A theoretical model to determine solar and diffuse irradiance in valleys

Solar Energy Vol. 35, No. 6, pp. 503-510, 1985

0038-092X/85 $3.00 + .00 © 1985PergamonPress Ltd.

Printed in the U.S.A.

A THEORETICAL MODEL TO DETERMINE SOLAR A N D D I F F U S E IRRADIANCE IN V A L L E Y S WILLIAM A. PETERSONt, INGE DIRMHIRN~, a n d REX L. HURST Utah State University, Logan, Utah

(Received for publication 11 June 1985) Abstract--A mathematical model to calculate global irradiance in valleys of any slope angle, orien-

tation, and albedo is presented. The model is designed to accept measured data of global irradiance at clear, cloudy, and overcast sky conditions from unobstructed locations. Diffuse irradiance is treated as nonisotropic. Several examples of realistic valley configuration are presented to show the strength of the model.

flected from the valley slopes (RS). Therefore, the total irradiance is

1. INTRODUCTION

Solar irradiance as a source for alternate energy can be determined more or less accurately from climatic values in plain terrain. The measured irradiance at one carefully selected location suffices frequently to derive the solar potential for an extended area up to about 100 miles radius. In the mountains, however, considerable differences in the solar irradiance can occur on small distances. Obstruction of the direct solar beam and the scattered irradiance from the sky by, sometimes substantially, elevated horizon provides losses in the radiated energy. Some part of these losses can be compensated for by direct reflection from the valley slopes, and this compensation will be the more significant the higher the albedo of the ground is, that is during the winter snow cover in high elevations. Since solar irradiance is usually limited during these winter months, the additional energy reradiated from valley slopes is a welcome enhancement of the energy source. It would result in unwarranted expenses of time and money to measure the solar potential at each site. Hence, emphasis was soon directed to modeling the incoming energy. If the global (direct plus diffuse) irradiance at an undisturbed station, say, a mountain peak, is known, the amount of radiated energy can be calculated for any place within a valley by strictly geometrical derivations. The following approach is an example of a model for the irradiance potential in complicated terrain.

RAD = DB + D S + RD + R S

(1)

Each component will be calculated for an infinitely long valley oriented in an arbitrary direction. The geometry of the model is depicted in Fig. 1.

A. Global irradiance (G) Generally, solar detectors are placed with their absorbing surface horizontally observing the total horizontal irradiance. This term, also called global irradiance (G) is the sum of both, direct and diffuse, radiation on a horizontal surface. The unobstructed global irradiance can be written as G = Io cos 13 + diffuse

(2)

The first term on the right side of eqn (2) represents the direct horizontal component of irradiance. Where Io is the direct normal solar beam and 13 is the solar zenith angle (Fig. 1). The diffuse component is composed of two parts, a quasi-isotropic part (in short here called isotropic) and a circumsolar part that has the same general angular time variations as the direct component from the sun. The circumsolar part is assumed to be concentrated within a 15° radius around the sun and is treated as being superimposed on an isotropic part that is assumed to be distributed uniformly over the hemisphere of the sky. The diffuse irradiance over the entire hemisphere of the sky can then be expressed as

2. MATHEMATICALMODEL The energy components of solar irradiance within a valley are, in general, composed of the direct beam (DB), diffuse (DS), direct beam reflected from the valley slopes (RD), and the diffuse re-

Diffuse = gflo cos 13 + (1 - g)flo

t present affiliation: US Army Atmospheric Sciences Laboratory, White Sands Missile Range, NM 88002 ~: present affiliation: Universit~it for Bodenkultur, Institut fiir Meteorologie, Klimatologie und Grundlagen der Physik, Gregor Mendelstr. 33, A-I180 Wien, Austria

(3)

where the first term on the right side of eqn (3) represents the circumsolar contribution and the second term gives the uniformly isotropic irradiance. The parameter f is defined as the ratio of the unobstructed diffuse irradiance measured over a hemi-

503

W. A. PETERSONet al.

504

z

surface 1

surface 2

h

I

>Y

~J Fig. 1. Geometry of the mathematical model. z = 13 = 01 and 02 = 0 = = ¢q =

za = lowest point direct beam intersects z-axis ze = lowest point direct beam illuminates the surface in z direction h = height of detector along the zenith "Y2 = view angle from detector to top of valley in y-z plane for surface 2 Io = direct solar beam

direction of zenith solar zenith angle of direct beam radiance Io slope angles of surfaces 1 and 2 with zenith view angle for surface azimuthal view angle angle between normal to surface 1 and direct beam 1o

sphere b y a flat horizontal detector (pyranometer) to the direct solar beam, Io[1]. f -

diffuse

to G as COS 13 COS 13 + f g ( c o s 13 - 1) + f '

(4)

Io

ifZ~<

h

DB

(8) The p a r a m e t e r g is the fraction of f due to the circ u m s o l a r irradiance of the diffuse irradiance. Substituting for the diffuse irradiance eqn (3) into eqn (2) gives for G

G

O,

ffZ~>-h

where

G = Io[cos13 + fg(cos13 - 1) + f ] .

(5)

w cos 0: Z~ - - (tan 13 - tan 0) tan 13

Solving for Io

tan 0 = tan 01/sin d? G Io = cos 13 + f g ( c o s 13 - 1) + f "

(6)

B. Direct beam irradiance (DB) The direct b e a m irradiance depends on the relationship b e t w e e n z~ and h (Fig. 1). If z~ > h there will be no direct b e a m on the detector and DB = 0. If z~ < h, the direct normal irradiance on the detector is DB = Io cos 13.

(7)

Using eqns (6) a n d (7), DB can be expressed relative

and sin 4, = cos 8 sin to/sin 13 8 = declination angle of the earth to = hour angle.

C. Diffuse irradiance (DS) 1. lsotropic part o f diffuse irradiance. The radiance over the sky is a s s u m e d to be uniform a n d isotropic. If L is the diffuse radiance of the sky, then the diffuse irradiance, E, over the entire sky

505

Solar and diffuse irradiance in valleys hemisphere without obstruction will be

and

E = ,'frL.

EV G

(1 - g)f cos 13 + f g ( c o s 13 - 1) + f

EV, the irradiance for the valley for the isotropic part, can be expressed as E V = 2L f0ar/2 fp.I Ix dIx dda 1

+ 2L

fow/2f / ~ Ix dIx d~b,

Ix = cos 0.

Ixt and IX2 are functions of the azimuthal angle ~b and the view angles ~/~ and 72, respectively. Integrating over Ix we have

x (sin 0, + sin 02)/2

2. Circumsolar part of diffuse irradiance. With the circumsolar part (CS) having the same general angular time variation as the direct irradiance, we can from eqn (3) express it as CS = gflo cos 13.

#rL

EV = -~- (sin ~/i + sin ~2)

CS ~-

DB g f --~-.

(15)

(9)

G

or

-

EV

G

CS

+ --.

G

(16)

D. Direct beam multiple reflection (RD)

E

E V = ~ ( s i n ~ + sin ~2).

(10)

From eqns (3) and (6) G(1 - g ) f cos 13 + fg(cos13 - 1) + f '

LI

fz . . . . o, f~

I--.2F - cos 01

To determine the reflected irradiance from the valley surfaces we need to calculate the irradiance from surface to surface and from the surface to the detector. The irradiance from surface 1 to surface 2 can be expressed by the following relationships (Lambert's Cosine Law)

ZlZ2 COS 01Cos 02 (tan 01 + tan02)EdxldZl

3 - ~ [(xl -- x 2 7 ~--(~1 t-al~'01 "~ z2 tan 02)2 + (Zl S Z2)2] 2"

Therefore,

EV G

=

The total diffuse irradiance relative to G is equal to the sum of eqns (11) and (15) DS

E =

(14)

Also, from eqn (8)

E V = L f S 2 (1 - Ix2) d~b + L f o ~/2 (1 - IX2)d~b, where IX~ = sin 2 d~/(tan2 ~/1 + sin 2 ~b). Therefore,

(13)

(1

-

LI is the radiance of surface 1 after initial reflection of radiation. Performing the indicated integration of eqn (17), we derive

g)f

cos 13 + f g ( c o s 13 - 1) + f

× (sin ~ + sin ~te)/2.

F = Lllv,_2(Z) 1----2 (11)

where

~r [z2cos01- zlcosO2cos(Ot

Im-~(Z) = "2

(az~ + bz, + c) 1/2

F r o m Fig. 1, it follows w sin 01 sin '~l = [W2 sin 2 01 + (w cos 01 - h)2] 1/2"

If h = O, eqn (12) becomes sin ~/i = sin Ot

(17)

+o2)i'~co~

(18) Ol

and (12)

a = COS2 02 b = - 2 z 2 cos 01 cos 02 cos(01 + 02)

C ~--- Z2 COS2 01. lvl_2(z) of eqn (18) is the view factor of surface 2

506

W. A.

PETERSON et

due to surface 1. Also, for the view factor of surface 1 due to surface 2, Zpz_,(z), we let 1 go to 2 and 2 go to 1 in eqn (18). Therefore,

F = LzZpz--t,(z).

al.

iffl+m S even

=

$

zp,+z(d12-d(h)

S even

=

2 n=l

(19)

2-I

The irradiance can be calculated

from the surface to the detector from the following expression or

wzcos 01 F

= L,h tan 8,

1-d

7Fu2’%~p,4(Z)~2-d(h) 7r2 -

h) d.x, dzl [x! + z: tan 8, + (21 - h)*12' (z, -

Performing the integration F

over xl and z1 we find

= ZI ,Zd (z)

1-d

*

~l~2L,wp,,(o)

Similarly for the odd number reflections ceive ~1~2z,,,,(z)z,,,(o)z,,a(h)

S

lr2 -

~la2L,(O)zp,,(0)

where zl-+d(t)

37 tan (3, 2 [z2 tan2 8, +” (z - Zz)2]1’2

=

(23 we re-

1 (24 .

Multiple reflections of direct beam with one side partially illuminated at distance z and detector at height h is

1. (20)

w cos 8, - [w* + 2hw cos 8, + /22],‘2

RD

Letting z = h and h = 0

Z(O)

,+d

F (1

=

-

sin 0,).

(21)

Z(z),&, of eqn (20) is the view factor of the detector of surface 1. Replacing 1 by 2 in eqn (20) represents the view factor for surface 2. To arrive at expressions for direct beam multiple reflection, we proceed as follows: Let L, be the radiance of a surface after the first reflection of the direct beam. Then

a,z,cos

L, =

ci,

IT

(22)

where a, = albedo of surface 1 and

z=

a,G cos a, IT

[cosp +

fg(cos p - 1) + f] .

We now sum the irradiance for multiple reflections due to surface 1. Summing the even number of reflections, gives =

$! X

+

&v,,

a, cos a, 7r[cos p + fg(cos

x

{~,--.dk)

+

~z2hd(h)1b2

+

p -

1) + f]

~2&,4(4 -

bl&,,(O)zl+d(h)

u,u2z,,,,(o)z,,,,(o)1}.

(25) If side 1 is fully illuminated then RD -= G

by the direct beam,

a, cos (Y, ?r[cos p + fg(cos

x

{II-d(h)

+

~z2+d(h)1/b2

+

p -

a2&u4(0) -

1) + f b,l,~:;%d(~)

~l~,z,,,,(o)z,,,(o)1}. (26)

cos p + fg(cos p - 1) + f

L, =

“eve”

&dd

G

so

s

RD -= G

=

12-d@)

1 +

y

&M(Z)

Z,,_,(O)

o1 and o2 are the angles between the direct beam and the normal to the surfaces 1 and 2, respectively. Cos (Y,and cos a2 can be related to the slope angles of the surfaces and the zenith angle of the sun. E. Diffuse multiple rejlections of irrudiunce (RS) 1. The multiple reflection of the circumsolar part of the diffuse irradiance (CSR), which has the same general time variations as the direct irradiante, can be treated similar to the direct irradiance utilizing eqns (25) and (26). Therefore,

Z,*.+,(O)

CSR = gf %.

(27)

2. Because of the assumed nondirectional property of the isotropic diffuse radiance, the initial irradiance on a surface can be expressed as f(1 -

Solar and diffuse irradiance in valleys

albedo

507

a2

, surface

aI de

)Y J

0

x

Fig. 2. Geometry for the multiple reflection between slopes. # Reflections

Irradiance

4

ala2Ll Y lp,~2(Z) 1p2~1(0) /2~d(h) Ip,~z(O) 'rr

a2Ll le~2(z) 12~a(h) ,it

5

a~a~ ~4 L,lpt~,(z) I2p2~,(0) l,~d(h) Ip,~(O)

alazL~ ~r2 lp,~(z) lp2~,(O) ll~d(h)

6

" 3 a'fa2 -~ L, lm_z(z) 122.,(0) 12~a(h) Iz2~2(0)

LIII~d(Z)

1

2

g)/o. Therefore, the radiance of surface 1 after one reflection is L-

ajIof(1

-

g)

F. Temporal variations o f irradiance in valleys For a valley oriented in any arbitrary direction, the geometric relationships between any particular orientation relative to the earth at any one time and the incoming direct solar radiation can be described as (Fig. 1)

or replacing/o from eqn (6) cos a = sin 8 sin ~b cos s L =

alGf(1 - g) ~r[cos 13 + fg(cos 13 -

1) + f ] "

(28)

-

sin ~ cos ~b sin s cos ~,

+ cos ~ cos qb cos s cos to

Equation (26) can be used to calculate the isotropic multiple reflection irradiance (IM) by replacing the factor

+ cos ~ sin d~ sin s cos -/cos to + cos 8 sin s sin ~ sin to

(30)

where al cos Otl

7r[cos13 + fg(cos 13 -

1) + y]

in eqn (26) by eqn (28). The diffuse multiple reflection of irradiance in a valley can now be expressed as R S = I M + CSR.

(29)

The multiple reflection between surface and sky (clouds) is not considered here, because it is included in the (already higher) value of the global irradiance measured at the station with free horizon that is used as input for the equation.

= the surface azimuth angle from the local meridian, zero point due south, east positive, and west negative s = slope angle of valley surface with horizontal to = hour angle = declination angle = latitude of location on earth a = angle of incidence of solar beam measured between the beam and the normal to the surface. For example, a valley oriented in an East-West

5O8

W. A. PETERSON et al.

I'd

u

Z <~

f._

0.9

................

.

W/m 2

0.8

..--

Io,oooF

n- 0 . 7 n."

9000 I-

/"

8000~

.u 0 . 6 < m 0.5 o _J o 0.4 o ~- 0.3

/ /

zOOOl-

// \

i

Lg

\...

I

\

0.2

> I-

<

O.t

n.

0

..J LU

\, O~

6

7

8

9

10 11 12 13 14 15 16 17 18 SOLAR TIME (o)

6

1

7

I

8

9

I

I

10 11 12 13 14 t.5 16 17 18 SOLAR TIME (b)

Fig. 3. N - S oriented valley of 65 and 45 degrees slope angle (from vertical) with and without snowcover. M a r c h 21, cloudless sky. a) relative values for each component: . . . . . direct solar irradiance;

reflected solar irradiance at snow cover (60% albedo); (the asymmetry stems from the higher reflection from the steeper E-oriented slope during morning); . . . . . diffuse irradiance from the sky; . . . . . . . reflected diffuse irradiance at snowcover (60% albedo). The reflected solar and diffuse irradiance from the slopes without snowcover (10% albedo) are negligible and not shown, b) global irradiance outside and in the valley: --- at snowcover; without snowcover.

direction with slope angle s cos a = cos(6 -+ s) cos ~ cos 0~ + sin(dp _ s ) s i n 8.

(31)

The + and - sign in eqn (31) stand for North and South facing slope, respectively. Also, for a valley oriented in a N o r t h - S o u t h direction cos a = sin ~ sin dp cos s + cos ~ cos d~ cos s cos ~ -+ cos ~ sin s sin ~. (32) In eqn (32) the + stands for an East facing surface and the - for a West facing surface. In summary, the total irradiance in a valley can be determined by eqns (8), (16), (25), (26), and (29). Equation (30) controls the temporal variations, such as date and time of day. The model is numerical and is applicable to a valley of any shape, orientation, ground albedo, and relation of diffuse to direct normal solar irradiance. It works well with overcast sky by using a ratio, f , of 100 or more. With scattered clouds, however, considerable errors can result from the distribution of clouds over the sky and from bright and dark areas within them. A computer p r o g r a m t was developed and run for several realistic cases, as can be expected for valleys in the intermountain region. Input parameters for the computer program are the following: Julian t" T h e c o m p u t e r p r o g r a m is available from R e x L. H u r s t , U t a h State University, Logan, U t a h 84322.

date, location (geographical latitude), azimuth of the valley, two independent slope incidence angles, two independent albedos, ratio of diffuse to direct normal solar irradiance ( f ) , ratio of circumsolar to total diffuse irradiance (g). The model then calculates the global irradiance every hour for a horizontal receiver surface in the valley. 3. SPECIFIC EXAMPLES

A valley of 65 ° and 45 ° slope angles from the zenith was chosen, comparable to an average and steep mountain slope. F o r cloudless sky, f was taken as 0.0811] and g = 0.3[2]. In Figs. 3 - 6 the global irradiance at the unobstructed horizon is taken from results from measurements at approximately 1500 m elevation in the Rocky Mountains[3]. The left hand part of the figures (a) represents the relative values to the global irradiance for each component as calculated from the model, while the right hand part (b) shows the actual total (multiplied by global) irradiance, in absolute units. Figure 3 shows the irradiance in a N - S oriented valley with the steeper slope facing East, on March 21. Sunrise is delayed from 6 a.m. to approximately 8:30 a.m. After this time the direct solar irradiance is not affected by the valley configuration c o m p ~ e d to a location with free horizon until sunset, which occurs considerably earlier, approximately 2:30 p.m., due to the steeper W-slope. The diffuse irradiance is only slightly affected by the valley. A minor loss is visible as double wave around sunrise and sunset, due to the contribution of the circumsolar part. With low albedo (soil, of approximately 0.1) the contribution due to direct reflection from the slopes

Solar and diffuse irradiance in valleys

509

of either component, direct as well as diffuse, is S W - N E oriented valley, when the steeper slope is negligible. However, when the slopes are snow cov- SE facing (Fig. 4) and NW-facing (Fig. 5), respecered to assume an average albedo of approximately tively. Also in this case, a visible effect in spring 0.6, the contribution due to direct reflection be- (March 21) is detectable only with snow cover (alcomes noticeable. This contribution is asymmetric bedo 0.6). around solar noon, as the steeper slope reflects For the summer months (not shown), a valley of more on a horizontal surface during the morning the 450/65 ° slope just loses irradiance due to the hours than does the more shallow slope in the af- delayed sunrise and early sunset; only low albedo ternoon. can be expected during this season. The reflection of the diffuse component throughIn Fig. 6, a case for mid-winter is chosen to demout a cloudless day is negligible. onstrate the strength of the model for overcast sky The right side of Fig. 3 shows that a horizontal (f = 100). The E - W oriented valley is shown once surface in a N - S valley of 45 ° and 65 ° slope angle with the steeper and once with the shallower slope from the zenith, receives considerably less global facing south. The bright area of clouds close to the irradiance during the early and late hours of the day sun, that partly appear above the skyline when the when the slopes block the direct solar irradiance, shallower slope is south, is well detected by the but an increased amount during the time the sun model. In case of high albedo (0.6), that is most is above the horizon, when the slopes are snow- probable during this time of the year in 1500 m covered. elevation, the contribution by multiple reflection Figures 4 and 5 show the peculiar effects for a from the slopes is considerable.

hi u Z

0.9 0.8

W/m

n." 0.7 n,,' . 0.6

10,00C 9000 8O0O 7000

m 0.5

O _J o

//"

6O00 5O00 400O

0.4

O ~- 0.3 W

> 0.2 ,z ..J O.t IM er 0

2

6

7

8

9

10

11

12

13

14 15 16 17 18

3000 2000 1000 0

/ / ./

/

/

"%~k\%\x%%%~% %,

/

//"

/ /

6 7

8

9

SOLAR TIME (o)

10 11

12 13 14 15

16 17 18

SOLAR TIME (b)

Fig. 4. Same valley slope angles as in Fig. 3, but valley orientation SW-NE. Steeper slope SE facing, (a) and (b) like in Fig. 3. (Notice: Inserts of Fig. 4 and 5 have to be exchanged.)

LIJ 0 0.9 Z 5 0.8

/

W/m2 10,000,

/

I

0.7

f /

0.6

JUd n-

i

82~45

i I i i i i

/

i i

m oJ 0.5 (.9 0.4 o 03 ud > 0.2 0.~ 6

7

i

6ooo,9°00'3ooo4O00i //" ,"/'// e°°2ooo5OOO7°°

j

t I I

--.~ j 8

9

10 11

~x

°

OT

f-?

0

..--"

12 13

SOLAR TIME (a)

14 15 16 t 7

18

IOOOi o6

7

8

9

10

11

xx%%%k

12 13 14 t5

16

SOLAR TIME (b)

Fig. 5. Same orientation and slope angles as in Fig. 4, but shallow slope SE facing. (a) and (b) like in Fig. 3.

17

18

510 b.I 0 Z

,,¢ < rr t~

W. A. PETERSONet al. 0.8 0.70.6-

.J

< m Q ,_1

0.5

o I-

0.3

uJ >

0.2

p.. < .J b.I ar

01

W/m 2

0,x 45.

0.4

3000

r

2000r

0 5

I 6

I 7

I 8

I 9

i 10

SOLAR

I 11

I 12

I 13

I 14

I 15

I 16

TIME (o)

o7

, ~"

8

9

10

11

12

SOLAR

13

14

15

16

t7

TIME

(b)

Fig. 6. E-W oriented valley of same slope angles as in Fig. 3 for overcast sky and albedo 0.6 (upper curves) and 0.1. - - steeper slope slope facing N, - ' - ' steeper slope facing South. Presently, this mathematical model is being tested with regard to an experimental simulation model[4]. Acknowledgement--This paper is part of a study funded by the Department of Energy under Grant No. EG-77-S07-1656. REFERENCES

1. W. A. Peterson and I. Dirmhirn, The ratio of diffuse to direct solar irradiance (perpendicular to the sun's

rays) with clear skies--a conserved quantity throughout the day. J. Appl. Met, 20, 826-828 (1981). 2. A. D. Watt, On the nature and distribution of solar radiation. U.S. Dept. of Energy, Rep. HCP/T2552-01 (1978). Circumsolar radiation. Sandia Nat. Lab. Report SAND 80-7009 (1980). 3. I, Dirmhirn, Solar energy potential, ultraviolet radiation, temperature, and wind conditions in mountainous regions. DoE Grant no. EG-77-S-07-1656.Final Report (1982). 4. T. Diniz, D. A. S. A., Solar Energy Availability Study under Distinct Topographic and Cloudiness Conditions. Thesis, Utah State University, Logan, Utah (1978).