4.5 Decoupling Beam and Diffuse: Clearness and Clear Sky Indices

The availability of solar data is very important when calculating the amount of radiation incident on a collector. Engineers and designers commonly make use of average hourly, daily, and monthly local data. However, the most common measurement available is the Global Horizontal Irradiance (GHI), which is then integrated through a data logger into hourly irradiation, or minute irradiation.

Estimation is an effective tool that involves the use of empirical models that were developed over the last 4-5 decades. The only tools we need are the equations for calculating hourly and daily extraterrestrial irradiance (Air Mass Zero, or AM0) and the integrated energy density (J/m²) gathered from a horizontally mounted pyranometer. These empirical methods to decouple beam and diffuse horizontal components are termed Liu and Jordan transformations, after the initial paper in 1960.

The Clearness Index

The linkage between the two data for horizontal orientation are the clearness indices (k_T, K_T, and ${\overline{K}}_T$). This index is simply a measure of the ratio of measured irradiation in a locale relative to the extraterrestrial irradiation calculated (AMo) at the given locale.

• $k_T=\frac{I}{I_0}$: the hourly clearness index for Total or global irradiation (that's what the "T" is for). This is a ratio of measured energy density against energy density for extraterrestrial solar in one hour.
• $K_T=\frac{H}{H_0}$: the daily clearness index for Total irradiation. This is a ratio of measured energy density against energy density for extraterrestrial solar in one day.
• ${\overline{K}}_T=\frac{\overline{H}}{\overline{H_0}}$: the monthly average daily clearness index for Total irradiation. This is a ratio of measured energy density averaged over the month as one day, against the energy density for extraterrestrial solar for an average day.

For K_T →1: atmosphere is clear. For K_T →0: atmosphere is cloudy. However, this measure incorporates both light scattering and light absorption. Keep in mind that a fraction is not a percentage, and in our case for a cumulative distribution, it is a decimal value between 0--1.

The Clear Sky Index

There is also an alternate indicator for the way that the atmosphere attenuates light on an hour to hour or day to day basis. This is the "clear sky index" (k_c). Mathematically, the clear sky index is defined as

$$kc=\frac{\text{measured}}{\text{calculated clear sky}}$$

and it has been proposed that 1-k_c is a very good indicator of the degree of "cloudiness" in the sky.

So, why do we use either the clearness index or the clear sky index? The answer at the moment is persistence. While it is likely that the clear sky index is more useful than the older clearness index in the long term, all the core research for the empirical calculations used in softwares like TRNSYS, Energy+, and SAM was based on k_T.

The Historical Backdrop: the Clearness Index

In the 1960s, Liu and Jordan found that for different US locations with the same value of ${\overline{K}}_T$ , the cumulative distribution curves of K_T were identical, almost irrespective of latitude and elevation.\marginnote{A cumulative distribution describes the frequency or fraction of occurrence of days in the month below a given daily clearness index, K_T}. This work was expanded into equations by Bendt et al.,\cite{Bendt81} using 20 years of real measurements in 90 locations in the USA. However, it was determined that the data sets were not so similar from region to region (e.g., the tropics had different correlations than the temperate USA, India was different from Africa, etc.) This work was followed by Hawas and Muneer for India and Lloyd for the UK, among others.\cite{Hawas85,Lloyd82}

Remember this! K_T distributions are not universal---they are regional and empirically derived. For all of our future work, we will only rely on hourly k_T values, and the manner in which k_T is used to back out a value of I_b, the hourly beam irradiation component on a horizontal surface.