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This work focuses on modeling the impact of desert aerosols on a mini central solar photovoltaic (PV). Our studied physical model is comparable to a multilayer. We have described and discretized the mathematical equations which govern the physical model. Also, we analyzed the influence of the parameters τa and X on the solar radiation received at the surface of solar PV modules. The results of the study taken
from** Figures 6(a)-(d)** representing
the variations of the global solar radiation on the solstices and equinoxes as well as the 21 of the months of the year days understood show that: if τ
_{a }
= 0 and
X
= 0, I
_{C }
= 67.87%; if τ
_{a }
= 0.5 and
X
= 0.5, I
_{C }
= 21%; if τ
_{a }
= 0.8 and
X
= 0.8, I
_{C }
= 12% and if τ
_{a }
= 1.5 and
X
= 1.5 then I
_{C }
= 4%. These results show that desert aerosols significantly influence the global solar radiation received. Unfortunately, this influence lowers the productivity of the central solar PV in general.

The desert aerosols are soil particles suspended in the atmosphere in regions with easily degradable arid soils, sparse vegetation and strong winds [_{a} and X related to the state of the atmosphere and to the surface of the PV field. Many experimental and numerical works had been carried out on the subject. It is known, for example, that dust can cause attenuation of solar PV radiation of around 90% [

The physical model studied is represented in

The physical model studied includes the optical thickness of desert aerosols in the upper atmosphere (τ_{a}), the thickness of the various aerosol deposits (X) and finally a glass surface representing the solar PV field (S = 234 m^{2}). The principle consists in observing τ_{a} and X on the electrical production of the mini central solar PV. Thus, the optical properties for each layer are shown in

The solar radiation which encounters aerosol particles undergoes transformations either by absorption, diffusion and emission [

Thus, _{0} passes through layer 1 at an incident angle θ, of optical air mass m_{a} and optical depth of aerosols τ_{a}. Layer 1 and 2 respectively represent the suspension of desert aerosols in the atmosphere [

thickness on the PV solar field is X. Layer 2 is crossed by a solar intensity I_{0} at an incident angle θ.

We have established the mathematical equations that govern our studied physical model. Thus, the clear sky radiation reaching the surface of the Earth (normal to the rays) is given by a commonly used model by treating the attenuation as an exponential decay function. The transmittance due to this multilayer is given as follows:

F ( τ a , X ) = α = exp ( − τ a m a ) × exp ( − X ) (1)

Knowing the transmisttance (Equation (1)) caused by the suspension and deposits of desert aerosols, direct radiation can be written:

I B C = I 0 × cos θ × exp ( − m a τ a ) × exp ( − X ) (2)

The final equation for direct solar radiation in a multilayer is:

I B C = I 0 × [ A sin ( H S R ) + B × cos ( H S R ) + C ] × exp ( − m a τ a ) × exp ( − X ) (3)

with the following coefficients:

cos θ = A × sin ( H S R ) + B × cos ( H S R ) + C ;

A = cos δ × cos ( 90 − Σ ) × sin γ ;

B = cos δ ( cos γ × cos ( 90 − Σ ) × sin φ × sin ( 90 − Σ ) × cos φ ) ;

C = sin δ ( cos γ × cos ( 90 − Σ ) × cos φ + sin ( 90 − Σ ) × sin φ ) ;

γ = tan δ ( ( sin φ / sin Σ ) × tan Σ − cos φ / tan Σ ) ;

The expression of diffuse solar radiation is:

I D C = F d × I 0 × ( 1 + cos Σ / 2 ) × exp ( − m a τ a ) × exp ( − X ) (4)

The expression of reflected solar radiation is:

I R C = ρ × I 0 × ( sin β + F d ) ( 1 − cos Σ / 2 ) × exp ( − m a τ a ) × exp ( − X ) (5)

Finally, the global solar radiation is:

I C = I 0 × ( I d + I f + I r ) × exp ( m a τ a ) × exp ( − X ) (6)

with:

I 0 = C S [ 1 + 0.034 × cos ( 360 × n / 365 ) ] ;

m a = P / 101325 [ sin ( h s ) + 0.15 ( h s + 3.885 ) − 1 , 253 ] − 1 ;

P = 101325 × exp ( − 0.0001184 β ) ;

h s = cos φ × cos δ × cos ( H S R ) + sin φ × sin δ ;

I d = I 0 [ A × sin ( H S R ) + B × cos ( H S R ) + C ] ;

I f = F d × I 0 ( 1 + cos Σ / 2 ) ;

I r = ρ × I 0 ( sin β + F d ) ( 1 − cos Σ / 2 ) ;

I g = I 0 ( I d + I f + I r ) .

The Equations (3)-(6) are solved using the Finish Differences Method (FDM). The discretization of the equations is:

( I i , j t + 1 − I i , j t ) / Δ t = α ( I i + 1 , j + 1 t − I i + 1 , j − 1 t − I i − 1 , j + 1 t + I i − 1 , j − 1 t ) / 4 Δ x Δ y (7)

I i , j t + 1 = α Δ t ( I i + 1 , j + 1 t − I i + 1 , j − 1 t − I i − 1 , j + 1 t + I i − 1 , j − 1 t ) / 4 Δ x Δ y + I i , j t

I i , j t + 1 = [ α Δ t ( I i + 1 , j + 1 t − I i + 1 , j − 1 t − I i − 1 , j + 1 t + I i − 1 , j − 1 t ) + 4 Δ x Δ y I i , j t ] / 4 Δ x Δ y (8)

Direct solar radiation:

I B C | i , j t + 1 = [ α Δ t ( I d | i + 1 , j + 1 t − I d | i + 1 , j − 1 t − I d | i − 1 , j + 1 t + I d | i − 1 , j − 1 t ) + 4 Δ x Δ y I d | i , j t ] / 4 Δ x Δ y (9)

Diffuse solar radiation:

I D C | i , j t + 1 = [ α Δ t ( I f | i + 1 , j + 1 t − I f | i + 1 , j − 1 t − I f | i − 1 , j + 1 t + I f | i − 1 , j − 1 t ) + 4 Δ x Δ y I f | i , j t ] / 4 Δ x Δ y (10)

Reflected solar radiation:

I R C | i , j t + 1 = [ α Δ t ( I r | i + 1 , j + 1 t − I r | i + 1 , j − 1 t − I r | i − 1 , j + 1 t + I r | i − 1 , j − 1 t ) + 4 Δ x Δ y I r | i , j t ] / 4 Δ x Δ y (11)

Global solar radiation:

I C | i , j t + 1 = [ α Δ t ( I g | i + 1 , j + 1 t − I g | i + 1 , j − 1 t − I g | i − 1 , j + 1 t + I g | i − 1 , j − 1 t ) + 4 Δ x Δ y I g | i , j t ] / 4 Δ x Δ y (12)

We present the numerical results of the code developed on MATLAB [

evolution of solar radiation (incident, direct, diffuse reflected and global) on the solar PV field. When the solar radiation is about 1400 W/m^{2} on the solar field, we contact the color of the curves pulls dark red and when it is weak it tends towards 0, the color of the curves pulls towards dark blue.

The curves of _{0}) is a function of the incident solar radiation and the angle of incidence (θ). For θ = 0 ˚ , the incident solar radiation is equal to the extraterrestrial solar radiation. The incident solar radiation decreases as the angle of incidence increases and at θ = 90 ˚ the incident radiation is 0.

We present the variations of direct, diffuse and reflected solar radiation on the solar PV field. The curves of Figures 4(a)-(c) show that they depend on the parameters τ_{a} and X. We find that the solar radiation decreases when the main parameters increase by attenuating the extraterrestrial solar radiation. Indeed, if τ a = 0 μ m and X = 0 μ m this corresponds to a clear blue sky, the maximum direct solar radiation is 1000 W/m^{2}. When the parameters τ a > 0 μ m and X > 0 μ m , the direct solar radiation (I_{BC}) decreases and tends towards 0 (^{2} when τ a = 0 μ m and X = 0 μ m . If τ a = 1.5 μ m and X = 1.5 μ m we contact that the diffuse solar radiation (I_{DC}) decreases to 0 (_{RC}), if τ a = 0 μ m , X = 0 μ m , we notice that the reflected solar radiation varies and reaches a value of 140 W/m^{2}. The reflected solar radiation tends towards 0 if τ a = 1.5 μ m and X = 1.5 μ m (

^{2}.

The curves in Figures 6(a)-(d) represent the variations of the global solar radiation on the solstices and equinoxes as well as the days 21 of the months of the year, under the influence of the parameters τ_{a} and X. By varying τ_{a} and X with a thickness of: ( τ a = 0 , X = 0 ; τ a = 0.5 , X = 0.5 ; τ a = 0.8 , X = 0.8 and τ a = 1.5 , X = 1.5 ), we note that the global solar radiation (I_{C}) drops considerably from an average value of 935 W/m^{2}: for τ a = 0 , X = 0 (^{2} at τ a = 1.5 , X = 1.5 (_{a} and X have a negative impact on the global solar radiation of the mini central solar PV.

In summary, for the two parameters (τ_{a} and X) of our study, we give the results of the different solar radiations which cross the surface of the solar PV field. Based on the geographic data of the mini central solar PV site. We have an average daily solar time of 12.25 hours or an annual solar time of 4476 hours. The

Solar radiations | Extraterrestrial | Direct | Diffuse | Reflected | Global |
---|---|---|---|---|---|

W/m^{2} | 1400 | 1000 | 120 | 140 | 1260 |

Finally, the quantitative study shows that out of 100% (1400 W/m^{2}) of extraterrestrial solar radiation in the direction of the PV solar field, only 71% of this radiation is converted into direct solar radiation (I_{BC}), 10% into solar radiation reflected (I_{RC}), 9% in diffuse solar radiation (I_{DC}) and 10% of solar radiation are considered lost. Thus, global solar radiation is 90% of extraterrestrial solar radiation. According to a variable angle of incidence (θ) on the solstices and equinoxes of the year, the average global solar radiation on the solar PV field is: if τ a = 0 and X = 0 , I C = 67.87 % ; if τ a = 0.5 and X = 0.5 , I C = 21 % ; if τ a = 0.8 and X = 0.8 , I C = 12 % and if τ a = 1.5 and X = 1.5 then I C = 4 %

At the end of our study on the influence of desert aerosols on the mini central solar PV, we described the mathematical equations which govern the system of the studied physical model. Then, these equations are discretized using a Finite Difference Method and solved by the Gaussian algorithm. The numerical results show that if τ a = 0 and X = 0 , with a cloudless sky; we have a maximum global solar radiation at the surface of the PV solar field of 1260 W/m^{2}. This value of global solar radiation indicates a very significant improvement in the productivity of the mini central solar. And if τ a = 1.5 μ m and X = 1.5 μ m , the global solar radiation tends towards very low values (<10 W/m^{2}). This shows a low productivity of the mini central solar PV. Finally, the parameters τ a and X considerably influence the operation of the mini central solar PV. We showed that our numerical results on the desert aerosols can be to influence the central solar PV productivity in West Africa countries. From a theoretical point of view, this modeling research has enabled us to establish a new model for evaluating the impacts of desert aerosols on the radiation received by a solar PV system. It is a true model that solves all cases of issues that deal with mineral dust deposits and suspensions on solar PV systems. Our numerical method consisted of the desert aerosols on global solar radiation, another digital method could have consisted of showing the impact of gaseous aerosols on solar radiation.

The authors declare no conflicts of interest regarding the publication of this paper.

Ilboudo, W.D.A., Ouedraogo, I., Koumbem, W.N.D. and Kieno, P.F. (2021) Modeling the Impact of Desert Aerosols on the Solar Radiation of a Mini Solar Central Photovoltaic (PV). Energy and Power Engineering, 13, 261-271. https://doi.org/10.4236/epe.2021.137018