Elsevier

Desalination

Volume 501, 1 April 2021, 114905
Desalination

Effect of pressure and temperature on solvent transport across nanofiltration and reverse osmosis membranes: An activity-derived transport model

https://doi.org/10.1016/j.desal.2020.114905Get rights and content

Highlights

  • The activity model accounting for the compressibility and vapor pressure was applied to solvent/solute (water, alcohols, and hydrocarbons) transport in NF/RO/OSN membranes.

  • This activity derived model provides a realistic framework to quantify the effects of the applied pressure and temperature in extensive pure solvent flux measurements.

  • The physical chemistry of the solvents was separated from the membrane specific effects.

  • Effective molar volumes of the solutes (Vm) in NF/RO/OSN membranes were estimated for a number of pure solvents.

  • The estimated energy of activation (Ea) for diffusion of water was identified within a polyamide membrane.

Abstract

The activity coefficients and equations for transport through nanofiltration (NF) and reverse osmosis (RO) membranes were derived to determine the contribution of the heat of vaporization, molar volume, activation energy (diffusion), thermal expansion, and liquid- and vapor-phase compression. These equations have been derived for pure substances; hence, the measured deviations were obtained from the membrane itself. The inclusion of these effects provides a more accurate prediction of membrane transport than that provided by current models. The model presented herein accurately determined fluxes over temperature and pressure ranges for four different membranes, which the Merten model failed to determine. The transport activity model was applied to water, ethanol, methanol, and n-hexane. The molar volume of water for commercial, brackish-water membranes was approximately equal to that of bulk water, while high-flux brackish-water and NF membranes exhibited significantly lower molar volumes, as compared to that of the bulk water. Surprisingly, slight changes in the molar volumes of water, ethanol, and n-hexane in organic NF membranes were observed. Significant changes in the physicochemical properties of water were identified in the Desal 5 membrane. These results indicate that the molar volume and compressibility enabled the pressure dependence of the RO and NF membranes. For all the membrane types, the resistance to flow, heat of vaporization, and diffusion coefficients controlled the magnitude of changes in flux due to pressure and temperature.

Introduction

Membrane transport models are important for understanding solute and solvent transport in pressure-driven, membrane-based processes, including reverse osmosis (RO), nanofiltration (NF), and organic solvent nanofiltration (OSN). Currently, these processes are extensively employed in desalination, purification and separation [[1], [2], [3], [4], [5], [6]]. A fundamental understanding of the transport mechanism is significant for determining membrane performance and designing novel, advanced membrane materials for various new applications and for addressing the increasing demands for separations.

Mass transport that governs any atomic or molecular transport across an interface in one direction is expressed in Eq. (1).Ji=DdCidxwhere Ji is the flux of component i, D is the coefficient of proportionality between the flux and the driving force, Ci is the concentration gradient of component i, and x is the thickness of the membrane.

The forward and backward diffusion rates across an interface are expressed in Eq. (2).Ji=DifCifDimCimwhere the subscript denotes the solute, and the superscript denotes the phase.

In the kinetic transport model, the static equilibrium partitioning across any membrane interface occurs when the activities of each solute is equal (aif = aim). Steady state equilibrium occurs when the chemical potential across the membrane interface is equal. This is achieved in both the Merten and activity models by including the chemical potential in further transport analysis. Activity is defined as the concentration multiplied by the activity coefficient (ai = miγi), and the analytical thermodynamic relationship between the activity and chemical potential modifies the activity. This ensures all the effects due to the chemical potential are included in the activity model, which results in the following general membrane flux equation (Eq. (3a)) [7].Ji=DφγimγifCifγipCip=BaγifCifγipCip

Here, Ba is the solute permeability, which considers the activity as well as concentration. This equation is useful for salts and any other solute or solvent. The activities of salts and solutes are pressure-, temperature-, and concentration-dependent. The advantage of using a transport equation such as Eq. (3a) is that the activity coefficient can be separated into several physical components, such as the non-linear effects due to vapor compressibility, liquid compressibility, and changes in molar volume. The relationship between activity and chemical potential correlates the solution diffusion approach with the activity model. The water-specific relationship is represented in Eq. (3b) [8].μw=μwo+RTlnawwhere μ is the chemical potential, T is temperature, and R is the gas constant.

The classical solution-diffusion (SD) theory was developed by Merten [9] and Lonsdale [10]. This model was initially employed to consider chemical potential transport and then to calculate the mass transfer. In this model, the overall driving force producing the transport of a component through a dense membrane is the chemical potential gradient, as described in Eq. (4).Ji=Lidμidxwhere Ji is the flux of component i, Li is the coefficient of proportionality between the flux and the driving force, μi is the chemical potential of component i, and x is the thickness of the membrane.

Various transport models have been reformulated and employed in practical applications over the years [[11], [12], [13]]. By assuming that the chemical potential is linear with the solute concentration, transport has been modified to derive the solvent (Eq. (5a)) and solute (Eq. (6)) transport equations.Jv=Apπ

Here, A is the solvent permeability coefficient, and Jv represents the solvent flux driven by the pressure gradient across the interface (p) and osmotic pressure (∆π). If A = AD D, where AD is the solvent permeability without the diffusion coefficient (D), the following equation (Eq. (5b)) is obtained:Jv=ADDpπJs=BCwhere Js is the solute flux driven by a concentration gradient (∆C) across the membrane, and B is the solute permeability coefficient.

There are several disadvantages involved in this simplified SD model [12]. According to Eqs. (5a), (5b), the flux is expected to increase linearly with the operating pressure without any limitation. There are several experimental observations that indicate non-linearities. The activity of water in salt solutions and the activity when mixed with several other solutes are non-linear [7]. The activity of water and the concentration dependence of the osmotic coefficient can be calculated with high accuracy in aqueous systems [[14], [15], [16]]. The osmotic coefficients and activity coefficients within the membrane are unknown and are expected to differ significantly from those of the bulk solution [17,18].

This discrepancy originates from simplified assumptions. The expansion of the chemical potential with pressure and other solute concentrations can be written in terms of activity and pressure as follows (Eq. (7)) [7,9,11,12]:dμi=RTlnai+p0pVmidp=RTlnγiCi+p0pVmidpwhere p is the pressure, and Ci, ai, γi, and Vmi are the concentration, activity, activity coefficient, and molar volume of component i, respectively.

It can be concluded from Eq. (7) that any change in the activity (including changes in the activity coefficient due to changes in concentration), molar volume, or pressure will alter the chemical potential. For desalination applications, where water is the most common solvent and salt solutions are effectively incompressible, the molar volume is considered to be relatively constant within the range of operating pressures; thus, this effect was considered to be negligible in cases where the range was sufficiently small to minimize non-linearities. These assumptions are not applicable in several cases, and this study uses the known non-linearities of salts and solutes to develop a superior model for the transport of water and other solutes.

The physicochemical properties of water within polyamides can vary significantly from the bulk properties, and there can be various states of water within a polyamide [17,18]. Each of these states exhibit different physical parameters. The molar volume of water in various polymers varies significantly, and in certain solutions, it can be zero or negative [19]. When water binds to a polymer, its packing becomes disrupted. The polymer composition, the volume excluded by the polymer, and specific interactions can significantly affect the effective molar volume and the heat of vaporization of water and other solutes within polymeric solutions. However, it is difficult to measure using experimental instruments.

Generally, the linear assumptions included in the SD model of RO or NF membrane operation might not be appropriate if the solvent or solute is compressible and exhibits non-linear volatility or significant thermal expansion. These non-linearities would cause the partial molar volume to change significantly with pressure or temperature [15]. Paul [20] reported that a water system with a molar volume of 18 mL mol−1 deviated from linearity by less than 5% until the pressure exceeded 138 bar. This amount of pressure is rarely observed, even in RO processes; therefore, non-linearity is not expected from this source. The same estimation of 5% deviation from a linear flux–pressure relationship was reported by Paul [20] for organic solvents as well, and the pressure required for an organic solvent with a molar volume of 100 mL mol−1 was reported to be 24.8 bar. In contrast, if the same organic solvent had a molar volume of 300 mL mol−1 the same compression would require only 8.3 bars of pressure. Organic NF often operates at higher pressures. In certain cases, the effect of changes in molar volume on the flux should be included. This report confirms that multiple, non-linear physical effects can combine to increase or decrease minor isolated effects.

Another discrepancy observed in the simplified SD model represented in Eqs. (5a), (5b), (6) is the large non-linearities of the solute and solvent activities. For decades, the activity coefficients of the feed solutions (usually salt solutions) have been regarded as a constant or unity to simplify the study of membrane filtration transport models. However, the non-ideal, thermodynamic properties of the solution cannot be neglected, especially for electrolytes that possess concentration-dependent activity coefficients and solutes with non-negligible, pressure- and temperature-dependent vapor pressures [14]. Moreover, the solute concentration varies as the partial molar volume changes via thermal expansion or isothermal compression under different pressures or temperatures. The changes in both the concentration and/or the activity coefficient contribute to the variation in the activity and result in changes in the chemical potential.

Therefore, the driving force of the chemical potential gradient originates from several parameters that are related to concentration, temperature, and pressure. These are physicochemical properties of the feed and permeate solutions. However, there are a few reports regarding the relative magnitudes of these effects on solvent and solute transport. Most models assume that coefficients A and B (Eqs. (5a), (5b), (6)) are completely membrane-specific properties. We report that these constants also possess the variable properties of the solutions being studied.

The ratio of two salt permeabilities (Ba/B, where Ba is the corrected activity and B is the Merten Model B) was used to determine the amount of Merten permeability due to the varying properties of the salt solution. The Ba/B of salts within the membranes with concentrations of 200 mM to 350 mM varied from 0.7 to approximately 2 [21]. Similar results were observed for different membranes and salts [22]. At higher salt concentrations, the Ba/B ratios will be substantially larger [7]. This result indicated that the previous models of salt transport in membranes inadvertently included a large portion of their coefficients from the physicochemical properties of electrolytes, not that of the membrane. Although this model was initially used for electrolytes, this model is also appropriate for solutes that involve temperature- and pressure-dependent vapor pressures.

There are numerous studies that explain how changes in membranes might result in solute specificity and non-linearities in transport. However, complex membrane transport models are beyond the scope of this study. Any non-linearities not attributed to physicochemical properties of the testing solutions could be due to polymer swelling and polymer deswelling, coupled transport within a membrane, polymer solute interaction, such as electrostatic interactions, or various non-linearities in the transport within complex polymer assemblies. In this study, we eliminated irreversible compaction by pre-compacting the membranes before the tests and confirmed that the flux is constant after retesting. This report describes a model that is complementary to the complexities of membrane transport, which demonstrates how non-linearities in the activities of water and other solvents across an interface can affect transport. This report uses a quantitative model of the activity and diffusion of solutions across an interface. Therefore, any deviations from these calculated effects are due to the membrane behavior and would require further modeling. A previous report briefly describes how the activity across a membrane can be managed.

To investigate the effects of the physicochemical properties of the solution on the solvent and solute transport, excluding the effects of membrane properties, the activity model established by Mickols [7] was employed for salt transport [21,22]. In this study, we utilized this model of solutes with measurable vapor pressures and compressibility, which enabled the distinction of different physical effects. This was performed by estimating the solute activity coefficients with significant vapor pressures. This includes the effects of pressure and/or temperature on the solvent molar volume and vapor pressure (e.g., thermal expansion, isothermal compression, and the heat of vaporization). First, each of the effects was modeled from the pure solvent, and the magnitude of these effects was evaluated in several membrane systems. The activity model [7,21,22] was compared with the Merten model [9], and the differences in aqueous and non-aqueous systems were elucidated. Subsequently, the coefficients of these effects from a pure solvent system were applied to the membrane system to analyze the impact of the physicochemical properties of the solute on its transport across the membrane.

New data related to the pure water permeability of three NF membranes were used in this study, and the different physical effects of pure water were investigated under different pressures and temperatures. Two published data sets of pure water permeability [21,23] were included. Furthermore, the activity model was also used with published transport data from three non-aqueous solvents (methanol, ethanol, and n-hexane) with different OSN membranes [24]. The schematic illustration and the thermodynamic calculations are presented in Fig. 1, Fig. 2, respectively. To the best of our knowledge, thus far, no study has focused on the transport mechanisms involved in the RO, NF, or OSN systems related to the bulk solution properties and those found in the membrane

Section snippets

Theory

Fig. 3 shows the stepwise calculations of the complete driving force of the activity model used in this study, which is applied to solutes that are compressible or involve non-negligible vapor pressures.

To include the effects of pressure, temperature, and concentration, as well as the presence of other solutes, a completely general representation of the activity, activity coefficient, and diffusion coefficient was developed. This model represents solutions that contain solutes, such as salts

Membranes

Three types of commercial NF membranes, Filmtec™ NF90, Filmtec™ NF270, and Fortilife XC-N, were kindly provided by Dow/Filmtec (Minneapolis, USA). The properties of the membranes used in this study and those of the six reference membranes are listed in Table 1.

According to the manufacturer, the three NF membranes (not including the reference membranes) used in this study are thin-film composite (TFC) membranes. The selective layer of NF90 consists of fully aromatic polyamide. NF270 possesses a

Comparing the activity coefficient of water in Eq. (18) to that in the Pitzer model

Fitting the flux equation to zero results in the activity of both vessels being equal (Eq. (28)). However, the concentrations are not necessarily equal if the activity coefficient compensates for the changes in concentration.γ1,wp1C1,w=γ2,wp2C2,w

Here, one side of the equation represents salt while the other represents pure water under pressure. The Pitzer theory of electrolytes as a function of temperature and concentration [33] was used to calculate the water activity and osmotic pressure for

Discussion

The activity model was expanded to include uncharged solutes that can be compressible and can have measurable vapor pressures. These include water, alcohols, hydrocarbons, and several other solutes. Deviations in the measured solute transport from the expected transport (using the activity transport model) were used to identify membrane-specific changes in the solutes within the membrane. The effects of changes in molar volume, compressibility, thermal expansion, heat of vaporization, and

Conclusions

The activity model of transport was expanded to include solutes that exhibit substantial vapor pressures and compressibility. This model was used to identify the magnitude of the effects of molar volume, isothermal compressibility, thermal expansion, diffusion coefficient, and the heat of vaporization across an interface. In the four membranes studied, the most variable coefficient vs membrane type was the resistance to flow. The largest effects governing flux (in the order of magnitude) were

CRediT authorship contribution statement

William Mickols, Conceptualization, Methodology, Formal Analysis, Original draft.

ZhaohuanMai, Investigation, Validation, Visualization, Writing Review and Editing.

Bart van der Bruggen, Resources, Supervision, Project Administration, Funding Acquisition, Writing Review and Editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 21567009) and the Science and Technology Department of Jiangxi Province (20192BAB213013). We want to thank Dow Water Solutions and Derik Stevens PhD. for kindly supplying flat sheet membrane samples for this study.

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