Optimum inclinations of south-facing solar collectors during the heating season in China

Enersy Vol. 14, No. 3, pp. 123-129, 1989 Printed in Great Britain. All rights reserved

0360-5442/89 $3.00 + 0.00 Copyright @ 1989 Pergamon Press plc

OPTIMUM INCLINATIONS OF SOUTH-FACING SOLAR COLLECTORS DURING THE HEATING SEASON IN CHINA LIN WENXIAN Solar Energy Research Institute, Yunnan Teachers’ University, Kunming, Yunnan Province, People’s Republic of China (Received 7 July 1988)

Abstract-Based

on Hay’s anisotropic,

diffuse-sky insolation

model, theoretical equations slopes of tilted, south-facing surfaces during the heating season in the northern hemisphere. The effects of latitude, declination, ground reflectivity, concentration ratio, direct-to-global insolation ratio, and clearness index on optimum slopes are discussed and analyzed. The optimum slopes depend mainly on latitude, declination, direct-to-global insolation ratio and the general correlations for estimating optimum slopes of tilted, south-facing surfaces during the heating season in China are obtained from data of solar insolation measured in major cities. The linear regression technique is used. The general correlations developed are valid anywhere in China and are accurate (with a maximum residual standard error of s = 1.05) and convenient to use in practice.

are derived, which can be used to estimate the optimum

INTRODUCTION

surfaces receiving solar insolation, such as flat-plate collectors and concentrating (non-tracking) devices with low concentration ratio are installed at fixed slopes for a long time (for a month, half a year or even many years). The determination of optimum surface slopes is very important. At present, the optimum slopes are determined primarily by using an empirical or intuitive approach.’ Some theoretical equations have been derived for estimating optimum slopes, 2-4 but these are not precise and are inconvenient to use because the equations are based on the isotropic diffuse-sky insolation model that is not very accurate. Furthermore, the optimum slopes are affected by very many parameters. It is the purpose of our work to obtain general correlations for estimating optimum slopes that are precise and convenient to use. They are developed especially for the determination of optimum slopes for a tilted, south-facing surface during the heating season anywhere in China. Most

THEORETICAL

ANALYSIS

The total insolation received by a tilted surface that faces south during the day is given by Ht = HbRb + NcHdRd + RHR,IC,

(1)

where C = 1 applies for the case of a flat-plate collector. For a concentrating remains fixed f& a long time (more than a month), C must be ~5.’ Rabl has proposed an equation for calculating N,?

device that

(2)

N, = Kb + (1 - &J/C. Klein has proposed an equation for calculating Rb, viz.6 R,, = [cos(F - S) cos D sin W: + P/18OWI

sin(F

- S) sin D]/T

(P = 4,

(3)

where T = cos F cos D + P/lSOW,

sin F sin D

and D = 23.45” sin[(284 + n)360/365]. 123

(4)

1n the northern

hemisphere,

the latitude is positive. Since our analysis is limited to the March of the following year), it follows that

heating season (21 September-22

W: = Ws = cos-‘(-tan

F tan 0).

(5)

Hence, Eq. (3) becomes R,, = [cos(F - D) cos D sin W, + P/18OW sin(F - D) sin D]/T.

(6)

The diffuse-sky insolation is anisotropic.7-9 It is therefore incorrect to treat it as isotropic, i.e., one should not use Rd = (I+ cos S)/2. Nevertheless, this simplification was used in many of the previously published papers.2*3 Although Koronakis obtained improved results with an anisotropic model, namely, Rd = (2 + cos S)/3, his results are also not satisfactory.4 Ma and Iqbalr’ made statistical comparisons of the isotropic and two anisotropic models. They found that Hay’s anisotropic, diffuse-sky insolation model is best and recommended it for use.” We have therefore employed Hay’s model in the present work to estimate Rd. In Hay’s model, it is assumed that the diffuse insolation on a horizontal surface is composed of a circumsolar component coming directly from the direction of the sun and an isotropicallydistributed, diffuse component from the rest of the skydome. These two contributions are weighted by writing” Rd = XbKtRb + (1 - K&)(1

+ cos 5)/2,

(7)

R, = (1 - cos S)/2.

(8)

Hence, Eq. (I) has the form FZt= H[KbRb + N,(l - Kb)Rd f RR&]. The optimum slope is then determined

(9)

from d&/dS = 0.

The theoretical form

equation

(10)

for estimating the optimum slope S, is next derived and has the

,S, = tan-l[M(cos

I) sin F sin W, - P/18O‘w, sin L) cos F)/V],

(11)

where

V = cos D sin F sin W, -k P/18OW, sin D cos F, I = K,,[l -t N&(1

- Kb)] + [N,(l - K,)(l - K&)/2]

Using monthly average daily values for K b, K,, optimum slope for any month in the heating season For the sake of simplicity and convenience, the was assumed to be delayed from 21 September-22 change has little effect on S,. If the surface is fixed during the entire heating total insolation becomes Ht, = 2 [HbiRbi + RiP&(l -

COS

R, and can be heating March

- R/(2C).

D, as recommended by Klein,$ the determined precisely from Eq. (11). season in the northern hemisphere to 1 October-31 March. This small

season, the received seasonal average daily

S)/(2C)]/6

(12) where i = 1,2, . . . ,6 represent, respectively, October, November, . . . , March. Using Eqs. (11) and (12), the seasonal optimum slope is found to have the form

St = tan-’ 7

{ARbi[l

+ Ncik;,i(l-

I(bi)]/E},

(13)

Solar collectors in China

125

90

75

60

0

I

I

I

I

15

30

45

60

F

Fig. 1. The relation between S,, and F for different D.

where A = (COS D sin F sin Wsi- P/lSOWsi sin D cos F)/T, E =

C

Kbi)] + Nci(l - K,i)(l - KbiKti)/2- RJ(2C)I*

+ NciKti(l -

{K,i[l

We note from Eqs. (11) and (13) that the optimum slope S, depends on F, D, R, Kb, K,, and C. Figures l-6 show the relations between S, and F, D, R, Kb, K,, and C, respectively. Figure 1 shows that the higher the latitude, the greater the optimum slope, with the correlation curve being almost linear and with large slope, which implies that F has a large effect on S,. This result follows because, as F increases or the location moves further away from the equator, the incidence angle for solar radiation decreases and, therefore, the inclination of the surface must be increased to obtain the maximum possible solar insolation. We see from Fig. 2 that the declination D has an effect similar to that of F on S,, and for similar reasons. Figure 3 shows that S, increases with Kb relatively rapidly when Kb < 0.2, but this increase is gradually reduced as Kb approaches 1.0; the curve is almost linear for Kb > 0.2.

60

-

20 t

K, = 0.5

Kt = 0.5

c

F

-2

-30

10 t

0

L

I

I

I

I

-5

-10

-15

-20

D

Fig. 2. The relation between SOand D for different R.

126

LIN WENXIAN

60

0.2

0.4

0.6

0.6

Kb

Fig. 3. The relation between SOand K, for different D.

60 0.6

Kb -0.5

D.-l5

c

F=

-3

30

10 I 0.2

0

I 0.6

I 0.4

I 0.6

K1

Fig. 4. The relation between SOand K, for different R.

.4 0.2 0.0

40 so

R

20

01 1

Kb

=0.5

Kt

=0.5

F

=30

0

=-15

I

I

I

2

3

4

C

Fig. 5. The relation between SOand C for different R.

I 5

Solar collectors in China

127

60

0

0.4

0.2

0.6

0.8

R

Fig. 6. The relation between S, and R for different D.

Figure 4 shows the relation between S, and the clearness index Kf. It may be seen that k; has a small effect on S,, which increases with K,. The curve is almost linear but with a very small slope. Because Xr, influences the second term of Eq. (9), this terms ~ntributes only a small fraction to Ht. For similar reasons, the concentration ratio C and the ground reflectivity R also have small effects on S,, as can be readily seen from Figs. 5 and 6. In summary, the main factors affecting S, are the latitude F, the declination D, and the direct-to-global insolation ratio &,, whereas the concentration ratio C, the ground reflectivity R, and the clearness index Kt only exert weak influences on S,.

GENERAL

CORRELATIONS

Although the theoretical equations are precise, they are complex and inconvenient to use. For this reason, we develop precise and simple correlations. Since C, lu, and R have little effect on S,, it is reasonable to assume that K, =0.5. Accordingly the value of R for China can be represented by two approximations: for the southern part of China, R = 0.2 during the heating season (there are no snow covers); for the northern part, R = 0.5 during the heating season because there are partial snow covers. The dividing line is the Yangtze river, with the northern part of China defined to lie to the north and the southern part to the south of the Yangtze river. Our analyses have been carried out for C=l and C22. Since S, varies approximately linearly with F, D, and lu, and K,, is larger than 0.2,*‘** it is reasonabIe to use the following correlation: Soi = U + blF + bzDi + b3Kbi.

(14)

Muitiv~able linear regression analysis shows that the effect of F on Soi in Eq. (14) may be ignored compared to the effects of Di and Kbi on &i. Hence, Eq. (14) can be written as Soi = U + blDi + bzKbi.

(15)

The optimum slope S,, for the entire heating season is then calculated from the correlation S0, = c + d,D, + dzF + dJKbt. However, Dt is a constant and we therefore

(16)

include it in c, i.e.,

S,, = c + d,F + d&T,,.

(17)

LIN WENXLW

128 Table 1. The meteorological

Southern

Part

Northern

Part

I

stations in China which provided insolation data.

I

I

L

Using Eqs. (15) and (17) and measured data for the solar insolation, the regression coefficients may be determined. The meteorological stations involved in the present work are listed in Table 1. These stations are distributed all over China. Using measured data for the solar insolation from the stations were derived for estimating optimum surface listed in Table 1,11 general correlations inclinations for south-facing solar collectors during the heating season in China, based on the use of the multivariable regression technique. The results are listed in Table 2. We note from Table 2 that, using stations distributed all over China with various climatic and meteorological conditions, the developed correlations are very precise, with the smallest correlation coefficient being r = 0.978 and a maximum residual standard error of s = 1.05. Therefore, our correlations provide very good estimates anywhere in China for the optimum Table 2. The general correlations of outimum inclinations for solar collectors in Cikina. Sector

Southern Part

Northern Part

station

Latitude(N)

Longitude(

(degree)

(degree)

E

Height (m)

Kunming

25.01

102.68

1981.4

Guiyang

26.58

106.72

1071.2 505.9

Chengdu

30.66

104.01

Nanlin

22.82

108.35

72.2

Zhenzhou

34.72

113.65

109.0

Nanjing

32.00

118.92

a.9

Shanghai

31.17

121.48

4.5

Xian

34.30

108.92

396.9

Lanzhou

36.05

103.88

1517.2

Guangzhou

23.13

113.31

6.3

Harbin

45.68

126.61

171.7

Changchun

43.90

125.21

236.8

Shenyang

41.76

123.44

41.6

Beijing

39.80

116.46

31.2

Taiyuen

37.78

112.55

777.9

Datong

40.10

113.33

1067.6

Xilin

36.75

101.64

2295.2

Jinan

36.68

116.9’3

51.6

Yantai

37.s4

121.40

46.7

Solar collectors in China

129

slopes of tilted, south-facing surfaces during the heating season. These general correlations more precise and more convenient to use than other available equations.

are

REFERENCES

1. J. A. Duffie and W. A. Beckman, Solar Engineering of Thermal Processes, Wiley, New York, NY (1980). 2. J. Kern and I. Harris, Sol. Energy 17, 97 (1975). 3. J. P. Chiou and M. M. El-Naggar, Sol. Energy 36, 471 (1986). 4. P. S. Koronakis, Sol. Energy 36, 217 (1986). 5. Solar Energy Technology Handbook (Part A), W. C. Dickinson and P. N. Chermisinoff eds., Marcel Dekker, New York, NY (1980). 6. S. A. Klein, Sol. Energy 19, 325 (1977). 7. R. Perez, R. Stewart, C. Arbogast, R. Seals, and J. Scott, Sol. Energy 36, 481 (1986). 8. V. M. Puri, R. Jimenz, and M. Menzer, Sol. Energy 25, 85 (1980). 9. R. Stewart and R. Perez, Proc. ASES, p. 639, Anaheim, CA (1984). 10. C. C. Y. Ma and M. Iqbal, Sol. Energy 31, 313 (1983). 11. Solar Radiation Data for China (1961-1977), Weather Publishing House, Beijing (1981).

NOMENCLATURE

a, bl, b2,b3, c, d,, d,, d, = Regression coefficients = Concentration ratio c (dimensionless) = Declination (degrees) Q Q> Dt = Latitude (degrees) F = Monthly average daily total inH, Hi solation on a horizontal surface

(MJ/m’) = Monthly average daily direct insolation on a horizontal surface Hd, Hdi

Ho, HOi

H,

HU

KI,, K,i

(MJ/m*) = Monthly average daily diffuse insolation on a horizontal surface (MJ/m’) = Monthly average daily extraterrestrial insolation on a horizontal surface (MJ/m*) = Monthly average daily total insolation on a tilted surface

(MJ/m*) = Seasonal average daily total insolation on a tilted surface (MJ/m’) = Monthly average daily direct-toglobal insolation ratio (dimensionless)

= Seasonal average daily direct-toglobal insolation ratio