Calculating the solar energy of a flat plate collector

Undergraduate Journal of
Mathematical Modeling: One + Two
Volume 6 | 2014 Fall Issue 1 | Article 1
Calculating the Solar Energy of a Flat Plate
Collector
Ariane Rosario
University of South Florida
Advisors:
Brian Curtin, Mathematics and Statistics
Scott Campbell, Chemical and Biomedical Engineering
Problem Suggested By: Scott Campbell
Abstract. The amount of solar energy that could be obtained by a flat plate solar collector of one square meter
dimension is calculated in three different locations: Tampa FL, Fairbanks AL, and Pontianak Indonesia, considering the
varying sunset time for each day of the year. The results show that if the collectors are placed near the equator, more
total energy could be obtained. In fact, by placing a solar collector in Pontianak, Indonesia 12.42% more solar energy
can be obtained than by placing it in Tampa and 96.9% more solar energy than Alaska.
Keywords. solar energy, renewable energy, solar energy collector
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Recommended Citation
Rosario, Ariane (2014) "Calculating the Solar Energy of a Flat Plate Collector," Undergraduate Journal of Mathematical Modeling: One +
Two: Vol. 6: Iss. 1, Article 1.
DOI: http://dx.doi.org/10.5038/2326-3652.6.1.4857
Available at: http://scholarcommons.usf.edu/ujmm/vol6/iss1/1

Calculating the Solar Energy of a Flat Plate Collector
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vol6/iss1/1

3
Keywords: Solar energy, Renewable energy, Solar energy collector
PROBLEM STATEMENT
The objective of this paper is to calculate the amount of solar energy that could be
achieved in Tampa for one year and compare it to two different locations. This has been
achieved by considering a flat plate solar collector placed in three different locations and
calculating and comparing their absorbed energy through a year.
MOTIVATION
The demand for energy has been growing exponentially in the recent years along with the
increase in global population. Based on a report by the US Energy Information Administration,
90% percent of the United States and 85% percent of global energy needs are satisfied by non-
renewable resources that are quickly diminishing (U.S. Energy Information Administration;
2013). Although experts predict that coal and natural gas resources are expected to last another
two centuries, petroleum, which accounts for over 50% of energy needs in the U.S. will be
depleted within 30-40 years in the U.S. (Greiner and Semmler; 2008). Figure 1compares the
production and consumption of various energy sources in the United States in 2012. The world is
facing an energy crisis that can only be resolved by finding and utilizing clean, renewable energy
sources (Combs; 2008).
This paper considers the possibility of solar power as a future energy resource. The use of
solar energy has been increasing steadily, becoming the fastest growing power technology in the
world. Photovoltaic (PV) cell solar collectors are able to absorb energy with almost no carbon
emissions by converting solar energy into hydrogen fuels (Reddy; 2012). However, solar panels
Rosario: Calculating the Solar Energy of a Flat Plate Collector
Produced by The Berkeley Electronic Press, 2014

4 ARIANE ROSARIO
are not effective everywhere. In areas with more sunlight, more solar energy can be obtained.
Due to the tilt and rotation of the earth, the amount of energy from the sun reaching a solar
collector decreases as the latitude from the equator increases. This also causes changes in length
of day and the temperature of the location. The objective of this paper is to calculate the
efficiency of solar energy collectors in three different locations throughout the world. Ideal
places to set up solar collectors can be determined by comparing the energy collected by each
panel throughout a year in these varying locations. This information is useful for governments,
businesses, and home-owners to establish renewable and more efficient energy resources for the
future.
Figure 1: US Energy Consumption and Production in 2012
MATHEMATICAL DESCRIPTION AND SOLUTION APPROACH
2.09
1.65
1.54
0.77 0.72
2.85
2.52
1.54
0.77 0.72
0
0.5
1
1.5
2
2.5
3
Petroleum Natural Gas Coal Crude Oil
and NGPL
Renewable
Energy
Nuclear
Electric
Power
QuadrillionBTU
Energy Source
Production
Consumption
Undergraduate Journal of Mathematical Modeling: One + Two, Vol. 6, Iss. 1 [2014], Art. 1
http://scholarcommons.usf.edu/ujmm/vol6/iss1/1

CALCULATING THE SOLAR ENERGY OF A FLAT PLATE COLLECTOR 5
To determine the amount of solar energy absorbed by a solar collector, several key pieces
of information are needed. The days are assumed to be perfectly clear for this research. The
amount of solar energy reaching earth from the sun on a daily basis is given by the following
formula:
𝑆 = 𝐼𝐸 𝑜 (1)
where 𝐼 equals 1000 W/m2
on the earth’s surface and 𝐸 𝑜 is a correcting factor accounting for
Earth’s elliptical orbit. The value of 𝐸 𝑜 which depend on the day of the year, 𝑛 (𝑛 = 1 for
January 1st
), is given by:
𝐸 𝑜 = 1 + 0.033 cos (
2𝜋𝑛
365
) (2)
where the cosine term is in radians. The total energy per unit of area falling onto a solar collector
in one day can be computed by:
𝐸 = 2 ∫ 𝑆 cos(𝜃𝑧) 𝑑𝑡
𝜏
0
(3)
where 𝜏 is the sunset time and the two doubles this value to account for the entire period of the
day (𝑡 = 0 corresponds to solar noon), 𝜃𝑧 is the zenith angle given by
cos 𝜃𝑧 = cos(𝜑) cos( 𝜔) cos(𝛿) + sin(𝜑) sin(𝛿)
(4)
where 𝜑 is the latitude of the collector’s location, 𝜔 is the hour-angle, which corresponds to 𝑡 by
𝜔 = 0.2618 𝑡
(5)
and 𝛿 is the angle of the Earth’s declination, in radians, given by
𝛿 =
𝜋
7.6759
sin (
2𝜋(𝑛 + 284)
365
)
(6)
Combining equations (1), (3) and (4) and simplifying, the total absorbed energy can be written as

6 ARIANE ROSARIO
𝐸 = 2𝐼𝐸 𝑜 [cos(𝜑) cos(𝛿) ∫ cos (ω) 𝑑𝑡
𝜏
0
+ sin(𝜑) sin(𝛿) ∫ 𝑑𝑡
𝜏
0
]
(7)
The sunset time can be solved for by substituting equation (5) into equation (4) after
setting cos 𝜃𝑧 = 0. Therefore, an equation to find 𝜏 can be formulated as the following
𝜏 = 𝑐𝑜𝑠−1
(
− sin(𝜑) sin(𝛿)
cos(𝜑) cos(𝛿)
)
(8)
Finally, by solving the two integrals in (7), the solution can be derived as
𝐸 = 2𝐼𝐸 𝑜 [cos(𝜑) cos(𝛿)
sin(0.2618 𝑡)
0.2618
+ sin(𝜑) sin(𝛿) (𝜏)]
(9)
This equation represents the solar energy obtained each day by changing values of 𝑛. An
excel spreadsheet calculator was set up to calculate 𝐸 for all values of 𝑛 ranging from 1 to 365,
and for three locations with different latitudes as presented in Table 4 of the appendix.
DISCUSSION
The objective of this project was to calculate the amount of solar energy that could be
obtained by a square meter solar collector in three locations throughout a year. The results are
presented in Figure 2.

Figure 2: Possible Energy obtained each day throughout one year for the three locations
As expected, the results verify that the panels on the equator can absorb the most sunlight
over the course of one year, while the panels near the poles absorb the least total energy. This is
due to the tilt of the earth and the decrease in temperature that accompanies this tilt. As the
incidence angle increases, the light has to travel more through the atmosphere over a greater
distance and therefore the panel will absorb less total energy. The possible maximum value for
the Tampa collector, with a latitude location of 27.947, is around the Summer Solstice, on June
18, with a total energy of 8.315 kW-hrs/m2
(Solstice; 2013). The possible minimum value is on
December 20, a day from the Winter Solstice, with a value of 4.25 kW-hrs/m2
. The difference
between these possible accumulated energies is 4.065 kW-hrs/m2
, showing a 95% difference in
efficiency. The total solar energy that could be obtained is 2,378 kW-hrs/m2
. This information is
represented in Table 1, Table 2, and Table 3.
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1
14
27
40
53
66
79
92
105
118
131
144
157
170
183
196
209
222
235
248
261
274
287
300
313
326
339
352
365
WattsHoursperm2Absorbed
Day of Year (1=Jan 1st)
Tampa
Fairbanks
Pontianak

8 ARIANE ROSARIO
In Fairbanks, Alaska, the possible obtainable energies are spread throughout a greater
range: 8.42 kW-hrs/m2
on June 21 and 0.063 kW-hrs/m2 on December 21. The difference
between these values is 8.357 kW-hrs/m2
, which reflects a 13365% difference. These possible
values make sense because Alaska is closer to the North Pole and it will undergo a more
dramatic shift in the amount of daylight due to the tilt of the Earth. As it tilts towards the sun,
Fairbanks experiences almost completely sunny days and extremely short days as it tilts away.
That is why Fairbanks, Alaska has the largest energy absorption range, but the lowest total
energy of the three, with 1,357 kW-hrs/m2
. Figure 3 compares the sunset times of the three
locations throughout a year. As the latitude increases, the length of the day experiences greater
change, leading to less possible sunlight absorption on average.
Figure 3: Sunset time of the three locations throughout a Year
0
2
4
6
8
10
12
-35 15 65 115 165 215 265 315 365
HourofSunset(hrs)
Day of Year (1=Jan 1st)
Tampa
Fairbanks
Pontianak

In Pontianak, Indonesia, the energy levels remain relatively constant through a year, with
two possible maximum attainable values in the curve and minimum possible during the Summer
Solstice. The energy flux reaches its possible maximum of 7.71 kW-hrs/m2
on March 10th
, a
possible local maximum of 7.62 kW-hrs/m2
on October 1st
, and its possible minimum of 6.78
kW-hrs/m2
on June 22nd
. The difference between these values is a mere 0.93 kW-hrs/m2
,
or 13.7%. These values make sense since the equator is least prone to the effects of the Earth’s
tilt. During the two possible attainable maximum values, the spring equinox and the fall equinox,
the Earth’s axis is neither leaning towards the sun nor away, so the equator is exposed directly to
the sun’s light: it always experiences the same sunset hour (March Equinox: Time and Date;
2013). The elliptical orbit of the Earth also accounts for the slight variations in energy, as the
Earth’s proximity to the sun varies throughout its orbit.
In 2011, the average nuclear power plant in the United States generated approximately
12.2 billion kWh of energy (U.S. Energy Information Administration). In order to recreate this
sort of intake with a solar panel placed on the equator, one would need 4,564,450 square meters
of solar collectors, assuming there are only clear days. In Tampa that number grows to 5,132,519
square meters and in Fairbanks, 8,990,420 square meters of solar collectors are needed to
generate the equivalent amount of energy as the average nuclear power plant. This requires about
3.471 square miles, or 2,221 acres, verses 1.762 square miles and 1,128 acres in Pontianak and
1.982 square miles and 1,268 acres in Tampa. These values, however, do not take into account
the space needed between the panels for maintenance and cleaning, or the generators required to
store the final energy to distribute to the population (U.S. Energy Administration; 2012).
An average American household uses about 940 kW-hrs of energy per month (U.S.
Energy Administration). If a family in Tampa installed a one square meter solar energy collector

10 ARIANE ROSARIO
onto to their roof, they could provide an additional 142 kW-hrs of energy in January and 249
kW-hrs in June. With several of these panels, that household’s energy costs could easily drop by
as much as 25% each month to the clean, renewable energy source provided by solar panels.
With several of these panels, an average household would have little to no energy costs.
Solar energy is a feasible alternative for the future. Although the efficiency of the solar
panels is not yet ideal and the prices are still relatively high compared to the electric grid,
advances in technology show a promising outlook for this resource. The price of solar collectors
is expected to drop to a quarter of the current value by 2015 and because of the abundance of
silicon in the crust, the primary material for creating most solar collectors, panels can be
produced at relatively low costs (Reddy; 2012). As the fastest growing power technology in the
world, solar collectors can easily become a staple source of energy. By placing these solar
collectors in the ideal locations, the renewable energy of the sun can be harnessed and distributed
throughout the world.
CONCLUSION AND RECOMMENDATIONS
The results show that solar panels on or near the equator are most efficient in their collection of
energy. Overall, the solar collector in Pontianak, Indonesia was the most efficient, generating
12.42% more energy than Tampa, Florida and 96.9% more energy than Fairbanks, Alaska. This
result suggests that both Pontianak and Tampa would be ideal locations for solar energy
collectors, while panels in Alaska are less likely to be as effective.
Several aspects of this research could have been done differently to obtain a greater
accuracy in the results. Throughout the calculations, the sky and panels were assumed to be free

of obstructions such as weather, debris, or dust that may have decreased the efficiency of the
solar energy collector, something that could very likely happen in a real-life setting. This can
reduce performance of the collector by as much as 7% per year (Dowd; 2008). Additionally,
solar panels are sensitive to damage from weather fluctuations or electricity that may cause some
discrepancies in the data, especially in locations with more extreme weather patterns.
The calculations could also be improved by increasing the number of time intervals
throughout the year. Rather than using units consisting of days, perhaps calculating the energy by
hours, or even minutes, would give a more precise estimation of the solar energy absorbed
during that time.

12 ARIANE ROSARIO
NOMENCLATURE
W Watts J/s
E Energy kW-hrs/m2
I Solar Constant W/m2
S Solar Energy W/ m2
t Sunset Time Hours
𝜃𝑧 Zenith Angle Radians
𝜔 Hour Angle Radians
T Time hours
𝜑 Latitude °
D Angle of Declination Radians
Btu British Thermal Units J
REFERENCES
Combs, Susan. "Overview: Non-Renewable Fuels, Chapter 3." 21 April 2008.
http://www.window.state.tx.us/specialrpt/energy/. 16 April 2013.
Dowd, Dean. CalFinder: Nationwide Home Solar Power Contractors and Information. 19 March 2008.
14 April 2013.
Greiner, Alfred and Willi Semmler. The Global Environment, Natural Resources, and Economic Growth.
Oxford Scholarship Online, 2008.
"March Equinox: Time and Date." 20 March 2013. http://www.timeanddate.com/calendar/march-
equinox.html. April 16 2013.
Reddy, Jayarama P. "Solar Power Generation: Technology, New Concepts, and Policy." Boca Raton:
Hoboken: CRC Press, 2012. 2-5.
Solstice. "Encyclopædia Britannica Online Academic Edition." n.d.
http://www.britannica.com/EBchecked/topic/553654/solstice. 13 April 2013.
"U.S. Energy Administration." 27 July 2012. How much Energy does a Typical Nuclear Power Plant
Generate? 14 April 2013.
"U.S. Energy Information Administration." March 2013.
http://www.eia.gov/totalenergy/data/monthly/pdf/sec1.pdf.

APPENDICES
Table 1: Locations of Solar collectors and their Latitude
City Latitude (°N)
Tampa, Florida 27.947
Fairbanks, Alaska 64.82
Pontianak, Indonesia 0.00
Table 2: Total Energy Absorbed by Solar Collectors throughout Year
Location kW-h/m2
Tampa, Florida 2377.554396
Fairbanks, Alaska 1357.3644
Pontianak, Indonesia 2672.834101
Table 3: Maximum and Minimum values of Solar Flux throughout year with Day
Pontianak Indonesia Value (KW-hrs/m2
) Month and Day
Max 1 7.7067 10-Mar
Max 2 7.6200 1-Oct
Min 6.7802 22-Jun
Tampa, FL
Max 8.3158 18-Jun
Min 4.2515 20-Dec
Fairbanks, AL
Max 8.4213 21-Jun
Min 0.0627 21-Dec

14 ARIANE ROSARIO
Table 4: Excerpt of Excel Spreadsheet Calculator
Day Constants Delta Cos Sin cos(𝝎)
Sunset
Time
First
Integral
Final Value
1 2065.9902 -0.401629 0.8130867 -0.183206 0.2253212 5.13187 3.7214841 4309.03872
2 2065.9609 -0.400214 0.8135747 -0.182595 0.2244354 5.135342 3.7222649 4319.20654
3 2065.912 -0.39868 0.8141017 -0.181933 0.2234766 5.1391 3.7231065 4330.18624
4 2065.8436 -0.397028 0.8146671 -0.181219 0.2224454 5.143141 3.7240075 4341.9732
5 2065.7557 -0.395258 0.8152703 -0.180454 0.2213422 5.147463 3.7249664 4354.56242
6 2065.6483 -0.393371 0.8159107 -0.179637 0.2201679 5.152062 3.7259816 4367.94853
7 2065.5214 -0.391368 0.8165874 -0.17877 0.218923 5.156936 3.7270517 4382.12579
8 2065.3751 -0.389248 0.8172997 -0.177851 0.2176083 5.162081 3.7281748 4397.08805
9 2065.2095 -0.387014 0.8180468 -0.176882 0.2162246 5.167496 3.7293493 4412.82884
10 2065.0245 -0.384664 0.8188279 -0.175862 0.2147725 5.173176 3.7305734 4429.34126
11 2064.8203 -0.382201 0.819642 -0.174791 0.2132529 5.179118 3.731845 4446.61807
12 2064.5968 -0.379624 0.8204882 -0.17367 0.2116667 5.185318 3.7331623 4464.65165
13 2064.3543 -0.376935 0.8213655 -0.172499 0.2100148 5.191773 3.7345234 4483.43399
14 2064.0926 -0.374134 0.822273 -0.171278 0.2082979 5.19848 3.735926 4502.95672
15 2063.812 -0.371223 0.8232095 -0.170007 0.2065171 5.205433 3.7373682 4523.21111
16 2063.5124 -0.368201 0.8241741 -0.168686 0.2046733 5.21263 3.7388478 4544.18804
17 2063.194 -0.36507 0.8251655 -0.167317 0.2027674 5.220065 3.7403626 4565.87804
18 2062.8569 -0.361831 0.8261827 -0.165898 0.2008004 5.227736 3.7419105 4588.27126
19 2062.5012 -0.358485 0.8272244 -0.16443 0.1987734 5.235638 3.7434892 4611.35751
20 2062.1269 -0.355033 0.8282895 -0.162914 0.1966874 5.243767 3.7450964 4635.1262
21 2061.7343 -0.351475 0.8293768 -0.16135 0.1945434 5.252118 3.74673 4659.56643
22 2061.3233 -0.347813 0.8304849 -0.159738 0.1923425 5.260686 3.7483876 4684.66691
23 2060.8942 -0.344048 0.8316126 -0.158078 0.1900857 5.269469 3.7500669 4710.41604
24 2060.447 -0.340182 0.8327585 -0.15637 0.1877741 5.27846 3.7517656 4736.80184
25 2059.982 -0.336214 0.8339214 -0.154616 0.1854088 5.287656 3.7534816 4763.812
26 2059.4991 -0.332147 0.8350999 -0.152816 0.1829909 5.297053 3.7552124 4791.43389
27 2058.9986 -0.327981 0.8362925 -0.150969 0.1805216 5.306645 3.7569558 4819.65453
28 2058.4807 -0.323718 0.837498 -0.149076 0.1780018 5.316428 3.7587095 4848.46064
29 2057.9454 -0.31936 0.8387148 -0.147138 0.1754328 5.326398 3.7604714 4877.8386
30 2057.3929 -0.314906 0.8399416 -0.145155 0.1728157 5.33655 3.762239 4907.77451
31 2056.8234 -0.310359 0.8411769 -0.143128 0.1701515 5.346879 3.7640103 4938.25413
32 2056.2371 -0.305721 0.8424193 -0.141056 0.1674415 5.357381 3.7657831 4969.26296
33 2055.6341 -0.300992 0.8436673 -0.138941 0.1646867 5.368052 3.7675551 5000.7862
34 2055.0147 -0.296173 0.8449194 -0.136782 0.1618882 5.378887 3.7693243 5032.80876
35 2054.3789 -0.291267 0.8461742 -0.134582 0.1590472 5.389881 3.7710885 5065.31531
36 2053.727 -0.286274 0.8474301 -0.132339 0.1561648 5.40103 3.7728457 5098.29023
37 2053.0592 -0.281197 0.8486858 -0.130054 0.1532421 5.41233 3.7745939 5131.71769
38 2052.3757 -0.276036 0.8499396 -0.127729 0.1502802 5.423776 3.776331 5165.58158
39 2051.6767 -0.270794 0.8511901 -0.125363 0.1472802 5.435364 3.778055 5199.86559
40 2050.9623 -0.265471 0.8524358 -0.122958 0.1442432 5.44709 3.7797642 5234.55319
41 2050.2329 -0.26007 0.8536752 -0.120514 0.1411703 5.458949 3.7814566 5269.62763
42 2049.4885 -0.254591 0.8549068 -0.118031 0.1380626 5.470937 3.7831303 5305.07199

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