Modeling Solar Energy Transfer through Roof Material in Africa Sub-Saharan Regions

Abstract

As a result of the global warming, the atmospheric temperature in sub-Saharan regions of Africa may drastically increase, thus worsening the poor living conditions already experienced by people in those regions. Roof’s thermal insulation capacity may play key role in reducing indoor thermal comfort cost. In the present study, effort is put to model heat transfer through roofs in south Saharan regions. Validation of the model was achieved using the slightly sloppy galvanized aluminum-iron sheet roof. Atmospheric data were hourly measured during April and June in Ouagadougou, Burkina Faso. Solar energy values increase from W/m2 in the morning to a maximum of W/m2 in the early afternoon. Ambient temperature follows the same trend as solar radiation with a maximum at °C. Wind speed varies from 0.5 to m/s. The measured roof inner wall temperatures agreed excellently with the developed model with a Nash-Sutcliffe Coefficient of Efficiency of 0.988. Energy flux entering the room through the roof varies from W/m2 earlier in the morning to a maximum of W/m2 in the earlier afternoon. These results shall help to better design human habitat under changing climate conditions in the sub-Saharan regions.

Full text

Hindawi Publishing Corporation
ISRN Renewable Energy
Volume , Article ID , pages
http://dx.doi.org/.//
Research Article
Modeling Solar Energy Transfer through Roof Material in
Africa Sub-Saharan Regions
Julien G. Adounkpe,1A. Emmanuel Lawin,2Clément Ahouannou,3
Rufin Offin Lié Akiyo,4and Brice A. Sinsin1
1Laboratory of Applied Ecology, Faculty of Agronomic Sciences, University of Abomey Calavi, 03 BP 3908 Cotonou, Benin
2Laboratory of Applied Hydrology, Faculty of Sciences and Technologies, University of Abomey Calavi UAC, BP 526, Benin
3Laboratory of Applied Energetic and Mechanics Ecole Polytechnique d’Abomey Calavi, University of Abomey Calavi,
03P 1175 Cotonou, Benin
4Department of Geography, University of Parakou, BP 123, Benin
Correspondence should be addressed to Julien G. Adounkpe; julvictoire@yahoo.com
Received  July ; Accepted  August 
Academic Editors: F. E. Little and M. Souliotis
Copyright ©  Julien G. Adounkpe et al. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
As a result of the global warming, the atmospheric temperature in sub-Saharan regions of Africa may drastically increase, thus
worsening the poor living conditions already experienced by people in those regions. Roof’s thermal insulation capacity may
play key role in reducing indoor thermal comfort cost. In the present study, eort is put to model heat transfer through roofs
in south Saharan regions. Validation of the model was achieved using the slightly sloppy galvanized aluminum-iron sheet roof.
Atmospheric data were hourly measured during April and June in Ouagadougou, Burkina Faso. Solar energy values increase from
24.50± 0.50W/m2in the morning to a maximum of 900.1±0.8W/m2in the early aernoon. Ambient temperature follows the
same trend as solar radiation with a maximum at 40.0±0.2∘C. Wind speed varies from . to 4.0±0.1m/s. e measured roof inner
wall temperatures agreed excellently with the developed model with a Nash-Sutclie Coecient of Eciency of .. Energy ux
entering the room through the roof varies from 63.1±0.3W/m2earlier in the morning to a maximum of 115.3±0.5W/m2in the
earlier aernoon. ese results shall help to better design human habitat under changing climate conditions in the sub-Saharan
regions.
1. Introduction
Ever since the appearance of human beings on earth, beside
food and other basic needs, shelters, or dwelling places have
been of major preoccupation. Human beings have set up
their homes utilizing materials from the nature. To protect
themselves from rain, heat, wind, cold, snows, or any sort
of enemies, human beings have invented, at very early ages,
habitat which has evolved from the caves, natural physical
dwellings to the modern houses known today []. One of the
most important elements of a house is its roof. Indeed, in
sub-Saharancountriesandmostcountriesontheearth,roof
must stand rainy seasons and during periods of elevated heat
must provide certain comfort []. e eectiveness of the roof
in terms of comfort and sustainability requires the thermal
insulation capacity and the mechanical strength of materials
employed.
e shapes of roof are adapted to the type of climate
of a given region [].Overtheworld,therearemorethan
 types of roof []. Moreover, many factors contribute to
the dierences among the types of roof. e most important
factors are the technology, the materials, the environment,
and the mere habit []. For instance, plat roof dominates in
dry regions, while cone shape roof dominates in semi-dry
regions such as some parts of Africa. Short eaves gable roof
type is widely observed both in Europe and North America,
whiledeepeavesaredominantinMassonAsia[].
Over ages, as human shelters, roofs have evolved from
what was then called traditional roong to what is known
nowadays to be the modern roong. In most sub-Saharan

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countries, traditional roong is still at use most importantly
in rural areas.
Traditional roofs made of indigenous or local materials
such as woods for stands or frames and straws (Figure ),
leaves or herbs, and so forth, for covertures, have thermal
isolation properties. eir main drawback is their relatively
short life which forces the users to rebuild basically almost
aer every rainy season [].
Modern roofs are made of metal sheets (Figure ), rein-
forced concrete, and so forth. Iron, aluminum, and zinc, most
oen protected against wear are used as roof coverings and
have mechanical resistances characterized by their relatively
long life. Galvanized iron sheet nds use in developing world,
basically where high rainfall is registered annually []. e
main handicap of the modern roong is their low isolation
capacity.
Inthelightofclimatechangeandallitsadverseeectsand
givingthefactthatincreasingnumberofhousesareadopting
the modern roong along with high thermal comfort cost,
combining traditional roong and modern roong at one
hand and modifying modern existing roofs on the other hand
to get indoor comfort at an aordable cost are of interest
[].
Green or vegetative roofs are in use in several European
and American countries []. Several problems, notably the
thermal isolation leading to the reduced energy consumption
and energy conservation [] and noise reduction, are solved
by green roong []. However, those benets linked to green
roofs are more signicant for extremely compact tropical
cities with severe shortage of ground-level green spaces and
intense Urban Heat Island eects [,].
Moreover, green roong is not much in use in south
Saharan regions for several reasons. First, those regions have
no space problem, and there are few compact cities where
such a technique could be deployed. Second, the construction
and the maintenance costs of those green roong are way
beyond what people in those regions could bear.
Climate change is a challenging situation to human
habitat as far as their thermal comfort in houses. In a
dry and hot climate of Mexicali in Mexico, a city of same
climate characteristics as Ougagougou where the present
study was conducted, a eld study was done, consisting on
the determination of electrical energy consumption of low
income dwellers and their perception of thermal comfort in
accordance with the design and the building material [].
Sustainable buildings have been designed by architects for
the improvement of indoor environmental quality, basically
including the provision of comfortable temperatures and
humidities. e users’ perceptions of thermal comfort had
been surveyed in  sustainable commercial and institutional
buildings in  countries []. e focus of those studies
among others was the overall thermal comfort in buildings.
However, few studies deal with the specic thermal contribu-
tion of the roof to the house. e possibility of roof light for
indoor comfort was explored in order to design an innovative
roongsystemforthetropicswithplentyofsunshineand
high humidity [].
In recent years, great interests have been noticed in
the eld of solar radiation modeling. Models have been
F : Traditional roong in South Sahara Africa. e roof cover
is made of straws and needs to be renewed aer each rainy season.
F : Typical modern roof type in Sub-Sahara Africa. Most of
theroofsaremadeofmetalsheets,basicallyironsheetwithslight
slope and oriented south.
developedtoevaluatebothglobalanddiusesolarradiations.
Employing a sensor FLA-GS that gives direct current
proportional to global radiation, the global solar radiance was
evaluated []. In , an estimation of diuse solar radiation
on horizontal plane was performed in three cities of Nigeria
in west Africa. Correlation between the clearance index and
the diuse to global solar radiation ratio was employed in
the estimation of the diuse radiation. e model eciency
is proven worldwide according to the authors []. Further-
more, real time monitoring of global solar radiation was
achieved designing a model of TCP/IP with an embedded
internet based data acquisition system []. Even though
scientists in the eld of solar energy have developed models
to monitor solar radiation, in the literature, to the best of our
knowledge, it was not possible to see eort related to roof
modeling linked to solar energy in the south Saharan region
of Africa.
e present study aims at modeling heat transfer through
roofs in sub-Saharan region in order to predict the roof
contribution to the overall heat of the room and to select
roof materials that lower comfort cost in those regions where
about%oftheroomoverallheatcomesfromtheroof[].
2. The Roof Thermal Model
As it is known, a model is meant to be the simplest way of
looking at a complex phenomenon. e conceptualization of

ISRN Renewable Energy 
e roof thermal model
Solar
radiation
Sun (E)
eConduction
Radiation (T
s)
Convection (T
a)
T
pe
T
pi
T
r
F : Model of heat ux exchanged between the roof and its
surroundings, where =thickofthesheetorroof,r=temperature
of the room, a= ambient temperature, s=temperatureofthe
sky, pe = the roof outer wall temperature, pi = the roof inner wall
temperature, and = solar radiation.
the heat ux through the roof of a house can be expressed as
follows (Figure ):
=incoming ux −out going ux
=
pe −pi, ()
where:
is the heat ux through the roof,
, the thermal conductivity of the sheet (roof ),
is the thick of the sheet,
pe and pi the external and internal wall temperature,
respectively.
In fact, pe and pi are obtained through studious math-
ematical calculations that require an energy balance to be
established at the interfaces atmosphere-roof and roof-room.
A roof exposed to the solar radiation is the center of
three modes of heat transfer: conduction, convection, and
radiation.
When there is between two bodies a temperature gra-
dient, heat travels from the hotter to the colder; thus the
dierence tends to resolve spontaneously. Essentially trans-
fers between two bodies implement three distinct processes,
simultaneous or not: conduction, convection, and radiation.
2.1. Conduction. Two bodies at dierent temperatures
brought into contact exchange heat by conduction where the
heat ows from the hottest to the coldest. In the case of the
present study, heat ows within the roof body from the outer
wall to the inner wall of the roof due to the temperature
gradient between the external and the internal walls of the
roof.
2.2. Convection. Whenabodyandauid(water,air,etc.),at
dierent temperatures, are in contact, there is heat exchange
by forced convection (when the speed of circulation of the
uid is imposed) or natural (otherwise).
2.3. Radiation. Any living or inanimate object whose temper-
ature is above absolute zero (−.∘C) emits electromag-
netic radiations that carry a certain amount of energy. Two
remoteobjectsexchangeheatbyemissionorradiation.e
sun is object of such a radiation towards the earth and the
other planets.
A roof exposed to the solar radiation is the center of
thermal exchanges characterized by the following thermal
coecients.
(i) Coecient of ermal Conductivity .It represents the
amount of heat which passes through the body per meter in a
second for a temperature dierence of ∘C.ItsunitisW/m/
∘C.
A material will be more heat conductor as its coecient
of thermal conductivity is high.
(ii) Absorption Coecient po.It represents the fraction of
solar radiation absorbed by the roof. po is unitless.
(iii) Emissivity. e roof has the property of emitting part
ofradiationreceivedtowardstheskyandtowardstheroom
characterized, respectively, by emissivity coecients 𝑒and 𝑖.
ese coecients characterize the radiation in the infrared.
(iv) Coecient of Convection h. It characterizes the heat
exchange and represents the ow per unit area of contact
and degree (W/m2/∘C). 𝑖for internal convection, and 𝑒for
external convection.
2.4. Adjacent Fluids to the Roof. At both sides of the roof,
there is heat exchange between the roof and the room air and
the atmospheric air characterized by physical parameters (𝑒,
pe,𝑒,𝑒), and a ow velocity.
3. Energetic Balance of the Roof
e roof exchanges heat with its surroundings as it receives
solar radiation .
(i) At the Roof External Wall. Consider
abs =𝑒⋅,
lost,conv 𝑒=𝑒⋅pe −a,
lost,rad 𝑒=⋅𝑒⋅4
pe −4
S.
()
(ii) Heat Recovered from the Roof (Material) by Conduction.
Consider
cond =
⋅pe −pi.()
(iii) At the Roof Internal Wall. Consider
lost,conv 𝑖=𝑖⋅pi −room,
lost,rad 𝑖= ⋅𝑖⋅⋅4
pi −4
room. ()

ISRN Renewable Energy
(iv) Heat Balance on the Roof. One has
𝑒=abs =lost,conv 𝑒+lost,rad 𝑒+cond,()
𝑒⋅=𝑒⋅pe −a+⋅𝑒⋅4
pe −4
S
+
⋅pe −pi, ()
𝑠=recover room =𝑔+lost,conv 𝑖+lost,rad 𝑖. ()
Knowing that 𝑔=,
recover room =cond =lost,conv 𝑖+lost,rad 𝑖,()

⋅pe −pi=𝑖⋅pi −ro om+⋅𝑖⋅4
pi −4
room.
()
Combining ()and() leads to the system of two
equations where the unknowns are pe and pi;bothare
raised to power  and :
𝑒⋅=𝑒⋅pe −a+⋅𝑒⋅4
pe −4
S
+
⋅pe −pi,

⋅pe −pi=𝑖⋅pi −ro om
+⋅𝑖⋅4
pe −4
room.
()
3.1. Modeling 𝑒.𝑒is the external convection coecient
between the outside ambient air and the roof. We made
the assumption that this exchange is governed by forced
convection because for simplicity, the external wind speed
is maintained constant for a dened time slot. us the
followingformulaswereusedtoevaluate
𝑒=5.7+3.8 for ≤4m/s,
𝑒=7.50.8 for >4m/s,()
where is the velocity of the wind.
3.2. Modeling 𝑖.e roof internal side exchanges heat with
the room by natural convection with air inside the room with
hi which is the convection coecient given by
𝑖=Nu ⋅𝑒
,()
where Nu is Nusselt number, is length of the roof and 𝑒is
thermal conductivity of the air. Out of these three parameters,
only canbeeasilyknown.eNusseltnumbervalueisgiven
by the correlation:
Nu =(Gr ⋅Pr)𝑚,()
where Gr is Grashof number, Pr is Prandtl number, and 
and are constants. Pr and Gr given for air are, respectively,
Pr ≈0.7, and Grashof number can be calculed by
Gr =2
𝑒3
2
𝑒,()
where istheinverseoftheaveragetemperatureoftheair
lm, is temperature dierence between the air lm and
the roof internal wall, 𝑒is thermal conductivity of the air, 𝑒
is density of air lm (Kg⋅m−3), 𝑒is dynamic viscosity of the
air (N⋅m−2⋅S−1), and =.m/s
2is gravity intensity.
𝑒,𝑒,and𝑒aretabulatedvaluesoftheairfora
given air temperature lm work. For the convenience of the
numerization, the air characteristics were linearized, and the
relationshipsfoundaredirectlyusedintheprogram.
3.3. Modeling 𝑒,𝑒,and𝑒.Here we give the linear relation-
ships obtained aer linear regression in regression domains
quite correct ( → 1,−1,−0.99,0.99) (in Kg/m3):
𝑒
=




18.25×10−4 +81.5×10−6; for 200≤<300
35.00×10−4 +75.8×10−6; for 300≤<350
46.90×10−4 +72.4×10−6; for 350≤≤400,
𝑒
=




2,9300−59,10×10−4; for 200≤<300
2,2583−35,88×10−4; for 300≤<350
1,8058−23,08×10−4; for 350≤≤400,
𝑒
=




29.81×10−7 +51.73×10−9; for 200≤<300
47.34×10−7 +45.76×10−9; for 300≤<350
59.80×10−7 +42.2×10−9; for 350≤≤400,
()
where (inK)isthetemperatureoftheairlmestimatedat
=
r+10during day time and =
r−5at night.
3.4. Modeling s, the Temperature of the Sky. Estimation of
thetemperatureoftheskycanbeachievedthroughthree
dierent formulae depending on the atmospheric conditions:
s=
a−12 or s=
a1/4.()
is emissivity of the sky which can take two forms:
=1−0.161⋅exp −0.00077a−2732
or =0.787−0.764ln v
273, ()
where vis the vapor temperature.
Ofthosethreeformulas,thesecondonewasfoundtot
the best the overall model taking into account the level of

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error acceptability. Hence, to evaluate the temperature of the
sky the present model uses the following sequation:
s= 1−0.161⋅exp −0.00077a−2732a1/4.()
4. Method and Material
4.1. Study Area
4.1.1. Climate. e study was conducted near the University
of Ougadougou, Ouagadougou, Burkina Faso, a city known
to be located in the Soudano-Sahelian zone. e climate is
characterized by a unimodal precipitation regime, with a
rainfall of about  mm per year. Rainfall events are mainly
generated by the dynamic of the West African Monsoon
(WAM). e rainy season stretches from mid-May to mid-
October, with an average temperature of ∘C. But the key
periodisknowntobeJune-July-August-September(JJAS)
which totalized more than % of the seasonal rainfall. e
cold season runs from December to January, with a minimum
temperature of ∘C. e maximum temperature during the
hot season, which runs from March to May, can reach ∘C.
e harmattan, a dry and cold wind, from late November
to mid-February, blows north-east with a high temperature
gradient between daytime and night that can reach. Relative
air humidity varies from % in March during dry season to
% in August during rainy season with a mean of % [].
is provides thermal comfort even with high temperatures.
4.1.2. Study Sites. e study was undertaken at a site located
inside the campus of the University of Ouagadougou where
the study materials were installed. e geographic character-
istics of the site are as follows: ∘󸀠.󸀠󸀠N, ∘󸀠.󸀠󸀠O,
and elevation  m.
4.2. Material. e experimental boxes are made of wood on
top of which galvanized iron sheets cover is placed to serve
as the roof. e boxes were put at a height of m above the
ground at points where no inuence from building shade,
trees, or other can aect the measurements.
4.3. Solar Radiation Measurement. For the solar radiation
measurement, several equipments are available depending on
the type of solar radiation of interest, either global, diuse,
or direct. Global solar radiation is the sum of the direct and
the diuse. e versatile CM- pyranometer was used to
measure the global radiation in Finland, Estonia, Poland, and
Sweden, the pyranometer PP- for the diuse one, and the
actinometer AT- for the direct solar radiation in Latvia
and Lithuania []. ose are very high technology that
was not available to us at the moment of the experimenta-
tion.However,weusedaCMP-pyranometerthatfullsthe
requirements of IEC  and IEC -X for the accurate
measurement of irradiance for photovoltaic and thermal solar
devices. at pyranometer was used as reference instruments
to test and certify PV cells for power plant projects. It was
proven very accurately and ecient. at instrument was
placedclosetotheboxatthesameheightfromtheground.
T : Solar radiation SR (W/m), wind speed V(m/s) and
ambient temperature are recorded on an hourly basis.
Time Solar
radiation (E)
Wind speed
(Vm/s)
Ambient
temperature
(a∘C)
: . . .
: . . .
: . . .
: . . .
: . . .
: . . .
: . . .
: . . .
: . . .
: . . .
: . . .
: . . .
4.4. Temperature Gauges. Temperature gauges were posi-
tioned everywhere needed: the ambient temperature, the
room temperature, the external and internal roof wall tem-
peratures,andsoforth.eambienttemperatureismeasured
ataheightofmabovetheground.egaugeswere
electrically connected to the temperature sensor that gives
a simultaneous read-out of all temperatures with decimal
absolute error.
4.5. Wind Speed Measurement. Two mai n t ypes of ins t r u-
ments that measure wind speed are the rotating cup
anemometer and the propeller anemometer. Both types of
anemometers consist of two subassemblies, the sensor and
the transducer. e sensor is the device that rotates by the
forceofthewind.etransduceristhedevicethatgenerates
the signal suitable for recording. Most of the time, the signal
needs to be conditioned by signal conditioner and displayed
by loggers and recorder. In the present study, the rotating cup
anemometer was employed.
5. Results and Discussion
e atmospheric parameters recorded were solar irradiation,
wind speed, and ambient temperature. ose parameters
were recorded hourly during April and June, on clear sky
days. A typical set of measurements is reported in Tab l e  .
It is very important to understand that the measurements
were done on a regular basis of one hour interval for  hours
aday.edatashowninTa b l e  and the following are just an
extraction of the one for which solar energy was not null, and
that the solar inuence can be observed on the temperatures.
In fact from  p.m. to  a.m., solar radiation was zero.
e bands between ∘and ∘north and south around
the earth receive the greatest amount of solar energy [],
but the equatorial belt between ∘Nand
∘Slatitudeisthe
most irradiated. Burkina Faso, Mali, Niger, and the northern
part of Benin, Togo, and Nigeria in West Africa are located in

ISRN Renewable Energy
T : Ambient temperature (a), solar irradiance (SR), wind
speed (V) and the galvanized ion sheet roof room temperature (r),
the roof inner wall temperature measured (pi m), and simulated
(pis) and energy ux entering the room were reported.
Time
(H) a
(∘C)
SR
(W/m)
V
(m/s)
Al-Fe
rpim pis Flux
: . . . . . . .
: . . . . . . .
: . . . . . . .
: . . . . . . .
: . . . . . . .
: . . . . . . .
: . . . . . . .
: . . . . . . .
: . . . . . . .
: . . . . . . .
20
25
30
35
40
45
0
100
200
300
400
500
600
700
800
900
1000
Time (hour)
Solar direct radiation and ambient temperature versus time
Solar radiation
Ambient temperature
Ambient temperature (∘C)
07:00
08:00
09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
Solar direct radiation (W/m2)
F : Solar direct radiation (W/m2) and ambient temperature
(∘C) recorded at a typical clear sky day of April.
these regions []. Consequently, those countries are subject
to important solar irradiance and thus have high solar energy
potentials.
e solar radiation variation observed and registered in
the present study is in accordance with the one observed
in various cities of Nigeria []andelsewhere[–], even
though solar irradiance is latitude dependent. At higher
latitudes >than ∘, high radiations can be observed as proof
of high potential solar energy availability for various solar
energy applications. For example, a -year cumulative annual
global solar radiation was found to be  Kwh/m2/day
in Spain (latitude ∘N, .∘W) and  Kwh/m2/day in
Malta (latitude ∘N, .∘E)whereJuneandJulyarethe
months that receive the greater sunshine [], giving the
proof that solar radiation decreases with the latitude and
determines the regional houses architecture []. Radiation
data recorded at  sites around the central part of the
Baltic Sea (latitude varies from ∘N, ∘Eto
∘N, ∘E)
from  to  showed June to be the month of highest
solar radiation in most part of the region with an average
daily total of . MJ/m2at Visby (.∘N, .∘E) [].
However, hourly measurements literature data are the ones to
compare those of the present study with. Hourly monitoring
of ambient temperature, solar global or direct radiation is
an important mean for engineers and building designers to
select construction materials that correspond to the specic
climate. Such reports are rather scarce, while yearly, monthly,
and daily mean values are the most reported in the literature
[,,]. In a study conducted in Ajaccio France, ambient
temperature, solar irradiance, wind speed and direction, and
so forth were recorded every minute. A perfect relationship
between the instantaneous increases of ambient temperature
and solar irradiance is observed (Tabl e  ). Both curves,
ambient temperature and solar irradiance, well shaped in
a Gaussian like form present their respective maximum at
about :p.m. [] unlike their counterparts of the present
study, where there is a gap of about  hours between the
maximum of the solar irradiance (:p.m.) and the ambient
temperature (:p.m.). is can be explained by the presence
of dust (ne particles) in the air hindering the ambient
airtocooldownasthesunissetting,causingadelayof
approximately  hours between the solar irradiance and the
ambient temperature.
Hourly mean temperatures of modern roofs are reported
[]. However, the solar irradiance on which depends the
temperature behavior was not reported. Interesting enough,
theGaussianshapewasobservedandthemaximumindoor
temperatures for various roofs were reached by  p.m. by
some roofs and  p.m. by others. ese results are in line with
what we obtained for the validation of the model employing
thegalvanizedironsheet(Figure ). e room temperature
increases from .∘C at : a.m. to reach a maximum
value of .∘C by : p.m. and decreases to .∘Cby
: p.m. having the same trend as the solar irradiance which
increases from  W/m2by : a.m. to reach a maximum of
. W/m2by noon and decreases to  W/m2.e-hours
gap between the maximums of the ambient temperature, and
the solar irradiance was observed. However, this gap was not
observed between the room temperature, the roof inner wall
temperature and the solar irradiance. A temperature gradient
of about ∘C was observed between the room temperature
and the roof inner wall temperature, giving the proof that
a substantial gradient of temperature can be achieved if
subsequent ceiling was made in buildings.
Asimilarstudyconducted[] shows similar relation-
ships between solar irradiance and ambient temperature
where the maximal solar irradiance of  W/m2attained
at : p.m. shi with the ambient temperature maximum
of .∘C observed at : p.m. In contrary, while the room
temperature in the present study is quite dierent from the
ambient temperature, the one from []observedwassimilar
to the ambient temperature. e dierence comes from the
ventilation of the room, while in the present, no ventilation

ISRN Renewable Energy 
Simulated temperatures versus measured temperature
30
35
40
45
50
55
60
65
70
75
80
08:30
09:30
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
Time (hour)
Temperature of the room
Measured roof inner wall temperature
Simulated roof inner wall temperature
Temperatures (∘C)
F : is graph shows a perfect agreement between simulated
roof inner wall temperatures in green (pis ) and measured roof inner
wall temperatures in red (pim ); the temperature of the room is in
blue.
is done. As it is known, ventilation reduces temperature. is
explains the observed phenomenon.
More importantly, the measured and simulated tem-
peratures of the roof inner temperature were compared.
Figure  not only shows a gap of ∘Cmaximumbetweenthe
rooftemperatureandtheroomtemperaturebutalsomore
importantly the similarity between simulated and measured
roof inner wall temperatures.
5.1. Model Calibration and Verication for Validation. A
model is always a simplication of the complex reality of
natural phenomena. is simplication has the merit of both
representing the reality in all its extent and keeping acces-
sibility and understandings to those not very well acknowl-
edgeable of the given eld. However, the constructed model
may not be capable of producing a response that is entirely
in line with the behavior of the observed phenomenon [].
erefore a model, aer being elaborated, needs to be tested
with the whole set of data that are available is in short the
model needs to be calibrated. e calibration of a model
requires that the overall complexity of the mechanism under
investigation to be taken into account [].
For the calibration of the present model, a suitable range
of error acceptability has been adopted, aer several trials.
e linearization of the tabulated temperatures to ll in the
sowaremadeforthemodel,thetemperatureoftheair
lm, the sky temperature, and other parameters in a repeated
iterative process were achieved. Even the parameters assumed
to be invariant were changed to see how well the model
adjusts to them [].
e verication procedure tests the entire reality of the
phenomenon. e nal step is the validation which entails a
broadersetofdataandpossiblyallowscomparisonwithother
models dealing with the same observations. Model validation
is related to the process of decision making. us the current
model has been validated and can serve as a successful tool
in the hands of experienced acknowledgeable persons in the
eld.
e qualitative model performance assessment is
achieved employing some statistical criteria to compare the
data from the measurements and the one produced by the
model. ese statistical criteria are considered objective and
provideunbiasedindicatorsofthemodelperformance[].
e lower the Mean Absolute Error or the Bias Percentage
value, the better the performance of the model. However,
theRootMeanSquareError(RMSE)whichmeasuresthe
scatteroftheresidualsortheRelativeRootMeanSquare
Error which is the normalized RMSE is more expressive than
the two others. In our study, the RMSE is calculated using
the following formula:
RMSE =
∑𝑁
𝑖=1 mod
𝑖−meas
𝑖2


1/2.()
eRMSEvaluecomputedshowstheclosenessofthemodel
to experimental data.
Furthermore,theNash-SutclieCoecientofEciency
(NSE) dened as follows:
=1−∑𝑛
𝑖=1 obs,𝑖 −model2
∑𝑛
𝑖=1 obs,𝑖 −obs 2,()
where obs is observed values and model is modeled values
at time , is generally used to quantitatively describe the
accuracy of model outputs. e closer the model eciency
is to , the more accurate the model is. e computed NSE
value for the present model eciency (.) conrms that
the theoretical formulation proposed here for the roof inner
temperature is realistic.
6. Conclusion
epresentstudyhasthemeritofestablishing,forthe
rst time in West Africa, relationships between direct solar
radiation at ground level and atmospheric temperature and
between atmospheric temperature and indoor temperature.
e model that is developed will be of great interest to
tropical buildings designers, climate engineers, and anyone
interested in indoor comfort cost-benet analysis, in the
line of global warming. Further studies are underway to test
various materials used as roof covers in sub-Saharan region
of Africa.
Acknowledgments
e authors are thankful to la Cooperation Francaise for
the partial support to this study and the former Ecole Inter

ISRN Renewable Energy
Etats des Ingenieurs de l’Equipement Rural for providing the
comfortable working environment.
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