Radiation Parameters and Conversions

Overview of Lecture 3: Practicing Problems

• The lecture focuses on calculating 9 radiation parameters using concepts from previous lectures.
• The goal is to calculate parameters and convert horizontal surface calculations to tilted surfaces.
• A flat plate collector is tilted at 30 degrees (β = 30°), located at latitude 19°7' N (φ) and longitude 72°51' E (λ).
• The surface azimuth angle (γ) is 0 degrees, pointing due south.
• The aim is to calculate the angle made by beam radiation with the normal to the flat plate collector on April 1st at 10:00 Local Apparent Time (LAT).
• The lecture also covers how to calculate LAT from Indian Standard Time (IST), based on longitude 82.50° E.
• The second problem involves calculating monthly average daily global and diffusive radiation on horizontal and tilted planes. The location is latitude 19°7' N, average sunshine hours are 7.2, and the elevation is 14 meters above sea level.

Calculation of Angle of Incidence

• The angle of incidence (θ) converts beam flux from the sun to an equivalent value normal to the surface.
• θ is the angle between incident beam flux (Ibn) and the normal to the plane surface.
• The equivalent flux normal to the surface is given by Ibn * cos(θ).
• θ is a function of: latitude φ, slope β, surface azimuth angle γ, declination angle δ, and hour angle ω.
• Latitude (φ) is the angle between the line joining a position to the Earth's center and its projection on the equatorial plane.
• Slope (β) is the angle between the horizontal plane and the tilted plane.
• Surface azimuth angle (γ) is the angle between the projection of the normal to the tilted plane on the horizontal plane and the horizontal line due south.
• Declination angle (δ) is the angle between the line joining the centers of the Sun and Earth and its projection on the equatorial plane.
• Hour angle (ω) is the angular measure of time, equivalent to 15 degrees per hour, varying from +180 to -180.
• Zenith angle is the angle between the line of sight to the sun and the normal to the horizontal plane.
• Solar altitude angle is complementary to the zenith angle.
• Solar azimuth angle is the angle between the projection of the line of sight on the horizontal plane and the horizontal line due south.
• April 1st is the 91st day of the year, β = 30°, γ = 0°, and local apparent time is 10:00.
• Converting latitude and longitude to decimal degrees: 19°7' N becomes 19.12°, 7' / 60' = 0.12° , 72°51' E becomes 72.85°, 51' / 60' = 0.85°.
• Declination angle (δ) is calculated as 23.45° * sin(360/365 * (284 + N)), and for the 91st day, it is 4.02°.
• Hour angle (ω) is calculated as (solar time - 12) * 15°, and for 10:00 LAT, it is -30°.
• cos θ = sin φ sin δ cos β + cos δ cos γ cos ω cos β + cos φ cos δ cos ω sin β - sin δ cos γ sin β

Alternative Formulas for Angle of Incidence

• The angle of incidence can also be calculated using zenith angle (θz): cos θz = sin φ sin δ + cos δ cos ω cos φ.
• In this formula, φ is replaced with (φ - β) for tilted surfaces.

Calculation of Hour Angle During Sunrise & Sunset & Day Length

• During sunrise and sunset, the zenith angle (θz) is 90°, thus cos(90°) = 0.
• The hour angle at sunrise or sunset (ωs) is given by cos ωs = -tan φ * tan δ for a horizontal surface.
• For a tilted surface, the hour angle at sunrise or sunset (ωst) is given by cos ωst = -tan (φ - β) * tan δ.
• Using the horizontal surface formula for tilted surfaces overpredicts from March 21st to September 22nd when the declination angle is positive.
• The maximum possible sunshine hours (Smax) is calculated as Smax = (2/15) * ωs, where ωs is in degrees.

Calculation of Local Apparent Time

• Local apparent time (LAT) is calculated using the formula: LAT = standard time +/- 4 * (standard time longitude - longitude of location) + equation of time correction (EOT).
• The equation of time correction (EOT) is given by a complex formula involving the day of the year (n).
• If the calculation is based on the eastern hemisphere, a negative symbol is used.

Calculation of Average Radiation Data

• Average radiation data is ideally measured over time at a specific location or nearby with similar geography and climate.
• If measured data is unavailable, empirical relationships connect radiation values with meteorological parameters like sunshine hours, cloud cover, and precipitation.
• Sunshine hours predict radiation values most accurately.
• Monthly average sunshine hours (S bar) and monthly average of maximum possible sunshine hours per day (S bar max) are important parameters.
• Day length on a horizontal surface can be calculated as S max = (2 / 15)*ωs.

Calculating Monthly Average of Daily Global Radiation

• The formula used is H g / H0 = a + b * (S bar/ S bar max), where a and b are fitting constants.
• The constants a = 0.31 and b = 0.43 were defined by Modi et al. (1979).
• Clear days cannot be defined by any parameter.
• H c is replaced by extraterrestrial radiation (H0); on specific days (e.g., April 15th), the daily extraterrestrial radiation H0 closely equals the monthly average H0 bar.

Calculating H0

• April 15th corresponds to the 105th day of the year, which is used to calculate the declination angle (δ) as 9.42°.
• Omega (WS) is calculated using horizontal surface formula.
• Omega (WS) is converted to radians.
• The formula to calculate H0 is H0 = [2436001.367/ pi][1+0.033 cos (360105/365)]*[cos (19.28) cos (9.42) sin 93.32 +pi/180 *93.32 cos (93.32)].

Solar Radiation Calculation

• The calculation involves finding H0 bar, equivalent to 37957 kilojoules per meter square day for April 15th.
• Formula: Hg / H0 bar = a + b * (S bar / S max bar).
• The values of a and b (0.31 and 0.43, respectively) are fitting constants, derived from Modi et al.
• The goal is to determine Hg (monthly average of daily global radiation).
• Hg is calculated by multiplying the result of the formula a + b * (S bar / S max bar) by 37957, equalling 21213 kilojoules per meter square day.
• Adjustments to the correlation can be made using elevation data, as per Gopinathan's correlations from 1988.
• Gopinathan's correlations require the height of the location above sea level.
• Remember to convert elevation to kilometers (e.g., 14 meters = 0.014 kilometers) when using Gopinathan's correlations.
• Ensure consistent units for parameters when using correlations.
• For Gopinathan's method, a = -0.309 + 0.539 * cos(19.28) - 0.0693 * EL + 0.290 * (S bar / S max bar), resulting in approximately 0.3666.
• For Gopinathan's method, b = 1.527 - 1.027 * cos(19.28) + 0.0926 * 0.014 - 0.359 * (7.2 / 12.44), resulting in approximately 0.3511.
• Using Gopinathan's a and b values, Hg bar = H0 bar * (a + b * S bar / S bar max); Hg calculates to 21627 kilojoules per meter square day.

Diffusive Radiation Calculation

• Monthly average daily diffusive radiation (Hd bar) can be calculated with three available correlations.
• Calculation uses Hg, derived from Modi et al., which equals 21213 kilojoules meter square day, and H0, which equals 37957 kilojoules meter square day; Hd = Hg * (1.411 - 1.696 * Hg / H0).
• Using Modi et al., Hd is calculated to be 9825 kilojoules meter square day.
• The correlation proposed by Garg et al. uses S bar / S bar max to predict Hd, which results in: 21213*(0.8677 - 0.7365 * 7.2 / 12.44)
• Using Garg et al., Hd is calculated to be 9364 kilojoules meter square day.
• The correlation by Gopinathan and Soler (1995) uses both Hg bar / H0 bar and S bar / S max bar to calculate Hd.
• Using Gopinathan and Soler's correlation, Hd is calculated as 8171 kilojoules meter square day, which seems to be an under prediction.

Hourly Global Radiation

• Calculations involve I0, declination angle, and hour angle (omega).
• Solar time of 9 to 10 hours is considered, with 9:30 AM used to calculate the hour angle.
• For 9:30 AM, the hour angle (omega) is 37.5 degrees; declination angle (delta) for April 15th is 9.42 degrees.
• The solar constant Isc is given as 1.367 kW/m^2, or 1.367 * 3600 kJ/m^2/hour.
• I0 is calculated using Isc, the day of the year, latitude (phi), declination (delta), and hour angle (omega), resulting in 3871 kilojoules per meter square hour.
• Constants a and b are calculated using equations involving omega s (93.32 degrees).
• Constant a is determined to be 0.6845, and constant b is 0.3990.
• A correction factor fc, introduced by Gueymard (1986), is included to align the calculated Ig value with experimental data.
• In Mumbai city, fc is approximately 1, suggesting that the Collares and Pereira and Rabi correlation is suitable for calculating hourly global radiation.
• Ig is calculated using Hg (21213), I0 (3871), H0 (37957), a (0.6845), b (0.3990), omega (37.5), and fc (0.9924), resulting in approximately 2182 kilojoules per meter square hour.
• Always remember when using references I(g), H(g), I(0), and H(0) are always referring to monthly averages