ACKNOWLEDGEMENTS The authors thank Canadian Water Network, GE Water and Process Technologies, Ontario Centres of Excellence, and Canadian Foundation of Innovation for the support on this research.
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REFERENCES Belfort, G., Davis, R. H. & Zydney, A. L. The behaviour of suspensions and macromolecular solutions in crossflow microfiltration. Journal of Membrane Science 96, 1–58. Bowen, W. R. & Jenner, F. Dynamic ultrafiltration model for charged colloidal dispersions: a Wegner-Seitz cell approach. Chemical Engineering Science 50 (11), 1707– 1736. Chen, J. C., Elimelech, M. & Kim, A. S. Monte Carlo simulation of colloidal membrane filtration: model development with application to characterization of colloidal phase transition. Journal of Membrane Science 255, 291–305. Flory, P. J. Principles of Polymer Chemistry. Cornell University Press, Ithaca, New York. Foley, G. A review of factors affecting filter cake properties in dead-end microfiltration of microbial suspensions. Journal of membrane Science 274, 38–46. Johnson, M., Kamm, R., Ethier, C. R. & Pedley, T. Scaling laws and the effects of concentration polarization on the permeability of hyaluronic acid. PCH PhysicoChemical Hydrodynamics 9 (3–4), 427–441. Koseoglu, H., Yigit, N. O., Iversen, V., Drews, S., Kitis, K., Lesjean, B. & Kraume, M. Effects of several different flux enhancing chemicals on filterability and fouling reduction of membrane bioreactor (MBR) mixed liquors. Journal of Membrane Science 320 (1–2), 57–64. Le Clech, P., Jefferson, B., Chang, I. S. & Judd, S. J. Critical flux determination by the flux step method in a submerged membrane bioreactor. Journal of Membrane Science 227, 81–93. Li, J., Zhang, X., Cheng, F. & Liu, Y. New insights into membrane fouling in submerged MBR under sub-critical flux condition. Bioresource Technology 137, 404–408. Navaratna, D. & Jegatheesan, V. Implications of short and long term critical flux experiments for laboratory-scale MBR operations. Bioresource Technology 102 (9), 5361– 5369. Petsev, D. N., Starov, M. V. & Ivanov, I. B. Concentrated dispersions of charged colloidal particles: sedimentation, ultrafiltration and diffusion. Colloids and Surface A: Physicochemical and Engineering Aspects 81, 65–81. Teraoka, I. Polymer Solutions: An Introduction to Physical Properties. Wiley-Interscience, Electronic. Wang, X. M., Li, X. Y. & Waite, T. D. Quantification of solid pressure in the concentration polarization (CP) layer of colloidal particles and its impact on ultrafiltration. Journal of Colloid and Interface Science 358, 290–300. Wu, Z., Wang, Z., Huang, S., Mai, S., Yang, C., Wang, X. & Zhou, Z. Effects of various factors on critical flux in submerged membrane bioreactors for municipal wastewater treatment. Separation and Purification Technology 62 (1), 56–63.
First received 27 September 2013; accepted in revised form 10 January 2014. Available online 24 January 2014
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© IWA Publishing 2014 Water Science & Technology
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A novel approach to analyzing concentration polarization of polysaccharide solutions S. Chang, S. Jamal, H. Chen, H. Zhou, Y. Hong and N. Adams
ABSTRACT This paper describes a novel approach to analyzing the concentration polarization (CP) in the shorttime constant flux filtration of polysaccharide solutions. With application of the pearl-necklace model based polymer solution permeability correlation, a transmembrane pressure (TMP) increase rate model was developed to relate the linear TMP increase rate measured in the short-time constant flux filtration to the solid concentration at the membrane surface, the polymer critical overlapping concentration, and the radius of gyration of the polymer molecules. Establishment of such a theoretical model allows analysis of the gelation propensity of the CP layer formed by polysaccharides based on the experimentally determined linear TMP increase rate. Key words
| concentration polarization, membrane bioreactor (MBR), membrane filtration, membrane fouling, polysaccharides
S. Chang (corresponding author) S. Jamal H. Chen H. Zhou School of Engineering, University of Guelph, 50 Stone Rd E. Guelph, Ontario, Canada N1G 2W1 E-mail: [email protected] Y. Hong N. Adams GE Water and Process Technologies, 3239, Dundas St. W, Oakville, Ontario, Canada L6J 4Z3
INTRODUCTION Gaining further understanding of membrane fouling is critical for the optimization and advancement of membrane bioreactors (MBRs). The short-time constant flux filtration is a widely used method to determine the critical flux and the filterability of MBR mixed liquor. Many researchers showed that the transmembrane pressure (TMP) rise in the short-time constant flux filtration of MBR mixed liquor can be characterized by a non-linear initial TMP increase followed by a linear TMP rise stage (Le Clech et al. ; Koseoglu et al. ; Wu et al. ; Navaratna & Jegatheesan ; Li et al. ). The linear TMP increase rate measured in the short-time constant flux filtration is usually used as an indicator to determine the critical flux and the fouling propensity of MBR mixed liquor. For the filtration of solutions containing polysaccharides, determination of the concentration polarization (CP) profile at the membrane surface is essential to develop deep understanding of the gelation propensity of the CP layer, the effect of water chemistry, and the interactions between the CP layer and the membrane. Dynamic modeling is usually used to study the CP at the membrane surface by solving a group of time-dependent equations, including the mass transport equation, the filtration equation, the permeability–concentration correlation, and the pressurespecific resistance correlations (Petsev et al. ; Bowen & Jenner ; Chen et al. ; Wang et al. ). doi: 10.2166/wst.2014.038
In this study, a model was developed to establish the relationship between the linear TMP increase rate measured in the short-time constant flux filtration, the solid concentration at the membrane surface, and the critical polymer properties affecting gelling behavior of the CP layer. The current common approach to analyzing the CP or cake properties is to link the filtration resistance to the specific resistance and the solid fraction in the CP or cake layer by using the Carmen–Kozeny or Happel correlation (Belfort et al. ; Foley ). Since the Carmen–Kozeny or Happel correlation treats the retained species as solid particles, the current filtration theory does not reflect the effect of molecular configuration and physical properties of polymer molecules in the CP layer. Such an inadequacy of the current filtration theory limits its efficiency in analyzing the CP layer formed by polysaccharides where the physical properties of molecules could play a critical role in determining the resistance characteristics of the CP layer. The model presented in this paper demonstrates the effect of the radius gyration and the critical overlapping concentration of macromolecules on the TMP increase rate and provides a practical approach to analyzing the gelling propensity of macromolecules in the CP layer based on the linear TMP increase rate measured in the short-time constant flux filtration. By using alginate as a model polysaccharide, we illustrated the methodology to analyse the CP of polysaccharide solution
S. Chang et al.
1345
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A novel approach to analyzing CP of polysaccharide solutions
based on the experimentally measured TMP increase rate. Establishment of such an approach is of significance to understand the resistance mechanisms, the gelation behavior, and the interactions between the macromolecules and the membrane surface. The developed model also provides a theoretical basis to determine the critical flux based on the gelling propensity of the CP layer.
MATERIALS AND METHODS The filtration system used in this study, as shown in Figure 1, consisted of a one-litre feed tank (1), a single hollow-fiber loop module (2), a pressure transducer with a pressure range of 14.5 to 15 psi (Cole Parmer, 68075-32) (3), a peristaltic permeate pump (Minipuls 3, Gilson) (4), a balance (CPA6235, Sartorius) for the permeate flow rate measurement (5), and a computer data logging system (6). The hollow fibers used in this study, which had a normalized pore size 0.04 μm, inner diameter 0.9 mm, and outer diameter 1.8 mm, were provided by GE Water Process Technologies (Oakville, Ontario, Canada). The fiber loop was 22 cm long with a total filtration area of 12.4 cm2. The filtration experiments were carried out in a dead-end constant flux mode with the permeate recycled back to the membrane tank. Filtration experiments were conducted at 10, 15, 20, 30, 40, and 50 L/(m2·h) (LMH) for 10 and 20 ppm alginate solution with a fresh hollow-fiber loop used for each of the filtration experiments. Sodium alginate used in this study was produced by Acros Organics (Fisher Scientific) and the alginate solutions used in the filtration experiments were made by adding 5 g/L concentrated alginate stock solution into 0.1 M sodium nitrate buffer solution to achieve a desired concentration. The pH of alginate solution was 8.33. An advanced tetra-detector size exclusion chromatograph (SEC) (Viscotek GPCmax, Malvern) was used to
Figure 1
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determine the molecular weight and intrinsic viscosity of the alginate molecules tested. The tetra-detector SEC used in this study was equipped with a multiple detector system that included a refractive index (RI), a photo-diode UV detector, a right and lower angle light scattering detector, and a viscometric detector. The tetra-detector SEC can directly determine the molecular weight by using the light scattering detectors without relying on the conventional SEC calibration, and measure the intrinsic viscosity of polymer using the equipped viscometric detector. The SEC columns used in this study were PolyAnalytik aqueous columns and the main SEC conditions included: mobile phase: 0.1 N KNO3; mobile phase flow rate: 1 mL/min; sample concentration: 5 mg/mL; injection volume: 100 μL; and temperature: 30 C. W
RESULTS AND DISCUSSION Filtration behavior of alginate Figure 2 shows an example of linear TMP increase profile a obtained in the dead-end filtration of 10 ppm alginate solution at different fluxes. From this figure, it can be seen that the TMP time-profiles obtained in the dead-end constant flux filtration of the alginate solution were characterized by a nonlinear initial TMP rise, followed by a linear TMP increase phase. Figure 3 shows the linear TMP increase rates measured under different flux conditions for the 10 and 20 ppm alginate solutions. Other experiments conducted in our group also showed that such a linear TMP increase stage occurred in the filtration of alginate solutions with addition of calcium and protein (data are not presented in this paper).
Schematics of the experimental set-up. (1) feed tank; (2) hollow-fiber membrane; (3) pressure transducer; (4) permeate pump; (5) balance; (6) computer data log system.
Figure 2
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TMP–time profiles of filtration of sodium alginate solutions of 10 ppm sodium alginate solution.
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TMP ¼ ηJRm þ ηJRCt þ ηJRC0 þ Δπ pb
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(1)
where J is the permeate flux (m/s); RCt is the hydraulic resistance caused by the bottom ‘dense’ CP layer, which changes with time (1/m); RC0 is the resistance caused by the concentration distribution beyond the bottom ‘dense layer’, which is time-independent at the linear TMP increase stage (1/m); Rm is the time–independent membrane resistance (1/m); Δπpb is the osmotic pressure difference between the permeate and the feed (Pa), and η is the viscosity of water (Pa s). The Δπbp can be formulated as Figure 3
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Linear TMP increase rate measured in the filtration of the 10 and 20 ppm
Δπ pb ¼ Δπ pc Δπ bc
(2)
sodium alginate solutions.
Linear TMP increase rate model Dynamic simulations of CP of charged colloidal solutions (Petsev et al. ; Bowen & Jenner ) showed that the solid fraction of a CP layer at the membrane surface reaches its maximum value when the long-range inter-molecule forces are no longer able to sustain the solid compression caused by the hydraulic drag induced by the permeate flow. Hence, the linear TMP increase phase observed in the short-time constant flux filtration can be assumed to be associated with the concentration profile described in Figure 4, which consists of a bottom CP layer with an equilibrium solid concentration and, beyond that, a concentration distribution zone with the concentration changing from the equilibrium concentration to the bulk concentration. The concentration distribution profile beyond the bottom ‘dense’ layer does not change with time for the constant flux filtration and the filtration resistance increase is caused by the increase of the thickness of the ‘condensed’ CP layer (Petsev et al. ; Bowen & Jenner ). For the concentration profiles described in Figure 4, the overall TMP at the linear TMP increase stage can be formulated as
where Δπbc is the osmotic pressure difference between the bulk solution and the CP layer and Δπpc is the osmotic pressure difference between the permeate and the CP layer. For a dilute polymer solution, Δπpb ≈ 0, so the TMP can be formulated as TMP ¼ ηJRm þ ηJRCt þ ηJRC0
(3)
Based on Equation (3), the TMP change rate under the constant flux condition can be expressed as dTMP dRCt ¼ ηJ dt dt
(4)
Introducing the thickness-based specific resistance, Equation (4) becomes dTMP dδ ¼ ηJαL dt dt
(5)
where δ is the thickness of the bottom CP layer (m), αL is the thickness-based specific resistance (1/m2). The growth of the equilibrium CP layer can be related to the mass deposition rate based on the mass balance ϕc dδ ¼ J(ϕb ϕp )dt
(6)
where φb is the bulk solid volume fraction; φp is the solid volume fraction in the permeate; φc is the solid volume fraction of the bottom CP layer. Combining Equations (5) and (6), the TMP increase rate is formulated as
Figure 4
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Schematics of the CP profile in the linear TMP increase phase.
dTMP ϕ ¼ ηJ 2 αL b ϕp dt ϕc
(7)
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where φp is the membrane rejection to the species retained at the membrane surface. In this study, the membrane rejection to the alginate tested was determined to be around 98%. TMP increase rate model based on the permeability correlation The concentration of polymer solutions can be characterized by three different regimes: the dilute solution, the semi-dilute solution, and the concentrated regimes. In the dilute concentration regime, polymer molecules are separated from each other and there is a negligible overlapping between the polymer molecule chains; in the semi-dilute regime, polymer molecular chains are overlapped and entangled; and in the concentrated regime, polymer chains are closely packed so that no segment of a molecule chain can claim its own territory (Teraoka ). The semi-dilute concentration regime covers a wide range of concentration with its upper limit around 0.2 to 0.3 g/mL (Teraoka ). Johnson et al. () derived a model based on the pearl-necklace model (Flory ), as shown below, to calculate the permeability of polymer solution in the semi-dilute concentration regime 3=2 1 ϕ K ¼ R2G c 9 ϕ
(8)
where K is the permeability that is the reciprocal of the specific resistance αL; RG is the radius of gyration of the polymer; φ* is the critical overlapping concentration of the polymer, which is the low end boundary of the semidilution concentration regime. Then, combining Equations (8) and (7), the following equation can be derived 1=2
dTMP 9ηJ 2 ϕb ϕp ϕc ¼ dt R2G ϕ(3=2)
(9)
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flux filtration. To implement such an application, it is necessary to determine the critical overlapping concentration and the radius of gyration of the polysaccharides retained in the CP layer. In this study, such properties of the alginate molecules tested were measured by using the tetra-detector SEC, which can directly measure the molecule weight distribution by using the light-scattering detector, and the intrinsic viscosity of the molecules by using the equipped viscometer. Based on the determined molecular weight and intrinsic viscosity, the equipped tetra-detector SEC software can determine the radius of gyration using the theory of polymer physics. Thus, the average number-based molecular weight, the intrinsic viscosity, and the radius of gyration of the tested alginate molecules were determined by the tetra-detector measurement as 130.8 kDa, 9.7 dL/g, and 34.5 nm, respectively. The critical overlapping concentration of polymer solution can be calculated by (Teraoka ) ϕ
4π 3 Mn RG ¼ 3 NA ρp
(11)
where NA is the Avogadro number, ρp is the specific gravity of polymer, and Mn is the number averaged molecular weight. Based on Equation (11) and the SEC determined molecular properties, the critical overlapping volume fraction for the alginate tested in this study was determined as 0.00079. Figure 5 shows φc values estimated using Equation (10) based on the TMP increase rates shown in Figure 3. For the alginate solution tested in this study, the calculated solid volume fraction of the CP layer formed at the flux range of 15 to 50 LMH is higher than its critical overlapping concentration (0.00079), so it is in the semi-dilute concentration range, while the values calculated for the filtration at 10 LMH is still in the dilute concentration regime. It is also interesting to see that the effect of the flux on the CP varies
Then, φc, the solid concentration at the membrane surface, can be expressed as
ϕc ¼
!2 R2G ϕ(3=2) dTMP 9ηJ 2 ϕb ϕp dt
(10)
Analysis of CP of alginate solution One of the main applications of Equation (10) is to analyse the CP of polysaccharides at the membrane surface based on the linear TMP increase rate measured in the short-time constant
Figure 5
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Solid volume fraction of the CP layer calculated based on Equation (10) for the filtration of 10 and 20 ppm alginate solutions.
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S. Chang et al.
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A novel approach to analyzing CP of polysaccharide solutions
in different flux ranges. For the flux range of 10 to 30 LMH, the estimated solid fraction of the CP layer evidently increases with the flux, while a less sensitive and an opposite trend is shown in the flux range of 30 to 50 LMH. It is easy to understand that the solid fraction of the CP layer increases with the flux because of the increased hydraulic compression at the higher fluxes. The slightly reduced solid fraction with the increased flux in the flux range of 30 to 50 LMH, however, cannot be elucidated by the existing filtration theory without considering the effect of the molecular configurations of alginates. Taking account of such a factor, the slightly reduced concentration of the CP layer at the higher fluxes in the flux range of 30 to 50 LMH could be attributed to the change in the packing randomness and molecular configuration under the effect of the increased permeate flow. Figure 5 also shows that there is an insignificant difference in the solid fraction of the CP layer for the filtration with the 10 and 20 ppm alginate solutions under the same flux condition, indicating that the evident higher TMP increase rate observed in the filtration of the 20 ppm solution (Figure 3) was mainly attributed to the increased growth rate of the CP layer at the higher feed concentration rather than to a more compressed CP layer.
CONCLUSIONS With application of the polymer solution permeability correlation based on the pearl-necklace model, a TMP increase rate model was developed to relate the linear TMP increase rate measured in the short-time constant flux filtration to the solid concentration at the membrane surface, the polymer critical overlapping concentration, and the radius of gyration of the polymer molecules. Establishment of such a theoretical model allows analysis of the gelation propensity of the concentration polarization layer formed by polysaccharides based on the experimentally determined linear TMP increase rate.
ACKNOWLEDGEMENTS The authors thank Canadian Water Network, GE Water and Process Technologies, Ontario Centres of Excellence, and Canadian Foundation of Innovation for the support on this research.
Water Science & Technology
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69.6
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2014
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First received 27 September 2013; accepted in revised form 10 January 2014. Available online 24 January 2014
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