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SOLAR ENGINEERING
OF THERMAL PROCESSES
SOLAR ENGINEERING
OF THERMAL PROCESSES
Second Edition
JOHN A, DUFFIE
Emeritus Professor of Chemical Engineering
WILLIAM A. BECKMAN
Professor of Mechanical Engineering
Solar Energy Laboratory
University of Wisconsin-Madison
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A Wiley-Interscience Publication
JOHN WILEY & SONS, INC.
New York Chichester Brisbane Toronto Singapore
xdy Prefaçe to the First Edition
several drafis of the book which have been typed by our student helpers at the
laboratory: it has often been difficult work, and their persistence, skill and good
humor have been tremendous.
Not the least, we thank our patient families for their forbearance during the
lengthy process of putting this book together.
JOHN A. DUFFIE
WILLIAM A. BECKMAN
Madison, Wisconsin
June 1980
CONTENTS |
PART 1 - FUNDAMENTALS 1
Introduction, 1
1. SOLAR RADIATION , 3
1] TheSun,3 [
1.2 The Solar Constant, 5 '
1.3 Spectral Distribution of Extraterrestrial Radistion, 7
1.4 Variation of Extraterrestrial Radiation, 9 |
1.5 Definitions, JO
1.6 Direction of Beam Radiation, 13
1.7 Angles for Tracking Surfaces, 21
1.8 Ratio of Beam Radiation on Tilted Surface tothat on
Horizontal Surface, 25 !
1.9 Shading, 32
1.10 Extraterrestrial Radiation on a Horizontal Surfice, 39
111 Summary, 44
References, 44
|
2. AVAILABLE SOLAR RADIATION : 46
2.1 Definitions, 46 ;
22 Pytheliometers and Pyrheliomenrio Scales, 47|
2.3 Pyranometers, 51
2.4 Measurements of Duration of Sunshine, 57
2.5 Solar Radiation Data, 58
2.6 Atmospheric Attenuation of Solar Radiation,
2.7 Estimation of Average Solar Radiation, 68
28 Estimation of Clear Sky Radiation, 73
2.9 Distribution of Clear and Cloudy Days and flours, 77 1
2.10 Beam and Diffuse Components of Hourly Radiation, 80
2.11 Beam and Diffuse Components of Daily Rafiation, 83 |
2.12 Beam and Diffuse Components of MonthlyRadiation, 8
2.13 Estimation of Hourly Radiation from Daily Data, 87
em no
Contente
2.14 Radiation on Sloped Surfaces, 91
2.15 Radiation on Sloped Surfaces - Isotropic Sky, 94
2.16 Radiation on Sloped Surfaces - Anisotropic Sky, 96
2.17 Radiation Augmentation, 102
2.18 Beam Radiation on Moving Surfaces, 107
2.19 Average Radiation on Sloped Surfaces - Isotropic Sky, 109
2.20 Average Radiation on Sloped Surfaces - The K-T Method, 113
2.21 Effects of Receiving Surface Orientation, 119
2.22 Utilizability, 122
2.23 Generalized Utiizability, 125
2.24 Daily Utilizability, 134
225 Summary, 141
References, 141
SELECTED HEAT TRANSFER TOPICS 147
3.1 The Electromagnetic Spectrum, 147
3.2 Photon Radiation, 148
3.3 The Blackbody - a Perfect Absorber and Emitter, 148
3.4 Planck's Law and Wien's Displacement Law, 149
3.5 Stefan-Bolizmann Equation, 151
3.6 Radiation Tables, 15)
3.7 Radiation Intensity and Flux, 154
3.8 Infrared Radiation Exchange Between Gray Surfaces, 156
3.9 Sky Radiation, 157
3.10 Radiation Heat Transfer Coefficient, 158
3.11 Natural Convection Between Flat Parallel Plates, 160
3.12 Convection Suppression, 164
3.13 Vee-Corugated Enclosures, 167
3.14 Heat Transfer Relations for Internal Flow, 169
3.15 Wind Convection Coefficients, 173
3.16 Heat Transfer and Pressure Drop in Packed Beds, 176
3.17 Effectiveness - NTU Calculations for Heat Exchangers, 178
References, 181
RADIATION CHARACTERISTICS OF OPAQUE MATERIALS
184
4.1 Absorptance and Emittance, 185
4.2 Kirchotfs Law, 187
43 Reflectance of Surfaces, 189
4.4 Relationships Among Absorptance, Emittance, and Reflectance, 193
4.5 Broadband Emíttance and Absorptance, 194
4.6 Calculation of Emittance and Absorptance, 196
47 Measurement of Surface Radiation Properties, 198
48 Selective Surfaces, 199
Contents xvit
4.9 Mechanisms of Selectivity, 204
4.10 Optimum Properties, 208
4.tt Angular Dependence of Sotar Absorptance, 209
4.12. Absorptance of Cavity Receivers, 210
4.13 Specularly Reflecting Surfaces, 211
References, 213
5. RADIATION TRANSMISSION THROUGH GLAZIN
ABSORBED RADIATION 216
5.1 Reflection of Radiation, 216
5.2 Absorption by Glazing, 220
5.3 Optical Properties of Cover Systems, 221
5.4 Transmittance for Diffuse Radiation, 226
5.5 Transmittance-Absorptance Product, 228
5.6 Angular Dependence of (to), 230
5.7 — Spectral Dependence of Transmittance, 231
5.8 Effects of Surface Layers on Transmittance, 234
5.9 Absorbed Solar Radiation, 235
S.J0 Monthly Average Absorbed Radiation, 239
5.11 Absorptance of Rooms, 246
5.12 Summary, 248
References, 248
6. FLAT-PLATE COLLECTORS 250
6.1 Description of Fiat-Plate Collectors, 250
6.2 Basic Flat-Plate Encrgy Balance Equation, 251
6.3 Temperature Distributions im Flat-Plate Collectors, 252
64 Collector Overall Heat Loss Coefficient, 254
6.5 Temperature Distribution Between Tubes and the Collector Efficiency
Factor, 268
6.6 Temperature Distribution in Flow Direction, 276
6.7 | Coltector Heat Removal Factor and Flow Factor, 277
6.8 Critical Radiation Level, 281
6.9 Mean Fluid and Plate Temperature, 282
6.10 Effective Transmittance-Absorptance Product, 283
6.11 Effects of Dust and Shading, 286
6.12 Heat Capacity Effects in Flat-Plate Collectors, 287
6:13 Liquid Heater Plate Gcometries, 291
6.14 Air Heaters, 296
6.15 Measurements of Collector Performance, 301
6.16 Collector Characterizations, 302
6.17 Collector Tests: Efficiency, Incidence Angle Madifier, and Time
Constant, 303
6.18 Test Data, 312
Contents
6.19 Thermal Test Data Conversion, 316
6.20 Flow Rate Corrections to Fa(To), and FpU,, 317
6.21 Flow Distribution in Collectors, 320
6.22 In Situ Coliector Performance, 322
6.23 Practical Considerations for Flat Plate Collectors, 323
6.24 Summary, 326
References, 327
CONCENTRATING COLLECTORS 330
7.1 Collector Configurations, 331
7.2 Concentration Ratio, 333
7.3 Thermal Performance of Concentrating Collectors, 336
7.4 Optical Performance of Concentrating Collectors, 341
7.5 Cylindrical Absorber Arrays, 343
7.6 Optical Characteristics of Nonimaging Collectors, 345
7.7 Orientation and Absorbed Energy for CPC Collectors, 351
7.8 Performance of CPC Collectors, 356
7.9 Linear Imaging Concentrators + Geometry, 358
7.10 Images Formed by Perfect Linear Concentrators, 362
7.11 Images from Imperfect Linear Concentrators, 367
7.12 Ray-Trace Methods for Evaluating Concentrators, 369
7.13 Incidence Angle Modifiers and Energy Balances, 370
7.14 Paraboloidai Concentrators, 376
7.15 Central Receiver Collectors, 377
7.16 Practical Considerations, 378
References, 379
ENERGY STORAGE 382
8.1 Process Loads and Solar Collector Outputs, 383
8.2 Energy Storage in Solar Process Systems, 384
8.3 Water Storage, 385
8.4 —Stratification in Storage Tanks, 388
8.5 Packed Bed Storage, 393
8.6 Storage Walls, 400
8.7 Seasonal Storage, 403
8.8 Phase Change Energy Storage, 405
8.9 Chemical Energy Storage, 410
References, 412
SOLAR PROCESS LOADS 414
9.1 Examples of Hourly Loads, 415
9.2 Hot Water Loads, 416
9.3 Space Heating Loads; Degree Days; Balance Temperature, 417
Contents ata
LO.
1.
9.4 Building Loss Coefiicients, 420
9.5 Building Energy Storage Capacity, 422
9.6 Cooling Loads, 423
9,7 Swimming Pool Heating Loads, 423
References, 426
SYSTEM THERMAL CALCULATIONS 427
10.1 Component Modeis, 428
10.2 Collector Heat Exchanger Factor, 429
10.3 Duct and Pipe Loss Factors, 43!
10.4 Controls, 434
10.5 Collector Arrays; Series Connections, 436
10.6 Performance of Partially Shaded Collectors, 439
10.7 Series Atrays with Sections with Different Orientation, 441
10.8 Use of Modified Collector Equations, 444
10.9. System Models, 447
10.10 Solar Fraction, 450
10.11 Summary, 45!
References, 452
SOLAR PROCESS ECONOMICS 453
1.1 Costs of Solar Process Systems, 454
11.2 Design Variables, 457
11.3 Economic Figures of Merit, 458
11.4 Discounting and Inflation, 461
1.5 Present Worth Factor, 463
11.6 Life Cycle Savings Method, 466
11.7 Evaluation of Other Economic Indicators, 471
118 The ?,, P; Method, 474
11.9 Uncertainties in Economic Analyses, 480
11.10 Summary, 483
References, 484
PART II - APPLICATIONS
12.
Introduction, 485
SOLAR WATER HEATING - ACTIVE AND PASSIVE 487
12.1 Water Heating Systems, 487
12.2 Freezing and Boiling, 492
12.3 Auxiliary Energy, 494
Part I
FUNDAMENTALS
In Part I, we Ireat the basic ideas and calculation procedures that must be understood
in order to appreciate how solar processes work and how their performance can be
predicted. The first five chapters are basic to the material in Chapter 6. In Chapter 6
we develop equations for a collector which give the usefu! output in terms of the
available solar radiation and the losses. An enesgy balance is developed which says,
in essence, that the useful gain is the (positive) difference between the absorbed solar
cnergy and the thermal losses.
The first chapter is concemed with the nature of the radiation emitted by the sun
and incident on the earth's atmosphere. This includes geometric considerations, i.e.,
the direction from which beam solar radiation is received and its angle of incidence on
various surfaces and the quantity of radiation received over various time spans. The
next chapter covers lhe effects of the atmosphere on the solar radiation, the radiation
data that are available, and how those data can be processed to get the information thai
we ultimately want — the radiation incident on surfaces of various orientations.
Chapter 3 notes a set of heat transfer problems that arise in solar energy
processes and is part of the basis for analysis of coltectors, storage units, and other
components.
The next two chapters treat interaction of radiation and opaque and transparent
materials, i.e., emission, absorptíon, reflection, and transmission of solar and long
wave radiation. These first five chapters lead to Chapter 6, a detailed discussion and
analysis of the performance of flat-plate collectors. Chapter 7 is concemed with
concentrating collectors and 8 with energy storage in various media. Chapter 9 is a
brief discussion of the loads imposed on solar processes and the kinds of information
that must be known in order to analyze the process.
Chapter 10 is the point a1 which the discussions of individual components are
brought together to show how solar process systems function and how their long-
term performance can be determined by simulations. The object is to be able to
quantitatively predict system performance; this is the point at which we proceed from
components to systems and see how transient system behavior can be calculated.
The last chapter in Part 1 is on solar process economics. ft concludes with a
method for combining the large number of economic parameters into two which can
be used to optimize thermal design and assess the eifects of uncertainties in an
economic analysis.
Chapter 1
SOLAR RADIATION
The sun's structure and characteristics determine the nature of the energy it radiates
into space, The first major topic in this chapter concemns the characteristics of this
energy outside of the earth's atmosphere, its intensity and spectrat distribution. We
will be concemed primariiy with radiation in à wavelength range of 0.25 to 3.0 um,
the portion of the electromagnetic radiation that includes most of the energy radiated
by the sun.
“The second major topic in this chapter is solar geometry, i.e., the position of the
sun in the sky, the direction in which beam radiation is incident on surfaces of
various orientations, and shading. The third topic is extraterrestrial radiation on a
horizontal surface, which represents the theoretical upper limit of solar radiation
available at the earth's surface.
An understanding of the nature of extraterrestriat radiation, the effects of
orientation of a receiving surface, and the theoretically possible radiation at the earth's
surface is important in understanding and using solar radiation data.
11 THE SUN
The sun is a sphere of intensely hot gaseous matter with a diameter of 1.39 x 10º m
and is, on the average, 1.5 x 10!! m from the earth. As seen from the carth, the sun
rotates on its axis about once every four weeks. However, it does not rotate as a
solid body; the equator takes about 27 days and the polar regions take about 30 days
for cach rotation.
“The sun has an effective blackbody temperature of 5777 K.! The temperature
in the central interior regions is variously estimated at 8 x 106 to 40 x 108 K and the
density is estimated to be about 100 times that of water. The sun is, in effect, a
continuous fusion reactor with its constituent gases as the “containing vessel" retained
by gravitational forces. Several fusion reactions have been suggested to supply the
Urhe effective blackbody temperature of 5777 K is the temperature of a blackbody radiating the same
amount of energy as does the sun. Other effective temperatures can be defined, for example, that
comesponding to the blackbody temperature giving the same wavelengih of maximum radiation as
solar radiation (about 6300 K).
;
:
4 Solar Radiation
energy radiated by the sun. The one considered the most important is a process in
which hydrogen (i.e., four protons) combines to form helium (i.e., one helium
nucleus): the mass of the helium nuclcus is less than that of the four protons, mass
having been lost in the reaction and converted to energy.
“The energy produced in the interior of the solar sphere at temperatures of many
millions of degrees must be transferred out to the surface and then be radiated into
space, A succession of radiative and convective processes occur with successive
emission, absorption, and reradiation; the radiation in the sun's core is in the x-ray
and gamma-ray paris of the spectrum, with the wavelengths of the radiation
increasing as the temperature drops at larger radial distances.
A schematic structure of the sun is shown ir Figure 1.1.1. It is estimated that
90% of the energy is generated in the region of O to 0,238 (where R is the radius of
the sun), which contains 40% of the mass of the sun. At a distance 0.7R from the
center, the temperature has dropped to about 130,000 K and the density has dropped
to 70 kg/m?; here convection processes begin to become important, and the zone from
0.7 to 1.0R is known as the convective zone. Within this zone the temperature
drops to about 5000 K and the density to about 105 kg/n?,
“The sun's surface appears to be composed of granules (irregular convection
cells), with dimensions from 1000 to 3000 km and with cell lifetime of a few
Corana T = — 108k
2 = very low
Chromosphere 7 = 5000 K +
Reversing layer
Hundred ot =
SA Phatosphere
iupper laver
of the convective
zone, saures of
most solar
/ TT âo% ot mass radiation)
18% 6! vamo
So at rear
| gerettud
0.23R 7
O a
OR T=-B40XIOK
p= 10kgim?
mu
Figure 1.1.1 The structure of the sun.
1.2 The Solar Constant 5
minutes. Other features of the solar surface are small dark arcas called pores, which
are cf the same order of magnitude as the convective cells, and larger dark areas
called sunspots, which vary in size. The outer layer of the convective zone is called
the photosphere. The edge of the photosphere is sharply defined, even though it is
of low density (about 10-4 that of air at sea level). It is essentially opaque, as the
gases of which it is composcd are strongly ionized and able to absorb and emit a
continuous spectrum of radiation. The photosphere is the source of most solar
radiation.
Outside of the photosphere is a more or less transparent solar atmosphere,
observable during total solar eclipse or by instruments that occult the solar disk.
Above the photosphere is a layer of cooler gases several hundred kilometers deep
called the reversing layer. Outside of that is a layer referred to as the chromo-
sphere, with a depth of about 10,000 km. This is a gaseous layer with temperatures
somewhat higher than that of the photosphere but with lower density. Still further
out is the corona, à region of very low density and of very high (108 K) temper-
ature. For further information on the sun's structure see Thomas (1958) or Robinson
(1966).
This símplified picture of the sun, its physical structure, and its temperature and
density gradients will serve as a basis for appreciating that the sun does not, in fact,
function as a blackbody radiator at à fixed temperature. Rather, the emitted solar
radiation is the composite result of the several layers that emit and absorb radiation of
various wavelengths. The resulting extraterrestrial solar radiation and its spectral
distribution have now been measured by various methods in several experiments; the
results are noted in the following two sections.
12 THE SOLAR CONSTANT
Figure 1.2.1 shows schematically he geometry of the sun-carth relationships. The
eccentricity of the carth's orbit is such that the distance between the sun and the earth
Sun 127x10'm
7900 mi
Sojar constant
| =1367 Wim?
Gocl = 483 But hr
92 Mm? hr
= 1495 x 10!!m
=9.3x10/mi
Distance al 17%
Figure 1.2.1 Sun-earth relationships.
10 Solar Radiation
Variation of the earth-sun distance, however, does lead to variation of
extraterrestrial radiation flux in the range of + 3%. The dependence of extraterrestrial
radiation on time of year is indicated by Equation 1.4.1 and is shown in Figure 14.1.
Gon = Gec[1+ 0.033 cos 3604) (1.4.1)
365
where G,, is the extraterrestrial radiation, measured on the plane normal to the
radiation on the ath day of the ycar.
15 DEFINITIONS
Several definitions will be useful in understanding the balance of this chapter.
Air Mass m The ratio of the mass of atmosphere through which beam
radiation passes to the mass it would pass through if the sun were at the zenith (i
directly overhead). Thus at sea level, m = 1 when the sun is at the zenith, and m =
2 for à zenith angle 8, of 60º. For zenith angles from 0º to 70º at sea level, to a
close approximation,
m= ljcos 8, (1.5.1)
For higher zenith angles, the effect of the earth's curvalure becomes sigaificant and
must be taken into account. For a more complete discussion of air mass, see
Robinson (1966), Kondratyev (1969), or Garg (1982).
Beam Radiation The solar radiation received [rom the sun without having
been scattered by the atmosphere. (Beam radiation is often referred to as dircct solar
radiation; to avoid confusion between subscripts for direct and dilfuse, we use the
term beam radiation)
Diffuse Radiation The solar radiation received from the sun after its
direction has been changed by scattering by the atmosphere. (Diffuse radiation is
referred to in some meteorological literature as sky radiation or solar sky radiation:
the definition used here wil) distinguish the diffuse solar radiation from infrared
radiation emitted by the atmosphere.)
Total Solar Radiation The sum of the beam and the diffuse solar radiation
on a surface.3 (The most common measurements of solar radiation are total
radiation on a horizontal surface, often referred to as global radiation on the
surface.)
Irradiance, W/m? The rate at which radiant energy is incident on à surface,
per unit area of surface, The symbol G is used for solar irradiance, with appropriate
subscripts for beam, diffuse, or spectral radiation.
3 Total solas radiation is sometimes used to indicate quantities integrated over all wavelengths of the
solar spectrum.
1.5 Definitions u
Irradiation or Radiant Exposure, Jjm? The incident energy per unit area
on a surface, found by integration of irradiance over à specified time, usually an hour
or a day. Insolation is a term applying specifically to solar energy irradiation. The
symbol /f is used for insolation for a day. The symbol / is used for insolation for
an hour (or other period if specified). The symbols H and ! can represent beam,
diffuse, or total and can be on surfaces of any orientation.
Subscripts on G, H, and 4 are as follows: o refers to radiation above the
carth's atmosphere, referred to as extraterrestrial radiation; b and d refer to beam
and diffuse radiation; T and n refer to radiation on a tilted plane and on a plane
normal to the direction of propagation. If neither 7 nor » appear, the radiation is on
a horizontal plane.
Radiosity or Radiant Exitance, W/m? The rate at which radiant energy
leaves a surface, per unit area, by combined emission, reflection, and transmission.
Emissive Power or Radiant Seif-Exitance, W/m? The rate at which
radiant energy leaves a surface per unit area, by emission onty.
Any of these radiation terms, except insolation, can apply to any specified
wavelength range (such as the solar energy spectrum) or to monochromatic radiation.
Insolation refers only to irradiation in the solar energy spectrum.
Solar Time Time based on the apparent angular motion of the sun across the
sky, wilh solar noon the time the sun crosses the meridian of the observer.
Solar time is the time used in al] of the sun-angle relationships; it does not
coincide with local clock time, Iris necessary to convert standard time to solar time
by appiying two corrections. First, there is a constant correction for the difference in
longitude between the observer's meridian (longitude) and the meridian on which the
local standard time is based.é The sun takes 4 minutes to transverse 1º of
longitude, The second correction is from the equation of time, which takes into
account the perturbations in the earth's rate of rotation which affect the time the sun
crosses the observer's meridian. The difference in minutes between solar time and
standard time is
Solar time — standard time = 4(Lg — Lic) 4 E (1.5.2)
where Ly i» lhe standard meridian for he local time zone. L,, is the longitude of
the location in question. and longitudes are in degrees west. i.e.. 0º <L < 360º,
The equation of time E (in minutes) is determined from Figure 1.5.1 or Equation
1.5.35 [from Spencer (1971). as cited by Igbal (19833).
E = 229,20.000075 + 0.001868 cos B — 0.032077 sin B
— 0014615 cos 28 — 0.04089 sin 28 ) (1.5.3)
2 Standard meridians for continental U.S. time zones are:
105ºW; and Pacific, 1209W.
5 AI equations use degrees, not radians.
ste. 75ºW; Central, 90ºW; Mountain.
Solar Radiation
Equation of time, min
o |
—0
NA
DJ FE MA MOIS ASOÕÃGNÕD
Month
Figure 15.1. The equation of time £ in minutes, às à function of time of year
where B=(n- 38 (1.536)
365
and n = day of the year. Thus 1 Sn < 365.
Note that the corrections for equations of time and displacement from the
standard meridian are in minutes and that there is a 60 minute difference between
daylight saving time and standard time. Time is usually specified in hours and
minutes. Care must be exercised in applying the corrections, which can total more
than 60 minutes.
Example 1.5.1
At Madison, WI, what is the solar time corresponding to 10:30 AM central time on
February 3?
Solution
In Madison, where the longitude is 89 4º, Equation 1.5.2 gives
tandard time + 4(90 — 89.4) + E
tandard time + 2.4 + E
Solar time
1.6 Direction of Beam Radiation 13
On February 3. n = 34, and from Equation 1.5.3 or Figure 1.5.1, E is —13.5
minutes, so the correction to standard time is —11 minutes. Thus 10:30 AM central
standard time is 10:19 AM solar time,
In this book all times are assumed to be solar times unless indication is given
otherwise.
1.6 DIRECTION OF BEAM RADIATION
The geometric relationships between a plane of any particular orientation relative to
the earth at any time (whether that plane is fixed or moving relative to the earth) and
the incoming bcam solar radiation, that is, the position of the sun relative to thai
Plane, can be described in terms of several angles [Benford and Bock (1939)]. Some
of the angles arc indicated in Figure 1.6.1. The angles are as follows:
é Latitude, ihe angular location north or south of the equaior, north positive;
90 < gs 90º.
ô — Declination, the angular position of the sun at solar noon (i.e,, when the sun
is on the local meridian) with respect to the plane of the equator, north positive;
-23,45º 5 6< 2345º,
B Slope, the angle between the plane of the surface in question and the
horizontal; OS Bs 180º. (> 90º means that the surface has a downward
facing component.)
Y Surface azimuth angle, the deviation of the projection on a horizontal plane
of the normal to the surface from the local meridia, with zero due south, east
negative, and west positive; —180º < y< 180º
«Hour angle, the angular displacement of the sun east or west of the local
meridian due to rotation of the earth on its axis at 15º per hour, morming
negative, aftemoon positive,
8 — Angle of incidence, the angle between the beam radiation on a surface and
the normal to that surface.
Additional angles are definc that describe the position of the sun in the sky:
8. Zenith angle, the angle between the vertical and the line to the sun,
angle of incidence of beam radiation on a horizontal surface,
&, Solar altitude angle, the angle between the horizontal and the line to the sun,
ie. the complement of the zenith angle.
Y Solar azimuth angle, the angular displacement from south of the projection
of beam radiation on the horizontal plane, shown in Figure 1.6.1.
Displacements east of south are negative and west of south are positive.
.e., the
The deciination 8 can de found from the equation of Cooper (1969):
- 8=23.45 sin(a60268.2-0)
(1.6.1)
a Solar Radistion
f
Zenith
Normaito
horizonta! surtace
Sun
Figure 1.6.1 (a) Zenith angle, slope, surface azimuth angle, and solar azimuth angle for a tilted
surface, (o) Plan view showing solar azimuth angle.
Table 1.6.1 Recommended Average Days for Months and
values of n by Monthst
For the Average Day of the Month
a forith
Month DayofMonth Date mn Dayof Year — à Declination
January i 17 7 -209
February a+i 16 47 30
March s9+i 16 7 -24
April 90+1 15 105 94
Mey 10+i 15 135 18.8
June 151415 n 162 Bl
July 18141 17 198 212
August n2+i 16 228 13,5
September 24341 is 258
October mm+i 15 288
November 30 +1 14 318
December 3341 10 344
à From Klein (1977)
The day of the year n can be conveniently obtained with the help of Table 1,6.1.
Note that declination is a continuous fanction of time. The maximum rate o change
of declination is at the equinoxes, when it is about 0,5º/day. For most engineering
calculations, the assumption of an integer n to represent a day results in a satisfactory
calculation of declination.
8.6 Direction of Beam Radiation 15
There is à set of usefu) relationships among these angles. Equations relasing the
angle of incidence of beum radiation on a surface, 8, to the other angles are
cos 8=sin ôsin gcos
-sin 8cos gsin fcos y
+ cos Sos gos fcos w
+cos Gsin sin ficos ycos o
+cos 6sin Bsin ysin o (1.6.2)
and cos 8 = cos &, cos + sin 8, sin B cos(
-y (1.6.3)
Example 1.6.1
Calculate the angie of incidence vf beam radiation on a surface located at Madison,
Wi at 10:30 (selar time) on February 13, if the surface is tilted 45º from the
horizontal and pointed 15º west of south.
Solution
Under these conditions, n is 44, lhe declination 8 from Equation 1.6.3 is —14º, the
hour angle co is -22.5º (15º per hour times 1.5 hours before noon), and the surface
azimuth angle is 15º. Using a slope 8 of 45º and the latitude & of Madison of
43ºN, Equation 1.6.2 is
cos 8= sint-t4) sin 43 cos 45
—sin(-14) cos 43 sin 45 cos 15
+ cos(-14) cos 43 cos 45 cos(-22.,5)
+ COS(-14) sin 43 sin 45 cos 15 cos(-22.5)
+ cos(-14) sin 45 sin 15 sin(-22,5)
cos 8=- 0.117 +0.121 + 0,464 + 0418 — 0.068 =0.817
B=35º E
There are several commonty occurring cases for which Equation 1.6,2 is
simplificd. For fixed surfaces sloped toward the south or north, that is, with à
surface azimuth angle of 0º or 180º (a very common situation for fixed flat-plate
coltectors), the last term drops out.
For vertical surfaces, = 90º and the equation becomes
cos 6
sin Écos cos y+ cos ôsin pros ycos «
+cos ôsin ysin «o (1.6.4)
; Solar Radiation
”
declination leads directly to times of sunrise and sunset and day length, for either
isphere. . ,
pemisçãe additional angle of interest is the profile angle of beam radiation on à
receiver plane R that has a surface azimuth angle of y. It is the projection s the
solar altitude angle on a vertical plane perpendicular to the plane in Susie
Expressed another way, it is the angle through which a plane that is initially
À is in the plane of the surface in question in
horizontal must be rotated about an axis 1% 1| e
order to include ihe sun. The solar altitude angle o, tie, £BAC), and the ei
angle a, (ie., ZDEF), for the plane R are shown in Figure 1.6.4. The plane A 8,
includes the sun. Note that the solar altitude and profile angle are the same when u E
sun is in a plane perpendicular to the surface R (e.g., at solar noon for a surfáce wi
à surface azimuth angle of 0º or 180º). The profile angle is useful in calculating
shading by overhangs. It can be calculated from
tino, (1.6.12)
cos(k—
Example 1.6.3
; O west of
Calculate the solar altitude, zenith, and profile angles for a surface facing 25º west of
i o.
south at 4:00 PM solar time on March 16 at a latitude of 43º.
Solution o
The solar altitude angle «3, is a function only of time of day and dectination. E
March 16, from Equation 1.6.1 (or Table 1.6.1), dis 2.49, At 4:00 PM, qo = 60º.
From Equation 1.6.5, recognizing that cos 8, = sin(90 — 8,) = sin o,
sin &, = cos 43 cos(-2.4) cos 60 + sin 43 sin(-2.4) = 0,337
a, = 19.7º
a EF) fe
Figure 2.6.4 The solar altitude angio x, (LBAC), and the profile angie a, (DEF) for a
surface R,
1.7 Angles for Tracking Surfaces a
The solar azimuth angle for this time can be caiculated with Equation 1.6.6:
tangj= im =2330,
Do “= 66,80
—cos 43 tan (15.4) % = 66.8
COS Mew=tan(-2.46)ftan 43 = 0045, ag = 9260
Thus C,, Cy and Cy are all Land y, = 3 = 66.89,
The profite angie for the surface with y= 25º is calculated with Equation
1.6.12:
tan co = — lan 19,
=25.7º E
TOs(66.8 = 23) 0.480, Op = 25.7
in various types nf charts. Examples of these are the Sun Angie Calculator (1954) and
the solar position charts (plots of «x, or 8, vs. % for various &, 8, and 09) in
Section 1.9 and Appendix H. Care is necescary in interpreting information from
other sources, since nomenclature, defini
itions, and sign conventions may vary from
those used here,
17 ANGLES FOR TRACKING SURFACES
Some soar collectors "track" the sun by moving in prescribed ways to minimize the
angle of incidence of beam radiation on their surfaces and thus maximize the incident
beam radiation. The angtes of incidence and the surface azirmuth angles are needed
for these coliectors. The relationships in this section will be useful in radiation
calculations for these moving surfaces. For further information see Eibling et al.
(1953) and Braun and Mitchel] (1983).
Tracking systems are classified by their morions, Rotation can be about a single
axis (which could have any ocientation but which in practice is usually horizontal
cast-west, horizontal north-south, vertical, or parallel 10 the earth's axis) or it can be
about two axes, The following sets of equations (except for Eguations 1.7.4) are for
surfaces that rotate on axes that are parallel to the surfaces.
For a plane rotated about a hor
izontal east-west axis with a single daily
adjustment so that the beam radiation is
normal to the surface at noon each day,
cos6= sin? 8 + cos28 cos q (17.19)
nz Solar Radiation
The slope of this surface wil] be fixed for each day and will be
B=ho-d (1.7.1b)
The surface azimuth angle for a day will be 0º or 180º depending on the latitude and
declination:
H(4-9>0,7=0 GO
H(g-9)<0, y= 180º
For a plane rotated about a horizontal east-west axis with continuous adjustment
to minimize the angle of incidence,
cosg=(1--cos26 sina)? (1.7.28)
The stope of this surface is given by
tan = tan 8. fos 4] (1.7.2b)
The surface azimuth angle for this mode of orientation will change between 0º and
180º if the solar azimuth angle passes through £ 90º, For either hemisphere,
Fbd<90, y=0º (1.7.20)
Hlj>90, y= 180º
For a plane rotated about a horizontal north-south axis with continuous
adjustment to minimize the angle of incidence,
cos B=(cos28, + cos? Bsin? q)? (1.732)
The slope is given by
tan B=1an ô.jcos (7— 4) €1.7.3b)
The surface azimuth angle y will be 90º or -90º depending on the sign of the solar
azimuth angle:
Hx>0, 7=90º (1.730)
IE 4<0, y=-90º
For a plane with a fixed slope rotated about a vertical axis, the angle of
incidence is minimized when the surface azimuth and solar azimuth angles are equal.
17 Angles for Tracking Suríaces
From Equation 1,6.3, the angle of incidence is
cos = cos 8. cos B + sin 8. sin 8
The slope is fixed, so
B= constant
The surface azimurh angfe is
Y=%
(1.7.43)
(1.7.4b)
€1.7.40)
For a plane rotated about a north-south axis parallel to the earth's axis with
continuous adjustment (o minimize 8,
cos B=cos é
The slope varies continuously and is
tan p= DO
cos y
The surface azimuth angle is
gs
p= tam! CSM A, gore,
cos 6'sin q
where cos = cos 8. cos À + sin 8. sin q
b if [ue Dest), %=0
c- cos &' sin q,
1 otherwise
1 if%20
Ge Nha
= ifk<o
(17.59)
(17.5)
(17.5)
(17.59)
(17.5)
(1.758)
Fora plane that is continuously tracking about two axes to minimize the angle
of incidence,
cos 8
(1.769)
7
“1
"|
|
Mu
Solar Radintion
za
B=a (1.7.6b)
v=% (1.7.60)
Example 1.7.1
Caleulate the angle of incidence of beam radiation, the slope of the surface, and the
surface azimuth angle for a surfaceat a = 40º, 8= 21º, and q = 30º (2:00 PM)
and b 4=40º, 8=21º, and 0= 100º if it is continuously rotated about an east-
west axis to minimize 6.
Solution
a Use Equations 1.7.2 for a surface moved in this way, First caloulate the angle
of incidence:
8=cost(l- cos? 21 sin? 30)? = 27.8?
Next calculate 6, from Equation 1.6.5:
8. = cos-t(cos 40 cos 21 cos 30 + sin 40 sin 21)=31.8º
we now need 1, Using Equations 1.6.6g, cos (2, = (tan 21)/tan 40) and 6d, =
62.80. Thus C; = 1,C)= 1.C3= 1, and y,=y. So
= cm! Si 30.205 21 = 62,49
HE ingl é
Then fom Equation 1.7.2b
B = tan-I(tan 31.8 cos 62.4) = 16.0º
From Equation 1.7.2, with X, < 90,7=0.
b The procedure is the same as in a:
6 = cosl - cos? 21 sin? 100)? = 66.89
8, = cos-Ifcos 40 cos 21 cos 100 + sin 40 sin 21) = 83.9º
The value of 09, is still 62.8º, from part a. In this case, the constanis in Equation
dem
L66aareC;=-1,C,= 1, and C;=1.
net si 100 cos 21 =1124º
= sin! sin 100 GO 2 + 180
18 Ratio of Beam Kadiatlon on Tiied Surface to That on Horizontal Surface 2
The slope is lhen
B=tan-! tan 83.9kos 12.4/= 74.30
And since Lg] > 90, 7 will be 180º. (Note that these results can be checked using
Equation 1.6.5.)
1.8 RATIO OF BEAM RADIATION ON TILTED SURFACE TO
THAT ON HORIZONTAL SURFACE
For purposes of solar process design and performance calculations, it is often
necessary to calculate the hourly radiation on a tilted surface of a collector from
meusurements or estimates of solar radiation on a horizontal surface. The most
commonly available data are total radiation for hours or days on the horizontal
surface, whereas the need is for beam and diffuse radiation on the plane of à
collector.
The geometric factor R,, the ratio of beam radiation on the tiited surface to that
on a horizontal surface at any time, can be calculated exactly by appropriate use of
Equation 1.6.2. Figure 1.8.1 indicates the angle of incidence of beam radiation on
the horizontal and tilted surfaces. The ratio G,4/G, is given byó
- Cor
p= LE
Gh Ghacos
Goncos 6 cos6
(1.8.1)
cos 8,
and cos 8 and cos 8, are both determined from Equation 1.6.2 (or from equations
derived from Equation 1.6.2).
Sb
Sim »
Figure 1.8.1 Beam radiation on horizontal and tilted surfaces.
É The symbol G is used in this book to denote rates, while 7 is used for energy quantities integrated
over an tour. The original development of Ry by Hoitel and Woeriz (1942) was for hourly periods;
for an hour (using angles at the midpoint of the hout), Rp = 174/p-
je Solar Radiation
Two situations arise, for positive values or for negative values of (q — f).
For positive values, the charts are used directly. If (6 P) is negative (which
frequently occurs when collectors are sloped for optimum performance in winter or
with vertical collectors), the procedure is modified. Determine cos 8, as before.
Determine cos 6 from the appropriate absolute value of ( — £) using the curve
for the other hemisphere, that is, with the sign on the dectination reversed.
Example 1.8.2
Calculate R, for a surface at tatitude 40ºN at a tilt 30º toward the south for the hour
9to 10 solar time on February 16.
Solution
Use Figure 1.8.2c) for the hour £2.5 hours from noon as representative of the hour
from 9 to 10. To find cos 6, enter at a latitude of 40º for the north latitude date of
February 16. Cos 6, = 0.45. To find cos 8, enter at a latitude of 6 — B= LOS tor
the same date. Cos 8=0.73. Then
Ry = 08
bÊ cos 8,
The ratio can also be calculated using Equation 1.8.2. The declination on February
16is-13º:
Rey = 28 10 cos 13) cos(37.5) sin 0 sini3) , gy E
“cos 40 cos(-13) cos(-37.5) + sin JO sin(-IB) —
Example 1.8.3
Calculate R, for a latitude 40ºN at a tilt of 50º toward the south for the hour 9 to LO
solar time on February 16.
Solution
Cos 8, is found as in the previous example and is 0.45, To find cos 8, enter at an
abscissa of 10º, using the curve for February 16 for south latitudes. The value of cos
8 from the curve is 0.80. Thus R, = 0.80/0,45 = 1.78. Equation 1.8.2 can also be
used:
= cos(-10) cos(-3) cos(-37.5) + sint-10) sin(-13) |
cos 40 cos(-13) cos(-37.5) + sin 40 sin(-13)
ro Má
1 is possible, using Equation 1.8.2 05 Figures 1.8.2, to construct plots
showing the effects of coilector tilt on &, for various times of the year and day.
Figure 1.8.3 shows such a plot for a latitude of 40º and a slope of 50º. Ht illustrates
1.8 Ratio of Beam Radiation on Tiited Surface to That on Horizontal Surface
cosbicos 8.
Ro =
Declination, degrees
Figure 1.8.3 Ratio &,
for a surface jo
from solar noon. > dit slopo 50
to south at latitude 46º for various hours
that very large gains in incident bea:
Im radiati ti a
surface toward the equator. ion are to be had by tilting a Teceiving
for a plane cotated continuousty
beam radiation on th: from E: 7.2a, the ratio of beam radiation on the
e plane, from Equation 1.7.2a, the rati
j e «7. 2a, the ratie ati
Plane to that on a horizontal surface at any time is
Ry= A co8ô sinta)'?
£Os fcos deos + sin Gsin é (1.8.4)
a
|
|
|
|
2 Solar Radiation
19 SHADING
Three types of shading problems occur so frequently that meshods are needed to cope
with them. The first is shading of a collector, window, or ather receiver by near-by
trees, buildings or other obstructions. The geomenies may be irregular, and
systematic calculations of shading of the receiver in question may be difficult.
Recourse is made to diagrams of the position of the sun ia the sky, e.g., plots of
solar altitude a, vs. solar azimuth , on which shapes of obstructions (shading
profiles) can be superimposed to determine when the path from the sun to the point in
question is blocked. The second type includes shading of collectors in other than the
first row of muitirow arrays by the collectors on the ajoining row. The third includes
shading of windows by overhangs and wingwalls. Where the geometries are regular,
shading is amenabie to calculation, and the results can be presented in general form.
This will be treated in Chapter 14.
At any point in time and at a particular latitude, q, 6, and 6 are fixed. From
the equations in Section L.6, the zenith angle 8, or solar altitude angle 0x, and the
solar azimuth angle 7, can be calculated. A solar position plot of 8, and q, vs. 7,
for latitudes of £45º is shown in Figure 1.9.1. Lines of constant declination are
labeled by dates of mean days of the months from Table 1,6.1. Lines of constant
hour angies labeled by hours are also shown. Plots for tatitudes from O to *70º are
included in Appendix H.
The angular position of buildings, vingwalls, overhangs or other obstructions
can be entered on the same plot. For example, as observed by Mazria (1979) and
Solar Altitude Angla ais
so 100 st O
o
Solar Azimuth Angle, 7s
Figure 1.9.1 Sotar position plot for £ 45º latitude. Solar attitude angle and solar azimuth angle
are functions of declination and hour angle, indicated on the plots by dates and times. The dates
shown are for norihem hemisphere; for southern hemisphere use the corresponding dates as indicated
in Figure 1.8.2, See Appendix H for other latitudes.
19 Shuding
33
A - .
nderson (1982), if à building or other obstruction of known dimensions and
orient, q
cleo rio ea FM Me it time ie te ecc
amd O poe é e angular coordinates corresponding to altitude and azimuth
amgle a) can be caleulated fo oo CORCE seimuth angle 7, and object aftinde
Example 5 1.9.1 and 192 a trigonometric considerations. This is illustrated in
azimuih angles may be made : po cn Vely, measurements of object altitude and
'y be made at the site of à Proposed receiver and the angles plorted
On the solar position j :
ã Plot. A variet s ;
angles. 'y Of instruments are available to measure the
Example 1.9.1
Om to the north of à long wall that shades itwhen
is af uniform height of 2.5 m above the center of
this wall on a solar position chart with a the wal]
all oriented on a southeast to northwest axis displaced
À proposed collector site a S is 10:
the sun is low in the sky. The wall
the proposed colector area. Show
Oriented cast-west and b the w
20º from east-wesr.
Solution
In each case, we pi i
+ We pick several points on the to) of a i i
it piso P Of the wall to establish the coordinates
& — Take three points indicated bs
YA, B, and C on the di i
and Band C westor A, Points B' and C' are taken lo the eus DER vi eo
object altitude angles as B a; ih bee ai angina Same
pri nd C and with object azimuth angles changed only in
Solar Radiation
For point A, the object azimuth %, is 0º. The object altitude angle is
tan 0,4 = 25/10, Gy = 140º
For point B, SB =(102+ 102)? = 14.Lm,
tan Gg = 2.5/14.1, Gog = 10.0º
tan %g = 1010, og = 450º
For point C, SC =(102+302)2 =31.6m,
=4.,52º
1.6º
im = 25816,
tan %ç=3000, Ke =
There are points coresponding to Band C butto the cast of A; these will tao
bject azimuth angles except with negative signs. The shading prof e
teinod b je, Ttis shown by the solid
i inates dent of latitud
determined by these coordinates is independes
line on the plot for à = 45º. Note that at object azimuth angles of 90º, the object
distance becomes infinity and the object altitude anglo becomes 0?
o
E
10
8)
20
7o|
E 30
s
E ao
<s a
É s
És
ç o
E 3
ê 70
20)
6a
so
10
so
-50 o E 100 west
Solar Azimuth Angle 75
The sun is obscured by the wall only during times shown on the diagram. The
wall does not cast a shadow on point S at any time of day from tate March to mid-
September. For December 10, it casts a shadow on point S before 9:00 AM and after
3:00 PM.
19 Shading 5
b. The obstruction of the sky does not show cast-west symmetry in this case, so
five points have been chosen as shown to cover the desirable range. Point A is the
same as before, ie., 0, = 1409, % = 0º.
Arbitrasily select points on the wall for the caleulation. In this case the
caleulations are easier if we select values of the object azimuth angle and calculate
from them the corresponding distances from the point to the site and the
corresponding €,. In this case we can select values of 7, for points B, €, D, and E
of 45º, 90º, 30º, and —60º.
For point B, with £,g = 45º, the distance SB can be calculated from the law of
sines:
sin 79 — Sin (18045 - 70)
SB 10 + SB=104m
tan Og = 2.5/10.4, Gon = 13.5º
For the point D, with %,p = -30º, the calculation is
sin LIO HSC180- NO 30)
SD ,
To D=146m
tan Gp = 2.5/14.6, mp =9.7º
The caleulations for points C and E give O,c =5.2ºat fc = 9º ando, =
2.6º ut Yg = 600º.
The shading profile determined by these coordinates is plotted on the solar
position chart for É = 45º and is shown as the dashed line. In this case, the object
altitude angle goes to zero at azimuth angles of -70º and 110º. In cither case, the
area under the curves represents the wall, and the times when the wall would obstruct
the beam radiation are those times (declination and hour angles) in the areas under the
curves. E
There may be some freedom in sclecting points to be used in plotting object
coordinates, and the calculation may be made easier (as in the preceding example) by
selecting the most appropriate points. Applications of trigonometey wil! always
provide the necessary information. For obstructions such as buildings, the points
selected must.jnclude comers or limits that define the extent of obstruction. It may or
may not be necessary to select intermediate points to fully define shading. This is
illustrated in the following example.
Example 1.9.2
It is proposed to install a solar collector at a level 4.0 m above the ground, A
rectangular building 30 m bigh is focated 45 m to the south, has its long dimension om
an east-west axis, and has dimensions shown in the sketch, The latitude is 45º.
a Sotar Radiation
At any point in time, the solar radiation incident on a horizontal plane outside of
the atmosphere is the normal incident solar radiation as given by Equation 1.4.1
divided by R,;
Go= Get + 0.033 cos “ln cos 8. (40.1
where G. is the solar constant and 2 is the day of the year. Combining Equation
1.6.5 for cos 8, with Equation 1.10.1 gives G, for a horizontal surface at any time
between sunrise and sunset.
G= Gl + 0.033 cos 38ônicos ócos ôcos o + sin ésin 5) (1.10.2)
It is often necessary for calculation of daily solar radiation to have the integrated
daily extraterrestriaf radiation on a horizontal surface, H,. This is obtained by
integrating Equation 1.10.2 over the period from suniise to sunset. IF G, is in watts
per square meter, H, in joules per square meter is
24X 3600Gx
e 7
(1 40.033 cos 60
x (cos cos sin co, + Ho sin Qin) (1103)
where 1», is the sunser hour angle, in degrees, from Equation 1.6.10.
The monthly mean! daily extraterrestrial radiatiom, Ma, is à uscful quantity.
For latitudes in the range +60 to -60 it can be calculated with Equation 1.10.3 using
n and 6 for the mean day of the month!! from Table 1.6.1. 1, is plotted as a
function of latitude for the northem and southem hemispheres in Figure 1.10.1. The.
curves are for dates that give the mean radiation for the month and thus show fo.
Values of H, for any day can be estimated by interpotation. Exact values of H, for
all latitudes are given in Table L.10.1,
Example 1.10.
What is H,, the day's solar radiation on a horizontal surface in the absence of the
atmosphere, at latitude 43ºN on April 15?
Solution
For these circumstances, n = 105 (from Table 1.6.1), 6 = 9,4º (from Equation
1.6.1), and = 43º, From Equation 1.6.10
cos «3, = tan 43 tan 9.4 and (,= 98.9º
10 An overbar is used throughout the book to indicate a monthly average quantity.
1lThe men day is the day having H, closest to FZ,.
LO Extraterrestrial Radiation on u Horizontal Surface
so
“0
430 DU
É Pd q
ia nes 3
1 O
Sep. 15 q
Ma 16d
no Oct. 15 O
o - Alia AS SINE
20 30 0 50 so 70 so so
North latitude, degrees
so
Ho. daily Mim?
É
8
8
June 19
Juy 17
May 35
Aug. 16
30 40 50 o 7 ao so
South latitude, degrees
Figure 1.10.1 Extraterrestrial daily radiation on a horizontal surface.
mean days of the month from Table 1.6.1. The curves are for the
41
í
| 1.10 Extraterrestrial Radiation on a Horizontal Surface
“ Solar Radiation
a
Table 1.10.1 Monthly Average Daily Extraterrestrial Radiation, MJ/m? Then from Equation 110,3, with Go, = 1367 Wim?
é Jan Feb Ma Apr May Jun Jul Aug Sep Ot Nov De
TOTO | 24x 3600 x 1367
H, = AX 3600 x 136 360 x 105
% 00 00 12 193 372 448 54 00 00 00 | e x -[1+0.033 cos 365 —
[ss 00 00 22 192 370 47 64 00 00 00 i
fm 00 00 47 196 366 442 90 06 00 00 i ) xx 989
: b y “ ” ! 1 X(cos 43 cos Es s
00 07 78 210 359 433 nº 22 00 006 ! cos 9.4 sin 98.9 + ão SD 43 sin 9,
7 O 27 109 231 353 421 148 49 03 00 i
6 12 54 139 254 357 410 7 78 20 04 3.8 MJm?
O 35 83 169 2716 366 410 38A 309 205 108 45 23 From E
5 62 13 198 296 376 394 26 241 134 73 48 T le tgure 1.10.a, for the curve for April, we read H = 34.0 MJ/m?, and from
50 91 144 225 315 385 415 400 341 255 167 103 77 i able L.10.1 we obtain 44, = 33.8 MJjm? by interpotation. ! ”
as 122 174 251 332 292 417 404 353 278 196 133 102 i .
4 153 203 274 346 397 417 406 364 208 224 164 137 | His also of interest to caleulate the extraterrestrial radiation on horizontal
ce a : . a a horizontal
35 183 231 296 358 400 415 406 373 7 250 193 168 | sur face for an hour Peciod. Integrating Equation 1.10.2 for a period between hour
3 213 257 315 368 400 411 404 378 332 274 222 199 angles co, and 4 which define an hour (where “% is the larger),
25 22 282 332 315 398 404 400 382 246 296 250 229 .
1% 20 305 347 379 393 305 393 382 356 316 217 25% 12x 3600
i 15 296 326 359 380 385 384 383 380 364 334 301 285 lo= z — Gs (1 + 0.033 cos 380 a)
10 320 M4 368 379 375 370 31 375 370 350 324 31 |
5 3M2 360 375 IA 363 353 356 367 372 363 345 335 d . . . n(a - q
O 362 374 3718 367 348 335 340 357 372 373 363 357 Picos ôsin 0» —sin mn) + a sin êsin 3) €1.10.4)
-5 380 385 379 356 350 34 321 344 369 380 379 376
0 395 393 377 345 3LL 292 299 329 363 385 303 394 (The limits «o, and co, may define a time other than an hour.)
-I5 408 398 372 330 289 268 276 31 354 387 404 409 .
-20 418 400 364 313 266 22 252 201 33 386 412 d21 Example 1.10,2
-25 425 400 354 293 21 25 26 2710 329 382 dit 431 Wiat is che solar radiati .
DO 430 397 340 W2 HA 187 199 246 H2 6 420 48 . a e o ar radiation on a horizontal surface in the absence of the atmosphere at
35 432 391 325 248 186 158 170 221 293 366 420 442 atitude 43ºN on April [5 between the hours of 10 and nº
“0 431 382 306 223 158 120 142 194 272 355 417 445 Solution
-A5 428 371 286 196 129 100 113 166 249 340 412 445
-S0 423 357 263 168 100 72 84 138 224 324 405 443 The deelination is 9.49 (from th ious .
-55 417 341 239 139 72 45 57 109 198 305 306 440 Equation [.10.4 with o nd o pg PD. Bor April 15,15 105, Using
IO a
-S0 410 324 212 109 45 22 31 80 470 284 387 437 30º and q, = 15º,
65 405 306 185 79 21 03 10 52 141 262 378 437
O 408 288 156 50 04 00 00 26 11 240 314 449 ho,
75 49 W6 126 24 00 00 04 OR 80 29 361 462
-80 427 274 97 06 DO 00 O 00 50 206 388 471 |
x eos 43 cos 9.445; a z
85 432 77 72 00 00 00 00 00 24 203 393 476 cos 43 cos 9.4 sin (-15) — sin (-30)] +7
-0 433 28 62 00 00 00 00 00 14 W4 394 478
- 12 x 3600 x 567
360 x 105
x pedia
( + 0.033 cos
5
15 —(-30)
ao » sin 43 sin 9.
=3,79 MJm?
n
The hourty extraterrestrial radiation can also be approximated by writing
A tion 1.10.2 in terms of 4, evaluating « at the midpoint of the hour. For te
Sircumstançes 9f Example 1.10.2, the hour's Tadiation so estimated is 3 80 Mk ê
Differences between the hourly radiation calcutated by these two methods ill bo
44 Solar Radiation
slightly larger at times near sunrise and sunset but are stitl small. For larger time
spans, the differences become larger. For example, for the same circumstances as in
Example 1.10.2 but for the two-hour span from 7:00 to 9:00, the use of Equation
1.104 gives 4.58 MJ/m?, and Equation 1.16.2 for 8:00 gives 4.61 MJ/m?,
111 SUMMARY
In this chapter we have ouilined the basic characteristics of the sun and the radiation it
emits, noting that the solar constant, the mean radiation flux density outside of the
earth's atmosphere, is 1367 W/m? (within +1%), with most of the radiation in a
wavelength range of 0.3 to 3 um. This radiation has directional characteristics that
are defined by a set of angles that determine the angle of incidence of the radiation on
a surface. We have included in this chapter those topics that are based on
extraterrestrial radiation and the geometry of the earth and sun. This is background
information for Chapter 2, which is concerned with effects of the atmosphere,
radiation measurements, and data manipulation.
REFERENCES
Anderson, E. E. Fundamentais of Solar Energy Conversion, Addison-Wesley Publishing Co..
Reading, MA (1982).
Benford, F. and J. E. Bock, Trans. cf the American Hlumination Engineering Soc. 34, 200
(1939), “A Time Analysis of Sunshine.”
Braun, J. E. and . C. Mitchell, Solar Energy. 31, 439 (1983). "Solar Geometry for Fixed and
Tracking Surfaces."
Cooper, PL. Solar Energy, 12, 3 (1969). "The Absorprion of Solar Radiation in Solar Stills”
Coulson, K. L., Solar and Terrestrial Radiation, Academic Press, Nes York (1975).
Duncan, €. H., R. €. Wilson, 3. M. Kendall, R. G. Harsison, and J. R. Hickey, Solar Energy,
28, 385 (1982). “Latest Rocket Measurements of the Solar Constant.”
Eibling, 1. A R. E. Thomas, and B. A. Landry. Report to the Office of Saline Water, U.S.
Department of the Interior (1953). “An Investigation of Multiple-Effect Evaporation of Saline
Waters by Steam from Solar Radiatior
Froblich. C.. in The Solar Output and tts Variation (O. R. White, ed.), Colorado Associated
University Press, Boulder (1977). “Contemporary Measures of the Solar Constant”
Garg. H. P, Treatise on Solar Energy, Vol. 1, Wiley-Interscience, Chichester (1982).
Hickey, 5. Ro B. M. Ailton, F. 4, Griffin, H. Jacobowitz, P. Pelligrino, R. E. Maschhoff, E. A.
Smith, and T. H. Vonder Haar, Solar Energy, 28, 443 (1982). “Extraterrestrial Solar Irradiance
Variability: Two and One-Half Years of Measurements from Nimbus 7."
Hotel, H. C. and B. B. Woertz, Trans. ASME, 64, 91 (1942). "Performance of Flat-Plate Solar
Heat Collectors." t
Igbal, M., An Introduction to Solar Radiation, Academic Press, Toronto (1983).
References as
Jobnson, E.
+ 4. 0f Meteorology, Li, 431 (1954), "The Solar Constant.”
Jones, R. E., Solar Energy, 24, 305 (1980), "Eltecis of Overhang Shading of Windows having
Arbitrary Azimuth,”
Klein, 8. A., Solur Energy 19, 325 (1977)
Tilted Surfacos
“Calculation of Monthly Average Insolation on
Kendraiyev, K. Y., Radiation in the Atmosphere, Academic Press, New York and London (1969),
Mucria, E., The Passive Solar Energy Book, Rondale Press, Emmaus, PA (1979),
NASA SP-8055, National Aeronautics and Space Administration, May (1971).
Electromagnetic Radiation.”
"Solar
Robinson, N. (ed.), Solur Radiation,
Isevier, Amsterdam (1966).
Sun Angie Calculator, Libby-Owens-Ford Glass Company (1951).
Spencer, 3. W. Search, 2 (5), 172 (1971), "Fourier Series Representation of the Position of the
Sun”
Thekaekara, M. P. and A. 3. Drummond, National Physical Science, 229, 6 ([971). “Standard
Values for the Solar Constant and lts Spectral Components.”
Thekackara, M. P., Solar Energy, 18, 309 (1976). "Solar Radiation Measurement: Techniques and
Instrumentation.”
Thomas. R. N., Trans. ef the Conference on Use of Solar Energy, 1, 1, University of Arizona
Press (1958). "Features of the Solar Spectrum as Iimposed by the Physics of the Sun."
U.S. Hiydrographic Office Publications No. 214 (1940). "Tables of Computed Altitude and
Avimuth,,
Whillier, A., Notes on Solar Energy prepared at MeGill University (19654).
Whillier, A... Solar Energy, 9, 164 (1965b). “Solar Radiation Graph.”
Wihillier, A., Personal communication (1975 and 1979),
Wilson, R. €.. S. Gulkis, M. Janssen, H, S, Hudson, and G. A. Chapman, Science, 211, 700
(1981). "Observations of Solar Irradiance Variability.
so Available Solar Radiution
Output
connector
Window
je Filter detemt
Model EG
wire-wouna
thermopile
Figure 22.2 Cross section of the Eppley Normal Incidence pyrheliometer. Courtesy The
Eppley Laboratory.
cross-section of a recent model of the Eppley is shown in Figure 2.2.2. The
instrument mounted on a tracking mechanism is shown in Figure 2.2.3. The detector
is at the end of the collimating tube, which contains several diaphragmis and which is
blackened on the inside. The detector is a mukijunction thermopile couted with
Parsons optical black. Temperature compensation to minimize sensitivity to
vanations in ambient temperature is provided. The aperture angie of the instrumem
ts 5.7º, so the detector receives radiation from the sun and from an area of the
circumsotar sky two orders of magnitude larger than that of the sim.
The Kipp & Zonen actinometer is based on the Linke-Feussacr design and uses
a 40-junction constantan-manganin thermopile with hot junctions heated by radiation
and cold junctions in good thermal contact with the case. In this instrumem the
assembly of copper diaphragms and case has very large thermal capacity, orders of
magnitude more than the hot junctions. On exposure to solar radiation the hot
junctions rise quickly to temperatures above the cold junction; the difference in lhe
temperatures provides a measure of the radiation. Other pyrheliometers were
designed by Moll-Gorczynski, Yanishevskiy, and Michelson.
The dimensions of the cellimating systems are such that the detectors are
exposed to radiation from the sun and from a portion of the sky around the sun.
Since the detectors do not distinguish between forward-scattered radiation, which
comes from the circumsolar sky, and beam radiation, the instruments are, in effect,
defining beam radiation. An experimenta! study by Jeys and Vant-Hul! (1976),
which utilized several lengths of coltimating tubes so that the aperturc angles were
reduced in step from 5.72º ta 2,02º, indicated that for cioudiess conditions this
reduction in aperture angle resulted in insignificant changes in the measurements of
beam radiation. On a day of thin uniform cloud cover, however, with solar altitude
angle of less than 32º, as much as 11% of the measured intensity was received from
the circumsolar sky between aperture angles of 5,72º and 2.02º.
1
'
23 Pyranometers
es —
* Pyrheliometer (NIP) ou an altazimuth tracking
Figure 2.2.3 An Eppley Normal incidenc
mount. Courtesy The Eppley Laboratory,
generalize from the few data available, bot it appcars th:
affect the angular distribution of radiation within the
Pyrheliometers. The World Meteorological Organization
Salibration of pyrheliometers only be undertaken on days in
mects or exceeds a minimum value.
at thin clouds or haze can
field of view of standard
(WMO) recommends that
whtich atmospheric clarity
23 PYRANOMETERS
Instruments for measuring total (beam plus diffuse) radi
Pyranometers, and it is from these instruments that most of ti
tadiation arc obtained. The detectors for these instrume:
independent of Wavelengih of radiation over the solar ene:
they should have a response independent of the angle of incidence of the sola:
radintion, The detectors of most Pyranometers are covered with one or tuo
hemispherical glass covers to Protect them from wind and other extraneous effects,
te covers must be very uniform in thickness so as not to Cause uneven distribution of
dons om the detectors. These factors are discussed in more detail by Coulson
Commonty used pyranometers in the United States are the E ley and
Spectrolab insttuments, in Europe the Moll-Gorezynski, in the USSR m
Yanishevskiy, and in Australia the Trickeit- Norris (Groiss) pyra ne
ation are referred to as
lhe available data on solar
DIS must have 2 response
T8y spectrum. In addition,
jometer.
si
|
|
|
|
s Availahte Solar Radiation
The Eppley 180º pyranometer was the most common instrument in ihe United
States. Tt used a detector consisting of two concentric silver rings; the outer ring
was coated with magnesium oxide, which has a high reflectance for radiation in the
solar energy spectrum, and the inner ring was coated with Parson's black. which has
a very high absorptance for solar radiation. The temperature difference between these
tings was detected by a thermopile and was à measure of absorbed solar radiation.
The circular symmetry of the detector minimized the effects of the surface azimuth
angle on instrument response. The detector assembly was placed in a nearly spherica!
glass bulb, which has a transmiftance greater than (1.90 over most of the solar
radiation spectrum, and the instrument response was nearly independent of
wavelength except at the extremes of the spectrum. The response of this Eppley was
dependent on ambient temperature, with sensitivity decreasing by 0.05 to 0,15%/C
[Coulson (1975)); much of the published data taken with these instruments was not
corrected for temperature variations. It is possible to add temperature compensation
to the external circuit and remove this source of error. It is estimated that carefully
used Eppleys of this type could produce data with less lhan 5% errors but that esrors
of twice this could be expected from poorly maintained instruments. The theory df
this instrument has been carefully studied by MacDonald (1951).
The Eppley 180º pyranometes is no longer manufactured and has been replaced
by other instruments. The Eppley Black and White pyranometer utilizes Parson's--
biack- and barium-sulfate-coated hot and cold thermopile junctions and has better
angular (cosine) response. It uses an optically ground glass envelope and
temperature compensation to maintain cahibration within £1.5% over a temperature
range of -20 to +40 €. Itis shown in Figure 2.3.1.
The Eppley Precision Spectral Pyranometer (PSP) utilizes a thermopile detector.
two concentric hemispherical optically ground covers, and temperature compensation
Figure 2.3.1 The Eppley Black and White pyranometer. Courtesy The Eppley Laboratory.
terms pe crimes meme
23 Pyranometers 5
that resuls in temperature dependence of 0.54% from -20 to +40 C. (Measurements
of irradiance in spectral bands can be made by usc of bandpass filters; the PSP can
be fitted with hemispherical domes of fiter glass for (hi purpose, See Stewart et al
(1985) for information and references.) It is shown in Figure 2.3.2.
The Moll-Gorczynski pyranometer uscs a Moll thermopile to measure the
temperature difference of the black detector surface and the housing of the instrument.
The thermopile assembly is covered with two concentric glass hemisphericat domes
to protect it from weather and is rectangular in configuration with the thermocouples
aligned in a row (which results in some sensitivity to the azimuth angle of the
radiation).
Pyranometers are usually calibrated against standard pyrheliometers, A
standard method has been set forth in the Annals of the Intemational Geophysical
Year HGY, (1958)), which tequires that readings be taken at times of clear skies,
with the pyranometer shaded and unshaded at the same time as readings are taken
wilh the pyrhetiometer. Shading is recommended to be accomplished by means ofa
dlise held 1 m from the pyranometer with the disc just large enough to shade the glass
envelope. The calibration constant is then the ratio of the difference in the output of
She shaded and unshaded pyranometer 10 the cuxtpat of the pyrheliometer multiplied by
lhe calibration constant of the pycheliometer and cos 8., the angle of incidence of
beam radiation on the horizontal pyranometer. Care and precision are required in
thesc calibrations.
It is also possible, as described by Norris (1973), tu calibrate pyranometers
against a secondary standard pyranometer suçh as he Eppley precision pyranometer.
This secondary standard pyranometer is thought to be good to +1% when calibrated
Against a standard pyrheliometer. Dircet comparison of the precision Eppley and field
instruments car be made to determine the calibration constant of the fiefd instruments.
Figure 2.3.2 The Eppley Precision Spectral pyranometer. Couresy The Eppley Laboratory.
E Avuilable Solar Radiation
A pyranometer (or pyrheliometer) produces a voltage from the thermopile
detectors that is a function of the incident radiation. is necessary to use a
potentiometer to detect and record this output. Radiation data usuaily must be
integrated over some period of time, such as an hour or a day. Integration can be
done by means of planimetry or by electronic integralors. Tt has been estimated that
with careful use and reasonably frequent pyranometer calibration, radiation
measurements should be good within +5%; integration errors would increase this
number. Much of the available radiation data prior to 1975 is probabiy not this good,
largely because of infrequent cafibrution and in some instances because of inadequate
integration procedures.
Another class of pyranometers, originally designed by Robitzsch, utilizes
detectors that are bimetallic elements heated by solar radiation; mechanical motion of
the element is transferred by a linkage to an indicator or recorder pen. These
instruments have the advantage of being entirely spring driven and thus require no
electrical energy. Variations of the basic design are manufactured by several
European fisms (Fuess, Caselia, and SIAP). They are widely used in isolated
stations and are a major source of the solar radiation data that are availabte for
locations outside ví Europe, Australia, Japan, and North America. Data from thesc
instruments are generalty not as accurate as that from thermopile-type pyranometers.
Another type of pyranometer is based on photovoltaic (solar cell) detectors.
Examples are the LICOR LI-200SA pyranometer and the Yellor Solarimeter. They
are less precise instraments than the thermopile instruments and have some limitations
on their use, They are also less expensive than lhermopile instruments and are easy to
use.
The main disadvantage of photovoliaic detectors is their spectrally selectivg
response. Figure 2.3.3 shows à typical terrestrial solar spectrum and the spectral
response of a silicon solar cell. If the spectral distribution of incidem radiation was
T T T T T
Salar radiation
€ 2000 +
ã Solar cel
E response
Ê ã
E 1000 + &
ê E
8
ot I 4
03 05 07 10 20 20
Wavelength, gm
Figure 2.43 Spectral distribution of extraterrestrial solar radiation and spectral response of a
silicon solar cell. From Coulson, Solar and Terrestrial Radiation, Academic Press, New York
1975).
23 Pyranometers ss
fixed, a calibration could be established that would remain constant; however, there
are some variations in spectral
stribution? with clouds and atmospheric water
vapor. LI-COR estimates that the ertor introduced because of spectral response is
+59 maximum under most conditions of natural dayliglt and is 3% under typical
conditions, .
Photovoltaic detectors have additional characteristics of interest. Their response
to changing radiation levels is essentially instantancous and is linear with radiation.
The temperature dependence is +0.15%/C maximum. The LI-COR instrument is
fited with an acrylic dilfuscr which substantially removes the dependence of
response on the angle of incidence of the radiation. The response of the detectors is
independent of its orientation. but reflected radiation from the ground or other
sutroundings will in general have a different spectral distribution than global
horizontal radiation, and measurements on surtaces receiving significant amonnts of
reflected radiation will be subject to additional errors.
The preceding discusston dealt entirely with measurements of total radiation on
a horizontal surface. Two additional kinds of measurements arc made with
pyranometers: measurements of diffuse radiation on horizontal surfaces and
measurement of solar radiation on inclined surfaces.
Measurements of diffuse radiation can be made with pyranometers by shading
the instrument from beam radiation. This is usually done by means of a shading ring,
as shown in Figure 2.3.4, The ring is used to allow continuous recording of difluse
Figure 2.3.4 Pyranometer with shading ring 10 eliminate beam radiation. Courtesy The Eppley
rt
2 This will be discussed in Section 2.6.
so Available Solar Radiation
all data are based on the WRR pyrheliometric scale. The Atlas includes” tables that
show averages, maxima, minima, cxlraterrestria! radiation, and sunshine hours.
Appendix G includes some data from the Atlas.
Average daily solar radiation data are also available from maps that indicate
general trends. For example, a world map is shown in Figure 2.5.2 |Lóf et al,
(1966a,b)].9 In some geographical areas where climate does not change abrupuy
with distance (i.e., away from major influences such as mountains or large industria!
cities), maps can be used as a source of average radiation if data are not available.
However, large-scale maps must be used with care becausc they do not show local
physical or climatological conditions that may greatly affect local solar energy
availability.
For calculating the dynamic behavior of solar energy equipment and processes
and for simulations of long-term process operation, more detailed solar radiation data
(and related meteorological informatiun) are needed. An example of this type of dala
(hourly integrated radiation, ambient temperature, and wind speed) is showr im Table
2.5.) for a January week in Boulder, Colorado. Additional information may also be
included in these records, such as wet bulb temperature and wind direction.
In the United States there has been a network of stations recording solar
radíation on a horizontal surface and reporting it as daily values. Some of these
stations also reported hourly radiation. In the 1970s, the U.S. National Occanic and
Atmospheric Administration (NOAA) undertook a program to upgrade the number
and quality of the radiation measuring stations, to rehabilitate past data (to account for
sensor deterioration, calibration errors, and changes in pyrhcliometric scales), and to
make these data available (with related meteorological data) on magnetic tapes. In
1978, corrected data tapes of hourly meteorolegical information (including soar
radiation on a horizontal surface based on the Solar Constant Reference Scale) for 26
stations over a period of 23 years became available. These tapes are reterved to as the
SOLMET tapes and are described in detail in the SOLMET Manual (1978). Fhe
monthly average data for the United States stations shown in Appendix G are derived
from the SOLMET data.
In the jate 1970s, the U.S. federal govemment funded thc development and
operation of a national sotar radiation network (SOLRAD), Measurements of hourly
total horizontal and direct normal radiation were made at the 38 stations that were part
ofthe network. Eleven of the stations also measured diffuse radiation. Data for 1977
ta 1980 were checked for quality and are available from the National Climatic Data
Center. Funding for much of the program was reduced in 1981, and by 1985 the
network was shut down. Since then, some additional funding has become available
to upgrade the insuumentation at many of the stations to automate data acquisition and
recalibrate pyranometers.
1
É Monthly average daily radiation on surfaces other than horizontal are in Volume IL of the Atas.
é Figure 2.5.2 is reproduced fram deJong (1973), who redrew maps originally published hy Lóf et
al. (19663). deJang has compiled maps and radiation Gata from many sources.
Solor
)
ontar
Daity Means of Tojal
--— Radiation (Begm + D
Incident en q Horia
Surtaca, catsemt
E
Z
i
able 2.51 Hourly Radiation for the Hour Ending at the Indicated Time, Air
Temperature, and Wind Speed Data for a January Week, Bouider, Colorado.
MEN
Day Hour 1 Ta Y Day Tour 1 To Y — ——Nnn—N——
km? e ms Mr c ms Day Hour ' Ta v
lt —
Hm Cm
8 L o 8 õ nos
E 2 o 8 va 1252 H $ : n 19 o +47 4
4 ot s& “ q 5
pas ss E
$ 5 o 2 u 46 foto no 56 54 un Do Sa 2
5 5 a 6 non 38 56 94 no 9 89 36
o R » u 12 163 se Bo 2 9 4 40
8 ? u ss 19 0 . no o «NI 3
8 8 17 8 2» q
2 9 4 su o Ro So 740 po o 3 s8
8 10 33 s 2 o b i o e a Dou am 2a
- .z 12 y
RR: Lo Pliml Lia as
x 12 5 0 167 63 12 7 3 22
9 1 o 9 Bo ng ú éo o a sá 12 18 o E
, =I72 6 12 19 ;
is tm PLS Ea
29 no 3 Ra > +0
9 4 o 9 16 389 RW 3 2.2 0 59
180 RR! 8.9 '
ros is E O
: 5 9 POR BL 56 76 5 ê ss
9 7 o 2019 a 2.2 o sa
9 3 4 920 0
3 92 u a o ! 5 1 “ 40 Voo 1926 56 54
2 10 155 PR”: o i 3 : 9 54 13 MIO 72 45
9 43 su o i 0 4s 3 o 15 130 83
9 12 402 9º u q : á Ê 9 3 3º 16 %3 is
õ 5 o 3.6 2» 5 é a
. 13 6 o 31 7 54
10 1 o 10 13 1872 13 7 0 6 5 v 44 36
10 2 o Do 7 13 $ 17 40 a po º no 36
10 3 0 o 15 138 13 9 314 5 » 9 oo 31
10 4 o 1 16 Ts 3 to a 63 5 b º 22 67
10 5 o 1 1” 205 3 u 1809 sa 5 B 9 28 72
10 6 o 10 18 4 1 1 2299 63 o 17 80
10 7 o 10 19 o o DB 2 o 17º 58
10 8 33 10 20 o : 14 ' o 42 14 a
10 9 419 10 2 o : 14 2 o 16 14 a 1968 s7
10 10 1047 o» 2 0 i 4 à o és 1733 67
10 u 1570 10 n uv i 14 4 o 7 14 15 1331 72
10 1 1805 0 2 o i 14 5 14 16 837 67
í és 5 42 so % 2
u 1 a " B 138 50 69 já ; 0 é 8 14 18 4 3327
i n 2 º ti 14 96 39 61 À 14 k x o ú 19 o 0 36
| “u 3 o " 15 o 44 76 14 9 452 49 1 20 o 1954
| u 4 o Dow 2 38 63 Modo no as ú 0 3» 36
| nos o ; non 4 50 63 oO ar 168 8 M La Do 39 58
| n 6 o n 18 0 56 45 Mo 12º 18 36 0 6154
| Mot 383 “ou 0 +47 63
o :
es
Available Solar Radiation
26 ATMOSPHERIC ATTENUATION OF SOLAR RADIATION
Solar radiation at normal incidence received at the surface of the earth is subject to
variations due to change in the extraterrestriai radiation as noted in Chapter 1 and to
two additional and more significant phenomena. (t) atmospheric scatiering by air
molecules, water, and dust and (2) atmospheric absorption by Os. H,O, and CO.
Igbal (1983) reviews these matters in considerable detail.
Scattering of radiation as it passes ihrough the atmosphere is caused by
interaction of the radiation with air molecules, water (vapor and droplets), and dust.
“The degree to which scattering occurs is a function of the number of partictes through
which the radiation must pass and of the size of thc particles relative to À, the
wavelength of the radiation. The path length of the radiation through air molecules is
described by the air mass. The particles of water and dust encountered by the
radiation depends on air mass and on the time- and location-dependent quantities of
dust and moisture present in the atmosphere.
Air molecules are very small relative to the wavelength of the solar radiation,
and scattering occurs in accordance with the theory of Rayleigh (i.c., ihe scanering
coefficient varies with 4-4). Rayleigh scattering is significant only at short
wavelengths; abavc À = 0.6 jum it has little effect or atmospheni transmittance.
Dust and water in lhe atmosphere (end to be in larger particle sizes due to
aggregation of water molecules and condensation of water on dust particles of various
sizes, These effects are more difficult to treat than the effects of Rayleigh scattering
by air molecules, as the nature and extent of dust and moisture particles in the
atmosphere are highly variabie with location and time. “Two approaches have been
used to treat this problem. Moon (1940) developed a transmission cocfficient for
precipitable water [the amount of water (vapor plus liquid) in the air column above ihe
observer] that is a function of 72 and a coefficient for dust that is a function of
A-2:75, Thus these transmittances are less sensitive to wavelength than is the
Rayleigh scaitering. The overall transmittance due 10 seattering 1s the product of lhree
transmitiances, which are three different functions of À.
The second approach lo estimation of cffcets of scattering by dust und water is
by use of Ângstrôm's turbidity equation. An equation for atmospheric transmittance
due to aerosols, based on this equation, can be written as
Wa expl-pa"Sm) (26)
where is the Ângstrôm turbidity coefficient, « is a single lumped wavelength
exponent, and À is the wavelengih in micrometers. Thus there are two parameters,
Band o, that describe the atmospheric turbidity and its wavelength dependence; 8
varies from O to 0.4 for very clean to very turbid atmospheres, «a depends on the
size distribution of the aerosols (a value of 1.3 is commonly used), and 8 and &
will vary wilh time as atmospheric conditions change.
More detailed discussions of scatiering are provided by Fritz (1958) who
included effects of clouds, by Thekaekara (1974) in a review, and by Igbal (1983).
2.6 Almospheric Attenuation of Solar Radistion
65
À increases above 0.29 à
sa -29 um, until at 0,35 i i
weak ozone ahsorption band near À = 06um ste no abrovptim, eres also a
Water vapor absorhs stron; nds à
. R igly in bands in the infrared pa
Egeu, with Sirong absorption bands centered at 10,14, and um von
25 dm me rtemision Of the atmosphere is very low due to absorption by Ho
2 He energy in the extraterrestrial spect à i 5
Of the total solar spectrum, am ve a te groaoa a o tan 5
ate Pe 4 AM energy received at the ground at À > 2.5 um is very
n =» . R
and E ffects of Rayleigh Seattering by air molecules and absorption by O,, H,0,
ne > one spectral distribution of beam irradiance are shown in Figure 26 1 R :
am atmosphere with = O and 2 em of preci entra
here g precipitable water, w. TI
Do ni sistibution is shown as a reference. The Rayleigh Seeing ira
Y the difterence berween the extraterrestrial cu )
e s d the curve a
shaded arcas; its effect become: seleng o D
lise s small at wavelengths greater th
The cara absorption bands arc shown by the shaded areas “e about Dó um,
"he eflcct of air mass is ilustrated in Fi, vi
. e s Ir mas in Figure 2.6.2, which shows th
distribution of beam imadiance for air masses of O (the extraterrestrial NDT
and 5 for an atmosphere of low urbidity. Teto 3
The spectral distribution of 1 ati
The 1 otal radiation depends aiso on th S
distribution of the diffuse radiation. Some measurements are avaiablo im ta
sale and visible portions of the spectrum [see Robinson (1966) and Kondra '
(1969), which have ted to the conclusion that in the wavelength range (0,35 to 080
/
|
|
|
|
Clean Atmosphera, & = 0
Air Mass = 1
r Extraterrestrial
| y Rayieigh Attenuation
»
8
ê
250]
Direct Normal Spectra! Irradiance, Wim? um
B
s
10 15 20 25 30 as 40
Wavelengih, um
Figure 24 a
sure 2.6.1 An example of lhe cffecis of Rayleigh Scaltering and atmos)
the spectral distribution oFbeam irrudianee, Adapted From Igbal (1983), Pen abeerprca on
Available Sotar Radiation
Values of F, can be calculated from Equation L. 10,3 using day umbers om
Table 1.6.1 for the mean days of the month, or it can be obtaincd em eiti pia
110.1 or Figure 1.10.1. The average day length À cun be calculated tom ut n
1.6.41 or it can be obtained from Figure 1.6.3, for the meu day of the no Vas
indicated in Table 1.6.1. Lof et al. (19663) developed sets ot o a a a
various climate types and locations based on radiation data then avail table. 5
given in Table 2.7.2.
Table 2.72 Climatic Constants for Use in Equation 2.7.2
Sunshine Hours
in Percentage
of Possible
Location Climated Veg Range A bo
Location Cmã Do
7 os 03
jue, NM BsBW E 6sas 4
Auta GA cr M 4811 E 0.38 9 %
Blue Hill, MA Dr D 42-60 2 º 2 os
Brownsville, TX Bs GDsp 4780 6 3 om
Buenos Aires, Arg cr q atos so 0.26 É
Charleston, SC cr E 6075 e7 dam 009
Darien. Manchuria Dw D 58-81 s7 03% 0,23
El Paso, TX sw Dã 78.88 sa 0.54 020
Ely NV Bw Bi 61-89 n 0.54 o 8
am 22
Hamburg, Germany cr D va x : “ o Eu
I Ar G 71 $
gt Dt M 40.72 ss “Io o a
q ou
Malange, Angola AwBS GD aits 58 o. % om
Miami, FL Av EGD 5671 65 q 2
Nice, France €s SE 49-76 61 o q
: x 0,30
Poona, India (Monsoon) Am s 25.49 a” sam
(Dm a5-49 81 a
Kisanganá, Zaire Af B 34-56 a :
Tamanrasset, Algeria BW Dp Te .
à Climatic classification bused on Trewartha's map (1954, 1961), where climate
types are:
AÉ Tropical forest climate, constantly mois; reinfal all through der
Am Tropical forest climate, monsoon rain; shom dry seasoo, but total rainfatl suflhcient
to support rain forest.
Aw— “ropical forest climate, dy season ia winter
BS — Steppe or semiarid climate.
BW Desert or arid climate. , |
CS Mesothermal forest climate: constartly mois raintal alt through the year.
K winter.
Cs Mesotherma! forest climate: dry season in .
Dt Microthermal snow forest climate: constanely moist; rainfdl al through the year.
Dow Microtherma! snow forest climate; dry season jm winter
2.7 Estimation of Average Solar Radintion n
The following example is based on Madison data (although the procedure is not
recommended for a station where there are data) and includes comparisons of the
estimated radiation wilh SOLMET data and estimates for Madison based on the Blue
Hill constants (those which might have been used im the absence of constants for
Madison).
Example 2.7.1
Estimate the monthly averages of lotal solar radiation on a horizontal surface for
Madison, Wisconsin, latitude 43º, based on the average duration of sunshine hour
duta of Table 2.7.1.
Solution
The estimates
re based on Equation 2.7.2 using constants a = 0.30 and 5 = 0.34
from Table 2.7.2. Values of H, are obtained from Figure 1.10.1 and day lengths
from Equation 1.6.1], each for the mean days of the month. The desired estimates
are obtained in the following table, which shows daily 4 in MI/m2. (For
Footnotes for Table 2.7.2 (continued)
D Vegetation classification based on Kiichler's map, where vegetation Iypes are:
B Droadicafevergreentees.
Broudleaf evergreen, shrubform. minimom height 3 feet, growth singly or in
groups or patches.
D Broadlleaf deciduous trees.
Ds Broadfeaf deciduous. shrubform. minimum height 3 feet, plants sufficiently far
apar that they frequently do not touch.
Dsg — Broadleat deciduous, sbrubform, minimum height 3 feet, growth singhy or in
proups or patches.
E Noedleicaf exergreen trees.
G Grass and other herbaccous plants,
SD Grass and ather herbacevus plants; broadizaf deciduous trees.
CiDsp Grass and other herbaceous plants; broadieaf deciduous, shrubforms, minimum
height 3 feet, growth singly ox in groups or patches
M Mixed broadleaf deciduous and necdleleaf evergreen trees.
S — Semideciduous: broadleaf evergreen and broadleaf deciduous trees. .
SE — Semideciduous: broadleat evergreen and broadleaf deciduous trees: needielear
evergreen trees.
Note: These constants are based ou radiation data available before 1966 and do not reflect
mprovements in data processing and interpretation made since then. The results of
estimations for United States stations will be at variance with SOLMET data. Iris
recommended that these correlations be uscd only when there are no radiation data
available.
- Available Sokar Radiation
comparison, data for Madison from Appendix G are shown, and in (he last colums
estimates of Madison radiation determined by using constants « und & for Blue
Hill)
mesh
Ha N nJN
Month Mm? dr Mt? Mutmé MJjm?
1136 43 0.49 nã so 2
18.80 10.2 055 92 91 9a
26.05 17 sy 138 129 1,4
335 12 057 1646 154 4
May 3939 144 63 19,8 no
June 4174 154 0.07 221 2
tuiy 4052 14,8 0.66 224 na
Aug 358% = 073 19.4 2
Sept. 2877 3 oa ue 16.4
Ou 4 10% 047 10.3 nó
Nov 146! “s vas 57 às
Dee UEM 29 n44 aa 53
& From SOLMET data
* Using constants for Blue Bill n
The agreement between measured and caleulated radiation is reasonably good,
even though the constants a and 4 for Madison were derived from a different data
base ftom the measured data. If we did not have constants for Madison and had to
choose a climate close to that of Madison, Blue Hill would be a reasonabte choice.
The estimated averages using the Blue Hilf constants are shown in the last column.
The trends are shown, but the agreement is not as good. This js the more typical
situation in the use of Equation 2.7.2.
Data are also available on mean monthly cloud cover É, expressed as tenths of
the sky obscured by clouds. Empiricat relationships have been derived to relate
monthly average daily radiation /7 to monthly average cloud cover C. These are
usually of the form
Ecasro 213)
o
Norris (1968) reviews several attempts to develop such a correlation. Bennett (1965)
compared correlations of // / dt, with C, with 4/N, and with a combination of the
two variables and found the best correlation to be with Hi /N, that is, Equation 2.7.2.
Cloud cover data arc estimated visualty, and there is not necessarily a direct
relationship between the presence of partial cloud cover and solar radiation at any
particular time. Thus (here may not be as gol a stalistical relationship between
H1 Ho and C as there is between 7 (ff, and n/N. Many surveys of solar radiation
data [e.g., Bennett (1965) and Lôf et al. (1966a,b)) bave been based on corretations
28 Esti tation of Clear Sky Radiation
3
Of radiation with sunshine h
"our data, However, Paltri
used cloud cover data to modify clear sky data si tolo (1976 have
Monthly averages of 7£, which are in 90d agreement with meas vê ta.
ges of Ff, whicl goad agy nt with measured average da
28 ESTIMATION OF CLEAR SKY RADIATION
Phere and for four climate types. The atmospherie
transmittançe fe ati i 6, Ti) e
ce for beam radiation t, is ConlGon (OF GyriG,y) and is given in th
o
form
W=a0+ arexpl-ticos 8)
(2.8.1)
The constants “or 44, and X for the standard at
are toa Ra 'mospherc with 23 km visibility
and kº, which are given for altitudes less than 2.5 km by
ap=0,4237- D.00821(6 AP
(28.10)
ai = 0.5055 +0.00595(6.5 - Ap Qêto)
Ate
E =02711+00185825 AP (28.19)
where À is the altitude ofihe observer in kilometers. (Hotic] al:
20: 1, and º for a standard atmosphere with 5 km visibilit
Correction factors are applied to aa, oo
types. The correction factors r= ne
2.8.1. Thus, the transmittance of
SO gives equations for
and &º to allow for changes |
0 ai, and 4 ges in climate
“potão ri=ai/al, and sy = kk are given in Table
|s standard atmosphere for beam radiation can be
+ '
able 28.1 Correction Factors for Climate Types?
Climate Ty
Yre + “
rk
Tropical
Midiatitude summer 6» 055 ia
Subarctic summer 0.99 095 vor
Mixitatitude winter 1 vol 1
103 LOL 109
* From Hotel (1976).
Availabie Solar Radintion
74
determined for any zemith angle and any altitude up to 2.5 fam. The clear sky beam
normal radiation is then
(2.8.2)
Grab = Gon
i M ar sky horizontal beam radiation
where G,, is obtained from Equation 1.4.1. The clear sky
is
28.3)
Goo = Gonto cos 8; (
For periods of an hour, the clear sky horizontal beam radiation is
(2.84)
leo = Ion th COS 8:
Example 2.8.1
beam radiation of the standard clear atmosphere at
ansmittance for e ]
a tlítuo st 22 at 11:30 AM solar time. Estimate the intensity
Madison (altitude 270 m) on Augu: r '
of beam radiation at that time and its component ema horizontal surface. —
, wo: HS
drãco
a
: Vora *
Solution Psi hs TOS N
On August 22, n is 234, the deelination is 1].4º, and from Equation 1.6.4 the cosin
of the zenith angle is 0,846.
fficients for Equation 2.8.1. First, the values for *
ae oba ns 2.8.1b to 2.8.1d for an altitude
the standard atmosphere are obtained from Equatioi
of 0.27 km:
ap = 0.4237 - 0.00821(6 — 027P=0.154
af = 0.5055 + 0.00595(6.5 — 0.27P = 0.736
k* = 0.2711 + 0.01858(2.5 -0.27) = 0.363
The climate-type correction factors are obtained from Table 2.8.! for midiatitude
summer. Equation 2.8.1 becomes
1 = 0.51440.97) + 0:735(0.99) exp(-0.363 x 1.02/0.846) = 0.62
“The extraterrestrial radiation is 1339 Wim? from Equation 3.4.1. The beam radiation
is then
Gony = 1339 x 0.62 = 830 Wim?
cem
2.8 Estlmation of Clear Sky Radintion 7]
“The component on a horizontal plane is
830 x 0,846 = 702 Wim? n
His also necessary to estimate the clear sky diffuse radiation on à horizontal
surface to get the total radiation, Liu and Jordan (1960) developed an empirical
relationship between the transmission coefficient for beam and diffuse radiation for
clear days:
u=(4=0 271 0.294%
(2.8.5)
where Ty is GG, (ot 1/f,), the ratio of diffuse radiation to the extraterrestrial
(beam) radiation on the horizontal plane, The equation is based on data for three
stations. The data used by Liu and Jordan predated that used by Hottel and may not
be entirely consistent with it; until better information becomes available, it is
suggested that Equation 2.8.5 be used to estimate diffuse clear sky radiation, which
can be added to the beam radiation predicted by Hottel's method to obtain a clear
hours total, These calculations can be repeated for each hour of the day, based on
the midpoints of the hours, to obtain a standard clear day's radiation, H,.
Example 2.8.2
Estimate the standard clear day radiation en a horizontal surface for Madison on
August 22.
Solution
For cach hour, based on the midpoints of the hour, the transmittances of the
atmosphere Tor beam und diffusc radiation are estimated. The calculation of £, is
illustrated for the hour [1 to 12 (ie, at 11:30) in Example 2.8.1, and the bcam
radiation for a horizontal surface for the hour is 2.53 MJfm? (702 Wim? for the
hour).
The calculation of 7, is based on Equation 2.8.5:
= 0.271 - 0.29440.62) = 0.089
Next Gu Calculated by Equation 1.4.5, às 1339 Wim?
so that
Then Ga is Gucos 0,
Gog= 1339 x 0.846 x 0.089 = 101 Wim?
Then the diffuse radiation for the hour is 0.36 MJ/m?, The total radiation on a
horizontal plane for the hour is 2.53 + 0.36 = 2.89 MJ/m?. These calculations are
s Available Solar Radiation
Solving for y in this cquation is not convenient, and Herzog (1985) gives an explicit
equation for 7 from a curve fit.
-2 -
1.184E-- 27.182 expl- 2.56) 25.64)
1.498 +
Kr, mas — Kr.min
and (2.9.6b)
A value 0f Ky cin Of 0.05 was recommended by Bendt ex al. Hollands and Huget
(1983) recommend that & ax be estimated from
Krmax = 0.6313 + 0.267k7 — 11.9(Ky-- 0.758 (2.9.6)
The universality of the Liu and Jordan distributions has becn questioncd,
particularly as applied to tropical climates. Saunier et al. (1987) propose an
alternative expression for the distributions for tropical climates, A brief review of
papers on the distributions is included in Knight et al. (1991).
Símilar distribution functions have been developed for houriy radiation.
Whillier (1953) observed that when the houriy and daily curves for a location are
ploited, the curves are very similar, Thus the distribution curves of daily occurrences
of Ky can also be applied ta hourly cleamess indexes. The ordinate in Figure 2.9.2
can be replaced by kp and the curves will approximate the cumulative distribution of
hourly cleamess. Thus for a climate with K7 = 0.4, 0.493 of the hours will have kz
equai to or less than 0.40.
2.10 BEAM AND DIFFUSE COMPONENTS OF HOURLY
RADIATION
In this and the following two sections we review methods for estimation of the
fractions of total horizontal radiation that are diffuse and beam. The questions of the
best methads for doing these calculations are not fully settled. A broader data base
and improved understanding of the data wil! probably cad to improved methods. In
each section we review methods that have been published and then suggest one for
use. The suggested correlations are in substantial agreement with other correlations,
and the set is mutually consistent.
The split cf total solar radiation on a horizontal surface into its diffuse and beam
components is of interest in twa contexts. First, methods for calculating total
radiation on surfaces of other orientation from data on a horizontal surface require
separate treatments of beam and diftuse radiation (see Section 2.15), Second,
estimates of the Jong-time performance of most concentrating collectors must be
based on estimates of availability of beam radiation, The present methods for
2-H Beam and Diffuse Components of Houriy Radiation
81
10
08
06
tatt
os
02
[o
o
a
00 o2 04 [a 08
Figure 2,10.€ À sample of difíuso fra
etion vs, cl
ads go sleamess index data irom Cape Canaveral, FL,
b é » Wilh kr, the houriy ch i
10.1 shows a plot of diffuse fraction /,/1 vs, kr Far Cape Cana nt
=» Lo In
order 19 obtain 1,/ vs. &
o 1º! Vs. ky correlations, data fr i
NA ! 1 ms, om many locations simila E
qua conure 2.10.1 are divided into “bins," or ranges of values ER a a
dean ese "bin are averaged ta obtain a point on the plot. A set of these oito o
ne correlation. Within each of the bins therc is a distibutiom of poi is
ints;
ak of (15 may be i
k produced by skies with hi i
Siffuse fraction, or by skies thai ave clear for O Bs e gh
Part of the hour, teading to a low diffuse
Fepresent a particular hour very closely
adequately represents the diftuse fraction, '
Eis Oreilt and Hollands (1977) have used data of this
“bs etal. (1982) have used data from four U. S
Reindl et al. (19904) have used do
Europe, The three correlation a
identical, although they were deri
correlation, Figure 2.10,3, is10
type from Canadian Stations,
one Australian station, and
am independent data set from United States and
Cpo in Figure 2.10,2, They are essentialiy
vet from three separate data bases. The Erts et al
+ produces results thas are for
ractica
- ANA is represemted by the following equation Ee
10-
u J O 0.249k7 forkr<0
7 oa Lbdiy for 035 <ky < 0,75
o.t77 forky>0.75
Available Solar Radiation
tab!
DAL Orgil& Haliands
—- - Etbs, etal,
oz Reid, ota.
kr
i it
2.18.2 The ratio 4,17 as a function of hourly cleamess index, kr. Showing the Drgi
lado (97, Erbs et al. (1982), and Reindt et at. (1990) correlarions.
os
06
04
o2
99
Figure 2.10.3 The satio / (1 as à f n of hourly clearness index, ky, from Erbs et al.
sato 4/1 nei y clearnes: 1, from Erbs
gt
(1982).
1.0- 0.09% forkr<0.22
09511 — 0.1604ky + 4.388k7
— 16.638 kj + 12.336 k$
(210.1)
= for 022<ky 50.80
1
0.165 for kz > 0.80
For values of kr greater than 0.8, there arc very few data. Some S ne qa
that are available show increasing diffuse fraction as ky increases nove no ps
ise 1 iffuse fraction is probably due to reflection of radiati
apparent rise in the diffuse r din from
ing ti cured but when there are clou:
s during times when the sun is unobscure: e )
o de sun to the observer. The use of a diffuse fraction of 0.165 às
recommended in this region.
erp
2:11 Beam and Diffuse Components of Daily Radintlon as
In a related approach described by Boes (1975), values of 4,14 from corre-
Iations are modified by a restricted random number that adds a statístical variation tg
the corretation.
211 BEAM AND DIFFUSE COMPONENTS OF DAILY RADIATION
Studies of available daily radiation data have shown thai the average fraction which is
ditfuse, Hy/H, is à function of Ky, the day's cleamess index. The ariginal
Somelation of Liu and Jordan (1960) is shown in Figure 2.11.1; the data vero for
Blue Hill, Massachusetts. Also shown un the graph are plots of data for Israet from
Stanhil (1966), for New Delhi from Choudhury (1963), for Canadian stations from
Ruth and Chant (1976) and Tuller (1976), for Highett (Melbournc), Australia, ftom
Bamnister (1969), and from Collares-Pereira and Rabi (19793) for feur US, stations.
There is some disagrecmen, with differences probably due in part to instrumental
Figure 2.10.2) is shown in Figure 2.11,2. A seasonal dependence is shown; the
spring, summer, and fal] data are essentially the same, while the winter data shows
somewhat lower diffusc fractions for hígh values of Ky. The season is indicated by
as
ê
giÊ 08
ii
ES =" Liu & Jordan
ÉS gafo - Ruth & Chant
E = — — Highot
> -— Stanhi)
* ==" Tuer I
02
ee Choudhuy
Coliares-Perisra & Fabi
º .
º 02 04 08 oa 1.0
Key - Daiiytotatradiaton my
"É Daily extratomastrial? E,
Figure 2.111 Correlations of daity ditíuse fraction with daily cleamess index. Adapted from
Kiein and Duffic (1978).
Available Solar Radintion
E 2)
Figure 2.1.2 Suggested correlation of daily diffuse fraction with Kg. From Erbs et ab. (1982)
the sunset hour angle O,. Equations representing this set of correlations are as
follows'!:
For q, <814º
| 10-0.2727K7 +2M95KF Kr<0715
19H14 KG + 9.3879 Ky
(ta)
(ou for Kr 20.715
and for ay, > 814º
2
10+02832K7-25557K? rg, 0702
+ 0 8448Kj
Ha. (2.11.1b)
H
los for Ky 20.722
1 The Collares-Pereira and Rabi correlation is
0.99 for kr 0.17
VABB=227KrA DADA (ar 017 <K7< 0.75
Ha 2) -N865K)+ 14 64BKT
—0.54K7 + 0.632 tor 075 <Kr< 0.80
02 for kr 20.80
2.42 Beam and Diftuse Components of Monthly Radiation ss
Example 2.11.1
The day's total radiation on a horizontal surface for St, Louis. Missouri (latitude
38.6º) on September 3 is 23,0 MJ/m?. Estimate the fraction and amount that is
diffuse,
Solution
For September 3, the declination is 7º. From Equation 1.6.10, the sunset hour angle
is 95.6º. From Equation 1.10.3, the day's extraterrestria] radiation is 33.3 MJ/m',
Then
From Figure 2.11.2 or Equation 2.11.1b, Ha/H is 0.26, so an estimated 26% of the
day's radidtion is diffuse, The diffuse energy is 0.26 x 23.0 = 6.0 MJfm?, "
2.12 BEAM AND DIFFUSE COMPONENTS OF MONTHLY
RADIATION
Charts similar to Figures 2.11.] and 2.11.2 have becn derived to show the
distribution of monthly average daily radiation into its beam and diffuse components.
En this case, the monthly fraction that is diffuse, Hy/H, is plotted as a function of
monthly average clearness index, K7 (= j7/H.). The data for these plots can be
obtained from daily data in cither of two ways. First, monthly data can be plotted by
summing (he daily data diffusc and total radiation. Second, as shown by Liu and
Jordan, a gencralized daily 4,/H versus Ky curve can be used with a knowledge of
the distribution of good and bad days (the cumulative distribution curves of Figure
2.9.2) to develop the manthly average relationships,
Figure 2,121 shows several correlations of 4q/!! versus K7. The curves of
Page (1964) and Collares-Pereira and Rabl (19793) are based on summations of daily
total and diflusc radiation. The original curve of Liu and Jordan (modified to correct
for a small error in H4/H at low X7) and those labeled Highett, Stanhil!, Choudhury.
Ruth and Chant, and Tuller are based on daily correlations by the various authors (as
in Figure 2.11.1) and on the distribution of days with various Ky as shown in Figure
2.9.2. The Collares-Pereira and Rabl curve in Figure 2.12.1 is for their all-year
correlation; they found a seasonal dependence of the relationship which they
expressed in terms of the sunset hour angle of the mean day of the month. There is
significant disagreement among the various correlations of Figure 2.12.1.
Instrumental problems and atmospheric variables (air mass, season, or other) may
contribute to the differences.
Erbs et al. developed monthly average diffuse fraction correlations from the
daily diffuse correlation of Figure 2.11.2. As with the daily correlations. there is a
%” Available Solar Radialion
0.20
Í
218
6 Hour from solar neon
o
3 0.14 3
zo
2 E De A ia
5
é ao DS ma) ==
i o
5 oo
É jp
Ê So
"
> 006]
Too
1
2 004
“3 LT
002
ELA
o
8 8 10 " 12 13 ta 15 16
Hours from sunrise to sunset
Lpjid!2isdLrrLLos +
3 7 so 105 120
Sunset hour angt, «u,, Degreos
Figure 2.132 Relationship berwecn hourly diffuse and daily diffuse radiation on a horizontal
surface as a function of day length. Adapted form Liu and Jordan (1960).
Figure 2.13.2 shows a retated set of curves for r,, the ratio of hourly diffuse 10
daily diffuse radiation, as a function of time and day length. In conjunction with
Figure 2.11.2, it can be used to estimate hourly averages of diffusc radiation if the
average daily total radiation is known:
nus fé (2.133)
These curves are based on the assumption of Liu and Jordan (1960) bat 4,4Hy is the
2.14 Radiation on Sloped Surfnces
9
same as (,//f,, and are represented by the following equation
NE cos o o A (213,4)
“E sin am AEE cos ax
Example 2.13.3
From Appendix G, the average daily June total radiation on a horizontal plane in
Madison is 22.1 MJ/m?. Estimate the average diffuse, the average beam, and the
average total radiation for the hours 10 to LI and Lto 2.
Solution
The mean daily extraterrestrial radiation JT, for June for Madison is 41.7 MJ/m2
(from Table 1.10.1 or Equation 1.10.3 with 2 = 162), 0) = 113º, and the day length
is 15.1 hours (from Equation 1.6.11). Then (as in Example 2.12.1), Ry = 0.53.
From Equation 2.12.1, Ay/H' = 0.40, and the average daily diffuse radiation is (0.40 x
22.1 = 8.84 MJ/m?, Entering Figure 2.13.2 for an average day length of 15.1 hours
and for 1.5 hours from solar noon, we find rq = 0.102. (Or Equation 2.13.3 can be
used with «o = 22.5º and 6), = 113º to obtain r; = 0.102.) Thus the average diffuse
for those hours is 0.102 x 8.84 = 0.90 MJ/m?.
From Figure 2.13.1 (or from Equations 2.13,1 and 2.13.2) from the curve for
1.5 hours from solar tioon, for an average day length of 15,1 hours, r, = 00.108, and
average hourly total radiation is 0.108 x 22,1 = 2.38 MJfm?. The average beam
radiation is the difference between the total and diffuse, or 2.38 — 0.90 = [.48
MIfm?. =
214 RADIATION ON SLOPED SURFACES
We tum next to the gencrat problem of calculation of radiation on tilted surfaces when
only the total radiation on a horizontal surface is known, For this we need the
directions from which the beam and diffuse components reach the surface in
question. Section [.8 deait with the geometric probtem of the direction of beam
radiation. “The direction from which diffuse radiation is received, ie, its distribution
over the sky dome, is à function of conditions of cloudiness and atmospheric clarity,
whích are highly variable, Some data are available, for example, from Kondratyev
(1969) and Coulson (1975). Figure 2.14.1, from Coulson, shows profiles of diffuse
radiation across the sky as à function of angular elevation from the horizon in a plane
that includes the sun, The data are for clear sky and for smog conditions.
Clear day data such as that in Figure 2.14.1 have led to à description of the
diffuse radiation as being composed of three parts. The first is an isotropic par,
teceived uniformiy from all of the sky dome. The second is circumsolar diffuse,
resulting from forward scartering of solar radiation and concentrated in the part of the
” Available Solar Radiation
= 0.965 um
ve intensity
am l l l L L l l
E o 40 20 o 2 40 60 BO
Zenith angle, degrees.
Figure 214.1 Kelative intensity of solar sadiation (at À = 0.365 um) as a foncrion of elevation
angle in the principal plane that includes the sun, for Los Angeles, for clear sky and for smog,
Adapted from Conison. Solar and Terrestrial Radiation, Academic Pross, Nes York (1975).
sky around the sun. The third, referred to as horizon brightening, is concentrated
near the horizon, and is most pronounced in clear skies. Figure 2.14.2 shows
schematically these three parts of ihe diffuse radiation. .
The angular distribution of diffuse is to some degree a function of the
reflectance py (the albedo) of the ground. A high reflectance (such as that of fresh
snow, with pg approximately 0.7) results in reflection of solar radiation back to the
sky, which in tum may be scattered to account for horizon brightening.
Gircumsolar
O
“- Horizon “Area!, Apz
Figure 2.142. Schematio of the distribution of diffuse radiation over the sky dome, showing the
circumsolar and horizon brightening components added to the isotropic component. Adapted from
Pescz et al. (1988).
2.14 Radiation on Sloped Suríaces 3
Sky models, in the context used here, are mathematical representations of the
diffuse radiation. When beam and reflected radiation are added, they provide the
means of calculating radiation on a tilied surface from measurements on the
horizontal. Many sky models have been devised. A review of some of them is
provided by Hay and McKay (1985), Since 1985, others have been developed. For
purposes of this book, three of the most useful of these models are presented. The
isotropic model is in Section 2.15, and two anisotropic models are in Section 2.16.
The differences among them are in the way they treat the three parts of the diffuse
radiation.
K is necessary to know or to be able to estimate the solar radiation incident on
tilted surfaces such as flat-plate collectors, windows, or other passive system
receivers. The incident solar radiation is the sum of a set of radiation sueams
including beam radiation, the three components of diffuse radiation from the sky, and
radiation reilected from the various surfaces “secn” by the tilted surface. The total
incident radiation vo this surface can be written as?
Ir= try + Iráiso + Erdes + Irah: + Irrep (2.14.1)
where the subscripts iso, cs. hz, and ref? refer to the isotropic, circumsolar,
horizon, and reflected radiation streams.
For a surface (a collecior) of area A, the total incident radiation can be
expressed in terms of the bcam and diffusc radiation on thc horizontal surface and the
total radiation on the surfaçes that reflect to the Lilted surface, The terms in Equation
2.14.] become
Adr = 1Ryà ct Laço As e + les ÃO + Lapa po hoc
+ DhpiAiFre (Quis)
The first tem is the beam contribution. The second is the isotropic diffuse term
which includes the product of sky area A, tan undefined area) and the radiation view
factor from lhe sky to the collector F,... The third is the circumsolar difluse, which
is treated as coming from the same direction as the beam. The fourth term is the
contribution of the diffuse from the horizon from a band with another undefined area
Ay; The fifth term is the set of reflected radiation streams from the buildings,
fields, etc., to which the tilted surface is exposed. The symbol i refers to each of the
reflected streams: f, is lhe solar radiation incident on the ith surface, p; is the
diffuse reflectance of that surface, and Fis lhe vicw factor from the ith surface to
the bilted surface, Tt is assumed that the reflecting surfaces are diffuse reflectors;
specular reflectors require a different treatment.
13 This and following equations are written in terms of / for an hour. They could also be serilten
in terms of G, the irradiance.
Ea Available Solar Radiation
In general, it is not possible (o calculate the reflected encrgy term in detail, to
account for buildings, trees, etc. the changing solar radiation incident on thern, and
their changing reflectances. Standard practice is to assume thal there is one surface, a
horizontal, diffusely reflecting ground, large in extent, contributing to this term. In
this case, is simply / and p; becomes p,, à composite "ground" reflectance.
Equation 2.14,2 can be rewritten in a useful form by interchanging areas and
view factors (since the view factor reciprocity relation requires that, for example,
Asc = AF os). This eliminales the undefined arcas A, and Ag, The arcaA,
appears in each term in the equation and cancels. The result is an equation that gives
Ein terms cf parameters that can be determined either thecretically or empirically.
tr= Rb + laisoP os + lgsRo + lapeFeha + IPgD og (2,14.3)
This equation, with variations, is the basis for methods of caleulating 4, that are
presented in the following sections.
When 4, has been determined, the ratio of total radiation on the tilted surface to
that on the horizontal surface can be determined. By definition,
- - Total radiation on the tilted surface 17
dia ! (2.14.4)
Total radiation on a horizontal surface 1
Many models have been developed, vf varying complication, as the basis for
calculating /r. The differences are largely in the way that the difíuse terms are
treated. The simplest model is based on the assumptions tha! the beam radiation
predominates (when it matters) and that the diffusc (and ground-reflected radiation)
are effectively concentrated in the area of the sun. Then R = R, and all radiation is
treated as beam. This leads to substantial overestimation of £,, and the procedure is
not recommended. Preferred methods are given in the following two sections.
2.15 RADIATION ON SLOPED SURFACES - ISOTROPIC SKY
It can be assumed fas suggested by Hotel and Woeriz (1942)] that ihe combination of
diffuse and ground-reflected radiation is isotropic. With this assumption, the sum of
the diffuse from the sky and the ground-reflected radiation on the tilted surface is the
same regardless of orientation, and the total radiation on the tilted surface is the sum
of the beam contribution calculated as /,R, and the diffuse on a horizontal surface,
Fy. This represents an improvement over the assumption that all radiation can be
treated as beam, but better methods are available,
An improvement on these models, thc isotropic diffuse model, was derived
by Liu and Jordan (1963). The radiation on the tilted surface was considered to
include three components: beam, isotropic diffuse, and solar radiation diffuscly
reflected from the ground. The third and fourth terms in Equation 2.14.3 arc taken as
zero as all diffuse is assumed to be isotropic. A surface tiltcd at slope B from the
2.15 Radiation on Sloped Surfaces--Isotropic Sky 95
horizontal has a view factor to the sky ,., which is given by (1 + cos f)/2. (If the
diffuse radiation is isotropic, this is also R,, lhe ratio of diffuse on the tilted surface
ta that on the horizontal surface.) The surface has a view factor to the ground Fegof
(1 cos By/2, and if the surroundings have a diffuse reflectance of p, for the total
solar radiation, the reflected radiation from the surroundings on the surface will be
tpg(1 - cos f)/2. Thus Equation 2.14.3 is modificd to give the total solar radiation
on the tilted surface for an hour as the sum of three terms:
L+e I-
tes tuto + SEE tpg tro f (2.15.1)
and by the definition of &,
f, ujl+cosB i-cos
atado eo
fr NR E (2.152)
Example 2.15.1
Using the isotropic diffuse model, estimate the beam, diffuse, and ground-reflected
components of solar radiation and the total radiation on a surface sloped 60º toward
the south at a latitude of 40ºN for the hour 9 to 10 AM on February 20. Here ! =
1.04 MI/m? and p, = 0.60.
Solution
For this hour, 4, = 2.3] MJjm2, so kz = 1.04/2,31 = 0.45. From the Erbs
correlation, Equation 2.10.1, 14] = 0.757. Thus
14=0.757x 1.04 = 0.787 MJ/m?
1,=0.243x LOM = 0.253 MJjm?
The hour angle «o for the midpoint of the hour is -37.5º. The declination ô is
—11.6º. Then
— cos(40 - 60) cos(- 1.6) cos(-37.5) + sin(40 — 60) sinf-11.6)
Rh
» cos 40 cos(-11.6) cos(-37.5) + sin 40 sin(- 11.6)
= 979
0.466 tm
Equation 2.15.1 gives the three radiation streams and the total:
Ir=0.253x 1.71 + 0787(LE59860) + 1,04 x 0.60/=sos 60)
= 0.433 + 0.590 + 0.156 = 1.18 Mm?
| 100 Available Solar Radi
Table 2.16.1 Brighiness Coefficients for Perez et al, Anisotropic Sky>
Range of e tu to ta fm ta fa
0-1.665 0,196 1084-0006 Olá O180 0019
1.065 - 1.230 0.236 Os19 -0.180 00] 002 0038
1.330 - 1.500 qas4 032] 0255 UM 008 0046
1.500 - 1.950 0866 DIB] 035 0203 0403 0.049
1.950 - 2.800 1026 OH 0426 0273 0602 0061
2.800 - 4.500 0978 0986 -0350 0280 0915 DOM
4.500 - 6.200 0748 0913 0,236 0173 1045 0065
6200 -1 O318 -0757 0103 0062 1698 023%
& From Perez et al, (1988).
This set of equations altows calculation of all of the three diffuse components on the
tilted surface. It remains to add the bcam and ground-reflected contributions. The
total radiation on the tilted surface includes five terms: the beam, the isotropic
diffuse, the cireumsolar diffuse, the diffusc from the horizon, and the ground-
reflected term (which parallel the terms in Equation 2.14.3):
1+
redes tato
(216.14)
+laFasinB+ ted
Equations 2.16.8 through 2.16.14, with Table 2.16.1, constitute 4 working version
of the Perez et al. model, Tts use is ilustrated in the following example.
Example 2.16.2
Do Example 2.15.1 using the Perez et al. method,
Solution
From Example 2.15.1: 4, = 2.31 MJ/m?, £ = 1.04 MJjm?, 1, = 0.253 Mym?,
lg = 0.787 MJ/m2, cos O = 0.799, B = 37.0º, cos 8, = 0.466, 6, = 62.2º, and
R$ = 171.
To use Equation 2.16.14, we need 4,b,E, und 4 in addition to the
quantities already calculated:
a = maxf0, cos 37.0] = 0.799
b=maxicos 85, cos 62.2] = 11.466
and atb = 0.799/0.466= 1.71 (the same as R, in Example 2.15.1)
2.16 Radintion on Sloped Surfaces-- Anisotropie Sky ro
Next calcuiate 4. The air mass , from Equation 1.5.1, is
ma= Ifcos 62.2 = 1/0466 = 2.144
We also need Usc Equation 14.t withn= 51:
ln = asa + 0033006 Ei SU s005 Mim?
From the defining equation for 4, Equation 2.16.11,
4=0.787 x 2.144/5.025 = 0,336
We next calculate € from Eguation 2.16.10. Thus Z, = Iyjcos 8,
é cabe ns s O; = 0.253%
= 0.544 MJm?, and E 3jcos 62.2
8. Sis0 ide , 5.535 x 1046227
—Uudd. > > =130
145.535 x 104622)
With this we can go to the table of coefficients needed in the calculation of F, and
Fo. These are, for the third & range,
fi =0454 to
0.321 fa =-0.255
fu=0072 fo=-0008 fo =-0046
So
Fr maxlO, (0454 + 0.321 x 0,336 + 62.2/7-0.255)/180)] = 0,285
Fo = 0.072 + (0.098) x 0,336 + 62.274-0,046)/180 = 0.011
We now have everything needed to use
uation 2.16.14 4 i
the sloped surface: Eq O ger the total radiation on
1r=0.253x 1,7140787 - 0.285/I-teos 60) +0.787x 0.285 x 1.71
=0.,011 x 0.787 sin 60 + 1.04x 0.60 (t-ses co)
2
=0.433+0.422 +0.384- 0.007+0.156= 1.39MJfm?
This is about 8% higher than the result of the H
ibout 8º IDKR model and ab i
dlian the isotropie model for this example. né about 16% higher
uz Available Soiar Radiation
The next question is which of these models for total radiation on the tilted
surface should be used. The isotropic model is the simplest, gives the most
conservative estimates of radíation on the tilted surface, and has been widely used.
The HDKR model is almost as simple to use as the isotropic and produces results that
are closer to measured values. For surfaces sloped toward the equator, these models
are suggested. The Perez model is more complex to use and generaliy predicts
slightly higher total radiation on the tiltcd surface; it is lhus the least conservative of
the three methods. K agrees the best by a small margin with measurements. !é For
surfaces with y far from 0º in the northern hemisphere or [80º in the southerm
hemisphere, the Perez model is suggested. (In examples to be shown in later
chapters, the isotropic and HDKR methods will be used, as they are more amenable
to hand calculation.)
2.17 RADIATION AUGMENTATION
K is possible to increase the radiation incident on an absorber by use of planar
reflectors. In the models discussed in Sections 2.15 and 2.16, ground-reflected
radiation was taken into account in the last term, with the ground assumed to be a
horizonial diffuse reflector infinite in extent, and there was only one term in the
summation in Equation 2.15.2. With ground reflectances narmally of the order of
0.2 and low collector siopes, the contributions of ground-refiected radiation are
smali. However, with ground reflectances of 0.6 to 0.7 typical of snow and with
high siopes,!? the contribution of reflected radiation of surfaces may be substantial.
In this section we show a more general casc of the effects vt diffuse reflectors of
finite size.
Consider the geometry sketched in Figure 2.17.1. Consider two intersecting
planes, the receiving surface c (i.e., a solar collector or passive absorber) and à
diffuse reflector r. The angle between the planes is y. The angle y is 180º — £
if the reflector is horizontal, but the analysis is not restricted to à horizontal reflector.
The length of the assembly is m. The other dimensions of the coltector and reflector
are n and p, as shown.
If the reflector is horizontal, Equation 2. 14.3 becomes
dr = 16Rh + hdP ist Be cr + IpgE cg (7)
where F.., is again (1 + cos f)/2. The view factor F,. is obtained from Figure
16 The HDKR method yields slightiy bexer results than either the isotropic mode! or the Perez
model in predicting utilizable radiation when the critical radiation levels are significant. See Sections
2.20 10 2.22 for notes on utilizable energy.
FT Ata slope of 459, a flat surface sees 85% sky and 15% ground. Al a stope of 90º, it sees 50%
sky and 50% ground.
2.17 Rodintion Augmentatiun
103
Figure 2.471 Gicometric relationship of an energy receiving surface c and reflecting surface 1.
217.2, Fis obtained from he reciprocity relationship A,
san be obtained from the summation rule, Ft E, + EE
Fc is showa in Fig a Of the ratios
908, 120º, and 150º,
=AF and Fo,
. 1. The view factor
ure 2.17.2 as à function of the ratios t/m and plm for W of
Example 2.17.1
À vertical window receiver in a passive heating system is 3.0 m high and 6.0 m k
There is deployed in front of it a horizontal, diffuse reflector of the same Tema
extending out2.4m. What is the view factor from the reflector to the window?
What is the view factor from the window to the reflector? What is the view f: tor
from the window to the ground beyond the reflector? certos
Solution
For the given dimensions, nm is 3 0/6.0 à
: o 30/60 = 0,5, py 4/60 =
Figure 2.17.2a, the view factor E, , is 0.27. Pim bs 24/60 = 0 and tom
The area of the window is 18,0 m? and the F
e s 18.0 m?, e area of the reflcctor i 2
From the reciprocity relationship, P., = (14,4 x 0.27)/18.0 = 0.22, Ts tt
The view factor from window 19 sky, E )
. “ecsviS (1 + cos 90)/2,
View factor from collector o ground is then 1 = (0.50 4 0.22) = om or tdo ES
am Hthe surfaces e and r are very long in extent (ie., m is large relative to n
p as might be the case with long arrays of collectors for large-scale solar
applications), Hottel's "crossed-string” method gives the view factor as
Fa=ttP
1 2» (2.172)
Available Solar Radiation
ç
é
B
s
E
z
ê
s
“01 02 03 05 10 23.5 10 20 -
nfm
01 02 03 05
0.06,
e
&
04
06
View factor, Fr-c
e
2
0.08 im = 1.0
o 0702 05 10 23 5 1 20
ter
Figure 2.172 View factor £,., as a function of the relative dimensions of the collecting and
reflecting surfaces. Adapted from Hamilton and Morgan (1952).
ion Angmentulion 105
re 2.17,3 Section of reflector and colector surfaces,
where 5 is the distance from the upper edge af the collector to the vuter edge of the
refletor, measured in a plane perpendicular to planes c and 7, as shown in Figure
2.17.3, This can be determined from
(n2+p2-2npcos 2 (2173)
[For a collector array as in Example 2.17.1 but very long in extent, » = (302 +
24205 =3.84 mand F,, = (342.4 3.84/4.8 = 0.33]
It is necessary to know the incident radiation on the plane of the reflector. The
team component is calculated by use of Rpy for the orientation of the reflector
Surface, The ditlusc component must be estimated from the view factor F,,. For
any orientation of the surface »,
Est Fic +Eng= (2174)
where the vicw factors are from surface r to sky, to surface «:, and to ground. The
vicw factor F, , is determined as noted abyvc and E v.g Will be zero for a horizontal
reflector and will be small for collectors that ure long in extent. Thus as a first
approximation, F,.,= | F,. for many practical cases (where there are no other
obstructions).
There remains the questions of the angle of incidence of radiation reflecicd from
surface 1 on surface c. As an approximation. an average angle of incidence can be
taken as that of the radiation from the midpoint of surface to the midpoint of surface
Sa shown in Figure 2.17.3,/8 The average angle of incidence 8, is piven by
sin 8, Y (2.475)
The total radiation reflected from surfaçe r with area A,to surface c with area 4, if
18 As the refletor area becomes vers; large, the angle of incidence becomes that given by the ground-
Teflectunce curve of Pigure 5.4.1, where the angle y beiween the reflector and the coltector is /, the
abscissa om the figure,
e-B=18
35 45 ss
25 35 as ss Latitude
Latitude & 1
1)
50 pre reeeeotieoemerreererem
E vertical / ã
asp
s0F 3
E Dec. / / 4
s Jon
> ba E
30Ê 7
F
25
29f
15E
106
LAILA
A |
=
CASADA
EA
uu e o aee sut
2s 38 25 38 E
tomado é Latitude é
e to
Figure 2.19. Estimated Rh for surtaces facing the equator as à function of latitude tor various tg - f). by months.
tb B=1SPiqbi to = 0% (e) (4 B)= 158: (4) B=909, For the soulhem hemisphere, interchange
months as shown on Figure 1.8.2, and use lhe absolute value of latitude, From Beckman etal. (1977).
nr
Po
112 Available Solar Radiation
For surfaces in the southem hemisphere sloped toward the equator, with 7=
180º, the equations are
R= costg-+ f) cos ôsin ai + (2/180) oi sinhg-+ B) sin & (2.19.49)
cos ficos Bsin ay + (17/180)0 sin Qsin Ô
cosil- tan êtan à)
and o; = min (2.19.4b)
[aos Blum à) |
“The numerator of Equation 2.19.3a or 2.19.4a is the extraterrestrial radiation on
the tilted surface, and the denominator is that on the horizontal surface. Fach of these
is obtained by integration of Equation 1.6.2 over the appropriate time period, from
true sanrise to sunset for the horizonta! surface and from apparent sunrise Lo apparent
sunset an the tilted surface. For convenience, piots of R, as à function of latitude for
various slopes for = 0º (or 180º in the southem hemisphere) are showa in Figure
2.19.1, and corresponding tables of Ry are in Appendix D. These values of É, can
be used for surface azimuth angtes of 0º (or RO?) + 15º with lintte error.
The following example illustrates the kind of calculations that will be used in
estimating monthly radiation on collectors as part Of heating system design
procedures.
Example 2.19.1
A collector is to be installed in Madison, latitude 43º, at a slope of 60º to the south.
Average daily radiation data are shown in Appendix G. The ground rellectance is 0.2
for ali months exccpt December and March (g, = 0.4) and January and February
(pg = 8.7. Using the isorropic diffuse assumption, estimate the monthly average
radiation incident on the collector.
Solution
“The caleulation is detailed below for January, and the results for the year are indicuted
in a table. The basic equation to be used is Equation 2.19.1, The first steps are to
obtain HH and Rb. The ratio Hyff is à function of Ky and can be obtained from
Equation 2.12.1 or Figure 2.12.2.
For the mean January day, the 17th, from Table 1.6.1, 1 = 17, O = 20.99
The sunset hour angle is caleulated trom Equation 1.6.10 and is 69.19. With n = 17
and «3, = 69.1º, from Equation 1.10.3 (or Figure 1.10.1 or Table 1.10.1), Ho is
13.36 MJ/m?. Then Ky = 5.85/13.36 = 0.44.
The Etbs correlation, Equation 2.12.1a. is used to calculate HH com Ky and
aa, and gives HH = 0.45. The calculation of R, requires the sunset hour angle on
the sloped collector. From Equations 2.19.3
cosH-tan(43 — 60) tan(-20.8)] = 96.7º
2.20 Average Radiation on Sloped Surfaces-—The K-T Method n3
The angle q), was calculated as 69.1º and is less than 96.7º. so 3, = 69.1º. Then
p= vost 17) cost-20.9) sin 69.1 + (7 x 69.1/180) sin(-17) sin(-20.9)
p = =2,79
cos 43 cost-20.9) sin 69.1 + (rx 69.1/180) sin 43 sinç--20.9)
The equation for Jfy, Equation 2.19.1, can now be solved:
Hir= 58501 - 0.45)2.79 + 585 x 045(L+ c05 00) + 5.85 x 07! cos 80)
=8.87 + 2.004 1.02= 11.89 Mm?
The results for the 12 months are shown in the table. Energy quantities are in
megajoules per square meter. The effects of sloping the receiving plane 60º to the
south on the average radiation (and thus on the total radiation through the winter
season) are large indeed. The monthly results are shown in the table. The 7, values
are shown to à tenth of a megajoule per square meter. The last place is uncertain due
to the combined uncertainties in lhe data and the correlations for H3/H and R. TU is
difficult to put limits of accuracy on them; they are probably no better than 10%. |
Month " Ho Kr Haftt Ro p» ty
Jan. ses 1337 Uda 0.46 2.79 07 o
Feb, 9.13 1881 0.49 0.41 2.04 87 155
Mar. 12,89 2603 o.so u43 142 os 15%
Apr. 1588 2378 047 146 1.96 82 us
May 19.79 142 est UA om “2 153
Jume 221 4178 0:53 0.40 0.62 “2 159
July 2196 40.55 (54 11.39 0.66 02 164
Aug. 19,39 as? 0.54 0.39 0.84 [ 16.6
Sep. 14.75 2880 0.51 va 1] o2 158
Out 16,34 20.90 0.0 11.40 E] 02 149
Nov sn 14.63 039 1.54 2.56 “2 96
Dec. as ER 0.37 0.54 3.06 ns £s
n
2.20 AVERAGE RADIATION ON SLOPED SURFACES - THE K-T
METHOD
An alternative approach to calculation of average radiation on sloped surfaces has
been developed by Klein and Theilacker (1981). K is a bit more cumbersome to use
but shows improved results over the isotropic method when compared with hourly
caleulations based on many years of radiation data. The method is first outlined
114 Available Solar Radiation
below in a form restricted to surfaces facing the equator and then in general form for
surfaces of any oricniation, As with Equations 2.19.1 and 2.19.2, it is based on the
assumption that both diffuse and ground-reflected radiation streams are isotropic.
The long-term value of R can be calculated by integrating Gy and G from
sunrise to sunset for all days over many years of data for a single month and
summing. (For example, data for all days in January for 10 years should represent
the long-term average for January.)
$ [orar
Tate
=dyrtt o (2.20.1)
$ fica
day=1 tor
The denominator is NH. To evaluate the numerator, it is convenient to replace Gy
by 1; and exchange the order of lhe integration and summation. Using Equation
2,15.1, the radiation at any time of the day (i.e., for any hour) for N days is
nir= nf- abs + ESB) tp; [I-c8
| (2.20.2)
where the 7 and 1; are long-term averages of the total and the diffusc radiation,
obtained by summing the values of f and 1, over N days for cach particular hour
and dividing by A. Equation 2.20.1 then becomes?!
[leiam dese deseo
o (2.20.3)
or
Equations 2.13,1 and 2.13,3 define thc ratios of hourly to daily total and hourty
to daity difíuse radiation, and Equations 2.13.2 and 2.13,4 relate r, and r; to time
«» and sunset hour angle 0, Combining these with Equation 2.0.3 leads, for the
case of south-facing surfaces in the northem hemisphere, to
(2.20.4a)
21 The developracnt of this equation assumes that the day length does not change during the month,
220 Average Radiation on Sloped Surfuces—The K-T Method uns
where qo, is again given by
comi | Cosan gtan 6)
aj= no) (2.20.4b)
cos-[an(g - Ban 6]
and 3) = cos-[an(g - 8) tan 6] (2.20.40)
Also, « and b are given by Equarions 2.13.2b and 2.13.2€, and dis given by
d=sin 0, - Es
sin 03 FR cos (3, (2.20,49)
Equations 2.20.4 can be used in thc southem he phere for north-faci by
mis
y ' s) for north-facing surfaces
Example 2.20.1
Redo Example 2.19.1 for the month of January. using the Klein-Theilacker method.
Solution
For January. from Example 2.19.1, Ff, = 13.36 MUm?, FLgH = 0.45. and for the
mean day of the month (x = 17), q = 6); = 69.1º, Forthe mean day,
a = 0.409 + 0.5016 sin(69.1 — 60) = 0.488
& = 0.6609 — 0.4767 sin(69.1 — 60) = 0.586
d=sin 69,1 EX62
Tão “Os 69.1 = 0.504
cos L-tan(43 — 60) tan(-20.9)] = 96.7º
Using Equation 2,20.4a,
R = Cos43 60) nx691
0504 cos 43 [o-se - vasifin 69.1 -— ao Sos x)
Tx 1
+ 586 erga + sin 69.Icos 69.1 --2 cos 96 )
+045[
1 +cos 60 1 -cos 60
500) + 0.7(L 280660 4.449 40.338 +0.075= 195
So monthly average radiation on the colector would be
He=IÍR=585x195= 114 Mm?
120 Available Solar Radiation
20
3
'r. MU/m? per day
HandH
LiILSILAILIILAS
dan. Feb. Mar. Apr. May Jun. Jul Aug Sept Oct. Nov, Dec
Figure 2.244 Variation in estimated average daily radission on surfaces of various stopes às a
function of time of ycar for a latitude of 45º, Ky of 0.50 surface azimuth angle of 0º, and a ground
reflectançe 090.20.
temperate climates. Figure 2.2].1 shows the varialions of Hr (and H) ihrough the
year and shows the marked differences in energy received by surfaces of various
in summer and winter. .
do 2.214,24) shows the total annual energy received as a function of slope
and indicates a maximum at approximately 8 =$. The maximum is a broad one,
and the changes in total annual energy are less than 5% for slopes of 20º more or less
than the optimum. Figure 2.21.2(a) also shows total “winter” energy, taken as the
total energy for the months of December, January, February, and March, which
would represent the time of the year when most space heating loads would occur.
The siope corresponding to the maximum estimated total winter energy is
approximately 60º, or 6 + 15º. A 15º change im the slope ot the collector frara the
optimum means a reduction of approximately 5% in the incident radiation. The
dashed portion of the winter total curve is estimated assuming that there is substantial
snow cover in January and February that results in a mean ground reftectance of 0.6
for those 2 months. Under this assumption, the total winter energy is less sensitive
to slope than with p, = 0.2. The vertical surface receives 8% less energy than does
the 60º surínce if p, = 0.6, and 11% lessifp,=0.2. —
Calculation of total anmual energy for $= 45º, Er = 0.50, and p, = 0.20 for
surfaces of slopes 30º and 60º are shown as a functios of surface azimuth angle in
2.21 Effects of Receiving Surface Orientation ui
8
E E
g 3 é
ê 5» 6%
ê É ê
5 s 2
3 5 q
ê FÊ
5 E 5a
g z g
é 2 B&
so o 15 30 as
Slope Surface azimuth angie, y
ta (o)
Figure 2.21.2 (a) Variation of tutal annual energy and total winter (December to March) energy
35 a funelion of surface slope for a Iutitude of 45º, Kr of 0.50, and surface aziminh angle of 0º
Ground reflectance is 0.20 except for the dashed curve where it is taken às 0.60 for January and
February, (b) Variation of total annual energy with surface azimuth angie for slopes of 30º and 60º,
Tatilude of 45º, Kr0f 0.50, and ground reflectance of 0.20,
Figure 2.21.2(b). Note the expanded scale. The reduction in annual energy is small
for thesc examples, and the generalization can be made ihat facing collectors 10º to
20º cast or west of south should make little difference in the annual energy received.
(Not shown by annual radiation figures is the effect of azimuth angle on the diumal
distribution of radiation on the surface, Each shift of y of 15º will shift the daily
maximum of available energy by roughly an hour toward moming if y is negative and
toward afternoon if y is positive, This could affect the Performance of a system for
Which there are regular diurnal variations in energy demands on the process.) Note
that there is implicit in these calculations the assumption that the days are symmetrical
about sotar noon.
Similar conclusions have been reached by others, for example, Morse and
Csamecki (3958), who estimated the relative total annual beam radiation on surfaces
of variable slope and azimuth angle.
From studies of this kind, gencral “rules of thumb” can be stated, For
maximum annual energy avaitability, a surface slope equal to atitude is best. For
maximum summer availability, slope should be approximately 20º to 15º Jess than
the latitude. For maximum winter energy availability, slope should be approximately
10º to 15º more than the latitude. The slopes are not critical, deviations of 15º result
in reduction of (he order of 5%, The expected presence of a reflective ground cover
such as snow leads to higher slopes for maximizing wintertime energy availability.
The best surface azimuth angles for maximum incident radiation are 0º in the northem
hemisphere or 180º in the southem hemisphere, that is, the surfaces should face the
equator. Deviations in azimuth angles of 10º or 20º have small effect on total annual
energy availabilicy.
m Available Solar Radiation
222 UTILIZABILITY qu No
he concepts of utilizabffity arc developed. The
atlove a critical or threshold intensity is. useful,
then we can define a radiation statistic, called utilizability, as he fraction of the total
sadiation that is received at an intensity higher than the critical evel. We can then
oz the period by this fraction to find the total ulizable
multiply the average radiation £
energy. In these sections we define utlizability and show for any critical level how it
can be calculated from radiation data or estimated from K7.
Tu this section we present the concept of monthly average hourly utilizabitity
(the $ concept) as developed by Whillier (1953) and Hotel and Whillier (1958).
Then in Section 2.23 we show how Liu and Jordan (1963) gencralized Whillier's É
curves. Tn Section 2.24 we show an extension of the hourly utilizability to montbly
average daily utilizability (the é concept) by Klein (1978). Collares-Pereira and Rabl
(1979a,b) independently extended hourly utilizability to daily utilizability. Evans et
al. (1982) have developed a modified and somewhat simplified general method for
calculating monthly average daily utilizability.
ta Chapter 6 we develop in detail an cnergy balance equarion to represent the
performance of a solar collector. The energy balance says, in essence, that the useful
gain at any time is the difference between ihe solar energy absorbed and the thermat
losses from the collector. The losses depend on the difference in temperature
between the collector plate and the ambient temperature and on a heat loss coefficient.
Given a coefficient, a coilector temperature, and an ambient temperature (Le., a loss
per unit area), there is a value of incident radiation that is just enough so that the
avsorbed radiation equals the losses. This value of incident radiation is the critical
radiation level, fr., for that colector operating under those conditions.
H€ the incident radiation on the tilted surface of the collector /r is equal ta fr.
all of the absorbed energy will be lost and there will be no useful gain. Tí the incident
radiation excecds Ir, there will be useful gain and the collector should be operated.
JE Ly às Jess than Jr. no useful gain is possible and the collector should not be
operated. The utilizable energy for any hou” is thus (17 = Iro)*, where the
superscript + indicates that the utilizable energy can be zero or positive but not
in this and the following two sections
basis is a simple one; if only radiation
negative.
The fraction of an hour's total en:
utilizability for that particular hour:
= fr=ind. 222.1)
Ei
ergy that is above the critical level is the
where $, can have values from zera to unity. The hour's utilizability is the ratio of
the shaded area (Ly = 1rç) to the total avea (Fr) under the radiation curve for the hour
as shown in Figure 2.22.1. (Utilizability contd be defined on the basis of rates, ie.
using Gy and Gr.. but as à practical mater, radiation data are available on an hourly
basis and that is the basis in use.)
222 Utilizability
13
6r
Time
Figure 2.224 Gy vs. time for a day. For wn, E is the area under the G7 curve,
' ti y the hour shown, fp T
otris
tro is the area under the constant critical radiation tevel curve. ,
ie uiizabity for à single hour is not useful, However, utilizability for a
articular hour for a month of N days (e.g. 10 to 1i in Jam in wii ,
average radiation is /7 is useful. It can be found from ver) a th
“
0= 15 Vr-irt
N z Ih (2.22.2)
The monthly average utilizabic enei i f
. " 1 rgy for the hour is the product /
cateuaion can be done for individual hours (10 to Li, LI to 12, di )or Eno
and the result summed to get the months utilizable enet e app
] d to g le energy. If the application i
that the conditions of critical radiatioi | rat
n level and incident radiation are s; i
about solar noon, the calculations can be done for hour-pairs (e.g., 10 to and Tn ;
Zo0r9 to EO and 2 to 3) and the amount of calculations halved. º
" Ge dou average radiation data by months and a critical radiation level, the
ext step is to determine q. This is done by processin) ation da
! 3 the h a ;
las outlined by Whillier (1953) as follows: Me holy slim a ir
fai Dor a given location, hour, month. and collector orientation, plot a cumulative
a tuo curve of /r/fr. An example for a vertical south-facing surface at Blue
: Ea A. or January is shown in Figure 2.22.2 for the hour-pair 11 ta 12 and 12 to
doi is provides E picture of the frequency of occurrence of clear, party cloudy, or
ly skies in that hour for the month. For example, fi o i a
À - s he hour- f Fi
2.22.2, for f of 0.20, 20% of the da) jaion char a 2a 10% of ho
1 .20, ys have radiation that is less than 10%
average. and for f of 0.80, 20% of the days h: jon ii E a
Acuede OE ie aver y5 have radiation in that hour-pair that
A dimensionless critical radiation is defined as
(2.223)
Ansemple is shown as the horizontal line in Figure 2.22.2 where X, = 0.75 and f.
04. The shaded area represents the monthly utilizability, that is, the fraction of
nthly energy That is above (he enitical level. Integrating hourly utilizability over
a
14 Available Solar Radiation
25 T T T T
Biue Hi Observatory
South-facing vertical surface
Januaty, 1953-1056
zo HI-12 am. ars 12-1 pum,
x - Critical ad.
* Average raú.
fe I
Q .2 0.4 85 [E] 19
Fractlonal time, f, during which radiation < 3,
Figure 2,222 Cumutative distribution curve for hourly radiation on a south-facing vertical
surface in Blue Hill, MA. Adupted from Liu and Jordan (1963).
all values off, gives é for that critical radiation level:
1
*[ du df (2.224)
Hc
e
As the critical radiation level is varicd, f, varies, and graphical integrations of
the curve give utilizability À as a function of critical radiation ratio X,.. An example
derived from Figure 2.22.2 is shown ir Figure 2.223.
Wihillier (1953) and later Liu and Jordan (1963) have shown that in a particular
location for a I-month period, à is essentially the same for all hours. Thus,
although the curve of Figure 2.22.3 was derived for the hour-pair 11 to 12 and 12 t0
1, it is useful for all hour-paírs for the vertical surface at Blue Hill,
The line labeled "imiting curve of identical days” in Figure 2.22.3 would result
from a cumulative distribution curve that is à horizontal line at a value of the ordinate
of 1.0 in Figure 2.22.2. In other words, every day of the month looks like the
average day. The difference between the actual q curve and this limiting case
represents the error in ulilizable energy that would be made by using a single average
day to represent a whole month,
Example 2.22.1
Caleulate the utilizabie energy on a south-facing vertical sofar collcctor in Blue Hil!,
MA, for the month of January when the critical radiation level on the collector is 1.97
Mimi, The averages of January solar radiation on a vertical surface are 1.52, 1,15,
and 0.68 MJ/m? for the hour-pairs 0.5, 1.5, and 2.5 hours from solar noon.
2.23 Generalized Utilizabitity
ns
10
oa
o
a
Usilizabinty, é
o3| Limeting
curve of
identical days.
04 08 12 16 O.
Critical radiation ratio, X, = yu/fy
Figure 2.22.3 Utilizability curve derived by icalhy à i
y numericaliy inté F
a iz y Integrating Figure 2.2.2. Adapted
Solution
For the hour-pair 11 to 12 and 1210 1, the dimensionless critical radiation ratio X is
e
107.
1.52 9:70
x
and the utilizability, from Figure 2.22.3, is 0.54. The uti
, 22.3, is 0,54, The utiliz:
collector during this hour is tabs energy om the
Irô= 1.52 x 0.54 = 0.82 MJjmê
For the hour-pair 10 to 11 and 1 to 2 the valve of X, is 0.93, gs 0.43, and heis
DAS. For the hour-pair 9 to 10 and 2 to 3, X, is 1.57, is 0.15, and rg is 0.10
The average utilizable energy for the month of January is then ,
ND ir$=31x2(0.82+04940.10)=875 MI/m? E
hes
223 GENERALIZED UTILIZABILITY
We now have a way of calculating q for specific locations and specific orientations,
For most locations the necessary data are not available, but it is possible to make use
of lhe observed statistical nature of solar radiation to develop generalized q curves
Ihat depend enly on É, latitude, and collector slope. As noted above, $ curves are
Do Available Solar Radíntion
sunset hour angle and the day length of February 16, the mean day of the month, are
78.9º and 10.5 hours, respectively. The monthly average ratio /Ly/H is 0.39 from
Figure 2.12.2, and ) = 7.5º. The fatios +, and rg from Figures 2.13. and 2.13.2
are 0.158 and 0.146. For the mean day in February and from Equation 1.8.2, R, is
1.62. Then fr ftom Equation 2.23.4 is
E = 0.5 x 20.5[(0.158 — 0.39 x 0.146)1.62 + 0.39 x 0.146 + 0.88
+ 0.740.158 + 0.12] = 2.33 MJm?
The critical radiation rate for this hour-pair is
eim 1280
“kh 233 0.55
From the figure of Example 2.23.1, é is 0.50. The utilizable energy for the month
for this hour-pair is
UE=233x 0.50x 2x 28 = 65.2 Mm? n
Liu and Jordan (1963) have generalized the calculations of Example 2.231.
They found that the shape of the À curves was not strongly dependent on the ground
reflectance or the view factors from the collector to the sky and ground.
Consequently, they were able to construct a set of $ curves for a fixed value of Er.
The effect of tilt was taken into account by using the monthly average ratio of beam
radiation on a tilted surface to month!y average beam radiation on a horizontal surface
Ro as a parameter. The generalized é curves arc shown in Figures 2.23.1 for values
of Er of 0.3, 0.4, 0.5, 0.6, and 0.7. The method of constructing these curves is
exactly like Example 2.23.1, except thal the tilt used in their calculations was 47º and
tie ground reflectance was 0.2. À comparison of the é curve from Example 2.23.1,
in which the tilt was 40º and the ground reflectance was 0.7 with the gencralized é ,
curve for Er = 0.5 and R$ = 1.79, shows that the two are nearly identical.
With the generalized à curves, it is possible to predict the utilizable energy ata
constant critical level by knowing only the long-term average radiation. This
procedure was illustrated (for one hour-pair) in Example 2.23.2. Rather than use the
é curve calculated in Example 2.23.1, the generalized 6 curve could have been
uscd. The onty additionat calculation is determining R, so that the proper curve can
be selected. In Example 2.23.2, X, = 0.55. From Table D-7 in Appendix D, Ry =
179. Figure 2.23.1e is used to obtain 6; it is approximately 0.50.
Xt is convenient for computations to have an analytical representation of the
utilizability function. Clark et al. (1983) have developed a simple algorithm to
represent the generalized É functions. Curves of q vs. X, derived from long-term
2.23 Generalized Udliizability m
1
Usilizabitity, o
o
0
Usilizabilicy, é
Figure
(1363).
“ANTT To
|
: 1
Beco Tifted surface
| po Bs
H Ny 1
TT
—
1
E itenbca dava" Horizon ERR q
[SE
2
E Limitingcune gado =
oc liia N bias EE
0 04 08 12 16 20º 24 28 32
Critica radiation ratio, X,
(a)
NUTTTO T TTT
DB T | +
05 — N | /-Limiting curve, Rg + os | q
! Titgd surto, à,
O | a!
E [7-2 30,6 +
oa |
N +
ting curve “ a
ntical days] 4
oa
0 04 08 12 1.6 zo 24 28 32
Critical radiation ratio, Xp
ca
2.231 Generalized q curves for south-facing surfaces. Adapted from Liu and Jordan
| 1» Available Solar Radintion 223 Generalized Utilizabilicy 133
| VETO TT weather data are representeu by
| Rç=05 4
Va i >
os 4 | o EX 2X
| - 3 05 (1-XolXmP EX =2 (2.23.54)
! HH | .
tas Ling cure em 4 | lgt-Le2+(1 +24) 2.74)? otherwise
ZE — Tilted suríace, É, 2
E 29 4 wire 2X -DQ-X,) (2.23.5b)
5 mu F Horizontal surface A
F | NK 3 , Xm = 1.85 + 0.169RyR? - 0.0696 cos B/k? - 0.981Ír/c0528 (2.23.5€)
o2- + q o
E Tente da] 3 The monthty average hourly cicamess index E is defined as
Guliuid Lupilisiiiss o
% 0 06 12 18 20 24 28 a? Rel (2.23.6)
Cica radiation ati, Xe i
teh
é Kt can be estimated using Equations 2.13.2 and 2. 13.4:
.
nà KoF, .
GL = Eir= Kofa + heos 0) (2237)
where a and b are given by Equations 2.13.2b and 2.13.2e,
The remaining term in Equation 2.23.5 is Ry, the ratio of monthly averae
hously radiation on the tilted surface 10 that on a horizontal surtace.
Ra=bil=tri(ttr) (2.23.8)
“OTTO TOTO NOT TITTITO
| Rr=08 R=:070 + Example 2.23.3
- t +
08) 4 | 4 Repeat Example 2.23.2 using the Clark et al. equations.
E 4 I
. É N | 4 Solution
= 08 Limiting curve Limiting curve .
E p>e | Ryu q The calculations to be made are Ry, É. XX, 8, and finally 9. Intermediate
Ê , 5 results from Example 2.23.2 that are useful here are / = 2.33 MJ/m2; 1, = 0.158;
E surteco a 00, = 789%: w= 7.59; and X, = 0.549:
02 Montana boricomal Ryetro 2.33 =144
[O identical dove 4 a NI 4 nH o 0.158x20.3x 0.50
Limitir
oETEPSTL NO LUSA 110 o | ,
0 04 0.8 12 0 04 08 12 16 To calculate kr we need the constants q and b in Equation 2.23.7;
ta Critical radiation catio, X, te)
ás à = 0.409 + 0.5016sin(78.9 - 60) = 0.571
Figure 223.1 (continued)
and b=0,6609 - 0,4767sin(78.9 — 60) = 0.506
14 Available Solar Radiatlon
Thus Er = 0.50(0.571 + 0.506 cos 7.5)= 0.536
Next calculate X,, with Equation 2.23.8e;
0 (0.981 x 0.536
= 0.169 Lé4. 0.0696 €05 = 1,942
Xm= 1.85 + 0.536? 0.536? co
The last steps are to calculate g and é with Equations 2.23.5b and 2.23.5a:
g=(1.942- 12 1.942) = 16.24
5492
Then o=[1624 - [16.24 +(1+2x 16.24)1- assa ] =0.52
a
This is neasly the same $ as that from Example 2.23.2.
“The é charts graphicaliy illustrate why a single average day should not be used
to predict system performance under most conditions, The difference in utilizability
as indicated by the limiting curve of identical days and the appropriate $ curve is the
extor that is incurred by basing performance on an average day. Only if Kr is high or
if the critical level is very low do all 6 curves approach the limiting curve. For many
siluations the error in using one average day to predict performance is substantial.
The é curves must be used hously, even though a single 6 curve applies for a
given collector orientation, critical level, and month. This means that three to six
hourly calculations must be made per month if hour-pairs are used. For surjaces
facing the equator, where hour-pairs can be used, the concept of monthly average
daily utilizability, 9, provides a more convenient way of calculating useful energy.
However, for processes that have critical radiation levels that vary in repeatabie ways
through the days of a month and for surfaces that do not face the equator, the
generalized é curves must be used for each hour.
2.24 DAILY UTILIZABILITY
The amount of calculations in the use of é curves led Klein (1978) ta develop the
concept of monthly average daily utilizabiliy, É. This daily utifizabiliy is defined as
the sum for a month (over all hours and days) of the radiation on a tilted surface that
is above a critical level divided by the monthly radiation. In equation form,
DL lir-ir
Era (2.24.1)
HiN
224 Daily Utilizability 135
where the critical level is similar to that used in the $ concept?! The monthly
utilizable energy is then the product HjN' The concept of daily utilizability is
illustrated in Figure 2.24.1, Considering either of the two sequences of days, Gis the
ratio of the sum of the shaded arcas to Lhe total areas under the curves.
The value of 6 for a month depends on the distribution of hourly values of
radiation in the month. 1fit is assumed that all days are symmetrical about solar noon
and that the hourly distributions are as shown in Figures 2.13.1 and 2.13.2, then &
depends on the distribution of daily total radiation, that is, on the relative frequency of
occurrence of below average, average, and above average daily radiation values.”
Figure 2.24.1 illustrates this point. The days in the top sequence are all average days;
for the low critical radiation level represented by the solid horizontal line, the shaded
areas show utilizable energy, whereas for the high critical level represented by the
dotted line, there is no utilizable energy. The bottom sequence shows three days of
varying radiation with the same average as before; utilizable energy for the low
critical radiation level is nearly the same as for the first set, but there is utilizable
energy above the high critical level for the nonuniform set of days and none for the
Sequence À
Radiation
Day 1 Dy2 Day3
Figure 2.24. Two sequences of days with the same average radiation levels on the plane of the
colector. From Klein (1978).
24 The critical level for & is based on monthly average "optical efficiency” and temperatures rather
than on values for particular hours, This wil! be discussed in Chapter 21.
25 Klein assumed symmetrical days in his development of $. It can be shown that depanure from
Symmetry within days (e.g., if aftenoons are brighter than mormings) wil! lead tg increases in 8;
thus a & calculateá from the corretations of this section is somewhat conservative.
References
42
Available Solar Radiation ;
: 140
From Equation 1.8.2, Ry, is 1.38, Then R, can be calculaicd using Equation
irom Equation 1.8.2, Ryn
2.24.2:
| 9134 062) 1384
0.146
rosie fiz
0.134 x 062 (1 sos 60)
Toi 1
Equation 2.19.2 is used to calculate R. From Figure 2.19.1 or Appendis » Ri .
1.42. From Figure 2.12.2 HalH is 0.43 at Ky = 0,50, (See Example 2.19.
more details.) Then
1 - cos 60
60 J=1.23
R (041.42 + 043(-+60860) + 04f o
and Ra!R = 1.12/1.23 = 0.91
From Equation 2.24.3 the dimensionless average critical radiation level is
245 x 3600 -
0.146 x 1.12 x 12.89 x 109
$ i ions 2.24.4,
We can now get the utilizability 9 from Figure 2.24.2c or from Equatons 24
With Er = 0.50, a = 0.685, b = 0.819, ande = 0.411, $= 0.68.
The months utilizable energy is thus
EN $= HRNô= 1289x 1.23x 31 x 0.68 = 334 MJjm? E
The $ depend on R and R,, which in tum depend on the aivisioa of toi
radiation into beam and diffuse components. As noted in Section z A ace
sustenta! uncertainties in determining this division, The correlation 5 a Mes
ii IC N
Er of Li 1960) was used by Klein (1978) to genorate the 1
a nos of Pá aa (1976), which indicates significantly Dieter actions
of diffuse radiation, was also used to generate & charts, and de real Is vers a
significantly different from those of Figure 2.24.2. A ground rei e ane o a
ed in generating the charts, but a value of 0.7 was also use: : made no
E ificant difference. Conscquentiy, even if the diffuse-to-total correi ion 5
cam ea a result of new experimental evidence, the É curves will remain vali a i
pia using different correlations will change the predictions of radiation on a tiltes
course, j '
à MI change the performance estimates. . o
tio em te thought Of as a radiation statistic that has built into i cia
? be applied to a variety ef design
iation levels. The $ and q concepts can e
ema for heating systems, combined sotar energy-heat pump Systems, and a
dibers The concept of utilizability has been extended to en to pessive sã ei
p ii that cannot
ildi excess energy (unutilizable energy) a
ii sue can be estimated. The unutilizability ídea can also apply 10
uildin)
photovoltaic systems with limited storage capacity.
215 SUMMARY
In this chapter we have described the instruments (pyrheliometers and Pyranometers)
used to measure solar radiation, Radiation data are available in several forms, with
the most widoty available being pyranometer measurements of total (beam plus
diffuse) radiation on horizontal surfaces, These data are available on an hourly basis
from a limited number ot Stations and on a daily basis for Many stations.
Solar radiation information is needed in severa! different forms, depending on
the kinds of calculations that are to be done, These calculations fall into two major
categories. First (and most detailed), we may wish to calculate on an hous-by-hour
basis Lhe long-time performance of a solar Process system; for this we want honrly
information of solar radiation and other meteorological measurements, Second,
monthly average solar radiation is useful in estimating long-term performance of
some kinds of solar processes. 1 not possible to predict what solar radiation will
be in the future, and recourse is made to use of past data to predict what solar
processes will do.
time distribution of radiation ir a day, and radiation on surfaces other than horizontal.
We introduced the concept of utilizabitity, a solar radiation statistic based on levels of
sadiation available above critical levels, Determination of eriti
collectors will be treated in Chapters 6 and 7, and the utilizability concepts will be the
dasis for most of Part LI], on desiga of solar energy processes.
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Algorithm for Calculating Monthly-Average Radiation on Inclined Surfaces *
Knight, K. MS. A. Klein, and 1. A. Duffie, Solar Energy, 46, 109 (1991). “A Methodology for
the Synthesis of Houriy Weather Data,”
Kondrayev, K. Y., Acrinometey (iranslated from Russian), NASA TT P.9712 (1965); also
Radiation in the Armosphere, Academio Press, New York (1969).
Xiucher, T. M.. Solar Energy. 23,111 (1979).
“Evaluating Models to Predica Insolation on Tilted
Surfaçes.”
Lutimer, 3. R.. in Proc. First Canadian Solar Radiation Data Workshop (3, E. Hay and T. K
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Monitoring Solar Radiation.”
Liu, B. Y. H. and R, C. Jordan, Solar Energy, 4(3), E (1960)
Chagpeteristie Distribution of Direct, Diffuse and Total Solar Radiat
- “The Interrelationship and
14 Available Solar Radiation
Liu, B. Y. H. and R. €. Jordan, ASHRAE tournal, 3 (0), 53 (19625. “Daily Insolatioo on
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Liu, B. Y. H. and R. C. Jordan, Solar Energy, 7, 53 (1963). “The Long-Term Average
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Liu, B. Y. H. and R. €. Jordan, in Application of Solar Energy for Heating und Cooling of
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of Solar Energy."
MeDaniels, D. K,, D. H. Lowndes, H. Mathew, J. Reynolds, and R. Gray. Solar Enerps
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17,277
ctor
Combinations.”
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Flal-Plate Solar Absorbers:
Morse, R. N. and 3. T. Cramexki, Repor F.E.6 of Engineering Secti
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References
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. 8
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are Dei
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150 Selected Neat Transfer Toplos
equating to zero, the wavelength corresponding to the maximum of the distribution
can be derived. This leads to Wien's displacement law, which can be written as
AqmasT = 2897.8 umk 43
Planck's law and Wien's displacement law are ilustrated in Figure 34.1,
which shows spectral radiation distribution for blackbody radiation from sources at
6000, 1000, and 400 K. The shape of the distribution and the displacement of the
wavelengih of maximum intensity is clearly showa. Nose that 6000 K represents an
approximation of the surface temperature of the sun so the distribution shown for that
temperature is an approximation of the distribution of solar radiation outside the
carih's atmosphere. The other two temperatures are representative of those
encountered in low- and high-temperature solar-heated surfaces.
The same information shown in Figure 3.4.1 has been replotted on à
normalized linear scale in Figure 3.4.2. The ordinate on this figure, which ranges
from sero to one, is the ratio of the spectral emissive power to the maximum value at
he same temperature. This clearly shoves the wavelength division between à 6000 K
suurce and lower temperature sources at 1000 and 400 K.
o!
Wimê um
Yefocus ot
maximo
Spectral emissive pomer, Exa.
5
Wovelength, À, um
Figure 3.41. Spectral distribution of blackbody radiation.
36 Radiation Tables
Is
VT ART 1
: [=
oeliia IS
3 %
HI8 E
«06|[ |
É
A oa 1
oz
ol )
am ra
RR RR RR RR
Wavelength, am
Figure 3.4.2 Normalized spectral distribution of blackbody radiation.
3.5 STEFAN-BOLTZMANN EQUATION
Pane 5 law gives the spectral distribution of radiation from a blackbody, but i
ngineering calculations the total energy is often of more interest. By integrar
Planck's law over all wavelengths, the total energy es | fe
s, tha
otal energy emitted by a blackbody is fornd
E =| En dA=or!
[ à (3.5.1)
mer Gis the Stefan-Boltzmann constant and is equal ta 5.6697 x 108 Wim2K4.
ls constant appears in cssentially all radiation equations. TE
36 RADIATION TABLES
Starti ! 5
ing with Planck's law (Equation 3.4.1) of the spectral distribution of blackbody
radiation, Dunkle (1954) has
mn, presented i i
aenfadons Pancho Ro va Doom a method for simplifying blackbody
Ep=— A
A'Texplcyar)-1] Gsm
Equation 3.6.1 can be intej i i
. grated to give the radiation betw imii
The total cmitted from zero to any wavelength À is given dy amy vaveng im
A
Ea dÃ
[ (3.6.2)
Eoab
| o Selected Heat Transter Toplos 36 Radintion Tables a
i Fº, the
Substituting Equation 3.6.1 into 3.6.2 and noting that by dividing by O) Table 3.6.1b Fraction af Blackbudy Radiation Energy between Zero und
| | integral can be made to be only a function of 27. AT for Even Fraclional Increments
Ara
! | ar F 1 Tm
0-Ar 7, mk midpoint b
' fo. AT.umK — midpoint
' foro É = adam 0.63 — o focdr o Mtmk miópin
art o otarflexplca anj-1l 0.05 1880 1660 oss 4410 “425
0J0 200 2050 0.60 4740 “sm
5 » am 2320 065 5130 4930
able 36.1a Fraction of Blackbody Radiont Energy betweca Lero a so 2560 am o ss
and àT for Even Increments of AT 030 3120 ata ao Si50 5850
DO RMR fom AGHMK four . a 0.80 ea6o caso
AT,umk fo.ar AT.pmk fo-ar ATumkK fo.ar ass 3350 3230 085 7850 310
070 [00 [0700 40 3580 3460
0.90 9380 as10
todo 0.0003 4000.5643 8000 — 0.856% | 5 a 3830 sro 095 12500 1
1100 0.0009 s600 0.5793 g100 O.8601 ? ano 3970 100 oo iesoo
1200 09021 4700 05937 amo 08639
1300 00043 4800 0.605 s300 08676 |
1400 0.0077 4900 0.6209 8400 0871 The value of this integral is the fraction of the blackbody energy between zero and
1500 00128 so 0.637 Bs gas AT. Sargent (1972) has caleulated values for convenient intervals and the results are
1600 0.0197 s100 6461 8600 — 0.8778 l given in able 3.6.1. (Note that when the upper limit of integration of Equation
voo 005 5200 046579 so om 3.63 is infinity, dhe value of the integral is unity)
1508 qs21 Sm Ee eso 0887 36 For use in a computer, the following polynomial approximations to Equation
2000 0.0667 5500 04909 s000 08899 e inte been given by Pivovonsky and Nagel (1961). For y greater than or
2100 0.0830 5600 0010 9100 08927 equal to
2200 0.1009 sm 07107 9200 08954
2300 0.1200 sso0 0.720 9300 0.8980 fre Foaro dá em
2400 0.1402 syga 0.729] 9490 0.9005 Car as a La at [rm 74 3)m pr6]my+ 6) (3.6.4)
2500 01613 6000 07378 9500 09030
2600 0.1831 sumo 07461 9600 09054 For less
2700 0.2053 e200 07541 go 090% Yless than 2
2800 0.2279 6300 07618 900 09099
2900 0.2506 6ao 07692 9900 09120 5 5 sl f y
3000 om esmo Om 10000 09141 DS SOaO * SIETOO 13305 600)
3100 02958 00 0981 nojo 0938 sv
3200 03181 6700 07897 12000 09450, . 6.
3300 0.340 6800 0.796 13000 09550 here vo
3400 03617 eo 08022 14000 09628 where y=Cy/AT.
3500 0.3829 00 08080 15000 09689
3600 0.4036 7100 08137 16000 0.937 Example 3,6.1
a 8191 17000 09776 o .
Í E gua 0 08244 iodo 09807 Assume that he sun is a blackbody at 8777 K. a What is the wavelengrh at which the
3900 0.4624 7400 08295 19000 0.9833 4 maximum monochromatic emissive power occurs? b What is the energy from this
| 4900 0.4809 7500 0.8343 20000 09855 source that is in the visible part of the electromagnetic spectrum (0.38 to 0.78 um)? 4
É ; a100 0.4987 7600 083% 30000 08952 ! 2.
Po 4200 0.5160 mo 0836 40000 09978 Solution
| 4300 0.5327 7800 08479 500 09988
! 4400 o.sus8 m00 08521 o
a Thé value of AT at which thc maximum monochromatic emissive power occurs is
2897.8 um K, so the desired wavelength is 2897.8/5777, or 0.502 um. b From
MTL ——"T
184 Selected Hent Transfer Topics
Table 3.6.1 the fraction of encrgy between zero and AT = 0.78 x 5777 = 4506 pm
is 56%, and the fraction of the energy between zero and AT = 0.38 x 5777 = 2195
pmk is 10%. The fraction of the energy in the visible is then 565% mínus 10%, or
46%. These numbers are close to the values obtained from the actual distribulion of
energy from the sun as calculated in Example 1.3.1. n
3.7 RADIATION INTENSITY AND FLUX
Thus far we have considered the radiation leaving a black surface in all directions;
however, it is often necessary to describe the dircctional characteristics of a general
radiation fieid in space. The radiation intensity is used for this purpose and is defined
as the energy passing through an imaginary plane per unit area per unit time and per
unit solid angle whose central direction is perpendicular to the imaginary plane.
Thus, in Figure 3.7.1, if AE represents the energy per unit time passing through
AA and remaining within Aq, then intensity is?
I= lim E. (3.71)
MA 50 AAAW
Aw0
“The intensity 1 has both a magnitude and a direction and can be considered as a
vector quantity. For a given imaginary plane in space, we can consider two intensity
vectors that are in opposite directions. These two vectors are often distinguished by
the symbo! ** and 1.
The radiation flux is closely related to the intensity and is defined as the energy
passing through an imaginary plane per unit area per unit time and in all directions on
one side of the imaginary plane. Note that the difference between intensity and flux is
that the differential area for intensity is perpendicular ta the direction of propagation,
whereus the differential area for flux lies in a plane that forms the base of a
hemisphere through which the radiation is passing.
The intensity can be used to determine the flux through any plane. Consider an
elemental area AA on an imaginary plane covered by a hemisphere of radius 7 as
aa NORMAL TO PLANE
|
du
IMAGINARY PLANE
Figure 3.7.1 Schematic of radiation intensity,
3 The symbol 1 is used for intensity when presenting basic radiation hes€ transfer ideas and for solar
tadiatios integrated over an hour period when presenting sotar radiation ídeas. The Iwo will seldom
be used together.
Plane. Second, the radiation Flux witl hay
two possible directions of the normal to th
emphasize which of the two
the superscript + or —
directions,
3.7 Radiation Intensity and Flux
+72 Schematic of radiation Flux.
shown in Figure 3,7.2, The energy
per unit ti E t
the surface of the hemisphere from th equal o UE Sm tea A4 on
e arga AM is equal to
AQ=IAA( cos dA”
4 (3.7.2)
Where AAr? is the solid angle between AM am
Perpendicujar to the intensity vector.
f direction can then be defined as
id AA! and SA cos Bis the arca
The energy fiux per unit sotid angle in the 6,
Sg= im Lico 6 MA
r
SAS0 AA
(3.73)
The radiation flux is then found b;
, integrati isphe)
ee a Y inlegrating over the hemisphere, The sphere
n terms of the angles and óso that
í nt2
g= Icos O si
j [ cos 8 sin 6 dg dg (3.7.4)
Kis convenient to define d=cos 8su that
ar
q= taud
Í [ duda Q.7.5)
Two important Points concerring
the radiation flux j
ddiation flux is, in general, a functi nO red Pi
ion Of the orientation of the chosen imaginary
e two valucs corresponding to cach of the
e Imaginary plane. When it is necessary to
té vo possible values of the radiation flux is being considere
an be used along with a definition of the posiive and negarive
Selected Hezt Transfer Topics
160
3.1t NATURAL CONVECTION BETWEEN FLAT PARALLEL
PLATES
The rate of heat transfer between twa plates inclined at some angle to he nero pa
obvious importance in the performance of flat-plate coilectors. Free comes Ton het
transfer data arc usually correlated in terms of two or three dimension! es pura e as
the Nusselt number Ny; the Rayleigh number Ra, and the Pranda pum W dr
Some authors correlate data in terms of the Grashof number, which is the ratio
Rayleigh number to the Prandil number.
” Ea Nusselt, Rayleigh, and Prandel numbers are given byé
3.1.1)
Nu AL t
esfrart? 41.7)
Re= a
311.3)
Pr=g é
where
ha = heat transfer coefficient
L
late spacing
ermal conductivity
= E tant
pe nano coefficiem of expansion (for an ideal gas, B' = 1/7)
AT = temperature difference between plates
v= kinematic viscosity
mr = thermal diffusivity
For parallel plates the Nusselt number is the ratio of a pure comtuction
resistance to a convection resistance [ie., Nu = (L/k)/(1/h)] so that a Nussel
it resents pure conduction. .
do (1958) examinça the published results of à number of investi tioos má
concluded that the most reliablc data for usc in solar colector calculations “ 35)
were contained in Repon 32 published by the U.S. Home Finance is É : e
In a more recent experimental study using air. Hollands et al. y pel e
relationship between the Nusselt number and Rayleigh number tor tilt angtes froi
to 75º as
13
á no Ra cos 8
7-8 ) fe =
RR Eh | SO |
Ga
' il It at the mean
5 Fluid properties in the convection relationships of this chapter should be evaluated at
temperature.
3-H Natural Convectio belween Flaz Paralle! Plates 161
where the meaning of the + exponent is that only positive values of the terms in the
square brackets are to be used (ie. use zero if the term is negative).
For horizontal surfaces, the results presented by Tabor compare favorably with
the correlation of Equation 3.11.4. For vertical surfaces the data from Tabor
approximate the 75º tilt data of Hollands et a]. Actual callector performance will
always ditter from analysis, but a consistent set of data is necessary to predict the
trends to be expected from design changes. Since a common purpose of this type of
data is to evaluate collector desiga changes, the correlation of Hollangs e; al is
considered to be the most reliable.
Equation 3.114 is plotted in Figure 3.11.1. Tn addition to the Nusselt number,
there às a second scale on the ordinate giving the value of the hear transfer coefficient
times the plate spacing for a mean temperature of 10 C, The scale of this ordinate is
not dimensionless but is mmWim2C, For temperatures other than 10 €, a factor E,
the ratio of the thermal) conductivity of air at 10 C to that at any other temperature, has
been plotted as a function of temperature in Figure 3.112. Thus to find hi at any
temperature other thar 10 €, it is only necessary to divide Eshl as read from the
chart by E) at the appropriate temperature,6
The abscissa also has an extra scale, FATE, To find ATP at temperatures
other than 10 C, it is only necessary 10 divide FLAT by Es, where Ey is the ratio
of L/Tya at the desired temperature to 1/Tverat 10 C. The ratio Fi is also plotted
in Figure 3.11.2.
ESTE mm! C taie ontyj
not 10 108
ETTA Trama
Correlation nf Holiands et al. (19764
AAA no É
Ê pr “| 4 a
f o É
í %
ê o É
é z
? T 5 =
E
| Ê
concuetio td
30
1
ot not od 108
fayleigh number
Figure 3.211 Nusselt Number às a fonction of Rayieigh number for free convection heat
transfer between paralel flat plates at various slopes.
É The towercase letter ! is used as a reminder that the units are millimeters instead of meters.
E Selected Heat Transfer Topics
19
os
og
ED]
02]
a
2
Zo
a &
os
04
L
es | Ra
n6 85 50
TEMPERATURE, C
Figure 3.112 Air propesty corrctions F, and > for use with Figure 3 1L.L. From Tabor
(1958).
Example 3.11.1
Find the convection heat transfer coefficient between two parallel plates separated by
25 mm with a 45º till. The lower plate is at 70 € and the upper plate is at 50 €.
Solution
029 WimkK, T=333K
mis. (Property data
At the mean air temperature of 60 C air properties are k=
so |' = 1/333, v= 1.88 x 10-5 mêfs, and q = 2.69 x 107
are from Appendix E.) The Rayleigh number is
9.81x 20x 00
Ra=— = -= 1.82x 10º
333 x 1.88 x 105 x 2.69 x 105
From Equation 3.11.4 or Figure 3.11.1 he Nusselt number is 2.4. The heat transfer
coefficient is found from
2.4 x 0.029
enuk-tIÊ US 22,78 Wim2kK
h=Nut=""0025
As an alternative, the dimensional scales of Figure 3.11.! cam be used with the
property corrections from Figure 3.11.2. At 60 C, F| = 6. 49 and Fa = 0.86.
Therefore, F;ATÉ = 0.49 x 20x 25º = 1.53 x 10º mmiC. From the 45º curve in
Figure 3.1.1, Fab! = 59. Finally, h = 5940.86 x 25) = 2.74 Wink.
Even with the substantially reduced radiation heat transfer resulting from the
low emittance in Example 3.10.1, the radiation heat transfer is about one-half of the
convection heat transfer.
3.11 Natural Convection between Flat Parallel Plates 163
Fra TP, mar? E (air only)
o 10%
E TT pr TT qi TT”
Reporr3z
St —-— Raithby etal;A = 5, 60, «
Randall tal, | 4 z
af —..— De Grafi and Van Der Hetd 100 5
E Vertical A £
ês 41.
LÊ So E
Ê E
é IE
2 3)»
ã
/ o é
. 4
.< A »
PA,
| Ji Í
1º ot 1 E
Ravicigh number
Lit
Figure 3.113 Nusselt number as a fanction of Rayleigh number for free canvection heat
transfer between vertical flat plates,
Mis recommended that the 75º correlation of Figure 3.11.1 be used for vertical
surfaces. The correlation given by Equation 3.11.1 does not cover the range from 75
to 90º, but comparisons with other correlations suggest that the 75º represents the
vertical case adeguately. Raithby ex al. (1977) have examined vertical surface
convection data from a wide range of experimental investigations. They propose a
correlation that includes the influence of aspect ratio À, that is, the ratio of plate
height to spacing. Their correlation is plotted in Figure 3.11.3 for aspect ratios of 5,
60, and infinity. For comparison, other correlations that do not show an aspect ratio
effect are also plotted on this figure and correspond approximately to the Raithby et
al. correlation with an aspect ratio of between 10 and 20.
Most of the experiments utilize a guarded hat-plate technique that measures the
heat transfer only at the center of the test region. Consequently the end cffects arc
largely excluded. However, Randall ct al. (1977) uscd un interferometric technique
that allowed determination of local heat transfer coefficients from which averages
were determined; they could not find an aspect ratio effect although a range of aspect
ratios from 9 to 36 was covered. The Raithby et al. (1977) correlation also includes
an angular correction for angles from 70 to 110º which shows a slight increase in
Nusselt number over this range of tilt angles consistent with the trends of Figure
3.11.1 [see Randall et al. 1977].
tis unusual to find a collcctor sloped at angles between 75 and 90º; if they are
to be that stcep, they will probably be vertical. Windows and collector-storage walis
are essentially always vertical. For vertical surfaces the four correlation shown in
Selected Heat Transfer Topics
jure 3,11,3 (with A & 15 for the Raithby et al. result) agree within approximately
with the 75º correlation of Hollands et al. in Figure 3.11.1. Vertical solar
lectors will have an aspect ratio on the order of 60, but at this aspect ratio the
fithby et af. result fails well below other correlations. Conseguently, the 75º
krelation of Figure 3,11.1 will give reasonable or conservative predictions for
rtical surfaces.
12 CONVECTION SUPPRESSION
me of the objectives in designing solar collectors is to reduce the heas loss through
1e covers. This has led to studies of convection suppression by Hollands (1965),
idwards (1969), Buchberg et al. (1976), Amold ct as. (1977, 1978), Meyer ct al.
1978), and others. In these studies the space between two plates, with one plate
ieated, is filed with a transparent or specularty reflecting honeycomb to suppress the
»nset of fluid motion. Without fluid motion the heat transfer between the plates is by
zonduction and radiation. Care must be exexcised since improper desig can lead to
increased rather than decreased convection losses, as was first shown experimentally
by Charters and Peterson (1972) and later verified by others.
For slats, as shown in Figure 3.12.1, the results of Meyer et al. (1978) can be
expressed as the maximum of two numbers as
Nu= max[1.1Ci Co Ra$38, 3] (3.321)
Figure 342.1 Siats for suppression of convection. From Meyer et aí. (1978)
312 Convection Suppression
165
where Cj and C, ure given in Figure 3 2.2 and the subscript £ indicutes that the
a ure given in Pigure 3.12, à cul a
Plaie spacing Z is the characteristic engih. Note that the coefficient € has a
nhas
maximum near 2n aspect ratio of 2.
To assess the ma i
e con ses e magnido SE he convection suppression with stars, tis possible
compare E ton 5 “L vit the correlation of Randall ez a). (1977) obtained
form dita to Same equipment. Although the Randall correfation use;
22-29 on the Raylcigh number, the correlation cam be slightly modified o
have a e
ave an exponent of 0,28. The ratio of the two correiations is then
Sto o mexÊttcyCotapos 1]
Mtmo stais max foi3rap “lcos(g- as)P 1]
oug
* Mean value
T Rongs
as
3a [ 2.3
Aspect rata, /f,
E,
Ga ,
E o 76
TILT ANGLE £, degrees ”
(312.23
Figure 3,122 Coéfficiems €
Bare 2.12.2 Coêfficiens C' and Cy for use in Equation 2.12.1. From Meyer et al, (1978).
Selected Heat Transfer Topics
vo
Their data is well represented by an equation of the form
alRe Pr Dill! (3.14,5)
De = Mim ty (Re Pr Di
iven in Fable 3.19.1.
ale 1958) e copored by Rocnon and Chi (196). present ae
Nussele numbers for the case of constant wall temperature. e a E nu
numbers of 0.7, 5, and infinity arc shown in Figure 314.1, “ s ri nu
also be represented by an equation of the form of Equation 3.14.
dra b,m,n, and Nic, given in Table 3.14.2.
Example 3.14.1
e! h he
What is the heat transfer coefficient inside the tubes of a solar collector in unica e
tubes are 10 mm in diameter and separated by a distance 100 mm. The cm E
15 m wide and 3 m long, and has total flow rate of water of 0,075 kg/s. The wat
isat80C.
ion 3.14.5 for
Table 3.14.1 Constants Tor Equation 3. :
Calcuiation of Local Nu for Circular Tubes with
Constant Hest Rate.
Prandd Number a bom
07 000398 ootId L66 112
10 0.00236 000857 166 113
o 000172 001 166 129
Nu = 44
TI
3 TT Timm
1
Laminar low
Circular tubes
Constant wall temperature
+
A "|
Rm
1000
pi tos Lito ,
Re PrDhIL
Figure 3,14.1 Average Nusselt numbers in short tubes for various Prandil numbers.
314 Heat Transfer Relations for Internal Flow 1m
Table 3.242 Constants for Equation 3.145 for
Culeulation of Average Nu for Circular Tubes with
Constant Wall Temperuture.
Prandil Number a dom A
=— [———
8? 0.079] 00331 Lis 0.82
s 00534 00335 115 082
o 0.0461 ema 115 08
Solution
The coliector has 15 tubes so that the flow rate per tube is 0.005 kgfs. The Reynolds
number is
which indicates laminas Now, The Prandtl number is 2.2 so that
RePrDIL=1800x22x001/3= 13
From Figure 3,14.1 the average Nusselt number is 4.6 s0 the average heat transfer
coefficient is
h=NukiD=46 x 0.600] =3 10 Wing ”
In the study of solar air heaters and colieetor-storage walls it is necessary to
know Lhe forced convection heat transfer coefficient berween two flat plates. For air
the following correlation cam be derived from the datu of Kays and Crawford (1980)
for fully developed turbulent flow syith One side heated and the other side insulated:
Na =0.0158 Re9:8 (.14.6)
where the characteristic Tength is the hydraulic diameter (twice the plate spacing). For
flow situations in which LD is 10, Kays indicates that the average Nusselt number
is approximately 16% higher than that given by Equation 3.14.6. At Z/D, equal to
30, Equation 3, 14.6 still underpredicis by 5%. At 1/D, equal to 100, the effect of
be entrance region has largcly disappeared.
Tan and Charters (1970) have experimentally studied flow of air between
parallel plates with smalt aspect ratios for use in solar air heaters, Their results give
higher heat Iransfer coefficients by about 10% than those given by Kays with an
infinite aspect ratio.
Selected Hest Transfer Fopies
m
Table 4.143 Constants for Equation 3.14.5 for
Calculation of Local Nu for Infinite Flat Plates, One
Side Insulated and Constant Heat Flux on Other Side,
Prande Number a o "
07 000190 0.00563 17 117
1 0.00041 0.00156 212 1.59
o 0.00021 0.00060 224 1.
Nu, 254
3 [TRT Tr[Trem LS Ri
20 Laminar fiow
Parallel fiat plates.
One wall constant temperature
Other adiabatio.
T
Average Nusselt Number
os
TT
8 j pare
As +
a
sli
pts pila +
3 10 109 900
RePrDpfL
Figure 3.142 Average Nusselt numbers in short ducis with onc side insulated and one side at
constant wall temperature for various Prandtt numbers.
The loca! Nusselt number for laminar flow between two flat Plates with one side
insulated and the other subjected to a constant hcai flux has been obtained by e
etal. (1964). The results have becn correlated in the form of Equation 3.14.5 witl
the constants given in Table 3.14,3. .
For the case of parallel plates with constant temperature on one side and
insulated on the other side, Mercer et al. (1967) obtained the average Nus
numbers shown in Figure 3.14.2. They also correlated these data into the form o!
Equation 3,14.7 for (LI < Pr< 10;
49 0.0606(Re Pr DyWL)? (3.147)
Mu=dd+, + 0.0909(Re Pr Dyt 7 Pr?
The results of Sparrow (1955) indicate that for Re Pr Dy/L < 1000, the Pr = 10
Nusseft numbers are essentially the same as for the case when the hydrodynamic
profile is fully developed.
218 Wind Convection Coefficients 13
Example 3.14,2
à Determine the convective heat transfer coefficient for air flow in a channel | m
side by 2 m long. The channel thickness is 15 mm and the air flow rate is 0.03 kg/s.
The average air temperature is 35 C.
b If the channel thickness is halved, what is the heat transfer coefficient?
€ Jfthe Now rate is halveg, what is the heat transfer coefi ficient?
Solution
2 Ata temperature of 35 C: the viscosity is 1.88 x 10-5 m2/s and the thermal
conductivity is 0.0268 WimC. The hydruulio diameter D, is twice the plate spacing
4 and the Reynolds number can be expressed in terms of the flow tate per umit width
mfW. The Reynolds number is then
2x 0,03
—ÍX00S 3200
1x 1.88 x 105
so that the flow is tubulent. From Equation 3.14.6 the Nusselt number is
Nu = 0.0158(3200)08 = 10,1
and the heat transfer coefficient is h = Nu k/D, = 9 Win?C. Since DL is less
than 100, 9 Wim2C is probably a few percent too low.
b If the channel thickness is halved, the Reynolds number remains the same
but the heat transfer coefficient will double 10 18 Win?
€ Ff the flow rate is halved, the Reynolds number will be 1600, indicating
laminar ow. Equation 3.14.7 or Figure 13,14.2 should be used. The value of Re
Pr DyfL is 1600 x 0.7 x 0.03/2 = 16,8 so the Nusselt number is 6.0 and the heat
transfer coefficient is 6.2 Wim2C. “
315 WIND CONVECTION COEFFICIENTS
The heat loss fram flat plates exposed to outside winds is important in the study of
solar collectors. Sparrow et al. (1979) did wind tunnel studies on rectangular plates
“! various orieutations and found the following correlation over the Reynolds number
range of 2x 101109 x 108:
Nu = 0,86 RelV2ppi/3 (3.15,1)
where the characterístic length is four times the Plate area divided by the plate
Perimeter. For laminar flow (ie., Reynolds numbers less than 106, the critical Re
for flow over a flat plate) aver a very wide flat plate at zero angie of attack, the
14 Selected Heat Transfer Topics
analysis of Pohlhausen [see Kays and Crawford (1980)] yields a coefficient for
Equation 3.15.1 of 0,94.º
This agreement at low Reynolds numbers suggests that Equation 3.15. may be
valid at Reynolds numbers up to 106 where direct experimental evidence is lacking.
This extrapolation is necessary since u solar collector array 2 m by 5 m has a
characteristic length of 2.9 m and Reynolds number of 9.4 x 105 in a 5 m/s wind.
From Equation 3.15.1, the heat transfer coefficient under these conditions is
approximately 7 WimiC.
McAdams (1954) reports the data of Jurges for a 0,5 m? plate in which the
convection coefficient is given by the dimensional equation
h=5.7+3.8V (3.15.2)
where V is the wind speed in m/s and À is in W/m2C, It is probable that the effects
of free convection and radiation are included in this equation. For this reason
Watmuff et al. (1977) report that this equation should be
h=28+3.0V (3.15.3)
For a 0.5 mê plate, Equation 3.15.1 yields a heat transfer coefficient of 16 Wim?C at
a 5 m/s wind speed and a temperature of 25 C. Equation 3.15.3 yields a value of 18
Wim2C at these conditions. Thus there is agreement berwcen the two at a
characteristic length of 0.5 m. It is not reasonable to assume that Equation 3.15,3 is
valid at other plate lengths.
The flow over a collector mounted on a house is not always well] represented by
wind tunnel tests of isolated plates. The collectors will sometimes be exposcd
directly to the wind and other times will be in thc wake region. The roof itself will
certainly influence the flow pattems. Also, nearby trees and buildings will greatly
effect local flow conditions. Mitchell (1976) investigated the heat transfer from
various shapes (actually animal shapes) and showed that many shapes were well
represented by a sphere when the equivalent sphere diameter is the cube root of the
volume. The hcat transfer obtained in this manner is an average that includes
stagnation regions and wake regions. A similar situation might be anticipated to
occur in solar systems. Mitchell suggesis that lhe wind tunnel results of thesc animal
tests should be increased by approximately 15% for outdoor conditions. Thus,
assuming a house to be a sphere, the Nusseit number can be expressed as
Nu =0.42 Ret é (3.15.4)
where the characteristic length is the cube root of the house volume.
9 To be consistent with Equation 3.15.1 the characteristic length in the Poblhausen solution must be
changed to twice the plate Jength. “This changes lhe familiar coefficient of 0.564 to 0.94
3.15 Wind Convection CoetTcients
175
When the wind speed is very low, free convecti ai i
h 5 , ection conditions may dominate.
Free convection data for hot inclined flat plates facing upward are not available
However results are available for horizontal and vertical flat plates. For hoi
orizuntal flat plates with aspect ratios uy : 4) gi
p to 7:!, Lloyd and
following eguations: ” Moran (1978) pve the
Nie = 0.76 Ra for Ii Ra IO? (3.15.5)
Nu =0.15Ra!8 for 10] Ra <3x 10º (3.15.6)
where the characteristic Jength às four times the arca divi i
her a ided by the perimeter. (The
original reference used A/P.) For vertical plates McAdams gives Ni
Ni = 0.59 Ra 8 for 103< Ra <t09 (3.157)
Nu =0.13 Ra! for 109 < Re <JQ!2 (.15.8)
where the characteristic length is the plate height.
For large Rayleigh numbers, as would be found in most solar coliector
systems, Equations 3.15.6 and 3.15,8 apply and the characteristic Tength drops out of
the calculation of the heat transfer coefficient. The heat transfer coefficients from
these two cquations are nearly the same since the coefficient on the Rayleigh numbers
difter only slightly. This megas that horizontal and vertical collectors have a
minimum heat transfer coefficient (i.c., under free convection conditions) of about S
Wim2C for a 25 € temperature difference and a valuc of about 4 Wim2C ata
temperature difference of 10C.
From the preceding discussion it is apparent that thc calculation of wind-
induced heat transfer coefficients is not well established. Until additional
experimental evidence becomes available, the following guidelines arc recommended.
When frec and forced convection occur simultancousiy, Me Adams recommends that
bath values be calculated and the larger value used in calculations. Counseguenty, it
appears that a minimum value of approximately 5 Wim? occurs in solar collectors
under still air conditions. For forced conve ici ildii
I ' ection conditions over buildings the 1:
of Mitchel] can be expressed as ve results
hu = 8.600.8
£94 (3.15.9)
The feat transfer coefficient Cir Wim2C) f
*C) for flush-mounted collectors
Pi besta rs can then be
- 600.6
ds max, á6408 (3.15.10)