Transport Mechanisms in Membranes Used for Desalination Applications

Abstract

Membranes have been widely utilized for different applications in various fields such as desalination and separation processes. The membrane-based desalination technologies, accounting for over 65% of desalination plants, share the common objective of extracting freshwater from various feed sources including seawater, high total dissolved solids brine extracted from deep saline aquifers, brackish water, and industrial effluent. However, the transport mechanisms within the membranes of these aforementioned techniques are significantly different in nature. These differences arise from the distinct specifications of membranes used in each technique, as well as the driving forces applied in their respective processes. Therefore, this chapter focuses on analyzing the transport mechanisms involved in the membranes of three common membrane-based desalination processes: reverse osmosis (RO), membrane distillation (MD), and electrodialysis (ED). In addition to the transport mechanisms occurring within the membranes, this chapter examines the driving forces, the required specifications of the membranes, and the performance parameters and energy consumption associated with the processes mentioned above.

1. Introduction

Due to the growing demand for freshwater across various sectors such as drinking, agriculture, and industries arising from water scarcity occurring globally even in locations currently considered as water-rich regions, addressing these problems has become urgent, calling for a tremendous amount of research for identifying robust solutions [1, 2, 3]. Desalination of various water sources, such as seawater, brackish water, high total dissolved solid (TDS) brine extracted from deep saline aquifers, industrial effluent, etc., has emerged as a viable solution for freshwater production. These desalination technologies have witnessed significant advancements in recent decades. In the past decade, there has been a notable shift in focus toward energy-efficient desalination techniques that offer reduced costs and lower energy consumption, garnering increased interest when compared to conventional desalination technologies [4]. In general, conventional desalination methods can be broadly classified into two categories: membrane-based and thermal-based methods. Among the thermal methods, Multi-Stage Flash (MSF) and Multi-Effect Desalination (MED) have emerged as the most prevalent techniques for freshwater production from saltwater, utilizing the evaporation and condensation process. Thermal-based desalination technologies, being known as energy-intensive methods, are mostly common in Gulf regions [1, 2, 3, 5].

Membranes, both naturally and synthetically, act as porous mediums and hold significant importance in various fields, from the operations of living organisms to separation technologies. Naturally, the skin, lungs, and kidneys are acknowledged as crucial membranes within the human body [3]. Membrane-based desalination and separation techniques, both conventional and emerging, have gained widespread utilization globally, constituting a majority share of over 65% in desalination plants [6]. Various membrane-based desalination technologies have emerged, employing diverse driving forces, including pressure-driven, temperature-driven, and electrical-driven mechanisms. Reverse osmosis (RO) stands as the leading first-generation membrane-based desalination technology and is commonly utilized in desalination plants worldwide [7]. As a pressure-driven membrane-based technique, RO functions by applying an operating pressure that surpasses the summation of osmosis pressure and friction loss [8, 9]. In contrast to RO, membrane distillation (MD) has emerged as a prominent second-generation membrane-based desalination technology that utilizes the vapor pressure difference, primarily arising from temperature differences in streams, as the driving force for the desalination process [10]. The operational advantages of MD lie not only in its utilization of low-grade heat, which makes it feasible to integrate with renewable energy and waste heat sources but also in its lower sensitivity when effectively treating feed water with high concentrations [11, 12]. These factors contribute to the increased attractiveness of MD systems when compared to other desalination technologies. Electrodialysis (ED) is a practical method for desalination that utilizes an electric field to separate salt and other ions from water [13]. This electrical-driven technique employs a membrane stack comprising alternating anion/cation-exchange membranes, which selectively allow the passage of ions based on their charge.

While all the previously mentioned membrane-based desalination technologies share the common objective of extracting freshwater from the feed water, the transport scenarios within the membranes of these techniques exhibit significant differences. These differences arise from the distinct specifications of membranes used in each technique, as well as the driving forces applied in their respective processes. To develop the performance of the membrane-based desalination technologies, a multidisciplinary endeavor is imperative, centered on fabricating highly suitable membranes with improved specifications as well as optimizing the operating parameters and module geometries. Such a comprehensive approach is essential for fostering progress and innovation in the field of membrane-based technologies for diverse applications such as desalination.

In this regard and based on the results obtained by our research group in the past decade, this chapter comprehensively elucidates the transport mechanisms involved in the membranes employed in three prevalent and emerging membrane-based desalination processes, i.e., RO, MD, and electrical potential difference (ED). Furthermore, this chapter provides a comprehensive and in-depth overview of the aforementioned membrane-based desalination techniques. It delves into the historical background and development of these techniques, the basic technical concept of processes, the desired properties of the membranes utilized, and the key performance parameters associated with them.

2. Reverse osmosis (RO)

2.1 Introduction and historical background

The first generation of membrane-based separation processes, i.e., reverse osmosis (RO), nanofiltration (NF), ultrafiltration (UF), and microfiltration (MF), mainly relies on pressure-driven mechanisms [8]. MF membranes possess the largest pore size among other types, enabling them to effectively only reject large particles and various microorganisms [3, 14]. UF membranes have smaller pores compared to MF membranes. Consequently, UF membranes not only reject large particles and microorganisms but also demonstrate the capability to exclude bacteria and soluble macromolecules such as proteins [8]. NF membranes, characterized by porous structures, exhibit enhanced separation performance in comparison with UF membranes due to their small pore sizes being on the order of 1 nm. Lastly, RO membranes excel at excluding particles, as well as many low molar mass species such as salt ions and organics. The figure presented below illustrates the spectrum of nominal membrane pore sizes (Figure 1) [8].

Figure 1.

Range of nominal pore diameter for first-generation membrane-based separation processes.

There are two inherent distinctions between RO and other filtration processes, namely, MF, UF, and NF. First, filtration processes rely on a sieving mechanism for separation, wherein the membrane allows the passage of smaller particles while retaining larger ones [3]. Conversely, RO membranes selectively permit the permeation of the solvent solely, effectively retaining the solute. As the second distinction, MF, UF, and NF processes are employed for the separation of suspended materials, whereas RO is utilized specifically for the separation of dissolved solids. Indeed, RO entails the use of a selectively permeable (semipermeable) membrane with the capability to effectively eliminate all low-molecular-weight organic compounds as well as monovalent and multivalent ions through rejection. While MF, UF, and NF are highly efficient as pretreatment stages, the increasingly stringent standards for purified water necessitate the prominent utilization of RO in the desalination of feed water [15]. RO, which is currently employed in almost 60% of desalination plants worldwide, has a rich historical background that turned it into the most predominant and commercially established technique for freshwater production [3]. The conceptualization of RO originated in the early 1950s at the University of California, Los Angeles (UCLA), and was spearheaded by Professor Samuel Yuster [16]. Professor Yuster’s idea involved employing Gibbs’ adsorption equation as a framework to explore methods for generating freshwater from brackish and seawater sources. According to Gibb’s equation, a distinct layer of relatively pure water should exist at the interface of brine and air, or any other hydrophobic surface, thereby presenting an opportunity to extract freshwater [16]. The initial concept aimed to allow the formation of pure water at the solid-liquid interface and subsequently harvest this water by compelling it to flow through the pores of a solid material under high pressure. In 1958, a groundbreaking milestone was achieved with the successful implementation of a flat plastic film, supported by a porous plate, in an experimental setup at UCLA [16, 17, 18]. Coincidentally, during the same period, independent efforts were underway by Breton and Reid at the University of Florida, who achieved the first successful RO tests using cellulose acetate membranes for seawater [19]. The commercial viability of RO took a significant leap forward in 1958 when Loeb and Sourirajan developed the first asymmetric cellulose acetate membrane, thereby revolutionizing the field [20]. Following this achievement, a substantial body of research studies was performed, centering on the development of novel membrane materials and the comprehensive evaluation of their performance. These investigations have significantly contributed to the advancement of RO technology and its application in desalination. In the mid-1970s, the introduction of cellulose triacetate hollow fiber membranes by Dow Chemical Company marked a significant milestone in the field. This achievement resulted in a remarkable enhancement in the produced flux of RO systems. In the past five decades, RO commercially stands as the most favored and widely adopted membrane-based technology for seawater desalination, primarily owing to its notably lower energy consumption rate. Extensive research and studies have consistently demonstrated that RO processes typically necessitate an energy input ranging from 4 to 6${\mathit{kWh}}_{\mathit{\text{elec}}}/{\mathit{m}}^{\mathrm{3}}$ for the treatment of seawater, while it ranges from 1.5 to 2.5${\mathit{kWh}}_{\mathit{\text{elec}}}/{\mathit{m}}^{\mathrm{3}}$ for brackish water [12, 21, 22]. These findings highlight the energy efficiency and practicality of RO in addressing the pressing global need for seawater treatment and freshwater production. Although RO technology is known for its lower energy consumption compared to other desalination methods, it is not without its challenges. Treating high TDS feed water using RO can still encounter significant issues, particularly related to scaling and fouling. These concerns pose important considerations in the successful application of RO for such brine treatment processes.

2.2 RO basic concept and process

Figure 2 illustrates a schematic representation of the osmosis and RO phenomena. As illustrated in Figure 2, the direction of solvent flow is determined by its chemical potential, which depends on factors such as pressure, temperature, and the concentration of dissolved solids. When pure water is in contact with an ideal semipermeable membrane on both sides, under equal pressure and temperature conditions, there is no net flux of water across the membrane because the chemical potential is balanced on both sides. However, when a soluble salt is introduced on one side, the presence of the salt solution reduces the chemical potential of the water. As a result, osmotic flow occurs from the pure water side to the salt solution side (Figure 2(a)), with the aim of restoring the equilibrium of chemical potential [3, 8, 9].

Figure 2.

Schematic representation of: (a) osmosis phenomena and (b) RO process.

It is worth noting that osmotic pressure is a property of the solution itself and remains independent of the membrane properties. Alternatively, the establishment of equilibrium can also be achieved by applying external pressure, which is equal to the osmotic pressure, to the concentrated feed side. Any additional pressure beyond this point will elevate the chemical potential of the concentrated feed, prompting solvent flow toward the pure water side due to the lower chemical potential it possesses. In fact, RO utilizes the semipermeable membrane to effectively eliminate all low-molecular-weight organic compounds as well as monovalent and multivalent ions. Overall, understanding these principles is crucial for comprehending the intricacies of osmosis and RO processes.

2.3 RO membranes

2.3.1 Membrane materials

The synthetic organic polymers are the main materials used for the fabrication of RO membranes. Generally, the polymers used in RO membranes are either cellulose acetate (CA) or polysulfone coated with aromatic polyamides [8, 9, 14]. Cellulose-based membranes initially held the dominant position in the market for RO membranes. These membranes, which often consist of a rather dense layer of polymer on the surface of an increasingly porous layer, exhibit homogeneity in terms of chemical composition but not in their structural arrangement. CA membranes are crafted from acetylated cellulose, a naturally occurring polymer abundant in plants like cotton. The degree of acetylation signifies the extent to which acetyl groups (CH3COO) replace the hydroxyl (OH) groups dangling from the cellulose chain, in the realm of commercial CA membranes used for RO, a typical degree of acetylation hovers around 2.7. CA membranes have several advantages that distinguish them from other RO membranes currently available. They are characterized by their relatively straightforward manufacturing process and exceptional mechanical properties. Additionally, CA membranes exhibit notable resilience against chlorine-induced deterioration. These membranes can withstand chlorine concentrations of up to 5 ppm, a remarkable threshold that surpasses the tolerance levels of other membrane types, such as those based on aromatic polyamides. Based on aromatic polyamides, thin film composite (TFC) membranes have higher flux and salt rejection in comparison with CA membranes. Most TFC membranes are made with a porous, highly permeable support such as polysulfone, which is coated with a cross-linked aromatic polyamide thin film. The coating provides the salt rejection properties of the membrane. TFC membranes have chemical and structural heterogeneity. In addition to high rejection and high flux, TFC membranes offer other advantages over CA membranes. For one, they can also reject some low-molecular-weight organics. They are also stable over a larger pH range and at higher temperatures than CA. However, one drawback of TFC membranes is their sensitivity to chlorine. TFC membranes are highly susceptible to attack by chlorine [8].

In addition to organic materials, membranes can also be fabricated using inorganic materials like ceramics or metals. Ceramic membranes, known for their microporous structure, exhibit excellent thermal stability and chemical resistance. However, their widespread adoption has been hindered due to certain drawbacks, including high cost and mechanical fragility, which limit their extensive use [8].

2.3.2 Membrane fabrication methods

In contrast to isotropic membranes, which are typically employed in MF and UF processes and have pores ranging between 50 and 150 nm, RO membranes are commonly anisotropic. Anisotropic membranes are mostly fabricated using phase separation and thin film composite (TPC) methods [8]. CA membranes are generally fabricated by casting a film from a solution of polymer and solvent and immersing the cast film in a nonsolvent for the polymer. TFC membranes are typically composed of a porous substrate that is coated with a thin, impermeable film made from a different polymer. The fabrication of these membranes can be achieved through various techniques, including interfacial polymerization (IP) as the most widely used approach, as well as techniques such as solution coating, plasma polymerization, and surface treatment. In general, the polymerization reaction occurs at the interface of two immiscible liquid phases. To create an extremely thin active layer of polyamide (PA) on top of a supporting membrane, a typical approach involves immersing the substrate into an aqueous solution containing amine monomers (usually ranging from 0.1 to 1% w/v). Subsequently, the substrate is immersed in a second organic solution containing acyl chloride monomers (typically ranging from 0.05 to 0.2% w/v). To enhance the polymerization properties of the PA layer and improve its adhesion to the support membrane surface, the membrane is subjected to heat treatment at temperatures between 70 and 90°C. The IP technique offers significant advantages in independently optimizing the characteristics of the skin layer and the microporous substrate layer.

2.4 Transport mechanisms through RO membranes

To conduct a comprehensive analysis of the transport mechanisms involved in the desalination/separation process, it is crucial to possess a thorough understanding of all potential scenarios of mass transfer that can take place across membranes. It is worth noting that membrane fouling significantly impacts the transport resistances in desalination/separation processes. However, this section investigates the mass transfer occurring through membranes in the absence of fouling, aiming to gain a better understanding of the transport mechanisms involved. There are typically two scenarios that account for transport across the membranes in desalination/separation processes: diffusion through the membrane and pressure-driven convection through pores. These scenarios can be mathematically described by the following equation [8, 9].

$${\stackrel{\rightharpoonup }{\mathit{J}}}_{\mathit{i}}={\rho }_{\mathit{i}}\stackrel{\rightharpoonup }{\mathit{v}}-{\mathit{D}}_{\mathit{im}}\frac{\stackrel{\rightharpoonup }{\nabla }{\rho }_{\mathit{i}}}{\mathit{l}}$$E1

where J_i [kg/(m².s)], ${\mathit{\rho }}_{\mathit{i}}$[kg/m³], v [m/s], D_im [m²/s], l [m], and $\mathrm{\nabla }{\mathit{\rho }}_{\mathit{i}}$ [kg/m³] are the mass flux of component i through the membrane, density of component i, velocity of solution, effective diffusion coefficient of component i in the membrane, characteristic length, and density gradient, respectively. In cases where pore flow plays a substantial role in flux, it is common to employ Darcy’s Law as a means to calculate the mass average velocity [8, 23].

$$\stackrel{\rightharpoonup }{\mathit{v}}=-\frac{\mathit{k}}{\mu \mathit{l}}\left(\stackrel{\rightharpoonup }{\nabla }\mathit{p}-\rho \stackrel{\rightharpoonup }{\mathit{g}}\right)$$E2

here, k [m²], $\mathit{\mu }$ [Pa.s], p [Pa], $\mathit{\rho }$ [kg/m³], and g [m/s²] refer to the Darcy permeability constant, solution viscosity, pressure, solution density, and gravity, respectively. By incorporating Eq. (2) into Eq. (1), while considering transport solely in the z-direction (which is typically perpendicular to the membrane surface) and disregarding the effects of gravity, we obtain the following expression:

$${\mathit{J}}_{\mathit{i},\mathit{z}}=\frac{{\rho }_{\mathit{i}}\mathit{k}}{\mu }\frac{\mathit{dp}}{\mathit{dz}}-{\mathit{D}}_{\mathit{im}}\frac{\mathit{d}{\rho }_{\mathit{i}}}{\mathit{dz}}$$E3

In Eq. (3), the first term illustrates the mass flux resulting from pressure-driven convection through pores, while the second term represents the flux arising from diffusion. Due to the nonporous nature of RO membranes, the transport of molecules across the membrane is primarily governed by diffusion. Consequently, the flux across the membrane is predominantly regulated by the second term of Eq. (3). The second term, known as Fick’s first law, establishes a relationship between diffusive flux and mass density corresponding to the concentration, assuming a state of equilibrium. According to this law, flux occurs from regions of high concentration to regions of low concentration, and its magnitude is directly proportional to the concentration gradient. In this mechanism, water molecules are absorbed into the upstream face of the membrane, diffuse across the membrane following the chemical potential gradient, and then desorb from the downstream face of the membrane. This mechanism of mass transport across membranes is often known as the “solution-diffusion” model. Moving toward a broader perspective that considers mass transport driven by chemical potential gradients instead of solely concentration gradients, the solution-diffusion transport equation for RO can be mathematically expressed as follows [23, 24]:

$${\mathit{J}}_{\mathit{w}}={\mathit{c}}_{\mathit{m}}\left(\mathrm{\Delta }\mathit{p}-\mathrm{\Delta }\pi \right)$$E4

Where J_w [kg/(m².s)], c_m [kg/(Pa.m².s)], $\mathrm{∆}\mathit{p}$[Pa], and $\mathrm{∆}\mathit{\pi }$ [Pa] are water flux through the membrane, membrane coefficient, the hydrostatic operating pressure difference between two sides of membrane, and the osmotic pressure difference between two solutions on both sides of membrane, respectively. A comprehensive introduction to osmotic pressure will be provided in the upcoming section. While several empirical models, such as the Baker and Wijmans model, Paul model, and others, exist to calculate the membrane coefficient, it is commonly accepted to employ the following equation as a means to describe transport in nonporous membranes [24].

$${\mathit{c}}_{\mathit{m}}=\frac{{\mathit{D}}_{\mathit{w}}{\mathit{S}}_{\mathit{w}}{\mathit{V}}_{\mathit{m}}}{\mathit{RTl}}$$E5

in which, D_w [m²/s], R [J/(mol.K)], S_w [kg/m³], V_m [m³/mol], and T [K] refer to the diffusion coefficient of water in the dense membrane, universal gas constant, water solubility in the membrane, molar volume of water, and temperature, respectively. As already mentioned, l [m] also exhibits the membrane characteristic length, which is equal to the membrane thickness.

It is important to highlight that the underlying mechanism in membrane-based filtration processes, such as MF and UF, differs significantly. As previously mentioned, the fundamental mechanism driving filtration processes like MF and UF is sieving, where molecules larger than the size of the membrane’s pores are effectively rejected. In these membrane-based filtration methods having porous membranes, the contribution of diffusion through the pores is generally considered insignificant compared to convection. When considering the convective as the dominant term in Eq. (3) for transport across UF and MF membranes, the Darcy permeability depends on various membrane structure factors, such as porosity and tortuosity, often in a complex manner.

2.5 Performance parameters of RO process

The evaluation of fabricated RO membranes and module configurations is conducted through the assessment of their performance and operating parameters. The performance of the RO process is characterized by several key variables, which are defined in this section.

• Osmotic and operating pressure

The operating pressure, along with the osmotic pressure, plays a crucial role in determining the efficiency and effectiveness of the RO process. These pressures directly determine the driving force for water permeation through the membrane and, as a result, influence the overall separation performance. The osmotic pressure of a feed solution can be determined through experimental measurements of the concentration of dissolved salts in the solution, while keeping the temperature and pressure constant. The following equation is commonly used to quantify the osmotic pressure of a solution [3, 8]:

$$\pi =\mathit{RT}\sum {\mathit{c}}_{\mathit{i}}$$E6

where $\mathit{\pi }$ [Pa], R [J/(mol.K)], T [K], and c_i [mol/m³] are osmotic pressure, universal gas constant, temperature, and concentration of each composition in feed water, respectively. To overcome their adverse impacts, the operating pressure in RO is adjusted to be higher than the sum of osmotic pressure, friction losses, membrane resistance, and permeate pressure.

• Salt rejection

The ability of the RO membrane to reject dissolved solutes is a critical parameter for assessing its effectiveness in desalination applications. This parameter is calculated as follows [3]:

$$\mathit{SR}=\left(1-\frac{{\mathit{C}}_{\mathit{p}}}{{\mathit{C}}_{\mathit{f}}}\right)\times 100$$E7

where SR [%], C_p [mol/m³], and C_f [mol/m³] refer to salt rejection, permeate concentration, and concentration of feed solution, respectively. Higher salt rejection rates indicate a membrane’s capability to effectively remove more dissolved solutes from the feed solution, resulting in the production of higher-quality purified water.

• Permeate recovery

Permeate recovery refers to the ratio of produced freshwater to the feed solution. It is an important parameter to determine the overall efficiency and sustainability of the RO process. This parameter is defined as follows [3, 8, 15]:

$$\mathit{PR}=\left(\frac{{\mathit{m}}_{\mathit{p}}}{{\mathit{m}}_{\mathit{f}}}\right)\times 100$$E8

in which PR [%], m_p [kg/(m².s)], and m_f [kg/(m².s)] are permeate recovery, produced permeate flow rate, and feed flow rate, respectively. Higher permeate recovery rates indicate more efficient feed utilization to produce more freshwater.

3. Membrane distillation (MD)

3.1 Introduction and historical background

In response to growing global concerns surrounding water and energy scarcity, there has been a significant increase in the focus on low-energy desalination technologies [25, 26]. One such technology that has garnered considerable attention is membrane distillation (MD), which offers notable advantages not only in desalination and wastewater treatment but also in diverse applications such as industrial recycling, as well as separation processes in the pharmaceutical, medical, and food industries [27, 28]. Unlike most membrane-based separation processes, which are typically isothermal and rely on transmembrane chemical potential differences (such as hydrostatic pressure, concentration, and electrical potential), MD, as a pioneer of the second-generation of membrane-based desalination technologies, operates as a nonisothermal process [11, 25]. The MD process possesses distinct advantages in comparison with alternative processes. MD is driven by the partial pressure difference which can be even supplied by a low-temperature difference. This lower operating temperature can be provided by eco-friendly sources like renewable energy and the harvesting of waste heat [11]. In addition to the advantages of reduced operating temperature, which translates to lower energy costs, and the simplicity of equipment, the process safety of the MD technique is higher than other membrane-based desalination technologies, owing to its lower operating pressure [29, 30]. It is worth noting that the MD has a long-standing history with a significant chronological gap. The first patent for MD systems was introduced by Bodell in 1963 [31]. Subsequently, Weyl filed the second MD patent in the United States in 1967 [32]. His patent claimed an enhanced method and apparatus for extracting freshwater from saline water, utilizing direct contact membrane distillation (DCMD). In this improved approach, two types of water, whether stationary or in motion relative to the membrane, were employed, and the process was conducted in a single stage or through multiple stages. Weyl employed a polytetrafluoroethylene (PTFE) membrane, characterized by a porosity of 42%, a thickness of 3175 μm, and an average pore size of 9 μm. In 1967, Findley conducted experiments using the DCMD configuration and investigated the performance of different membrane materials such as paper hot cup, gumwood, aluminum foil, cellophane, glass fibers, paper plate, diatomaceous earth mat, and nylon [33]. To achieve the necessary hydrophobicity of the membranes, coatings made of silicone and Teflon were employed. Findley analyzed the MD experimental results and identified the membrane characteristics that are most suitable for successful MD operation. Building upon this work, Bodel, in 1968, introduced a novel system and method for converting nonpotable aqueous solutions into potable water [34]. His second MD patent involved the utilization of a parallel array of tubular silicone membranes. Without a break, researchers worldwide introduced and modeled various configurations of MD. Despite the initial rapid progress in MD, the interest in this process waned, gradually losing its allure. This decline can be attributed, in part, to the observed lower produced flux of MD when compared to that of the RO process. In the 1980s, the attention of academic communities was paid back again to the MD systems when novel membranes with better characteristics became available [35, 36]. These membranes, typically composed of materials, such as PTFE, PP, and PVDF, possessed remarkable attributes, including reduced thickness and porosity levels of up to 90%. Given that ongoing efforts are underway to further enhance membrane performance through modifications, MD has emerged as a compelling and energy-efficient desalination technology based on membrane principles.

Based on the available literature [11, 12, 21], MD has been reported to require an average energy input of 1.25${\mathit{kWh}}_{\mathit{\text{elec}},\mathit{\text{equi}}}/{\mathit{m}}^3$, encompassing both thermal and electrical energy, for a large-scale desalination plant. In contrast to the RO process, which relies solely on electrical energy, MD requires the additional input of thermal energy. Remarkably, approximately 90% of the total energy consumption in MD can be attributed to this heat energy demand. To mitigate the expenses associated with MD, consideration can be given to alternative and cost-effective energy sources. For instance, utilizing solar energy, which is readily available and relatively inexpensive, could prove economically advantageous. By harnessing solar energy as the primary energy source and implementing a continuous energy recovery system, the MD process emerges as an economically viable technology. Moreover, the utilization of waste heat as an energy source presents another promising avenue for reducing MD costs in desalination operations. In such cases, the MD process becomes a favorable alternative when compared to the traditional RO process from an economic standpoint.

Moreover, previous research has identified the potential for integrating MD technology into large-scale seawater RO desalination plants, with the MD process effectively operating within the brine stream. This is due to MD low sensitivity to the feed concentration. This integration offers a promising approach to enhance overall desalination efficiency and cost-effectiveness in the field of water treatment and supply.

3.2 MD basic concept and process

As mentioned in the previous section, MD represents a thermally driven separation process employing membranes well-suited for treating aqueous feed solutions. In this method, a highly porous and hydrophobic membrane serves as the barrier between the feed solution and permeate [10, 27]. In all MD configurations, the feed solution is in direct contact with the membrane surface, whereas the positioning of permeate varies across different configurations. The transportation of volatile compounds from the feed channel to the permeate channel in various MD configurations, such as DCMD, air gap membrane distillation (AGMD), vacuum membrane distillation (VMD), and osmotic membrane distillation (OMD), is primarily driven by the vapor pressure gradient [25, 26]. To establish the required vapor pressure gradient across the membrane, different techniques, like temperature difference as the most common method, are employed in different MD setups. In continuous operations, the feed and permeate aqueous streams, maintained at varying temperature conditions, are tangentially circulated around the MD membrane. Under such temperature conditions, water molecules evaporate at the feed-membrane interface. Subsequently, the vapor diffuses through the membrane pores and condenses at the permeate-membrane interface. Figure 3 depicts the schematic representation of the MD process for DCMD configuration, consisting of a feed channel, membrane, and permeate channel [25].

Figure 3.

Principle of heat and mass transfer in MD continuous process.

The hydrophobic membrane utilized in MD systems serves as a physical barrier, effectively sustaining the formation of liquid-vapor interfaces at the entrances of the membrane pores. This membrane’s hydrophobic properties effectively prevent the penetration of liquid water (brine) into the membrane pores from the feed channel. Under the influence of surface tension, the liquid-vapor interface can be maintained up to a certain pressure threshold. Beyond this threshold, liquid water permeates the membrane pores. Termed as the liquid entry pressure (LEP), this pressure can be determined through various experimental tests or theoretically calculated using the Young-Laplace equation [10, 26]:

$$\mathit{LEP}=\frac{-4\sigma cos\theta }{{\mathit{d}}_{\mathit{p}}}$$E9

In the context of this equation, $\mathit{\sigma }$[N/m] denotes the surface tension of the liquid phase, $\mathit{\theta }$[$°$] represents the contact angle formed between the fluid and the membrane, and d_p [m] signifies the average diameter of the membrane pores.

3.3 MD membranes

3.3.1 Membrane materials

Selecting an optimal membrane for MD applications involves considering factors like low thermal conductivity, high porosity, and minimal tortuosity. Additionally, desirable characteristics include a narrow pore size distribution, superhydrophobic properties with high LEP, longevity, thermal and chemical stability, and an optimized membrane thickness for balancing mass and heat transfer. These criteria collectively contribute to achieving efficient and effective MD performance [11, 25]. Figures 4(a) and (b) displays the top surface and cross-section scanning electron microscope (SEM) images of an industrially available membrane, denoted as MS-3010/0.45 μm, manufactured by the Membrane Solution Company [37, 38]. Also, Figure 4(c) shows the 3D structure of MS-3010/0.45 μm, generated by the virtual reconstruction approach.

Figure 4.

(a): Top section SEM image; (b): Cross-section SEM image; (c): 3D structure, generated by the virtual reconstruction approach [27], of MS-3010/0.45 μm membrane.

In the fabrication of MD membranes, the materials employed can be broadly categorized into two groups: polymeric and inorganic materials. The frequently utilized polymeric materials for MD membrane fabrication are polytetrafluoroethylene (PTFE), polypropylene (PP), and polyvinylidene fluoride (PVDF), respectively [39, 40, 41, 42, 43]. Among the materials introduced above, PVDF has garnered the most attention in research studies due to its exceptional mechanical strength, impressive chemical resistance, favorable thermal stability, and excellent aging resistance. The thermal conductivity of PVDF is measured at 0.17–0.21 W/(m.K). The prevailing method for fabricating PVDF membranes is through phase separation [39, 41, 44]. Alternatively, PTFE, known for its high hydrophobicity, has demonstrated remarkable efficiency as an MD membrane material. It boasts excellent chemical resistance and thermal stability, and its thermal conductivity ranges from approximately 0.25 to 0.29 W/(m.K). However, the fabrication techniques for PTFE membranes are somewhat limited, involving stretching of thin films and sintering of fine PTFE powders [42, 45]. PP, a crystalline polymer, is also characterized by its high hydrophobicity and thermal conductivity of 0.1–0.22 W/(m.K). Nonetheless, its solubility in most solvents at ambient temperatures is minimal. Unfortunately, PP membranes do not possess the same level of hydrophobicity as PTFE, and their fabrication is comparatively more challenging than that of PVDF membranes. As a result, the utilization of PP as a material for MD membranes has been less prevalent compared to PVDF and PTFE [39, 42, 44].

In addition to polymeric materials, mineral substances find utility in the fabrication of MD membranes. Notably, minerals exhibit superior chemical and thermal stability compared to their polymeric counterparts. Indeed, four types of minerals commonly employed for membrane production are metals, zeolites, carbons, and ceramics [46, 47, 48]. The mineral membranes are commonly fabricated using the sol-gel process. However, it is essential to acknowledge that ceramic and other inorganic membranes might not be cost-effective under typical thermal and chemical conditions. This aspect warrants further consideration when evaluating their feasibility for specific applications [46, 47, 48, 49, 50].

3.3.2 Membrane fabrication methods

The most common techniques employed for the fabrication of MD membranes are sintering, stretching, phase inversion, sol-gel, and electrospinning. Notably, PVDF, PP, and PTFE membranes are commonly manufactured using the phase inversion method, stretching, and either sintering or stretching processes, respectively. Also, sol-gel is the common technique used in the fabrication of mineral membranes, like ceramic membranes. Here, we present concise explanations of these methods widely utilized in membrane production.

• Sintering technique: Powder of polymeric particles is pressed into a film or plate and then sintered just below its melting point [42, 51].

• Stretching technique: In the realm of film production, a significant technique involves the extrusion of a polymer at temperatures near its melting point, alongside a rapid drawdown process. This particular method leads to the alignment of crystallites within the polymer in the direction of the drawing. Subsequently, through annealing and cooling, mechanical stress is applied perpendicular to the aforementioned drawing direction. The outcome of this manufacturing process results in the creation of a remarkably uniform porous structure, characterized by a porosity of approximately 90% [52, 53, 54].

• Phase inversion technique: Phase inversion technique involves the dissolution of the polymer in a suitable solvent, followed by casting the resulting solution to a thickness of 20-200 mm onto a suitable support material. The homogeneous solution is then subjected to a coagulant bath, which induces a solvent-nonsolvent exchange, leading to the separation of the solution into two distinct phases: one rich in polymer content and the other rich in liquid content. This process results in the formation of a cast film with desirable properties for various applications [55, 56, 57].

• Electrospinning: In this particular technique, an elevated voltage is administered to a polymer solution, prompting the generation of a fine jet that subsequently solidifies into nanofibers [58, 59].

• Sol-gel: These techniques involve the conversion of inorganic precursors into sol (a colloidal suspension) followed by gelation, resulting in the formation of a ceramic matrix. The process allows for precise control over the membrane’s composition and microstructure, leading to membranes with enhanced properties for various applications, such as filtration and separation processes [46, 47, 48, 49, 50].

3.4 Transport mechanisms through MD membranes

3.4.1 Transmembrane mass transfer theory

In the porous medium of MD, a nonisothermal membrane-based desalination method, the mass transfer flux through the membrane pores is determined by the following equation [4, 12, 25].

$${\mathit{J}}_{\mathit{w}}={\mathit{c}}_{\mathit{m}}\left({\mathit{P}}_{\mathit{mf}}-{\mathit{P}}_{\mathit{mp}}\right)$$E10

in which J_w [kg/(m².s)], c_m [kg/(Pa.m².s)], P_mf [Pa], and P_mp [Pa] refer to the mass transfer flux, mass transfer coefficient in the membrane, water vapor partial pressure at the feed-membrane interface, and water vapor partial pressure at the permeate-membrane interface, respectively. The Antoine equation can be used to calculate the water vapor partial pressures at the local temperatures of the membrane interfaces. Incorporating various transport mechanisms can describe the transportation of volatile compounds through membrane pores in MD. The first effective mechanism is the Maxwell–Stefan mechanism, which corresponds to ordinary molecular diffusion, indicating the transport effect of molecule-molecule collisions. Through this mechanism, water vapor molecules diffuse through the air trapped inside the larger pores of the membrane. The driving force behind this diffusion is the concentration difference between the two interfaces of the membrane. The following is used to calculate the ordinary molecular mass transfer coefficient [4, 25].

$${\mathit{C}}_{\mathit{om}}=\frac{1.895{\mathit{e}}^{-5}\times {\left(\mathit{T}\right)}^{2.072}{\epsilon }_{\mathit{m}}{\mathit{M}}_{\mathit{w}}}{\mathit{RT}{\tau }_{\mathit{m}}{\delta }_{\mathit{m}}\left(\frac{{\mathit{P}}_{\mathit{air},\mathit{mf}}-{\mathit{P}}_{\mathit{air},\mathit{mp}}}{ln\left(\frac{{\mathit{P}}_{\mathit{air},\mathit{mf}}}{{\mathit{P}}_{\mathit{air},\mathit{mp}}}\right)}\right)}$$E11

where T [K], ${\mathit{\epsilon }}_{\mathit{m}}$, M_w [kg/mol], R [J/(mol.K)],${\mathit{\tau }}_{\mathit{m}}$, and ${\mathit{\delta }}_{\mathit{m}}$ [m] refer to the temperature, membrane porosity, molecular weight of water, universal gas constant, membrane tortuosity, and membrane thickness, respectively. In addition, P_air,mf [Pa] and P_air,mp [Pa] are the partial pressures of air at both interfaces of the membrane.

The second mechanism is known as Hagen–Poiseuille, which is characterized by viscous flow. This mode occurs when a total pressure difference exists across the larger pores of the membrane. Through this mechanism, water vapor is transported through these pores via viscous flow. The following equation is defined to determine the Hagen–Poiseuille mass transfer coefficient [11, 25].

$${\mathit{C}}_{\mathit{H}-\mathit{P}}=0.03125\frac{{\epsilon }_{\mathit{m}}{\mathit{d}}_{\mathit{p}}^2{\mathit{P}}_{\mathit{m}}{\mathit{M}}_{\mathit{w}}}{{\tau }_{\mathit{m}}{\delta }_{\mathit{m}}\mathit{RT}{\mu }_{\mathit{v}}}$$E12

here, C_H-P [kg/(Pa.m².s)], d_p [m], and ${\mathit{\mu }}_{\mathit{v}}$ [Pa.s] are the Hagen–Poiseuille mass transfer coefficient, mean pore diameter of membrane, and water vapor dynamic viscosity, respectively.

Lastly, the third mechanism is Knudsen diffusion, which occurs within the nanopores of the membrane. This mode comes into play when the characteristic length of the nanopores is comparable to or smaller than the mean free path of water molecules at the given temperature and pressure, indicating the transport effect of molecular-wall collisions. Knudsen diffusion is driven by the partial pressure difference between the two interfaces of the membrane. The Knudsen mass transfer coefficient is mathematically formulated in the following equation [4, 12].

$${\mathit{C}}_{\mathit{kn}}=\frac{8}{3}\frac{{\epsilon }_{\mathit{m}}{\mathit{d}}_{\mathit{m}}}{2{\tau }_{\mathit{m}}{\delta }_{\mathit{m}}}\sqrt{\frac{1}{2\pi {\mathit{M}}_{\mathit{w}}\mathit{RT}}}$$E13

Although there are various empirical equations, i.e., the Bruggeman equation, the Millington equation, and Macki–Meares equation, to estimate the pore tortuosity, these correlations cannot be generalized for all kinds of porous medium with various structures. For this reason, according to the membrane structure, it is proposed to employ the following two equations for two types of domains [11]:

1. In loose-packed domains:

   $${\tau }_{\mathit{m}}=\frac{1}{{\epsilon }_{\mathit{m}}}$$E14

2. In distances between closed domains:

   $${\tau }_{\mathit{m}}=\frac{{\left(2-{\epsilon }_{\mathit{m}}\right)}^2}{{\epsilon }_{\mathit{m}}}$$E15

To assess the prevalence of the aforementioned diffusion mechanisms, ordinary molecular and Knudsen diffusions, the Knudsen number serves as a pertinent indicator for each distinct membrane, considering various specifications like the mean pore diameter. By employing this index, we can effectively compare the dominance of these diffusion mechanisms for a membrane. Knudsen number is defined as follows [10, 25]:

$$\mathit{Kn}=\frac{\lambda }{{\mathit{d}}_{\mathit{p}}}$$E16

in which $\mathit{\lambda }$ [m] refers to the mean free path of water molecules. At atmospheric pressure, when dealing with the common pore diameters of commercial MD membranes, which typically range between 10 nm and 1 μm, the Knudsen number can be estimated to fall within the range of 0.1 < Kn < 10. Consequently, one can expect a superposition of both ordinary molecular diffusion and Knudsen diffusion phenomena to occur simultaneously.

Among the models frequently employed in the literature, the dusty gas (DGM), Schofield, and KM-P models stand out as the most widely utilized ones [4, 25, 60]. In general, both models involve three mass transfer mechanisms explained above. As depicted in Figure 5, the resistance of ordinary molecular diffusion is in series with the parallel combination of the resistances of Knudsen diffusion and Hagen–Poiseuille (viscous flow) in the Schofield model. In contrast, the series combination of the resistances of Knudsen and ordinary molecular diffusions is in parallel with the resistance of Hagen–Poiseuille (viscous flow) in DGM. As for the KM-P model, the resistance of Hagen–Poiseuille (viscous flow) is placed in series with the parallel combination of the resistances from ordinary molecular and Knudsen diffusions.

Figure 5.

Commonly adopted models for the prediction of the mass transfer flux across the hydrophobic membrane in MD.

3.4.2 Transmembrane heat transfer theory

There are several various heat transfer regions and mechanisms present in the whole module of MD, depending on their particular configuration. However, the overall heat transfer across the membrane occurs through only two mechanisms: (1) conduction within the solid structure of the membrane and its gas-filled pores and (2) the latent heat carried by water molecules generated during evaporation. The following equation expressed these mechanisms in mathematical representation [4, 12, 25].

$${\mathit{Q}}_{\mathit{m}}={\mathit{Q}}_{\mathit{c}}+{\mathit{Q}}_{\mathit{v}}=-\frac{{\mathit{k}}_{\mathit{m}}}{{\delta }_{\mathit{m}}}\left({\mathit{T}}_{\mathit{mf}}-{\mathit{T}}_{\mathit{mp}}\right)+{\mathit{J}}_{\mathit{w}}{\mathit{h}}_{\mathit{fg}}$$E17

here, Q_c [W/m²] refers to the heat loss by thermal conduction, Q_v [W/m²] is the heat transferred by water vapor flux, k_m [W/(m.K)] stands for the thermal conductivity of the membrane, T_mf [K] denotes the temperature at the feed-membrane interface, T_mp [K] represents the temperature at the permeate-membrane interface, and h_fg [J/kg] corresponds to the enthalpy of water vaporization. The mass transfer flux through the membrane, J_w, is found using transmembrane mass transfer theory, discussed in the previous section. Indeed, the thermal conductivity of a membrane is influenced by both the solid material’s thermal conductivity and the gas species’ thermal conductivity within its pores. Researchers have proposed several empirical models to calculate the membrane’s thermal conductivity. Notably, among these models, the Isostrain, Isostress, and flux low models, as expressed in Eqs. (18)–(20), respectively, have been widely employed in the literature [27].

$${\mathit{k}}_{\mathit{m}}={\epsilon }_{\mathit{m}}{\mathit{k}}_{\mathit{w}}+\left(1-{\epsilon }_{\mathit{m}}\right){\mathit{k}}_{\mathit{\text{Polymer}}}$$E18

$${\mathit{k}}_{\mathit{m}}={\left(\frac{{\epsilon }_{\mathit{m}}}{{\mathit{k}}_{\mathit{w}}}+\frac{\left(1-{\epsilon }_{\mathit{m}}\right)}{{\mathit{k}}_{\mathit{\text{Polymer}}}}\right)}^{-1}$$E19

$${\mathit{k}}_{\mathit{m}}={\mathit{k}}_{\mathit{w}}\left[\frac{1+\left(1-{\epsilon }_{\mathit{m}}\right)\beta }{1-\left(1-{\epsilon }_{\mathit{m}}\right)\beta }\right],\beta =\frac{\frac{{\mathit{k}}_{\mathit{\text{Polymer}}}}{{\mathit{k}}_{\mathit{w}}}-1}{\frac{{\mathit{k}}_{\mathit{\text{Polymer}}}}{{\mathit{k}}_{\mathit{w}}}+2}$$E20

where k_w [W/(m.K)] and k_Polymer [W/(m.K)] are the thermal conductivity of water vapor which pass through the membrane pores and thermal conductivity of polymer, respectively. Based on the findings of combined microstructure and pore-scale modeling for calculating transport properties of MD membranes [27], the Isostrain model is more accurate compared to the Isostress and flux low models for estimating the thermal conductivity of membranes commonly used in MD systems.

3.5 Performance parameters of MD process

This section delves into the performance parameters of the MD system, encompassing both the membrane and its overall efficiency. The thermal efficiency ($\mathit{\eta }$), which quantifies the ratio of heat loss to the heat utilized for water vapor production, is expressed by the following equation [4, 12].

$$\eta =\frac{{\mathit{Q}}_{\mathit{v}}}{{\mathit{Q}}_{\mathit{c}}+{\mathit{Q}}_{\mathit{v}}}$$E21

Q_c [W/m²] and Q_v [W/m²] have been already introduced in the previous section. The gain output ratio (GOR) stands as another crucial performance parameter employed to assess and compare different designs of MD modules. Essentially, GOR represents the proportion between the heat generated due to vapor transfer through the membrane and the energy input provided. This metric plays a significant role in evaluating the efficiency and effectiveness of various MD modules [12, 25].

$$\mathit{GOR}=\frac{{\mathit{J}}_{\mathit{w}}{\mathit{h}}_{\mathit{fg}}{\mathit{A}}_{\mathit{m}}}{{\dot{\mathit{m}}}_{\mathit{f}}{\mathit{C}}_{\mathit{p}}\left({\mathit{T}}_{\mathit{fi}}-{\mathit{T}}_{\mathit{po}}\right)}$$E22

where A_m [m²], C_p [J/(kg.K)], T_fi [K]_, and T_po [K] are the effective membrane area, specific heat capacity at constant pressure, feed inlet temperature, and permeate outlet temperature, respectively.

4. Electrodialysis (ED)

4.1 Introduction and historical background

Electrodialysis (ED) stands as a membrane-based desalination/separation process wherein ions undergo transportation through ion-selective membranes, propelled from one solution to another under the compelling force of an electric field [61]. This intricate transportation phenomenon gives rise to the creation of two distinct solutions: a concentrated one with higher ion content than the original and another comprising nearly pure water. It should be noted that ED only removes ions. Therefore, any bacteria, colloidal material, or silica present in the feed water stream will remain in the product stream. The significance of ED lies notably in its application as an advanced environmental technology, facilitating the development of pristine treatment sequences for water recovery in industrial processes [13, 62].

The timeline of the most important developments for ED is illustrated in Figure 6. The concept of ED was first discovered in 1889 by Maigrot and Sabates [63]. They devised an early unit that combined electrolysis and dialysis to eliminate harmful substances that were interfering with sugar manufacturing. In 1890, they successfully demineralized sugar syrup using carbon electrodes, a permanganate paper membrane, and an electric current supplied by a dynamo. The term “electrodialysis” was later coined by Schollmeyer around 10 years after these initial experiments [63]. In 1900, Schollmeyer patented the purification of sugar syrup using ED, employing iron as the anode [64, 65]. However, it was not until 1911, when Donnan introduced his exclusion principle, that the theory of ED began to take shape [65]. Building upon the Donnan exclusion principle, the development of polymeric ion-selective membranes started, initially in granule form and gained momentum after 1930 [63, 64]. In 1939, Manegold and Kalauch utilized cation- and anion-selective membranes to construct a 3-compartment ED system [63]. To address energy loss concerns, Meyer and Strauss proposed a significant advancement in 1940—the multicomponent ED stack. This innovative design involved arranging multiple ion-selective membranes in an alternating pattern between the cathode and anode [66]. The significant advancement of ED for desalination and demineralization applications on an industrial scale can be attributed to key developments in membrane technology. In 1950, Juda and McRae [67], followed by Wyllie and Patnode [68], achieved a breakthrough in fabricating low electric resistance, stable, and highly selective membranes. Similarly, Winger et al. [69] at Rohm made crucial contributions in this field in 1953. Prior to 1954, ED had not been employed on an industrial scale. It was not until this year that the inaugural desalination plant, incorporating ED technology, was constructed in Aramco, located in Saudi Arabia [65, 67]. This historical development marked the beginning of ED’s utilization in large-scale industrial applications.

Figure 6.

Timeline of the most important developments for ED.

Another pivotal moment came with the invention of bipolar membranes, which propelled the progress of ED and paved the way for the emergence of bipolar ED. In 1954, Sollner et al. [70] proposed the possibility of fabricating layered membranes, a concept that sparked new possibilities in the field. Building on this idea, just 2 years later, Frilette [71] successfully developed a bipolar membrane consisting of an anion-exchange membrane (AEM) on one side and a cation-exchange membrane (CEM) on the other side. The concept of electrodialysis reversal (EDR) emerged in 1974 as a promising method for periodically reversing the current, effectively mitigating membrane fouling issues [65, 72]. Subsequently, in 1980, Tomas A. Davis proposed the concept of employing bipolar membranes electrodialysis (BMED) specifically for the production of acids and bases [73]. These breakthroughs in membrane design and technology not only revolutionized ED processes but also opened up new strategies for industrial-scale desalination and demineralization, setting the stage for further advancements in the field, as introduced in Figure 6.

The energy demand of ED increases notably with rising feed salinity, making it a high-energy-intensive option compared to other membrane-based desalination methods for concentrated feeds. However, ED proves to be particularly advantageous and favorable when applied to low TDS feed water desalination scenarios. In prior research, it has been established that ED processes typically demand electrical energy ranging from 2.64 to 5.5${\mathit{kWh}}_{\mathit{\text{elec}}}/{\mathit{m}}^3$ for the treatment of mid-concentrated feed, whereas for low TDS feed, the energy requirement falls within the range of 0.7 to 2.5${\mathit{kWh}}_{\mathit{\text{elec}}}/{\mathit{m}}^3$. It has been reported that the energy required for desalinating a brine with a TDS concentration of 70,000 ppm was found to be approximately 49.7${\mathit{kWh}}_{\mathit{\text{elec}}}/{\mathit{m}}^3$, while for a higher TDS brine containing 250,000 ppm, the energy consumption was measured at 175.7${\mathit{kWh}}_{\mathit{\text{elec}}}/{\mathit{m}}^3$ [21, 61, 62].

4.2 ED basic concept and process

4.2.1 Assembling ED cells and stacks

ED is an electrochemical separation technique that relies on the migration of anions and cations within ion-exchange membranes (IEMs) under an applied electrical field, creating the driving force for ion transport [62]. For this purpose, the AEM and CEM, with positive and negative fixed charges, respectively, are used to selectively allow the penetration of specific ionic species [13]. Figure 7 depicts an ED cell pair consisting of dilute and concentrate channels, along with AEM, CEM, anode, and cathode. In practical ED units, often referred to as “stacks,” several hundred membranes may be utilized. The principle behind this stack configuration is that one Faraday passing through a membrane pair can transport one equivalent gram of electrolyte from a diluted compartment to a concentrated one. Hence, incorporating multiple pairs of membranes can enhance the process yield proportionally. When an electrical voltage is applied, cations and anions migrate toward the cathode (negatively charged electrode) and the anode (positively charged electrode), respectively [13]. The cations permeate through the CEM and move into the concentrate base channel, while anions pass through the AEM and enter the concentrate acid channel. In Figure 7, the cations and anions of Na⁺ and Cl⁻, respectively, are employed to depict the ED process.

Figure 7.

Schematic representation of an ED process for desalination application.

4.2.2 ED process configurations

The illustration of a single-cell ED system can be found in Figure 7 in the previous section. Depending on the various feeding methods employed to supply the inlet streams, single-cell EDs are categorized into two main configurations: single-pass (continuous) and batch setups. Furthermore, the utilization of bipolar membranes enables the integration of multiple EDs into a series arrangement, resulting in what is known as BMED. This innovative approach enhances the capabilities of the ED process.

• Single-pass (continuous) configuration:

In this setup, the outlet solutions from each compartment of the ED unit are collected and stored in separate tanks without being recirculated back into the ED system. Consequently, the concentrations of the inlet solutions remain constant over time and do not undergo any variations during the process. This configuration ensures a continuous flow of solutions through the ED compartments, maintaining a steady state throughout the operation.

• Batch configuration:

Within the batch configuration, a dynamic process occurs, involving the continuous circulation of dilute and concentrate solutions between the ED’s channels and their respective tanks. This intricate exchange continues until the desired concentration level is achieved. Throughout this iterative procedure, the inlet solutions’ concentrations exhibit temporal variations, adding complexity to the overall system dynamics.

• Bipolar membrane ED (BMED)

In BMED, an innovative approach combines conventional electrodialysis with bipolar membranes to produce acids and bases from corresponding salts, while effectively eliminating unwanted gas production, such as O₂ and H₂. At the heart of BMEDs lies the crucial bipolar membrane, a layered IEM comprising a CEM and an AEM. The success of this technology hinges on several key characteristics, including a high capacity for water dissociation, remarkable stability in highly concentrated acidic and alkaline solutions, superior permselectivity, and minimal electrical resistance.

The water-splitting process within the bipolar membrane, wherein water molecules are divided into proton and hydroxide ions (2H₂O → H₃O⁺ + OH⁻), takes place at the interface of the anion and cation permeable layers. This unique arrangement accelerates the generation of hydroxide and proton ions, leading to efficient acid and base production [74]. BMEDs offer distinct advantages over conventional EDs, particularly in terms of lower energy consumption. However, it should be noted that the operational costs of BMEDs are comparatively higher [75, 76]. To provide a visual representation, Figure 8 depicts the configuration of the BMED stack employed for acids and base production. The stack comprises repeating units, each consisting of a CEM, an AEM, and a bipolar membrane, all arranged between two electrodes. The dilute salt solution enters the central channels of the repeating units, while the production of acidic and alkaline solutions occurs in the anode and cathode channels, respectively.

Figure 8.

Schematic representation of BMED stack for acid ad base production.

4.3 ED ion-exchange membranes

4.3.1 Membrane materials

IEMs play a vital role in ED, as sheets of highly swollen ion-exchange resin containing cationic or anionic groups. These membranes are made up of thin polymeric films such as polyethylene, polysulfone, polystyrene, etc., and have charged groups. They exhibit two types of electrical conductivity (charged groups), namely CEMs and AEMs. The AEMs possess a positive charge and are primarily composed of secondary amines ($\mathit{-}{\mathit{NRH}}_2^{\mathit{+}}$), tertiary amines ($\mathit{-}\mathit{N}{\mathit{R}}_2{\mathit{H}}^{\mathit{+}}$), quaternary amines ($\mathit{-}{\mathit{NR}}_3^{\mathit{+}}$), as well as ammonium groups ($\mathit{-}{\mathit{NH}}_3^{\mathit{+}}$,${\mathit{SR}}_2^{\mathit{+}}$, ${\mathit{PR}}_3^{\mathit{+}}$), all of which are chemically bonded to the polymer membrane’s structure. This configuration effectively blocks the passage of positively charged ions while selectively allowing the transit of negatively charged ions. In contrast, the CEMs feature the presence of negatively charged functional groups, including carboxylic acid ($\mathit{-}\mathit{CO}{\mathit{O}}^{\mathit{-}}$), sulfonic acid ($\mathit{-}\mathit{S}{\mathit{O}}_3^{\mathit{-}}$, ${\mathit{C}}_6{\mathit{H}}_4{\mathit{O}}^{\mathit{-}}$), phosphonic acid ($\mathit{-}\mathit{P}{\mathit{O}}_3{\mathit{H}}^{\mathit{-}}$), and phosphoryl ($\mathit{-}\mathit{P}{\mathit{O}}_3^{2\mathit{-}}$). These functional groups endow CEMs with the capability to perform functions that are distinctly different from their AEM counterparts [61, 62, 77, 78]. The ions that can freely pass through CEMs and AEMs, representing positively charged cations and negatively charged anions, are referred to as counter-ions. On the other hand, co-ions, which possess the same charge as the membrane, are impermeable due to the phenomenon of Donnan exclusion [78]. As illustrated in Figure 9, the fixed ions remain embedded, while both counter-ions and co-ions are mobile within the polymer matrix. CEMs maintain electrical equilibrium between mobile cations and fixed negative charges, whereas mobile co-ions remain excluded from the polymer matrix. CEMs and AEMs share several key characteristics, including but not limited to low electrical resistance, insolubility in aqueous solutions, semi-rigidity for convenient handling during stack assembly, resistance to osmotic swelling, pH resistance ranging from 1 to 10, long life expectancy, resistance to fouling, and ease of hand-washing.

Figure 9.

Schematic representation of a CEM structure.

4.3.2 Membrane fabrication methods

Given the properties of IEMs such as active surface charge distribution, hydrophobicity, resistance to electricity, surface roughness, along with the manufacturing methods of both AEM and CEM, they are classified into homogeneous and heterogeneous IEMs. Homogeneous membranes exhibit a uniform distribution of charge throughout their structure, while heterogeneous membranes contain fixed charges distributed nonuniformly within the polymer matrix. Consequently, homogeneous membranes boast superior electrochemical properties, whereas heterogeneous ones excel in mechanical characteristics. Given the distinct structural and inherent characteristics of these two membrane types, they typically require distinct fabrication methodologies. In the fabrication of homogeneous membranes, common techniques encompass processes such as polymerization, polycondensation of functional monomers, and post-sulfonation of polymers. In addition, heterogeneous membrane preparation entails methods such as dispersing ion-exchange resin within a polymer solution or the fusion and compression of dry ion-exchange resin alongside granulated polymer material.

4.4 Concentration polarization and limiting current density

Electric currents in ED cells are carried by both cations and anions, with similar transfer numbers in aqueous solutions. However, within IEMs, the current transfer is primarily conducted by counter-ions, with a transfer number near unity. The variation in ion transfer numbers between the solution and an IEM results in a decrease in electrolyte concentration at the dilute side of the IEM interface, with a corresponding concentration increase on the concentrate side. Consequently, this leads to the establishment of a concentration gradient in the solution between the membrane surface and the bulk, known as concentration polarization. The concentration profile of ions in the dilute and concentration solutions adjacent to the CEM is illustrated in Figure 10 [79].

Figure 10.

Concentration polarization at the boundary layer on both sides of a CEM.

According to the theory of concentration polarization, the concentration of ions decreases at the dilute side of the membrane surface. When the electrolyte concentration at the dilute interface approaches zero, there are no more ions available to carry the electric current, resulting in the achievement of the limited current density. This condition corresponds to the increment of voltage drop across the boundary layer, resulting in a higher energy consumption and generation of water dissociation. The consequence of water dissociation is a loss of current utilization and significant pH shifts. This results in a higher pH at the AEM surface and a lower pH at the CEM surface. The pH increase may cause the precipitation of multivalent ions on the membrane, while the pH decrease can harm the membranes themselves. Consequently, in practical ED applications, it is essential to minimize concentration polarization effects and prevent water dissociation [79].

4.5 Transport mechanisms through ED membranes

In electrolyte solutions, the concentrations and velocities of ions vary significantly along the channels. These variations occur due to the continuous flux of ions and water through the IEMs. To comprehend these changes, material balance equations are employed. As the transfer of ions and water molecules transpires from the dilute channel to the concentrate channels, there is a noticeable increase in the concentration and velocity of the solution in the concentrate channels. Conversely, downstream in the outlet of the dilute channel, these quantities experience a decrease. Eq. (23) is used to quantify the transmembrane water flux through the IEMs in all compartments [80].

$${\mathit{J}}_{\mathit{w}}=\frac{{\rho }_{\mathit{w}}\mathit{W}\left({\mathit{u}}_{\mathit{\text{in}}}-{\mathit{u}}_{\mathit{\text{out}}}\right)}{\mathit{H}}$$E23

in which, the parameters W [m] and H [m] represent the width and height of the channels. Here, J_w [kg/(m².s)] and ${\rho }_w$ [kg/m³] denote the transmembrane water flux and density, while u_in [m/s] and u_out [m/s] refer to the inlet and outlet velocities, respectively. In the context of the dilute channel, J_w represents the collective flow of water molecules passing through both the anion and cation-exchange membranes. The following equation shows the relationship between the transmembrane ion (species) flux through the IEMs and the transmembrane water flux, expressed by Eq. (23) [80].

$${\mathit{J}}_{\mathit{i}}=\frac{\mathit{W}\left({\mathit{c}}_{\mathit{i},\mathit{\text{in}}}{\mathit{u}}_{\mathit{\text{in}}}-{\mathit{c}}_{\mathit{i},\mathit{\text{out}}}{\mathit{u}}_{\mathit{\text{out}}}\right)}{\mathit{H}}$$E24

here, J_i [mol/(m².s)], c_i,in [mol/m³], and c_i,out [mol/m³] refer to the transmembrane flux of species i, and concentration of species i at the inlet and outlet of channels, respectively.

Various transport mechanisms are responsible for the transportation of ions from solutions through the IEMs. The first effective mechanism is the migration mechanism. Through this mechanism, counter-ions are transferred through the corresponding IEMs, using the electrical field driving force. The second mechanism is diffusion. The driving force behind this diffusion is the ion concentration difference across the IEMs. The convection term constitutes the third transport mechanism involved in the ED process through IEMs. This term arises from two distinct phenomena: osmotic pressure and electro-osmosis. However, it is important to highlight that in ED processes, the influence of osmotic pressure is generally negligible compared to the significant impact of electro-osmotic phenomena. The Nernst-Planck equation is used to explain the roles of the aforementioned transport mechanisms in determining the transmembrane ion flux through IEMs. This equation is mathematically defined as follows [81, 82]:

$${\mathit{J}}_{\mathit{i}}=-{\mathit{z}}_{\mathit{i}}{\mathit{M}}_{\mathit{i}}^{\mathit{m}}{\mathit{c}}_{\mathit{i}}^{\mathit{m}}\frac{\mathrm{\Delta }{\phi }_{\mathit{m}}}{{\mathit{l}}_{\mathit{m}}}-{\mathit{D}}_{\mathit{i}}^{\mathit{m}}\frac{\mathrm{\Delta }{\mathit{c}}_{\mathit{i}}^{\mathit{m}}}{{\mathit{l}}_{\mathit{m}}}+{\mathit{u}}_{\mathit{m}}{\mathit{c}}_{\mathit{i}}^{\mathit{m}}$$E25

where the first, second, and third terms on the right-hand side are the migration, diffusion, and convection fluxes, respectively. In this equation, z_i, M_i^m [m²/(V.s)], and c_i^m [mol/m³] refer to the valence of ions, mobility of species i at membrane medium, and concentration of ion i at IEM interfaces, respectively. Also, $∆{\phi }_m$ [V], l_m [m], D_i^m [m²/s], $∆c_i^m$ [mol/m³], and u_m [m/s], are the voltage difference across IEMs, thickness of the membrane, diffusion coefficient of species i at membrane medium, ion concentration difference across IEMs, and water velocity through IEMs, respectively. The mobility of species i can be estimated using the Nernst-Einstein equation as follows [82]:

$${\mathit{M}}_{\mathit{i}}^{\mathit{m}}=\frac{{\mathit{FD}}_{\mathit{i}}^{\mathit{m}}}{\mathit{RT}}$$E26

here, F [96485.33 (A.s)/mol], R [J/(mol.K)], and T [K] are the Faraday’s constant, universal gas constant, and temperature, respectively. Then, Faraday’s law can be employed to calculate the current density (I) [62].

$$\mathit{I}=\mathit{F}\sum {\mathit{z}}_{\mathit{i}}{\mathit{J}}_{\mathit{i}}$$E27

In the literature [83], Eq. (28) is repeatedly suggested to calculate the diffusion coefficients of ions at IEMs by leveraging their diffusivity in the feed solutions in channels.

$${\mathit{D}}_{\mathit{i}}^{\mathit{m}}={\left(\frac{{\mathit{VF}}_{\mathit{w}}}{2-{\mathit{VF}}_{\mathit{w}}}\right)}^2{\mathit{D}}_{\mathit{i}}^{\mathit{s}}$$E28

in which, VF_w and D_i^s [m²/s] are volume fraction of pores filled by water in IEMs and ion diffusivity in the feed solutions in channels, respectively. As discussed earlier, the movement of water molecules driven by osmotic pressure across the membrane is found to be ignorable when compared to the dominant electro-osmotic phenomenon. Electro-osmotic water transport occurs due to the transport of hydrated ions under the influence of an electrical potential gradient. Therefore, in the context of determining the transmembrane water velocity mathematically defined in Eq. (29), it is considered that water molecules are primarily drawn into IEMs through the force of electro-osmosis. This force depends on both the applied electrical potential and the friction encountered by ions in the solution [79, 81, 84].

$${\mathit{u}}_{\mathit{m}}=\frac{\mathit{\epsilon \zeta }}{4\mathit{\pi \mu }}\frac{\mathrm{\Delta }{\phi }_{\mathit{m}}}{{\mathit{l}}_{\mathit{m}}}$$E29

here, $\zeta$ [V], $\epsilon$ [F/m]_, and $\mu$ [Pa.s] are the zeta potential, solvent permittivity, and dynamic viscosity, respectively. Given the effective membrane surface area (A_m), the transmembrane water flux, expressed in Eq. (23), is calculated as follows:

$${\mathit{J}}_{\mathit{w}}={\mathit{A}}_{\mathit{m}}{\mathit{u}}_{\mathit{m}}$$E30

The Donnan potential equation describes the potential difference across an interface between an IEM and a solution. This potential difference is influenced by the ion concentrations on both sides of the IEM-solution interface, as well as the pressure difference across the membrane. However, for the specific scenario under consideration, it is assumed that the effect of pressure difference is negligible. Consequently, the simplified form of the Donnan potential equation can be expressed as follows in the context of this study [85, 86, 87].

$${\phi }^{\mathit{m}}-{\phi }^{\mathit{s}}=\frac{\mathit{RT}}{{\mathit{z}}_{\mathit{i}}\mathit{F}}ln\left(\frac{{\mathit{c}}_{\mathit{i}}^{\mathit{s}}}{{\mathit{c}}_{\mathit{i}}^{\mathit{m}}}\right)$$E31

where superscripts m and s refer to IEM and solution sides of IEM-solution interfaces. Considering the Helmholtz double-layer model [88], the concentration of ion i at IEM interfaces in each channel can be determined as follows:

$${\mathit{c}}_{\mathit{i}}^{\mathit{m}}={\mathit{c}}_{\mathit{i}}^{\mathit{s}}exp\left(\pm \frac{{\mathit{z}}_{\mathit{i}}{\mathit{l}}_{\mathit{i}}^{\mathit{d}}}{\epsilon \mathit{RT}}{\mathit{F}}^2\left(2{\mathit{r}}_{\mathit{i}}\right){\mathit{c}}_{\mathit{i}}^{\mathit{s}}\right)$$E32

in which, positive and negative signs are allocated to AEM and CEM. Also, l_i^d [m] and r_i [m] refer to the thickness of the double layer and radius of counter-ion i moving within the IEM.

4.6 Performance parameters of ED process

The assessment of desalination and treatment efficiency through ED involves the computation of two crucial parameters: percent extraction and current efficiency. These metrics play a pivotal role in evaluating the overall effectiveness of the ED process in purifying and desalinating water.

4.6.1 ED percent extraction

The percentage extraction for each ion in the feed solution for continuous mode is calculated using Eq. (33) [61, 89].

$$\mathit{PE}=\left(\frac{{\mathit{c}}_{\mathit{i},\mathit{\text{out}}}-{\mathit{c}}_{\mathit{i},\mathit{\text{in}}}}{{\mathit{c}}_{\mathit{i},\mathit{\text{in}}}}\right)\times 100$$E33

4.6.2 ED current efficiency

In the practical ED processes, it is not possible to fully utilize the current passing through the stack for the purpose of desalinating the feed solution. Besides the electro-osmosis phenomena, there are some other reasons, such as water dissociation, parallel current, inappropriate IEMs, contributing to the current loss. Thus, the total current efficiency denotes the amount of current needed to produce a specific quantity of product water with a designated level of quality. This efficiency is determined by dividing the actual current consumption by the theoretically ideal current required for the process. To express this mathematically, the current efficiency can be defined as follows [61, 62, 90]:

$$\mathit{CE}=\frac{\mathit{F}}{\mathit{I}\mathrm{\Delta }\mathit{t}}\left({\mathit{c}}_{\mathit{i},\mathit{\text{in}}}{\mathit{u}}_{\mathit{\text{in}}}-{\mathit{c}}_{\mathit{i},\mathit{\text{out}}}{\mathit{u}}_{\mathit{\text{out}}}\right)$$E34

where the symbol I represents the electric current density flowing through the system over a specific time interval, denoted as ∆t.

5. Conclusion

This chapter offered an in-depth and comprehensive overview of three common membrane-based desalination processes of RO, MD, and ED. As the main purpose of this chapter, a thorough exploration of the transport mechanisms involved in the membranes utilized in the aforementioned membrane-based desalination processes was presented and reviewed. Moreover, this chapter comprehensively presented the historical background and development of these techniques, the basic technical concept of processes, the desired properties of the membranes utilized in each technique, and the key performance parameters associated with them. Table 1 summarily presents and compares the transport and separation mechanisms occurring within the membranes of each of the aforementioned membrane-based desalination techniques. Furthermore, this table provides a comprehensive analysis of the driving forces, membrane properties, and energy consumption associated with the introduced desalination techniques.

Membrane-based desalination technique	Transport mechanism in membrane	Driving force	Membrane properties	Energy consumption (Electrical equivalent) ${\mathit{kWh}}_{\mathit{\text{elec}},\mathit{\text{equi}}}/{\mathit{m}}^{\mathit{3}}$
RO	Diffusion (dominant) + Convection	Total pressure difference (15-80 bar)	Hydrophilic, Dense, High mechanical strength to withstand the higher operational pressure, Resistant to chemical degradation and biological fouling, High fouling resistance, High stability over pH changes, High thermal resistance	1.5-2.5 (BW); 4-6 (SW)
MD	Mass transfer: Ordinary molecular diffusion + Knudsen diffusion + Poiseuille (viscous flow); Heat transfer: Conduction + Latent heat associated with the water vapor molecules	Temperature difference (Vapor pressure difference)	High hydrophobicity, High porosity, High chemical and thermal stability, Low thermal conductivity, Low tortuosity, Low thermal resistance, High scaling resistance	$\approx$1.25
ED	Migration + Diffusion + Convection	Electrical potential difference	Ion-exchange membranes, Low electrical resistance, High fouling resistance, Suitable rigidity, High resistance over pH changes, Long life, High thermal resistance	0.7-2.5 (BW); 2.64-5.5 (MW); $\approx$49.7 (High TDS)

Table 1.

Side-by-side assessment of common membrane-based desalination technologies.

BW: Brackish water ($<3000\mathrm{ppm}$); MW: Moderate water (3000 ppm to 10,000 ppm); SW: Seawater (10,000 ppm to 50,000 ppm); High TDS: Seawater ($>50000\mathrm{ppm}$).

RO is a membrane-based separation process driven by the total pressure difference. It is extensively employed for the separation of salts and small organic molecules from liquid streams. The RO membranes typically consist of dense active layers, and the preferred transport mechanisms in these membranes are commonly attributed to two processes: diffusion, which is more dominant and convection. The operating pressure applied in RO, typically ranging from 15 to 80 bar, overcomes the summation of osmosis pressure and friction loss. Despite being a widely adopted and energy-intensive desalination technology, RO does not come without its share of challenges. For instance, when dealing with high TDS feed water, the utilization of RO may still encounter notable issues, primarily concerning scaling and fouling phenomena on the membrane surfaces of RO.

In MD, three mass transport mechanisms of Knudsen diffusion, ordinary molecular diffusion, and Poiseuille (viscous flow) are generally in charge of transporting the water vapor molecules through the membranes. Furthermore, the transfer of heat across the MD membranes transpires through two distinct mechanisms. Firstly, heat is conducted within the solid structure of membrane and its gas-filled pores. Secondly, the latent heat is conveyed through the membrane pores associated with the water vapor molecules. MD membranes should exhibit not only low resistance to mass transfer but also possess high resistance to heat transfer, which aligns with their low thermal conductivity. Moreover, an ideal characteristic of these MD membranes lies in their higher hydrophobicity, which leads to a higher LEP. Additionally, to enhance the effectiveness of MD membranes, it is preferable for them to have lower tortuosity factors and higher porosity. These attributes contribute to a smoother path for mass transfer, thereby enhancing the overall efficiency of the membrane-based desalination process. As shown in Table 1, MD stands as a moderately energy-intensive technology for desalination. However, the MD process exhibits a unique advantage as it can effectively utilize low-grade heat as its driving force. This advantage makes MD a promising and competitive technology, especially when renewable energy sources and waste heat are harnessed to provide the necessary thermal energy input.

ED, driven by an electrical potential difference, involves the transportation of ions through ion-selective membranes, which are compelled from one solution to another under the influence of an electric field. In the ED process, both CEM and AEM share common characteristics, such as low electrical resistance, resistance to fouling, solidity to facilitate handling during stack assembly, resistance over pH changes, long life expectancies, suitability for operation in high temperatures, and the convenience of being hand washable. The energy consumption in ED is notably influenced by the concentration of the feed water. This characteristic makes ED a compelling and competitive technology for desalinating brackish water and moderately saline water sources.

Acknowledgments

The authors are grateful for the funding support provided by Key R&D Program of Inner Mongolia Province of China (2021ZD0034) as well as the Science and Technology Plan Project of Sichuan Province (2022YFSY0018). Additionally, Hesam Bazargan Harandi expresses gratitude to Prof. Liwei Zhang for providing the funding for this research.

Nomenclature

Ji	transmembrane flux of species, i mol/(m^2.s) or kg/(m^2.s)
${\rho }_i$	density of species, i kg/m^3
v	velocity of solution, m/s
Dim	effective diffusion coefficient of component i in the membrane, m^2/s
$\nabla {\rho }_i$	density gradient, kg/m^3
k	Darcy permeability constant, m^2
$\mu$	solution viscosity, Pa.s
g	gravitational acceleration, m/s^2
Jw	transmembrane water flux, kg/(m^2.s)
cm	membrane coefficient, kg/(Pa.m^2.s)
$∆p$	hydrostatic operating pressure difference between two sides of membrane, Pa
$∆\pi$	osmotic pressure difference between two solutions on both sides of membrane, Pa
Dw	diffusion coefficient of water in the dense membrane, m^2/s
R	universal gas constant, J/(mol.K)
Sw	water solubility in the membrane, kg/m^3
Vm	molar volume of water, m^3/mol
T	temperature, K
ci	concentration of species, i mol/m^3
SR	salt rejection, %
Cp	permeate concentration, mol/m^3
Cf	feed concentration, mol/m^3
PR	permeate recovery, %
mp	produced permeate flow rate, kg/(m^2.s)
mf	feed flow rate, kg/(m^2.s)
σ	surface tension, N/m
θ	contact angle formed between the fluid and the membrane, $°$
dp	average diameter of the membrane pores, m
Pmf	water vapor partial pressure at the feed-membrane interface, Pa
Pmp	water vapor partial pressure at the permeate-membrane interface, Pa
Com	ordinary molecular mass transfer coefficient, kg/(Pa.m^2.s)
Mw	molecular weight of water, kg/mol
${\epsilon }_m$	membrane porosity, −
${\tau }_m$	membrane tortuosity, −
${\delta }_m$	membrane thickness, m
Pair,mf	partial pressures of air at the feed-membrane interface, Pa
Pair,mp	partial pressures of air at the permeate-membrane interface, Pa
CH-P	Hagen–Poiseuille mass transfer coefficient, kg/(Pa.m^2.s)
${\mu }_v$	water vapor dynamic viscosity, Pa.s
CKn	Knudsen mass transfer coefficient, kg/(Pa.m^2.s)
$\lambda$	mean free path of water molecules, m
Qc	heat loss by thermal conduction, W/m^2
Qv	heat transferred by water vapor flux, W/m^2
km	thermal conductivity of the membrane, W/(m.K)
Tmf	temperature at the feed-membrane interface, K
Tmp	temperature at the permeate-membrane interface, K
kw	thermal conductivity of water vapor, W/(m.K)
kPolymer	thermal conductivity of polymer, W/(m.K)
$\eta$	thermal efficiency, −
GOR	gain output ratio, −
Am	effective membrane area, m^2
Cp	specific heat capacity at constant pressure, J/(kg.K)
Tfi	feed inlet temperature, K
Tpo	permeate outlet temperature, K
W	width of channels, m
H	height of channels, m
uin	inlet velocity, m/s
uout	outlet velocity, m/s
ci,in	concentration of species i at the inlet of channels, mol/m^3
ci,out	concentration of species i at the outlet of channels, mol/m^3
zi	valence of ions, −
Mi^m	mobility of species i at membrane medium, m^2/(V.s)
ci^m	concentration of ion i at IEM interfaces, mol/m^3
$∆{\phi }_m$	voltage difference across IEMs, V
lm	thickness of the membrane, m
Di^m	diffusion coefficient of species i at membrane medium, m^2/s
$∆c_i^m$	ion concentration difference across IEMs, mol/m^3
um	average water velocity through IEMs, m/s
F	Faraday’s constant 96485.33, (A.s)/mol
I	electric current density, A
VFw	volume fraction of pores filled by water in IEMs, −
Di^s	ion diffusivity in the feed solutions in channels, m^2/s
ri	radius of counter-ion i moving within the IEM, m
${\sigma }_m$	membrane surface charge density, C/m^2
li^d	thickness of the double layer, m
$\epsilon$	permittivity, F/m