*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.


#### **Table 1.**

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

#### *1.1.1.1 Feed side*

At feed side, hot bulk solution come in contact with the MD membrane. Convective heat transfer *<sup>Q</sup> <sup>f</sup>* (W�m�<sup>2</sup> ) takes place and can be expressed as per Newton's law of cooling [14]:

$$Q\_f = h\_f \* \left(T\_{fb} - T\_{f,m}\right) \tag{1}$$

Where *<sup>h</sup> <sup>f</sup>* is the feed convective heat transfer coefficient (W�m�<sup>2</sup> K�<sup>1</sup> ), *Tfb* is the bulk feed solution temperature (K), *T <sup>f</sup>*,*<sup>m</sup>* 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 *Qv* at pores mouth and second is conduction heat transfer *Qc*. The *Qc* is the conduction heat loss and does not contributes to the vapor mass. Hence, for an efficient system, it is required to minimize *Qc* 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 *Qm* (W�m�<sup>2</sup> ) is the sum of both heat transfers and can be expressed as [2, 5]<sup>1</sup> :

$$Q\_m = Q\_c + Q\_v = \frac{k\_m}{\delta\_m} \ast \left( T\_{\,\,f,m} - T\_{p,m} \right) + f \ast h\_{\rm fg} \tag{2}$$

$$h\_m = \frac{k\_m}{\delta\_m} \tag{3}$$

Where *Qc* is the conduction heat transfer (W�m�<sup>2</sup> ), *km* is the membrane thermal conductivity (W�m�<sup>1</sup> �K�<sup>1</sup> ), *δ<sup>m</sup>* is the membrane thickness (m), *J* is the distillate flux (kg�m�<sup>2</sup> h�<sup>1</sup> ), *hfg* is the latent heat of evaporation (kJ�kg�<sup>1</sup> ), *Tp*,*<sup>m</sup>* and *T <sup>f</sup>*,*<sup>m</sup>* are the temperature of permeate-membrane interface and feed-membrane interface respectively.

The thermal conductivity *km* (W�m�<sup>1</sup> K�<sup>1</sup> ) of the membrane can be calculated from the thermal conductivity of vapor *kg* and the polymer material *kp* as per following relations [18]:

$$k\_m = \varepsilon k\_\mathrm{g} + (1 - \varepsilon)k\_p \tag{4}$$

Where *ε* 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 *Qp* (W�m�<sup>2</sup> ) can be expressed as [15, 18]:

$$Q\_p = h\_p \* \left(T\_{pb} - T\_{p,m}\right) \tag{5}$$

<sup>1</sup> The influence of mass transfer on heat transfer was ignored.

Where *hp* is the permeate convective heat transfer coefficient (W�m�<sup>2</sup> K�<sup>1</sup> ), *Tpb* is the bulk permeate solution temperature, *Tp*,*<sup>m</sup>* 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 *<sup>Q</sup>* (W�m�<sup>2</sup> ) can be expressed in terms of universal (overall) heat transfer coefficient as *U* [2, 18]

$$\begin{aligned} \mathbf{Q} &= \mathbf{Q}\_{f} = \mathbf{Q}\_{m} = \mathbf{Q}\_{p} \\ &= h\_{f} \ast \left( T\_{fb} - T\_{f,m} \right) = \frac{k\_{m}}{\delta\_{m}} \ast \left( T\_{f,m} - T\_{p,m} \right) + f \ast h\_{f\xi} = h\_{p} \ast \left( T\_{pb} - T\_{p,m} \right) \\ &= U \ast \left( T\_{f,m} - T\_{p,m} \right) \end{aligned} \tag{6}$$

Where *<sup>U</sup>* (W�m�<sup>2</sup> ) can be expressed as:

$$\frac{1}{U} = \frac{1}{h\_f} + \frac{1}{\frac{k\_m}{\delta\_m} \* \frac{Jh\_{\text{fr}}}{\left(T\_{\,\,f,m} - T\_{p,m}\right)}} + \frac{1}{h\_p} \tag{7}$$

The membrane interface temperatures, i.e. *T <sup>f</sup>*,*<sup>m</sup>* and *Tp*,*<sup>m</sup>* 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]:

$$T\_{\,f,m} = T\_{fb} - \frac{J \, h\_{f\text{g}} + \frac{k\_m}{\delta\_m} \left( T\_{\,f,m} - T\_{p,m} \right)}{h\_f} \tag{8}$$

$$T\_{p,m} = T\_{pb} - \frac{J \ h\_{f\text{g}} + \frac{k\_m}{\delta\_m} \left( T\_{f,m} - T\_{p,m} \right)}{h\_p} \tag{9}$$

The heat transfer coefficients *h <sup>f</sup>* and *hp* can be calculated using Nusselt number relation as shown in Eq. (10), while the *hm* can be calculated from Eq. (3). The Nusselt number correlations are available in the literature [2, 4, 5].

#### **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.*

*Desalination by Membrane Distillation DOI: http://dx.doi.org/10.5772/intechopen.101457*

$$\text{Heat transfer coefficient} = \frac{Nu \ast k\_m}{D\_p} \tag{10}$$

Where *hm* 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]:

$$J = \mathbb{C}\_m \* \left( p\_{\,^f, m} - p\_{p, m} \right) \tag{11}$$

#### **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.*

Where *Cm* is the membrane mass transfer coefficient (kg�m�<sup>2</sup> h�<sup>2</sup> �Pa�<sup>2</sup> ), *p <sup>f</sup>* and *pp* 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):

$$p\_v = \exp\left[23.19 - \left(\frac{3816.44}{T\_{f,m} - 46.13}\right)\right] \tag{12}$$

where *pv* is the vapor pressure of water (Pa) and *T <sup>f</sup>*,*<sup>m</sup>* 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]:

$$p\_{(\mathbf{x})} = (\mathbf{1} - \mathbf{x}\_{\mathbf{s}}) \* p\_{w}^{\boldsymbol{\sigma}} \tag{13}$$

Where *xs* mole fraction of non-volatile solute in feed water and *p<sup>o</sup> <sup>w</sup>* is the partial pressure of pure water.

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

$$
\dot{m}\_{\,f,(x+dx)}\;h\_{\,fb} = \dot{m}\_{\,f,(x)}\;h\_{\,fb} - \left(\int h\_{\rm fg} + q\_m \right)dA\tag{14}
$$

**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.*

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

$$J = k\_f \cdot \rho\_f \quad \ln \frac{\mathcal{X}\_{f,m}}{\mathcal{X}\_f} \tag{15}$$

Where *k <sup>f</sup>* is the mass transfer coefficient and can be calculated using Sherwood number as follow [4]:

$$Sh = \frac{k\_f \cdot d\_h}{D\_s} \tag{16}$$

Where *dh* is the hydraulic diameter (m), *Ds* is the solute diffusion coefficient in bulk feed (m<sup>2</sup> �s �1 ). The Sherwood number can be calculated using following semi-empirical relationship [4]:

$$\mathbf{S}\mathbf{h} = \mathbf{a} \ \mathbf{R}\mathbf{e}^{\beta} \ \mathbf{S}\mathbf{c}^{\gamma} \tag{17}$$

Where *α*, *β*, and *γ* are the coefficients calculated from experiments. Sc is the Schmidt number and Re is the Reynolds number, which can be expressed as [2]:

$$\text{Sc} = \frac{\eta\_f}{\rho\_f \, D\_s} \tag{18}$$

$$Re = \frac{\rho\_f \,\upsilon\_f \,d\_h}{\eta\_f} \tag{19}$$

Where *η <sup>f</sup>* is the viscosity of the bulk fluid (Pa�s), *ρ <sup>f</sup>* is the density of the bulk fluid (kg�m�<sup>3</sup> ), and *v <sup>f</sup>* 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 Δ*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* ∝ *Dp <sup>ε</sup> χ δ<sup>m</sup>* [1].


*Distillation Processes - From Solar and Membrane Distillation to Reactive Distillation…*

The vapor transport mechanism through MD membrane pores is governed by three basic mechanisms known as Knudsen diffusion (*Kn*), 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]:

$$J = \mathcal{C}\_m \ast \frac{dP}{dT} \left( T\_{f,m} - T\_{p,m} \right) \tag{20}$$

The Pressure term *dP dT* can be calculated from Clausius-Clapeyron equation as:

$$\frac{dP}{dT} = \frac{\Delta H}{RT^2} \quad P\_{av}(T\_m) \tag{21}$$

The pressure *Pav* 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 *Cm*. 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 (*Kn*) determines the type of governing mechanism of mass transport inside the membrane pore and can be expressed as [1]:

$$K\_n = \frac{\lambda}{D\_p} \tag{22}$$

Where *Dp* is the pore diameter (m) *λ* is the mean free path (m) of the vapor molecule and can be written as [19]:

$$
\lambda = \frac{k\_B T\_r}{\sqrt{2} \pi P\_m \sigma\_v^2} \tag{23}
$$

where *kB* is the Boltzmann constant (1.38 � <sup>10</sup>�23�J�K�<sup>1</sup> ), *Pm* is the mean average pressure in membrane pores (Pa), *Tr* is the average temperature in the pore, and *σ<sup>v</sup>* is the water vapor collision diameter (0.2641 nm).

Depending on the value of *Kn*, three possible mass transfer modes exist as follows: (a) Knudsen diffusion (*Kn* >1) in which molecular collisions with the walls dominate as compared to the molecule-molecule collisions, (b) molecular diffusion (*Kn* < 0*:*01) in which the frequency of gas molecule collisions is much higher than those with the pore walls, and (c) Knudsen-molecular diffusion 0ð Þ *:*01<*Kn* <1 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.

i. *Knudsen diffusion:* If the mean free path of water molecules is greater than the pore diameter (*λ*> > *Dp*), the frequency of collision between water molecules are less than the molecule–pore wall. The Knudsen number in this regime is ≥ 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.

### *Desalination by Membrane Distillation DOI: http://dx.doi.org/10.5772/intechopen.101457*

#### **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.*

Therefore, Knudsen diffusion decreases with the increase of temperature. In this case, the membrane mass transfer coefficient is shown in **Table 2**.



#### **Table 2.**

*Mass transfer mechanism within the membrane pores follow a specific regime based on the Knudsen number. The mass transfer coefficient C*ð Þ *<sup>m</sup> of each mechanism is shown [1, 2, 4, 5, 19, 23, 24].*

> Knudsen, and molecular diffusion [5]. The membrane mass transfer coefficient is shown in **Table 2**.

#### *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:

$$k\_s = \frac{P\_{av}}{P\_a} \frac{\varepsilon}{\delta} \frac{D}{\gamma + b} \frac{M\_w}{R} \frac{M\_w}{T\_m} \tag{24}$$

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 [26–28]. 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].

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

*Desalination by Membrane Distillation DOI: http://dx.doi.org/10.5772/intechopen.101457*

#### **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* Δ*Tm and bulk* Δ*Tb is shown to determine the polarization coefficient.*

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 Θ is expressed as [19]:

*Distillation Processes - From Solar and Membrane Distillation to Reactive Distillation…*

$$
\Theta = \frac{\Delta T\_m}{\Delta T\_b} = \frac{\left(T\_{f,m} - T\_{p,m}\right)}{\left(T\_{fb} - T\_{pb}\right)}\tag{25}
$$

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).

$$
\Theta = \frac{T\_{f,m}}{T\_{fb}} \tag{26}
$$

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, 31–34]. 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**.


#### **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].*

and permeate channel respectively. The turbulences increase the heat and mass transfer coefficient (*h <sup>f</sup>* , *hp*) 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].


**Figure 9.** *Vapor flux variation with respect to the modular dimensions of various MD flow cell sizes.*

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 *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.

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

*Desalination by Membrane Distillation DOI: http://dx.doi.org/10.5772/intechopen.101457*

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:


**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.*

