Numerical Analysis of Entropy Generation in a Double Stage Triangular Solar Still Using CNT-Nanofluid under Double-Diffusive Natural Convection

Abstract

The analysis of entropy generation provides valuable information for the design and optimization of thermal systems. Solar stills are used for water desalination and purification. Using renewable energies, they provide a sustainable solution for drinking water supply in remote areas and off-grid situations. This work focuses on the 3D numerical study of entropy generation in a two-stage solar still subjected to the natural double diffusion convection phenomenon in the presence of CNT nanoparticles. The effects of Rayleigh number, buoyancy ratio, and nanofluid concentration on thermal, solutal, and viscous irreversibilities and flow structure were studied. The results show that increasing the buoyancy ratio leads to an increase in thermal and solutal entropy generation. The results of this study also show that total entropy is minimal for positive volume force ratios, N, at a nanoparticle volume fraction of around 3%, and for negative N ratios, at a volume fraction of around 4%.

1. Introduction

Water scarcity appears to be an increasingly alarming prospect for the human race. The number of countries experiencing water stress is growing while the demand for water is on the rise. In light of this situation, a range of technical solutions has been developed, including seawater desalination. This practice has recently experienced significant growth due to continuous improvement of techniques and decreased costs. Desalination systems that utilize solar energy may be a revolutionary solution for regions facing water scarcity or energy shortages. Solar-powered desalination options, such as solar stills, can offer investment and operational cost savings of over 65% compared to traditional systems, as they require no energy costs or fossil fuels.

Numerous research endeavors have been undertaken to study the operational performance of solar stills, as documented in references [1,2,3,4,5,6,7,8,9,10]. Several techniques and designs have been introduced for thermal systems to improve energy efficiency [11]. The optimization of energy efficiency in thermal systems is related to the reduction of entropy generation. Within the context of solar stills, the intricate dynamics of flow, heat transfer, and mass transfer are governed by the phenomenon of double-diffusive convection, where the driving fluid is an air-vapor mixture. Consequently, the irreversibilities arising in solar stills stem from thermal, mass, and frictional influences. Various scientific investigations have explored the generation and mitigation of irreversibilities in cavities employing diverse techniques. In one such study, Dhivagar et al. [12] considered a solar still equipped with crushed gravel sand and a biomass evaporator. The researchers observed that the maximum entropy observed in this system was 61% higher than that of a conventional solar still. Ghachem et al. [13] and Borjini et al. [14] conducted comprehensive studies on three-dimensional double diffusion convection and irreversibilities production within cavities. Specifically, Ghachem et al. [13] focused their investigation on solar stills, while their subsequent study [15] explored entropy production during free convection in a cubic enclosure containing an adiabatic baffle. Maatki et al. [16] delved into examining entropy generation in the presence of discrete hot surfaces. Das et al. [17] analyzed entropy generation via free convection in various triangular cavity configurations, employing a discrete heating strategy. Their findings revealed that the configuration featuring two equally sized heating elements located in the middle and bottom halves exhibited optimal performance. Zafar et al. [18] performed an irreversibility analysis on the triple diffusive flow occurring in a porous cavity while considering the chemical reaction effect. The research demonstrated that an increase in Darcy and Lewis numbers led to a reduction in entropy generation rates. Avellaneda et al. [19] employed variational methods with weighting factors to optimize the heat transfer rate and minimize convective irreversibility production in a gas flow channel. The study showed that the optimized velocity fields led to lower entropy generations, accompanied by decreased temperatures of the heated plate. Tayebi [20] explored entropy production due to the convective flow in a porous square cavity, considering the influence of local thermal equilibrium. Chen and Jian [21] analyzed entropy generated by the convective heat transfer in a cavity involving two immiscible fluids. Surprisingly, their findings revealed that the use of two immiscible fluids generates less entropy compared to that of a single fluid. Mayeli and Sheard [22] studied the entropy generation in a shallow enclosure. The investigation showed that beyond a particular threshold of the Gay-Lussac number exceeding 0.5, the irreversibilities induced by conduction and convection manifest comparable magnitudes. Vijaybabu [23] explored the entropy production during the MDH double-diffusive convection within a porous cylinder. The author observed that the irreversibility increases with the intensity of the magnetic field when its direction aligns with the flow direction. Furthermore, the study revealed that augmenting the buoyancy ratio tends to suppress irreversibilities arising from fluid friction and the magnetic field. Kumar et al. [24] endeavored to model and numerically analyze the frictional entropy generation, as well as heat and mass flux diffusion, under the effect of a magnetic field within a saturated porous cavity containing doubly stratified fluid. Their investigation unveiled that magnetohydrodynamic forces exert a significant reduction in entropy generation resulting from viscous dissipation. Finally, Maatki et al. [25] discovered an intriguing phenomenon where intensifying the magnetic field diminishes the overall entropy generated by double-diffusive convection within a cavity.

In an intricate investigation, Munawar et al. [26] conducted a comprehensive examination of free convection occurring within a heated triangular corrugated ring, wherein a fluid was filled and augmented with a hybrid nanofluid. Astonishingly, their findings unveiled that the proposed design not only mitigates entropy production but also retains its effectiveness even when employing a high thermal conductivity hybrid nanofluid. Moreover, the study demonstrated that by incorporating corrugated walls instead of flat walls, the overall entropy is minimized, presenting a remarkable improvement in system performance. Several esteemed researchers have undertaken numerical investigations to delve into the intriguing realm of entropy generation induced by magneto-convection in the presence of nanofluids, as documented in references [27,28,29,30]. Korei et al. [28] meticulously explored a lidded cavity scenario wherein two rounded corners were filled with a nanofluid. Their study highlighted a captivating finding, i.e., that by reducing the radius of the corners, they were able to enhance heat transfer while simultaneously diminishing irreversibility. The authors further unveiled that heat transfer rate and irreversibility both exhibited a downward trend with an increase in magnetic field strength, alongside a decrease in the influence of mixed convection. Akhter et al. [29] embarked on an investigation of entropy production of a magnetoconvective flow within a porous cavity filled with a hybrid nanofluid in the presence of an external magnetic field. Their investigation disclosed intriguing correlations, revealing that the rate of heat transfers and average entropy generation intensified with an escalation in the Rayleigh number, concurrently accelerated by an augmentation in cavity porosity and nanofluid volume fraction. Sáchica et al. [30] performed meticulous numerical simulations to unravel the MHD transient mixed convection of a nanofluid in a 2D cavity having an internal hot cylinder. The authors astutely elucidated the impact of buoyancy and magnetic parameters on minimizing entropy generation within the nanofluid flow. Notably, their study highlighted the predominant influence of irreversibilities due to heat transfer in the overall entropy generation. In a distinct investigation, Aich et al. [31] meticulously explored entropy generation within a hybrid nanofluid-filled C-shaped enclosure. Chen et al. [32] undertook a comprehensive parametric study, scrutinizing entropy generation within natural double-diffusive nanofluids flow confined within a 2D cavity, specifically focusing on solar thermal systems. Their findings revealed that entropy generation intensified notably when the flow regime transitioned to turbulence. Additionally, they unveiled that the total entropy generation reached its minimum value when the buoyancy force ratio equaled 1 and exhibited a decreasing trend with an increase in nanoparticle volume fraction. Alsarraf et al. [33] conducted an insightful examination, unraveling the entropy production within turbulent forced convection of a two-phase fluid flow, accentuating the presence of various nanofluid shapes within a flat plate solar collector. Their study displayed a remarkable revelation: the case with a nanofluid concentration of ϕ = 4%, taking the shape of a blade, yielded the minimum entropy production, thus demonstrating the potential of this specific configuration. Other interesting papers related to the subject can be found in the literature [34,35,36,37].

Upon meticulous examination of the aforementioned literature review, it became apparent that a dearth of studies focusing on entropy generation within three-dimensional (3D) solar stills exists. Consequently, the present study endeavors to bridge this knowledge gap by employing the finite element method to assess entropy generation within a 3D double-stage triangular solar still. The novelty in this work lies in the study of the efficiency and entropy generation of a solar still featuring a two-stage configuration at the evaporation surface. The seawater feed zone is thus designed with two stages. The upper-level water feed plate overflows centrally, causing the water to flow from both sides towards the lower plates, thereby increasing the evaporation surface area. The surface of both feeder stages is coated with a thin film of CNT nanoparticles. Moreover, the investigation explores the influence of incorporating carbon nanotube (CNT) nanoparticles on various forms of irreversibilities. The set of equations governing the considered configuration is adeptly converted to the dimensionless form and subsequently solved using the COMSOL Multiphysics software version 5.5 (Comsol AB, Stockholm, Suède).

The outcomes of this numerical investigation are meticulously portrayed, encompassing a diverse range of variables. The depiction of the flow structure offers a detailed visualization of the intricate patterns and dynamics within the system. The temperature and concentration fields provide a profound understanding of the distribution and variation of these essential parameters throughout the domain. Furthermore, the local entropy generation fields offer valuable insights into the irreversible processes occurring within the system, shedding light on the sources and magnitude of entropy production. Additionally, the averaged entropy generations serve as crucial metrics, summarizing the overall system performance and providing a comprehensive assessment of its thermodynamic efficiency. Through the meticulous application of numerical techniques and the astute analysis of the results, a comprehensive depiction of the system’s behavior is attained. The intricate interplay between flow, temperature, concentration, and entropy generation is unraveled, offering researchers and experts a profound understanding of the system’s characteristics and enabling informed decisions and further advancements in the field.

2. Studied Configuration

When solar energy is absorbed by the solar still, temperature and concentration gradients create heat and mass fluxes that promote the evaporation of saltwater and the condensation of pure water vapor to obtain fresh water. The region of the solar still bounded by the first formed vapor layer (at the evaporation surface) and the last layer that disappears through condensation at the cold surface is characterized by an air-vapor mixture. At the evaporation surface, the vapor concentration is maximum, while at the condensation surface, the vapor concentration is minimum. This is how the phenomenon of double-diffusive convection of the air-vapor mixture appears in the solar still.

Increasing the heat exchange surface enhances the performance of the solar still. This led to the introduction of a triangular geometric design, which provides a larger condensation surface than the rectangular solar still. Additionally, a new design for the evaporation surface was achieved by adding a second stage in the center, allowing the flow of saltwater on both sides of this stage towards the primary stage, resulting in an increased evaporation surface.

The studied configuration (Figure 1) consists of a 3D triangular solar still equipped with a double stage at the bottom and filled with humid air (air-vapor mixture). A cold temperature (Tc) and low concentration (Cl) are imposed at the left and right inclined walls, where condensation occurs. On the double-stage surface, a thin film of nanofluid (CNT-water) exists (gray domain in Figure 1), and it is maintained at a hot temperature (T_h) and high concentration (C_h) where the evaporation occurs. The remaining walls are considered thermally insulated and impermeable. The properties of CNT and water are presented in

Table 1. Thermophysical Properties of CNT and water [31].

Physical Properties	CNT-Nanoparticles	Water
$\rho$ (kg/m3)	2600	997.1
Cp (kJ/kg·K)	0.425	4.179
k (kW/m·K)	6.600	0.613 × 10−3
$\beta$ (K−1)	1.6 × 10−6	21 × 10−5

.

3. Mathematical Formulation

3.1. Assumptions and Governing Equations

This study delves into the intricate dynamics of the laminar, Newtonian, and incompressible flow of moist air, considering the interplay between thermal and solutal forces. Furthermore, it conjectures that the buoyancy forces operate in a negative direction along the y-axis. The properties of the humid air and the nanofluid remain constant throughout the analysis, except for the density term, which is subject to variation. To incorporate solutal and thermal buoyancy forces, the density term is approximated using the Boussinesq approximation.

The Soret and Dufour effects, while pertinent in other scenarios, are deliberately neglected in this analysis. Within the context of solar still operation, a continuous feed flow is assumed to be present to replenish evaporated water, sustain the volume fraction of the nanofluid, and curtail the potential risks associated with sedimentation.

The region occupied by the CNT water nanofluid is commonly mentioned as the “CNT water nanofluid zone” in this study. This region is characterized by carbon nanotubes (CNTs) suspended in water, which leads to unique thermophysical properties that differ from those of pure water. The CNT water nanofluid is utilized in the operation of the solar still, and its concentration is continuously maintained through a feed flow to prevent sedimentation and ensure the system’s efficient functioning.

The mathematical model employed in this study comprises five equations, namely the equation of continuity, the equation of momentum, the equation of energy governing the region of humid air, the equation of energy governing the region of nanofluid, and the equation of diffusion governing the species.

$$\nabla \times \overrightarrow{V}{}^{\prime }=0$$

(1)

$$\frac{\partial \overrightarrow{V}{}^{\prime }}{\partial t^{\prime }}+\left(\overrightarrow{V}{}^{\prime }\times \overrightarrow{\nabla }\right)\overrightarrow{V}{}^{\prime }=-\frac{1}{\rho }\overrightarrow{\nabla }{P}^{\prime }+\nu \Delta \overrightarrow{V}{}^{\prime }+[1-{\beta }_t\left(T^{\prime }-T_0\right)-{\beta }_c\left(C^{\prime }-C_0\right)]\overrightarrow{g}$$

(2)

$$\frac{\partial T^{\prime }}{\partial t^{\prime }}+\overrightarrow{V}{}^{\prime }\times \nabla T^{\prime }=\alpha {\nabla }^2{T}^{\prime }$$

(3)

$$\frac{\partial T^{\prime }}{\partial t^{\prime }}={\alpha }_{nf}{\nabla }^2{T}^{\prime }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{I}\mathrm{n} \mathrm{C}\mathrm{N}\mathrm{T}-\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{z}\mathrm{o}\mathrm{n}\mathrm{e}$$

(4)

$$\frac{\partial C^{\prime }}{\partial t^{\prime }}+\overrightarrow{V}{}^{\prime }\times \nabla C^{\prime }=D{\nabla }^2{C}^{\prime }$$

(5)

The equations are subject to further transformation to get the dimensionless form based on specific parameters. These parameters are crucial in the analysis of the system’s behavior and its physical properties. The following transformations are used to get the dimensionless governing equations, where the dimensionless numbers (Pr, Le, Ra, and N) appear.

$$t=\frac{t^{\prime }}{\frac{{l^{\prime }}^2}{\alpha }};\left(x,y,z\right)=\frac{{\left(x,y,z\right)}^{\prime }}{l^{\prime }};\left(V_x,V_y,V_z\right)=\frac{{\left(V_x,V_y,V_z\right)}^{\prime }}{\alpha /l^{\prime }};P=\frac{P^{\prime }}{\rho \left(\frac{\alpha }{l^{\prime }}\right)};C=\frac{C^{\prime }-C_l}{C_h-C_l} \mathrm{and} T=\frac{T^{\prime }-T_c}{T_h-T_c}$$

$$\nabla \times \overrightarrow{V}=0$$

(6)

$$\frac{\partial \overrightarrow{V}}{\partial t}+\left(\overrightarrow{V}\times \overrightarrow{\nabla }\right)\overrightarrow{V}=-\overrightarrow{\nabla }P+Pr\Delta \overrightarrow{V}+\left[\begin{array}{c}0\\ RaPr\left(T+NC\right)\\ 0\end{array}\right]$$

(7)

$$\frac{\partial T}{\partial t}+\overrightarrow{V}\times \nabla T={\nabla }^{2}T$$

(8)

$$\frac{\partial T}{\partial t}={\frac{{\alpha }_{nf}}{\alpha }\nabla }^{2}T\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{I}\mathrm{n} \mathrm{C}\mathrm{N}\mathrm{T}-\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{z}\mathrm{o}\mathrm{n}\mathrm{e}$$

(9)

$$\frac{\partial C}{\partial t}+\overrightarrow{V}\times \nabla C=\frac{1}{Le}{\nabla }^{2}C$$

(10)

The mathematical expressions presented in Equations (6)–(10) have been transformed into dimensionless forms by utilizing certain dimensionless parameters. These equations are characterized by the appearance of four dimensionless numbers:

$$Pr=\frac{v}{\alpha },Ra=\frac{g{\beta }_t(T_h-T_c){l^{\prime }}^3}{v\alpha },N=\frac{{\beta }_c{(C}_h-C_l)}{{\beta }_t(T_h-T_c)}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{a}\mathrm{n}\mathrm{d} Le=\frac{\alpha }{D}$$

(11)

In this study, the effective properties of the nanofluid, namely density, heat capacity, and thermal conductivity, are evaluated using Equations (12)–(14).

The effective density is given by Equation (12):

$${\rho }_{nf}=\left(1-\mathsf{\varphi }\right){\rho }_f+\mathsf{\varphi }{\rho }_s$$

(12)

The heat capacity is expressed by Equation (13):

$${\left(\rho C_p\right)}_{nf}=\left(1-\mathsf{\varphi }\right){\left(\rho C_p\right)}_f+\mathsf{\varphi }{\left(\rho C_p\right)}_s$$

(13)

The effective thermal conductivity is determined by Equation (14), which is based on the Xue model.

$$\frac{k_{nf}}{k_f}=\frac{1-\mathsf{\varphi }+2\mathsf{\varphi }\frac{k_s}{k_s-k_f}ln\left(\frac{k_s+k_f}{2k_f}\right)}{1-\mathsf{\varphi }+2\mathsf{\varphi }\frac{k_f}{k_s-k_f}ln\left(\frac{k_s+k_f}{2k_f}\right)}.$$

(14)

3.2. Entropy Generation Equations

The thermal and solutal gradients near the active cavity walls also cause entropy generation in the system. The local entropy generation in a three-dimensional flow is given by:

$$\begin{array}{c}{S^{\prime }}_{gen}=\left\{\frac{k_f}{{T_0}^2}\left[{\left(\frac{\partial T^{\prime }}{\partial x^{\prime }}\right)}^2+{\left(\frac{\partial T^{\prime }}{\partial y^{\prime }}\right)}^2+{\left(\frac{\partial T^{\prime }}{\partial z^{\prime }}\right)}^2\right]\right\}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{{\mu }_f}{T_0}\left\{2\left[{\left(\frac{{\partial V^{\prime }}_x}{\partial x^{\prime }}\right)}^2+{\left(\frac{{\partial V^{\prime }}_y}{\partial y^{\prime }}\right)}^2+{\left(\frac{{\partial V^{\prime }}_z}{\partial z^{\prime }}\right)}^2\right]+{\left(\frac{{\partial V^{\prime }}_y}{\partial x^{\prime }}+\frac{{\partial V^{\prime }}_x}{\partial y^{\prime }}\right)}^2+{\left(\frac{{\partial V^{\prime }}_z}{\partial y^{\prime }}+\frac{{\partial V^{\prime }}_y}{\partial z^{\prime }}\right)}^2+{\left(\frac{{\partial V^{\prime }}_x}{\partial z^{\prime }}+\frac{{\partial V^{\prime }}_z}{\partial x^{\prime }}\right)}^2\right\}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left\{\frac{RD}{C_0}\left[{\left(\frac{\partial C^{\prime }}{\partial x^{\prime }}\right)}^2+{\left(\frac{\partial C^{\prime }}{\partial y^{\prime }}\right)}^2+{\left(\frac{\partial C^{\prime }}{\partial z^{\prime }}\right)}^2\right]+\frac{RD}{T_0}\left[\left(\frac{\partial T^{\prime }}{\partial x^{\prime }}\right)\left(\frac{\partial C^{\prime }}{\partial x^{\prime }}\right)+\left(\frac{\partial T^{\prime }}{\partial y^{\prime }}\right)\left(\frac{\partial C^{\prime }}{\partial y^{\prime }}\right)+\left(\frac{\partial T^{\prime }}{\partial z^{\prime }}\right)\left(\frac{\partial C^{\prime }}{\partial z^{\prime }}\right)\right]\right\}\hfill \end{array}$$

where C₀ and T₀ are the reference concentration and temperature, respectively. R is the universal gas constant.

The dimensionless local entropy generation can be expressed as:

$$\begin{array}{l}{N}_s=\left[{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2+{\left(\frac{\partial T}{\partial z}\right)}^2\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +I_{s1}\left\{2\left[{\left(\frac{\partial V_x}{\partial x}\right)}^2+{\left(\frac{\partial V_y}{\partial y}\right)}^2+{\left(\frac{\partial V_z}{\partial z}\right)}^2\right]+\left[{\left(\frac{\partial V_y}{\partial x}+\frac{\partial V_x}{\partial y}\right)}^2+{\left(\frac{\partial V_z}{\partial y}+\frac{\partial V_y}{\partial z}\right)}^2+{\left(\frac{\partial V_x}{\partial z}+\frac{\partial V_z}{\partial x}\right)}^2\right]\right\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +I_{s2}\left[{\left(\frac{\partial C}{\partial x}\right)}^2+{\left(\frac{\partial C}{\partial y}\right)}^2+{\left(\frac{\partial C}{\partial z}\right)}^2\right]+I_{s3}\left[\left(\frac{\partial T}{\partial x}\right)\left(\frac{\partial C}{\partial x}\right)+\left(\frac{\partial T}{\partial y}\right)\left(\frac{\partial C}{\partial y}\right)+\left(\frac{\partial T}{\partial z}\right)\left(\frac{\partial C}{\partial z}\right)\right]\end{array}$$

The dimensionless ratios $I_{s1}$, $I_{s2},$ and $I_{s3}$ are associated with irreversibility distribution ratios for velocity gradients, concentration gradients, and the product of concentration and temperature gradients, respectively.

The irreversibility distribution ratios are given respectively by:

$$I_{s1}=\frac{{\mu }_f{\alpha }^2}{k{l}^2\Delta T^{\prime }}{T}_0, I_{s2}=\frac{RD{T}_0}{k{C}_0}{\left(\frac{\Delta C^{\prime }}{\Delta T^{\prime }}\right)}^2, I_{s3}=\frac{RD}{k}\left(\frac{\Delta C^{\prime }}{\Delta T^{\prime }}\right)$$

3.3. Boundary Conditions

The following are the definitions of the dimensionless boundary conditions:

✓

Temperature: at $y=0$ $T=1$, at inclined wall $T=0$; $\frac{\partial T}{\partial n}=0$ on remaining walls.

✓

Concentration: at $y=0.1 C=1$, at inclined wall $C=0$; $\frac{\partial C}{\partial n}=0$ on remaining walls.

✓

Velocity: $V_x=V_y=V_z=0$ on all walls.

At the humid air-nanofluid interface, the continuity condition for both temperature and heat flux is expressed as follows:

$${\left(\frac{\partial T}{\partial n}\right)}_f=R_c{\left(\frac{\partial T}{\partial n}\right)}_i$$

(15)

where $R_c=k_{nf}/k$.

The local Nusselt and Sherwood numbers are:

$$Nu={\frac{\partial T}{\partial y}|}_{y=0}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Sh={\frac{\partial C}{\partial y}|}_{y=0.1}$$

(16)

The averaged Nusselt and Sherwood numbers are as follows:

$$N{u}_{av}={\int }_0^1{\int }_0^{1}Nu.\partial y\partial z\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}S{h}_{av}={\int }_0^1{\int }_0^{1}Sh.\partial y\partial z$$

(17)

4. Numerical Method and Validation

A computational model was developed using COMSOL Multiphysics 5.5 (Tegnérgatan 23, 111 40 Stockholm, Suède) to investigate the flow structure and entropy generation in a double-stage solar still. COMSOL Multiphysics uses the finite element method. In keeping with the software’s Multiphysics concept, four coupled “application modules” are used to model the phenomenon of three-dimensional diffusive double convection: fluid flow, heat transfer, mass transfer, and chemical engineering. Each of these models corresponds to a physical phenomenon, such as fluid flow, heat or energy transfer, and vapor transport in the air-humid mixture, and also has a differential equation (Equations (1)–(3) and (5)). We specify the model region and boundary conditions, as described in Section 3.3. We have defined the lower region corresponding to the two evaporation stages as being formed by water, and a thin film of nanofluid is applied; heat transfer is assumed to be by conduction in this zone. The upper region is formed by the air-vapor mixture and is where double-diffusive convection takes place. The simulation employed a second-order upwind discretization technique and implemented a species transport model for the air-vapor mixture. We have opted to use the default quadratic Lagrange shape functions on a mesh composed of triangular finite elements for all variables as our discretization method.

The simulation converged on the chosen mesh for most parameter variations, presented below, using the solver’s default options. We also opted for transient simulations to check the stability of the steady-state results. When the steady state is established, the entropy generation related to each irreversibility cause: friction, thermal, and solute is calculated. The total entropy production of all causes of irreversibility is then also calculated.

The convergence of the solution was determined by the scaled residuals for the energy equation, which were required to be less than 10⁻⁶, and 10⁻⁵ for the other equations. This rigorous approach ensures the accuracy and reliability of the simulation results.

Figure 2 compares the 3D flow structure with the results of Kolsi et al. [34] for Pr = 0.7, Le = 0.85, N = −1, and Ra = 10⁵. The authors studied the effect of conductive fins on the double-diffusive convection in a trapezoidal solar still, where the bottom wall is maintained at a hot temperature and the top inclined wall is kept cold. A very good concordance between the results ensures the validity of the present numerical code.

5. Results and Discussions

The results presented in this study pertain to examining intricate interactions and patterns within an air-vapor mixture that arise when simultaneous gradients of temperature and concentration exist in a double-stage water-feed solar still. The surface of both feeder stages is coated with a thin film of CNT nanoparticles. Iso-concentration and iso-temperature profiles are employed to delineate the dispersion characteristics of the vapor within the moist air mixture. Furthermore, examining local entropy production fields offers invaluable insights into the irreversible processes transpiring within the system, thereby shedding light on the origins and magnitude of entropy generation. The outcomes were derived within specified ranges of guiding parameters, encompassing Rayleigh numbers spanning from 10⁴ to 10⁶, buoyancy values ranging from −2 to 2, and nanofluid concentrations varying between 0 and 0.45. The Prandtl number was rigorously set to Pr = 0.7, while the Lewis number was obtained as Le = 1.2.

Figure 3 shows the particle trajectories (left side) and the irreversibilities generated by viscous effects for Ra = 10⁵, ϕ = 0%, and different values of buoyancy ratio, N = 0, N = 1, and N = −1. The particle trajectory defining the flow structure in the solar still is characterized by four cells for the three values of buoyancy forces. When N = 0, the cells are governed by the thermal volume forces, with the cells on the right rotating in the clockwise direction and those on the left rotating counterclockwise. When N = 1, the thermal and solutal volume forces cooperate to increase the intensity of the particle velocity in the solar still. The maximum dimensionless velocity increases from 7.28 for N = 0 to 22.6 for N = 1. For N = −1, there is an equitable competition between the thermal and solute volume forces, resulting in a reversal of the direction of rotation of the cells in the still compared to the N = 0 and N = 1 cases. The particle velocity in the solar still is therefore reduced and weakened. The viscous entropy generation is related to the velocity component’s gradients. For N = 1, the maximum values of viscous entropy generation occur at the corners of the solar still due to the important friction, especially between the fluid and the wall at these regions. For N = 1, a similar field of the viscous entropy generation is encountered but with higher values. In this case, the increase of the viscous entropy is justified by the cooperative behavior of the thermal and solutal forces that lead to higher velocities and, thus, more friction. For N = −1, due to the competitive interplay between the solutal and thermal forces, the flow intensity is reduced, and lower viscous entropy is generated.

Figure 4 illustrates the iso-surfaces of temperature (left side) and iso-surfaces of the thermal entropy generation (right side) for Ra = 10⁵, ϕ = 0%, and various buoyancy ratio values. For all the considered values of N, the iso-surfaces of temperature are parallel to the active wall with distortions at the interface separating the humid air and the nanofluid film. These distortions are due to the difference in the thermal conductivity between the humid air and the water film. For N = −1, the temperature field is characterized by a vertical stratification at the core region of the cavity, indicating the dominance of the conductive heat transfer mode. This is due to the opposition between the thermal and solutal forces that reduce the flow intensity, as discussed in Figure 3. For higher values of N, some distortions of the iso-surface of temperature are encountered at the center of the cavity due to the increase of the fluid velocities. It is also to be mentioned that at the left and right corners of the solar still, the iso-surfaces of temperature are piled up due to the existence of the collector that connects the hot and cold actives walls. The iso-surfaces of entropy generation are mainly in the central and lower regions of the cavity. This is due to the important temperature gradients in these regions. In fact, the top region is mainly delimited by two cold walls, and thus, the temperature gradients are very low. It is also to be mentioned that the maximum values of the thermal entropy generation are at the left and right corners. In fact, these regions exist close to the connections between the hot and cold walls, where the density variations are very important and cause higher temperature gradients.

Figure 5 depicts the effects of the buoyancy ratio on the iso-surfaces of concentration and solutal entropy generation for Ra = 10⁵ and ϕ = 0%. Due to the imposed constant concentration boundary condition, the iso-surfaces of concentration are parallel to the active wall. In the central region of the cavity, the concentration field is characterized by vertical stratification. The concentration gradients are more important close to the cold and hot wall walls, especially close to the corner of the cavity. This fact leads to the generation of higher solutal entropy generation values at these regions, as presented in the right column of the figure. It is also to be mentioned that the concentration gradients are more important for N = 1 and N = 0 compared to N = −1, and thus, lower solutal entropy generation occurs for N = −1. This is due to the equilibrium between the solutal and thermal buoyancy forces at this specific value of the buoyancy ratio.

Figure 6 illustrates the iso-surfaces of total entropy generation for Ra = 10⁵, ϕ = 0%, and different values of buoyancy ratio. As described in Figure 4 and Figure 5, the solutal and thermal entropy generations are concentrated close to the active walls, and their lowest values occur for N = −1. Obviously, the total entropy generation follows the same behavior.

Figure 7 presents the effect of the nanoparticles’ volume fraction on the flow structures (top rows) and frictional entropy generation (bottom row) for Ra = 10⁶ and different values of the buoyancy ratio. For N = 1, the flow structure is characterized by four vortexes, and compared to Ra = 10⁵ (Figure 3), it is noticed that due to the intensification of the flow, the particle trajectories are no more symmetrical. The increase of the nanoparticles’ volume fraction does not cause a considerable change in the flow structure and leads only to an increase in the velocity magnitude. This increase in the velocity causes an increase in the viscous effect entropy generation, as presented in the second row of the figure. It also noticed that compared to Ra = 10⁵ (Figure 3), the viscous entropy generation becomes distributed on the whole domain, but still, the maximum values occur at the corners. For N = −1, the flow structure is again symmetrical due to the reduction of the flow intensity caused by the concurrence between the solutal and thermal forces. A considerable reduction in the velocity magnitude is encountered compared to N = 1 cases, and by increasing the nanoparticles’ volume fraction, it becomes lower. This reduction in the velocity causes an obvious reduction in the frictionally generated entropy generation.

Figure 8 is plotted to present the effect of the nanoparticles’ volume fraction on the temperature field and thermal entropy generation for Ra = 10⁶ and different values of the buoyancy ratio. Similarly, Figure 9 depicts the same effect on the concentration field and solutal entropy generation. Due to the fixed Lewis number (Le = 0.85) corresponding to the air vapor mixture, which is close to 1, the temperature and concentration fields are similar and the same for the thermal and solutal entropy generations. For N = −1, the temperature (concentration) field is characterized by a central vertical stratification. For N = 1, the iso-surfaces of temperature (concentration) at the central region become distorted due to the intensification of the convective flow, as can be noticed for the velocity magnitude in Figure 7. This intensification is due to cooperative solutal and thermal buoyancy forces for positive N values. The addition of nanoparticles enhances the thermos-physical properties of the fluid and leads to more intense flow, and thus, the iso-surface distortions become more pronounced. The thermal and solutal entropy generation is mainly concentrated close to the active walls for N = −1. For N = 1, due to the flow intensification, the hot and highly concentrated fluid is carried away toward the central region. Thus, the thermal and solutal entropy generations become distributed in the whole domain. The increase in the nanoparticle volume fraction leads to an increase in the thermal and solutal entropy generations.

Figure 10 presents the iso-surfaces of total entropy generation for Ra = 10⁶. Similar to the viscous and thermal entropy generations, the iso-surfaces of total entropy generation are mainly concentrated close to the active walls. It is also to be mentioned that for N = −1, the values of the produced entropy generation are the lowest due to the equilibrium between the thermal and solutal forces. In addition, the increase in the CNT volume fraction leads to an increase in the total entropy generation.

Figure 11 presents the variations of the total entropy generation versus the nanoparticles volume fraction for various buoyancy ratios and Rayleigh number values. It is obvious that the increase of Ra increases the produced irreversibilities. For Ra = 10⁴ (Figure 11a), the heat transfer regime is mainly conductive, and the buoyancy ratio is not very effective; thus, the curves are grouped into two groups; positive N values and negative N values (including N = 0). For Ra = 10⁵ (Figure 11b), the convective effects become more important, and the curves of total entropy generation become no longer superposed. It is also to be mentioned that for negative N values, the maximum entropy generation occurs for ϕ = 0.03, and the minimum occurs for ϕ = 0.04. For positive N values, the maximum entropy generation occurs for ϕ = 0.02, and the minimum occurs for ϕ = 0.03. This result is very interesting when the aim is to enhance the heat and mass transfer while producing less irreversibility leading to higher performances and lower energy losses.

6. Conclusions

The present study deals with the investigation of the effects of using CNT-nanofluid on the entropy generation in a 3D double-stage solar still. A numerical model based on the FEM is established using COMSOL Multiphysics 5.5 (Comsol AB, Stockholm, Suède) to solve the dimensionless governing equations. The effects of the governing parameters such as Rayleigh number, buoyancy ratio, and CNT volume fraction on the flow structure, temperature and concentration field, and entropy generations are investigated. The main findings can be summarized as follows:

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The increase of the buoyancy ratio, Rayleigh number, and CNT volume fraction leads to the intensification of the flow;

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Due to the equal competitively between the solutal and thermal forces, the lower fluid velocity magnitudes and viscous entropy generation occur for N = −1;

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The maximum values of viscous entropy generation occur close to the corners of the solar still;

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The increase of the buoyancy ratio leads to the increase of the thermal and solutal entropy generations;

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The volume fraction of CNT added in a film bonded to the two water supply stages of a Solar still improves heat transfer performance at the evaporation surface, which can be used to optimize the entropy generated;

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Depending on the values of Ra and N, an optimum value occurs on all the variations of the entropy generations versus the volume fraction. For our two-stage water supply design, the total entropy is minimal for positive volume force ratios, N, at a nanoparticle volume fraction of around 3%, and for negative N ratios, at a volume fraction of around 4%.