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CHAPTER 49
SOLAR ENERGY APPLICATIONS
Jan E Kreider
Jan F. Kreider and Associates, Inc.
and Joint Center for Energy Management
University of Colorado
Boulder, Colorado
49.1 SOLAR ENERGY
AVAILABILITY 15 49
49.1.1 Solar Geometry 15 49
49.1.2 Sunrise and Sunset 15 52
49. 1 .3 Quantitative Solar Flux
Availability
49.3.2 Mechanical Solar Space
Heating Systems 15 69
49.3.3 Passive Solar Space
Heating Systems 15 71
49.3.4 Solar Ponds
15 71
15 54
49.3.5 Industrial Process
Applications
15 75
49.2 SOLAR THERMAL
COLLECTORS 15 60
49.2. 1 Flat-Plate Collectors 1 560
49.2.2 Concentrating Collectors 1564
49.2.3 Collector Testing 15 68
49.3.6 Solar Thermal Power
Production
15 75
49.3.7 Other Thermal
Applications 15 76
49.3.8 Performance Prediction for
Solar Thermal Processes 1576
49.3 SOLAR THERMAL
APPLICATIONS 15 69
49.3.1 Solar Water Heating 15 69
49.4 NONTHERMAL SOLAR
ENERGY APPLICATIONS 15 77
49.1 SOLAR ENERGY AVAILABILITY
Solar energy is defined as that radiant energy transmitted by the sun and intercepted by earth. It is
transmitted through space to earth by electromagnetic radiation with wavelengths ranging between
0.20 and 15 microns. The availability of solar flux for terrestrial applications varies with season, time
of day, location, and collecting surface orientation. In this chapter we shall treat these matters
analytically.
49.1.1 Solar Geometry
Two motions of the earth relative to the sun are important in determining the intensity of solar flux
at any time—the earth's rotation about its axis and the annual motion of the earth and its axis about
the sun. The earth rotates about its axis once each day. A solar day is defined as the time that elapses
between two successive crossings of the local meridian by the sun. The local meridian at any point
is the plane formed by projecting a north-south longitude line through the point out into space from
the center of the earth. The length of a solar day on the average is slightly less than 24 hr, owing to
the forward motion of the earth in its solar orbit. Any given day will also differ from the average
day owing to orbital eccentricity, axis precession, and other secondary effects embodied in the equa-
tion of time described below.
Declination and Hour Angle
The earth's orbit about the sun is elliptical with eccentricity of 0.0167. This results in variation of
solar flux on the outer atmosphere of about 7% over the course of a year. Of more importance is the
variation of solar intensity caused by the inclination of the earth's axis relative to the ecliptic plane
of the earth's orbit. The angle between the ecliptic plane and the earth's equatorial plane is 23.45°.
Figure 49.1 shows this inclination schematically.
Mechanical Engineers' Handbook, 2nd ed., Edited by Myer Kutz.
ISBN 0-471-13007-9 © 1998 John Wiley & Sons, Inc.
815043747.003.png 815043747.004.png
Fig. 49.1 (a) Motion of the earth about the sun. (b) Location of tropics. Note that the sun is so
far from the earth that all the rays of the sun may be considered as parallel to one another
when they reach the earth.
The earth's motion is quantified by two angles varying with season and time of day. The angle
varying on a seasonal basis that is used to characterize the earth's location in its orbit is called the
solar "declination." It is the angle between the earth-sun line and the equatorial plane as shown in
Fig. 49.2. The declination 8S is taken to be positive when the earth-sun line is north of the equator
and negative otherwise. The declination varies between +23.45° on the summer solstice (June 21 or
22) and -23.45° on the winter solstice (December 21 or 22). The declination is given by
sin 8S = 0.398 cos [0.986(7V - 173)]
(49.1)
in which N is the day number.
The second angle used to locate the sun is the solar-hour angle. Its value is based on the nominal
360° rotation of the earth occurring in 24 hr. Therefore, 1 hr is equivalent to an angle of 15°. The
hour angle is measured from zero at solar noon. It is denoted by hs and is positive before solar noon
and negative after noon in accordance with the right-hand rule. For example 2:00 PM corresponds to
hs = -30° and 7:00 AM corresponds to hs = +75°.
Solar time, as determined by the position of the sun, and clock time differ for two reasons. First,
the length of a day varies because of the ellipticity of the earth's orbit; and second, standard time is
determined by the standard meridian passing through the approximate center of each time zone. Any
position away from the standard meridian has a difference between solar and clock time given by
[(local longitude - standard meridian longitude)/15) in units of hours. Therefore, solar time and
local standard time (LST) are related by
solar time = LST - EoT - (local longitude - standard meridian longitude)/15 (49.2)
815043747.005.png
Fig. 49.2 Definition of solar-hour angle hs (CND), solar declination ds (VOD), and latitude L
(POC): P, site of interest. (Modified from J. F. Kreider and F. Kreith, Solar Heating and Cooling,
revised 1st ed., Hemisphere, Washington, DC, 1977.)
in units of hours. EoT is the equation of time which accounts for difference in day length through a
year and is given by
EoT =12 + 0.1236 sin x - 0.0043 cos x + 0.1538 sin 2x + 0.0608 cos 2x (49.3)
in units of hours. The parameter x is
360(JV - 1)
X = -*MT
(49'4)
where N is the day number counted from January 1 as N = 1.
Solar Position
The sun is imagined to move on the celestial sphere, an imaginary surface centered at the earth's
center and having a large but unspecified radius. Of course, it is the earth that moves, not the sun,
but the analysis is simplified if one uses this Ptolemaic approach. No error is introduced by the
moving sun assumption, since the relative motion is the only motion of interest. Since the sun moves
on a spherical surface, two angles are sufficient to locate the sun at any instant. The two most
commonly used angles are the solar-altitude and azimuth angles (see Fig. 49.3) denoted by a and as,
respectively. Occasionally, the solar-zenith angle, defined as the complement of the altitude angle, is
used instead of the altitude angle.
The solar-altitude angle is related to the previously defined declination and hour angles by
sin a. = cos L cos 8S cos hs + sin L sin 8S
(49.5)
in which L is the latitude, taken positive for sites north of the equator and negative for sites south
of the equator. The altitude angle is found by taking the inverse sine function of Eq. (49.5).
The solar-azimuth angle is given by1
cos 8S sin hs
^
sin a, =
(49.6)
cos a
815043747.006.png
Fig. 49.3 Diagram showing solar-altitude angle a. and solar-azimuth angle as.
To find the value of as, the location of the sun relative to the east-west line through the site must be
known. This is accounted for by the following two expressions for the azimuth angle:
. , /cos £„ sin h\ tan 6_
a- = sin I-^T"} ™h*>^i
(49J)
aj=180°-sin-(C°Sg-SinM, cos*,<^
(49.8)
\ cos a /
tan L
Table 49.1 lists typical values of altitude and azimuth angles for latitude L = 40°. Complete tables
are contained in Refs. 1 and 2.
49.1.2 Sunrise and Sunset
Sunrise and sunset occur when the altitude angle a = 0. As indicated in Fig. 49.4, this occurs when
the center of the sun intersects the horizon plane. The hour angle for sunrise and sunset can be found
from Eq. (49.5) by equating a to zero. If this is done, the hour angles for sunrise and sunset are
found to be
hsr = cos^C-tan L tan ds) = -hss
(49.9)
in which hsr is the sunrise hour angle and hss is the sunset hour angle.
Figure 49.4 shows the path of the sun for the solstices and the equinoxes (length of day and night
are both 12 hr on the equinoxes). This drawing indicates the very different azimuth and altitude
angles that occur at different times of year at identical clock times. The sunrise and sunset hour
angles can be read from the figures where the sun paths intersect the horizon plane.
Solar Incidence Angle
For a number of reasons, many solar collection surfaces do not directly face the sun continuously.
The angle between the sun-earth line and the normal to any surface is called the incidence angle.
The intensity of off-normal solar radiation is proportional to the cosine of the incidence angle. For
example, Fig. 49.5 shows a fixed planar surface with solar radiation intersecting the plane at the
incidence angle i measured relative to the surface normal. The intensity of flux at the surface is lb X
cos i, where Ib is the beam radiation along the sun-earth line; Ib is called the direct, normal radiation.
For a fixed surface such as that in Fig. 49.5 facing the equator, the incidence angle is given by
cos i = sin ^(sin L cos j3 - cos L sin (3 cos aw)
+ cos 8S cos hs(cos L cos ft + sin L sin (3 cos aw)
(49.10)
+ cos ds sin j3 sin aw sin hs
815043747.001.png
Table 49.1 Solar Position for 40°N Latitude
Solar
Position
Alti- Azi-
tude muth
8.1 55.3
16.8 44.0
23.8 30.9
28.4 16.0
30.0 0.0
4.8 72.7
15.4 62.2
25.0 50.2
32.8 35.9
38.1 18.9
40.0 0.0
11.4 80.2
22.5 69.6
32.8 57.3
41.6 41.9
47.7 22.6
50.0 0.0
7.4 98.9
18.9 89.5
30.3 79.3
41.3 67.2
51.2 51.4
58.7 29.2
61.6 0.0
1.9 114.7
12.7 105.6
24.0 96.6
35.4 87.2
46.8 76.0
57.5 60.9
66.2 37.1
70.0 0.0
4.2 117.3
14.8 108.4
26.0 99.7
37.4 90.7
48.8 80.2
59.8 65.8
69.2 41.9
73.5 0.0
Solar
Position
Alti- Azi-
tude muth
2.3 115.2
13.1 106.1
24.3 97.2
35.8 87.8
47.2 76.7
57.9 61.7
66.7 37.9
70.6 0.0
7.9 99.5
19.3 90.9
30.7 79.9
41.8 67.9
51.7 52.1
59.3 29.7
62.3 0.0
11.4 80.2
22.5 69.6
32.8 57.3
41.6 41.9
47.7 22.6
50.0 0.0
4.5 72.3
15.0 61.9
24.5 49.8
32.4 35.6
37.6 18.7
39.5 0.0
8.2 55.4
17.0 44.1
24.0 31.0
28.6 16.1
30.2 0.0
5.5 53.0
14.0 41.9
20.0 29.4
25.0 15.2
26.6 0.0
Solar
Time
AM PM
8 4
9 3
10 2
11 1
12
7 5
8 4
9 3
10 2
11 1
12
7 5
8 4
9 3
10 2
11 1
12
6 6
7 5
8 4
9 3
10 2
11 1
12
5 7
6 6
7 5
8 4
9 3
10 2
11 1
12
5 7
6 6
7 5
8 4
9 3
10 2
11 1
12
Solar
Time
AM PM
5 7
6 6
7 5
8 4
9 3
10 2
11 1
12
6 6
7 5
8 4
9 3
10 2
11 1
12
7 5
8 4
9 3
10 2
11 1
12
7 5
8 4
9 3
10 2
11 1
12
8 4
9 3
10 2
11 1
12
8 4
9 3
10 2
11 1
12
Date
January 21
Date
July 21
February 21
August 21
March 21
September 21
April 21
October 21
May 21
November 21
June 21
December 21
in which aw is the "wall" azimuth angle and ft is the surface tilt angle relative to the horizontal
plane, both as shown in Fig. 49.5.
For fixed surfaces that face due south, the incidence angle expression simplifies to
cos i = sin(L - /3)sin 8S + cos(L - /3)cos 8S cos hs
(49.11)
A large class of solar collectors move in some fashion to track the sun's diurnal motion, thereby
improving the capture of solar energy. This is accomplished by reduced incidence angles for properly
tracking surfaces vis-a-vis a fixed surface for which large incidence angles occur in the early morning
and late afternoon (for generally equator-facing surfaces). Table 49.2 lists incidence angle expressions
for nine different types of tracking surfaces. The term "polar axis" in this table refers to an axis of
815043747.002.png

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