2024年3月19日发(作者：华为荣耀9i致命缺点)

Solar Energy, VoL 19, pp. 325-329. PergamonP ress 1977. Printed in Great Britain

REVIEW PAPER

CALCULATION OF MONTHLY AVERAGE

INSOLATION ON TILTED SURFACES

S. A. KLEIN

Solar Energy Laboratory, University of Wisconsin-Madison, Madison, WI 53706, U.S.A.

(Received

16

June

1976;

in revised form

28

October

1976)

Abstract--Several simplified design procedures for solar energy systems require monthly average meteorological

data. Monthly average daily totals of the solar radiation incident on a horizontal surface are available. However,

radiation data on tilted surfaces, required by the design procedures, are generally not available. A simple method of

estimating the average daily radiation for each calendar month on surfaces facing directly towards the equator has

been presented by Liu and Jordan[l]. This method is verified with experimental measurements and extended to

allow calculation of monthly average radiation on surfaces of a wide range of orientations.

INTRODUCTION

Estimates of the monthly average solar radiation incident

on surfaces of various orientations are required for solar

energy design procedures, heating load calculations, and

other applications. Monthly averages of the daily solar

radiation incident upon a horizontal surface are available

for many locations. However, radiation data on tilted

surfaces are generally not available.

A simple method of estimating the average daily

radiation for each calendar month on surfaces facing

directly towards the equator has been developed by Liu

and Jordan[l]. Their method is described here and com-

pared with the work of Page[2] and with additional

experimental measurements. The method is then exten-

ded so that it is applicable for surfaces oriented east or

west of south.

RADIATION ON SURFACES FACING

DIRECTLY TOWARDS THE EQUATOR

given for each month in Table 1, 6 is the latitude, and 8

is the solar declination which can be approximately

expressed

8 = 23.450 sin [360(284+ n)/365]

tos is the sunset hour angle

cos ~o~ =-tan ~b tan 8. (5)

(4)

ESTIMATIONO F AVERAGED AILY

/to can be conveniently estimated from eqn (3) by selec-

ting for each month, the day of the year for which the

daily extraterrestrial radiation is nearly the same as the

monthly mean value. Using the 16th day of each mouth

can lead to small errors in/~0, particularly for June and

December. Recommended days for each month are given

in Table 1./4o is tabulated for each month as a function

of latitude in Table 2. The value of the solar constant

used in the construction of Table 2 is 4871 kJ hr ~m 2,

Thekaekara and Drummond [3]], which is approximately 3

per cent lower than the value used by Liu and

Jordan[l, 4] and Page [2].

The average daily radiation on a tilted surface, Hr, can

The average daily radiation on a horizontal surface, H,

for each calendar month can be expressed by defining

/~r, the fraction of the mean daily extraterrestrial radi-

ation, Ho.

g7 =/3/Ho

_.___l___l m~

Ho = (m2 - mr) ~" (Ho),

n =ral

(1)

Table 1. Recommended average day for each

(2)

Month

Jan.

Feb.

Mar.

Apr.

May

June

Aug.

Sept.

Oct.

Nov.

Dec.

July

month

Day of the year

17

47

75

105

135

162

198

228

258

288

318

344

Date

17 Jan.

16 Feb.

16 Mar.

15 Apr.

15 May

11 June

17

July

16 Aug.

15 Sept.

15 Oct.

14 Nov.

10 Dec.

where ml and ms are, respectively, the days of the year

at the start and end of the month and (Ho), is the ex-

traterrestrial radiation on a horizontal surface on day n

of the year which is given by

(Ho).

= 24 1 [ 1 + 0.033 cos [360n~]

~" ~c [ 365/J

× [cos 6 cos 8 sin cos + (ws2~r/360) sin 6 sin 8]

(3)

where Lc is the solar constant, n is the day of the year

325

326 S.A. KLEIN

Table 2. Monthly average daily extraterrestrial radiation, kJ/m2

Lat.

25

30

35

40

45

50

55

Jan. Feb. Mar.

32,848

31,141

29,200

27,040

24,677

22,131

19,423

Apr. May June

40,046

40,706

41,129

41,328

41,322

41,147

40,863

July

39,606

40,071

40,292

40,281

40,055

39,644

39,100

Aug. Sept.

37,832 34,238

37,534 32,917

36,976 31,348

36,166 29,542

35,118 27,515

33,851 25,283

32,391 22,863

Oct.

29,413

27,213

24,820

22,255

19,541

16,705

13,778

Nov. Dec.

23,902 28,115

21,034 25,679

18,069 23,072

15,403 20,319

11,998 17,448

8987 14,490

6082 11,486

37,111 39,356

36,436 39,569

35,497 39,530

34,303 39,247

32,869 38,737

31,209 38,025

29,345 37,152

24,909 22,669

22,161 19,714

19,296 16,687

16,344 13,626

13,344 10,579

10,342 7605

7396 4791

be expressed

/4T ~-/~/4 =/~RTHo (6)

directly towards the equator,

Rb - cos (~ - s) cos 8 sin co'e+ ¢r/180o~'~s in (~ - s) sin

cos ~ cos ~ sin co, + ~r/180co, sin ~ sin 8

(8)

where ~o is the hour angle which is 15°×(hours from

solar noon), afternoons, positive, mornings negative and

co's is the sunset hour angle for the tilted surface which is

given by

~ol = min[oJ, arcos [-tan (~ - s) tan 8]]. (9)

where /~ is defined to be the ratio of the daily average

radiation on a tilted surface to that on a horizontal

surface for each month. /~ can be estimated by in:

dividually considering the beam, diffuse, and reflected

components of the radiation incidence on the tilted sur-

face. Assuming diffuse and reflected radiation to be

isotropic, Liu and Jordan[l] have proposed that/~ can be

expressed

/~ = (1

- ffIalft)Rb + ffta/ffI(l +

cos

s)/2 +

p(l - cos

s)/2

(7)

where Hn is the monthly average daily diffuse radiation,

/~b is the ratio of the average beam radiation on the tilted

surface to that on a horizontal surface for each month, s

is the tilt of the surface from horizontal, and p is the

ground reflectance. Liu and Jordan[4] suggest that p

varies from 0.2 to 0.7 depending upon the extent of snow

cover. /~b is a function of the transmittance of the

atmosphere (except during times of equinox) which

depends upon the atmospheric cloudiness, water vapor

and particulate concentration. However, Liu and Jordan

suggest that l~b can be estimated to be the ratio of

extraterrestrial radiation on the tilted surface to that on a

horizontal surface for the month. For surfaces facing

Page has calculated values of /~b for five surface

orientations of several latitudes by integrating the direct

radiation on the tilted and horizontal surface calculated

at hourly intervals for a standard direct radiation curve.

Values of /~b calculated from eqn (8) are in reasonably

good agreement with the values tabulated by Page as

seen in Table 3. Page's values are slightly more con-

servative, i.e. closer to unity.

Since measurements of H~, the monthly average daily

diffuse radiation are rarely available, Hd must be es-

timated from measurements of the average daily total

radiation. A number of investigators have found that the

diffuse radiation fraction,

HdH,

is a function of /(r.

Shown in Fig. 1 are the relationships reported by Liu and

Jordan, and Page which can be expressed

(10a)

(10b)

_~a= ~ 1.390 - 4,027/~T + 5.531/£~ -- 3.108/~r 3 [Liu & Jordan]

L

1.00 - 1.13/(r [Page].

Table 3. Comparison of values of i~b from Page[2] and eqn (8)

,~ = 30°

4~ - s = 0

Page

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

1.61

1.40

1.18

0.99

0.89

0.84

0.85

0.94

1.09

1.30

1.53

1.67

Eqn (8)

1.66

1.43

1.20

1.00

0.87

0.87

0.84

0.94

1.12

1.35

1.60

1.74

Page

1.49

1.06

0.64

0.29

0.13

0.06

0.09

0.21

0.45

0.88

1.33

1.61

Vertical

Eqn (8)

1.59

1.13

0.67

0.30

0.11

0.05

0.08

0.21

0.50

0.97

1.46

1.74

(~ - s = 0

Page

2.15

1.72

1.35

1.07

0.90

0.84

0.85

0.98

1.20

1.57

1.98

2.30

Eqn (8)

2.26

1.79

1.38

1.06

0.88

0.80

0.83

0.98

1.24

1.64

2.12

2.42

Page

2.11

1.50

0.93

0.48

0.27

0.19

0.22

0.37

0,69

1.24

1.86

2,36

d, =4O°

Vertical

Eqn (8)

2.32

1.59

0.%

0.48

0.25

0.17

0.21

0.37

0.74

1.36

2.10

2.58