Terrestrial Radiation

Introduction

The Earth’s atmosphere has several effects on terrestrial radiation. The figure below depicts the relative importance of atmospheric impacts on the sunlight striking the Earth’s surface.  The process poster also depicts the solar energy balance of the Earth-atmosphere system.

Terrestrial Radiation

The major impacts of the atmosphere on sunlight include:

  • A reduction in solar radiation and change in spectral content given atmospheric absorption;
  • A change in sunlight quality due to Raleigh scattering (e.g. the introduction of a diffuse sunlight); and
  • Cloud and surface albedo, which is the reflection and release of radiation following absorption.

The values in the process poster are dimensionless and relative quantities of energy for illustration purposes.  The radiation balance assumes that the sum of all sources minus all sinks equals zero.  For example, clouds are in radiative balance since they absorb and emit 64 units of radiation.  The components of solar radiation striking the Earth’s surface are defined below.

Direct Normal Irradiance (DNI)

DNI is the solar radiation on a surface element perpendicular or normal to the Sun’s rays, and excludes diffuse insolation. Empirical estimates of DNI begin with the estimated irradiance at the top of the atmosphere, as defined here.  The surface level DNI by wavelength, corrected for atmospheric filtering, F_{DNI}(\lambda), is given by:

(1)   \begin{equation*} F_{DNI}(\lambda) = F_{toa}(\lambda)e^{-\tau_{\lambda} AM(\theta_{Z})d_{R}(AM)} \end{equation*}

where F_{toa}(\lambda) is solar radiation by wavelength at the top of the atmosphere; \tau_{\lambda} is the atmospheric optical depth or the attenuation coefficient by wavelength on a cloudless day; AM(\theta_Z) is the optical air mass for the solar zenith angle Z; and d_{R}(AM) is the Raleigh optical thickness at air mass AM.

DNI varies depending on the time of year, time of day, atmospheric absorption, and Raleigh scattering. All of these parameters have been defined previously except Raleigh optical thickness, which can be estimated using the improved formula of Kasten (1996):1

For AM < 20

(2)   \begin{equation*} d_{R}(AM) = \frac{1}{6.6296 +1.7513AM - 0.1202AM^2 +0.0065AM^3-0.00013AM^4} \end{equation*}

For AM > 20

(3)   \begin{equation*} d_{R}(AM) = \frac{1}{10.4 + 0.718AM} \end{equation*}

DNI on Angled Surfaces

The direct beam irradiance on a horizontal surface F_{DNI}^{HS}(\lambda) in W/m2 is calculated as:

(4)   \begin{equation*} F_{DNI}^{HS}(\lambda) = F_{DNI}(\lambda) \sin h_{0} \end{equation*}

where h_{0} is the solar altitude angle (radians) and is defined by standard sun position equations.

The beam irradiance on an inclined tilted surface F_{DNI}^{TS}(\lambda) in W/m2 is calculated as:

(5)   \begin{equation*} $\small{F_{DNI}^{TS}(\lambda)$ = F_{DNI}(\lambda) d_{exp} \end{equation*}

where d_{exp} is the solar incidence angle measured between the Sun and an inclined surface as defined by the standard sun position equations.

DNI Measurement

DNI is measured in W/m2 or kilowatt-hours per square meter per day. The basic conversion between the two measures is shown below:

(6)   \begin{equation*} \frac{1 kWh}{m^2 per day} = \frac{1,000 W * 1 hour}{1 m^2 * 24 hours} = 41.67 \frac{W}{m^2} \end{equation*}

DNI observations can be measured directly via an absolute cavity radiometer (ACR). ACRs are considered the most accurate method, but the equipment is not designed for continuous, unattended, outdoor use. Instead, they are used to calibrate and validate more traditional instruments. An ACR connected to a tracker and digital data logger is shown below.

AbsoluteCavityRadiometer

ACR – Click to enlarge

pyrheliometer

Pyrheliometer – Click to enlarge

             

The primary field instrument for DNI field measurement is a pyrheliometer. Pyrheliometers employ thermopile sensors at the base of a light-collimating tube and a glass window face (or they are constructed with another photosensitive element in place of the thermopile).  The light-collimating tube limits the instrument field of view to 5° to ensure focus on the direct beam and circumsolar radiation. The small field of view requires that the pyrheliometer tracks perpendicular to the Sun’s path. An image of a pyrheliometer appears in the second image above.

Diffuse Horizontal Irradiance (DHI)

As the cloudless sky becomes more turbid with water vapor and aerosols, the diffuse irradiance increases while the beam irradiance decreases. The estimation of the diffuse component on a horizontal surface F_{DHI}^{HS}(\lambda) and measured in W/m2 is given by:2

(7)   \begin{equation*} F_{DHI}^{HS}(\lambda) = F_{toa}(\lambda)  T_{N}(\tau_{\lambda})  F_{d}(h_{0}) \end{equation*}

where

  • F_{toa}(\lambda) is irradiance at the top of the atmosphere
  • T_{n} is a diffuse transmission function dependent on aerosol optical depth \tau_{\lambda}, and
  • F_{d} a diffuse solar altitude function  dependent only on the solar altitude h_{0}

The estimate of the transmission function T_{n}(\tau_{\lambda}) gives the theoretical diffuse irradiance on a horizontal surface with the Sun vertically overhead for the AM(2) attenuation factor. The following second order polynomial expression is used:

(8)   \begin{equation*} T_{n}(\tau_{\lambda}) = -0.015843 + 0.030543\tau_{\lambda} + 0.0003797\tau_{\lambda}^2 \end{equation*}

The diffuse solar altitude function is determined by:

(9)   \begin{equation*} F_{d}(h_{0}) = A_{1} + A_{2}\sin h_{0} + A_{3} \sin^2 h_{0} \end{equation*}

where the values of the coefficients A_{1}A_{2} and A_{3} depend only on the attenuation coefficient \tau_{\lambda} as follows:

(10)   \begin{equation*} \begin{split} A_{1}^* &= 0.26463 - 0.061581 \tau_{\lambda} + 0.0031408 \tau_{\lambda}^2 \\ A_{1} &= \frac{0.0022}{T_{n}(\tau_{\lambda})} ~~~~~~~ if ~ A_{1}^*{T_{n}(\tau_{\lambda})} < 0.0022 \\ A_{1} &= A_{1}^* ~~~~~~~~~~~~ if ~ A_{1}^*{T_{n}(\tau_{\lambda})} \geq 0.0022 \\ A_{2} &= 2.04020 + 0.018945 \tau_{\lambda} - 0.011161 \tau_{\lambda}^2 \\ A_{3} &=  -1.3025 + 0.039231 \tau_{\lambda} + 0.0085079 \tau_{\lambda}^2 \end{split} \end{equation*}

DHI on Angled Surfaces

The model for estimating the clear-sky diffuse irradiance on an inclined tilted surface F_{DHI}^{TS}(\lambda) in W/m2 distinguishes between sunlit and shadowed surfaces. The equations are as follows:3

For sunlight surfaces, non-overcast skys and h_{0} in radians

if  h_{0} \geq 0.1 (or 5.7 degrees)

(11)   \begin{equation*} F_{DHI}^{TS}(\lambda) = F_{DHI}^{HS}(\lambda) [F(g_{N}) (1-K_{b}) + K_{b} \frac{\sin d_{exp}}{\sin h_{0}}] \end{equation*}

if h_{0} < 0.1

(12)   \begin{equation*} F_{DHI}^{TS}(\lambda) = F_{DHI}^{HS}(\lambda)[F(g_{N}) (1-K_{b}) + K_{b} \frac{\sin g_{N} \cos A_{LN}}{0.1 - 0.008 h_{0}}] \end{equation*}

where g_{N} is the surface or module tilt angle and A_{LN} depends on the azimuth angle A_{0} (horizontal angle between the Sun and meridian – measured from East) and A_{N} the azimuth aspect (an angle between the projection of the normal on the horizontal surface and East), as defined by standard sun position equations:

(13)   \begin{equation*} \begin{split} A_{LN}^* &= A_{0} - A_{N} \\ A_{LN} &= A_{LN}^* ~~~~~~~~~~~~if -\pi \leq A_{LN}^* \leq \pi \\ A_{LN} &= A_{LN}^* - 2\pi ~~~~if A_{LN}^* > \pi \\ A_{LN} &= A_{LN}^* + 2\pi ~~~~if A_{LN}^* < -\pi \end{split} \end{equation*}

The K_{b} variable is a measure beam irradiance on a horizontal surface expressed as a proportion of the extraterrestrial irradiance on a horizontal surface:

(14)   \begin{equation*} K_{b} = \frac{F_{DNI}^{HS}(\lambda)}{F_{toa}^{HS}(\lambda)} \end{equation*}

where

(15)   \begin{equation*} F_{toa}^{HS}(\lambda) = F_{toa}(\lambda) \sin h_{0} \end{equation*}

For surfaces in shadows (d_{exp} < 0 and h_{0} \geq 0)

(16)   \begin{equation*} F_{DHI}^{TS}(\lambda) = F_{DHI}^{HS}(\lambda) F(g_{N}) \end{equation*}

where (g_{N} in radians)

(17)   \begin{equation*} \begin{split} F(g_{N}) &= r_{i}(g_{N}) + N*(\sin g_{N} - g_{N}\cos g_{N} - \pi \sin^2 \frac{g_{N}}{2}) \\ r_{i}(g_{N}) &= \frac{1 + \cos g_{N}}{2} \\ N &= 0.00263 - 0.712K_{b} - 0.6883K_{b}^2 \end{split} \end{equation*}

 

Global Horizontal Irradiance (GHI)

 

Irradiance vs Insolation

UnderConstruction

B.Horn

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Show 3 footnotes

  1. Kasten F. (1996) “The Linke Turbidity Factor Based on Improved Values of the Integral Rayleigh Optical Thickness,” Solar Energy, Vol 56, N° 3, 239-244.
  2. Scharmer and Greif (2000),  The European Solar Radiation Atlas, Vol. 2: Database and Exploitation software. Paris (Les Presses de l’ École des Mines)
  3. Muneer, T. (1990), “Solar Radiation Model for Europe,” Building Services Engineering Research and Technology, 11, 4, 153-163.