22
A Rational Procedure for Predicting The Long-Term Average Performance of Flat-Plate Solar-Energy Collectors With Design Data for the U. S., Its Outlying Possessions and Canada* Benjamin Y. H. Liu and Richard C. Jordan Assistant Professor Professor and Head Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota T HE flat-plate collector is the simplest and one of the most effective means of collecting solar energy for use in systems that require thermal energy at com- paratively low temperatures. Flat-plate collectors have been used successfully for many years in the southern United States and elsewhere as water heaters, and potentially can be used for space heating, drying, refrigeration, power generation and similar purposes. The majority of these potential applications of solar energy have not been fully evaluated, but experi- ments ~-7 in solar space heating conducted during the past two decades have shown that space heating with solar energy is entirely feasible. It is now recognized that with the rapid depletion of fossil fuels, solar energy will be increasingly utilized. In comparison with collectors of the concentrating type, such as those used in high-temperature solar- furnace applications, flat-plate collectors offer these advantages: (1) no complicated mechanisms for fol- lowing the apparent diural motion of the sun are needed for their operation, (2) construction is simple and cost relatively low, and (3) diffuse as well as direct solar radiation is utilized. This last advantage is especially important in view of the fact that, of the total solar radiation received on the surface of the earth (46 percent of the extraterrestrial radiationS), approxi- mately 40 percent (18 percent of the extraterrestrial radiation 9) is diffuse radiation. Manuscript received August 1, 1962. Basic to the design of any solar-energy utilization system in which fiat-plate collectors are used is the long-term average performance of these collectors. The long-term average performance, instead of the instantaneous rate of energy collection, is needed since the latter is extremely variable due to differences in cloudiness; Since sufficient heat storage is usually provided, the average energy collection is also the useful energy collection. Methods for predicting the long-term average col- lector performance have been described by Hottel, Woertz, and Whillier. 1°, H, ,~ To apply these methods, it is necessary to have a detailed record of the radiation and temperature data of the locality of the collector. The fact that a large volume of meteorological radia- tion and temperature data must be analyzed has made the prediction of collector performance an extremely tedious and time-consunfing task. It is therefore de- sirable to study the available radiation and temperature data and to correlate and present these in such a form that further detailed analysis of these data becomes unnecessary in the prediction of collector performance. This objective is here achieved through the develop- ment of a set of "generalized utilizability curves" by means of which the performance of a collector of any angle of tilt at any locality can be predicted when the following two parameters are known: (1) the monthly-average daily total radiation on a horizontal * This paper is in part the result of researches sponsored by a grant from the National Science Foundation, Washington, D.C. Vol. 7, No. 2, 1963 53

Liu and Jordan 1963

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Page 1: Liu and Jordan 1963

A Rational Procedure for Predicting

The Long-Term Average Performance of Flat-Plate Solar-Energy Collectors

With Design Data for the U. S., Its Outlying Possessions and Canada*

Benjamin Y. H. Liu and R icha rd C. Jordan Assistant Professor Professor and Head

Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota

T HE flat-plate collector is the simplest and one of the most effective means of collecting solar energy

for use in systems that require thermal energy at com- paratively low temperatures. Flat-plate collectors have been used successfully for many years in the southern United States and elsewhere as water heaters, and potentially can be used for space heating, drying, refrigeration, power generation and similar purposes. The majority of these potential applications of solar energy have not been fully evaluated, but experi- ments ~-7 in solar space heating conducted during the past two decades have shown that space heating with solar energy is entirely feasible. I t is now recognized that with the rapid depletion of fossil fuels, solar energy will be increasingly utilized.

In comparison with collectors of the concentrating type, such as those used in high-temperature solar- furnace applications, flat-plate collectors offer these advantages: (1) no complicated mechanisms for fol- lowing the apparent diural motion of the sun are needed for their operation, (2) construction is simple and cost relatively low, and (3) diffuse as well as direct solar radiation is utilized. This last advantage is especially important in view of the fact that , of the total solar radiation received on the surface of the earth (46 percent of the extraterrestrial radiationS), approxi- mately 40 percent (18 percent of the extraterrestrial radiation 9) is diffuse radiation.

Manuscr ip t received August 1, 1962.

Basic to the design of any solar-energy utilization system in which fiat-plate collectors are used is the long-term average performance of these collectors. The long-term average performance, instead of the instantaneous rate of energy collection, is needed since the latter is extremely variable due to differences in cloudiness; Since sufficient heat storage is usually provided, the average energy collection is also the useful energy collection.

Methods for predicting the long-term average col- lector performance have been described by Hottel, Woertz, and Whillier. 1°, H, ,~ To apply these methods, it is necessary to have a detailed record of the radiation and temperature data of the locality of the collector. The fact that a large volume of meteorological radia- tion and temperature data must be analyzed has made the prediction of collector performance an extremely tedious and time-consunfing task. I t is therefore de- sirable to study the available radiation and temperature data and to correlate and present these in such a form that further detailed analysis of these data becomes unnecessary in the prediction of collector performance. This objective is here achieved through the develop- ment of a set of "generalized utilizability curves" by means of which the performance of a collector of any angle of tilt at any locality can be predicted when the following two parameters are known: (1) the monthly-average daily total radiation on a horizontal

* This paper is in pa r t the resul t of researches sponsored by a g ran t from the Nat iona l Science Founda t ion , Washington, D.C.

Vol. 7, No. 2, 1963 53

Page 2: Liu and Jordan 1963

surface and (2) the monthly-average day-time ambient temperature. A table giving these parameters for each of the twelve months at 80 localities is presented in the Appendix. The table is for localities in the U. S., its outlying possessions and Canada, and is compiled from data published by the weather bureaus of the two countries.

2.8

the entire blackened plate for the removal of the absorbed solar energy. I t will be assumed that : (1) no concentrators are used in conjunction with the collector, and (2) the collector is tilted toward the equator at a fixed angle from the horizontal surface during any month, but the angle of tilt can be different during different months.

2.4

to

< ~ : . ZO

<1[~ ,e¢"

,2

¢ Y

OA

0 I 00 2 0 0 3 0 0 4 0 0 500 600 7 0 0 800

DAILY TOTAL RADIATION ON A HORIZONTAL SURFACE, B T U / D A Y - F T z

FIG. 1- -Var ia t ion of r~diat ion conversion factors wi th solar in tens i ty .

9 0 0 I000

Solar collectors are generally tilted toward the equator to improve their performance during winter. However, solar radiation data are commonly available for horizontal surfaces only. Hence a radiation-con- version factor must be considered in predicting col- lector performance. This radiation-conversion factor is a function of, among other variables, the intensity of radiation as shown in Fig. 1 which is derived from experimental data of Blue Hill, Massachusetts. This fact has been neglected in the previous investiga- tions,10. ,i. 1~ but is taken into consideration in de- veloping the "generalized utilizability curves".

T h e Bas i c P e r f o r m a n c e E q u a t i o n

Various designs of fiat-plate collectors are shown in Fig. 2. Principally, they are composed of a blackened plate for absorbing solar energy, one or more trans- parent cover plates, insulation for reducing the heat loss through the back, supporting members and pro- vision for circulating liquid through tubes in good thermal contact with the blackened plate or air over

Under steady-state conditions, the rate of useful energy collection is the difference between the rate at which solar energy is absorbed by the blackened plate and the rate at which energy is lost due to the difference in temperature of the blackened plate and the ambient air. I t can be shown '3 that the useful energy collection rate per square foot of collector area is

q = F R [ I T t (7"~a) - - U ( t l - - to)] (1)

(A) (B) (C) ~.Ass ~ e~cK GAUZE7

BLACK

(o) (E) (F) BLACK PLATE 7 BLACK PLATE 7 BLACK PLATE 7

WATER L// WATER ~ W A T E R ~ /

FIG. 2--Solar collector design.

54 Solar Energy

Page 3: Liu and Jordan 1963

The heat removal efficiency, F R , is independent of the radiation intensity and the temperature to the extent that these factors do not affect the heat-transfer coefficients. When water is used for the removal of the absorbed solar energy, FR is of the order of 0.9. And when air is used, it is more nearly 0.8.

The collection of useful energy at a temperature higher than the ambient temperature is possible only during those hours when the intensity of radiation incident upon the collector surface is higher than the critical intensity,

I~ = U ( t ~ - t o ) / ( - ~ ) (2)

which is also the intensity of radiation at which the useful energy collection rate is zero. In terms of the critical intensity,

q = FR (r~)(Irt -- I~) (3)

Hence to prevent the fluid for energy-transportation from losing useful energy to the ambient air, the pump or blower for circulating the fluid through a collector should be so controlled that it is stopped when the in- tensity of radiation falls below the critical value.

The fact that only hourly radiation data, and not true instantaneous intensities of radiation, are avail- able for predicting collector performance requires that Eq. 3 be given a slightly different interpretation. Two interpretations are possible. If the hourly radiation data are considered as the hourly summation of the radiation, then q must also be considered as the hourly summation of useful energy collection. If the hourly radiation is interpreted as the hourly average radiation intensity, q should also be interpreted as the hourly- average rate of useful energy collection.

The prediction of the long-term average performance of a collector requires, in principle, the hour-to-hour evaluation of Eq. 3 by means of the hourly radiation and temperature data of a number of years sufficient to obtain a good statistical average. Since the process is extremely tedious, a simpler procedure, first intro- duced by Hottel and Whillier, 1~ will be used. According to this procedure one mouth is taken as a period during which the ambient air temperature and the position of the sun in the sky during a given hour of the day do not vary excessively so that both ( ~ ) and I~ can be assumed to remain constant during the same hour throughout the month. The long-term average hourly energy collection, ~, for a particular hour of day in a given month can then be obtained by means of the following summation,

= F, (r--~) (1/n))--] (ITt -- I~) + (4)

where n is the total number of hours of data used. The plus sign outside the parenthesis indicates that only positive values should be used in the summation. Equation 4 can also be written more compactly as follows,

= FR (7-~)~?T~O (5)

where i r t is the long-term average hourly radiation incident upon the collector surface and the symbol, ~, stands for the quanti ty,

= ( U n ) ~ [ ( X ~ J l r ~ ) - (L / i r~)] + (6)

Hottel and Whillier suggested that ~ be named "utiliza- bility", which they defined in terms of radiation on a horizontal surface. But the concept of utilizability is quite general and is applicable to a collector surface of any orientation. The more general definition of the utilizability given by Eq. 6 is used in the present study.

The meaning of the utilizability, which can also be considered as the dimensionless efficiency factor, ~ / i T t F R (r-~), is clear from Eq. 5. I t is the fraction of the incident radiation that can be collected or "utilized" by an idealized collector. The collector is idealized in the sense that it has a perfect heat-removal circuit of 100 percent efficiency (FR = 1), a perfectly black absorbing plate and a set of cover plates that are per- fectly transparent ((va) = 1). The fact that even such an idealized collector cannot "utilize" all of the radia- tion incident upon its surface is not surprising, however, since the fraction, (1 - ¢), that is not utilized either arrives at the collector surface with insufficient in- tensity for collection or is lost to the ambient due to the inability of the cover plates and back insulation to prevent such a heat loss. If the heat loss could be pre- vented, the utilizability would ahvays be equal to one.

Equation 5 shows that to calculate the long-term average performance of a collector it is necessary to determine four factors of which two, e.g. F , and (ra), are primarily physical characteristics of the collector and two, e.g. _Trt and ~, are characteristics of the solar weather. Since Fa is independent of the incidence angle of the sun, only the remaining three factors need be determined for each hour so that the long-term average daily useful energy collection, Q, can be determined. Methods for evaluating FR and (~-~a) have been ade- quately treated elsewhere. 1°-~4 The determination of the average hourly radiation and the utilizability for collectors at various angles of tilt from solar incidence data on a horizontal surface will be considered in detail in the following sections.

L o n g - T e r m Average H o u r l y T o t a l R a d i a t i o n o n a T i l t ed S u r f a c e

1 - - T h e o r e t i c a l C o n s i d e r a t i o n - - T h e determination of the long-term average hourly total radiation incident upon a tilted surface, i r t , requires a knowledge of the corresponding direct and diffuse (or total and diffuse) radiation on a horizontal surface and the reflectance of the ground for solar radiation. Because published radiation data generally consist of daily sums only, it is necessary to determine the hourly radiation from the daily radiation.

Vol. 7, No. 2, 1963 55

Page 4: Liu and Jordan 1963

N O M E N C L A T U R E

Ds f =

f~=

FR= H and H =

1 t o =

]c Idh

Ion

ITh

ITt

iTt

K r =

K T =

L = q =

~=

O=

rd =

rv =

R =

RD =

to =

U =

Oh

p

(~) =

D a n d / ) =

o~ s /

o j s

I

daily and monthly average daily diffuse radiat ion on a horizontal surface, B t u / day-ft ~ percent possible sunshine, dimensionless fractional time that radiation is less than or equal to a certain value value of f at the critical intensi ty of radia- tion heat removal efficiency, dimensionless daily and monthly average daily total radia- tion on a horizontal surface, B tu /day- f t 2 extraterrestr ial daily radiat ion on a hori- zontal surface, B tn /day - f t 2 critical intensi ty of radiation, B tu /h r - f t ~ long-term average hourly diffuse radiation on a horizontal surface, B tu /h r - f t ~ radiat ion at normal incidence outside the atmosphere of the earth, B tu /h r - f t 2 long-term average hourly total radiation on a horizontal surface, B tu /h r - f t 2 instantaneous intensi ty of total radiation on a t i l ted surface or the hourly total radia- tion on a t i l ted surface, B tu /h r - f t ~ long-term average hourly total radiation on a t i l ted surface, B tu /h r - f t ~ H / H o , dimensionless f I / H o , dimensionless lat i tude, degrees instantaneous rate of useful energy collec- tion, B tu /h r - f t 2 long-term average hourly useful energy collection, B tu /h r - f t 2 long-term average daily useful energy col- lection, B tu /day- f t 2 i.~/D

conversion factor for long-term average hourly total radiation, dimensionless eonversion factor for daily direct radiation, dimensionless temperature of fluid entering collector, deg F ambient temperature , deg F heat loss coefficient, Btu/hr-f t2-°F angle of til t of collector from horizontal, degrees tan-~(cos 0t/cos 0h), degrees solar declination, degrees incidence angle on horizontal surface, deg. incidenee angle on a t i l ted surface, degrees reflectance of ground for solar radiation, dimensionless overall t ransmiss ivi ty-absorpt ivi ty product = fraction of the total radiation arriving at the outer glass plate absorbed by the blackened plate of collector, dimensionless product of the effective glass t ransmissivi ty and the absorpt ivi ty of the blackened plate for direct solar radiation, dimensionless. utilizability, dimensionless sunset hour angle, radian sunset hour angle on the t i l ted collector surface, radian

I

I - -

I'q

" r

z ~ c

c

=.

" s s ,o ,, ,2 ,3 ,4 ,5 ,6 .OURS FROM SUNR,SE TO SUNSET

~o -/~ 9b ,65 tlo SUNSET HOUR ANGLE, DEGREES

FIG. 3--Relat ionship between daily radiat ion and hourly radiation on a horizontal surface.

The relationships between the hourly radiation and the daily radiation on a horizontal surface are shown in Fig. 3. 9 The relationships for total radiation are derived from data of 14 widely separated localities and represent the actual experimental points to accuracies better than =t=5 percent for all hours within =i=3 hours from solar noon. The relationships for diffuse radiation are computed from theoretical equations and are found to agree closely with the experimental data of Blue Hill, Massachusetts, and Helsingfors, Finland, for all hours of the day. The sunset hour angle, x~, the independent variable in Fig. 3, is given by the equation,

cos ~ = - t a n L tan ~ (7)

Since diffuse radiation data are extremely sparse, it is also necessary to estimate the diffuse radiation from the measured total radiation. Table 19 gives the ratio of the long-term monthly average daily diffuse radiation to the long-term monthly average daily total radiation on a horizontal surface as a function of the cloudiness index, /~T. /~r is the ratio of the long-term monthly average daily total radiation incident upon a horizontal surface to that incident upon a horizontal surface out- side the atmosphere of the earth and therefore repre- sents the fraction of the extraterrestrial radiation trans- mitted through the atmosphere. A large value of A'T indicates a clear atmosphere of low turbidi ty and cloudiness and a sm~ll value o f / ~ r indicates an atmos-

56 S o la r E n e r g y

Page 5: Liu and Jordan 1963

TABLE 1--The Ratio of the Long-Term Monthly Average I)ailv Diffuse Radiation on a Horizontal Surface, D, to the Long-Term Monthly Average ])ally Total Radiation, /4, on a Horizontal Surface

K T = -[l/Ho

D/H

O.3 [ 0.4

I 0.596 ! 0.457

i

0.5

O. 376

0.6

0. 290

0.7

0.213

0.75

0.167

phere of high turbidity and cloudiness. With only a few exceptions, the value o f / ( r is within the range from 0.3 to 0.75 (see Appendix).

The daily solar radiation incident upon a horizontal sm'face outside the atmosphere of the earth, Ho, is a function of the latitude and solar declination and is given by the equation,

Ho = (24/~r)Io,(eos L cos ~ sin o~ + ¢o~ sin L sin ~) (8)

The above relationships permit the hourly total and hourly diffuse radiation on a horizontal surface to be determined from the daily total radiation on a hori- zontal surface. To obtain the direct and diffuse (sky) radiation received by a tilted surface, the corresponding radiation on a horizontal surface should be multiplied respectively by the appropriate conversion factors. The conversion factor for direct radiation is cos 0t/cos 0h, where

cos oh = cos L cos ~ cos o~ + sin L sin ~ (9)

and for a surface tilted ~ degrees from the horizontal surface toward the equator,

cos ot = cos (L - /~) cos~tcos o~ q- sin (L -- fl) sin ~ (10)

The conversion factor for diffuse (sky) radiation is ½(1 q- cos/~) for a sky of uniform intensity. Furthermore, a tilted surface also receives solar radiation reflected from the ground. For a ground of infinite horizontal extent whose surface reflects solar radiation diffusely, the intensity of the ground-reflected solar radiation on a tilted surface is equal to a fraction, ½(1 - cos fl)p, of the intensity of the total radiation on the horizontal surface, where p is the reflectance of the ground for solar radiation. Adding these three components of solar radiation received by a tilted surface, we have,

it, = (cos 0gcos oh)(irh - i~h) + ½ (1 + cos ~)i~h (11)

-t- ½(1 -- cos ~)PiTh

Making use of the definitions of ra and r r , the conver- sion factor for the long-term average hourly total radiation is,

k = l r t / i r h = (COS Or/COS Oh)[1 -- ( r d / r T ) ( D / H ) ] (12)

q-½(1 q- COSl3)(rd/rT)(D/fI) q-- ½(1 -- COSfl)p

Finally, the hourly total radiation incident upon the tilted surface is given by

ir t = RrTH (13)

2 - - C o m p a r i s o n of R e s u l t s - - T o test the accuracy of the above method of computing the hourly total radia- tion on a tilted surface from the daily total radiation

TABLE 2--Fractional Time, c, When Snow is More Than One Inch Thick, and the Estimated Ground Reflectance, o, for Blue Hill, Massachusetts, September, 1952 to August, 1956

Month Jan. Feb. Mar.

O.40 i

I

Apr. May-Oct. [ Nov.

0.07 0 0.02 0.24 0.20 ] 0.21 I

Dec.

0.20 0.30

on a horizontal surface, the experimental radiation data for Blue Hill, Massachusetts, obtained during the period from September, 1952 to August, 1956 are used. Since the reflectance of the ground is needed in the computation and is unknown, it must be estimated independently. A value of 0.2 is assumed for all months during which the ground is free of snow. For the six- month period from November to April when snow covers the ground part of the time, the average re- flectance of the ground for each month is computed from the following equation,

p = 0.2 (1 - c) -k 0.7 c (14)

where c is the fractional time during the month when snow of more than one inch thickness is present. This is equivalent to assuming that the reflectance is 0.2 when the ground is covered with less than one inch of snow or no snow and is 0.7 when snow is more than one inch thick. The factor, c, can be determined from weather bureau records. I ts value, together with the computed value of p, for each month during the four year period is given in Table 2 for Blue Hill.

The hourly total radiatiml are computed by means of the above method for a south-facing vertical surface and an east or west-facing vertical surface at Blue Hill for the three hours nearest solar noon. The results, together with the measured radiation, are given in Tables 3 and 4. The measured radiation given in the tables are obtained by combining data for hours sym- metrical with respect to solar noon. Thus for the south- facing vertical surface, the data for the hours 11-12 A.M. and 12-1 P.M. are combined. Similarly, the data for the hour 11-12 A.M. oU the east-facing vertical surface are combined with the data for the hour 12-1 P.M. on the west-facing vertical surface. This is neces- sary because the ratios, r r and r~, were also obtained by combining data for symmetrical hour pairs and any asymmetry in the distribution of the hourly radiation cannot be accounted for by the use of these ratios.

Also given in the tables are the hourly total radiation on the vertical surfaces computed by means of Eq. 14 and an empirical equation for the conversion factor given by Brooks, n

R = tan[~(3-y D -k 17) q- ~D,] (15)

where

~/D = tan-1 (COS Ot/COS Oh)

D~ = percent possible sunshine, a value reported by the weather bureau. Since the empirical constants

Vol. 7, No. 2, 1968 57

Page 6: Liu and Jordan 1963

TABLE 3 - -Compar i son of Computed and Measured Hour ly Tota l Rad ia t ion on a Vertical Surface Facing South at Blue Hi l l - -Based Upon D a t a of September, 1952 to Augus t , 1956

Hourly Total Radiation, I T t , on a Vertical Surface Facing South, langley/hour*

Hour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-12 or 12-1 10-11 or 1-2 9-10 or 2-3

Month

J a n . Feb. Mar.

Apr. M a y June

Ju ly Aug. Sept.

Oct. Nov. Dec.

H, langley/ D s % day

134 43 206 48 293 49

367 44 465 52 529 60

510 60 433 59 356 63

238 54 157 47 126 49

Pres. method

36.9 41.4 41.6

33.8 31.2 30.2

31.0 34.0 39.8

38.2 36.2 38.6

Brooks method

37.1 42.1 40.6

32.5 30.0 29.6

31.0 35.1 43.7

42.3 39.1 39.8

Sum 433 436

Rat io , C o m p u t e d / M e a s u r e d 1.00 1.01

Measured

40.6 43.4 38.0

31.1 28.0 27.6

28.7 30.9 40.3

41.4 40.8 40.9

432

Pres. method

33.4 37.2 37.2

30.5 28.0 27.0

28.0 30.6 35.7

34.3 32.9 34.6

Brooks method

33.0 37.6 36.7

29.2 26.6 26.1

27.6 31.3 39.3

38.1 35.8 35.4

Measured

36.1 38.8 35.4

28.5 24.7 24.7

26.2 28.8 36.3

37.5 37.5 37.8

Pres. method

25.9 30.3 30.5

24.7 23.3 21.9

22.5 25.0 29.3

27.9 26.2 26.3

Brooks method Measured

26.2 27.5 31.2 31.0 30.3 28.0

23.4 21.2 20.5

21.5 25.3 32.3

31.7 28.8 27.2

23.0 19.4 19.3

21.3 23.7 29.6

31.4 30.4 29.4

389 397 392 314 320 315

0.99 1.01 - - 1.00 1.01 - -

* 1 langley = 3.687 Btu/f t%

in Brook's equation are those for a vertical surface, they should not be used for surfaces tilted at angles other than 90 degrees from the horizontal surface without further testing. The percent possible sun- shine, D~, in the equation of Brooks is included to account for the variation of the conversion factor with the variation of the cloudiness of the weather. It serves the same purpose as the index /~r , which is the param- eter used in Table 1 for estimating the ratio of daily diffuse to daily total radiation on a horizontal surface.

A comparison between the computed and measured

2.5

2.0

~) L5

In" ~ 0 , -x. . ~ ,CRITICAL RAD. r 'g,

0.5

O ~ 0 0.2 0.4 0.6 0.8 1.0

FRACTIONAL TIME, f , DURING WHICH RADIATION < I l t

FIG. 4--a and b--Radiat ion distribution and ut i ] izabi l i ty curves for hour ly radia t ion on a south- fac ing vert ical surface at Blue ]=Jill.

hourly total radiation in Tables 3 and 4 shows that the overall accuracies of the two methods are good and are comparable, while based upon a comparison of the individual items of the computed and measured hourly radiation, the results obtained by means of Brooks' equation are slightly more accurate--the mean percent deviations of the computed radiation from the meas- ured radiation are 7.6 percent for the present method and 6.3 percent for the method based upon Brooks' equation. This is to be expected, since Brooks' equation is primarily based upon the data of Blue Hill and is applicable to vertical surfaces only, while the present

O8

~ 0.6

0 0 0 . 4 0.8 1.2 1.6 2.0

CRITICAL INTENSITY RATIO,X c= IT'~t

58 Solar Energy

Page 7: Liu and Jordan 1963

TABLE 4--Comparison of Computed and Measured Hourly Total Radiation on an East-Facing (morning) or West-Facing (afternoon) Vertical Surface at Blue Hill--Based upon Data of September, 1952 to August 1956

Hourly Total Radiation, I T t , on a Vertical Surface Facing East (morning hour) or Facing West (afternoon hour), langley/hour*

Hour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-12 or 12-1 10-11 or 1-2 9-10 or 2-3

Month

J a n . Feb. Mar.

Apr. May June

July Aug. Sept.

Oct. Nov. Dec.

Pres. method

Ratio, Comp./Meas.

13.6 17.0 20.6

19.5 20.5 21.6

21.3 20.2 18.9

15.3 12.3 12.1

Sum 213

1.10

Brooks method

9.0 11.7 12.7

13.9 16.6 18.8

18.4 17.4 17.2

13.1 1O.1 9.5

- - ~ 8 - - - -

I

Measured Pres. Brooks I method method

18.9 I 23.s I 27.8

26.9

12.2 14.6 15.7

16.7 19.2 21.0

20.7 18.6 17.1

14.3 11.9 11.1

193

28.5 30.1

30.1 28.6 27.6

22.5 18.6 18.3

302

O .98

17.8 21.9 24.8

24.4 27.4 30.2

30.2 28.8 29.1

23.5 19.6 18.8

297

0.97

Measured

19.0 23.4 26.6

26.6 28.9 31.8

31.6 29.1 27.8

23.8 19.6 18.4

307

Pres. method

20.8 27.3 32.0

31.2 34.4 36.9

36.3 34.5 33.5

26.8 21. 20.2

-2/; 0.92

Brooks method

21.5 27.5 32.1

31.7 36.0 39.3

38.8 37.6 37.5

30.2 24.5 22.1

369

Measured

22.0 28.6 33.1

34.0 35.6 40.2

41.0 38.4 36.6

31.9 23.9 22.1

387

* 1 langley = 3.687 Btu/ft ~.

method makes comparatively little direct use of the data of Blue Hill and its application is not restricted to vertical surfaces.

Stat is t ical Dis tr ibut ion of Solar Radiat ion and Uti l izabi l i ty

Detailed records of hourly radiation oa the collector surface are needed for the exact evaluation of the utilizability defined in Eq. 6. When such records are available, such as for the south-facing vertical surface at Blue Hill, a cumulative statistical distribution curve can be plotted as in Fig. 43 in which the shaded area, given by the integral,

f l [(Irt/ir,) -- (L/iT,)] df (16)

is the utilizability, 4. In obtaining the distribution curve of Fig. 43, four years' data have been used and data for the morning and afternoon hours are sym- metrical with respect to solar noon have been combined.

The effect of varying the temperature of the entering fluid or the rmmber of collector cover plates is to vary the critical hourly radiation, I~, and consequently the ordinate of the horizontal line in Fig. 4a. The corre- sponding variation of the utilizability can be seen from Fig. 4b in which the utilizability is shown as a function of the critical intensity ratio,

X c = I c / i T t (17)

The straight line in Fig. 4b, the limiting curve of identical days, is that for a locality where the weather remains the same from day to day. The large difference between the actual ~b-curve and the straight line, espe- cially at high critical intensity ratios, shows that it is

inadequate, in predicting collector performance, to assume that the radiation intensity at a given hour of the day is equal to the long-term average and that there is no variation from this average from day to day.

The effect of a number of variables on the forms of the distribution and Q-curves will first be investigated in the following sections and a set of generalized Q-curves will then be developed that can be used to predict the performance of collectors of different angles of tilt and at different localities.

1--Dis tr ibut ion and Q-Curves Based on Hourly and Daily D a t a - - T h e variation of the distribution curves and the Q-curves from hour to hour is given in Figs. 53 and 5b where the curves for each of the three hours from solar noon are derived from the January data for the south-facing vertical surface at Blue Hill. Also shown in the figures are the distribution and Q-curves derived from the daily radiation data for the same vertical surface and locality. These curves illustrate an important statistical characteristic of solar radiation, first pointed out by Hottel and Whillier, which is quite general and not restricted to the specific data used in constructing these curves. They show that the hour-to- hour variation of the distribution curves and the Q-curves for the same month and locality is not large and that the curves based upon hourly radiation data are not substantially different from the corresponding curves based upon the daily data. Because the differ- ence between the values of 4) obtained from the daily Q-curve and the hourly Q-curves is small in the range of critical intensity ratios of interest in practical collector operation, the former can be used as an approximation to the latter in calculating collector performance. (The

Vol. 7, No. 2, 1963 59

Page 8: Liu and Jordan 1963

FIG. 5a--Hourly and daily radiat ion dis tr ibut ion curves for a south-facing vertical surface at Blue Hill.

_~z-

E)

n ~

2.8

2.4

2.0

1.6

1.2

0.8

0.4

0 0

1.0

0.2 0.4 0.6 0.8 1.0

FRACTIONAL TIME, f , DURING WHICH RADIATION ~ ITt or H t

0.8

O.6

I - - _ J

b,I

_ J ~ 0.4

0.2

FIG. 5b--Hour ly and daily uti l izabil i ty curves for a south-facing vertical surface at Blue Hill.

(34 O~ 1.2 1.6 2.0

CRITICAL "INTENSITY RATIO, XC= I I. or H, H

2.4 2.8

60 Solar Energy

Page 9: Liu and Jordan 1963

-e-

>: I - - ,._/

N ._1 I.--

1.0

Q8

0.6

0.4

0.2

0 0

\ \

\

I 1 t 1 1 1 1 1 1 1 1 BLUE HILL OBSERVATORY SOUTH FACING VERTICAL SURFACE JANUARY

), \ \ \ I - - ~ - t - - ~-, E \ i

; ~ v ~ , --I/~-- (H: II0 BTU/DAY-FT 2)

\ i ' % , , ! I

IDENTICAL DAYS~ ~ , ~ , .

0.4 0.8 1.2 1.6 2D CRITICAL INTENSITY RATIO, X c

i

2.4 2.8

FIG. 6 - -Var ia t ion of the u t i l izabi l i ty curve for a south-facing vert ical surface at Blue Hill from the period 1952-56 to 1957-59.

1.0

QB

0 6

>: I - - _1

04 <I N _I l--

oz ~--

\ I I I I 1 1 1 1 SCHENECTADY, NY

k HORIZONTAL SURFACE ~ . NOVEMBER

\ ,.--1951-1953 ( B= 476 BTUI DAY-FT=.) I \ , \ x \ ~ . ,i/_1954_1958 (R=413 BTU/DAY-FT ~) i \ \ / ( ~

\ \ , L LIMITING CURVE[~ ' \ \ \ \

0 0.4 0.8 1.2 1.6 2.0 2.4

L - -

CRITICAL INTENSITY RATIO, X c

FIG. 7 - -Var ia t ion of the u t i l izabi l i ty curve for a hor izonta l surface at Schenectady from the period 1951-53 to 1954-58.

somewhat larger difference between the daily C-curve and that for the hours 9-10 and 2-3 is of less signifi- cance because of the relatively small amount of energy collected during these hours in comparison with that collected during the other hours which are nearer to solar noon.) This approximation is also necessary in most cases, since the radiation data published by the

U. S. Weather Bureau are daily values only and the hourly radiation data are not readily available. In the subsequent development all the distribution and C-curves presented will be based upon daily radiation data only.

2--Variation of C-Curve from One Period to Another-- Four years' data were used in constructing the C-curves

Vol. 7, No. 2, 1963 61

Page 10: Liu and Jordan 1963

1.0

Q8

o- 0.6 )_-

b- "3

N

J 0.4 I - -

0.2

0

0 0.4 0,8 1.2 1.6 2.0 2.4 2_8

CRITICAL INTENSITY RATIO, X c

FIG. 8 - -Compar i son of the ut i l izabi l i ty curve for a sou th- with t h a t calculated from the exper imenta l da ta for a hori- facing vert ical surface cons t ruc ted from exper imenta l da ta zontal surface.

of Figs. 4 and 5. I t is evident that to obtain a bet ter statistical average data of a larger number of years could be used. The question then arises as to how many years ' data are adequate.

No unique answer can be found for this question since it depends upon the degree of accuracy of the Q-curves

TABLE 5 - -Th e Ra t io of the Dai ly Diffuse Rad ia t ion on a Horizontal Surface, D, to the Dai ly Tota l Rad ia t ion

on a Hor izonta l Surface, H

K T = H / H o

D/H

0 0.1 0.2 0.3 0.4 0.5 0.5 ] 0.7 0.75

0

required as well as on the particular type of weather of a locality. Nevertheless, the variat ion of the Q-curve from one period to another for the same month and the same locality can be seen by examining Figs. 6 and 7. The large difference between the two curves in each graph shows tha t for the two cases under consideration, it is inadequate to construct Q-curves from data of three to five years. Da ta of a larger number of years should be used. This is not true in general, however, since the Q-curves shown are those of localities with very cloudy weather and consequently the weather is more variable. The large difference is also approximately the max imum to be expected for any locality. These examples are selected to emphasize the fact tha t substantial error can and does arise when data of a finite number of years

are used to predict the long-term average performance of a collector. Similar curves for localities with clearer weather would show a much smaller variation. For most localities, especially those of interest to solar energy utilization, sufficiently accurate ~b-curves can be obtained from data of five years.

3--oh-Curves for Horizontal and Tilted Smfaces--The effect of varying the orientation of the surface oil which solar radiation is measured on the form of the Q-curve is apparent by comparing the two solid curves in Fig. 8 which are constructed from the January data on a horizontal surface and a south-facing vertical surface at Blue Hill. Since the difference between Q-curves is large, the q~-curve for a horizontal surface cannot, in general, be used directly to predict the performance of tilted collectors.

To construct the distribution and ~b-curves for a tilted surface from radiation data oil a horizontal surface it is Imcessary to convert each i tem of the hori- zontal data to the tilted surface. This can be accom- plished by multiplying each i tem of the horizontal data by an appropriate conversion factor. The conversion factor is for the daily total radiation and is the weighted average of the conversion factors for the daily direct, diffuse and ground reflected radia t ion--averaged ac- cording to the relative proportions of the daily direct, diffuse, and total radiation. This conversion factor is given by, I5

R = [1 - (D/H)]RD + ½(1 + cos~)(D/H) (18)

+ ½(1 -- cos f~)p

62 Solar Energy

Page 11: Liu and Jordan 1963

2.8

DAILY RADIATION , El. RATIO: AVE. DAILY RADIATION R

2.4

2.0

1.6

1.2

0.8

0.4

1.0

0 O 0.2 0.4 0.6 0.8 1.0

FRACTION TIME,f, DURING WHICH RADIATION ~-H

FIG. 9a--Comparison of radiat ion dis tr ibut ion curves of Schenectady, S, Ste. Marie, and Annet te . ( K T = 0.3)

0.8

0.6

>:

-J O.4 rr~

__N J

0.2

0 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8

CRITICAL INTENSITY RATIO, X c

FIG. 9b--Comparison of ut i l izabil i ty curves of the same localities of Fig. 9a. (lr~T = 0.3)

ol. 7, No . 2, 1963 63

Page 12: Liu and Jordan 1963

I=^Tin, DAILY RADIATION H ! t - - " . . . . AVE. DAILY RADIATION ' I~

2.0

Z HKt;#~NT[~~~AL SURFACE

. . . . ALBUOUEROUE, N.M.JULY, 1954-1958 1.6 - - ( £-- 2590 BTU/DAY-FT z, K1- = 0.703)

------ WAKE ISLAND, P A . , DEC., 1 9 5 4 - 1 9 5 8 (R = 1630 BTU/DAY-F 'F =, K.r = 0 .698 )

= . . . . . . LAS VEGAS, NEV. , JULY, 1 9 5 4 - 1 9 5 8 ' / ,~ 12 ---1 ( FI= 2 5 7 0 BTU/DAY-FT z, R.r = 0~698.~) , . ~ -;,-;' ": -

. .-

O8 / : O8 r

i, ! . ! i i

°*( 1 !

0 0 0.2 0.4 0.6 0.8 1.0

FRACTIONAL TIME, f , DURING WHICH RADIATION ~ H

FIG. 10a--Comparison of the radiation distribution curves of Albuquerque, Wake Island, and Los Vegas. (KT = 0.7)

1.0 \ \ I

\ i i 08 I O.6

i !! IkJ

LIMITING CURVE ~.~,..__~.~/'~ LBUQURQE i 'DENTICAL DAYS ---~ R).'. T/_._~--_._.I"-] WAKE ISLAND

, vEgAs

o I l l l l l i X 0 Q4 0.8 1.2 1.6

CRITICAL ITENSlTY RATIO, X c

:FIG. 10b--Comparison of utilizability curves of the same loca- tions as Fig. 10a. (Kr = 0.7)

>."

0.4 N .J h-

where RD, the conversion factor for daily direct radia- tion, can be computed from the following two equations:

cos (L - 3) sin w, - ~, cos o~/ RD - - when ~ < w,~'

cos n sin ~, -- w, cos ~ (19)

cos (L --/~) sin ~,' -- ~,' cos w~' RD when w~' < ~,

cos n sin w~ -- ws cos ~

and w/, the sunset hour angle for the tilted surface, is given by the expression,

cos ~ ' = - t a n (L - /~)tan ~ (20)

These expressions for RD are exact during the time of equinox but have been found to give satisfactory results when used for other times of the year. I t should be noted, however, that they are applicable to surfaces tilted toward the equator only.

The ratio of the daily diffuse radiation to the daily total radiation on a horizontal surface, D/H, in Eq. 18 varies from a maximum of 1.0 for a completely overcast day when the radiation received is all diffuse radiation, to a minimum of approximately 0.16 for a clear day. Betwemx these extremes, the ratio varies with the degree of atmospheric cloudiness. This variation is shown in Table 5, 9 which is derived from the data of Blue Hill but is believed to be applicable also to other localities.

Since the ratio, D/H, varies greatly from day to day due to changing atmospheric cloudiness, a large day-to- day variation of the conversion factor, R, for a surface of a fixed angle of tilt is also expected. This is indeed

the reason the two b-curves in Fig. 8 are different. Had the conversion factor been independent of atmospheric cloudiness, ~b-curves for a tilted surface and a horizontal surface would then be identical.

The dotted curve in Fig. 8 for the south-facing verti- cal surface at Blue Hill is obtained by first converting the horizontal data with the conversion factor given by Eq. 18 and then constructing the C-curve with the converted data. The ground reflectance used is that given in Table 2. The small difference between the b-curves based upon the computed and measured radia- tion on the south-facing vertical surface shows that for the present purpose, the method used for converting the daily radiation from a horizontal surface to a tilted surface is adequate. Similar curves for other months have also beett constructed and compared in the same way and the results are substantially the same as those shown in Fig. 8. These results, therefore, are not presented here.

4--Generalized Distribution Curves for a Horizontal Surface--When the distribution curves for horizontal surfaces at a large number of localities are constructed and compared according to the value of the index , /~r , a correlation between the form of the distribution curve and the index becomes apparent. Two sets of curves, for /~r = 0.3 and 0.7, are presented in Figs. 9a and 10a for illustration. The large differences between the curves with different values of the index and the simi- larity between the curves with the same value of the index should be noted. The only parameter used in the

64 Solar Energy

Page 13: Liu and Jordan 1963

above correlation is /7/r. The effects of solar declina- tion and latitude of the locality is small and has been neglected in this correlation. The C-curves correspond- ing to the distribution curves of Figs. 9a and 10a are shown in Figs. 9b and 10b. These ~-curves, of course, are useful only in predicting the performance of hori- zontal collectors.

A set of five "generalized distribution curves", shown in Fig. l l , for a horizontal swface and for /~'T = 0.3, 0.4, 0.5, 0.6 and 0.7 has been obtained from radiation data of 27 localities. Each curve is obtained by corn- bitting six distribution curves (of six different localities) with the same value of the index, /7/r, except that for / ~ = 0.3, which, due to the lack of data, is the com- t)ined curve of three localities.

5--Generalized ¢)-Cur~'es--The "generalized 0-curves" constructed from the generalized distribution curves of Fig. l l and the conversion factor for daily radiation given by Eq. 18 are given i~) Figs. 12a-e. These curves are obtained by assuming the cos ~ = 0.683 (average of cos 30 ° and cos 60 °) and p = 0.2. Since the form of the ~b-curves changes very slightly when other values of cos ~ and p are used in Eq. 18, these curves are also applicable with very little error" to surfaces of other angles of tilt and to localities where the ground re- fiectance for solar radiation is different from the assumed value of 0.2. This does not mean, however, that the form of the ~b-curve is not affected by tilting the surface, since the value of Ro , the conversion factor for daily direct radiation, is dependent upon the angle of tilt and can be a very sensitive function of the angle of tilt.

I t is interesting to note that as the value of / ( r in- creases, all ~b-curves approach the straight line, the limiting curve of identical days. W h e n / ~ r = 0.7, all curves are nearly identical. This is because that at a locality with such a high value o f / ~ r , the weather is nearly clear from day to day and that during any mouth there is almost no day-to-day variation of solar incidence on any surface of any orientation.

When the generalized O-curves are used ill calculating collector performance the error will, in general, be small except for localities with extremely cloudy weather where K r is small. This can be seen by examining the C-curves of Figs. 9b and 10b. In any case, the error due to the use of these generalized C-curves is expected to be no greater than, or at most of magnitude comparable to, that due to the use of data of a limited number (three to five) of years, as a comparison of Figs. 6 and 7 with Figs. 9b and 10b will show.

The generalized C-curves together with the table in the Appendix provide the complete weather informa- tion needed for determining the long-term average per- formance of fiat-plate collectors of any angle of tilt at all the localities in the table of the Appendix. The usual tedious process of data analysis is thus elimilmted. I t

#

Z8

?_4

2.0

1.6

1.2

0.8

0.4

RATIO= AVE. DAILY RADIATION R

o o 0.2 0.4 0.6 0.8 I.o

FRACTIONAL TIME,f , DURING WHICH RADIATION _< H

FIG. ll--The generalized radiation distribution curves for a horizontal surface.

should be noted that these procedures are also directly applicable to collectors with selective-absorbing sur- faces 16 (high absorptivity for solar radiation and low absorptivity or emissivity for long-wave thermal radi- ation), since this type of collectors differs from the conventional type with nollselective absorbing sur- faces only in the respect of a different (lower) heat loss coefficient, U, and a different (lower) absorptivity for solar radiation.

P e r f o r m a n c e o f Col lec tors

The performance of collectors at four latitudes, 20, 30, 40 and 50 deg N, during two months, January and July, is calculated by means of the above method and the results are shown in Figs. 13a-b. The collectors are covered with two panes of glass of good quality. The angles of tilt of the collectors from the horizontal are 15 deg greater than the corresponding latitudes for opt inmm winter performance. Consequently, the col- lector surfaces are all parallel. The necessary steps for calculating each of the curves are shown in Table 6. For this example the collector is on 40 deg N latitude and at a locality where /~r = 0.5 during January. I t is

Vol. 7, No. 2, 1963 65

Page 14: Liu and Jordan 1963

1.0

0.8

O.6

.~.-

- J en

'~ O.4 N

0.2

O 0 0.4 0.8 1.2 1.6 2_0 2.4 2.8 3.2

CRITICAL INTENSITY RATIO, X c

FIG. ]2a- -The generalized u t i l i zab i ] i t y curves for a hor izonta l surface and surfaees t i l ted toward the equator. (KT = 0.3)

1.0

0.8

0.6

>_-

. J

<

N O.4 ._,1 7-

0.2

0 O 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2

CRIT;CAL INTENSITY RATIO, X c

FIG. 12b--The generalized utilizability curves for a horizontal surface and surfaces t i l ted toward the equator. (KT = 0 . 4 )

t i l t ed a t an angle of fl = 40 + 15 = 55 deg f rom the hor izon ta l surface and is covered wi th two panes of glass of good qua l i ty , b u t the glass panes are not surface t r e a t ed for low ref lec t iv i ty . F u r t h e r i t is a ssumed t h a t : Col lec tor hea t loss coefficient, U = 0.7 B t u / h r - W - deg F , ( ~ ) for t o t a l r ad i a t i on = 0.98 (1 - - D) -

(1 -- S)(r~a) where (1 -- D) = d i r t f ac to r = 0.99, (1 -- S) = shad ing fac tor , n (tea) = p r o d u c t of effect ive t r a n s m i s s i v i t y and a b s o r p t i v i t y for d i rec t r a d i a t i o n y and ref lectance of g round for solar r ad ia t ion , p = 0.4.

F igu re 13, ca lcu la ted for col lectors wi th two panes of glass, can also be used to de t e rmine a p p r o x i m a t e l y t he

66 Solar Energy

Page 15: Liu and Jordan 1963

1.0

Q8

Q6

I , - .,.J t ' n

N Q4 - i W-

0.2

0 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 5.2_

CRITICAL INTENSITY RATIO, X c

FIG. 12c--The generalized utilizability curves for a horizontal surface and surfaces til ted toward the equator. (KT = 0.5)

1.0

0.8

Q6 .$ >_-

J

N_ O.4 ._1 I , -

Q2

0 0 0.4 0.8 1.2 o 0.4 O.S 1.2 1.6

CRITICAL INTENSITY RATIO, Xe

FIG. 12 d and e--The generalized utilizability curves for a horizontal surface and surfaces til ted toward the equator. (KT = 0.6 and 0.7)

pe r fo rmance of col lectors wi th o the r number s of glass panes. Th is call be done b y first m u l t i p l y i n g the ac tua l t e m p e r a t u r e difference, (h -- to), b y the ra t io , Ca = [U(rea)']/[U'(reoO] , and t hen m u l t i p l y i n g the va lue of Q / F , in Fig . 13 cor respond ing to th is " equ iva l en t t em- p e r a t u r e d i f ference" b y the ra t io , c2 = (rea)/(rea) ' .

The u n p r i m e d quan t i t i e s are those of the col lector whose per fo rmance is being de t e rmine d and the p r imed quan t i t i e s are those of a col lector wi th two panes of glass. The t r a n s m i s s i v i t y - a b s o r p t i v i t y p roduc ts , (r~a) and ( r ~ ) ' , are to be e v a l u a t e d a t the m i d p o i n t of the hour 10 11 or 1-2 ( incidence angle = 22.2 deg for

Vol. 7, No. 2, 1963 67

Page 16: Liu and Jordan 1963

J

1800

1500

1400

900

800

5 0 0

400.

FI(;. 1 3 a - - P e r f o r m a n c e of a co l lec tor w i t h two p a n e s of g lass

I JANoAR

~k< L-~= -15 ° !

' , . \ . ~ % * ~ " , x 1 I I ~ i ; i • -~ . \ ~ 'x . ' .x , \ x ~ . I i J 1 ~ " \ ~ x - ' ~ . l N , . - ' - N . ' \ , ~ ~ - - , 3 , " , 'K-" - - i : : ~ - 1 :

b . F ' \ "kL'-'L P - - : , . I I I I

_ _

~ - ~ . . ( ~ ~ ~ ,~. < -~ ~ .~-.-~.--:-~i~-'--+.~-"----~.. '~..~-..L~ ~-::Z-~ ~-.~--...-"~,-~ ~ .<..,

0 50 I00 150 200 250

tt - to ~ °F

in Janu '~ ry . (L = l '~t i tude, ~ = ang le of t i l t f rom the ho r i zon ta l )

January and 42.4 deg for July). This procedure follows from the assumption tha t the ratio of the values of -/F Q/ R of two collectors, one with two panes of glass and

the other a different number of glass panes, is the same as the ratio of the values of #t/FR of the same two col- lectors for the hour 10-11 or 1-2, when both collectors are operated at the same critical intensity ratio during the hour 10-11 or 1 2. This assumption has been found to give results within 4 percent of those obtained by the detailed and step by step calculation even for a collector with as m a n y as ten panes of glass. Typical values of Cl and c2 for collectors with one, two, and three panes of glass are given in Table 7.

To illustrate the use of Fig. 13 in calculating collector performance consider the following example.

E x a m p l e : C a l c u l a t e t he J a n u a r y p e r f o r m a n c e of co l lec tors wi th one, two, a n d t h r ee p a n e s of g lass a t F r e sno , Ca l i fo rn ia , Blue Hil l , M a s s a c h u s e t t s , a n d W i nn i peg , C a n a d a . T h e collec- to r s are t i l t ed a t ang le s wh i ch are 15 deg g r e a t e r t h a n t h e corre- s p o n d i n g l a t i t u d e s . T h e e n e r g y - t r a n s p o r t fluid en t e r s t h e col- l ec to rs a t a t e m p e r a t u r e of 110 deg F. So lu t ion :

For Fresno, CaIO'ornia--From t h e t ab le in t h e A p p e n d i x , L = 36°46 ' N, /£T = 0.462, to = 47 deg F. The re fo re , t h e col- l ec tor is t i l t ed a t an ang le of 15~6°46 ' = 51o46 ' f r o m tile hor i - zon ta l su r faee . T h e t e m p e r a t u r e difference, t, - l0 , = 110 - 47 = 63 deg F. F r o m Fig. 12a, Q/FR = 722 B t u per d a y - s q f t for t h e col lee tor w i t h two g lass panes .

To d e t e r m i n e t he p e r f o r m a n e e of t he col lector w h e n one p a n e of g lass is used , t h e t e m p e r a t u r e differenee of 63 deg F is m u l t i p l i e d by t he e o n s t a n t , e~ = 1.570, to o b t a i n t h e e q u i v , d e n t - t e m p e r a t u r e differenee of 99 (leg F. T h e v a l u e of O/Fa in Fig. 12a c o r r e s p o n d i n g to a t e m p e r a t u r e difference of 99 deg F is 414 B t u / d a y sq f t . T h u s for t he o n e - p a n e col lec tor , Q/F~ = (414) (c2) = (414) X (1.090) = 451 B t u / d a y - s q f t 2.

S imi la r ly for a col lee tor w i th t h r e e p a n e s of g lass , O/FR = 565 B t u / d a y - s q f t .

Tab l e 8 s u m m a r i z e s t he p e r f o r m a n e e of eo l lee tors a t t h e s e t h r ee loea l i t ies .

C O N C L U S I O N S

The above method of anMysis has reduced the meteor- ological data necessary for the determination of col- lector performance to a minimum, e.g. to the monthly average daily total radiation on a horizontal surface

6S Solar Energy

Page 17: Liu and Jordan 1963

1400

1 3 0 0

[ ] dULY

i : i I ! !

" I ! i ! J

• " I ! i I i ~I ~ ~ ~ ~ i

i- +VT! ! V " , . " 4 ~ % . N , i I i i ~ ~ i

\ " + , \ . K - - ' 2 K - , . ! ' ] l ~ i !

' ' -6' " N " ~ Q • ' l ! t

0 5 0 100 150 2 0 0 2 5 0

t i - tO , * F

FIG. 13b - -Pe r fo rmance of a collector wi th two panes of glass in Ju ly . (L = la t i tude , fl = angle of t i l t f rom the hor izonta l )

TABLE 6 - -Sample Collector Pe r fo rmance Calcu la t ion

1. w~ = 71.3 ° (Eq. 7). H o u r s be tween sunr ise and sunse t = (71.3)(2)/15 = 9.5 hours

2. Ho = 1364 B t u / d a y - f t 2, (Eq. 8) 3. ffI = ff~THo = 682 B t u / d a y - f t ~ 4. D / f [ = 0.376 (Table 1) 5. RD = 2.49 (Eq. 19)

Hour of Day

6. cos Oh (Eq. 9) 7. cos Ot (Eq. 10) 8. r r , (Fig. 3) 9. ra, (Fig. 3)

10. /~, (Eq. 12) 11. I v t , B t u / h r - f t ~ (Eq. 13) 12. 0t , degrees 13. (Tea), (Ref. 12)

14. (1 -- S), (Ref. 11) 15. (Ta), (see a s s u m p t i o n s )

11-12 12-1

0.477 0.986 0.172 0.160

1.711 201

9.6 0.819

0.990 0.787

10 1l 9 10 1-2 2 3

0.429 0.926 0.149 0.144

1.751 178

22.2 0.810

0.980 0.770

8-9 3-4

0.335 0.203 0.808 0.642 0.109 0.059 0.113 0.069

1.862 140 36.1

0.788

0.964 0.737

2.201 89 50.1

0.728

0.939 0.663

7-8 45

0.042 0.438 0.011 0.014

5.891 44 64.0

0.550

0.891 0.475

Ht or Q/FR, Btu/day-ft~

1304

(tl -- to) = 0°F

16. [ e , B t u / h r - f t ~ (Eq. 2) 0 0 0 0 0 17. X~, (Eq. 17) 0 0 0 0 0 956 18. ~, (Fig. 12e) 1.0 1.0 1.0 1.0 1.0 19. ?I/FR , B t u / h r -ft2 (Eq. 5) 158 137 103.5 58.7 21.0

(tl -- to) = 50°F

16. I~ 44.5 45.5 47.5 52.8 73.7 17. Xc 0. 222 0. 256 0. 339 0. 594 1.67 667 18. q~ 0.800 0. 765 0. 700 0. 500 0.02 19. ?l/Fn 126.4 104.8 72.5 29.4 0.4

(tz - to) = lO0°F, etc.

Vol. 7, No. 2, 1968 69

Page 18: Liu and Jordan 1963

TABLE 7--Values of c~ and c2 for Different Thickness of Glass

Glass panes

U ( B t u / hr-f t2-°F)

1.2 0.7 0.5

Jan . I6 J u l y 16

(at 22.2 ° )

0.883 11.570 0.810 1.000 0.746 0.776

1.090 1.000 O. 920

1.561 1.000 0.776

C1 C2

1.098 1.000 0.918

TABLE 8--Performance of Collectors in January at Fresno, California, Blue Hill, Massachusetts, and Winnipeg, Canada. Fluid enters the collectors at 110 deg F.

Fresno, Calif. Blue Hill, Mass. Winnipeg, Can.

36°46 ' 42°13 ' 49°54 '

Angle of T i l t

51°46 ' 57°13 ' 64054 '

K~

0.462 0. 445 0. 601

O_/FR, B t u / d a y - f t 2

to, °F 1 2 3

pane panes panes

47 451 551 563 28 299 409 440 3 418 580 624

and the m o n t h l y average day- t ime a m b i e n t tempera-

ture. Together with the table in the appendix, the generalized ut i l izabi l i ty curves provide the complete in format ion needed for predic t ing the long- term average performance of f iat-plate collectors. These da ta can be used wi th any flat-plate collector, inc luding the collector with select ive-absorbing surface, provided no special concen t ra t ing devices are used.

The results of calculat ion shown in Fig. 13 shows tha t the performance of collectors which are s imilar ly or ien ta ted with respect to the sun (same L - ¢0 is

p r imar i ly de termined by the two variables, K r a nd (t~ - to), and to much lesser extents by the la t i tude of the locality. Indeed the var ia t ion of the local cl imate

and the value of K r is so large from one locali ty to another t h a t l a t i tude is a re la t ively u n i m p o r t a n t factor to consider in solar-collector appl icat ion. Therefore, it would be erroneous to assume tha t a local i ty in the lower la t i tude is necessarily more favorable to solar collector opera t ion t h a n a locali ty in the higher la t i tude. This is also appa ren t by referring to Table 8 which shows tha t despite the low day- t ime t empera tu re and the high la t i tude of Winnipeg , collectors with two or three panes of glass are capable to collect more energy in J a n u a r y a t Winn ipeg t h a n ~t ei ther Blue Hill or Fresno.

A P P E N D I X

R a d i a t i o n and Other Data for 80 S e l e c t e d Cit ies o f t h e U n i t e d S t a t e s and C a n a d a

1--Values of H, the average daily total radiation on a hori- zontal surface, are in units of Btu/day-ft 2. They are derived primarily from data for the period, 1950-1960, published by the U. S. Weather Bureau in "Climatalogical Data, National Sum- mary". At least five years' data are used in the average except those values in the parenthesis, which are averages of three or

four years. Localities with less than three years' data are not included in the table.

2--KT is the ratio, ft/Ho , where Ho , the extraterrestrial daily radiation on a horizontal surface is computed from Eq. 8 using a solar constant of 442 Btu/hr-ft ~ and a solar declination on the 16th day of the nmnth, except February when the declination of February 15 is used. Solar declinations are taken from Solar Ephermeris for the year 1954.

3 The day-time ambient temperature, to, is computed from the following equation by Hottel, n

to = 0.3 tmax -~ 0.7 tm ... . . deg F

where t .... = average daily maximum temperature, deg F and tm~, = average daily mean temperature, deg F. tm~ and troop, are long-term (primarily 1921-1950) average values taken from the following sources: (1) "Local Climatalogical Data", Weather Bureau, U. S. Department of Commerce, (2) "Cli- matic Summary of the United States--Supplement for 1931 through 1952", Weather Bureau, U. S. Department of Con> merce, and (3) "Canada Year Book, 1960", Canada Year Book Section, Information Service Division, Dominion Bureau of Statistics, Canada. Where temperature data are unavailable for a station (indicated by an asterisk *), the data given are computed from those of a nearby station.

REFERENCES

1 Hesselachwerdt. A. J. Jr., "Performance of the M.I.T. Solar House", Space Heating with Solar Energy, Proceeding of a Course-Symposium Held at M.I.T., pp. 99-106, (1950)

2 LSf, G. O. G., "Solar House Heating--A Panel", Proceed- ings, World Symposium on Applied Solar Energy, pp. 131- 145, (1955)

3 Telkes, M., "Performance of the Solar-Heated House at Dover, Mass.", Space Heating with Solar Energy, Proceed- ings of a Course-Symposium Held at M.I.T., pp. 97-98 (195o)

4 Bliss, R. W. Jr., "Solar House Heating--A Panel", Proceed- ing s, World Symposium on Applied Solar Energy, pp. 151- 158, (1955)

5 Bridges, F. H., Paxton, D. D. and Haines, R. W., "Per- formance of a Solar-Heated Office Building," A S H A E Transaction, 64: 83-96, (1958)

6 Thomason, H. E., "Solar Space Heating and Air Condi- tioning the Thomason Home", Solar Energy, IV(4) : 11-19, (Oct., 1960)

7 Harper, E. Y., "Solar House Tests to Compare Heating Costs", The Sun at Work, III(3): 3-4, 8, (Sept., 1958)

8 Fritz, S., "Transmission of Solar Energy through the Earth's Clear and Cloudy Atmosphere", Transaction of the Conference on the Use of Solar Energy, The Scientific Basis, I, (pp. 17-36, 1955)

9 Liu, B. Y. H. and Jordan, R. C., "The Interrelationship and Characteristic Distribution of Direct, Diffuse and Total Solar Radiation", Solar Energy, IV(3): 1-19 (July, 1960)

10 Hottel, H. C. and Woertz, B. B., "The Performance of Flat-Plate Solar Heat Collectors", ASME Transaction, 64: 91-104, (1942)

11 Hottel, H. C., "Performance of Flat-Plate Solar Energy Collectors", Space Heating with Solar Energy, Proceeding of a Course-Symposium Held at M.I.T., pp. 58-71 (1950)

12 Hottel, H. C. and Whillier, A., "Evaluation of Flat-Plate Solar Collector Performance", Transaction of the Confer- ence on the Use of Solar Energy: The Scientific Basis, II, Part 1, Section A, pp. 74-104, (1955)

13 Whillier, A., "Solar Energy Collection and Its Utilization for House Heating", D.Sc. Thesis in Mechanical Engineer- ing, M.I.T., Cambridge, Mass., (May, 1953)

14 Bliss, R. W. Jr., "The Derivation of Several 'Plate-Effi- ciency Factors' Useful in the Design of Flat-Plate Solar Heat Collectors", Solar Energy, III(4), 55-64 (Dec., 1959)

15 Liu, B. Y. H. and Jordan, R. C., "Daily Insolation on Surfaces Tilted toward the Equator", A S H R A E Journal 3(10): 53-59 (Oct., 1961)

16 Tabor, H., "Selective Radiation", Transaction of the Con- ference on the Use of Solar Energy: The Scientific Basis, 3, Part I, Section A, pp. 24-33, (1955)

70 Solar Energy

Page 19: Liu and Jordan 1963

R a d i a t i o n a n d O t h e r D a t a f o r 80 L o c a t i o n s i n t h e U n i t e d S t a t e s a n d C a n a d a

( / t = M o n t h l y average daily to ta l r ad ia t ion on a hor izonta l surface, B tu /day- f t2 ; /~ t = the f rac t ion of the extra t e r res t r i a l ra- (liation t r a n s m i t t e d t h r ough the a tmosphere ; to = ambien t tempera ture , deg F.)

Albuquerque, N . M . Lat . 35°03'N. El. 5314 ft

Anne t t e Is., Alaska Lat . 55°02 'N. El. 110 ft

Apalachicola, Flor ida Lat . 29°45 , N. El. 35 ft

Astoria, Oregon Lat . 46°12 ' N. El. 8 ft

At lan ta , Georgia Lat . 33°39 ' N. El. 976 ft

Barrow, Alaska Lat . 71°20 ' N. El. 22 ft

Bismarck, N. D. Lat . 46°47 'N. El. 1660 ft

Blue Hill, Mass. Lat . 42013 , N. El. 629 ft

Boise, Idaho Lat . 43034 , N. El. 2844 ft

Boston, Mass. Lat . 42°22 ~ N. El. 29 ft.

Brownsvil le, Texas Lat . 25°55 ' N. El. 20 ft

Car ibou, Maine Lat . 46°52 ' N. El. 628 ft

Char les ton, S. C. Lat . 32054 ' N. El. 46 ft

Cleveland, Ohio Lat . 41024 , N. El. 805 ft.

Columbia, Mo. Lat . 38°58 'N. El. 785 ft

Columbus, Ohio Lat . 40°00 , N. El. 833 f t

Davis, Calif. Lat . 38°33 ' N. El. 51 ft

Dodge City, Kan . Lat . 37046 ' N. El. 2592 ft

Eas t Lansing, Michigan Lat . 42044 ' N. El. 856 ft

Jan

/4 1150.9 /£t 0.704 to 37.3

/ t 236.2 ~'~ 0.427

35.8

/4 1107 i / ~ 0.577

to 57.3

q 338.4 Kt 0.330

41.3

B 848 /£t 0.493

[ to 47.2

/ t 13.3

! to --13.2

B 587.4 /£t 0.594 to 12.4

/~ 555.3 Kt 0.445 to 28.3

H 518.8 Kt 0.446 to 29.5

H 505.5 Kt 0.410 to 31.4

H 1105.9 Kt 0.517

I to 63.3

B 497 /~t 0.504 to 11.5

H 946.1 Kt 0.541 to 53.6

H 466.8 Kt 0.361 to 30.8

H 651.3 Kt 0.458 to 32.5

/ ) 486.3 /~t 0.356 to 32.1

/ t 599.2 /£t 0.416 to 47.6

/~r 953.1 /~t 0.639 to 33.8

/ t 425.8 Kt 0.35 to 26.0

Apr May Jun Aug Sep ' Oct Feb Mar July Nov

1453.! 1925.4 2343.5] 2560.9] 2757.5 2561.2 2387.81 2120.3] 1639.8 0.691 0.719 0 .72210 .713 0.737 0.695 0.708 0 .72810 .711 0.684 ] 43.3 50.1 59.6 i 69.4 [ 79.1 82.8 80.6 73.6 62.1 47.8

428.4 883.4 1357.2 1634.7 1638.7 1632.1 1269 4' 962 ] 454 6 220.3 I 0.415 0.492 0.507 0.484 0.441 0.454 0.427 0.449 ~ 0.~47 0.304 ] 37.5 39.7 44.4 51.0 56.2 58.6 59.8 54.8 i 48.2 41.9

1378.2 1654.2 2040.9 2268.6 2195.9 1978.6 1912.9 1703.3 1544.61 1243.2 0 5 8 4 ] 0 . 5 7 6 0.612 0.630 0.594 0.542 0.558 0 . 5 5 9 ] 0 . 6 0 8 ] 0 . 5 7 4 59.0 ,! 62.9 69.5 76.4 81.8 83.1 ~ 83.1 80.6 73.2 / 6 3 . 7

607 iloo8.5 1401.5 1838.7 1753.5 2007.7 i 1721 11322.5 780.4 i413 .6 0.397 ]0.454 0.471 0.524 0.466 0.551 ]0.538 ]0.526 0.435 10.336 447 4 6 . 9 513 55o 593 626 636 622 557 ]48.5

1080.1 1426 91 1807 2018.1 2102.6 2002.9 1898.11 1519.2 1290.8 i 997.8 0.496 ! 0.522 I 0.551 0.561 0.564 0.545 ] 0.559 I 0.515 0 .543] 0.510 49.6 ]55 .9 65.0 73.2 80.9 82.4 81.6 I 77.4 66.5 54.8

143 .21713 .3 1491.5 1883 2055.3 1602.2] 953.5 428.4 1 5 2 . 4 1 2 2 . 9 0 . 7 7 6 ] 0 . 7 7 3 0.726 0.553 0.533 0 . 4 4 8 ] 0 . 3 7 7 0.315 0.35 ] - - --15.9 --12.7 2.1 20.5 35.4 41.6 40.0 31.7 18.6 2.6

934.3 ] 1328.4 1668.2 2056.1 2173.8 2305.5] 1929.1 1441.: 1018.1] 600.4 0 . 6 2 8 ] 0 . 6 0 5 0.565 0.588 0.579 0 . 6 3 4 ] 0 . 6 0 6 0.581 0 .58410 .510 15.9 29.7 46.6 58.6 67.9 76.1 I 73.5 61.6 49.6 31.4

797 1143.9 1438 1776.4 1943.9 1881.5 1622.1 1314 941 ]592.2 0.458 0.477 0.464 0.501 0.516 0.513 I 0.495 0.492 0 . 4 7 2 ] 0 . 4 0 6 28.3 36.9 46.9 58.5 67.2 72.3 70.6 64.2 54.1 43.3

884.9 1280.4 1814.4 2189.3 2376.7 2500.31 2149.4 1717.7 1128.4] 678.6 0.533 0.548 0.594 0.619 0.631 0.684 I 0.660 0.656 0 .58810 .494 36.5 45.0 53.5 62.1 69.3 79.6 77.2 66.7 56.3 42.3

738 1067.1 1355 1769 1864 1860.51 1570.1 1267.5 8 9 6 . 7 ] 6 3 5 . 8 0 .42640 .445 0.438 0.499 0.495 0 .507] 0.480 0.477 0 . 4 5 3 ] 0 . 3 7 2 31.4 39.9 49.5 60.4 69.8 74.5 73.8 66.8 57.4 46.6

1262.: 1505.9 1714 2092.2 2288.5 2345 I 2124 1774.9 1536.51 1104.8 0.500 0.505 0.509 0.584 0.627 0.650 I 0.617 0.566 0.570 I 0.468 66.7 70.7 76.2 81.4 85.1 86.5 86.9 84.1 78.9 170.7

861.6 1360.1 1495.9 1779.7 1779 71 1675.6 1254.6 793 1415.5 0.579 0.619 0.507 0.509 0.473 0.527 0.506 0 .45510 .352 12.8 24.4 37.3 51.8 61.6 65.0 56.2 44.7 i 31.3

11523 1352.4 1918.8 2063.41 2113.3! 1933.6 1557.2 1332.1i 1073.8 0.521 0.491 0.584 0 .57410 .567 0.569 0.525 0.554 0.539 55.2 60.6 67.8 74.8 80.9 82.3 79.1 69.8 59.8

681.9 1207 1443.9 1928.41 2102.61 1840.6 1410.3 997 526.6 0.383 0.497 0.464 0-543 I 0"559 i 0.559 0.524 0.491 0.351 30.9 39.4 50.2 62.4 72 7 i 75.1 68.5 57.4 44.0

i 2129.1 0.492 0.520 0.514 . ~ " i 0 .566] 0.588 0.606 0.562 0.510 36.5 45.9 57.7 75.9 79.4 71.9 61.4 46.1

746.5 1112.5 1480.8 1839.1 (2111) 2041.3] 1572.7 1189.3 919.5 479 0.401 0.447 0.470 0.515 (0.561) 0.555 0.475 0.433 0_441 0.302 33.7 ]42 .7 53.5 64.4 74.2 78 75.9 70.1 58 44.5

945 i 1504! 1959 2368.6 2619.2 2565.6 2287.8 1856.8 1288.5 795.6 0.490 ] 0.591 0.617 ] 0.662 0.697 0.697 0.687 0.664 0.598 0_477 52.1 56.8 63.1 69.6 75.7 81 79.4 76.7 67.8 57

1186.3] 1565.7 1975.6! 2126.5 2459.8 2400.7 2210.7 1841.7 1421 1065.3 o.598 o.6060618]059410 65572 o.652 o . 6 6 3 0 . 6 5 4 o.65o o.625

i38 .7 465 57.7 667 838 824 737 61.7 465

739.1 1086 1249.81 1732.81 1914 11884.5 1627.7 1303.3 891.5 473.1 0.431 0.456 0.406 0.489 0 .50810 .514 0.498 0.493 0.456 0.333 26.4 35.7 48.4 59.8 70.3 174.5 72.4 65.0 53.5 40.0

Dec

1051.6 0.704 39.4

152 0. 361 37.4

982.3 0.543 58.5

295.2 0. 332 43.9

751.6 0. 474 47.7

- 8 . 6

464.2 0. 547 18.4

482.3 0. 436 31.5

456.8 0. 442 33.1

442.8 0. 400 34.9

982.3 0. 488 65.2

398.9 0. 470 16.8

952 0. 586 54.0

427.3 0.371 32.8

590.4 0.457 35.8

430.2 0.351 34.0

550.5 0. 421 48.7

873.8 0. 652 36.8

379.7 0. 349 29.0

Vol. 7, No. 2, 1963 71

Page 20: Liu and Jordan 1963

East Wareham, Mass. Lat. 41°46 ' N. El. 18 ft

Edmonton, Alberta Lat. 53°35 ' N. El. 2219 ft

E1 Paso, Texas Lat. 31°48 ' N. El. 3916 ft

Ely, Nevada Lat. 39°17'N. El. 6262 ft

Fairbanks, Alaska Lat. 64049 ' N. El. 436 ft

For t Worth, Texas Lat. 32050 ' N. El. 544 ft.

Fresno, Calif. Lat. 36046 ' N. El. 331 ft.

Gainesville, Fla. Lat. 29039 , N. El. 165 ft

Glasgow, Mont . Lat. 48°13'N. El. 2277 ft

Grand Junct ion, Colorado Lat. 39007 , N. El. 4849 ft

Grand Lake, Colo. Lat. 40°15 ' N. El. 8389 ft

Great Falls, Mont . Lat. 47°29 ' N. El. 3664 ft

Greensboro, N. C. Lat. 36°05 , N. El. 891 ft

Griffin, Georgia Lat. 33015 , N. El. 980 ft

Hat teras , N. C. Lat. 35°13'N. El. 7 ft

Indianapolis, Ind. Lat. 39°44 , N. El. 793 ft

Inyokern, Calif. Lat. 35039 , N. El. 2440 ft

I thaca, N. Y. Lat. 42027 ' N. El. 950 ft

Lake Charles, La. Lat. 30°13 , N. El. 12 ft

Lander, Wyo. Lat. 42°48 , N. El. 5370 ft

i Jan

504.4 / ~ t 0.398

32.2

~t 331.7 0.529

to 10.4

H 1247.6 t 0.686

to 47.1

i 871.6 tK0 t I 0"618

27.3

/ t 66 /~t 0.639 to - 7 . 0

H 936.2 t 0.530

to 48.1

712.9 K t 0.462

47.3

/4 1036.9 Rt 0.535 to 62.1

/ t 572.7 Kt 0.621

13.3

B 848 /£t 0.597 to 26.9

/ t 735 /£t 0.541 to 18.5

B 524 /(~ 0.552 to 25.4

H 743.9 t 0.469

to 42.0

/ t 889.6 /~t 0.513 to 48.9

/ t 891.9 /~t 0.546 to 49.9

/ t 526.2 R~ 0.380 to 31.3

/ t !1148.7 R~ 0.716 l0 47.3

H 434.3 Kt 0.351 to 27.2

H 899.2 Kt 0.473 to 55.3

H 786.3 Kt 0.65 to 20.2

Feb

762.4 0.431 31.6

652.4 ! 0.585 14

1612.9 0.714 53.1

1255 0.660 32.1

283.4 0.556 0.3 !

M a r

1132.1 0.469 39.0

1165.3 0.624 26.3

2048.7 0.730 58.7

1749.8 0.692 39.5

860.5 0.674 13.0

1198.5' 1597.8 0 . 5 4 1 0 . 5 7 7 52.3 59.8

1652.8

Apr M a y

1392.611704.8 0 .449]0 .480 48.3 58.9

1541.71 1900.4 0 .56410.558 42.9 55.4

24472 . ~2673 0 . 7 4 1 0 . 7 4 3 67.3 75.7

2103.32322.1 0.664 0.649 48.3 57.0

1481.2 1806.2 0 . 6 4 7 6 . 5 4 6 32.2 50.5

1829.12105.1 0.556 0.585 68.8 !75.9

2049.412409.2 1116.6 0.551 53.9

1324.7 0.56 63.1

965.7 0.678 17.3

1210.7 0.633 35.0

1135.4 0.615 23.1

o.632io.638lo.672 I 59.1 ~i 65.6 73.5

0.568[0.587 87 67.5 ~ 72.8

1437.61 1741.3[ 2127.3 0 . 6 7 2 ! 0 . 5 9 7 1 0 . 6 1 1 311 i478 !59.3 1622.9 i 2002.2 2300.3 0.643 10 .632!0 .643 44.6 55.8 :!663

1579.31 1876.7 i 1974.9 0.637 : 0 .59710.553 28.5 39.1 !48.7

J u n J u l y i Aug

1958.311873.8[ 1607.4! 0 . 5 2 0 1 0 . 5 1 1 0 . 4 8 9 67.5 74.1 72.8

1914.4] 1964.9 0.514[ 0.549 61.3 66.6

2731 I 2391.1 0.733 t 0.652 84.2 84.9

2649 2417 0.704 0.656 65.4 74.5

1970.1 1702.9 0.529 0.485 62.4 63.8

2437.6 i 2293.3 0 .65410.624 84.0 87.7

2641.71 2512.2 0703 i 0.682 80.7 ~ 87.5 i 1960.911895.6 0.53 0.519 83.4 83.8

2 2 6 1 6 2 4 1 4 .7 1 9 8 4 .~ 0.69: 0.666 0.630 67.3 76 73.2

2 6 4 5 4 2 5 1 7 . 7 2 1 5 7 . 2 0.70 0.690 0.65 75.7 82.5 79.6

2 3 6 9 7 2 1 0 3 .3 1 7 0 8 .~ 0.63 0.572 0.516 56.6 62.8 (31.5

1528 0.506 63.2

2350.5 0.669 83.4

2307.7 0.695 72.3

1247.6 0.463 58.3

2216 .61880 .81476 0.653 0.634 0.612 88.6 81.3 71.5

2300.7 i 1897.811415.5 0.686 [0.665 ]0.635 84.9 78.6 68.7

1873.8 1615.1 1312.2 0.547 0.529 0.515 84.1 82 75.7

1531 0.629 61.2

1957.5 0.705 71.4

1715.8 0.626 55.5

Sep Oc t N o v ' _ _ _ _

1363.81 996 .7[636 .2 0 .50810 .49610 .431 (35.9 [56 146 1113.: 704.4 I 413.6 0 . 5 0 6 0 . 5 0 4 1 0 . 5 1 0 54.2 44.1 26.7

2o77 1324.7 0.693 0.647 78.5 69.0 56.0

1935 1 4 7 3 1078.6 0.696 0.691 0.658 63.7 52.1 39.9

699.6 323.6 104.1 0.419 0.416 0.47 47.1 29.6 5.5

1147.6 0.576 58.8

906.6 0.512 57.3

1169.7 0.537 67.2

997 574.9 0.593 I 0.516 49.2 31.0

1394.81 969.7 0 .65410 .59 58.3 42.0

1212.2 775.6 0.583 0.494 45.2 30.3

869.4 0.596 27.6

1031.7 0.499 44.2

1135.8 0.517 51.0

1184.1 0.563 49.5

797.4 0.424 33.9

1554.2 0.745 53.9

755 0.435 26.5

1145.7 0.492 58.7

1146.1 0.672 26.3

1369.7 i' 1621.41970.82179.3 0.565 0.580

47.7 64.3 35.60"63110"551 i 5 7 . 5 1

1323"21i~!! i~! i~ 5 2 1 1 1 '4 0.499 i 83 0.563 51.7 i 78.0

1450.91923.6 0.528 0.586 59.1 66.7

1590.42128 0.593 0.655 54.7 !61.5

i

1184.1 i 1481.2 0 .47210.47 43.0 !54.1

2136.9i 2594.8 °8°3i d 6 59.1

2163.1 0.601 74.6

2376.4 0.661 69.9

1828 0.511 64.9

2925.4 0.815 73.5

1074.9 1322.91 1779.3 ~645 0.428 [ 0.502

48.4 59.6

1487.4', 1801.81 2080.4 0.521 0.542 0.578 63.5 70.9 77.4

1638 1988.5 2114 0.691 0.647 0.597 34.7 45.5 56.0

2383 0.656 73.8

2033.9 0.552 80.2

2176 2064.9 0.583 0.562 81.2 83.0

2438 2334.3 0.652 0.634 77.2 80.0

2042 2039.5 0.543 0.554 74.8 79.6

3108.81 2908.8 0.830 ! 0.790 80.7 i 87.5

i !i8 2213.31968.6 0.597 0.538 83.4 84.8

2492.22438.4 0 . 6 6 2 0 . 6 6 5 65.4 74.6

1986.3 0.627 71.3

1810.3 0.538 78.9

1961.2 0.578 82.2

2085.~ 0.619 79.8

1832.1 0.552 77.4

2759.4 0.820 84.9

1736.9 0.530 71.9

1910.31678.2 0 .558]0 .553 85.0 81.5

~!~960.6471712"9

72.5 61.4

1536.5984.9 575.3 0.626 0.574 0.503 60.6 51.4 38.0

1517 .31202 .6908 .1 0.527 ~ 0.531 0.501 7 3 . 9 6 2 . 7 51.5

1605 .91352 .41073 .8 0.543 0.565 0.545 78.4 68 57.3

1758.31 1337.6 1053.5 0 .60510 .58 0.566 76.7 67.9 59.1

0.520 0.413 59.3 44.2

g4i 42 1819.21 1370.1 0 . 7 9 5 0 . 7 4 3

78.6 68.7 57.3

1320.3918.4 466.4 0.497 0.465 0.324 64.2 53.6 41.5

1505.51122.1 0.597 0.524 73.8 62.6

1301.8837.3 0.666 0.589 48.3 33.4

Dec

521 0.461 34.8

245 0.492 14.0

1051.6 0.626 48.5

814.8 0.64 31.1

20.3 0.458 --6.6

913.6 0.563 50.8

616.6 0.44 48.9

919.5 0. 508 62.4

428.4 0.548 18.6

793.4 0.621 31.4

660.5 0. 542 22.6

420.7 0.518 29.1

690.8 0.479 43.2

781.5 0.487 49.4

798.1 0.535 51.3

491.1 0.391 33.4

1094.4 0.742 48.9

370.8 0.337 29.6

875.6 0.494 56.9

694.8 0.643 23.8

72 Solar Energy

Page 21: Liu and Jordan 1963

' Jan Feb Mar Apr May Jun July Aug Sep

Las Vegas , Nev . / q 2799 2 2524 I 2342 I 2062 I La t . 36°05 ' N. /~ t 0.746 I 0.685 I 0.697 I 0.716 ] El. 2162 f t to 88.2 ! 95.0 92.9 85.4 !

Lemon t , I l l inois / t (590) 879 1255.7 1481.5 1866 2041.T,i 1990.8i 1836.9 1469.4 Lat . 41040 , N. Kt (0.464) 0.496 0.520 0.477 0.525 0.542 i 0.542 0.559 0.547

i El. 5 9 5 f t , to 28.9 30.3 39.5 49.7 59.2 70.8 ! 75.6 74.3 67.2 i

/~r/~t i _ _ - - 1834.7 2171.2 76.2 2246.51 2064.9 1775.6 Lat.Lexington'38°02 'Ky'N. / ) ! - - - - 0.575 0.606 0 . 6 1 0 1 0 . 6 1 9 0.631 H . 979 f t to ! 36.5 , 38.8 47.4 57.8 (}7.5 79.8 78.2 72.8

Lincoln , Neb . 712.5 ] 955.7 ] 1299.61 1587.81 1856.1 2040.61 2011.41 1902.6 1543 Lat . 40°51 ' N. i /~t 0.542 ] 0.528 ] 0.532 [ 0..507 ] 0.522 0.542 ] 0.547 ] 0.577 0.56~ El. 1189 ft to 27.8 3 2 . 1 ! 42.4 55.8 I 65.8 76.0 82.(3 80.2 71.5

L i t t l e Roek , Ark. H 704.4 974.2 1335.8 1669.4 1960.11 2091.5] 2081.21 1938.7 1640 Lat . 34°44 ' N. , 0.424 0.458 I 0.49(3 I 0.513 I 0.545 I 0.559 I 0.566 0.574 0.5(3 El. 265 ft. ~ 44.6 48.5 ] 56.0 ] (35.8 73.1 7(}.7 85.1 84.6 78.3

Los Angeles , Calif . (WBAS) / t 930.6 1284.1 1729.5:1948 219(3.7 2272.3] 2413.6] 2155.3 1898.1 Lat . 33056 , N. /~t 0.547 0.596 0.635 i 0.5951 (].61(] ! 0 608 0.657 I 0.635 0.641 El. 99 t0 56.2 5(3.9 59.2 ] 61.4 64.2 66.7 ] 69.6 70.2 69.1

Los Angeles , Calif . (WBO) H 911 8 1223 (] 1640 9 ~ 1866 8 2061 2: !2428.4] Lat . 34°03 ' N. / ( , ~J.5.~8 0.51~S 0.6&10.57i10.57,~i 2259

2198.9 1891.5 0.605 066 I o.648 0.643

to 57.9 59.2 (31.8 64.3 67.(3 70.7 75.8 76.1 74.2

Mad i son . Wis. It 564.6 812.2 1232.1 i 1455.3] 1745.41 2031.7] 2046.5] 1740.21 1443. / ] La t . 43008 , N. /~t 0.49 0.478 0.522 I 0.474 0.493 0.540 ] 0.559 ] 0.5:34 ] 0.549 El. 866 f t &, 21.8 24.6 35.3 149.0 161.0 17o.9 ! 7 6 . 8 74.4 65.6

M a t a n u s k a , Alaska /7 119.2 345 - - 1327.6 1(328.4 1727.6 1526.91 1169 I 737.3 Lat . 61030 ' N . / ~ 0.513 0.503 0.545 0.494 0 . 4 6 6 0.434 ] 0.419 ] 0.401 El. 180 ft to 13.9 21.0 27.4 , 38.6 50.3 57.6 60.1 ] 58.1 50.2

I ,at . 42°23 , N. /-it 0.353 0.4(34 0.527 0.584 0.625 0 . 6 4 8 0.710 .(') 0.628 El. 1329 ft t0 3,(].4 45.4 50.8 5(}.3 63.1 69.4 7(3.9 11118 (39.6

I

Mi'~mi, F lo r ida It 1292.2 1554.(! 1828.8! 2020.6 2068.(] 1991.5 1992.6 . . 1646.8 Lat . 25°47 ' N. K~ 0.604 0.(316 0.612 0.(300 0.578 0.545 ~ 0.552 0.54!) I 0.525 El. 9 ft t0 71.6 72.0 73.8 77.0 79.9 82.9 84.1 84.5 i 83.3

Mid l 'md , Texas / ) 10(3(3.4 1784.81 2036.1 2301.1 2317.71 2301.8 2193 i 1921.8 L,tt . 31°56 , N. El. 2854 ft

Nashvi l l e , Tenn . L,~t. 36007 , N. El. 605 ft

New1)ort , R. I. La t . 41°29 ' N. El. 60 f t

New York, N. Y. Lat . 40046 , N. El. 52 f t

Oak Ridge , Tenn . Lat . 36°01 ' N. El. 905 ft

Ok lahom a Ci ty , O k la homa La t . 35024 , N. El. 1304 ft

O t t awa , Onta r io Lat . 4 5 ° 2 0 ' N . El. 339 f t

Phoen ix , Ariz. Lat . 33026 , N. El. 1112 ft

P o r t l a n d , Ma ine La t . 43039 ' N. El. 63 f t

K t to

t 0

J~t to

B

t o

H /~ t t0

H /?, to

f/ 2

/~ t to

R, to

0. 587 47.9

589.7 0.37;3 42.6

565.7 0.438 29.5

539.5 0.40(] 35.0

(}04 0.382 41.9

938 0.580 40.1

539.1 0.499 14.6

1126.6 0.65 54.2

565.7 0.482 23.7

1345.7 0.596 52.8

907 0.440 45.1

856.4 0.482 32.0

790.8 0.435 34.9

895.9 0.435 44.2

1192.6 1534.3 0.571 0.57(3 45.0 53.2

852.4 1250.5 0.540 0.554 15.6 27.7

1514.7! 1967.1 0.691~ 0.71(3 58.8 6 4 . 7

874.5 i 1329.5 0.524 0.569 24.5 34.4

0. 638 1 60.0

1246.8 0.472 0.514 52.9 63.0

1231.7 1484.8 0.507 0.477 39.6 48.2

1189.4 142(3.2 0.480 0.455 43.1 52.3

1241.71 1689.6 0.471 0.524 51.7 61.4

0 . 6 1 7 0.(3;39 68.8 77.2

1(}(32.3 1997 0.55(3 71.4

1849 0. 520 58.6

1738.4 0.488 63.3

0. (322 0.628 83.9 85.7

2149.4 2079.7 0565

201:].2 832 1942.:

0 53(3 0.529 67.0 I 73.2

1994.1 1938.7 0.53 ~ 0. 528 72.2 7(}. 9

i 1849.4 i 2005.11 2355 0.570 I 0.558 I 0.629 63.6 80.(3 71.2 I

1506.61 1857.2:2084.5 0.502 0.529 i 0.554 43.3 57.5 67.5 ! 2388.2! 2709.6 2781.5 0.728 0.753 0.745 72.2 80.8 89.2

i 1528 4' 1923 21 2017.3 0.50010.54410.536 44.8 5 5 . 4 ! 65.1

1942.8 2066.4i 1972.3 0 . 5 4 1 1 0 . 5 5 1 0.536 69.8 177.8 80.2

2273.8 0.618 85.5

2045.4 0.560 71.9

2450.5 0.667 94.6

2095.6 0.572 71.1

0.643 85.0

I 18(32.7! 0.554 81.9

1687.1 0.513 72.3

i 1605.91 0.486 75.3

1795.6 0.534 78.8

2211 0.656 85.4

1752.4 0.546 69.8

2299.6 0.677 I 92.5 [

i 1799.2' 0.554 69.7

0. 642 78.9

1600.7 O. 556 76.6

1411.4 0.524 6(3.7

1349.4 0.500 69.5

1559.8 0.542 74.5

1819.2 0.628 77.4

1326.6 0.521 61.5

2131.3 0.722 87.4

1428.8 0.546 61.9

I Oct ! Nov Dec

1602.6 1190 964.2 0.704 0.657 0.668 71.7 57.8 50.2

1015.5 (639) (531) 0.506 (0.433) (0.467 57.6 43.0 30.6

1315.8 - - 681.5 0.604 - - 0.513 61.2 47.6 38.5

1215.8 773.4 643.2 0.596 0.508 0.545 59.9 43.2 31.8

1282.6 913.6 701.1 0.552 0.484 0.463 67.9 ! 5 4 . 7 46.7

1372.7 i 1082.3 901.1 0.574 I 0.551 0.566 66.1 ()2.6 58.7

1362.31 1053.1 877.8 0.578 I 0.548 0.566 69.6 I 65.4 60.2

993 555.7 495.9 0.510 0.396 0. 467 53.7 37.8 25.4

373.8 142.8 56.4 0.390 0.372 0.364 37.7 22.9 13.9

1043.7 558.7 346.5 0.526 0.384 0. 313 58.7 47.1 40.5

143(3.5 1321 1183.4 0. 534 0. 559 0. 588 80.2 75.6 72.6

1470.8 1244.3 1023.2 0.600 0.609 0.611 70.3 56.6 49.1

1223.(3 823.2 614.4 0.540 0.454 0.426 (35.4 52.3 44.3

1035.4 656.1 527.7 0.512 0.44 0.460 5(3.2 46.5 34.4

977.8 598.1 476 0.475 0. 397 0. 403 59.3 48.3 37.7

1194.1 796.3 (310 0.527 0.438 0.422 (32.7 50.4 42.5

1409.( 1085.6 897.4 0.614 0.588 0.608 66.5 52.2 43.1

826.9 458.7 408.5 0.450 0.359 0.436 48.9 35 19.6 i

1688.9 1290 1040.9 0.708 0.657 0.652 75.8 63.6 56.7

1035 591.5 507.7 0.539 0.431 0.491 51.8 40.3 28.0

Vol. 7, No. 2, I963 73

Page 22: Liu and Jordan 1963

Rapid City, S. D. Lat. 44°09 ' N. El. 3218 ft

Riverside, Calif. Lat. 33°57 ' N. El. 1020 ft

Saint Cloud, Minn. Lat. 45035 ' N. El. 1034 ft

Salt Lake City, Utah Lat. 40046 , N. El. 4227 ft

San Antonio, Tex. Lat. 29°32 ' N. El. 794 ft

Santa Maria, Calif. Lat. 34°54 , N. El. 238 ft

Sault Ste. Marie, Michigan Lat. 46°28 ' N. El. 724 ft

Sayville, N. Y. Lat. 40°30 ' N. El. 20 ft

Schenectady, N. Y. Lat. 42°50 ' N. El. 217 ft

Seattle, Wash. Lat. 47°27'N. El. 386 ft

Seattle, Wash. Lat. 47°36 ' N. El. 14 ft

Seabrook, N. J. Lat. 39°30'N. El. 100 ft

Spokane, Wash. Lat. 47°40'N. El. 1968

State College, Pa. Lat. 40048 , N. El. 1175 ft

Stillwater, Okla. Lat. 36009 , N. El. 910 ft

Tampa, Fla. Lat. 27°55 ' N. El. 11 ft

Toronto, Ontario Lat. 43°41'N. El. 379 ft

Tucson, Arizona Lat. 32°07 ' N. El. 2556 ft

Upton, N. Y. Lat. 40°52'N. El. 75 ft

Washington, D. C. (WBCO) Lat. 38°51 ' N. El. 64 ft

Winnipeg. Man. Lat. 49°54 ' N. El. 786 ft

Ht to

/7

to

/7 R~ to

/7 K~ to

/7 K~ to

/7 K~ to

/7 K~ to

K~

J a n

687.8 0.601 24.7

999.6 0.589 55.3

632.8 0.595 13.6

622.1 0.468 29.4

1045 0.541 53.7

983.8 0.595 54.1

Feb Mar

1032.51 1503.7 I 0.627 0.649 27.4 34.7

1335 1750.5 0.643 I 0.617 60.6

57.0

976.7 1383 0.629 0.614 16.9 29.8

986 1301.1 0.909 0.529 36.2 44.4

1299.2 1560.1 0.550 0.542 58.4 65.0

1296.3 1805.! 0.613 0.671 55.3 57.6

843.9 1336.~ 0.560 0.606 16.2 25.6

936.2 1259.~ 0.511 0.510 34.9 43.1

753.5 1026.( 0.441 0.433 24.6 34.9

520.6 992.2 0.355 0.456 45.0 48.9

471.6 917.3 0.324 0.423 42.9 46.9

854.2 1195A 0.453 0.476 37.6 43.9

837.6 1200 0.579 0.556 31.7 4O.5

749.1 1106.{ 0.413 0.451 31.4 39.8

1081.5 1463.8[ 0.527 0.555 I 45.6 53.8 I

1461.2 1771.9 i 0.600 0 6061 65.7 68.8

674.5 1088.9 0.406 0.467 I 26.0 34.2

1453. 0.646 57.3 I 872.7 1280 0.483 0.52: 34.9 43.1

901.5 1255 0.470 0.491 39.6 48.1

835.4 1354 0.636 0.66 7.1 21.3

Apr

1807 0.59 48.2

1943 0.59 65.0

1598 0.53 46.2

1813.3 0.578 53.9

1664.6 0.500 72.2

2067.9 0.636 59.5

1559.4 0.526 39.5

1560.5 0.498 52.3

1272.3 0.413 48.3

1507 0.510 54.1

1375.6 0.468 51.9

1518.8 0.481 54.7

1764.6 0.602 49.2

1399.2 0.448 51.3

1702.6 0.528 64.2

2016.2 0.602 74.3

1388.2 0.455 46.3

2434.7 0.738 69.7

1609.9 0.514 52.3

1600.4 0.504 57.5

1641.3 0.574 40.9

M a y

2028 I 0.574 I 58.3

2282.~ 0.635 69.4

1859.4 0.530 58.8

63~

2024.7 0.563 79.2

2375.¢ 0.661 61.2

1962.3 0.560 52.1

Jun July Aug Sep Oct

2193.71 2235.8 2019.9 1628 1179.3 0.58310.612 0.622 0.628 0.624 67.3 76.3 75.0 64.7 52.9

2492.61 2443.5 2263.8 1955.3 1509.6 0 .66710.665 0.668 0.665 0.639 74.0 81.0 81.0 78.5 71.0

2003.31 2087.81 1828.4] 1369.4 890.4 0.533 [ 0 . 5 7 3 ] 0 . 5 7 0 1 0 . 5 3 9 0.490 68.5 174.4 71.9 62.5 50.2

i

- - - - - - i 1689.3 1250.: 0.621 0.610

71~ 8 1 ~ 79~ 168.7 57.0

814.8 2364.2] 2185.2] 1844.61 1487.4 0.220 o.647Io.6371o.6o3]o.584 85.0 87.4 87.8 I 82.6 74.7

2599.6 2540.6 2293.3 1965.7 / 1566.4 0.695 0.690 0.678 0.67410.676 63.5 65.3 65.7 65.9 64.1

2064.2 2149.4 1767.9 1207 809.2 0.549 0.590 0.554 0.481 0.457 61.6 67.3 66.0 57.9 46.8

Nov

763.1 0.566 38.7

1169 0.606 63.1

545.4 0.435 32.1

42.5

1104.4 0.507 63.3

1169 0.624 60.8

392.2 0.323 33.4

1857.2 2123.2 2040.9i 1734.7 1446.8 i 1087.4 697.8 0.522 0.564 0.555 0.525 0.53010.527 0.450 63.3 72.2 76.9 75.3 69.5 i 59.3 48.3

1553.1 1687.8 1662.3 1494.8 1124 7 820.6 436.2 0.438 0.448 0.454 0.458 0.42610.420 0.309 61.7 70.8 76.9 73.7 64.6 53.1 40.1

1881.5 1909.9 2110.7 1688.5 1211.81 702.2 386.3 0.538 0.508 0.581 0.533 0.49210.407 0.336 59.8 64.4 68.4 67.9 63.3 56.3 48.4

1664.9'i 1724 1805.1 1617 1129.11 638 325.5 0.477 0.459 0.498 0.511 0.459 I 0.372 0.284 58.1 62.8 67.2 66.7 61.6 I 54.0 45.7

I

1800/ 1964.6 1949.8 1715 1445.71 1071.9 721.8 0.504 0.522 0.530 0.517 0.524 0.508 0.449 64.9 74.1 79.8 77.7 69.7 61.2 48.5

2104.4 2226.5 2479.7 2076 1511 844.6 486.3 0.603 0.593 0.684 0.656 i 0.616 0.494 0.428 57.9 64.6 73.4 71.7 i 62.7 51.5 37.4

1754.61 2027.6 1968.2 1690 1336.1 1017 580.1 i

0.493 I 0.539 ~ 0.536 0.51210.492 0.496 0.379 63.4 71.8 75.8 73.4 66.1 55.6 43.2

1879.3 2235.8 2224.3 2039.: I 1724.3 1314 991.5 0.523 0.596 0.604 0.607[ 0.599 0.581 0.548 71.6 81.1 85.9 85.9 77.5 67.6 52.6

2228 2146.5 1991.9 1845.~ I 1687.8 1493.3 1328.4 0.620 0.583 0.548 0.53710.546 0.572 0.590 79.4 83.0 84.0 84.4 182.9 77.2 69.6

I

1785.2 1941.; 1968.6 1622.51 1284.1 835 458.3 0_506 0.516 0.539 0 .50010 .493 0.438 0.336 58 68.4 73.8 71.8 64.3 52.6 40.9

2601.~ 2292.2 2179.7 2122.5 1640.9 1322.1 - - 0.698 0.625 0.6401 0.710 0.672 0.650

78.0 87.0 90.1 87.4 84.0 73.9 62.5

1891.5 2159 2044.6 1789.~ 1472.7 1102.6 686.7 0.532 0.574 0.557 0.542 0.542 0.538 0.448 63.3 72.2 76.9 75.3 69.5 59.3 48.3

1846.8 2080.81 1929.9 17122 1446.1 1083.4 763.5 0.516 0.553 / 0.524 0.516 0.520 0.506 0.464 67.7 76.2 79.9 77.9 ! 72.2 60.9 50.2

1904.4 1962 ] 2123.6 1761.2 1190.4 767.5 444.6 0.550 0.524 0.587 0.567 0.504 0.482 0.436

71.9 69.4 58.6 25.2 55.9 65.3 45.6

Dec

590.4 0.588 29.2

979.7 0.626 57.2

463.1 0.504 18.3

552.8 0.467 34.0

954.6 0.528 56.5

943.9 0.627 56.1

359.8 0.408 21.9

533.9 0.447 37.7

356.8 0.331 28.0

239.5 0.292 44.4

218.1 0.269 41.5

522.5 0.416 39.3

279 0.345 30.5

443.9 0.376 32.6

783 0.544 43.9

1119.5 0.589 65.5

352.8 0.346 30.2

1132.1 0.679 56.1

551.3 0.467 37.7

594.1 0.460 40.2

345 0.503 10.1

74 Solar Energy