Characterization of meso- and macroporous ceramic membranes in terms of flux measurement: A moment-based analysis

Journal of Membrane Science 302 (2007) 218–234

Characterization of meso- and macroporous ceramic membranes in terms of flux measurement: A moment-based analysis Marc Pera-Titus, Joan Llorens ∗ Chemical Engineering Department, University of Barcelona, 08028-Barcelona, Spain Received 20 February 2007; received in revised form 4 June 2007; accepted 23 June 2007 Available online 3 July 2007

Abstract This work addresses the determination of unimodal log-normal pore size distributions (PSDs) in porous symmetric membranes through a convenient selection of a set of statistical moment generating functions. Relevant information concerning such distributions can be obtained by performing three independent permeation experiments: (1) pure Knudsen single-gas diffusion permeance, (2) pure liquid permeability, and (3) non-hindered diffusion permeance of a target solute. In the special case of porous asymmetric UF, NF and MF ceramic membranes, the contribution of the support to the overall mass transfer tends to increase for permeation processes governed by lower moments of the PSD. For such membranes, a mean diameter of the top layer and the thickness-to-porosity ratio of the support can be accurately determined, which constitute two structural parameters relevant for a preliminary membrane selection. The application of moment theory following the guidelines of Mason allows the determination of upper and lower bounds on the cumulative PSD of active layers from measured moments. © 2007 Elsevier B.V. All rights reserved. Keywords: Membrane characterization; Ultrafiltration; Microfiltration; Pore size distribution; MWCO

1. Introduction The research field on membrane technology has grown worldwide in the last decade to become a one-billion dollar industry, as membrane separation processes have been increasingly used for industrial applications. Meso- and macroporous membranes such as those found in ultrafiltration (UF), nanofiltration (NF) and microfiltration (MF) processes are especially outstanding in light of their wide range of practical applications, covering more than 75% of the world membrane market [1]. Although mesoporous membranes are usually characterized by a molecular weight cut-off value (i.e. MWCO (kDa), the molecular weight of a solute for which 90% separation can be achieved [1]), this does not provide much information concerning neither their separation performance nor structure and might be only regarded as a useful tool for a preliminary membrane selection. In fact, the separation ability of porous membranes is strongly governed by its pore size distribution (PSD). Two membranes with the same MWCO, but with different PSD, can show quite different per-

∗

Corresponding author. Tel.: +34 934031304; fax: +34 934021291. E-mail address: [email protected] (J. Llorens).

0376-7388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2007.06.057

meation and separation behavior. Consequently, considerable efforts have been directed towards developing methods for the determination of PSDs in porous membranes. The currently available techniques for the characterization of PSDs and mean pore sizes in meso- and macroporous membranes cover a broad range of physical methods [2–4]. A comprehensive survey of some conventional and more incipient characterization methods is given in Table 1 . As can be seen, each method is specific for the characterization of a particular pore size range and presents some limitations. Although electron microscopy (SEM, TEM) and atomic force microscopy (AFM) have been widely investigated for the determination of PSDs and porosities from surface visualization, the translation of images into predictions of PSDs and separation performance is not always straightforward. In fact, active pores to permeation cannot be always easily distinguished from dead-end pores, thus providing an under- or overestimation of mean pore sizes. Other important shortcomings are related to their destructive character and low representativity. Other methods that rely on the relationship between pore size and pressure (gas/liquid, liquid/liquid porometry and permporometry) have been also used to characterize PSDs. While the former two involve the determination of a PSD from

Table 1 Classification of the available techniques for the determination of PSDs in meso- and macroporous membranes Group

Method

Principle

Pore sizea (nm)

Remarks

References

Methods related to permeation or rejection performance

Gas/liquid porometry or displacement Liquid/liquid porometry or displacement Mercury porosimetry

Measurement of the pressure required to blow air through a liquid-filled porous membrane Similar to bubble point method, but a non-wetting liquid is used to displace a wetting one Introduction of mercury in the membrane pores by applying an external pressure to overcome interfacial tension

>50

Inappropriate for UF and NF membranes, because very high pressures are required Non-destructive and fast technique, which only refers to pores open to flux Modified version of liquid/liquid porometry No distinction between active and dead end pores Destructive technique Use of Hg is forbidden in many countries

[5–7]

Permporometry

Variation of gas flux through a membrane due to a controlled pore blocking by capillary condensation

>5

Only pores active to permeation are detected It requires knowledge of contact angles

[14–17]

Liquid and gas flux measurement

Liquid and gas permeabilities are monitored as a function of the pressure drop across the membrane

All

Only rough information about the PSD can be obtained Cheap and easy-to-operate set-up

[1]

Solute rejection or sieving

Measurement of the retention of either non-ionic or charged solutes of different molecular weight

>2–3

Reflection coefficient is a function of the solute size (e.g., dextrans) Difficult interpretation of sieving data

[18–21]

Microscopy techniquesb : (SEM/FESEM/TEM/ HRTEM/AFM/STM)

Visualization of the top and cross section of a membrane and determination of PSD from further computerized image analysis

0.3–10

Presence of artifacts onto the surface

[7,13,22–26]

Gas adsorption/desorption

Determination of the PSD from the adsorption or the desorption isotherms

>0.1

Thermoporometry

Calorimetric study of the liquid/solid transformation of a capillary condensate that saturates the pores

2–30

Ultrasonic frequency domain reflectometry (UFDR) Light transmission

Determination of characteristic acoustic responses from a membrane

260–280c

Non-destructive technique

[29]

Determination of the light transmitivity through a porous membrane filled with a transparent liquid Variation of the refractive index of during adsorption and desorption of a gas in the porous structure

100–900c

Cheap and easy-to-operate set-up

[30]

1–2c

Non-destructive technique adapted to thin film characterization

[31]

Spectroscopic ellipsometry

>3

High electron beam energy in SEM and TEM may damage the samples Difficult interpretation of results in AFM and STM Destructive techniques No distinction between active and dead end pores No distinction between active and dead end pores Need of models to relate pore geometry with the form of the isotherms Possible deformation of the pore structure during the solidification process Dry samples are not necessary

[8–12] [1,13]

[9,26]

[27,28]

M. Pera-Titus, J. Llorens / Journal of Membrane Science 302 (2007) 218–234

Methods related to surface morphology

>2

219

Pore size range for which the technique is fully applicable. SEM: scanning electron microscopy; FESEM: field energy scanning electron microscopy; TEM: transmission electron microscopy; HRTEM: high resolution transmission electron microscopy; AFM: atomic force microscopy; STM: scanning tunneling microscopy. c Only the PSD range indicated is reported.

a

b

[35] Only simulation studies have been reported –

Small-angle neutron scattering (SANS) Electron spin resonance (ESR) Electrokinetic phenomena

Modification of the induced streaming and membrane zeta potential

[3] – –

[34] Non-destructive technique >1.9

Promising for polymeric thin film characterization 100–500c

Measurement of NMR spin-lattice relaxation times of water molecules condensed in the pores of a membrane Measurement of carrier diffusivity within the pores of the membrane by pulsed field gradient spin echo NMR (PGSE-NMR). Measurement of neutron scattering of water molecules retained in the membrane pores Measurement of spin transitions of unpaired electrons Nuclear magnetic resonance (NMR)

Remarks Pore sizea (nm) Principle Method Group

Table 1 (Continued)

[32,33]

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References

220

the flux of a gas or a non-wetting liquid through a membrane previously soaked with an immiscible wetting liquid, the latter is based on the variation of flux of a wetting gas through a membrane due to pore blocking by capillary condensation. Problems arise when interpreting experimental data from these techniques due to the uncertainty in the determination of contact angles as well as pore necking, which in practice only allows the measurement of pore sizes in the range of tens of nanometers. In addition, gas porometry-based techniques also present some technical limitations when excessively high pressures are needed, which might damage the membrane structure. In fact, gas/liquid porometry is usually restricted to the detection of defects such as cracks, pinholes or very large pores as a primary integrity test (bubble point method, ASTM standard—procedures F316-86 and Modified Boiling Point Method, MBM) [1]. Furthermore, most of the aforementioned methods are unsatisfactory for the determination of PSDs in asymmetric membranes, because they are unable to discriminate between the active and the porous support [4], since it only constitutes a very small fraction of its material. Because of the large difference in pore size and pore volume of the active layer and the support, the characterization of the selective layer typically lacks detail and resolution and the results are difficult to be interpreted. To avoid all the stated shortcomings, an alternative approach could focus on relating permeation measurements with the nature of the PSD. The characterization of PSDs via direct liquid, gas or solute fluxes might show the advantage of enabling the determination of PSDs under experimental conditions similar to those in which the membrane is likely to be used. In fact, the permeation mechanisms of a membrane provide a set of moment generating functions that allow the determination of a PSD without any previous assumption about its nature. Note that this strategy is analogous to that used in the determination of PSDs from solute sieving data [18–21] (see Table 1) and similar to that used in the characterization of the dynamical behavior of systems subjected to input pulses, such as that found in temporal analysis of products (TAP) in heterogeneous catalysis [36]. Furthermore, bounds on the cumulative PSD of porous symmetric membranes can be placed using moment theory without need of aprioristic assumptions about its nature following the method formulated by Knierim and Mason [37] and Knierim et al. [38]. In their original work, these authors proposed the determination of lower and upper bounds on the cumulative PSD using diffusion/convection and solute reflection moment generating functions. The information about a PSD is captured because the diffusion/convection and the sieving behavior of monodisperse solutes vary with the pore size. The former function takes advantage of the fact that diffusion and convection depend differently on the pore size, while the latter is based on the dependence of the reflection coefficient on the pore size for large monodisperse solutes at large solute Peclet numbers, NPe . More recently, Baltus [39,40] carried out theoretical calculations with model membranes containing log-normal distributions that revealed that narrow bounds can also be placed on the cumulative PSD by measuring their pure liquid permeability together with: (1) one or two sieving measurements at NPe  1

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(reflection of a solute), (2) one or two diffusion–convection fluxes of a small, non-hindered solute at NPe ∼ 1, or (3) one or two large solute hindered diffusivity measurements. All these sets of moment generating functions allow discrimination between membranes with the same mean pore size, but with different PSDs. In light of all these comments, it is the purpose of this work to devise the relationship existing between gas, liquid and solute permeabilities with the pore size in meso- and macroporous membranes to obtain information about their PSDs. In the special case of log-normal PSDs, two relevant mean diameters of the PSD can be obtained by knowing three independent permeances: (1) pure Knudsen gas diffusion permeance, (2) pure liquid permeability, and (3) non-hindered diffusion permeance of a solute. In its turn, these two calculated mean diameters can be used to infer the entire unimodal log-normal PSD. In the special case asymmetric of UF, NF and MF membranes, the method allows the determination of a mean diameter and the thickness-to-porosity ratio of the support, S /τ S εS , which constitute two structural parameters that might be useful for a preliminary membrane selection. Moreover, following the guidelines of Mason, lower and upper bounds can be placed on the cumulative PSD of the active layer using moment theory. 2. Moment analysis Meso- and macroporous membranes usually consist of an active top layer deposited or grown onto a much thicker porous support. In the case of ceramic membranes, this porous layer can be visualized as an array of non-tortuous, cylindrical and non-interconnected pores of size d (m). In a first approach, the analysis will focus on this active layer without taking into account any possible contribution of the support. In fact, the present considerations might be directly applied to self-supported or symmetric membranes and to composite membranes with very thin supports. In general terms, irrespective of its actual functional dependence on the pore size, a PSD can be defined in terms of a continuous density function, ε(d), in such a way that the fraction of pore diameters lying in the range d1 and d1 + δd is ε(d1 )δd. The function ε(d) is normalized to unity (Eq. (1)):  ∞ ε(d)δd = 1 (1) 0

Furthermore, the cumulative pore size distribution, E(d* ), accounts for the fraction of pore diameters for all the pores ranging from 0 to d* (Eq. (2)):  d∗ ε(d)δd (2) E(d ∗ ) = 0

The statistical moments di  of the function ε(d) (hereinafter simply referred to as PSD) can be generated by Eq. (3):  ∞  i d = d i ε(d)δd (mi ), 0

for i = −∞, . . . , −2, −1, 0, 1, 2, . . . , +∞,

(3)

221

A number of mean pore sizes, d¯ i (m), can be deduced from two consecutive statistical moments di  and di−1 , which allow the determination of the function ε(d) by Eq. (4):  i ∞ i d ε(d)δd d ¯di =   =  ∞0 (4) i−1 i−1 ε(d)δd d 0 d Of course, the knowledge of a higher number of mean pore sizes, d¯ i , allows a more accurate description of a PSD, which in its turn implies the knowledge of a higher number of its statistical moments. Our interest focuses on the possibility of relating these moments to some physical properties or permeation fluxes. 2.1. Statistical moments of a PSD 2.1.1. 0th moment The 0th moment of ε(d), d0 , always equals 1, since the distribution is assumed to be normalized:  ∞    ∞ d0 = d 0 ε(d)δd = ε(d)δd = 1 (5) 0

0

2.1.2. First and second moments In addition to the 0th moment, the first and second moments of ε(d), d1  and d2 , can be calculated, respectively, from pure Knudsen diffusion and pure pressure-driven viscous or Poiseuille fluxes, on the basis of their different functional dependence on the PSD of a membrane, since the involved mass transfer mechanisms are also different. The relative contribution of each mechanism to mass transfer hinges on the mean free path of the permeating molecules, λ (m), which is defined as the mean distance that a molecule runs between two consecutive collisions [41]. Pure Knudsen diffusion takes place preferentially when the pore size is much lower than λ, which implies a higher frequency of collisions between molecules and pore walls than with other molecules. On the other hand, viscous flux is predominant for pore sizes much higher than λ and involves strong intermolecular collisions. In practical applications, pure Knudsen and viscous fluxes are favored, respectively, when the conditions Pm d ≤ 0.01 Pa m and Pm d ≥ 0.1 Pa m are fulfilled [41]. Pure Knudsen and viscous fluxes, NGKn and NGV , respectively, for single-gas permeance across a membrane characterized by a narrow PSD can be accounted for, respectively, by the wellknown Eqs. (6) and (7):  8 εT d¯ εT DKn δP Kn NG = − = P (6) τ RT δz 3τ πMRT NGV = −

εT d¯ 2 Pm εT d¯ 2 Pm δP = P τ 32μG RT δz τ 32μG RT

(7)

where τ, εT ,  and P are, respectively, the tortuosity, the porosity, the membrane thickness and the transmembrane pressure, and d¯ is the mean diameter (d¯ = d¯ i , ∀i = 1, . . ., n). It should be stressed that Eq. (7) is restricted to membranes with low pore interconnectivity (e.g. porous ceramic membranes, but not chain-like polymeric membranes or filters), where the flux is hardly affected by turbulence. On the other hand, if ε(d) differs

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from a ␦-Dirac function, the first and second moments can be calculated, respectively, by Eqs. (8) and (9):     ∞ 3τ πMRT NGKn 1 1 d = d ε(d)δd = εT 8 P 0  3τ πMRT Kn QG (m) (8) = εT 8    d2 =

∞

d 2 ε(d)δd =

0

τ 32μG RT V QG = εT Pm

τ 32μG RT NGV εT Pm P (m2 )

(9)

V Kn where QKn and QV G = NG /P, both in G = NG /P −2 −1 −1 (mol m s Pa ), correspond, respectively, to pure Knudsen and viscous gas permeances. Note that the Knudsen permeance is independent of pressure. Pure Knudsen and viscous gas permeances can be rewritten in the general form:  ∞   1 QKn = ψ = ψ d d 1 ε(d)δd (10) G 0

   2 d = ΦP = ΦP QV m m G

∞

d 2 ε(d)δd

(11)

0

with ψ = (1/3(DKn /RT ))(εT /τ) (mol m−3 s−1 Pa−1 ) and Φ = (DV /RT )(εT /τ) (mol m−4 s−1 Pa−2 ). In the most general situation that Knudsen and viscous gas fluxes occur simultaneously, the overall mass transfer is described by the sum of Eqs. (10) and (11). In case of pure liquid permeation (e.g., water), only viscous flux can be attained. Accordingly, Eq. (9) is transformed into Eq. (12):    ∞ 32MμL NLV d2 = d 2 ε(d)δd = τ ρL P 0 = τ

32MμL V QL ρL

(m2 )

(12)

where NLV is the flux of the pure liquid (mol m−2 s−1 ) and QV L = NLV /P is the pure viscous liquid permeance or permeability (mol m−2 s−1 Pa−1 ). The Knudsen contribution to mass transfer of a single-gas across a membrane at Pm = 1.0 bar, defined as the ratio Ψ /Φ from Eq. (12), can be used as a preliminary assessment of the presence of defects in micro- and mesoporous membranes [42]. 2.1.3. kth moments (k = 0 → ∞) Additional moments of the density function ε(d) can be obtained from the hindered diffusion at P = 0 of a large solute through a membrane of comparable mean pore size. This process is ascribed to a hindered diffusivity, D (m2 s−1 ), whose dependence with the ratio dm /d can be modeled by means of the Renkin equation [43]. In NF applications, this equation is often approached to a ν-exponent power law, where ν usually takes a value about 4 (Eq. (13)): 

dm dm ␯ εT D (m2 s−1 ) (13) = D∞ 1 − d τ d

where D∞ (m2 s−1 ) is the diffusivity of a solute without interaction with the pore walls (i.e. dm /d → 0). In fact, the explicit dependence of the hindered diffusivity of a solute on the pore size allows its role as moment generating function, as was previously proposed by Baltus [39]. In the absence of electrostatic D (mol m−2 s−1 ) can gradients, the diffusive flux of a solute, Nm be expressed by Eq. (14):

 ∞

 dm C D Nm = D ε(d)δd d  0

 ∞ 

␯ εT dm = 1− D∞ ε(d)δd C (14) τ d 0 where C is the solute concentration difference across the membrane (mol m−3 ). Eq. (14) can be rewritten as a function of the moments of ε(d) by developing it in a McLaurin series around dm /d = 0:   

 ∞ ∞ v dm k εT D k Nm = (−1) ε(d)δd C D∞ τ d k 0 k=0 ∞      ν  ν εT k k −k = dm d C with D∞ (−1) τ k k k=0

= v!/k(v − k)!

(15)

As can be seen, the solute flux depends on the −kth moments of ε(d) and on a k-power function of dm . The weight of the kth k d−k . The 0th term in term depends on the relative value of dm Eq. (15) equals 1 and is dominant for a solute with dm /d → 0. 1 d−1  actually governs the Eq. (15) also reveals that the term dm hindering effect of the pore wall and the terms related to higher k values play a more relevant role for higher dm /d ratios. For instance, for the diffusion of a solute with dm /d < 0.1 and ν = 4.0 (1–dm /d)4 ≈ 1–4 (dm /d)1 with a truncation error < 10%. Thus, Eq. (15) approaches to Eq. (16):

 ∞  εT D 1 −1 Nm ≈ D∞ (1 − 4dm d )ε(d)δd C τ 0   εT 1 = d −1 ]C (16) D∞ [1 − 4dm τ In case of non-hindered solute diffusion (dm /d → 0), Eq. (16) can be rewritten to Eq. (17):

 ∞  εT εT D Nm ≈ D∞ 1ε(d)δd C = D∞ C (17) τ τ 0 which allows the direct determination of the term εT /τ (m−1 ). Moreover, Eq. (17) turns into Eq. (18) through the convenient definition of the permeance of the non-hindered solute, QD m = D / (RTC) (mol m−2 s−1 Pa−1 ), in a similar manner as was Nm pointed out by Masselin et al. (2000) [44]: εT RT D Q = τ D∞ m

(18)

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2.2. Resistances to mass transfer The above outlined Eqs. (9), (10) and (19) include relevant information concerning the form of the ε(d) function. For practical purposes, these equations can be rewritten using the set of resistances RiM (m(1−i) ), which can be experimentally determined for each mass transfer mechanism by Eqs. (19)–(21): 1 D τ   = ∞ D (m) R0M = (19) 0 RT Qm εT d  1 8 τ 1 1   = (20) RM = 3 πMRT QKn εT d 1 G R2M =

1 τ ρL 1 Pm  = = V 32μG RT QV 32Mμ εT d 2 Q L L G

(m−1 ) (21)

It is noteworthy that the latter resistance, R2M , for pressure-driven viscous flux can be experimentally determined either from a single-gas permeance for Pm d > 0.1 (Pa m) or from a pure liquid permeability. 2.3. Application to log-normal PSDs It is generally accepted that a real PSD in porous membranes follows a log-normal distribution [17,20]. In this case, the flux pattern of the membrane can be easily derived from the considerations above outlined. The normalized density function for a log-normal PSD can be described by Eq. (22):  2  d 1 1 (22) ln ε(d) = √ exp − β α πβd where parameters α (m) and β are, respectively, the peak value and the breath of the distribution. For log-normal PSDs, parameters α and β can be calculated by knowing only two mean diameters of the distribution (e.g. d¯ 1 and d¯ 2 ). This implies the determination of only two moments of the PSD (i.e.d1  and d2 ) from two independent permeance experiments: (1) pure Knudsen diffusion permeance of a single gas and (2) pure pressure-driven viscous flux of either a single gas or a pure liquid. A third experiment is required (i.e. permeance of a target solute by non-hindered diffusion) to obtain the ratio εM /τ M M of the membrane. Using the resistances defined by Eqs. (19)–(21) and a pure liquid flux for experiment (2), diameters d¯ 1 and d¯ 2 can be calculated, respectively, by Eqs. (23) and (24):  1 ∞ 1 d ε(d)δd d R0 = M d¯ 1 =  0  = 0∞ 0 d R1M 0 d ε(d)δd  πMRT QKn G = 3D∞ (m) (23) 8 QD m  2 ∞ 2 d ε(d)δd d R1M ¯d2 =   = 0∞ = 1 d1 R2M2 0 d ε(d)δd  8 32 MηL RT QV L = (m) 3 ρL πMRT QKn G

(24)

223

Note that the mean value of a distribution, μ (m), can be directly related to the mean diameter, d¯ 1 , while the variance, σ 2 (m), is a function of both diameters (i.e. σ 2 /μ = d 2 − d 1 ). 2.4. Lower and upper bounds on a cumulative PSD: moment theory Given the knowledge of a small number of moments, moment theory and the properties of Tchebycheff systems (see Appendix) allow the prediction of lower and upper bounds on the cumulative PSD of a membrane. Following the nomenclature of Baltus [39], the density function ε(d) can be approached to a discrete function or canonical representation, (d), which is a sum of δ-Dirac functions:  (d) = Wj δ(d − xj ) j ≤ number of moments (25) j

where Wj are the weighting factors and xj (m) the abscissa points. This representation is useful, because it contains the minimum number of terms needed to reproduce a density function. The lower and upper bounds on E(d* ) are summations involving the weights in the canonical representation:  Elower (d ∗ ) = Wj xj < d ∗ (26) j

Eupper (d ∗ ) = Elower (d ∗ ) + Wd ∗

(27)

Furthermore, the lower and upper principal representations on ε(d) are defined by Eqs. (28) and (29):  lower (d) = WL,j δ(d − xL,j ), j = number of moments j

upper (d) =



(28) WU,i δ(d − xU,j )

(29)

j

where the weighting factors WL,j and WU,j and the abscissa points xL,j and xU,j are determined from the given moments. As shown in the Appendix A, in the particular case of normalized lognormal PSD, lower and upper bounds on the cumulative PSD can be determined for the given set of moments {1,d1 ,d2 }. Fig. 1 displays more explicitly the lower and upper bounds on a cumulative distribution together with the theoretical density function that might be obtained for two known mean diameters d¯ 1 and d¯ 2 . For a given α, narrower bounds are expected at lower β values, since the distribution approaches a δ-Dirac function. 2.5. Characterization of porous asymmetric membranes Regarding now the case of asymmetric membranes with meso- or macroporous active layers showing low pore interconnectivity, such as those found in UF, NF and MF ceramic membranes, the permeation behavior can be strongly influenced by the porous nature of the support. Therefore, prior to determination of mean diameters, the contribution of the support to the overall permeances must be considered. For each permeation

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Fig. 1. Lower and upper bounds on a normalized log-normal PSD characterized by parameters α = 10 nm and β = 1. From left to right: (a) density function; (b) cumulative distribution (straigth line) and bounds (dashed lines). Mean diameters: d¯ M,1 = 12.8 nm; d¯ M,2 = 21.2 nm.

mechanism, the overall permeance that includes the contribution of both the top layer and the support can be expressed by Eq. (30): 1 1 1 = T + T QTi Qi,M Qi,S

(30)

where the subscript i refers to the gas (i = G), liquid (i = L) or non-hindered solute (i = m), and the subscripts M and S refer to the permeances related to the active layer and the support, respectively. Assuming that the support can be characterized by a narrow monodisperse PSD, the latter permeances can be computed by the set of Eqs. (31)–(34): V QTG,S = QKn G,S + QG,S  1 8 Pm 1 1 = + 3 πMRT R1S 32μG RT R2S

QTL,S = QV L,S =

1 ρL 32MμL R2S

QTm,S = QD m,S =

D∞ 1 RT R0S

(mol m−2 s−1 Pa−1 )

(mol m−2 s−1 Pa−1 )

(mol m−2 s−1 Pa−1 )

(31) (32) (33)

with RiS =

τS S εS d¯ Si

(m(1−i) )

(34)

where εS , τ S and S are, respectively, the porosity, tortuosity and thickness of the support, and D∞ is the molecular diffusivity of the solute. It should be emphasized that, in a single-gas permeance experiment, although the active top layer obeys to a pure Knudsen diffusion mechanism, the support can show contribution of the viscous mechanism due to the higher mean pore size of the latter. The contribution of the support to mass transfer is strongly dependent on the moment of the PSD of the active layer that governs the permeation mechanism. As a general rule, it can be stated that the lower the moment, the higher the contribution of the support. In this way, the permeation of a pure liquid through an asymmetric membrane is expected to be

less influenced by the support (contribution < 1%) than in nonhindered solute diffusion, where the support actually rules the permeation performance (contribution > 99%). In practice, nonhindered solute diffusion permeances can be only accurately determined for composite membranes with thicknesses of the active layer and support of the same order of magnitude. In any case, the method here presented allows the accurate determination of the mean diameter d¯ M,2 of the membrane layer and the ratio εS /τ S of the support, which themselves constitute two structural parameters useful to assess for the permeation behavior in practical solute separations as those found in the separation of heavy ions from ground and wastewaters by NF and RO. 3. Experimental section A collection of commercial UF, NF and MF titania (rutile) tubular membranes (TAMI, France) with 1–3 pores in their lumen, all asymmetric and characterized with nominal MWCO values in the range 1–150 kDa (NF and UF) and mean pore sizes 0.12–0.80 ␮m (MF), was used to evaluate the suitability of the characterization method presented here. The tubes (7 mm i.d. and 10 mm o.d.), were enameled at each ends, defining a permeation length of approximately 20 cm. The membranes were characterized by the set of three independent experiments using three different experimental set-ups. The related permeances are summarized in Table 2. 3.1. Single gas Knudsen diffusion permeance The experimental set-up used for the determination of single gas Knudsen diffusion permeances is schematically depicted in Fig. 2. For such experiments, the membrane, previously dried overnight at 373 K under vacuum (<0.1 kPa) was placed in a permeation module immersed in a temperature-controlled oil bath (298–333 K). The system was depressurized until ∼3 kPa using a membrane vacuum pump (Vacuumbrand Gmbh & Co, Germany). A depressurized reservoir (48 L) calibrated on our premises was used as high-volume vacuum reservoir at the per-

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225

Table 2 Commercial tubular asymmetric TiO2 membranes used in this study Membrane

MWCO (kDa)

Number of channels

Sin (cm2 )

QTG × 106a (mol m−2 s−1 Pa−1 )

QTL × 1011b (m3 m−2 s−1 Pa−1 )

QTHCl × 1011c (mol m−2 s−1 Pa−1 )

1 2 3 4 5 6 7 8 9

1 8 50 150 0.14 ␮m 0.20 ␮m 0.45 ␮m 0.80 ␮m 0.80 ␮m

3 3 3 3 3 3 3 3 1

95.2 95.2 95.2 95.2 95.2 95.2 95.2 95.2 51.5

4.34 ± 0.13 6.60 ± 0.07 6.60 ± 0.08 6.75 ± 0.09 – – 8.11 ± 0.30 – 9.41

2.96 ± 0.02 20.2 ± 0.1 58.1 ± 0.07 106 ± 3 417 528 722 ± 2 764 817 ± 5

95.5 ± 5.6 8.22 ± 0.02 7.99 ± 0.04 8.26 ± 0.12 – – 10.7 ± 0.16 – 14.8 ± 1.6

a b c

301–303 K. 288–298 K. 298–299 K.

meate side of the module. Each experiment began by closing valve (9) to define a fixed permeate volume and subsequently the needle valve (2) was opened, thus filling stepwise the retentate side of the membrane with the target gas (N2 , He, CO2 ) (Air Liquide, >99.999% quality) at a desired pressure (5–30 kPa). The pressures at both the retentate and permeate sides were monitored with a digital vacuum gauge with an accuracy of ±0.1 kPa (Schlee Gmbh & Co V-D3, Germany). In the end, a breakthrough curve of permeate pressure was obtained, from which the Knudsen gas permeance could be determined. The experiment was considered to end once the permeate pressure reached steadystate (retentate pressure). All the experiments were repeated two to seven times to ensure good reproducibility. The relationship between the permeate pressure and time is given by a mass balance equation in the permeate volume (Eq. (35)):

constant value in the course of each experiment, and P2 is the permeate pressure. The overall gas permeance that includes the contribution of both top layer and support is given by Eq. (36): 1 1 1 = Kn + Kn T QG QG,M QG,S + QV G,S =

(36) where P2S is the pressure at the active layer/support surface, R1M and R1S are the first order resistances of the layer and the support, and QG,1 and QG,2 are defined, respectively, by Eqs. (37) and (38): QG,1

δn2 V2 δP2 (t) = = QTG Sin [P1 − P2 (t)] (mol s−1 ) δt RT δt (35) where Sin is the geometrical area of the inner surface of the membrane tube, P1 is the retentate pressure (kPa), kept at a

1 1 + 1 1 QG,1 /RM (QG,1 /RS ) + (QG,1 QG,2 /R2S )(P2S + P2 )

1 = 3



8 πMRT

(37)

and QG,2

3 = 64μL



πM 8RT

(38)

Fig. 2. Set up used for the determination of single gas pure Knudsen diffusion flows. (1) N2 cylinder; (2) needle valve; (3) thermostatic bath; (4) module; (5) vacuum pressure gauge; (6) on/off valve; (7) vacuum lung; (8) vacuum pressure gauge; (9) on/off valve; (10) membrane vacuum pump.

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permeability with the inverse of the square nominal diameter of the active layer (Eq. (41)), while the resistance R0M of the active layer can be calculated from the slope of this representation: τM M 32μL τS S 32μL 1 = + T 2 ¯ εM dM,nom εS d¯ S2 QL   R0M 2 (Pa s m2 m−3 ) = 32ηL + RS 2 d¯ M

Fig. 3. Plot of ln[(P1 − P2 )/(P1 − P2 (0)] vs. time for the single-gas permeance of N2 through membrane 1 (1 kDa). The straight line refers to the linear fitting.

Because the permeate pressure does not change much in the course of an experiment, the overall single-gas permeance can be easily determined by integrating Eq. (35) for a mean permeate pressure, P 2 , at short times (initial condition: t = 0 → P2 = P2 (0)): 

Sin P1 − P¯ 2 (t) = − RTQTG t (39) ln P1 − P2 (0) V2

Regarding the linear fitting plotted in Fig. 4, the values of the resistances R0M = 6.94 × 10−5 m and R2S = 4.03 × 109 m−1 can be computed for all the layers and supports. Since all membranes five to eight were provided by the same manufacturer and the active layers were grown on the same kind of support, the resistance R2S can be used to characterize the support for the whole membrane set. 3.3. Non-hindered electrolyte diffusion permeance The non-hindered diffusion experiments were carried out using a strong acid (HCl, Panreac, Spain) or a strong alkali (NaOH and LiOH, both supplied by Aldrich, Germany) as permeating species due to their smaller sizes compared to the pore diameters. All the solutions were prepared in deionized water with a supporting electrolyte (100–500 mM), similar in nature to the permeating species (NaCl for HCl and NaOH and LiCl for

The resistance R1M can be calculated by Eq. (40):   1 ¯ ¯ T ¯ 2 ) + QG,1 (1 + 2d¯ S QG,2 P¯ 2 )2 + 2 d Q Q P − Q Q G,1 S G,1 G,2 1 G,1 4dS QG QG,2 RS (P1 − P 1 RM = 2d¯ S QTG QG,2 (P1 − P¯ 2 ) As an example, Fig. 3 plots the trend of ln[(P1 − P2 )/ (P1 − P2 (0))] versus time for an experiment performed with N2 for membrane 1 (1 kDa). 3.2. Pure liquid permeability The set-up used for the determination of pure liquid permeabilities consisted of a membrane module, a glass beaker (1 L) filled with deionized water (Millipore with 18 M cm−1 resistivity) to avoid the presence of fouling, a coupled centrifugal pump (Ismatec BVP-Z, Switzerland), a manometer and a regulation valve. Clean water was pumped from the beaker to the membrane feed at a 1 L min−1 flow rate at room temperature (288–299 K) and further recirculated. The desired transmembrane pressure (5–240 kPa) was adjusted by means of a regulation valve at the outlet of the module. The permeability of pure water, QTL (m3 m−2 s−1 Pa−1 ), calculated as the quotient of the water flux, NL (m3 m−2 s−1 ), by the applied transmembrane pressure, P (Pa), was monitored using a graduated cylinder and a chronometer for at least 3 h to ensure constancy in its value. Furthermore, to elucidate the contribution of the support, some water permeability tests were performed on the set of asymmetric MF membranes six to nine (see Table 2). Assuming that for these membranes the PSD of the active layer is very narrow, the resistance R2S of the support can be calculated from the intercept of the linear representation of the inverse of water

(41)

(40)

LiOH, all the salts were supplied by Panreac, Spain) to contract the double layer on the pore walls, thus reducing the ζ-potential. The concentration of all the electrolytes was determined potentiometrically by means of a glass electrode. Fig. 5 shows the schematic representation of the set-up used for the determination of non-hindered diffusion permeances. The set-up consisted of two closed retentate and permeate

2 for water permeability experiments for commerFig. 4. Plot of 1/QTL vs. 1/d¯ M cial TiO2 asymmetric MF membranes six to nine (see Table 2). The straight line refers to the linear fitting.

M. Pera-Titus, J. Llorens / Journal of Membrane Science 302 (2007) 218–234

227

Fig. 5. Set up used for the determination of the non-hindered electrolyte diffusivities. (1) magnetic stirrer; (2) feed tank; (3) pH meter; (4) electrode; (5) Peristaltic pump; (6) module; (7) magnetic pump; (8) electrode; (9) permeate volume; (10) pH meter.

circuits with known volumes. In the course of an experiment, the membrane, wetted overnight in the electrolyte solution (beyond 500 mM no variation of the calculated resistances R0S with the ionic strength was observed), was placed in the module and immersed in a temperature-controlled oil bath (298 K). The same solution used to wet the membrane was recirculated (1 L min−1 ) from the feed tank (2) in the retentate side of the membrane by the action of a peristaltic pump. On the basis of hydrostatic pressure differences, the closed permeate volume (V2 = 380 mL) (9) was filled with the electrolyte solution. When this volume reached half of the total value, the permeate solution began to be recirculated (300 mL min−1 ) by a magnetic pump (7) and the remaining volume was filled, taking care that no bubbles were present. After stabilization of the permeate volume (∼30 min), a known volume of acid (HCl) or alkali (LiOH, NaOH) measured with a volumetric pipette was supplied to the feed tank to achieve a target pH value, and the pH at both the permeate and retentate volumes was continuously monitored using glass electrodes (3) and (10). In the end, a breakthrough curve of pH was obtained. The experiments were considered to end once the pH in the permeate volume reached a steady-state value (retentate pH). All the experiments were repeated two to five times to ensure optimal reproducibility. Note that the high recirculation rates in the retentate together with the rather low observed electrolyte diffusion rates across the membrane ensure negligible contribution of boundary layer effects to mass transfer.

The relationship between the concentration of a target ion in the permeate volume and time is given by a mass balance equation (Eq. (42)) and the application of the Fick’s first law: δC2 (t) δn2 −1 = V2 = QD m,T Sin [C1 (t) − C2 (t)] (mol s ) δt δt

(42)

where C1 (t) and C2 (t) are the concentrations of the electrolyte in the feed and permeate volumes, respectively. Eq. (42) can be integrated using the initial condition: t = 0 → C1 (t) = C1 (0),

C2 (t) = C2 (0)

and a mass balance equation for the electrolyte: V1 C1 (0) + V2 C2 (0) = V1 C1 (t) + V2 C2 (t) thus giving Eq. (44):

1 1 D t ln(Ω) = −Qm,T Sin + V1 V2

(43)

(44)

with Ω=

C1 (0) + (V2 /V1 )C2 (0) − (1 + (V2 /V1 )) C2 (t) C1 (0) − C2 (0)

(45)

As an example, Fig. 6 plots the representation of Eq. (45) for an experiment performed with HCl for membrane 1 (1 kDa). Moreover, the thickness of the support, S , can be also determined from the resistance R0S and the time lag, θ (s), that is,

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the support, d¯ s , which in its turn allows the calculation of the resistance R1S :  R0S (m) → R1S = R2S d¯ S (47) d¯ s = R2S 3.4. Calculation strategy

Fig. 6. Plot of ln(Ω) vs. time for the diffusion of H+ (HCl) through membrane 1 (1 kDa), keeping both the feed and permeate solutions at I = 500 mM. The time lag (θ) for this experiment was 720 s.

the delay observed in the reception of the signal in the permeate volume. For plane sheet geometries (the supports might be regarded as flat because their thickness is far smaller compared to their length), Eq. (46) can be used [45]: θ=

τS 2S R0S S = 6Nh D± εS 6Nh D±

(s) → S =

6Nh D± θ R0S

(m) (46)

where N is the number of channels in the tubes. The thickness of the support allows the determination of the porosity-to-tortuosity ratio of the support, εS /τ S , through the definition of the resistance R0S (Eq. (34) for i = 0). On the other hand, the knowledge of resistances R0S and R2S of the support, the latter determined by Eq. (41), allows the determination of the mean pore size of

Fig. 7 shows the schematic representation of the procedure used for the determination of mean diameters d¯ M,1 and d¯ M,2 of the active layers for the set of commercial asymmetric membranes listed in Table 2. In general terms, the calculation strategy consists of the determination of resistances R0s , R1S and R2S related to the support, which allow the further determination of the resistances R1M and R2M of the active layer. The calculation strategy involves the following steps: 1. Determination of resistance R0s of the support from nonhindered diffusion of a target electrolyte (Eq. (33)). This resistance allows the calculation of the thickness of the support through the use of Eq. (46), and subsequently the determination of the ratio εS /τ S of the support by Eq. (34). 2. Determination of resistances R2S and R2M of the support and active layer, respectively, from pure water permeability experiments. In the case of MF membranes, the resistance R0M can be calculated from the slope of Eq. (41). The mean pore size of the support, d¯ S , can be obtained by the combination of resistances R2S and R0S through the use of Eq. (47). 3. Calculation of the resistance R21 of the support and determination of the resistance R1M of the active layer from single gas Knudsen diffusion permeances.

Fig. 7. Schematic representation of the calculation procedure used for the characterization of the NF and MF asymmetric membranes.

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Fig. 8. Contribution of the support to pure Knudsen N2 permeance (open cycles) and pure viscous water flux (open squares) vs. mean diameter d¯ M,2 . The straight lines refer to the observed trends.

4. Calculation of the mean diameters d¯ M,1 and d¯ M,2 by combining, respectively, resistances R1M and R0M and R2M and R0M . Note that, as shown in Fig. 8, the calculation of the mean diameter d¯ M,1 is only allowed for MF membranes, for which Eq. (41) applies. 4. Results and discussion 4.1. Contribution of the support to the permeation performance Table 3 shows the values of resistances R1M and R2M of the membrane layer and the resistance R0S related to the support together with the computed values of the structural parameters d¯ M,2 and εS /τ S for the set of membranes listed in Table 2. As expected, for all the surveyed membranes, the resistance R1M shows lower values than those of resistance R2M , since the latter depends on the second moment of the PSD, while the former depends on the first one. Furthermore, both resistances show decreasing trends with the mean diameter d¯ M,2 , namely the contribution of the support to the overall mass transfer increases. This trend is more clearly drawn in Fig. 8. As can be seen,

229

Fig. 9. Evolution of computed d¯ M,2 mean diameters with the MWCO for membranes one to four (see Table 3). The straight lines refer to predicted trends.

the contribution of the support in pure Knudsen gas permeance and viscous water permeability increases dramatically with the mean pore size, showing values higher than 90% for d¯ M,2 higher than 100 nm (i.e. for MF membranes). Otherwise, for the tested UF membranes, the contribution of the support is much lower and becomes practically negligible for pure water permeability. Therefore, while the permeation behavior of UF and NF membranes is strongly governed by the active layer, the support exerts a more relevant contribution to the overall mass transfer in MF membranes. 4.2. Characterization of commercial asymmetric UF, NF and MF membranes The results provided in Table 3 also reflect a positive trend between the mean diameter d¯ M,2 and the MWCO or nominal diameter for the set of commercial UF and NF membranes surveyed in this study, which is more clearly depicted in Fig. 9. As can be seen, for membranes one to four, the mean diameter d¯ M,2 shows a potential trend or scaling law with the MWCO. This trend is in agreement with that predicted from estimates using the kinetic diameter of macromolecules, obtained from intrinsic viscosity data, in the molecular weight range 1–150 kDa (i.e.

Table 3 Resistances RM,1 , RM,2 and RS,0 and structural parameters d¯ M,2 and εS /τ S for the set of commercial membranes listed in Table 2 Membranea

MWCO (kDa)

RM,2 b × 10−8 (m−1 )

RM,1 c × 10−3

RS,0 d,e × 102 (m1 )

d¯ M,2 (nm)

εs /τ s

1 2 3 4 7 9

1 8 50 150 0.45 ␮m 0.80 ␮m

11772 ± 65 1519 ± 7 492 ± 6 311 ± 1 4.72 ± 0.10 3.40 ± 0.21

7.07 ± 0.97 1.83 ± 0.11 1.82 ± 0.11 1.65 ± 0.13 0.21 ± 0.01 1.33 ± 0.01

1.91 ± 0.10 2.52 ± 0.13 2.59 ± 0.12 2.50 ± 0.05 2.10 ± 0.07 1.27 ± 0.01

6.0 ± 0.8 12.0 ± 0.8 37.0 ± 2.3 52.4 ± 3.3 454 ± 27 875 ± 42

0.16 ± 0.09 0.12 ± 0.01 0.12 ± 0.01 0.12 ± 0.03 0.15 ± 0.02 0.16 ± 0.01

a Support: R 9 −1 for membranes one to five (d ¯ s = 2.3 ␮m for membranes one to eight) RS,2 = 3.13 × 109 m−1 for membranes six (d¯ s = 2.0 ␮m for S,2 = 4.03 × 10 m membrane nine). b Determined from pure water permeability experiments. c Determined from N permeance experiments. 2 d Determined from the non-hindered diffusion of HCl (I = 500 mM). e Time lag: θ = 650–780 s for membranes one to eight and 480–600 s for membrane nine.

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Fig. 10. Log-normal density and cumulative PSDs computed from mean diameters d¯ M,1 and d¯ M,2 for: (a) membrane 7, nominal pore size = 450 nm, d¯ M,1 = 324 ± 20 nm, d¯ M,2 = 454 ± 27 nm; and (b) membrane 9, nominal pore size = 800 nm, d¯ M,1 = 673 ± 76 nm, d¯ M,2 = 875 ± 42 nm. The dashed lines in the cumulative PSDs representations refer to lower and upper bounds.

dextrans, polyoxyethylenes and polyvinylpyrrolidones). Furthermore, the exponent of the potential function fitted to the experimental trend slope in Fig. 9, 0.46 ± 0.18 for the given set of membranes, might be also regarded as a structural parameter that would provide relevant information concerning the sieving ability of the membrane in the absence of electrostatic gradients. On the other hand, regardless of the computed mean diameter d¯ M,2 , the porosity-to-tortuosity ratio of the support, εS /τ S shows values in the range 0.12–0.16 for all the tested membranes, which is in fairly good agreement with the value 0.16 ± 0.02 found for membrane nine from a water impregnation test. Moreover, for these membranes, no significant differences for this parameter were found when determined from non-hindered diffusion experiments of either HCl or LiOH electrolytes, keeping the ionic strength in both cases at a value >500 mM (Debye–H¨uckel ˚ to avoid double layer effects on the computed thickness <4.3 A) ratios. As illustrated in Fig. 7, the mean diameter d¯ M,1 cannot be accurately calculated from ionic non-hindered diffusion experiments for the tested UF and NF membranes (one to four) because of the much higher contribution of the resistance R0S of the support compared to that of the active layer, R0M . However, this latter resistance can be accurately determined for MF membranes from the slope in Eq. (42) for the set of membranes five to eight using

water permeability data. In the present study, the fitted values of 6.94 × 10−5 m and 2 × 10−4 m for the resistance R0M were used to estimate the mean diameter d¯ M,1 of membranes seven and nine, respectively, with corresponding nominal diameters of 450 and 800 nm. The mean diameters d¯ M,1 and d¯ M,2 allow the determination of the normalized and cumulative PSD through the use of Eq. (22) (see Fig. 10). The computed values of parameters α and β for both membranes are, respectively, 274 ± 63 (nm) and 0.82 ± 0.30, and 591 ± 116 (nm) and 0.72 ± 0.23. Fig. 10 shows that the nominal diameters of both membranes lie within the computed mean diameters. This observation supports the idea that PSDs of active layers in commercial ceramic membranes can be well approached by log-normal distributions. 4.3. Lower and upper bounds on the cumulative PSD in UF and NF membranes Although the PSD of the active layer in most commercial ceramic NF and UF membranes cannot be experimentally determined from permeation data due the restriction of the support in the determination of resistance R0M without breaking the membrane for microscopy analysis, moment analysis allows obtaining a good approach to the actual distribution. For such calculations, the value of 2.0 × 10−5 m for the resistance R0M

M. Pera-Titus, J. Llorens / Journal of Membrane Science 302 (2007) 218–234

Fig. 11. Computed lower and upper bounds on the cumulative PSD for membranes 1–4.

231

behavior of asymmetric membranes. Furthermore, for the sake of simplicity, the hindering term in the Renkin equation for solute diffusion has been approached to (1 − 4dm /d¯ M,2 ). A next point deals with the determination of defects in membrane layers. It is true that the methodology presented in this work is only applicable to layers with unimodal log-normal PSDs. Therefore, in principle, this method excludes the most general situation where a certain number of defects are present in the layer, thus implying the presence of at least bimodal distributions (i.e. one for the active pores and another displaced to higher pores for defects). However, the presence of defects might be accounted by a displacement of the mean diameter d¯ M,2 to much higher values than the nominal diameter and by an increase in the breath of the distribution, d¯ M,2 − d¯ M,1 , in the case of symmetric membranes. 5. Final remarks

was used for the set of membranes one to four. Fig. 11 shows the computed lower and upper bounds for these membranes. As expected, the estimated intervals tend to appear at higher diameters for membranes with higher MWCO or higher diameters d¯ M,2 . 4.4. Discussion Moment-based analysis of membrane fluxes constitutes a powerful methodology to determine a mean diameter (i.e. d¯ M,2 ) and the porosity-to-tortuosity ratio of the support, εS /τ S , in asymmetric ceramic membranes, together with lower and upper bounds on the cumulative PSD of the active layer using Mason’s formulation, from only three independent permeation experiments: (1) pure Knudsen gas permeance, (2) pure liquid permeability, and (3) non-hindered ionic diffusion. These two structural parameters are relevant to characterize the separation ability of porous membranes. In the most general situation, the flux across UF, NF and RO membranes can be represented by one equation for the solvent including the contribution of the osmotic pressure, Π (Pa), for systems with reflection coefficients, σ, different from 0, and n−1 Nernst–Plank-type equations for each solute m, that is: NLV =

δΠ 1 δP ρL +σ 2 32MμL RM δη δη

D Nm ≈

 F δϕ Dm,∞ (1 − 4dm /d¯ M,2 ) δCm + z C m m δη RT δη R0S + (1 − σ)

Cm V N , CT L

m = 1, . . . , n − 1

(48)

A method has been presented that allows the characterization of PSDs in meso- and macroporous symmetric membranes. The input data consist of three independent permeation experiments: (1) single gas diffusion permeance at low pressure, (2) pure liquid permeability, and (3) non-hindered diffusion of a species (e.g. an electrolyte). In the special case of asymmetric membranes (UF, NF and MF), the latter experiment only provides data related to the support because in practice it governs the overall mass transfer. Although for such membranes the mean diameter d¯ M,2 and the ratio εS /τ S of the support can be only experimentally determined, the moment theory allows the determination of lower and upper bounds on the cumulative PSD that provide an insight into the nature and breath of the distribution. The great advantage of the method relies on the possibility to characterize membranes in light of their permeation behavior instead of MWCO values, to overcome the shortcomings related to low sample representativity of microscopic techniques (e.g., SEM, TEM, FESEM and AFM) and to distinguish between dead-end and active pores to permeation, which cannot be usually distinguished by techniques such as permporometry and mercury porosimetry. On the other hand, compared to the latter two techniques, the method does not involve the use of the Kelvin equation, which shows a strong limitation for pores <5–10 nm. Therefore, pores in this range and even lower, in the nearby of the micropore range (2–10 nm) can be in principle subjected to characterization by the present method. Acknowledgements

(49)

where η is the dimensionless coordinate parallel to the membrane surface and ϕ is the electrostatic field (V). In Eqs. (48) and (49) is has been implicitly assumed that the resistances R2M of the active layer and R0S of the support constitute the only structural parameters that govern, respectively, the solvent permeability and the solute hindered diffusion across the membrane, which seems reasonable in light of the comments above stated concerning the relative contribution of the support to the permeation

The authors are grateful to the Spanish Ministry of Education and Science for funding support (project CTQ2005-08346-C0201). Appendix A Following the approach presented by Knierim et al. [37], bounds on the cumulative PSD can be placed if the set of moment generating factors, in our case {di |i = −∞, . . ., −2, −1, 0, 1, 2,

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. . ., +∞} according to Eq. (3), form a normalized Tchebycheff system. In practice, this constraint is accomplished if only the linear combination: C (d) = (n)

n 

W1,a → 1 − αi d

i

(A1)

i=0

W2,a →

for αj arbitrary has no more than n distinct roots. This condition is fulfilled by the aforementioned set of moment generating functions, since Eq. (A1) has utmost n roots.Using the set of moments 1, d1  and d2 , a density function can be approached to a discrete principal representation that contains two abscissa points, one of which is an end point: lower (d) = WL,1 δ(d − 0) + WL,2 δ(d − xL,2 )

(A2)

upper (d) = WU,1 δ(d − xU,1 ) + WU,2 δ(d − dmax )

(A3)

where dmax → ∞. The values of the weights WL,1 and WL,2 and the abscissa point xL,2 can be determined from the substitution of Eq. (A2) into the expressions of the three moments: WL,1

 1 2 d = 1 −  2 d

(A4)

 1 2 d WL,2 =  2  d  2 d xL,2 =  1  = d¯ 2 d

(A5)

(A6)

In a similar manner, the substitution of Eq. (A3) into the expressions of the three moments yields the values of the weights WU,1 and WU,2 and the abscissa point xU,1 : WU,1 → 1

(A7)

and WU,2 → 0   xU,1 → d 1 = d¯ 1

(A8) (A9)

The knowledge of the abscissa points xL,2 and xU,1 allows building up the canonical representation of the PSD. For three known moments, two canonical representations can be found, each valid for a different range of d* : a (d) = W1,a δ(d − 0) + W2,a δ(d − d ∗ ) + W3,a δ(d − dmax )

d¯ 1 d∗

xU,1 < d ∗ < xL,2

b (d) = W1,b δ(d − d ∗ ) + W2,b δ(d − d  )

d ∗ < xU,1

(A10) and (A11)

where d is unknown. In each case, the weighting factors and d can be determined by substituting the corresponding expression

(A12)

d¯ 1 d∗

(A13)

and W3,a → 0

(A14)

d¯ 2 − d¯ 1 W1,b → ¯ d2 + d ∗2 /d¯ 1 − 2d ∗

(A15)

d¯ 1 + d ∗2 /d¯ 1 − 2d ∗ W2,b → ¯ d2 + d ∗2 /d¯ 1 − 2d ∗

(A16)

and d →

d¯ 2 − d ∗ 1 − (d ∗ /d¯ 1 )

(A17)

The lower and upper bounds on the cumulative PSD can be calculated as follows: Elower (d ∗ ) = 0,

and

d ∗ > xL,2

of (d) into the three known moments. In the end, the following expressions are obtained for dmax → ∞:

Elower (d ∗ ) = W1,a ,

Eupper (d ∗ ) = W1,b Eupper (d ∗ ) = 1

d ∗ < xU,1

(A18)

xu,1 < d ∗ < xL,2 0 (A19)

Elower (d ∗ ) = W2,b ,

Eupper (d ∗ ) = 1

d ∗ > xL,2

(A20)

Nomenclature C d dm d¯ i dn  D E(d* ) F  M MWCO N N NPe P PSD Q QG,1 QG,2 R

concentration (mol m−3 ) pore diameter (m) kinetic diameter (m) mean pore i of the density function ε (d) (m) nth moment of the density function ε (d) (mn ) diffusivity (m2 s−1 ) cumulative pore size distribution Faraday constant (96,487 C mol−1 ) membrane thickness (m, ␮m) molecular weight (kg mol−1 ) molecular weight cutoff (kDa) flux (kg m−2 h−1 ; mol m−2 s−1 ) number of channels in the support Peclet number pressure (Pa) pore size distribution permeability (mol m−2 s−1 Pa−1 ) parameter defined by Eq. (38) (mol m−2 s−1 Pa−1 ) parameter defined by Eq. (39) (m−1 Pa−1 ) constant of gases (8.314 Pa m3 mol−1 K−1 )

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233

References Ri (d) S T V W x z

(m(1−i) )

resistance to mass transfer discrete canonical representation of ε(d) surface (m2 ) temperature (K) volume (m3 ) weighting factor in (d) Abscissa point in (d) (m) ionic charge

Greek symbols α, β characteristic parameters in a log-normal distribution εT porosity ε (d) density function (m−1 ) Φ parameter defined in Eq. (12) ϕ electrostatic field (V) λ mean free path (m) μ mean value (m); viscosity (kg m−1 s−1 ) η dimensionless coordinate Π osmotic pressure (Pa) θ time lag (s) ρ density (kg m−3 ) σ standard deviation (m); reflection coefficient τ tortuosity ν exponent in the Renkin equation (Eq. (14)) Ψ parameter defined in Eq. (11) Ω variable defined in Eq. (46) Subscripts G Gas in Inner Kn Knudsen L liquid; bound in lower (d) lower lower bound m mean; solute max maximum M membrane layer n number of moles nom nominal S support t time T total U bound in upper (d) upper upper bound ∞ bulk diffusion ionic ± Superscripts D diffusion Kn Knudsen S support T total V viscous

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