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

Energy Vol. 14, No. 11, pp. 123-726, 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 COOLING SEASON IN CHINA LIN WENXIAN Solar Energy Research Institute, Yunnan Teachers University, Kunming, Yunnan, People’s Republic of China (Received 16 Febrwry 1989) Abstract-Using Hay’s anisotropic, diffuse-sky insolation model, we derive a theoretical equation which can be used to estimate the optimum slopes of tilted, south-facing surfaces during the cooling season in the northern hemisphere. The equation yields implicit results and it is difficult to obtain optimum slopes. The major factors affecting the optimum slopes are the latitude and direct-to-global insolation ratio. With data on solar radiation for 30 of China’s major cities, a general correlation is developed by using the linear regression technique. The result is valid anywhere in China during the cooling season and is accurate (with a correlation coefficient r = 0.912 and a residual standard error s = 0.94) and convenient to use in practice.

INTRODUCTION

Many surfaces receiving solar insolation are installed at fixed slopes during the cooling season (23 March-20 September). Studies dealing with determinations of optimum surface slopes during this season are rare. Most investigations are focused on determinations of those slopes during the heating season (21 September-22 March).‘-’ The reason is that theoretical equations for estimating the optimum inclinations are complicated and difficult to analyze. It is therefore useful to develop a general correlation that is accurate. Such a correlation for estimating the optimum inclinations of south-facing solar collectors during the heating season in China was developed previously. It is our purpose in this work to develop a similar correlation for estimating the optimum inclinations of south-facing solar collectors during the cooling season in China.

THEORETICAL

ANALYSES

AND DISCUSSIONS

The total insolation received by a tilted surface during a day is given by Ht = H&dir

+ H&d,

+ RHR,,

(1)

where H, Hdir and Hdif are the daily total, direct and diffuse insolations on a horizontal surface (MJ/m*); Rdir, Rdif and R, are, respectively, the orientation ratios for the direct component, diffuse-sky component and diffuse-ground-reflected component (dimensionless). Also, R is the ground reflectivity (dimensionless). For south-facing surfaces, Klein has proposed an equation for calculating Rdir, viz.6 Rdir=

{[COS(F

-

s)](COS

~)(Sh

Wl) -I-n/180 Wi[sin(Ii - S)](sin D)}/T

where F, S, D, and Wl are the latitude (degree), surface slope (degree), and sunset hour angle of the tilted surface (degree), respectively. Also,

declination

(2) (degree),

T = (cos F)(cos D)(sin W,) + n/180 W,(sin F)(sin D),

(3)

D = 23.45”{sin[(284 + n)360/365]},

(4)

where W, and n represent, respectively, the sunset hour angle of a horizontal surface (degree) and the day number. 723

LIN WENXIAN

724

In the northern hemisphere, cooling season, it follows that

the latitude

is positive.

Since our analysis is limited in the

W, = cos-‘{ - [tan(F - S)](tan D)}.

(5)

The diffuse-sky insolation is anisotropic.7-9 According to the analysis of Ma and Iqbal,” Hay’s anisotropic, diffuse-sky insolation model should be employed to estimate Rdif. 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 &if = KdirKt&ir + (1 - K,&,)[l

+ (cos S)]/2,

where Kdir and K, are the daily direct-to-global insolation clearness index (dimensionless), respectively. Furthermore,

(6)

ratio (dimensionless)

R, = [l - (cos S)]/2.

and daily (7)

Using monthly average daily values for Kdirr K,, R, Zldir, Hdify and D as recommended by Klein6 the monthly average daily total insolation received by the tilted surface may be determined from Eq. (1). For the sake of simplicity and convenience, the cooling season in the northern hemisphere was assumed to be delayed from 23 March-20 September to 1 April-30 September. It has been proved that this small delay has little effect on the optimum slopes. The seasonal average daily total insolation received by the tilted surface during the entire cooling season (1 April-30 September) becomes H,t = C (Hdir,iRdir,i + Hdif,&ir,i + Ri&R& I

= C (AiRdi,.i + Bi + Ci COSS),

where i = 1, 2, . . . , 6 represent, Ai =

respectively,

(8)

April, May, . . . , September;

also,

Hdir,i + Kdir,iKt,iHdif,ir

(9)

Bi = [(1 - Kdir,&t,i)&if,i + R&]/2,

(10)

Ci = [(I - Kdir,iKt,i)Hdir,i - RiHi]/2*

(11)

The seasonal optimum slope is determined

from dH,,/dS = 0.

(12)

The theoretical equation for estimating the seasonal optimum slope So (degree) is next derived and has the implicit form So = sin-’ ([ 7 (Fi + GiV;)AilE,I/F

G) *

(13)

where Ei = (COS F)(COS D;)(sin W,,J + n/180 W,,i(sin F)(sin Di),

(14)

fi = (COS Di)[sin(F - S)](sin Wi,,) - n/180 W:,i[COS(F - S)](sin Di),

(15)

G, = [cos(F - S)](COS Di)(cos W(,J + n/180 W:,Jsin(F - S)](sin D;),

(16)

v=dW&/dS=

(17)

Xi = -[tan(F

-tan Di[l+ tan(F-S)]/(l - S)](tan Di).

-Xi),

(18)

From the preceding equations, we see that So must be determined by using some special techniques. They are very complicated and difficult to apply because the theoretical equation is inconvenient to use in practice. We must therefore develop some other equation that is more convenient to use in practice.

South-facing solar collectors in China Table

1. Data

Latitude(N)

for 30 meteorological stations in China. Longitude(E)

so

Height Kbt

Stat ion

Haikou Nan1 ing Guangzhou Kunr iag Fuzhou Gu I yang Changsa Lhasa Chengdu Y ichang Shanghai Nanj ing Xian Zhenzhou Lanzhou Jinan Xiling Yantai Ta i yuan Y I ngchung Tianj tng Eieljing Huhhot Shenyang Ulueqi Changchun Yillng tiarbln Jlami Halhe

725

(degrees)

tdrgrees)

20.00 22.81 23.13 25.60 26.10 26.60 28.20 29.10 30.61 30.70 31.17 32.00 34.30 34.70 36.05 36.70 36.75 37.53 37.80 38.30 39.10 39.80 40.80 41.80 43.80 43.90 43.95 45.70 46.80 50.25

110.62 108.37 113.32 102.70 119.30 106.70 113.10 91.13 104.00 111.10 121.43 118.80 108.93 113.65 103.90 117.00 101.63 121.40 112.55 106.20 117.17 lib.47 111.70 123.40 87.60 125.20 al.33 126.60 130.30 127.45

Ktt degrees)

(I) 14. I 72.2 6.3 1891.4 88.4 1071.2 44.9 3658.0 505.9 131.1 4.5 8.9 396.9 109.0 1517.5 51.6 2295.2 46.7 177.9 1111.5 3.3 31.2 1050.8 41.6 917.5 236.8 662.5 171.7 al.2 165.8

0.55 0.44 0.39 0.45 0.51 0.40 0.49 0.63 0.35 0.47 0.50 0.51 0.49 0.52 0.52 0.60 0.57

0.61 0.58 8.57 0.54 0.55 8.59 0.56 0.66 8.57 0.65 0.57 0.58 0.59

0.45 0.41 0.36 0.40 0.40 0.36 0.41 0.66 0.31 0.40 0.43 0.45 0.46 0.45 0.54 0.50 0.58 0.48 0.54 0.59 0.53 8.57 0.63 0.54 0.62 0.54 0.65 0.54 0.51 0.59

9

11 10 12 12 13 13 14 14 14 15 15 lb 17 17 ia 18 ia ia 19 19 19 20 20 22 21 22 22 23 24

In Table 1, we list the optimum inclinations of the south-facing surfaces during the cooling season in China’s 30 major cities. These values were calculated from the theoretical equation by using measured data for the solar insolation in these 30 cities.

GENERAL

CORRELATION

According to the analysis of Wenxian, 5 the major factors affecting S,, are the latitude F and the direct-to-global insolation ratio Kdir; Kt and R have only small effects on &. The dependence of S, on F and that on Kdir are approximately linear. These features may also be seen from Table 1. We therefore develop a correlation between So and F, Kdir. The correlation should have following form: 5’0= u + bF +

CKdir,t,

(19) a, b

where Kdir,t is the seasonal average daily direct-to-global insolation ratio (dimensionless); and c are regression coefficients. Using Eq. (19) and values of F and Kdir,t for China’s 30 major cities (Table l), with Kdir,t calculated from measured data for the solar insolation, we obtain the correlation ~=0.48F+4.6K,i,,t-2.5,

(20)

with the correlation coefficient (dimensionless) r = 0.912 and the residual standard error (degree) s = 0.94. The meteorological stations involved in the present work are listed in Table 1. These stations are distributed all over China and have various climatic and meteorological conditions. It is evident that our correlation is quite precise. Therefore, our correlation provides good estimates anywhere in China for the optimum slopes of tilted, south-facing surfaces during the cooling season. It is a general correlation for China and is more precise and convenient to use than other available equations. EGY14:11-O

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LIN WENXIAN

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. L. Wenxian, Energy 14, 123 (1989). 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, 86 (1980). 9. R. Stewart and R. Perez, Proc. ASES, p. 639, Anaheim, CA (1984). 10. Solar Radiation Data for China (1961-1977), Weather Publishing House, Beijing (1981).