Solar Energy Notes

The Sun

The Sun is the star at the center of our Solar System.
Its diameter is approximately 1,392,000 km, which is about 109 times the Earth's diameter.
Its mass is about 330,000 times the mass of the Earth.
The energy produced by the Sun's core is through nuclear fusion.
The average distance from the Sun to the Earth is about 150 million km.
Light travels at $3 \times 10^8$ m/s (300,000 km/s).
It takes light approximately 500 seconds (8 minutes and 20 seconds) to reach Earth from the Sun.
The surface of the Sun is called the photosphere.
The temperature of the photosphere is about 6000 K.

Solar Energy Reaching Earth

The fraction of solar energy reaching the Earth's surface is about 170 million gigawatts (GW).
This is significantly greater than the electrical power used on Earth.
The Sun provides enough energy to meet the world's needs, but the challenge lies in efficiently capturing and using this energy.

Black Body Temperature

A black body absorbs all radiation without emitting anything back into the atmosphere.
The more perfect the black body, the more perfect the absorption.
Gases absorb light.

Solar Radiation Spectrum

Solar radiation spans a spectrum from approximately 100 nm to 400 nm.
About 7% of the sun's emission is in the ultraviolet (UV) range.
Visible light has a wavelength between 400 nm to 750 nm, accounting for 43% of the sun's radiation.
About 50% of the sun's radiation falls in the infrared region.
The extraterrestrial irradiance averages 1353 W/m $^2$ , varying by ±3% as the Earth orbits the sun.

Importance of Energy Reaching a Panel

It is crucial to know the amount of energy that reaches a solar panel through all paths.

Factors Affecting Sunlight

As sunlight passes through the atmosphere, some of it is absorbed, scattered, and reflected by:
- Air molecules
- Water vapor
- Clouds
- Dust
- Pollutants
- Forest fires
Direct beam solar radiation is the solar radiation that reaches the Earth's surface without being diffused.

Air Mass (AM)

Air Mass refers to how much atmosphere solar radiation has to pass through before reaching the Earth.
A larger AM number indicates that radiation has to travel through more mass/length.
AM0 represents solar radiation outside the Earth’s atmosphere.
Air Mass = 1 when the sun is directly overhead at sea level (θ = 0°).
- If θ is 0°, then $AM = \frac{1}{\cos(0°)} = 1$
- This is denoted as AM1.
If θ is 48.2°, then $AM = \frac{1}{\cos(48.2°)} = 1.5$
- This is denoted as AM1.5.
Standard spectra are defined due to the variability in terrestrial radiation, such as AM1.5G.

AM1.5D and AM1.5G

D represents the direct radiation.
G represents the Global radiation, which is the sum of direct and diffused radiation.
As Air Mass (AM) increases, the intensity of radiation decreases.
The intensity of the direct component of sunlight throughout each day can be determined as a function of air mass:
- $I = 1353 \times 0.7^{(0.678)^{AM}}$
- AM1.5G is normalized to 1000 W/m $^2$ .

Earth's Motions and Solar Radiation

The major motions of the Earth affect the solar radiation received on any surface:
1. Rotation of the Earth about its polar axis
2. Orbit of the Earth around the sun
The amount of solar radiation at any spot on Earth varies according to:
- Latitude and Longitude of the geographic location
- Time of the day
- Season
- Local landscape
- Local weather
- Angle of tilt

Declination Angle (δ)

The declination angle (δ) varies seasonally due to the Earth's tilt on its axis and its rotation around the sun.
Maximum and minimum declination angles occur at summer and winter solstices, respectively.
The declination angle is 0 at equinoxes.
On June 21, the noontime sun is at its highest point, and δ = +23.45°. This is the summer solstice, marking the beginning of summer in the northern hemisphere.
The winter solstice occurs on December 22, when the northern hemisphere is tilted away from the sun, and δ = -23.45°.
On September 23 (autumnal equinox) and March 22 (vernal equinox), the line from the Earth to the sun lies on the equatorial plane, and δ = 0°.
Declination angle (δ) equation: $\delta = 23.45 \sin\left(360 \times \frac{284+n}{365}\right)$
Where n is the day of the year (e.g., n=1 on Jan 1st, n=173 on June 22nd).

Hour Angle (ω) and Azimuth Angle (ɣ)

Hour Angle (ω): the angular displacement of the sun east or west of the local meridian due to rotation of the earth on its axis at 15° per hour; morning negative, afternoon positive.
- $\omega = 15° \times (LST - 12)$
- The Hour Angle converts the local solar time (LST) into the number of degrees which the sun moves across the sky.
Azimuth Angle (ɣ): the angular displacement from south of the projection of beam radiation on the horizontal plane, ɣ
- \gamma = \cos^{-1} \left[ \frac{\cos(\delta) \times \cos(\phi) - \cos(\delta) \times \cos(\O) \times \cos(\omega)}{\sin(\delta) \times \sin(\phi)} \right]
- You need to first estimate the declination angle, latitude, hour angle, and solar elevation angle to find the azimuth angle.
- Azimuth = Azi, for LST <12 or ω < 0
- Azimuth = 360° - Azi, for LST > 12 or ω >0

Elevation and Zenith Angles

The elevation angle (or altitude angle) is the angular height of the sun in the sky measured from the horizontal.
- $\sin(\alpha) = \sin(\delta) \sin(\varphi) + \cos(\delta) \cos(\varphi) \cos(\omega)$
- Where, δ is declination and φ is the latitude, HRA= ω
The elevation angle varies throughout the day; it is 0° at sunrise and achieves its maximum value at solar noon.
The elevation angle at solar noon is largest at Summer Solstice (latitude plus maximum declination $23.45°$ ) and lowest at Winter Solstice (latitude minus maximum declination $23.45°$ ).
Zenith angle is measured from vertical rather than horizontal.
- $\theta = 90° - \alpha$

Examples (Zenith Angle)

If the solar elevation angle is 38.5°, what would be the zenith angle?
At what time of day is the solar elevation angle at a minimum?

Equation of Time (EoT)

Equation of time (EoT) (in minutes) is an empirical equation that corrects for the eccentricity of the Earth's orbit and the Earth's axial tilt.
It is a measure of the slight variation in the length of each day depending on the time of year.
Where $\ B = \frac{360}{365}(n-81)$
$\ E = 9.87 \sin(2B) - 7.53 \cos(B) - 1.5 \sin(B)$
The difference in minutes between solar time and standard time (Time correction factor) is
Where $ Time correction factor = 4(L{st} - L{Loc}) + E $
$L_{st}$ is the standard meridian for the local time zone.
$L_{Loc}$ is the longitude of the location in question.

Equation of Time (EoT) Example

At Muscat, Bawshar, what is the solar time corresponding to 10:30 AM central time on February 3?
- n= 34
- B= 32.5
- E=-13.5
- longitude is 58.42° and the standard meridian is 60°.
Solar Time= Standard time+ 4(60°-58.42°)+(-13.5)
Solar Time = 10:30 – 7.18(min)
Thus 10:30 AM Central Standard Time is 10:23 AM solar time.

Tilt Angle

A tilt angle of a solar plant is the angle formed between the horizontal plane and the pitch of our solar panels.
The solar panel angle affects the amount of solar electricity generated and is based on two factors: latitude and the season.
Optimizing and adjusting your solar panel tilt helps maximize solar power production.
The ideal tilt of a solar panel should be equal to the latitude of the location.
In the Northern Hemisphere, Solar PV Panels must be south-facing.
In the Southern Hemisphere, Solar PV Panels must be north-facing.

Sun Path Chart

A sun path diagram provides the track of the sun over a year's time for your local latitude and time zone.

Solar Window

The area inside a rectangle in the sky is called the solar window for a given location.
The solar window should be clear and not be shaded by trees or any obstacle between 9:00 a.m. and 3:00 p.m.

Terminologies of Solar Radiation

Solar Irradiance: is the sunlight intensity measured in watts or kW on one square meter (kW/m $^2$ ).
- It is an instant snapshot of power and changes throughout the day based on the movement of the sun and cloud cover.
- In optimal conditions (middle of a clear sunny day), 1000 watts per meter squared of energy is available.
Solar Insolation: is a measure of solar radiation energy received on a given surface area in a given time, expressed in kWh/m $^2$ .
- Irradiance multiplied by time equals insolation.

Peak Sun Hours (PSH)

Average daily insolation can also be given in terms of peak sun hours.
Take total insolation and express it as the equivalent number of hours of being under 1kWh/m $^2$ /day.
Peak Sun Hours (PSH) is used to calculate power generation of PV modules.
Why? Because PSH is equivalent to the number of hours per day that a PV array will operate at peak rated output levels at rated temperature.

Peak Sun Hours (PSH) Examples

The solar power incident on a surface averages 500 W/m $^2$ for 12 hours. How much solar energy is received?
The amount of solar energy collected on a surface over 8 hours is 5 KWh/m $^2$ . What is the average solar power received over the period?
A PV system produces an 8 kW/day dc output at peak sun and average operating temperatures. How much energy is produced from this system per day if the solar energy received on the array averages 4.5 peak sun hours?

Case Study

Calculate the declination, altitude, zenith, and solar azimuth angles and the EoT for Muscat at given latitude φ = 23.58° and Longitude= 58.42° at a) 9:30 AM on February 13 and b) 5:30 PM on July 1.

The declination angle is how much the Earth is tilted. It changes with the seasons because of Earth's tilt and movement around the sun. It's highest and lowest in summer and winter. It's zero in spring and fall. We can find it with this $equation: δ = 23.45 ewline sin(360imesrac{284+n}{365})$ , where n is the day of the year.