Dean Vortex Membrane Microfiltration Non-Newtonian Viscosity

The goals were to compare the performance of Dean vortex filtration for non-Newtonian fluids with that for Newtonian fluids and, using a shear-thicken...
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Ind. Eng. Chem. Res. 2002, 41, 494-504

Dean Vortex Membrane Microfiltration Non-Newtonian Viscosity Effects Maarten Schutyser† and Georges Belfort* Howard P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180

Many industrial feeds behave as non-Newtonian fluids, and little understanding exists as to their influence on cross-flow microfiltration (CMF) performance. The viscosity effects of a model non-Newtonian shear-thickening fluid were investigated in CMF with and without suspended silica particles in the feed. The goals were to compare the performance of Dean vortex filtration for non-Newtonian fluids with that for Newtonian fluids and, using a shear-thickening fluid, to establish, in a negative control experiment for equivalent axial flow rates, that the mean wall shear rates is higher in helical tube flow than in linear tube flow. Hence, the effect of viscosity on the formation of controlled centrifugal instabilities (Dean vortices) was studied. NonNewtonian behavior was imparted to the fluid through the addition of a cationic surfactant, tetradecyltrimethylammonium salicylate (TTASal), that could freely permeate the membrane and that exhibited little noticeable fouling. Because the membrane did not retain the micelles and freely passed the surfactant monomer, fouling behavior could be separately studied by adding silica particles to the micellar solution. Results were compared with a previously studied Newtonian system for both linear and helical CMF. Two new mass transfer correlations were obtained for a shear-thickening surfactant solution in the presence of a 0.1 w/w % silica concentration for laminar flow in linear (no vortices) and helical (with vortices) membrane pipes. The viscosity was described as a function of surfactant concentration and velocity. The correlation for Dean vortex laminar flow incorporated the effect of increasing viscosity on the stability of Dean vortices by adjusting the exponent of the Reynolds number in the expression Shhel ) 0.19(a/rc)0.07Re[0.55(η(v)0)/η(v)v))]Sc0.33 for Re < 800. Introduction Much attention has been paid to solute effects and the enhanced flux performance due to secondary flow on cake formation during cross-flow microfiltration (CMF). Recently, a study in our laboratory focused on the effect of viscosity on the formation and stability of Dean vortices1 and how it influences improved performance of systems with secondary flow. Experiments with varying polymer concentrations (and therefore varying viscosities) in the presence of suspended silica particles showed that, for relatively high viscosity (>12 mPa s), the flux advantages of the helical design (with secondary flow) as compared to the linear design (without secondary flow) were negligible. Aqueous solutions containing poly(ethylene glycol) (MW 10 kDa) and silica particles used in this earlier study exhibited Newtonian behavior, i.e., the viscosity was constant with varying flow or shear rates. This linear behavior between local shear stress and local rate of strain is often not present with complex industrial feeds (e.g., yeast broth),2 and therefore, it would be useful to analyze and predict filtration performance for non-Newtonian fluids. With non-Newtonian fluids, the viscosity can change significantly with flow rate, as is the case in this work. Several cationic surfactants in the presence of certain strongly binding counterions form micellar solutions * Corresponding author. Phone: 1(518)276-6948. Fax: 1(518)276-4030. E-mail: [email protected]. † Current address: Department of Food Technology and Nutritional Sciences, Wageningen University, Wageningen, The Netherlands.

with extraordinary rheological behavior.3-5 The surfactants are often long-chain quaternary ammonium or pyridinium ions, whereas the counterions are hydroxyor halo-substituted benzoates. The existence of micelles is especially dependent on the surfactant and counterion concentrations, the ratio between the cationic surfactant and counterion concentrations, the ionic strength, and the temperature of the solution. In addition to rodlike micelles, the existence of which depends on surfactant concentration, other micellar structures (e.g., different micellar networks) have been observed.6 Micellar structures are also strongly influenced by the ionic strength of the solution. At low salt concentrations, ionic micelles change their aggregation numbers only in a stepwise fashion, whereas at high salt concentrations, the micelles can also coalescence and break into pieces.4 One of the most-studied systems with dynamic wormlike micelles is with the cationic surfactant tetradecyltrimethylammonium and counterion salicylate (TTASal). The salicylate ion exhibits special adsorption and penetration abilities. Therefore, the tertiary structural characteristics of the micelles in solution will be altered as a result of changes in the surface potential and electrostatic double layer (Debye) thickness.7 In Figure 1, the chemical structures of the two compounds are shown. In this study, concentrations between 2 and 6 mmol/L of TTASal were used, and experiments were carried out at constant room temperature (25 ( 2 °C). In this dilute regime, the solution contained rodlike micelles with a diameter of approximately 30 Å and a length of about 500 Å.8 For TTASal, the “overlap” concentration was 5 mmol/L.3 Below this concentration,

10.1021/ie010448o CCC: $22.00 © 2002 American Chemical Society Published on Web 01/11/2002

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Figure 1. Chemical structures of the counterion salicylate (left) and the cationic surfactant tetradecyltrimethylammonium (right).

the solution is hydrodynamically dilute, and above this concentration, it behaves as a semidilute polymer solution in which micelle-micelle interactions play an important role. These increased interactions can result in clustered micelles and, at higher concentrations, even in a transistion to another phase, such as branched micelles. Another effect of clustering micelles is a steep rise in viscosity with increasing concentration. Using solutions with such a high viscosity should cause a significant drop in the permeate flux during filtration. In addition to the four factors mentioned above (concentration, concentration ratio, ionic strength, and temperature), nonlinear viscosity effects resulting from changes in the micellar structure are also important. Work on the rheological behavior of dilute micellar solutions has shown that shear-induced collisions between initially short micelles lead to micellar growth beyond the equilibrium sizes.9 This statement is based on four observations: (i) The stress grows over a period of time that is exceedingly long in comparison to the relaxation time of the equilibrium solution. (ii) Above a critical shear rate, shear-thickening solutions exist. (iii) The relaxation times of the shear-thickening solutions are substantially longer than those of equilibrium solutions. (iv) The induction period for stress growth is inversely proportional to the shear rate. The structures that exist during shear are called shear-induced structures (SISs). The SISs consist of small clusters of rodlike micelles. With increasing shear rate, the clusters have a tendency to grow during flow. Shear-induced phase transitions are processes that are, under all conditions, completely reversible.5 The effect of SISs during flow can also cause a drastic decrease of the turbulent drag and leads hence to a gain in energy.5 The reason for the decrease in drag can be explained by assuming that the SISs suppress smallscale turbulence by resisting rapid changes in alignment. Research into the phenomenon of SISs for TTASal has been conducted.3,5,8 Wunderlich et al.5 reported that, for TTASal, no SISs could be observed below a surfactant concentration of 1.5 mmol/L. Therefore, concentrations between 2 and 6 mmol/L were used in this study. In this work, we investigate the behavior of a nonNewtonian fluid (containing the cationic surfactant tetradecyltrimethylammonium salicylate, TTASal), with and without suspended silica particles (nominal size of 2.4 µm). TTASal was used to study the behavior of nonNewtonian flow without fouling during microfiltration experiments at constant transmembrane pressure (TMP) for linear and helical membrane modules (a device containing multiple membranes). Because shear rate and shear stress are not proportional for a non-Newtonian fluid, the flow behaviors in the two modules were different from that of ordinary Newtonian fluids. The occurrence of fouling was checked by measuring the viscosity of the retentate and permeate, and it was found to be negligible. Viscosity measurements were carried out with a viscometer with a cylindrical spindle and a visometer with a cone-and-plate configuration to characterize the non-Newtonian behavior of TTASal.

A possible complicating factor was the unwanted interaction between the surfactant additive (TTASal) and the induced-fouling additive (2.4-µm silica particles). To reduce this interaction, the silica particles were hydrophilized (treated via low-temperature plasma in the presence of water vapor). This allowed us to distinguish between viscosity effects through micelle formation and shear-rate dependency and fouling of the membrane through the addition of silica particles. The results of this study were compared with those of a similar study in which Newtonian fluids were used to vary the viscosity. Kluge et al.1 used 10-kDa poly(ethylene glycol) and silica particles (the same as those used here) to obtain these effects. Theory Modeling of the buildup of fouling layers during filtration is needed to understand and control the effects of materials that foul the membrane. Generally, a model predicting permeate flux for cross-flow membrane filtration has to account for10 (1) the transport of particles in the direction normal to the membrane under crossflow conditions and (2) the accumulation of materials on the membrane surface, which causes the permeate flux to decline. Two basic theoretical models are described here. The first is a phenomenological resistance model, which is based on the cake filtration model. When a cake layer is formed as a result of the accumulated deposition of particles, which are retained by the membrane, an additional resistance to filtration is created. The resulting permeate flux decline during microfiltration (MF) can be described by Darcy’s law

J)

∆pTMP 1 dVp ) Am dt η(Rm + Rc)

(1)

where J (m3 m-2 s-1) is the permeate flux, Vp (m3) is the total volume of permeate, ∆pTMP (Pa) is the transmembrane pressure, Am (m2) is the external (to the pores) membrane surface area, η (Pa s) is the viscosity of the suspending fluid, Rm (m-1) is the membrane resistance, and Rc (m-1) is the cake resistance. The transmembrane pressure is defined as

∆pTMP )

pi + po - pp 2

(2)

where pi (Pa) is the inlet pressure, po (Pa) is the outlet pressure, and pp (Pa) is the permeate pressure. The second model is the concentration-polarization model or back-flux model, which is based on the film theory and is path-length-independent

J ) k ln

( ) cwall cfeed

(3)

where k (D/δ) (m s-1) is the overall mass transfer coefficient of the net mass transfer of solute from bulk to membrane surface, D (m2 s-1) is the mutual diffusion coefficient, and δ (m) is the film thickness. The wall concentration, cwall, is the solute concentration at the membrane surface. This gel-polarization model is assumed to operate in the pressure-independent regime and at any position along the flow path. It has its limitations because neither membrane resistance nor

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applied TMP are explicitly taken into account. Despite this deficiency, the model mostly predicts the flux decline during cross-flow filtration mainly through the mass transfer coefficient, k. Mass Transfer. The mass transfer that occurs during CMF is mainly controlled by fluid flow conditions. To examine mass transfer in the two modules, a dimensionless Sherwood number (Sh) is defined as

Sh ≡

kdi D

(4)

where di (m) is the inner diameter of the hollow fiber. The solute diffusion coefficient is calculated with the Stokes-Einstein relationship

D)

k*T 6πηrs

(5)

where k* (1.38 × 10-23 J K-1 ) is the Boltzmann constant, T (K) is temperature, and rs (m) is the radius of the solute particle. The mass transfer coefficient for laminar flow in linear CMF can be estimated from11

klinear ) 0.816

( ) 8vm 2 D dil

1/3

(m s-1)

(6)

where vm (m s-1) is the average velocity of the feed solution in the fiber bore and l (m) is the active fiber length. The mass transfer coefficient in the helical module can be calculated assuming that the concentration-polarization model holds for microfiltration and that the values for ln(cwall/cfeed) are relatively similar in the two modules. Hence, klinear/khelical ∝ Jlinear/Jhelical, which results in

khelical ) klinear

Jhelical Jlinear

(7)

Using these equations, the Sherwood number can be calculated for both modules. From the literature, experimental mass transfer correlations of the Sherwood number as a function of the Reynolds number (Re) and Schmidt number (Sc) are given by12

Sh ) RRebScc

(8)

where R, b, and c are constants; Re ) vmFdi/η; and Sc ) η/FD. The constant R, b, and c have been determined under different mass transfer conditions.12 The parameter R is dependent on the geometry of the spacer. For membrane processes, the exponent of the Reynolds number, b, ranges from 0.33 for fully developed laminar flow to 0.875 for fully turbulent flow. The exponent of the Schmidt number, c, is often set to a value of 0.33. Gehlert et al.13 proposed a new correlation for ultrafiltration of Dextran T500 solutions for laminar flow in a helical hollow-fiber module

()

Sh ) R

a rc

0.07

RebScc

(9)

where R is 0.43, b is 0.55, c is 0.33, a (m) is the radius of the tube, and rc (m) is the mean radius of the curvature. (a/rc) is called the “curvature” for flow around a curved tube (see below). In this study, we seek a

similar correlation for microfiltration of a non-Newtonian shear-thickening fluid containing surfactant (TTASal) and suspended silica particles. Dean Vortices. To describe flow in a curved channel (tube or slit), the Dean number (De) is introduced as14

x

De ) Re

a rc

(10)

Equation 10 shows that both the flow velocity and the curvature of the channel are important for describing Dean vortex flow. Also, a critical value of De () Dec) determines the flow rate above which vortices are substantial and measurable. The critical Dean number depends on the module design.15 Materials and Methods Materials. Chemical and Feed Suspensions. The micellar (TTASal) system consisted of (1) tetradecyltrimethylammonium bromide (TTA) (MW 336.4 g/mol, Sigma, 99% pure) and (2) sodium salicylate (Sal) (MW 160.11 g/mol, Fluka, >99% pure). All solutions contained equimolar concentrations of the two compounds. The solutions were prepared by adding the two compounds to a flask, which was then filled with deionized (DI) water to a final volume of 4 L. Subsequently, the solution was stirred until the compounds dissolved completely. The solution was left standing for 2 days at room temperature to equilibrate.5 To study performance with a non-Newtonian aqueous solution and an added foulant, silica particles were added to the solution. In previous research on Newtonian systems, silica particles were also used as the foulant.1 The particle size distribution of the silica particles (SiO2, MW 60.08 g/mol, 99% pure, Sigma) was between 0.5 and 10 µm, with 80% between 1 and 5 µm. The average size was 2.4 µm. Because possible adsorption of TTASal onto the silica particles (as a result of attractive interactions) might change the rheological behavior of the system, viscosity measurements were also conducted on the combined system. The systems reverted to Newtonian behavior for the TTASal solution with added silica. Clearly, the surfactant interacted with the silica particles in such a way that their ability to form SISs was affected. To reduce this interaction, the surface of the silica was covered with hydroxyl (-OH) groups. The technique used to attach these functional groups to the surface of the silica particles was low-temperature plasma treatment with water vapor. The plasma reactor was evacuated for at least 30 min (to a final system pressure of 0.01 Torr). The gas pressure was set to 0.15 Torr by adjusting the nitrogen flow rate. Then, the system was purged with water vapor, and the pressure was increased to 0.3 Torr total gas pressure. Plasma was created using a power supply (RF5S) with matching network (AM5 and AMNPS-2A, 13.67 MHz; all made by RF Plasma Products, Inc., Marlton NJ), connected to a copper coil surrounding the reaction chamber. For additional details on the low-temperature systems, see Ulbricht et al.16 Membrane Modules. Two membrane modules were used in this study. Both were hollow-fiber designs and were produced by Millipore Corporation (prototype experimental module 082997-1/3, Bedford, MA) according to specifications. The only difference between the two modules was the geometry of the hollow fibers. The

Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002 497 Table 1. Specifications of the Membrane Modules propertya

linearb

helicalb

membrane material mean pore size (µm) number of fibers, m active fiber length, l (mm) inner diameter of the fiber, 2a ) di (mm) outer diameter of the fiber, do (mm) membrane surface area, Am (cm2) diameter of the rod, drod (mm) radius of curvature, rc (mm) radius ratio, a/rc radius ratio, η critical Reynolds number, Rec

polyethersulfone 0.1 6 342.9 0.84 1.57 54 -

polyethersulfone 0.1 6 317.5 0.84 1.57 50 6.35 4.45 0.094 0.822 45.8

a Parameters defined in the text and in Figure 2. More information on the calculation of the radius ratio and the critical Reynolds number can be found in Mallubhotla and Belfort.15 b Millipore designations #082997-4 and #082997-1 for the linear and helical modules, respectively.

Figure 2. Linear and helical hollow-fiber module configuration.

linear module consisted of relatively straight parallel fibers, whereas the helical module contained hollow fibers wound in a single-wrap helix around an acrylic rod, as shown in Figure 2. The purpose of the rod was to stabilize the membrane helix and establish the desired curvature (a/rc). Specifications of the modules are given in Table 1. Methods. Viscosity Measurements. Two viscometers, one with a large aspect ratio (>1) (model LV-DV III, Brookfield, Stoughton, MA) and one with a small aspect ratio (250 s-1), an increased collision rate between the micelles structures induced growth of SISs, which resulted in an increase in viscosity. This increase in viscosity with shear rate in the laminar regime will be used to demonstrate, in a negative control experiment, that, for equivalent axial flow rates, the mean wall shear rate is higher in helical tube flow than in linear tube flow. Rheological Behavior under Turbulent Flow. Viscosity measurements were also conducted with a Brookfield cone-and-plate viscometer in the turbulent flow regime, where the existence of SISs was unlikely because of the

Figure 6. Phase diagram of the TTASal concentration versus shear rate. The dotted lines represent phase transitions. The square represents the operating (shear rate) region for the CMF experiments.

very high mixing rates (Figure 5). This explains the difference in viscosity in the laminar and turbulent flow regimes as measured by the two viscometers. Thus, dilute TTASal solutions behave essentially as Newtonian fluids under turbulent conditions (at concentrations of 2, 3, and 4.5 mmol/L; not all data are shown). At 6 mmol/L, a slight decrease in viscosity with shear rate (or shear thinning behavior) was observed during turbulent flow. Above 5 mmol/L, micelle-micelle interactions began to play an important role in the quiescent state. Consequently, a decrease in viscosity with shear was observed, which is due to the decreased rod-rod interactions at higher shear rates.3 Phase Diagram. To summarize the rheological behavior of TTASal during laminar flow, a phase diagram was constructed where the concentration TTASal is plotted versus the shear rate (Figure 6). Dotted lines represent phase transitions, and the micellar structures present in solution are depicted in the phase diagram. The approximate operating concentration and shear rate range for the microfiltration experiments is also shown as a square. Rheological Behavior of Dilute TTASal Solutions with Suspended Silica Particles. The addition of silica to dilute TTASal solutions can substantially change the rheological properties of the solution. Because of the hydrophobic interactions between the silica particles and the surfactant, the concentration of surfactant in the bulk solution can be seriously affected. It is important that the rheological behavior of the bulk solution not be affected by the addition of the silica particles. Then, the combined effects of fouling and viscosity on CMF can be studied using a non-Newtonian solution with and without particles. Viscosity was measured as a function of shear rate for different TTASal solutions with suspended silica particles (data not shown). To increase the compatability with water and decrease the hydrophobic interactions between the silica and surfactant, silica particles were plasma-treated (pt) with water. For the case of a 0.1 w/w % silica concentration, no qualitative difference in rheological behavior was observed between TTASal solutions with and without suspended silica. For higher silica concentrations (e.g., 0.5 w/w %), the rheological behavior was siginificantly different for TTASal suspensions to which untreated or plasma-silica was added. The viscosity of the TTASal suspensions with untreated silica was nearly independent of the shear rate, sug-

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Figure 7. Permeate flux versus Reynolds number and Dean number ratio for different TTASal solutions. TMP ) 27.6 kPa (4 psi). Solid symbols, linear module; open symbols, helical module. T ) 25 ( 2 °C.

gesting Newtonian behavior. In addition, all values of the viscosity increased. The viscosity of the TTASal solutions with plasma-treated silica varied with shear rate and showed about the same trend and values as the viscosity of the solutions without silica. Thus, the addition of a small concentration of (hydrophylized) silica particles to the TTASal solutions did not significantly affect the characteristic rheological behavior of the surfactant solutions. Microfiltration of a non-Newtonian Surfactant Solution. Several microfiltration experiments with dilute (2-6 mmol/L) TTASal solutions were carried out. In Figure 7, the permeate flux is plotted as a function of the Reynolds number and the Dean ratio. The dashed vertical line represents the critical Dean or Reynolds number for the onset of Dean vortex flow in the helical module. The pure water fluxes were similar for the two modules (Jwater ≈ 3000 L m-2 h-1). The permeation fluxes of the linear and helical modules for similar concentrations of TTASal were different. Because retention of surfactant is negligible, different “local” viscosities can explain this difference. Because of the shearthickening characteristics of TTASal solutions, the local viscosity in the helical module was higher than in the linear module. The viscosity is local because it depends on the shear rates at the membrane-surface interface. Mallubhotla et al.19 showed that, because of the additional effect of secondary flow, the wall shear rates in the helical tube are significantly higher than those in the linear tube. They also vary nonlinearly as the Dean vortices bifuricate from two to four vortices with increasing Reynolds number. The permeate flux seemed to decrease slightly with increasing Reynolds number for both modules. This decline in flux was more pronounced at higher concentrations. The average flux decreases from Re ) 50 to

Re ) 400 were 2, 6, 14, 17, and 46% for DI water and for surfactant concentrations of 2, 3, 4.5, and 6 mmol/ L, respectively. These percentages suggest that the observed flux decline was significant. The flux decrease can be explained by increased shear rates at higher Reynolds numbers, resulting in increased SISs and hence higher viscosities. A similar flux decline was not observed for PEG solutions, which exhibit Newtonian behavior.1 Local viscosities that existed inside the fiber pores were not measured here. Because calculation of the Reynolds number requires knowledge of the viscosity, the Reynolds number had to be estimated. According to the resistance model, the permeate flux should be a linear function of the inverse viscosity (eq 1). Kluge et al.1 observed this linearity using aqueous PEG solutions (J ) 2.2η-1, r2 ) 0.95, MWPEG ) 10 kDa, TMP ) 27.6 kPa, T ) 25 ( 2 °C). Because the work here was conducted under exactly similar conditions and with the same membranes and modules, their results can be used to estimate the local viscosities from the measured permeate flux. Microfiltration of Dilute TTASal Solutions with Suspended Silica Particles. Two microfiltration experiments at constant TMP were conducted with TTASal solutions and suspended silica particles (0.1 w/w %). During the required stabilization period, the drop in permeate flux in the linear module was very high in comparison with that in the helical module (Figure 8). The initial permeate flux in the helical module was lower than that in the linear module. Concentration polarization and fouling of silica particles caused a decrease in the flux for both modules during the stabilization period. The final permeate flux in the helical module was higher than that in the linear module. Apparently, the positive effect of Dean vortex flow on the permeate flux (reducing

Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002 501

Figure 8. Permeate flux versus time during the stabilization period for 6 mmol/L TTASal with 0.1 w/w % pt-silica, TMP ) 27.6 kPa (4 psi). Solid symbols, linear module; open symbols, helical module. Reynolds number Re ≈ 150. T ) 25 ( 2 °C.

Figure 9. Permeate flux versus Reynolds number for two different TTASal solutions (3 and 6 mmol) with suspended plasmatreated silica (0.1 w/w %), TMP ) 27.6 kPa (4 psi). The lines are linear regression lines. Solid symbols, linear module; open symbols, helical module. T ) 25 ( 2 °C.

concentration polarization) overcompensated for the negative effect of shear thickening (viscosity effect). In Figure 9, the permeate flux (after 1.5-2 h of stabilization) is plotted against the Reynolds number. The decrease in permeate flux with increasing Reynolds number was significant in the helical module and became more pronounced with higher concentration. Hence, as the Reynolds number is increased, the fluxes for the two modules are expected to approach each other. Above the point of coincidence, the negative viscosity effect will be larger than the positive effect of the Dean vortex flow. This phenomenon of viscosity outweighing Dean vortex flow has previously been demonstrated by Kluge et al.1 using Newtonian PEG solutions with increasing viscosity. The flux improvement for the helical module over the linear module decreased with increasing TTASal concentration. The viscosity increase with increasing TTASal concentration obviously reduced the strength of the Dean vortices and hence the percentage flux improvement. This effect has been previously reported for PEG solutions.1 The increase in viscosity decreases the stability of the Dean vortices. This can finally result in curved tube flow without vortices and similar mass transfer characteristics for the two modules.

Figure 10. Viscosity versus axial velocity for the linear module (points are experimental results). The arrows indicate extrapolation of the viscosity to zero velocity (called the intrinsic viscosity). T ) 25 ( 2 °C.

These results suggest that it is unwise to operate at high Reynolds numbers, as the flux improvement decreases with increasing Reynolds number. For PEG solutions, the flux improvement increases with Reynolds number until reaching an asymptote.1 A reason for the absence of an initial increase in flux improvement is not clear for the non-Newtonian TTASal solutions. Near the critical Reynolds number, the Dean vortices are likely to weaken and finally disappear, although a decrease in flux performance is not observed here. It might therefore be optimal to operate at a low axial flow rate. At this point, the flux advantage will be highest, and the energy consumption will be the lowest. This result can be used in dealing with, for example, concentrated extracellular polysaccharide solutions, which are found to exhibit non-Newtonian (shear-thickening) behavior.20 To achieve optimal performance during CMF of concentrated yeast solutions (with a helical fiber configuration), operating at low Reynolds number might be an optimal option. Mass Transfer in a non-Newtonian Surfactant Solution. Before describing mass transfer with a general Sherwood correlation, we develop a model describing the equivalent viscosity of dilute TTASal solutions as a function of the TTASal concentration and velocity. The local viscosities for flow in the two modules were calculated from calibration curve determined by Kluge et al.1 for the permeate fluxes during microfiltration at constant TMP (see earlier). To describe viscosity as a function of concentration, the intrinsic viscosities (intercept values) at zero velocity, which were obtained by extrapolation of the best-fit linear line of calculated viscosity data to zero velocity (see arrows in Figure 10), were plotted against the TTASal concentration (Figure 11). The extrapolated (or intrinsic) viscosities at zero velocity were fitted with the following equation as a function of the TTASal concentration (valid only between 2 and 6 mmol/L)

η(v)0) )

1 (a - bc)

(11)

where c (mmol/L) is the surfactant concentration and, for the linear module, a ) 1220 and b ) 160 (r2 ) 0.96), and for the helical module, a ) 1.06 and b ) 0.14 (r2 ) 0.96).

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Figure 11. Intrinsic viscosity (at zero velocity) for the linear and the helical modules as a function of the concentration of TTASal. Equation 11 is fitted for the linear module (dashed line) and the helical module (straight line). T ) 25 ( 2 °C.

Figure 12. Viscosity versus velocity for the linear module. Equation 12 is shown for different concentrations as full lines. T ) 25 ( 2 °C.

The velocity dependency of the viscosity is described for both modules by

η(v)v) ) η(v)0)e0.19v for 0 < v < 2 m s-1

(12)

where η(v)0) is given by eq 11. Using this equation and the known surfactant concentration and velocity, the viscosity was predicted. In Figure 12, the viscosity data for the linear module are again plotted versus velocity. The lines represent the viscosity predicted by eq 12 at different concentrations and velocities. A similar figure was obtained for the helical module (not shown). The viscosity, as predicted by eq 12, was used to calculate the Sherwood, Reynolds, and Schmidt numbers at different velocities and concentrations of TTASal (3 and 6 mmol/L) in the presence of suspended silica particles (0.1 w/w %). For a given concentration of TTASal, all three dimensionless numbers changed with increasing velocity (as a result of changes in viscosity), whereas for a Newtonian fluid, the Schmidt number should remain constant. The classical Leveque correlation (R ) 0.23, b ) 0.33, and c ) 0.33, eq 8) fitted the data for the laminar flow of a shear-thickening fluid (TTASal) in the linear module.12 The measured Sherwood numbers (data points) and the predicted Sherwood numbers from eq 8 (lines) are plotted versus the Reynolds number in Figure 13. The data for the helical module could not be fitted with eq 9, which was used for Dextran T500 solutions during

Figure 13. Sherwood versus Reynolds number for TTASal solutions with plasma-treated silica particles (0.1 w/w %), TMP ) 27.6 kPa (4 psi), as calculated with eq 4. The data for the linear module are fitted with the Sherwood number predicted according to eq 8 with R ) 0.23, b ) 0.33, and c ) 0.33. The data for the helical module are fitted with the Sherwood number predicted according to eq 13 with R ) 0.19, b ) 0.55, and c ) 0.33. Solid symbols, linear module; open symbols, helical module. T ) 25 ( 2 °C.

ultrafiltration in the helical module.13 The data for a plot of Sherwood number versus the Reynolds number indicated that, at high TTASal concentration (6 mmol/ L), the Sherwood number decreased with Reynolds number at high Reynolds number. This result suggests that the exponent, b, of the Reynolds number is dependent on the concentration of TTASal and, hence, on the shear rate and viscosity. To account for this effect, the constant exponent b was replaced by a modified variable exponent, b′ ) b(η(v)0)/η(v)v)), which equals b when v ) 0 and is less than b when v > 0. The new correlation is then

()

Shhel ) R

a rc

0.07

Re[b(η(v)0)/η(v)v))]Scc for Re < 800 (13)

where R ) 0.19, b ) 0.55, and c ) 0.33 and where η(v)0) is the viscosity at zero velocity and η(v)v) is the viscosity at a given velocity (v). For the fit of eq 13 to the data in Figure 13, the variable exponent of the Reynolds number decreased for both concentrations from 0.55 to 0.45 for flow in the helical module. For the Reynolds numbers, the exponent is expected to decrease to 0.33 for laminar flow as a result of the loss of the Dean vortices. Conclusions The non-Newtonian behavior of a dilute aqueous surfactant solution (TTASal) was examined by measuring the viscosity as a function of shear rate. Microfiltration experiments were carried out with varying concentrations of TTASal. Finally, CMF experiments were conducted with TTASal solutions to which suspended silica particles had been added. Conclusions from these experiments are summarized below. The viscosity of dilute TTASal solutions (2-6 mmol/ L) containing rodlike micelles increased with increasing shear rate (shear-thickening behavior) during laminar flow. The increase in viscosity could be explained by the buildup of shear-induced structures (SISs). Applying turbulent flow to these micellar solutions prevents the buildup of SISs. It was shown that, at low silica contents (0.1 w/w %), interactions between the silica particles and the surfactant could be neglected. Hydrolyzed silica

Ind. Eng. Chem. Res., Vol. 41, No. 3, 2002 503 Table 2. Summary of Mass Transfer Correlations for Standard (Linear Module) and Dean Vortex (Helical Module) Filtration for Newtonian and Non-Newtonian Fluids linear Newtoniana

Shlin ) 0.071Re

non-Newtonianb a

Kluge et al.1

b

Shhel ) 0.23Re

helical

0.33

0.33

Sc

Sc

0.33

This paper.

particles exhibited even fewer interactions with the surfactant. These observations were useful in distinguishing between decreases in the permeate flux due to viscosity effects or to membrane fouling. The permeate flux was measured as a function of the Reynolds number at constant TMP (27.6 kPa) for different TTASal concentrations. The permeate flux decreased with increasing TTASal concentration and increasing Reynolds number. The flux decrease could be explained by an increase in viscosity in both cases. The lower permeate flux in the helical module was explained by a higher local viscosity resulting from the higher average and wall shear rates caused by the Dean vortex flow as compared to the linear module. Upon the addition silica particles to the TTASal solutions, a strong flux decline was observed for both modules. At the beginning of the stabilization period for the buildup of SISs, the flux of the linear module was higher than that of the helical module because of viscosity effects. However, the permeate flux of the helical module was higher than that of the linear module at the end of the stabilization period because the improved hydrodynamics and low fouling in the helical module compensated for the higher viscosity. At higher Reynolds numbers, the flux improvement decreased as a result of the weakening of the Dean vortex flow by viscosity effects. To model the mass transfer, the viscosity was described as a function of the surfactant concentration and velocity. Using newly derived correlations, the mass transfer in terms of the Sherwood number could be predicted as a function of Reynolds and Schmidt numbers for the shear-thickening TTASal solutions with suspended silica particles (0.1 w/w %). A new mass transfer correlation for the helical module was obtained, in which the destabilizing effect of increasing viscosity on the Dean vortices was incorporated in the exponent of the Reynolds number

Shhel ) 0.19

() a rc

() ()

a 0.07 0.55 0.61 Shhel ) 0.015 Re Sc rc a 0.07 [0.55(η(v)0)/η(v)v))] 0.33 Shhel ) 0.19 Re Sc rc

0.61

0.07

Re[0.55(η(v)0)/η(v)v))]Sc0.33 for Re < 800 (13)

For the case of cross-flow microfiltration, a summary of the mass transfer correlations for Newtonian1 and non-Newtonian (this work) fluid flow is given in Table 2. Although the new mass transfer correlation has only been tested on a few data, it is a first attempt to incorporate non-Newtonian behavior into a predictive correlation. In the future, a mass transfer correlation should be developed that can predict Sherwood numbers for varying solute concentration (silica particles) and perhaps varying particle size. In this way, a model can be developed that could be applicable for many types of feed. To accomplish this goal, many experiments with varying silica concentrations and varying particle sizes

still need to be carried out. To examine mass transfer for industrial applications, experiments with highly concentrated (non-Newtonian) fermentation broths should be conducted to confirm and revise the correlations developed here for non-Newtonian fluids containing surfactant (TTASal) and silica. List of Symbols Constants a ) inner radius of the tube (m) Am ) membrane surface area (m2) di ) inner diameter of the fiber (m) do ) outer diameter of fiber (m) drod ) diameter of the rod (m) k* ) Boltzmann constant (kg m2 s-2 K-1) l ) active fiber length (m) m ) number of fibers in module rc ) radius of curvature (m) rs ) radius of solute particle (m) η ) radius ratio F ) density (kg m-3) Variables c ) concentration (mmol L-1) D ) Dean number ratio De ) Dean number D ) diffusion coefficient (m2 s-1) E ) energy consumption (W) J ) permeate flux (L m-2 h-1) JW ) pure water flux (L m-2 h-1) k ) mass coefficient (m s-1) LP ) water permeability (L m-2 h-1 Pa-1) p ) pressure [Pa (or psi)] Q ) axial flow rate (m3 s-1) rc ) radius of curvature in a helix (m) R ) resistance (m-1) Re ) Reynolds number Sc ) Schmidt number Sh ) Sherwood number t ) time (s) T ) temperature (K or °C) v ) average velocity (m s-1) ∆pTMP ) transmembrane pressure (Pa) ∆px ) axial pressure drop (Pa) γ ) shear rate (s-1) η ) dynamic viscosity (Pa s)

Acknowledgment The authors acknowledge Steven Pearl, Steve Dzengeleski, and Ralf Kuriyel of Millipore Corporation for providing the membrane modules. Gareth H. McKinley

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and Robert K. Prud’homme are thanked for suggesting the use of the micellar (TTASal) system. Literature Cited (1) Kluge, T.; Kalra, A.; Belfort, G. Viscosity effects on Dean vortex membrane microfiltration. AIChE J. 1999, 45 (9), 1919. (2) Manno, P.; Moulin, P.; Rouch, J. C.; Clifton, M.; Aptel, P. Mass transfer in helically wound hollow fiber ultrafiltration modules yeast suspensions. Sep. Purif. Technol. 1998, 14, 175. (3) Prud’homme, R. K., Khan, S. A., Eds. Foams: Theory, Measurements, and Applications; Marcel Dekker: New York, 1995. (4) Rehage, H.; Hoffman, H. Rheological properties of viscoelastic surfactant systems. J. Phys. Chem. 1988, 92, 4712. (5) Wunderlich, I.; Hoffman, H.; Rehage, H. Flow birefringince and rheological measurements on shear induced micellar structures. Rheol. Acta 1997, 26, 532. (6) Lequeux, F.; Candeau, S. J. Dynamical properties of wormlike micelles deviations from the “classical” picture. In Structure and Flow in Surfactant Solutions; Herb, C. A., Prud’homme, R. K., Eds.; ACS Symposium Series 578; American Chemical Society: Washington, D.C., 1994; pp 51-62. (7) Imae, T. Spinnability of viscoelastic surfactant solutions and molecular assembly formation. In Structure and Flow in Surfactant Solutions; Herb, C. A., Prud’homme, R. K., Eds.; ACS Symposium Series 578; American Chemical Society: Washington, D.C., 1994, 140-142. (8) Ohlendorf, D.; Interthal, W.; Hoffmann, H. Surfactant systems for drag reduction: Physicochemical properties and rheological behavior. Rheol. Acta 986, 25, 468-486. (9) Wang, S.-Q.; Hu, Y.; Jamieson, A. M. Formation of nonequilibrium micelles in shear and elongational flow. In Structure and Flow in Surfactant Solutions; Herb, C. A., Prud’homme, R. K., Eds.; ACS Symposium Series 578; American Chemical Society: Washington, D.C., 1994, 278-287. (10) Lee, Y.; Clark, M. C. Modeling of flux decline during crossflow ultrafiltration of colloidal suspensions. J. Membr. Sci. 1988, 149, 181. (11) Henry J. D.; Corder, W.; Ho, W. S. W.; Hoglund, R. L.; Lemlich, R.; Li, N. N.; Moyers, C. G.; Newman, J.; Pohl, H. A.;

Pollock, K.; Prudich, M. E.; Spiegler, K. S.; Halle, E. V.; Perry’s Chemical Engineers’ Handbook, 6th ed.; McGraw-Hill: New York, 1984; Chapter 17. (12) Zeman, L. J.; Zydney, A. L. Microfiltration and Ultrafiltration: Principles and Applications: Marcel Dekker: New York, 1996. (13) Gehlert, G.; Luque, S.; Belfort, G. Comparison of ultra- and microfiltration in the presence and absence of secondary flow with polysaccharides, proteins and yeast suspensions. Biotechnol. Prog. 1998, 14, 931-942. (14) Dean, W. R. Fluid motion in a curved channel. Proc. R. Soc. A 1928, 121, 402-420. (15) Mallubhotla, H.; Hoffmann, S.; Schmidt, M.; Vente, J.; Belfort, G. Flux enhancement during Dean vortex membrane nanofiltration. 10. Design, construction and system characterization. J. Membr. Sci. 1998, 141, 183-195. (16) Ulbricht, M.; Belfort, G. Surface modification of ultrafiltration membranes by low-temperature plasma. II. Graft polymerization onto polyacrylonitrile and polysulfone. J. Membr. Sci. 1996, 111, 193-215. (17) Luque, S.; Mallubhotla, H.; Gehlert, G.; Kuriyel, R.; Dzengeleski, S.; Pearl, S.; Belfort, G. A new coiled hollow fiber design for enhanced microfiltration performance in biotechnology. Biotechnol. Bioeng. 1999, 65 (3), 247-257. (18) Mallubhotla, H.; Nunes, E.; Belfort, G. Microfiltration of yeast suspensions with self-cleaning spiral vortices: Possibilities for a new membrane module design. Biotechnol. Bioeng. 1995, 48, 375. (19) Mallubhotla, H.; Edelstein, W. A.; Earley, T. A.; Belfort, G. Magnetic resonance flow imaging and numerical analysis of curved tube flow: 16. Effect of curvature and flow rate on Dean vortex stability and bifurcation. AIChE J. 2001, 47 (5), 1126-1140. (20) Bailey, J. E.; Ollis, D. F. Biochemical Engineering Fundamentals, 2nd ed.; McGraw-Hill: New York, 1986; p 504.

Received for review May 16, 2001 Revised manuscript received December 4, 2001 Accepted December 4, 2001 IE010448O