# An Effective Standalone Solar Air Gap Membrane Distillation Plant for Saline Water Desalination: Mathematical Model, Optimization

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## Abstract

**:**

^{−2}·h

^{−1}achieved at 68 °C, a feed temperature. Moreover, gained output ratio (GOR) of the unit of thermal solar desalination was estimated to be about 4.6, which decreases with increasing hot water flow and temperature.

## 1. Introduction

^{−2}/day when the annual-mean-daily GSR reached 5656 Wh·m

^{−2}/day. This region has its place in the great solar deposit characterized by tremendous advances in a vast array of alternative renewable energy options, including solar power, which is a very important means for developing desalination technologies, such as membrane distillation (MD). Recently, the latter has encountered a specific interest in its applications in the water desalination field, particularly when paired with solar power [6].

^{2}. A feeding solution of NaCl with 1 and 35 g·L

^{−1}concentrations was used. Specific distillate flux was displayed by modules of AGMD with values up to a maximum of 6.5 L·m

^{−2}·h

^{−1}at 65 °C feedwater temperature and for 1 g·L

^{−1}of NaCl feed solution. Kubota et al. [21], in their study, reported distillate fluxes of about 4.7 L·h

^{−1}·m

^{−2}for a system made up of an AGMD module of 1.92 m

^{2}of membrane, a feed flow rate of 20 L·min

^{−1}, and 40 °C feed water temperature, while Banat et al. [10] reported that a distillate fluxes up to 2.5 L·m

^{−2}·h

^{−1}for a larger system composed of 4 MD modules of 10 m

^{2}of each membrane, a feed flow rate up to 21.2 L·min

^{−1}, and 74 °C feed temperature; in both investigations, they used real seawater.

^{−1}m

^{−2}for 34.5 °C of the temperature of the feed water with a salinity of 4 g·L

^{−1}.

^{−2}·h

^{−1}, MD hot and cold temperature difference must be kept between 40 °C and 45 °C, while it is necessary to adjust the flow rate from 6 to 7 L·min

^{−1}depending upon the operating season.

^{−1}and remains nearly constant on various days during the year by utilizing solar energy alone. Afterward, a pilot-scale experiment study was carried out to determine the performance of a 14.4 m

^{2}multichannel spiral-wound AGMD module, which can be used for water desalination with capacities up to 18 kg·h

^{−1}from the flow rate of distillate water. The AGMD system’s specific thermal energy consumption ranged from 158.83 to 346.55 kWh·m

^{−3}, and the maximum gain output ratio (GOR) achieves 4.4 at 52 °C, based on the inlet temperature of feed.

## 2. System Description

## 3. Numerical Modeling and Proposed Method

#### 3.1. AGMD Unit

#### 3.1.1. Heat Transfer

_{ch}is the coefficient of convective exchange and T

_{h}is the differential temperature between the hot solution temperature, and T

_{hm}is the temperature of the hot solution at the interface of the membrane. Equation (2) is used to determine the heat flow in the hot channel, where R

_{mT}is the membrane’s thermal resistance, T

_{mg}is the temperature at the membrane-air gap interface, J

_{v}is the vapor flux traveling through the membrane, and h

_{v}is the evaporation enthalpy [38].

_{ag}and T

_{p}are the air gap thermal resistance and the permeate temperature, respectively [8].

_{cc}and T

_{c}are the coefficient of heat transfer and the cold solution temperature in the cold channel, respectively. The heat flux in the cold channel’s boundary layer, as shown in Equation (5) [38]:

#### 3.1.2. Mass Transfer

_{w}) is proportional to the vapor pressure difference throughout the membrane matrix, and it can be calculated using Equation (6) [8,40], where P

_{hm}and P

_{p}are the vapor pressures at the membrane’s surface in the hot channel’s boundary layer and the air gap at the cooling plate’s surface, respectively. α is the activity coefficient and is the solution’s water fraction.

_{hm}and P

_{p}, the Antoine equation can be used and is given by Equation (7) [8]:

_{w}is given in Equation (8). Water molecular weight, gas constant, absolute membrane temperature, and total pressure inside the pores are all represented by M

_{w}, R, T

_{m}, and P, respectively. D

_{va}is the water vapor’s thermal diffusivity in the air gap, and τ is the membrane’s tortuosity [8].

#### 3.2. Heat Exchangers Model

#### 3.3. Solar Flat Plate Collector

_{eff}denoting the effective optical fraction of the absorbed energy, I

_{T}is the total amount of solar radiation incident on the collector surface in W/m

^{2}, and A

_{c}is the collector surface in m

^{2}.

_{0}is determined by the collector’s total heat transfer coefficient U

_{L}and its temperature. It can be expressed by Equation (15) [42]:

_{0}represents the heat loss in W, U

_{L}represents the heat loss coefficient W/K·m

^{2}, T

_{c}represents the collector’s average temperature in °C, and Ta represents the ambient temperature in °C. As a result, the rate at which the collector extracts useful energy, denoted as the extraction rate under stable state conditions, is proportional to the amount of useful energy absorbed by the collector minus the quantity lost by the collection. It is written as shown in Equation (16) [41]:

_{u}is derived as shown in Equation (18) [41,44]:

_{u}on incident solar energy corresponds to the collector’s efficiency, as shown in Equation (19) [43].

#### 3.4. System Performance Assessment

## 4. Results and Discussion

#### 4.1. Validation of the AGMD Model

^{−2}·h

^{−1}at a cooling temperature set at 15 °C and a cooling flow rate set at 5 L·min

^{−1}. This change in permeate flow may be the result of the water vapor’s increased transmembrane force. The outcomes are consistent with those that have been reported by many authors in the literature [47,48].

#### 4.2. Evaluation of Solar Potential

#### 4.3. Variation of the Temperature in AGMD Unit

_{hm}), the temperature at the interface between the membrane and the air gap (T

_{mg}), the temperature of permeate (T

_{p}), and the temperature at the interface between the cooling solution and the cold plate’s surface (T

_{pc}). The obtained results are given in Figure 7.

#### 4.4. Variation of the Saline Water Temperature in Flat Plate Collector

^{−1}mass flow rate.

#### 4.5. Variation of the Permeate Flux

^{−2}·h

^{−1}for a flow rate of 0.01 kg·s

^{−1}and is considered as the maximum value goshawk 12:00 h. These obtained results show the best matching with the other results cited in the literature. As can be observed from Figure 13, the permeate flux is very sensitive to the variation of the flow rate of seawater. It was marked that the permeate flux increases as the flow rate of seawater decreases. Permeate flux passes from 8 kg·m

^{−2}·h

^{−1}for a flow rate of 0.01 kg·s

^{−1}to 16 kg·m

^{−2}·h

^{−1}for 0.005 kg·s

^{−1}value of a flow, which means that the flow increased two times. From the obtained data and in order to guarantee the smooth running of the coupled system, 0.01 kg·s

^{−1}was chosen as the optimum flow rate value.

^{−1}, while the flow rates of feed and coolant were kept constant at 2 L·min

^{−1}. The increased air gap thickness in the module at the permeate side, caused by the higher mass transfer resistance, greatly reduces the permeate flux. Additionally, the performance of the AGMD process is directly influenced by the minimum air gap thickness.

#### 4.6. System Performance

## 5. Conclusions

- The variation of the temperature of the heat transfer fluid at the outlet of the solar SFPC increases gradually to reach 50 °C, which is the most used value in the literature for the AGMD process and the collector’s safety.
- An average distillate water production of 8 kg·m
^{−2}·h^{−1}could be achieved at 68 °C for a feed temperature and a flow rate of 1 L·min^{−1}. - Enhanced air gap thickness will be conducted to thermal and mass resistance, and thus a decrease in the mass flux and the thermal efficiency of the AGMD.
- The maximum GOR and PR values of 4.6 and 2, respectively, can be reached at 12:00 h corresponding to the minimum irreversible loss of the overall desalination system.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | surface area [m^{2}] | VMD | vacuum membrane distillation |

B_{w} | mass transfer coefficient [kg·m^{−2}·h^{−1}·Pa^{−1}] | Greek letters | |

C_{p} | thermal capacity [J·kg^{−1}·K^{−1}] | α | activity coefficient [−] |

D_{va} | thermal diffusivity of water vapour in air [m^{2}·s^{−1}] | β | water fraction [−] |

d_{h} | hydraulic diameter [m] | δ | thickness [m] |

F | correction factor [−] | ε | porosity [−] |

F_{R} | heat removal factor [−] | μ | dynamic viscosity [kg·m^{−1}·s^{−1}] |

GOR | gained output ratio [−] | ρ | density [kg·m^{−3}] |

hc_{c} | heat transfer coefficient [W·m^{−2}·K^{−1}] | φ | thermal flux [W·m^{−2}] |

h_{ch} | convective heat transfer coefficient [W·m^{−2}·K^{−1}] | τ | tortuosity [−] |

h_{v} | enthalpy [kJ·kg^{−1}] | Subscripts | |

J_{w} | permeate flux [kg·m^{−2}·s^{−1}] | 0 | reference state |

k | thermal conductivity [W·m^{−1}·K^{−1}] | a | air |

L | module length [m] | ag | air gap |

LMDT | logarithmic mean temperature difference [K] | c | cold |

M_{w} | Molar mass of water [kg·mol^{−1}] | f | feed |

$\dot{m}$ | mass flow rate [kg·s^{−1}] | h | hot |

MED | multiple-effect distillation | h_{m} | hot fluid—membrane interface |

MD | membrane distillation | in | inlet |

MSF | multi-stage flash distillation | m | membrane |

P | pressure [Pa] | mg | membrane—air gap interface |

PR | Performance Ratio [−] | out | outlet |

${Q}_{u}$ | useful energy delivered by the solar collector [kW] | p | plate |

$\phi $ | thermal flux [W·m^{−2}] | m_{g} | membrane—air gap interface |

R | thermal resistance [m^{2}·K·W^{−1}] | pc | cold fluid—plate interface |

RO | reverse osmosis | r | receiver |

s | salinity [g·kg^{−1}] | so | source |

T | temperature [°C] | sw | seawater |

t | time [s] | th | thermal |

U | heat transfer coefficient [W·m^{−2}·K^{−1}] | v | vapour |

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**Figure 4.**Evolution of permeate flux as a function of feed temperature [38].

**Figure 10.**Variation of the inlet and outlet of the heat transfer fluid temperatures from the flat plate collector with time.

MD Type | Membrane Type | Pore Size | Solution | Feed Teperature (°C) | J_{w} (kg·m^{−2}·h^{−1}) | Reference |
---|---|---|---|---|---|---|

AGMD | PVDF | 0.22 | Methanol/water | 50 | ≈3.9–4.6 | [49] |

AGMD | PTFE | 0.2 | NaCl | 65 | ≈7 | [8] |

AGMD | PTFE | 0.2 | NaCl | 80 | ≈6.5 | [47] |

VMD | PP | 0.2 | NaCl | 55 | ≈10.7–7.0 | [50] |

DCMD | PTFE | 0.2 | NaCl | 31 | ≈32.4–25.2 | [51] |

DCMD | PVDF | 0.4 | NaCl | 81 | ≈44–63 | [52] |

DCMD | PVDF | 0.22 | NaCl | 68 | ≈36–28.8 | [53] |

AGMD | PTFE | 0.2 | NaCl | 65 | 7.03 | Current study |

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**MDPI and ACS Style**

Mibarki, N.; Triki, Z.; Belhadj, A.-E.; Tahraoui, H.; Zamouche, M.; Kebir, M.; Amrane, A.; Zhang, J.; Mouni, L.
An Effective Standalone Solar Air Gap Membrane Distillation Plant for Saline Water Desalination: Mathematical Model, Optimization. *Water* **2023**, *15*, 1141.
https://doi.org/10.3390/w15061141

**AMA Style**

Mibarki N, Triki Z, Belhadj A-E, Tahraoui H, Zamouche M, Kebir M, Amrane A, Zhang J, Mouni L.
An Effective Standalone Solar Air Gap Membrane Distillation Plant for Saline Water Desalination: Mathematical Model, Optimization. *Water*. 2023; 15(6):1141.
https://doi.org/10.3390/w15061141

**Chicago/Turabian Style**

Mibarki, Nawel, Zakaria Triki, Abd-Elmouneïm Belhadj, Hichem Tahraoui, Meriem Zamouche, Mohammed Kebir, Abdeltif Amrane, Jie Zhang, and Lotfi Mouni.
2023. "An Effective Standalone Solar Air Gap Membrane Distillation Plant for Saline Water Desalination: Mathematical Model, Optimization" *Water* 15, no. 6: 1141.
https://doi.org/10.3390/w15061141