# Glancing Blow

Working with the incident angle modifier.

If you have been working with solar collectors, chances are you have seen one of the OG-100 solar collector rating sheets put out by the Solar Rating and Certification Corporation (SRCC). An example of such a rating sheet is shown in *Figure 1*.

These sheets are freely available for hundreds of rated solar collectors at www.solar-rating.org/ratings/index.html. Each rating sheet lists physical data for a specific collector including its dimensions, weight, area and fluid volume. It also gives the results from several types of thermal and hydraulic testing. A table near the top of the sheet lists the expected daily heat collection for the solar collector in five different operating scenarios. The latter is based on using the results of thermal performance tests in combination with computer simulation.

One of the least understood performance indexes given on the OG-100 rating sheet is called the incident angle modifier, which is the topic of this column.

NORMAL CONDITIONS

The test procedure used to determine the thermal efficiency values listed on the SRCC OG-100 rating sheet is ASHRAE standard 93-2010 Methods of Testing to Determine the Thermal Performance of Solar Collectors. One set of numbers generated by testing based on this standard are known as the “Y intercept” and “Slope.” Numerical values for both are listed in both Imperial and metric units near the bottom right of the SRCC OG-100 rating sheet. For example: the value listed for the Y intercept in Figure 1 is 0.706. The imperial units value listed for the slope is -0.865 Btu/(hr•ft^{2}•ºF). These values can be used to create a graph of instantaneous thermal efficiency versus the “inlet fluid parameter” as shown in *Figure 2*.

The instantaneous thermal efficiency of a solar collector is determined by the current values of the inlet temperature (Ti), the air temperature surrounding the collector (Ta) and the solar radiation intensity (I). Together these three factors make up the inlet fluid parameter, which is plotted along the horizontal axis. Any operating conditions that are favourable such as low fluid inlet temperature, higher air temperature, or intense solar radiation, will decrease the value of the inlet fluid parameter and increase the thermal efficiency of the collector. The converse is also true.

Any of the three values that make up the inlet fluid parameter can change quickly and as such the efficiency of the collector can vary over a wide range in a short time. Hence the word “instantaneous” should always be stressed when discussing this measurement of efficiency.

It is also important to understand that, under the ASHRAE testing standard, the Y-intercept and slope values are determined when the beam solar radiation is perpendicular, or very close to perpendicular, to the plane of the collector. Under this condition, the transmissivity of the collector’s glazing is maximized (the glazing allows the maximum percentage of incident solar radiation to pass through). Likewise, the optical property absorptivity, which describes the ability of the absorber plate surface to absorb rather than reflect incoming solar radiation, is also maximized when the solar radiation is perpendicular to the plane of the collector. The Y-intercept of the collector’s efficiency rating depends on both the transmissivity of the glazing and the absorptivity of the absorber plate surface. Thus, the Y-intercept value given on the SRCC OG-100 rating sheet is technically only valid when beam solar radiation is perpendicular to the plane of the collector. Under such a condition, the angle of incidence (the angle between the incoming solar radiation and a line perpendicular to the plane of the collector) is zero.

When installed in a typical “fixed” mounting, a solar collector spends very little time with an angle of incidence equal to zero. Instead, incoming solar radiation strikes the glazing and absorber plate at some angle, especially during early morning and late afternoon. Under such conditions, the effective Y-intercept of the collector is less than the Y-intercept reported in the SRCC rating sheet. This happens because of the optical properties of both the glazing and the absorber plate coating. Lower Y-intercept values imply lower efficiency and reduced energy collection.

The incident angle modifier was developed to correct the Y-intercept values determined by the ASHRAE testing procedure. This helps to more accurately predict instantaneous efficiency and daily energy collection.

The “corrected” collector efficiency equation now becomes:

**Formula 1 (see formula at right)**

Where:

n = instantaneous thermal efficiency of the collector (decimal percentage)

K = incident angle modifier (unitless)

Ti=inlet fluid temperature (ºF)

Ta = ambient air temperature (ºF)

I= solar radiation intensity onto plane of collector (Btuh/ft^{2})

Notice that K is a multiplier for the Y-intercept and not the slope.

The maximum value of K is 1.0. This only occurs when the angle of incidence = 0.

In theory, the value of K at other angles of incidence is given by ** Formula 2 (see formula at right)**:

Where:

K = incident angle modifier (unitless)

θ= angle of incidence (º)

b_{0}= a constant determined by testing (unitless)

The value of the constant (b_{0}) is reported in the SRCC OG-100 rating sheet. For the rating sheet shown in *Figure 1*, the value of K is listed as 0.20 (based on a linear fit of K versus the quantity [(1/cosθ)-1].

The graph in *Figure 3* shows the incident angle modifier (K) as a function of the angle of incidence based on a b_{0} value of 0.20. Notice that the value of K remains close to it maximum value of 1.0 until the angle of incidence reaches about 20º. K then decreases to about 0.9 when the angle of incidence reaches 50º and drops off rapidly at higher angles. In theory, the incident angle modifier is 0 when the angle of incidence is 90º.

The good news is that most single glazed flat plate collectors lose very minor amounts of solar radiation input during the peak solar collection times of 9:00 a.m. to 3:00 p.m. due to the effects of the incident angle. These effects are much more pronounced in earlier morning and later afternoon, when the typical clear day solar radiation intensity is much lower.

The net effect of the incident angle effect can be determined through calculations that use specific hourly values of solar radiation in the plane of the collector at a given location and orientation, as well as specific hourly values for angle of incidence. The ASHRAE 93-2010 standard shows an example of such calculations.

For the flat plate collector with the rating sheet given in *Figure 1*, the loss in total solar energy collected between 9:00 a.m. and 3:00 p.m., solar time, for a collector sloped at 42º and facing due south at a 32º northern latitude, is about 5.3 per cent.

As far as comparisons, the smaller the value of the constant b_{0}, listed in the SRCC rating sheet, the lower the optical losses due to solar radiation striking the collector at incident angles other than 90º. Lower optical losses imply higher thermal efficiency and great energy harvesting. Figure 4 shows the effect of the value b_{0} on the annual solar heating fraction for a typical two-collector domestic water heating system operating in Syracuse, NY. The data for this graph was generated using f-chart software (www.fchart.com).

In theory, if the optical properties of the glazing and absorber plate coating were unaffected by the angle at which solar radiation strikes them, the value of b_{0}) would always be

zero. For the collector represented by the rating sheet in Figure 1, the value of b0) is 0.2. Thus, according to Figure 4, the optical properties of this “real” collector cause the annual solar fraction to drop about 11 per cent relative to a collector with theoretically perfect optical characteristics.

No collector with fixed mounting will have a b0 value of 0. However, all other performance measures being equal, the lower the b0 value the better. So keep an eye on the incident angle modifier, as well as the other thermal performance numbers when comparing collectors on the SRCC OG-100 rating sheets. <>

John Siegenthaler, P.E., is a mechanical engineering graduate of Rensselaer Polytechnic Institute and a licensed professional engineer. He has over 34 years experience in designing modern hydronic heating systems. He is also an associate professor emeritus of engineering technology at Mohawk Valley Community College in Utica, NY.