POTSOL: Model to predict extraterrestrial and clear sky solar radiation

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  • Solar Energy Vol. 33, No. 6, pp. 485492, 1984 0038-092X/84 $3.00 + .0~ Printed in the U.S.A. 1985 Pergamon Press Ltd.

    POTSOL: MODEL TO PREDICT EXTRATERRESTRIAL AND CLEAR SKY

    SOLAR RADIATIONt

    RUSSELL BRINSFIELDJ~ Department of Agricultural Engineering, University of Maryland, Queenstown, MD 21658, U.S.A.

    and

    MELIH YARAMANOGLU and FREDRICK WHEATON Department of Agricultural Engineering, University of Maryland, College Park, MD 20740, U.S.A.

    (Received 19 August 1983; accepted 9 February 1984)

    A b s t r a c t - - A model was developed to predict potential and clear sky solar radiation for any latitude. The model (POTSOL) uses the fundamental geometric relationships between the earth and sun to predict the theoretical solar radiation outside the earth's atmosphere, clear sky solar radiation received at the earth's surface after accounting for atmospheric interference, and clear sky solar radiation on a panel with any tilt angle between 0 and 90 from the horizontal. The only model input parameters are latitude (PHI), clearness number (CN), and panel tilt angle (PT). The model was verified using weather data obtained from the National Climatic Center, Asheville, North Carolina for Ely, Nevada.

    INTRODUCTION

    Future acceptance of solar energy systems as viable alternatives to fossil fuel depends not only on design of efficient collection and storage systems, but also on a knowledge of the solar energy available at the site under consideration. Since detailed records for most sites do not exist, it is necessary to reliably predict solar radiation at the site using the fundamental relationships between the earth and sun.

    Several models exist which compute direct and diffuse radiation, but most require detailed mete- orological observations. One objective of this study was to develop a model which would adequately predict hourly and daily total potential and clear sky solar radiation using latitude and clearness number as the only input parameters. Model outputs include hourly and daily total potential and clear sky radi- ation and sunrise and sunset times for each day of the year. Options include both tabular and/or graphic outputs for each parameter. The model was verified using data from Ely, Nevada. A discussion of the model development and verification are presented below.

    LITERATURE REVIEW

    Potential solar radiation The average amount of solar radiation impinging

    on the earth's atmosphere is called the solar constant. Its measured value is !.353 Kw/m 2 or 81.44 Kj/m2[l]. However, since the sun-earth orbit is elliptical, the sun-earth distance varies by _+ 1.7 per cent during the year, causing potential solar radiation to vary slightly according to the inverse of the square of the distance.

    485

    As solar radiation passes through the earth's atmo- sphere, it is attenuated by local climatic interactions and air pollution. The solar radiation that reaches the earth's surface is of two forms: direct and diffuse. Direct radiation is collimated and capable of casting a shadow; diffuse radiation is dispersed or reflected by the earth's atmosphere and is not collimated.

    Potential solar radiation at any location is that which would occur in the absence of atmospheric interference. The flux density of potential radiation at any place and time can be calculated mathematically using known geometric relationships, and the solar constant.

    The daily total potential solar radiation on a horizontal surface is defined by Lee et al. [3] as

    Ipa = (lod/rZ)[cos (PHI) cos (DEL)]

    x (sin H - H cos H). (1)

    Daily values for r can be obtained from a solar ephemeris, or calculated by eqn (la), as developed by Lee et al. [3]

    r = (1 - e2)/[1 + e cos (77.5- (LAMBDA))] (la)

    ~'Scientific Article No. A-3735, Contribution No. 6711 of the Maryland Agricultural Experiment Station (Department of Agricultural Engineering).

    :~Head, Wye Research and Education Center, and Affiliate Assistant Professor.

    Assistant Professor. Professor.

  • 486 R. BRINSFIELD el al.

    Daily values of (LAMBDA) can be determined as follows

    (LAMBDA) = n(180/186) for n less than or

    equal to 186 (lb)

    (LAMBDA) = n - -6 for n greater than 186.

    (lc)

    The instantaneous flux density of potential solar radiation as defined by Lee et a/.[3] is

    Ip,, = ( lo,~/r2)[cos (PHI) cos ( D EL )]

    (cos h - cos H). (2)

    The total potential solar radiation for any time interval between sunrise and sunset can be obtained by integration of eqn (2); the total for any particular hourly interval is given by

    Iph = (loh/r 2)[cos (PHI) cos (DEL )]

    x (0.9972 cos h -- cos H). (3)

    CLEAR SKY RADIATION

    Direct normal solar radiation In passing through the earth's atmosphere, some

    potential solar radiation is reflected, scattered and absorbed by dust, gas molecules, ozone and water wt- por. The mechanisms of scattering and absorption by atmospheric constituents have been studied by several investigators[4-9]. The extent of radiation depletion and scattering at any time is determined by the atmospheric composition and the path length traversed by the sun's rays. The path length is expressed in terms of the air mass, m, the dimen- sionless ratio of the actual length of path of the sun's rays through the atmosphere to the length of path when the sun is in zenith position[10].

    The effects of atmospheric composition was quan- titatively defined by Moon [7] as the total atmospheric optical depth, (TA U). Values for (TA U) for average atmospheric conditions using a sea level location, a moderately dusty atmosphere and amounts of precip- itable water vapor representative of average values for the United States as a whole for each month are given by Imamura et al.[11].

    Deviations in (TAU] will result because atmo- spheric dust and water vapor content will vary with geographical region and weather conditions. To take this into account, a parameter called the atmospheric clearness number was introduced by Threlkeld and Jordan[10]. The clearness number (CN) is defined as the dimensionless ratio between the actual (mea- sured) clear sky normal incident direct global solar radiation at any given location and the calculated value using average values for (TAU) along with a given relative air mass. Average values of CN for the United States are given by Imamura et al.[ l l] .

    The direct normal solar radiation reaching the earth's surface for clear sky conditions is defined by Bouger's law[12] as

    I'd,= I p e x p ( - T A U sec(THETA)o). (4)

    The direct normal solar radiation is then multiplied by the clearness number CN to give

    Id ,=CN Ipexp[ - TAU sec(THETA)o ]. (5)

    Imamura et al.[11] defines the normal solar radiation reaching a horizontal panel on the earth's surface for clear sky conditions as

    ldn,h = [ CN Ip exp (-- TA U sec ( THETA )o) ]

    x cos (THETA)o (6)

    and the direct normal solar radiation reaching a panel tilted at an angle other than horizontal on the earth's surface for clear sky conditions as

    Idn,p [CNIpexp ( -TAUsec (THETA)o ) ]

    cos (THETA). (7)

    Diffi~se sky radiation Since diffuse sky radiation comes from all parts of

    the sky, its intensity and distribution is difficult to predict. According to Fritz[6] the diffuse sky radi- ation on a horizontal surface for an average clear sky is about 16 per cent of the total when the sun is high in the sky and about 37 per cent of the total when the solar altitude angle is about 10 .

    The theory of radiation scattering is rather in- volved and currently no theoretical method for deter- mining diffuse sky radiation is known. Several in- vestigators have measured diffuse sky radiation separately from direct radiation for clear sky condi- tions. Klein[13] gives a method of computing diffuse sky radiation in terms of atmospheric conditions, and Hand[14] has published information on the ratio of direct-to-diffuse sky radiation during typical winter cloudless days on horizontal surfaces. Parmelee [15] measured and plotted diffuse sky radiation versus solar altitude and solar azimuth of vertical surfaces. Threlkeld[10] extended Parmelee's work by deriving a dimensionless parameter C, which is equal to the ratio of diffuse horizontal radiation divided by direct normal radiation. Average monthly values of C for the United States are given by Imamura et a/.[l l] .

    Kusada[16] developed a relationship to predict the diffuse sky radiation (Ia) reaching a horizontal surface on the earth's surface for clear sky conditions as

    Id = (C/CN)Ip exp [-- TA U sec (THETA)o]. (8)

    Flat-plate collectors absorb direct beam and diffuse sky radiation. Although the angular correction for the direct beam component is rather straight forward,

  • POTSOL: model to predict extraterrestrial and clear sky solar radiation

    the correction factor for the diffuse component de- pends on its distribution over the sky, which is generally not known. Two "limiting cases" as defined by Duffle and Beckman[17] have been assumed as a basis for angular correction for diffuse sky radiation.

    First, it is assumed that most of the diffuse sky radiation comes from an apparent origin near the sun, that is, most of the scattering is forward scatter- ing. With this assumption Duffle and Beckman[17] suggest using eqn (8) to predict the diffuse sky solar radiation on a horizontal panel for clear sky condi- tions. Therefore

    /a,h = Id. (9)

    The second assumption is that the diffuse sky radiation is uniformly distributed over the sky. Then the diffuse radiation on a surface other than horizon- tal is dependent only on how much of the sky the surface sees.

    Liu and Jordan [18] derived a conversion factor by assuming that a panel tilted at a slope P T from the horizontal sees a portion of the sky dome given by

    [1 + cos (PT)]/2. (10)

    Equation (10) is the conversion factor for diffuse sky radiation on a panel other than horizontal for clear skies. Therefore, the diffuse sky solar radiation reach- ing a tilted panel on the surface of the earth for clear sky conditions becomes

    ld,p = Id[(1 + COS (PT))/2]. (11)

    Substituting eqn (8) into eqn (11) yields

    lap = ( C /CN)Ip exp [ - TA U sec ( T H E T A )o]

    {[(i + cos (PT)]/2} . (12)

    M O D E L D E V E L O P M E N T

    General The model Potential Solar Radiation (POTSOL) as

    developed by Brinsfield[19] uses the fundamental geometric relationships between the earth and sun as described by Kreith and Krieder[2] and clearness number to predict solar radiation outside the atmo- sphere (SP) , clear sky radiation on a horizontal surface at the earth's surface (SG) , and clear sky radiation on a solar collector at any panel tilt angle ( P T ) from the horizontal ( S G P T ) .

    487

    Posi t ional calculat ions POTSOL assumes the sun is constrained to move

    with two degrees of freedom on the celestial sphere. As a result, the solar altitude angle is measured from the horizontal plane upward to the center of the sun, the azimuth angle is measured in the horizontal plane between a due south line and the vertical projection of the site-to-sun line on the horizontal plane, and the zenith angle is measured between the beam from the sun and the vertical.

    Although the altitude, azimuth and zenith angles are not fundamental angles, POTSOL relates them to the fundamental angular quantities: hour angle, lati- tude and solar declination as defined by Kreith and Kreider[2].

    Potent ial solar radiation Hourly and daily totals of direct normal solar

    radiation reaching the surface of the earth are im- portant quantities in the prediction of solar system performance. These totals are given in general by

    ~t to + A t

    Ito t = lan(t) cos ( T H E T A ) ( t ) dt. (13)

    Generally it is not possible to employ eqn (13) to determine hourly or daily totals since la,(t) depends on the local weather which is not known a priori. It is possible to calculate the potential solar radiation outside the earth's atmosphere, however, since la,(t ) is simply the solar constant divided by the sun-earth radius vector r. Therefore the direct normal radiation in kJ/m 2 per unit time can be determined as

    lan(t ) = lo/r 2. (14)

    The potential solar radiation for any time interval can be determined from

    I r = (lo/r 2) sin ~ALP/-/A )(t) dt. o

    (15)

    Evaluating eqn (15), Lee et al.[3] developed the following relationship used in POTSOL to predict the daily total potential solar radiation as

    S P D = (lod/r2)(COS ( P H I ) cos ( D E L ))

    (sin H - H cos H) (16)

    Table 1. Statistical results comparing dai!y paired totals of potential radiation calculated with POTSOL with observed values

    Source Mean, kJ/m 2 Slope Intercept Corr

    Observed Calculated Calculated 28,530

    1.010 151.9 0.99 Observed 28,970

  • and the hourly total potential solar radiation as

    S P H R = (loh/r2)(cos (PHI) cos (DEL ))

    x (0.9972 cos h - cos H). (17)

    Clear sky radiation--direct beam component Hourly values of direct normal solar radiation on

    a horizontal surface are determined in POTSOL as

    I,I,h = CN S P H R e x p [ - TAU sec (THETA)o]. (18)

    Daily totals of direct normal solar radiation (la, a) are determined by summing the hourly values from sun- rise to sunset.

    Finally POTSOL predicts the hourly direct normal solar radiation reaching a panel with tilt angle P T from the horizontal for clear sky conditions as

    ld, h.p = { CN S P H R exp [ - TA U sec ( THETA )o]}

    x cos (THETA) . (19)

    Daily totals of direct normal solar radiation reaching a panel (Id,d,p) are determined by summing the hour- ly values from sunrise to sunset.

    50000

    Clear sky radiation--diffuse component POTSOL predicts the hourly diffuse sky radiation

    on a horizontal surface for clear sky conditions as

    lab = (C/CN) SPHR

    exp[- TAU sec (THETA)o]. (20)

    Daily totals of diffuse solar radiation on a horizontal surface (Idd) is determined by summing the hourly values from sunrise to sunset.

    The diffuse sky factor C is contained in an array in POTSOL, depending on day number. The hourly diffuse radiation on a panel with tilt angle P T above the horizontal for clear sky conditions is calculated [ l 1] as

    ldh,p = [( C /CN) S P H R exp ( - TA U sec ( THETA )o)]

    [(I + c o s ( e T ) ) / 2 ] .

    Daily totals of diffuse radiation on a panel (Idd~) is determined by summing the hourly values from sun- rise to sunset.

    Total solar radiation clear sky--direct and diffuse

    The hourly total solar radiation on a horizontal surface for clear sky conditions is determined in

    4 0 0 0 0 !

    / /

    /

    0

    Z 3 0 0 0 0 O H

    H Q

    E

    E

    j 2 0 0 0 0 O

    W J

    J H

    " 1 > I0000 - - P O T E N T I A L \

    C L E A R S K Y \ \

    \

    488 R. BRINSFIELD et al.

    0 ~ ~ L ~ I ~ L ) , I ~ ~ ~ ~ I ~ ~ ~ A I 0 : 1 0 0 2 0 0 : 3 0 0 , 4 0 0

    T I M E ( D A Y S )

    Fig. I. Potential and clear sky solar radiation as a function of time for Salisbury, Maryland.

  • POTSOL as

    POTSOL: model to predict extraterrestrial and clear sky solar radiation

    SGHR = la,,h + Iah. (22)

    Substituting eqns (18) and (20) into eqn (22) yields

    SGHR = [SPHR exp ( - TAU sec (THETA)o)]

    x [CN + (C/CN)]. (23)

    The daily total solar radiation (SGD) is determined by summing the hourly values from sunrise to sunset.

    Finally, POTSOL predicts the hourly total solar radiation on a panel with tilt angle PT from the horizontal for clear sky conditions as

    SGPTHR = l~.h, + ldh,~. (24)

    Substituting eqns (19) and (21) into eqn (24) yields

    SGPTHR = SPHR exp ( - TA U sec (THETA)o]

    x {CN cos (THETA) + (C/CN)

    x [(1 + cos (PT))/2]}. (25)

    The daily total solar radiation on a panel (SGPTD) is determined by summing the hourly values from sunrise to sunset.

    489

    VERIFICATION OF POTSOL

    Before discussing model verification, it is useful to briefly summarize the potential source of error or differences between observed and predicted solar radiation.

    The two main types of errors are those resulting from assumptions made in developing the models and errors in the meteorological data and analyses of that data. Errors associated with meteorological data and their analysis can have a significant impact. These errors include those associated with instrument cali- bration, errors due to deviations from the estimated values for the total optical depth (TAU), the diffuse sky factor C, and the clearness number CN. Verification of the model POTSOL and a discussion of these differences are presented below.

    POTSOL was verified by comparing daily potential solar radiation SPD at Ely, Nevada, calculated by POTSOL with values obtained from the National Climatic Center (NCC), Asheville, North Carolina. Statistical results correlating pairs of daily totals for 365 days are given in Table 1. Results indicate good agreement between the two data sets.

    RESULTS AND DISCUSSION

    Under clear skies, or constant atmospheric condi- tions, the flux density of solar radiation SG at any site

    50000

    4 0 0 0 0 A

    O

    Z 3 0 0 0 0 O H

    < H O

    < d 2 0 0 0 0 O

    W

    m < J H < > iO000 <

    / LEGEND \ / \

    ~ / P O T E N T I A L ~ . ~ . . ~ CLEAR SKY

    0 , , , , I , , , ~ I , , , , I , , , , I

    0 100 200 300 400

    TIME (DAYS)

    Fig. 2. Potential and clear sky solar radiation as a function of time for Ely, Nevada.

  • 490 R. BRINSFIELD el al.

    is highly correlated with the potential (extra- terrestrial) flux. Since potential solar radiation S P is a function of season, time of day, and latitude (math- ematical solutions discussed previously), these pa- rameters are useful in defining the upper limits of S P and serve as a point of departure in evaluating screening effects. In this study, clear sky solar radi- ation is estimated as a function of potential solar radiation and the atmospheric parameters C and CN.

    The model POTSOL was used to predict potential and clear sky solar radiation on a horizontal surface and clear sky solar radiation intercepting a panel set at various tilt angles. Figures 1 and 2 illustrate the annual cycles of daily totals of potential solar radiation and clear sky solar radiation as predicted by POTSOL for Salisbury, Maryland, and Ely, Nevada. On an annual basis the predicted clear sky solar radiation is approximately 78 per cent of the potential solar radia- tion for Salisbury, Maryland, and 82 per cent for Ely, Nevada. Clear sky solar radiation tends to be a larger fraction of the potential solar radiation during summer months than winter months for both locations. Atmos- pheric interference, particularly moisture, during summer months probably accounts for most of the differences indicated.

    The dependence of daily totals of clear sky solar radiation on a panel tilt angle for both locations are shown in Figs. 3 and 4. Five cases were considered:

    (1) a horizontal panel; (2) lat. minus 15; (3) lat.; (4) lat. plus 15"; and (5) a vertical panel. The cases considered show the general dependence of daily, seasonal and annual totals of clear sky solar radiation as a function of panel tilt angle for both locations. It is interesting to note that the horizontal panel re- ceives as much or more radiation than the optimum tilt angle for the months of May-August. Hence the latitude tilt angle is only optimum if a yearly period is considered. If seasonal periods are considered, the optimum tilt angle will vary.

    CONCLUSIONS

    The model POTSOL included in this study was developed using the fundamental geometric re- lationships between the sun and the earth to predict potential and clear sky solar radiation for any site given the latitude and clearness number. The model was verified using a data base from Ely, Nevada. The model can be used to estimate hourly, daily, weekly, and seasonal trends of potential and clear sky solar radiation for any site in the United States. Data generated by the model can serve as a point of departure for estimating trends of solar energy avail- ability for design purposes.

    Acknowledgements--The authors express their gratitude to the Computer Science Center, University of Maryland, for

    2 5 0 0 0

    20000

    0

    z 1 5 0 0 0 o H

    j Ioooo o

    w J m

    J H

    > < 5000

    i

    / ,7

    LEGEND

    - - HORIZONTAL

    - - LATITUDE-15

    LATITUDE

    LATITUDE+J5

    VERTICAL

    \

    \ , / , , / ~,',:~. "\, , ,'~-----

    0 0 100 200 300 4 0 0

    TIME (DAYS)

    Fig. 3. Clear sky solar radiation as a function of time for various panel tilt angles for Salisbury, Maryland.

  • POTSOL: model to predict extraterrestrial and clear sky solar radiation

    30000

    491

    2 ooo ; , , ,

    ~ " / x

    O X

    ,~ 20000

    z I 77 . . . . . . . " > , , ,/,/ I LE E.o I \ % H n H 15000 ([ rr

    rr < I

    . J I-4

    >

    5000

    0 0 ~00 200 300 400

    TIME (DAYS]

    Fig. 4. Clear sky solar radiation as a function of time for various panel tilt angles for Ely, Nevada.

    providing the necessary computer funds for the project. The authors also thank Barbara South and Donna Harding for their help in typing the manuscripts.

    NOMENCLATURE

    ( ' ratio of diffuse horizontal solar radiation to direct normal solar ration, dimensionless

    CN "clearness number," the ratio between actual (measured) clear sky normal incident direct global solar radiation at a given location and the calculated value using average values of T A U with a given relative air mass, dimension- less

    e eccentricity of the earth 's orbit h solar hour angle (h P I ) t / I 2), measured from

    the mid point of the hour in question, rad H solar hour angle (H - ( P I ) T / 1 2 ) , tad Id ditt'use sky solar radiation on a horizontal

    surface for clear sky conditions, k J /m 2 per unit time

    lad daily total diffuse sky solar radiation on a horizontal surface for clear sky conditions, k.I/m 2 day

    lab hourly total diffuse sky solar radiation on a horizontal surface for clear sky conditions, kJ/m-' hr

    la.h diffuse sky solar radiation on a horizontal panel for clear sky conditions, k J / m 2 per unit time

    Id.p diffuse sky solar radiation on a tilted panel for clear sky conditions, k J / m z per unit time

    IdJ., daily total diffuse sky solar radiation on a tilted panel for dea r sky conditions, k J / m 2 day

    Ida,p hourly total diffuse sky solar radiation on a tilted panel for clear sky conditions, k J /m 2 hr

    /j. direct normal solar radiation for clear sky conditions, k J /m 2 hr

    la.d daily total direct normal solar radiation on a horizontal surface for clear sky conditions, k J /m 2 day

    Id.h hourly total direct normal solar radiation on a horizontal surface for clear sky conditions, k J /m 2 hr

    Ido.h direct normal solar radiation on a horizontal panel for clear sky conditions, k J /m 2 per unit time

    la.p direct normal solar radiation on a tilted panel for clear sky conditions, k J /m 2 per unit time

    la.d.p daily total direct normal solar radiation on a tilted panel for clear sky conditions, k J /m 2 day

    Idh,p hourly total direct normal solar radiation on a tilted panel for clear sky conditions, k J /m 2 br

    Io solar constant = 81.44 kJ/m z I,,j solar constant, k J / m : day I,,h solar constant, k J /m 2 hr 1,,~ solar constant, k J /m 2 rain

    1, potential solar radiation, k J /m 2 per unit time lpd S P D - daily total potential solar radiation,

    k J /m 2 day lph S P H R hourly total potential solar radiation,

    k J /m 2 hr Ipm instantaneous potential solar radiation, k J /m 2

    rain n number of days following the vernal equinox

    (21 March)

  • 492 R. BRINSFIELD et al.

    PT

    r

    SG

    SGD

    SGttR

    S G P I

    SGPTD

    S G P Ttt R

    SP SPD

    SPHR

    t

    T

    ALPHA DEL

    THETA

    (THETA)o LAMBDA

    TA L; PHI

    panel tilt angle measured from the horizontal, radians radius vector (ratio of sun earth distance to mean sun earth distance), dimensionless clear sky solar radiation on a horizontal sur- face, kJ /m 3 per unit time daily total clear sky solar radiation on a hori- zontal surface, k J /m: day hourly total clear sky solar radiation on a horizontal surface, k J/re" hr clear sky solar radiation on a tilted panel for clear sky conditions, kJ /m ~' per unit time daily total solar radiation on a tilted panel for clear sky conditions, kJ /m 2 day hourly total solar radiation on a tilted panel ['or clear sky conditions, kJ /m 2 hr potential solar radiation, k.I/m 2 per unit time daily total potential solar radiation on a hori- zontal surface, kJ /m 2 day hourly total potential solar radiation on a hori- zontal surface, k J/re" hr absolute time interval between solar noon and the mid point of the hour in question, hr absolute time interval between solar noon and sunrise or sunset, hr solar altitude angle, rad solar declination angle (the angular position of the sun at solar noon with respect to the plane of the equator; north positive), rad solar incidence angle between the incoming solar rays and a line normal to the panel in question, tad solar zenith angle, rad solar longitude angle, rad total optical depth, dimensionless solar latitude angle (not positive), tad

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    20. H. M. Woolf, On the computation of solar elevation angles and the determination of sunrise and sunset times. NASA TMX-1646 (1963).

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