# Multiphysics Modeling and Analysis of a Solar Desalination Process Based on Vacuum Membrane Distillation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Exergy Balance

^{−1}feed flow rate was assumed along with a constant recovery ratio resulting in an effluent brine mass fraction of 0.1 (recovery ratio 0.65). This was done with the recognition that, because VMD is a modular process, the whole process can scale up or down to meet any desired capacity.

_{t}is the thermal exergy flux entering the system, and W

_{t}is the mechanical power added to the system [17]. A

^{t}is the total non flow exergy defined by [17]:

_{0}, V, T

_{0}are, respectively, the internal energy, dead state pressure, volume, dead state temperature, and entropy of the control volume. The dead state is typically defined as the environmental conditions and is the reference state for all calculations [16]. If a system is in equilibrium with the surroundings, it is said to be at a dead state and no work of any kind can be done by the process [16]. E

_{k}and E

_{p}are the kinetic and potential energy of the control volume and ṁ

_{i}is the mass flow rate of stream “i” leaving the control volume. The specific flow exergy of a stream “i” was defined as [17]:

_{i}and s

_{i}are the specific enthalpy and entropy of stream “i” and h

_{0}and s

_{0}are the specific enthalpy and entropy of the dead state. Ȓ

_{k}, x

_{k}, and x

_{k}

^{0}are the specific gas constant (J kg

^{−1}K

^{−1}), mole fraction, and dead state mole fraction of component “k” in stream “i”. $\widehat{N}$ is the number of streams entering or leaving the system. ξ

_{0}is the specific Gibb’s free energy of the dead state where ξ

_{0}= h

_{0}− Ts

_{0}and Ex

_{irr}is the total exergy destroyed in the process.

_{0}is a constant and the sum of the mass flow rates is zero at steady state). The dead state is defined as the conditions at which stream 1 (and 10) enters the system; thus ṁ

_{1}b

_{1}

^{t}= ṁ

_{10}b

_{10}

^{t}= 0 as well. Equation (1) then becomes:

_{t}in Equation (1) is replaced by Ex

_{solar}which denotes the exergy added to the process by the solar collector defined by [10,17]:

_{solar}is the solar energy required to raise the temperature of stream 3 to the desired feed temperature. T

_{sun}= 6000 K was assumed as the temperature of the Sun in Equation (5). To determine Q

_{solar}, as well as the various other parameters needed for the model, a mass and energy balance was performed assuming that stream 9 leaves the system as a saturated liquid at the defined vacuum pressure, and the temperature of stream 7 is 10 K higher than that of stream 1 (an estimation of the minimum temperature difference for the heat exchanger) [18]. Thermodynamic properties of the various streams were determined using publicly available steam tables and seawater properties [19,20,21]. With all terms defined, the exergy efficiency of the process can be defined as [16,17]:

#### 2.2. Geometry Definition

^{®}. Figure 2 outlines the geometric definitions used to create the model. The thickness of the membrane is δ

_{m}, R

_{i}is the inner radius of the hollow fiber (the radius of the lumen), R

_{o}is the outer radius of the hollow fiber, and a is the fiber spacing parameter defined by R

_{o}/a = 0.35 (this ratio can assume any number smaller than 0.5). This packing configuration is known as “close-packed” and allows the maximum number of fibers to be fit into a module with regular spacing. The close-packed configuration generates three planes of symmetry for each fiber, which can be used to reduce the overall computational domain to a single unit cell that is descriptive of the whole. Each hollow fiber membrane consists of two domains, the saline feed, and the permeate. The permeate domain is divided into two subdomains: the membrane and the vacuum. The model was further simplified with a few assumptions: (a) momentum transfer within the vacuum domain is negligible [22]; (b) heat transfer in the permeate domain was neglected—the vacuum domain is well insulated, and conductive heat transfer through the membrane is negligible [23,24]; and (c) mass transfer within the permeate was neglected and the mass fraction of water vapor everywhere assumed to be unity—for the latter to be valid, the feed stream is assumed to be degassed. With these assumptions, the permeate flux is determined using the total pressure drop across the membrane and Darcy’s law can be used to model the flow of permeate through the membrane [25]. Because the outlet pressure at the vacuum/membrane interface is constant, the driving force in Darcy’s law is only a function of saturation pressure, which only exists at the feed/membrane interface. The computational domain can then be limited to only the feed side of the membrane, and Darcy’s law applied as a boundary condition along the membrane/feed interface. It should be noted that for a shell side feed, axial symmetry cannot be assumed and the model must be three-dimensional. For the lumen side feed, axial symmetry holds; however, a three-dimensional model was used to keep definitions consistent between the domains.

#### 2.3. Momentum Transfer Governing Equations and Boundary Conditions

_{f}is the pressure of the feed fluid, µ

_{f}is the dynamic viscosity of the feed fluid, and $\stackrel{\rightharpoonup}{\stackrel{\rightharpoonup}{I}}$ is the identity tensor. At the fiber inlet, an average velocity was provided as the boundary condition: ${\stackrel{\rightharpoonup}{u}}_{f}\left(z=0\right)={\stackrel{\rightharpoonup}{u}}_{f,in}$. A no-slip boundary condition was provided at the membrane/feed interface: ${\stackrel{\rightharpoonup}{u}}_{f}\left(r={R}_{i},{R}_{o}\right)=\stackrel{\rightharpoonup}{0}\text{}\left({\mathrm{m}\text{}\mathrm{s}}^{-1}\right)$. An outlet pressure was defined at the outlet of the feed channel: P (z = L

_{m}) = 1 atm.

#### 2.4. Mass Transfer Governing Equations and Boundary Conditions

^{−2}s

^{−1}, LMH), and M

_{w}is the molar mass of water. The lumped diffusion coefficient (C

_{t}) is the sum of the Knudsen diffusion coefficient (C

_{1}) and the Poiseuille diffusion coefficient (C

_{2}). Each of these coefficients is determined using the membrane parameters (Equations (1)–(13)) [27].

_{p}is the pore diameter, δ

_{m}is the thickness of the membrane, τ is the membrane tortuosity, R is the ideal gas constant, T

_{f}is the temperature, and ϵ is the membrane porosity (void fraction). P

_{m}is the mean pressure within the membrane defined by [27]:

_{vac}is the vacuum pressure (5 kPa) which is in the normal range for high-production VMD [28]. P

_{sat}is the saturation pressure of water at temperature T

_{f}defined by [27]:

_{w}, is defined based on the mole fraction of salt (x

_{s}) [29]:

_{w}is a polynomial fitting that is accurate up to the saturation concentration of salt in water (about 350 g/L), which is valid from 30 °C to 100 °C [30]. At this concentration, a

_{w}takes on a constant value as salt spontaneously precipitates out of the solution. Finally, ΔP is defined simply as the difference between the saturation pressure and the vacuum pressure [27].

_{i}is the mass fraction of component “i” in the fluid [31].

_{i}

^{m}is the mixture averaged diffusion coefficient defined by [31]:

_{ik}is the binary diffusion coefficient for species “i” in species “k” and x

_{k}is the mole fraction of species “k”. M

_{n}is the mean molar mass of the mixture defined by [31]:

_{i}being the molar mass of species “i”. ${\stackrel{\rightharpoonup}{j}}_{ci}$ is the mixture diffusion correction term defined by [31]:

_{s}(z = 0) = ω

_{s,in}. The mass flux of water across the membrane/feed interface was defined by the model for mass flux presented in Equations (10)–(17): ${\stackrel{\rightharpoonup}{N}}_{w}\left(\text{}r={R}_{i},{R}_{o}\right)=N\left(\stackrel{\rightharpoonup}{n}\right)$ [27], where $\stackrel{\rightharpoonup}{n}$ is the normal vector pointing away from the feed stream. The inlet mass fraction of salt was defined for seawater ω

_{s,in}= 0.035, and the diffusion coefficient for salt in water was estimated as D

_{sw}= 10

^{−10}m

^{2}s

^{−1}[32,33]. Along the boundaries not described as having boundary conditions, planes of symmetry were defined in accordance with Figure 2.

#### 2.5. Heat-Transfer-Governing Equations and Boundary Conditions

_{p}is the heat capacity of the fluid, ${\stackrel{\rightharpoonup}{q}}_{f}$ is the conductive heat flux, and k

_{f}is the thermal conductivity of the feed. A feed temperature served as the boundary condition at the inlet to the fiber: T

_{f}(z = 0) = T

_{f,in}. Heat flux across the membrane/feed interface was defined based on the mass flux in Equation (10) [34]: q

_{m}(r = R

_{i},R

_{o}) = −N·H

_{vap}− h

_{m}(T

_{f,in}− T

_{m}), where q

_{m}is the heat flux across the boundary, H

_{vap}is the heat of vaporization of water (40 kJ mol

^{−1}), h

_{m}is the convective heat transfer coefficient, and T

_{m}is the temperature at the boundary. The convective term in the heat flux boundary condition was seen to be essentially zero and was neglected [23,24]. Table 1 shows all of the input parameters used for the geometric definitions and physics calculations.

## 3. Results

#### 3.1. Effects of Different Membrane Parameters

#### 3.1.1. Effect of Fiber Length

_{m}). The phenomena that lead to the boundary layer are less significant at lower temperatures, so the boundary layer is less pronounced. This becomes significant when we consider the energetic requirements of membrane distillation. MD relies on thermal energy provided to the system to evaporate water [13]. The energy requirement is significant due to both the high heat capacity of water (which makes raising the temperature difficult) and the high heat of vaporization [7]. This severely limits the efficiency of the process and makes fiber length a key parameter for design and optimization. These findings suggest that shorter membranes, which may be able to operate without a fully formed boundary layer, are desirable.

#### 3.1.2. Effect of Porosity

_{m}in Equations (10) and (13) are larger. The same change in porosity leads to a larger change in the overall slope determined by Equations (10)–(13). A slight concavity is present in the data sets due to polarization. Higher values of permeate flux at the inlet lead to more significant polarization and a degradation in average flux, though the initial behavior predicted by Equations (10)–(13) can be maintained.

#### 3.1.3. Effect of Pore Diameter

#### 3.1.4. Effect of Thickness

#### 3.1.5. Effect of Tortuosity

_{m}terms.

#### 3.2. Limiting Phenomena

_{sat}/a

_{w}). The red lines are only a function of temperature, while the black lines are a function of both temperature and salt concentration. The difference in the two functions is the effect of salt concentration.

^{−10}m

^{2}s

^{−1}[33], while the thermal diffusivity of water is on the order of 10

^{−7}m

^{2}s

^{−1}[36]. The three orders of magnitude difference between these numbers is reflected in the results shown in Figure 5, confirming salt concentration must be the limiting factor. Salt concentration is significant in another way. Scaling is known to be a problem in desalination systems [13]. As concentration in the boundary layer increases, so does scaling [11,23,24,28]. As crystalline salt forms on the surface of the membrane, it can block pores and inhibit permeate flux. Scaling also increases the wettability of a membrane, which means the process may need to be run at a higher pressure in the vacuum domain or risk contaminating the permeate [37]. The model presented here is limited in that it does not present a kinetic model for scaling and cannot take into account how scaling will change membrane performance. Unlike salt, temperature does not have a saturation value that limits its effect on membrane performance. As length increases, the temperature boundary layer becomes more apparent and will eventually become the determining factor in membrane performance.

#### 3.3. Effect of Baffling Design

#### 3.4. Exergy Efficiency

^{−4}). In comparison to this single pass recovery ratio, the overall recovery ratio of the system shown in Figure 1 is designed to be 0.65. To make up the difference between the single-pass recovery ratio and the overall recovery ratio, the streams that comprise the recycling (streams 3, 4, and 5) must be substantially larger than the streams outside of it. The magnitude of these streams means they dominate the mass and energy balance calculations. Another effect of the desired recovery ratio is that the feed stream (stream 1) cannot provide sufficient cooling duty to condense the permeate stream (stream 8). To condense the permeate, another stream (stream 10) with a higher flow rate must provide the cool water to condense the permeate. Finally, the magnitude of the recycle streams relative to other streams means that the properties of the recycle loop are roughly the same and are largely unaffected by the feed being added or the effluent or permeate removed.

_{solar}and Ex

_{solar}. The effect of temperature on the other three streams that determine exergy efficiency in Equation (6) is less significant. No change should occur in stream 7 because the recovery ratio is fixed and the outlet temperature of stream 7 is determined by the feed temperature of stream 1. Stream 11 should see an increase in temperature due to the increased temperature of stream 8. This should result in an increase in exergy efficiency but the flow rate of streams 3 and 4, which determine the heat duty, are orders of magnitude higher than these streams and therefore the effect of temperature is most pronounced on the increased heating duty required. The second phenomenon to be observed is how little effect membrane parameters has on exergy efficiency. The largest change occurs for thickness at 353 K. Around a 0.005% decrease is observed by increasing the thickness of the membrane. This again is likely due to the relative magnitude of Q

_{solar}to the membrane parameters and its dominating effect on Equation (6). The final phenomenon is the functionality of exergy efficiency with the respective membrane parameters. The trends are largely the same as those observed for permeate flux in Figure 4 albeit of a very depressed nature. This is probably the remaining effect of streams 8–11 observed above. Increasing flux increases the flow rate of streams 8 and 9 and with it the exergy leaving with stream 9. The increase in flow rate also leads to an increase in the temperature of stream 11 and its exergy.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Terms | Definition |

${\delta}_{m}$ | Membrane thickness $\left(\mathrm{m}\right)$ |

$\u03f5$ | Membrane porosity |

${\eta}_{II}$ | Exergy efficiency |

$\kappa $ | Membrane permeability $\left({\mathrm{m}}^{2}\right)$ |

$\mu $ | Dynamic viscosity of fluid $\left(\mathrm{Pa}\text{}\mathrm{s}\right)$ |

${\mu}_{f}$ | Dynamic viscosity of the feed stream $\left(\mathrm{Pa}\text{}\mathrm{s}\right)$ |

${\xi}_{0}$ | Specific Gibbs free energy of the dead state $\left({\mathrm{J}\text{}\mathrm{kg}}^{-1}\right)$ |

$\rho $ | Density of feed stream $\left({\mathrm{kg}\text{}\mathrm{m}}^{3}\right)$ |

$\tau $ | Membrane tortuosity |

${\omega}_{i}$ | Mass fraction of component $\text{\u201c}i\text{\u201d}$ |

${\omega}_{s,in}$ | Mass fraction of salt at the inlet |

$a$ | Fiber spacing parameter |

${a}_{w}$ | Activity coefficient of water |

${A}^{t}$ | Total non-flow exergy $\left(J\right)$ |

${b}_{i}^{t}$ | Specific flow exergy of stream $\text{\u201c}i\text{\u201d}$ $\left({\mathrm{J}\text{}\mathrm{kg}}^{-1}\right)$ |

${C}_{1}$ | Knudsen diffusion coefficient $\left({\mathrm{s}\text{}\mathrm{mol}\text{}\mathrm{kg}}^{-1}{\mathrm{m}}^{-1}\right)$ |

${C}_{2}$ | Poiseuille diffusion coefficient $\left({\mathrm{s}\text{}\mathrm{mol}\text{}\mathrm{kg}}^{-1}{\mathrm{m}}^{-1}\right)$ |

${C}_{p}$ | Heat capacity of fluid $\left({\mathrm{J}\text{}\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}\right)$ |

${C}_{t}$ | Total diffusion coefficient $\left({\mathrm{s}\text{}\mathrm{mol}\text{}\mathrm{kg}}^{-1}{\mathrm{m}}^{-1}\right)$ |

${d}_{p}$ | Mean pore diameter $\left(\mathrm{m}\right)$ |

${D}_{i}^{m}$ | Mixture average diffusion coefficient of component “i” $\left({\mathrm{m}}^{2}{\text{}\mathrm{s}}^{-1}\right)$ |

${D}_{sw}$ | Diffusion coefficient for salt in water $\left({\mathrm{m}}^{2}{\text{}\mathrm{s}}^{-1}\right)$ |

${E}_{k}$ | Kinetic energy $\left(\mathrm{J}\right)$ |

${E}_{p}$ | Potential energy $\left(\mathrm{J}\right)$ |

$E{x}_{irr}$ | Total exergy loss due to system irreversibility $\left(\mathrm{W}\right)$ |

$E{x}_{solar}$ | Solar exergy flux $\left(\mathrm{W}\right)$ |

$E{x}_{t}$ | Total Thermal Exergy Flux $\left(\mathrm{W}\right)$ |

${h}_{0}$ | Specific enthalpy of dead state $\left({\mathrm{J}\text{}\mathrm{kg}}^{-1}\right)$ |

${h}_{i}$ | Specific enthalpy of stream “i” $\left({\mathrm{J}\text{}\mathrm{kg}}^{-1}\right)$ |

$\widehat{{h}_{m}}$ | Convective heat transfer coefficient $\left({\mathrm{Wm}}^{-2}{\mathrm{K}}^{-1}\right)$ |

${H}_{vap}$ | Heat of vaporization of water $\left({\mathrm{kJ}\text{}\mathrm{mol}}^{-1}\right)$ |

$\stackrel{\rightharpoonup}{\stackrel{\rightharpoonup}{I}}$ | Identity tensor |

${\stackrel{\rightharpoonup}{j}}_{ci}$ | Mixture diffusion correction term $\left({\mathrm{kg}\text{}\mathrm{m}}^{-2}{\mathrm{s}}^{-1}\right)$ |

${\stackrel{\rightharpoonup}{j}}_{i}$ | Diffusive flux of component $\text{\u201c}i\text{\u201d}$ in the feed stream $\left({\mathrm{kg}\text{}\mathrm{m}}^{-2}{\mathrm{s}}^{-1}\right)$ |

${k}_{f}$ | Thermal conductivity of feed stream $\left({\mathrm{W}\text{}\mathrm{m}}^{-1}{\mathrm{K}}^{-1}\right)$ |

${L}_{m}$ | Length of membrane module $\left(\mathrm{m}\right)$ |

${\dot{m}}_{i}$ | Mass flowrate of stream $\u201ci\u201d$ $\left({\mathrm{kg}\text{}\mathrm{s}}^{-1}\right)$ |

${M}_{i}$ | Molar mass of component $\u201ci\u201d$ in membrane feed $\left({\mathrm{kg}\text{}\mathrm{mol}}^{-1}\right)$ |

${M}_{n}$ | Mean molar mass of the feed stream $\left({\mathrm{kg}\text{}\mathrm{mol}}^{-1}\right)$ |

$n$ | Number of fibers in a module |

$\stackrel{\rightharpoonup}{n}$ | Normal vector |

$N$ | Mass flux of water vapor across membrane $\left({\mathrm{kg}\text{}\mathrm{m}}^{-2}{\mathrm{s}}^{-1}\right)\text{}$ |

$\widehat{N}$ | Total Number of Streams entering and leaving |

${\stackrel{\rightharpoonup}{N}}_{i}$ | Total mass flux of component “i” within the membrane feed stream $\left({\mathrm{kg}\text{}\mathrm{m}}^{-2}{\mathrm{s}}^{-1}\right)$ |

${\stackrel{\rightharpoonup}{N}}_{w}$ | Total mass flux of water within the membrane feed stream $\left({\mathrm{kg}\text{}\mathrm{m}}^{-2}{\mathrm{s}}^{-1}\right)$ |

${p}_{0}$ | Dead state pressure $\left(\mathrm{Pa}\right)$ |

${P}_{f}$ | Pressure of the feed stream $\left(\mathrm{Pa}\right)$ |

$\mathsf{\Delta}P$ | Transmembrane change in pressure $\left(\mathrm{Pa}\right)$ |

${P}_{m}$ | Average pressure within the membrane $\left(\mathrm{Pa}\right)$ |

${P}_{sat}$ | Saturation pressure of water $\left(\mathrm{Pa}\right)$ |

${P}_{vac}$ | Vacuum pressure $\left(\mathrm{Pa}\right)$ |

$q$ | Volumetric flowrate $\left({\mathrm{m}}^{3}{\text{}\mathrm{s}}^{-1}\right)$ |

${\stackrel{\rightharpoonup}{q}}_{f}$ | Conductive heat flux in the membrane feed stream $\left({\mathrm{W}\text{}\mathrm{m}}^{-2}\right)$ |

${q}_{m}$ | Transmembrane heat flux $\left({\mathrm{W}\text{}\mathrm{m}}^{-2}\right)$ |

${Q}_{solar}$ | Solar energy $\left(\mathrm{W}\right)$ |

$r$ | Radial spatial variable $\left(\mathrm{m}\right)$ |

$R$ | Ideal gas constant $\left({\mathrm{J}\text{}\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}\right)$ |

${R}_{i}$ | Inner radius of the membrane $\left(\mathrm{m}\right)$ |

$\widehat{{R}_{k}}$ | Specific Gas constant for component “k” $\left({\mathrm{J}\text{}\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}\right)$ |

${R}_{o}$ | Outer radius of membrane $\left(\mathrm{m}\right)$ |

${s}_{i}$ | Specific entropy of stream “i” $\left({\mathrm{J}\text{}\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}\right)$ |

${s}_{0}$ | Dead state-specific entropy $\left({\mathrm{J}\text{}\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}\right)$ |

$S$ | Entropy $\left({\mathrm{J}\text{}\mathrm{K}}^{-1}\right)$ |

${T}_{0}$ | Dead State temperature $\left(\mathrm{K}\right)$ |

${T}_{f}$ | Feed stream temperature $\left(\mathrm{K}\right)$ |

${T}_{f,in}$ | Feed temperature at inlet $\left(\mathrm{K}\right)$ |

${T}_{m}$ | Feed stream temperature at membrane interface $\left(\mathrm{K}\right)$ |

${T}_{sun}$ | Temperature of the Sun $\left(\mathrm{K}\right)$ |

${\stackrel{\rightharpoonup}{u}}_{f}$ | Feed velocity $\left({\mathrm{m}\text{}\mathrm{s}}^{-1}\right)$ |

${\stackrel{\rightharpoonup}{u}}_{f,in}$ | Feed velocity at inlet $\left({\mathrm{m}\text{}\mathrm{s}}^{-1}\right)$ |

$U$ | Internal Energy $\left(\mathrm{J}\right)$ |

$V$ | Volume $\left({\mathrm{m}}^{3}\right)$ |

${W}_{t}$ | Mechanical Power $\left(\mathrm{W}\right)$ |

${x}_{k}$ | Mole fraction of component k |

${x}_{k}^{0}$ | Dead state mole fraction of component k |

${x}_{s}$ | Mole fraction of salt in feed stream |

$z$ | Axial spatial variable $\left(\mathrm{m}\right)$ |

## References

- Kummu, M.; Guillaume, J.H.A.; De Moel, H.; Eisner, S.; Flörke, M.; Porkka, M.; Siebert, S.; Veldkamp, T.I.E.; Ward, P.J. The world’s road to water scarcity: Shortage and stress in the 20th century and pathways towards sustainability. Sci. Rep.
**2016**, 6, 38495. [Google Scholar] [CrossRef] [Green Version] - Ng, P.J.H.; Teo, C. Singapore’s water challenges past to present. Int. J. Water Resour. Dev.
**2019**, 36, 269–277. [Google Scholar] [CrossRef] - El-Nashar, A.M.; Samad, M. The solar desalination plant in Abu Dhabi: 13 years of performance and operation history. Renew. Energy
**1998**, 14, 263–274. [Google Scholar] [CrossRef] - Ghaffour, N.; Bundschuh, J.; Mahmoudi, H.; Goosen, M.F. Renewable energy-driven desalination technologies: A comprehensive review on challenges and potential applications of integrated systems. Desalination
**2015**, 356, 94–114. [Google Scholar] [CrossRef] [Green Version] - Peñate, B.; Rodríguez, M.D.L.G. Current trends and future prospects in the design of seawater reverse osmosis desalination technology. Desalination
**2012**, 284, 1–8. [Google Scholar] [CrossRef] - Aboelmaaref, M.M.; Zayed, M.E.; Zhao, J.; Li, W.; Askalany, A.A.; Ahmed, M.S.; Ehab, S.A. Hybrid Solar Desalination Systems Driven by Parabolic Trough and Parabolic Dish CSP Technologies: Technology Categorization, Thermodynamic Performance and Economical Assessment. Energy Convers. Manag.
**2020**, 220, 33. [Google Scholar] [CrossRef] - Deshmukh, A.; Boo, C.; Karanikola, V.; Lin, S.; Straub, A.P.; Tong, T.; Warsinger, D.M.; Elimelech, M. Membrane distillation at the water-energy nexus: Limits, opportunities, and challenges. Energy Environ. Sci.
**2018**, 11, 1177–1196. [Google Scholar] [CrossRef] - Wang, P.; Chung, T.-S. Recent advances in membrane distillation processes: Membrane development, configuration design and application exploring. J. Membr. Sci.
**2015**, 474, 39–56. [Google Scholar] [CrossRef] - Drioli, E.; Ali, A.; Macedonio, F. Membrane distillation: Recent developments and perspectives. Desalination
**2015**, 356, 56–84. [Google Scholar] [CrossRef] - Ghaleni, M.M.; Al Balushi, A.; Bavarian, M.; Nejati, S. Omniphobic Hollow Fiber Membranes for Water Recovery and Desalination. ACS Appl. Polym. Mater.
**2020**, 2, 3034–3038. [Google Scholar] [CrossRef] - Warsinger, D.M.; Swaminathan, J.; Guillen-Burrieza, E.; Arafat, H.A. Scaling and fouling in membrane distillation for desalination applications: A review. Desalination
**2015**, 356, 294–313. [Google Scholar] [CrossRef] - Li, Q.; Omar, A.; Cha-Umpong, W.; Liu, Q.; Li, X.; Wen, J.; Wang, Y.; Razmjou, A.; Guan, J.; Taylor, R.A. The potential of hollow fiber vacuum multi-effect membrane distillation for brine treatment. Appl. Energy
**2020**, 276, 115437. [Google Scholar] [CrossRef] - El Kadi, K.; Janajreh, I.; Hashaikeh, R. Numerical simulation and evaluation of spacer-filled direct contact membrane distillation module. Appl. Water Sci.
**2020**, 10, 1–17. [Google Scholar] [CrossRef] - Lou, Y.; Gogar, R.; Hao, P.; Lipscomb, G.; Amo, K.; Kniep, J. Simulation of net spacers in membrane modules for carbon dioxide capture. Sep. Sci. Technol.
**2016**, 52, 168–185. [Google Scholar] [CrossRef] - Wang, Z.; Horseman, T.; Straub, A.P.; Yip, N.Y.; Li, D.; Elimelech, M.; Lin, S. Pathways and challenges for efficient solar-thermal desalination. Sci. Adv.
**2019**, 5, eaax0763. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Demirel, Y. Nonequilibrium Thermodynamics, 2nd ed.; Elsevier B.V.: Amsterdam, The Netherlands, 2007. [Google Scholar]
- Signorato, F.; Morciano, M.; Bergamasco, L.; Fasano, M.; Asinari, P. Exergy analysis of solar desalination systems based on passive multi-effect membrane distillation. Energy Rep.
**2020**, 6, 445–454. [Google Scholar] [CrossRef] - Kirk-Othmer Encyclopedia of Chemical Technology, 5th ed.; John Wiley & Sons: Hoboken, NJ, USA, 2004; Volume 13.
- Holmgren, M. X Steam, Thermodynamic Properties of Water and Steam. Available online: https://www.mathworks.com/matlabcentral/fileexchange/9817-x-steam-thermodynamic-properties-of-water-and-steam (accessed on 16 April 2021).
- Nayar, K.G.; Sharqawy, M.H.; Banchik, L.D. Thermophysical properties of seawater: A review and new correlations that include pressure dependence. Desalination
**2016**, 390, 1–24. [Google Scholar] [CrossRef] [Green Version] - Sharqawy, M.H.; Lienhard, J.H.; Zubair, S.M. Thermophysical properties of seawater: A review of existing correlations and data. Desalination Water Treat.
**2010**, 16, 354–380. [Google Scholar] [CrossRef] - Zhang, J.; Li, J.-D.; Duke, M.; Hoang, M.; Xie, Z.; Groth, A.; Tun, C.; Gray, S. Modelling of vacuum membrane distillation. J. Membr. Sci.
**2013**, 434, 1–9. [Google Scholar] [CrossRef] [Green Version] - Alklaibi, A.M.; Lior, N. Membrane-distillation desalination: Status and potential. Desalination
**2005**, 171, 111–131. [Google Scholar] [CrossRef] - Alkhudhiri, A.; Darwish, N.; Hilal, N. Membrane distillation: A comprehensive review. Desalination
**2012**, 287, 2–18. [Google Scholar] [CrossRef] - Wilkes, J.O. Fluid Mechanics for Chemical Engineers with Microfluidics and CFD, 2nd ed.; Pearson Education Inc.: Upper Saddle River, NJ, USA, 2006. [Google Scholar]
- Bird, R.B.; Stewart, W.E.; Lightfoot, E.N. Transport Phenomena; John Wiley & Sons: Hoboken, NJ, USA, 1960. [Google Scholar]
- Zhang, Y.; Peng, Y.; Ji, S.; Wang, S. Numerical simulation of 3D hollow-fiber vacuum membrane distillation by computational fluid dynamics. Chem. Eng. Sci.
**2016**, 152, 172–185. [Google Scholar] [CrossRef] - Abu-Zeid, M.A.E.-R.; Zhang, Y.; Dong, H.; Zhang, L.; Chen, H.-L.; Hou, L. A comprehensive review of vacuum membrane distillation technique. Desalination
**2015**, 356, 1–14. [Google Scholar] [CrossRef] - Lawson, K.W.; Lloyd, D.R. Membrane distillation. J. Membr. Sci.
**1997**, 124, 1–25. [Google Scholar] [CrossRef] - Schofield, R.W. Membrane Distillation. Ph.D. Thesis, Univeristy of New South Wales, Sydney, Australia, 1989. [Google Scholar]
- Chemical Reaction Engineering Module User’s Guide, COMSOL Multiphysics® v. 5.4; COMSOL AB: Stockholm, Sweden, 2018.
- Millero, F.J.; Feistel, R.; Wright, D.G.; McDougall, T.J. The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale. Deep. Sea Res. Part I Oceanogr. Res. Pap.
**2008**, 55, 50–72. [Google Scholar] [CrossRef] - Vitagliano, V.; Lyons, P.A. Diffusion Coefficients for Aqueous Solutions of Sodium Chloride and Barium Chloride. J. Am. Chem. Soc.
**1956**, 78, 1549–1552. [Google Scholar] [CrossRef] - Ghaleni, M.M.; Bavarian, M.; Nejati, S. Model-guided design of high-performance membrane distillation modules for water desalination. J. Membr. Sci.
**2018**, 563, 794–803. [Google Scholar] [CrossRef] - Alkhudhiri, A.; Hilal, N. Membrane Distillation—Principles, Applications, Configurations, Design, and Implementation. In Emerging Technologies for Sustainable Desalination Handbook; Gude, V.G., Ed.; Butterworth-Heinemann: Oxford, UK, 2008. [Google Scholar]
- Blumm, J.; Lindemann, A. Characterization of the thermophysical properties of molten plymers and liquids using the flash technique. High Temp. Press.
**2003**, 35/36, 627–632. [Google Scholar] [CrossRef] [Green Version] - Chang, Y.; Ooi, B.; Ahmad, A.; Leo, C.; Low, S. Vacuum membrane distillation for desalination: Scaling phenomena of brackish water at elevated temperature. Sep. Purif. Technol.
**2021**, 254, 117572. [Google Scholar] [CrossRef] - Reverter, J.A.; Talo, S.; Alday, J. Las Palmas III—The success story of brine staging. Desalination
**2001**, 138, 207–217. [Google Scholar] [CrossRef] - Magara, Y.; Kawasaki, M.; Sekino, M.; Yamamura, H. Development of reverse osmosis membrane seawater desalination in Japan. Water Sci. Technol.
**2000**, 41, 1–8. [Google Scholar] [CrossRef] - Tong, T.; Elimelech, M. The Global Rise of Zero Liquid Discharge for Wastewater Management: Drivers, Technologies, and Future Directions. Environ. Sci. Technol.
**2016**, 50, 6846–6855. [Google Scholar] [CrossRef] [PubMed] - Salmón, I.R.; Luis, P. Membrane crystallization via membrane distillation. Chem. Eng. Process. Process. Intensif.
**2018**, 123, 258–271. [Google Scholar] [CrossRef]

**Figure 1.**The solar thermal desalination scheme used for exergy analysis. Here, HEX I and HEX II refer to the heat exchanger on the feed and permeate stream, respectively. The MIX refers to a mixing tank used for thermal and concentration management of the solar-VMD loop. (

**A**) A single-stage process. (

**B**) A multi-stage process with 3 stages.

**Figure 2.**(

**A**) Packing configuration showing the planes of symmetry generated (dashed lines), and the resulting lumen and shell domains used for computation. (

**B**) Definition of variables for the computational domain in this work. Here, the gray-shaded areas represent porous membrane.

**Figure 3.**Localized permeate flux and average flux as a function of fiber length. Black lines indicate localized values for permeate flux at a certain distance from the fiber inlet. Red lines indicate the average permeate flux for a fiber with the length shown. The result is based on a shell-side feed, inner radius 350 µm, thickness 300 µm, R/a 0.35, feed velocity 5 m/s, vacuum pressure 5 kPa, tortuosity 2, pore diameter 400 nm, and porosity 0.5.

**Figure 4.**Average permeate flux as a function of (

**A**) porosity, (

**B**) nominal pore size, (

**C**) membrane thicknesses, and (

**D**) effective tortuosities. In (

**A**–

**C**) the feed was charged to shell side, (

**D**) Shows for both cases, charging feed into the shell and lumen. Here, the R/a ratio is fixed at 0.35, the membrane length is 5 cm, feed velocity is 5 m/s, and the permeate pressure is 5 kPa.

**Figure 5.**Saturation pressure as a function of fiber length. Black lines indicate saturation pressure as calculated using Equation (16). Red lines indicate saturation pressure as determined using Antoine’s equation This study is based on shell-side feed. Here, the inner radius of the hollow fiber is 350 µm and the thickness of media is 300 µm. The R/a was kept at 0.35, the fibers’ lengths are 5 cm, the feed is velocity 5 m/s, and the pressure in permeate channel was kept at 5 kPa. We assumed membrane tortuosity of 2, pore size diameter of 400 nm, and a porosity 0.5 of for the membrane media.

**Figure 6.**Baffling geometry that was used to reduce concentration polarization on the surface of membranes.

**Figure 7.**(

**A**) Average permeate flux as a function of fiber length with different baffling spacings. (

**B**) Salt concentration profile along he membrane surface for a normal (unbaffled) fiber and a fiber with baffling at 1 mm spacing. The result is based on the shell side feed, inner radius 350 µm, thickness 300 µm, R/a 0.35, feed velocity 5 m/s, vacuum pressure 5 kPa, tortuosity 2, pore diameter 400 nm, and porosity 0.5.

**Figure 8.**Exergy efficiency as a function of (

**A**) porosity, (

**B**) nominal pore size, (

**C**) membrane thicknesses, and (

**D**) effective tortuosities. In (

**A**–

**C**) the feed was charged to shell side, (

**D**) Shows for both cases, charging feed into the shell and lumen. Here, the R/a ratio is fixed at 0.35, the membrane length is 5 cm, feed velocity 5 m/s, and the channel in the permeate pressure is maintained at 5 kPa.

**Figure 9.**Exergy efficiency as fiber length changes at various baffling spacings. Shell-side feed. Inner radius 350 µm, thickness 300 µm, R/a 0.35, feed velocity 5 m/s, vacuum pressure 5 kPa, tortuosity 2, pore diameter 400 nm, porosity 0.5.

**Figure 10.**Exergy efficiency as the number of stages increases. Shell-side feed. Inner radius 350 um, thickness 300 um, R/a 0.35, feed velocity 5 m/s, vacuum pressure 5 kPa, tortuosity 2, pore diameter 400 nm, porosity 0.5, and a value for the ṁ

_{11}= 200 kg/hr. Legend indicates recovery ratio. Recovery ratio 0.65 corresponds to mass fraction of 0.1 for the salt in stream 7.

Parameter | Value | Parameter | Value |
---|---|---|---|

${T}_{f,in}$ | 333–353 K | ${d}_{p}$ | 200–500 nm |

${P}_{vac}$ | 5 kPa | $\tau $ | 2–4 |

${u}_{f,in}$ | 5 m s^{−1} | $\u03f5$ | 0.5–0.9 |

${R}_{i}$ | 350 μm | ${L}_{m}$ | 2.5–7.5 cm |

${\delta}_{m}$ | 150–400 μm | ${\omega}_{s,in}$ | 0.35 |

**Table 2.**Mass and energy balance, shell side feed. Inner radius 350 um, thickness 300 um, R/a 0.35, Length 5 cm, feed velocity 5 m/s, vacuum pressure 5 kPa, tortuosity 2, pore diameter 400 nm.

Scheme | Mass Flow Rate (kg h ^{−1}) | Mass Fraction of Salt % | Temperature (K) |
---|---|---|---|

1 | 1 | 3.5 | 288 |

2 | 1 | 3.5 | 307.2 |

3 | 2041.3 | 10 | 352.8 |

4 | 2041.3 | 10 | 353 |

5 | 2040.7 | 10.003 | 352.8 |

6 | 0.35 | 10 | 352.8 |

7 | 0.35 | 10 | 298 |

8 | 0.65 | 0 | 350.5 |

9 | 0.65 | 0 | 306 |

10 | 50 | 3.5 | 288 |

11 | 50 | 3.5 | 295.8 |

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**MDPI and ACS Style**

Shuldes, B.N.; Bavarian, M.; Nejati, S.
Multiphysics Modeling and Analysis of a Solar Desalination Process Based on Vacuum Membrane Distillation. *Membranes* **2021**, *11*, 386.
https://doi.org/10.3390/membranes11060386

**AMA Style**

Shuldes BN, Bavarian M, Nejati S.
Multiphysics Modeling and Analysis of a Solar Desalination Process Based on Vacuum Membrane Distillation. *Membranes*. 2021; 11(6):386.
https://doi.org/10.3390/membranes11060386

**Chicago/Turabian Style**

Shuldes, Benjamin N., Mona Bavarian, and Siamak Nejati.
2021. "Multiphysics Modeling and Analysis of a Solar Desalination Process Based on Vacuum Membrane Distillation" *Membranes* 11, no. 6: 386.
https://doi.org/10.3390/membranes11060386