Geog 410 Modeling of Environmental Systems
Lab 7 Modeling Incoming Solar Radiation
Due Date: 2nd, Nov, 2007
1 Objective
(1) Energy is the ultimate driving force for all ecosystem processes. The objective for
this lab is to calculate the amount of solar radiation available at a horizontal
surface at any place at any time under a typical clear sky condition.
(2) In this lab, we will learn how to use MATLAB for simple modeling purpose.
2 Theories
All energy on earth surface ultimately comes from the sun. The energy flux from the sun
at the top of the atmosphere per perpendicular to the sun beam is called the solar constant
(S0=1367 w/m2). However, the sunbeam must pass through a thick layer of atmosphere in
which it will be subject to attenuation (scattering and absorption), reducing the amount of
solar radiation received on the Earth surface. In the meantime, the beam solar radiation at
the top of the atmosphere will be broken into two components: the beam (or direct) and
the diffuse components. The total radiation at the earth surface is the sum of the beam
and diffuse radiation.
St  Sb  S d
(1)
Where St, Sb, and Sd are the total, beam and diffuse radiation on a horizontal surface,
respectively. The amount of attenuation of solar radiation is a function of atmospheric
conditions (e.g. presence of clouds, haziness etc.). The fraction of solar radiation that
arrives the Earth surface when the sun is at the zenith is called atmospheric transmittance.
In a typical cloudless clear day, the transmittance of the atmosphere is τ=0.75, meaning
75 percent of the direct radiation reach the earth surface when the sun is at the zenith.
Depending on the time of the day, the distance that the sunbeam passes through the
atmosphere is different. The attenuation amount accumulates as along the pass of
sunbeam. Thus on a horizontal surface, the direct sunbeam arriving at the Earth surface is
S b  cos( ) S 0 m
(2)
Where θ is the solar zenith angle and m is the air mass that the sunbeam passes.
m
pa
101.3 cos( )
(3)
1
Where pa is the atmospheric pressure at (kpa) at the point of interest. The constant, 101.3,
is the standard atmospheric pressure at the sea level in kpa. The diffuse component of the
radiation at the earth surface is
S d  0.3(1   m ) S 0 cos( )
(4)
Where cos(θ) is calculated as
cos( )  sin(  ) sin(  )  cos( ) cos( ) cos( )
(5)
Where φ is the latitude, and ω is the hour angle. δ is the sun declination angle which is
the latitude that the sun is on, and
 2 (284  J ) 

365


  23.5 sin 
(6)
where J is the Julian date, J=1 for January, 1 and J=365 for December 31.
3 Steps
1. Start the MATLAB program under windows.
2. Define the constants in MATLAB;
Latitude is 39.8 degrees, Julian date is 181 days, transmittance is 0.75, solar constant
is 1367 w/m2, air pressure is 100 kpa,
>>lati=39.8; % latitude of the place (degrees)
>>Julian_date=181; %Julina date of June 30, 2007.
>>t=0.75; % transmittance (unitless)
>>S=1367; %solar constant (w/m^2)
>>p=100; %air pressure (KPa)
3. Define the time at which we want to calculate the solar radiation;
>>hours=[7,8,9,10,11,12,13,14,15,16,17];
4. Calculate the hour angle;
>>hangle=(12.0-hours)*15.0*pi/180.0; % Note pi/180.0 is the factor to convert
angles in degrees in radiances.
5. Calculate the sun declination angle;
>>declangle=23.45*sin(2.0*pi*(284.0+Julian_date)/365.0)*pi/180.0;
6. Calculate the solar zenith angle;
>>cosz=sin(lati*pi/180.0)*sin(declangle)+cos(lati*pi/180.0)*cos(declangle)*cos(han
gle);
2
Notice: “cosz” is an array now containing the cosine of solar zenith angle for
each hour listed in the array “hours”;
7. Calculate optical air mass, m.
>>m=p./(101.3*cosz);
Notice: A constant divided by an array is not defined, if we add ‘.’ before ‘/’, that
means the constant is divided by each element in the array;
8. Calculate the beam radiation, diffuse radiation and total radiation.
>>Sb=cosz*S.*(t.^m); % Sb is the beam radiation on a horizontal surface
>>Sd=0.3*(1.0-t.^m)*S.*cosz; % Sd is the diffuse radiation on a horizontal surface
>>St=Sb+Sd; % St is the total radiation;
Notice: A array multiplied by an array, if we add ‘.’ before ‘*’, that means each
element in the first array is multiplied by the corresponding element in the
second array; if an array is on the index of the exponential function, we must
add ‘.’ before ‘^’ to show that for each element in the array, we calculate the
power of t.
9. Plot the radiation to a figure;
>>plot(hours,Sb,'-'); % Plot the beam radiation, ‘_’ indicates plotting with a solid line
>>hold on % Based on the current figure, you can plot new series on it and the
MATLAB does not generate a new figure
>>plot(hours,Sd,':'); % ‘:’ indicates plotting with dotted line
>>plot(hours,St,'--'); % ‘--‘ indicates plotting with dashed line
>>hold off % Close the function that you can add new series to the current figure.
>>xlabel('Hours of Day');
>>ylabel('Solar Radiation (w/m^2)');
>>legend('Beam radiation','Diffuse radiation','Total radiation','Location','Best'); %
Note the sequence of the label must be exact corresponding to what you have plotted.
'Location', 'Best', means MATLAB will place the legend to the best location the
software think it is. If you don’t like the location, you can always drag it to the place
where you like.
Now you have calculated the beam, diffuse and total radiation on a horizontal
surface at latitude 39.8 degrees North on Julian date 181 (June 30th) for
3
atmospheric transmittance at 0.75 and air pressure at 100 kpa. You may see the
radiation like this:
1200
solar radiation (w/m 2)
1000
800
600
Beam radiation
Diffuse radiation
Total radiation
400
200
0
7
8
9
10
11
12
13
hours of Day
14
15
16
17
10. We have put all the commands in a file and created a Matlab .m file named
“lab7_radiation.m” in the data/ directory. You can copy and paste it to your
student folder;
11. Run the m file to complete the following problems.
Problems
1. Create a graph showing the hourly patterns of beam, diffuse and total radiation.
Describe the patterns you see on the graph.
2. Create the same graph above for a place with latitude at 0 and 60 degrees.
3. Create the same graph above for a place with altitude at 3000 and 5000 meters.
4. Create the same graph above for January 1, March 21, June 21, and Sept 21.
5. Describe how the total solar radiation received on a horizontal surface changes
with respect to latitude, date of the year, and elevation.
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