Trends and errors in reverse osmosis membrane performance calculations stemming from test pressure and simplifying assumptions about concentration polarization and solute rejection

Abstract

A primary goal in the design of reverse osmosis (RO) membranes is to improve water-solute selectivity and water permeance. These transport properties are commonly calculated in the literature using the solution-diffusion model with selectivity (A/B, bar⁻¹) defined as the ratio between water permeance (A, L.m⁻².h⁻¹.bar⁻¹) and solute permeance (B, L.m⁻².h⁻¹). In calculating transport properties, researchers often use simplifying assumptions about concentration polarization (CP; i.e., assuming negligible CP or a certain extent of CP) and solute rejection (i.e., assuming solute rejection is approximately 1 to enable the explicit use of the CP modulus in solute permeance calculations). Although using these assumptions to calculate transport properties is common practice, we could not find a study that evaluated the errors associated with using them. The uncertainty in these errors could impede unequivocally identifying manufacturing approaches that break through the commonly plotted trade-off frontier between selectivity and water permeance (A/B vs. A); however, we did not find in the literature a study that quantified such errors. Accordingly, we aimed to: (1) quantify the error in transport properties (A, B, and A/B) calculated using common simplifying assumptions about CP and rejection; and (2) determine if using simplifying assumptions affects conclusions drawn about membrane performance or trends concerning the trade-off frontier. Results show that compared with the case where no simplifying assumptions were made, simplified calculations were least accurate at low pressures for water permeance (up to 78% overestimation) and high pressures for solute permeance (up to 188% overestimation). Accordingly, the corresponding selectivities were least accurate at low pressure (up to 111% overestimation) and high pressure (up to 66% underestimation), and conclusions drawn about membrane performance and trade-off trends were pressure-dependent. Importantly, even in the absence of simplifying assumptions, selectivity results were pressure-dependent, indicating the importance of standardizing test conditions for the continued use of current performance metrics (i.e., A/B and A). We propose a two-pressure approach—collecting data for A and B at a high and a low pressure, respectively—combined with simplifying assumptions for more accurate simplified estimations of selectivity (< 10% absolute error). Our work contributes to a better understanding of the effects of operating pressure and key simplifying assumptions commonly used in calculating RO membrane performance metrics and interpretation of corresponding results.

Graphical Abstract

1. Introduction

Reverse osmosis (RO) is a technology that uses semipermeable polymeric membranes and externally applied pressure to remove contaminants from water. RO is used in many applications, including desalination, brackish water treatment, wastewater reclamation, and municipal water treatment. A guiding goal in the design of RO membranes is to make a membrane with better water-solute selectivity (A/B, bar⁻¹) and water permeance (A, L.m⁻².h⁻¹.bar⁻¹) [1,2] than existing membranes; selectivity is defined as the ratio between water permeance and solute permeance (B, L.m⁻².h⁻¹). Authors commonly use plots of water-solute selectivity as a function of water permeance to compare the performance properties and trade-off trends of membranes within a study and more broadly with the literature [1-5]. For the construction of these ‘trade-off plots,’ water (A) and solute (B) permeance are calculated under the framework of the solution-diffusion (SD) model.

The SD model relates observed performance (i.e., permeate flux and solute rejection) to membrane transport properties (i.e., water and solute permeance) [6,7]. The SD model assumes that water and solute molecules independently partition into the membrane at the feed side, diffuse through the membrane due to a gradient in chemical potential, and partition out of the membrane at the permeate side. Because water molecules permeate through the membrane faster than solute molecules, concentration polarization (CP) occurs [8-10], in which the concentration of solutes at the membrane feed wall builds up, becoming greater than that in the bulk feed solution during RO operation. The ratio between these two concentrations is the CP modulus (β, unitless). In general, β is positively correlated with permeate flux (i.e., pressure) and solute rejection and inversely correlated with the mass transfer coefficient of the solute in the CP layer (k, L.m⁻².h⁻¹) [1,7,10,11]. Therefore, rigorous evaluation of A, B, and A/B parameters requires accounting for CP. While one can account for CP in water and solute permeance calculations by experimentally determining k, doing so accurately is resource-intensive [12-18]: one must perform a model fit of experimental permeate flux and solute rejection data recorded at several pressures (5–10). Instead, many studies only report permeate flux and solute rejection (not A, B, and A/B), and those that report water and solute permeance often make a simplifying assumption in their calculations by either neglecting CP altogether or assuming a certain extent of CP representative of practice or approximated by analytical equations for the mass transfer coefficient (see Sections 2, 2.4 for further details) [2,19]. However, depending on the extent of actual CP relative to the assumed CP (i.e., neglecting CP or assuming a representative value), water and solute permeance and the resulting water-solute selectivity may be over- or underestimated to an unknown extent.

A recent study by Yang et al. [2] postulated the upper bound of the trade-off between water-solute selectivity and water permeance using data gathered from the published literature. Their upper bound has some unknown error because the authors had to neglect CP due to a lack of relevant information in the literature [2]. As a result, they have asked that future studies include the CP modulus in their water and solute permeance calculations. The corresponding unreported (and in many cases unknown) errors in water-solute selectivity values could represent an impediment to unequivocally identifying manufacturing approaches that break through the trade-off frontier. However, we did not find a study in the literature quantifying the error in selectivity values associated with the different water and solute permeation quantification approaches (i.e., neglecting, assuming some extent of, or accounting for CP).

Accordingly, this study aimed to: (1) quantify the error in water permeance, solute permeance, and water-solute selectivity values calculated by implementing simplifying assumptions about CP and solute rejection; and (2) determine if using simplifying assumptions affects conclusions drawn about membrane performance or trends concerning trade-off frontiers of unmodified membranes compared with modified membranes. To accomplish our aims, we compared water permeance, solute permeance, and selectivity values obtained across three scenarios of differing simplification approaches with values obtained without simplifying assumptions and evaluated if differences across the three simplification approaches depended on test pressure, membrane type (i.e., high- or low-flux), or solute.

2. Theoretical background

2.1. Concentration polarization modulus

The CP modulus (β, dimensionless) is defined as the ratio between the solute concentration at the feed-side of the membrane wall (c_w, M) and that of the feed solution (c_f, M) as given by

$$\beta =\frac{c_w}{c_f}.$$ (1)

The solute concentration at the membrane wall can be related to the solute mass transfer coefficient (k, L.m⁻².h⁻¹) by [7]

$$\frac{C_w-C_p}{C_f-C_p}=e^{(J_v∕k)},$$ (2)

in which c_p (M) is the solute concentration in the permeate and J_v (L.m⁻².h⁻¹) is the permeate flux. Observed solute rejection (R, dimensionless) is defined as

$$R_{obs}=1-\frac{C_p}{C_f}=\frac{1}{1+\frac{C_p}{\left(C_f-C_p\right)}},$$ (3)

which can be rearranged to give

$$\frac{C_p}{C_f-C_p}=\frac{1-R_{obs}}{R_{obs}}.$$ (4)

Going forward, all rejection refers to observed rejection (as opposed to intrinsic rejection, see Section S1) and the subscript is dropped. By splitting the numerator on the left-hand side of Eq. (2) into two fractions, substituting in Eqs. (1) and (4) into (2), and rearranging terms, the CP modulus (β) can be expressed as a function of solute rejection (R), permeate flux (J_v), and the mass transfer coefficient (k) as [7]

$$\beta =e^{(J_v∕k)}R+(1-R).$$ (5)

2.2. Water permeance

Under the framework of the SD model [6], water permeance (A, L.m⁻².h⁻¹.bar⁻¹) is expressed as

$$A=\frac{J_v}{\Delta P-\Delta \pi },$$ (6)

in which ΔP (bar) is the difference between the feed and permeate hydraulic pressures and Δπ (bar) is the difference between the osmotic pressures at the membrane wall (π_w, bar) and in the permeate (π_p, bar),

$$\Delta \pi ={\pi }_w-{\pi }_p.$$ (7)

The CP modulus can be very closely approximated (less than 0.1% error) as the ratio of the osmotic pressures at the feed side of the membrane wall and in the feed solution [7],

$$\beta =\frac{C_w}{C_f}\approx \frac{{\pi }_w}{{\pi }_f},$$ (8)

because osmotic pressure scales linearly with concentration in the concentration range of our study [7]. Eq. (8) can be rearranged to express the osmotic pressure at the membrane wall as

$${\pi }_w=\beta {\pi }_f.$$ (9)

The CP modulus becomes apparent in the water permeance equation when Eq. (7) and (9)) are substituted into Eq. (6) to give

$$A=\frac{J_v}{\Delta P-(\beta {\pi }_f-{\pi }_p)}.$$ (10)

Further, Eq. (5) can be substituted into Eq. (10) to arrive at an equation for water permeance that accounts for CP directly through the mass transfer coefficient instead of the CP modulus

$$A=\frac{J_v}{\Delta P-\{[e^{(J_v∕k)}R+(1-R)]{\pi }_f-{\pi }_p\}}.$$ (11)

2.3. Solute permeance

In its simplest—and most common—formulation, the SD model neglects any advective transport through relatively large pores or imperfections (i.e., there is no advective term in the equations). This is the formulation typically used in the literature to obtain A, B, and A/B. However, to rigorously quantify CP from performance data (i.e., to obtain solute mass transfer coefficients, k, from permeate flux and solute rejection data), one must explicitly account for potential advective transport. Accordingly, extended SD models that include an advective transport parameter have been used in the literature that better describe solute rejection e.g., as a function of applied pressure [11-13,20,21]. Taking into account both diffusive and advective transport, solute transport is described by [12]

$$\frac{c_p}{c_f-c_p}=\left(\frac{\overline{B}}{1-\overline{\alpha }}\times \frac{1}{J_v}+\frac{\overline{\alpha }}{1-\overline{\alpha }}\right)e^{(J_v∕k)},$$ (12)

for which $\overline{B}$ (L.m⁻².h⁻¹) is the diffusive transport coefficient and $\overline{\alpha }$ (dimensionless) is the advective transport coefficient. In this study, Eq. (12) was used solely to determine k for our system and since we were able to determine the experimental k values relevant for our system, we did not use correlations of k from the literature in this study.

The use of Eq. (12) for analysis of permeate flux and solute rejection data is resource-intensive because for a given membrane and water quality conditions, it requires data collection at several pressures (i.e., we used eight) to solve for $\overline{B}$, $\overline{\alpha }$, and k as fitting parameters (or just $\overline{B}$ and $\overline{\alpha }$ if k is approximated through analytical equations [19,22-27]). Therefore, for thin-film composite membranes, a common simplifying assumption is to neglect advective transport, setting $\overline{\alpha }=0$. Doing so attributes all solute transport (both advective and diffusive) to the solute permeance coefficient (B, L.m⁻².h⁻¹). The resulting equation is that of the typical SD model [6,28]

$$\frac{c_p}{c_f-c_p}=\frac{e^{(J_v∕k)}B}{J_v}.$$ (13)

Solving for B gives

$$B=\frac{J_v{C}_p}{(C_f-C_p)e^{(J_v∕k)}}.$$ (14)

Eq. (14) can be expressed in terms of R by substituting in Eq. (4), giving [7]:

$$B=\frac{J_v(1-R)}{e^{(J_v∕k)}R}.$$ (15)

If R ≈ 1, Eq. (5) simplifies to

$$\beta =e^{(J_v∕k)},$$ (16)

which can be substituted into Eq. (15) as [7]

$$B=\frac{J_v(1-R)}{\beta R}.$$ (17)

As mentioned in the introduction and this section, Eq. (12) is not commonly used to analyze performance data because of its resource intensiveness. However, Eq. (12) provides the best approach for evaluating CP from permeate flux and solute rejection data because it enables one to fit the value of the mass transfer coefficient, k, under the actual distribution of solute transport mechanisms (i.e., diffusive and advective). A k value obtained using Eq. (13) does not describe membrane performance data as well as a k value obtained using Eq. (12) (see Section S1). Specifically, neglecting advection leads to errors in the fitted values of k.

2.4. Common approaches in the literature to obtain water and solute permeance

Most membrane studies only report permeate flux and solute rejection data [19]. Doing so aligns with industry practice but leaves out important information required to calculate, understand, and compare membrane transport properties. Studies that do present water (A) and solute (B) permeance metrics use Eqs. (10) and (17) to calculate them, respectively. In doing so, they either neglect CP completely (i.e., β = 1 [29-32]), estimate CP by approximating the mass transfer coefficient (k) with Sherwood relations [11,19,33-36], or choose a CP modulus (or mass transfer coefficient) representative of industry (e.g., β = 1.2 [7,37] or k = 100 L.m⁻².h⁻¹ [1,2,19]). Note that in using Eq. (17), it is implicitly understood that R ≈ 1 even though the observed R value is used as input to Eq. (17). This is because the derivation of Eq. (17) requires assuming R = 1 in the transition from Eq. (15) to Eq. (17) to enable approximating e^J_v/k as β. This approximation may be appropriate for the rejection of inorganic salts by seawater desalination membranes, but not for lesser performing membranes or other solutes like trivalent arsenic and boric acid.

3. Experimental

3.1. Chemicals and solutions

Ethanol (99.5%), hexane (GC Resolv™), methanol (≥99.9%), ammonium acetate (97.0%), boric acid (99.99%, metals basis), ethylenediaminetetraacetic acid disodium salt dihydrate (99+%), glacial acetic acid (ACS plus), hydrochloric acid (ACS plus), L-ascorbic acid (99%), and sodium chloride (≥99%) were purchased from Fisher Scientific (Pittsburgh, PA). Arsenous acid (+3 at 1,000 μg/mL in 2% HCl), arsenic acid (+5 at 1,000 μg/mL in H₂O), and 26 component trace metals in drinking water (in 2% HNO₃ + Tr HF, Mix A) were purchased from High Purity Standards (North Charleston, SC). Sodium hydroxide pellets (ACS) were purchased from Mallinckrodt (Phillipsburg, NJ). Ammonium carbonate (puriss.), ammonium acetate (≥99.99% trace metals basis), and m-phenylenediamine flakes (MPD, 99%) were purchased from Sigma-Aldrich (St. Louis, MO). Azomethine-H (>97%) and 1,3,5-benzenetricarbonyl trichloride (TMC, >98%) were purchased from TCI Chemicals (Philadelphia, PA). Laboratory grade water (LGW, ≥17.8 MΩ.cm) (Dracor, Durham, NC) was used throughout all experiments.

3.2. Membranes

The membranes used in this work were SWC4+, a seawater membrane, and ESPA3, a brackish water membrane, both donated by Hydranautics (Oceanside, CA). Membranes were cut to size (7.5×11 cm², unmodified or 22×14.5 cm², modified), thoroughly rinsed with LGW, and stored in two liters of LGW at 4 °C until use.

3.3. Membrane sample preparation

3.3.1. Modified membranes

Membranes were modified with a procedure adapted from a recent study [38] and was used solely to show conceptual differences in trade-off trends between modified and unmodified membranes. Briefly, commercial membranes were subjected to two sequential modification steps: 1) exposure to 0.35 wt% trimesoyl chloride (TMC) in methanol for 10 minutes, and 2) exposure to 3.5 wt% to m-phenylenediamine (MPD) in water (5 min and 10 min for SWC4+ and ESPA3, respectively). Additional details are provided in Section S2.

3.4. Cross-flow filtration experiments

Cross-flow experiments were performed with a custom-built cross-flow filtration apparatus described elsewhere [20,38]. The membranes were compacted with LGW for 24 hours at 33 bar, which was long enough to achieve stable performance (Section S3). Filtration (22 °C at pH 5.3 with 14 cm.s⁻¹ cross-flow velocity) of a feed solution (20 L) comprised of 2000 mg.L⁻¹ NaCl, 1 mg.L⁻¹ H₃AsO₃, 1 mg.L⁻¹ H₃AsO₄, and 200 mg.L⁻¹ H₃BO₃ was performed at eight pressures for three hours each. Duplicate feed, permeate, and permeate flux samples were collected at the end of three hours for all eight pressures (see Section S3 for details on the choice of permeate flux over pure water flux for water permeance calculations). Compaction was performed and filtration started at 33 bar so that any potential deformation of the membrane (see [39]) would be as consistent as possible throughout the experiment.

3.5. Solute quantification

Two charged solutes—sodium chloride and arsenic acid (As(V))—and two neutral solutes—boric acid and arsenous acid (As(III))—were used in permeation experiments. Conductivity was used as a surrogate for sodium chloride concentration; samples were measured with a conductivity probe. Boric acid was quantified by the azomethine-H spectroscopic method reported elsewhere [20]. The As(III) and As(V) species of the feed and permeate samples were quantified by high-performance liquid chromatography (HPLC) coupled to inductively coupled plasma mass spectrometry (ICP-MS) using an ammonium carbonate mobile phase (60 mM, 1% EtOH, pH 9 [40]).

3.6. Calculation of solute rejection, osmotic pressure, and mass transfer coefficient

Solute rejection was calculated as

$$R=\frac{c_f-c_p}{c_f}\times 100\%,$$ (18)

where C_f and C_p represent mass concentration for arsenous acid, arsenic acid, and boric acid or conductivity for sodium chloride.

Osmotic pressure (π, bar) was calculated using the van’t Hoff approximation [7]

$$\pi =C{R}_{u}T,$$ (19)

in which C is the solute concentration (M), R_u is the universal gas constant (L.bar.K⁻¹.mol⁻¹), and T is temperature (K).

The mass transfer coefficient (k) was obtained for each experiment, along with $\overline{B}$ and $\overline{\alpha }$, by performing nonlinear least-squares regression fitting (R v4.0.5) of R and permeate flux (J_v) data collected at eight pressures to Eq. (12) (see Section S1). The As(V) data was fitted first because it had the most CP. The k from the As(V) fitting was then used to calculate the k for the other solutes as given by [12]

$$k_s=k_{As(V)}{\left(\frac{D_s}{D_{As(V)}}\right)}^{2∕3},$$ (20)

where D is the diffusion coefficient, and the subscript s refers to solutes other than As(V).

3.7. Calculations of water permeance, solute permeance, and water-solute selectivity

For the calculation of water (A) and solute (B) permeance, we used a total of four approaches. Three of the approaches, named Scenario #1, #2, and #3 in Table 1, had varied simplifying assumptions about CP (i.e., an assumed extent of CP through the choice of a specific modulus value in Scenarios #1 and #2) and/or using β in place of e^J_v/k in Eq. (17) (Scenarios #1–3). The A and B values were calculated at each of the eight pressures used during filtration. We compared the results from the three simplification approaches with values that we termed “reference permeance” in Table 1, calculated without making assumptions about CP or using the e^J_v/k ≈ β approximation. These reference permeance values were calculated from the most rigorous accounting of CP and rejection; therefore, they are the closest estimation of the membrane transport properties available.

Table 1.

Equations used for the calculation of water (A) and solute (B) permeance. “Reference permeance” refers to the rigorous estimation of the solute mass transfer coefficient (k) to account for concentration polarization (CP) with no assumptions about solute rejection. Scenarios #1 through #3 refer to different simplifying assumptions about CP and approximating e^J_v/k as β in Eq. (17) (see Section 3.7).

	Concentration polarization		Water permeance, A (L.m−2.h−1.bar−1)		Solute permeance, B (L.m−2.h−1)
Reference permeance	k fitted via Eq. (12)		Slope of Jv vs. ΔP − (βπf − πp)*		$B=\frac{J_v(1-R)}{e^{(J_v∕k)}R}$	Eq. (15)
Scenario #1	β = 1		$A=\frac{J_v}{\Delta P-(\beta {\pi }_f-{\pi }_p)}$	Eq. (10)	$B=\frac{J_v(1-R)}{\beta R}$	Eq. (17)
Scenario #2	β = 1.2
Scenario #3	β = e(Jv/k) R + (1 − R)	Eq. (5)

Note: the (βπ_f − π_p) term in Eq. (10) accounts for the osmotic pressure contributed by all solutes.

^*β in this formulation takes the form of Eq. (5).

3.7.1. Reference water and solute permeance

The reference water and solute permeance of the membranes were determined from the entire dataset of permeate flux and rejection values at eight pressures without making any assumptions about CP nor approximating e^J_v/k as β (Table 1). The reference water permeance was obtained as the slope of the best-fit line to the data set of permeate flux as a function of net transmembrane pressure (see Eq. (6)), using the actual CP modulus calculated using Eq. (5) to determine the osmotic pressure at the membrane wall on the feed side at every pressure (see additional discussion in Section S4). Thus, the reference water permeance was a single value representative across all test pressures. Using the slope of the fit line to the flux versus pressure as the reference water permeance enabled us to have a reference metric that was robust and reproducible against experimental error. The reference solute permeance was determined using the solute permeance equation (Eq. (15)), where the solute mass transfer coefficient was rigorously obtained via Eq. (12) as described in Section 3.6 and Section S1.

3.7.2. Scenarios #1 and #2

These are the two approaches typically used in the literature. In both approaches, Eqs. (10) and (17) are used to obtain water and solute permeance, respectively. In using Eq. (17), one implicitly assumes that R ≈ 1 in Eq. (5), which allows for the substitution of e^(J_v/k) in Eq. (15) with β in Eq. (17); as mentioned previously, the observed R value is used as input to Eq. (17). Further, Scenarios #1 and #2 assume a specific extent of CP by using a specific CP modulus value: β = 1 and 1.2, respectively. These assumptions are used in the literature [1,2,7,19,29-32] as they enable (i) the estimation of A and B without the need to quantify CP, and (ii) the use of one data point (i.e., water flux and solute rejection at a single applied pressure). Assuming β = 1 (Scenario #1) is equivalent to completely neglecting CP (i.e., k = ∞ in Eq. (2)). A value of β = 1.2 (Scenario #2) is also commonly assumed because it is typical of operational conditions in the industry [7].

3.7.3. Scenario #3

As in Scenarios #1 and #2, Eqs. (10) and (17) were used in Scenario #3 for the calculation of A and B, respectively. However, instead of assuming a specific extent of CP through a specific value of β, we used the actual β calculated according to Eq. (5). This scenario is not representative of a common or convenient way to analyze performance data because it requires fitting Eq. (12) (using data at multiple pressures) for k to find the actual β. Instead, this scenario represents a case in which one did not know the k for their system, so chose the same equation typically used in the literature (Eq. (17)) and coincidentally correctly guessed the actual CP modulus. However, because Scenario #3 uses Eq. (17), not Eq. (15), it carries the implicit assumption that R ≈ 1 (even though the observed R value is used for the calculations) to enable the substitution of β for e^(J_v/k). Therefore, Scenario #3 isolates the error associated with substituting β for e^J_v/k without the additional error of guessing an incorrect β (i.e., uses Eq. (17) with the actual β).

3.7.4. Water-solute selectivity

Water-solute selectivity (A/B, bar⁻¹) for each reference and scenario case, was calculated according to

$$A_{scenario}∕B_{scenario}=\frac{A_{avg,scenario}}{B_{avg,scenario}},$$ (21)

where A_avg,scenario and B_avg,scenario refer to the average of water and solute permeance values, respectively, obtained from duplicate filtration experiments.

3.7.5. Errors incurred by simplifying assumptions

Percent errors between reference A, B, and A/B values and those obtained via Scenarios #1–3 were calculated using

$$Percenterror=\frac{P_{scenario}-P_{reference}}{P_{reference}}\times 100\%,$$ (22)

where P refers to A, B, or A/B as defined in Table 1 and Eq. (21). A negative, positive, and zero value indicate an underestimation, overestimation, and accurate estimation by the scenario, respectively.

4. Results and discussion

4.1. Water permeance

4.1.1. Trends in water permeance

We determined the reference and Scenarios #1–3 water permeances (Table 1) for two membrane types: a high-flux membrane (ESPA3) and a low-flux membrane (SWC4+) (Fig. 1). For each membrane, the reference water permeance is a single value equivalent to the slope of the J_v vs. ΔP − (βπ_f − π_p) plot (Table 1). Similarly, results show that the simplified water permeances (i.e., Scenarios #1–3) are relatively constant with pressure, except at the lowest pressure. All three scenarios estimated water permeance similarly, with the error bars generally overlapping at each pressure. For both membrane types, all scenarios overestimated water permeance the most at the lowest pressure (4 bar). For the rest of the pressures (ΔP > 4 bar), the reference water permeance was consistently underestimated the most by Scenario #1, over- and underestimated by Scenario #2 depending on permeate flux and pressure, and most closely matched by Scenario #3. The water permeance results for Scenario #3 were generally between those for Scenarios #1 and #2 at low pressures and higher than those for Scenarios #1 and #2 at high pressures. Scenarios #1 and #2 were less accurate for the high-flux membrane than for the low-flux membrane.

Fig. 1.

Water permeance is overestimated at the lowest (a) pressure and (b) permeate flux for both high-flux (ESPA3, diamonds) and low-flux (SWC4+, triangles) membranes. The dashed line represents the reference water permeance determined from all eight pressures. Each data point is the average of duplicate filtration experiments, and the error bars are the standard deviation.

The differences in water permeance across scenarios are explained by the corresponding osmotic pressure (Δπ) differences (see Eq. (6)), which in turn—and as expected—are explained by the CP moduli (β) differences (see Eqs. (7), (9), Figs. 2, S5). Specifically, Scenario #1 resulted in the lowest water permeance because it had the lowest Δπ, Scenario #2 resulted in higher water permeance than Scenario #1 because it had a correspondingly higher Δπ, and Scenario #3 resulted in water permeances closest to the reference water permeance (at ΔP > 4 bar) because it had the actual Δπ. In turn, the Δπ values for Scenario #1 (β = 1) are the lowest because it has the lowest β (i.e., it underestimates π_w at every pressure), the Δπ values of Scenario #2 (β = 1.2) are shifted up the y-axis from Scenario #1 by a factor of approximately 1.2, and the Δπ values of Scenario #3 (β = f(k)) are greater than those of Scenario #1 by a factor of approximately the actual CP modulus at each pressure. Therefore, CP assumptions directly affect the calculated value of water permeance.

Fig. 2.

Actual concentration polarization (CP) modulus (β) increased with pressure for both high-flux (ESPA3, diamonds) and low-flux (SWC4+, triangles) membranes for (a) NaCl, (b) As(III), (c) As(V), and (d) boron. The solid line represents no CP (Scenario #1), and the dotted line represents an estimated CP modulus of 1.2 (Scenario #2) for reference. The actual CP moduli as a function of permeate flux are presented in Fig. S6. Symbols represent the average of duplicate filtration experiments; error bars are the standard deviation.

4.1.2. Errors in water permeance

Fig. S7 shows how the percent errors of each scenario relative to the reference water permeance varied with pressure and permeate flux. Considering both membrane types and Scenarios #1–3, water permeance was overestimated by 19–78% at the lowest pressure (4 bar) and moderately estimated by −11–7% for the remaining pressures (8–33 bar). The substantial error observed across scenarios at the lowest pressure, 4 bar, is at least partially explained by the greater effect that experimental error has on low-magnitude permeate fluxes than high-magnitude permeate fluxes (see discussion in Section S5) and potential decompaction (i.e., swelling) of the membrane at the lowest pressure. Since the reference water permeance is a fitted constant representative of the greater pressure range, the scenario water permeances which were calculated at individual pressures, reflected experimental deviations more acutely, especially at the lowest pressure (4 bar). Scenario #2 had the highest error because it had the highest β (β = 1.2) and Scenario #1 had the lowest error because it had the lowest β (β = 1). The error from Scenario #3 was intermediate because at this pressure it had an intermediate β, between Scenarios #1 and #2. At higher pressures, the experimental error and error associated with the CP assumption contributed minimally to the observed water permeance error.

Although all scenarios estimated the reference water permeance with similar accuracy for ΔP > 4 bar (i.e., overlapping standard deviations), we rationalize the relative differences between the average percent errors of each scenario by the corresponding differences between their CP moduli and the actual CP moduli at each pressure (Fig. 2). For pressures greater than 4 bar, Scenario #1 generally underestimated water permeance (−5 to −11% and −6 to 1% for the high- and low-flux membranes, respectively) because CP starts to occur as soon as the membrane produces a permeate flux. Scenario #2 overestimated (maximum 3% and 7% for the high- and low-flux membrane, respectively), closely matched (−2% and 0.1%, respectively), and underestimated (maximum −9% and −1%, respectively) reference water permeance for pressures that resulted in actual CP moduli (Fig. 2) less than 1.2, close to 1.2, and greater than 1.2, respectively. Scenario #2 most closely matched the reference water permeance of the high- and low-flux membranes at 12 bar (40 L.m⁻².h⁻¹) and 28 bar (41 L.m⁻².h⁻¹), respectively. Scenario #3 most closely matched the reference water permeance (< ±5% and < ±4% error for the high- and low-flux membrane, respectively) because its formulation contains the actual CP modulus at every pressure.

In terms of differences across membranes, the water permeances of the low-flux membrane for Scenarios #1 and #2 had less error than the corresponding water permeances of the high-flux membrane both as a function of pressure and permeate flux. Both membranes had similar errors with Scenario #3.

Overall, the water permeance values of all three scenarios varied with pressure, showing large overestimations (up to 78%) at the lowest applied pressure and relatively well-matched estimations (within ~11%) at applied pressures greater than 4 bar. The error at each pressure varied with the scenario and was well explained by the assumptions about the CP modulus with Scenario #3 resulting in the lowest errors (< 5% at > 4 bar) because it uses the actual CP modulus. Additionally, experimental error in permeate flux measurements contributed substantially to the water permeance error at the lowest pressure (4 bar) and minimally at all other pressures. These results show that operating pressure is an important factor to consider when making simplifying assumptions for water permeance calculations because (1) experimental error can be significant at low permeate fluxes, and (2) any educated guess about the CP modulus value will not be representative for both high and low permeate fluxes. For one-pressure experiments, collecting data at a moderate-to-high pressure to calculate water permeance will result in less water permeance error than collecting it at a low pressure.

4.2. Solute permeance

4.2.1. Trends in solute permeance

We calculated the permeance of four solutes for a high- (Fig. 3) and low-flux (Fig. 4) membrane. We show the results as a function of permeate flux (Figs. 3, 4) and pressure (Figs. S9, S10). Unlike the reference water permeance—a single value across all permeate fluxes (pressures)—the reference solute permeance generally increased with permeate flux (pressure) for both membrane types. This is notable because it shows that when solute permeance is defined using the SD model approach, solute permeance, and therefore water-solute selectivity, depend on the operating pressure even when CP is accounted for in the most accurate manner possible. This result may stem from the inadequacy of the SD model to account explicitly for advective transport, either as an intrinsic property of the polymer or a result of defects, or other material considerations, such as membrane deformation [39,41]. A detailed discussion on advective transport in RO membranes was outside the scope of this study. The reader is referred to recent studies discussing this topic [42,43].

Fig. 3.

Solute permeance, B, for (a) NaCl, (b) As(III), (c) As(V), and (d) boron increases as a function of permeate flux for the high-flux (ESPA3) membranes. We reproduce the results for the low-flux membrane in Fig. 4 at a greater scale. Reference solute permeance (filled-in squares) and solute permeances calculated with Scenarios #1 (empty symbols), #2 (dotted symbols), and #3 (hatched symbols) are defined in Table 1. The difference between the reference permeance and Scenarios #1 and #2 is the error due to concentration polarization and e^J_v/k ≈ β approximation, whereas the difference between the reference permeance and Scenario #3 is the error due to the e^J_v/k ≈ β approximation. Dashed lines guide the eye. Symbols represent the average of duplicate filtration experiments; error bars are the standard deviation.

Fig. 4.

Solute permeance, B, for (a) NaCl, (b) As(III), (c) As(V), and (d) Boron increases as a function of permeate flux for the low-flux membrane, SWC4+. Reference solute permeance (filled-in squares) and solute permeances calculated with Scenarios #1 (empty triangles), #2 (dotted triangles), and #3 (hatched triangles) are defined in Table 1. The difference between the reference permeance and Scenarios #1 and #2 is the error due to concentration polarization and the e^J_v/k ≈ β approximation , whereas the difference between the reference permeance and Scenario #3 is the error due to the e^J_v/k ≈ β approximation. Lines guide the eye. Symbols represent the average of duplicate filtration experiments; error bars are the standard deviation.

As was the case for the reference solute permeance, simplified solute permeances (i.e., Scenarios #1–3) generally increased with permeate flux (pressure) which contrasted with the relatively constant value that simplified water permeance had as a function of permeate flux (pressure). Scenario #1 generally resulted in the greatest solute permeance overestimations relative to reference solute permeance values. The difference between Scenario #1 and the reference solute permeance is attributable to neglecting CP (as indicated in Figs. 3, 4), such that the solute permeance from Scenario #1 is greater than the reference solute permeance by a factor of e^(J_v/k) (see Table 1). For both membrane types, Scenario #1 matched the reference permeance more closely than Scenario #2 at permeate fluxes less than 22 L.m⁻².h⁻¹ because actual CP moduli were closer to β = 1 than β = 1.2 for those conditions (see Figs. 2, S6). Likewise, Scenario #2 was more accurate than Scenario #1 at permeate fluxes greater than 22 L.m⁻².h⁻¹ because actual CP moduli increased with permeate flux (pressure) and were therefore closer to β = 1.2. than β = 1 (see Figs. 2, S6). Since Scenario #3 incorporates the actual CP modulus, it was the most accurate scenario for both membrane types, with permeance results very closely matching the reference permeance in most cases.

Results also showed that the high-flux membrane was more subject to overestimations of solute permeance at any given applied pressure than the low-flux membrane, and conversely, the low-flux membrane was more subject to underestimations (Figs. 3, 4). These results are explained by the greater extent of CP experienced by the high-flux membrane than by the low-flux membrane (Figs. 2, S6). For example, neglecting CP (i.e., Scenario #1) resulted in a substantially greater overestimation of solute permeance for the high-flux membrane (Fig. 3) because the actual CP modulus for the high-flux membrane always had a greater deviation from unity (Fig. 2), which is the assumed value in Scenario #1. Similarly, assuming β = 1.2 (i.e., Scenario #2) resulted in underestimations of solute permeance for the low-flux membrane for a greater range of pressures than for the high-flux membrane (Fig. 3, 4) because of the corresponding greater range of pressures for which β = 1.2 overestimated the actual CP modulus for the low-flux membrane (Fig. 2). For both membranes, Scenario #2 closely matched the reference solute permeance at permeate fluxes (pressures) where the assumed CP modulus of 1.2 was close to the actual CP modulus (Fig. 2, S6).

4.2.2. Errors in solute permeance

We calculated the percent error of Scenarios #1–3 relative to the reference solute permeance for both membrane types and all solutes at every pressure and permeate flux (Figs. 5, S11-S13). For any given scenario at any given permeate flux, we found that the error was similar for the two membranes (e.g., Scenario #1 for the high-flux membrane compared with Scenario #1 for the low-flux membrane, Figs. S12, S13). Consistent with the observations in Section 4.2.1, Scenario #1 had less error (2–12%) than Scenario #2 (−15 to −7%) at low permeate fluxes (< 22 L.m⁻².h⁻¹) and Scenario #2 had less error (−8 to 140%) than Scenario #1 (11 to 188%) at high permeate fluxes (≥ 22 L.m⁻².h⁻¹). As expected, Scenario #3 was the most accurate of the three scenarios for both membrane types (i.e., < 5% in all cases except boron in the high-flux membrane) because it accounted for the actual CP modulus, though it still had some error associated with the assumption of R ≈ 1 used to approximate β to e^(J_v/k) in Eq. (17).

Fig. 5.

Percent error of B for Scenarios #1–#3 relative to the reference solute permeance of (a and c) NaCl and (b and d) boron increases as a function of increasing pressure for (a and b) a high-flux membrane (ESPA3, diamonds) and (c and d) a low-flux membrane (SWC4+, triangles). Symbols represent the average of duplicate filtration experiments; error bars are the standard deviation. Corresponding plots for As(V) and As(III) are in Fig. S11. Plots of percent error of B as a function of permeate flux are in Figs. S12-S13.

In general, at any given pressure, Scenarios #1–3 had more error for the high-flux membrane than the low-flux membrane, except for Scenario #2 at ΔP ≤ 12 bar (Fig. 5). For Scenarios #1 and #2, the higher error of the high-flux membrane is due to its larger actual CP modulus relative to that of the low-flux membrane (Fig. 2). Scenario #1 calculations resulted in overestimations of solute permeance from 5–188% for the high-flux membrane and 2–44% for the low-flux membrane, with the smallest overestimation associated with the lowest pressures and the biggest with the highest pressures, consistent with smaller and larger actual CP moduli, respectively. Scenario #2 calculations resulted in underestimations of solute permeance at low pressures (up to −12% and −15% for the high- and low-flux membrane, respectively) and overestimations at high pressures (up to 140% and 20%, respectively). The error from Scenario #2 was minimal at the permeate flux that resulted in a CP modulus equal to the assumed 1.2 value (≈ 40 L.m⁻².h⁻¹, see Figs. S12, S13); this permeate flux corresponded to a relatively low pressure (≈ 12 bar) for the high-flux membrane and a relatively high pressure (≈ 28 bar) for the low-flux membrane. Because CP increases with pressure for the low-flux membrane at a lesser rate than for the high-flux membrane (Fig. 2), the assumption of β = 1 from Scenario #1 was relatively accurate at several pressures (i.e., generally ≤12 bar) for the low-flux membrane but only at the lowest pressure (4 bar) for the high-flux membrane.

Scenario #3 uses the actual CP modulus (Eq. (17)), so the error in solute permeance for Scenario #3 stems solely from assuming R ≈ 1 in Eq. (5) to enable approximating β to e^(J_v/k) (Table 1). Accordingly, Scenario #3 resulted in lower error for the low-flux membrane (0.01–2.6%) than for the high-flux membrane (0.1–26%) because the solute rejections of the low-flux membrane were higher (i.e., closer to 1) than those of the high-flux membrane. Similarly, relative differences in solute rejection were an excellent predictor of relative differences in Scenario #3 error (Fig. 6b), where higher solute rejections resulted in lower Scenario #3 errors. For example, the error for NaCl (R > 97.9%) ranged from 0.01–0.49%, while the error for boron (R = 35–90%) ranged from 0.96–25.9%. It is important to note that higher permeate fluxes (pressures) resulted in greater error because while rejection is greater at greater permeate flux, the approximation e^(J_v/k) ≈ β becomes less accurate. Even though membrane type, solute, and pressure played a role in the resulting solute permeance error because all of them affect rejection, it is notable that the Scenario #3 error was always less than 5%, except for the case that combined the high-flux membrane and the lowest rejected solute (i.e., boron). Therefore, the error incurred by approximating e^(J_v/k) as β in Eq. (17) was generally relatively small unless a confluence of relatively high flux and low rejection occurred.

Fig. 6.

(a) Scenario #1 percent errors of solute permeance, (b) Scenario #3 percent errors of solute permeance, (c) difference between percent errors of solute permeance of Scenarios #1 and #3, and (d) rejection of NaCl, As(V), As(III), and boron by the high-flux membrane (ESPA3). The inset in panel (c) shows the actual CP moduli for each solute with the high-flux membrane. Corresponding plots for the low-flux membrane (SWC4+) are in Fig. S14. Values in (a), (b), and the inset in (c) correspond to Figs. 2, 5. The dashed lines in the inset in (c) and panel (d) mark β = 1.2 and 100% rejection, respectively. Markers represent the average of duplicate filtration experiments; error bars in (a), (b), and (d) are the standard deviation, and in (c) are propagated from (a) and (b).

4.2.3. Ranked contribution of simplifying assumptions to the solute permeance error

As discussed above, one or both of the following two assumptions contributed to the error in solute permeance in Scenarios #1–3: (1) the CP assumption (i.e., how close the assumed modulus is to the actual modulus), and (2) the e^J_v/k ≈ β approximation. To evaluate the quantitative, ranked contribution of these two assumptions to the solute permeance error, we performed a detailed evaluation of Figs. 6 and S14 in Section S7. Overall, examining the individual contributions to the overall error by the CP assumption (Fig. 6c) and the e^J_v/k ≈β approximation (Fig. 6b) showed that mis-accounting for CP contributes most of the error, while the e^J_v/k ≈ β approximation generally contributes minimally to the overall error.

4.3. Trends and errors in water-solute selectivity

The reference water-solute selectivity generally varied with permeate flux or pressure (Fig. 7). Because the reference water permeance was a single value across all permeate fluxes or pressures (Fig. 2) and selectivity is the quotient of water and solute permeance (Eq. 21), the observed trend in the reference selectivity is due to the solute permeance trends (Fig. 3). More specifically, for all solutes except As(III), the reference water-solute selectivity decreased as a function of permeate flux because the reference solute permeance increased with permeate flux (Fig. 4). For the case of As(III), the reference selectivity increased as a function of permeate flux for the low-flux membrane because the reference As(III) permeance decreased, and the reference selectivity was relatively constant as a function of permeate flux for the high-flux membrane because of the correspondingly constant As(III) permeance. It is important to note that for any given solute-membrane pair, the variation in reference selectivity across the range of permeate fluxes (pressures) tested was substantial: 107% (NaCl), 162% (As(V)), 9% (As(III)), and 53% (boron) for the high-pressure membrane, and 89% (NaCl), 160% (As(V)), 112% (As(III)), and 29% (boron) for the low-pressure membrane. Therefore, when water and solute permeance are defined based on the SD model (Eq. (6), (15)), the reference water-solute selectivity for any given solute-membrane pair depends on the applied pressure, even in the absence of assumptions about CP or the e^J_v/k ≈ β approximation.

Fig. 7.

Water-solute selectivity (A/B) for (a) NaCl, (b) As(III), (c) As(V), and (d) boron generally decreased with increasing permeate flux for both high-flux (ESPA3, purple) and low-flux (SWC4+, pink) membranes. The filled-in squares are reference selectivity, empty markers are Scenario #1, dotted markers are Scenario #2, and hatched markers are Scenario #3. Symbols represent the average of duplicate filtration experiments; error bars are the standard deviation. Corresponding plots of selectivity as a function of pressure are in Fig. S15.

We also calculated the water-solute selectivity for each scenario and compared it with the reference selectivity (Fig. 7). Similar to the reference selectivity, the Scenarios #1–3 selectivities varied with permeate flux (pressure) because of the corresponding trends in water and solute permeance. Specifically, for all scenarios, the selectivities for water over solutes generally decreased with increasing permeate flux because of the increasing trend in corresponding solute permeances (Figs. 3, S9). For any given solute-membrane pair, the variation in selectivity across the range of permeate fluxes (pressure) tested was generally greater for each scenario than for the reference selectivity (i.e., maximum of 166% (NaCl), 191% (As(V)), 123% (As(III)), and 148% (boron) for the high-pressure membrane, and 141% (NaCl), 183% (As(V)), 96% (As(III)), and 103% (boron) for the low-pressure membrane). Therefore, water-solute selectivities calculated using the approaches commonly used in the literature (i.e., Scenarios #1 or #2) depend on permeate flux (pressure), with greater permeate fluxes (pressures) generally resulting in substantially lower selectivities for a given solute-membrane pair.

Regarding the errors in Scenario #1–3 selectivities relative to reference selectivities (Figs. 8, S16-S18), results show that all three scenarios overestimated selectivity (i.e., maximum of 22%, 111%, and 28% error for Scenarios #1, #2 and #3, respectively) at the lowest pressure (4 bar) (Fig. 7). These overestimations result from the corresponding substantial overestimations in water permeance (Figs. 1, S7). At all other pressures (i.e., ΔP > 4 bar), Scenario #1 generally underestimated selectivity (i.e., maximum of 66% and 31% for the high-flux and low-flux membrane, respectively) because it overestimated solute permeance (Figs. 3, 4) and Scenario #3 was closest to the reference selectivity (i.e., 2 to −19% and 3 to −6% error for the high-flux and low-flux membrane, respectively) because it had the closest estimate of solute permeance (Figs. 3, 4). Consistent with its solute permeance errors (Figs. 3, 4), Scenario #2 overestimated, closely matched, and underestimated (i.e., maximum of 92% and 111% error for the high-flux and low-flux membrane, respectively) water-solute selectivity at relatively low, intermediate, and relatively high pressures, respectively.

Fig. 8.

Percent error of Scenarios #1–3 relative to the reference water-solute selectivity as a function of applied pressure for (a and c) NaCl and (b and d) boron for (a and b) a high-flux (ESPA3, diamonds) and (c and d) low-flux (SWC4+, triangles) membrane. Symbols represent the percent error of average selectivity for duplicate filtration experiments; error bars are the standard deviation. Corresponding plots for As(V) and As(III) are in Fig. S16. Plots of percent error of A/B as a function of permeate flux are in Figs. S17-S18.

Considering the results together, when calculating selectivity, neglecting CP always results in underestimation, assuming a certain extent of CP can be accurate (within 5%) for intermediate to high pressures for a low-flux membrane but only at a very narrow range of pressures for a high-flux membrane, and accounting for CP through the determination of the mass transfer coefficient is the most accurate (within 5%, except at the lowest pressure of 4 bar) for both membrane types.

4.4. Errors in water permeability (A), solute permeability (B), and water-solute selectivity (A/B) at an operating pressure typical in the literature

To better understand the errors of transport properties reported in the literature, we tabulated the errors associated with neglecting CP and the e^J_v/k ≈ β approximation (i.e., Scenarios #1–3) for NaCl at an operating pressure typical of the literature: 16 bar (Table 2) [19]. All scenarios were more accurate for the low-flux membrane than the high-flux membrane. For all three transport properties at 16 bar, neglecting CP was less accurate than assuming β = 1.2, and accounting for CP through the mass transfer coefficient was the most accurate. The error contribution to the calculated water-NaCl selectivity by the e^J_v/k ≈ β approximation alone (i.e., Scenario #3) was < 2% for both membranes. By contrast, the corresponding error contributions from neglecting CP (i.e., Scenario #1 minus Scenario #3) were 30% and 11% for the high- and low-flux membranes, respectively. Therefore, most error at 16 bar is contributed by the CP assumption. Selectivity results in the literature that follow the assumptions of Scenarios #1 or #2 (both of which are common) are likely to have substantial errors compared with the values that would have been obtained if no CP nor e^J_v/k ≈ β approximation had been made.

Table 2.

Errors of water permeance, NaCl permeance, and water-NaCl selectivity for Scenarios #1–3 for the high- (low-) flux membranes at 16 bar, typical operating pressure [19].

Scenario	#1 (β = 1)	#2 (β = 1.2)	#3 (β = f(k))
Water permeance (A)	−7.5% (−2.9%)	−3.9% (0.9%)	−1.5% (−0.9%)
Solute permeance (B)	34% (11%)	12% (−7.9%)	0.2% (0.02%)
Water-solute selectivity (A/B)	−32% (−12%)	−15% (9.5%)	−1.7% (−1.0%)

4.5. Performance assessment via traditional selectivity plots

We made traditional selectivity plots for the reference and Scenario #1–2 performance values for unmodified and modified membranes (Fig. 9) to help understand how the different calculations might affect conclusions drawn about perceived membrane performance and performance changes after membrane modification. We do not include Scenario #3 in the discussion because it is not a scenario used in the literature to assess performance. We present the selectivity plots for NaCl and As(III) at a representative low pressure (4 bar, Fig. 9a, b) and a representative high pressure (33 bar, Fig. 9c, d). The selectivity plots for As(V) and boron (Fig. S19) follow the same trends discussed here. In this discussion, “better performance” refers to both water permeance and water-solute selectivity being higher than the referenced condition, or one of them being higher and the other one the same. Similarly, “worse performance” refers to water permeance and selectivity being lower than the referenced condition, or one of them being lower and the other one the same. We will describe explicitly situations in which water permeance or selectivity was higher and the other one lower.

Fig. 9.

Water-solute selectivity (A/B) as a function of water permeance for (a) NaCl at 4.14 bar, (b) As(III) at 4.14 bar, (c) NaCl at 33 bar, and (d) As(III) at 33 bar for a high-flux (ESPA3) and low-flux (SWC4+) membrane. The insets are the trade-off trends of a modified membrane relative to an unmodified membrane for the reference performance and when neglecting concentration polarization. Symbols represent the average of duplicate filtration experiments; error bars are the standard deviation. The corresponding plots for As(V) and boron are in Fig. S19 and follow the same trends of NaCl and As(III).

4.5.1. Performance of unmodified membranes

Comparing results at the two pressures, the trends for selectivity and water permeance follow those previously discussed. In most cases, the difference relative to the reference performance was substantial for both scenarios, such as at low pressure for both membranes and high pressure for the high-flux membrane (Fig. 9, S19). Relative to the reference membrane performance, Scenarios #1–2 showed better performance at the low pressure and worse performance (i.e., lower selectivity with similar water permeance) at the high pressure. Scenario #2 always showed better performance than Scenario #1, including substantially better at low pressure. Therefore, when using simplifying assumptions about CP and e^J_v/k ≈ β, perceived performance is better at low pressure and worse at high pressure than the reference performance, with the difference being substantial in many cases. Further, assuming a certain extent of CP (as opposed to just neglecting CP) exacerbated the difference with the reference performance at low pressure but decreased it at high pressure.

4.5.2. Performance of modified membranes relative to unmodified membranes

We used the performance results for modified membranes (Fig. 9) to identify qualitative trends of performance change upon membrane modification (i.e., trade-off trends). We obtained the trade-off trends by drawing an arrow from the unmodified membrane marker (pink or purple) to the modified membrane marker (teal or green, respectively) Fig. 9 (see insets). Results helped understand (1) the effect of applied pressure on the reference trade-off trend and (2) the effect of the CP assumption and the e^J_v/k ≈ β approximation on the observed trade-off trends.

Regarding the effect of applied pressure on the reference trade-off trend, we observed similar trends (i.e., the general direction of the arrows in the insets of Fig. 9) at both the low and high pressures for NaCl, As(V), and boron for the low-flux membrane and for NaCl and As(III) for the high-flux membrane. By contrast, for the high-flux membrane, the decline in modified membrane performance observed for As(V) at the low pressure was substantially steeper than that observed at the high pressure. Similarly, also for the high-flux membrane, while a decline in performance was observed for boron at the low pressure, an improvement in performance was observed at the high pressure. Therefore, while pressure does not appear to have a major effect on the reference trade-off trend observed for the commonly used test solute NaCl, it may substantially affect other solutes, leading to disparate trends at different pressures (e.g., the lesser rejected boron).

Regarding the effect of the CP and the e^J_v/k ≈ β approximation on the observed trade-off trends, Scenarios #1–2 sometimes represented the reference trade-off trends well and other times did not; applied pressure played a role in the outcome. For both membrane types, the trends of the reference performance and Scenarios #1–2 were similar to each other at the high pressure for all solutes (Fig. 9c-d and S19c-d). By contrast, at the low pressure, Scenarios #1–2 showed a steeper decline in performance or less steep incline (Figs. 9a-b and S19a-b) compared with the reference performance. It is important to remember that performance plots such as Fig. 9 are on a logarithmic scale, so differences in trends are larger than they appear.

Overall, in addition to the important role pressure had for all solutes in the reference membrane performance—as defined by the membrane position in the selectivity-permeance plot—pressure also affected the performance observed in modified membranes relative to unmodified counterparts (i.e., trade-off trends) for solutes other than NaCl for the high-flux membrane. Further, pressure had a role in the differences observed between reference trade-off trends and those represented by Scenarios #1–2, where low pressure resulted in Scenarios #1–2 misrepresenting the reference trade-off trends.

4.6. New experimental design for more accurate performance evaluation

We showed in previous sections that the CP assumption and the e^J_v/k ≈ β approximation typically used in the literature (i.e., Scenarios #1–2) to calculate representative values of membrane performance (i.e., water-solute selectivity and water permeance) result in errors with reference to the values that would be calculated in the absence of such assumptions. Further, results also showed that water-solute selectivity values depend on the applied pressure of permeation tests. To address these drawbacks of current membrane characterization practice—while allowing for the use of simplifying assumptions—we propose an experimental design that enables obtaining values for these performance metrics that are both true to the membrane performance and comparable across studies, when resources are limited, and a rigorous evaluation of CP is not possible or warranted. The design is as follows:

1. Obtain water permeance (A) from data at a high pressure using Scenario #1 (i.e., β = 1)

2. Obtain solute permeance (B) from data at a low pressure using Scenario #1 (i.e., β = 1)

3. Obtain water-solute selectivity (A/B) from the A and B obtained in Steps 1 and 2.

The rationale for the above procedure is as follows: (i) the reference water permeance is a robust and reproducible metric representative of water productivity across all pressures, and the Scenario #1 water permeance closely matches this value when measured at high pressure (Figs. 1, S7); and (ii) the reference solute permeance increases with pressure, and the Scenario #1 solute permeance at low pressure closely matches the reference solute permeance at low pressure (Figs. 3, 5). An additional consideration of choosing a low pressure to collect filtration data for the calculation of solute permeance is the lesser contribution of any potential advective transport to overall transport, which is not accounted for explicitly in the SD model. Therefore, the corresponding water-solute selectivity will match the reference water-solute selectivity at low pressure, and the resulting performance data point in the trade-off plot (A; A/B) will match the reference performance point at low pressure (Fig. 9). We chose to use Scenario #1 for these calculations because it does not require assuming a certain extent of CP, it is similarly accurate to Scenario #2 at high pressure for calculating water permeance, and it is more accurate than Scenario #2 at low pressures.

Performance values calculated with our proposed two-pressure approach closely matched the reference performance at low pressure (Fig. 9). NaCl selectivity of an unmodified membrane was underestimated by 10% and 2% for the high- and low-flux membranes, respectively, compared with the reference selectivity at 4 bar. The two-pressure scenario was similarly accurate for the other solutes: −11% (−2.4%), −12% (−2.9%), and −11% (−2.4%) for As(III), As(V), and boron, respectively, for the high- (low-) flux membranes. These errors are substantially less than those calculated at a typical operating pressure in the literature (16 bar, see Table 2): −32% and −12% for the high- and low-flux membrane, respectively, for NaCl when CP is neglected (β = 1). The errors obtained with the two-pressure approach were also substantially lower than those obtained with Scenarios #1–2 at 4 bar (Fig. 9) in all cases (i.e., unmodified and modified membranes, all solutes), except for Scenario 1 with the modified high-pressure membrane which had a similar error, albeit an overestimation instead of an underestimation. Additionally, the trade-off trends (i.e., comparing the performance of a modified membrane with an unmodified counterpart) of the two-pressure approach exactly matched the reference trade-off trends for all conditions.

Typically, in the literature, researchers perform cross-flow experiments at one pressure (e.g., 16 bar) unless there is a need for the mass transfer coefficient, for which several pressures (5–10) are required to achieve an appropriate number of degrees of freedom in the parameter fitting of Eq. (12). Given the results discussed here, there is an opportunity to neglect CP while achieving more accurate performance results than achievable with a one-pressure experiment and requiring fewer resources than when rigorously accounting for CP. Performing permeation tests and calculations at two pressures mitigates the overestimation of water and solute permeance at low and high pressures, respectively (i.e., underestimation of selectivity), and results in an accurate estimation of membrane performance as given by its positioning in a traditional selectivity plot (Fig. 9). Employing this approach would enable better comparisons of membranes across the literature.

5. Conclusions

We investigated the error in water permeance (A), solute permeance (B), and water-solute selectivity (A/B) values associated with common simplifying assumptions used in their calculations: neglecting or assuming a certain extent of concentration polarization (CP), and assuming a solute rejection of 100% to enable approximating the CP modulus to e^(J_v/k). The error was defined in reference to corresponding results obtained in the absence of simplifying assumptions. We quantified this error as a function of pressure for high- and low-flux membranes and four solutes of varied physical properties commonly studied in RO literature. Our analysis led to the following conclusions:

• Even when no simplifying assumptions are made, solute permeance (B) and water-solute selectivity (A/B) are pressure-dependent.

• Assumptions regarding CP had a substantially greater contribution to the error than the e^J_v/k ≈ β approximation.

• Factors that affected the accuracy of either the CP assumptions or the e^J_v/k ≈ β approximation affected the observed error in calculated A, B, and/or A/B: pressure, permeate flux, assumed extent of CP, solute identity, and membrane type.

• Simplified transport properties of the low-flux membrane were more accurate than those for the high-flux membrane due to its overall low permeate flux and correspondingly lower extent of CP and higher solute rejection.

• Errors for water permeance were much lower than errors for solute permeance; thus, errors for water-solute selectivity were heavily affected by solute permeance error.

• Errors for water permeance were highest at the lowest pressure (4 bar) and moderate or low at all other pressures.

• In general, errors for solute permeance were highest with (i) high permeate fluxes, (ii) high pressures, (iii) high actual CP moduli, and (iv) lower solute rejection.

• Errors for simplified selectivity values (A/B) were greater at higher pressures.

• Neglecting CP (β = 1) yielded better perceived membrane performance in traditional selectivity plots (A/B vs. A) at low pressure and worse perceived performance at high pressure.

• Performance trade-off trends (i.e., the performance of a modified membrane relative to its unmodified counterpart) are pressure-dependent, and using simplifying assumptions can lead to opposing trends at high and low pressures, where the trend at the low pressure is more accurate than that at the high pressure.

• A proposed two-pressure approach that measures solute permeance at a low pressure (e.g., 4 bar) and water permeance at a high pressure (e.g., ≥ 33 bar), and uses common simplifying assumptions in the calculations of A, B, and A/B, resulted in more accurate selectivity values and trade-off trends for use in a selectivity plot than single-pressure values.

Our work will help membrane scientists quantify the error resulting from simplifying assumptions used to calculate membrane transport parameters and proposes a solution for obtaining more accurate performance results. Using two pressures to calculate water-solute selectivity along with convenient simplifying assumptions is a good compromise between the resource-intensive approach of evaluating performance at multiple pressures to accurately quantify CP and the traditional approach of using one pressure, which we showed results in inaccurate values.