Research ArticleModeling the Incident Solar Radiation of the City of N’Djamena(Chad) by the Capderou Method

Mahamat Hassane Babikir ,1 Donatien Njomo,1 Mahamat Barka,2

Mahamoud Youssouf Khayal,2 Deli Goron,1 Venant Sorel Chara-Dackou,1

Tefouet Tiague Martial,1 Kamta Legue Daniel Romeo,1 Gram-shou Jean Paul ,1

and Nzadi Siwe Elie1

1Environmental Energy Technologies Laboratory (EETL), Department of Physics, Faculty of Science, University of Yaounde I,P.O. Box 812, Yaounde, Cameroon2Department of Physics, Faculty of Science, University of N’Djamena, P.O. Box 1027, N’Djamena, Chad

Correspondence should be addressed to Mahamat Hassane Babikir; hassanemahamat6@gmail.com

Received 6 September 2019; Accepted 14 December 2019; Published 8 January 2020

Guest Editor: Marco Rivera

Copyright © 2020MahamatHassaneBabikir et al. This is an open access article distributed under theCreativeCommonsAttributionLicense, which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

Chad is like many African countries with no meteorological station at the moment to measure solar radiation throughout thecountry. Thus, theoretical models are used to estimate incident solar radiation. These models are established in correlation form.Our objective was to present a model, which allows the determination of the solar component on two surfaces (horizontal andinclined). This model allowed us to determine, over time, the component of global, direct, and diffuse solar radiation over aperiod that will cover the different seasons of the year. The calculation is done according to Klein’s days over all the months of theyear. The hourly results of the global, direct, and diffuse radiation obtained for all the planes going from January to December aresatisfactory compared to the results of the other authors quoted in the literature, which give the maximum and minimum valuesvery close to theirs. These results allowed us to validate the applicability of this model in a climate other than the desert climate.

1. Introduction

The knowledge of temporal distribution of solar irradiation onthe ground and across the atmosphere is necessary for themodeling of local warming of the atmosphere as well as theplanning of energy production systems. Solar energy is abun-dant, free, and environmentally friendly [1]; unpredictableintermittence is the only drawback. Indeed, solar energy beingthe most important source of extraterrestrial energy controlsmost of the atmospheric phenomena that ultimately have animpact on the productivity on land (control of the flow of therivers, exchange of quantities of materials between the surfaceand atmosphere). Ground measurements provide accurateinformation on the estimation of the solar component neces-sary for the proper design of an energy conversion system [2].The parameters of the atmosphere and solar irradiation alsoplay a very important role in climate monitoring, environmen-

tal control, and human activities [3]. They can be obtained bydirect measurement [4], remote sensing or simulation [5], arti-ficial neural network [6–8], and satellite data [9, 10]. It is thislack of information on the solar energy potential of Chad thatmotivated this work. It is therefore necessary to carry out aninvestigation on it. Many research studies are being done todevelop empirical, theoretical, analytical, and statistical models[11]. These models are correlations depending on several fac-tors (latitude, atmospheric disorder, and seasonal rhythm ofthe year) that vary according to the localities. Numerousmethods have been proposed by different researchers to esti-mate solar radiation on any given plane; these models take intoaccount astronomical meteorological parameters in situ [12,13]. The work published by [14] used five models to estimatesolar radiation in the south Pakistan regions, based on meteo-rology and geographic parameters in situ. To evaluate their effi-cacy, the author calculated the average prediction errors and

HindawiInternational Journal of PhotoenergyVolume 2020, Article ID 6292147, 10 pageshttps://doi.org/10.1155/2020/6292147

showed that the efficacy of the model varies with respect to theregions. The works done by [15] used empirical models to esti-mate the solar component on an inclined plane of 11° (degree).They concluded that only Liu Jordan’s model is best suited forpredicting solar radiation on an inclined plane from their chosensites. In their work, [16] believes that the DYB (day of the year-based) Gaussian model is favorable compared to the othermodels of daily solar radiation; they then enumerate that thismodel is highly variable to the climatic conditions of the studyzone. Hence, [17] developed Pro-Energy algorithms based on achronologic series to predict sunshine errors during variablehours. Ref. [18] tested many theoretical models to estimate thesolar time potential of Nairobi (Kenya) for 10 years, and theresult showed that for most of the tested models that estimatedglobal daily monthly average radiation, the Akinoglu and Ecevitmodel had the best estimation. In Ref. [19], authors have evalu-ated 12 models of solar radiations based on the meteorologicdata, obtained from 21 meteorologic stations in China. Theauthors showed in their results a satisfied correlation with a sta-tistical value less than 0.01.

Authors like in Ref. [20] included seven statistical Tarp-ley models calculating horizontal global solar radiation on ahorizontal plane based on astronomical and meteorologicaldata from seven stations in Uruguay. The work publishedby Ref. [16] states that the McClear model accurately esti-mates global radiation. Researchers such as Ref. [21] used lin-ear regression models to estimate the monthly and annualhourly mean of the global solar radiation of the city ofTroyes-Barberey. Then, they compared the models with thedata measured over 3 years and finally showed that theregression models used to estimate the global radiation givesatisfactory results where the errors between the measuredand calculated values are almost negligible.

Some empirical models link the components of solar radi-ation to major meteorological parameters to estimate globalradiation, which has been the subject of much work in the lit-erature. Thus, in this work, Ref. [22] developed a model thatallows estimating the global irradiation on a horizontal planefrom the insolation duration and relative humidity. Ref. [23]developed a model taking into account six astronomical andmeteorological parameters to calculate global radiation.Another model was developed by Ref. [24] to estimate globalirradiation, but this one is based on measurements of dailymonthly averages, absolute humidity, and duration of insola-tion. Ref. [25] revealed that solar radiation could also be esti-mated by means of the correction factor without applyingmeteorological parameters on locations in which the longi-tude varies from 70 to 125. Artificial neuron network hasrecently attracted a lot of attention in the domains of solarenergy prediction. Based on this, Ref. [26] used two functionsto predict solar radiation on a year by subdividing a year intofour seasons. The author was able to show that BPNN andRBFNN network are less performant during a cloudy dayand efficient during variable meteorological conditions.

The knowledge of solar resource is a major challenge inChad due to the unavailability of meteorological station mea-surements to ensure the collection and popularization of mete-orological information in Chad. It becomes imperative to findalternatives for the determination of the solar climate of the city

of N’Djamena located in the Sahelian region with geographiccoordinates presented in Table 1. The approach advocated hereis to find a well-adapted model compatible with our region; thismodel will allow us to make a numerical modeling of the solarresource to estimate the temporal incident solar radiation com-ponent of the city of N’Djamena (Sahelian climate). In orderto carry out this investigation in a year, we choose the mostappropriate days of the month as recommended by S. A. Kleinpresented in Table 2, where we calculated the correspondingrelative daily insolation. The model chosen to estimate the inci-dent radiation is that of Capderou, which is widely used in Alge-ria (desert climate) [27, 28]. This model links solar radiationcomponents to major weather parameters, such as sunstrokeduration, and astronomical parameters, such as maximum day-light duration, sun declination, earth-to-sun distance variation,and solar irradiation at the limit of the atmosphere.

2. Site Location

The region of N’Djamena is located in the Sahelian climatewith a total area of 395 km2. The average value of the normaldirect component per hour available on any plan varies from5 to 6 kWh/m2 for the favorable month and from 2.5 to3 kWh/m2 for the unfavorable month according to a studymade on the analysis of estimation of the direct componentof solar radiation by Ref. [29]. Table 1 presents the geo-graphic coordinates of the town of N’Djamena.

The choice of the convenient day is made according tothe days of Klein’s year [30].

3. Calculation of Different Components ofNormal Direct Solar Radiation

The calculation of incident solar energy depends on the geo-metric position of the sun vector with respect to the surface.

Table 1: Geographic coordinates of the town of N’Djamena [21].

Positionstudied

Latitude(°)

Longitude(°)

Altitude(m)

Climate Albedo

N’Djamena 12.07 15.04 295 Sahelian 0.2

Table 2: Representative day of each month after Klein [22].

Months Number agenda, X, in the year Dated

January 17 17th January

February 47 16th February

March 75 16th March

April 105 15th April

May 135 15th May

June 62 11th June

July 198 17th July

August 228 16th August

September 258 15th September

October 288 15th October

November 318 14th November

December 344 10th December

2 International Journal of Photoenergy

The calculation procedure begins by estimating the positionof the sun in space and calculating azimuth angles and eleva-tion of the sun. These angles are used to calculate angles ofincidence on the study surface.

3.1. Calculation of Solar Declination. It is the angle formed bythe direction of the sun with the equatorial plane; it variesduring the year between −23.45° and +23.45°. This inclina-tion has as effect the presence of seasons and also is the causeof the longest or shorter hours of light within the seasons andis calculated at any day of the year.

d =180aπ

sin 0:369 sin n − 82ð Þ 360365

� ����

+ 2 sinn − 2ð Þπ180

360365

� �� ��π

180

� ���:

ð1Þ

3.2. Horizontal Coordinates. The sun is located with respectto the horizontal plane of the altitude by two angles, theheight, and azimuth of the sun.

3.3. Calculation of the Hour Angle. It is the angle formed bythe projection of the sun on the equatorial plane at a givenmoment and the projection of the sun on the same plane attrue midday. The hour angle of the sun increases approxi-mately 360° in 24 hours (about 15° per hour); it is measurednegatively in the morning and positively in the afternoon.

H =15π180

TSV − 12ð Þ� �

,

TSV = TL −DE +ET + 4λ

60,

ET = 9:87 sin2n2π180

� �−7:35n2π180

− 1:5 sinn2π180

� �,

ð2Þ

with

n2 =n − 81ð Þ360

365, DE = 1: ð3Þ

3.4. Calculation of the Height of the Sun. The height of thesun is the angle made by the direction of the sun withits projection on the horizontal plane of the place. Theheight evolves at each moment of the day according tothe following expression:

h = arcsin cos dð Þ cos φð Þ cos Hð Þ + sin φð Þ sin dð Þ½ �:ð4Þ

3.5. Calculation of the Angle of Incidence on any Plane.This is the angle between the direction of the sun andthe normal of the plane. This angle is determined by theknowledge of the incident ray cosine direction and thenormal in cylindrical coordinate.

β = 15,

ω =90 − βð Þπ180

,

α = 0,

cos ið Þ = sin αð Þ cos ωð Þ sin Hð Þ cos dð Þ + cos αð Þ× cos ωð Þ cos Hð Þ cos dð Þ sin φð Þ − sin dð Þ cos φð Þ½ �+ sin ωð Þ cos Hð Þ cos dð Þ cos φð Þ + sin dð Þ sin φð Þ:

ð5Þ

4. Variations of Solar Irradiation Incident onthe Ground

Solar radiation is the fraction of solar energy emitted in theform of electromagnetic energy passing through the earth’satmosphere; the latter attenuates part of this radiation byabsorption, reflection, or diffusion by particles in suspension,water vapor, ozone content, and atmospheric pressure ofsolar geometry. Thus, in general, the component of globalsolar radiation has been defined as the sum of these variouscomponents which are the component of direct solar radia-tion, the component of diffuse radiation, and the componentof reflected radiation.

IG = ID + Id + Ir: ð6Þ

4.1. Capderou Model. The Capderou model is based on anatmospheric disorder balance due to absorption and diffu-sion caused by the constituents of the atmosphere and canbe expressed by the cloud factor to determine the compo-nents of solar radiation on a surface. This factor can beobtained from the empirical formula developed by Capderouin 1987 in the Solar Atlas of Algeria in the case of a clear sky.

4.2. Calculation of the Link Atmospheric Factor. The atmo-sphere is the layer of air consisting mainly of O2, O3, H2O,and CO2 that surround the planet earth. Solar radiation alongits path to the surface is modified as a result of interactionswith atmospheric components. This disturbance factor char-acterizes the atmospheric disturbance due to water vapor,mist, fumes, and dust from a clear or slightly obscured sky.This factor makes it possible to take into account the atmo-spheric transmission as a function of the altitude of the pointand turbidity coefficient, related to the particular microcli-mate of the site resulting from its development. This factoris the sum of gaseous absorption disorder, aerosol diffusiondisorder, and aerosol scattering disorder.

4.2.1. Disorder due to Gas Absorption. This factor can bedefined as the atmospheric disorder that depends only onthe geoastronomical parameter of the atmosphere and thefixed constituents of the atmosphere: ozone and water vaporare determined from the following equation:

T0 = 2:4 − 0:9 sin φð Þ + 0:1 2 + sin φð Þð ÞAhe − 0:2z− 1:22 + 0:14Aheð Þ 1 − sin hð Þð Þ, ð7Þ

3International Journal of Photoenergy

where

Ahe = sin360365

� �N − 121ð Þ π

180

� �� �: ð8Þ

4.2.2. Turbidity due to Both Gas Absorption (O2, CO2, and O3)and Molecular Diffusion of Rayleigh. The calculation of thisparameter is given by

T1 = 0:89z: ð9Þ

4.2.3. Aerosol Diffusion Disorder Calculation. The aerosol dif-fusion disorder is determined from

T2 = 0:63z 0:9 + 0:4Aheð Þ: ð10Þ

The link trouble factor calculation is the sum of thesethree atmospheric components:

TL = T0 + T1 + T2: ð11Þ

4.2.4. Calculation of Direct Irradiation on a Horizontal Planeby Clear Sky. The direct solar radiation component repre-sents the solar flux that has directly touched a surface. Itdepends on the height of the sun and angle of exposure ofthe wall to the sun at the moment considered, but beforeentry into the Earth’s atmosphere, which is composed of amixture of dry air and clean (gas), water vapor, and aerosols,the latter attenuates. The following equations below are usedto calculate the direct solar radiation incident on any plane atground level [31]:

Idir = I0ψ sin hð Þ exp −TL

0:9 + 9:4/0:89zð Þ sin hð Þ� �

, ð12Þ

where

ψ = 1 + 0:033 cos360365

� �n − 3ð Þ π

180

� �� �, I0 = 1367

Wm2 :

ð13Þ

4.2.5. Calculation of Diffuse Irradiation on a HorizontalPlane by Clear Sky. A diffuse radiation comes from mul-tiple diffractions and reflections from all other directions.This fraction of diffuse radiation depends at everymoment on meteorological and astronomical phenomenaexpressed by

Idif = I0ψ exp −1 + 1:06 log sin hð Þ + a −ffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 + b2

ph i� �,

ð14Þ

where

a = 1:1, b = log T1 − T0ð Þ − 2:8 + 1:02 1 − sin hð Þð Þ2:ð15Þ

4.2.6. Calculation of the Global Irradiation on aHorizontal Planed Clear Sky. The global radiation on ahorizontal plane is the sum of two components on thesame horizontal plane given by

Ig = Idir + Idif : ð16Þ

5. Solar Radiation on an Inclined Plane

For any inclined surface at an angle of inclination i with thehorizontal plane and the azimuth angle counted from thesouth direction, the radiation is equal to the product of theradiation on the normal plane and the cosine of the incidentangle i.

5.1. Direct Irradiation on an Inclined Clear Sky. For aninclined surface at ground level, the direct irradiation isexpressed mathematically as

Iinc = Inor cos ið Þ, ð17Þ

where

Inor = I0ψ exp −0:9T1 +0:89z

9:4 sin hð Þ� �

: ð18Þ

5.2. Calculation of Diffuse Irradiation on a Clear Sky InclinedPlane. To calculate the diffuse radiation, we will use the equa-tions described below:

σd = I0ψ exp −2:48 sin hð Þ + a1 −ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 + 4b21

q� �, ð19Þ

where

a1 = 3:1 − 0:4b1,

b1 = log T1 − T0ð Þ − 2:28 − 0:5 log sin hð Þð Þ:ð20Þ

(i) The isotropic component corresponds to a sky of uni-form luminance:

σi = Idif − σd sin hð Þ ð21Þ

(ii) Component of the circle of the horizon comes from ahorizon band of a height of 6°; it seems associatedwith an accumulation of aerosols in the atmosphericlower layers:

σh = −0:02ψI0 exp sin hð Þð Þ

a22 + a2b2 + 1:8, ð22Þ

4 International Journal of Photoenergy

where

a2 = log T1 − T0ð Þ − 3:1 − log sin hð Þð Þ,b2 = exp 0:2 + 1:75 log sin hð Þ½ �ð Þ:

ð23Þ

5.3. Diffuse Irradiation of the Sky.

dcl = σd cos ið Þ + σi2

1 + sin ωð Þð Þ + σh cos ωð Þ: ð24Þ

5.4. Diffuse Soil Irradiation.

dsol =σs

2 1 + sin hð Þð Þ , ð25Þ

where

σs = ρIgh: ð26Þ

5.5. Backscattered.

dre = 0:9 ρ − 0:2ð ÞIgh exp −4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

T1 − T0p

!: ð27Þ

5.6. Calculation of Diffuse Irradiation Incident on anInclined Plane. The diffused component received on aninclined plane is divided into three components, the dif-fused sky, diffusion of the ground, and the backscattereddiffusion which is given by

Idin = dsol + dcl +dre2

1 + sin ωð Þð Þ: ð28Þ

5.7. Calculation of the Global Irradiation Incident on anInclined Plane. Global solar radiation is evaluated as thesum of the diffused and direct solar radiation with respectto any angle of incidence.

IGin = Iinc + Idin: ð29Þ

6. Results

The theoretical results of the model thus described arepresented in Figures 1(a)–1(l), according to the days ofKlein for the horizontal plane, and Figures 2(a)–2(l) forthe inclined plane representing the sunshine of the cityof N’Djamena. Each panel represents the daily variationof temporal distribution of the incidental illuminationson a plane; in order to distinguish the curves, we chosethe global in blue, the direct in green, and the diffusionin red. We noted that the amplitude variations are strongfor the months of March, April, and May that characterize

the dry season. Also noted is that the minimum values arein July and August, characterizing the rainy season. Allcurves have the same paces, and the maximum valuereached by the global, direct, and diffusion in a horizontalplane for the favorable month exceeds 1000 Wh/m2 forthe total, 900Wh/m2 for direct, and 200Wh/m2 for diffu-sion. The same observations were made for the inclinedsurface. Though Chad does not have a solar radiationmeasuring station, it is preferable to model the solar radi-ation incident on a sensor to estimate solar radiation insitu. To validate an estimation of the various componentsof solar radiation to the ground, it is necessary to comparethe calculated values with the measured values. In thisinvestigation, it is not the case because we do not havethese data, but we compared our theoretical results withthe results of Refs. [28, 32, 33] in Algeria using the samemodel. In their work, they compared the theoretical resultswith the experimental one and observed a relatively smallerror (about 5%). We also compared our results with thoseof Ref. [29] which are almost the same.

This work enabled us to compare our theoreticalresults of the model applied to our site with those ofthe aforementioned authors (theoretical and experimen-tal). We note that our results are almost similar togetherwith the observation made on the minimum amplitudeand maximum magnitudes for each month of the year.Thus, it can be deduced from this that the Capderoumodel can also be applied in a semidesert (Sahelian) cli-mate to estimate the solar radiation incident in the caseof a clear sky, giving appreciable results. Our work is, likethat of several other writers in the literature, a contribu-tion to the evaluation of the solar field of our country.The different models established by the researchers andthe close dependence of coefficients of these relationswith the climatic conditions specific to the localitieswhere these relations have been established make thechoice of the relevant relation to be used to evaluatethe solar field. This model is interesting to carry outthe study on the sizing of solar systems (thermal photo-voltaic). These results confirm those listed by the directorgeneral of SNE (National Electricity Company) at aninternational forum of renewable energies in N’Djamenain February 2012. According to the director, Chad hasenormous potential in solar energy; from north to south,respectively, the sun shines 2750 to 3250Wh/year whichgives on average 4 to 6 kWh/m2/day. This model can beused as a tool to help size the energy system for a coun-try like Chad which has enormous potential in solarenergy which is almost not exploited.

Figure 3 shows us the variation of the maximum ofglobal, direct, and diffuse irradiations for the considereddays in both Figures 1 and 2. In this figure, the dashedcurves are obtained when we consider a horizontal planeand the continuous curves are obtained when we consid-ered an inclined plane. Globally, we see that the irradia-tions for the horizontal plane are most important in themiddle of the year (April to September), while at thebeginning and at the end of the year, irradiations are moreimportant for the inclined plane.

5International Journal of Photoenergy

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Figure 1: Temporal distribution of incident illuminations on a horizontal plane.

6 International Journal of Photoenergy

7. Conclusion

The Capderou model presented in this article has made itpossible to evaluate the component of solar radiation(global, direct, and diffusion) on horizontal and inclinedplanes of solar irradiance in the city of N’Djamena. This

model is implemented by Capderou using meteorologicaland astronomical data often based on atmospheric distur-bances due to absorption by gases and diffusion andabsorption by solid particles (aerosols) and liquid (clouds).A comparison is made with the results of the other authors(theoretical, experimental, and satellite). Our theoretical

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Figure 2: Temporal distribution of incident shifts on an inclined plane.

7International Journal of Photoenergy

results of calculation thus obtained are satisfactory for allthe months of the year, and consequently, we can say thatthe model of Capderou can also be applied to a type ofsemidesert climate.

Nomenclature

d: Solar declinationH: Hour angleTSV: True solar timeTL: Local timeET: Correction of the equation of timeZ: Altitude of placeDE: Time difference from the meridianω: Hour angleT0: Disturbance factor due to gaseous absorption (ozone

and water vapor)

T2: Angstrom trouble factorIdr: Horizontal direct radiationdsl: Diffuse groundd: BackscatterIDin: Direct inclinedIdin: Inclined diffusionh: Height of the sunφ: Latitude of the locationi: Angle of incidenceβ: Plane inclination with respect to the horizontalα: Azimuth of the planeρ: Albedoψ: Earth-sun correctionTL: Link trouble factorT1: Disturbance factor due to absorption by atmospheric

gases (O2, CO2, O3, and Rayleigh scattering)Ig: Global horizontal radiation

Month

J F M A M J J A S O N D

Glo

bal i

rrad

iatio

n max

(W/m

2 )

750

800

850

900

950

1000

1050

1100

Inclined planeHorizontal plane

(a)

Month

J F M A M J J A S O N D

Dire

ct v

irrad

iatio

n max

(W/m

2 )

650

700

750

800

850

900

950

1000

Inclined planeHorizontal plane

(b)

Inclined planeHorizontal plane

Month

J F M A M J J A S O N D

Diff

use i

rrad

iatio

n max

(W/m

2 )

90

100

110

120

130

130

140

150

(c)

Figure 3: Maximum of global (in (a)), direct (in (b)), and diffuse (in (c)) irradiations during the year for both inclined and horizontal planes.

8 International Journal of Photoenergy

Idif : Horizontal diffuse radiationdcl: Diffusion from the skyIGin: Inclined globalI0: Solar constant.

Data Availability

The data we used are just geographique coordinates of thetown of N’Djamena: latitude, longitude, altitude, and albedoas found in Table 1. Based on these, we carried out our calcu-lations before doing numerical simulations. They are foundand accessible at the general direction meteorological centerof N’Djamena.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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