Upper- and Lower-Bound Estimates of Flux for Gas-Sparged

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Ind. Eng. Chem. Res. 2005, 44, 7684-7695

Upper- and Lower-Bound Estimates of Flux for Gas-Sparged Ultrafiltration with Hollow Fiber Membranes S. R. Smith, Z. F. Cui, and R. W. Field* Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, U.K.

Cross-flow ultrafiltration is an important industrial process, which is inherently limited by concentration polarization and fouling. The introduction of gas bubbles into the filtration feed improves flux. However, for design purposes, accurate models of the process are required. In the first part of this paper a mechanism of flux enhancement is proposed for gas-sparged hollow fiber membrane ultrafiltration. This mechanism is determined from previously published experimental and computational fluid dynamics studies of the capillary slug flow process, as well as from dimensional analysis of the process. A physicochemical model for flux prediction is designed around the postulated enhancement mechanism. The flux-prediction model enables estimation of upper and lower bounds. The model comprises an approximate solution of the flow problem, assuming controlled gas bubble distribution, coupled with a one-dimensional, integral method boundary layer analysis and flux models. The boundary layer analysis is adapted to include the effect of wall suction. In the second part of the paper, this model was evaluated against experimental flux results. In certain cases, it was found that flux estimates for the equivalent one-phase filtration process was higher than the lower bound flux estimate for the two-phase process. This highlights that gas sparging can be detrimental to the hollow fiber membrane ultrafiltration process under certain sparging conditions. The predicted upper and lower bound flux values were correctly found to encapsulate experimental values and should therefore assist process design. 1. Introduction Cross-flow ultrafiltration (UF) is widely used in the dairy, pharmaceutical, and water production industries. Cui1 first showed that injection of long gas slugs (a few tube diameters in length) into the feed stream of tubular UF membranes improved filtration flux. This technique has also been applied to hollow fiber (HF) membranes and numerous experimental studies2-4 subsequently confirmed the ability of gas sparging to enhance flux in these systems. To make the gas-sparged process available for industrial application, the ability to quantify and hence understand the process is imperative. Modeling attempts by Ghosh and Cui,5 Taha and Cui,6 and Smith and Cui7 showed reasonable success. However, the approaches of Ghosh and Cui5 and Taha and Cui6 were largely underestimations as these employed the classic, one-dimensional nonporous boundary layer (BL) analysis. The approach followed in Smith and Cui7 employed a simplified molecular-mass transport and hydrodynamic model and gave overestimations of the flux. The model proposed in Smith and Cui7 is a basic variant of the modeling technique that is expected to yield a general model for gas-sparged filtration. A more rigorous solution strategy using a similar approach to that of Smith and Cui7 was developed by Wiley and Fletcher8 for single-phase UF. However, rigorous models are still lacking for gas-sparged UF. The present work proposes a mechanism of flux enhancement for gassparged HF membrane UF based on filtration experimental results, two-phase flow analysis, and computational fluid dynamics (CFD) models of capillary tube slug flow. A phenomenological flux prediction model * To whom correspondence should be addressed. E-mail: [email protected]. Tel: +441865273814. Fax: +441865283273.

Figure 1. A diagrammatic representation of the capillary tube slug flow pattern, highlighting the three distinct regions of flux enhancement. The single-headed arrows indicate direction and magnitude of shear stress, while the double-headed arrows indicate velocity magnitude and direction.

based on the proposed enhancement mechanism is subsequently developed. The model employs a onedimensional BL approach for prediction of the upper and lower flux bounds for gas-sparged HF membrane UF. 2. Hydrodynamics and Enhancement of Gas-Sparged UF A number of models exist for hydrodynamic analysis of gas-liquid slug flow in capillary-type tubes. Initial studies by Bretherton9 were mainly analytical, but were limited in applicability. Availability of modern computing power has enabled the advent of numerical methods for solutions of the full Navier-Stokes equations across the gas and liquid phases, e.g., Taha and Cui.10 In this paper the focus is on proper incorporation of the flux enhancement mechanism into a phenomenological flux prediction model for gas-sparged HF UF. For this reason all hydrodynamic and mass-transfer models will be kept deliberately simple. The capillary slug flow pattern comprises alternating flows of long bullet-shaped bubbles, nearly filling the tube cross section, and liquid slugs (Figure 1). The flow

10.1021/ie048989b CCC: $30.25 © 2005 American Chemical Society Published on Web 05/20/2005

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Figure 2. Comparison of the typical shear stress profiles for gas-sparged HF (left-hand side) 11 and tubular 6 (right-hand side) membranes UF.

pattern is divided into three hydrodynamically distinct regions: the liquid slug, bubble nose-and-tail, and the bubble film region. The liquid slug region is a one-phase liquid only region, where the flow is either developing or fully developed, while the nose and tail regions are the points of intersection of the gas and liquid phases. The film region is where the bubble diameter and liquid film velocity are expected to be constant. 2.1. Flux Enhancement. The two-phase system hydrodynamics reflect the sources of flux enhancement. The mechanism of flux enhancement in HF UF membranes is believed to comprise three factors: (i) Enhanced mass transport in the bubble nose and tail regions (Figure 1, region I) due to the physical effects of flow reversal in these regions where liquid flow changes direction, or at least decelerates from upward flow to zero flow. This is due to the piercing motion of the upward rising bubble moving faster than the liquid ahead of it. The leading liquid is overtaken and pushed downstream (Figure 1). Flow reversal physically abolishes the momentum transfer BL, causing it to be rebuilt and, in so doing, reduces the thickness of the masstransfer BL, increasing flux. (ii) Increased shear stress or shear stress spikes at the membrane wall due to flow compression (bubble nose) and flow expansion (bubble tail) as the flow changes direction (region I). Flow reversal, described in point (i), enhances flux by physically removing boundary layers and is not yet well-quantified. However, increased shear stress can be related to boundary layer thinning by boundary layer analysis. Increased shear stress enhances mass transfer by reducing the thickness of concentration or mass-transfer BLs built up in the liquid slug region.10,11 This reduced mass-transfer resistance leads to higher mass-transfer coefficients and larger mass transport. Together (i) and (ii) result in positionvarying shear rate and flux enhancement within the nose and tail regions. (iii) Enhanced bulk mixing due to fast moving bubbles pushing fluid from the liquid slug ahead of itself down to the trailing liquid slug. Bulk mixing reduces high bulk concentration pockets and hence results in a much lower chance of excessive concentration buildup near any one portion of the membrane surface. Such buildup could lead to gellike concentration polarization and/or fouling, both of which are stronger, more permanent forms of membrane efficiency reduction. As seen in Figure 2, HF membranes do not experience enhanced wall shear stress in the liquid film region,

Figure 3. Effect of sparging frequency on permeability, q; flow rate ) 27 mL/min; feed concentration, Cb ) 10 g/L dextran, with percentage enhancement over one-phase shown.

unlike tubular membranes.5,6 In fact HF membrane film shear stress, like its film velocity, is lower than that of the liquid slug.10,11 Tubular film shear stress increases with bubble length, Figure 2. This implied that longer bubbles were better for flux enhancement in tubular membranes, but worse for HF membranes. This was clearly displayed by the tubular flux prediction work of Taha and Cui.6 This work clearly points out the monotonic relationship between average void fraction and flux enhancement in tubular systems. This simplicity is not conferred to the HF counterpart, making process optimization more challenging. From Figure 2 it can be seen that the nose and tail regions, and not the film itself, enhances flux in HF membranes. Hence, as a simple preliminary design perspective, it can be concluded that shorter, more frequently sparged bubbles would be important for optimizing HF UF enhancement via slug flow. Higher sparging frequency improves flux and increases void fraction. Longer bubbles also increases void fraction, but decreases flux. This competitive effect was studied by Smith and Cui,12 where only the sparging frequency was altered, Figure 3. This study showed that as bubbling frequency decreased, two-phase HF UF flux could drop to values below the one-phase counterpart. This has not been previously reported in any low-frequency gas-sparged tubular membrane experiments, e.g., Li et al.13 and Sur et al.14 This observation is highly likely linked with the long bubbles (31 mm, ∼ 35 tube diameters in length) employed in the study of Smith and Cui12 and also the

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Figure 4. A symmetric cut-through view of the bubble film, nose (equivalently tail), and liquid slug regions in capillary tube slug flow (R ) tube radius; δo ) terminal film thickness and δf ) position-varying film thickness). The regions are numbered consistently.

fact that HFs of bore 0.89 mm were employed. This is an important consideration for process optimization in HF UF. Further differences between the shear stress profiles of HF and tubular sparged membranes include a much more exaggerated shear pulse at the bubble nose, when compared to the liquid slug shear, in HF membranes. A significant tail pulse is also observed in HF membranes, but no wake is present. These tail and nose shear spikes are expected to result from the large shear stress forces present in these regions and this is discussed further in the subsequent force balance section. Tubular membranes, on the other hand, exhibit mainly large wake and film region shear stress enhancement. It can thus be proposed that membrane processes employing a hybrid of the tubular and HF characteristics, e.g., 4-8 mm circular channels for airwater like fluids, may exhibit optimal flux enhancement as it may display aspects of the capillary and tubular processes. However, investigations into such processes are required to draw reliable conclusions. 2.2. Force Balance Analysis. To understand capillary slug flow properly, a quantitative force balance which includes surface tension and viscous and inertial forces is required to analyze the system. However, solutions of the Navier-Stokes momentum equations across two phases are not trivial.10 For creeping slug flow in capillary tubes, analytical force balances have been proposed by Bretherton.9 However, analytical solutions are not attainable for slug-flow hydrodynamics when convective terms are not negligible and this problem is still under active research in a number of applied mathematical investigations, e.g., Heil.15 A dimensional analysis of two-phase flow16 shows that Eotvos number, Eo ) ∆Fgr2eff/σ and Weber number, We ) FU2mreff/σ are important for describing the process. In these equations, reff is the effective system radius or size, σ is the surface tension, and ∆F is the difference between the liquid and gas densities. If one uses δf from Figure 4 as the local reff value in these dimensionless groups, one can approximately evaluate the force balance for the two-phase system. Eo is indicative of the ratio of gravity or buoyancy to surface tension forces. Eo is very low (Eo < 0.04 because δf < 0.5 mm) for the water-air system studied here and hence buoyancy is negligible compared to the surface tension forces in this system. This was also observed by Bretherton,9 who stated that the main forces comprising the force balance for capillary tubes (nominal bore < 3.6 mm) for airwater flows, using the method of Tung and Parlange17 would be inertial forces and capillary or surface tension forces, not buoyancy forces. In larger tubes (say bore > 3.6 mm) the buoyancy will have an impact on the force balance, causing the film region to have a higher velocity than that in capillary tubes. This gives one explanation

why the tubular membranes (nominal bore g 3.6 mm) enjoy a heightened shear rate enhancement in the film region.5 Weber number, We is the ratio of inertial-to-surface tension forces and since buoyancy is negligible, these are the main forces comprising the force balance.9 Taking δf as the local reff value, one sees from Figure 4 that (i) at the tip of the nose and tail region, reff ) δf ) R and so We ∼ 7, and so inertial forces are slightly larger than surface tension; (ii) at the start of the film region, reff ) δf ) δo and so We ∼ 0.5 (using data from Table 2), and so surface tension is slightly larger than inertia, yielding the constant terminal film thickness of δo. In the nose and tail regions where δo < reff < R and hence 0.5 < We < 7 (Figure 4), the surface tension and inertial forces compete in strength, resulting in the observed bubble shape. This leads to a second reason tubular membranes have increased shear stress in the film region: in larger tubes, with a much larger reff and We in the film region, the capillary forces are less important than the inertial forces in the film region. Inertial forces in the film region increase the film velocity to values above the free falling film velocity that would occur under gravity only, even causing films to accelerate.6 For this same reason, longer bubble noses are observed at higher We. In HF membranes, the capillary forces are thus significant and do in fact compete with the inertial forces in the bubble nose and tail region, resulting in the three distinctive force balance regions outlined earlier and now summarized: (i) Inertial force (and possibly buoyancy in larger capillaries) dominated region in the liquid slug; (ii) competitive region in the bubble nose and tail regions, where the inertial and surface tension forces compete and neither one dominates totally, resulting in the bubble head and tail shape being altered as a result of the force balance. These competing forces also result in the shear stress spikes in this region, which are characteristic to HF membranes. (iii) Capillary force dominated film region, where the size of the liquid region is relatively small so that inertial forces are outsized; no competition between forces implies the film thickness (δo) is constant in this region. The capillary number, Ca, is another key dimensionless group of the capillary tube slug flow process.16 Ca is defined as

Ca )

µUtb σ

(1)

Utb is the gas slug/Taylor bubble rise velocity, µ is the liquid viscosity, and σ is the surface tension for the airliquid interface. Ca is the ratio of viscous-to-surface tension forces and in general (e.g., see Bretherton,9 Smith et al.,18 and Taylor19) it serves as a good parameter by which to correlate shape and velocity data for the capillary tube slug flow system. Such correlative data are useful for determining estimates of bubble diameter and liquid slug length based on measured Utb. Utb is, however, related to Um, the liquid cross-flow velocity via empirical correlations of the form Utb ) C1Um + Uo (see some examples in Thulasidas et al.20). C1 is a constant and Uo is the rise velocity of the bubble in a stagnant liquid, which is zero in capillary tubes. Hence, Utb ) C1Um and this highlights that Um is a crucial factor in deciding the hydrodynamics of the


capillary bubble train flow process. Surface tension is assumed to be a constant value for this study, which would be the case for a pure solvent, since the experimental study employed aqueous dextran, which is not expected to significantly alter the water surface tension (0.0728 N/m). Dynamic surface tension effects would have to be included in a more rigorous study for filtration systems with surface-active agents in the feed stream. Studies of capillary tube slug flow with varying bubble volume, with both CFD (Figure 2) and experimental work,11,18 showed that bubble volume had little effect on the bubble shear stress profile, nose shape, and tail shape, but simply altered the length of the bubble film region. As mentioned before, the film region shear stress was lower than that of the liquid slug10,11 in HF membranes. Therefore, above a certain threshold volume (about bubble/diameter ratio of 2), a larger bubble volume increases only the length of the film region present in a membrane module and hence the amount of the lower shear stress region present in the membrane. These results support the theory that shorter bubbles are a better enhancement source for sparged HF UF. 2.3. Two-Phase Hydrodynamics and Flux Enhancement. All bubble length, velocity, and diameter data were measured using the infrared technique outlined in Smith and Cui.12 The liquid slug velocity (Uls) was determined from the measured Utb, using the data from Taylor.19 Uls for cases 6-8 (Table 2) were taken as the average over the three calculations as these had the same one-phase liquid supply rate. This same paper by Taylor19 also provides data for determining bubble diameter (db) from the measured Utb, which allows all data required for the two-phase flux prediction (assuming that gas and liquid slug length are controlled parameters) to be inferred from the known bubble velocity. (Utb can easily be measured using a visual stopwatch method or estimated from the drift-flux model with known gas and liquid flow rates.) These data are reported in Table 2. Based on these slug and bubble lengths and the shear stress profiles reported in Taha and Cui,10 it was assumed that the shear stress spikes in the bubble nose and tail region contribute minimally to the overall flux enhancement as these spikes are small in length-weight compared to the shear rate in the liquid slug and film region. It is proposed that enhancement is mainly derived from the continuous breakdown or thinning of the mass-transfer BL due to the position-varying shear stress which occurs in the bubble nose and tail regions (Figure 5). Furthermore, closer investigation of the shear stress profiles presented in Taha and Cui10 also show that the film shear stress is essentially zero. This simplifying assumption is employed throughout this work. Figure 5 also shows five scenarios of possible behavior of the mass-transfer BL as the bubble passes through the HF UF system. It must be noted that the exact behavior of the mass-transfer BL in the position-varying shear region is not well-understood. In fact, it is expected that none but Scenario 5 bears any resemblance to the real process. Investigations of this phenomenon are critical to accurate modeling and for understanding the flux enhancement mechanism. In Scenario 1 flow, reversal at the nose and tail regions completely remove the mass-transfer BL. Scenario 1 may represent an upper bound of flux enhancement.

Figure 5. The various models of the effect of flow reversal on the mass-transfer BL thickness for the gas-sparged HF UF process. Solid-lined arrows indicate direction of flow. Note the flow reversal at the bubble nose and tail.

Scenarios 2&3 show how one could lump the enhancement from both the nose and tail into one, complete theoretical removal of the mass-transfer BL at one position, either nose or tail, within the gas-liquid unit. Scenarios 2&3 yield a single point of enhancement, due to flow reversal, per gas bubble-liquid slug unit and are the most commonly employed solutions (e.g., see Ghosh and Cui5 and Taha and Cui6), as it seems natural to model the average shear rate across one unit, assuming repeated units, each starting from δc ) 0. Scenario 4 is the case where there is no flow reversal effect on the mass-transfer BL thickness and the BL grows continuously throughout the length of the membrane, at different paces when in the film, nose, tail, or liquid slug regions. Scenario 4 is conservative as far as mass-transfer enhancement is concerned and represents the lower bound of flux enhancement. Scenario 5 is what is believed to be closest to the real shape of the masstransfer BL. In Scenario 5, flow reversal does reduce the BL thickness, but does not completely remove it. Furthermore, the growth and repeated thinning due to flow reversal are expected to reach a steady pattern, yielding the shape of the BL shown in Scenario 5. It is complex to model exactly what degree of BL thinning occurs at the flow reversal points as would be required in order to model Scenario 5. In this work the goal is to develop reliable upper and lower bounds for the flux estimate. Future work should aim to derive more exact flux estimates. For general solutions of Scenario 1, the simple hydrodynamic model consists of (a) the bubble region modeled as an equivalent length of film (region II), ignoring the shear rate contributions from the nose and tail regions (region I) and (b) a liquid film and liquid slug (region III) modeled as a one-phase fully developed flow system (Figure 4). This model may slightly underestimate the shear rate and flux as it ignores the spikes in the shear rate in the nose and tail region and also ignores the fact that the liquid slug exhibits developing flow with a higher shear rate. However, by solving the BL model for the modified film-only bubble (regions I and II combined) and liquid slug (region III) separately, assuming zero initial mass-transfer BL thickness for each region, one accounts for the flow reversal which occurs in the bubble nose and tail regions. The specific solution employed here uses the simplifying assumption mentioned above for region II; an effectively zero shear rate is assumed to lead to a negligible flux contribution for this region. The validity of this assumption is discussed later.

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For Scenarios 2&3, the length-weighted average velocity over the film (region II) and liquid slug (region III) regions was used for modeling of each gas-liquid unit, with the assumption that δc ) 0 at the start of each unit. To model Scenario 4, it was simply decided to ignore the flow reversal induced by the bubble and base the film mass-transfer BL growth on the lengthweighted average liquid slug (region III) and film (region II) velocity. The solutions for Scenarios 1 and 4 respectively are expected to be the extreme maxima and minima of the flux predictions, which should encapsulate the real solution. However, the zero film region (region II) velocity and hence zero shear rate assumption causes a zero flux assumption as per the flux models of this work. This subsequently reduces weighted flux estimates produced from the assumed Scenario 1 model, making Scenarios 2&3 a more suitable, albeit stricter upper bound. With this simple model, the shear rate is determined as γw ) 8Um/dt at the known respective length-weighted average velocities for Scenarios 2&3 and Scenario 4. These shear rates can then be employed in the BL and flux models to generate flux estimates. For Scenarios 2&3 the flux in one gas-liquid unit is that for the membrane and so calculations are performed with a growth length for the BL equal to that of a repeating unit. For Scenario 4 the growth is over the entire membrane, unaffected by the presence of bubbles, and so the growth length of the BL is the membrane length. For all single-phase UF, shear rate is also calculated from γw ) 8Um/dt as this equation applies for laminar fully developed pipe flow. The application of this equation to two-phase calculations may result in slight underpredictions of shear rates, as shear rates are lower in developed flow than in developing flow.

so the Schmidt number, Sc, of these systems are exceptionally high, on the order of 104. Sc is defined as

Sc )

One-dimensional nonporous BL analysis approaches for modeling UF often ignored thinning of the mass and momentum BLs by neglecting suction effects.21-23 Hence, this approach overestimates BL thicknesses and masstransfer resistances, resulting in underestimates of mass-transfer coefficients and filtration flux. Approximate suction inclusion will be employed in this work to improve flux estimates. To predict flux via a BL approach, the nonporous system hydrodynamics is quantified, then using analogy theory, hydrodynamics is related to mass transfer. The mass-transfer BL thickness is directly related to the mass-transfer coefficient (stagnant film model) and, finally, using flux models, filtration flux is determined. In this work, the nonporous single-phase hydrodynamics are quantified by assuming a fully developed velocity profile in the membranes. The two-phase hydrodynamics are quantified based on the geometry of the gas slugs used for sparging as discussed before. The mass-transfer BL thickness (δc) determined from the nonporous hydrodynamics is modified to include suction effects before being employed for flux prediction. The stagnant film and osmotic pressure models are employed for flux prediction. We aim to also show the importance of suction inclusion in flux prediction. 3.1. Nonsuction BL Analysis. Due to their pore sizes, UF membranes are mainly used for separating and concentrating macromolecules such as proteins, DNA, RNA, and biopolymers. The diffusivity (D) of these molecules is very low, on the order of 10-10 m2 s-1 and

(2)

In this equation D is the diffusion coefficient of the dissolved macromolecule in the carrier fluid of density F and viscosity µ. At large Sc, δc is relatively low and is thus totally immersed in the momentum-transfer BL. The mass-transfer BL can thus be modeled as being in the near-wall velocity profile of the momentum-transfer BL. The near-wall momentum BL velocity profile is linear, regardless of whether the flow is turbulent, laminar, developing, or fully developed. The important characteristic is the slope of the linear velocity profile or, equivalently, the wall shear rate (eq 3). Shear rate is lower for fully developed flow than for developing flow and higher for turbulent flow than for laminar flow. Hence in UF, with its large Sc, it is assumed, with little error, that the mass-transfer BL is situated within a linear velocity profile region within the momentum BL, near the membrane wall. This rationale is in fact employed for fully developed laminar flow as the basis of the Sieder-Tate equation (see Kay and Nedderman,24 pp 410-412). This analysis may also apply at lower Sc numbers (Sc > 1), for most practical systems with laminar, parabolic flow profiles (i.e., laminar pipe, parallel plate, and free surface flow) as the linear nearwall velocity profile extends further into the bulk region in such systems. The linear, near-wall BL velocity profile, obeying the nonslip boundary condition, can be written as

U) 3. One-Dimensional Mass-Transfer BL Analysis

µ FD

|

∂U ‚y ) γwy ∂y w

(3)

In this equation U is the near-wall liquid velocity, y is the distance from the membrane wall, and γw is the wall shear rate. The wall shear rate is determined from the hydrodynamic models. Using the flat plate BL analysis outlined in section 16.4 of Kay and Nedderman24 (pp 410-412) and correcting the flat plate analysis for the curvature of a circular conduit or pipe flow as proposed in Smith and Cui,25 one can relate δc to γw as

δc ) 2.897

( ) ( ) Dx γw

δc,ave ) 2.173

1/3

Dl γw

1/3

(4) (4a)

where D is the diffusion coefficient, x is the distance from the leading edge of the BL, and l is the length parameter. Equation 4 is used throughout this work to determine the nonsuction mass-transfer BL thickness used for flux prediction. 3.2. Approximate Inclusion of Suction into BL Analysis. 3.2.1. Empirical Suction Correction. A number of methods exist to include suction into the BL analysis. The most rigorous method would include rederivation of the mass and momentum BL equations with suction as part of the material balances. Chapter 11 of Schlichting and Gersten26 reported such work for the momentum-transfer BL. The analytical solution investigated in ref 26 looked solely at the effect of


suction on the momentum-transfer BL and did not incorporate a simultaneous mass- or heat-transfer BL development in the analysis. Such studies have been important for the field of aerodynamics as it is wellknown that suction applied on a BL can delay the onset of turbulence and flow separation, which result in undesired high-pressure drop. Other seminal work in this area includes the vorticity transport equation27 and that of Berman28 and Yuan and Finkelstein29 who employed a perturbation solution to alter the fully developed laminar velocity profile to include wall suction effects on the momentum BL. If a solute were completely transmitted by a membrane, the analyses of Schlichting and Gersten,26 Panton,27 Berman,28 and Yuan and Finkelstein29 could directly be employed for masstransfer analysis for the nonretentive filtration process. However, with partial and total retention of the solute species, two main characteristics of the momentum- and mass-transfer problems arise which render the solutions of refs 26-29 impractical for this problem: (i) Some or all of the solute species are stopped at the membrane active surface, but the carrier fluid continues to permeate, albeit at a slower rate, through the membrane. (ii) The diffusivity, viscosity, and density of the solution are increased as the solute is concentrated at the membrane surface. This phenomenon couples the flow and mass-transfer problems inseparably. Simultaneous solutions of the hydrodynamics and masstransfer problems are complex and in this paper we attempt this link empirically using an adapted integral method BL analysis. The unique property of the retentive UF process is that solute may be retained and thus this process has no practical heat-transfer analogue to employ for this analysis. However, it is in fact the retention of the solute which reduces wall suction significantly and so prompted previous workers, e.g., Blatt et al.,21 Porter,22 and Henry,23 to attempt the UF mass-transfer analysis with a nonsuction integral BL analysis. However, this approach often underpredicted experimental flux by an order of magnitude.30 Since the wall suction velocity was low compared to the cross-flow velocity in UF, its effect was minimal on the momentum BL, but seemingly dramatic on the mass-transfer BL. A number of approaches to rigorously solve the suction mass-transfer problem for single-phase UF were reported.30-38 These solutions attempted to solve the flow and mass-transfer problems simultaneously, but were all designed for single-phase UF, which already proved to be numerically intensive. Incorporation of the two-phase flow problem, and any other complex flow pattern, into such methods would prove a challenge. Furthermore, often computations require in-depth knowledge of the fluidsolute mixtures behavior under shear stress; e.g., shear or friction-affected diffusion or diffusivity coefficients are required for most models. Such data may not always be available. For the two-phase slug flow problem at hand, it is mainly the complex flow pattern which makes these rigorous solutions unattainable and so the blackbox BL method will be followed in this work. Here, the aim is simply to modify the existing Von Karman integral solution of the one-dimensional BL problem21-23 to include suction effects and then use this BL model to derive a mass-transfer coefficient for use with flux models (eqs 7 and 8) for flux prediction. We write an empirical relationship for the ratio of the mass-transfer

BL thickness with (δc,suc) and without (δc) suction (throughout the paper the subscripts will follow this format, where “suc” refers to suction inclusive results) for the retentive UF process as

δc,suc ) a(1 + Vp/Um)b δc

(5)

In this equation a and b are constants to be determined experimentally. The form of eq 5 was selected for its monotonicity and also to give an expected value of δc,suc/ δc ) 1 at Vp ) 0 and hence the value of a was taken as unity during the fit of the equation. Equation 5 relates the boundary layer thickness decrease due to suction, directly to the dimensionless suction velocity. To evaluate b, corresponding values of suction (δc,suc) and nonsuction (δc) boundary layer thicknesses were required along with the associated liquid velocity (Um) and permeation velocity (Vp). The nonsuction analysis of Smith and Cui25 was then employed to determine δc (the nonsuction value). The experimental flux and corresponding transmembrane pressure (TMP) values and the membrane permeability constant (Rm) were employed in a two-step approach with flux models (eqs 7 and 8) to determine δc,suc (including suction). First, measured flux and TMP and known Rm were employed in the osmotic pressure model (eq 8) to determine solute wall concentration, Cw. Then Cw is employed in the film theory model (eq 7) to determine δc,suc. The value of b was determined over the single-phase data sets from Smith and Cui12 reported in Table 2 and Li et al.,13 Sur et al.,14 and Cui and Wright39 using the units of m/s for Um and µm/s for Vp. With these units a value was determined as b ) -0.63 (a ) 1). This value was taken from four single-phase experimental data sets, containing 18 duplicate totally retentive, nonfouling, laminar flow experiments, of which three sets were removed as outliers. Caution as to the applicability of the fit is thus to be exercised. Further nonfouling UF data where membrane resistance, fluid rheology, and hydrodynamics are known could be incorporated into the correlation. The main weakness of this correlation revolves around the lack of turbulent flow data where the velocity and concentration BL profiles (one-seventh profile) do not resemble those from laminar flow (parabolic) and so applicability of the correlation to turbulent flow is not supported. It is expected that the equation will also hold for highly retained species as the degree of freedom in mass-transfer BL thickness will account for the partial or total rejection of the species. It must be pointed out that selecting a monotonic form for eq 5 and forcing it to match the nonsuction condition (δc,suc ) δc at Vp ) 0; essentially an extra data point) makes the equation more reliable outside of its derived range of experimental Vp/Um values. The accuracy of fit to determine b was evaluated by the value of the square of the Pearson product moment correlation coefficient or the R-squared value. R-squared indicated variance in the fit data. The best fit gave an R-squared value of 91%. However, this fit only employed the 10 best data sets (67% of the data sets, b ) -0.62), but confirmed the suitability of eq 5 for this analysis. An R-squared value of 86% was obtained with the best 12 data sets (80% of data sets, b ) -0.60), and an R-squared value of 72% was attained over all 15 data sets employed (b ) -0.63). Since the form of the equation (R-squared of 88%) was felt to be accurate, it

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improvement of the classical BL analysis for filtration modeling and flux prediction. 4. Prediction of Permeate Flux

Figure 6. Schematic representation of the assumed suction mechanism employed in deriving the semiempirical suction correction factor.

was decided to employ the fit with all 15 data sets in order to cover the widest range of suction conditions (3.8 e Vp/Um [µm/m] e 22) with the available data. Flux estimates with and without this correction factor will, however, be compared in this study. 3.2.2. Semiempirical Suction Correction. Others, e.g., De and Bhattacharya,38 have attempted to derive Sherwood number relations from first principles to account for the effect of suction. However, the unique property of UF is that the mass flux due to diffusion changes throughout the depth of the concentration BL unlike conventional mass-transfer processes. This poses a challenge to the first principles approach. Near the membrane the mass flux by diffusion is Jv(Cw - Cp), where Jv is the membrane flux, Cp the permeate, and Cw the wall solute concentration, respectively. At the edge of the concentration BL this mass flux has dropped to Jv(Cb - Cp), where Cb is the bulk concentration. Thus, equations of the form kc(Cw - Cb) ) D(dC/dy)wall, where kc is the mass-transfer coefficient and y is the depth through the BL, do not apply if kc is taken to be representative of the entire BL, e.g., De and Bhattacharya.38 Instead kc in this equation should be treated as a near wall value, kc,wall. If kc is not treated as a local value near the membrane wall, the mass flux equation subsequently derived will only hold if Jv/kc is low, i.e., for low concentration polarization cases only and will not provide a general equation. Here we test a simplified explicit analytical approach for including the effect of suction. This suction correction follows the assumption of a linear near-wall velocity profile as previously discussed, where the gradient is equal to the shear rate, γw. It is then postulated that a layer of material near the membrane wall of thickness t is removed by suction and that the remaining material moves up to the wall to replace the fluid sucked away. Essentially, the nonsuction mass-transfer BL thickness, δc is reduced by t to give the nonsuction mass-transfer BL thickness, δc,suc, as shown in Figure 6. Based on this mechanism of suction, one can write the following material balance: volume flow through annular region of thickness t ) volume sucked out through membrane wall, giving (πdtt)(γwt/2) ) Jv(πdtL), which, when assuming fully developed laminar flow (γw ) 8Um/dt) gives

t)

x

2JvL ) γw

x

JvdtL 4Um

(6)

In this equation, t ) δc - δc,suc and Jv ) membrane flux. This equation provides a more theoretical approach for suction inclusion and will be compared to the empirical approach previously discussed. These two methods of suction inclusion (eqs 5 and 6) into the BL analysis are initial attempts, but could provide a priori means to include suction into and

Two-phase flow data required for flux prediction, e.g., bubble (Ltb) and liquid slug length (Lls) data, are listed in Table 2. Flux is determined from simultaneous solutions of the stagnant film (eq 7) and osmotic pressure models (eq 8).

Jv ) kc ln Jv )

( ) Cw Cb

(TMP - ∆π) Rm

(7)

(8)

In these equations, kc is the mass-transfer coefficient (D/δc), which is determined from the hydrodynamics (eqs 4a and 5); Cw is the wall concentration, Cb the bulk concentration, TMP the transmembrane pressure, ∆π the osmotic pressure difference across the membrane wall, and Rm the intrinsic membrane resistance (a constant determined at 22 °C as 0.0234 h‚bar‚m2/L). Furthermore, the authors follow the assumption made by Davis and Sherwood,30 Bacchin et al.,31 and Song and Elimelech40 that the bulk concentration does not vary axially along the membrane length. This assumption is justified because of the low diffusivities of macromolecules upon which UF is applied and the extremely low permeation rates (Jv , 1% of Um) of these systems, hence resulting in the thin mass-transfer boundary layers. This assumption is true also for the film region of the two-phase UF where calculated δc values are lower than the film thickness (δo). Hence, outside the mass-transfer BL of the film region, the constant axial bulk concentration assumption still holds. For total rejection, the permeate side osmotic pressure is zero and so ∆π ) πw. TMP, Cb, and Rm are known and πw (wall-side osmotic pressure) is determined for 144 kDa dextran as a function of Cw as5

) π ) 0.1 exp(4.311 + 1.892C0.3048 w

(9)

The BL and analogy theory and hydrodynamics parts of the model, discussed earlier, determines δc,ave and hence kc for the membrane. Knowing kc, one can determine Cw, and flux (Jv) from eqs 7 and 8. All fluid physical properties were evaluated at the average concentration across the mass BL to improve the accuracy of the BL analysis. Bulk fluid properties were not found to be representative for BL calculations. The average concentration was evaluated by assuming a parabolic concentration profile within the masstransfer BL, as this shape was congruent to the parabolic velocity profile required for the momentum BL to generate the fully developed, laminar flow parabolic profile. The average concentration for a parabolic concentration profile is evaluated by integration over the BL to give

2 1 Cave ) Cb + Cw 3 3

(10)

In this equation Cb is the bulk and Cw the wall concentration. With the fluid properties (eqs 11 -13) used for BL calculations evaluated at Cave, the entire

Ind. Eng. Chem. Res., Vol. 44, No. 20, 2005 7691 Table 1. Summary of the Mass Transfer, Suction Correction, and Flux Prediction Models Employed for Single-Phase and Two-Phase (Scenarios 2&3 and Scenario 4) Systems aspect of solution

basic assumption

( )

equations employed

( )

1. mass-momentum transfer analogy under assumption of zero wall permeability

Sc f ∞, thus δc , δ and momentum-transfer BL growth can be assumed to be independent of mass-transfer BL growth

δc ) 2.897

2. wall-suction inclusion

suction thins mass-transfer BL and two relationships were devised to model this phenomenon

δc,suc Jv[µm/s] ) 1 + δc Um[m/s]

no fouling only concentration polarization; solute diffusivity changes with concentration and needs to be predicted at the average BL concentration during flux prediction:

simultaneous solution of eqs 4a-13; eqs 11-13 are the rheology relationships Cw (TMP - ∆π) Jv ) kc ln (7) Jv ) (8) Cb Rm

2 1 Cave ) Cb + Cw (10) 3 3 fully developed, laminar, parabolic velocity profile

) (9) π ) 0.1 exp(4.311 + 1.892C0.3048 w

5. two-phase shear rate

assume zero film velocity (Uf) and that liquid slug exhibits fully developed laminar flow

dU shear rate determined as: γw ) dr

5a. Scenarios 2&3 shear rate

lump the BL breakdown from the nose and tail regions into an effective single breakdown in each gas-liquid slug unit

take length-weighted* average velocity to determine shear rate for each gas-liquid slug unit; get single flux value for the unit; this is the same flux as for the entire membrane

5b. Scenario 4 shear rate

ignore BL breakdown completely and treat in a way similar to the single-phase calculation

take length-weighted average velocity for a unit and use this over the entire membrane length for the flux estimate

3. filtration flux prediction

4. one-phase shear rate

flux prediction becomes iterative in Cw. The diffusivity (eq 11) was determined using the method outlined by Clifton et al.41 using the form of their 46 kDa dextran equation:

D ) Do(1.47 + 0.52 tanh(28.4Xm - 1.491)) (11) In this equation D is the diffusion coefficient, Do is the diffusion coefficient at infinite dilution, and Xm is the mass fraction of dextran in the aqueous solution. Do was estimated as 2.235 × 10-11 m2/s for 144 kDa dextran using the molecular weight (MW) scaling method of Pradanos et al.42 with the Do value (4.045 × 10-11 m2/s) of 46 kDa dextran.41 The viscosity estimates were taken from an experimentally determined correlation for 100200 kDa dextran as shown in eq 12.

µ ) µaq(1 + 0.0451‚C)

(12)

where C is the concentration of dextran in kg/m3 and µ and µaq are the viscosity of the dextran-rich fluid and pure water, respectively. Density values were calculated based on the assumption of zero volume contributions from the dissolved dextran such that

F ) Faq + C

(13)

where C is the dextran concentration [kg/m3] and F and Faq are the density [kg/m3] of the dextran-rich fluid and pure water, respectively. Surface tension variation was negligible. Table 1 gives a summary of the overall mass-transfer and flux prediction model. 5. Results and Discussion In this section we employ the experimental results from Smith and Cui12 to test the derived model. Gassparged (6 Hz bubbling frequency) and one-phase UF experiments were performed using an aqueous 144 kDa dextran test solution in a purpose-built membrane unit with 0.89 mm bore polysulfone HF membranes with lengths of 1.18 m and a MWCO of 50 kDa (dextran

Dx γw

(

1/3

δc,ave ) 2.173

(4)

)

Dl γw

1/3

(4a)

-0.63

(5)

δc,suc ) δc -

x

JvdtL (6) 4Um

( )

shear rate determined as: γw )

| |

dU dr

)

8Um dt

)

8Um dt

r)R

r)R

marker). Total rejection is expected in these experiments. The two-phase experiments were designed with relatively accurate control of both the gas and liquid flow rates in each individual HF as well as control of the bubble length and frequency of sparging. Simultaneous measurement of the hydrodynamic characteristics of the gas-liquid two-phase flow, such as Utb, bubble length, and diameter (hence liquid film thickness, δo) and bubbling frequency was carried out using an infrared (IR) based system.12,18 The data obtained from the simultaneous UF and hydrodynamic experiments provided the data required for flux prediction of the twophase process. A detailed report of the above-mentioned HF UF experiments is given by Smith and Cui.12 The salient one-phase (cases 1-5) and two-phase (cases 6-10) experimental results from the study by12 are summarized in Tables 2 and 3. Cases 1-5 were corresponding one-phase cases of cases 6-10, respectively. From Figure 7 it can be seen that the empirical suction model (Method 1) is overpredictive compared to the always underpredicting nonsuction model. The semiempirical suction-inclusion method (Method 2) was employed with the experimental flux value used directly in eq 6 as convergence could not be attained for the a priori approach. Method 2 showed some overpredictions and some underpredictions, but always had flux estimates larger than the underpredictive classical flux. These single-phase cases (1-5) are hydrodynamically simple to model and so Figure 7 gives an accurate account of the flux prediction models as largely overpredictive. The mass-transfer BL thicknesses with and without suction for the single-phase cases are reported in Table 2. Nonsuction thicknesses for the single-phase experiments range between 3 and 4 times that of the suction concentration BL thicknesses. This implies that the ratio of nonsuction-to-suction Sherwood number is in the range 3-4. When the method of De and Bhattacharya38 was employed, the ratio of nonsuction-tosuction Sherwood numbers were found to be in the range of 2-4 as shown in Table 2. Thus, the approach of De and Bhattacharya38 is not completely dissimilar from suction Method 1 (eq 5), although diffusive flux

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Table 2. Experimental12 and Predicted Flux: Nonsuction (δc) and Suction (δc,suc, By Method 1, Eq 5) case no. Cb (g/L) TMP (bar) δo (µm) Utb (m/s) Uls (m/s) Jv,exp (L/m2‚h) γwall (s-1) Jv,suc (L/m2‚h) δc,suc (µm) δc (µm) Cw,suc (g/L) % error in Jv b δc/δc,suc38

1

2

3

4

5

5 0.40

10 0.48

30 0.62

10 0.24

10 0.76

0.72 11.92 6472 16.28 10.01 37.49 34.5 +36.6 4

0.72 9.90 6472 17.7 9.94 38.67 66.2 +78.8 3

0.72 9.83 6472 14.56 12.18 41.08 124.4 +48.1 2.5

0.38 5.88 3416 9.35 12.02 45.95 36.4 +59.0 2.2

0.99 13.85 8899 25.97 8.87 35.87 96.9 +87.5 4

6

7

8

9

10

5 0.59 37.8 1.13 0.90 24.94 2119 22.47 10.37 38.61 65.4 -9.9

10 0.61 44.5 1.20 0.90 20.06 2119 18.25 13.38 44.48 104.7 -9.0

30 0.81 62.9 1.32 0.90 15.14 2119 9.85 23.69 57.31 164.3 -34.9

10 0.42 31.2 0.61 0.52 17.33 1224 16.74 7.68 32.72 42.9 -3.4

10 0.94 55.1 1.75 1.23 25.96 2896 19.2 17.30 50.3 154.6 -26.0

a Average BL thicknesses and wall concentration under suction (C ) are also shown. Two-phase predictions from Scenario 4. (6 Hz w bubbling frequency; Ltb ) 31 mm; Lls ) 11 mm; Lm ) 1.18 m; 144 kDa Dextran; 50 kDa HF MWCO with 0.89 mm i.d.) b %flux prediction error ) 100 × (Jv,model - Jv,exp)/Jv,exp.

Table 3. Experimental12 and Predicted Two-Phase Average BL Thickness and Flux Values for Scenario 1 (with Eq 5 Suction Correction), Scenario 4 (Nonsuction), and Scenarios 2&3 (without and with Eqs 5 and 6 Suction Corrections) case no. Jv,exp Jv,suca % error Jv,modela δc,suca(µm) Jva % error Jva δca(µm) Jvb % error Jvb δcb(µm) Jv,succ (Jv,film/Jv,liquid slug),succ a

eq 5 eq 6 (eq 5) (eq 6) (eq 5) (eq 6)

6

7

8

9

10

24.94 24.93 23.44 -0.04 -6.0 4.29 8.85 17.24 -30.9 17.87 9.62 -61.4 17.87 6.58 0.99

20.06 25.24 21.25 +25.8 +5.9 4.32 9.86 14.88 -25.8 18.36 7.22 -64.0 18.36 6.76 0.69

15.14 28.9 16.54 +90.9 +9.3 4.37 12.62 11.54 -23.8 19.73 4.34 -71.3 19.73 8.64 0.27

17.33 17.44 14.79 +0.6 -14.7 4.66 12.35 10.94 -36.9 21.56 8.04 -53.6 21.56 4.64 0.97

25.95 38.63 30.97 +48.8 +19.3 3.69 8.3 19.37 -25.4 17.11 7.52 -71.0 17.11 10.44 0.53

Scenarios 2&3. b Scenario 4. c Scenario 1.

Figure 7. Evaluation of the suction-inclusion equations using single-phase HF UF data 12. Method 1 uses eq 5 and Method 2 uses eq 6.

for ref 38 was incorrectly assumed to be constant through the concentration BL thickness. In Table 2 the Scenario 4 model of the two-phase cases, with suction (Method 1), was reported. This model, as expected, always underpredicts flux. It is important to note how accurate the Scenario 4 model is with low polarization (case 9, Table 2), but how heavily underpredictive the model is for cases with higher polarization (cases 8 and 10, Table 2). The absolute values and ratio of wall-to-bulk concentration is indica-

Figure 8. Comparison of two-phase experimental data 12 with estimates using different scenarios, all including suction effects.

tive of the degree of polarization. The calculated wall concentration, Cw, for each case is reported in Table 2 to show the degree of polarization. Case 9 in Table 2 suggests that flow reversal is not critical to enhancement at low polarization and that flux enhancement due to introduction of gas slugs is mainly important for highly polarized HF UF. One can thus achieve accurate flux prediction for low-polarization HF UF studies without accounting for the flow reversal effect on the mass-transfer BL (note that suction has been accounted for in this model). Modeling of only low-polarization UF experiments could thus lead to the misconception that flow reversal inclusion into models is not critical in deriving a general model for flux prediction. Higher polarization models prove otherwise. It can be seen in Figure 8 that the experimental flux values always lie between the conservative flux estimate of Scenario 4 and the overestimation of Scenarios 2&3 (including suction Method 1 for both models). In fact, the average of these upper and lower bound flux estimates (using Method 1 for suction inclusion) seemed to produce a fairly reliable flux estimate as expected and shown in Figure 8. This is a useful tool for process design along with the upper and lower bound flux estimate. In Figure 8 and Table 3, the suction Method 2, Scenarios 2&3 predictions were determined a priori. In the case of low polarization, case 9, the flux estimates from Scenarios 2&3 and Scenario 4 (suction Method 1) are similar due to the thinness of the mass-transfer BL and thus assuming a zero BL thickness at either the nose or tail, as in Scenarios 2&3, only makes a slight difference in the flux prediction from Scenario 4 for this


case. In contrast, in the higher polarization cases (cases 8 and 10), the Scenarios 2&3 and Scenario 4 solutions differ considerably as the flow reversal or positionvarying shear stress effect on the mass-transfer BL thickness is more pronounced in such cases. A detailed study of the effect of position-varying shear stress on the mass-transfer BL is thus critical for developing a general model for flux prediction for gas-sparged HF UF. The approach followed by Smith and Cui7 is a simple model to try to quantify the local hydrodynamics effect on the mass-transfer BL thickness and this is one approach which could be pursued in developing an understanding of this important phenomenon. From Table 3 it can be seen that using Scenarios 2&3 (severe mass-transfer BL thinning) while ignoring the suction effect in the mass-transfer calculations, one can in fact predict flux reasonably well, especially for cases 8 and 10. In fact the predictions tend to be more accurate than the Scenarios 2&3 model with suction inclusion (Table 3), prompting one to conclude that all models which include one complete removal of the masstransfer BL by flow reversal and ignores suction effects will give accurate flux predictions. This is an erred approach: overprediction of flux due to including complete removal of the mass-transfer BL at the nose or tail cancels the underprediction resulting from ignoring the thinning effect of suction on the mass-transfer BL. This results in a false belief that suction inclusion is not critical in modeling HF UF, while flow reversal only is important. Support for the above arguments is found in the significant differences in the suction-inclusive mass-transfer BL thicknesses reported in Tables 2 and 3. Here the two-phase Scenarios 2&3 BL thicknesses with suction are in some cases nearly an order of magnitude thinner than those for Scenario 4 without suction. This is a clear indication that position-varying shear stress phenomena are an important consideration. The film shear and flux contribution were assumed zero throughout this study. However, this assumption penalized the Scenario 1 flux estimate (Table 3) most severely due to the large length of zero film flux included in its length-weighted flux calculation. This caused Scenario 1 to predict a lower flux than that of even Scenario 4 and so Scenario 1 was not a generous upper bound as expected. In essence, the Scenario 1 flux estimates (Table 3) point out that film flux cannot truly be zero. In fact, the film flux in some Scenario 1 cases was calculated as equal (cases 6 and 9, Table 3) to that of the liquid slug, based on the required experimental flux. Film flux was not found to be zero for any of cases 6-10 and the lowest ratio of film-to-liquid slug flux was 0.27 (case 8, Table 3). This observation brings to the fore the issue of the dynamicity of the entire system and how it seems plausible to base the liquid slug on a quasistatic BL model, but not the liquid film. The real system does not have stagnant BLs, but comprises continuously moving bubbles where even the front-most part of the film region and the region just trailing the bubble are never free of phase change for more than 0.05 s (case 9: largest Ltb/Utb ) 31 mm/0.61 m‚s-1). The zero-flux static film region model is thus clearly severely underpredictive for this dynamic system. It is important to observe that for corresponding onephase/two-phase experiments, cases 3&8 and 5&10, the suction-inclusive (Method 1) flux estimates are larger for the one-phase cases than the lower bound of the twophase cases (Scenario 4). Essentially, this implies that

these particular experiments, which coincidentally are higher polarization experiments, had a possibility of being disadvantaged by the introduction of bubbles, if sparging was perhaps not optimized. This is an important aspect of gas-sparged HF UF to observe. A more general model would have to (a) investigate inclusion of the permeation velocity into the boundary layer analysis in a more integrated and fundamental manner and (b) perform a more detailed analysis of the effect of position-varying and dynamic shear stress on the mass-transfer BL. The study of these two important phenomena also play an important role in the development of predictive models for a number of other enhanced mass-transfer UF systems, such as rotating disk and vibrating membrane systems43 as well as pulsatile flow systems and in systems where scouring balls were added for enhancement.44 A possible approach for quantifying the mass transfer under position-varying and dynamic shear stress could build upon surface renewal theory as developed by Dankwerts45 as this method could allow for various speeds and frequencies of gas sparging. 6. Conclusions 1. A model, based on simplified hydrodynamics and one-dimensional boundary layer and analogy theory, was constructed for predicting filtration flux for gassparged and single-phase HF UF. The model also accounted for the effect of suction through the membrane walls in the boundary layer analysis. The predicted flux compared reasonably well with the experimental values for the single-phase case, highlighting that suction inclusion was important for predicting flux from standard mass-transfer correlations. The empirical suction-inclusion method was reliable for all flux estimates, one- and two-phase. The semiempirical suctioninclusion method gave good flux estimates for systems with modest repeating unit lengths, e.g., Scenarios 2&3. 2. The two-phase experimental flux values were correctly found to always lie between the extremes of the conservative (Scenario 4) and generous (Scenario 2&3) flux estimates proposed in this paper based on the empirical suction-inclusion method. These thus provided reliable upper and lower bound flux estimates. The average of the upper and lower bound flux values gave a good approximation of the experimental flux, presenting a useful tool for process design, along with the actual flux bounds. The flux estimates from Scenarios 2&3 using the semiempirical suction method did not give an actual flux upper bound as did the empirical method, but instead produced a reasonably accurate flux estimate overall. 3. The two-phase hydrodynamics model ignored shear rate contributions from the film region as well as shear rate spikes at the bubble nose and tail, but included flow reversal effects experienced in the nose and tail regions. The latter was proposed, in this study, to be the major source of flux enhancement. The reliable performance of the flux prediction model built around this proposal supports this hypothesis. 4. A general model for flux prediction can be derived using the approach outlined in this paper, but the boundary layer analysis would have to include suction in a more rigorous manner. Furthermore, a more accurate analysis of the breakdown of the mass-transfer BL by flow reversal (position-varying shear) and dynamic shear needs to be developed, e.g., via surface

7694


renewal theory. These studies are important for a number of other enhanced UF processes, e.g., vibrating and rotating disk membranes. Acknowledgment The authors thank the U.K. EPSRC (GR/N44638) for partial funding of this project. Nomenclature Ca ) capillary number, µUtb/σ Cave ) average boundary layer concentration [kg/m3] Cb ) bulk or feed (dextran) concentration [kg/m3] Cw ) wall concentration [kg/m3] D ) diffusion coefficient [m2/s] db ) film region bubble diameter [m] Do ) diffusion coefficient at infinite dilution [m2/s] dt ) tube diameter [m] 2 Eo ) Eotvos number, ∆Fgreff /σ Jv ) calculated volumetric filtration flux without suction effects [L/m2‚h] Jv,suc ) calculated volumetric filtration flux with suction effects [L/m2‚h] Jv,exp ) experimental volumetric filtration flux [L/m2‚h] kc ) mass-transfer coefficient [s-1] Lls ) liquid slug length [m] Lm ) membrane length [m] Ltb ) Taylor bubble length [m] q ) liquid flow rate [mL/min] reff ) effective radius [m] Rm ) intrinsic membrane resistance constant (22 °C) [h‚ bar‚m2/L] Sc ) Schmidt number [dimensionless: µ/FD] TMP ) transmembrane pressure [Pa] U ) near wall liquid velocity [m/s] Uf ) film velocity [m/s] Um ) liquid cross-flow or superficial velocity [m/s] Utb ) bubble rise velocity [m/s] Vp ) permeate velocity ≡ Jv [m/s] We ) Weber number, FUm2reff/σ x ) axial position or distance [m] l ) length [m] Xm ) mass fraction y ) radial position or distance from wall [m] Greek Letters δc ) calculated mass-transfer boundary layer thickness without suction effects [m] δc,suc ) calculated mass-transfer boundary layer thickness with suction effects [m] δc,ave ) average calculated mass-transfer boundary layer thickness [m] γw ) wall shear rate [s-1] µ ) solution viscosity [Pa‚s] µaq ) pure water viscosity [Pa‚s] πw ) wall osmotic pressure [Pa] F ) solution density [kg/m3] Faq ) pure water density [kg/m3] σ ) liquid-air interfacial tension or surface tension [N/m]

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Received for review October 17, 2004 Revised manuscript received March 27, 2005 Accepted March 28, 2005 IE048989B

Upper- and Lower-Bound Estimates of Flux for Gas-Sparged

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