Numerical Investigations on Heat and Mass Transport in Passive Solar Evaporators with Non-Uniform Surface Temperature

Abstract

Passive solar desalination with no discharge promises great potential for sustainable desalination. Herein, we provide a comprehensive modelling scheme for the investigation of coupled heat and mass transport in passive desalination devices. Our modelling approach integrates mass, momentum, species, and energy transport models to study the coupled phenomena of wicking, solar-driven evaporation, and salt precipitation. Our numerical model can predict the impact of spatiotemporal variation in temperature, salt concentration, and wicking velocity on the evaporation flux and thermal efficiency of solar evaporators. The impact of the evaporator’s shape, solar flux, salt concentration, and light reflection by salt crystals has been studied on the evaporator’s performance. We observed a two-fold increase in evaporation flux when solar irradiance increases from 1000 W/m² to 2500 W/m². A reduction in the thermal efficiency of the evaporators is predicted at higher solar fluxes. The modelled evaporator can achieve an evaporation flux of over 0.5 kg/m²h under 1000 W/m² for 3.5 wt.% saline water. The salt concentration along the z-position of the evaporator exhibited a double arch-shaped profile, which influences its evaporation performance. These findings provide vital guidelines for the design of high-throughput solar desalination systems.

1. Introduction

Water scarcity is one of the major challenges of the current century [1], with rapid economic development and population growth [2]. Conventional desalination methods based on fossil fuels have been employed to produce fresh water, which has serious environmental concerns [3]. For instance, brine discharge into the marine environment has endangered aquatic life due to increased seawater salinity [4]. Moreover, the emission of greenhouse gases from fossil-fuel-based desalination plants has aggravated the situation. Passive solar desalination is an environment-friendly approach to clean water production with no carbon footprint [5]. Like trees and plants [6], it relies on porous wicks to pull water from ground level to the evaporating interface without external forces [7]. Porous wicks are hydrophilic structures that can transport water against gravity via capillary forces [8]. On a macroscopic scale, capillary forces depend on surface wettability and the radius of the capillary channel. Porous structures in practice do not have uniform channels, and a pore radius equivalent to that of a capillary tube is used [9,10]. The ability of a porous structure to allow passage of a fluid is characterized by its permeability. The higher the permeability, the easier the fluid passage [11,12].

Attention has been given to various structural designs and material innovations for solar desalination, including aerogel [13], bio-based [14], anti-bacterial hydrogel [15], wood-based [16], leaf-like [17], 3D printed [18], and seashell-inspired [19] evaporators. Each desalination device includes a solar absorber to harvest solar energy and an evaporator for distillation. In some cases, the evaporator has a dual role: acting as both a light absorption and evaporation surface. Usually, the solar absorber is coated with black material to efficiently harvest solar energy [20]. Advancements in material research have enabled solar evaporators to achieve a photothermal efficiency of over 90% [19]. For instance, Su et al. [21] achieved a record absorption rate of over 97% for a quasi-optical microcavity absorber for the wavelength range of 300–2500 nm. In some cases, the remarkable absorptivity of 99% has been achieved for porous structures incorporating plasmonic nanoparticles [22].

Photothermal heating in solar evaporators takes place at the air-water interface of the absorber, known as interfacial heating. The thermal efficiency of the evaporators with interfacial heating is higher than that of conventional solar stills due to the localized heating. Interfacial heating also decreases start-up time for vapor generation [23]. Various thermal management strategies have been implemented to minimize heat losses from the evaporator and increase water production [24]. Passive solar desalination systems may or may not involve salt precipitation. In the former case, the salt is precipitated on the evaporating surface of the desalination device, which can be actively or passively harvested [25]. The salt is diffused back into the bulk feed water, and precipitation is avoided in the latter types of systems. Solar desalination with salt precipitation and zero liquid discharge has shown promising potential for resource recovery from seawater [26]. A comprehensive discussion on physical principles and various design strategies in capillary-fed solar evaporators can be found in [27,28].

The evaporation of water from the air-water interface of the porous structures depends on the mass transfer coefficient of the porous interface, the salinity of the salt water, solar irradiance, and environmental conditions (i.e., ambient humidity and temperature). Research in the area of heat and mass transport in solar evaporators has been somewhat limited, which considers the impact of the above-mentioned parameters to model and evaluate the performance of solar desalination systems. Fries et al. [29] provided an analytical model for capillary transport in metallic wicks. They investigated the impact of porous media properties on wicking characteristics with and without evaporation; however, their work was limited to pure water. Lazhar et al. [30] studied the wicking and salt precipitation process to understand soil salinization and salt weathering of buildings. They employed glazed ceramic as a porous material to study the evaporation and salt crystallization under natural conditions. Mae et al. [31] developed an analytical model to predict the salt concentration profile at the porous wick and saturation time for salt crystallization. Their study was mainly focused on salt transport and the prediction of precipitation locus without considering the impact of non-uniform temperature and evaporation rate.

Herein, we propose a modelling approach for coupled liquid transport, evaporation, and precipitation of salt in porous solar evaporators. The effect of various controlling parameters, such as solar flux and brine salinity, wicking velocity, and salt concentration, is investigated on the evaporation performance of the evaporator, particularly under varying surface temperatures. Insights into the underlying physics of various phenomena are presented to better understand the evaporation and precipitation process. Spatiotemporal variation in salt concentration, evaporation flux, brine velocity, and evaporation rate can easily be predicted with our model. The proposed numerical model provides comprehensive guidelines for the energy-efficient design of passive solar desalination systems.

2. Mathematical Formulation

2.1. Physical Model and Simulation Domain

Various heat and mass transport processes involved in the operation of solar evaporators are indicated in the schematic diagram (Figure 1). The porous evaporator is assumed to be composed of hydrophilic spherical particles of titanium with a contact angle < 90° [32]. The particle size of the evaporator is taken to be uniform, resulting in a homogeneous porous medium. The thickness, width, and height of the evaporator are represented with b, w, and H, respectively, as illustrated in Figure 1. The various properties of the evaporator are listed in

Table 1. Values of parameters used for the simulations.

Parameter	Symbol/ Equation	Value(s)
Width of the evaporator	w	10 mm
Thickness of the evaporator	b	0.5, 5 mm
Height of the evaporator	H	50 mm
Solar area of the evaporator	Asol = wH	500 mm2
Cross-sectional area of the evaporator	Ac = bw	5, 50 mm2
Total surface area of the evaporator	Asurf = (2wH) + (2bH) + (bw)	1055, 1550 mm2
Evaporation area of the evaporator	Aevap = Asurf	1055, 1550 mm2
Permeability of the evaporator	${\tilde{k}}_{\mathrm{p}}$	30 µm2
Volumetric porosity of the evaporator	ϕv	0.4
Surface porosity of the evaporator	ϕs	1
Diameter of the spherical particles	$d_{\mathrm{p}}$	175 µm
Empirical parameter	δ	0.004 m
Standard diffusion coefficient of NaCl	D	1.5 × ${10}^{-9}$ m2/s
Absorption coefficient of the evaporator	$\alpha$	0.97
Emissivity of the evaporator	ε	0.97
Stefan–Boltzmann constant	σ	5.67 × ${10}^{-8}$ W/m2 K4
Thermal conductivity of porous matrix	$k_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$	11.4 W/m K
Specific heat capacity of porous matrix	$c_{\mathrm{p},\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$	523 J/kg K
Density of porous matrix	${\rho }_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$	4500 kg/m3
Gravitational acceleration	g	−9.81 m/s2
Molar mass of NaCl	$M_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}$	58.44 g/mol
Relative humidity of air	φ	0.5
Ambient temperature	Tamb	298.15 K
Inlet temperature of the brine	Tin	298.15 K

. Note that the total surface area of the evaporator is represented by A_surf, the solar area by A_sol, the cross-sectional area by A_c, and the total evaporation area by A_eva. However, the total evaporation area is the same as the total surface area. These different types of areas as a function of thickness, width, and height of the evaporator are listed in Table 1.

The saline water is pumped up by the porous evaporator through capillary forces from surface 1. The evaporation process takes place at surfaces 2 to 6, which causes salt precipitation on the evaporator’s free surface (known as salt efflorescence). The evaporation is driven by solar energy (which is received on surface 5), as illustrated in Figure 1. The evaporation and heat loss from the evaporator in the form of conduction, convection, and radiation happen on surfaces 2 to 6. The physical model shown in Figure 1 also serves as a computational domain for the numerical analysis.

2.2. Evaporative Crystallisation

The evaporation of water from the evaporator surface results in an increase in salt concentration. Supersaturation takes place when the concentration of salt in the solution is higher than its solubility limit. The saturated molality (mol/kg) as a function of temperature $T_{\mathrm{s}}$ (K) is defined by the following correlation [33]:

$$m_{\mathrm{s}\mathrm{a}\mathrm{t}}=6.044+\left({\displaystyle \frac{2.8}{{10}^3}}\left(T_{\mathrm{s}}-273.15\right)\right)+\left({\displaystyle \frac{3.6}{{10}^5}}{\left(T_{\mathrm{s}}-273.15\right)}^2\right)$$

(1)

The saturated weight concentration ($c_{\mathrm{s}\mathrm{a}\mathrm{t}}$) and saturated molality ($m_{\mathrm{s}\mathrm{a}\mathrm{t}}$) are related as:

$$c_{\mathrm{s}\mathrm{a}\mathrm{t}}={\displaystyle \frac{m_{\mathrm{s}\mathrm{a}\mathrm{t}}{M}_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}}{1+m_{\mathrm{s}\mathrm{a}\mathrm{t}}{M}_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}}}}$$

(2)

For a temperature of 298.15 K, the saturated concentration is found to be ~26.3 wt.%. Once the solution becomes supersaturated, the chemical potential of salt in the H₂O/NaCl solution (${\tilde{\mu }}_{\mathrm{l}})$ becomes higher than its equilibrium value (${\tilde{\mu }}_{\mathrm{l},\mathrm{e}\mathrm{q}})$. The chemical potential difference (${\mathsf{\Delta }\tilde{\mu }={\tilde{\mu }}_{\mathrm{l}}-\tilde{\mu }}_{\mathrm{l},\mathrm{e}\mathrm{q}}$) can be related to salt concentration (c) by Equation (3) for evaporative crystallization at low supersaturation [34]:

$$\mathsf{\Delta }\tilde{\mu }=R{T}_{\mathrm{s}}{\displaystyle \frac{\mathsf{\Delta }c}{c_{\mathrm{e}\mathrm{q}}}}$$

(3)

where $\mathsf{\Delta }c=c-c_{\mathrm{e}\mathrm{q}}$, and $c_{\mathrm{e}\mathrm{q}}$ refers to the equilibrium concentration at the solubility limit. $T_{\mathrm{s}}$ is the surface temperature of the evaporator (K), and R is the ideal gas constant (J/K⋅mol). According to classical nucleation theory, the energy barrier for crystallization is reduced with the increase in chemical potential difference, which depends on salt concentration as described by Equation (3). Therefore, the precipitation is likely to happen when the solution is supersaturated (i.e., $c>c_{\mathrm{e}\mathrm{q}}$). Hence, it is reasonable to assume that the locations at which salt concentration reaches supersaturation are most likely to have salt crystallization/precipitation. From now on, we will refer to the locations with $c\ge c_{\mathrm{e}\mathrm{q}}$ as precipitation zones and $c<c_{\mathrm{e}\mathrm{q}}$ as precipitation-free zones.

2.3. Governing Equations

We employ coupled mass, species (i.e., salt), and energy transport equations in COMSOL Multiphysics (https://www.comsol.com/) to model solar desalination. The brine transport through the porous evaporator is governed by Darcy’s law as given below:

$$\mathit{u}=-{\displaystyle \frac{{\tilde{k}}_{\mathrm{p}}}{{\mu }_{\mathrm{l}}}}\left(\nabla \mathrm{p}-{\mathsf{\rho }}_{\mathrm{l}}\mathbf{g}\right)$$

(4)

In the above formulation, u is the Darcy’s velocity (m/s), k_p is the permeability of the porous evaporator (m²), ${\mu }_{\mathrm{l}}$ is liquid viscosity (Pa·s), p is pressure (Pa),${\mathsf{\rho }}_{\mathrm{l}}$ is liquid density (kg/m³), and $\mathbf{g}=(0,0,g)$ is the gravitational acceleration vector (with $g=-9.81$ m/s²), respectively. The permeability of the porous medium composed of spherical particles depends on porosity and particle size as described by Equation (5) [35]:

$${\tilde{k}}_{\mathrm{p}}={\displaystyle \frac{{{\varphi }_{\mathrm{v}}}^3}{180{(1-{\varphi }_{\mathrm{v}})}^2}}{d}_{\mathrm{p}}^2$$

(5)

The symbol $d_{\mathrm{p}}$ represents the diameter of the spherical particles, and ${\varphi }_{\mathrm{v}}$ is the volumetric porosity of the evaporator, which may vary from 0.37 to 0.43 for packed spheres [36]. For $d_{\mathrm{p}}=175$ µm and ${\varphi }_{\mathrm{v}}$ = 0.4, the permeability turns out to be 30 µm². Darcy’s law is coupled with the continuity equation for the conservation of mass as given below:

$${\displaystyle \frac{\partial }{\partial t}}\left({\mathsf{\rho }}_{\mathrm{l}}{\varphi }_{\mathrm{v}}\right)+\nabla ·\left({\mathsf{\rho }}_{\mathrm{l}}\mathbf{u}\right)=0$$

(6)

The species transport equation for the transport of salt during solar-driven evaporation is given as:

$${\varphi }_{\mathrm{v}}{\displaystyle \frac{\partial }{\partial t}}\left(c\right)+\nabla ·\left(-D_{\mathrm{s}}\nabla c\right)+\mathbf{u}·\nabla c=S_{\mathrm{c}}$$

(7)

The first term represents the temporal variation in salt concentration (c). The second and third terms represent the diffusive and advective transport of salt during the evaporation process. D_s is the effective diffusion coefficient of salt in the porous medium (m²/s), which is calculated through the Millington and Quirk model as: D_s = D${\varphi }_{\mathrm{v}}$^4/3 [37,38]. D is the standard diffusion coefficient of the salt (i.e., 1.5 × ${10}^{-9}$ m²/s [39]). The volumetric source term for salt generation (S_c) in terms of salt concentration (c), evaporator’s volume ($V_{\mathrm{e}\mathrm{v}\mathrm{a}}$), surface area ($A_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$), liquid density (${\mathsf{\rho }}_{\mathrm{l}}$), and evaporation flux ($J_{\mathrm{v}}$) is written as [40]:

$$S_c={\displaystyle \frac{c}{{{\mathsf{\rho }}_{\mathrm{l}}V}_{\mathrm{e}\mathrm{v}\mathrm{a}}}}{A}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}{J}_{\mathrm{v}}$$

(8)

The above correlation can be modified to obtain salt generation at the evaporator surface ($J_c$) as:

$$J_c={\displaystyle \frac{c}{{\mathsf{\rho }}_{\mathrm{l}}}}{J}_{\mathrm{v}}$$

(9)

It is pertinent to mention that generated salt does not leave the evaporation surface, unlike water molecules. Also, note that Equations (8) and (9) assume that the surface of the evaporator is fully wet with a surface porosity (ϕ_s) of 1. For heat transfer in porous media, the time-dependent energy equation is given by Equation (10):

$${\left(\rho c_{\mathrm{p}}\right)}_{\mathrm{e}\mathrm{f}\mathrm{f}}{\displaystyle \frac{\partial T}{\partial t}}+\nabla ·\left(-k_{\mathrm{e}\mathrm{f}\mathrm{f}}\nabla \mathrm{T}\right)+{\mathsf{\rho }}_{\mathrm{l}}{C}_{\mathrm{p}}\left(\mathbf{u}·\nabla \mathrm{T}\right)=0$$

(10)

The subscript eff represents effective properties of the porous media and term ${\left(\rho c_p\right)}_{eff}$ can be determined from Equation (11) [9]:

$${\left(\rho c_{\mathrm{p}}\right)}_{\mathrm{e}\mathrm{f}\mathrm{f}}={\varphi }_{\mathrm{v}}{\rho }_{\mathrm{l}}{c}_{p,\mathrm{l}}+\left(1-{\varphi }_{\mathrm{v}}\right){\rho }_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}{c}_{\mathrm{p},\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$$

(11)

${\rho }_{\mathrm{l}}$ and $c_{\mathrm{p},\mathrm{l}}$ are the density and specific heat of the liquid, while ${\rho }_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$ and $c_{\mathrm{p},\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$ are the density and specific heat of the porous matrix of the evaporator. Similarly, the effective thermal conductivity of the evaporator in terms of thermal conductivities of liquid ($k_l$) and porous matrix ($k_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$) takes the form:

$$k_{\mathrm{e}\mathrm{f}\mathrm{f}}={\varphi }_{\mathrm{v}}{k}_{\mathrm{l}}+\left(1-{\varphi }_{\mathrm{v}}\right)k_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$$

(12)

The absorbed solar energy (${\dot{Q}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$) as a function of absorption coefficient ($\alpha$) [41,42], solar irradiance ($q_{\mathrm{s}\mathrm{o}\mathrm{l}}$), and solar area ($A_{\mathrm{s}\mathrm{o}\mathrm{l}}=wH$) can be written as: ${\dot{Q}}_{\mathrm{s}\mathrm{o}\mathrm{l}}=\alpha q_{\mathrm{s}\mathrm{o}\mathrm{l}}wH$. Note that the solar irradiance is assumed perpendicular to the evaporator’s surface 5. The heat loss in the form of natural convection from the evaporator’s surface to the surrounding air is given as:

$${\dot{Q}}_{\mathrm{c}\mathrm{o}\mathrm{n}}=h_{\mathrm{c}\mathrm{o}\mathrm{n}}{A}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}\left(T_{\mathrm{s}}-T_{\mathrm{\infty }}\right)$$

(13)

where $h_{\mathrm{c}\mathrm{o}\mathrm{n}}$ is the heat transfer coefficient for natural convection, which depends on the orientation of the surface. The correlations given in [43] are employed to calculate the heat transfer coefficient. Similarly, A_surf is the surface area, T_s is the evaporator’s surface temperature, and $T_{\mathrm{\infty }}$ is the ambient temperature. The radiative heat transfer from the surface to the ambient environment in terms of emissivity and Stefan–Boltzmann constant (σ = 5.67 × ${10}^{-8}$ W/m² K⁴) can be written as:

$${\dot{Q}}_{\mathrm{r}\mathrm{a}\mathrm{d}}=\mathsf{\sigma }\mathsf{\epsilon }{A}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}\left(T_{\mathrm{s}}^4-T_{\mathrm{\infty }}^4\right)$$

(14)

The conduction heat transfer from the evaporator to bulk liquid is calculated in terms of effective thermal conductivity ($k_{\mathrm{e}\mathrm{f}\mathrm{f}}$) and cross-section area ($A_{\mathrm{c}}=wb$) of the evaporator as:

$${\dot{Q}}_{\mathrm{c}\mathrm{n}\mathrm{d}}=k_{\mathrm{e}\mathrm{f}\mathrm{f}}{A}_{\mathrm{c}}{\displaystyle \frac{dT}{dz}}$$

(15)

The evaporation flux (J_v) is the ratio of mass loss ($\dot{m}$) in kg/s to surface area (A_surf) of the evaporator, defined as: J_v = $\dot{m}$/A_surf. The energy transfer associated with the mass loss (${\dot{Q}}_{mass}$) depends on the enthalpy of vaporization (h_fg) and evaporation flux (J_v): ${\dot{Q}}_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}$ = J_v A_surf h_fg. The enthalpy of vaporization and other properties of the saline water can be calculated based on the correlations given in [44,45]. The thermal efficiency (η) of the evaporator can be calculated as: η = $\dot{m}$h_fg/${\dot{Q}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$. The evaporation flux based on Fick’s law and the theoretical model proposed in [45] is given below:

$$J_v={\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}{D}_{\mathrm{v}}\left({\gamma }_{\mathrm{s}\mathrm{a}\mathrm{t}}-{\gamma }_{\mathrm{a}\mathrm{m}\mathrm{b}}\right)/\delta$$

(16)

The formulations for the air-vapor mixture density (${\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}$), diffusion coefficient of vapor in air ($D_{\mathrm{v}}$), mass fractions of vapor at the air-water interface of the evaporator (${\gamma }_{\mathrm{s}\mathrm{a}\mathrm{t}}$), and in ambient (${\gamma }_{\mathrm{a}\mathrm{m}\mathrm{b}}$) are given in [45]. $\delta$ is an empirical parameter for evaporation, and it depends on the geometry of the evaporator and operating conditions [46].

2.4. Initial and Boundary Conditions

Mass flux (${-\mathit{n}·{\rho }_{\mathrm{l}}\mathit{u}=J}_v$) and salt generation (${-\mathit{n}·\mathit{J}=J}_{\mathrm{c}}$) at boundaries 2 to 6 are implemented as Neumann-type boundary conditions. Fixed salt concentration ($c\left(z=0,t\right)=c_0$) is applied at surface 1 as a Dirichlet-type boundary condition. In addition, pressure is also set to 0 Pa at surface 1 while the reference pressure is 1 atm. Similarly, the temperature is also fixed to 298.15 K at surface 1.

The convective (${\dot{Q}}_{\mathrm{c}\mathrm{o}\mathrm{n}}$) and radiative (${\dot{Q}}_{\mathrm{r}\mathrm{a}\mathrm{d}}$) heat losses are implemented for surfaces 2 to 6. The evaporative energy transfer (${\dot{Q}}_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}$) due to mass loss is also applicable to these surfaces. However, solar flux (${\dot{Q}}_{\mathrm{s}\mathrm{o}\mathrm{l}}$) is considered for surface 5 only. The initial conditions for pressure, concentration and temperature of the domain are expressed as: $P\left(x,y,z,t=0\right)=P_0(x,y,z)$, $c\left(x,y,z,t=0\right)=c_0(x,y,z)$ and $T\left(x,y,z,t=0\right)=T_0(x,y,z)$. $P_0$ is 0 Pa, $c_0$ for different cases varies from 3.5 wt.% to 20 wt.%, and $T_0$ is 298.15 K, respectively.

2.5. Model Validation

Before model validation, a mesh independence study was carried out to ensure that numerical results are independent of the grid size (however, for brevity, these results are not included here). For model validation, 1D simulations are performed, and the results for variation in salt concentration are compared with the corresponding theoretical results. The location at which salt concentration reaches saturation is calculated based on the following Equation (17) [45]:

$${\displaystyle \frac{1}{c_{\mathrm{s}\mathrm{a}\mathrm{t}}}}=1+\left[\left({\displaystyle \frac{1-c_{\mathrm{o}}}{c_{\mathrm{o}}}}\right)\left(1-{\displaystyle \frac{z_{\mathrm{s}\mathrm{a}\mathrm{t}}}{H}}\right)\right]$$

(17)

where $z_{\mathrm{s}\mathrm{a}\mathrm{t}}$ is the evaporator’s location where saturation of salt happens, H is the evaporator height, $c_{\mathrm{o}}$ is the inlet, and $c_{\mathrm{s}\mathrm{a}\mathrm{t}}$ is the saturation concentration. Note that the above Equation neglects the role of back diffusion. In addition, Equation (17) does not involve the term evaporation flux as it assumes that the evaporation flux is uniform over the surface. For a fair comparison, we have also implemented the above-mentioned assumptions in the numerical model. The comparison of the results (in

Table 2. Comparison of theoretical and simulation results for the height of the precipitation-free zone (z_sat).

Inlet Concentration, co	zsat (Theoretical)	zsat (Simulation)
3.5 wt.%	44.9 mm	43.7 mm
10 wt.%	34.4 mm	33.5 mm

) shows that the simulation results are in agreement with the numerical results, which validates our numerical formulation.

3. Results and Discussion

3.1. Effect of the Salt Concentration on Evaporation Performance

Figure 2 shows variations in brine concentration along with the evaporation surface of the evaporator. The blue color in concentration maps (shown in Figure 2) represents the lowest salinity, while dark red shows the highest salinity. The black curved contour line shows the position of the precipitation front at which brine concentration reaches saturation (i.e., 26.3 wt.%). The regions below and above the black contour line in Figure 2 are termed precipitation-free and precipitation zones, respectively. In the precipitation zone, the salt concentration is equal to the concentration of saturated brine (i.e., 26.3 wt.%). The salt first precipitates at the top edge of the evaporator, and the curved interface recedes downward toward the evaporator inlet (as shown in Figure 2 at t = 0.25 h and 0.5 h). The temporal and spatial variations in the concentration of saline water given in Figure 3 support these observations. The profiles at t = 2 h and t = 3 h overlap with each other, which means the concentration profile has reached a steady state.

Convective and diffusive transports, along with precipitation rate, are the main factors that govern the transport and precipitation of brine. During the evaporation process, the salt is transported from the inlet of the vertical evaporator and precipitates at the air-water interface. The brine concentration at the top edge of the evaporator is higher than the bottom one (Figure 3), and the concentration gradient results in back diffusion of salt. The ratio of advective to diffusive transport of salt is represented by the Peclet number (Pe) defined below [47]:

$$Pe=H{\displaystyle \frac{U}{D_{\mathrm{s}}}}$$

(18)

where $U$ is the magnitude of the Darcy’s velocity of brine, H is the height of the evaporator, and D_s is the effective diffusion coefficient of the salt. The variation in Darcy’s velocity ($U$) for an inlet concentration of 3.5 wt.% can be seen in Figure 4 at t = 3 h. The velocity does not have a linear profile owing to non-uniform evaporation flux at the evaporator’s surface. The maximum velocity occurs at the inlet of the evaporator with a gradual reduction along its height. Note that the profile of Peclet number is similar to the profile of Darcy’s velocity (Figure 4), which implies that convective salt flux dominates the diffusive one.

For Pe >> 1, the convective transport dominates the diffusive transport, which results in salt precipitation. On the other hand, Pe << 1, diffusive transport overcomes the convective one, and no precipitation of salt happens at the evaporator. The ratio of precipitation rate to diffusive salt transport is known as the Damkhöler number (Da) [30]. Herein, we assume that Da > 1 and salt keep growing, albeit with a decreased rate, owing to light reflection by the salt precipitates. As the evaporation is driven by solar energy, the precipitated salt will reflect the solar irradiation, leading to a reduction in absorbed solar energy in the regions of c/c_sat ≥ 1. This will result in lower surface temperature and evaporation rate (as well as lower precipitation rate) in the regions of precipitated salt, as shown in Figure 5.

The effect of brine concentration on average evaporation flux and height of the salt-free zone is shown in Figure 6. With the increase in concentration, there is a significant decrease in average evaporation flux. The evaporation fluxes of 0.5, 0.44, 0.32, 0.22, and 0.16 kg/m²h are observed at salt concentrations of 0, 3.5, 8, 14, and 20 wt.%, respectively. Note that 0.68 kg/m²h is the theoretical limit in our case for the evaporation flux of water under 1000 W/m². The evaporation flux decreases due to an increase in salt concentration. This is due to the reduced vapor pressure of the saline water [48]. Additionally, for solar desalination with salt precipitation, an increase in concentration leads to a larger precipitation zone. The more precipitation region causes more light reflection by the white salt precipitates. Ultimately, this results in a reduction of the evaporation flux from the evaporator. The area of the precipitation-free zone of the evaporator reduces as the concentration at the inlet of the evaporator increases. For instance, at an inlet concentration of 3.5 wt.%, 17% of the evaporator’s surface is predicted to be covered with salt precipitates (as visualized in Figure 6 at t = 3 h). It is important to note that once the salt precipitates at the edge of the evaporator, the precipitation may continue to take place on already precipitated salt instead of on the porous evaporator. This phenomenon has been reported in the literature [40]. It will cause a higher salt exclusion zone than the predicted one. Therefore, the evaporation rate predicted by the numerical model may be lower as compared to the experimental one.

3.2. Effect of Solar Flux on Evaporation Performance

The vapor flux of solar evaporators is not very high at 1000 W/m². High yield can only be achieved by concentrating solar irradiation through an optical concentrator, as discussed in [49,50]. Accordingly, solar irradiance is varied from 500 W/m² to 2500 W/m² in order to investigate the impact of optical concentration on the thermal performance of the solar evaporator.

The temperature maps for different solar irradiances are shown in Figure 7 at t = 3 h. The trend is non-monotonic as temperature is lower near both bottom and top surfaces as compared to the middle of the evaporator. Since the temperature at the bottom surface (i.e., boundary 1) is fixed, it can be considered a thermal source. The conduction heat transfer to the bulk saline water reduces the surface temperature. Similarly, the temperature at the top surface is also low due to less light absorption by the salt crystals. At 2000 and 2500 W/m², the contour plots at a temperature of 312.15 K confirm the non-linear temperature distributions. The surface temperature at 2500 W/m² is higher than that of 2000 W/m², as shown in Figure 7. The temperature distribution is also non-linear in the lateral direction, as confirmed by the curved shape of the contour lines at 312.15 K.

The average temperature of the evaporator increases with the increase in solar flux. The higher temperature favors the evaporation process; however, heat loss from the evaporator surface also increases. The contribution of heat losses under different irradiances is shown in Figure 8 at t = 3 h. For 3.5 wt.% brine and solar flux of 1000 W/m², the radiative, convective, and conductive heat transfer losses are 10%, 8.5%, and 14% of the total absorbed solar energy, respectively (Figure 8). The radiative and convective heat losses from the evaporator depend on ambient and surface temperatures. With the increase in solar flux, increased surface temperature enhances heat losses (Figure 8) and adversely affects thermal efficiency.

However, the literature suggests that convective and radiative losses from the evaporator are not significant. There exist heated vapors over the air-water interface of the evaporator. Therefore, the evaporator does not directly exchange thermal energy (through convection and radiation) with the environment owing to the presence of heated vapors. Therefore, convective and radiative energy losses are reduced due to this phenomenon. For instance, Liu et al. [51] observed a thermal efficiency of ~90% for their system, which was significantly higher than the theoretically estimated thermal efficiency of 62.1%. The record high thermal efficiency was attributed to reduced convective and radiative thermal losses, which were experimentally found to be 2.6% and 1.8%, respectively. Therefore, these two losses are not significant and can be neglected. Figure 8 and Figure 9 show average evaporation flux and thermal efficiency at t = 3 h for two cases: with and without convective and radiative heat losses. The lower black curve (in Figure 9 and Figure 10) corresponds to the former case, while the upper red curves are related to the latter case. For both cases, the evaporation flux varies linearly with the solar irradiance. Almost a two-fold increase in evaporation flux (i.e., from 0.44 kg/m² h to 0.99 kg/m² h) is found when all heat losses are considered and irradiance is increased from 1000 to 2500 W/m². Neglecting radiative and convective heat losses results in a 27% enhancement in evaporation flux while a 26% increase in thermal efficiency of the evaporator under 1000 W/m² as depicted in Figure 10. Therefore, the evaporator can achieve a high thermal efficiency of 83% under 1000 W/m². On the other hand, the thermal efficiency of the solar evaporator decreases with the increase in solar irradiance, as shown in Figure 10.

3.3. Effect of Evaporator’s Shape on Evaporation Performance

The shape of the evaporator also affects its performance and locus of salt precipitation. Herein, we simulate solar-driven evaporation of saline water from a three-dimensional evaporator with a thickness (b) of 5 mm while keeping width (w) and height (H) unchanged. Figure 11a shows the volumetric distribution of temperature across the evaporator’s surface at t = 10 h under 1000 W/m² for 3.5 wt.% salt concentration while taking all heat losses into account. The temperature of the region close to boundary 1 (i.e., inlet) is the lowest, while it is highest at the middle surface. The average surface temperature of the evaporator is around 300.8 K, which is lower than that of the evaporator with a thickness of 0.5 mm, as discussed in Section 3.1. The contour line in black at 301 K visualizes the temperature gradient along the yz and xz planes. It also implies that the temperature at surface 5 is higher than that of surface 6, since only surface 5 is exposed to solar flux. This can also be confirmed from the temperature contours at the xy plane (z = 25 mm) shown in Figure 11a. The three-dimensional spatial distribution of salt concentration at t = 10 h is shown in Figure 11b. The concentration profiles at the front (i.e., surface 5) and left sides (i.e., surface 3) of the three-dimensional evaporator are qualitatively similar to the one reported in Figure 2. The dark contour line (with c = ~26.3 wt.%) at the front and left surfaces of the evaporator illustrates the double arch-shaped pattern of salt distribution. There is an elliptical profile at the xy plane (z = 25 mm), which shows that the salt concentration is the highest at the sharp edges of the evaporator.

The evaporation flux shown in Figure 12a mainly follows the salt concentration behavior rather than the surface temperature. This is owing to the fact that the variation in salt concentration along the z-position of the evaporator is higher than the corresponding variation in temperature. Therefore, the salt concentration plays a greater role in shaping the profile of evaporation flux. It is pertinent to mention that the evaporation flux at the edges of the evaporator is lower than that of the central areas (Figure 12a, right), primarily because of the higher salt concentration at the edges. The average evaporation flux is found to be 0.28 kg/m² h at 1000 W/m². Therefore, a 10-fold increase in the thickness (with a 47% increase in surface area) of the evaporator results in a 37% decrease in evaporation flux.

The evaporation process drives the brine from the bottom of the evaporator (i.e., z = 0 mm) to the top surface (i.e., at z = 50 mm), where salt precipitation happens when water evaporates. Based on mass balance, the average evaporation flux (J_v) and average Darcy’s velocity at the inlet ($U$_in) can be related as:

$$U_{in}={\displaystyle \frac{J_{\mathrm{v}}{\mathrm{A}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}}{{\rho }_{\mathrm{l}}{A}_{\mathrm{c}}}}$$

(19)

The higher the evaporation flux, the higher the velocity at the inlet of the evaporator. Therefore, high inlet velocity of the brine is likely under high evaporation rates. The contour map for the magnitude of Darcy’s velocity is depicted in Figure 12b. The average velocity at the inlet of the evaporator is 2.2 µm/s, which gradually decreases with the evaporator’s height. The velocity contours for the xy plane at z = 25 mm, given in Figure 12b, show that it follows a stretched elliptical profile. The velocity is higher at the evaporator’s free surface than its interior, owing to mass loss at free surface.

4. Conclusions

The rational design of solar evaporators is a first step toward achieving sustainable desalination practices. Understanding various transport processes and physics involved in the operation of three-dimensional evaporators requires robust three-dimensional simulations. In this study, we presented an all-inclusive simulation model to evaluate the evaporation performance and thermal efficiency of solar evaporators under different operating conditions. The model can take into account the effect of variations in brine concentration, solar irradiance, and the evaporator’s shape on its evaporation performance. Simulation results show that the evaporation flux and precipitation-free region of the evaporator decreased with the increase in inlet salt concentration. For instance, the evaporation flux became half when salt concentrations increased from 3.5 wt.% to 14 wt.%. The temperature and evaporation flux distribution along the z-position of the evaporator exhibited highly non-linear and non-monotonic trends. The peak evaporation flux occurred at mid-height (i.e., ~25 mm) of the evaporator for 3.5 wt.% concentration at 1000 W/m². Moreover, the competing roles of advective, diffusive, and precipitation rates of salt were considered to find the locus of salt precipitation and its impact on light absorption and evaporation flux. The conduction heat loss was higher than the convective and radiative heat losses. The studied evaporator can achieve a thermal efficiency of over 80% for saline water (3.5 wt.%). The presented model is applicable to any type of evaporator and promises significant potential for the eco-friendly and efficient design of solar desalination systems.