SOLAR RADIATION AVAILABILITY ON SURFICES IN THE UNITED STATES AS AFFECTED BY SEASON, ORIENTATION, LATITUDE, ALTITUDE AND CLOUDINESS By Clarence Frederick Becker AN ABSTRACT Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering Approved by V-' ProQ uest Number: 10008519 AH rights reserved INFO RM ATIO N TO ALL USERS The quality o f this reproduction is dependent upon the quality of the copy subm itted. In the unlikely event that the author did not send a com plete m anuscript and there are m issing pages, these will be noted. Also, if m aterial had to be removed, a note will indicate the deletion. uest ProQ uest 10008519 Published by ProQ uest LLC (2016). C opyright o f the Dissertation is held by the Author. All rights reserved. This w ork is protected against unauthorized copying under Title 17, United States Code M icroform Edition © ProQ uest LLC. ProQ uest LLC. 789 East Eisenhow er Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346 Clarence Frederick Becker 1 Radiation from the sun is the primary source of all energy used by mankind. The life expectancy of fossil and nuclear fuel reserves are difficult to determine with ac­ curacy; however, the earth*s supply of fuels is limited and severe shortages of this form of fuel are possible within a century. There is a need to learn how to use directly the daily supply of energy from the sun before stockpiles of other energies are exhausted. Some of the potential uses for solar radiation on the farm, such as final drying of grain, hay and other agricultural crops can utilize solar radiation with little disadvantage due to the intermittency of the radiation. It is necessary to know the availability of solar radi­ ation at the surface of the earth if it is to be used in engineering applications. The United States has a network of Weather Bureau stations which measure and record solar radiation incident upon a horizontal surface. Records of radiation Incident on vertical surfaces are available for one United States station. Methods for estimating the quantity of solar radiation available on surfaces with various orientations in the United States are presented in this thesis. Calculations of hourly and daily total cloudless day radiation incident on surfaces with various orientations Clarence Frederick Becker 2 and located at different north latitudes with elevations at or near sea level are presented. Seasonal curves showing the availability of the solar radiation on surfaces with various orientations have been constructed and polynomials have been determined for the curves representing the avail­ ability of solar radiation on a horizontal surface. Optimum, tilt angles of solar collectors for the various seasons and latitudes are presented, and the ratios of solar radiation incident upon tilted surfaces with various tilt angles to that incident on horizontal surfaces are presented in the form of curves for which polynomials have been developed. The calculated ratios can be multiplied by the values given for a horizontal surface to give the corresponding value for a tilted surface at various tilt angles. Curves and polynomials giving the percentage increase in solar radiation with altitude are shown which can be used to correct the values for conditions at or near sea level. Comparisons between the calculated cloudless day radi­ ation determined from the curves and polynomials developed above and recorded clear day solar radiation are presented. The correlation between the ratio of observed to cal­ culated cloudless day radiation and percentage of possible sunshine is developed. The resulting regression equ at I on gives a method for correcting the cloudless day values for cloudiness by relating it to a parameter which is measured Clarence PrederIck Becker 3 at many places and for which long time averages are avail­ able . The variability of the availability of solar radiation is discussed and the distribution of days with various amounts of solar radiation and the mean total solar radiation in vari­ ous categories are shown for Madison, Wisconsin based on data beginning July of 191^1 • The sequence of dark days for the various months is determined for the same same period. station during the OLAR RADIATION AVAILABILITY ON SURFACES IN THE UNITED STATE AS AFFECTED BY SEASON, ORIENTATION, LATITUDE, ALTITUDE AND CLOUDINESS By Clarence Frederick Becker A THESIS Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1956 ACKN CWLhDGKEK T S The author wishes to express his sincere thanks x'cr the guidance and constant supervision of Doctor Janes Boyd of the Department of Agricultural Bngineering of Michigan State University, He is also greatly indebted to Doctors J, T. Andersen, W. D* Baton, W, M. Carleton, B, Hi, Stewart, L. h. Turk, and D. J. Montgomery for their helpful suggestions during the preparation of the manuscript. Sincere thanks is extended Doctor Arthur Farrall, iieaa of the Department of Agricultural, ijnpineeririg, i ox* appro­ priation of the funds necessary for carrying on the work. Acknowledgment is due Doctor Frederick Due low of the Department of Agricultural Engineering for helpful sugges­ tions offered during various aspects of the stuay, The author is grateful to his wife, Irene, who typed and helped prepare the rough copy of the manuscript. i VITA Clarence Frederick Becker candidate for the degree cf Doctor of Philosophy Final Examination: Dissertat I o n : June H 4., 1956, 2:50 P.M., Boom 216, Agricultural Engineering Building Solar Radiation Availability on Surfaces in the United States as Affected by Season, Orientation, Latitude, Altitude ana C-loudin ess Outline of Studies Major Subject: Minor Subject: Agricultural Engineering Statistics Biographical Items Born: March 27, 1920, Ro ge r s , Berth Dakota Undergraduate Studies: Graduate Studies: Experience: North Dakota Agricultural College, 1938-1+3. E.S., 191+3. North Dakota Agricultural College, 19 /4.6 -4 7 , M.S., 1949; University of Wyoming, 1952-55; Michigan State University, 1955-56. U. S. Air Force, 1943-46; Graduate Research Assistant, North Dakota Agri­ cultural College, 1946-4?; District Representative, John Bean Mfg. Co., 1948-49; Assistant Agricultural Engineer, University of Wyoming, 1949-1956. Honorary Societies: Alpna Beta Phi Kappa Phi Blue Key Sigma PI Sigma Professional Societies: American Society of Agricultural Engine e.rs ii TABLE OP CONTENTS PAGE INTRODUCTION 1 Statement of the Problem Objective of Thesis • • • • • • • * . . . . 1 ...................... * * 4 History of the Use of Solar R a d i a t i o n ......... 5 Solar Energy Outside the Atmosphere 6 * • • • • • ........................ Solar Constant Spectral Distribution ......................... 6 7 Position of Earth Relative to the S u n ......... 8 INTENSITY OP SOLAR RADIATION UPON SURFACES LOCATED AT THE SURFACE OP THE EARTH DURING CLOUDLESS DAYS Direct Radiation ............. . . • • * • • • • • Direction Radiation during Cloudless Days • ♦ Direct Solar Radiation Incident upon a Surface Perpendicular to the S u n ’s Rays • • Solar Angles * Angle of Incidence for Direct Radiation • . . Horizontal surface South-facing vertical surface South-facing tilted surface 9 9 9 10 11 14 14 15 15 Sky R a d i a t i o n ................* .................... 15 Total Solar R a d i a t i o n ..............................17 DAILY TOTAL SOLAR RADIATION AVAILABLE AT THE SURFACE OF THE EARTH DURING CLOUDLESS DAYS FOR LOW ELEVATION AREAS 19 Procedure for Calculating Hourly Rates ......... 19 Procedure for Calculating Daily Total Radiation. 20 iii TiibLL OF COHTEHuS (Cont.) Pa^e Results 20 Horizontal Surface ........................ South-Facing Vertical Surface . . . . . . South-Facing Tilteu Surface Optimum Tilt Angles for Sclar Collectors « 21 21+ 25 * Ratio cf Tilted to Horizontal Surface Incident R a a i a t i on .................................. 31 Cor;pari Fori between Calculated Cloudless Dav ana ................ Recorded Clear Day Radiation 35 Procedure Results . . . . . . . . . . . . . . . . . Come arisen b e t w e m calculated a no Observed Ratio of Vertical to Horizcrtal Surface Incident Relation Procedure Results ............................... .. .......................... VAR IAT I OH IN SOLAR RADIATI01 IHTENSITY Review of Literature b_'THALTITUDE . . . . . . . 35 37 Lb i-o l\.t 50 50 Procedure for Calculating Corrections due to Altitude 55- Percentage Increase of Solar Radiation Intensity with A l t i t u d e ............................. 55 Comparison between Calculated Cloudless Day and Recorded Clear Day Radiation for Two High Altitude Stations ........................... EFFECT OF CLOUDS 01. SOLAR RADI All OH ih TENSITY . . . Review cf Literature Procedure and Results ........................... .......... . ......... DISTRI BUTTON OF DATS *fITII VARIOUS AitOUNTS OF SOL,2 RADIATION . Iv 5b 6: cC TABLE OP CONTENTS (Gont.) Pag© Average Number of Days in Various Categories for Madison, W i s c o n s i n .............................61j. Mean Total Solar Radiation in th.e Various ............. Categories 68 Sequence of Dark D a y s ...................... 69 CONCLUSIONS AND PROPOSED FUTURE RESEARCH APPENDIX I APPENDIX II ............. 71 T A B L E S ........................................ 75 SAMPLE COMPUTATION ...................... 101 GLOSSARY 1C4 BIBLIOGRAPHY 105 v LIST OF FIGURES FIGURE 1. PAGE Direct solar radiation incident upon a surface perpendicular to sun's rays at sea level on the earth during cloudless days. 12 2. Definition of solar angles. 13 3* Sky solar radiation (Irfh) for various values of direct radiation (1^) upon a horizontal surface during cloudless days with clearness ratio equal to one. 18 Sky solar radiation (I^v ) upon a vertical surface for various solar altitudes (b) and wall solar azimuth (a) during cloudless days with clearness ratio equal to one. 18 Daily total direct and sky radiation incident upon a horizontal surface at various north latitudes during cloudless days. 22 Daily total direct and sky radiation incident upon a vertical south-facing surface at various north latitudes during cloudless days. 25 Daily total direct and sky radiation incident upon a south-facing surface sloped 3 0 ° from vertical at various north latitudes during cloudless days. 26 Daily total direct and sky radiation incident upon a south-facing surface sloped 6 0 ° from vertical at various north latitudes during cloudless days. 27 I4.. 5* 6. 7* 8. 9. 10. 11. Number of degrees of tilt south-facing surface from vertical to make it perpendicular to sun's rays at solar noon. 30 Ratio of direct and sky radiation incident upon south-facing surfaces of various tilt angles to radiation incident upon a horizontal surface. 32 Ratio of direct and sky radiation Incident upon south-facing surfaces of various tilt angles to radiation incident upon a horizontal surface. 33 vi LIST OP FIGURES (Cont.) Page 12. x3. 11*. 15. 16 . 17. Comparison between recorded and calculated direct and sky radiation incident upon a horizontal surface at Madison, Wisconsin during days with 0 - 3 tenths cloud cover, 1950 through 1954• 38 Comparison between recorded and calculated direct and sky radiation incident upon a horizontal surface at Lincoln, Nebraska during days with 0 - 3 tenths cloud cover, 1950 through 1954* 39 Comparison between recorded and calculated direct and sky radiation Incident upon a horizontal surface at Blue Hill, Massachusetts during days with 0 - 3 tenths cloud cover, 1950 through 1954* 40 Comparison between recorded and calculated direct and sky radiation incident upon a horizontal surface at East Lansing, Michigan during days with 0 - 3 tenths cloud cover, 1950 through 1951)-. 1+1 Comparison between recorded and calculated direct and sky radiation incident upon a vertical surface at Blue Hill, Massachusetts during days with 0 - 3 tenths cloud cover, 1950 through 1954* 45 Comparison between actual and calculated ratio of total radiation incident upon a south-facing vertical surface to that incident upon a horizon­ tal surface at Blue Hill, Massachusetts during days with 0 - 7 tenths cloud cover, 1950 through 1951*. . 18 19. 20 . 14-7 Mean summer and winter transmission coefficients with cloudless sky at high level stations as a function of altitude. 52 Percentage increase of solar radiation intensity with altitude. 53 Comparison between recorded and calculated direct and sky radiation Incident upon a horizontal surface at Lander, Wyoming during days with 0 - 3 tenths cloud cover, 1950 through 1954* 57 vii LIST OF FIGURES (Cont.) Page 21. 22 Comparison between recorded and calculated direct and sky radiation incident upon a horizontal surface at Albuquerque, New Mexico, during days with 0 - 3 tenths cloud cover, 1950 through 195^N between ratio of . Relationship cloudless day radiation (i/i possible sunshine. actual to calculated > and percentage of ° viii 58 63 LIST OP TABLES TABLE I* II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. PAGE Coefficients of the polynomial, Y * a cX , (m), for the various curves of 10 and 11 + bX ♦ Figures 3k Average distribution of days with various categories of radiation as received on a horizontal surface for Madison, Wisconsin 19^4-1 through 1955 ........................... 66 Maximum sequence of days in category . . 70 Calculated solar radiation per square foot of surface during cloudless days 3 0 ° north latitude .......................... 76 Calculated solar radiation per square foot of surface during cloudless days 35° north .................................. latitude 80 Calculated solar radiation per square foot of surface during cloudless days i^0° north latitude .................................. 81p Calculated solar radiation per square foot of surface during cloudless days k S ° north latitude • • • • • • • • • • • • • • • • 88 Ratio of horizontal to south-facing tilted surface incident radiation ................ 92 Average number of clear, partly cloudy, and ............................... cloudy days 93 Percentage of possible sunshine .............. 95 Relationship between the ratio of observed (I) to calculated cloudless day radiation (I ) and percentage of possible sunshine (S) . • 96 IV . Monthly mean precipitable water in the United States ............................. ix 100 1 INTRODUCTION Radiation from the sun Is essentially the primary source of all energy used by mankind* are most easily utilized, Energy supplies that such as the fossil fuels, have undergone natural concentration* water vapor above the earth, Energy from the sun lifts and this energy is partially recoverable in the form of water power. Energy for moving masses of the atmosphere over the surface of the earth, or the production of wind, comes from the sun. Statement of the Problem The life expectancy of fossil fuel reserves is diffi­ cult to determine with accuracy, for it involves not only the uncertainty of estimates of the quantity of fuel present, but also the prediction of rates of production and of de­ mand (2)• In addition, there are coal, oil, and gas in the earth*s crust that may never be used because it would not pay to dig to the depths necessary to recover them. The finding of large deposits of uranium in recent years will alleviate the energy situation, however, this type of fuel, like fossil fuels, cannot be considered inexhaustible. 2 It is evident that the earth*s supply of fuels, wood, gas, oil and coal is limited and that severe shortages of this form of energy is possible within a century. There is a need to learn how to use the daily supply of energy from the sun directly before the stockpile of other energies is exhausted. Solar radiation energy is immense in quantity but relatively low in intensity. Threlkeld (28) estimates that the solar radiation intercepted by the earth each day equals about £600 million million BTU. Because of the vast­ ness of the supply of solar energy, the direct conversion of solar radiation into useful forms of energy has great attrac­ tiveness. Solar radiation is currently being used on a limited basis for distillation of salt water, for manufacture of salt by solar evaporation, for water heating in Southern United States, and for generation of electricity by thermo­ piles, the photogalvanlc effect, or by the solar battery utilizing a silicon p-n junction. Research is under way to determine other methods of transforming and storing solar energy. Work is being done in photosynthesis in an attempt to shorten the time necessary for completion of the process by which petroleum was naturally formed (22); this work is coupled with the culture of algae and the subsequent pro­ duction of carbohydrates with the most desirable properties. 3 The utilization of solar energy by the above-mentioned methods is still very inefficient* However, the direct col­ lection of solar energy as heat is much more efficient and the greatest progress is likely to take place in this direc­ tion in the near future. heating others Experimental solar collectors for air have been built by Buelow (5), Telkes (9), and collectors for heating water by Hottel (26) and have been built (12) and the University of Florida (9)* These collectors show efficiencies as high as $0 to 80 percent* The possibility of utilizing solar energy on the farm has been investigated only to a very limited extent, though there are many potential uses. even In many respects it seems to be the logical place to start, for one of the dis­ advantages of solar radiation, namely, its intermittency, is not critical for such uses as final drying of grain, hay and other agricultural crops. to be used for space heating, Where solar energy is some means of heat storage at relatively high temperatures is necessary. of collection is thereby reduced, The efficiency since threshold intensity of radiation must be reached before energy can be collected and stored. Solar radiation of relatively low Intensity could be utilized for drying agricultural crops, where any amount of heating of air would be advantageous. A solar energy collection and storage system could also supply supplemental heat to farm buildings* There is a network of Weather Bureau Stations in the United States which measure and record solar radiation inci­ dent upon a horizontal surface. The first solar station in the United States was started at Madison, Wisconsin, in 1911* The number of stations has increased from ten in 19i(-0 to twenty-five in 191^9, and to seventy-five today* Daily total values of solar radiation incident upon a horizontal surface are published by the Weather Bureau (31)* In addition to the data for radiation on horizontal surfaces, a station of the Weather Bureau at Blue Hill, Massachusetts, measures and publishes data for radiation incident upon vertical surfaces facing north, south, east and west. Objective of Thesis To utilize solar energy at the surface of the earth for engineering application, it is necessary to determine the availability of the supply. In many cases it may be desirable to orient surfaces other than horizontally for more efficient collection of solar energy or for structural reasons. Frequently the col­ lecting surface will be incorporated into the wall or roof of a house or farm building in which case the horizontal orientation may not be practical. It is the purpose of this thesis to construct curves and develop equations useful for estimating quickly the 5 quantity of solar radiation energy available on surfaces with various orientations anywhere in the United States* History of the Use of Solar Radiation Investigations on potential and actual use of solar radiation are not new since the sun was recognized as a source of energy hundreds of years ago. Green (9) made a brief summary of the history of solar utilization which is quoted as followsJ In 1902 H. E. Willsie, American Engineer, built a binary-vapor pumping plant run by a collector. Heated water was stored and used to boil off sulphur dioxide which ran a small pumping engine. The heat­ er and collector had an asphalt bottom, wooden sides, and was covered with glass. On later more efficient collectors he used a double glass cover and tilted the collector so that it was perpendicular to the s u n ’s rays. Several such plants were built after 1909""by Frank Shuman, Philadelphia engineer. The oldest method of collecting solar energy at high tempersTbure^was by concentrating the rays of the sun upon a surface by the means of mirrors. Archimedes is said to have useclthis method as early as 21i|_ B.C. to set fire to Roman ships while they were at considerable distance from the shores of the island of Sicily. In 1878 August Mouchot first generated power from the reflected and concentrated rays of the sun. He used a huge reflector shaped like a lamp shade and lined with burnished silver to reflect and concentrate rays of the sun upon a boiler located at the focal axis of the reflector. By using a large reflector area as compared to boiler area, he was able to get a steam pressure of 75 pounds per square inch (325 F approximately) and run a one-half horsepower engine. The diameter of the conoidal reflector was eight and one-half feet. The noted engineer, John Ericson, later experi­ mented with this type of equipment, using a parabolic 6 trough lined with small mirrors to reflect the rays of the sun upon a small tubular boiler located along the focal axis of the reflector. He obtained as high as 210 BTU per square foot of sunshine collector. About 1901 the English inventor, A. G-. Eneas, built several reflectors of the truncated-cone type that ab­ sorbed as high as 223 BTU per square foot of sunlight collected at their Arizona location. In 1913 a successful steam plant using parabolictrough reflectors which were geared to an engine so as to be kept facing the sun continually, was built by Frank Shuman and Professor Boys at Meadi, Egypt. This plant developed 50 horsepower and collects 13*269 square feet of sunshine. (...) The University of Florida built a parabolictrough heater 3 x 1+ feet in 193U- with one and onequarter inch pipe placed inside a 2-inch pipe located at the focus. Water circulated between the two pipes by gravity, provisions were made to measure the quan­ tity of water and temperature rise; results showed as high as 128 BTU per square foot per hour of sun­ shine collected was possible. Operating temperatures as high as 220 F. were readily obtained. When filled with water and with all outlets closed, the steam pressure rose to 50 pounds per square inch gage, in­ dicating a temperature in the absorber of approxi­ mately 300 F. (...) Solar Radiation Outside the Atmosphere Solar Constant The solar constant is the energy incident upon a unit area located at mean distance of the earth from the sun and oriented perpendicular to the sun's rays outside the atmos­ phere. The most recent data available on the value of the solar constant is presented by Johnson (18). His value is 2.00 + O.Olj. calories per minute per square centimeter or about )|)|0 BTU per square foot per hour. The latest value given by the Smithsonian Institute Is 1.95 calories per 7 minute per square centimeter. The difference between these two values lies in a rediscussion of the corrections applied to the measured data for the ultra violet and infrared spec­ tral regions lying outside the observed region. Spectral Distribution The spectral distribution of solar energy outside the atmosphere is such that for all practical purposes, the energy lies between the limits of 0*22 and 7 microns. About 0,U|_percent of the energy has wave length greater than 7 microns and 0,02 percent of it has wave length les3 than 0,22 microns. According to Fritz (7), the energy in the ultra violet below 0 mk. microns comprises about 9 percent of the total incident energy; the energy in the visible range tains the peak at 0,1^6 microns) (which con­ is L|.l percent and that in the infrared beyond 0,72 microns contains about 5>0 percent. In tracing the solar spectrum down t hrough the atmosphere, the short wave constituents are absorbed high in the ionos­ phere, principally by ozone. The absorption coefficient is such that the spectrum of solar energy at the ground is cut off below 0,29 microns. There appears to be irregular changes in solar activity which change the solar constant. However, the effect on the integrated energy available for non-selective power sources is very small (11), 8 Position of the Earth Relative to the Sun The angle at which the s u n ’s rays reach a surface on the earth changes from day to day and from hour to hour owing to the changing position of the earth relative to the sun and the rotation of theearth about its axis. As the earth moves about the sun in its seasonal orbit, its axis maintains a nearly constant angle of 66.5 degrees with the plane of its orbit. On June 21, the summer solstice, the sun is vertically overhead at solar noon at the Tropic of Cancer and has the greatest noon elevation in all north latitudes greater than 23-5 degrees. winter solstice, sun and the axis of the earth points away from the the angle which tal surface On December 21, the thes u n ’s rays make with a horizon­ on the earth is least in northern latitudes. The sun is directly overhead at noon at the equator on March 21, the vernal equinox, and on September 21, the autumnal equinox. Days and nights are of equal length throughout the world at the time of the equinox. The sun is slightly off the center of the elliptical path which the earth makes around the sun. As a result, at aphelion (July 1), the distance of the earth from the sun is 1.03J+ that at perihelion (December l). Hence, the intensity of sunshine outside the atmosphere on December 1, other things being equal, is approximately 1.069 that on July 1. 9 INTENSITY OP SOLAR RADIATION UPON SURFACES LOCATED AT THE SURFACE OP THE EARTH DURING CLOUDLESS DAYS Direct Radiation Direct Radiation during Cloudless Days The quantity of direct solar radiation incident upon a surface located on the earth*s surface is given by the formula: (see Glossary, page 101}., for definition of terms) If = (JQ/ r 2 )tm cos i where ........ (a) 1^ = the solar energy incident upon the surface JQ = the solar constant r - the radius vector of the earth t — the transmission coefficient of the atmosphere m * the air mass i = the angle of incidence* In tracing the solar spectrum down through the atmosphere during cloudless days, depletion of the direct beam takes pJLace by scattering and absorption (7, 21), Scattering is caused primarily by air molecules, dust, and to a certain extent, by water vapor along the path of the sun's rays to the surface of the earth. The principal absorbing agents are water vapor, ozone and cloud particles. 10 Direct Solar Radiation Incident upon a Surface Perpendicular to the Sun's Rays In a very important paper, Moon (23) has calculated the direct solar radiation incident upon a surface normal to the s u n fs rays during cloudless days. His calculations of the spectral distribution of the energy at sea level are based on an assumed atmospheric condition of 2 centimeters of precipitable water vapor, 300 dust particles per cubic centimeter and 0.28 centimeters of ozone at 760 millimeters of mercury pressure and 0 C. air masses of 0, 1, 2, 3* U The calculations were made for anci Curves constructed by Moon from the calculations show the spectrum of solar energy cut off below 0.29 microns wave length, and a large vertical sep­ aration of the curves for short wave lengths. This vertical separation shows the influence of ozone absorption as the sun passes through longer and longer paths. The absorption bands of water vapor are very evident in the infrared region; no energy reaches the surface of the earth in the stronger water vapor bands. Moon's data have been taken as standard for cloudless day summer conditions by the American Society of Heating and Ventilation Engineers. The integrated direct solar radiation as a function of the solar altitude appears in the Heating, Ventilating and Air Conditioning Guide (1). P. W. Hutchinson and W. P. Chapman (lU) have applied Moon's calculations to a standard winter atmosphere with 11 assumed atmospheric conditions of 33 F dew-point, 300 dust particles per cubic centimeter, and 0*23 centimeters of ozone* son^ The winter curve in Figure 1 is a copy of Hutchin­ data which show direct solar radiation at normal in­ cidence as a function of the solar altitude angle. The sum­ mer curve is the result of a plot of the information presented in the Heating, Ventilating and Air Conditioning Guide ci t,)* (op. It is noted that for a given solar altitude, the summer value is lower than the winter value* The summer value is smaller because of an assumed higher moisture con­ tent and due to a greater solar distance* Solar Angles The sun as seen by an observer on the surface of the earth follows a circular arc from horizon to horizon* position can be defined by the solar altitude solar azimuth (a), Figure 2* This (b), and the These angles vary continuously from sunrise to sunset and are different for various days of the year. The diurnal variation is symetrical with respect to a north-south line and therefore with respect to solar noon* The sun's altitude and azimuth can be secured from the United States Hydrographic Office Tables 2 II4. (29) If the latitude, known. local hour angle, and declination of the sun are The declination of the sun can be secured from any 12 d S) (*<*H) Jsd jQ j;g ‘ 90U 6PTO U I p.B u o f ^ .B fP 'B H >P Cl cfl aj aj CD d d p CL cd CO d = CD CO CD O CD rH DO Cd i—i rH CO CO o d^ 3-0®>iTQ bO d d CD d d o > COI—1 d • *H o d— d O rH cO CD d •H CD i—1 > •rH d rd d p —■*• P d o d o d d CD P o o > d CO CO o *H rH d d •H d 0 d p cO > rd co CO •H d CD O i—1 P bOd d d d cd cO ctf K H W rH CD p O co >>aJ d ca d CD d p cd p cd cO d o CD CO co i_r\ co d S>d CT' P •H cO rH ca d e o d d CO d •rl CD • *-

p O bO d d d t H o d CDCL, pt« p o CO5^ 13 4 -> S Q • OJ m bQ •H 114- table of ephemeris. For all practical purposes, the declin­ ation is the same on January 21 and November 21, February 21 and October 21, March 21 and September 21, April 21 and August 21, and Tor May 21 and July 21; therefore, the solar geometry will be nearly the same on the dates with equal declinations. using The local hour angle is expressed in degrees 0 degrees 1 p.m., for solar noon,15 degrees for 11 a.m. and 30 degrees for 10 a.m. and 2 p.m., etc. Angle of Incidence for Direct Radiation A formula for determining the angle of incidence of direct solar radiation upon a surface is given by Brown and Marco (i+); their formula was changed to the following form by trigonometric substitution: cos i = sin b sin e +cos b cos a cos where e •••• (b) i = angle of incidence a = wall solar azimuth b = solar altitude e = angular tilt of surface Figure from vertical. 2 defines the solar angles used. Horizontal surface. For a horizontal surface, e in equation (b) is 90 degrees and the equation becomes: cos i = sin b Ijk = IQ sin b ....... .. (c) (d) 15 where 1^ *= direct solar radiation incident upon a horizontal surface, Ic » direct solar radiation incident upon a surface normal to the sun's rays (Figure l ) • South-facing vertical surface. e inequation b For a vertical surface, is 0 degrees and the equation cos i = cos b cos a becomes: •••.•••• (e) ...••••• (f) or Iv = IG cos b cos a where Jv = direct solar radiation incident upon a vertical surface♦ South-facing tilted surface* For a south-facing tilted surface: cos i = sin b sin e + cos b cos a cos e ... (b) I_k * I0 (sin b sin e + cos b cos a cos e) (g) ♦ Sky Radiation It was stated earlier that as the solar radiation passes through the atmosphere to the surface of the earth, the depletion of the direct solar radiation is due partially to scattering. A portion of the scattered radiation will re­ turn to space, but some of it will reach the surface of the earth as sky radiation. 16 According to Fritz (7)> Tor an average clear sky, the diffuse sky radiation on a horizontal surface is about 16 percent of the total when the sun is high in the sky and about 37 percent of the total when the solar elevation is about 10 degrees. In addition to diffuse radiation from the sky, some of the radiation reflected from the ground may reach a surface. The amount of the reflected radiation that reaches a surface will depend on the orientation of the surface. The theory of radiation scattering is rather involved and at this time no theoretical method for calculating the quantity of sky radiation is known. Several investigators have measured sky radiation separately from the direct radi­ ation. Klien (2 1 ) gives a method for computing sky radiation In terms of atmospheric conditions and hand (10) has published information on the ratio of direct to sky radiation during typical winter cloudless days for horizontal surfaces. Possibly the best information on sky radiation avail­ ability at this time has been presented by Parmelee (2l|-). He has measured sky radiation on horizontal and vertical sur­ faces. ation In the paper cited, Parmelee plotted sky solar radi­ (Iaft) versus 1^ for a horizontal surface and sky solar radiation (I^y) versus solar altitude (b) and wall solar azimuth (a) for the vertical surface; each was plotted for atmospheres with clearness ratios of 1.0, 0.8 and 0.6. He 17 has defined the ratio of the observed intensity of direct radiation to that computed from M o o n ’s data (Figure l) as the clearness ratio because he noted that the direct radi­ ation varied for a given solar altitude for cloudless skies and that as the intensity of direct radiation decreased, the intensity of the sky radiation increased. Figures 3 and ip show P a r m e l e e ’s data for a clearness ratio of one. Total Solar Radiation The total solar radiation during cloudless days inci­ dent upon a surface is found by adding the direct and sky radiation upon the surface. per 10 BTU (Hr.)(Sq. Ft. Radiation 30 Sky (1^) 18 0 20 BTU Fig. 3. Sky solar radiation (lrfb) for various values of direct radiation (1^) upon a horizontal surface during cloudless days with clearness ratio equal to one. All values from Parmelee (p l\ ) . 30 Sky Radiation (I^v ) BTU per (Hr.)(Sq. Ft.) Direct Radiation upon Horizontal Surface, per (Hr.)(Sq. f t. ) 20 90-180 10 0 10 20 30 Solar Altitude, 60 70 90 b, Degrees Fig. Ij.. Sky solar radiation (I, ) upon a vertical surface for various solSr altitudes (b) and wall solar azimuths(a) during cloudless days with clearness ratio equal to one. 19 DAILY TOTAL SOLAR RADIATION AVAILABLE AT THE SURFACE OF THE EARTH DURING CLOUDLESS DAYS FOR LOW ELEVATION AREAS Procedure for Calculating Hourly Rates The calculation of hourly rates of direct and sky radi­ ation was carried out for the twenty-first of each month using the following procedure: (1) The solar declination was secured for the twentyfirst of each month from the Ephemeris (2) With the solar declination known, (30)• the solar alti­ tude and the solar azimuth were secured for each daylight hour of the day at 3 0 ° , 3 5 °> ^4-0 ° and I4.5 0 north latitude from the United States Hydrographic Office Tables Number 21i|- (29) • (3) With the solar altitude for each hour known, the direct solar radiation at normal incidence (I ) o was secured from Figure 1, using the winter curve for the months of September through March and the summer curve for the remainder of the months* (Ij.) The hourly rate of direct solar radiation Inci­ dent upon a horizontal surface, south-facing vertical surface, and south-facing surfaces tilted 30° and 60° from vertical, was calculated by using formulas d, f, and g, respectively. 20 (5) With hourly values for the solar altitude, the wall solar azimuth, and the direct radiation in­ cident upon a horizontal surface known, Figures 2 and 3 were used to determine the hourly rate of sky radiation incident upon a horizontal sur­ face (1^ ) and vertical surface (X^)* Since there is no information on sky radi­ ation incident upon a tilted surface (I,.)# this dt value was estimated by linear interpolation utilizing the equation: Tdt - Id* - The total hourly rate of solar radiation was determined by adding the direct and the sky com­ ponents* Procedure for Calculating Dally Total Radiation To determine the daily total radiation incident on the various surfaces, it is necessary to integrate the hourly values over the hours of the day. This integration was accomplished by using Simpson's Rule. Results Tables IV through VII show the results of the com­ putations for 30°, 35° » and north latitude. The 21 hourly rates of direct solar radiation agree fairly well with those presented by Hutchinson (15* 16, 17) except for the tilted surfaces, where he did not allow for any incident energy when the solar azimuth angle was greater than 90 de­ grees. Horizontal Surface Curves showing the daily total solar radiation incident upon a horizontal surface at 3 0 °, 3 5 ° » ^4-0 ° and l|5 ° north latitude during cloudless days are shown in Figure 5* A comparison of the curves in Figure 5 shows that the horizontal surface intercepts a maximum of radiation in summer, with the amount being nearly the same for all four latitudes in late June and early July at about 2650 BTU per day per square foot. This near equality is explained by the fact that the effect of greater solar altitudes near the middle of the day for the southern latitudes is coun­ terbalanced by more daylight hours and by greater solar altitudes for the northern latitudes during the morning and later afternoon hours (see Tables IV through VII). The cloudless day radiation decreases from June to December to a value of about 50 percent of the summer value at 3 0 ° lati­ tude and 25 percent of the summer value at 1^5 ° latitude. The decrease from summer to winter is explained by the greater angle of incidence owing to lower solar altitudes 22 X - ± X X cm ♦ co O' O- rH CM vO • •___ crt rH O O ao c\j I *^<3©g rcS •rH • o £ CO •rH ►>» cd £ O •rH £/J W P cd 0 •rH rH xJ x* cd 3 O rH O rH O CO IT O U CM CO to c X> *H C cd 3 XJ -p o CO 0 0 £h XJ •H 3 Xl -p rH •rH •H P £ o ex. 3 |^ a OJ rH cd P O -P >> rH ♦H cd Q p cd rH ^ P SH O £ ca 3 O • •rH 1A cd • > bO •H p P*H cd 3000 o o o o CM O O O 23 and the smaller number of daylight hours for a given lati­ tude in winter# The more pronounced change for northern latitudes in the number of daylight hours from summer to winter explains the larger decrease from summer to winter for latitude as compared with 3 0 ° north latitude# The shape of the curves was such that it suggested the possibility of being able to write equations for them# A method for determining the coefficients for a fourth de­ gree polynomial suggested by Baten (3 ) was used in obtaining equations I, j, k, and 1 in Figure 5# In computing these polynomials, the origin (X * 0) was located at June 21# X Is the number of months, or factions thereof, from June 21# By using this origin and using X = 6 twice in the computation, fourth degree polynomials with exponents of the X and X^ terms equal to zero were secured# A sample computation using the computed data for 35° latitude is shown in Appendix II. The overall average de­ viation between the values determined from the polynomials and the corresponding calculated value for the 2 1st of each month is 1? BTU per square foot per day# amounts to less than 1 percent# The deviation 2k South-Facing Vertical Surface Figure 6 shows the daily total solar radiation incident upon a south-facing vertical surface during cloudless days. The curves of Figure 6 show that the soler radiation incident upon a south-facing vertical surface is at a maximum during the winter months. It is a maximum then because the lower solar altitudes have a more favorable angle of incidence on a vertical surface as shown by equa­ tion e . The normally higher values for northern lati­ tudes are also explained by consistently lower solar al­ titudes than for southern latitudes. The dip in the curve for the more northern latitudes in December, which occurs even though the angles of Incidence are more favor­ able then, is due to the increased depletion of the atmos­ phere and fewer daylight hours. It is noted that the curves for the four latitudes peak at between 1800 and 1900 BTU per day per square foot. The peaks are very nearly equal because the more favorable angle of incidence for the northern latitude is offset by more daylight hours and less depletion by the atmosphere for the southern latitudes. South-Facing Tilted Surfaces Curves for daily total solar radiation incident upon a south-facing surface tilted 30 degrees and 60 degrees from vertical are shown in Figures 7 and 8 , respectively. 25 1 0 0 XI 4-> 3 O co -p i— i cd o •H -P J h 0 • C O t5"# cd cd cd £ O Cl, C O P rH -P cd > 00 0 p P O cd •H O £ •H |H 0 o aO G •H h a P o P •H cd -p a 01 0 •rH cd cd cd P JH p> •rH j >>-p cd C Oi— 1 CJ cd G -P cd P h o -p g o C O P-t P •H o cd -H G i— ! cd cd > -p o -p -p aS 0 0 i— 1o •rH cd cd Q h G P m • bO g 3000 • •rH ttf) o •H cd o o o CM (•q.£ *bs)(^BG) o o O 11X9 ‘uDpqjsxp^H qxiQpxoui o 26 7 / r / / I-AOM i #q.oo \ 4> \ S)(-fi«a) Jsd flJH ‘uoT^Bfp^H q.u©ppoui I*U8f Fig* 7* Daily total direct and sky radiation incident upon a south-facing surface tilted 30 degrees from vertical at various north latitudes during cloudless days* i*o ©a 27 bQ £ o0 £ O H cd £

*P £ 03 0 > d £ £ cd o 0 0 £ -p <£ o 0 0 £ 0 tH 0 d £ b0 rH 0 cd d p o o P vO • >sd w H D h •rH P C d cd rH id CH *H p 0 0 •0 0 ao o rH cd d ♦ <£ £ bO £ O •rH d 1—I bn W O 3000 O o o CM (•q-til o O O rH fltLS ^uoT^B-fp^a ^u©ppoui 28 These curves reveal the importance of proper orientation of solar collector surfaces for the most efficient collection of solar energy* It is to be noted, for example, that for a surface tilted 30 degrees from the vertical, as compared with the vertical surface, the tilted surface shows a slight increase of incident cloudless day energy during mid-winter, and a substantial increase to a maximum of slightly more than 2200 BTU per day per square foot during the spring and fall months for north latitudes of 30°, 3 5 ° !p5 ° latitude, 9 and lj.0°* For the tilted surface shows increased incident energy during spring, summer and fall, and a slight de­ crease during mid-winter* A comparison of the curves for a south-facing surface tilted 60 degrees from vertical with curves for a horizontal surface shows that a slight attenuation occurs during the mid-summer period with small increases during the other sum­ mer months and much higher values for the remainder of the year • Optimum Tilt Angles for Solar Collectors It Is apparent that the optimum tilt angle for a col­ lector of solar radiation will depend on the latitude and on seasonal demand based on the uses to which the collected energy is to be put* In some cases it may be necessary to 29 determine the optimum tilt angle for a period o f time rather than Tor a particular time because a solar collector on a farm, for example, will usually be incorporated into, or made a part of, a farm structure. The daily total radiation incident upon a surface, for all practical purposes, will be a maximum if a south-facing surface is tilted so that the sun's rays are perpendicular to it at solar noon. A south-facing surface will be perpen­ dicular to the sun’s rays at solar noon if th© tilt angle from vertical is equal to the solar altitude for that time. The solar altitudes for the 21st of each month at 30°, 35°> it.0° and ip5° north latitude are given in Tables IV through VII. Figure 9 shows curves which give the number of degrees to tilt a south-facing surface from vertical to make it per­ pendicular to the sun's rays at solar noon and thus, for all practical purposes, the optimum tilt angle from vertical for maximum incident solar radiation for various times of the year. However, there is a slight difference to be noted. Calculations using formula g for June 21 showed a slight increase in daily total incident energy with small increases In the tilt angle from the apparent optimum angle. The small Increase occurs because the increase in incident energy during early morning and late afternoon hours due to additional tilting of the surface more than counter­ balances the decrease during mid-day. The above situation 30 90 80 at at 70 Degrees at 30 20 10 rH rH U © bO -P CO > O o Fig* 9* Number of degrees to tilt a south-facing surface from vertical to make it perpendicular to the sun's rays at solar noon* 31 is true only during mid-summer when the solar azimuth angles are relatively small during the early morning and late after­ noon hours* It should be noted that the above situation would be even more pronounced at more northerly latitudes* Ratio of Tilted to Horizontal Surface Incident Radiation The ratios of the daily total solar radiation incident upon a south-facing vertical surface and upon south-facing surfaces tilted 30 degrees and 60 degrees from vertical to that incident upon a horizontal surface for the latitudes mentioned before, were calculated for 21st of each month* The calculated ratios are shown in Table VIII* Figures 10 and 11 show a plot of the ratio of tilted to horizontal surface incident radiation* of the form shown in equation method of least squares m below, was fitted by the to each of the curves* Y * a +b X + c X 2 where A polynomial, ....... (m) X s the slope of the south-facing surface in degrees from vertical Y = the ratio of tilted to horizontal surface incident radiation* Table I shows a tabulation of the coefficients of equation m m for various latitudes and seasons. Equation can be used to secure factors that can be multiplied by o •rl P ?H o P -P CM CM P P p O C> O CM i —I o CM o O 1A 0 o > o fn s P| P CM O rH •H o o O O O rH CM CM O o O 0 a cd Pi 3 GQ Pi O 0 a o i —i GO O 0 ai 0 SO 0 Q cd P o *iH > 0 Pi a o u 0 O JH 3 cd Pi CO Pi O 0 Qa O rH 17D 10. Ratio of direct and sky radiation incident upon south-facing surfaces various tilt angles to radiation incident upon a horizontal surface. cd Fig. with 32 CQ a) 0 SO 0 O O C\J • C\J o • C\J • o• • C\J C\J i—I uoi^Btp^y aoBjjng xB^uoz-faon O I —I CO # o• r—! rH o • rH o Cu O • CD co (D O O LA O 4> i-l C\J O CA O "LA O * po^Tll, 3° °T3^H o Fig. 11. Ratio of direct and sky radiation incident upon south-facing surfaces with various tilt angles to radiation incident upon a horizontal surface• 33 CD CD ui 4-> 34 TABLE I COEFFICIENTS OF THE POLYNOMIAL, Y = a + bX + cX , (m), FOR THE VARIOUS CURVES OF FIGURES 10 AND 11 Date Latitude Coefficient a b c Dec* 21 30° 35° 40 ° 45° 1.496 1.734 2.132 2.499 0.012657 .014313 0.12366 .010621 -0.0002025 .0002502 .0002774 .0003033 Jan. 21 and Nov. 21 30° 35° 40° 45° 1.322 1.310 1.827 2.154 0.014395 .013339 .013236 .012117 -0.0001999 .0002112 .0002500 .0002776 Feb. 21 . and Oct. 21 30° 33° 40° 43° 0.997 1.141 1.312 1.500 0.014531 .015173 .014250 .013619 -O.OOOI613 .0001860 .0001972 .0002139 M a r • 21 and Sept. 21 30° 33° 43° 0.638 .768 .372 1.011 0.016279 .015605 .015237 .015365 -0.0001361 .0001445 .0001532 .0001722 A p r • 21 and Aug. 21 30° 33° 40° 43° 0.314 .400 .492 .571 0.017447 .017659 .016878 .018014 -0.0001083 .0001221 .0001249 .0001472 May 21 and July 21 30° 35° 43° 0.186 .244 .345 .399 0.016350 .016964 .015827 .016884 -0.0000806 .0000944 .0000944 .0001132 30° 33° 4o° 43° 0.105 .205 .277 .326 0.016696 .016145 .016064 .017028 -0.0000744 .OOOO805 .0000889 .0001055 June 21 4o° 4o° values of Figure 5* or by the interpolated values of equa­ tions i, j, k and 1, to secure the daily total cloud­ less day solar radiation incident upon a south-facing sur­ face at any tilt angle between horizontal and vertical* Ratios for any day of the year can be secured by interpola­ tion between the values secured for the 21st of the months between which the desired day comes* Figures 10 and 11 show the importance of seasonal de­ mand on proper orientation of a collector surface* It is noted that little can be gained by an orientation other than horizontal during the mid-summer months. The curves show the distinct advantage of a surface that is vertical, or very nearly vertical, during mid-winter months. The tilt angles for maximum incident radiation for the various latitudes agree with those given in Figure 9. Comparison between Calculated Cloudless Day and Recorded Clear Day Radiation Procedure The information published in Climatogical Data, National Summary (31) on recorded total solar radiation was utilized to check the calculations previously presented. The informa­ tion available is restricted to horizontal surfaces, except for the measurements made at Blue Hill, Massachusetts on vertical surfaces oriented at the cardinal compass points. Supplements of Local Climatological Data (33) were se­ cured from the United States Weather Bureau for Lincoln* Nebraska; Madison, Wisconsin; Boston, Massachusetts; Lander, Wyoming; Albuquerque, New Mexico and East Lansing, Michigan for the five-year period, 1950 through 195il* The average sky cover from sunrise to sunset in tenths was recorded for each day in the supplements mentioned# When the average sky cover from sunrise to sunset is three-tenths or less, the day was recorded as clear; four-tenths to seven-tenths, partly cloudy; and eight-tenths or more, cloudy. Technical Paper No. 12 of the United States Weather Bureau (32) sum­ marizes the average number of clear, partly cloudy, and cloudy days from sunrise to sunset for over 200 United States Weather Bureau Stations; the Summaries included data for periods up to 77 years. Table IX shows a copy of the data for the stations used in connection with this study. Dates when the average sky cover from sunrise to sun­ set was three-tenths or less were recorded for each of the stations mentioned for the five-year period, 195^-* 1950 through Daily total Incident radiation was then secured for the recorded clear days from Climatological Data, National Summary (.qg. c i t .) for a horizontal surface at each of the stations and for a south-facing vertical surface at Blue Hill, Massachusetts. Average sky cover data was not avail­ able at the same location that solar radiation data was taken 37 for Blue Hill, Massachusetts, so the sky cover data were taken from records made at the Logan International Airport, Boston, Massachusetts* The solar radiation data was recorded in units of gram-calories per square centimeter per minute; the units were converted to BTU per square foot per hour by multi­ plying by 3*68. Calculated solar radiation during cloudless days in­ cident upon a horizontal surface for the 21st of each month was determined for each of the stations by interpolating be­ tween values secured from the equations in Figure 5 for the latitude of each station* Calculated cloudless day radiation was determined for a south-facing vertical surface at Blue Hill, Massachusetts by Interpolation of values given in Figure 6. Curves showing the calculated incident cloudless day radiation were constructed for each of the stations* Plots of points representing the recorded radiation during clear days were made in order to make a .comparison between the re­ corded and the calculated cloudless day values for the various days of the year. Results Figures 12, 13 $ lU- lf> show the comparison between calculated cloudless day and recorded clear day solar radi­ ation incident upon a horizontal surface at Madison, Wisconsin; 3000 I *3-30 xrLJLh. o O O o o OJ •bsM^a) ^ 6 MS O I £i cj • ••• I *J'QR o o o )— ! «uoxq.BTP^H ^uopioui o Pig. 12. Comparison between recorded and calculated direct and sk; radiation incident upon a horizontal surface at Madison, Wisconsin during days with 0 - 3 tenths cloud cover, 1950 through 1951|* 3 > • 4• 3000 o o (•Id • f r s H ^ a ) o o m g 'uo-fiBfpBH ^uepjotii o Fig. 13. Comparison between recorded and calculated direct and sky radiation incident upon a horizontal surface at Lincoln, Nebraska, during days with 0 - 3 tenths cloud cover, 1950 through 195^. 3000 I ld©s i i I *JT3W o o C\t *bs)(^Ba) O o iH MS #u o t i b t p t 2h duapToui o Fig. ll|.. C omparison between recorded and calculated direct and sky radiation incident upon a horizontal surface at Blue Hill, Massachusetts, during days with 0 - 3 tenths cloud cover, 1950 through 195^* 1+0 3000 I -^W O o o (•*,£ * b s ) ( ^ a ) (\J O o o me *uoxr®TP»H ^U9p-[0UX I— ( O Fig# 15• Comparison between recorded and calculated direct and sky radiation incident upon a horizontal surface at East Lansing, Michigan during days with 0 - 3 tenths cloud cover, 1950 through 1955* 41 1+2 Lincoln, Nebraska; Blue Hill, Massachusetts and Bast Lansing, Michigan, respectively. It is evident that the calculated curves, based on assumea standard atmospheric and sea level conditions, gave reasonably good correlation with the ob­ served radiation at Madison, Lincoln, and Blue Hill which have elevations of 672, 9 3 8, and II8I4. feet above sea level, respectively. It is noted, In each case, that the calculated values are low during the spring and early summer period and high during the last part of October and first part of November. This difference is possibly due to higher at­ mospheric moisture content than assumed during the OctoberNovember period and a lower atmospheric moisture content than assumed during the spring months. Table XII shows values of monthly mean precipitable water for all days in the United States published by the Weather Bureau (3U) an8 values of precipitable water for cloudless days used for the computa­ tion of direct radiation at direct incidence (Figure 1). It Is to be expected that cloudless day values will be less than the mean recorded values. However, the data indicates a tendency for low values of precipitable water during the spring months as compared with the fall months. The dashed curve of Figure 12 Is a plausible calculated curve for actual precipitable water during cloudless days. k3 Figure llj_, Tor Blue Hill, shows a great deal more scatter of the recorded radiation than do Figures 12 and 13 for Madi­ son and Lincoln, respectively* One possible explanation of the greater scatter could be the location of the measuring pyrheliometer• Hand (10), in a report on solar radiation measuring stations in the United States, indicated that the Blue Hill pyrheliometer is located 10 miles south of Boston and that there is almost no smoke or dust except with winds of a northerly component which carry smoke from Boston* He also indicated that the pyrheliometer at Madison is located on top of a building located at the University of Wisconsin and that little smoke interference is noted. The Lincoln pyrheliome ter is located in downtown Lincoln (since 19/fO ) • Figure 15* which shows the comparison for East Lansing, Michigan, indicates that the computed values are consistently too high, varying from about 25 percent to 10 percent too high for winter and summer, respectively. The tendency for com­ paratively higher recorded values during spring is also noted for East Lansing. A possible explanation for the consistently low recorded values at East Lansing may be due to the fact that Michigan is surrounded, to a great extent, by large bodies of water over which the masses of air must move in their predominant easterly movement. The bodies of water and the large industrial activity in the whole area undoubtedly tend to increase the haziness and smoke content of the atmosphere above the assumed amount for a normal atmosphere. of* work by Fritz Results (8), who has constructed isolines of cloud­ less day solar radiation for horizontal surfaces in the United States for each month of the year, based on close analysis of recorded data supplemented by computed values where there were no recording stations, also showed con­ sistently lower values for the Great Lakes Region, and in particular for Michigan, than for other parts of the coun­ try at the same latitude. Crabb (6) also noted the relative low amount of sola.r radiation at East Lansing as compared with other stations in the United States. The variation noted for all stations can be attributed, in part, to varying cloud amounts because the recorded values used were for days with average cloud cover between zero and three-tenths. Additional variation may be caused by occasional presence of dust or moisture on the glass cover of, or by improper leveling of, the measuring pyrheliometer. Figure 16 shows the comparison between calculated and recorded values of incident radiation on a south-facing vertical surface at Blue Hill, Massachusetts. good average correlation is again shown. A great deal of variation in the recorded radiation is noted, during the winter months. Reasonably particularly In addition to the explanation given for the variation of solar radiation on a horizontal surface at Blue Hill, the vertical surface will be subjected k5 to P p 0 ^ CO CO 3 ^1 a as Od O aj co • •V' CO _^J- P a*lA o S cr> J h O d d o 0 0 rd ad !nH d O aj O O O rH 0 *H O Eh - P Eh CO £ 0^ 0 >P 0 C 5 ai 0 P p 0 5 d! P O rp a, £ 3 i o to P o > EH h ^ 0 PMi cd ad P Cii *f— f *rH £ o £ o £ o -H co !>s a ai o ad _ 'H t—I P tiO bOad Eh cd C • »H *d HJH o O O iH ^uoTq.^xp^H ^uopxoui o 46 to varying amounts of* reflected radiation from the ground# The high recorded values can reasonably be attributed to increased ground reflection when the ground is covered with snow# Comparison between Calculated and Observed Ratio of Vertical to Horizontal Surface Incident Radiation Procedure The only possible check of the calculated ratios of solar energy incident upon south-facing surfaces to that incident upon a horizontal surface, as presented In Figures 10 and 11, is to check the ratio of south-facing vertical to horizontal surface incident radiation for Blue Hill, Mas sachusetts• The calculated ratio of south-facing vertical to hori­ zontal incident radiation for Blue Hill was determined for the 21st of each month by interpolation between the values given for k5° and 1+0° north latitude in Table VIII for Ip2° — X3 *» which is the latitude of Blue Hill. The curve in Figure 17 was constructed from the points determined by the method m e n ­ tioned above. Supplements of Local Climatological Data for Boston, Massachusetts (33) were used to determine the dates during the five-year period, 1950 through 195k* when the average sky cover from sunrise to sunset was classified as clear or -fit CM xt3^ 02!*10!! °3- T130T%'19A o 9 «—I J° °TT®H Fig. 17* Comparison between actual and calculated ratio of total radiation incident upon a south-facing vertical surface to that incident upon a horizontal surface at Blue Hill, Massachusetts, during days with 0 - 7 tenths cloud cover, 1950 through 1951±. 1*7 <»• o 48 partly cloudy (0-7 tenths cloud cover, inclusively)• Daily total solar radiation measured and recorded during the clear and partly cloudy days determined above, was secured from Climatological Data, National Summary (3 1 ) for the horizontal and south-facing vertical surface at Blue Hill. The ratio of the vertical to horizontal surface incident raaiation was calculated and the values for the various days were plotted on Figure 17. Result s Figure 17 shows good average correlation between calculated and observed ratio of daily total solar radiation incident upon a south-facing vertical surface to that incident upon a horizontal surface at Blue Hill, Massachusetts for clear and partly cloudy days. Only clear and partly cloudy days were used for this comparison because only a very small portion of the total radiation is available during cloudy days. It is noted that there is consideraoly more scatter of points during winter months than during summer months. The very high ratios during the winter can be attributed to the Increased radiation incident upon the south-facing vertical surface owing to increased reflection from the ground caused by snow cover. This increased reflection will affect the vertical surface more than the horizontal surface and con­ sequently, increase the observed ratio. The observed ratios 49 which Tall considerably below the calculated ratios oceurr on the relatively cloudy days because with increased cloud ness, the radiation tends to become equal in all direction or the ratio tends toward unity. 50 VARIATION IN SOLAR RADIATION INTENSITY WITH ALTITUDE Review of Literature Preliminary comparisons between calculated cloudless day and recorded clear day solar radiation for Lander, Wyoming (5#563 ft.) anci Albuquerque, New Mexico (5#310 ft.) indicated that the calculated results, based on the atmos­ pheric and sea level conditions which were assumed, gave values that were much too low. The higher recorded values for Lander and Albuquerque are to be expected because the solar radiation has a shorter path through the atmosphere and less chance of depletion in reaching a surface at high altitude * The earliest known information on the variation in solar radiation intensity with altitude was published in 1919 by Kimball (19). Prior to that time, in cooperation with the Weather Bureau and the Smithsonian Institute, he made studies on the increase in solar radiation intensity with altitude westward from the Atlantic Coast of the United States. Records of solar radiation intensity were taken at various places near sea level, in the Great Plains, and at various places at high altitude; such as Hump Mountain, North Carolina; Mount Wilson, California; Cheyenne, 51 W y o m i n g ; Flagstaff*, Arizona; and Santa Fe, New Mexico* From these studies, he arrived at a monthly mean increase in solar radiation with altitude as shown by the yearly curve of Figure 19* This work was continued and in 19^6, Klein (21) summarized work which by that time had included results from 56 plateau and mountain stations supplemented by information from balloon ascents* He noted that the variation of the transmission of solar radiation with altitude depended on the season of the year and the length of the path of the s u n 1s rays (air mass)* His work indicated that the variation with altitude was logarithmic for the various air masses* (See Glossary for definition of Air Mass.) Figure 18 shows the results of his findings for summer conditions with air masses of one and two and for winter conditions with air mass of two* It is noted that the winter and summer lines, for air masses of two, come very nearly to being equal at four kilometers. Below this level, atmospheric transmission of solar radiation is lower during summer, which is as ex­ pected due to higher moisture content of the atmosphere* Both K l e i n fs (i bi d*) and Kimball's (loc * cit*) data show the increase in atmospheric transmission of solar radiation with altitude as compared with that at threetenths of a kilometer ( 9 ft.) elevation because at lower elevations no definite relationship between transmission and 5h al CD rH O Summe r with Air Mafcs of 1 c o o •H Cm © O O •H Summer with Air Mass of 1.3> a .70 O •H Winter w Air Mass CQ CO Summer witJb. Air Mass of o •H 60 CD rCj fDu CQ i -p •^i 0.3 ►. 2.0 Altitude in Kilometers Pig. 18. Mean summer and winter transmission coefficients wit h cloudless sky at high level stations as a function of altitude, from data by Klein (21)• 53 13 w o c\l r—I CM CM C\J \A CM CM O O EH TO TO CM O CM 1A i —I O H GSTssaouj ^ u e o j Q j 51+ elevation could be established and differences appeared to be more a function of local conditions. Procedure for Calculating Corrections Due to Altitude Solar altitudes for June 21 at JLpO0 north latitude for the various daylight hours using Simpson's Rule; (Table VI) were integrated by this integration resulted in an inte­ grated mean solar altitude of l± 7 * 7 degrees. A mean solar altitude of 1|_7*7 degrees corresponds to an integrated mean air mass of 1.35 as the secant of the zenith angle of the sun is a good approximation of the air mass. Interpola­ tion for an air mass of 1.35 was then made between values of atmospheric transmission for air masses of one and two for summer conditions as shown in Figure 1 8 . Using the same method used for June 21, the integrated air mass for December 21 turned out to b e about three; but since data were not available for air masses greater than two, the data shown in Figure 18 for a winter air mass of two were used to determine approximate corrections for altitude in winter. 55 Percentage Increase of Solar Radiation Intensity with Altitude The percentage increase of solar radiation intensity with altitude was determined for winter and summer condi­ tions according to the increase in the atmospheric trans­ mission coefficients shown in Figure 18 for winter air mass of two and summer air mass of 1.35# respectively. The per­ centage increase for winter and summer is shown in Figure 19* The yearly mean increase with altitude, determined by Kimball (l o c . c i t .) is also shown in Figure 19* It is of interest to note how closely the curve for yearly mean Increase ap­ proximates being an average between the summer and winter increases• The curves were drawn to show no increase In solar radiation intensity with elevation up to 1000 feet above sea level because no definite relationship between atmospheric transmission of radiation and elevations to 1000 feet could be established from Klein's (loc. cit.) and Kimball's (loc . c i t .) data, and also due to the good correlation between the calculated and recorded values previously discussed for Blue Hill, Massachusetts; Lincoln, Nebraska; and Madison, Wisconsin which have altitudes above sea level of 672, II8I4., and 938 feet, respectively. The method of least squares was used to determine equa­ tions n and o of Figure 19 for the summer and winter 56 curves, respectively. These equations can be used between altitudes of4 2000 and 10,000 feet above sea level* Points calculated by the equations for 1000 feet intervals between 2000 and 10,000 feet fit the points used in plotting the curve w it h average deviations of 0.25 and 0.10 percent for summer and winter, respectively. The equations can be used to approximate the percentage increase of solar radiation with altitude for December 21 and June 21. Percentage in­ creases for any other time of the year can be approximated by interpolation between the values given by the two equations. Comparison between Calculated Cloudless Day and Recorded Clear Day Radiation for Two High Altitude Stations Figures 20 and 21 show the comparison between calculated cloudless day and recorded clear day radiation incident upon a horizontal surface for Lander, Wyoming and Albuquerque, New Mexico, respectively. The period covered for the compari­ son was from April of 1950 through 195U Albuquerque be­ cause April 1 was the beginning date for publication of re­ corded solar radiation data in Climatological Data, National Summary (31) for Albuquerque. Publication of recorded data for Lander began July of 1950 and was interrupted during the first eleven months of 1951 and in January of 1952, which explains the relatively few points for the first months of the year. t ^d©s I Sny 4^ I unp rH rH i—I O o o o o o oJ O o o •£>g)(^a) J0d ilia ‘uopq.BTpBH ^ u q p t o u i o Fig* 20. Comparison between recorded and calculated direct and sky radiation incident upon a horizontal surface at Lander, Wyoming during days with 0 - 3 tenths cloud cover, 1950 through 195i|. 57 58 • # y,2/ r * X.v •jrr.*: m - / / 'v' ‘t v* • i • a */ _ V •/ •T r •v r /. • . yt * •* * / . * a• *; / /* * • •* • V • / •■ **•* * rH © > © I'^O I aS rH CO X . *• -:r xJ A o o cd O /_ O \ •** •*V.* * •A .•Ov ' ♦ \ * X •• * * . * • X ©imp pc; • .ate -V I*AOJa © > o i —i p © cd > © -P 4) 4> cd © CO O ,ate * X* 09(3 * / * ■/ . * -X •jfr* / •/V* ’. * 'A y* • ». / /. CJF **m/ V, v: *"• ••/ * < :** / I cd X #Jdv •f V f X • iAt/ • * \ •• X* « W \ : •vS .• v N *»v * O o o c°> o o o CO \ N \ ',v . \ ••• . \ o o o rH (•Id *t>S)(^13(l) J9d HUH #uoxq.*xP«H quopioui x*u^r Pig* 21* Comparison between recorded and calculated direct and sky radiation incident upon a horizontal surface at Albuquerque, New Mexico, during days with 0 - 3 tenths cloud cover, 1950 through 195 b v:ta* ' <:‘p, l •Jf" * 59 The procedure used in making this comparison was the same as the one use d Tor Blue Hill, Madison, Lincoln and East Lansing except that corrections for altitudes of 55&3 and 5 3 TO feet above sea level were made for Lander and Albu­ querque, respectively. The dashed curves in each case repre­ sent the calculated dail^ total solar radiation for sea level conditions for the latitudes of the respective stations. Again, reasonably good average correlation between re­ corded and calculated values is noted, except, during July and August for Albuquerque when the calculated values are high. A good explanation for the latter is now known, except that it could possibly be due to the numerous thunderstorms at Albuquerque during those months. 60 EFFECT OF CLOUDS ON SOLAR RADIATION INTENSITY Review of Literature So far the discussion presented for calculated solar radiation intensity at the surface of the earth has been for cloudless skies, and the observed values referred to have been for clear days, or days when the average percent­ age of cloud cover was 30 percent or less. ever, is not always cloudless; during some seasons, The sky, how­ in fact, in some places cloudy conditions are very prevalent. Because of the great effect clouds have on solar radiation, it is one of the most Important considerations in deter­ mining the availability of solar energy. The effect of clouds is also one of the most difficult to determine. Fritz (7) states that if we look at the earth as a whole, the planet reflects about 35 percent of the solar radiation incident upon it back to space and that clouds are the major cause for this reflection. Haurwitz (ll) has reported that the ratio of total radiation with complete overcast to total radiation with cloudless skies varies from about 0.83 for cirroform type clouds to 0.18 for fog. This ratio depends not only on cloud type, but also on the mean free path of the sun's rays through the cloud, drop size and 61 distribution, and liquid water content of the clouds. Be­ cause of the difficulty involved in approximating the para­ meters mentioned, which are needed to determine the effect of clouds, and owing to the non-homogeneous nature of clouds, it is more practical to correlate the measurements of solar radiation at a few stations with some other parameter, such as average cloudiness or percentage of possible sunshine, which are observed in many places. Fritz (o p » c i t .), Kimball (20) and others, suggest cor­ relating the ratio of average daily solar radiation to cloud­ less day radiation with percentage of possible sunshine. Percentage of possible sunshine is suggested in favor of average percentage of cloud cover because the photoelectric cell which is used to measure the minutes of sunshine only records when the intensity of solar radiation is more than 82 BTU per square foot per hour. Therefore, some of the very thin cirroform clouds will be ignored by the sunshine recorder as they reduce solar energy by relatively small amounts. United States Weather Bureau Technical Paper No. 12 (3 2 ) gives long time means, based on up to 58 years of data, of hours and percentage of possible sunshine for nearly 200 United States Stations. Table X shows a copy of this information for the stations used in this study. Monthly means of the percentage of possible sunshine are currently recorded in the United States Weather Bureau Climatological Data, National Summary. 62 Procedure and Results The mean daily recorded solar x^adiation on a horizontal surface and the mean percentage of possible sunshine were secured from Climatological Data, National Summary (3 1 ) for Albuquerque, Blue Hill, Madison, Lincoln and Lander for each month during the period, 195>0 through 195k* Calculated total radiation incident upon a horizontal surface for cloudless days was calculated for each station for the 15th of each month using the equations in Figure 5 and interpolating for the latitude of the respective stations. The values secured from the computation for Lander and Albuquerque were corrected for the altitude of the respective stations by use of the equations.in Figure 19* Table XI shows a compilation of the percentage of possible sunshine (S), mean daily recorded radiation (I) and the ratio of recorded to calculated cloudless day radi­ ation (l/lQ ), for the stations mentioned. The ratio, l/lQ , for each month was plotted as a function of the percentage of possible sunshine in each case as shown in Figure 22. The method of least squares was used to determine the linear equation p . The coefficient of correlation between the ratio and percentage of possible sunshine is 0,82 and the standard error of estimate is percent. 63 o rH CO CO CD O o i —I 3 o o 03 rd i —t XI •H CO CO O o* < h o O « bfl cd -P £ 6> o CD PM O cd • rH CD 3 a a •H rH ,Cj cd 00 o £ d o co -p CD I —)i —1 cd x> P 00 o CO cd o a* o Cd *H P cd • —i C\J 'd OJ cd G O OJ O i— ! • U bO >> •H cd O o o o CO o O ^u©oa©j uf o cd i/i O OJ O O 614- DISTRIBUTION OF DAYS WITH VARIOUS AMOUNTS OF SOLAR RADIATION Average Number of Days in Various Categories for Madison, Wisconsin The discussion so far presents a method for approxi­ mating the many-year average availability of solar radiation. Prediction of the day to day availability of radiation accu­ rately is not possible because of the variation In cloudi­ ness* The need for day to day predicting is great when energy is to be used for house heating, or for some other purpose, where solar energy is to be stored for use during night-time hours or during periods of cloudy weather, because the solar collector performance depends on solar in­ tensity. Telkes (2 6 ) states that from the standpoint of solar house heating, the most important information to de­ rive from solar statistics is the sequence of clear, partly cloudy and cloudy d a y s . The fact that there were relatively few stations in the United States with solar statistics extending over a very long period of time was pointed out in the introduction. In addition, until July of 1941, only the weekly mean of daily total solar ra diation recorded by the Weather Burean were published; it has been only since that time that daily totals have been published* 65 Average distribution of days with various categories of solar radiation incident upon a horizontal surface at Madison, Wisconsin during the period July of 191^1 through October of 1955 was determined as presented in Table II. Madison, Wis­ consin was selected because it had the first solar station in the United States and because the records published for that station were fairly complete. Category I includes days wit h daily total radiation equal to, or greater than, the calculated cloudless day radiation from the formulas in Figure 5; category II is from the lower limit of category I to the calculated mean determined from the formulas in Figure 5 corrected for cloudiness by use of equation p and the JLp3 — year average percentage of possible sunshine from Table X; category III is from the lower limit of category II to 25 percent of the calculated cloudless day radiation; and cate­ gory IV includes days with radiation less than the lower limit of category III. The range of the number of days in each category during the period studied is also shown. The wide range of the number of days which fall into each cate­ gory points out the large variability in the day-to-day availability of solar radiation, about which nothing can be done except to derive probabilities on the number of, and sequences of, days in various categories when sufficient recorded data are available to do so. Generally speaking, solar radiation statistics of the kind mentioned would TABLE I I AVERAGE DISTRIBUTION OF DAYS WITH VARIOUS CATEGORIES-::- OB’ RADIATION AS RECEIVED ON A HORIZONTAL SURFACE FOR MADISOIJ, WISCONSIN 1941 THROUGH 1955 Month Category Number of Days Mean January I IX III IV February I 7.4 10.1 9.7 3.8 Range 3 7 r-' > 1 Total Monthly Radiation BTU per (Ft^)(Month) Percent: of Total -- 12 -- I k -- 15 -. 8 6,624 6,582 3,457 378 38.8 38.6 20.2 2.2 II III IV 6.6 9.5 C 3.1 3 -- 13 7 ■• I k k -- 12 1 -. 6 8,212 8,957 4,716 120 37.3 40.7 21.4 00.5 March I II III IV 8.5 9.6 7.9 5-9 3 -- 13 6 -■ 16 k -* 11 2 -- 10 14,450 12,924 5,94° 982 42.1 37.6 17.3 2.3 April I II III IV 9.2 6.2 9.6 2.9 5 ^ 6 0 -- 16 ■ 13 -■ 15 -■ 8 lc ,302 14*284 9,373 738 44.8 32.3 21.2 1.6 May I II III IV 7.1 9.5 11.7 2.6 1 3 7 0 -- 13 -■ 15 -■ 20 -- 5 18,207 19,981 13,870 809 34 •4 37.7 26.2 1.5 June I II III IV 5.2 11.7 10.9 2.1 2 7 7 0 -■ 9 -• 15 -* 15 -■ 5 14,383 27,003 14,445' 6?3 25.4 47.7 25.5 1.2 July I II III IV 5.5 13.2 11.6 .7 2 9 9 0 -. 8 -■ 17 -* 15 -■ 1 14,575 30,733 15,732 229 23.7 50.1 25.6 00.3 67 TABLE Month Category ^ Number of Days Mean August September October November December .8 14.2 10.8 1.1 1 8 8 0 - I II III IV 6.3 12.3 8.5 3.3 3 7 5 0 — — I II III IV 1^.9 lk.3 8.8 3.1* l 10 5 0 I II III IV 2.1 9.6 9.8 6.7 I II III IV 3.5 1 0 .k 9.3 5.3 0 7 lj3 1 7 7 0 — - - •> - _ _ _ _ _ (C ont.) Total Monthly 5 o^ ^ Haaiation vgTU per (Ft2 )(Month) Percentage Q£ m ni.Q-, rotal 9 19 14 5 IX ,280 28,793 12,653 325 2X.2 54.2 23.8 00.6 10 18 13 7 X2,027 19,451 7,548 757 30.2 48.8 18.9 1.9 XI 17 12 7 6,804 16,715 5,725 587 22.8 56.0 19.1 1.9 5 18 1410 1,966 7,310 4,008 786 13.9 51.9 28.4 5.5 10 18 14 11 2.800 6,X58 2 ,88)4. 485 22.7 49.9 23.3 3.9 Range I II III IV Overall Mean I II III IV XI 29.8 45.5 22.6 2.0 '"'Category I includes days with daily total radiation equal to or more than calculated cloudless day radiation from the formulas in Figure 5; Category II is from the lower limit of category I to the calculated mean determined from the formulas in Figure 5 cor­ rected for cloudiness by use of equation p and percentage of possible sunshine from Table XI ; Category III is from the lower limit of category II to.25 percent of calculated cloudless day radiation and category IV includes radiation less than the lower limit of category III. 68 aP P 1y a relatively small area immediately adjacent to the measuring station. It is also possible that with advanced meteorological techniques, short range forecasts and relatively long range trends in solar radiation availability could be made by r e ­ lating solar energy availability to some other meteorological factors which can be forecast. Mean Total Solar Radiation in the Various Categories The approximate means total radiation, in the various categories, mentioned in the last section, was determined for Madison, Wisconsin by multiplying the mean number of days by the most probable average radiation for each cate­ gory. The percentage of the total radiation of each month in the various categories is also shown. that, for the year, The latter shows about 75 percent of the total radiation is available on days with above average solar radiation. Where solar energy is to be collected and stored at rela­ tively high temperatures, the relative importance of the clear days, or days with above-average solar raaiation, will be even greater. by Telkes The importance of the latter was pointed out (27) in a report on the Dover Solar House, where it was shown that almost 9k percent of the energy collected during the month of February 19^9 occurred during days with above average radiation. 69 Sequence of Dark Days The sequence of days with low amounts of solar radiation, during which little or no energy can be collected, is impor­ tant, particularly where storage of energy is involved* Table III shows the maximum sequence of days in category IV during the various months between July of 191^.1 and October of 1955 f o r Madison, Wisconsin* The importance of statistics of this kind is apparent from an examination of data for December which shows that the sequence of days in this cate­ gory varies from none to five during the period of years studied. For solar collector and energy storage design, statistics of this kind would be needed, based on a threshold amount of solar energy below which the amount of energy col­ lected and stored would be negligible. 70 'LA CM o 02 Q AJ °A O C\J c^CM • CM CM r^ vO > o I i—I cA CM CM l A I CM I I-d'CM H H r ^ | CM OJ | • c M CM (D -P cd i —t d o i —i cd o tA ■P O o CM H H H I rArH H H (\1 O O C\J A O +5 a, CD CO !>• rH c m o j o j h h h o c m c j c m c m h o o j h -d* • i— 1 o -p d cD o <~A M b0 d b « o <*! d • O H C M r l r l r l O O O tMCVIHOOO o 0 O a. rH 13 M cq <1 o s M CO ■LA CM 0- 02 d • r l H H O H O r l H r l H O r l r l O -P O r l r l C M r l r l H O > cd a d cd ,d o o• © £3 O O -p rH 02 02 cA rH © C7' • i— I i— • rH O i— I i— I i— t t— I i— ! ,d O ■P •H £ o (h § 02 rH ed Tj • 04 & CM -< m vO 3 CO d cd a rO 0) tii CA r—I C ArH \ A H H C M r l C M H C M r H C M o • rH i— 1 cd rH i— 1 d A- t>» 1A d cd ♦ rd d d o CM rH rH rH CM CM O rH «A rH OJ CM CM CM d cd • rH O qO © 02 40 02 cd * rH CM c A - ^ f l A O O-CO 0 s O rH CM OA-^ftA 0 N C7> 0N CT'0> C7x 0>‘0^Cr'0N CACr' O' O v CA rH rH rH rH rH i— I i— I rH rH rH rH rH rH i— I i— 1 d o 0 cd o a 1 rH O 71 CONCLUSIONS AND PROPOSED FUTURE RESEARCH The amount of solar radiation available is very large but has the disadvantages of being intermittent and relatively low in intensity# Prediction of the day to day useful energy collection of a given solar collector is not possible owing to the random variation of meteorological factors, such as cloudiness, which are interrelated with solar radiation in­ tensity . The relative importance of cloudless day solar radiation is great because approximately 7? percent of the total solar radiation is available on days with above-average radiation intensity# In addition, the efficiency of collection with a solar collector is a function of solar intensity in such a way that even more importance is attached to the cloudless day solar radiation# The cloudless day solar radiation intercepted by a horizontal surface is at a maximum and has little latitudinal variation during mid-summer in the United States. The amount of radiation incident upon a horizontal surface during mid­ winter decreases to about percent and percent of the summer value at 30° and I45 ° north latitude, respectively# 72 The increase in solar radiation intensity with altitude above sea level as we progress from east to west in the United States is significant with the increase being greater in summer than in winter. The amount of solar radiation incident on a surface is affected to a very great extent by the orientation of the surface. The proper tilt angle of a south-facing surface is a function of the latitude and season of the year. Little increase in incident solar radiation is noted for an orienta­ tion other than horizontal during mid-summer while during m i d ­ winter the distinct advantage of a surface which is nearly vertical is evident. A south-facing surface with optimum tilt angle will have approximately 5 percent and l£ percent more incident cloudless day radiation than a vertical sur­ face on December 21 at ly. l\S ° and 30° north latitude, respective­ The optimum tilt angle from vertical increases from December to June when the optimum surface would be very nearly horizo nta l. The curves and polynomials developed for estimating: (1) the amount of solar radiation available on horizontal surfaces during cloudless days, (2) the ratio of radiation incident upon a south—facing tilted surface to that incident on a horizontal surface, and (3) the increase in radiation with altitude, show reasonably good correlation with actual 73 recorded data ©xccpt for the Great Lakes region where tlie calculated values are 10 percent to 25> percent high., depending on the season* The regression equation relating the ratio of actual to calculated cloudless day radiation and percentage of possible sunshine, provides a method for estimating the average solar radiation for any season at a great many places* The percentage of possible sunshine is a parameter measured and recorded at many stations; for the parameter are available* long-time averages The estimates apply only to average values and are subject to large deviations in individual cases due to local variations in conditions such as, atmospheric pollution, ground reflection, snow cover and local variations due to large cities and industrial areas* There is a need for verification of the ratio of solar radiation incident upon tilted surfaces to that incident on horizontal surfaces further than that presented in this thesis, as the only verification used was the ratio of verti­ cal to horizontal incident radiation at one station. also conceivable It is that these ratios could be refined further by determining the affect of cloudiness upon them as it would be expected that the ratios would tend toward unity with increased cloudiness* There is also a need for an intensive study of atmos­ pheric moisture conditions and an extension of the data for 7^ direct solar radiation at direct incidence, as shown in Figure 1, to include data Tor a wider range of atmospheric moisture conditions. solar radiation Methods of It would then be possible to predict intensity more accurately. describing the variability of solar radia­ tion availability in terms of average distribution of, and probabilities of, numbers and sequences of days with various categories of incident radiation are needed. The categories of incident radiation would conceivably be based on the per­ formance factors of a solar collector design as affected by solar radiation intensity. using data from one of the older recording stations from Madison, Wisconsin. A method could be devised by such as The method could then be used for other localities as more recorded data becomes available. APPENDIX TABLES 76 TABLE IV CALCULATED SOLAR RADIATION PER SQUARE FOOT OF SURFACE DURING CLOUDLESS DAYS 3 0 ° Latitude Solar Time 21st day of month Dec 8 am Lj.pm 9 am 3Pm 10am 2pm 11am 1pm 12 noon Sun Azimuth Degrees Sun Altitude Degrees T Horizontal Surface per hour '' cos i per . hour per hour 136 II48 163 180 11 21 29 35 37 196 263 288 300 302 0.192 •35'8 .1+85 .571+ .602 38 91+ 11+0 172 182 17 26 30 31 32 123 lb 133 D46 162 180 zb 32 38 bo 230 27b 295 305 308 0.21+2 .1+07 .530 112 156 .616 188 •61+3 198 56 20 28 30 32 32 170 203 21b 0.577 .672 .71+2 .783 .799 76 II4.0 186 220 230 0.529 .623 .703 .71x9 .766 1)412 107 116 126 U4O 158 180 7 19 30 bo bi b9 114.0 256 290 309 315 316 0.122 .326 .500 •61+3 .731 .755 0.121 .1+12 .5 09 .587 230 30 32 33 29 110 175 231 263 .632 239 3b 273 •656 17 81+ 11+5 199 12 26 Total BTU /day Mar Sept 7am 5pm 8 am If.pm 9am 3pm 10am 2pm 11am 1pm 12 noon Total BTU /day 55 120 1268 Total BTU /day Feb Oct 7am 5pm Bam 1+pm 9am 3pm 10am 2pm 11am 1pm 12 noon 4J-----r :| COS 1 I 126 Total BTU /day Jan Nov 8am Ippm 9am 3pm 10am 2pm 11am 1pm 12 noon 1th BTU per hour 1696 13 225 26 38 280 98 106 117 131 b9 152 57 180 60 305 316 322 321+ 0.225 .1+38 .616 .755 .839 .866 51 20 123 28 188 239 270 281 32 3b 35 35 71 0.135 .21+8 151 220 .358 .1+30 213 : .1+81 305 316 1 .500 2090 77 TABLE IV (Cont *) Vertical Surface! BTU per nour. 113 177 214 235 241 Zdv BTU per nour 12 26 32 43 43 ^■tv BTU per nour 125 2 0 J4. 246 278 28k Surface Sloped 30° From Vertical cos i BTU per nour 0.596 .761 .885 .965 .993 117 200 255 289 300 BTXJ per hour 11}. 26 31 39 39 1866 122 171 207 228 236 16 29 37 45 45 138 200 244 273 281 4 19 33 14-2 46 46 21 124 181 223 245 253 0.579 -743 ..874 .957 .984 133 20lp 258 292 303 17 29 35 14-1 lj.1 11 24 34 41 46 47 41 93 143 174 201 219 1345 131 22o 286 328 339 cos i BTU per nour 0.14-55 •6 I4.6 .791 .889 •921 ■^dt BTU per nour 150 233 292 333 Skk 0 •166 .520 .691 .829 .912 .914-5 23 133 200 256 28 7 299 10 20 32 39 42 k? 33 153 232 295 329 31+1 15 101*. 26 228 31 35 36 195 259 267 278 51 121 .618 188 .714-9 236 269 231 .836 .866 14 25 33 39 42 k3 65 146 221 275 311 32142092 302 3 H 4- 1950 O.I4.7 I4- 109 •66 I4. 182 .811 239 .908 277 .914-0 289 19 28 32 38 38 128 210 271 313 325 2059 0 •166 .I4.88 .688 .850 .9149 .982 23 125 199 263 299 310 6 23 31 35 37 38 29 II4.8 230 298 336 3 I4-8 2197 2193 0.229 .I4.3 I4 tt BTU per nour 89 169 2229 1677 30 69 109 136 155 162 BTU per hour 2173 1867 17 105 lk8 181 199 207 Surface Sloped 60° From Vertical 0.263 .503 .712 .869 .967 1.000 59 llj.0 217 2714311 321}. 17 27 33 36 39 39 76 167 250 31 0 350 363 2366 78 TABLE Sun Solar Time Azimuth 21st day of month Be ore e s Apr Aug 6 &)ii 8 pm 7 am 8 pm c a m Up in 9 air, 3 pm 10 am 2 pm 11am 1pm 12 n oon Sun Altitude Degrees IV I ■*-o BTU per hour 61 6 80 39 9/ 103 116 139 l30 19 32 55 57 67 72 195 250 266 281 288 Oi '*' d yyj (G o n tJ horizontal Surface I-a cos i BTU per nour 0 .io5 .326 •530 •707 .339 .920 •951 ^dh BTU per hour 6 5 13 65 127 22 86 29 32 35 35 35 156 220 270 188 -\«~v/ < ^30 265 276 To t a1 5TU /hay July Bay oan 6 pm 7 am 9 pm 8 am 1+pm 9am 3 pm 10 am 2 pm 11am 1pm 12 noon Tot a.1 BTU /day 300 311 cos i 1-0,155 .017 .059 .159 .239 .295 .309 2501 72 79 35 93 103 122 130 10 23 35 bo 61 73 8c 122 210 2go 278 28lp 291 293 0.175 21 •391 .575 82 152 • 375 .956 •985 ■>-1 2iy9 278 239 15 25 30 335 35 35 Tot a 1 BTU /day June 6 am 8 pm 7am 3 pm 3am iipm 9am 3pm 10 am 2pm 11am 1 pm 12 noon ■*"th BTU per hour 35 107 172 233 283 313 325 - 0.305 - .176 - .071 .935 .109 .155 .175 2589 70 75 82 88 97 112 l80 11 25 37 59 62 75 83 150 215 251 271 23Z| 291 293 0.191 •507 .602 •735 .883 •966 .993 27 87 151 205 251 281 291 15 25 30 32 35 35 35 52 112 181 237 285 316 326 261j.l -0.336 - .237 - .111 - .023 .057 .097 .122 79 TABLE dv BTU per hour Ifv BTU per hour 10 ib ¥ 67 65 89 3b 37 76 39 106 b3 128 133 23 1+5 xt cos 1 BTU per hour | i - 0.114.8 3 (C ont.) Surface Sloped 30 5 From. Vertical Vertical Surface BTU per hour IV 1 .316 .14-91 •626 .715 .71+3 29 76 131 176 206 216 b &5 per hour b 11+ 25 Surface Sloped 6 0 ° From Vertical xtt BTU per hour cos I BTU per hour b b3 bo 101 16 I4. 1 ! 213 214.6 ! 1+2 258 33 37 1 0.013 .273 .14-89 .692 10 51*. 117 181+ •81+6 238 .91+14- 272 .977 283 per hour b 16 27 33 36 38 38 BT h* per hour 11+ 72 ll+i+ 217 271+ 310 32 1 , 1 1802 799 2379 j 9 31 1+5 51 5 12 16 32 37 k3 kb 16 i+i 68 88 95 0.0^3 .225 •14-01 .531 .612 ♦ 6^3 9 56 108 151 178 188 8 16 21 32 36 8 25 77 lLO 187 bo 218 1+1 229 - 0.250 .14.61 ii .661 .813 .906 •91+0 52 178 231 2614. 275 11 21 25 32 35 38 38 11 73 139 210 266 302 313 1j 1532 508 28 5 12 16 20 30 1+2 20 53 70 36 bb 80 16 301 0.205 .357 .14-90 .567 .602 51 97 139 165 176 8 16 21 2b 31 bo i+i 8 16 72 121 170 205 217 1385 2317 1 1 1-0.233 " .1+65 i .61(.2 .793 ; .885 1 .921 50 117 171+ 225 257 270 12 21 25 28 33 37 38 12 71 11+2 202 258 291430 8 2260 30 TABLE V CALCULATED SOLAR RADIATION PER SQUARE FOOT OF SURFACE DURING CLOUDLESS DAYS 35° Latitude Solar Time 21st day of month Dec * 8 am ippm 9am 3pm 10am 2pm 11am 1pm 12 noon Sun Azimuth Degrees 126 137 150 161+ 180 Sun Alt i tude Degree s 8 18 25 30 32 | Horizontal Surface lo BTU ^th Idh per ; cos i Ih cos 1 BTU BTU BTU t hour I ' per per per hour hour hour ! 160 253 278 290 295 0.11+0 22 .310 78 .1+23 118 .500 H+5 .530 156 lip 23 28 29 30 Total BTU /day Jan Nov 8am ippm 9am 3pm 10am 2pm 11am 1pm 12 noon 121+ 135 XI4.8 163 180 11 22 29 33 35 196 261 288 29 8 302 0.392 38 .371+ 98 .1+85 11+0 •51+5 163 •575 171+ 17 27 31 32 33 Total BTU /day 0.51+9 .655 .71+2 .802 .819 55 125 171 195 207 1258 ! I 1 ! 108 117 128 li+3 160 180 6 17 27 36 Ip2 kb 120 2!p8 283 301 310 312 0.105 .293 .1+51+ .588 .669 .695 13 lb 129 177 207 217 10 23 29 31 33 3ip Total BTU /day Mar Sept 7am 5pm Sam Ippm 9am 3pm 10am 2pm 11am 1pm 12 noon : 0.582 ! .695 .785 ' .832 .81+8 1076 Total BTU /day Feb Oct 7am 5pm Sam ippm 9am 3pm 10am 2pm 11am 1pm 12 noon 36 101 lij.6 17lp 186 23 97 158 208 21+0 251 1527 98 108 120 135 155 180 12 2U 35 bS 52 55 208 0.209 271p .1+07 300 ! .571+ .707 313 319 : .788 320 ; .819 ip3 112 172 221 251 262 18 28 31 33 3k 3b 61 11+0 Z0J 251+ 285 296 I9J4.O 0.129 .1+31+ .51+1+ .61+6 .698 .719 : 1 0.136 .282 i .1+10 I .500 .558 .571+ 81 TABLE Vertical Surface Tv T dv T tv BTU per hour BTU per hour BTU per . hour 93 176 218 21+1 250 9 25 33 1+1 ^1 102 201 251 282 291 V (C ont.) Surface Sloped 30 From Vertical ^dt ^tt cos i BTU BTU per per per hour hour hour 92 192 .898 250 .971 28 2 .999 295 0 .5 7 k .7 5 9 11 25 32 38 38 12 29 39 k2 kk 120 200 2514280 291 0.571 .755 .886 .960 .990 112 197 255 286 299 Ik 28 38 39 k l 3 17 32 1+0 k5 k5 19 125 186 231+ 261 269 0.173 .525 .716 .853 .939 .970 21 130 203 257 291 303 5 19 31 37 1*2 1*2 10 21+ 31 1+1 1+6 1+7 38 101 X5k 19 8 221+ 231 11+81 29 3k 3k 126 225 291 325 3i+0 26 11*9 23k 29k 333 3k5 0.223 .1+1+8 .61+2 .787 .888 .907 1+6 123 193 21+6 283 290 13 25 31 39 1+2 1+3 59 11+8 221+ 285 325 333 211+9 78 l8l 2k3 280 291+ 1802 o.l+l+l .6 35 .791 .900 .939 86 166 228 268 283 15 27 33 35 36 101 193 261 303 319 1952 ! 0.156 .1*71* .695 .832 .928 .962 18 118 197 250 288 300 7 21 30 3k 37 38 25 139 227 281+ 325 33 8 2107 2186 171+6 28 77 123 157 178 181+ 12 2k 2166 1891+ 16 108 151+ 191+ 216 22l+ : 0.1+12 66 .620 157 .771 2 1 k 2k6 ■ l! * i! .882 260 2100 1885 108 171 215 238 21+7 103 217 282 320 333 Surface Sloped 60 6 From Vertical tt cos i t BTU BTU per per per hour hour hour 0.21+9 .1+93 .702 .862 .961 .996 52 135 211 270 307 318 15 27 31 35 38 38 67 162 21+2 305 31+5 356 2311 82 T A b L ji SolarTime 21st day of month Apr Aug 6am 6pm 7 am 5 pm Sam Lppm 9am 3pro 10am 2pm 11am 1pm 12 noon Sun Azimuth Degree s 82 08 97 108 123 1/.6 18C Sun Altitude Degree s 7 19 31 43 54 V (C ont.) !| Horizontal Surface Io BTU per a#tj .cos i b ??u hour 1 per per per • hour nour hour ■■■ ■■ 80 c. 1 2 ? 195 .326 .515 236 262 63 278 285 67 288 .632 .809 .891 .921 10 63 122 179 225 254 265 8 21 29 31 33 34 34 Total BTU / day May Jul 6am 6pm 7 ami 5 pm Qam 4pm 9am 3pm 10am 2pm 11am 1pm 12 noon To tal BTU /day | cos i 1 i !1 18 34 151 210 258 288 299 ..M - 0.138 - .038 .105 .226 .320 .376 .391 2310 73 82 89 98 111 134 180 11 23 36 48 140 210 250 270 60 203 70 75 290 291 0.192 .391 .588 •743 •866 .940 .966 27 11 82 147 25 30 32 34 35 35 201 245 273 281 Total^ BTU /'day June 6 am 6 pm 7 am 5 pm Sam 5 pm 9am 3p^ 10am 2pm 11am 1pm 12 noon ! 1 38 1C7 177 233 279 308 316 -0.286 - .128 0 .093 .179 .238 .259 2573 71 79 85 94 13 25 37 49 155 220 251 271 0.225 .423 .607 .755 106 62 204 .863 127 73 78 291 292 .956 .976 180 35 93 151 205 251 278 286 16 26 30 32 34 35 35 237 285 313 -0.317 - .173 - .069 .046 .129 .176 321 .208 51 119 181 2655 ; 83 TABLE V (Cont.) *tv BTU per hour Surface Sloped 30° Surface Sloped 60° Prom Vertical From Vertical ^dt *tt 1 Idt t cos i BTU cos i xBTU BTU BTU BTU per per per per per per hour hour hour !r hour hour hour .... i ~ ■— — I 11 1+9 92 130 152 159 0 .131+ •31+9 .337 : .682 ! .772 , .800 916 1: Vertical Surface BTU per hour ^dv BTU per hour k 25 59 89 107 113 11 21+ 33 1+1 US 1+6 23 31 69 73 13 31 83 109 120 5 13 21+ 32 38 1+1 1+3 3 39 106 173 228 261 27 3 ; 0.037 3 ’ .263 51 1 .1+99 118 •70i+ 181+ .861 239 .960 2 7 k .991+ 286 7 17 27 31 35 37 38 10 68 345 215 27k 3 11 322+ ' ' 2367 1893 i+ 11 13 26 31+ 1+0 J+3 26 82 lL+1 190 220 230 0.083 .291+ 18 71+ 122 166 196 206 6 16 20 28 33 38 1+2 6 31+ 91+ 150 199 231121+8 , 0.022 3 .273 38 ! .388 D+7 ! .690 186 .828 231+ ! .933 271 ! 1 .967 281 qy 20 25 30 31+ 36 38 12 78 172 216 268 307 319 ! 21fll 1676 630 1 12 37 51 61 7 12 15 33 39 1+3 1+5 15 2+5 76 92+ 106 333 0.062 •2l|l|- Ik 61 113 157 183 195 10 13 20 33 37 1+0 1+2 10 27 81 11+6 191+ 223 237 1588 0.036 .279 .1+91 .677 .830 .916 •951 6 61 123 183 236 267 278 13 21 25 32 36 38 38 19 82 11+8 215 272 305 316 2390 % TABLE V I C A LC U LA T E D SOLAR R A D IA T IO N D U R IN G PER SQUARE FOOT OF SURFACE C LO U D LE S S DAYS 1+0° Latitude Solar t line 21st day of month Sun Azimuth Degrees Sun Altitude Degrees 11' 1 Dec 8 am Ippm 9am 3pm 10am 2pm flam 1pm 12 noon Total BTU /day Jan Nov Bam Ippm 9 am 3 pm 10am 2pm 11am 1pm 12 noon "- Horizontal Surface DIU I 1 per : cos i cos 1 B^U hour per per ■ per " ■'hour hour hour ■] \\ 127 138 151 165 180 5 1^ 21 25 26 110 0.087 •2lp2 23 k : .358 263 278 .Ip23 280 .k 3 & 10 56 95 113 123 8 13 26 28 29 81+8 1 125 136 !Jp9 l6!p 180 8 17 2il 2Q 30 160 2ip8 27ip 287 290 0.139 .292 •Ip06 Jp70 .500 22 72 111 135 U5 15 23 28 30 30 Total BTU /day Feb Oct 8am Ippm 9am 3 pro 10am 2pm 11am 1pm 12 noon 37 95 139 165 175 0.568 .687 .783 .31+9 .866 1021 ,, 118 130 ilp5 161 180 lip 2lp 31 37 39 230 27ip 292 302 308 0 •2ip2 .Ip06 .515 .602 .629 56 ill 190 182 191+ 20 28 30 32 32 Total BTU /day Mar Sept 7 am 5 pm Bam Ippm 9am 3pm 10am 2pm 11am 1pm 12 noon Total BTU /day 18 0.600 .721 69 121 .817 ll+fa .873 152 , .899 76 139 180 211+ 226 0 .1+56 .587 .702 .756 .777 1383 100 110 123 138 157 180 11 23 33 k2 kQ 5o 196 272 297 310 316 317 0.191 .391 •5i+5 .669 •7ip3 .766 37 106 162 207 235 21+3 17 27 30 33 31+ 3k 51+ 133 192 21+0 269 277 1831 0.171 •315 .1+57 .552 .615 .61+3 85 TABLE VI (Cont.) b ¥u per nour per hour 66 166 215 21+3 252 6 17 33 36 37 *tv BTU per hour 183 214.8 279 289 cos i B^U per . ■ hour i ' 0.520 ' .7U5 ! .886 i .968 ' .998 57 17k 233 269 279 ■^dt BTU per hour 1 -hfc B T U *s per hour 7 16 31 3k 35 61+ 190 261+ 303 311+ 10 22 32 39 39 101 192 2k7 283 290 1858 Surface Sloped 6 0 ° Prom Vertical cos i I t BTC per hour 0.375 -571 .719 .801+ .829 131+ 189 221+ 232 ■^dt BTU per hour 7 ik 28 30 31 0.571 .751 .881 .970 1.000 91 186 214-1 278 290 11 22 31 36 36 101 208 272 311+ 326 : 0.1+01+ .597 ! •71+1+ .832 ' ; .866 65 11+8 201+ 239 251 ik 23 29 33 33 i 201+0 ! : Itt BTU per hour ik 8 217 25k 263 1596 1920 l80k 91 170 215 21+1+ 251 I Surface Sloped 30° Prom Vertical Vertical Surface 79 171 233 272 28k 1736 1 1 105 161 205 228 239 16 29 37 £2 k2 121 190 21+2 270 281 ; 0.516 .711 .865 .956 .987 119 195 253 289 30ip 17 29 36 39 39 2178 1818 31+ 86 136 171 19k 20k 7 21 3k kl iT**’ k5 1+6 0.21+3 1+1 107 ; .1+68 .668 170 .812 212 .901+ 239 250 , •91+0 1590 136 221+ 289 328 31+3 k8 127 198 252 286 298 10 23 33 39 k2 k2 58 150 231 291 328 31+0 2188 0.1+37 .61+6 .797 .899 •931+ 101 176 233 271 288 19 28 32 35 35 2010 i 1! i! j! |j ; 120 20k 265 306 323 0.251 .1+97 .700 .855 .951 .985 1+9 135 208 265 300 312 ik 25 31 35 37 33 63 160 239 300 337 3ko 226k 86 TABLE Solar Time 21st day of month Sun Azimuth Degrees 6am 6pm 7 am 5 pm Sam ippm 9am 3pm 10am 2pm 11am 1pm 12 noon 83 90 100 112 128 150 180 Sun Altitude Degrees 8 19 30 1+1 51 59 62 Io BTU per hour 110 195 23k 260 275 282 2Qk VI (C ont.) Horizontal Surface -Lh cos i BTU per hour .0 .IlpO ; .326 .50 0 . 656 .777 .857 .883 15 6k 117, 171 21k 2k 2 251 ^dh BTU per hour 11 21 28 31 33 3k 3k Total BTU /day May < July 6 am 6pm 7 am 5 pm Bam l+pm 9am 3 pm 10am 2 pm 11am 1pm 12 noon Total BTU /day 26 85 345 202 2k 7 276 285 cos i -0.121 .0 .151 .283 .387 •kk^ .14-70 2232 72 83 92 101+ 119 11+2 180 13 2k 36 14-7 56 66 70 160 215 250 269 281 288 290 0.225 .407 .588 .731 .81+8 .913 .914-0 36 88 lk 7 197 238 263 273 17 25 30 32 53 113 177 229 3k 212 35 35 298 308 Total BTU /day June 6 am 6 pm 7 am 5 pm 8am ]+pm 9 am 3 pm 10am 2pm 11am 1pm 12 noon ^th BTU per hour -0.301 -0.111 .028 .165 .257 .321 .31+2 251+7 72 80 89 100 a 180 15 26 37 k9 60 69 73 170 220 251 271 283 289 291 0.259 .1+38 .602 .755 .866 •934 .956 kk 97 151 205 21+5 271 279 18 27 28 33 31+ 36 36 62 ; -0.299 -0.156 12k .0 179 238 . .111+ .279 5 .205 •266 307 .29'2 315 87 TABLE VI (Cont.) X» BTU per hour- B$f per . hour Surface Sloped "6"o°~" Prom Vertical Surface Sloped 30° Prom Vertical Vertical Surface cos i per hour b Iu per hour per. hour cos i B$U per nour per hour B&$ per nour b I5 per hour • 9 18 26 31 34 37 38 9 73 122 216 272 309 322 i k 35 74 106 126 133 10 22 33 37 43 45 57 107 143 169 178 I 0.163 .381 .573 .723 .81I4. .848 32 89 049 199 230 241 7 14 24 33 36 40 42 7 46 113 182 235 270 233 1 1986 10 8 k 7 12 7 hk 72 92 99 34 Co 44 k7 7 12 31 78 112 136 llj.6 ■0.107 .318 .508 .647 .734 .766 23 80 137 182 211 222 10 16 26 3^38 41 43 31 58 77 11 85 17 66 96 118 129 716 55 96 185 238 272 284 1 1 0.296 1 .523 ! .716 .863 .952 .985 2342 64 131 192 243 274 286 14 21 28 32 36 38 39 i !0.084 .341 •496 .611 .697 .731 18 86 134 173 201 213 11 17 20 34 37 39 41 11 35 106 168 210 240 255 1781 14 85 169 224 279 312 325 2481 1850 895 8 12 17 35 38 kl H ""4* 44 10 39 106 171 220 252 265 : 0.282 1 .409 ! *710 1 .867 i .965 1.000 0.301 .521 .711 j .853 i .942 '! .974 66 131 193 241 272 283 17 22 24 33 35 37 38 17 88 155 226 276 309 321 2463 88 TABLE C A LC U LA T E D SOLAR R A D IA T IO N D U R IN G V II PER SQUARE ROOT OF SURFACE C LO U D LE S S DAYS 25° Latitude | Solo r Sun Time Azimuth 21st day of month Degree s Dec 9am 3 pm 10 am 2 pm 11 am 1pm 12 nc on 139 151 165 180 Sun Altitude Degrees 10 15 20 22 Io BTU per hour 185 241 260 266 Hori zont al Surf ace cos i per hour 0 .172 .276 .329 .382 32 67 91 102 per hour 16 23 26 27 Total BTU / day Jan Nov 8am Ippm 9am 3 pm 10 am 2 pm 11 am 1pm 12 nc on 125 137 150 165 180 5 13 19 22 25 110 225 256 272 276 0 .037 •22y .326 •229 •23- 10 51 83 115 121 8 19 25 26 29 Total BTU /day 28 90 117 129 0 .744 \ .641 .908 .927 108 0 .$86 .712 .818 123 150 .906 18 70 .883 822 119 132 lip6 162 180 12 21 26 33 32 20 6 263 267 297 293 0 .208 .366 .269 .525 i •-ej'erc/ 23 96 135 162 167 18 61 26 30 31 31 122 165 -193 196 Total BTU / day Far Sept 7 am 8 pm 8 am ippm 9 am 3 pm 1C am 2 pm 11 am 1pm 12 noon 1 cos i 626 Tots: ■'a .y Ulu r1e b Oct 8am 2pm 9am 3 pm 10 am 2 pin 11am 1pm 12 nc on bW per xhour 0 .474 .625 .732 .798 .829 12.32 107 112 ^ 0B.ry _Lt lipl 159 160 11 21 30 38 23 kS 196 263 290 305 311 313 0 .191 .366 .500 .616 .682 .707 37 96 125 188 212 221 17 26 30 32 33 33 0 .287 122 .359 .509 175 220 . .612 .683 225 .707 252 52 1678 89 TABLE VII Surface Sloped 30'° From Vertical Vertical Surface Iv BTU per , hour 137 203 236 247 ^■dv BTU per hour 13 23 31 33 ^tv BTU per hour xt cos i BTU per hour 150 226 267 280 0.731 •866 .961 •991). 135 209 250 264 BTU per. hour 14 23 30 31 4 18 27 37 37 68 178 236 279 289 1772 99 164 210 237 21*7 12 26 36 41 4l 6 16 31 40 45 k5 .730 .871 .9 7 5 .999 62 110 179 227 257 265 1698 61 164 223 267 278 5 18 27 34 34 ! 111 190 246 27 8 288 1843 56 94 146 167 212 221 0 .5 5 1 I Surface Sloped 6 0 ° Prom Vertical Jtt BTU per nour cos 1 BTU per. hour Idt BTU per hour per hour 114.9 232 280 293 0.522 97 172 .7114.756 197 .976 212 15 23 28 29 112 195 225 241 1651 1601 64 160 209 21*2 252 (Cont.) 66 182 250 301 312 i i i ! 1 0.368 .551 | .691 •801). .891 1320 40 124 177 220 248 7 19 26 31 31 1539 187 5 0.514 .724 .875 .964 .998 107 190 251 286 297 14 26 34 38 38 ; 0 .14.15 ; .629 .772 1 .871 1 .898 86 165 222 259 268 16 26 32 34 34 68 130 200 256 290 302 9 19 31 37 4l 41 77 1 0.308 •l)-96 U 4.9 .687 231 .839 293 .932 331 .965 31)-3 2210 102 191 254 293 302 1900 2125 ; ! 0.345 I .494 ! .691 ! .838 i .932 •966 121 216 285 32 k 335 47 143 203 251 279 60 130 199 256 290 302 13 23 30 34 37 37 73 153 229 290 327 339 2198 90 TABLE VII (Cont.) TJ Solar Time 21st day of month Apr Aug 6am 6pm 7 am 5 pm Gam L|_pm 9am 3pm 10am 2pm 11am 1pm 1 2 noon Sun Azimuth Degrees Sun Altitude Degrees Io BTU per hour | i' 1 ! Horizontal Surface ' cos i B*?U per hour per hour 82 93 103 116 133 151+ 180 8 19 30 39 1+8 55 57 110 195 231+ 0.139 .326 .500 .629 .71+3 .819 .839 25k 270 279 281 15 6k 117 160 201 229 236 11 22 27 30 32 33 3k 26 -0.138 86 .0I+9 .195 H+4 190 .353 .1+56 233 262 .516 270 ' .51+5 76 3k 96 109 125 11+8 180 Ik 25 35 1^6 55 62 65 0.21+2 .1+23 .57k .719 .819 .883 .906 160 220 2W 268 279 2Bk 287 39 93 li+2 193 229 251 260 17 26 30 32 33 3k 3k 56 119 172 22 5 262 285 294 /day -0.235 -0.063 .056 .231 .338 .1+00 .1+23 21+91 76 85 93 105 121 il+5 180 20 27 37 kQ 58 65 69 200 220 251 270 281 287 289 O.3I+2 .k5k .602 .743 . 8/4.8 i .906 .93k 68 100 151 201 238 260 270 22 26 30 32 3k 3k 35 90 126 181 233 272 291+ 305 T otal BTU . 1 2136 : Total BTU /day June Gam 6pm 7 ara 5 pm Gam Ifpm 9am 3pm 10am 2pm 11am 1pm 12 noon cos i ■ Total BTU /day May July 6am 6pm 7am 5pm Gam i|.pm 9am 3pm 10am 2pm 11am 1pm 12 noon B® per hour 2610 -0.227 !;-0.078 j .01+2 ! .173 i .273 .31+6 | .358 ! 91 TABLE VII Vertical Surface *v BTU per hour Idv BTU per hour 10 1+6 90 123 ii+i+ 153 U 10 26 32 14-0 ktbl US ■J-tv BTU per hour 20 72 122 163 188 198 (Cont.) Surface Sloped 60° Surface Sloped 30° | From Vertical From Vertical iat Xt Idt *tt *tt cos i BTU cos i BTU BTU BTU BTU BTU per per per per per per hour hour hour hour hour hour i j |0.205 ' .U19 .621 1 .767 ! .857 11 .892 bo 98 158 207 239 251 6 13 26 31 37 U1 1+2 6 53 121+ 189 21+1+ 280 293 i ; 1 j ' !| 0.091 .303 .530 .721 .871 .967 .999 10 59 124 183 235 270 281 9 18 27 31 31+ 37 38 19 77 151 211+ 269 307 319 i ll+ 62 91+ 111+ 121 8 12 27 3b bo 10+ b5 i+i 96 131+ 1^8 166 0.517 .335 .560 .703 .788 .819 35 83 150 196 22b 235 11 17 28 33 38 Ui 1+1 11 14 »-7 77 99 103 30 81+ ii5 11+2 11+8 875 11 52 in 183 231+ 265 276 0.093 •331+ .525 .738 .878 i ! .965 ! .996 ;\ 15 74 130 198 21+5 271+ 286 H| 21 29 33 35 37 38 0*159 .337 1 .522 •660 ! .753 1 .777 35 85 1U1 185 216 225 11+ 17 23 35 37 1+0 1+2 0.183 11+ 52 .359 108 • 542 176 .729 .870 222 256 .9 58 .988 267 |i 1919 29 95 159 231 280 311 321+ 2521 1986 991+ 10 12 20 37 38 U3 05 2381 2083 ! i I l 1186 37 79 136 197 21+1+ 275 286 18 21 27 31+ 35 37 38 55 100 163 231 279 312 321+ 2558 92 TABLE VIII RATIO OP HORIZONTAL TO SOUTH-FACING TILTED SURFACE INCIDENT RADIATION O Latitude 1*0 0 35° 1A °o rn Angle of Surface from Vertical Degrees 1 Dec* 21 0 30 60 1.1*9 1.71 1.51* 1.75 1.95 1.68 2.13 2.26 1.87 2 .1*8 2.55 2 .01* Jan* 21 and Nov* 21 0 30 60 1.32 1.58 1.1*6 1.51 1.72 1.55 1.82 2.02 1.70 2.15 2.28 1.87 Feb. 21 and Oct • 21 0 30 60 0.99 1.29 1.29 l.ll* 1.11-3 1.38 1.31 1.57 1.1*5 1.50 1.73 1.51* Mar • 21 and Sept * 21 0 30 60 0.61* 1.00 1.13 0.77 1.10 1.19 0.87 1.19 1.21* 1.01 1.32 1.31 Apr • 21 and Aug. 21 0 30 60 0.32 .72 .99 O.kO .82 1.02 0.1*9 .89 1.05 0.57 .98 1.12 May 21 and July 21 0 30 60 0.19 .59 .89 0.25 .65 .91+ 0.35 .72 .97 O.kO .80 1.01 0 30 60 0.11 .52 .85 0.21 .60 .90 0.28 .67 .93 0.33 .73 .98 June 21 93 TABLE IX AVERAGE NUMBER OF CLEAR, PARTLY CLOUDY, AND CLOUDY DAYS** Boston Yrs. of Data Lansing Lincoln Albuquerque Madison Lander 63 37 51 53 69 56 Jan. C le ar 9 Partly Cloudy 9 Cloudy 13 k 11 8 12 18 7 6 8 12 9 13 Month Feb. Clear 10 Partly Cloudy 8 Cloudy 10 Mar. Clear 10 Partly Cloudy 9 Cloudy 12 April Clear 9 Partly Cloudy 10 Cloudy 11 May Clear 9 Partly Cloudy 10 Cloudy 12 June 10 Clear Partly Cloudy 9 Cloudy 11 July Clear 9 Partly Cloudy 13 Cloudy 9 Aug. 11 Clear Partly Cloudy 11 Cloudy 9 1 20 5 8 15 11+ 9 7 8 11 8 13 6 11 12 5 10 9 12 16 8 10 9 111 10 1k 7 9 13 9 9 12 15 10 7 10 13 13 9 9 10 8 11 17 10 12 12 k 9 13 8 10 7 12 12 8 18 10 2 13 11 5 12 13 12 10 13 6 12 6 15 Ik 7 13 15 7 11 12 12 12 10 13 13 7 9 15 8 13 7 12 7 5 5 k 15 k 8 8 11 12 8 k 91*. TABLE IX (Cont.) Boston 63 Dans ing 37 51- 53 69 Lander 56 Sept. Clear 12 Partly Cloudy 9 Cloudy 9 10 10 10 1k 17 9 k 10 10 10 11 5 Oct. Clear 11 Partly Cloudy 10 10 Cloudy 10 9 12 15 7 9 20 7 4 10 9 12 u* 11 6 l*ov. Clear 9 Partly Cloudy 9 12 Cloudy 6 19 11 8 11 20 6 7 9 k Ik 11 13 6 k 10 9 12 18 7 6 6 8 17 12 13/ 6 9k 132 113 120 197 iii 51+ 98 122 1U5 li+0 155 170 Yrs. of data Dec. Clear 9 Partly Cloudy 9 Cloudy 13 Clear 118 Partly Cloudy118 Cloudy 129 6 21 113 158 Lin co In 8 8 Albuquerque Madison ^Technical Paper No. 12, U. S. Department of Commerce Weather Bureau 95 TABLE X PERCENTAGE OP POSSIBLE SUNSHINE^ Yrs* of Data Boston 51+ Lansing 37 Lincoln 1+3 Albuquerque 25 Madison 1+3 Lander 1+5 Month Jan* 1+9 33 57 71 1+3 66 Feb. 56 kb 59 70 1+8 70 Mar • 58 53 60 71+ 51 71 Apr. 57 58 60 76 53 65 May 59 63 62 80 56 65 June 62 69 69 81+ 62 71+ July 53 75 76 77 70 76 Aug. 63 66 70 77 61+ 75 Sept • 61 59 66 79 57 71 Oct • 58 52 61+ 80 52 66 Nov • 1+8 35 57 77 39 60 Dec. 1+8 28 51+ 72 36 62 57 53 63 77 53 69 Annual '"'Prom Technical Paper No. 12, U. S. Department of Commerce Weather Bureau 96 TABLE X I RELATIONSHIP BETWEEN THE RATIO OP OBSERVED (I) TO CALCULATED CLOUDLESS DAY RADIATION (I0) AND PERCENTAGE OF POSSIBLE SUNSHINE (S) Dat 0 Blue Hill Radi ation L y ./Day I 1A 0 Io S . Jan* 236 *51 *52 ’53 325 *5i *53 ' 5k Mar. K O *51 *52 *53 *5}+ U5-2 A p r . »5o ’5i !52 •53 561+ '5k May June »5o *51 '52 '53 *5k 244 .81 «51 30 38 37 1+0 33 183 215 220 226 1 92 .56 .66 .68 .69 .59 37 39 51 92 1+8 344 380 .66 .58 .62 .60 .73 53 30 51 38 57 461 257 271+ 265 323 3 37 1+2L 376 351+ 391+ .69 kk 57 6 .75 .67 .63 .70 50 11+2 191 121 '5k Feb. 120 150 66k »5o 713 *51 : *52 1 *53 i '5k ! Line oln Radiation ./Day I0 I VIG .51 •61+ .60 1+97 500 1+50 1+56 390 .75 .75 ,68 .59 1+6 1+0 602 1+37 555 655 1+79 .81). .68 .78 .92 .67 61+ 1+3 65 71+ 53 .69 7+9 c"-' 51+ 73 53 59 59 217 2S>8 E i » 285 274 .78 .73 - 63 51 50 58 71 306 303 345 338 338 .7 4 •75 - 55 55 1+3 55 59 434 314 285 332 290 323 366 .67 .75 55 ‘ 555 l+l 51 52 67 337 431 363 380 .75 • i> — .76 .72 61 r‘6/ ^ 68 61+ 61+ 664 542 498 414 482 468 .92 .62 - 83 1+1 80 76 81 713 574 503 498 589 548 188 185 •71+ .61 50k — 510 465 50 713 .73 .73 350 — 434 669 .77 .33 426 388 39 50 136 168 .76 .89 — .77 .76 186 218 — kkr S Madison Radi ation Ly.,/Day I Io 657 440 — 568 594 .80 .83 142 124 152 208 189 206 210 204 97 TABLE XI Madison (Cont.) Albuquerque Radiation Ly./Day Lander Radiation Ly./Day o H H S *0 257 •63 .77 .65 .57 .70 45 41 36 I|.6 .68 .62 .67 .69 .67 56 33 ^2 50 59 3*68 .72 .66 .76 .67 55 14.96 1 V x 0 220 .86 226 .88 S 56 58 62 69 352 267 307 302 3i+X 312 .76 .87 .85 .97 .88 68 75 69 81 67 J+55 373 3 88 388 i+09 456 .82 .85 .85 •90 1.00 71 73 70 70 92 583 428 485 505 534 502 .74 .83 .86 .91 .86 73 68 68 73 72 700 601 600 619 639 668 •86 .86 .88 .91 .95 71 72 73 87 615 674 671 724 708 .78 .85 .85 .92 .90 683 .82 .93 .88 6k 32h .88 72 35 8 .97 320 .87 a 66 75 442 .89 56 h92 .97 •7k 68 k3 7 .88 65 60 68 78 60 .66 .61 .78 .65 .68 kB 600 •93 69 582 589 559 .90 •91 .87 6^ 67 79 .82 .75 .62 .73 .70 75 59 59 60 57 786 595 .76 65 63 61 61 71 789 .81 .71 .70 .83 .77 75 SI4-O 78 76 79 86 73 8^0 6I4.2 35 *k Ij»2 51 5k 60 72 69 595 594 632 .76 .76 .80 716 .85 747 738 679 .88 .81 .89 S 1 778 732 745 777 .89 .93 Qk 88 7k 75 76 79 87 91 86 81 90 98 TABLE TABLE X I I [ July *5o *5l •52 '53 ’5i+ ■ i J ! 699 i A u g . '5o 623 '51 1 '52 '53 i '51+ 1 \ Sept '5o \ S o i '51 '52 '53 1 "51+ 1 Get. I Blue Hill Radiation Ly ./Day Date '50 ; 385 '51 '52 i '53 '54 Nov • '50 27Ip '51 '52 J '53 '51+ 1 D e c . •50 :206 '51 '52 '53 ’51+ VI0 s .75 .61 .92 .95.85 65 1+2 83 87 1+96 356 370 1+11 1+1+2 331 31+9 360 351 237 .88 .91 82 51 82 83 5-7 381+ 1+2 1 279 50 5-55-7 51 218 .78 .76 .66 53 59 71 57 76 257 217 176 — 169 166 160 67 51+ 66 67 65 201 53 52 57 63 52 3+3 .68 .72 .75 .79 .51 1+5 60 69 69 5-9 .78 .68 .68 .62 .58 63 1+6 67 52 50 .63 .58 .57 .514.51+ 127 .62 5-2 50 86 .69 .1+2 173 156 11+8 12+8 121+ 110 I 1+63 1+07 1+71 501+ 1+73 .75. .66 .68 .73 .62 158 Xo 623 1+61+ 1+13 1+23 1+52 389 21+0 225 .68 .75 - s 62 62 63 81+ 62 .72 302 221+ 262 I/-Ic 599 529 576 523 511+ 55 56 75 61 60 378 1+03 258 1 Radiation j Ly./Day I 1 699 .79 .72 366 i1"" Madi son i Radiation Ly./Day Line oln 5-9 61+ 80 73 80 553 501 577 _ 501+ -83 (C ont.) .60 5-1 53 .53 5.1 699 1 1 1 i l 525 626 1 ! 515 569 500 .81 .83 518 1+79 1+71 552 1+1+9 .77 .75 .88 .72 386 315 1+71+ 1+82 1+37 ! I | 1 5-77 396 j 1 213 185 229 .83 .89 .60 .82 .81 .78 .77 •71+ 81 271+ 232 312 291 211 171 170 161 127 153 117 93 118 - 99 TABLE XI (Cont.) 67 80 67 61 61 78 81 59 593 .73 .62 .81+ .78 .56 65 i+3l* 52 81 75 55 .1+9 1+5 .76 • Iw 1+3 .58 37 mb, 6 36 39 40 .5 9 298 i 1 1 .82 .81+ .8 3 .8 8 67 71+ 76 70 81 759 649 620 640 671 645 .8 6 .82 *8k .86 .85 79 69 72 78 80 491 600 555 619 568 .76 .93 .88 73 91 82 95 82 1*51 U52 435 471 471 .88 93 •88 .85 .92 .92 93 86 86 79 83 85 71+ 69 1*15 — 561 528 - .70 — .95 .89 - 63 69 82 81+ 85 61*5 365 ■1+05 .81+ — .93 .79 - 82 55 82 68 512 .78 — .77 .82 .80 56 61 51+ 62 .76 _ .88 53 231+ 231 2I+6 239 225 679 691 688 690 724 573 585 610 31+I+ l+l 52 52 52 826 75 S — .78 .80 .83 56 .72 .75 .83 .89 .67 •66 .66 .62 611 72 .8 8 .82 .8 6 1 1 !H 1\ 1H 0 732 CD 65 637 — 719 670 702 Io KjJ 818 s . -'si 76 72 71+ 61+ 73 I CD *0 . •71+ .65 .76 .81 .76 s Albuquerque Radiation Ly./Day 0 H .86 .76 .82 .75 •71+ Lander Radiation Ly./Day H H H O Madison 171 199 190 195 .81+ .87 68 77 87 80 396 77 55 61+ ! 71 83 325 .81+ . 86 .96 .76 •8{jr .85 299 335 33 6 348 374 •91+ 250 269 284 296 295 .82 .87 .91 .90 • 88 .77 81 91 71+ 71 80 93 89 61+ 62 72 71+ 100 TABLE XII MONTHLY MEAN PRECIPITABLE WATER IN THE UNITED STATES Mean Precipitable Water Centimeters Month R e c o r d e d for All Days Cloudless Day Value Used in Computation January 1.12 1.05 February l.llt- 1.05 March 1 .214- 1.05 April 1.62 2.00 May 2. ll|. 2.00 June 2.83 July 3.32 0 0 0♦ 0 • OJ August 3.32 2.00 September 2.78 1.05 October 2.08 1.05 November 1.50 1.05 December 1.35 1.05 ■»Prom Technical Paper Mo. 10. United States Department of Commerce Weather Bureau APPENDIX II SAMPLE COMPUTATION 102 Sample Computation of Fourth Degree Polynomial Tor Solar Radiation Curves of Figure 5 For the sample computation shown below, Table V for 35° Latitude is used. the data from X is equal to the number of months from June 21 and Y is equal to the calculated cloudless daily total solar radiation incident upon a hori­ zontal surface. Y* is the value of Y computed from the fourth degree polynomial. Date Dec. 21 Jan. 21 Feb. 21 Mar. 21 A p r • 21 May 21 June 21 July 21 Aug. 21 S e p t •21 Oct. 21 Nov. 21 Dec. 21 X Y -6 -5 -14-3 -2 -1 0 1 2 3 145 6 1076 1258 1527 1914-0 2310 2573 2655 2573 2310 1914-0 1527 1258 1076 Y' 1091 1218 1550 195142319 2566 2657 2566 2319 1951+ 1550 1218 1091 For the above data: SX SY SXY SXfY SX2Y Sx'+Y SX2 0 = 2 k , 05k = 0 = 214-7,825 = 0 = 5,537,104.9 * 182 SX:3 SX^ SX? sx° sxl SX8 0 k,55o o 1 3 k ,3 k 2 0 I4.,285,190 1C3 The following normal equations are secured from the above data: 13a Oa 182a Oa 14-,550a + + + + + Ob 182b Ob Ij.,550b Ob + + + + + 182c Oc 4,550c Oc 134,342c + + + + + Od + i+,55Of * 2k>05k + Of = 0 Od + 13lj.,342f = 247,825 134,352d + Of = 0 Od + l4.,285,190f = 5,537,1449 k*550& The solution of the above normal equations yields: a b c d e = 2656.9 = 0 = 89.671 = 0 = 1 .28214. Therefore the polynomial is: Y = 2656.9 - 89.671X2 + 1.2824X^ (j) The average deviation between the values calculated by the polynomial ( Y f) and the corresponding Y value is 16. 101* GLOSSARY Definition of Terns Air M a s s » m . Path length of light through the atmosphere, considering the vertical path at sea level as unity. The air mass is approximately the secant of the angle of inci­ dence for a horizontal surface. Angle of Incidence , i . Number of degrees between actual direction of the sun's rays and a normal to the surface. Radius Vec t o r . Actual distance between earth and sun, considering the mean distance between earth and sun as unitv. * Solar A l t i t u d e . The angle, in a vertical plane, the sun's rays and the horizontal. between Solar A z i m u t h . The angle, in a horizontal plane, from north to the horizontal projection of the sun's ray. Solar C o n stant. The energy Incident upon a unit area located at mean distance of the earth from the sun and oriented perpendicular to the sun's rays outside the at­ mosphere. The value of the solar constant is 2.00 + O.Oi^ calories per minute per square centimeter or IpipO BTU per square foot per hour. Solar D e c lination. The angular distance of the sun north or south of the celestial equator. Solar N o o n . The time for any day when the sunreaches maximum altitude for that day. its Transmission Coefficient. Portion of the solar energy incident at the top of the atmosphere which reaches the surface of the earth. Wall A z i m u t h . The angle, in a horizontal plane, between normal to the surface and north. Wall Solar A z i m u t h . The angle, in a horizontal plane, between the s u n 's rays and a normal to the surface. Zenith Angle. 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