Solar Radiation and Angles

Solar Radiation

• Renewable energy engineering covers solar, wind, and biomass energy systems.

Solar Angles

• Angles useful in solar radiation analysis include:
  • Latitude of location (Φ)
  • Declination (δ)
  • Hour angle (ω)
  • Solar altitude angle (αa)
  • Zenith angle (Өz)
  • Surface azimuth angle (γ)
  • Solar azimuth angle (γs)
  • Slope (β)
  • Angle of incidence (Ө)

Latitude Angle

• Latitude (Φ) is the angle made by the radial line joining a location to the Earth's center, with the projection of the line on the equatorial plane.
• By convention, latitude is positive in the northern hemisphere.
• Latitude ranges from -90 to 90 degrees, where 0° is at the equator and 90° at the poles.
• Latitude represents the angular distance north or south of the equator, measured from the center of the Earth.

Declination Angle

• Declination is defined as the angle made by the line joining the centers of the Sun and Earth with its projection on the equatorial plane.
• Declination varies from +23.45° on June 21 to -23.45° on December 21.
• It is zero on the two equinox days, March 21 and September 22.
• The formula to calculate Declination is: δ (in degrees) = 23.45 × sin [360 / 365 × (284 + N)]

Hour Angle

• The hour angle measures time, equivalent to 15° per hour, ranging from +180° to -180°.
• It has a positive value in the morning and a negative value in the evening.
• The hour angle is calculated as: ω = [Solar Time-12:00] hrs x 15 degrees.

Surface Azimuth Angle

• Surface azimuth angle is the angle made in the horizontal surface between the line due south and the projection of the normal to the surface on the horizontal surface.
• It ranges from -180° to +180°.
• It is positive if the normal is east of the south and negative if the normal is west of the south.
• For a south-facing surface, γ = 0°, and for a north-facing surface, γ = 180°.

Angle of Incidence

• General equation for the angle of incidence (θ) is given.

Vertical Angle of Incidence

• Equations for the angle of incidence for vertical surfaces and horizontal surfaces are given.

Surfaces Facing South Angle of Incidence

• Equations for the angle of incidence for surfaces facing south is given.

Slope

• Slope is the angle made by the plane surface with the horizontal.
• It is positive for surfaces sloping towards the South and negative for surfaces sloping towards the North.
• The slope varies from 0 to 180°.

Solar Altitude Angle

• The solar altitude angle is the vertical angle between the projection of the Sun's rays on the horizontal plane on Earth's surface and the direction of the Sun's ray.
• The altitude angle is maximum at solar noon.
• It is related to the solar zenith angle, with the equation: θz + α = 90
• Ranges from 0° ≤ α ≤ 90°

Solar Zenith Angle

• The solar zenith angle is the vertical angle between the Sun’s rays and a line perpendicular to the horizontal plane through a point of interest.
• It's the complimentary angle of the solar altitude angle.

Horizontal Surface Angle of Incidence

• Angle of incidence (0) on a horizontal surface (β = 0) pointing due south (γ = 0) with the equation: cos 0z = sin Φ sin δ + cos Φ cos δ cos ω

Solar Azimuth Angle

• The solar azimuth angle is in degrees along the horizon east or west of North.
• Measured horizontally from North to the horizontal projection of the sun's rays.
• It is positive if the projection of the line of sight is in east of the south and negative if it is in the west of the south.

Sunrise, Sunset and Day Length

• At sunrise, sun's rays are parallel to the horizontal surface, the angle of incidence is Ө = Өz = 90°.
• The corresponding hour angle, ω, is cos Ө = 0 = cos Փ cos δ cos ω + sin δ sin Փ
• The equation for this simplifies to: ω = cos-1(-tan Փ tan δ) and ωst = cos-1([tan(Փ - β) tan δ]
• The angle between sunrise and sunset is: 2ω = 2cos-1(-tan Փ tan δ)
• Duration of sunshine hours or daylight hours, td, with 15° hour angle equivalent to 1 hour.
• Equation: td = (2/15) cos-1(-tan Փ tan δ)

Sun Path Diagram

• Sun path diagrams show the position of the Sun with respect to any geographical location on Earth by day/time.
• They determine shadow, building position, and solar panel installation.

Shadow Determination

• The relationship between the sun elevation e, object height H, and shadow length L is: e = tan-1(H / L)

Solar Constant

• The solar intensity (solar radiation/solar irradiance) in the extraterrestrial region has been measured (by NASA through satellite).
• Solar constant is the radiant solar (energy) flux received in the extraterrestrial region on a plane of unit area.
• This area must remain perpendicular to the solar radiation at the mean Sun-Earth distance.
• The value of the solar constant is 1367 W/m².
• The equation for the nth day of the year, the solar intensity on a plane perpendicular to the direction of solar radiation:
  • Iext = Isc * [1.0 + 0.033cos(360n/365)]
  • For June 22, 2013, n = 173, Iext = 1322.49 W/m²
  • For December 21, 2013, n = 355, Iext = 1411.43 W/m²
• Due to the Earth's elliptical orbit, Sun-Earth distance varies up to 1.7%.

Solar Time

• Solar time measures the passage of time with reference to the position of the Sun.
• It is divided amongst 2 types: apparent solar time and mean solar time (clock time or standard time).
• Solar Time = Standard Time ± 4(Lst – Lloc) + E
  • Lst is the standard meridian for the local time zone.
  • Lloc is the longitude of the location (in degrees west).
  • E is the equation of time in minutes. The negative sign applies to the eastern hemisphere. The positive sign is applicable for the western hemisphere.
  • For India, Lst is 82°30′ E.
• It can be given by the expression: E = 229.18 × (0.000075 + 0.001868cos B – 0.032077 sin B - 0.032077 sin B 0.014615cos2B - 0.04089 sin 2B). Where B = (n-1)×360/365 and n = nth day of year.

Solar Radiation on Horizontal Surface

• Relations between solar and other radiations is given using equations

Solar Radiation on Inclined Surface

• Inclined surfaces have beam, diffuse, and reflected solar radiation.
• The conversion factor for beam radiation (Rb) with equation:
  • Rb = I'b/Ib = cos θi / cos θz where I is Intensity and i and z are angles.
  • ( Ib = IN × cos θz)
  • I'b = IN × cos θi
• IN is the intensity of normal irradiance/solar radiation incidence to the inclined surface.
• Өz and Өi are the angles of incidence on the horizontal and inclined surfaces, respectively.

Solar Radiation, Diffuse Radiation

• Conversion factor for diffuse radiation (Rd) is the ratio of diffuse radiation on an inclined surface in W/m² to that on a horizontal surface in W/m² and is governed by the equation:
  • Rd = (1 + cos β) / 2

Solar Radiation, Reflected Solar

• Reflected solar radiations is reflected from nearby objects. The factor for reflected solar radiation (Rr) is:
  • Rr = ρ * (1 - cos β) / 2

Solar Radiation, Total

• Total solar radiation is calculated with the following equation:
  • IT = Ib * Rb + Id * Rd + Rr * (Ib + Id)
  • The typical coefficient ρ = 0.2 for ordinary ground, ρ=0.6 for snow-covered ground.