# Productivity and Thermal Performance Enhancements of Hollow Fiber Water Gap Membrane Distillation Modules Using Helical Fiber Configuration: 3D Computational Fluid Dynamics Modeling

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}h) of water flux was produced by the WGMD system, while only 24 L/(m

^{2}h) of water flux was produced by the AGMD system, with around 12.5% of outperformance for WGMD at 80/20 °C feed/coolant inlet temperatures and tap water salinity levels. Gao et al. [14] derived experimental work on an HF-WGMD module. The module consisted of eight fibers inserted inside eight cooling tubes, all inside a shell with a 25 mm diameter and a 425 mm length. The authors used water at a 10,000 ppm salinity level for the feed and coolant channels to provide a recovery system for the evaporation’s latent heat from the feed side. The module produced water fluxes of up to 9.14 L/(m

^{2}h) at 70/20 °C feed/coolant inlet temperatures while the amount of thermal energy consumed was around 6 kWh per unit kg of produced water at the same feed/coolant inlet conditions. On the other hand, they tested the same module with the same operating conditions but in the form of the HF-DCMD module. The HF-DCMD module achieved superiority in terms of flux, yet the HF-WGMD system outperformed HF-DCMD system in terms of energy efficiency by around 14%. This emphasizes the requirement to boost the HF-WGMD module’s flux.

^{2}h) and 4.97 L/(m

^{2}h) of distillate flux that represented 34.3% and 39% more water flux enhancements compared to that of the no-wire module at 60 and 40 °C feed inlet temperatures and seawater salinity, respectively. Additionally, some 3D CFD models of hollow fiber MD modules were introduced for DCMD and VMD configurations [23,24,25]. Zhang et al. [24] introduced a 3D CFD model to mimic the aquatic NaCl solution in the HF-VMD process. The CFD study included the effects of the membrane thickness, feed temperature, and pressure on the boundary layer development over membrane surfaces. It was observed that the majority of changes occurred on the boundary layer in the membrane silk. Yang et al. [25] provided an optimization study on the HF-DCMD desalination module. They introduced a 3D CFD study aimed at exploring the ability of HF designs to enhance the DCMD system’s performance. Hollow fibers with different geometries were included in the study such as wavy and gear-shaped cross sections to compare its performance with the original straight fibers at the same operating conditions. The authors observed a 66% enhancement in terms of the mass flux for the gear-shaped designs over the original straight fibers, followed by the wavy designs. Additionally, the CFD study included the results of the module’s hydraulic energy consumption, where the gear-shaped designs achieved the highest productivity and the lowest hydraulic energy consumption, followed by the wavy designs.

^{2}h) when the feed inlet temperature was varied from 40 to 70 °C with around −21.9% and 1.7% deviations from the experimental fluxes, respectively. The model predicted that the water flux from the graphite-filled MGMD module was 11% to 22% higher than that of the WGMD module, while other MGMD modules such as the silica gel and zeolite MGMD modules had 17% to 24% and 18% to 27% water output fluxes, respectively, which were lower than that of the WGMD module, and they all had a 40 to 70 °C feed inlet temperature range.

^{3}/day in the case of using 25/5 °C feed/coolant inlet temperatures, while the product rate could be enhanced to 12.1 m

^{3}/day for 45/25 °C feed/coolant inlet temperatures; both rates are per cubic meter of desalination unit.

## 2. HF-WGMD System Description

## 3. Mathematical Model

- The feed and coolant flows are laminar and in a counter flow pattern.
- The flow is stagnant in the water gap.
- The HF membrane porosity is uniformly distributed.
- There is no pore wetting of the membrane.
- The MD module’s exterior heat losses are disregarded.
- No fouling occurs on the feed–membrane interface.

#### 3.1. Governing Equations

#### 3.1.1. Mass Transport Phenomena

#### 3.1.2. Momentum Transport Phenomena

#### 3.1.3. Thermal Energy Transport Phenomena

#### 3.2. Boundary Conditions

#### 3.3. Solution Procedure

#### 3.4. Performance Evaluation Parameters

_{w}

^{3}/(m

_{du}

^{3}day) and can be determined as follows:

^{2}h), ${A}_{m}$ is the total HF membrane’s effective area inside the cooling tube in m

^{2}, and ${P}_{t}$ and $L$ are the cooling tube spacing and module’s effective length in m, respectively, as illustrated in Figure 3.

_{w}and can be determined as follows:

#### 3.5. Grid Independence Test

## 4. Results and Discussion

#### 4.1. Experimental Validation of CFD Model

#### 4.2. CFD Simulation of Helical HF-WGMD

#### 4.2.1. Effect of Feed Inlet Velocity on Feed Salinity and Temperature

#### 4.2.2. Effect of Coolant Inlet Velocity on Water Gap Temperature

#### 4.2.3. Effect of Using Helical Hollow Fibers on the Water Gap Temperature

^{3}and 4.116 mol/m

^{3}are attained in the case of single and double fibers, respectively. Due to the significant accumulated heating load caused by the double fiber configuration, the peak concentration difference in the case of a single fiber is bigger than that of double fibers.

#### 4.2.4. Effect of Feed Water Inlet Temperature on the Vapor Concentration on Both Sides of the HF Membrane

^{3}at a 40 °C feed temperature and enhances by 54, 129.5, and 232.2% at 50, 60, and 70 °C feed inlet temperatures, respectively.

^{3}to 1.485, 2.465, and 3.709 mol/m

^{3}when increasing the feed inlet temperature from 40 °C to 50, 60, and 70 °C, respectively, as depicted in Table 9.

#### 4.3. Parametric Studies

#### 4.3.1. Effect of Feed Flow Velocity on the Water Output Flux

^{2}h), the authors choose to forego an additional 5.3% of flux at 1.45 m/s to prevent an additional 22% of pressure loss per unit of produced flux.

#### 4.3.2. Effect of Coolant Flow Velocity and Number of Helical Turns on the Water Output Flux

^{2}h) for a single straight fiber to a peak value of 10.98 L/(m

^{2}h) for a single helical fiber with 20 turns at a velocity of 0.05 m/s in the coolant inlet, which can be explained by the helical fiber approaching from the module’s cooling tube in a helical configuration, as discussed in Section 4.2.3. At 50 helical turns, the flux then decreases once again to 9.92 L/(m

^{2}h) as a result of the greater heating loads imposed on the WG by the longer fiber length. The same pattern is observed for double fibers, where the water flux is increased by 10.4% at 20 helical turns at a 0.05 m/s coolant velocity and lowered once again at higher numbers of helical turns (Figure 11b). At the minimum velocity of 0.00031 m/s, however, the straight HF fiber module produces the highest flux.

^{2}h) for single fibers and 8.82 L/(m

^{2}h) for double fibers, but at a 0.21 m/s coolant inlet velocity, only 4% to 8% of the flux can be added. So in the performance evaluation of the desalination units in the next sections, 0.05 m/s will be set as the coolant inlet velocity.

#### 4.3.3. Effect of Feed Temperature on the Flux and Productivity

^{2}h) at a feed temperature of 40 °C to maximums of 4.04, 6.87, and 10.64 L/(m

^{2}h) at feed temperatures of 50, 60, and 70 °C, respectively. At higher feed temperatures, the effect of the number of turns on the produced flux is more obvious. When comparing the flux at 20 turns with the reference module of a straight HF membrane at 70 °C, the highest increase of 11.4% is attained, as noticed in Figure 13d, and only a 7.9% flux increase is achieved at a 40 °C feed temperature, as noticed in Figure 13a. The flow in the turning pass associated with the helical HF design increases the flow mixing and reduces the temperature and salt concentration polarizations at the boundary layer, which, in turn, can enhance the produced flux. However, the accumulated heat in the water gap channel works on eliminating these enhancements in the flux, especially at higher numbers of turns.

_{w}

^{3}/day per cubic meter of desalination unit is produced, as shown in Figure 13d. This demonstrates the significant effect of increasing the fiber length using the proposed helical configuration without affecting the total volume of the module.

^{2}h) is reached, which is smaller than the corresponding one with a single helical case of 8.07%.

_{w}

^{3}/(m

_{du}

^{3}day), which is 40.3% higher than the specific productivity of the double straight configuration at the same feed temperature. When compared to the single helical design under the identical conditions, as shown in Figure 13d and Figure 14d, this maximum productivity is significantly greater by 78%. Due to the noticeable reduction in fluxes produced by double configurations brought on by the increased heating load applied to the water gap, as mentioned in Section 4.2.3, the desalination unit’s specific production of a double helical design is observed not to be doubled over the modules of a single helical design. However, the double helical design will be preferable when the desalination unit’s compactness is the primary consideration.

#### 4.3.4. The Effect of Feed Temperature on the Thermal Performance

_{w}is attained at a 70 °C feed temperature and 50 single helical turns with a positive reduction of 35% in comparison to the reference case, as illustrated in Figure 15d. When the source of the feed is limited to 40 °C [30], the use of 50 single helical turns favorably reduces the STEC by 32.6%, as shown in Figure 15a. Similar findings are obtained for the STEC values in the case of the double helical design, as illustrated in Figure 16. This demonstrates the practical uses for HF in a helical form in the cooling tubes of the WGMD systems to lower the energy consumption and unit size.

#### 4.3.5. The Effect of Multi-Stages Arranged in Series on the System Thermal Performance

_{w}for the single-stage unit to 1.8 kWh/kg

_{w}for the three-stage unit, representing a 53.8% decrease in the STEC.

## 5. Practical Considerations and Future Prospects

## 6. Conclusions

_{w}

^{3}/(m

_{du}

^{3}·day) is obtained from 50 turns in the case of single helical modules at a feed temperature of 70 °C, which represents a 46.2% improvement over the typical module of a single straight HF membrane. When using a double helical design under the same circumstances, this maximum productivity can be enhanced once more by 78% to be 45 m

_{w}

^{3}/(m

_{du}

^{3}·day).

_{w}is achieved, providing a positive reduction in the energy consumption of 35% from the reference straight fiber case.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${a}_{w}$ | Water activity coefficient |

${C}_{p}$ | Specific heat at constant pressure (J/kg K) |

$c$ | Concentration (mol/m^{3}) |

$D$ | Diffusion coefficient (m^{2}/s) |

${D}_{Kn}$ | Knudsen diffusion coefficient (m^{2}/s) |

${D}_{Or}$ | Ordinary molecular diffusion coefficient (m^{2}/s) |

$d$ | Diameter (m) |

${d}_{p}$ | Membrane pore diameter (m) |

$G$ | Diffusive flux (kg/(m^{2} s)) |

GOR | Gain output ratio |

${h}_{fg}$ | Latent heat of vaporization (J/kg) |

$J$ | Flux (L/(m^{2} h)) |

$Kn$ | Knudsen number |

$k$ | Thermal conductivity (W/m K) |

$L$ | Module effective length (m) |

$M$ | Molecular mass (g/mol) |

$p$ | Pressure (Pa) |

${P}_{t}$ | Cooling tube spacing (m) |

$ppm$ | Concentration in parts per millions |

$q$ | Heat flux (W/m^{2}) |

$\overline{R}$ | Universal gas constant (J/(mol K)) |

$Re$ | $\frac{\rho {U}_{fi}{d}_{mi}}{\mu}$ |

$SP$ | Specific productivity (m_{w}^{3}/(m_{du}^{3} day)) |

$STEC$ | Specific thermal energy consumption (kWh/kg_{w}) |

$T$ | Temperature (°C) |

$TER$ | Thermal energy recovered (%) |

U | Flow stream mean velocity |

$\stackrel{\u20d1}{u}$ | Flow velocity field vector (m/s) |

$v$ | Specific volume (m^{3}/kg) |

$W$ | Water vapor content (kg_{v}/kg_{a}) |

${x}_{NaCl}$ | Salt mole fraction |

${x}_{w}$ | Water mole fraction |

$\mathrm{S}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{t}\mathrm{s}$ | |

$a$ | Air |

$atm$ | Atmospheric |

$c$ | Coolant channel |

$du$ | Desalination unit |

$f$ | Feed channel |

$g$ | Water gap |

$i$ | Inlet |

$m$ | Membrane |

$NaCl$ | NaCl salt |

$o$ | Outlet |

$sat$ | Saturation |

$t$ | Cooling tube |

$v$ | Vapor |

$w$ | Water |

$\mathrm{G}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{k}\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{b}\mathrm{o}\mathrm{l}\mathrm{s}$ | |

$\epsilon $ | Membrane porosity |

$\mu $ | Fluid dynamic viscosity (Pa s) |

$\rho $ | Fluid density (kg/m^{3}) |

$\sigma $ | Molecule collision diameter (m) |

$\tau $ | Membrane pore tortuosity |

## Appendix A

#### Appendix A.1. Proposal of Helical Fibers’ Support Tube

**Figure A1.**Schematic diagrams of (

**a**) perforated support tube, (

**b**) single helical fiber supported by the perforated tube, and (

**c**) double helical fibers supported by the perforated tube.

#### Appendix A.2. Mathematical Model

- (1)
- Water–salt diffusion coefficient

^{3}/mol.

- (2)
- HF membrane diffusion coefficient

- (3)
- HF membrane thermal conductivity

- (4)
- Heat source and heat sink boundary conditions

- (5)
- Membrane interfaces’ boundary conditions

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**Figure 2.**The simulated 3D HF-WGMD units with (

**a**) single straight fiber, (

**b**) single helical fiber, (

**c**) double straight fibers, and (

**d**) double helical fibers, all inserted inside the unit cooling tube.

**Figure 3.**A representation diagram of the 3D simulated HF-WGMD unit. (1) Feed channel, (2) HF membrane, (3) water gap, (4) cooling tube, and (5) coolant channel.

**Figure 4.**Experimental validation. (

**a**) Numerical flux calculated using the 3D CFD model against flux produced by different experimental modules [14,28] at the same operating conditions. (

**b**) Output flux comparison between the current 3D CFD model and the 1D model proposed by [28] for the same modules with multi-fibers inside the cooling tube.

**Figure 5.**Feed water salinity contours at middle of $xy$ half section in the feed channel of desalination unit with single straight fiber at ${U}_{ci}=0.21\mathrm{m}/\mathrm{s}and{T}_{fi}=70\xb0\mathrm{C}$; (

**a**) ${U}_{fi}=0.29\mathrm{m}/\mathrm{s}$, (

**b**) ${U}_{fi}=0.58\mathrm{m}/\mathrm{s}$, (

**c**) ${U}_{fi}=0.87\mathrm{m}/\mathrm{s}$, (

**d**) ${U}_{fi}=1.16\mathrm{m}/\mathrm{s}$, and (

**e**) ${U}_{fi}=1.45\mathrm{m}/\mathrm{s}$.

**Figure 6.**Feed water temperature contours at $zx$ cross section at middle of fiber length $(y=250\mathrm{m}\mathrm{m})$ of desalination unit with single straight fiber at ${U}_{ci}=0.21\mathrm{m}/\mathrm{s}and{T}_{fi}=70\xb0\mathrm{C}$; (

**a**) ${U}_{fi}=0.29\mathrm{m}/\mathrm{s}$, (

**b**) ${U}_{fi}=0.58\mathrm{m}/\mathrm{s}$, (

**c**) ${U}_{fi}=0.87\mathrm{m}/\mathrm{s}$, (

**d**) ${U}_{fi}=1.16\mathrm{m}/\mathrm{s}$, and (

**e**) ${U}_{fi}=1.45\mathrm{m}/\mathrm{s}$.

**Figure 7.**Temperature contours at middle of $xy$ section in desalination unit at ${U}_{fi}=1.16\mathrm{m}/\mathrm{s}and{T}_{fi}=70\xb0\mathrm{C}$ with single fiber [(

**a**) ${U}_{ci}=0.0031\mathrm{m}/\mathrm{s}$, (

**b**) ${U}_{ci}=0.0125\mathrm{m}/\mathrm{s}$, (

**c**) ${U}_{ci}=0.05\mathrm{m}/\mathrm{s}$ and (

**d**) ${U}_{ci}=0.21\mathrm{m}/\mathrm{s}$], and double straight fibers [(

**e**) ${U}_{ci}=0.0031\mathrm{m}/\mathrm{s}$, (

**f**) ${U}_{ci}=0.0125\mathrm{m}/\mathrm{s}$, (

**g**) ${U}_{ci}=0.05\mathrm{m}/\mathrm{s}$ and (

**h**) ${U}_{ci}=0.21\mathrm{m}/\mathrm{s}$]. The arrows represent the axial velocity magnitudes relative to each other.

**Figure 8.**Water gap temperature contours at middle of $xy$ section at ${U}_{fi}=1.16\mathrm{m}/\mathrm{s},{U}_{ci}=0.05\mathrm{m}/\mathrm{s},and{T}_{fi}=70\xb0\mathrm{C}$ for desalination unit with single (

**a**) straight fiber with (

**b**) $10$, (

**c**) $20$, (

**d**) $30$, (

**e**) $40$, and (

**f**) $50$ turns of helical fibers and double (

**g**) straight fibers with (

**h**) $10$, (

**i**) $20$, (

**j**) $30$, (

**k**) $40$, and (

**l**) $50$ turns of helical fibers.

**Figure 9.**Water vapor concentration contours on both HF membrane interfaces with $50$ turns of double helical fibers at ${U}_{fi}=1.16\mathrm{m}/\mathrm{s}and{U}_{ci}=0.05\mathrm{m}/\mathrm{s}$. (

**a**,

**b**) ${T}_{fi}=40\xb0\mathrm{C}$, (

**c**,

**d**) ${T}_{fi}=50\xb0\mathrm{C}$, (

**e**,

**f**) ${T}_{fi}=60\xb0\mathrm{C}$, and (

**g**,

**h**) ${T}_{fi}=70\xb0\mathrm{C}$.

**Figure 10.**Water output flux and pressure drop per unit flux against feed inlet velocity with single straight fiber module at ${T}_{fi}=70\xb0\mathrm{C}$ and ${U}_{ci}=0.21\mathrm{m}/\mathrm{s}$.

**Figure 11.**Water output flux against number of HF helical turns at different coolant inlet velocities and ${U}_{fi}=1.16\mathrm{m}/\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{d}{T}_{fi}=70\xb0\mathrm{C}$ for desalination units with (

**a**) single helical fiber and (

**b**) double helical fibers.

**Figure 12.**Water output flux against coolant inlet velocity at ${U}_{fi}=1.16\mathrm{m}/\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{d}{T}_{fi}=70\xb0\mathrm{C}$ for a desalination unit with $50$ turns of helical fiber; (

**a**) single and (

**b**) double.

**Figure 13.**Water output flux and specific productivity against number of HF helical turns for desalination unit with single fiber at ${U}_{fi}=1.16\mathrm{m}/\mathrm{s}and{U}_{ci}=0.05\mathrm{m}/\mathrm{s}$; (

**a**) ${T}_{fi}=40\xb0\mathrm{C}$, (

**b**) ${T}_{fi}=50\xb0\mathrm{C}$, (

**c**) ${T}_{fi}=60\xb0\mathrm{C}$, and (

**d**) ${T}_{fi}=70\xb0\mathrm{C}$.

**Figure 14.**Water output flux and specific productivity against number of HF helical turns for desalination unit with double fibers at ${U}_{fi}=1.16\mathrm{m}/\mathrm{s}and{U}_{ci}=0.05\mathrm{m}/\mathrm{s}$; (

**a**) ${T}_{fi}=40\xb0\mathrm{C}$, (

**b**) ${T}_{fi}=50\xb0\mathrm{C}$, (

**c**) ${T}_{fi}=60\xb0\mathrm{C}$, and (

**d**) ${T}_{fi}=70\xb0\mathrm{C}$.

**Figure 15.**STEC and %TER against number of HF helical turns for desalination unit with single fiber at ${U}_{fi}=1.16\mathrm{m}/\mathrm{s}and{U}_{ci}=0.05\mathrm{m}/\mathrm{s}$; (

**a**) ${T}_{fi}=40\xb0\mathrm{C}$, (

**b**) ${T}_{fi}=50\xb0\mathrm{C}$, (

**c**) ${T}_{fi}=60\xb0\mathrm{C}$, and (

**d**) ${T}_{fi}=70\xb0\mathrm{C}$.

**Figure 16.**STEC and %TER against number of HF helical turns for desalination unit with double fibers at ${U}_{fi}=1.16\mathrm{m}/\mathrm{s}and{U}_{ci}=0.05\mathrm{m}/\mathrm{s}$; (

**a**) ${T}_{fi}=40\xb0\mathrm{C}$, (

**b**) ${T}_{fi}=50\xb0\mathrm{C}$, (

**c**) ${T}_{fi}=60\xb0\mathrm{C}$, and (

**d**) ${T}_{fi}=70\xb0\mathrm{C}$.

**Figure 17.**The effect of multi-stages in series on the GOR and STEC of $50\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{n}\mathrm{s}$ of single helical fiber desalination units at ${U}_{fi}=1.16\mathrm{m}/\mathrm{s},{U}_{ci}=0.05\mathrm{m}/\mathrm{s}$, and ${T}_{fi}=70\xb0\mathrm{C}$.

**Figure 18.**Percentage of thermal energy recovered of (

**a**) one, (

**b**) two, and (

**c**) three single helical fiber desalination units with 50 turns fitted in series at ${U}_{fi}=1.16\mathrm{m}/\mathrm{s},{U}_{ci}=0.05\mathrm{m}/\mathrm{s}$, and ${T}_{fi}=70\xb0\mathrm{C}$.

Parameter | Symbol | Value | Unit |
---|---|---|---|

HF membrane inner diameter | ${d}_{mi}$ | 0.8 | mm |

HF membrane outer diameter | ${d}_{mo}$ | 1.16 | mm |

Cooling tube inner diameter | ${d}_{ti}$ | 5 | mm |

Cooling tube outer diameter | ${d}_{ti}$ | 5.56 | mm |

Cooling tube spacing | ${P}_{t}$ | 6.95 | mm |

Module effective length | $L$ | 500 | mm |

Feed inlet salinity | - | 35,000 | ppm |

Water gap salinity | - | 0.0 | ppm |

Coolant inlet temperature | ${T}_{ci}$ | 20 | °C |

Feed water thermal conductivity | ${k}_{f}$ | 0.64 | W/m K |

Membrane thermal conductivity | ${k}_{m}$ | 0.07 | W/m K |

Cooling tube thermal conductivity | ${k}_{t}$ | 0.445 | W/m K |

Membrane porosity | $\epsilon $ | 82 | % |

Membrane pore tortuosity | $\tau $ | 1.698 | - |

Membrane pore diameter | ${d}_{p}$ | 0.16 | μm |

Water vapor molar mass | ${M}_{w}$ | 18 | g/mol |

Salt molar mass | ${M}_{NaCl}$ | 58.4 | g/mol |

Domain | Position | Boundary Conditions | ||
---|---|---|---|---|

Mass | Momentum | Energy | ||

Feed channel | $y=0$ | ${c}_{f}={c}_{fi}$ | $\stackrel{\u20d1}{{u}_{f}}=(0,{U}_{fi},0)$ | ${T}_{f}={T}_{fi}$ |

$y=L$ | $\frac{\partial {c}_{f}}{\partial y}=0$ | ${p}_{f}={p}_{atm}$ | $\frac{\partial {T}_{f}}{\partial y}=0$ | |

HF membrane | $y=0$ | $\frac{\partial {c}_{m}}{\partial y}=0$ | - | $\frac{\partial {T}_{m}}{\partial y}=0$ |

$y=L$ | $\frac{\partial {c}_{m}}{\partial y}=0$ | - | $\frac{\partial {T}_{m}}{\partial y}=0$ | |

Water gap | $y=0$ | - | - | $\frac{\partial {T}_{g}}{\partial y}=0$ |

$y=L$ | - | - | $\frac{\partial {T}_{g}}{\partial y}=0$ | |

Cooling tube | $y=0$ | - | - | $\frac{\partial {T}_{t}}{\partial y}=0$ |

$y=L$ | - | - | $\frac{\partial {T}_{t}}{\partial y}=0$ | |

Cooling channel | $x=0andx={P}_{t}$ | - | $\frac{\partial \stackrel{\u20d1}{{u}_{c}}}{\partial x}=\stackrel{\u20d1}{0}$ | $\frac{\partial {T}_{c}}{\partial x}=0$ |

$z=0andz={P}_{t}$ | - | $\frac{\partial \stackrel{\u20d1}{{u}_{c}}}{\partial z}=\stackrel{\u20d1}{0}$ | $\frac{\partial {T}_{c}}{\partial z}=0$ | |

$y=0$ | - | ${p}_{c}={p}_{atm}$ | $\frac{\partial {T}_{c}}{\partial y}=0$ | |

$y=L$ | - | $\stackrel{\u20d1}{{u}_{c}}=(0,-{U}_{ci},0)$ | ${T}_{c}={T}_{ci}$ |

**Table 3.**Grid independence study on a case of single straight fiber with ${T}_{fi}=70\xb0\mathrm{C}$ and ${U}_{fi}=0.81\mathrm{m}/\mathrm{s}$.

Number of Grid Elements | Water Flux (L/(m^{2} h)) | %Variation in Water Flux | Feed Outlet Temperature (°C) | % Variation in Feed Outlet Temperature | |
---|---|---|---|---|---|

Grid 1 | 1,203,853 | 8.83 | - | 59.131 | - |

Grid 2 | 2,411,325 | 9.05 | 2.492 | 59.349 | 0.369 |

Grid 3 | 5,168,202 | 9 | −0.552 | 59.438 | 0.15 |

**Table 4.**Operational and geometrical parameters of experimental modules used in CFD model validation.

Reference | Experimental Module | Number of Fibers per Tube | Module Length (mm) | Feed Inlet Temperature (°C) | Feed Inlet Velocity (m/s) |
---|---|---|---|---|---|

[28] | Module 1 | 1 | 350 | 40, 50, 60, and 70 | 0.69 |

Module 2 | 2 | ||||

Module 3 | 3 | ||||

[14] | Variable feed inlet temperatures | 1 | 425 | 0.81 | |

Variable feed inlet velocities | 70 | 0.28, 0.4, 0.53, 0.69, and 0.81 |

**Table 5.**Salinity and temperature at feed–membrane interface at fiber middle length $(y=250\mathrm{m}\mathrm{m})$ of desalination unit with single straight fiber at different feed inlet velocities.

$\mathbf{Feed}\mathbf{Inlet}\mathbf{Velocity}\left({\mathit{U}}_{\mathit{f}\mathit{i}}\right)$ (m/s) | |||||
---|---|---|---|---|---|

0.29 | 0.58 | 0.87 | 1.16 | 1.45 | |

Salinity (ppm) | 65,401 | 62,941 | 61,552 | 60,269 | 58,945 |

Temperature (°C) | 51.9 | 59 | 61.8 | 63.3 | 64.2 |

**Table 6.**Water gap average temperature at different cross sections along desalination unit with single and double straight fibers at different coolant inlet velocities.

$\mathit{y}=125\mathbf{m}\mathbf{m}$ | $\mathit{y}=250\mathbf{m}\mathbf{m}$ | $\mathit{y}=375\mathbf{m}\mathbf{m}$ | ||
---|---|---|---|---|

WG Average Temperature (°C) | ||||

${\mathit{U}}_{\mathit{c}\mathit{i}}=0.0031\mathrm{m}/\mathrm{s}$ | Single | 61.3 | 56.4 | 48.2 |

Double | 67.5 | 64.7 | 58.6 | |

${\mathit{U}}_{\mathit{c}\mathit{i}}=0.0125\mathrm{m}/\mathrm{s}$ | Single | 44.6 | 41.6 | 37.7 |

Double | 55.2 | 50.6 | 46.3 | |

${\mathit{U}}_{\mathit{c}\mathit{i}}=0.05\mathrm{m}/\mathrm{s}$ | Single | 36.1 | 35.2 | 33.5 |

Double | 44.6 | 41.4 | 39.7 | |

${\mathit{U}}_{\mathit{c}\mathit{i}}=0.21\mathrm{m}/\mathrm{s}$ | Single | 33.4 | 33.2 | 32.2 |

Double | 40.4 | 38.1 | 37.5 |

WG Average Temperature (°C) | ||||||
---|---|---|---|---|---|---|

Straight | $10$ Turns | $20$ Turns | $30$ Turns | $40$ Turns | $50$ Turns | |

Single | 31.4 | 30 | 30.5 | 31.2 | 32 | 32.9 |

Double | 38.2 | 38.4 | 39.2 | 40.4 | 41.5 | 42.9 |

**Table 8.**Average water vapor concentration differences across the HF membrane with single and double fibers.

Average Concentration Difference (mol/m^{3}) | ||||||
---|---|---|---|---|---|---|

Straight | $10$ Turns | $20$ Turns | $30$ Turns | $40$ Turns | $50$ Turns | |

Single | 4.116 | 4.435 | 4.51 | 4.439 | 4.293 | 4.206 |

Double | 3.792 | 4.069 | 4.116 | 4.009 | 3.837 | 3.709 |

Feed Inlet Temperature (°C) | ||||
---|---|---|---|---|

40 | 50 | 60 | 70 | |

Feed–membrane interface | 2.445 | 3.765 | 5.612 | 8.122 |

Membrane–WG interface | 1.682 | 2.28 | 3.147 | 4.413 |

Concentration difference | 0.763 | 1.485 | 2.465 | 3.709 |

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## Share and Cite

**MDPI and ACS Style**

Elbessomy, M.O.; Elsheniti, M.B.; Elsherbiny, S.M.; Rezk, A.; Elsamni, O.A.
Productivity and Thermal Performance Enhancements of Hollow Fiber Water Gap Membrane Distillation Modules Using Helical Fiber Configuration: 3D Computational Fluid Dynamics Modeling. *Membranes* **2023**, *13*, 843.
https://doi.org/10.3390/membranes13100843

**AMA Style**

Elbessomy MO, Elsheniti MB, Elsherbiny SM, Rezk A, Elsamni OA.
Productivity and Thermal Performance Enhancements of Hollow Fiber Water Gap Membrane Distillation Modules Using Helical Fiber Configuration: 3D Computational Fluid Dynamics Modeling. *Membranes*. 2023; 13(10):843.
https://doi.org/10.3390/membranes13100843

**Chicago/Turabian Style**

Elbessomy, Mohamed O., Mahmoud B. Elsheniti, Samy M. Elsherbiny, Ahmed Rezk, and Osama A. Elsamni.
2023. "Productivity and Thermal Performance Enhancements of Hollow Fiber Water Gap Membrane Distillation Modules Using Helical Fiber Configuration: 3D Computational Fluid Dynamics Modeling" *Membranes* 13, no. 10: 843.
https://doi.org/10.3390/membranes13100843