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Mathematical modeling and experimental validation for square pyramid solar still

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Abstract

This study presents a detailed mathematical model of the pyramid solar still which is not performed in most of the previous studies. Computer programs (in Pascal language) are developed for both studying thermal performance of the square pyramid solar still and estimating the hourly variations of the solar intensity incident on the tilted still’s covers. Comparisons between the obtained theoretical results and experimental results (from previous work) are performed to validate the proposed mathematical model. The daily productivity of the pyramid solar still (Pd) varies from 4.22 to 4.43 kg/m2 day with values of the glass cover’s tilt angle of 10°–60°. Pd decreases from 3.88 to 1.52 kg/m2 day with increasing the depth of the still’s water (dw) from 0.01 to 0.30 m. Values of the top losses of the still decrease from 8.8064 to 8.2304 W/m2 K with increasing glass cover surface area from 0.063 to 0.125 m2, which correspond to values of tilt angles of the still covers changing from 10° to 60°, respectively.

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Abbreviations

A ge :

Surface area of the eastern glass cover (m2)

A p :

Surface area of the absorber plate (m2)

b :

Width of the pyramid solar still (m)

C w :

Specific heat of water (J/kg K)

Gr :

Grasshof number (dimensionless)

h 1e :

Total heat transfer coefficient between the basin water and the inner surface of the eastern glass cover (W/m2 K)

h 1w :

Total heat transfer coefficient between the basin water and the inner surface of the western glass cover (W/m2 K)

h 1s :

Total heat transfer coefficient between the basin water and the inner surface of the southern glass cover (W/m2 K)

h 1n :

Total heat transfer coefficient between the basin water and the inner surface of the northern glass cover (W/m2 K)

h cga :

Convective heat transfer coefficient between the four sides of the glass cover and the ambient air (W/m2 K)

h cwge :

Convective heat transfer coefficient between the basin water and the inner surface of the eastern glass cover (W/m2 K)

h ewge :

Evaporative heat transfer coefficient between the basin water and the inner surface of the eastern glass cover (W/m2 K)

h rge :

Radiative heat transfer coefficient between the eastern glass cover and the ambient air (W/m2 K)

h rwge :

Radiative heat transfer coefficient between the basin water and the inner surface of the eastern glass cover (W/m2 K)

I :

Solar radiation incident on a horizontal surface (W/m2)

I b :

Beam solar radiation (W/m2)

I d :

Diffuse solar radiation (W/m2)

I e :

Solar radiation incident on the eastern glass cover (W/m2)

I o :

Extraterrestrial solar radiation incident on a horizontal surface (W/m2)

I t :

Total solar radiation incident on a tilted surface (W/m2)

k w :

Thermal conductivity of the basin water (W/m K)

L :

Length of the still (m)

L ave :

Daily average of the latent heat of vaporization (J/kg)

L w :

Latent heat of vaporization (J/kg)

m w :

Mass of the basin water (kg)

Nu :

Nusselt number (dimensionless)

P :

Accumulated distillate (kg/m2 h)

P d :

Daily productivity (kg/m2 day)

P ge :

Partial vapor pressure at temperature of the eastern glass cover (N/m2)

P h :

Total hourly productivity (kg/m2 h)

P he :

Hourly productivity from the eastern glass cover (kg/m2 h)

P hn :

Hourly productivity from the northern glass cover (kg/m2 h)

P hs :

Hourly productivity from the southern glass cover (kg/m2 h)

P hw :

Hourly productivity from the western glass cover (kg/m2 h)

Pr :

Prandtl number (dimensionless)

P w :

Partial vapor pressure at temperature of the basin water (N/m2)

T a :

Ambient air temperature (°C)

T ge :

Temperature of the eastern glass cover (°C)

T p :

Temperature of the basin liner (°C)

T w :

Temperature of the basin water (°C)

T s :

Sky temperature (°C)

T w :

Still water temperature (°C)

∆t :

Selected time interval (s)

U b :

Bottom loss coefficient (W/m2 K)

V :

Wind speed (m/s)

α g :

Absorptivity of the glass cover

α p :

Absorptivity of the absorber plate

α w :

Absorptivity of the still water

β :

Tilt angle with horizontal (°)

θ :

Incident angle of beam radiation on tilted surface (°)

θ z :

Incident angle of beam radiation on horizontal surface or zenith angle (°)

δ :

Solar declination angle (°)

ε g :

Emissivity of the glass cover

η d :

Daily collection efficiency of the still (%)

φ:

Latitude (°)

τ g :

Transmissivity of the glass cover

τ w :

Transmissivity of the still water

σ :

Stefan-Boltzmann’s constant (W/m2 K4)

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Correspondence to Abd El-Monem Khallaf.

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Appendix A1

Appendix A1

$$ {\displaystyle \begin{array}{cc}{R}_1=\frac{1}{U_b}=\frac{X_b}{K_b}& {R}_2=\frac{1}{h_{cpw}}\\ {}{R}_3=\frac{1}{\left({h}_{cwge}+{h}_{rwge}+{h}_{ewge}\right)}& {R}_4=\frac{1}{\left({h}_{cga}+{h}_{rge}\right)}\\ {}{R}_5=\frac{1}{\left({h}_{cwgw}+{h}_{rwgw}+{h}_{ewgw}\right)}& {R}_6=\frac{1}{\left({h}_{cga}+{h}_{rgw}\right)}\\ {}{R}_7=\frac{1}{\left({h}_{cwgs}+{h}_{rwgs}+{h}_{ewgs}\right)}& {R}_8=\frac{1}{\left({h}_{cga}+{h}_{rgs}\right)}\\ {}{R}_9=\frac{1}{\left({h}_{cwgn}+{h}_{rwgn}+{h}_{ewgn}\right)}& {R}_{10}=\frac{1}{\left({h}_{cga}+{h}_{rgn}\right)}\end{array}} $$

Appendix A2

Values of the coefficients in Eq. 24 are

$$ M={m}_w{C}_w $$
$$ {\displaystyle \begin{array}{c}a={h}_{cpw}{A}_p+{h}_{1e}{A}_{ge}+{h}_{1w}{A}_{gw}+{h}_{1s}{A}_{gs}+{h}_{1n}{A}_{gn}+{U}_s{A}_s-\frac{h_{cpw}^2{A}_p}{h_{cpw}+{U}_b}-\frac{h_{1e}^2{A}_{ge}}{h_{1e}+{h}_{cga}+{h}_{rge}}-\\ {}-\frac{h_{1w}^2{A}_{gw}}{h_{1w}+{h}_{cga}+{h}_{rgw}}-\frac{h_{1s}^2{A}_{gs}}{h_{1s}+{h}_{cga}+{h}_{rgs}}-\frac{h_{1n}^2{A}_{gn}}{h_{1n}+{h}_{cga}+{h}_{rgn}}\end{array}} $$

and

$$ {\displaystyle \begin{array}{c}\overline{f(t)}={I\tau}_g{\alpha}_w{A}_p+\frac{{I\tau}_g{\tau}_w{\alpha}_p{h}_{cpw}{A}_p}{h_{cpw}+{U}_b}+\frac{I_e{\alpha}_g{h}_{1e}{A}_{ge}}{h_{1e}+{h}_{cga}+{h}_{rge}}+\frac{I_w{\alpha}_g{h}_{1w}{A}_{gw}}{h_{1w}+{h}_{cga}+{h}_{rgw}}+\\ {}+\frac{I_s{\alpha}_g{h}_{1s}{A}_{gs}}{h_{1s}+{h}_{cga}+{h}_{rgs}}+\frac{I_n{\alpha}_g{h}_{1n}{A}_{gn}}{h_{1n}+{h}_{cga}+{h}_{rgn}}+{T}_a\left[{U}_s{A}_s+\frac{h_{cpw}{U}_b{A}_p}{h_{cpw}+{U}_b}\right.+\\ {}\left.+\frac{h_{1e}{h}_{cga}{A}_{ge}}{h_{1e}+{h}_{cga}+{h}_{rge}}+\frac{h_{1w}{h}_{cga}{A}_{gw}}{h_{1w}+{h}_{cga}+{h}_{rgw}}+\frac{h_{1s}{h}_{cga}{A}_{gs}}{h_{1s}+{h}_{cga}+{h}_{rgs}}+\frac{h_{1n}{h}_{cga}{A}_{gn}}{h_{1n}+{h}_{cga}+{h}_{rgn}}\right]+\\ {}+{T}_s\left[\frac{h_{1e}{h}_{rge}{A}_{ge}}{h_{1e}+{h}_{cga}+{h}_{rge}}+\frac{h_{1w}{h}_{rgw}{A}_{gw}}{h_{1w}+{h}_{cga}+{h}_{rgw}}+\frac{h_{1s}{h}_{rgs}{A}_{gs}}{h_{1s}+{h}_{cga}+{h}_{rgs}}+\frac{h_{1n}{h}_{rgn}{A}_{gn}}{h_{1n}+{h}_{cga}+{h}_{rgn}}\right]\end{array}} $$

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El-Sebaii, A., Khallaf, A.E. Mathematical modeling and experimental validation for square pyramid solar still. Environ Sci Pollut Res (2020). https://doi.org/10.1007/s11356-019-07587-5

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Keywords

  • Square pyramid solar still
  • Heat transfer coefficient
  • Partial pressure
  • Efficiency
  • Productivity