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Influence of concentration polarization and thermodynamic non-ideality on salt transport in reverse osmosis membranes
Author’s Accepted Manuscript Influence of Concentration Polarization and Thermodynamic Non-ideality on Salt Transport in Reverse Osmosis Membranes Eui-Soung Jang, William Mickols, Rahul Sujanani, Alysha Helenic, Theodore J. Dilenschneider, Jovan Kamcev, Donald R. Paul, Benny D. Freeman www.elsevier.com/locate/memsci
PII: DOI: Reference:
S0376-7388(18)32399-8 https://doi.org/10.1016/j.memsci.2018.11.006 MEMSCI16608
To appear in: Journal of Membrane Science Received date: 29 August 2018 Revised date: 29 October 2018 Accepted date: 4 November 2018 Cite this article as: Eui-Soung Jang, William Mickols, Rahul Sujanani, Alysha Helenic, Theodore J. Dilenschneider, Jovan Kamcev, Donald R. Paul and Benny D. Freeman, Influence of Concentration Polarization and Thermodynamic Nonideality on Salt Transport in Reverse Osmosis Membranes, Journal of Membrane Science, https://doi.org/10.1016/j.memsci.2018.11.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Influence of Concentration Polarization and Thermodynamic Non-ideality on Salt Transport in Reverse Osmosis Membranes
Eui-Soung Janga, William Mickolsb, Rahul Sujanania, Alysha Helenica, Theodore J. Dilenschneidera, Jovan Kamceva, Donald R. Paula, Benny D. Freemana*
a
Department of Chemical Engineering, Texas Materials Institute, Center for Energy and Environmental Resources, and Center for Research in Water Resources
The University of Texas at Austin, 10100 Burnet Road, Bldg. 133 – CEER, Austin, TX 78758 USA b
Mickols Consulting LLC, 3318 175th Lane SW , Tenino, Washington 98589 USA
*
To whom correspondence should be addressed: Tel: +1-512-232-2802; fax: +1-512-232-2807, [email protected]
Abstract The classic Merten and Lonsdale transport model for reverse osmosis membranes was reformulated to explicitly demonstrate the effects of concentration polarization and solution phase thermodynamic non-idealities on salt transport. A framework presented here accounts for the concentration dependence of ion activity coefficients in salt solutions, which was not explicitly included in the classic model. This approach was applied to four salt solutions, NaCl, MgCl2, CaCl2, and Na2SO4, tested in cross-flow conditions for a commercial RO membrane,
Dow FilmtecTM BW30XFR. Salt transport coefficients corrected for concentration polarization and non-ideal thermodynamic effects,
, were calculated as a function of permeate flux and
compared with apparent salt transport coefficients, B. These corrections were significant, resulting in
values greater than B values by a factor of 1.3~2.1 for 2:1 and 1:2 salts (i.e.,
MgCl2, CaCl2, and Na2SO4).
values for NaCl (a 1:1 salt), however, were similar to or
somewhat smaller than B values.
Keywords: Reverse osmosis (RO); membranes; desalination; the Merten and Lonsdale model; salt transport
1. Introduction Desalination by membrane-based reverse osmosis was proposed more than 50 years ago, and many desalination plants around the world now produce fresh water based on this concept [1-5]. Fundamental understanding of water and salt transport in polymer membranes, however, is still incomplete [6-8]. The most commonly used model to describe transport in reverse osmosis membranes was developed by Merten and Lonsdale based on the solution-diffusion mechanism [3, 9-11]. This model uses the salt concentration gradient across a membrane as the driving force for salt transport, assuming that ions exhibit ideal thermodynamic behavior [9, 10]. However, reverse osmosis, nanofiltration, and forward osmosis membranes treat water containing substantial amounts of salt, so salt activity coefficients can be far from unity and change significantly with salt concentration [12, 13]. Because the Merten and Lonsdale model does not explicitly account for concentration polarization and solution thermodynamic non-idealities, water and salt transport coefficients
2
calculated from this model are not purely material properties, frustrating the ability to construct fundamental membrane structure/property correlations. In this study, we quantify concentration polarization and non-ideal thermodynamic effects on salt transport coefficients in desalination membranes. Recently, we proposed a model accounting for the effects of electrolyte solution thermodynamics on salt transport in desalination membranes [13]. Salt activity coefficients in aqueous solution were calculated using the Pitzer model [14, 15]. Concentration polarization effects were quantified using the Sutzkover et al. model [16]. In the current study, this framework was used to correct apparent salt transport coefficients for these two effects using four salt solutions commonly found in desalination membrane processes (i.e., NaCl, MgCl2, CaCl2, and Na2SO4). A commercial desalination membrane (i.e., Dow FilmtecTM BW30XFR) was used for these studies. 2. Background In the Merten and Lonsdale transport model, steady state water transport is given by [9, 10]: (1) where
is water flux (L/(m2 h), or LMH),
permeance in units of LMH/bar), membrane (bar), and
is the water transport coefficient (or water
is the difference in hydrostatic pressure across the
is the difference in osmotic pressure across the membrane (bar).
In the Merten and Lonsdale model, steady state salt flux through a membrane is given by [9, 10]: (2)
3
where
is the salt flux (mol/(m2 h)),
the salt concentration in the membrane,
is the salt diffusion coefficient in the membrane, is position, and is the membrane thickness.
is and
are salt concentrations in the membrane at the upstream (0) and downstream (l) faces, respectively, as shown in Fig. 1, and and permeate solutions, respectively. where
and
are salt concentrations (mol/L) in the bulk feed
is the salt sorption coefficients (i.e.,
is the salt concentration in the external solution).
upstream and downstream faces (i.e.,
⁄
,
is assumed to be constant at the
) [9]. B (called “apparent
value” in this study)
is the salt transport coefficient (or salt permeance) with units of LMH (L/(m2 h)) [9].
Figure 1. Pressure, chemical potential, concentration, and solute activity profile in a reverse osmosis membrane according to the solution-diffusion model [11, 13].
The permeate salt concentration,
, depends on water and salt flux as follows [17]:
4
(3)
Using this relation, the apparent B value is given by [17]: (4)
Generally, desalination membranes transport water rapidly and selectively, leading to concentration polarization, which results in the salt concentration at the upstream membrane surface being higher than that in the bulk feed solution [17, 18]. In this study, the extent of concentration polarization was estimated using the Sutzkover et al. model [16]. In this model, membrane water permeance, , is taken to be independent of feed composition over narrow ranges of feed composition [16, 19]. This approximation is generally believed to be valid for current RO membranes. However, for highly charged or highly water-swollen membranes, effects like osmotic deswelling could influence the membrane water content, which would cause A (and B) to vary with salt concentration [20, 21]. Additionally, because some novel materials being considered for desalination are non-equilibrium, glassy polymers, their history of exposure to various salt solutions and temperatures can alter water and salt transport [20, 22]. For a pure water feed (i.e.,
) Eq. (1) gives: (5)
where
is the steady state pure water flux. Water flux in a feed containing salt is given by (
5
)
(6)
where
and
are the osmotic pressures of the feed solution at the membrane surface and
in the bulk permeate solution, respectively. The subscript “0(m)” represents values at the upstream membrane surface, to distinguish them from valued in the bulk feed solution. If
is
constant, Eqs. (5) and (6) yield [16]:
(
(7)
)
In this way, the osmotic pressure of the feed solution at the membrane surface, same
, can be calculated from experimental data by measuring
given feed salt concentration, and Then,
at a given
, and the ,
at a
from the measured salt concentration of the permeate.
is converted to the salt concentration at the membrane surface,
Pitzer model. Afterwards, the concentration polarization factor,
, using the
, may be calculated as follows
[14, 17]:
(8)
The salt transport coefficient corrected for concentration polarization, B*, is defined as follows: (9)
More rigorously, diffusive salt flux can be written using the salt chemical potential gradient as the driving force for salt transport [23]:
6
(10)
where
is the gas constant,
is absolute temperature, and
is the salt chemical potential in
the membrane. This formalism inherently includes effects of salt thermodynamic non-idealities on the driving force for salt transport. Here,
is the apparent salt diffusion coefficient and
includes the impact of any convective transport of salt in the membrane [6, 24]. The salt chemical potential in the membrane can be expressed as follows [25]: (11) where
is the standard state chemical potential,
is the activity of salt in the membrane,
the partial molar volume of salt in the membrane, and
is the hydrostatic pressure. Since
is and
are constant in the membrane phase, combining Eqs. (10) and (11) yields [26]: (12)
The activity of salt at any location activity coefficient,
in the membrane,
, and salt concentration,
, is expressed as the product of the salt
, at that location [25]: (13)
During steady-state operation, chemical potential equilibrium is established at the upstream and downstream faces of the membrane (i.e., state chemical potentials, molar volume,
and
) [17, 27]. The standard
, are equal in the solution and membrane phases [21, 26]. The partial
, and hydrostatic pressure, , of salt do not change appreciably between the
membrane and solution phases [21, 26]. Therefore, chemical potential equilibrium at the
7
membrane surface reduces to equality of activity at the membrane surfaces and the contiguous solutions (i.e.,
and
).
Eq. (12) can be integrated across the membrane thickness:
∫
(14)
∫
to yield:
〈
where 〈
〉 (
)
〈
〉 (
)
〈
〉 (
)
(15)
〉 is the ratio of the effective salt diffusion to the salt activity coefficient in the
membrane averaged over the membrane thickness. This derivation is shown in more detail in the Supporting Information. This expression describes salt transport driven by the gradient in salt chemical potential across the membrane [13]. The original Merten and Lonsdale formulation used concentration gradient, rather than chemical potential gradient, as the driving force for salt transport [9, 10]. When salt activity coefficients in the membrane and solution are equal to one (i.e., ideal solution), these two approaches are equivalent. Here, we define the salt transport coefficient,
, as follows [13]: (16)
This expression includes contributions from both concentration polarization and non-ideal thermodynamic effects on salt transport in the membrane. According to the definition of activity (Eq. (13)), the salt sorption coefficient can be expressed as a ratio of activity coefficients (i.e., 8
). If
is constant at the
upstream and downstream faces of the membrane (i.e.,
) and the salt diffusion
coefficient is constant across the membrane (i.e., independent of salt concentration):
〈
〉
〈
〉
(17)
Using this expression, Eq. (15) clearly reduces to the original form of the Merten and Lonsdale model: (18)
Fig. 2 presents definitions and expressions for the various salt transport coefficients (i.e., B values) described above.
Figure 2. Definitions and expressions for salt transport coefficients. B denotes the salt transport coefficient used in the original Merten and Lonsdale model. B* denotes the salt transport coefficient corrected for concentration polarization.
denotes the salt transport coefficient
corrected for concentration polarization and solution phase thermodynamic non-idealities.
9
3. Experimental 3.1. Membranes and materials Commercial flat-sheet reverse osmosis membranes, Dow FilmtecTM BW30XFR, from Dow Water Solutions were used for this study. BW30XFR membranes are thin film composite membranes with a polyamide separation layer formed via interfacial polymerization [19]. The manufacturer reported a water flux of 48.4 L/(m2 h) (LMH) with minimum stabilized NaCl rejection values between 99.4 and 99.65 %, respectively, in a 2000 ppm NaCl feed at 225 psig (15.5 barg) feed pressure at 25 oC and a pH of 8 [28]. Flux, rejection and area are reported by the manufacturer and are average apparent values from tests on whole modules. As such, they are not corrected for concentration polarization or thermodynamic non-idealities. All membranes were purchased from the manufacturer as dry flat-sheet membranes and stored in a cool, dark place before use. Chemical aging may occur when polyamide-based membranes are exposed to ambient air due to oxidation of residual functional groups on the membrane surface [19]. Color changes (from white to yellow) on the membrane surface and a reduction in salt rejection can be observed. Thus, only fresh membrane samples with no color change were used for these studies. The membrane samples were pretreated according to a procedure reported previously [19]. Flat-sheet membrane samples were cut into the size that fits the active area (9.2 cm x 4.6 cm) of the crossflow test cells. The samples were soaked in 25 vol% aqueous isopropanol solutions for 20 min to assist in removing residual production chemicals and glycerol from the membrane. Then, the samples were soaked in de-ionized (DI) water at least 24 h before testing, and the DI water was changed twice during this period. The containers used for sample pretreatment and storage were covered with aluminum foil before testing to limit exposure to light, which can cause chemical degradation of the polyamide layer. DI water was generated by a Millipore RiOS and
10
A10 water purification system (Billerica, MA) and had an electrical resistivity of at least 18.2 M -cm and less than 5.4 ppb organics based on total organic carbon (TOC). Sodium chloride, magnesium chloride, calcium chloride, and sodium sulfate were purchased from Sigma-Aldrich (St. Louis, MO) and used as received to make feed salt solutions with DI water.
3.2. Crossflow system and measurement Salt rejection and water flux were measured using a custom-built crossflow system (Fig. 3). Crossflow experiments simulate tangential-flow in commercial desalination membrane applications. The membranes were clamped into commercial bench-scale crossflow cells (CF042, Sterlitech, Kent, WA). Three cells were connected in series to a 30 L feed tank, a pump and pulse dampener, a back pressure regulator, bypass valve, a pressure gauge, and a flow meter. The feed tank was filled with at least 20 L of salt solution, and the feed concentration did not change appreciably during the experiment. The feed solution was passed through a filter (Hytrex GX059-7/8 (pore size: 5 m), Big Brand Filter, Chatsworth, CA) to prevent bacterial growth and particulate fouling on the membrane surface. The feed solution temperature was maintained at 25 o
C using a stainless steel coil placed in the feed tank and connected to a heater/chiller (Thermo
Neslab RTE-10 Digital One Refrigerated Bath, Thermo Fisher Scientific, Waltham, MA). The flowrate was 1 gallon per minute to produce turbulent flow (Reynolds number = 3220) [19]. In the Reynolds number calculation, the hydraulic diameter was calculated as 2 × the channel height (2.3 mm), according to the parallel plate approximation for a flat channel (width/height >3-4) [29, 30]. The viscosity of pure water was used in the Reynolds number calculation. No spacer was used in the feed channel. The crossflow system was cleaned before each experiment by circulating a 200 ppm bleach solution (3.4 g Clorox®/L) through the system for 30 min. Then, the crossflow system 11
was thoroughly rinsed by circulating fresh DI water four times to remove residual bleach, since the polyamide layer on desalination membranes can be damaged by aqueous chlorine [17]. For each rinse cycle, DI water was circulated for at least 10 min and then drained. After four rinse cycles, removal of chlorine and residual salts from previous tests was confirmed by the feed solution pH (~6, the pH of water equilibrated with atmospheric CO2 [31]) and the conductivity (< 15 μS/cm).
Figure 3. Schematic of crossflow filtration system. Adapted from [19] with permission from Elsevier.
Multiple applied feed pressures were chosen around a standard brackish water desalination condition (
225 psig). The applied pressure ranged from 125 to 375 psig (8.62
to 25.9 barg) in intervals of 50 psig. Pure water flux was first measured at each pressure to obtain pure water permeance. Then, a mass of salt corresponding to 0.1 M was added to the feed solution. The salt concentrations in the feed and permeate solutions were determined by 12
measuring the conductivity of those solutions with a handheld conductivity meter (Oakton CON 110, Oakton Instruments, Vernon Hills, IL). Individual calibration curves for the four salt solutions considered in this study (e.g., NaCl, MgCl2, CaCl2 and Na2SO4) were established and used for converting conductivity to salt concentration. Apparent salt rejection,
, was
calculated as follows:
(
where
and
(19)
)
are the salt concentrations in the bulk feed and permeate solutions,
respectively. The steady-state permeate flux (L/(m2 h), or LMH),
, was calculated as follows: (20)
where
is the membrane area (38 cm2), and
is the volume of permeate solution,
is the
sample collection time. The permeate sample mass was measured and converted to volume using the density of water (
g/mL). To ensure steady state operation, sufficient time (typically
1~2 min) was given for permeation before measurements, and two sets of data for permeate flux and salt rejection were successively collected and compared with each other.
4. Results and discussion 4.1. Salt activity coefficients Salt activity coefficients in aqueous solution are presented as a function of molar concentration (M or mol/L solution) in Fig. 4. Activity coefficients were calculated using the Pitzer model [14, 15]. This model originally used molal concentration, whereas experiments 13
conducted in this study used molar concentration. These two values are essentially equal within the concentration ranges considered here (as shown in the Supporting Information), so molar concentration was used in the following discussion. Activity coefficients in dilute solutions (i.e., near ~0 M) approach unity. These values decrease significantly over the concentration range from 0 to 0.3 M. The magnitude of decrease in activity coefficient varies depending on the salt.
Figure 4. Activity coefficients of salts in aqueous solution at 25 oC as a function of molar concentration (mol/L solution) calculated according to the Pitzer model [14, 15].
The difference between concentration and activity (where activity,
, is given by
)
can be significant for these salts. For example, the activity coefficient of Na2SO4 at 0.1 M is 0.45, which makes the thermodynamic activity equal to 0.045 for an 0.1 M Na2SO4 solution. The same analysis for 0.1 M MgCl2 and CaCl2 gives 47% and 48% differences between solution activity 14
and concentration, respectively. Because the driving force for salt transport is the salt activity gradient across the membrane, rather than the concentration gradient, not accounting for such thermodynamic non-idealities causes the resulting B value to depend on both the membrane properties (i.e., salt diffusion coefficient, partition coefficient, and membrane thickness) as well as the thermodynamic properties of the feed solution. As a result, B values do not simply reflect properties of the membrane unless this effect is accounted for. Because the feed solutions, particularly at higher salt concentrations, are not ideal, osmotic pressure values will also deviate from ideal solution theory. Osmotic pressure of salt solutions considered in this study were also calculated using the Pitzer model [14, 15]. To provide an estimate of this effect, a comparison between osmotic pressures calculated from the van’t Hoff relation (i.e., ideal solution) and the Pitzer model is provided in the Supporting Information.
4.2. Crossflow test results The results of crossflow experiments for BW30 XFR membranes with an 0.1 M NaCl feed solution are presented in Fig 5. Apparent salt rejection increased with increasing feed pressure, consistent with previous observations [19]. Salt concentration in the permeate solution is related to the ratio of salt flux to water flux as indicated in Eq. (3). Water flux,
, increases as
applied feed pressure increases (cf. Eq. (1)). Salt flux, , is approximately independent of feed pressure. Thus, increasing feed pressure leads to higher water flux, resulting in lower salt concentration in the permeate solution,
, thereby increasing apparent salt rejection. Data for
other salts considered in this study (i.e., MgCl2, CaCl2, and Na2SO4) showed similar trends and are recorded in the Supporting Information for brevity.
15
Figure 5. Influence of feed pressure on permeate water flux and apparent salt rejection in a 0.1 M NaCl feed for BW30XFR membranes. Each data point represents the average of at least three individual measurements, and uncertainties are reported as one standard deviation from this average. The permeate pressure was atmospheric in all cases.
4.3. Concentration polarization Concentration polarization was calculated using the model proposed by Sutzkover et al. [16] In this model, the water permeance of a membrane is taken to be constant and independent of feed composition [16, 19]. The model accounts for changes in driving force in terms of osmotic pressure due to concentration changes at the membrane surface. The osmotic pressure at the membrane surface,
, was obtained from Eq. (7), and it was used to calculate the salt
concentration at the membrane surface,
, using the Pitzer model. Then, the salt 16
concentration at the membrane surface,
, was divided by the bulk feed salt concentration,
, to obtain the concentration polarization factor, CP, as indicated in Eq. (8). Calculated concentration polarization factors for BW30XFR membranes are presented as a function of permeate flux in Fig. 6. According to Wijmans, CP is expected to increase as permeate flux increases when solutes are highly rejected by the membrane [18], and we observe this trend for all salts considered. The osmotic pressures of 2:1 (MgCl2 and CaCl2) and 1:2 (Na2SO4) salt solutions are inherently higher (and, therefore, water fluxes are inherently lower) than those of the 1:1 (NaCl) salt solution at the same feed concentration (i.e., 0.1 M). Thus, CP values for NaCl were higher than those of the other salts due, in large measure, to the higher driving force for water flux (i.e.,
) in the NaCl containing solutions. Thus, NaCl
solutions had higher permeate flux values at the same applied transmembrane pressure than those of the other salt solutions considered. The osmotic pressures of the salt solutions considered in this study are recorded in the Supporting Information.
17
Figure 6. Influence of permeate flux on concentration polarization factor, CP, in 0.1 M feed solutions of NaCl, MgCl2, CaCl2, and Na2SO4 for BW30XFR membranes.
4.4. Effective feed concentration/activity Concentration polarization-corrected feed concentrations,
were calculated
according to Eq. (7) and are presented as a function of permeate flux in Fig. 7. These values, labeled “Polarization” in Fig. 7, represent the calculated salt concentration at the upstream membrane surface. While the bulk feed concentration was constant at 0.1 M, the feed salt concentration at the membrane surface increased significantly with increasing permeate flux. For example, concentration polarization-corrected feed concentrations for NaCl change from 0.12 to 0.22 M as permeate flux increased. The other salts exhibited similar trends with permeate flux.
18
Using the calculated surface,
values, feed solution activity values at the upstream membrane
, were calculated using the Pitzer model (labeled “Activity” in Fig. 7) [14, 15].
Figure 7. Influence of permeate flux on concentration polarization-corrected feed concentration and activity in 0.1 M: (a) NaCl, (b) MgCl2, (C) CaCl2, and (d) Na2SO4 feed solutions. Bulk and polarization corrected concentrations are
and
19
, respectively.
4.5. Salt transport coefficients Salt transport coefficients corrected for concentration polarization effects, calculated using Eq. (9).
, were
values are presented as a function of permeate flux in Fig. 8, and
apparent salt transport coefficients (i.e., B values) are also presented for comparison. For the salts considered,
values were always less than or equal to the corresponding B values. The
difference between B and
increased as permeate flux (i.e., applied feed pressure) increased
due to increasing concentration polarization (cf. Fig. 6). For NaCl (i.e., a 1:1 salt), differences between
and B values were noticeable even at low permeate fluxes, whereas these differences
were negligible (i.e.,
B) at very low permeate fluxes for the 2:1 and 1:2 salts (i.e., MgCl2,
CaCl2, and Na2SO4) for the reasons discussed above [18]. Salt transport coefficients corrected for both concentration polarization and non-ideal thermodynamic effects, 8.
, were calculated using Eq. (16) [13]. The results are presented in Fig.
values are always greater than
values because the salt activity coefficients in solution
are less than 1 (cf. Figs. 2 and 4). Interestingly, for NaCl values. In contrast, for MgCl2, CaCl2, and Na2SO4
values were generally lower than B
values were higher than the corresponding
B values. This result is a combination of correcting for both concentration polarization and nonideal thermodynamic effects. Effective salt concentration at the upstream membrane surface was greater than the salt concentration in the bulk feed solution (cf. Fig. 7), which tends to decrease values relative to B. On the other hand, salt activity coefficients were less than 1 at the upstream face of the membrane (cf. Fig, 4), which tends to increase the divalent salts, the latter effect dominates over the former effect.
20
values relative to B. In
Figure 8. Influence of permeate flux on salt transport coefficients (i.e., B,
, and
values) in
0.1 M: (a) NaCl, (b) MgCl2, (C) CaCl2, and (d) Na2SO4 feed solutions for BW30XFR membranes. B is the apparent salt transport coefficient calculated from the Merten and Lonsdale model [9, 10].
is the salt transport coefficient corrected for concentration polarization.
is
the salt transport coefficient corrected for both concentration polarization and solution phase thermodynamic non-idealities.
21
The ratio of
to B is presented as a function of permeate flux in Fig. 9.
for NaCl
ranges from 0.8 to ~1.0, suggesting that concentration polarization and solution phase thermodynamic non-idealities corrections to salt transport coefficients nearly cancel one another. For MgCl2 and CaCl2, activity coefficient values are similar, resulting in similar values for For Na2SO4,
ranged from ~1.7 to ~2.1, which was the highest among the four salts
considered. This behavior is due to the greater departure from ideal solution activity coefficients in Na2SO4 than in the other salts, as shown in Fig. 4. The approach to calculate
values used in
this study was applied to the nanofiltration membrane salt transport data reported by Bason et al. [32] The results of this calculation are also presented in Fig. 9.
values from Bason et al.’s
study also fall in a very similar range to those calculated for the RO membranes in this study.
22
Figure 9. The ratio of
to B (i.e., salt transport coefficient corrected for concentration
polarization and thermodynamic non-ideality, function of permeate flux.
, to apparent salt transport coefficient, B) as a
values for various salts in an NF membrane were taken from Bason
et al. for comparison [32].
23
5. Conclusions Concentration polarization and solution phase thermodynamic non-idealities can significantly influence salt transport coefficients in desalination membranes. Concentration polarization effects were evaluated using pure water permeance, water flux from salt solutions, and salt rejection data. Osmotic pressure and salt concentration at the upstream membrane surface were higher than in the bulk feed solution due to concentration polarization. Concentration polarization effects were more significant at higher feed pressures due to the increasing water flux at higher feed pressures. Concentration polarization corrected salt transport coefficients (i.e.,
values) were equal to or lower than uncorrected (i.e., B) values due to higher
salt concentration at the upstream membrane surface than in the bulk feed solution. The influence of solution phase thermodynamic non-idealities on salt transport coefficients was evaluated using salt activity coefficients in the feed and permeate solutions. Corrections for both concentration polarization and non-ideal thermodynamic effects (i.e., resulting in
values) were significant,
values that were greater than B values by a factor of 1.3~2.1 for 2:1 and 1:2 salts
(MgCl2, CaCl2, and Na2SO4).
values for NaCl (a 1:1 salt), however, were similar to or slightly
less than corresponding B values. For the salt solutions considered in this study (e.g., NaCl, MgCl2, CaCl2, and Na2SO4),
values varied significantly depending on the departure of
solution phase thermodynamics from ideality. Finally, while reporting B values facilitate comparisons with existing literature data, it is also useful to report Ba values since these represent the performance of the membrane material itself.
Acknowledgements This material is based upon work supported in part by the Welch Foundation Grant No. F-192420170325. A.H. was supported in part by the National Science Foundation (NSF) Graduate 24
Research Fellowship under Grant No. DGE-1610403. Additionally, B.D.F.’s contributions to the preparation of this manuscript were supported as part of the Center for Materials for Water and Energy Systems (M-WET), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DESC0019272.
References [1] C. Reid, E. Breton, Water and ion flow across cellulosic membranes, J Appl Polym Sci, 1 (1959) 133-143. [2] K.P. Lee, T.C. Arnot, D. Mattia, A review of reverse osmosis membrane materials for desalination—Development to date and future potential, J Membr Sci, 370 (2011) 1-22. [3] D. Paul, Reformulation of the solution-diffusion theory of reverse osmosis, J Membr Sci, 241 (2004) 371-386. [4] L.F. Greenlee, D.F. Lawler, B.D. Freeman, B. Marrot, P. Moulin, Reverse osmosis desalination: water sources, technology, and today's challenges, Water Res, 43 (2009) 2317-2348. [5] R.J. Petersen, Composite reverse osmosis and nanofiltration membranes, J Membr Sci, 83 (1993) 81-150. [6] G.M. Geise, H.S. Lee, D.J. Miller, B.D. Freeman, J.E. McGrath, D.R. Paul, Water purification by membranes: the role of polymer science, J Polym Sci, Part B: Polym Phys, 48 (2010) 1685-1718. [7] G.M. Geise, D.R. Paul, B.D. Freeman, Fundamental water and salt transport properties of polymeric materials, Prog Polym Sci, 39 (2014) 1-42. [8] H.B. Park, J. Kamcev, L.M. Robeson, M. Elimelech, B.D. Freeman, Maximizing the right stuff: The trade-off between membrane permeability and selectivity, Science, 356 (2017). [9] H. Lonsdale, U. Merten, R. Riley, Transport properties of cellulose acetate osmotic membranes, J Appl Polym Sci, 9 (1965) 1341-1362. [10] U. Merten, Flow Relationships in Reverse Osmosis, Industrial & Engineering Chemistry Fundamentals, 2 (1963) 229-232. [11] J. Wijmans, R. Baker, The solution-diffusion model: a review, J Membr Sci, 107 (1995) 121. [12] R.A. Robinson, R.H. Stokes, Electrolyte solutions, Courier Dover Publications, 2002. [13] W. Mickols, Substantial Changes in the Transport Model of Reverse Osmosis and Nanofiltration by Incorporating Accurate Activity Data of Electrolytes, Industrial & Engineering Chemistry Research, 55 (2016) 11139-11149. [14] K.S. Pitzer, Thermodynamics of electrolytes. I. Theoretical basis and general equations, J Chem Phys, 77 (1973) 268-277. [15] K.S. Pitzer, G. Mayorga, Thermodynamics of electrolytes. II. Activity and osmotic coefficients for strong electrolytes with one or both ions univalent, J Chem Phys, 77 (1973) 2300-2308. [16] I. Sutzkover, D. Hasson, R. Semiat, Simple technique for measuring the concentration polarization level in a reverse osmosis system, Desalination, 131 (2000) 117-127. 25
[17] R.W. Baker, Membrane technology, Wiley Online Library, 2000. [18] H. Wijmans, Concentration polarization, Encyclopaedia of Separation Sciences, Academic Press, San Diego, 4 (2000) 1682. [19] E.M. Van Wagner, A.C. Sagle, M.M. Sharma, B.D. Freeman, Effect of crossflow testing conditions, including feed pH and continuous feed filtration, on commercial reverse osmosis membrane performance, J Membr Sci, 345 (2009) 97-109. [20] D.M. Stevens, B. Mickols, C.V. Funk, Asymmetric reverse osmosis sulfonated poly (arylene ether sulfone) copolymer membranes, J Membr Sci, 452 (2014) 193-202. [21] J. Kamcev, D.R. Paul, B.D. Freeman, Ion activity coefficients in ion exchange polymers: applicability of Manning’s counterion condensation theory, Macromolecules, 48 (2015) 80118024. [22] W. Xie, G.M. Geise, B.D. Freeman, C.H. Lee, J.E. McGrath, Influence of processing history on water and salt transport properties of disulfonated polysulfone random copolymers, Polymer, 53 (2012) 1581-1592. [23] N. Lakshminarayanaiah, Transport phenomena in membranes, Academic Press, London, 1969. [24] J. Kamcev, D.R. Paul, G.S. Manning, B.D. Freeman, Accounting for frame of reference and thermodynamic non-idealities when calculating salt diffusion coefficients in ion exchange membranes, J Membr Sci, 537 (2017) 396-406. [25] A.J. Bard, L.R. Faulkner, Electrochemical Methods: Fundamentals and Applications, Wiley New York, 1980. [26] J. Mackie, P. Meares, The diffusion of electrolytes in a cation-exchange resin membrane. I. Theoretical, Proc R Soc London, Ser A, 232 (1955) 498-509. [27] J.G. Wijmans, R.W. Baker, The solution-diffusion model: a review, J Membr Sci, 107 (1995) 1-21. [28] DOW Water Solutions BW30XFR-400/34i Product Specification, https://www.dow.com/en-us/markets-and solutions/products/DOWFILMTECBrackishWater ReverseOsmosis8Elements/DOWFILMTECBW30XFR40034i. [29] S. Jankhah, Hydrodynamic conditions in bench-scale membrane flow-cells used to mimic conditions present in full-scale spiral-wound elements, Membrane Technology, 2017 (2017) 7-13. [30] G. Schock, A. Miquel, Mass transfer and pressure loss in spiral wound modules, Desalination, 64 (1987) 339-352. [31] V. Snoeyink, D. Jenkins, Water chemistry, John Wiley & Sons, New York, 1980. [32] S. Bason, V. Freger, Phenomenological analysis of transport of mono-and divalent ions in nanofiltration, J Membr Sci, 360 (2010) 389-396.
26
Highlights
A theoretical framework accounting for concentration polarization and non-ideal thermodynamic effects on salt transport in RO membranes is proposed.
The influence of concentration polarization and thermodynamic non-idealities on salt transport coefficients (i.e., B values) in RO membranes was quantified.
This framework was applied to four common salt solutions, NaCl, MgCl2, CaCl2, and Na2SO4.
Corrections for concentration polarization and non-ideal thermodynamic effects (i.e., values) were significant, resulting in
values that were greater than B values by a factor
of 1.3~2.1 for MgCl2, CaCl2, and Na2SO4 solutions.
27
PII: DOI: Reference:
S0376-7388(18)32399-8 https://doi.org/10.1016/j.memsci.2018.11.006 MEMSCI16608
To appear in: Journal of Membrane Science Received date: 29 August 2018 Revised date: 29 October 2018 Accepted date: 4 November 2018 Cite this article as: Eui-Soung Jang, William Mickols, Rahul Sujanani, Alysha Helenic, Theodore J. Dilenschneider, Jovan Kamcev, Donald R. Paul and Benny D. Freeman, Influence of Concentration Polarization and Thermodynamic Nonideality on Salt Transport in Reverse Osmosis Membranes, Journal of Membrane Science, https://doi.org/10.1016/j.memsci.2018.11.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Influence of Concentration Polarization and Thermodynamic Non-ideality on Salt Transport in Reverse Osmosis Membranes
Eui-Soung Janga, William Mickolsb, Rahul Sujanania, Alysha Helenica, Theodore J. Dilenschneidera, Jovan Kamceva, Donald R. Paula, Benny D. Freemana*
a
Department of Chemical Engineering, Texas Materials Institute, Center for Energy and Environmental Resources, and Center for Research in Water Resources
The University of Texas at Austin, 10100 Burnet Road, Bldg. 133 – CEER, Austin, TX 78758 USA b
Mickols Consulting LLC, 3318 175th Lane SW , Tenino, Washington 98589 USA
*
To whom correspondence should be addressed: Tel: +1-512-232-2802; fax: +1-512-232-2807, [email protected]
Abstract The classic Merten and Lonsdale transport model for reverse osmosis membranes was reformulated to explicitly demonstrate the effects of concentration polarization and solution phase thermodynamic non-idealities on salt transport. A framework presented here accounts for the concentration dependence of ion activity coefficients in salt solutions, which was not explicitly included in the classic model. This approach was applied to four salt solutions, NaCl, MgCl2, CaCl2, and Na2SO4, tested in cross-flow conditions for a commercial RO membrane,
Dow FilmtecTM BW30XFR. Salt transport coefficients corrected for concentration polarization and non-ideal thermodynamic effects,
, were calculated as a function of permeate flux and
compared with apparent salt transport coefficients, B. These corrections were significant, resulting in
values greater than B values by a factor of 1.3~2.1 for 2:1 and 1:2 salts (i.e.,
MgCl2, CaCl2, and Na2SO4).
values for NaCl (a 1:1 salt), however, were similar to or
somewhat smaller than B values.
Keywords: Reverse osmosis (RO); membranes; desalination; the Merten and Lonsdale model; salt transport
1. Introduction Desalination by membrane-based reverse osmosis was proposed more than 50 years ago, and many desalination plants around the world now produce fresh water based on this concept [1-5]. Fundamental understanding of water and salt transport in polymer membranes, however, is still incomplete [6-8]. The most commonly used model to describe transport in reverse osmosis membranes was developed by Merten and Lonsdale based on the solution-diffusion mechanism [3, 9-11]. This model uses the salt concentration gradient across a membrane as the driving force for salt transport, assuming that ions exhibit ideal thermodynamic behavior [9, 10]. However, reverse osmosis, nanofiltration, and forward osmosis membranes treat water containing substantial amounts of salt, so salt activity coefficients can be far from unity and change significantly with salt concentration [12, 13]. Because the Merten and Lonsdale model does not explicitly account for concentration polarization and solution thermodynamic non-idealities, water and salt transport coefficients
2
calculated from this model are not purely material properties, frustrating the ability to construct fundamental membrane structure/property correlations. In this study, we quantify concentration polarization and non-ideal thermodynamic effects on salt transport coefficients in desalination membranes. Recently, we proposed a model accounting for the effects of electrolyte solution thermodynamics on salt transport in desalination membranes [13]. Salt activity coefficients in aqueous solution were calculated using the Pitzer model [14, 15]. Concentration polarization effects were quantified using the Sutzkover et al. model [16]. In the current study, this framework was used to correct apparent salt transport coefficients for these two effects using four salt solutions commonly found in desalination membrane processes (i.e., NaCl, MgCl2, CaCl2, and Na2SO4). A commercial desalination membrane (i.e., Dow FilmtecTM BW30XFR) was used for these studies. 2. Background In the Merten and Lonsdale transport model, steady state water transport is given by [9, 10]: (1) where
is water flux (L/(m2 h), or LMH),
permeance in units of LMH/bar), membrane (bar), and
is the water transport coefficient (or water
is the difference in hydrostatic pressure across the
is the difference in osmotic pressure across the membrane (bar).
In the Merten and Lonsdale model, steady state salt flux through a membrane is given by [9, 10]: (2)
3
where
is the salt flux (mol/(m2 h)),
the salt concentration in the membrane,
is the salt diffusion coefficient in the membrane, is position, and is the membrane thickness.
is and
are salt concentrations in the membrane at the upstream (0) and downstream (l) faces, respectively, as shown in Fig. 1, and and permeate solutions, respectively. where
and
are salt concentrations (mol/L) in the bulk feed
is the salt sorption coefficients (i.e.,
is the salt concentration in the external solution).
upstream and downstream faces (i.e.,
⁄
,
is assumed to be constant at the
) [9]. B (called “apparent
value” in this study)
is the salt transport coefficient (or salt permeance) with units of LMH (L/(m2 h)) [9].
Figure 1. Pressure, chemical potential, concentration, and solute activity profile in a reverse osmosis membrane according to the solution-diffusion model [11, 13].
The permeate salt concentration,
, depends on water and salt flux as follows [17]:
4
(3)
Using this relation, the apparent B value is given by [17]: (4)
Generally, desalination membranes transport water rapidly and selectively, leading to concentration polarization, which results in the salt concentration at the upstream membrane surface being higher than that in the bulk feed solution [17, 18]. In this study, the extent of concentration polarization was estimated using the Sutzkover et al. model [16]. In this model, membrane water permeance, , is taken to be independent of feed composition over narrow ranges of feed composition [16, 19]. This approximation is generally believed to be valid for current RO membranes. However, for highly charged or highly water-swollen membranes, effects like osmotic deswelling could influence the membrane water content, which would cause A (and B) to vary with salt concentration [20, 21]. Additionally, because some novel materials being considered for desalination are non-equilibrium, glassy polymers, their history of exposure to various salt solutions and temperatures can alter water and salt transport [20, 22]. For a pure water feed (i.e.,
) Eq. (1) gives: (5)
where
is the steady state pure water flux. Water flux in a feed containing salt is given by (
5
)
(6)
where
and
are the osmotic pressures of the feed solution at the membrane surface and
in the bulk permeate solution, respectively. The subscript “0(m)” represents values at the upstream membrane surface, to distinguish them from valued in the bulk feed solution. If
is
constant, Eqs. (5) and (6) yield [16]:
(
(7)
)
In this way, the osmotic pressure of the feed solution at the membrane surface, same
, can be calculated from experimental data by measuring
given feed salt concentration, and Then,
at a given
, and the ,
at a
from the measured salt concentration of the permeate.
is converted to the salt concentration at the membrane surface,
Pitzer model. Afterwards, the concentration polarization factor,
, using the
, may be calculated as follows
[14, 17]:
(8)
The salt transport coefficient corrected for concentration polarization, B*, is defined as follows: (9)
More rigorously, diffusive salt flux can be written using the salt chemical potential gradient as the driving force for salt transport [23]:
6
(10)
where
is the gas constant,
is absolute temperature, and
is the salt chemical potential in
the membrane. This formalism inherently includes effects of salt thermodynamic non-idealities on the driving force for salt transport. Here,
is the apparent salt diffusion coefficient and
includes the impact of any convective transport of salt in the membrane [6, 24]. The salt chemical potential in the membrane can be expressed as follows [25]: (11) where
is the standard state chemical potential,
is the activity of salt in the membrane,
the partial molar volume of salt in the membrane, and
is the hydrostatic pressure. Since
is and
are constant in the membrane phase, combining Eqs. (10) and (11) yields [26]: (12)
The activity of salt at any location activity coefficient,
in the membrane,
, and salt concentration,
, is expressed as the product of the salt
, at that location [25]: (13)
During steady-state operation, chemical potential equilibrium is established at the upstream and downstream faces of the membrane (i.e., state chemical potentials, molar volume,
and
) [17, 27]. The standard
, are equal in the solution and membrane phases [21, 26]. The partial
, and hydrostatic pressure, , of salt do not change appreciably between the
membrane and solution phases [21, 26]. Therefore, chemical potential equilibrium at the
7
membrane surface reduces to equality of activity at the membrane surfaces and the contiguous solutions (i.e.,
and
).
Eq. (12) can be integrated across the membrane thickness:
∫
(14)
∫
to yield:
〈
where 〈
〉 (
)
〈
〉 (
)
〈
〉 (
)
(15)
〉 is the ratio of the effective salt diffusion to the salt activity coefficient in the
membrane averaged over the membrane thickness. This derivation is shown in more detail in the Supporting Information. This expression describes salt transport driven by the gradient in salt chemical potential across the membrane [13]. The original Merten and Lonsdale formulation used concentration gradient, rather than chemical potential gradient, as the driving force for salt transport [9, 10]. When salt activity coefficients in the membrane and solution are equal to one (i.e., ideal solution), these two approaches are equivalent. Here, we define the salt transport coefficient,
, as follows [13]: (16)
This expression includes contributions from both concentration polarization and non-ideal thermodynamic effects on salt transport in the membrane. According to the definition of activity (Eq. (13)), the salt sorption coefficient can be expressed as a ratio of activity coefficients (i.e., 8
). If
is constant at the
upstream and downstream faces of the membrane (i.e.,
) and the salt diffusion
coefficient is constant across the membrane (i.e., independent of salt concentration):
〈
〉
〈
〉
(17)
Using this expression, Eq. (15) clearly reduces to the original form of the Merten and Lonsdale model: (18)
Fig. 2 presents definitions and expressions for the various salt transport coefficients (i.e., B values) described above.
Figure 2. Definitions and expressions for salt transport coefficients. B denotes the salt transport coefficient used in the original Merten and Lonsdale model. B* denotes the salt transport coefficient corrected for concentration polarization.
denotes the salt transport coefficient
corrected for concentration polarization and solution phase thermodynamic non-idealities.
9
3. Experimental 3.1. Membranes and materials Commercial flat-sheet reverse osmosis membranes, Dow FilmtecTM BW30XFR, from Dow Water Solutions were used for this study. BW30XFR membranes are thin film composite membranes with a polyamide separation layer formed via interfacial polymerization [19]. The manufacturer reported a water flux of 48.4 L/(m2 h) (LMH) with minimum stabilized NaCl rejection values between 99.4 and 99.65 %, respectively, in a 2000 ppm NaCl feed at 225 psig (15.5 barg) feed pressure at 25 oC and a pH of 8 [28]. Flux, rejection and area are reported by the manufacturer and are average apparent values from tests on whole modules. As such, they are not corrected for concentration polarization or thermodynamic non-idealities. All membranes were purchased from the manufacturer as dry flat-sheet membranes and stored in a cool, dark place before use. Chemical aging may occur when polyamide-based membranes are exposed to ambient air due to oxidation of residual functional groups on the membrane surface [19]. Color changes (from white to yellow) on the membrane surface and a reduction in salt rejection can be observed. Thus, only fresh membrane samples with no color change were used for these studies. The membrane samples were pretreated according to a procedure reported previously [19]. Flat-sheet membrane samples were cut into the size that fits the active area (9.2 cm x 4.6 cm) of the crossflow test cells. The samples were soaked in 25 vol% aqueous isopropanol solutions for 20 min to assist in removing residual production chemicals and glycerol from the membrane. Then, the samples were soaked in de-ionized (DI) water at least 24 h before testing, and the DI water was changed twice during this period. The containers used for sample pretreatment and storage were covered with aluminum foil before testing to limit exposure to light, which can cause chemical degradation of the polyamide layer. DI water was generated by a Millipore RiOS and
10
A10 water purification system (Billerica, MA) and had an electrical resistivity of at least 18.2 M -cm and less than 5.4 ppb organics based on total organic carbon (TOC). Sodium chloride, magnesium chloride, calcium chloride, and sodium sulfate were purchased from Sigma-Aldrich (St. Louis, MO) and used as received to make feed salt solutions with DI water.
3.2. Crossflow system and measurement Salt rejection and water flux were measured using a custom-built crossflow system (Fig. 3). Crossflow experiments simulate tangential-flow in commercial desalination membrane applications. The membranes were clamped into commercial bench-scale crossflow cells (CF042, Sterlitech, Kent, WA). Three cells were connected in series to a 30 L feed tank, a pump and pulse dampener, a back pressure regulator, bypass valve, a pressure gauge, and a flow meter. The feed tank was filled with at least 20 L of salt solution, and the feed concentration did not change appreciably during the experiment. The feed solution was passed through a filter (Hytrex GX059-7/8 (pore size: 5 m), Big Brand Filter, Chatsworth, CA) to prevent bacterial growth and particulate fouling on the membrane surface. The feed solution temperature was maintained at 25 o
C using a stainless steel coil placed in the feed tank and connected to a heater/chiller (Thermo
Neslab RTE-10 Digital One Refrigerated Bath, Thermo Fisher Scientific, Waltham, MA). The flowrate was 1 gallon per minute to produce turbulent flow (Reynolds number = 3220) [19]. In the Reynolds number calculation, the hydraulic diameter was calculated as 2 × the channel height (2.3 mm), according to the parallel plate approximation for a flat channel (width/height >3-4) [29, 30]. The viscosity of pure water was used in the Reynolds number calculation. No spacer was used in the feed channel. The crossflow system was cleaned before each experiment by circulating a 200 ppm bleach solution (3.4 g Clorox®/L) through the system for 30 min. Then, the crossflow system 11
was thoroughly rinsed by circulating fresh DI water four times to remove residual bleach, since the polyamide layer on desalination membranes can be damaged by aqueous chlorine [17]. For each rinse cycle, DI water was circulated for at least 10 min and then drained. After four rinse cycles, removal of chlorine and residual salts from previous tests was confirmed by the feed solution pH (~6, the pH of water equilibrated with atmospheric CO2 [31]) and the conductivity (< 15 μS/cm).
Figure 3. Schematic of crossflow filtration system. Adapted from [19] with permission from Elsevier.
Multiple applied feed pressures were chosen around a standard brackish water desalination condition (
225 psig). The applied pressure ranged from 125 to 375 psig (8.62
to 25.9 barg) in intervals of 50 psig. Pure water flux was first measured at each pressure to obtain pure water permeance. Then, a mass of salt corresponding to 0.1 M was added to the feed solution. The salt concentrations in the feed and permeate solutions were determined by 12
measuring the conductivity of those solutions with a handheld conductivity meter (Oakton CON 110, Oakton Instruments, Vernon Hills, IL). Individual calibration curves for the four salt solutions considered in this study (e.g., NaCl, MgCl2, CaCl2 and Na2SO4) were established and used for converting conductivity to salt concentration. Apparent salt rejection,
, was
calculated as follows:
(
where
and
(19)
)
are the salt concentrations in the bulk feed and permeate solutions,
respectively. The steady-state permeate flux (L/(m2 h), or LMH),
, was calculated as follows: (20)
where
is the membrane area (38 cm2), and
is the volume of permeate solution,
is the
sample collection time. The permeate sample mass was measured and converted to volume using the density of water (
g/mL). To ensure steady state operation, sufficient time (typically
1~2 min) was given for permeation before measurements, and two sets of data for permeate flux and salt rejection were successively collected and compared with each other.
4. Results and discussion 4.1. Salt activity coefficients Salt activity coefficients in aqueous solution are presented as a function of molar concentration (M or mol/L solution) in Fig. 4. Activity coefficients were calculated using the Pitzer model [14, 15]. This model originally used molal concentration, whereas experiments 13
conducted in this study used molar concentration. These two values are essentially equal within the concentration ranges considered here (as shown in the Supporting Information), so molar concentration was used in the following discussion. Activity coefficients in dilute solutions (i.e., near ~0 M) approach unity. These values decrease significantly over the concentration range from 0 to 0.3 M. The magnitude of decrease in activity coefficient varies depending on the salt.
Figure 4. Activity coefficients of salts in aqueous solution at 25 oC as a function of molar concentration (mol/L solution) calculated according to the Pitzer model [14, 15].
The difference between concentration and activity (where activity,
, is given by
)
can be significant for these salts. For example, the activity coefficient of Na2SO4 at 0.1 M is 0.45, which makes the thermodynamic activity equal to 0.045 for an 0.1 M Na2SO4 solution. The same analysis for 0.1 M MgCl2 and CaCl2 gives 47% and 48% differences between solution activity 14
and concentration, respectively. Because the driving force for salt transport is the salt activity gradient across the membrane, rather than the concentration gradient, not accounting for such thermodynamic non-idealities causes the resulting B value to depend on both the membrane properties (i.e., salt diffusion coefficient, partition coefficient, and membrane thickness) as well as the thermodynamic properties of the feed solution. As a result, B values do not simply reflect properties of the membrane unless this effect is accounted for. Because the feed solutions, particularly at higher salt concentrations, are not ideal, osmotic pressure values will also deviate from ideal solution theory. Osmotic pressure of salt solutions considered in this study were also calculated using the Pitzer model [14, 15]. To provide an estimate of this effect, a comparison between osmotic pressures calculated from the van’t Hoff relation (i.e., ideal solution) and the Pitzer model is provided in the Supporting Information.
4.2. Crossflow test results The results of crossflow experiments for BW30 XFR membranes with an 0.1 M NaCl feed solution are presented in Fig 5. Apparent salt rejection increased with increasing feed pressure, consistent with previous observations [19]. Salt concentration in the permeate solution is related to the ratio of salt flux to water flux as indicated in Eq. (3). Water flux,
, increases as
applied feed pressure increases (cf. Eq. (1)). Salt flux, , is approximately independent of feed pressure. Thus, increasing feed pressure leads to higher water flux, resulting in lower salt concentration in the permeate solution,
, thereby increasing apparent salt rejection. Data for
other salts considered in this study (i.e., MgCl2, CaCl2, and Na2SO4) showed similar trends and are recorded in the Supporting Information for brevity.
15
Figure 5. Influence of feed pressure on permeate water flux and apparent salt rejection in a 0.1 M NaCl feed for BW30XFR membranes. Each data point represents the average of at least three individual measurements, and uncertainties are reported as one standard deviation from this average. The permeate pressure was atmospheric in all cases.
4.3. Concentration polarization Concentration polarization was calculated using the model proposed by Sutzkover et al. [16] In this model, the water permeance of a membrane is taken to be constant and independent of feed composition [16, 19]. The model accounts for changes in driving force in terms of osmotic pressure due to concentration changes at the membrane surface. The osmotic pressure at the membrane surface,
, was obtained from Eq. (7), and it was used to calculate the salt
concentration at the membrane surface,
, using the Pitzer model. Then, the salt 16
concentration at the membrane surface,
, was divided by the bulk feed salt concentration,
, to obtain the concentration polarization factor, CP, as indicated in Eq. (8). Calculated concentration polarization factors for BW30XFR membranes are presented as a function of permeate flux in Fig. 6. According to Wijmans, CP is expected to increase as permeate flux increases when solutes are highly rejected by the membrane [18], and we observe this trend for all salts considered. The osmotic pressures of 2:1 (MgCl2 and CaCl2) and 1:2 (Na2SO4) salt solutions are inherently higher (and, therefore, water fluxes are inherently lower) than those of the 1:1 (NaCl) salt solution at the same feed concentration (i.e., 0.1 M). Thus, CP values for NaCl were higher than those of the other salts due, in large measure, to the higher driving force for water flux (i.e.,
) in the NaCl containing solutions. Thus, NaCl
solutions had higher permeate flux values at the same applied transmembrane pressure than those of the other salt solutions considered. The osmotic pressures of the salt solutions considered in this study are recorded in the Supporting Information.
17
Figure 6. Influence of permeate flux on concentration polarization factor, CP, in 0.1 M feed solutions of NaCl, MgCl2, CaCl2, and Na2SO4 for BW30XFR membranes.
4.4. Effective feed concentration/activity Concentration polarization-corrected feed concentrations,
were calculated
according to Eq. (7) and are presented as a function of permeate flux in Fig. 7. These values, labeled “Polarization” in Fig. 7, represent the calculated salt concentration at the upstream membrane surface. While the bulk feed concentration was constant at 0.1 M, the feed salt concentration at the membrane surface increased significantly with increasing permeate flux. For example, concentration polarization-corrected feed concentrations for NaCl change from 0.12 to 0.22 M as permeate flux increased. The other salts exhibited similar trends with permeate flux.
18
Using the calculated surface,
values, feed solution activity values at the upstream membrane
, were calculated using the Pitzer model (labeled “Activity” in Fig. 7) [14, 15].
Figure 7. Influence of permeate flux on concentration polarization-corrected feed concentration and activity in 0.1 M: (a) NaCl, (b) MgCl2, (C) CaCl2, and (d) Na2SO4 feed solutions. Bulk and polarization corrected concentrations are
and
19
, respectively.
4.5. Salt transport coefficients Salt transport coefficients corrected for concentration polarization effects, calculated using Eq. (9).
, were
values are presented as a function of permeate flux in Fig. 8, and
apparent salt transport coefficients (i.e., B values) are also presented for comparison. For the salts considered,
values were always less than or equal to the corresponding B values. The
difference between B and
increased as permeate flux (i.e., applied feed pressure) increased
due to increasing concentration polarization (cf. Fig. 6). For NaCl (i.e., a 1:1 salt), differences between
and B values were noticeable even at low permeate fluxes, whereas these differences
were negligible (i.e.,
B) at very low permeate fluxes for the 2:1 and 1:2 salts (i.e., MgCl2,
CaCl2, and Na2SO4) for the reasons discussed above [18]. Salt transport coefficients corrected for both concentration polarization and non-ideal thermodynamic effects, 8.
, were calculated using Eq. (16) [13]. The results are presented in Fig.
values are always greater than
values because the salt activity coefficients in solution
are less than 1 (cf. Figs. 2 and 4). Interestingly, for NaCl values. In contrast, for MgCl2, CaCl2, and Na2SO4
values were generally lower than B
values were higher than the corresponding
B values. This result is a combination of correcting for both concentration polarization and nonideal thermodynamic effects. Effective salt concentration at the upstream membrane surface was greater than the salt concentration in the bulk feed solution (cf. Fig. 7), which tends to decrease values relative to B. On the other hand, salt activity coefficients were less than 1 at the upstream face of the membrane (cf. Fig, 4), which tends to increase the divalent salts, the latter effect dominates over the former effect.
20
values relative to B. In
Figure 8. Influence of permeate flux on salt transport coefficients (i.e., B,
, and
values) in
0.1 M: (a) NaCl, (b) MgCl2, (C) CaCl2, and (d) Na2SO4 feed solutions for BW30XFR membranes. B is the apparent salt transport coefficient calculated from the Merten and Lonsdale model [9, 10].
is the salt transport coefficient corrected for concentration polarization.
is
the salt transport coefficient corrected for both concentration polarization and solution phase thermodynamic non-idealities.
21
The ratio of
to B is presented as a function of permeate flux in Fig. 9.
for NaCl
ranges from 0.8 to ~1.0, suggesting that concentration polarization and solution phase thermodynamic non-idealities corrections to salt transport coefficients nearly cancel one another. For MgCl2 and CaCl2, activity coefficient values are similar, resulting in similar values for For Na2SO4,
ranged from ~1.7 to ~2.1, which was the highest among the four salts
considered. This behavior is due to the greater departure from ideal solution activity coefficients in Na2SO4 than in the other salts, as shown in Fig. 4. The approach to calculate
values used in
this study was applied to the nanofiltration membrane salt transport data reported by Bason et al. [32] The results of this calculation are also presented in Fig. 9.
values from Bason et al.’s
study also fall in a very similar range to those calculated for the RO membranes in this study.
22
Figure 9. The ratio of
to B (i.e., salt transport coefficient corrected for concentration
polarization and thermodynamic non-ideality, function of permeate flux.
, to apparent salt transport coefficient, B) as a
values for various salts in an NF membrane were taken from Bason
et al. for comparison [32].
23
5. Conclusions Concentration polarization and solution phase thermodynamic non-idealities can significantly influence salt transport coefficients in desalination membranes. Concentration polarization effects were evaluated using pure water permeance, water flux from salt solutions, and salt rejection data. Osmotic pressure and salt concentration at the upstream membrane surface were higher than in the bulk feed solution due to concentration polarization. Concentration polarization effects were more significant at higher feed pressures due to the increasing water flux at higher feed pressures. Concentration polarization corrected salt transport coefficients (i.e.,
values) were equal to or lower than uncorrected (i.e., B) values due to higher
salt concentration at the upstream membrane surface than in the bulk feed solution. The influence of solution phase thermodynamic non-idealities on salt transport coefficients was evaluated using salt activity coefficients in the feed and permeate solutions. Corrections for both concentration polarization and non-ideal thermodynamic effects (i.e., resulting in
values) were significant,
values that were greater than B values by a factor of 1.3~2.1 for 2:1 and 1:2 salts
(MgCl2, CaCl2, and Na2SO4).
values for NaCl (a 1:1 salt), however, were similar to or slightly
less than corresponding B values. For the salt solutions considered in this study (e.g., NaCl, MgCl2, CaCl2, and Na2SO4),
values varied significantly depending on the departure of
solution phase thermodynamics from ideality. Finally, while reporting B values facilitate comparisons with existing literature data, it is also useful to report Ba values since these represent the performance of the membrane material itself.
Acknowledgements This material is based upon work supported in part by the Welch Foundation Grant No. F-192420170325. A.H. was supported in part by the National Science Foundation (NSF) Graduate 24
Research Fellowship under Grant No. DGE-1610403. Additionally, B.D.F.’s contributions to the preparation of this manuscript were supported as part of the Center for Materials for Water and Energy Systems (M-WET), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DESC0019272.
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Highlights
A theoretical framework accounting for concentration polarization and non-ideal thermodynamic effects on salt transport in RO membranes is proposed.
The influence of concentration polarization and thermodynamic non-idealities on salt transport coefficients (i.e., B values) in RO membranes was quantified.
This framework was applied to four common salt solutions, NaCl, MgCl2, CaCl2, and Na2SO4.
Corrections for concentration polarization and non-ideal thermodynamic effects (i.e., values) were significant, resulting in
values that were greater than B values by a factor
of 1.3~2.1 for MgCl2, CaCl2, and Na2SO4 solutions.
27