Comparison

The mode of tracking affects the amount of incident radiation falling on the collector surface in proportion to the cosine of the incidence angle. The amount of energy falling on a surface per unit area for the four modes of tracking for the summer and winter solstices and the equinoxes are shown in Table 2.2. This analysis has been performed with the same radiation model used to plot the solar flux figures in this section. Again, the type of the model used here is not important, because it is used for comparison purposes only. The performance of the various modes of tracking is compared to the full tracking, which collects the maximum amount of solar energy, shown as 100% in Table 2.2. From this table it is obvious that the polar and the N-S horizontal modes are the most suitable for one-axis tracking, since their performance is very close to the full tracking, provided that the low winter performance of the latter is not a problem.

2.2.2 Sun Path Diagrams

For practical purposes, instead of using the preceding equations, it is convenient to have the sun's path plotted on a horizontal plane, called a sun path

Solar Angles 69 Table 2.2 Comparison of Energy Received for Various Modes of Tracking

Tracking mode

Solar energy received (kWh/m2)

Percentage to full tracking

E

SS

WS

E

SS

WS

Full tracking

8.43

1G.6G

5.7G

1GG

1GG

1GG

E-W polar

8.43

9.73

5.23

1GG

91.7

91.7

N-S horizontal

7.51

1G.36

4.47

89.1

97.7

6G.9

E-W horizontal

6.22

7.85

4.91

73.8

74.G

86.2

Notes: E = equinoxes, SS = summer solstice, WS = winter solstice.

e)

90

re

80

gr

e (d

70

le

60

gl

50

n

a

e

40

d

tu

30

alt

20

lar ol

10

S

Declination curves in steps of 5°

Latitude = 35"N

Declination curves in steps of 5°

Noon +23 45

Morning hours 11»-

Afternoon hours

10

7 jj///y/x

-23.45'^^^oiS^y

wvo* 1/

§J/ff//j//y/

\VV\\V<18

1 ■ ■ 1 1 1 ■ ■ f

Solar azimuth angle (degree) figure 2.17 Sun path diagram for 35°N latitude.

Solar azimuth angle (degree) figure 2.17 Sun path diagram for 35°N latitude.

diagram, and to use the diagram to find the position of the sun in the sky at any time of the year. As can be seen from Eqs. (2.12) and (2.13), the solar altitude angle, a, and the solar azimuth angle, z, are functions of latitude, L, hour angle, h, and declination, 6. In a two-dimensional plot, only two independent parameters can be used to correlate the other parameters; therefore, it is usual to plot different sun path diagrams for different latitudes. Such diagrams show the complete variations of hour angle and declination for a full year. Figure 2.17 shows the sun path diagram for 35°N latitude. Lines of constant declination are labeled by the value of the angles. Points of constant hour angles are clearly indicated. This figure is used in combination with Figure 2.7 or Eqs. (2.5)-(2.7); i.e., for a day in a year, Figure 2.7 or the equations can be used to estimate declination, which is then entered together with the time of day and converted to solar time using Eq. (2.3) in Figure 2.17 to estimate solar altitude and azimuth angles. It should be noted that Figure 2.17 applies for the Northern Hemisphere. For the Southern Hemisphere, the sign of the declination should be reversed. Figures A3.2 through A3.4 in Appendix 3 show the sun path diagrams for 30°, 40°, and 50°N latitudes.

2.2.3 Shadow Determination

In the design of many solar energy systems, it is often required to estimate the possibility of the shading of solar collectors or the windows of a building by surrounding structures. To determine the shading, it is necessary to know the shadow cast as a function of time during every day of the year. Although mathematical models can be used for this purpose, a simpler graphical method is presented here, which is suitable for quick, practical applications. This method is usually sufficient, since the objective is usually not to estimate exactly the amount of shading but to determine whether a position suggested for the placement of collectors is suitable or not.

Shadow determination is facilitated by the determination of a surface-oriented solar angle, called the solar profile angle. As shown in Figure 2.18, the solar profile angle, p, is the angle between the normal to a surface and the projection of the sun's rays on a plane normal to the surface. In terms of the solar altitude angle, a, solar azimuth angle, z, and the surface azimuth angle, Zs, the solar profile angle p is given by the equation tan( p) = tan(a) (2.30a)

A simplified equation is obtained when the surface faces due south, i.e., Zs = 0°, given by tan( p) = (2.30b)

The sun path diagram is often very useful in determining the period of the year and hours of day when shading will take place at a particular location. This is illustrated in the following example.

Solar Profile Angle

Example 2.8

A building is located at 35°N latitude and its side of interest is located 15° east of south. We want to investigate the time of the year that point x on the building will be shaded, as shown in Figure 2.19.

Elevation
Solar Profile Angle

Solution

The upper limit of profile angle for shading point x is 35° and 15° west of true south. This is point A drawn on the sun path diagram, as shown in Figure 2.20. In this case, the solar profile angle is the solar altitude angle. Distance x-B is (72 + 122)1/2 = 13.9 m. For the point B, the altitude angle is tan(a) = 8/13.9 ^ a = 29.9°. Similarly, distance x-C is (42 + 122)1/2 = 12.6 m, and for point C, the altitude angle is tan(a) = 8/12.6 ^ a = 32.4°. Both points are as indicated on the sun path diagram in Figure 2.20.

Latitude = 35"N Declination curves in steps of 5°

Latitude = 35"N Declination curves in steps of 5°

Solar Azimuth Angle

Solar azimuth angle (degree) FiGuRE 2.20 Sun path diagram for Example 2.8.

Solar azimuth angle (degree) FiGuRE 2.20 Sun path diagram for Example 2.8.

Therefore, point x on the wall of interest is shaded during the period indicated by the curve BAC in Figure 2.20. It is straightforward to determine the hours that shading occurs, whereas the time of year is determined by the declination.

Solar collectors are usually installed in multi-rows facing the true south. There is, hence, a need to estimate the possibility of shading by the front rows of the second and subsequent rows. The maximum shading, in this case, occurs at local solar noon, and this can easily be estimated by finding the noon altitude, a„, as given by Eq. (2.14) and checking whether the shadow formed shades the second or subsequent collector rows.

Example 2.9

Find the equation to estimate the shading caused by a fin on a window. Solution

The fin and window assembly are shown in Figure 2.21.

Window

Window

South

Perpendicular to window FIGuRE 2.21 Fin and window assembly for Example 2.9.

South

From triangle ABC, the sides AB = D, BC = w, and angle A is z - Zs. Therefore, distance w is estimated by w = D tan(z - Zs).

Shadow calculations for overhangs are examined in more detail in Chapter 6, Section 6.2.5.

2.3 SoLAR RADiATioN 2.3.1 General

All substances, solid bodies as well as liquids and gases above the absolute zero temperature, emit energy in the form of electromagnetic waves.

The radiation that is important to solar energy applications is that emitted by the sun within the ultraviolet, visible, and infrared regions. Therefore, the radiation wavelength that is important to solar energy applications is between 0.15 and 3.0 |m. The wavelengths in the visible region lie between 0.38 and 0.72 |im.

This section initially examines issues related to thermal radiation, which includes basic concepts, radiation from real surfaces, and radiation exchanges between two surfaces. This is followed by the variation of extraterrestrial radiation, atmospheric attenuation, terrestrial irradiation, and total radiation received on sloped surfaces. Finally, it briefly describes radiation measuring equipment.

2.3.2 Thermal Radiation

Thermal radiation is a form of energy emission and transmission that depends entirely on the temperature characteristics of the emissive surface. There is no intervening carrier, as in the other modes of heat transmission, i.e., conduction and convection. Thermal radiation is in fact an electromagnetic wave that travels at the speed of light (C = 300,000km/s in a vacuum). This speed is related to the wavelength (X) and frequency (v) of the radiation as given by the equation:

When a beam of thermal radiation is incident on the surface of a body, part of it is reflected away from the surface, part is absorbed by the body, and part is transmitted through the body. The various properties associated with this phenomenon are the fraction of radiation reflected, called reflectivity (p); the fraction of radiation absorbed, called absorptivity (a); and the fraction of radiation transmitted, called transmissivity (t). The three quantities are related by the following equation:

It should be noted that the radiation properties just defined are not only functions of the surface itself but also of the direction and wavelength of the incident radiation. Therefore, Eq. (2.32) is valid for the average properties over the entire wavelength spectrum. The following equation is used to express the dependence of these properties on the wavelength:

where pX = spectral reflectivity. aX = spectral absorptivity. tx = spectral transmissivity.

The angular variation of absorptance for black paint is illustrated in Table 2.3 for incidence angles of 0-90°. The absorptance for diffuse radiation is approximately 0.90 (Lof and Tybout, 1972).

Most solid bodies are opaque, so that t = 0 and p + a = 1. If a body absorbs all the impinging thermal radiation such that t = 0, p = 0, and a = 1,

Table 2.3 Angular Variation of Absorptance for Black Paint (Reprinted from Löf and Tybout (1972) with Permission from ASME).

Angle of incidence (°)

Absorptance

0-30

0.96

30-40

0.95

40-50

0.93

50-60

0.91

60-70

0.88

70-80

0.81

80-90

0.66

regardless of the spectral character or directional preference of the incident radiation, it is called a blackbody. This is a hypothetical idealization that does not exist in reality.

A blackbody is not only a perfect absorber, it is also characterized by an upper limit to the emission of thermal radiation. The energy emitted by a black-body is a function of its temperature and is not evenly distributed over all wavelengths. The rate of energy emission per unit area at a particular wavelength is termed the monochromatic emissive power. Max Planck was the first to derive a functional relation for the monochromatic emissive power of a blackbody in terms of temperature and wavelength. This was done by using the quantum theory, and the resulting equation, called Planck's equation for blackbody radiation, is given by

where

EbX = monochromatic emissive power of a blackbody (W/m2-pm). T = temperature of the body (K). X = wavelength (pm). C = constant = 3.74 X 108 W-pm4/m2. C2 = constant = 1.44 X 104 pm-K.

By differentiating Eq. (2.34) and equating to 0, the wavelength corresponding to the maximum of the distribution can be obtained and is equal to XmaxT = 2897.8 pm-K. This is known as Wien's displacement law. Figure 2.22 shows the spectral radiation distribution for blackbody radiation at three temperature sources. The curves have been obtained by using the Planck's equation.

Comparison Wien Planck Radiation

Wavelength, |m

FÍGuRE 2.22 Spectral distribution of blackbody radiation.

Wavelength, |m

FÍGuRE 2.22 Spectral distribution of blackbody radiation.

The total emissive power, Eb, and the monochromatic emissive power, EbX, of a blackbody are related by

Substituting Eq. (2.34) into Eq. (2.35) and performing the integration results in the Stefan-Boltzmann law:

where a = the Stefan-Boltzmann constant = 5.6697 X 10~8 W/m2-K4.

In many cases, it is necessary to know the amount of radiation emitted by a blackbody in a specific wavelength band X1 ^ X2. This is done by modifying Eq. (2.35) as x

Since the value of EbX depends on both X and T, it is better to use both variables as

AT e

Therefore, for the wavelength band of Xi — X2, we get

which results in -b(0 —> XjT) - —b (0 —X2T). Table 2.4 presents a tabulation of —b(0 — XT) as a fraction of the total emissive power, Eb = ctT4, for various values of XT. The values are not rounded, because the original table, suggested by Dunkle (1954), recorded XT in micrometer-degrees Rankine (|m-°R), which were converted to micrometer-Kelvins (|m-K) in Table 2.4.

A blackbody is also a perfect diffuse emitter, so its intensity of radiation, Ib, is a constant in all directions, given by

Of course, real surfaces emit less energy than corresponding blackbodies. The ratio of the total emissive power, E, of a real surface to the total emissive power, Eb, of a blackbody, both at the same temperature, is called the emissivity (e) of a real surface; that is,

The emissivity of a surface is not only a function of surface temperature but depends also on wavelength and direction. In fact, the emissivity given by Eq. (2.38) is the average value over the entire wavelength range in all directions, and it is often referred as the total or hemispherical emissivity. Similar to Eq. (2.38), to express the dependence on wavelength, the monochromatic or spectral emissivity, eX, is defined as the ratio of the monochromatic emissive power, EX, of a real surface to the monochromatic emissive power, EbX, of a blackbody, both at the same wavelength and temperature:

Kirchoff's law of radiation states that, for any surface in thermal equilibrium, monochromatic emissivity is equal to monochromatic absorptivity:

The temperature (T) is used in Eq. (2.40) to emphasize that this equation applies only when the temperatures of the source of the incident radiation and the body itself are the same. It should be noted, therefore, that the emissivity of a body on earth (at normal temperature) cannot be equal to solar radiation (emitted from the sun at T = 5760 K). Equation (2.40) can be generalized as e(T) = a(T) (2.41)

Equation (2.41) relates the total emissivity and absorptivity over the entire wavelength. This generalization, however, is strictly valid only if the incident and emitted radiation have, in addition to the temperature equilibrium at the surfaces, the same spectral distribution. Such conditions are rarely met in real life; to simplify the analysis of radiation problems, however, the assumption that monochromatic properties are constant over all wavelengths is often made. Such a body with these characteristics is called a graybody.

table 2.4 Fraction of Blackbody Radiation as a Function of XT

XT (p,m-K)

Eb(G ^ XT)/aT4

XT (p,m-K)

Eb(G ^ XT)/aT4

XT (p,m-K)

Eb(G ^ XT)/aT4

555.6

1.7GE-GS

4GGG.G

G.4SGS5

7444.4

G.S3166

666.7

7.56E-G7

4111.1

G.5GG66

7555.6

G.S369S

777.S

1.G6E-G5

4222.2

G.51974

7666.7

G.S42G9

SSS.9

7.3SE-G5

4333.3

G.53SG9

7777.S

G.S4699

1GGG.G

3.21E-G4

4444.4

G.55573

7SSS.9

G.S5171

1111.1

G.GG1G1

4555.6

G.57267

SGGG.G

G.S5624

1222.2

G.GG252

4666.7

G.5SS91

Slll.l

G.S6G59

1333.3

G.GG531

4777.S

G.6G449

S222.2

G.S6477

1444.4

G.GG9S3

4SSS.9

G.61941

S333.3

G.S6SSG

1555.6

G.G1643

5GGG.G

G.63371

SSSS.9

G.SS677

1666.7

G.G2537

5111.1

G.6474G

9444.4

G.9G16S

1777.S

G.G3677

5222.2

G.66G51

1GGGG.G

G.91414

1SSS.9

G.G5G59

5333.3

G.673G5

1G555.6

G.92462

2GGG.G

G.G6672

5444.4

G.6S5G6

11111.1

G.93349

2111.1

G.GS496

5555.6

G.69655

11666.7

G.941G4

2222.2

G.1G5G3

5666.7

G.7G754

12222.2

G.94751

2333.3

G.12665

5777.S

G.71SG6

12777.S

G.953G7

2444.4

G.14953

5SSS.9

G.72S13

13333.3

G.957SS

2555.5

G.17337

6GGG.G

G.73777

13SSS.9

G.962G7

2666.7

G.197S9

6111.1

G.747GG

14444.4

G.96572

2777.S

G.222S5

6222.1

G.755S3

15GGG.G

G.96S92

2SSS.9

G.24SG3

6333.3

G.76429

15555.6

G.97174

3GGG.G

G.27322

6444.4

G.7723S

16111.1

G.97423

3111.1

G.29S25

6555.6

G.7SG14

16666.7

G.97644

3222.2

G.323GG

6666.7

G.7S757

22222.2

G.9S915

3333.3

G.34734

6777.S

G.79469

22777.S

G.99414

3444.4

G.3711S

6SSS.9

G.SG152

33333.3

G.99649

3555.6

G.39445

7GGG.G

G.SGSG6

33SSS.9

G.99773

3666.7

G.417GS

7111.1

G.S1433

44444.4

G.99S45

3777.S

G.439G5

7222.2

G.S2G35

5GGGG.G

G.99SS9

3SSS.9

G.46G31

7333.3

G.S2612

55555.6

G.9991S

Similar to Eq. (2.37) for a real surface, the radiant energy leaving the surface includes its original emission and any reflected rays. The rate of total radiant energy leaving a surface per unit surface area is called the radiosity (J), given by

where

Eb = blackbody emissive power per unit surface area (W/m2). H = irradiation incident on the surface per unit surface area (W/m2). e = emissivity of the surface. p = reflectivity of the surface.

There are two idealized limiting cases of radiation reflection: The reflection is called specular if the reflected ray leaves at an angle with the normal to the surface equal to the angle made by the incident ray, and it is called diffuse if the incident ray is reflected uniformly in all directions. Real surfaces are neither perfectly specular nor perfectly diffuse. Rough industrial surfaces, however, are often considered as diffuse reflectors in engineering calculations.

A real surface is both a diffuse emitter and a diffuse reflector and hence, it has diffuse radiosity; i.e., the intensity of radiation from this surface (I) is constant in all directions. Therefore, the following equation is used for a real surface:

Example 2.10

A glass with transmissivity of 0.92 is used in a certain application for wavelengths 0.3 and 3.0 pm. The glass is opaque to all other wavelengths. Assuming that the sun is a blackbody at 5760K and neglecting atmospheric attenuation, determine the percent of incident solar energy transmitted through the glass. If the interior of the application is assumed to be a blackbody at 373 K, determine the percent of radiation emitted from the interior and transmitted out through the glass.

Solution

For the incoming solar radiation at 5760 K, we have

A1T = 0.3 X 5760 = 1728|m-K A2T = 3 X 5760 = 17280 pm-K

From Table 2.4 by interpolation, we get

oT 4

oT 4

Therefore, the percent of solar radiation incident on the glass in the wavelength range 0.3-3 |m is

aT 4

In addition, the percentage of radiation transmitted through the glass is 0.92 X 94.61 =87.04%.

For the outgoing infrared radiation at 373 K, we have

XjT = 0.3 X 373 = 111.9 |m-K X2T = 3 X 373 = 1119.0^m-K

From Table 2.4, we get

aT 4

oT 4

The percent of outgoing infrared radiation incident on the glass in the wavelength 0.3-3 |im is 0.1%, and the percent of this radiation transmitted out through the glass is only 0.92 X 0.1 = 0.092%. This example, in fact, demonstrates the principle of the greenhouse effect; i.e., once the solar energy is absorbed by the interior objects, it is effectively trapped.

Example 2.11

A surface has a spectral emissivity of 0.87 at wavelengths less than 1.5 |m, 0.65 at wavelengths between 1.5 and 2.5 |im, and 0.4 at wavelengths longer than 2.5 |im. If the surface is at 1000 K, determine the average emissivity over the entire wavelength and the total emissive power of the surface.

Solution

From the data given, we have

From Table 2.4 by interpolation, we get

and aT 4

oT 4

0.01313

0.16144

Therefore, and

oT 4

The average emissive power over the entire wavelength is given by e = 0.87 X 0.01313 + 0.65 X 0.14831 + 0.4 X 0.83856 = 0.4432 and the total emissive power of the surface is

E = eaT4 = 0.4432 X 5.67 X 10~8 X 10004 = 25129.4W/m2

2.3.3 Transparent Plates

When a beam of radiation strikes the surface of a transparent plate at angle 0!, called the incidence angle, as shown in Figure 2.23, part of the incident radiation is reflected and the remainder is refracted, or bent, to angle 02, called the refraction angle, as it passes through the interface. Angle 0! is also equal to the angle at which the beam is specularly reflected from the surface. Angles 0! and 02 are not equal when the density of the plane is different from that of the medium through which the radiation travels. Additionally, refraction causes the transmitted beam to be bent toward the perpendicular to the surface of higher density. The two angles are related by the Snell's law:

sin 01 sin 02

where n! and n2 are the refraction indices and n is the ratio of refraction index for the two media forming the interface. The refraction index is the determining factor for the reflection losses at the interface. A typical value of the refraction index is 1.000 for air, 1.526 for glass, and 1.33 for water.

Incident beam

Reflected beam

\01

Medium 1 \

/ ni

1 Medium 2

\ "2 l*

?

\ ^^^ Refracted beam

Transmitted beam

FiGuRE 2.23 Incident and refraction angles for a beam passing from a medium with refraction index n1 to a medium with refraction index n2.

FiGuRE 2.23 Incident and refraction angles for a beam passing from a medium with refraction index n1 to a medium with refraction index n2.

Expressions for perpendicular and parallel components of radiation for smooth surfaces were derived by Fresnel as

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