In a previous subsection, the water flux for PRO was derived as given in (7). One should note that the water flux is based on an ideal assumption that $\Delta \pi $ is only represented by the solute concentration difference between the bulk feed solution (${C}_{F,b})$ and bulk draw solution (${C}_{D,b})$. Although such an ideal assumption often provides convenience in observing the overall tendency of a PRO process, the values resulting from the water flux with an ideal assumption are usually incorrect due to the lack of the reflection of the concentration polarization. Concentration polarization (CP) is a phenomenon which is led by an encounter between the water flux and the salt flux, occurring in the inside and near-outside of a membrane. To understand CP, the basic transport phenomena around a membrane first should be acknowledged.

#### 2.2.1. Internal Concentration Polarization

As the solvents in the solution travel across a membrane, the solutes in the solution cross a membrane due to the solute concentration difference between the feed and draw sides. This phenomenon of solute transport across a membrane is called salt flux. The physical meaning of salt flux is the mass rate of solute penetrating per unit membrane area (

$\frac{g}{{m}^{2}\xb7h})$. Understanding the factor of salt flux is imperative for understanding the system dynamics and CP in membrane-based desalting processes. As shown in

Figure 3, the water flux and the salt flux in a PRO system, as well as in a FO system, tend to flow in opposite directions.

That is, while the water flux in a PRO system tends to stream from the feed side to the draw side, the salt flux streams from the draw side to the feed side. This contrary tendency results from the difference of media which drive the flow of each flux. The opposite directions of each flux, first of all, result in the isolation of salts inside a membrane—imagine a situation in which the salts try to reach the feed side but fail because of the water flux serving as shoving the salts back to the draw side. Consequently, some amounts of salts are stranded in a membrane permanently and form a new concentration located inside a membrane,

${C}_{ICP}$. This phenomenon is called internal concentration polarization (ICP). ICP is one of the major burdens imposed on FO and PRO processes, as it strongly aggravates the performance of those processes. As ICP is taken into consideration, the substantial osmotic pressure difference changes from the one estimated with

${C}_{F,b}$ and

${C}_{D,b}$ to the one estimated with

${C}_{ICP}$ and

${C}_{D,b}$. Such a change in the substantial osmotic pressure difference is directly linked to the decrease in the driving force. Naturally, the performance of a PRO process decreases, as well. If the composition of solutes is not just confined to a pure salt, the impact of ICP grows even larger. As the composition of solutes gets intricate, the interactions among the solute compounds or organisms aggravate the performance of a membrane. For example, the microbes in the feed and draw solutions produce byproducts such as polysaccharides and proteins [

13]. Such byproducts get accumulated inside a membrane and result in a severer ICP. In addition, the increased microbes also clog the pores in a membrane so that the performance of a process essentially goes down. One of the ways to relieve such an impact of fouling propensity worsened by ICP is to switch the membrane orientation [

13,

31]. A PRO process usually sets the membrane orientation of a system as

Figure 3a, which is called the AL-DS (Active Layer–Draw Solution) mode, due to its higher initial water flux than that of

Figure 3b, which is called the AL-FS (Active Layer–Feed Solution) mode [

32,

33]. However, the AL-DS mode is more vulnerable to the fouling propensity incurred inside a membrane than the AL-FS mode since the porous structure of the support layer plays a role to trap the foulants from the feed solution more sensitively [

34]. Therefore, while the same amount of foulants is accumulated on the surface of a membrane, the water flux in the AL-DS mode declines more rapidly than that in the AL-FS mode [

13,

31]. However, a sequence in which ICP induces the fouling further or the fouling induces ICP further depends on the composition of foulants in the raw water [

13,

17,

35,

36]. Thus, an operator should choose whether to apply the AL-DS mode or the AL-FS mode for a given system in accordance with the compound composition of the intake water. As the orientation of a PRO membrane is switched, the equations of the models are also slightly changed since the solute concentration distribution inside a membrane changes. In the current paper, the equations of the AL-FS mode will not be represented additionally because most of the research works regarding a PRO process are based on the AL-DS mode.

ICP is an inevitable phenomenon in the FO and PRO processes since the directions of water flux and salt flux in the processes are always oriented reversely. To analyze ICP mathematically, the mathematical expression of the salt flux should be noted first [

37,

38].

Here,

${J}_{s}$,

$B$,

${C}_{D,m}$, and

${C}_{F,m}$ represent the salt flux, the salt permeability, the solute concentration of the draw solution at the adjacent area of the membrane, and the solute concentration of feed solution at the adjacent area of the membrane, respectively. Lee et al. [

37] developed a theoretical model for the PRO process based on the conservation of the mass resulting from the water flux and the salt flux. Assuming, for the sake of simplicity, that there is only one solute (i.e., salt) in the draw solution, the mass transfer of the salt entering the membrane support layer at each boundary layer is always equal to the sum of the salinity displacement due to convection and diffusion caused by the salinity difference. Thus, at a steady-state, the conservation of mass in a system can be described as follows, [

39]

As noted in a previous subsection,

$C\left(x\right)$ is the solute concentration at position

$x$; and

${D}_{s,l}$ is the diffusion coefficient at the support layer of a membrane (

Figure 3), which is defined as [

40],

Here,

$D$ is the diffusion coefficient, which is commonly referred to as diffusivity, in the bulk solution;

$\epsilon $ is the porosity of the support layer; and

$\tau $ is the tortuosity of the support layer. A detailed description of these parameters will be covered a bit later. By rearranging Equation (13) appropriately, the following relation is derived,

If (15) is integrated along with the boundary conditions shown in Equation (16), the solute concentration of ICP (

${C}_{icp}$) comes out as given in Equation (17) [

14],

where

$x$ is the distance from the support layer to the active layer boundary; and

${t}_{s}$ is the thickness of the support layer of a membrane (

Figure 3). A new term that emerges in Equation (17),

$K$, represents the solute resistivity in the support layer and is defined as [

41]

As shown in Equation (18), $\frac{\tau {t}_{s}}{\epsilon}$ is generally expressed as $S$, structure parameter. As mentioned before, three membrane parameters, $A$ (water permeability), $B$ (salt permeability), and $S$ (structure parameter), are considered the main parameters displaying the properties of a membrane. The roles of $A$ and $B$ are relatively clear. In order to enhance the performance of a system, the operators of a PRO process should increase the water flux by augmenting the value of $A$ as much as possible: see Equation (7). In the meantime, the salt flux should be decreased as much as possible, so that the value of $B$ must be repressed: see Equation (12). However, understanding the role of $S$ is a bit more complicated than understanding those of $A$ and $B$. When considering Equation (18), it is apparent that $K$ is proportional to the value of $S$. Considering that ${C}_{icp}$ increases as $K$ increases, the value of $S$ should be kept minimal to maintain a high substantial osmotic pressure difference. However, the reduction of $S$ must be in a balance with the tortuosity and the thickness of the support layer because excessively low values of those terms can have harmful impacts on the mechanical strength of a membrane. In short, recalling that $S=\frac{\tau {t}_{s}}{\epsilon}$, the thickness of the support layer and the tortuosity should be as low as possible while the porosity is kept high. Therefore, satisfying these features should be the priority when fabricating a PRO membrane.

#### 2.2.2. External Concentration Polarization

ICP is a phenomenon resulting from a lump of concentration located inside a membrane. Unlike ICP, there are two more CPs that occur outside of a membrane, known as external concentration polarization (ECP). The names of ECP differ according to regions where CP occurs. If ECP occurs at an interface region between the draw solution and a membrane, such a CP is referred to as dilutive ECP because the interface region becomes diluted by water which has crossed a membrane. By contrast, ECPs occurring at an interface region between the feed solution and a membrane is called concentrative ECP due to the impact of salt shifting from the draw side to the feed side. Much like the term ICP was derived, the terms for each ECP phenomenon are derived based upon the interaction with boundary conditions. The boundary conditions and the solute concentration at the interface region of the draw side are given as follows (

Figure 3):

If the integration of Equation (15) is carried out according to the boundary conditions as in the case of ICP, then the solute concentration at the interfacial region of the membrane and the draw solution (

${C}_{D,m}$) comes as follows:

Meanwhile, the boundary conditions and the solute concentration at the interface region of the feed side are given as follows:

In the same manner, the concentration at the interfacial region of the membrane and the feed solution (

${C}_{F,m}$) can be obtained with the integration as follows:

Here,

$\delta $ and

$k$ represent the thickness of the solution boundary layer and the mass transfer coefficient, respectively. The subscripts appended to the notation each indicates a different region: subscript

$F$ indicates the feed solution;

$D,$ the draw solution. It should be noted that the thickness of the boundary layer and mass transfer coefficient are interrelated based on the famous stagnant film theory [

41,

42,

43,

44,

45],

In other words, the thickness of the solution boundary layer diminishes as the mass transfer coefficient of a given system increases. Thus, understanding the characteristics of the mass transfer coefficient can aid in designing a complete system. In membrane-based desalting systems, the mass transfer coefficient can be calculated in two different ways: one is based on a dimensionless number concerned with an empirical method, and the other is based on an equation derived by the diffusion-convection equation. The first empirical method is as follows [

14]:

Here,

${d}_{h}$ and

$Sh$ represent the hydraulic diameter and the Sherwood number, respectively.

$Sh$ is a dimensionless number which represents the ratio of the convective mass transfer to the rate of mass diffusion. In the PRO system, empirical relations for

$Sh$ are given as follows [

14,

46,

47]:

It should be noted that the dimensionless constants in the relations (i.e.,

${a}_{1}$,

${a}_{2}$,

$b$, and

$c$) are determined in experimental procedures by separating a turbulence state (Equation (25)), and a laminar state (Equation (26)). Two newly emerged terms in Equations (25) and (26),

$Re$ and

$Sc$, also represent dimensionless numbers, which are called the Reynolds number and the Schmidt number, respectively. These dimensionless numbers illustrate the physical and motional characteristics of fluids in a given system. For example, when the value of

$Re$ is more than 2000 for a completely stabilized fluid, the fluid motion of the system is generally classified as a turbulence state [

48]. Such properties of

$Re$ and

$Sc$ imply that

$Sh$ also can illustrate the characteristics of fluids in a given system, given that

$Sh$ is expressed by a linear relation of

$Re$ and

$Sc$. When considering this sequential logical flow, a final conclusion can be drawn: that the area of the ECP boundary layer will decrease as the value of

$k$ increases: see Equations (23) and (24). Although the diffusivity and the hydraulic pressure are engaged when calculating

$k$, the effects of those parameters upon

$k$ are relatively small unless a dramatic change in the given system occurs [

33,

49,

50]. Furthermore,

$D$ is canceled out if Equations (23) and (24) are incorporated. Thus it can be concluded that the motional characteristics of fluids in the system play a primary role in determining the impact of ECP. The motional characteristics of fluids are mainly controlled by the operation conditions such as the velocity of inlet solutions or the hydraulic pressure. However, the motional characteristics of fluids can be also controlled to some extent with a membrane spacer or other auxiliary parts. The membrane spacer contributes to promoting the displacement of fluids by enhancing the mass transfer of the system (i.e., increasing the value of

$k$) [

51,

52]. Furthermore, it was shown that the value of

$Sh$ can be significantly varied as the geometry of a membrane spacer and the frequency of the spacer channels change, according to the previous studies [

51,

52,

53]. That is, the way a membrane spacer takes its own shape can also be a critical factor to relieve ECP.

The fact that the impact of ECP is highly dependent upon the motional characteristics of fluids is intriguing given that by contrast, the impact of ICP is mainly dependent on the parameters of a membrane. During the nascent period of the PRO process, ECP was often dismissed because of its small impact compared to ICP. However, as the scale of the process became sizable, it was found that the impact of ECP could no longer be overlooked. Consequently, to address the optimization of a membrane structure and the control over the fluid dynamics simultaneously became a significant goal for researchers.

The second way to find the mass transfer coefficient in the membrane-based desalting process is to harness an equation derived from the diffusion-convection differential equation. The equation is given as,

known as Brian’s equation, which describes a relationship between the mass transfer coefficient and the water flux [

54,

55,

56]. The rearranged form of Equation (27) leads to the advent of a new dimensionless number, called

$Pe$, as follows,

$Pe$ is the Péclet number, which is defined as the ratio of the advection rate to the diffusion rate. As

$Sh$ was, so

$Pe$ is represented by the linear relation of

$Re$ and

$Sc$: in other words,

$Pe$ represents the motional characteristics of a system as

$Sh$ does. In theory, this implies that the motional characteristics of fluids can be described only with the concentration deployments in a given system, or the ratio of the water flux to the mass transfer coefficient (

Figure 4). Thus, the importance of the mass transfer coefficient is again apparent, given that it can contribute to illustrating the overall tendency of a system.