4.10 Learning Activity: Clearness Index

The purpose of this activity is to learn how the clearness index ($k_T$ ) can be determined for a specific day and time based on collected meteorological data and knowledge of the extraterrestrial solar irradiance. By definition, $k_T$ is essentially the attenuation factor of the atmosphere, showing the ratio between the solar radiation incoming into the Earth atmosphere and that reaching the ground:

where $I$ is the energy density measured at the horizontal surface at a locale, and $Io$ is the energy density just outside the Earth’s atmosphere at AM0. Of course, there are multiple natural phenomena that are responsible for scattering and reflection losses.

The clearness index was developed by the researchers Liu and Jordan at the University of Minnesota in the 1960s, and to this day this metric is still used by various models and empirical correlations for quantifying components of solar light.

In this activity, you will need to calculate the hourly $k_T$ values for two different hours on July 31^st, 2007 at Penn State SURFRAD location (Rock Springs). This activity builds upon irradiance data you were treating in Lesson 3, so you will use the same SURFRAD file for the clearness index calculations.

Part 1. Calculation of hourly irradiation (I) from SURFRAD data

Extract GHI data from the SURFRAD file (Penn State location) for July 31^st, 2007 for two different hours: (a) 8-9 am and (b) 1-2 pm.

Convert GHI data (measured in $W/m^2=J/s.m^2$ ) to energy density values (in $J/m^2$ ) by multiplying them by time. Note: SURFRAD data are recorded at 3 min step.

Find the total solar energy density (irradiation) delivered per square meter over each hour period. This is the measured $I$ value. Present your results in $MJ/m^2$ .

Part 2. Calculation of the extraterrestrial irradiation (I_o)

1. Calculate the hourly extraterrestrial GHI (AM0) for each of the hours of 8-9 am and 1-2 pm hours on July 31, 2007 using the equation below:
   \[{I_o} = \frac{{12 \cdot 3600}}{\pi } \cdot {G_{0,n}} \cdot \left[ {\frac{\pi }{{180}}({\omega _2} - {\omega _1})\sin \phi \sin \delta + \cos \phi \cos \delta (\sin {\omega _2} - \sin {\omega _1})} \right]\]
   In this equation: ${\omega }_i$ is hour angle for hourly endpoints 1 (beginning) and 2 (end); $\varphi$ is latitude, $\delta$ is declination for the day of interest, and $G_{0,n}$ is normal irradiance at AM0. All these parameters can be taken from the Bird model for State College location you used in Lesson 3. You only need data for day #212 in this calculation.
2. Present your $I_o$ value in $MJ/m^2$ . The results for $I_0$ are expected to be on the order of 0.5-5 $MJ/m^2$ . This will provide you with the denominator value in the $k_T$ expression.

Part 3. Calculation of $k_T$

Plug in your $I$ and $I_o$ values into the $k_T$ ratio and obtain values for both 8-9 am and 1-2 pm hours. Your result should be a number between 0 and 1.

Provide a brief discussion of results. What are the reasons for clearness index to change during the day?

Submitting Your Work

Your report should include the following: (a) Data tables with GHI for specific hours (8-9 am and 1-2 pm on 7/31/07) and corresponding irradiation values; (b) hourly energy density $I$ in $MJ/m^2$ ; (c) calculation of extraterrestrial irradiation shown; (d) hourly extraterrestrial energy density $I_o$ in $MJ/m^2$ ; (e) $k_T$ values for each of two hours; (f) brief discussion of the obtained values.

Upload your report file to Canvas (Lesson 4 Learning Activity DropBox) in PDF or docx format.

Grading Criteria

This activity is graded out of 30 points. 

Deadline

See the Calendar tab in Canvas for specific due dates.