Diffuse solar radiation on a horizontal surface for a clear sky

Solar Energy, Vol. 21, pp. 503-509 0038-092X/78/1201-.0503/$02.00/0 © PePgamon Press Ltd., 1978. Printed in Great Britain

DIFFUSE SOLAR RADIATION ON A HORIZONTAL SURFACE FOR A CLEAR SKY

R. O. BUCKIUS? and R. KtNG~: Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champmgn, Urbana,

IL 61801, U.S.A.

(Received 25 Apr/l 1978; accepted 6 July 1978)

Abstract--A model of the diffuse or sky radiation as a function of measurable optical properties of the earth's atmosphere is presented. The radiant energy incident upon a horizontal plane at the earth's surface is expressed in an approximate closed-form. The dependence of the sky radiation upon climate type, air mass, and surface retlectivity is given and comparisons with existing empirical correlations show that the model accurately predicts these characteristics.

1. iI~OI~CI'ION

In the design and assessment of solar collection systems, the knowledge of the diffuse or sky radiation is needed. This scattered radiation is composed of the radiant energy incident at the earth's surface from the complete sky hemisphere except at the direct solar beam angle. This portion of the total radiation incident upon a horizontal surface can be quite significant. Therefore, it is essential to accurately quantify the diffuse or scattered radiation incident at the earth's surface.

Most existing studies of the sky radiation either present correlations of experimental data or theoretical analyses of the atmospheric layer. The classic work of Liu and Jordan[l] presented correlations of the data for the sky and total radiation on a horizontal surface. Their results have been used extensively although their correlations are based upon data collected at a single location. Similar analyses have been performed recently by Ruth and Chant[2] and Orgill and Hollands[3] for various locations in Canada and by Brnno[4] for Hamburg, West Germany. All of these correlations show similar variations yet they are different in magnitude. Ruth and Chant further indicate a latitude dependence in their results while Orgill and Hollands have presented correlations based upon hourly rather than daily data. One common feature among all of these studies is that the correlations represent only mean values while the actual observed data show a large scatter about these values [3, 5]. The other difficulty in utilizing these results is the need for the total radiation incident at the earth's surface which may not be available for the location of interest.

Numerous theoretical results have been presented based upon specific atmospheric models. Chandrasekhar[6] has presented exact numerical solutions for the radiation transfer in a planar medium. Specific numerical results for the earth's atmosphere

?Assistant Professor of Mechanical Engineering. ~Graduate Research Assistant.

have been presented for various wavelengths and haze models [7,8]. The effect of surface reflection was presented and was shown to be important [7]. Dave[9] has also performed numerical calculations based upon specific optical depths and haze models and has shown that the isotropic distribution assumption underestimates the magnitude of the sky radiation. All these numerical results involve large computing efforts and are specific to the model employed in the calculations. These compu- tations are, therefore, difficult to use in solar collection system modeling.

Various simplified approaches for planar scattering systems have been proposed. Irving[l~12] has compared and discussed various approximate solutions for azimuthally symmetric radiation fields. The small angle approximation, Eddington approximation, and modified two-stream approximation are evaluated and their limitations are presented. An alternate approximate technique for the radiation transfer and its directional distribution was presented for a conservative anisotropic scattering media [13].

The purpose of the present work is to develop an accurate model for the diffuse or sky radiation based upon measurable optical properties of the earth's atmosphere. A closed-form solution for the radiant energy incident on a horizontal plane at the earth's surface is presented and the dependence upon climate model, air mass, and surface reflection is shown.

2. ANAL¥~IS

2.1 Formulation The earth's atmosphere is modeled as a planar medium

with finite dimensions in the vertical direction and infinite dimensions in the horizontal direction (Fig. 1). The solar energy at the top of the atmosphere is incident at an angle of (0o, 4'0) and the scattering is generally anisotropic so that the resulting radiation field is dependent upon the azimuthal angle. The present model includes diffuse reflection at the earth's surface and atmospheric emission is neglected. The equation of trans-

503

504 R. O . BUCKIUS and R. K~s~

\ \ K , / ~ , ~ , ) , / ~ > 0

- - ~),/J.<O Ix ,.,o, I

/ / / / / ~ / / / / / / / / / / / / / / / / / / / / / / / / / Fig. I. C o o r d i n a t e sy s t em.

fer is

/ / /I / / / //I

d~(K,/~, ~b) + f(K,/z, 4,) /~ dK

~o r2" /"

= T~ Jo J-t/'(K' ~ ' ' 4,')p(~' 4,; ~ ' ' 4,')

x dt~' d4,' = S(K, I~, ~b)

a n d t h e b o u n d a r y c o n d i t i o n s a r e

(1)

is significant at larger wavelengths and the resulting incident radiation is therefore small. This absorption is further increased by the scattering due to the resulting increase in pathlength. The scattering contribution in the low wavelength region is much greater than the aerosol and gaseous absorption so that the scattering albedo is large [16, 17]. The present model assumes that the atmospheric layer is conservative so that the scattering albedo is unity.

The scattering phase function must include the anisotropic distribution of the scattered energy that results from the air molecules and aerosol in the atmosphere. The molecular components of the atmosphere are characterized by the Rayleigh phase function. This phase function is symmetric in forward and backward directions and can be considered as isotropic, especially for energy transfer calculation [18, 19]. The aerosol phase function calculated from Mie theory has a very sharp peak in the forward direction [7]. Various authors [11,18,20] have shown that this sharp forward diffraction peak can be treated as directly transmitted past the particles thereby significantly reducing the complexity of the phase function. Based upon these works, the phase function is approximated in the linear anisotropic form as

t+(0, ~, 4,) = i+(0), ~ > 0

f-(xc, ~, 4,) = io~(~ - tzda(4, - 4,o), i t < 0 (2)

where f(K,/~, 4,) is the azimuthally dependent intensity, the cosine of the zenith angle, 4, the azimuthal angle, the optical depth, ~ the scattering albedo, p(/~, 4'; ~', 4,') the scattering phase function, and S(K, #, ~b) denotes the source function. The resulting solutions for the positive and negative directed intensities subject to the boundary conditions in eqn (2) are

f+(K, p., 4,) = i÷(0) exp ( - Kip)

+ ~o " S(K', U, 4,) exp [ - (K - ~')/u]

× (dx'/~), ~ >0

t-(x, iz, 4,) = io8(~ - ~o)8(4, - 4,0)

f , L x e×p [(~L -- K)I~] -- S(K', #, 4,)

X exp [ - (~.- K')/~I d~'/~, ~ < 0. (3)

The surface boundary condition is independent of p and 4, and is, therefore, represented as i+(0). It is dependent upon the ground reflectivity, p, and the incident intensity at the earth's surface, i'-(0, ~, 4,). Typical values for the surface reflectivity are 0.1, 0.15, and 0.75 which represent water surfaces, soil, and snow covered surfaces, respectively [14].

The spectral region of interest for the sky component of the incident solar radiation is from 0 to 0.9 t~m [15]. The sky radiation that is incident at the earth's surface beyond 0.9 ~m is neglected since the gaseous absorption

p(,a, ~;/z', 4,') = 1 + affztz' + at(1 - tz2) u2

x (1 - ,~ ) t~ cos (4' - 4,'). (4)

Rayleigh scattering is approximated by al = 0 and the aerosol phase function is considered by at = + 1. There- fore, the optical properties of the atmosphere are characterized by the optical depth of the layer, xL, which includes the concentration distributions of the scattering constituents.

In general, the intensity is composed of the collimated or directly transmitted portion and the sky or scattered portion given as

i'CK,., 4,) = tc(X,., 4,) + t.(x, ~, 4,). (5)

The equation of transfer for the collimated portion is

di~(K,/~, ~b)+ re(K,/~, 4,) = 0 (6) /~ dK

and the boundary conditions are

Cc + (0, ~, 4,) = 0

re-(K,_, ~, 4,) = io8(~ -- ~)8(4, -- 4'0). (7)

The solutions are given as

t~+(x, t~, 40 = 0

~-(K, ~, 4,) = ioS(~ - ~o)8(4, - 4,0) exp [(~L -- ~)lt~] (8)

which represent the directly transmitted portion of the solar energy through the atmosphere. These solutions are then substituted into the sky portion resulting in a new

Diffuse solar radiation on a horizontal surface

source function for the sky radiation. The equation of transfer for the sky component is then given by

io dt,(K,/,, ¢) + f, tK, t~, 4') = ~ P(P, $, - /~o, $o) /z dK

xexpt-(XL-X)/,d+ f;" ¢,') x p(/~, $; #', $') d/z' d~'. (9)

505

q(K) is independent of K for conservative scattering since there is no absorption by the atmosphere and is hereafter denoted by q. The boundary conditions for eqn (12) are

i.+o(O,/z) = i*(0) 05)

i;O(KL, IX) = O.

The resulting boundary conditions for the sky component are

~+0 i. ( , t~,~)=i+(0), # > 0

i ' f(~L, IZ, ~) = O, t~<0. (lO)

The intensity for the sky component can be expressed in a two-term cosine series expression for this phase function as

fs(K, IZ, ~) = i,o(K, lZ) + is,(K, #) COS ($ -- ~bo). (11)

This expression is then substituted into eqn (9) which results in two azimuthally independent integrodifferential equations. Since the present work seeks the incident radiation upon a horizontal surface and not the dependence, the i.~(K, tz) term is not needed. The governing equation for i.o(K, ~) is

q(~) (12) # ~ + i ' d K ' / z ) = ~ ) +a'l* 4~r

where functions G(K) and q(K) are defined as

G(r)=f;"f~/(K,l~',$')d#'d$' (13a)

q(~)= J [ F,f(.,~,,,~,)~,, do' d~b'. (13b)

The intensities in eqn (13) are replaced by the collimated component and the two-term representation for the scattered component. The resulting expressions for G(K) and q(K) after this substitution are

f G(K) = io exp [ - (KL -- K)/~£O] + 2# i.dK, it') d/z' I

= G~(r) + G,(K) (14a)

and

q(~) = -io#o exp [ - - (KL -- K)/$,~O] + 2"h" f_ll iso(K, t L ' ) # ! d o t

= qc(K) + q,(K). (14b)

The collimated and sky incident energy per unit area are represented by G~(x) and G,(r) and the collimated and sky radiant heat flux are given by qc(K) and q,(K) while the sum represents the total contributions. The heat flux,

The solutions of eqn (12) subject to the boundary conditions in eqn (15) are then substituted into eqn (14) which results in a set of coupled integral equations as

G(~¢) = G~(~:) + i+(0)E~(,)+ r ra, , , , , | - ~ / ~ , . , E,(IK - ,'1) 27r 2¢r .,o t ,,~r

+ ~-~-sgn (K- K')Ez(IK- *r'l) ] dK' (16a)

q _ q,(K)+ + f f ~ [a (~ ' ) . ,. 2~r - '2~r i+(0)E3(K) I.'4-~'-~ sgn ¢,K -- K J

, + a . q E,( I~ - K'I)] d~' x Ez(IK - r ) ~ (16b)

where E.(x) denotes the exponential integral function of argument x.

The sky radiation is represented as the radiant energy incident on a horizontal plane at the earth's surface and is expressed in terms of the incident flux, q,-(0), at the earth's surface. This radiation transmission includes all radiant energy incident at the earth's surface from all directions excluding the collimated portion. The incident radiant heat flux is evaluated from eqn (13b) with

f(0,/~, $) = i+(0) (17)

and the definition of the reflectivity

i+(O)=pf-(O,t~,40. (18)

The resulting expression is

q= q+(O)-q-(O) = ~o2" fo ti*(O)/~' dr.' d$'

f~,, fo i+fll~

*Jo = ~i+(0) - Iri+(O)/p. (19)

The two terms in eqn (19) represent the positive and negative directed radiant heat flux at the boundary, respectively.The resulting expression for the incident energy is then expressed as

~-(o)= q o - 1" (20)

Sky radiation observed at the earth's surface is simply the difference between the negative directed radiant heat flux and the collimated radiant heat flux,

q,-(0)= q-(0)-qc-(0). (21)

5O6

Therefore, the incident sky radiative heat flux is given as

R. O. BUCKIUS and R. KING

q,-(0) = ~ - p.oio exp ( - ~cL/~o). (22)

2.2 Approximate solution The kernel substitution method [19,21] is used to

obtain an approximate closed-form solution to the governing equations. In the solution of eqns (16a) and (16b) for G(K) and 6, the dependence of the exponential integrals is taken as

E2(t) = exp (- at). (23)

Various authors have recommended different values for the coefficients of the exponential functions [21]. Comparisons between the approximate closed-form solutions given below and available exact solutions have been made and the recommended value is a = 2 [13].

These approximate forms of the exponential integral functions are substituted into eqns (16). The integrals are eliminated by double differentiation and the equations for G(K) and q are decoupled. The resulting differential equations are

d2G(K) : [ 1 2\ r = ~ O ~ ° -- a } exp t - (~L - K)/~to] (24)

~ = o. (25)

The solutions are expressed as

G0c) = io(1 - a 2/.to 2) exp [ - (KL -- K)I~O] + A~K + A2 (26)

and q is an unknown constant. The unknown coefficients in eqn (26) and q are obtained from the boundary conditions contained in eqns (16) by substituting these equations back into eqns (16) and equating the various powers of K. The resulting expressions for the constants are

2io/~o[ 1 + 2~o+ (1 - 2~o) exp ( - KgtZo)] - 4~ri+(0) q = a, tel -- 4KL - 4

(27)

At = (al - 4)6 (28)

A2 = 2/o(1 + 2Fo)tto - (aIKL -- 4KL -- 2)6. (29)

expression for 6 in terms of fundamental The parameters is obtained by eliminating the unknown i+(0) and by combining eqn (19) and eqn (27). This results in

2iop~[1 + 25o+ (1 - 2/~o) exp ( - KtJ/to)] 6 = aIKL--4KL--4+4p/(p-- 1) . . . . " (30)

The resulting expression for q,-(0) after substituting eqn (30) into eqn (22) is

q,-(0) = 2io/to[l + 2tto+ (1 - 21zo) exp ( - K~JFo)] (p - I)(a~KL -- 4KL -- 4) + 4p

-- iottO exp (- KU~o). (31)

3. IRSULTS AND DlSCU~ON

The optical characteristics of a specific atmosphere are expressed in terms of the mean optical thickness, ICL, and this must be evaluated to compare with existing correlations. As indicated previously, the wavelength region of interest is from 0 to 0.9/tm where the depletion of the collimated or direct solar incident energy is primarily due to Rayleigh and aerosol scattering with very little absorption. Thus, the directly transmitted radiation which is the source of all sky radiation is expressed as [22]

fO°'9isun exp (- 0.0089 p/A 4 -- ~/A '~) dA

f: .9 isu, dA

where p(atm) is the surface pressure, A(/~m) the wavelength,/3 and a the AngstrSm turbidity parameters [23], and is,, the solar spectrum [24]. The mean optical thickness being employed in the present analysis is based upon this source of radiant energy as

iron9 i,,m exp (-0.0089 p/A4-/J/Aa)dA]

KL = -- In ~O.9 is,, dA J" J U

(32)

The transmission of sky radiation to a horizontal surface due to incidence from all wavelengths is given as

T~ - q,-(0) _ io [2[1 + 2~to + (1 - 2~o) exp ( - ~p.o)] -/~oi~c - z,~-~ [ ( p - I)(alKL--4KL--4)+4~

- exp (- KLJ~O)I J (33)

where i,c is the solar constant. Since most of the incident energy at wavelengths greater than 0.9/~m is absorbed, the only portion of the solar spectrum that contributes to the sky radiation is below this wavelength and is represented by io. The fraction of the solar spectrum which is in the wavelength region is ioli, c---0.634 [24]. The expression employed for evaluating the direct solar radiation for a clear sky is [22]

~'D = {5.228[exp (- 0.0002254 rap) - exp (- 0.1409 rap)]

x exp [-m~S/(0.6489)a]}/mp - 0.2022 m/3/2.875 a

+ 0.00165 mp + 0.2022 - [0.1055 + 0.07053

x log (m U + 0.7854)] exp (- m~/1.519 ~) (34)

where U(cm) is the precipitable water in the atmosphere, m = lifo. Any alternate direct solar transmittance expression could be employed. Then the total radiation transmitted through the atmosphere and incident upon a horizontal surface is

Tr = ~D + T~. (35)

The effect of climate model, air mass, and surface

Diffuse solar radiation on a horizontal

i i ooo ol 0'2 0'3 0'4 0'5 o'6 o7 o18 o19 io

TT

Fig. 2. Effect of air mass, precipitable water, and surface reflectivity on the diffuse radiation for KL = 0.375 and a~ = 1.

0.20

F-.¢O i5

0.10

0 3C , , , , , , ,

0.25

i 0 O5

0 0 0 , ,

oo d, d2 o'3 o.' o'5 o7 08 d9 ,o T~

Fig. 3. Effect of air mass, precipitable water, and surface reflectivity on the diffuse radiation for ICL ---- 0.79 and at = 1.

o3° : _ ' , I

0 1 ~

O r O

t ,, I I I

OOCO0 Q, d2 03 0 4 05 06 07 08 09 IO TT

Fig.-:4. Effect of climate type (or optical depth) and surface reflectivity on the diffuse radiation.

corresponds responds to significantly energy and

surface 507

for a 23 inn visibility while / /= 0.41 cor- a 5 km visibility. Since the water vapor effects the direct transmission of solar has a negli~'ble effect on the scattered

portion, the precipitable water, U, is an independent parameter varying from 0.2cm to lOcm in the calculations of ~ The results are presented for surface reflections of 0.0 and 0.15. Therefore, the enclosed area on the figures represent a specific climate model and ground reflectivity with the limiting horizontal edges representing air masses of 1.0 and 5.0 and the other bounding edges representing precipitable water concen- trations of 0.2 and 10.0 cm.

The results for the sky radiation as a function of total radiation show that an increase in optical depth increases the sky portion of the incident radiation until the total radiation at the earth's surface is between 0.4 and 0.5 and then this trend is reversed. The increase is a result of the increase in the number of scatterers as KL increases yet when the atmosphere becomes optically dense, a further increase inhibits the radiation transfer. These characteristics are exhibited for variations in air mass for a fixed climate type (Figs. 3 and 4) since as air mass increases, the optical pathlength increases. The surface reflectivity always tends to increase the sky radiation with all other conditions fixed (Fig. 5). This is a result of an added radiation contribution to the scattering medium for non-zero reflectivities which can be scattered into the surface considered. Although not indicated in the figures, the resultant increase in the sky radiation for snow (O = 0.75) is significant.

The empirical relation between the sky and total radiation presented by Liu and Jordan[l] is also shown in Figs. 2--4, with most experimental values within -+ 10 per cent of these mean values. The values of Ruth and Chant[2l at different locations have also been presented and their correlations lie considerably above the results given by Liu and Jordan. The present model lies between these correlations with all three models predicting similar variations. Ruth and Chant suggest that the correlations for different locations have similar trends with increasing

ond HOLLANDS [3]

reflectivity is presented in Figs. 2--4. The surface pressure is taken as I arm and the wavelength variation of the turbidity is taken as a = 1.3. The concentration dependence of the turbidity is 0.0, 0.12, 0.20, 0.30, and 0.41,

06

05

O 4

0 3

O~

OI

LIU and JORDAN [I]

RUTH and CHANT [2]

oc o'J o'2 0'3 o!4 o'5 o~ 0~7 d8 oL9 JJo representing variations between an extremely clear at- T T

mosphere (pure Rayleigh scattering) and a very turbid Fig. 5. Patio of diffuse-to-total'radiation on a horizontal surface atmosphere. These values of the turbidity correspond to (the shaded area represents the predictions of the present model optical depths evaluated from eqn (32) of 0.19, 0.375, for0.t9~KL<_0.79,0.0<_p<0.15, l < m < 5 and0.2<_U (era)_< 0.53, 0.72, and 0.79, respectively. The values of/3 = 0.12 I0.0).

508 R. O. Bucxlus and R. KING

values of T, with increasing latitude. This effect might be explained in terms of the ground reflectivity and air mass since the present result shows that with higher values of p and m, there is a shift to higher values of T,. For snow case, p = 0.75, the values of 7", range from 0.27 to 0.45 which is in agreement with the data presented by Ruth and Chant at Resolute Bay which is at a latitude of 74 ° 33'. Thus, the latitude dependence indicated by Ruth and Chant might be due to the ground reflectivity coupled with the larger air mass.

One common feature among all the data is that the sky radiation always has a minimum non-zero value. The clear atmosphere without aerosols is presented in Fig. 4 (KL = 0.19) where the minimum value for the sky radiation transmission is 0.054. The experimental values for a typical clear atmosphere are 0.08, 0.07, and 0.12 which were presented by Liu and Jordan[l], Bngler[5], and Ruth and Chant[2], respectively, and are in agreement with the present model. The present model is based upon the assumption of a cloudless sky and, therefore, is unable to predict values for TT < 0.32. This is consistent with the experimental observations of Liu and Jordan[l] for such conditions.

The variations of the ratio T, IT¢ with respect to TT are shown in Fig. 5. The shaded area represents the range of values predicted by the present model including variations in climate models, surface reflectivity, air mass, and water vapor. Also shown are the correlations of the experimentally observed data which indicates the accuracy of the present approach. The results for snow (p = 0.75) lie above this shaded area yet are within the range of observed data presented by Ruth and Chant[2].

4, ~ J M M A R Y

A model for the sky radiation based upon measurable optical properties of the atmosphere has been developed. The radiant energy incident upon horizontal surface is expressed as

T, = q,-(0) a ~aAJ "2[1 + 2/~o+ (1 - 2/zo) exp ( - rL//ZO)] Izoi, c = . . . . [ (p-- I)(a,KL--4rL--4)+4p

exp ( - Kd/~o)~ (36) J

where iZo = llm is the cosine of zenith angle (reciprocal air mass), KL the atmospheric optical depth, p the surface reflectivity, and a~ the anisotropic scattering parameter. The value 0.634 is the fraction of the solar energy in the wavelength region from 0.0 to 0.9/zm. The optical depth is evaluated from eqn (32) with turbidity measurements or visibility [22] and the solar spectrum [241.

Acknowledgement--This work was supported in part by the University of Illinois Research Board.

NOMIff~ClATUllg Aj, A2 coefficients defined in eqn (26)

aj anisotropic scattering coefficient defined by eqn (4) a coefficient defined in eqn (23)

En(x) exponential integral function of argument x

G(~) average incident intensity per unit area i intensity

i,c solar constant i,,~ normal spectral intensity of sun m air mass

p(p, ~;/z', ~k') scattering phase function p pressure q radiative heat flux

S(K,/L, ~) source function T radiation incident at the earth's surface U precipitable water a wavelength dependence for aerosol scattering

turbidity coefficient 8 Dirac delta function 0 polar angle

optical depth A wavelength # cos 0 p surface reflectivity

transmittance azimuthal angle

~o scattering albedo

Supercripts " azimuthally dependent quantities

+ positive direction - negative direction

Subscripts 0 incident quantities at K = KL C collimated L layer thickness s scattered T total

R I ~ £ R E ~ C I ¢ ~

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Diffuse solar radiation on a horizontal surface 509

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Diffuse solar radiation on a horizontal surface for a clear sky