**2. Methodology**

The methodology of this research is composed of the selection of site in Pakistan for solar power desalination following the mathematical modeling of the single slope of solar stills and employs modern software for the solution of mathematical model. The governing equations for mathematical model of the solar stills are based on the law of conservation of mass and law of conservation of energy for the system. The equations for convective heat transfer coefficients and radiation heat transfer coefficients are based on the Dunkle's model. The MATLAB r2019 is employed to solve the equations.

#### **2.1 Site selection in Pakistan**

The availability of **1900–2200 kWh/m<sup>2</sup>** annual global irradiance makes Pakistan highly favorable for solar power-based desalination [5]. The Balochistan and Sindh province of Pakistan is rich in solar energy with an average daily direct normal irradiance of 5.3–5.6 kWh/m2 and 2.5–3.0 kWh/m<sup>2</sup> with sunshine duration of 8–8.5 hours a day [6]. Therefore, in this research paper the site at Lyari River Karachi has been selected for the modeling the single-slope solar stills.

#### **2.2 Mathematical modeling of solar stills**

The basic assumptions while modeling the solar stills take negligible temperature stratification within the evaporator basin. Temperature is uniform within each still component. Temperature is time dependent. The evaporated water is assumed only pure water; that is, the evaporated water has no dissolved salt or ion. The stills have no vapor leakages. The governing equations are based on law of conservation of mass and law of conservation of energy. The schematic of single slope solar stills is shown in **Figure 1**.

The law of conservation of mass can be written as [7].

$$
\dot{m}\_{sw} = \dot{m}\_{ev} + \dot{m}\_b \tag{1}
$$

If **Xsw** is the concentration of salt in the feed saline water, **Xb** is the concentration of salt in the cbrine within the basin. Then, the salt balance is [7].

$$
\dot{m}\_{sw}X\_{sw} = \dot{m}\_bX\_b\tag{2}
$$

The solubility of salt determines the salt content in the brine. The salt content in the brine is important in practice to avoid the problem of forming layer and blockage. The factor *fc* for concentration is defined as the ratio of brine concentration to feed concentration.

$$f\mathbf{c} = \frac{\mathbf{X}\_b}{\mathbf{X}\_{sw}}\tag{3}$$

This factor is used to fix a threshold limit to not exceed during evaporation and condensation. By solving Eq. (2)) and Eq. (3), we have the following equation.

$$
\dot{m}\_b = \frac{1}{f\varepsilon} \dot{m}\_{sw} \tag{4}
$$

**Figure 1.** *Schematic of single-slope solar stills.*

*Distillation Processes - From Solar and Membrane Distillation to Reactive Distillation…*

$$
\dot{m}\_{ev} = \frac{fc - 1}{fc} \dot{m}\_{sw} \tag{5}
$$

Eq. (5) is for the stationary conditions, the rate of evaporated water as a function of rate of feed saline water. Now, the distillated water or recovery rate can be defined as

$$
\rho = \frac{fc - 1}{fc} \tag{6}
$$

The recovery rate is an important parameter, which indicates the possible amount of distillate water from the saline feed water without scaling [7]. It means that only 40% of saline water can be transformed into distillate water without encrustation and blockage.

The law of conservation of energy gives the following set of equations for the respective components in the solar stills.

The energy balance equation for the outer of the transparent glass cover is as follows [7]:

$$\frac{\rho\_{\rm g} V\_{\rm g} C\_{p\rm g}}{2A\_{\rm g}} \frac{dT\_{\rm g\epsilon}}{dt} = \frac{\lambda\_{\rm g}}{e\_{\rm g}} \left( T\_{\rm g\epsilon} - T\_{\rm g\epsilon} \right) - hc\_{\rm g\epsilon - am} \left( T\_{\rm g\epsilon} - T\_{amb} \right) - hr\_{\rm g\epsilon - b\rm y} \left( T\_{\rm g\epsilon} - T\_{\rm aky} \right) \tag{7}$$

The energy balance equation for the inner of the transparent glass cover is as follows:

$$\frac{\rho\_{\rm g} V\_{\rm g} \mathbf{C}\_{p\rm g}}{2A\_{\rm g}} \frac{dT\_{\rm ge}}{dt} = \dot{a}\_{\rm g} I(t) + \left( hc\_{\rm sw-gi} + hr\_{\rm sw-gi} \right) \left( T\_{\rm sw} - T\_{\rm gi} \right) + \frac{\dot{m}\_{\rm ev} L\_{\rm v}}{A\_{\rm g}} - \frac{\lambda\_{\rm g}}{c\_{\rm g}} \left( T\_{\rm gi} - T\_{\rm ge} \right) \tag{8}$$

The energy balance for the seawater inside the basin of solar still is as follows [7]:

$$\frac{\rho\_{sw}V\_{sw}C\_{p,sw}}{A\_{sw}}\frac{dT\_{sw}}{dt} = $$
 
$$\begin{split} \dot{m}\_{sw}I(t) + hc\_{c-sw}(T\_c - T\_{sw}) + \frac{\rho\_{sw}D\_{sw}}{A\_{sw}}\left(c\_{p,sw}T\_{sw,in} - c\_{p,b}T\_{b,out}\right) - \frac{\dot{m}\_{ev}}{A\_{sw}}\left(h\_L - c\_{p,b}T\_{b,out}\right) \\ - \left(hc\_{m-\dot{g}} - hr\_{m-\dot{g}}\right)\left(T\_{sw} - T\_{\dot{g}}\right) \end{split} \tag{9}$$

The convective heat transfer coefficient of between the outer of the transparent glass cover and the ambient temperature depends on the wind velocity. According to McAdams correlation [8], this coefficient is approximated by the following equation.

$$hc\_{\mathcal{Y}^{t-amb}} = \begin{cases} 5.621 + \frac{1151.2v}{T\_{amb}} \dot{q}\dot{v} < 4.88 \text{ ms}^{-1} \\\\ 604.29 \left(\frac{v}{T\_{amb}}\right)^{0.78} \dot{q}\prime 4.88 \le v < 30.48 \text{ ms}^{-1} \end{cases} \tag{10}$$

The heat transfer coefficient between the saline water and the inner of the transparent glass cover is given by the second form of Dunkle's model and can be written as [9].

*Modeling of Solar-Powered Desalination DOI: http://dx.doi.org/10.5772/intechopen.103934*

$$hc\_{sw-\text{g}i} = 0.884 \left( T\_{sw} - T\_{\text{g}i} + \frac{\left( p\_{sw} - p\_{\text{g}i} \right) T\_{sw}}{2.689 \times 10^{5} - p\_{sw}} \right)^{\frac{1}{5}} \tag{11}$$

The radiation heat transfer coefficient between the outer of the transparent glass cover and the sky is given by

$$
\delta hr\_{\text{ge}-sky} = \varepsilon\_{\nu} \sigma \left( T^2\_{\text{ge}} + T^2\_{\text{sky}} \right) \left( T\_{\text{ge}} + T\_{\text{sky}} \right) \tag{12}
$$

*σ* is the Steffen Boltzman constant. The sky temperature is determined by [10].

$$T\_{sky} = T\_{amb} \left( \mathbf{0.74} + \mathbf{0.006} \theta \right)^{0.25} \tag{13}$$

Where Ɵ is the dew point temperature given by [11].

$$\theta = \frac{273.3}{17.27 - In\varepsilon + \frac{17.27T\_{amb} - 4061}{T\_{amb} - 35.85}} \left( In\varepsilon + \frac{17.27T\_{amb} - 4061}{T\_{amb} - 35.85} \right) \tag{14}$$

*ε* is the relative humidity.

The radiation heat transfer coefficient between the saline water and the sky is expressed as

$$
\delta hr\_{\text{gt}-sky} = \varepsilon\_{\text{eff}} \sigma \left( T^2\_{\text{sw}} + T^2\_{\text{gj}} \right) \left( T\_{\text{sw}} + T\_{\text{gj}} \right) \tag{15}
$$

Emissivity is given by,

$$
\varepsilon\_{\rm eff} = \left(\frac{\mathbf{1}}{\varepsilon\_{\rm sw}} + \frac{\mathbf{1}}{\varepsilon\_{\rm g}} - \mathbf{1}\right)^{-1} \tag{16}
$$

The equation in the second part of the (Eq. (9)) in given by Dunkle's model as [9].

$$
\dot{m}\_{ev} h\_L = h\_{ev} A\_{sw} \left( T\_{sw} - T\_{gi} \right) \tag{17}
$$

The latent heat of vaporization **hL** is given by [9].

$$h\_L = \mathbf{3146} - \mathbf{2.36}T\_{sw} \tag{18}$$

The evaporative heat transfer coefficient hev is given by [9].

$$h\_{ev} = 0.016273 \, hcv\_{sw-gi} \frac{p\_{sw} - p\_{gi}}{T\_{sw} - T\_{gi}} \tag{19}$$

The evaporative heat transfer coefficients Eq. (11) and Eq. (19) can only be estimated through correlations when the following conditions is satisfied: the aspect ratio 2*:*5≤*a*≤5*:*5, the inclination angle 10<sup>0</sup> <sup>≤</sup>*<sup>i</sup>* <sup>≤</sup>300, and Rayleigh number 5 � <sup>10</sup><sup>6</sup> <sup>≤</sup> *Ra*≤<sup>5</sup> � 107 .

If the above conditions are not fulfilled, then it could be done either experimentally or by using 2D modeling of the problem such as considered in some other systems [12, 13].

The Nusselt number is obtained through a correlation in the form by to express the convective heat transfer coefficients [14].

$$\left(\mathrm{Nu} = \mathfrak{c}(\mathrm{Ra})\right)^{n} = \mathfrak{c}(GrPr)^{n} \tag{20}$$

The Grashof and Prandtl number is given by,

$$Gr = \frac{\beta \text{g} \rho^2 L^3 \triangle T}{\mu^2} \tag{21}$$

$$Pr = \frac{\mu c\_{\rho}}{\lambda} \tag{22}$$

The correlation that gives Nusselt number is [14].

$$Nu = \begin{cases} 1\,\text{j}\text{\'}Gr < 10^5\\ 0.5(Ra)^{0.25}\text{\'}\text{\'}10^5 < Gr < 2 \times 10^7\\ 0.15(Ra)^{0.33}\text{\'}\text{\'}Gr > 2 \times 10^7 \end{cases} \tag{23}$$

When the Nusselt number is known, the heat transfer coefficient between the basin liner plate and the saline water can be calculated for an active solar still as

$$hcv\_{c-sw} = \frac{Nu\_{c-sw}\lambda\_{sw}}{L} \tag{24}$$

The convective heat transfer coefficient between the fluid and the plate for an active solar still is calculated as [15].

$$hcv\_{h-c} = \frac{Nu\_{h-c}\lambda\_h}{L} \tag{25}$$

The heat loss coefficient is approximated by the following equation [15].

$$U\_{loss} = \frac{\lambda\_{is}}{\mathcal{e}\_{it}}\tag{26}$$

Tb,out can be assumed to be equal to that of the plate Tp for larger length of basin liner for an active solar stills [15].

The MATLAB's solver for ordinary differential equations (ODEs), MATLAB ode45 function, has been employed for the efficient computation of the differential equations.

The material properties and dimensions of solar still are given in **Table 1** and thermophysical properties of glass, basin, and insulation are given in **Table 2** in Appendix A.

#### **3. Results**

The hourly production of distill water in *kg=hour* at Lyari River, Karachi, with and without Fresnel (FRL) lens. The results are depicted in **Figure 2**.

**Figure 2.** *Result of mathematical model of single slope solar still.*

The maximum ambient temperature and sky temperature on the hottest day is 39.5°C and 14.7°C. The result is showing that the maximum water temperature with and without Fresnel (FRL) lens is 82.3°C and 47.2°C. And also the maximum glass temperature with and without FRL lens is found to be 80°C and 39.5°C. The production of water is calculated using the temperatures. The maximum water production with and without FRL lens is 8 kg/hour and 1 kg/hour. Using FRL lens, the production of water is 330% more than without using FRL lens.
