### Reading Assignment

- J.R. Brownson,
*Solar Energy Conversion Systems*(SECS),**Chapter 8**: Measure & Estimation of the Solar Resource (Focus on Empirical Correlation for Components.) - Reindl, Beckman, and Duffie (1990) Diffuse Fraction Correlations. Solar Energy J. 45(1) 1-7.
- [repeated] C. A. Gueymard (2008) From Global Horizontal to Global Tilted Irradiance: How accurate are solar energy engineering predictions in practice? Solar 2008 Conference, San Diego, CA, American Solar Energy Society
**Optional**: Liu and Jordan (1960) The Interrelationship and Characteristic Distribution of Direct, Diffuse, and Total Solar Radiation. Solar Energy J. 4(3), 1–19.

Please make sure you read all of Ch 8 in SECS for this lesson, again maintaining focus on the same section "Empirical Correlation for Components" and this page content. In the two additional readings, it is OK to scan the *Reindl et al.* paper and the Gueymard paper for key elements that are parallel with the page content.

System designers do not always have the benefit of designing SECS with horizontal surfaces. Many times, these surfaces are tilted at various angles and have various orientations. In such situations, designers and engineers must make estimations for tilted surfaces based on data for horizontal surfaces.

In order to estimate, we first have to break apart the beam horizontal component from the diffuse horizontal components. This has been achieved historically by a methodology established in the 1950s and 60s by Profs. Ben Liu and Richard Jordan (our supplemental reading that is included to add context and the entire line of research that has been applied from then until now).

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/m ^{2}*) 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*, 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.

_{T}- ${k}_{T}=\frac{I}{{I}_{0}}$: 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*→0: atmosphere is cloudy. However, this measure incorporates both light scattering and light absorption. Keep in mind that a

_{T}*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*}. 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}

_{T}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*values, and the manner in which

_{T}*k*is used to back out a value of

_{T}*I*, the hourly beam irradiation component on a horizontal surface.

_{b}#### Self-check questions:

1. What is the ratio for the hourly clearness index?

Click for answer.

*k*

_{T}_{ }is the ratio of measured hourly horizontal irradiance

*(I)*per calculated hourly horizontal irradiation estimated for Air Mass Zero (AM0), or

*I*.

_{0}2. What is the ratio for the hourly clear sky index?

Click for answer.

*k*is the ratio of measured hourly horizontal irradiance

_{c}*(I)*per calculated hourly horizontal irradiation estimated for clear sky conditions (via REST2, or Bird, or otherwise).

3. Which ratio is an argument in the typical Liu-Jordan style empirical correlations?

Click for answer.

*k*, the hourly clearness index

_{T}4. What are the results of using the clearness indices in Liu-Jordan empirical correlations?

Click for answer.

*I*fractions.

_{d }5. Can we determine the contribution of the ground components or horizon diffuse components in this correlation step?

Click for answer.