Desalination by Membrane Distillation

Abstract

At present, around 25% of water desalination processes are based on distillation. Similar to classical distillation, membrane distillation is a phased-change process in which a hydrophobic membrane separates two phases. Membrane distillation is considered an emerging player in the desalination, food processing and water treatment market. Due to its high salt rejection, less fouling propensity, operating at moderate temperature and pressure, membrane distillation is considered as a future sustainable desalination technology. The distillation process is quite well known in desalination. However, membrane distillation emerged a few decades ago, and a thorough understanding is needed to adapt this technique in the near future. This review chapter introduces the classical distillation and membrane distillation as an emerging technology in the desalination arena. Heat and mass transfer and thermodynamics in membrane distillation, characteristics of the performance metrics of membrane distillation are also described. Finally, the performance evaluation of MD is presented. The possibility of using low-grade heat in membrane distillation allows it to integrate directly to solar energy and industrial waste heat.

1. Introduction

Distillation is a thermal process in which a component separates out from a multi-component mixture by a phase-change process. When a membrane is put in between the feed and condensing solution, the feed can be vaporized and condensed at the membrane interface (thickness $\approx$ 200 $\mathit{\mu m}$). Figure 1 shows a schematic of a single-stage distillation and membrane distillation setup. The membrane distillation (MD) process combines the use of conventional distillation and membranes processes. It is a hybrid technology that uses the advantages of membrane separation and thermal distillation processes. In MD, the separation process employs a porous hydrophobic membrane between feed and permeate, allowing only solvent vapor to pass through, retaining the liquid/solid phase. Although the membrane provides a mass transfer resistance to the vapors, its employment allows water to condense within a minimal distance. This gives an advantage in creating a large partial pressure gradient across the membrane.

Figure 1.

A schematic of a typical single stage distillation system and the membrane distillation system. In MD, the vapors generate at the feed-membrane interface, and move to the permeate side and membrane allows only vapors to pass through.

In a typical MD setup, the trans-membrane temperature difference (temperature difference between two sides of the membrane) creates the vapor pressure difference, which drives mass transport [1, 2, 3, 4, 5]. The vapor is generated at the feed side and moves through the membrane pores to condense/get collected at the permeate side of the membrane. The salts, due to their non-volatile nature, remain in the feed solution.

MD has advantages over conventional desalination techniques. These include [6]:

• Low feed temperature in the range 50–90°C.

• Low fouling propensity.

• Capable to treat high saline water.

• High product water quality.

• Good compromise between specific enthalpy and efficiency.

Based on the distillate collection methods, MD can be classified into four broad configurations: (1) air gap membrane distillation (AGMD); (2) sweeping gas membrane distillation (SGMD); (3) vacuum membrane distillation (VMD); and (4) direct contact membrane distillation (DCMD) [7]. In all configurations, the feed solution remains in direct contact with one side of the membrane, while on the other side, distillate collection differs based on the type of the variant.

1. Air gap membrane distillation: In AGMD, the permeate side has an air gap followed by a condensing plate. A typical AGMD set up is shown in Figure 2(A). The vapor transfers from the hot feed to the air gap and eventually condenses at the cooling plate. The cooling plate can be cooled with multiple ways such as liquid cooled, air cooled, evaporative cooled; and gives the flexibility to use any coolant liquid as it does not mix with the condensate. AGMD is the most practical configuration in water production. However, due to high mass transfer resistance at permeate side, the flux is less as compared with other configurations. The air gap reduces the conduction heat loss from feed to permeate [8, 9, 10].

2. Sweeping gas membrane distillation: The SGMD uses a stream of gas to strip off the vapors from the permeate channel, after which they are either condensed externally or discarded as waste. Figure 2(B) shows a typical SGMD set up. SGMD shows less conduction heat loss than DCMD due to less conductivity of gases at the permeate side. The pore wetting possibility is very less as the flow of gas takes off the distillate without condensing it [11]. The main disadvantage of this configuration is the high gas flow to sweep a unit volume of the permeate product. SGMD can be used in desalination and concentration of non-volatile liquids [7, 8].

3. Vacuum membrane distillation: The VMD and SGMD are very much similar in stripping out the permeate product. The VMD uses vacuum suction to strip off the vapors from permeate channel and either condense those externally or discarding as waste. Figure 2(C) shows a typical VMD configuration. The main advantage of this configuration is to separate the volatile organics from the main stream. However, it requires an additional pump and a condenser, increasing both CAPEX and OPEX. Also, the possibility of wetting the membrane is high due to the negative pressure at the distillate side [5, 8].

4. Direct contact membrane distillation: The DCMD is the most widely used MD configurations due to its simplicity, low capital cost and comparatively higher flux [12, 13, 14]. In DCMD, as the name suggests, the membrane remains in direct contact with both solutions, i.e., feed and permeate. Figure 2(D) shows a schematic of a typical DCMD set up. The vapors transfer through the membrane and condense in the circulating coolant at the other side of the membrane. To achieve high thermal efficiency, the feed and coolant stream runs in counter-current directions. DCMD is best suited for applications where water is a major permeate, such as desalination and concentration of fruit juices. Also, the condensation step is carried out inside the module itself. High conduction losses across the membrane are the main disadvantage of DCMD [8, 15].

Figure 2.

A schematic of the four different MD configuration (A–D). The feed water flows tangentially in a cross flow regime at the feed side. At the permeate side, distillate taken off through various mechanisms and membrane allows only vapors to pass through.

Table 1 summarizes the applications, advantages and disadvantages of different MD configurations [4, 6, 7, 8, 9, 11, 15, 16, 17].

1.1 Transport phenomena

The temperature gradient between the feed and distillate sides of the MD system result in heat transfer from hot feed to cold distillate accompanied by mass transfer.

1.1.1 Heat transfer

MD is a non-isothermal process. Due to the difference in temperature at feed and permeate sides, three main heat transfer mechanisms take place: convective heat transfer, conduction heat transfer, and latent heat transfer. At the feed side and permeate side, convective and latent heat transfer take place. Across the membrane, conductive and convective heat transfer takes place. The two interfaces show convection with the bulk fluid, and membrane pores demonstrate the conduction phenomenon associated with vapor heat transfer. Due to a positive temperature gradient, it is clear that the heat transfer takes place only from feed to permeate.

1.1.1.1 Feed side

At feed side, hot bulk solution come in contact with the MD membrane. Convective heat transfer $Q_f$ (W·m⁻²) takes place and can be expressed as per Newton’s law of cooling [14]:

$$Q_f=h_f\ast \left(T_{\mathit{fb}}-T_{f,m}\right)$$E1

Where $h_f$ is the feed convective heat transfer coefficient (W·m⁻²K⁻¹), $T_{\mathit{fb}}$ is the bulk feed solution temperature (K), $T_{f,m}$ is the temperature (K) at feed-membrane interface.

1.1.1.2 Membrane pore

Heat transfer through membrane occurs in two parallel routes: first is latent heat transfer $Q_v$ at pores mouth and second is conduction heat transfer $Q_c$. The $Q_c$ is the conduction heat loss and does not contributes to the vapor mass. Hence, for an efficient system, it is required to minimize $Q_c$ as low as possible. It can be expressed by the Fourier’s law of conduction. Therefore, thicker membrane contributes less to conduction heat transfer. The $Q_m$ (W·m⁻²) is the sum of both heat transfers and can be expressed as [2, 5]¹:

$$Q_m=Q_c+Q_v=\frac{k_m}{{\delta }_m}\ast \left(T_{f,m}-T_{p,m}\right)+J\ast h_{\mathit{fg}}$$E2

$$h_m=\frac{k_m}{{\delta }_m}$$E3

Where $Q_c$ is the conduction heat transfer (W·m⁻²), $k_m$ is the membrane thermal conductivity (W·m⁻¹·K⁻¹), ${\delta }_m$ is the membrane thickness (m), $J$ is the distillate flux (kg·m⁻²h⁻¹), $h_{\mathit{fg}}$ is the latent heat of evaporation (kJ·kg⁻¹), $T_{p,m}$ and $T_{f,m}$ are the temperature of permeate-membrane interface and feed-membrane interface respectively.

The thermal conductivity $k_m$ (W·m⁻¹K⁻¹) of the membrane can be calculated from the thermal conductivity of vapor $k_g$ and the polymer material $k_p$ as per following relations [18]:

$$k_m=\epsilon k_g+\left(1-\epsilon \right)k_p$$E4

Where $\epsilon$ is the porosity of the membrane.

1.1.1.3 Permeate side

Vapor that pass through membrane pores condense at the permeate side at the expense of latent heat of condensation. The heat transfer at the permeate side $Q_p$ (W·m⁻²) can be expressed as [15, 18]:

$$Q_p=h_p\ast \left(T_{\mathit{pb}}-T_{p,m}\right)$$E5

Where $h_p$ is the permeate convective heat transfer coefficient (W·m⁻²K⁻¹), $T_{\mathit{pb}}$ is the bulk permeate solution temperature, $T_{p,m}$ is the temperature (K) at permeate-membrane interface.

1.1.1.4 Overall heat transfer

At steady state, heat transfer from feed to membrane, across the membrane and from membrane to permeate are equal. Figure 3 shows with the electrical analogy of local heat transfer coefficients and their relationship with the overall heat transfer coefficient. The overall heat transfer $Q$ (W·m⁻²) can be expressed in terms of universal (overall) heat transfer coefficient as $U$ [2, 18]

Figure 3.

Electrical equivalent of heat transfer resistance in MD process. An overall heat transfer resistance is an equivalent of all three heat transfer resistance of feed, membrane and permeate.

$$\begin{array}{l}Q=Q_f=Q_m=Q_p\\ =h_f\ast \left(T_{\mathit{fb}}-T_{f,m}\right)=\frac{k_m}{{\delta }_m}\ast \left(T_{f,m}-T_{p,m}\right)+J\ast h_{\mathit{fg}}=h_p\ast \left(T_{\mathit{pb}}-T_{p,m}\right)\\ =U\ast \left(T_{f,m}-T_{p,m}\right)\end{array}$$E6

Where U (W·m⁻²) can be expressed as:

$$\frac{1}{U}=\frac{1}{h_f}+\frac{1}{\frac{k_m}{{\delta }_m}\ast \frac{{\mathit{Jh}}_{\mathit{fg}}}{\left(T_{f,m}-T_{p,m}\right)}}+\frac{1}{h_p}$$E7

The membrane interface temperatures, i.e. $T_{f,m}$ and $T_{p,m}$ can not be measured experimentally. Therefore, mathematical iterative procedure is generally used to evaluate both interface temperatures. The membrane interface temperature can be derived from the Eq. (6) and expressed as [15, 19]:

$$T_{f,m}=T_{\mathit{fb}}-\frac{J{h}_{\mathit{fg}}+\frac{k_m}{{\delta }_m}\left(T_{f,m}-T_{p,m}\right)}{h_f}$$E8

$$T_{p,m}=T_{\mathit{pb}}-\frac{J{h}_{\mathit{fg}}+\frac{k_m}{{\delta }_m}\left(T_{f,m}-T_{p,m}\right)}{h_p}$$E9

The heat transfer coefficients $h_f$ and $h_p$ can be calculated using Nusselt number relation as shown in Eq. (10), while the $h_m$ can be calculated from Eq. (3). The Nusselt number correlations are available in the literature [2, 4, 5].

$$\text{Heat\hspace{0.17em}transfer\hspace{0.17em}coefficient}=\frac{\mathit{Nu}\ast k_m}{D_p}$$E10

Where $h_m$ is defined in Eq. (3)

1.1.2 Mass transfer

Mass transport in MD can be explained by three sequential stages: vapor generation, vapor transport, and vapor condensation, which respectively take place at the feed-membrane interface, membrane pores, and permeate-membrane interface. Figure 4 shows the mass transfer process, including vapor generation at the feed side, vapor transport through the porous membrane, and finally, vapor condensation at the permeate side. There is resistance to mass transfer at each stage, which is defined by a correlation explained in the sequel. The overall mass transfer in the MD system is often expressed using Darcy’s law i.e., the vapor pressure difference between two sides of the membrane, and is given as [1, 2]:

Figure 4.

A schematic of vapor transfer from vapor generation at the feed side, to the permeate side through the membrane pores. Mass transfer is the transport mechanism within the membrane pores. Finally, the distillate condenses at the permeate side.

$$J=C_m\ast \left(p_{f,m}-p_{p,m}\right)$$E11

Where $C_m$ is the membrane mass transfer coefficient (kg·m⁻²h⁻²·Pa⁻²), $p_f$ and $p_p$ are the partial pressures of water at feed and permeate sides, respectively. The three stages of mass transport are explained as below:

1.1.2.1 Stage 1 (Vapor generation)

At the feed side of the membrane, liquid feed remains in contact with the membrane pores. The vapors are generated at the feed-membrane pore interface when the hot feed solution passes over the hydrophobic MD membrane. The partial pressure of the vapor generated is directly proportional to the temperature as per Antoine’s equation [19]. In the case of water, Antoine equation can be rewritten as Eq. (12):

$$p_v=\mathit{exp}\left[23.19-\left(\frac{3816.44}{T_{f,m}-46.13}\right)\right]$$E12

where $p_v$ is the vapor pressure of water (Pa) and $T_{f,m}$ is the temperature at the feed-membrane interface (K).

The presence of non-volatile solute in the feed water decreases the vapor pressure of the feed solution. Therefore, the vapor pressure is a function of mole fraction of that component and can be expressed as [4]:

$$p_{\left(x\right)}=\left(1-x_s\right)\ast p_w^o$$E13

Where $x_s$ mole fraction of non-volatile solute in feed water and $p_w^o$ is the partial pressure of pure water.

Figure 5 shows a control volume of heat and mass transfer of length $\mathit{dz}$. A feed of mass flow rate $\dot{m}$ fed into the feed channel, and heat and mass transfer takes place along the membrane surface. The width $\mathit{dy}$ is taken as unity. The mass balance at feed side can be explained mathematically as [13, 20]:

Figure 5.

A schematic of the mass balance in a control volume at the feed side and permeate side. The mass transfer takes place through the porous membrane along the vapor pressure gradient.

$${\dot{m}}_{f,\left(z+\mathit{dz}\right)}{h}_{\mathit{fb}}={\dot{m}}_{f,\left(z\right)}{h}_{\mathit{fb}}-\left(J{h}_{\mathit{fg}}+q_m\right)\mathit{dA}$$E14

The mass flux of water that passes through the membrane can be expressed as:

$$J=k_f{\rho }_{f}ln\frac{x_{f,m}}{x_f}$$E15

Where $k_f$ is the mass transfer coefficient and can be calculated using Sherwood number as follow [4]:

$$\mathit{Sh}=\frac{k_f{d}_h}{D_s}$$E16

Where $d_h$ is the hydraulic diameter (m), $D_s$ is the solute diffusion coefficient in bulk feed (m²·s⁻¹). The Sherwood number can be calculated using following semi-empirical relationship [4]:

$$\mathit{Sh}=\alpha {\mathit{Re}}^{\beta }{\mathit{Sc}}^{\gamma }$$E17

Where $\alpha$, $\beta$, and $\gamma$ are the coefficients calculated from experiments. Sc is the Schmidt number and Re is the Reynolds number, which can be expressed as [2]:

$$\mathit{Sc}=\frac{{\eta }_f}{{\rho }_f{D}_s}$$E18

$$\mathit{Re}=\frac{{\rho }_f{v}_f{d}_h}{{\eta }_f}$$E19

Where ${\eta }_f$ is the viscosity of the bulk fluid (Pa·s), ${\rho }_f$ is the density of the bulk fluid (kg·m⁻³), and $v_f$ is the fluid flow velocity (m·s⁻¹). Different correlations used for Sherwood number are listed in Appendix 1A.

1.1.2.2 Stage 2 (Vapor transport)

Two major factors control the vapor transfer in MD membrane pores. One is the vapor pressure difference $\mathrm{\Delta }p$, and the second is the mass transfer coefficient of the membrane. The vapor transfer through the membrane may be the limiting step for mass transfer in MD, which is influenced by the physical properties of the membrane and other characteristics which expresses with the relation $J\propto \frac{D_p\epsilon }{\chi {\delta }_m}$ [1].

1. Porosity ($\epsilon$): In general, regardless of the type of MD configuration, membranes with high porosity have more distillate flux as well as lower conductive heat loss. The porosity of MD membranes lies between 30 and 85% [4, 21].

2. Tortuosity ($\chi$): In a simple assumption, membrane pores are considered as straight cylindrical channels. However, they possess many curved paths. High tortuosity value leads to lower distillate flux due to vapor permeation through tortuous paths. Therefore, membrane permeability is inversely proportional to the membrane tortuosity. In most theoretical models in MD studies, a tortuosity value of 2 is frequently considered to predict the transmembrane flux [15].

3. Pore size: Large pore size allows more vapor to pass through. However, after a certain point, it will limit the applied pressure to avoid pore wetting. On the other hand, small pore size enables working at high pressures but at the cost of lower fluxes [4].

The vapor transport mechanism through MD membrane pores is governed by three basic mechanisms known as Knudsen diffusion ($K_n$), Molecular-diffusion, and viscous (poiseuille flow). The mass transport through the membrane is described by the simple Darcy’s law, expressed in Eq. (11). For dilute solutions, it can be written as [7, 22]:

$$J=C_m\ast \frac{\mathit{dP}}{\mathit{dT}}\left(T_{f,m}-T_{p,m}\right)$$E20

The Pressure term $\frac{\mathit{dP}}{\mathit{dT}}$ can be calculated from Clausius-Clapeyron equation as:

$$\frac{\mathit{dP}}{\mathit{dT}}=\frac{\mathrm{\Delta }H}{{\mathit{RT}}^2}{P}_{\mathit{av}}\left(T_m\right)$$E21

The pressure $P_{\mathit{av}}$ for non-ideal aqueous solution can be calculated from Raoult’s law as expressed by Eq. (13).

As per Darcy’s law Eq. (11), membrane mass flux is mainly governed by the partial pressure difference between feed and permeate side and the membrane mass transfer coefficient $C_m$. The membrane mass transfer coefficient is primarily a function of membrane properties (thickness, pore size, tortuosity) and the process conditions (temperature and pressure). Its value depends upon the mass transfer mechanism inside the membrane pore.

The air molecules (particles) act as a medium in the pores. Knudsen number ($K_n$) determines the type of governing mechanism of mass transport inside the membrane pore and can be expressed as [1]:

$$K_n=\frac{\lambda }{D_p}$$E22

Where $D_p$ is the pore diameter (m) $\lambda$ is the mean free path (m) of the vapor molecule and can be written as [19]:

$$\lambda =\frac{k_B{T}_r}{\sqrt{2}\pi P_m{\sigma }_v^2}$$E23

where $k_B$ is the Boltzmann constant (1.38 × 10⁻²³·J·K⁻¹), $P_m$ is the mean average pressure in membrane pores (Pa), $T_r$ is the average temperature in the pore, and ${\sigma }_v$ is the water vapor collision diameter (0.2641 nm).

Depending on the value of $K_n$, three possible mass transfer modes exist as follows: (a) Knudsen diffusion ($K_n>1$) in which molecular collisions with the walls dominate as compared to the molecule-molecule collisions, (b) molecular diffusion ($K_n<0.01$) in which the frequency of gas molecule collisions is much higher than those with the pore walls, and (c) Knudsen-molecular diffusion $\left(0.01<K_n<1\right)$ in which the frequency of molecular collisions with the pore walls is similar to that of the gas-gas collisions (often referred to as “transitional regime”) [4, 19]. Figure 6 shows a schematic of three possible mass transfer mechanisms through the pores of an MD membrane.

1. Knudsen diffusion: If the mean free path of water molecules is greater than the pore diameter ($\lambda >>D_p$), the frequency of collision between water molecules are less than the molecule–pore wall. The Knudsen number in this regime is $\ge$ 1 [19]. Therefore, it is the dominating process of mass transport in DCMD. The mass transfer resistance arises from the momentum transfer of vapor molecules with the sidewalls of the pores. Therefore, Knudsen diffusion decreases with the increase of temperature. In this case, the membrane mass transfer coefficient is shown in Table 2.

2. Molecular diffusion: Molecular diffusion occurs when the mean free path of vapor molecule ($\lambda <<D_p$) is much shorter than the pore diameter. The Knudsen number in this case is below 0.01 [5, 7]. Therefore, the intramolecular (within own molecules) and intermolecular (with the air molecules) collisions prevail vapor molecule-wall collisions. The mass transfer resistance comes from the collision of vapor molecules with the air molecules entrapped inside the membrane pores. In this case, the membrane mass transfer coefficient and mass transfer mechanism are shown in Table 2 and Figure 6 respectively. Since the feed and permeate in DCMD is deaerated, therefore, molecular diffusion in DCMD is minimum [2]. Similarly, in the case of VMD, the vacuum in the permeate side remove the entrapped air in the pore. Therefore, molecular diffusion is neglected there as well.

3. Poiseuille flow: In poiseuille flow (viscous flow), vapor molecule acts as a continuous fluid flow under a pressure gradient between two sides of membrane. It occurs in deaerated membrane pores under a stream vacuum. It applies when the mean free molecular path of vapor molecule is smaller that the pore size($d_p>100\lambda$) [2, 25]. The membrane mass transfer resistance is due to momentum transfer to the pore walls through viscous drag and its coefficient is shown in Table 2.

4. Knudsen-molecular diffusion: Transport of water vapors is a complex phenomenon and can encounter mass transfer resistance from both air molecule and the pore walls. In practical, there exist more than one type of transport phenomenon, therefore, called transition flow. This type of transport occurs when the mean free path of the vapor molecule is in the range (1–100) $\lambda$ and the flow is a kind of transition flow between pure Knudsen, and molecular diffusion [5]. The membrane mass transfer coefficient is shown in Table 2.

Figure 6.

(A) Various mechanisms of vapor transport through membrane pores are shown. (B) Mass transport regime based on the pore size. The transition regime occurs in the intermediate values of the Knudsen number. (C) Mass transport resistance and their electrical circuit analogy.

MD configuration	Application area	Advantage	Disadvantage
DCMD	Seawater desalination Industrial wastewater Dye effluents	High distillate flux Simple design and operation	High conductive heat loss High heat transfer coefficient
AGMD	Seawater desalination Industrial wastewater	Low conductive loss Low TP Less chances of distillate contamination High thermal efficiency	Low permeate flux High mass transfer resistance
SGMD	Brackish water desalination Azeotropic mixture separation VOC removal	Low conductive loss Less pore wettability No in-system condensation	High gas volume needed
VMD	Seawater desalination Aroma recovery Industrial effluents VOC removal	Low conductive loss High permeate flux Condensate obtained externally	Membrane wetting High pump power Heat recovery is difficult

Table 1.

Summary of the applications, advantages and disadvantages of different MD configurations [4, 6, 7, 8, 9, 11, 15, 16, 17].

Knudsen number	Type of transport	Condition	Mass transfer coefficient $C_m$
$>1$	Knudsen diffusion	$\lambda$$>>D_p$	$\frac{2\epsilon r}{3\chi \delta }\sqrt{\frac{8M_w}{\pi RT}}$
$0.01-1$	Knudsen-molecular diffusion	$\lambda <D_p<100\lambda$	${\left[\left(\frac{\chi \delta }{\epsilon }\ast \frac{R{T}_m}{M_w}\frac{P_a}{P_T}\frac{1}{D_w}\right)+\left(\frac{\left(3\chi \delta \right)}{\left(2\epsilon r\right)}\sqrt{\frac{\pi RT}{8M_w}}\right)\right]}^{-1}$
$<0.01$	Molecular diffusion	$\lambda$$<<D_p$	$\left(\frac{\epsilon }{\chi \delta }\frac{M_w}{R{T}_m}\frac{P_T}{P_a}\left(D_w\right)\right)$
$<0.01$	Poiseuille flow	$\lambda$$<<D_p$	$\left(\frac{1}{8}\frac{r_2\epsilon }{\chi \delta \mu }\frac{M_w}{R{T}_m}{P}_m\right)$

Table 2.

Mass transfer mechanism within the membrane pores follow a specific regime based on the Knudsen number. The mass transfer coefficient $\left(C_m\right)$ of each mechanism is shown [1, 2, 4, 5, 19, 23, 24].

1.1.2.3 Stage 3 (Vapor condensation)

At permeate side, each configuration of MD uses different methods to collect permeate. In some cases, the permeate is the waste and is discarded, while in other instances, permeate is the product and collected using various methods based on the configuration. In DCMD, the permeate is stripped by circulating pure water. Therefore, it offers no mass transfer resistance. In the case of VMD, the presence of vacuum also negates the existence of mass transport resistance.

However, AGMD show a significant resistance due to the presence of an air gap between the membrane and the condensing plate. In AGMD, the mass transfer occurs through molecular diffusion between the membrane pore surface to the condenser plate. The mass transfer resistance can be expressed by:

$$k_s=\frac{P_{\mathit{av}}}{P_a}\frac{\epsilon }{\delta \gamma +b}\frac{D_w{M}_w}{R{T}_m}$$E24

Where $b$ is the thickness of the air gap (m) [2, 4].

1.1.3 Temperature polarization

The phenomenon of evaporation at the feed-membrane interface and condensation at the permeate-membrane interface creates a temperature gradient with the bulk solution. Temperature polarization (TP) is a condition when the temperature at the membrane interface differs from its bulk solution [26, 27, 28]. TP is considered a critical factor that impacts the vapor flux of an MD system. The evaporation phenomenon at the liquid-air interface draws the latent heat from the bulk solution. Similarly, at the permeate side, the liquid-membrane interface releases heat of condensation to the coolant liquid. This creates a temperature difference between the bulk solution and the membrane interface. Figure 7 shows a schematic representation of temperature profile across the membrane [28].

Figure 7.

A profile of temperature deviation at interfaces with the bulk fluid on both sides of the membrane. The temperature change at the membrane interface $\mathrm{\Delta }{T}_m$ and bulk $\mathrm{\Delta }{T}_b$ is shown to determine the polarization coefficient.

There exist a thermal boundary layer at each side of the membrane. However, this thermal boundary layer does not have a significant effect in the case of AGMD, VMD, and SGMD. Moreover, the salt concentration at the feed-membrane interface increases due to mass transfer, leading to the concentration polarization (CP) phenomenon. However, the effect of CP is negligible in MD [18, 29]. The operation parameters such as fluid velocity, concentration, and temperature of feed solution affect the TP. Higher velocity increases the heat transfer by creating turbulences locally, hence diminishing the thermal boundary layer. Additionally, the lower concentration and higher temperature of the feed solutions produce high vapor pressure as per Rault’s law and Antoine’s Eq. (12) respectively. All the above parameters decrease the effect of TP. In addition, the membrane also plays an important role in TP due to the heat transfer across it. High porosity decreases the TP, while higher thermally conductive polymers show high TP. However, high thickness decreases the heat loss across the membrane as per Fourier’s law Eq. (2) and hence decreases TP. It is observed that TP decreases the vapor flux significantly [21].

1.1.3.1 Temperature polarization coefficient

The quantification of TP is expressed in temperature polarization coefficient (TPC). Figure 7 shows the temperature difference at the membrane interfaces with its bulk. TPC represented as $\mathrm{\Theta }$ is expressed as [19]:

$$\mathrm{\Theta }=\frac{\mathrm{\Delta }{T}_m}{\mathrm{\Delta }{T}_b}=\frac{\left(T_{f,m}-T_{p,m}\right)}{\left(T_{\mathit{fb}}-T_{\mathit{pb}}\right)}$$E25

The thermal boundary layer at both sides of the membrane acts as resistances to heat transfer. In other words, TPC is the ratio of thermal boundary layer resistance to the total heat transfer resistance. In the case of VMD, TPC can be simplified as the ratio of feed-membrane interface temperature to the bulk feed temperature [1], as expressed in Eq. (26).

$$\mathrm{\Theta }=\frac{T_{f,m}}{T_{\mathit{fb}}}$$E26

The TPC value, lies between 0–1 and most of the literature reports results between 0.4 and 0.7 [1, 3, 5, 30]. When the TPC value approaches zero, the system is limited by heat transfer at the feed side, which indicates an inefficient design. In contrast, TPC value 1 indicates that the system is affected by mass transfer resistance.

Several studies have been proposed to mitigate TP by using turbulence promotors such as spacers. However, this approach creates additional energy requirements [27, 31, 32, 33, 34]. Considering that TP and conduction heat loss is an intrinsic process deficiency that cannot be fully mitigated, it is highly desirable to seek alternative approaches to alleviate heat loss and achieve a sustainable MD performance.

2. Parameters that affect MD vapor flux

Most of the MD research is focused on maximizing vapor flux. However, taking vapor flux as a matrix to evaluate the thermal performance, may not be a correct approach since vapor flux depends upon many factors including MD system configurations, active membrane area, type of energy input, heat recovery from exiting feed etc. Therefore, the highest flux may not lead to the best thermal efficiency. Following are the parameters affecting the vapor flux:

1. Effect of feed temperature: The feed temperature is the most important factor and has a major effect on the vapor flux of an MD system. An exponential increase in vapor flux was observed with the increase in inlet feed temperature. This is due to the exponential increase of vapor pressure with temperature resulting in increase in driving force. Figure 8 shows the effect of feed temperature on permeate vapor flux on DCMD.

   Further, the conduction heat transfer is also higher at higher feed temperatures, leading to higher temperature polarization. Though, at high inlet feed temperature, the thermal efficiency is higher due to higher vapor flux, as shown in the Figure 8.

2. Effect of permeate temperature: In most of lab scale MD systems, the permeate temperature varies from 10 to 40°C. Increase in permeate temperature resulted in decrease in trans-membrane driving force, hence, vapor flux decreases. The effect varies from linear to exponential depending upon the membrane module characteristics [15, 35].

3. Effect of feed and permeate flow rate: The feed and permeate velocity influence the temperature polarization by creating the hydraulic turbulences in the feed and permeate channel respectively. The turbulences increase the heat and mass transfer coefficient ($h_f,h_p$) at the MD membrane’s feed and permeate boundary layer. The increased flow rate decreases the effect of concentration polarization. The vapor flux increases to an asymptotic value with an increase in feed velocity. Further, the thermal boundary layer thickness decreases at higher feed velocities. Kubota et al. (2020) revealed that the permeate flux increases with the feed velocity until it reaches a maximum and then decreases [7, 36].

4. Effect of the presence of non-condensable gasses: Non-condensable gasses such as CO₂ present in the feedwater stream. These gasses may mask the membrane pores, producing mass transfer resistance. Feedwater is generally aerated with N₂ gas before feeding in the MD system. However, this effect does not affect the VMD process. The process of deaeration consumes energy, and hence it decreases the overall performance [36].

5. Effect of solute concentration: High salt concentration decreases the vapor flux due to decrease in vapor pressure as per Roult’s law. Additionally, concentration polarization may develop at the vicinity of the membrane, producing mass transfer resistance. A reduction in coefficient of polarization ($\theta$) and thermal efficiency were observed with the increase in salt concentration [37]. Martinez et al. demonstrated that vapor flux decreased only by 1.15 fold by increasing the salt concentration by fivefold [38]. Therefore, MD can be used to treat high saline water with a slight decrease in productivity [1, 39].

6. Effect of module geometry: The configuration arrangement and the modular dimensions affect the TP in an MD system. Longer channel (keeping the width constant) leads to higher water production capacity, however, the flux decreases due to continuous decrease in feed water temperature along the length. The higher water production capacity is due to high water residence time with the MD membrane [40].

Figure 8.

Graph showing temperature effect on vapor flux of a DCMD configuration. The Line graph shows the variation in thermal efficiency with feed inlet temperature. Adapted from [15].

As the feed water moves along the length (with constant width), it loses heat due to latent heat and heat of conduction, which results in temperature decrease. Therefore, the permeate flux decreases. However, due to more residence time, the total water capacity increases. Therefore, there is an optimum length of the feed channel length. The water flux variation with the length, length/width ratio, surface area have been investigated to underline the affect. [40] presented a mathematical model simulation which shows that the total residence time of feed water in the feed channel has a positive effect until a limit, after that it started decreasing. The peak of the curve shows the optimum length of the feed channel.

Figure 9 shows the vapor flux data of four MD module types having surface area 4.5, 9.3, 19.5, 54, and 231 ${\mathit{cm}}^2$. The vapor flux was plotted with various variables. Figure 9(A) It is observed that the small active surface area membranes performance is higher than large surface area. This was a general consideration because the MD types were of different length and width. This can be deduced that regardless of width and length, the flux of lower active membrane surface area is higher than the larger ones, which is shown in the Figure 9(B). This shows a negative exponential correlation of the flux with the surface area.

Figure 9.

Vapor flux variation with respect to the modular dimensions of various MD flow cell sizes.

The length has a negative effect on the vapor flux as demonstrated in a simulation by Lee et al. [40]. Our experimental data verifies this relation as shown in Figure 9(C). If the length was kept constant, increasing the width was a positive effect on vapor flux [40]. The length to width ratio versus flux is plotted in the Figure 9(D) shows a positive slop.

As the length increases, the temperature of the feed drops due to the effect of latent heat and conduction heat loss along the length. The temperature drop increases. This was confirmed by the simulated results demonstrated by [40].

3. Membrane properties

Unlike Reverse Osmosis, MD membranes are not chemically involved in the mass transfer phenomenon. However, they are involved in the heat transfer phenomenon. MD membranes require some specific physical and chemical characteristics to perform well in the MD process. Mainly two types of membranes are used: Polyvinylidene Fluoride (PVDF) and Polytetrafluoroethylene (PTFE). Figure 10(A) shows the required characteristics of an ideal MD membrane, and Figure 10(B–C) shows an SEM topography of typical PVDF and PTFE membranes. PVDF membranes have round pores that could be seen on the top surface, while PTFE has pores strangled between the polymer fibers. The required physical and chemical characteristics of an MD membrane include:

1. High Porosity ($\epsilon$): Membrane porosity is the volume of pores to the total volume of the membrane. Membranes with high porosity produce more distillate flux due to more available channels for mass transfer and lower conductive heat loss. However, it decreases the mechanical strength and make membrane prone to crack under mild pressure. Porosity of MD membranes lies between 30 and 85% [21].

2. Adequate thickness (${\delta }_m$): The membrane thickness offsets both membrane permeability and heat transfer. Thicker membranes are suitable to prevent heat loss but will affect water vapor permeability [21].

3. Lower tortuosity ($\chi$): Tortuosity is the deviation of pores from a straight cylindrical path in porous media. In a simple assumption, membrane pores can be considered as straight cylindrical. However, this assumption is often far from reality. High tortuosity values lead to lower distillate flux due to vapor permeation through tortuous paths. Therefore, membrane permeability is in inverse relation to membrane tortuosity. In most theoretical models in MD studies, a value of 2 is frequently considered to predict the transmembrane flux [15].

4. Low thermal conductivity$\left(k_m\right)$: The high thermal conductivities facilitate sensible heat transfer, which leads to a decrease in temperature difference across the two sides of the membrane. Hence, vapor flux decreases. Therefore, low thermal conductivity is desired for MD membranes [22].

5. Low surface energy: Low surface energy correlates with high hydrophobicity. Materials with high hydrophobicity can transform to membranes with larger pores, which eventually increases the vapor flux. Also, high hydrophobic materials allow membrane processes to operate at high pressure for a given pore size [22].

Figure 10.

(A): The typical characteristics of an ideal MD membrane. (B–C): A surface topography of PTFE and PVDF membranes through SEM imaging. The pores in a typical PVDF and PTFE membranes are also depicted.

3.1 MD module design

DCMD is one of the most used MD configurations due to its simplicity. Different MD modules have been developed to address a variety of process requirements. Based on membrane type and design principle, MD modules have been developed into three different designs:

1. Flat sheet module: In the flat sheet module, a flat sheet membrane is put tangentially with the feed flow. Flat sheet membranes offer simple fabrication and assembly, easy cleaning and maintenance. The main disadvantages of the flat sheet are its low packing density, and high propensity to wetting [41]. Figure 11(A) shows a typical set up of flat sheet DCMD.

2. Hollow fiber module: In this type of module, hollow fiber membranes are used. Membranes are set in parallel bundles in the shell of a cylindrical casing. The main advantage of the hollow fiber module is the high packing density (surface area to volume ratio) and lack of membrane support. However, the permeate flux of the hollow fiber module is lower to that of the flat sheet. Additionally, the cleaning process and the membrane replacement is difficult in the hollow fiber module [4, 41]. Figure 11(B) shows a typical set up of hollow fiber DCMD.

3. Spiral wound: Flat sheets are spirally coiled in a cylindrically wound shape. The membrane is rolled with the spacer between the feed and coolant fluids, creating evaporator and condenser channels. Such type of system was first commercialized by Solar Springs as Oryx unit as shown in Figure 11(C) [42]. The condensation process first preheats the cold feed as a condenser stream in the spiral wound design. Then, preheated feed is heated to a required temperature before entering from its center. The feed solution flows axially in the evaporator channel, and the permeate which passes through the membrane flows spirally. The condenser channel is placed toward the shell side while feed towards the core side to get the advantage of counter-current circulation [6].

Figure 11.

(A): Flat sheet membrane module, showing a rectangular membrane between two channels. (B): A schematic of Hollow Fiber module set up, comprising hollow fiber membrane cased in a housing. (C): A spiral wound DCMD module commercialized by Solar Springs, adapted from [42].

4. Performance evaluation

The MD performance metrics can be divided into two thermodynamic categories: local and system level. Local metrics are impacted by local properties such as porosity, pore size, thickness, membrane conductivity etc. These include permeate flux and thermal efficiency. On the other hand, the system-level metrics are impacted by the process parameters such as temperature, energy flow, etc. These can be divided into first law efficiencies (GOR, SEC) and second law efficiencies (thermal efficiency-II) [43]. In the context of MD, these can be described as follows:

1. Permeate flux: The permeate flux $J$ (kg·m⁻²h⁻²) is the amount of distillate transported through a unit membrane area. It is the most significant parameter to evaluate the performance of an MD system. It can be expressed as:

   $$J=\frac{{\dot{m}}_d}{A_m}$$E27

   Where $A_m$ is the active surface area (m²) of the membrane, and ${\dot{m}}_d$ is the amount of distillate that passes through the membrane (kg·s${}^{-1}$).

2. Gained output ratio: Gained output ratio (GOR) is the first law efficiency of a thermal desalination system and is often used to quantify energy efficiency. It is defined as the ratio of thermal energy required to vaporize the distillate mass to actual heat input. Mathematically, it can be expressed as [44, 45]:

   $$\mathit{GOR}=\frac{{\dot{m}}_d\ast h_{\mathit{fg}}}{Q_{\mathit{\text{in}}}}$$E28

   Where

   $$Q_{\mathit{\text{in}}}={\dot{m}}_f\ast c_p\ast \mathrm{\Delta }{T}_{12}$$E29

   Where $c_p$ is the specific heat capacity of fluid (Jkg⁻¹K⁻¹). For a system, with no heat energy recovery, GOR is simply a thermal efficiency without a heat exchanger. The GOR is a dimension-less quantity and its value lies between 0 and 1 for a single pass system without any heat recovery, and more than 1 if the evaporation and condensation heat is reused. In other words, GOR tells how many times the enthalpy of evaporation is reused. For most of the commercial systems and large distillation units, the condensation heat is utilized, therefore, shows values greater than 1.

3. Specific energy consumption: Specific energy consumption (SEC) is the energy consumed to produce a unit amount of distillate volume [42, 44, 46]. It is expressed as:

   $$\mathit{SEC}=\frac{Q_{\mathit{\text{in}}}}{m_d}=\frac{{\dot{m}}_f\ast c_p\ast \mathrm{\Delta }{T}_{12}}{m_d}$$E30

   Where ${\dot{m}}_f$, $c_p$, $\mathrm{\Delta }{T}_{12}$, $Q_{\mathit{\text{in}}}$, and $m_d$ are the mass flow rate of feed solution (kg·s⁻¹), specific heat of water (kJ·kg⁻¹K⁻¹), the temperature difference of inlet and outlet feed streams (K), the total input energy (kWh) consumed by the circulating feed, and the total mass of distillate produced (m³), respectively.

4. Thermal efficiency: Thermal efficiency ${\eta }_{\mathit{th}}$ in MD is the ratio of energy of distillate to that of actual energy used [44]. It is also called first law efficiency. It can be expressed as:

   $${\eta }_{\mathit{th}}=\frac{Q_v}{Q_v+Q_c}$$E31
   $${\eta }_{\mathit{th}}=\frac{J\ast A_m\ast h_{\mathit{fg}}}{Q_{\mathit{\text{in}}}}$$E32

   Where $J,A_m,h_{\mathit{fg}},Q_m,Q_v,\mathit{\text{and}}{Q}_c$ are distillate flux (kg·m⁻²h⁻¹), membrane active area (m), latent heat of vaporization (kJ·kg⁻¹K⁻¹), heat flux through membrane (W·m⁻²), heat energy of vaporization (kJ·kg⁻¹K⁻¹) and heat of condensation (kJ·kg⁻¹K⁻¹), respectively. The $Q_{\mathit{\text{in}}}$ can be expressed as shown in Eq. (29)

   To improve the thermal efficiency, the heat conduction $Q_c$ should be minimized by using higher thickness membranes, an air gap etc. However, this may require optimizing other parameters as well. Thermal efficiency matches with the GOR if there is no heat recovery [43].

5. Second law efficiency: First law efficiency generally used to compare systems with same energy source. However, if the energy source is different, the two systems can not be compared fairly. Using second law efficiency, exergies are compared instead. Therefore, systems can be compared regardless of their energy source. Exergy is the maximum available work extracted from the system when the system moves from its initial state to equilibrium. Figure 12 shows a representation of second law efficiency. The first law expression can be deduced as [47]:

   $${\dot{Q}}_H+{\left(\dot{m}.h\right)}_{\mathit{sw}}={\dot{Q}}_0+{\left(\dot{m}.h\right)}_d+{\left(\dot{m}.h\right)}_{\mathit{br}}$$E33
   $${\dot{Q}}_H-{\dot{Q}}_0={\left(\dot{m}.h\right)}_d+{\left(\dot{m}.h\right)}_{\mathit{br}}-{\left(\dot{m}.h\right)}_{\mathit{sw}}$$E34

   The second law can be expressed as:

   $$\frac{{\dot{Q}}_H}{T_0}-\frac{{\dot{Q}}_0}{T_0}={\left(\dot{m}.s\right)}_d+{\left(\dot{m}.s\right)}_{\mathit{br}}-{\left(\dot{m}.s\right)}_{\mathit{sw}}+{\dot{s}}_{\mathit{gen}}$$E35

   Multiplying second law equation Eq. (35) by $T_0$ and subtracting from Eq. (34):

   $$\left(1-\frac{T_0}{T_H}\right)\frac{{\dot{Q}}_H}{{\dot{m}}_d}=\left(G_d-G_{\mathit{br}}\right)-\frac{1}{\mathit{RR}}\left(G_{\mathit{sw}}-G_{\mathit{br}}\right)+T_0\frac{{\dot{S}}_{\mathit{gen}}}{{\dot{m}}_d}=\frac{{\dot{W}}_{\mathit{sep}}}{{\dot{m}}_d}$$E36

   Where $\mathit{RR}$ is the recovery ratio $\frac{{\dot{m}}_p}{{\dot{m}}_{\mathit{sw}}}$, G is the Gibbs free energy ($h-T.s$). The second law efficiency can be expressed as:

   $${\eta }_{\mathit{II}}=\frac{{\dot{W}}_{\mathit{\text{separation}}}}{{\dot{W}}_{\mathit{\text{used}}}}$$E37

6. Recovery ratio: Recovery rate (RR) is the distillate production relative to the input feed stream flow in the MD system [48]. It is expressed as:

$$\mathit{RR}=\frac{{\dot{m}}_d}{{\dot{m}}_f}\ast 100\%$$E38

Figure 12.

A schematic diagram of a black-box desalination system with heat transfer occurring from an external heat source at temperature $T_H$, and to the environment at temperature $T_0$.

A high recovery means a high distillate flow rate is obtained by a given feed flow. In a single pass MD system, feed recovery is very low as compared to other membrane systems. It is reported that the maximum recovery attained in single pass MD reached 10% even if 100% thermal efficiency is attained [49].

In the latest developments, hybridization of MD with existing technologies is used to improve the energy efficiency of MD. The MD has been successfully integrated with other processes as hybrid such as RO, MED, and FO. MD unit has been successfully realized to treat RO brine and FO draw solution which is a challenging part of the process. A comprehensive review of different hybrid technologies with the MD is described by Ghaffour et al. [50]. Additionally, in-situ heating of the feed water inside the MD module, so-called localized heating, has been introduced recently. Localized heating has shown a decreased TP effect which ensures the delivery of heat energy at the site of the feed-membrane interface. It eliminates the circulation heat loss associated with the conventional bulk heating [17, 51]. Photothermal energy source is the prime consideration due to its renewable and ever-existing energy source where the heat is delivered directly to the MD membrane. Politano et al. introduced the photothermal concept in MD first time using surface plasmon effect of silver nanoparticles [52, 53]. Various organic, inorganic, and polymeric materials have also been investigated as photothermal materials in the MD system [6, 54, 55, 56].

Similarly, Joule heating elements and spacers have been used to deliver localized heating to the feed channel [57, 58]. Ahmed et al [17] recently demonstrated the TP reduction by using the electrothermal property of carbon nanostructure. They obtained a decrease of SEC by 58%. Hence, localized heating provides a relatively simple infrastructure for small-scale clean water generation in remote off-grid regions.

5. Conclusions

Membrane distillation (MD) is a promising technology for the separation and purification industry. It is a specific distillation process in which vapor molecules travel through a hydrophobic membrane. MD has several advantages, including low-grade heat input, less fouling propensity, ability to treat high saline water. Four typical membrane configurations, membrane characteristics, membrane modules, heat and mass transfer mechanisms, thermal efficiency, and operating parameters have been presented. The most important limitation that has to be considered with membrane distillation is temperature polarization, which reduces the trans-membrane temperature difference, hence the performance. MD is found to be most suitable when the input energy source is solar or waste heat, due to energy intensive nature of distillation process. The recent development in membrane technology allows MD to run in compact modular configurations such as spiral module configurations. The performance of MD is expressed in terms of flux output and the specific energy. Although flux is an important but not the only factor to demonstrate the performance of an MD system, energy input also plays an important role. The recent advances in localized heating make the MD more promising to operate on a bigger scale.

Acknowledgments

The authors would like to acknowledge King Abdullah University of Science and Technology (KAUST) for supporting the authors in preparing this manuscript.

Conflict of interest

The authors declare no competing financial interests.

Nomenclature

AGMD

Air gap membrane distillation

CAPEX

Capital expenditure

DCMD

Direct contact membrane disitillation

EE

Energy efficiency

FS

Flat sheet

GOR

Gain output ratio

MD

Membrane distillation

OPEX

Operational expenditure

RO

Reverse osmosis

RR

Recovery ratio

SEC

Specific energy consumption

SEM

Scanning electron microscope

SGMD

Sweeping gas membrane distillation

TP

Temperature polarization

TDS

Total dissolved solids

TPC

Temperature polarization coefficient

VMD

Vacuum membrane distillation

Symbols

$A_m$

membrane surface area (m²)

$C$

concentration (mol·L⁻¹)

$C_m$

membrane mass transfer coefficient (kg·m⁻²·s⁻¹·Pa⁻¹)

$c_P$

specific heat capacity (J·kg⁻¹·K⁻¹)

$\alpha$

water activity ($-$)

$\chi$

membrane tortuosity ($-$)

$\delta$

membrane thickness (M)

$\dot{m}$

mass flow rate (kg·s⁻¹)

$\dot{Q}$

heat transfer rate (W)

$\eta$

efficiency ($-$)

$\gamma$

activity coefficient ($-$)

$\lambda$

molecular mean free path (m)

$\mu$

viscocity (Pa·s)

$\pi$

osmotic pressure (Pa)

$\mathrm{\Pi }$

pi (3.14) ($-$)

$\rho$

density (kg·m⁻³)

$\sigma$

molecular collision diameter (m)

$\mathrm{\Theta }$

temperature polarization ($-$)

$\epsilon$

membrane porosity ($-$)

$D_p$

diameter of pore (m)

$D_w$

diffusion coefficient of water (m²·s⁻¹)

$G$

Gibbs free energy (kJ·kg⁻¹)

$h$

heat transfer coefficient (W·m⁻²K⁻¹)

$h_{\mathit{fg}}$

latent heat of vaporization (kJ·kg⁻¹)

$J$

vapor flux (kg·m⁻²·h⁻¹)

$k_B$

Boltzmann constant (J·K⁻¹)

$k_m$

thermal conductivity (W·m⁻¹K⁻¹)

$K_n$

Knudsen number ($-$)

$M$

molecular weight (kg·mol⁻¹)

$\dot{m}$

mass flow rate (mg·s⁻¹)

$N_A$

Avagadro’s number $\left(6.023\times {10}^{23}\right)$ (mol⁻¹)

$\mathit{Nu}$

Nusselt number ($-$)

$p$

partial pressure (Pa)

$P$

total pressure (Pa)

$\mathit{Pr}$

Prandtl number ($-$)

$q$

heat flux (W·m²)

$\dot{Q}$

heat transfer rate (kW)

$r$

pore radius (m)

$R$

universal gas constant (J·mol⁻¹K⁻¹)

$\mathit{Sc}$

Schmidt number ($-$)

$s$

specific entropy generation (kJ·kg⁻¹K⁻¹)

$T$

temperature (K)

$t$

time (s)

$U$

overall heat transfer coefficient (W·m⁻²·K⁻¹)

$v$

specific volume (m³·kg⁻¹)

$W$

work (kW)

$X$

mole fraction ($-$)

Subscripts

$0$

natural state ($-$)

$12$

between input and output ($-$)

$\mathit{av}$

average ($-$)

$b$

bulk ($-$)

$\mathit{br}$

brine ($-$)

$c$

conduction ($-$)

$d$

distillate ($-$)

$f$

feed ($-$)

$\mathit{fb}$

bulk feed ($-$)

$\mathit{fg}$

fluid-gas ($-$)

$g$

gaseous state ($-$)

$\mathit{gen}$

generation ($-$)

$H$

heat ($-$)

$m$

membrane ($-$)

$p$

permeate ($-$)

$\mathit{pb}$

bulk permeate ($-$)

$\mathit{sw}$

seawater ($-$)

$\mathit{sat}$

saturation ($-$)

$v$

vapor ($-$)