High-throughput premix membrane emulsification using nickel sieves having straight-through pores

Journal of Membrane Science 383 (2011) 116–123

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High-throughput premix membrane emulsification using nickel sieves having straight-through pores Akmal Nazir ∗ , Karin Schroën, Remko Boom Wageningen University, Food Process Engineering Group, Bomenweg 2, 6703 HD Wageningen, The Netherlands

a r t i c l e

i n f o

Article history: Received 18 July 2011 Received in revised form 17 August 2011 Accepted 18 August 2011 Available online 26 August 2011 Keywords: Premix membrane emulsification Metallic sieves Straight-through pore Oil-in-water emulsion Fouling

a b s t r a c t We report on the use of nickel sieves, having a uniform pore size (typically 10 ␮m × 300 ␮m), for (oilin-water) premix emulsification at relatively low transmembrane pressures. The droplet break-up was found to be based on elongation and recompression of droplets typical of high-pressure homogenization. The dependence on the transmembrane pressure indicated at least partial turbulent conditions. In line with this, the transmembrane fluxes were very high, while a reasonable span (around 1) of the droplet size was found. There was no indication of fouling in the process, even after 5 passes, which indicates that the process is tolerant to product and conditions. A master curve was found for the Weber number as function of the transmembrane pressure normalized on the Laplace pressure of the emulsion before emulsification, which is helpful in further scale-up of this process. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Emulsions are of great significance in our daily life, as they are the basis of many products, from e.g. food, pharmaceutical, cosmetics, and chemical industries. For stable emulsions, the droplet size needs to be small, and ideally uniform. Traditional emulsification techniques are known to be rather energy consuming; in some cases only 1% of the applied energy is used to form emulsion droplets, and the rest is dissipated as heat leading to temperature increase of the product, which may influence the ingredients negatively [1]. A lot of research has been carried out to make the emulsification process more efficient. In the last decades, a number of new emulsification techniques have become available which are based on microstructured systems, such as membranes and microfluidic devices. With these systems, emulsion droplets may be formed directly by extrusion through pores, or larger droplets from a coarse premix may be broken up into smaller ones. These techniques have been extensively investigated, and are compared for example in [2]. As introduced by Suzuki [3], premix membrane emulsification, along with other latest emulsification techniques, has established itself as a promising technique for the production of small sized and relatively monodisperse emulsions at relatively low energy inputs as compared to traditional emulsification techniques. As the name implies, the process starts with a coarse premix emulsion

∗ Corresponding author. Tel.: +31 317 482240; fax: +31 317 482237. E-mail address: [email protected] (A. Nazir). 0376-7388/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2011.08.051

which is then pushed under pressure through a membrane. This results in a (more or less) homogenized emulsion. The emulsion may be passed through the membrane repeatedly depending on the desired level of homogenization [4]. The simplicity of premix emulsification, and the fact that higher dispersed phase fractions can be made makes it an attractive technique for large-scale production of dispersions compared to e.g. membrane emulsification [5]. On the other hand, cross-flow membrane emulsification, and straight-through emulsification with microfluidic devices is known to lead to more monodisperse emulsions, even though the process is only feasible for relatively low disperse phase fractions. In this paper, we focus on premix emulsification. From the work of Vladisavljevic et al. [6] it is known that the membrane type has a great influence on the emulsion droplets; Shirasu Porous Glass (SPG) membrane is the most popular membrane of choice, not just for premix emulsification but also for cross-flow emulsification. Various membrane related factors have been reported in literature to influence droplet size and monodispersity, such as pore size, porosity, thickness, tortuosity, and membrane surface properties. A hydrophilic surface is normally required to produce oil-in-water (O/W) emulsion whereas for water-in-oil (W/O) emulsion the surface should be hydrophobic. Alternatively, phase inversion can be induced in premix emulsification to get high disperse phase fraction [7]. For example, by using a hydrophilic membrane with a W/O premix emulsion, the resulting emulsion would become O/W with high disperse phase fraction. It should be noted that phase inversion can only be applied with specific systems. In premix emulsification, besides the membrane properties, the transmembrane pressure, and also the properties of the various

A. Nazir et al. / Journal of Membrane Science 383 (2011) 116–123

ingredients e.g. the viscosity ratio of the phases (as is the case in traditional emulsification driven by elongation or local shear), surfactant, temperature (related also to viscosity and adsorption rate), and pH (in case charged components are present) influence the resulting emulsion. These aspects have already been considered in literature [4,8–15] (as summarized in our previous review [2]), but simultaneous optimization of all these factors is far from trivial. Ideally, scaling relations are used for this, as are available for a number of microfluidic devices such as T-shaped [16–18] and Yshaped junctions [19], co-flow devices [20], microchannels [21–23] and EDGE devices [24]. However, given the many different droplet break-up mechanisms that occur simultaneously as e.g. described in a microfluidic study by van der Zwan et al. [25], they are not readily available for premix emulsification. Compared to other novel production methods like cross-flow membrane emulsification and microchannel emulsification, premix emulsification is an attractive technique because of its high production rate per m2 membrane area [2]. However, its sensitivity to depth fouling of the membrane is a major obstacle in its commercialization with systems containing for example proteins [26]. A potential solution is the use of membranes that are very thin and do not have long, narrow interconnected pores. We report here on the use of metal (nickel) sieves, which have relatively short, straight-through pores and are available with different pore sizes and shapes. The effects of operating conditions like transmembrane pressure and number of homogenization cycles were evaluated in terms of their influence on emulsion properties and transmembrane flux. Besides, all data points were compiled into a single master curve based on dimensionless numbers. Finally, the energy usage and fouling tendency were quantified and compared to more traditional techniques. 2. Theory

w,p =

8c J εdp

(1)

where c is the continuous phase viscosity (the emulsions used here have low dispersed phase fraction, and therefore this simplification is allowed), J is the transmembrane flux,  is the membrane tortuosity (=1 for straight-through pores), ε is the membrane porosity, and dp is characteristic pore dimension (pore width in this case), respectively. The Reynolds number, Re, is given by: vdh Re = c

(2)

where  is the emulsion density, v is the emulsion velocity inside the pore (v = J/ε), and dh is the pore hydraulic diameter, defined for rectangular pores as: dh =

2lw l+w

(3)

where l and w are the pore length and width, respectively. The process of droplet break-up in the turbulent regime has been described in literature by using the dimensionless Weber number, We, which is a ratio between the inertial forces (as a result of local pressure fluctuations) and interfacial tension forces [27]. For our emulsification setup, we defined Weber number as follows: We =

d32 v2 2

The friction coefficient, f, inside the pore can be calculated by the following expression: f =

p w 1/2pv2 h

(4)

where d32 is the Sauter diameter of droplet produced, and  is the interfacial tension.

(5)

where p is the pressure drop across the sieve, and h is the depth of the pore. Vankova et al. [28] used the same relation to calculate the f for narrow-gap homogenizer. Following the approach of van der Zwan and co-workers [26], the pressure drop across the sieve was estimated with the following second order polynomial equation: p = ˛J 2 + ˇJ

(6)

where ˛ and ˇ are the fit parameters. For the first pass through the sieve, the pressure drop can be used instead of energy density, Ev : Ev =

p pv = = p v v

(7)

where P is the power input, and Фv is the volume flow. For more than one pass, the energy densities of all the passes are cumulative. The pratio , which is an indicative of the minimum amount of energy needed to deform the droplet and the applied energy, can be defined as a ratio of the transmembrane pressure and the droplet Laplace pressure: pratio =

rp 2

(8)

where r is the droplet radius before entering the pore. Karbstein and Schubert [29] derived an expression for the Sauter diameter of a homogenized emulsion and the energy density, Ev , for continuous mechanical emulsification processes (avoiding recoalescence) where the residence time of droplets in the dispersing zone lies in the order of milliseconds to tenths of a second: d32 = ˛dr Ev−ˇdr

The average wall shear stress,  w,p , inside the membrane pores can be defined as [9]:

117

(9)

Here ˛dr and ˇdr are constants. ˛dr depends on the efficiency of droplet disruption and is effected by dispersed phase viscosity, whereas, ˇdr is affected by the flow conditions. In case of multiple passes, Eq. (9) can be extended to incorporate the number of cycles, N, with dr as another fit parameter [26]: d32 = ˛dr Ev−ˇdr N dr

(10)

The droplet diameter is usually normalized by dividing the droplet diameter with the pore diameter (pore width, w, here), known as dratio : dratio =

d32 w

(11)

The droplet span (an indicator of droplet uniformity), ı, is calculated with: ı=

d90 − d10 d50

(12)

where dx is the droplet diameter corresponding to x vol.% on a cumulative droplet volume curve. 3. Experimental 3.1. O/W premix preparation An O/W premix consisting of 5% n-hexadecane (99% for synthesis, MERCK) in milliQ water having 0.5 vol.% Tween-20 (for synthesis, MERCK) as a surfactant, was prepared with an Ultraturrax homogenizer (IKA® T-18 basic) operated at 3500 rpm for 10 min (unless otherwise mentioned); this leads to reproducible starting emulsions for our experiments. The droplet size was typically around 27 ␮m with a span of 0.9.

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Droplet diameter, d32 [μm]

35 30 25 20 15 10 5 0 0

50

100

150

200

250

Transmembrane pressure, Δp [kPa] Fig. 2. Effect of transmembrane pressure on droplet diameter: (♦) 1st, () 3rd, and () 5th pass through the sieve (13.2 ␮m).

The results were compared and fitted to the relations presented in the theory section. All fits were conducted with MS Excel using solver function, and the correlation coefficients and 95% confidence intervals were determined. 3.4. Interfacial tension measurement

Fig. 1. Schematic representation of experimental setup.

3.2. Nickel sieves used for emulsification The different nickel sieves (Stork, Eerbeek, the Netherlands) used in this study have long rectangular pores as shown in Table 1. Around the pores, the sieves may have a raised supporting structure, either on one side or on both sides. The overall thickness of the sieves (including the supporting structure) was 80, 200, 350 and 400 ␮m, and the pore width (determined by SEM imaging) was 13.2, 12.8, 11.6 and 10.6 ␮m, respectively. The pore depth (without supporting structure) was typically around 80 ␮m. The SEM images were obtained without coating, in a field emission scanning electron microscope (Magellan 400, FEI, Eindhoven, the Netherlands). The samples were fitted on SEM sample holders by carbon adhesive tabs (EMS Washington, USA) and subsequently analyzed at 2 KV at room temperature, and the images were digitally recorded. 3.3. Emulsification setup The emulsification setup is shown in Fig. 1. The pressure vessel containing the premix emulsion was connected to the module (a Plexiglas column) having a metal sieve at the bottom with an effective surface area of 1.43 cm2 . The sieve was held in place by two rubber O-rings (above and below) at the bottom junction of the column. The experiment was started by pressurizing the emulsion vessel (containing about 500 ml of emulsion) with air, keeping the valve opened and connected to the column. The emulsification was started by opening the outlet valve of the column and collecting the homogenized emulsion in a flask placed on a balance for recording the flux. The cycle was repeated up to five times; no significant differences in droplet size were observed when more cycles were used. The coarse emulsion and all the homogenized emulsion samples were analyzed for droplet size and size distribution with light scattering (Malvern Mastersizer 2000). Three readings were taken, and provided the results were close together, the average droplet diameter and span was determined.

The equilibrium interfacial tension at the hexadecane and surfactant-solution interface was measured using a drop profile analysis tensiometer PAT-1 (SINTERFACE Technologies, Germany) at a controlled temperature of 23 ◦ C. A surfactant-solution droplet was formed at the tip of a capillary immersed in hexadecane. The measurements were started soon after the droplet formation and continued till a constant (equilibrium) interfacial tension was obtained. The equilibrium interfacial tension at the hexadecane water (0.5 vol.% Tween-20) interface was found to be 5.8 mN m−1 , and was used as such in calculations. 4. Results and discussion At Reynolds numbers larger than 50, the friction coefficient remains nearly constant (f = 1), which is around 7 times higher than the constant value obtained with a narrow-gap homogenizer [28], and around 2.4 times higher than obtained with a static mixer [30]. A higher friction coefficient in case of these nickel sieves suggests that in spite of relatively low Reynolds numbers (Re ≤ 220), the emulsification regime is turbulent because of the specific pore geometry, possibly leading to droplet break-up after passage of the pore [31]. As indicated in the theory, various factors are expected to influence the resulting droplet size after premix emulsification. Here we start by discussing the effect of transmembrane pressure on droplet size (distribution) and flux, and relate the results to various dimensionless numbers introduced in the theory section. 4.1. Effects of transmembrane pressure Emulsions containing 5% n-hexadecane were repeatedly pressed through the 13.2 ␮m sieve at different transmembrane pressures. Higher transmembrane pressure resulted in much smaller droplets (see Fig. 2), as could be expected since the actual shear stress on the droplets increased. The final droplet diameter at 200 kPa was around 6 ␮m after fifth pass through the sieve which was much lower than the pore width of the sieve (13.2 ␮m, as indicated by the dotted line). All sieves showed similar droplet size behavior as function of pressure and number of passes; here we limit the discussion to only this sieve in this section.

A. Nazir et al. / Journal of Membrane Science 383 (2011) 116–123

119

Table 1 SEM images of all nickel sieves used in this study along with their specification. Sieve specification

Front view

Back view

Pore size: 13.2 m × 336.9 m Thickness: 80 m Porosity: 4.79% Plain surface on both sides

Pore size: 12.8 m × 329.3 m Thickness: 200 m Porosity: 4.37% Supporting structure on back side

Pore size: 11.6 m × 331.1 m Thickness: 350 m Porosity: 3.95% Supporting structure on back side

Pore size: 10.6 m × 330.2 m Thickness: 400 m Porosity: 3.62% Supporting structure on both sides

The dratio (ratio between droplet size and pore width) of emulsions prepared with cross-flow membrane emulsification are typically between 2 and 10. As comparison, the droplet size of the coarse premix emulsion (prior to emulsification) was between 2.2 and 2.3 times the pore size, and was further reduced through repeated passage. The size reduction during the first pass was maximum, and in general after the 3rd pass, the droplet size did not essentially decrease anymore, indicating that the limit of the system is reached. After the fifth pass, the droplet size was around 0.4 times the pore size. The Weber number (Eq. (4)) was used to compare the results. The We number is based on the size of the droplets that are produced, where it is also expected that the droplet size of the feed

emulsion (premix) will have a decisive role in the droplet size reduction. Therefore, in Fig. 3, we used Weber number against pratio (the applied pressure over the minimum pressure required to deform droplets of certain size) calculated using Eq. (8). A high pratio value corresponds to a surplus of applied energy. Smaller droplets have a high Laplace pressure, and on the other hand due to a decrease in the effective surface area, smaller droplets experience less disruptive forces, i.e. a decrease in d32 v2 term in Eq. (4). This results in a decrease in Weber number as a result of different passes through the sieve. All data collapsed into one master curve that can be used to link premix size and applied pressure to the actual droplet size that can be obtained under specified experimental conditions. Keep in mind

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1.6 1.4

60

Droplet span, δ [-]

Weber number, We [-]

75

45 30 15

1.2 1.0 0.8 0.6 0.4 0.2 0.0

0 0

200

400

600

800

0

1000

50

prao [-] Fig. 3. Weber number as a function of pratio : (♦) 1st, () 2nd, () 3rd, () 4th, and () 5th pass through the sieve (13.2 ␮m).

Transmembrane flux, J [m3m-2hr-1]

100

150

200

250

Transmembrane pressure, Δp [kPa] Fig. 5. Effect of transmembrane pressure on droplet size distribution: (♦) 1st, () 3rd, and () 5th pass through the sieve (13.2 ␮m). Table 2 Measured values and standard deviations of fit parameters of Eq. (10).

1750 1500 1250

˛dr ˇdr dr

1000

13.2 ␮m

12.8 ␮m

11.6 ␮m

10.6 ␮m

16.59 ± 0.55 0.67 ± 0.04 0.40 ± 0.05

17.32 ± 0.66 0.66 ± 0.05 0.37 ± 0.06

19.18 ± 1.03 0.66 ± 0.07 0.43 ± 0.09

17.39 ± 0.76 0.66 ± 0.05 0.35 ± 0.07

750 500 250 0 0

50

100

150

200

250

Transmembrane pressure, Δp [kPa] Fig. 4. Dependence of the flux on the transmembrane pressure: () clean water flux, () 1st, and () 5th pass through the sieve (13.2 ␮m).

that in all experiments, the pratio values were high, indicating that the amount of energy used largely surpasses the minimum amount of energy needed to deform feed droplets (which is not equal to the energy needed to make them). From Fig. 3, also information on different break-up mechanisms was obtained, especially at very low Weber number, where it is expected that the droplet is deformed by the pore, and may or may not snap off depending on the local conditions. A droplet breaks spontaneously due to Laplace pressure differences if the Weber number exceeds a certain value, the so-called critical Weber number, Wecr , and most probably, we are very close to this critical value at We = 5, below which there was hardly any reduction in droplet size. This type of droplet break-up has also been reported for microchannel emulsification [32]. The flux was not completely linearly dependent on the transmembrane pressure, as shown in Fig. 4. The convex curve is probably an indication on the development of (more) turbulent flow conditions at higher pressures; alternatively this could also be an indication that pressure losses occured between our pressure vessel and the module, and that this effect was larger at higher applied pressures, although we expect the pressure losses in the sieve to be far greater.

Despite their low porosity (0.05), the transmembrane flux of the nickel sieves used here was more than 100 times higher compared to premix emulsification studies by Vladisavlijevic et al. [11]. There was no significant difference between pure water flux and emulsion fluxes at first and last cycles. This implies that droplet break-up did not contribute significantly to the friction loss in the sieve, and that there was no appreciable membrane fouling throughout the experiments. The droplet uniformity, expressed as the span of the distribution (Eq. (12)), was typically in the order of 1.0–1.4 for all emulsions, which increased somewhat at higher transmembrane pressures (see Fig. 5). At higher transmembrane pressure (v = 8–10 m s−1 at 200 kPa), droplet breakdown is expected to be accelerated, but recoalescence is more probable due to insufficient stabilization by surfactant. 4.2. Membrane characterization based on energy density The experimental data was fitted to the energy density Eq. (10) extended to take in the influence of homogenization cycles. In Fig. 6, the measured droplet Sauter diameter is plotted against the fitted results of Eq. (10) for all the sieves. It is clear that the suggested equations gave reasonable descriptions, although a slight curved behavior is visible, which could be related to a change in droplet break-up mechanism, possibly, relatively more turbulent conditions. The estimated values of fit parameters ˛dr , ˇdr and dr using Eq. (10) are shown in Table 2 along with their standard deviations. The values of ˛dr decreased somewhat with increasing pore size; the others were more or less constant, except for the 10.6 ␮m sieve. This may be because of the different morphology of the membrane (see Table 1).

Table 3 Correlation coefficients between fit parameters of Eq. (10). 13.2 ␮m ˛dr ˛dr ˇdr dr

12.8 ␮m ˇdr −0.30

−0.30 −0.69

0.80

dr −0.69 0.80

˛dr −0.28 −0.70

11.6 ␮m ˇdr

dr

−0.28

−0.70 0.77

0.77

˛dr −0.26 −0.66

10.6 ␮m ˇdr

dr

−0.26

−0.66 0.82

0.82

˛dr −0.313 −0.705

ˇdr

dr

−0.313

−0.705 0.776

0.776

A. Nazir et al. / Journal of Membrane Science 383 (2011) 116–123

12.8 μm

40

40

30

30 d32 [μm]

d32 [μm]

13.2 μm

20

20

10

10

0

0 0

10

20 30 d32 (Ev, N) [μm]

40

0

10

11.6 μm 40

40

30

30

20

10

0

0 10

20 30 d32 (Ev, N) [μm]

40

20

10

0

20 30 d32 (Ev, N) [μm]

10.6 μm

d32 [μm]

d32 [μm]

121

0

40

10

20 30 d32 (Ev, N) [μm]

40

Fig. 6. Experimental values of droplet Sauter diameter plotted against estimated values using Eq. (10).

4.3. Effect of disperse phase fraction, including a comparison with SPG membrane Premix emulsification allows the preparation of more concentrated emulsions; also this aspect was evaluated through experiments carried out at 25 and 50 vol.% dispersed phase (100 kPa transmembrane pressure and 0.5% Tween-20 concentration). In both cases, similar trends were obtained as for 5% of dispersed phase, but the residual droplet size was considerably larger. The closest available comparison for our results can be found in the work of Vladisavlijevic et al. [10], who used a SPG membrane (8.5 ␮m pore diameter, 0.8 mm wall thickness compared to our sieve 12.8 ␮m pore width and 0.08 mm wall thickness), to produce a 40 vol.% corn oil in water emulsion.

a

8 7

drao [-]

6 5 4 3 2 1 0 0

b Transmembrane flux, J [m3 m-2 hr-1]

The value of ˇdr contains information about the droplet disruption mechanism. For high-pressure homogenizers, values are found between 0.35 (turbulent inertial forces) and 1 (laminar shear or elongational flow) [33], whereas, for membrane emulsification it may be greater than one [34]. As for our experiments, ˇdr was in the range of homogenization, with at least partly turbulent flow. As mentioned before, we expect that the droplets may pass the constriction, but will break-up after passing the pore, as has been described by Harvie et al. [31] for microfluidic circuits in which turbulence after a constriction was identified as the reason for droplet break-up. This could also contribute to the retention of the high fluxes upon repeated passage of the membrane since droplet break-up does not require contact between droplet and membrane. The correlation coefficients between the three fit parameters of Eq. (10) are shown in Table 3; that shows that they are not correlated.

1

2

3

4

5

6

No. of passes, N [-] 1000 800 600 400 200 0 0

1

2

3

4

5

6

No. of passes, N [-] Fig. 7. (a) Dimensionless droplet diameter (using d50 ) and (b) transmembrane flux as a function of different passes through the sieve (12.8 ␮m) at a transmembrane pressure of 100 kPa: () 5%, (♦) 25%, () 50%, and () 40% taken from Fig. 11 of Vladisavlijevic et al. [10].

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The emulsification conditions in the work of Vladisavlijevic were similar to our study, and it is clear that the SPG membrane is eventually more effective in reducing the droplet size as shown in Fig. 7. The internal structure of the SPG membrane, which has many long interconnected pores, is expected to lead to different droplet break-up mechanisms compared to our sieves which only have one constriction point (i.e. the pore). The sieves operate in similar fashion as a high pressure homogenizer (fluxes are much higher than for SPG), where inside an SPG membrane, the droplets may breakup due to Laplace instabilities that are expected at lower fluxes. In case of nickel sieves, the major droplet reduction took place after the first pass and after that, the reduction was quite gradual; while in case of SPG membrane, the reduction may continue up to the third pass. It is clear that membrane emulsification with metal sieves shows distinctive behavior, which makes it a valuable addition to the range of emulsification techniques that are currently available. Especially for the preparation of large amounts of emulsions with droplet sizes of around 5–10 ␮m that do not need to be perfectly monodispersed, the technique presented here could be of interest. When technological developments lead to production of sieves with smaller pores, also smaller emulsion droplet sizes come within reach.

P pratio

power input (W) ratio of transmembrane pressure to droplet Laplace pressure r droplet radius (m) Reynolds number Re v emulsion velocity inside the pore (m s−1 ) w pore width (m) Weber number We Greek letters ˛ and ˇ constants in Eq. (6) ˛dr , ˇdr , and dr constants in Eqs. (9) and (10) ı droplet span p pressure drop across the sieve (Pa) ε membrane porosity continuous phase viscosity (Pa s) c  emulsion density (kg m−3 )  interfacial tension (N m−1 )  w,p average wall shear stress inside the pore (Pa)  membrane tortuosity Фv volume flow (m3 s−1 )

5. Conclusions

References

Premix membrane emulsification with nickel microsieves was found to be based on elongation and recompression of droplets, and thereby it is comparable to high-pressure homogenization. The transmembrane fluxes were very large, while a reasonable span of the droplet size was found; the dependence on the transmembrane pressure indicated at least partial turbulent conditions. There was no indication of fouling in the process, even after five passes, which indicates that the process is tolerant to product and conditions. A master curve was derived for the Weber number as a function of the transmembrane pressure normalized on the Laplace pressure of the emulsion before emulsification. This curve comprises all process parameters and allows estimation of the emulsion droplet size.

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Acknowledgments The authors thank Stork Veco (Eerbeek, the Netherlands) for providing the metal sieves, and Adriaan van Aelst (Wageningen Electron Microscopy Centre, WUR) for his help with the SEM. A. Nazir is supported by a scholarship program of Higher Education Commission of Pakistan, and by VLAG graduate school.

Nomenclature d32 dh dp dratio dx Ev f h J l N

droplet Sauter diameter (m) pore hydraulic diameter (m) characteristic pore dimension (m) ratio of droplet diameter to pore width droplet diameter corresponding to x vol.% on a cumulative droplet volume curve (m) energy density (J m−3 ) friction coefficient pore depth (m) transmembrane flux (m3 m−2 s−1 = m s−1 ) pore length (m) number of passes through the sieve

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