Thursday, March 19, 2015

HP Prime and Casio Prizm: Solar Irradiance

Solar Irradiance

The program IRRAD will calculate two properties:

(1)  The solar angle of incidence given the angular elevation and azimuth (from south going “counterclockwise”:  east-north-west) of both the sun and panel.
(2)  The irradiance given by the solar panel. 

Formulas

Angle of Incidence Given Azimuth

Degree mode is assumed, the angle of incidence (θ) is found in the following equation:

cos θ = cos(ep)*sin(es) + sin(ep)*cos(es)*cos(as-ap)

Where:
θ = angle of incidence
es = elevation of the sun
ep = elevation of the panel
as = azimuth of the sun, from south headed towards east, then north, then west

ap = azimuth of the panel, from south headed towards east, then north, then west

Angle of Incidence

Azimuth and Elevation of the Sun and Solar Panel
Calculating Flux Density of Solar Radiation on a Surface

Calculating the flux density (energy) of solar radiation on a surface, Lambert’s Cosine Law is used.  Lambert’s Cosine Law states that the relation between the irradiance of the sun and the angle of incidence. The result is the irradiance of the surface.  Irradiance is the rate of energy (power) over a unit area.  In SI units, irradiance is measured in Watts per square meter (W/m^2).

ls = lb * cos θ

Where:
lb = the sun’s power or irradiance.  Often this is treated as a constant, which is approximately 1367 W/m^2 for extraterrestrial solar power, or approximately 1000 W/m^2 when we are dealing with the Earth’s surface (taking scattering of light into account)
ls = the panel’s power or irradiance


Casio Prizm IRRAD

Deg
“SUN: ELEL, AZI(S)”
?->A:?->B
“PANEL: ELEV, AZI(S)”
?->C:?->D
“IRRADIANCE OF THE SUN”
“(W÷M²)”
cos C * sin A + cos A * sin C * cos(B-D)
cos¹ Ans -> θ
I * cos θ -> S
“INCIDENCE ANGLE”
θ
“IRRADIANCE OF SUN”
S

HP Prime IRRAD

EXPORT IRRAD()
BEGIN
LOCAL es,as,ep,ap,θinc;
LOCAL ib,is;
HAngle:=1;
INPUT({es,as},”Sun”,{“Elev.:”,”Azi (S):”});
INPUT({ep,ap},”Panel”,{“Elev.:”,”Azi (S):”});
INPUT(ib,”Sun’s Irradiance”,”I:”,”W/m^2”);
// angle of incidence
θinc:=ACOS(COS(ep)*SIN(es)+SIN(ep)*COS(es)*COS(as-ap));
MSGBOX(“Incidence Angle = “+θinc);
// Lambert’s Cosine Law
Is:=ib*COS(θinc);
MSGBOX(“Surface Irradiance = “+is);
RETURN {θinc,is};
END;

Example

Data:

Sun: 
Elevation:  55⁰24’21” ≈ 55.40583⁰
Azimuth:  175⁰15’44” ≈ 175.26222⁰

Panel:
Elevation:  40⁰
Azimuth:  90⁰ (panel is facing due east)

Irradiance of the Sun:  1000 W/m^2

Output:
Incidence Angle ≈ 48.643169⁰
Surface Radiance ≈ 660.746510 W/m^2


Sources:

Baldocchi, Dennis “Lecture 7, Solar Radiation, Part 3, Earth-Sun Geometry”  Biometeorogy, ESPM 129  University of California, Berkeley.
Retrieved February 17, 2015. 

Mortimer, David  “Lambert’s Cosine Law”  30 January 2014.  The Solar Bucket.
Retrieved March 18, 2015

University of Oregon Solar Radiation Monitoring Laboratory  “Solar Radiation Basics”  University of Oregon.  http://solardat.uoregon.edu/SolarRadiationBasics.html   Retrieved February 10, 2015



This blog is property of Edward Shore – 2015. 

1 comment:

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