4th Year Project

MATHEMATICAL MODELLING OF SOLAR
RADIATION INCIDENT ON A HORIZONTAL
SURFACE
• Md. Mizanur Rahman (Roll- 1418)
• Syed Abu Sayeed Motaleb(Roll- 1420)
• Adiba Ibnat (Roll- 1508)
• Md. Tariqul Islam (Roll- 1526)

PURPOSE
• Out of all renewable energy resources, solar energy is one of
the most feasible alternative and sustainable energy
resources in the world.
• The purpose of this project is to develop a mathematical
model that can estimate solar radiation on a horizontal
surface per day.

SUN
 Sun is the star which is at the center of
the solar system.
 It is the only source of all energy of our
solar system.
 Sun is nothing but a large ball of
hydrogen and helium undergoing a
constant nuclear fusion that releases a
large amount of energy in the form of
radiation.

RADIATION
 Radiation is energy that comes from a source and travels through space.
 Generally refers to electromagnetic radiation.
 Sunlight, microwaves are some types of radiation.

SOLAR RADIATION
 Solar radiation is the radiation or
energy that we get from the sun.
 It is also known as short-wave
radiation.
 It comes in many forms such as
visible and invisible light, x-rays, ultra
violet rays.

WHY SOLAR RADIATION?
 Reduce the use of fossil fuels
 A renewable energy source
 Minimal impact on environment
 Eco-friendly

EARTH SUN ASTRONOMICAL
RELATIONSHIP
• Latitude & Longitude
• Earth Sun Distance
• Declination Angle
• Hour Angle
• Sun’s Position Relative to The Observer

Latitude & Longitude
• Latitude and longitude are angles that uniquely define points on a
sphere. They are actually imaginary lines on earth.
• Latitude is the distance from the equator towards north or south
• Longitude is the distance from the prime meridian towards east or west

EARTH SUN DISTANCE
• Amount of solar radiation reaching the earth is inversely
proportional to the square of its distance from the sun.
• Measured by astronomical unit (AU)
• 1 AU = 1.496 x 108 km
• E0 = 1.000110 + 0.034221 cosΓ + 0.001280 sinΓ+
0.000719 cos2Γ +0.000077 sin 2Γ
• Where, the day angle , Γ = 2π (dn - l) /365

SOLAR DECLINATION
 If a line is drawn from between the centre of the earth and sun ,
the angle between this line and earths equatorial plan is called
declination angle. It may vary from -23.5 to +23.5 degree.
 Calculated by ,δ = (0.006918 - 0.399912 cos Γ + 0.070257 sin Γ-
0.006758 cos 2Γ + 0.000907 sin 2Γ- 0.002697
cos 3Γ+ 0.00148 sin 3Γ) (180/π)

HOUR ANGLE
 Hour angle is an expression describing the difference between local solar
time and solar noon. It is expressed in degrees.
 Local solar time relative to position of sun in a particular location.
 Solar noon is usually defined as 12’o clock in the daytime.

SOLAR ANGLES RELATIVE TO OBSERVER
 Zenith Angle is the angle between the local zenith and the line
joining the observer and the sun.
 Altitude Angle is the sun's angular height above the observer's
position.
 Azimuth Angle is an angle that defines the position of the sun with
respect to horizon.

RADIATION TRANSFER FROM SUN TO EARTH
 Solar Constant
 Irradiance or Insolation
 Atmospheric Effects
 Radiation Categories
 Rough Calculation

SOLAR CONSTANT
The average amount of solar radiation received by the
Earth's atmosphere, per unit area, when the Earth is at its
mean distance from the Sun
Average Value- 1360 W/m2

IRRADIANCE OR INSOLATION
 The intensity of solar radiation falling on a surface is called irradiance
 Solar constant is used to calculate irradiance
 𝐼 𝑜𝑛 = 𝐼𝑠𝑐[1 + 0.033 𝑐𝑜𝑠
2𝜋𝑑 𝑛
365
)

ATMOSPHERIC EFFECTS
 Water Vapor
 Clouds
 Dust
 Air Molecules
 Pollutants

RADIATION CATEGORIES
 Beam or Direct Solar Radiation
 coming straight through the atmosphere to hit the plane
 Diffused Radiation
 scattered in all direction in the atmosphere and then some arrives at
the plane on the Earth’s surface

RADIATION CATEGORIES
 Reflected Solar Radiation
 beam and diffused radiation that hits the Earth’s surface and is
reflected onto the plane

ROUGH CALCULATION
The Earth presents a disc of area πR2 to the sun
total amount of extraterrestrial insolation incident on the
Earth is ISC × πR2
Then divided by half the surface areas of the Earth, which
gives 684 W/m2
assume that 30% of the Sun’s energy is lost in the
atmosphere and that the day is an average of 12 hours long
at any location
H = 0.7×684×12 = 5.75 kWh/day
And if the day is an average of 6 hours a day
H = 0.7×684×6 = 2.88 kWh/day

IRRADIATION CALCULATION
Extraterrestrial irradiance
IO = ISC EO (sin δ sin Φ + cos δ cos Φ cos ω)
Irradiation energy falling on a plane surface
Hoh = 2
0
ss
I0 dt
Final equation
HOh=
24
𝜋
ISC EO [(π/180) ωssin δ sin Φ + (cos δ cos Φ sin ωs)]

MATHEMATICAL MODEL FOR ESTIMATING
SOLAR RADIATION
Angstrom proposed the following relationship to predict insolation:
H/Ho =a1+ (1-a1)*S =a1+b1*S
Where
Ho = is the perfectly clear day horizontal insolation
S= the monthly mean daily fraction of possible sunshine obtained
from
S=
𝑛
N
The symbol 𝑛 is the monthly average number of instrument-recorded
bright sunshine hours per day, and N is the average day length.

SOLAR RADIATION
Limitation of 𝐀ngstr 𝐨m Model:
• 𝐴ngstr 𝑜m’s proposed equation is for perfectly
sunny day
• But this is not practical

SOLAR RADIATION
• Due to this limitation, the original Angstrom type regression equation which
is modified by Prescott, which is , (we will use this for our project)
H/Ho = a + b (n/N)
Where,
H = monthly average daily radiation on a horizontal surface
Ho = Monthly average daily extraterrestrial radiation
a,b =Empirical constants
n =Monthly average daily hours of bright sunshine
N = day length.

SOLAR RADIATION
In this project,
We follow 4 steps
1. Collect solar radiation data
2. Evaluation of Regression Coefficients ‘a’ and ‘b’, using
Angstrom- Prescott method. (for Dhaka )
3. Using these coefficients to predict the monthly average
daily global solar radiation (H).
4. A comparison of various model and our model

SOLAR RADIATION
Presenting the source of data

MATHEMATICAL MODEL FOR ESTIMATING SOLAR
RADIATION
Before going further we need to recall the two equation
1. On a given day, let HOh be the extraterrestrial irradiation
Then , HOh =
24
𝜋
ISC EO [(π/180) ωs sin δ sin Φ + (cos δ cos Φ sin ωs) ]
Where ωs = cos-1 [ - tan Φ tan δ ] … … in degree.
2. Day length ,Nd =(2/15) cos-1( — tan Φ tan δ )

SOLAR RADIATION
Months Radiation
Measured
𝐇 (Mj/m2)
Extra
terrestrial
Radiation
𝐇𝐨 (Mj/m2)
Average
Day length,
N
Relative sunshine
hours,
𝐧/ 𝐍
January 14.011 25.692 11.066 0.63845
February 17.953 29.848 11.613 0.7069
March 20.934 34.739 12.254 0.652
April 19.530 38.893 12.999 0.652
May 20.419 40.989 13.593 0.56205
June 15.311 41.631 13.909 0.34225
July 12.020 41.155 13.752 0.33955
August 13.885 39.463 13.227 0.37385
September 15.736 36.109 12.533 0.4045
October 16.596 31.122 11.793 0.57365
November 15.682 26.559 11.203 0.6949
December 13.558 24.354 10.906 0.635
Collected data for Dhaka in a table

SOLAR RADIATION
 The regression constants a & b in equation have been calculated from the values of H/Ho &
(n/N),
 The method of least squares was used to obtain the constants a & b as follows:
a=
H/Ho
𝑚
− 𝑏 ∗
n/N
𝑚
b=
𝑚∗ ( n/N )(H/Ho) −(
n
N
)∗( H/Ho)
𝑚∗
n
N
2
−( n/N)2

SOLAR RADIATION
 The regression coefficient model that we found, is through MATLAB
software.
 We used Curve Fitting app to solve the regression model
 where we let 𝐻/𝐻0= y and 𝑛/ 𝑵 = x and import them in the Curve
Fitting app to get the values of a & b.
 Command, >> cftool(x,y) ,

SOLAR RADIATION

SOLAR RADIATION
And Finally we get our regression model
𝐻/𝐻0 = .11 + .70 ( 𝑛/ 𝑵 )

SOLAR RADIATION
 Using all the data, the estimated radiation is presented below--
Months Radiation
Measured
𝐇 (Mj/m2)
Extra
terrestrial
Radiation
𝐇𝐨 (Mj/m2)
Average
Day length,
N
Climate
index ,KT
𝐇/ 𝐇𝐨
Relative
sunshine
hours,
𝐧/ 𝐍
Radiation
Estimated
(from eq.)
January 14.011 25.692 11.066 .5453 0.63845 14.22
February 17.953 29.848 11.613 .6015 0.7069 17.95
March 20.934 34.739 12.254 .6026 0.652 19.55
April 19.530 38.893 12.999 .5021 0.652 21.89
May 20.419 40.989 13.593 .4982 0.56205 20.49
June 15.311 41.631 13.909 .3678 0.34225 14.38
July 12.020 41.155 13.752 .2921 0.33955 14.13
August 13.885 39.463 13.227 .3518 0.37385 14.51
September 15.736 36.109 12.533 .4358 0.4045 14.05
October 16.596 31.122 11.793 .5333 0.57365 15.81
November 15.682 26.559 11.203 .5905 0.6949 15.75
December 13.558 24.354 10.906 .5567 0.635 13.42

SOLAR RADIATION
10
12
14
16
18
20
22
24
0 2 4 6 8 10 12
Globalradiation
Months
Comparison between H (measured & estimated)
H measured H estimated

SOLAR RADIATION
• Reitveld’s model : H/Ho = .18 + .62 (𝑛/N)
• Glover and McCulloch : H/Ho = .29*cos Φ + .52 (𝑛/N)
• Fagbenle’s model : H/Ho = .28 + .39 (𝑛/N)
• Turton’smodel : H/Ho = .30 + .40 (𝑛/N)
• Our model: H/Ho = .11 + .70 (n/N)
 The five model listed above were applied to the sunshine data at Dhaka.

SOLAR RADIATION
 The five model listed above were applied to the sunshine data at Dhaka
MODEL a b %MBE %RMSE r SSE
REITVELT .18 .62 5.62 10.6 .89 1.94
GLOVER .27 .52 2.76 19.45 .83 1.98
FEGBENLE .28 .39 -7.56 12.52 .76 2.3
TURTON .3 .4 -2.64 10.97 0.75 2.01
OUR MODEL .11 .70 4.42 3.41 .95 .0116

 The MBE is an indication of the average deviation of the predicted values from the measured
values. It is defined by
MBE= 𝑖=1
𝑁
(𝑦 𝑖−𝑥 𝑖)
𝑁
 The RMSE is a measure of the variation of predicted values around the measured values. It is
defined as follows:
RMSE = 𝑖=1
𝑁 (𝑦 𝑖−𝑥 𝑖)2
𝑁
 The CC is a test of the linear relationship between the calculated and measured values. It is
defined by
CC = 𝑖=1
𝑁
𝑦 𝑖− 𝑦 𝑥 𝑖− 𝑥
𝑖=1
𝑁 𝑦 𝑖− 𝑦 2 [ 𝑖=1
𝑁 𝑥 𝑖− 𝑥 2]
MATHEMATICAL MODEL FOR ESTIMATING SOLAR
RADIATION

CONCLUSION
The model proposed for Dhaka, Bangladesh in this study can
be a great help in future for estimation of solar radiation .

Reference
 "How Round is the Sun?” NASA. 2 October 2008. Retrieved 7 March 2011.
 Pandey, C. K., and A. K. Katiyar. 2013. “Solar Radiation Models and Measurement Techniques.” Review Article,
Hindwai publishing Corporation, Journal of Energy.
 Abdul Qayoom Jakhrani, Al-Khalid Othman, Andrew R.H. Rigit, Saleem Raza Samo,Shakeel Ahmed Kamboh.
“Estimation of Incident Solar Radiation on Tilted Surface by Different Empirical Models”.International Journal of
Scientific and Research Publications, Volume 2, Issue 12, December 2012 1 ISSN 2250-3153.
 http://www.itacanet.org/the-sun-as-a-source-of-energy/part-1-solar-astronomy/
 http://www.itacanet.org/the-sun-as-a-source-of-energy/part-2-solar-energy-reaching-the-earths-surface/
 Iqbal M. An introduction to solar radiation. London: Academic Press; 1983.
 Debazit Datta,Bimal Kumar Datta,"EMPIRICAL MODEL FOR THE ESTIMATION OF GLOBAL SOLAR RADIATION IN
DHAKA, BANGLADESH".IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-
1163|pISSN: 2321-7308.
 NASA Prediction of Worldwide Energy Resource (POWER),
Higher Resolution Daily Time Series by Location. ( https://power.larc.nasa.gov/cgi-bin/hirestimeser.cgi)
 Assessment of Renewable Energy Resources of Bangladesh- by Mazharul Islam.
http://www.sdnbd.org/sdi/issues/energy/publications/shakti-ebook1.pdf

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