# 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 :

I d :

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)

## References

1. Ahmed HM, Alshutal FS, Ibrahim G (2014) Impact of different configurations on solar still productivity. J Adv Sci Eng Res 4(2):118–126

2. Altarawneh I, Rawadieh S, Batiha M, Al-Makhadmeh L, Alrowwad S, Tarawneh M (2017) Experimental and numerical performance analysis and optimization of single slope, double slope and pyramidal shaped solar stills. Desalination 423:124–134

3. Arunkumar T, Jayaprakash R, Prakash A, Suneesh PU, Karthik M, Kumar S (2010) Study of thermo physical properties and an improvement in production of distillate yield in pyramid solar still with boosting mirror. Indian J Sci Technol 3(8)

4. Duffie JA, Beckman WA (2013) Solar engineering of thermal processes, 4th edn. John Wiley & Sons, New York

5. Dunkle RV (1961) Solar water distillation: the roof type still and a multiple effect diffusion still. International developments in heat transfer. Univ Colorado Part 5:895–902

6. Dwivedi VK, Tiwari GN (2009) Comparison of internal heat transfer coefficients in passive solar stills by different thermal models: an experimental validation. Desalination 246:304–318

7. Eze JI, Ojike O (2012) Comparative evaluation of rectangular and pyramid-shaped solar stills using saline water. Int J Phys Sci 7(31):5202–5208

8. Fath HES, El-Samanoudy M, Fahmy K, Hassabou A (2003) Thermal-economic analysis and comparison between pyramid shaped and single-slope solar still configurations. Desalination 159:69–79

9. Jamil B, Siddiqui AT, Akhtar N (2016) Estimation of solar radiation and optimum tilt angles for south-facing surfaces in humid subtropical climatic region of India. Eng Sci Technol Int J 19:1826–1835

10. Kabeel AE (2009) Performance of solar still with a concave wick evaporation surface. Energy 34:1504–1509

11. Kabeel AE, Mohamed Abdelgaied, Nouaf Almulla (2016) Performances of pyramid-shaped solar still with different glass cover angles: experimental study. 7th International Renewable Energy Congress (IREC)

12. Kabeel AE, Mohamed A, Teamah MA, Abdel Aziz GB (2017) Modified pyramid solar still with v-corrugated absorber plate and PCM as a thermal storage medium. J Clean Prod 161:881–887

13. Kalaivani S, Radhakrishnan ST (2013) Heat mass transfer and thermophysical analysis for pyramid type solar still. Int J Sci Res 2(9):181–184

14. Kianifar A, Heris SZ, Mahian O (2012) Exergy and economic analysis of a pyramid-shaped solar water purification system: active and passive cases. Energy 38:31–36

15. Mahian O, Kianifar A (2011) Mathematical modelling and experimental study of a solar distillation system. Proc Int Mech Eng C J Mech Eng Sci 225:1203–1212

16. Maleki SAM, Hizam H, Gomes C (2017) Estimation of hourly, daily and monthly global solar radiation on inclined surfaces: models re-visited. Energies 2017(10):134

17. Manokar AM, Taamneh Y, Kabeel AE, Sathyamurthy R, Winston DP, Chamkha AJ (2018) Review of different methods employed in pyramidal solar still desalination to augment the yield of freshwater. Desalin Water Treat 136:20–30

18. Nayi KH, Modi KV (2018) Pyramid solar still: a comprehensive review. Renew Sust Energ Rev 81:136–148

19. Ossie NM (1988) Heat transfer. McGraw-Hill, New York

20. Patel SG, Bhatnagar S, Vardia J, Ameta SC (2006) Use of photocatalysts in solar desalination. Desalination 189:287–291

21. Prakash A, Jayaprakash R, Kumar S (2016) Experimental analysis of wick type pyramid shape solar still. Int J Sci Eng Res 7(4)

22. Sathyamurthy R, Nagarajan PK, Subramani J, Vijayakumar D, Mohammed Ashraf Ali K (2014a) Effect of water mass on triangular pyramid solar still using phase change material as storage medium. Energy Procedia 61:2224–2228

23. Sathyamurthy R, Kennady HJ, Nagarajan PK, Ahsan A (2014b) Factors affecting the performance of triangular pyramid solar still. Desalination 344:383–390

24. Sathyamurthy R, Harris Samuel DG, Nagarajan PK, Arunkumar T (2016) Geometrical variations in solar stills for improving the fresh water yield—a review. Desalin Water Treat 57(45):21145–21159

25. Senthil Rajan A, Raja K, Marimuthu P (2015) Increasing the productivity of pyramid solar still augmented with biomass heat source and analytical validation using RSM. Desalin Water Treat:1–14

26. Sharma VB, Mullick SC (1991) Estimation of heat transfer coefficients, the upward heat flow and evaporation in a solar still. Trans ASME Solar Energy Eng 113:36–41

27. Taamneh Y, Taamneh M (2012) Performance of pyramid-shaped solar still: experimental study. Desalination 291:65–68

28. Tiwari GN, Tiwari A, Shyam (2016) Handbook of solar energy: theory, analysis and applications. Springer Science+Business Media, Singapore

29. Vala S, Kanabar B (2017) Solar distillation based on pyramid shape solar still—a review. J Res 3(3)

30. Yadav YP, Kumar A (1991) Transient analytical investigations on a single basin solar still with water flow in the basin. Energy Convers Manag 31:27–38

## Author information

Correspondence to Abd El-Monem Khallaf.

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Responsible editor: Philippe Garrigues

## Appendix A1

### Appendix A1

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

### Appendix A2

Values of the coefficients in Eq. 24 are

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

and

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

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