International Journal of Engineering Trends and Technology (IJETT) – Volume 26 Number 4- August 2015
Mathematical Modelling of Different Parameters of Solar
Umbrella
Sonal Chavan #1, Dr. R. J. Patil *2
#
Dr.D.Y.Patil Institute of Engineering and Technology, Pune, Maharashtra, India
Dr.D.Y.Patil Institute of Engineering and Technology, Pune, Maharashtra, India
*
Abstract— The performance of solar umbrella has
been studied. The household umbrella is used to serve
the dual purpose as providing shades and operating
various appliances, making outdoor experience more
enjoyable. One can use this umbrella to charge mobile
or tab while roaming outside during day time. Same
umbrella can be used as a table umbrella and can
able to operate various electrical appliances using
storage battery (12v) and MPS during night hours.
Here performance analysis is done on two types of
system. In first type solar umbrella is loaded with four
solar panels (6v, 200mA) and secondly with eight
solar panels (6v, 200mA).The effects of ambient
temperature, solar radiation and wind speed on
performance analysis of both the system have been
analysed in particular location called Khopoli, India.
Mathematical modelling of both systems has studied
by observing above parameters which affected the
performance of solar umbrella.
Keywords— Ambient Temperature, Mathematical
Modelling, Solar Radiation, Solar Panels, Solar
Umbrella, Wind Speed
I.
INTRODUCTION
Solar energy is radiant light and heat from
the sun harnessed using a range of ever- evolving
technologies such as solar heating, solar architecture
and artificial photosynthesis. To convert solar energy
into electricity solar panels are used. When light
shines on the solar cell it creates an electric field
across the layers. The stronger the sunshine, the more
electricity is produced. Group of cells are mounted
together in panels that can be mounted on solar
umbrella. In this paper efforts have been made to
present effective use of electricity which is obtained
from solar energy by using solar umbrella. Many
appliances are used to convert solar energy into
electricity. In the literature few works are referred to
solar umbrella which is provided with illumination,
lighting system and could able to operator various
appliances since the use of energy has become an
integral part of our life, its supply should be secure
and sustainable. The energy requirement of the world
is ever increasing. The increasing energy demands put
a lot of pressure on conventional energy sources.
Therefore there is need for alternative energy sources
which can provide us energy in a sustainable manner.
The obvious choice of a clean energy source, which is
abundant and could provide security for the future
development and growth, is the sun’s energy. Recently,
there has been an enormous increase in the
understating devices, which led to a rapid increase in
ISSN: 2231-5381
the power conversion efficiencies of such devices.
Solar cells vary under temperature changes which will
affect the power output of the cell. Solar Modules
work best in certain weather conditions, but since the
weather is always changing, most solar photovoltaic
modules do not operate under normal operating
conditions. The performance of a PV system depends
not only on its basic characteristics but also on the
environmental issues. The environmental issue like
ambient temperature, radiations plays an important
role in the photovoltaic conversion process.
In the present work authors used the big size
household umbrella which can be able to recharge
mobile or tab while roaming outside. Same umbrella
can be used as a small table umbrella. The paper is
focused on the performance analysis of solar umbrella
by studying mathematical modelling done on its
different parameters, at Khopoli in India.
II.
DESIGN CONCEPT OF SOLAR
UMBRELLA
Umbrella consists of 8 individual flaps which are
connected from one side to a octagon centre structure.
The dimension of each flap approximately 700 mm x
440mm and solar panel has dimension 165 mm x 70
mm, fixed on the upper side of flap. So that panel can
receive more rays coming from sun. On each flap of
umbrella solar panel is placed. Solar panels are
connected in series and output power is stored in 12v
battery. A single monocrystalline silicon solar panel
(5v) is attached separately. While roaming outside it is
possible to remove 8 solar panels system to avoid
carrying extra load. Only one solar panel (5v) is
attached and one can able to charge mobile or tab
using single solar panel power system. Figure 1 shows
the Experimental Setup of Solar umbrella and Figure 2
shows top view of Solar Umbrella.
Fig.1: Experimental Setup of Solar Umbrella
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International Journal of Engineering Trends and Technology (IJETT) – Volume 26 Number 4- August 2015
0.072 m2 and resistance of battery is taken as 0.021
ohm.
A sample observation is shown in Table I and Table II.
TABLE 1: Readings of Parameters of Solar Umbrella
for 4 panels system (Date- 23 May 2015)
Fig. 2: Solar Panels placed on Umbrella Canopy
III.
TESTING PROCEDURE AND
EXPERIMENTATION
The experimentation was carried in month of May
2015. This month covers summer in India. In the
present work two systems have been studied. In first
system solar umbrella is loaded with four solar panels.
Before starting the experimentation the battery should
be discharged completely so that first reading in the
voltage table should be 0V. Reading has been taken at
the interval of 4 minutes. Corresponding reading of 12
volt battery is taken. Other parameters like
Temperature, Solar Radiations and Wind Speed which
affected performance of solar panels are noted down.
Multimeter was used to measure voltage which has
range 0v to 400v. Wind speed is measured by battery
operated digital anemometer. It has range of 0 km/hr
to 90 km/hr. Same anemometer had a facility to show
ambient temperature. Solar power meter is used to
measure solar radiations. It has a range of 0 w/m2 to
2000 w/m2. In second system solar umbrella is loaded
with eight solar panels. Again output from 12V battery
is noted down. Like previous care should be taken to
discharge the battery before experimentation. In
experimentation we used two similar 12v batteries for
our convenience. Corresponding readings of other
parameters like temperature, radiations and wind
speed are taken. The readings were taken at different
time zones between 10 am to 4 pm for 15 days in
period 17/05/2015 to 31/05/2015. All readings were
taken at 4 minutes time interval. More than 800
observations were recorded. Every time care was
taken while adjusting the angle of solar panel. A
mathematical model on above solar systems is done
by using different parameters like Solar Radiation,
Area of Solar Panel, Time, Resistance, Temperature,
and Wind Speed. A special support is fabricated to
maintained vertical opened position of solar umbrella.
A Velcro material is pasted on umbrella by using
special adhesive. Solar panels are attached to this
Velcro material so as to maintain its fixed position.
Author could able to attach or detach the solar panels
system by using Velcro. Area of solar panel is taken as
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Sr.
no.
Time
Hour
volt
v
Temp.
0
C
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
10.00
10.04
10.08
10.12
10.16
10.20
10.24
10.28
10.32
10.36
10.40
10.44
10.48
10.52
10.56
11.00
11.04
11.08
11.12
11.16
11.20
0
1
2
3
4
6
6
7
7
7
8
8
8
9
9
9
10
10
10
10
11
37.0
37.3
37.5
37.5
38
38.5
38.5
39
39.1
39.4
39.6
39.9
40
40
40.2
40.2
40.3
40.3
40.5
40.5
40.5
Wind
speed
Km/hr
3.1
3.5
4.5
3.5
2.1
2.9
5.2
6.9
7.2
5.2
3.9
2.1
2.3
2.9
3.2
3.1
3.5
3.9
4.2
4.5
4.7
Radiation
falling w/m2
985
985
990
1002
1025
1045
1095
1112
1140
1145
1145
1152
1159
1169
1175
1201
1209
1211
1215
1211
1225
Table 2: Readings of Parameters of Solar Umbrella for
8 panels system (Date- 23 May 2015)
Sr.
no.
Time
Hour
volt
v
Temp.
0
C
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1.30
1.34
1.38
1.42
1.46
1.50
1.54
1.58
2.02
2.06
2.10
2.14
2.18
2.22
2.26
2.30
0
2
4
6
6
6
7
7
8
8
9
9
9
10
10
11
40.3
40.3
40.5
40.5
40.5
40.7
41
41.2
41.5
41.7
41.7
42
42
42
42.2
42.5
Wind
speed
Km/hr
2.1
2.3
2.5
1.9
1.9
2.2
2.9
3.1
2.9
3.5
4.5
2.9
2.1
2.3
2.2
2.5
Radiation
falling w/m2
1225
1201
1279
1295
1301
1307
1315
1321
1324
1321
1330
1339
1341
1349
1352
1355
IV.
MODEL FORMULATION
Identification of Variables
The term variables are used in a very general
sense to apply any physical quantity that undergoes
change. If a physical quantity can be changed
independent of the other quantities, then it is an
independent variable, If a physical quantity changes in
response to the variation of one or more number of
variables, then it is termed as dependent or response
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International Journal of Engineering Trends and Technology (IJETT) – Volume 26 Number 4- August 2015
variable. If a physical quantity that affects our test is
changing in random and uncontrolled manner, then it
is called an extraneous variable. The variables
affecting the effectiveness of the phenomenon under
consideration are solar radiation, area, time resistance,
temperature, wind speed. The dependent or the
response variables in this case is voltage.
Reduction of independent variables/dimensional
analysis
There are several quite simple ways in which
a given test can be made compact in operating plan
without loss in generality or control. The best known
and the most powerful of these is dimensional analysis.
In the past, dimensional analysis was primarily used as
an experimental tool whereby several experimental
variables could be combined to form one. The field of
fluid mechanics and heat transfer were greatly
benefited from the application of this tool. Almost
every major experiment in this area was planned with
its help. Using this principle, modern experiments can
substantially improve their working techniques and be
made shorter requiring less time without loss of
control. Deducing the dimensional equation for a
phenomenon reduces the number of independent
variables in the experiments. The exact mathematical
form of this dimensional equation is the targeted
model. This is achieved by applying Buckingham’s π
theorem (Hibert, 1961). When we apply this thereon
to a system involving n independent variables, (n
minus number of primary dimensions viz. L, M, T and
Ø) i.e. (n-4) numbers of π terms are formed. When n is
large, even by applying this theorem number of π
terms will not be reduced significantly than of all
independent variable. Thus much reduction in number
of variables is not achieved. It is evident that, if we
take the product of the terms it will also be
dimensionless number and hence a π term. This
property is used to achieve further reduction of the
number of variables. Dimensional analysis is used to
reduce the variables and following Pi terms we
evolved out of it (Modak & Bapat, 1994).
Buckingham theorem in brief.
Voltage (v) = f (area, wind speed, solar radiation, time,
temperature, resistance)
v = f (A, W, b, ø, R)
f (A, W, ℓb, ø, R) = 0
φ ( π1, π 2, π 3, π 4) = 0
Since n = 7 and m = 3
No of π terms (n-m) = 7-3 = 4
Repeating variables are
A[L2], W[LT -1], ℓb [MT-3], V
π 1 = Aa Wb ℓbc V
o o o
[M L T ] = [L2] a [LT-1] b [MT-3] c [M ½ L½ T-1]
Equating power of MLT on both sides we get
0 = 2a + b + (1/2)
0 = c + (1/2)
0 = -b-(3c)-1
b=½
a = -½
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π 1 = A-½ W½ ℓb-½ V
π 1= W½ V
A½ ℓb½
Similarly π 2 =
Wø
A 3/2 ℓb
π 3 = WT
A½
π4 = W R
Now π 1 = f (π 2, π 3, π 4)
π 1 = K x π 2a x π 3b x π 4c
Taking log on both side
Log π 1 = Log [K x π 2a x π 3b x π 4c]
Log π 1 = Log K + a Log π 2 + b Log π 3 + c Log π 4
Putting
Log K = k
Log π = y
Log π = A
Log π 3 = B
Log π 4 = C
A set of equations are formed from number of
readings taken during experimentation, therefore
taking summation of these for n values,
The equation become,
y
=
k + aA + bB +
cC
Multiplying above equation by A, B, C on both side
respectively.
Ay
=
kA + aA2 + bAB + CAC
By =
kB + aAB + bB2 + cBC
Cy
=
kC + aAC + bBC + cC2
To solve these equations, reducing it to matrix form
y
Ay
By
Cy
=
n
A
B
C
A
A2
AB
AC
B
AB
B2
BC
C
AC
BC
C2
X
k
q
b
c
By putting the values for various parameters in the
matrices shown above the following matrix is obtained.
The various mathematical models for different
systems of solar umbrella are stated below:
For 4 panels system
π 1 = 1.627665 (π 2)-0.0635283 (π 3)0.18979586 (π 4)0.14734078
For 8 panels system
π 1 = 0.27461 (π 2)-0.00661768 (π 3)0.24301104 (π 4)-0.21705355
V. CONCLUSIONS
The mathematical model is formulated which can be
used to analyse the data and to establish relationship
between different variables of Solar Umbrella. This
type of model is not prepared before by researchers in
this area and thus model will be useful to find and
optimize the parameters affecting output of the
phenomenon. From the model author concluded that
Time is most dominant parameter as compare to other
parameters.
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International Journal of Engineering Trends and Technology (IJETT) – Volume 26 Number 4- August 2015
ACKNOWLEDGMENTS
Thanks to Dr R. J. Patil, Principal Dr. D. Y. Patil
Institute of Engineering and Technology Ambi, Pune,
for his valuable contribution in developing research
article.
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