This Appendix describes the methods used to calculate the monthly and yearly averages of incident and transmitted solar radiation and the illuminanace diurnal profiles. It also describes how data uncertainities were determined and how the climatic information was derived.
The incident solar radiation for a horizontal window and vertical windows facing north, east, south, and west was determined using models and hourly data from the 1961-1990 National Solar Radiation Data Base (NSRDB).
Global solar radiation. The incident global solar radiation (I) received by a surface, such as a window, is a combination of direct beam radiation (), sky radiation (), and radiation reflected from the ground in front of the surface (). The following equation can be used to calculate incident global solar radiation:
equation (1)
where is the incident angle of the sun's rays to the surface.
The incident angle is a function of the sun's position in the sky and the orientation of the surface. Algorithms presented by Menicucci and Fernandez (1988) were used to compute incident angles. Hourly values of direct beam solar radiation from the NSRDB were used to determine the direct beam contribution () for each hour. Except for the first and last daylight hour, incident angles were calculated at the midpoint of the hour. For the first and last daylight hour, incident angles were calculated at the midpoint of the period during the hour when the sun was above the horizon.
The sky radiation () received by the surface was calculated using an anisotropic diffuse radiation model developed by Perez et al. (1990). The model determined the sky radiation striking the surface using hourly values (from the NSRDB) of diffuse horizontal and direct beam solar radiation. Other inputs to the model included the sun's incident angle to the surface, the surface tilt angle from horizontal, and the sun's zenith angle. The Perez et al. model is an improved and refined version of their original model that was recommended by the International Energy Agency for calculating diffuse radiation for tilted surfaces (Hay and McKay 1988). The following equation is the Perez et al. model for diffuse sky radiation for a surface:
equation (2)
The model coefficients and are organized as an array of values that are selected for use depending on the solar zenith angle, the sky's clearness, and the sky's brightness. Perez et al.(1990) describe completely the manner in which this is done.
The ground-reflected radiation () received by a surface is assumed isotropic and is a function of the global horizontal radiation (), the tilt of the surface from the horizontal (), and the ground reflectivity or albedo ().
equation (3)
For the data in this manual, an albedo of 0.2 was used. This albedo is a nominal value for green vegetation and some soil types. The effect of other albedo values can be determined by adding an adjustment
equation (4)
where
Diffuse solar radiation.The incident diffuse solar radiation () received by a surface is the sum of the sky radiation () and the radiation reflected from the ground in front of the surface (), both of which are considered diffuse.
equation (5)
Clear-day global solar radiation.Incident clear-day global solar radiation represents the global radiation obtainable under clear skies. It was calculated as above, but using clear sky values of dirext beam and diffuse horizontal solar radiation. The clear sky values of direct beam and diffuse horizontal solar radiation were modeled using METSTAT (NSRDB - Vol.2,1995), the same model used to model solar radiation for the NSRDB. Inputs to METSTAT included cloud cover values of zero; average monthly values of aerosol optical depth, precipitable water, albedo, and ozone; and the day of the month of which the solar declination equals the monthly average. Average precipitable water values were multiplied by 80% to compensate for expected clear-day precipitable water compared to the mean for all weather conditions.
As solar radiation passes through a window, some of it is reflected or absorbed by the glass. Consequently, the solar radiation transmitted into the living space is less than the radiation incident on the outside of the window. The losses are dependent on the type of window. For this manual, the transmitted solar radiation data are for two layers of single-strength clear glass. Each glass is 3.18 mm (0.125 inch) thick and has an index of refraction of 1.526 and an extinction coefficient of 32/meter (for glass with greenish appearance).
Transmitted solar radiation values were determined for windows with and without external shading.
Transmitted solar radiation for unshaded windows.For windows without external shading, transmittance coefficients were applied to the hourly values of incident solar radiation to obtain the transmitted solar radiation (I_{t}).
equation (6)
where
= transmittance for reflectance of radiation
I = incident solar radiation
As presented by Duffie and Beckman (1991), Bouguer's law can be used to determine the transmittance for absorption, and Fresnel equations can be used to determine the transmittance for reflectance. Both transmittance coefficients depend on the incident angle of the solar radiation. Because of their differences in incident angles, diffuse and direct beam radiation were treated separately when determining the transmitted solar radiation.
If incident isotropic diffuse sky radiation is integrated over all angles, it has been shown to have an effective incidence angle of approximately 60 degrees for vertical and horizontal surfaces (Duffie and Beckman 1991). Isotropic ground-reflected diffuse radiation for vertical surfaces also has an effective incidence angle of approximately 60 degrees. To simplify the analysis, and because diffuse radiation from the horizon is a small part of the total radiation, it is treated in the same manner as isotropic sky and ground-reflected diffuse radiation.
To determine the transmitted solar radiation, an incidence angle of 60 degrees was used for all sky radiation (including diffuse radiation from the horizon but not circumsolar diffuse radiation) and ground-reflected radiation values. For direct beam and circumsolar diffuse radiation, the incident angle was the angle of incidence for the direct beam radiation.
Transmitted solar radiation for shaded windows.Windows externally shaded by a roof overhang were treated differently than unshaded windows in two ways. First, the direct beam radiation component and the circumsolar diffuse radiation were reduced if portions of the window were shaded. Second, the isotropic diffuse sky radiation were reduced to account for the reduced field of view of the sky because of the roof overhang. The presence of the roof overhang was assumed to have no effect on the transmitted diffuse radiation from the horizon and on the ground-reflected diffuse radiation.
The shading geometry selected for each station balanced the need for maximum solar heat gain for south-facing windows during the heating season without creating unreasonable solar heat gain during the cooling season. For each station, the same shading geometry was used for all vertical windows, and the roof overhang was assumed to extend and infinite distance with respect to the window width.
Two angles ( and ) describe the shading geometry. These angles determine the amount of shading of south-facing windows at solar noon throughout the year. If the sun elevation at solar noon is greater than (summer), then the window is completely shaded. If the sun elevation at solar noon is less than but greater than (spring and fall), then the window is partically shaded. If the sun elevation at solar noon is less than (winter), then the window is completely unshaded.
For most stations in this manual, = 108 degrees -latitude and = 71 degrees -latitude. This shading geometry provides no shading of the south-facing window from November 17 to January 25, and provides complete shading of the window (at solar noon) from May 12 to August 2.
Stations in southern states can benefit from more summertime shading; therefore, the shading geometry was modified to accommodate a longer shading period. Their monthly heating degree day (base 65 degrees Fahrenheit) requirements were examined to find the first fall month wiht a value greater than zero. If this month was October or later, then = 92 degrees Fahrenheit -latitude and =66.5 degrees -latitude.
For these stations, this provides no shading of the southfacing window only on December 21, and provides complete shading of the window (at solar noon) from March 26 to September 18.
Other exceptions to me shading geometry were also made. Hawaii, Guam, and Puerto Rico have zero heating degree days; consequently, their shading geometry provides complete shading of south-facing windows at noon throughtout the year. Shading geometries that provide complete shading of south-facing windows at noon throughout the year were also used for stations that had more cooling degree days than heating degree days in December. This included stations in southern Florida. Alaskan stations, with no summer cooling loads, have shading geometries that do not shade south-facing windows at noon throughout the year. For the situations described in this paragraph, the roof overhang width was calculated using 1.0 for the window height and 0.2 for the vertical distance from the window to the overhang.
For each hour, the shading geometry and trigonometric relationships (ASHRAE 1993) were used to determine the fraction of window that was not shaded. This fraction was then multiplied by the transmitted direct beam component for unshaded windows to determine the transmitted beam radiation for shaded windows. Circumsolar diffuse radiation was treated in the same manner.
The transmitted isotropic diffuse sky radiation was reduced to account for the reduced field of view of the sky because of the roof overhang. The fraction of the sky viewed by a vertical window with a roof overhang to that viewed by a vertical window without an overhang can be determined using methods presented by Iqbal (1983). These methods determined to be , where varies from (top of window) to (bottom of window).
An average fraction for the window was calculated by dividing the window into 100 equal horizontal segments and finding the average , where is based on the midpoint of each segment. This average fraction was multiplied by the transmitted isotropic diffuse sky radiation for unshaded windows to determine the transmitted isotropic diffuse sky radiation for shaded windows.
Hourly values of the total transmitted solar radiation were determined by summing the transmitted component values for direct beam radiation, circumsolar diffuse radiation, isotropic sky diffuse radiation, horizon diffuse radiation, and ground-reflected diffuse radiation.
Incident illuminance for the horizontal window and vertical windows facing north, east, south, and west was determined using equation 1,2,and 3 and inputs of global horizontal illuminance, direct beam illuminance, and diffuse horizontal illuminance instead of their solar radiation counterparts. When used to calculate the diffuse illuminance for a tilted surface, equation 2 uses a different array of values for model coefficients and than when it is used to calculate the diffuse solar radiation for a tilted surface. The input illuminance values were calculated using luminous efficacy models developed by Perez et al. (1990).Inputs to the luminous efficacy models are global horizontal radiation, direct beam radiation, diffuse horizontal radiation, and dew point temperature.
For each station location and window orientation, hourly values of solar radiation and illuminanace for the windows were calculated. Monthly and yearly averages for solar radiation and hourly average profiles for illuminance for 4 months of the year were then determined for the period 1961-1990. Illuminance profiles were determined for mostly clear and mostly cloudy conditions by calculating separate averages for hourly illuminances when the hourly total cloud cover was less than 50% (mostly clear) and when the hourly total cloud cover was equal to or greater than 50% (mostly cloudy).
For a few stations, the averages do not include data for 1989, 1990, or both because NSRDB data did not include those station years. The stations with less than 30 years of NSRDB data and their period of record are listed below:
Eagle, CO 1961-1988
Minot, ND 1961-1988
Miles City, MT 1961-1989
Cut Bank, MT 1961-1988
Burns, OR 1961-1988
The solar radiation and illuminance values were calculated using improved models and data. The estimated data uncertainities assigned to the calculated values show how they might compare with true values. They were determined using the uncertainity method of Abernethy and Ringhiser (1985).This root-sum-square method defines an uncertainty, in which 95% of the time, the true value will be within plus or minus the uncertainity of the calculated value.
equation (7)
where
Random and bias error.The two types of errors that contribe to uncertainties are random errors and bias errors. Random errors usually follow statistical distributions and result in values both above and below the true values. Random errors tend to cancel when individual values are used to determine an average. For example, a 30-year monthly average of solar radiation may use 10,800 hourly values (assuming 30 days per month and 12 hours of sunlight per day) to determine the average monthly solar radiation. The random error of the average is reduced by a factor of 10,800^{1/2}, or approximately 100. For the hourly averages of illuminance, each hourly average is based on approximately 900 hourly values. The random error of their average is reduced by a factor of 900^{1/2}, or 30. Consequently, random error sources do not contribute significantly to the uncertainly of the solar radiation and illuminance averages.
Bias errors, however, are not reduced by averaging. Bias errors, which are often referred to as fixed or systematic errors, cause values to be in error by about the same amount and direction. The reason for bias errors, as well as their magnitude and direction, may be unknown; otherwise, corrections such as changes in calibration factor can be made. When detailed information is not known about the bias errors, reasonable estimates of the bias error magnitude can be made using procedures similar to those described in this section.
For the solar radiation and illuminance averages, we evaluated the three major bias errors: (1) errors in direct beam radiation and direct beam illuminance incident on the window because of errors in NSRDB direct beam radiation data, (2) errors in diffuse radiation and diffuse illuminance incident on the window because of errors in NSRDB diffuse horizontal radiation, and (3) errors in diffuse radiation and diffuse illuminance incident onthe window because of errors from modeling the diffuse solar radiation or diffuse illuminance for the window. Climate change could also bias monthly average solar radiation and illuminance values but was not considered a major source or error for this work.
The analysis beginning in the next paragraph pertains to the solar radiation incident on the windows, but an analysis for incident illuminance would yield similar results because: (1) errors in the direct beam illuminance and the diffuse horizontal illuminance are predominately the result of errors in the direct normal radiation and diffuse horizontal radiation from which they are modeled, and the errors are the same relative magnitude, and (2) errors from modeling the diffuse solar radiation and diffuse illuminance for the window for the window are approximately equal.
The root-sum-square of the individual bias errors yields the total bias error and, because the random error is negligible, is the same as the total uncertainty of the monthly averages. Consequently, the uncertainty, , can be expressed as:
equation (8)
where
The bias errors for direct beam and diffuse horizontal radiation were extracted from the NSRDB daily statistic files for each station. The NSRDB daily statistic files include, among other information, 30-year averages and their uncertainties for direct beam and diffuse horizontal radiation. An integer number represents an uncertainity range. Examples of uncertainty ranges for the monthly averages are from 6% to 9%, from 9% to 13%, and from 13% to 18% of the monthly average.
For 30-year averages, most of the stations have direct beam radiation uncertainties in the 9% to 13% range. The remaining stations have direct beam radiation uncertainties in the 6% to 9% range and diffuse horizontal radiation uncertainties in the 9% to 13% range. The remaining stations have direct beam radiation uncertainties in the 9% to 13% range and diffuse horizontal radiation uncertainties in the 13% to 18% range. For the purpose of extracting the bias errors from the daily statistic files, a single integer value near the midpoint of the range was used (8% for the 6% to 9% range, 11% for the 9% to 13% range, and 16% for the 13% to 18% range).
The bias error for modeling the window radiation is attributed to the diffuse solar radiation model because the direct beam component is considered an exact solution (). An evaluation of the original Perez model by Hay and McKay (1988) provided information whereby the bias error was estimated to be about 5% of the total window radiation of the applications.
The uncertainty, , can be expressed as a percentage of the total window radiation by the following equation:
equation (9)
where
Uncertainty values in tables. Because of the large number of solar radiation and illuminance values presented in the manual, it was judged impractical with respect to space limitations to present uncertainty values for each solar radiation and illuminance value. Rather, a simplifying assumption was made so that only one uncertainty value was presented for all windows. The assumption was that the direct beam radiation and diffuse radiation incident on the window were of equal weight. The uncertainties of the diffuse horizontal and direct beam radiation have about the same value, so this assumption did not create large changes in calculated uncertainties for window radiation.
Over a range of direct beam radiation to diffuse radiation ratios (30/70 to 90/10), the assumption yield uncertainties within 1% or 2% of uncertainties calculated using the exact proportions of direct beam radiation and diffuse radiation (uncertainty of 8% or 10% instead of 9%, and so on). This was judged acceptable, considering that there are uncertainties associated with the uncertainty values used for the average monthly diffuse horizontal radiation, and the solar radiation modeling for tilted surfaces. As a conservative measure, the calculated uncertainties were rounded to the next highest integer value.
For most of the stations in the data manual, uncertainties of 9% were assigned to the solar radiation and illuminance data. The few stations with higher uncertainties for direct beam and diffuse horizontal radiation were assigned uncertainties of 11%.
The climatic data were derived using both data from the NSRDB and from climatic data sets provided by the National Climatic Data Center (NCDC), Asheville, North Carolina.
Climatic data pertaining to average temperature, average daily minimum temperature, average daily maximum temperature, average heating degree days base 65^{o}F (18.3^{o}C), and average cooling degree days base 65^{o}F (18.3^{o}C) were extracted from NCDC's data tape "1961-1990 Monthly Station Normals All Elements." This data tape includes temperature and degree day normals for about 4775 stations in the United States and its territories. The normals are average computed by NCDC for the period of 1961-1990.
For this data set, NCDC used procedures, when possible, to estimate missing data and to correct for other inconsistencies by using data from neighboring stations. For one of the stations in this data manual, data were not available on NCDC's data tape. For this station, in Arcata, California, the averages were computed using NSRDB data, but no attempt was made to estimate missing data or to correct for other inconsistencies.
NSRDB hourly data were used to calculate the average wind speed, average clearness index, and average humidity ratio. The average humidity ratio was determined by first calculating the average station pressure (p) and average partial pressure of water vapor (). Partial pressure of water vapor values were determined using psychometric relationships (ASHRAE 1993) and NSRDB dew point temperatures. The humidity ratio (W) was then calculated using the following equations:
equation (10)
This approach gives a more accurate portrayal of the average water vapor content in the air than would be given by averaging hourly values of the humidity ratio or relative humidity (Linacre 1992).
Record minimum and maximum temperatures were obtained primarily from NCDC's data diskette "Comparative Climatic Data Tables-1991." This data diskette contains, among other useful parameters, record minimum and maximum temperatures for about 90% of the stations in this manual and spans periods of records back to 1948 and earlier. For the remaining 10% of the stations, record minimum and maximum temperatures are based on NSRDB data.
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