# Numerical Study of Inclination Effect of the Floating Solar Still Fitted with a Baffle in 3D Double Diffusive Natural Convection

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}/day. Although the freshwater yield of floating solar panels was significantly improved in their study, it is still much lower than that of land-based solar panels. Wang et al. [12] studied the design of a solar still with a floating solar desalination film that uses concentrated light. The authors performed an optical simulation of the light concentration process and examined the heat and mass transfer. They observed that the water production per unit area of this test system reached 1.38 kg/m

^{2}/day. Chen et al. [13] studied the seawater supply to a bionic floating solar still inspired by the water uptake of plant roots. The authors observed a decrease in downward heat loss and uniformity of water uptake. They obtained a daily freshwater yield of 1.5 kg/m

^{2}/day.

^{4}and 6 × 10

^{6}. They observed that when the aspect ratio decreases, the multicellular pattern decreases, and the convective heat and mass transfer increases up to 41%. They also showed that as the inclination angle rose, the Sherwood and Nusselt numbers increased up to 3.8%. A computational study of three-dimensional double-diffusive natural convection in a tilted rectangular solar still has been established by Ghachem et al. [16]. The effects of buoyancy ratio and cavity inclination are investigated. They showed that as N increases, a reversal of the main flow rotation occurs, resulting in an increasing dominance of the solute over the thermal thrust. They also showed that the maximum of Nusselt and Sherwood numbers occurs at an angle of 30° to the horizontal. Recently, Maatki [17] has numerically studied the triangular solar still and proposed an improvement of the evaporation surface. By increasing the number of feeder connections, the author showed an improvement in the performance of the solar still.

## 2. Physical Model and Governing Equations

#### 2.1. Physical Model

_{1}= 0.1 × H, case 1 is flat also of length L

_{2}= 0.3 × H, and cases 2 and 3 have the same length as case 2 and are triangular and curvilinear, respectively. The other walls (vertical walls and distillate collection panels) are considered adiabatic and impermeable.

#### 2.2. Governing Equations, Assymtions, and Boundary Conditions

- Concentration and temperature: $C\left(x,o,z\right)=1$, $T\left(x,o,z\right)=1$.

- Velocity: ${u}_{x}={u}_{y}={u}_{z}=0$ on all walls.

- Vorticity and vector potential:

- Vector potential: $\frac{\partial {\psi}_{x}}{\partial x}={\psi}_{y}={\psi}_{z}=0$ at $x=0$ and 1, ${\psi}_{x}=\frac{\partial {\psi}_{y}}{\partial y}={\psi}_{z}=0$ at $y=0$ and 1,${\psi}_{x}={\psi}_{y}=\frac{\partial {\psi}_{z}}{\partial z}=0$ at $z=0$ and 1.

## 3. Numerical Method, Validation, and Grid Sensitivity

#### 3.1. Numerical Method

^{−4}.

- ○
- Step 1: Initializing;
- ○
- Step 2: Resolution of the energy equation;
- ○
- Step 3: Resolution of the concentration equation
- ○
- Step 4: Resolution of the vorticity equation;
- ○
- Step 5: Resolution of the potential vector equation.

^{−5}.

#### 3.2. Validation

#### 3.3. Grid Sensitivity

^{4}. Four distinct mesh sizes were checked (41

^{3}, 51

^{3}, 61

^{3}, and 71

^{3}). The average Sherwood number is taken as the sensible variable. The incremental rise in Sh

_{av}between grid sizes 61

^{3}and 71

^{3}is about 0.14%. Hence, by considering the economy of computation and accuracy, a 61

^{3}mesh size is used.

## 4. Results and Discussion

_{1}= 0.1 × H, case 1 is also flat with length L

_{2}= 0.3 × H, cases 2 and 3 have the same length as case 2 and are triangular and curvilinear respectively. A mixture of air and water vapor is assumed to be perfect and the flow is laminar. Rayleigh numbers between 10

^{3}≤ Ra ≤ 5 × 10

^{4}and a positive buoyancy ratio between 0 ≤ N ≤ 2 are considered in the computations. All thermo-physical properties of the air vapor are taken as constant. The tilt angles are in the range of 0° ≤ θ ≤ 25° with an increment of 5°. The results of the three-dimensional numerical study of double-diffusive convection in the floating solar still are presented in the patterns of streamlines, particle trajectories, iso-temperatures, and iso-concentrations, Nusselt and Sherwood averages.

#### 4.1. Flow Structure, Iso-Temperatures, and Iso-Concentrations

^{3}, the flow strength is weak, and the thermal and mass buoyancy forces do not allow the fluid to reach the cold walls. In this case, the fluid escapes directly from the outlet windows by pressure gradient without being cooled.

^{4}, the intensity of the flow increases, in fact, the maximum velocity increases from 3.02 to 26.15. The development of two symmetrical vortices rotating in opposite directions is observed slightly below the baffles. The two vortices have a three-dimensional character by spiral effect. The temperature iso-surfaces show the transition to the convective mode. Thermal gradients increase at the bottom and top of the cavity. A slight attenuation of the horizontal stratification of the concentration iso-surfaces is observed. The concentration gradient increases mainly near the cold surfaces. When Ra = 2 × 10

^{4}, the flow becomes three-dimensional mainly in the upper region of the cavity, the vapor cooling region. The two vortices below the baffles decrease in size but increase in intensity. The temperature iso-surfaces show the dominance of convective effects. The shape of the temperature iso-surfaces becomes parabolic throughout the cavity. The thermal gradients are intensified near the active walls. The structure of the concentration iso-surfaces highlights the significance of the three-dimensional character of the flow. Mass convection becomes increasingly intense at this Rayleigh number.

^{3}and θ = 5°, the flow structure shows an asymmetric flow of air-vapor mixture from the base surface to the cooling surfaces. Two small vortices are developed under the baffles. The particle trajectories show that there is a leakage of the uncooled air-vapor mixture to the right distillate outlet window. When θ = 15°, the right vortex disappears while the left one expands. A quantity of hot air-vapor mixture located in the right part of the cavity is trapped in the lower zone and no longer reaches the cooling zone. At θ = 25°, the vortex expands more and more and occupies the central part of the cavity. The cooling of the mixture becomes essentially localized at the left cold surface. A slight rise in flow intensity is observed with increasing tilt angle. When Ra = 5 × 10

^{4}, the transition from θ = 5° to θ = 15° is characterized by the appearance of a vortex rotating counter-clockwise at the left central zone of the cavity. The maximum intensity of the velocity increases.

^{3}, but an uncooled air-vapor leakage is still observed from the right distillate outlet window.

^{3}, the concentration iso-surfaces exhibit a dominant diffusive regime. Increasing cavity tilt indicates a slight variation in the solutal gradient at the bottom surface. When Ra = 5 × 10

^{4}and θ = 5°, the concentration iso-surfaces show a higher solutal gradient on the left side of the low wall compared to the right side. While it is higher on the right side of the cooling zone. The concentration iso-surfaces have a slightly inclined parabolic shape. At θ = 15° and 25°, the iso-concentration structure is S-shaped in the central region of the cavity. The solutal gradients increasingly rise near the left and right low cooling active walls at θ = 25°. While the solutal gradient decreases near the left cooling wall.

^{3}. The conductive regime is dominant. When Ra = 5 × 10

^{4}, at θ = 5°, the temperature iso-contours form parabolic patterns surrounding the hot impermeable surfaces. When passing to θ = 15°, the left thermal gradient of the cold wall decreases. At θ = 25°, the thermal gradient at both cold surfaces decreases significantly.

^{3}), the particle trajectory shows uncooled air-vapor leakage to the distillate outlet window on the right and vortex development on the left. At this Rayleigh number, the three types of baffles designs presented in Figure 6a, show the absence of this vortex development. The particle trajectory for case 1, also shows uncooled air-vapor leakage to the distillate outlet window on the right. This problem disappears in cases 2 and 3. When Ra = 5 × 10

^{4}, the flow structure in case 1 is characterized by the development of two vortices near the left baffle. The maximum velocity, in this case, is lower than in the reference case. In cases 2 and 3, the flow structure is characterized by a single vortex and the maximum velocity intensity decreases further. It is noted also that the maximum velocity intensity is minimal in case 2. The shape of the baffles in cases 2 and 3 assisted the air-vapor mixture to cool before the exit.

^{4}. In the reference case, the air-vapor convection takes an upward movement from the right zone with a high solutal gradient in the lower left part of the cavity. However, in cases 1, 2, and 3, the air-vapor convection is almost uniformly in the lower part of the solar still. The solutal gradient in the cooling zone has decreased in cases 2 and 3 compared to the reference case at this Rayleigh number. An improvement of the air-vapor convection is observed in the heating zone for all the new baffle designs studied.

^{4}. The temperature iso-contours show that the thermal gradient has decreased in the lower part of the cavity in cases 1, 2, and 3. On the other hand, the impact of thermal convection in the central part of the cavity is improved. The thermal gradient is enhanced in the cooling zone mainly in the middle of the cold surfaces for cases 2 and 3.

#### 4.2. Heat and Masse Transfer Rates

^{3}to 10

^{4}, showing a shift from the conductive to the convective regime for all cases studied. An improvement of 50% is observed when the Rayleigh number is moved from 10

^{4}to 5 × 10

^{4}. Case 2 shows that the heat transfer rate is maximum for Ra = 2 × 10

^{4}. While for Ra = 5 × 10

^{4}, case 3 shows the maximum heat transfer rate.

^{4}and θ = 0°. The average heat transfer rate rises with increasing N. The thermal and solutal buoyancy forces cooperate for positive values of the buoyancy ratio. From N = 0.5, the average heat transfer rate is highest in case 2. When N is greater than 1, a 12% improvement in the average heat transfer rate compared to the reference case is observed. Cases 1 and 3 have almost equal average Nusselt values with a 6% improvement over the reference case.

^{4}, cases 0 and 1 have nearly equal mass transfer rate values and are the highest compared to the other cases.

^{4}and θ = 0° is shown in Figure 8b. Case 3 displays the minimum values of the average Sherwood for all values of N. Cases 0 and 2 exhibit the maximum values of the mass transfer rate with approximately equal values. When the buoyancy ratio is changed from N = 0 to 2, a significant enhancement of about 150% is seen for all cases studied.

^{4}, N = 0, and N = 1. When N = 0, for the reference case, inclination caused a decrease in the average heat transfer rate. From θ = 0° to θ = 10°, a decrease of 12% is observed. The average Nusselt number shows constant values in the range of 10° to 20°, then undergoes another decrease of 10% from θ = 25°. When the inclination is less than 18°, all cases studied have higher average Nusselt values than the reference case. And case 2 has higher values of average heat transfer rate. When N = 1, the evolution of the average Nusselt as a function of the inclination presented is completely different from that of N = 0. Indeed, the heat transfer rate for cases 1,2, and 3 is higher than the reference case only when θ is lower than 5°. A considerable decrease of 45% of the average transfer rate is observed for these three cases when moving from θ = 5° to θ = 10° while it is 10% for the reference case.

^{4}, N = 0 and N = 1. When N = 0, for cases 0 and 1, the increase of the tilt angle decreases the average Sherwood. For cases 2 and 3, the evolution is parabolic with a maximum of the average transfer rate observed for θ = 5°. For all angles studied, the average mass transfer rate for case 3 is the lowest of the cases. In the range of angles between 5° and 20°, case 2 has the highest average mass transfer rate.

#### 4.3. Sensitivity Analysis

^{3}, 5 × 10

^{4}], N = [0, 2] and θ = [0, 25°]. For the baffle design factor, four configurations were performed and labeled as 0: reference case, 1: plane, 2: triangular, and 3: curvilinear.

_{av}= 0.647 + 1.31 × Ra − 0.247 × N − 0.0738 × Baffle design − 0.152 × θ

_{av}= −3.38 + 4.18 × Ra + 1.38 × N − 0.666 × Baffle design − 0.0835 × θ

^{2}value of the heat transfer rate model was 92.5% while that of the mass transfer rate model was 87.8%.

## 5. Conclusions

- ○
- Uncooled air-vapor leakage was observed during tilting for the solar still equipped with a small flat baffle (reference case) at the cooling zone.
- ○
- The triangular and curvilinear baffle design assisted the air-vapor mixture to cool down before the exit.
- ○
- When Ra = 2 × 10
^{4}and θ = 0°, from N = 0.5, the average heat transfer rate is highest in case 2, for triangular baffle. A 12% improvement in the average heat transfer rate compared to the reference case is observed. - ○
- When Ra = 5 × 10
^{4}and N = 0, an improvement of the air-vapor convection is observed in the heating zone for all the new baffles designs studied at θ = 15°. The thermal gradient is enhanced in the cooling zone mainly in the middle of the cold surfaces for cases 2 and 3. When the inclination is less than 18°, all cases studied have higher average Nusselt values than the reference case. And case 2 has higher values of average heat transfer rate. For cases 2 and 3, triangular and curvilinear baffle design, the evolution is parabolic with a maximum of the average transfer rate observed for θ = 5°. In the range of angles between 5° and 20°, case 2 has the highest average mass transfer rate. - ○
- When Ra = 5 × 10
^{4}and N = 1, the heat transfer rate for cases 1, 2, and 3 is higher than the reference case only when θ is lower than 5°. Cases 2 and 3 exhibit higher average Sherwood values than the reference case from θ = 12°. Increasing the angle from 0° to 25° exhibits for case 2 an increase in the average mass transfer rate of 35%.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

C | Dimensionless concentration |

D | Mass diffusivity (m^{2}/s) |

Gr | Grashof number |

K | Thermal conductivity (W/m·K) |

H | Height from the base surface to the top (m) |

L | Baffle width (m) |

Le | Lewis number |

N | Buoyancy ratio |

Nu | Nusselt number |

Pr | Prandtl number |

Ra | Rayleigh number |

Sh | Sherwood number |

T | Dimensionless time (=${t}^{\prime}.\alpha /{H}^{2}$) |

T | Dimensionless temperature |

$\overrightarrow{V}$ | Dimensionless velocity vector (=$\overrightarrow{V}{}^{\prime}.H/\alpha $) |

Greek symbols | |

$\alpha $ | Thermal diffusivity (m^{2}/s) |

${\beta}_{t}$ | Coefficient of thermal expansion (K^{−1}) |

${\beta}_{c}$ | Coefficient of solutal expansion (K^{−1}) |

$\mu $ | Dynamic viscosity (kg/m.s) |

$\nu $ | Kinematics viscosity (m^{2}/s) |

$\overrightarrow{\psi}$ | Dimensionless vector potential ($\overrightarrow{\psi}{}^{\prime}/\alpha $) |

$\overrightarrow{\omega}$ | Dimensionless vorticity (=$\overrightarrow{\omega}{}^{\prime}.\alpha /{H}^{2}$) |

Θ | Angle of inclination of still |

Subscripts | |

x, y, z | Cartesian coordinates |

h | hot, high |

c | Cold |

l | Low |

av | average |

0 | reference |

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**Figure 1.**Physical Models and different designs of baffles studied. (

**a**) shows the model of the floating solar still filled with an air-vapor mixture considered in this work; (

**b**) shows the pyramidal solar still which is designed to float on the surface of the salt water, absorb and convert the incident solar flux into thermal energy, and transfer this heat to the saltwater below for steam production; (

**c**–

**f**) illustrates the four different designs of baffles studied.

**Figure 3.**Comparison between the results of the average Nusselt and Sherwood numbers in the current study with the results of Rahman et al. [24].

**Figure 4.**Particle trajectories, iso-surfaces of temperature and concentration for N = 0 and θ = 0° (case 0) .

**Figure 5.**Particle trajectories (

**a**), iso-contours of concentration (

**b**) and temperature (

**c**) for N = 0 and z = 0.5 (case 0).

**Figure 6.**Particle trajectories (

**a**), Iso-concentration (

**b**) and iso-temperature (

**c**), for z = 0.5 N = 0, θ = 15° and Ra = 5 × 10

^{4}.

**Figure 7.**Effect of Rayleigh number (

**a**) and buoyancy ratio for Ra = 2 × 10

^{4}(

**b**) on the average heat transfer rate in all cases for θ = 0°.

**Figure 8.**Effect of Rayleigh number (

**a**) and buoyancy ratio for Ra = 2 × 10

^{4}(

**b**) on the average mass transfer rate in all cases for θ = 0°.

**Figure 9.**Effect of inclination on the average Nusselt (

**a**) and Sherwood (

**b**) in the all cases for Ra = 5 × 10

^{4}, N = 0 and N = 1.

Mesh Size | Sh_{av} | Percentage Increase | Incremental Increase |
---|---|---|---|

41^{3} | 2.697 | - | - |

51^{3} | 2.733 | 1.33481646 | - |

61^{3} | 2.854 | 5.82128291 | 4.48646644 |

71^{3} | 2.8578 | 5.9621802 | 0.14089729 |

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**MDPI and ACS Style**

Almeshaal, M.A.; Maatki, C.
Numerical Study of Inclination Effect of the Floating Solar Still Fitted with a Baffle in 3D Double Diffusive Natural Convection. *Processes* **2022**, *10*, 1607.
https://doi.org/10.3390/pr10081607

**AMA Style**

Almeshaal MA, Maatki C.
Numerical Study of Inclination Effect of the Floating Solar Still Fitted with a Baffle in 3D Double Diffusive Natural Convection. *Processes*. 2022; 10(8):1607.
https://doi.org/10.3390/pr10081607

**Chicago/Turabian Style**

Almeshaal, Mohammed A., and Chemseddine Maatki.
2022. "Numerical Study of Inclination Effect of the Floating Solar Still Fitted with a Baffle in 3D Double Diffusive Natural Convection" *Processes* 10, no. 8: 1607.
https://doi.org/10.3390/pr10081607