The determination of hourly insolation on an inclined plane using a diffuse irradiance model based on hourly measured global horizontal insolation

$olarEnergy, Vol. 19, pp. 477-491. Pergamon Press 1977. Printed in Great Britain

THE DETERMINATION OF HOURLY INSOLATION ON AN INCLINED PLANE USING A DIFFUSE

IRRADIANCE MODEL BASED ON HOURLY MEASURED GLOBAL HORIZONTAL INSOLATION

J. W. BUGLER Capricornia Institute of Advanced Education, Rockhampton, Australia 4700

(Received 28 November 1975; in revised form 28 October 1976)

Ahstraet--Using only measured hourly values of global insolation on a horizontal surface, a method has been developed for computing the corresponding hourly values of insolation on a surface inclined at any angle and oriented in any direction. The method uses a solar radiation model in which the diffuse component is calculated from global horizontal radiation using three different relationships; the appropriate equation is selected according to the value of the ratio of measured hourly global insolation to hourly global insolation computed for clear sky conditions. The method has been checked using measured hourly values in Melbourne over a 5-yr period of insolation on both a horizontal surface and a plane inclined at 38 ° to the horizontal facing north. The differences between the computed hourly values and the measured hourly values are found to be approximately normally distributed about zero with a standard deviation of 0.16 MJ m -2. This method is particularly useful for predicting the heat output of inclined solar fiat plate collectors when only measured global horizontal insolation is available, which is often the case. Good agreement was found between the predicted output of a typical collector using measured 38 ° insolation and the computed hourly values using this method. Since the method has been checked only against Melbourne data it should be applied elsewhere with caution, but it is believed to have general application.

1. INTRODUCTION

Solar radiation data are a necessary basis for the design of solar energy conversion devices and for feasibility studies into possible uses of solar energy.

The total solar energy incidence at a given locality can conveniently be considered to be made up of the direct beam radiation and the diffuse radiation; the direct radiation is received solely along the Sun-Earth vector, whilst the diffuse radiation is distributed over the whole sky hemisphere. Since the direct and diffuse radiations have quite distinct features, it is considered necessary for many purposes to measure the two parts separately, and this necessitates two more or less separate measuring systems. Furthermore, although it is evident that the solar energy received, and utilizable, at a given location will change with the inclination and orientation of the irradiated surface, any attempt to encompass measurements over a range of such inclinations and orientations will be prohibitively expensive except for one or two specialized sites. For this reason, measurements are normally taken only on a horizontal plane and, because of lack of funds and for simplicity in operation, it is common for only one measuring system to be employed at a given location sensing total (i.e. global) irradiance and recording radiant exposure.t

It is the purpose of this present study to establish an

~Author's note. Throughout this paper the word radiation has been used as the generic term for the Sun's energy, insolation has been used as a general term to describe the amount of energy received at a location per unit area over a period of time. Where precision of terminology has been required, irradiance is the solar radiant energy received per unit area per unit time and radiant exposure is the integration of irradiance over a specified time interval.

acceptable method of determining the total solar radiation incident upon an inclined plane from data of global radiation only in the horizontal plane at the same (or near) location for short time intervals, typically 1 hr.

A model of clear sky solar radiation may be considered to be reasonably well established, within the degree of accuracy of present day routine radiation measurements (say, -+ 5%), through the work of Threlkeld and Jordan[l], Rao and Seshadri[2], Loudon and Petherbridge[3], and others. The primary problem still facing anyone wishing to apply this clear sky model is the need for atmospheric water vapour and dust content data, which may not be available for the required locality. Nevertheless, the situation is such that publication of tables of clear sky solar radiation for a centre of population, such as those for Melbourne by Spencer[4], can be confidently produced and applied to, say, the design of air con- ditioning systems in that area.

The fundamental difficulty with modelling non-clear sky conditions is that diffuse radiation can no longer be related solely to solar altitude (plus, of course, atmospheric attenuation); clouds play an overwhelming role in modifying diffuse radiation, generally increasing it. However, the view is taken that, within the context of the currently accepted accuracy of solar radiation measurement, a correlation between diffuse radiation, solar altitude, atmospheric attenuation and some form of cloudiness index could prove to be fruitful. The "clas- sical" work of Liu and Jordan[5], which since 1960 has been the basis of most solar radiation empirical analysis, adopts this approach; however, their work is largely applicable to daily average radiation values, and where they offer hourly data reduction thek method gives average, not actual, values.

SE Vol. 19, No. 5--D 477

478 J . W . BUGLER

Heywood [6], has presented a technique for calculating total radiation on inclined surfaces based upon empirical data of total radiation on the horizontal. However, the method seems to ignore even the simple solar radiation model of distinct direct and diffuse components with distinguishably different properties, thus appearing to rely upon empirical correlations for each region to which it is applied. The present study, on the other hand, sets out to derive a universally applicable empirical correlation of diffuse and global irradiance.

~. MODELS OF SOLAR IRRADIANCE

2.1 Direct radiation The Earth's atmosphere depletes the extraterrestrial

solar irradiance by the scattering effects of air molecules and particulate material, and the absorption effects of ozone, water vapour and other gases. Based upon earlier work by a number of authors, Rao and Seshadri[2] produced curves of normal solar irradiance at the Earth's surface as functions of atmospheric precipitable water content and dust content. These curves, in the matrix form suitable for computer usage by Spencer[7], have been used as the direct solar radiation model in this study. For Melbourne, the particular dust content of 300 particles cm -3 and mean monthly precipitable water vapour figures for all days from Pierrehumbert[8]--from January to December, respectively: 19, 20, 19, 16, 15, 13, 12, 12, 13, 15, 17, 18 mm--have been assumed.

2.2 Diffuse radiation The currently accepted method of modelling clear sky

diffuse radiation is to assume a generally uniform irradiance from the sky hemisphere, together with an amount located in the immediate vicinity of the Sun (known as the circumsolar radiation) which, for the purposes of this study, is included as a small addition to the direct radiation; the relative proportions of these two parts is not yet clearly established. The clear sky diffuse irradiance is, however, of a low absolute order of magnitude rarely exceeding 150Wm -2, and, for the purposes of this study, may be modelled in a com- paratively crude manner.

The values given in the IHVE Guide (1970)[9] have been used for the uniform component, for which the following simple curve fit (accurate to within -+2.5% of the IHVE values) has been derived

/) = 16.0 ot °'5-0.4 a Wm -2 (2.1)

where a is the solar altitude in degrees. The circumsolar irradiance has so far received very

little separate attention from researchers and it is consequently difficult to model. Grether, Nelson and Wahlig[10] have reported recently that, from measurements made at Berkeley, California, circumsolar irradiance within 3 ° of the Sun-Earth vector was typically 1.75 per cent of the direct irradiance, although factors ranging from zero to 10 per cent are likely, depending upon atmospheric turbidity, etc. From subsequent analyses within this study it has been found that the developed radiation model is insensitive to a wide range of

circumsolar radiation factors, as might have been foreshadowed since the model is based on a measure of the total radiation received; in this paper, all radiation computations have allowed for circumsolar irradiance as 5 per cent of the direct normal irradiance.

The determination of a model for cloudy sky diffuse radiation presents many difficulties. Most of these difficulties are related to the lack of uniformity of the radiation over the sky hemisphere; even for completely overcast conditions Norris [11] has reported a degree (sic) of anisotropy up to 21 per cent. For the commonly encountered partly cloudy condition anisotropy of diffuse radiation obviously prevails; but it appears unlikely in the foreseeable future that anything but a uniform modelling of such radiation will be attempted, based either upon a detailed analysis of cloud data or upon some sort of cloudiness index.

The data most readily available for this analysis were global and diffuse radiant exposures for hourly intervals on a horizontal surface at Highett (Melbourne) for the years 1966-70. These records were obtained by the CSIRO Division of Mechanical Engineering and were conveniently available in computer compatible form.

Two forms of cloudiness index were used as arguments for numerous plots in the search for correlation between diffuse and global insolation: (i) the ratio of global to extraterrestial hourly exposure, G/E, and (ii) the ratio of global to clear sky global (as modelled) hourly exposure, G/T. The solar altitude angle (or, alternatively, atmospheric air mass factor) was another important argument. Figures 1-7 illustrate the plots of D/E vs G/E for nominal solar altitudes of 10, 20, 30, 40, 50, 60 and 70 °, for the year 1%6; values of altitude within -+1 ° of the nominal value were used in these plots. Figure 8 is a set of the curves of best fit from Figs. 1-7.

The outstanding feature of Figs. 1-8 is the very good correlation between D and G for the cloudy to overcast states, i.e. for 0 ~< G/E <~ 0.35; only when the solar altitude is less than about 20 ° is the linearity limit of G/E slightly reduced. What is, perhaps, surprisingly from this data, is not so much the strong linear relationship in this region but that there is no detectable tendency for D/G to go to unity as G/E tends to zero as one might expect; in fact, by a least squares analysis

For 0 ~< GIE <~ 0.35, D = 0.94G. (2.2)

These features of insolation under heavily clouded skies closely resemble the findings of Liu and Jordan[5] for daily totals. No doubt this is an effect of integrating irradiance, and it appears that, in this respect, hourly integration does not differ significantly from daily integration; a true irradiance model must surely have / ) = G as G tends to zero. From the practical point of view, however, there is no great significance in this as the absolute value o f / ) is quite small under these circum- stances.

The dependence of the correlation on altitude at values of G/E above 0.35 is clearly evident in spite of the much greater scatter of data points in this region due, of course, to the variety of cloudiness states for a given value of

The determination of hourly insolation on an inclined plane 479

0.6

0,5

0,4

D/E 0.3

0.2

0.1

NOMINqL i0 [EC4:~F_S S~L~q_TITLI[£

HIGHETT I n

o

8 ~

.% o o oo ~o

0'.1 01.2 0',3 0'.~ 0 I.S OI,B (].7 0:8 G/E

Fig. 1. For nominal 10" solar altitude.

L 019 1,0

0,6

0.5

0,4

D/E

0,3

0,2

0.i

NOMINq_ 20 ~GFEES S~_~lqLTITLQE

HIGI-LTT 1966

o o ~o° o

o. o o,~:oO

:°0~ ° o o

~ l ,1 Ui2 013 014 OL5 DIG O'./ BIB U',9 I.'U G/E

Fig. 2. For nominal 20* solar altitude.

0,6

0.5

0.4

D/E

0.3

03

o

0.1 o,: : o

o

0 0 ,1

N~MINqL 30 DEGREES S~L~ALTITUOE

HIDETT 19F~5

,b o o °o ° o ~ °° o=oo

i i i i i ~ i i i i

0,2 0.3 0,4 0,5 0,6 0.7 0,8 0.9 1,0 WE

Fig. 3. For nominal 30 ° solar altitude.

Figs. 1-7. Correlation plots of Diffuse vs Global hourly radiant exposure in the horizontal, normalised by the corresponding Extraterrestrial exposure, for Highett (Australia) throughout 1966.

GIE. Plots of D/G vs GIE and (}IT were then considered. As would be expected, dependence of the D/G vs GIE correlation upon altitude is still evident, as is shown in typical plots given in Figs. 9 and 10; however, the scatter of points in these plots is sutticiently confined to

encourage the study of the D/G vs G/T plots, in which the use of T as a normaliser in the abscissa should go a long way towards eliminating the altitude dependence of the correlation.

Figures 11-17 illustrate the plots of D/G vs GIT for

480 J.W. BUOLER

O.G

0.5

D/E

0,3

0.2

0,1

ixO'41JxFI_ 40 (]EC, REES S~LP~ ~_TITL;Z]E

HIGHETT 19f~6 :o o °

: ,~ °2 - ° ° o

~ . ° 0%° oo o o°q~ o

%o o

o

% o° %00 o o

° ~Oo

i I I i I I i I

0,1 0.2 0.3 O,q 0.5 0.6 0,7 0,8 G/E

Fig. 4. For nominal 400 solar altitude.

0.9 1,0

0,6

0.5

0.4

0.3

0.2

0.1

NOMINq_ SO DEGIqEES SOLFIqCLTITLUE

HI(]~ETT 1966

o

I I

0,i 0,2

D°

Do

o°~°o o % o g °o° Oo # o # o °~

i l l E i

0.3 0,4 0,5 0.6 0,7 0.8 G/E


0',9 i',0

0.6

0.5

0.4

I)/E

0.3

0.2

0.1

~INFL 60 I]EC.REES S~L~CI_TITLI]E

HI(I~ETT 1966

0

i I

0 0.i 0,2

° ° % o

o

o

i i , , i

0.3 0.4 0.5 0.6 0,7

Fig. 6. F o r n o m i n ~ 6 ~ s o l a r a l t i tude .

i i i

0,8 0.9 1.0

nominal solar altitudes of 10-70 ° for the year 1966 from the same data as used in the previous figures. There undoubtedly remains some degree of dependence on solar altitude in these plots, as particularly indicated in Table 1, however, the dependence on altitude rapidly decreases as

altitude increases. Since the absolute value of diffuse irradiance decreases as solar altitude decreases, it was considered likely that a correlation of the data in these plots, independent of solar altitude, could be established which would prove adequate for practical applications.


0.6

NOMIiqq_ 70 [ICPEES SOLP~P,_TITLI]E

HIGFETT 1966

0,5

0.4

D/E

0.3

0.2

0,1

0,6

D D

O

D

DO

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~ o D

I I I I I ~ I I I I

0 0,i 0.2 0,3 0.4 0.5 0,6 0.7 0,8 0,9 1,0

G/E


0.5

O.q

D/E

0.3

0,2

0.1 ~o*

I t i , I i I I !

0 O.l 0.2 0.3 0.4 0.5 0,6 0,7 0,8 0,9 G/E

Fig. 8. Curves of best fit through the plotted points of Figs. 1-7 for solar altitudes of nominal 10, 20 . . . . . 70".

Table 1. Solar data for Highett, Victoria, from the clear sky model of Section 2

Solar altitude Mean value of Mean value of Meal value of degrees D/G when GIT = 1 G/T when GIE = 0.35 GIE when GIT = 0.4

10 0.37 0.66 0.21 20 0.21 0.53 0.26 30 0.15 0.48 0.29 40 0.12 0.46 0.30 50 0.11 0.44 0.32 60 0.11 0.44 0.32 70 0.10 0.43 0.32

Note: The need for listing "mean" values arises from the dependence of E and T upon time of year.

Reference to Table 1 columns 3 and 4, and the Figures, indicates a value of G/T=0.4 at which the linearity between D and G (from zero values) ceases, instead of the limit GIE = 0.35 as tentatively suggested in eqn (2.2). An improvement on eqn (2.2) would be, then, for heavily clouded and overcast skies

For 0 < (3[ T <~ 0.4, D = 0.94G. (2.3)

At the high end of the GIT scale it may be seen that points in Figs. 11-17 tend to cluster around the computed values of D/G for G/T = 1.0, as listed in Table 1 column 2 for the respective solar altitudes. If independence from solar altitude is to be preserved here, a single point must be selected. Once again, with the greater ablolute relevance of the diffuse irradiance values at higher solar altitudes in mind, together with the visual aid of Figs.

482 J . W . BUOLER

1.0

0.9

0.8

0.7

0.6

D/G 0.5

0,4

0,3

0,2

0,1

0

L' b/~ ~ ;J ~#

o o

.'I!N'EH IgE~ ~,~ o

N~:~A 33 DE~i~FkS ~ ~.TlrdI ~ o 'i~ ~o% °o

O.1 O . 2 0 . 3 O . 4 0 , 5 0 . 6 O . ? 0 . 8 0 . 9 1 .O

G/E


Figs. 9-10. Correlation plots of the ratio diffuse to global hourly radiant exposure in the horizontal vs the ratio global to Extraterrestrial hourly radiant exposure in the horizontal for

Higher (Australia) throughout 1966.

11-17, the common point of D/G = 0.15 when GIT = 1.0 was selected.

Over the range of party clouded conditions, there is such a scattering of points that a straight line between points (0.94, 0.40) and (0.15, 1.00) appears to be a resonable proposition; however, a clear indication of

1.0

0.9

0.8

0 . 7

0 . 6

D/G 0 . 5

0 . 4

0 . 3

0 . 2

O. I

O

,.%a

oo qj O

8 =J

F IIC.J~E 10

~o

O.1 0 . 2 O . 3 0 . 4 0 . 5 0 . 6 O . 7 0 . ~ 0 . 9 1 .O


curvature is revealed upon closer inspection. An intervening point on such a curve was obtained by averaging all data points between 0.62< GIT <0.68 to yield the point D/G = 0.66 at GIT = 0.65, whence a simple three point curve fit gave the relationship

For 0.4 < G! T <~ 1.0,

DIG = [1.29- 1.19(GlT)]/[1.OO-O.334(GlT)]. (2.4)

1.1

1.0

0,9

0,8

0.7

O,b

D/G

0,5

0,4

0.3

0,2

0,1

mu

o o o a

o

o

o o

o o ° o

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0 D

u o

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oo o o o

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o

L k h i i i i i I i 0.1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

GIT

F i g . I1 . F o r n o m i n a l 10 . s o l a r a l t i t u d e .

F i g s . 1 1 - 1 7 . C o r r e l a t i o n p l o t s o f D I G v s GIT f o r H i g h e t t ( A u s t r a l i a ) t h r o u g h o u t 1966.

l

1,1


1.l

1.0

0 ,9

0,8

0 , 7

0.6

D/6

0,5

0,4

0.3

0.2

0,1

oZ o ~o o o o

o °o o

o

° ° o~= o ° o°~ °°= % o

o

= o

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=%

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G/T


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o ~,L o~ ~ ~ o

01,8 0[9 i/0 i.'i

1.0

0 ,9

1,1

0 ,8

0 .7

0.6

DIG

0.5

0.4

0.3

0.2

0.1

o o o o o o

o o

0 o ~D o

%

~o o Q o o

5

o

o

o

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o ° O

° o

o o

I r I I

~.i 0.2 0',3 014 0,5' OJ,6 0.7 0,8 OJ.9 ii0 i',I G/T


484 J .W. BUGLER

1.1

1.0

0.9

0.8

0.7

0.6

D/G

0,5

0.4

0,3

0,2

0.1

O ~

° 2

ra Qo u o o

% o

H [ ~ 7 T I',366

N~}'vIP,QL4O OESREESSSI rqR ~FITLOE

O

D

u

o

o

o° o o

a

i , p

0,1 012 0.3 0.4 01s o16 0,7 o18 0'.9 GIT

Fig. 14. For nominal 40 o solar altitude.

Go

%

u o o o

5

o a

~ ou

~o~

if0 i l l

1,1

1.0

0.9

0,8

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0,5

0.4

0,3

0.2

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0 D 0 0

ra

Duc

HI(~ETT 195S

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0,1 0.'2 0,3 014 OJ. 5 0,6 0,7 018 0',9 1',0 2,1 GIT



1,1

1.0

0,9

0.8

0,7

0,6

U/G

0,5

0.4

0,3

0,2

0.1

m

c

HI E~'ETT 19r-:~

N~ill','cL60 []EGF~Tt~50LF~ g L T I ~

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O D

o ~-:=

018 0~,9 1'.0 111

1.1

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0,8

0,7

0.6

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0.3

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0,1 0.2 0.3 0,4 0.5 0.6 0.7 GtT


I

0,8 I

0,9 I

1,0

I

1,1

486 J. W. BUGLER

Included in this analysis, to complete the full range of G[T values encountered, was the relationship

For G / T > 1.0, D--- 0.15G. (2.5)

Using hourly average values, and bearing in mind that when the Sun is not shining the direct irradiance makes no contribution to global insolation, eqn (3.3) may be written

It is accepted that eqn (2.5) is somewhat arbitrarily chosen; however, it is a compromise of a number of factors in the light of practical application, the most important being the desire for independence of solar altitude.

Figures 18 and 19 show the plots of 1%6 Highett data for high and low solar altitudes respectively, with the diffuse insolation model of eqns (2.3)-(2.5) superposed.

Although eqns (2.3)--(2.5) have been established from data for Highett in 1%6, it is suggested that their validity is far more general. Included in these relationships is the normalising quantity T, the computed clear sky total horizintal radiation for a given place at a given time, which includes the effect of local atmospheric turbidity and water vapour content. Moreover, the year-round weather at Highett (Melbourne) is renowned for its wide variety of cloud type and amount. So, to the extent that the clear sky model for solar irradiance is generally accepted, and to the extent that Melbourne's cloud states cover worldwide conditions, these three equations should have commensurate validity elsewhere.

3. THE COMPUTATIONAL TEOINIQUE

In the previous section, relationships have been empirically established whereby the diffuse exposure, D~, over one hour may be deduced from the global exposure, GH, over that hour through the intermediary of a computation of the clear sky global horizontal exposure, T, over that hour for the same time and place. Since the diffuse radiation model is one of uniform hemispherical radiation and continues uninterrupted for the full hour of integration, if an average value of diffuse irradiance at mid-hour is justified the eqns (2.3)-(2.5) are effectively a diffuse irradiance model--albeit, a discontinuous one.

Now, in general, the total radiant exposure on an inclined surface during a short period of time to is the result of four contributing components, the direct, the uniform diffuse, the ground reflected and the circumsolar diffuse radiations; i.e.

~o t° • Go = ( Io+Do+Jo+d;o)d t . (3.1)

As previously inferred, the circumsolar irradiance will here be considered proportional to the direct irradiance, in the form

] + 6' = ki. (3.2)

Using standard relationships, then

f0 t° • Go = [kin cos i +/)u(1 + cos 0)12

+ Gup(1 - cos 0)/2] dt. (3.3)

Go = kiNt cos i +/)Hto(1 + cos 0)/2+ G.top(1 - cos 0)12

or

Go = kiNt cos i + DH(1 + cos 0)/2+ Gup(1 - cos 0)/2 (3.4)

where t is the period of sunshine within the period to. For the horizontal surface, in particular

Gn = kiNt sin a + DH. (3.5)

With a measured value of Gm eqn (3.5) may be used to determine t which, when substituted in eqn (3.4), yields a value of Go. Supporting this computation are values of IN modelled according to Section 2.1, values of DH modelled according to eqns (2.3)-(2.5), and the appropriate tri- gonometrical relationships.

4. VERIFICATION OF THE MODEL

Verification of this model is seriously hampered by a lack of reliable diffuse radiation measurements and of measurements of insolation on an inclined surface for hourly or half-hourly intervals over a long period of time. The data recorded at the CSIRO Division of Mechanical Engineering, Highett, Victoria, between 1%6 and 1970 was available, and has been used, for this purpose.

The two methods of verification used may be referred to as a fundamental method and an applied method; in the former, direct comparison between computed and measured data has been made and statistically analysed, whilst in the latter a comparison is made of the heat outputs of a solar collector using the computed data on the one hand and the measured data on the other. Of course, the two methods are closely related, but both are presented to satisfy readers with different interests.

4.1 Fundamental veriIication A computer program has been developed in which the

solar radiation model of Section 2, together with the technique of Section 3, enables hourly values of radiant exposure on an inclined plane at any angle to the horizontal and at any azimuth angle to be computed from the corresponding hourly values of global horizontal exposure. Applying this program to the hourly global horizontal data for Highett to compute the hourly total radiant exposure (without ground-reflection) on a nor- therly facing surface at 38 o to the horizontal, and comparing these values with the measured values (which exclude ground reflection) for such a surface, errors of prediction in MJ m 2 were determined. This routine was carried out for each hour the Sun was above the horizon every day for the years 1%6-70 individually and


i , i

1,0

0,9

0.8

0.7

0.6

I~/G

0.5

0,4

0,3

0,2

0.1

1,1

© c

- : = =~ = o = = = = m e o G : =

_ = = = t, = =~=~e - =

c ~ - a - = o = =

u = ~ = o: = =gz . =

HIG;-fTT 1956 = == 0

z

011 012 013 014 015 0,6 O.l U.8 0.9 1.0 G/T

Fig. 18. For nominal solar altitudes 10, 20 and 30 °.

1,1

1,0

0.9

0.8

0,7

0,6

b/G 0.5

0,4

0,3

0,2

O,1

o o

o o D o o a

n ~ . = ~ o n o o

o D O ~ D =n O~ a

n m ~ n

D n i l

o ~ ~ n a o n o

HIt3-ETF 1966 o °

50"R~70 []EO:~EESSOL~ FLTITUDE ~° °'J ~ ~ °

OI 2 I t I I b I ~ ~ I J

0 011 0 3 0,4 0.5 0.6 0.7 0.8 0.9 1,0 1,1 G/T

Fig. 19. For nominal solar altitudes 50, 60 and 70 ° .

Figs, 18-19. The superposition of eqns (2.3)-(2.5) onto the correlation plots of D/G vs G/T for Highett (Australia) throughout 1%6.

488 J.W. BUGLER

Table 2. Prediction of hourly radiant exposure on an inclined surface at 380 to horizontal facing north at Highett, Victoria (Lat. 38 ° S). Error frequency analysis (Go oo~p,,o~-

G~ ....... d)

Hourly radiant exposure Cumulative differences in MJ m 2 1966 1967 1968 1969 1970 1%6-70

LT -1.00 1 0 18t 12t 2 33t -1.00 to -0.90 0 0 5 3 0 8 -0.90 to -0.80 1 0 5 3 0 9 -0.80 to -0.70 2 0 3 1 0 6 -0.70 to -0.60 2 0 7 12 0 21 -0.60 to -0.50 5 0 8 20 1 34 -0.50 to -0.40 10 8 21 90 12 141 -0.40 to -0.30 24 25 64 136 58 307 -0.30 to -0.20 98 146 180 246 216 886 -0.20 to -0.10 326 406 442 375 405 1954 -0.10 to 0.00 1390 1510 1820 1835 1895 8450

0.00 to 0.10 2120 2041 1740 1430 1719 9050 0.10 to 0.20 384 322 234 386 452 1778 0.20 to 0.30 105 93 76 196 175 618 0.30 to 0.40 42 24 15 99 65 245 0.40 to 0.50 11 12 9 42 16 90 0.50 to 0.60 12 11 11 37 17 88 0.60 to 0.70 8 11 10 17 8 54 0.70 to 0.80 6 5 1 13 9 34 0.80 to 0.90 3 2 4 6 2 17 0.90 to 1.00 2 0 0 3 1 6

GT 1.00 2 2 13t 15t 0 32~"

Mean error 0.015 0.005 -0.020 -0.005 0.008 0.001 Standard deviation 0.125 0.114 0.174 0.211 0.131 0.156

~These values are largely attributable to known anomalous measurements.

cumulatively, and the prediction errors analysed statistically. The results of these analyses are given in Table 2. An encouraging feature of Table 2 is that whereas the diffuse radiation model originated from the 1966 Highett data, the analyses for the subsequent years show similar, and sometimes better, results.

The prediction errors in these analyses have been expressed in absolute terms of the energy per unit area, and not as percentages. This is considered more appropriate because users of this method will almost invariably be concerned with the energy obtainable from insolation; and, in that context, a percentage error analysis would be meaningless because a 50 per cent error in an hour's exposure of, say, 0.1 MJ m -2 is of less significance than a 10 per cent error in a typically good hour's exposure of 3.5 MJ m -2.

4.2 Verification by heat output of a flat plate collector A major application of the method outlined in this

paper will be in the design of fiat plate solar energy collector systems; it is appropriate, then, to use a calculation of the heat output of a typical flat plate collector as a gauge of the usefulness of this method for insolation computation on inclined plates. Proctor [12] has described a method for calculating the daily quantities of heat collected by a flat plate tilted at an appropriate angle, in which the required input data are hourly values of ambient temperature, wind speed and radiant exposure on the plane of the collector. Using this method with hourly

data for Highett (Melbourne) over the years 1%6-70, a comparison was made of the heat output of a given solar collector mounted at 380 to the horizontal facing north, using firstly CSIRO Division of Mechanical Engineering data for insolation (without ground reflection) on the plane of the collector, and secondly, using hourly insolation values on the plane of the collector computed by the technique of this study from the measured insolation values on the horizontal at the same location. Common to both calculations was the Meteorology Bureau data for ambient temperature and wind speed. The tabulated results of these calculations for water outlet temperatures of 30, 50 and 70°C are given in Table 3.

It is readily apparent from a study of Table 3 that:

(a) The yearly average values of daily heat output compare by the two methods well within any practically acceptable degree of accuracy, namely +-1 per cent;

(b) The mean monthly values of daily heat production compare within -+ 10 per cent, the percentage error increasing as the outlet water temperature from the collector increases--as would be expected; the collector in question was designed to operate in the region of 35-55°C.

(c) There is a tendency for the mean monthly values of daily heat production to be overpredicted by the technique of this paper in summer and underpredicted in winter. This phenomenon might be attributed to one of the atmospheric variables in the radiation model which is

The determination of hourly insolation on an inclined plane

Table 3. Comparison of the daily heat output of a typical fiat plate solar collector

489

Tout °C Jail. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Average

Mean monthly values of heat output in MJ m -2. Using Highett radiation data measured on 38 ° surface over the years 1966-1970

30 13.36 12.81 11.05 9.04 5.60 5.02 5.21 6.70 8.60 10.03 11.34 11.82 9.21 50 11.16 10.68 9.16 7.29 4.21 3.70 3.77 5.13 6.73 8.04 9.17 9.64 7.39 70 8.85 8.46 7.24 5.60 2.92 2.55 2.47 3.65 4.89 6.05 6.96 7.46 5.59

Mean monthly values of heat output in MJ m -2. Using radiation on 380 surface computed by the method of this paper from Highett horizontal measured data over the 5 yr 1966--1970

30 13.84 12.94 10.59 8.52 5.57 5.13 5.13 6.56 8.20 10.04 11.74 12.36 9.22 50 11.66 10.83 8.71 6.76 4.11 3.68 3.62 4.94 6.31 8.04 9.56 10.18 7.37 70 9.34 8.61 6.82 5.08 2.77 2.42 2.27 3.41 4.46 6.02 7.32 7.98 5.54

Percentage of overprediction of daily heat output using the computational technique of this paper

30 +3.6 + 1.0 -4.2 -5.8 -0.5 +2.2 - 1.5 -2.1 -4.7 +0.1 +3.5 +4.6 + 0.1 50 +4.5 +1.4 -4.9 -7.3 -2.4 +0.5 -4.0 -3.7 -6.2 0.0 +4.3 +5.6 -0.3 70 +5.5 + 1.8 -5.8 -9.3 -5.1 -5.1 -8.1 -6.6 -8.8 -0.5 +5.2 -7.0 -0.9

Values are given for a standard single glazed, selective surface, flat plate collector with nominal dimensions 1220 mm × 610 mm inclined at 38 ° to the horizontal in Melbourne, facing north, neglecting ground reflected radiation. Water outlet temperature Tou~ held

constant by varying the mass flow rate.

seasonably variable, such as atmospheric precipitable water and solar altitude, or even to the radiation measurements themselves. However, the effect of solar altitude on the diffuse radiation model has already been considered (see Figs. 18 and 19) and, within the limits of the evidence available, no clearly identifiable relationship revealed; and the effect of precipitable water values on the total radiation model is unlikely to be seasonally significant, unless the contribution of this factor in the Rao and Seshadri clear sky radiation model itself is considerably modified.

5. CONCLUDING REMARKS

A method has been presented whereby values of solar irradiance integrated over short time intervals (typically 1 hr) may be computed for an inclined plane, of any azimuth angle, from measured values of total solar irradiance on the horizontal plane integrated over the same time interval for the selection location.

Using a developed insolation model, it has been shown that values of hourly radiant exposure on north facing planes inclined at latitude angle at Highett computed from the corresponding measured horizontal data over the years 1966-70 are given to a practically satisfactory degree of precision, the quantitative evidence for which is as follows:

(a) A statistical analysis of the differences between the computed values and the measured values over the 5 years yields an approximately normal distribution about zero with a standard deviation of 0.16 MJ m 2;

(b) Calculation of the yearly average daily heat output of a typical flat plate solar collector mounted at latitude angle agreed to within 1 per cent using the computed data on the one hand and the measured data on the other for

colector outlet temperatures up to 70°C: the monthly averages agreed to within 10 per cent.

In Australia, an established network of some 20 Meteorological Bureau stations are currently recording half-hourly global horizontal radiant exposure con- tinuously, and have been doing so for up to 8 yr in some locations. In other countries similar networks of solar radiation recording stations exist, predominantly recording global horizontal insolation. The results of this study open up the possibility of deriving from these measurements a prolonged series of half-hourly or hourly radiant exposures upon any inclined surface wherever such measurements have been made. In particular, it is possible to compute the hourly radiation data needed for the design of fixed fiat plate collectors at any orientation, movable flat plate collectors and concentrating collectors-a l l depending, of course, on the availability of the necessary short period horizontal insolation data. There remains a requirement for atmospheric dust content and precipitable water data for the locations in question, but the sensitivity of the computations to these data would appear to be quite low.

Because of the lack of suitable solar radiation measurements, this method has been checked only against Melbourne data and should be applied with caution elsewhere, but is believed to have general application.

The method of insolation computation outlined in this paper would appear to satisfy a pressing need in the solar energy applications field, but its validation at other locations and at various azimuth angles is certainly needed. It is, however, pertinent to point out that this technique has been established intentionally upon the simplest possible practical basis (i.e. the commonly measured global horizontal insolation) and mindful of the accuracies of the current radiation measurements upon which any such computation must be based.

490 J .W. BUGLER

Acknowledgements--The use of hourly insolation data ac- cumulated over a number of years with great care at the CSIRO Division of Mechanical Engineering has been essential to this work, and is hereby gratefully acknowledged. This work was

,carried out during a period of study leave spent by the author with the Solar Energy Studies group of CSIRO, and the assistance and encouragement of the Director of this group, Mr. R. N. Morse, is gratefully acknowledged.

NOMENCLATURE

circumsolar irradiance, Wm -2 D diffuse radiant exposure, Jm -2 /) diffuse irradiance, Wm -2 d declination angle, north positive, south negative E extraterrestrial radiant exposure on horizontal surface,

jm -2 /~ extraterrestiral irradiance on horizontal surface, Wm -2 e equation of time, rain

G total radiant exposure (Global), Jm -2 total irradiance (Global), Wm 2

h solar hour angle, zero at noon, negative a.m., positive p.m.

I direct radiant exposure (Beam), Jm -2 j direct irradiance (Beam), Wm ~ i angle of incidence of solar vector to plane normal J albedo radiant exposure, Jm z j albedo irradiance, Win-: k =l+Cli I latitude angle, north positive, south negative

n day number (1 January = 1, 31 December = 365) T total radiant exposure on horizontal plane for clear sky

conditions, Jm -~ t time

to time interval of irradiance integration w precipitable water content of atmosphere, mm a solar altitude above horizontal y azimuth angle of inclined plane, i.e. deviation of normal to

plane from the local meridian, east positive, west negative

p albedo factor of surface surroundings 0 inclination of plane above horizontal

Suffices H on horizontal plane N on plane normal to Sun-Earth vector 0 on plane at angle 0 to horizontal

~ C E S

1. J. L. Threlkeld and R. C. Jordan, Direct solar radiation available on clear days. Heat. Pip. Air Condit. 29(12), 135-45 (1957).

2. K. R. Rao and T. N. Seshadri, Solar insolation curves, Indian J. Met. Geophys. 12(2), 267-72 (1961).

3. A. G. Loudon and P. Pentherbridge, Solar radiation on inclined surfaces. Nature, Lond. 206, 603-4 0965).

4. J. W. Spencer, Melbourne solar tables. CSIRO, Division of Building Research, Tech. Paper (2nd Series) No. 7 0974).

5. B. Y. H. Liu and R. C. Jordan, The Interrelationship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4(3), 1-19 (1960).

6. H. Heywood, A general equation for calculating total radiation on inclined surfaces. Int. Solar Energy Soc. Conf. Melb. Paper 3/21. 0970).

7. J. W. Spencer, Computer estimation of direct solar radiation on clear days. Solar Energy 13(4), 437-438. (1972).

8. C. L. Pierrehumbert, Precipitable water statistics, Australia. Comm. Bureau of Met., Aust., Met. Summary (1972); and private communication. I.H.V.E. Guide; Book A--Design Data. Institution of Heating and Ventilating Engineers, London (1970). D. Grether, J. Nelson and M. Wahlig, Measurement of Circumsolar Radiation. Paper No. 20[4 presented at I.S.E.S. Conference, Los Angeles 0975). D. J. Norris, Diffuse sky radiation. Int. Solar Energy Soc. Conf. Paper 3/37. Melbourne (1970). D. Proctor, Climatic Design Data for Solar Collector Performance. Paper presented at meeting of A.N.Z. Section of I.S.E.S., Melbourne 0975).

9.

10.

11.

12.

Resumen--Usando s61o mediciones de valores horarios de la radiaci6n global en una superficie horizontal, ha sido desarroUado un mttodo para computar los valores correspondientes horarios de insolaci6n en una superficie inclinada en cualquier fingulo y orientada en cualquier direcci6n. E1 mttodo usa un modelo en el cual la componente difusa es calculada desde la global horizontal usando tres relaciones diferentes. La ecuaci6n apropiada es seleccionada acorde con el valor de la relaci6n de la insolaci6n global horaria medida con la computada para las condiciones de cielo claro. E1 mttodo ha sido comprobado usando valores horarios medidos en Melbourne por cinco afios sobre superficies horizontales y a 38* respecto a la horizontal mirando al uorte. Las diferencias entre los valores horarios computados y los medidos se han encontrado aproximadamente normalmente distribufdas alrededor de cero con una desviaci6n standard de 0.16 MJ m -2. Este mEtodo es particularmente 6til para predecir la salida tErmica en colectores planos inclinados donde s61o la insolaci6n medida en superficies horizontales es obtenible, lo que es comfin. Se ha hallado una buena concordancia entre la salida prevista de un colector tipico que usa insolaci6n medida a 380 y los valores horarios computados usando este mEtodo. Dado que el mttodo ha sido contrastado s61o con los datos de Melbourne, debe ser aplicado con precauci6n en otros lugares, pero se estima que tiene aplicaci6n general.

R~sumt--En utilisant uniquement les valeurs horaires du rayonnement global mesurtes sur une surface horizontale, on a dtvelopp6 une mtthode de calcul des valeurs horaires correspondantes du rayonnement sur une surface inclinte d'un angle quelconque et orientte dans n'importe quelle direction. La mtthode utilise un modtle de rayonnement solaire pour lequel la composante diffuse est calculte g partir du rayonnement global horizontal raide de 3 relations difftrentes. L'tquation voulue est choisie en tenant compte de la valeur du rapport du rayonnement global horaire mesur6 au rayonnement global horaire calcul6 pour un ciel clair. La mtthode a 6t6 controlEe en utilisant les valeurs horaires, mesurEes ~ Melbourne pendant une pEriode de cinq ans, du rayonnement h la fois sur une surface horizontale et sur un plan incline de 38 ° par rapport ~ I'horizontale et orientE au Nord. On a trouvE clue les difftrences entre les valeurs horaires calculEes et les valeurs horaires mesurEes Etaient apprnximativement distribuEes normalement autour de zero avec un 6cart type de O.16MJm -~. Cette mEthode est particulitrement utile pour prtdire la quantit6 de chaleur extraite de collecteurs solaires plans inclints lorsqu'on dispose seulement de la mesure du rayonnement global horizontal, ce qui est souvent le cas. On a trouv6


un bon accord entre la pr6vision de la chaleur extraite d'un collecteur type en mesurant rinsolation h 380 et les valeurs horaires calcul6es en se servant de cette m~thode. Puisque la m6thode a ~t6 control6e seulement pour des donn6es a Melbourne, elle devrait ~tre appliqu6e ailleurs avec prudence, mais on pense qu'elle a une application g6n6rale.

Documents

The determination of hourly insolation on an inclined plane using a diffuse irradiance model based on hourly measured global horizontal insolation