Chen C.J. Physics of Solar Energy

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102  Tracking Sunlight

Equation 4.75 can be used to convert standard time to solar time. Standard time

is deﬁned by an oﬀset Δ from UT, almost always an integer number of hours. For

example, Eastern Standard Time (EST) is deﬁned by UT–5h, or Δ = −5; Eastern

Daylight Saving Time (EDT) is deﬁned by UT–4h, or Δ = −4. The solar time t



=UT+

λ − Δ+ET, (4.76)

where λ is the longitude of the observer, each hour corresponds to 15

◦

degree; and ET

is the equation of time given by Eq. 4.73.

By setting t



= 0, the standard time for solar noon, T

, can be determined from

Eq. 4.76,

=Δ−

λ − ET. (4.77)

As an example, New York City, where λ = −73

◦



, corresponds to minus 4 h

and 56 min. For EST, Δ = −5. At EST noon, the mean solar time is 12:04 in the

afternoon. In mid-November, Eq. 4.75 gives ET = 16 min, which means the real Sun

is 16 min faster than the mean Sun. Therefore, at 12:00 EST, solar time is 12:20 in the

afternoon. The Sun is 5

◦

west of the meridian.

4.4.7 Declination of the Sun

In Section 4.3.1, we presented an approximate equation for the declination of the Sun

(Eq. 4.27). Here, we derive a more accurate equation for the declination δ of the Sun.

The reference point of the Sun’s declination is the March equinox, where by deﬁnition

the declination is zero and the ecliptic longitude is also zero. If the Sun’s longitude

runs linear over time, the sine formula gives

sin δ =sinε sin l. (4.78)

However, the ecliptic motion of Earth gives rise to an additional term,

sin δ =sinε sin {l +2e [sin(l − l

) − sin l

]}. (4.79)

Consequently,

δ = arcsin (sin ε sin {l +2e [sin(l − l

) − sin l

]}) . (4.80)

4.4.8 Analemma

As a combined result of Equation of Time and the variation of declination, the apparent

position of the Sun at a given time of the day varies over the date in a year, which forms

a well-deﬁned trajectory on the sky, the analemma. The trajectory can be recorded

by ﬁxing a camera towards the Southern sky (in the Northern hemisphere), taking one

picture at the same time of the day everyday, then superimpose the pictures of the

sunny days. A “8”-like pattern is revealed. Plate 2 is compiled by Greek astronomer

Anthony Ayiomamitis from 47 clear-day photos taken in 2003 near the Temple of

Apollo, Corinth, Greece.

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Problems

4.1. For a place in the frigid zone with latitude φ that satisﬁes φ>π/2 −ε, determine

the starting day and ending day in a year that the Sun never sets.

4.2. For a place in the frigid zone with latitude φ that satisﬁes φ>π/2 −ε, determine

the starting day in a year and ending day in the next year that the Sun never rises.

4.3. For a south-facing window of area A, calculate the total solar radiation for any

day in the year.

Hint: For days after the vernal equinox and before the autumnal equinox, the time

of radiation is between the two points where the Sun crosses the east–west great circle.

Otherwise the Sun is at the west side of the building.

4.4. On gage C3 of the 2009 Astronomical Almanac, the leading terms of the equation

of time in units of seconds are (using our notations)

ET = −108.5sinl + 596.0sin2l − 428.2cosl, (4.81)

where

l = 279

◦

.791 + 0.985647N, (4.82)

where N is the number of days counted from January 1, and l is the mean longitude

of the Sun.

Questions:

1. What is the meaning of the number 0.985647?

2. What is the meaning of the phase angle 279

◦

.791?

4.5. Write the sine and cosine terms in Eq. 4.81 in the form of sin (l −l

), where l

a constant phase angle. Explain the meaning of the constant l

4.6. Determine the sunset time (in solar time) and the length of daytime of New York

City on New Years Day, Memorial Day, Labor Day, and Thanksgiving Day.

4.7. What is the ratio of solar power on a vertical surface facing south, a horizontal

surface, and a latitude-tilt surface, at solar noon of an equinox (vernal equinox or

autumnal equinox) for New York City?

4.8. In New York City, where the latitude is 40

◦

47’, on Memorial Day (May 25) and

Thanksgiving Day (Nov 27) of 2009, at solar noon, determine the declination of the

Sun, the height of the Sun, and the power density of direct sunlight in watts per square

meter on a horizontal surface at that time.

4.9. In New York City, where the latitude is 40

◦

47’ and longitude is -73

◦

58’, on Memo-

rial Day (May 25) and Thanksgiving Day (Nov 27) of 2009, determine the civil time

(EST, or EDT if necessary) of solar noon (the time the Sun passes the local meridian).

Problems  103

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104  Tracking Sunlight

4.10. For a surface of arbitrary orientation β and γ, ﬁnd out the starting time and

ending time of solar radiation on the surface.

4.11. For a surface facing south, γ = 0 but β = 0, ﬁnd out the starting time and

ending time of solar radiation on the surface.

4.12. For a surface facing south, γ = 0 but β = 0, ﬁnd out the direct daily radiation.

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Chapter 5

Interaction of Sunlight with Earth

In this chapter, we study the interaction of sunlight with Earth. Due to the eﬀect of

the atmosphere, about one-half of sunlight is reﬂected, scattered, or absorbed before

reaching the ground. The available sunlight varies with location and time. Furthermore,

the sunlight absorbed by the ground penetrates into Earth, is stored as heat, and

becomes shallow geothermal energy, a signiﬁcant component of renewable energy.

5.1 Interaction of Radiation with Matter

In this section, the general physical phenomena of the interaction of radiation with

matter are described.

5.1.1 Absorptivity, Reﬂectivity, and Transmittivity

When a ray of radiation falls on a piece of matter, in general, part of the radiation is

reﬂected, another part is absorbed, and yet another part is transmitted. To describe and

characterize the interaction of radiation with matter, the following three dimensionless

coeﬃcients are introduced:

A(λ), absorptivity or absorptance: fraction of incident radiation of wavelength λ

absorbed

R(λ), reﬂectivity or reﬂectance : fraction of incident radiation of wavelength λ re-

ﬂected

Figure 5.1 Absorptivity, reﬂectivity,

and transmittivity. Part of the radi-

ation falling on a piece of matter is ab-

sorbed, another part is reﬂected, and the

rest is transmitted. Three dimensionless

coeﬃcients are introduced: absorptivity

A(λ), reﬂectivity R(λ), and transmittivity

T (λ). Conservation of energy requires that

A(λ)+R(λ)+T (λ)=1.

105Physics of Solar Energy C. Julian Chen

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106  Interaction of Sunlight with Earth

T (λ), transmittivity or transmittance: fraction of incident radiation of wavelength

λ transmitted

If the nature of the radiation does not change or its wavelength stays unchanged, conser-

vation of energy requires that incident radiation be absorbed, reﬂected, or transmitted;

see Fig. 5.1. Therefore

A(λ)+R(λ)+T (λ)=1. (5.1)

For opaque surfaces, the transmittivity is zero. Conservation of energy requires that

incident radiation be either absorbed or reﬂected,

A(λ)+R(λ)=1. (5.2)

5.1.2 Emissivity and Kirchhoﬀ’s Law

When matter is heated, it will radiate according to Planck’s law (Fig. 5.2). The ac-

tual power density of radiation also depends on the nature of the surface. However, it

never exceeds that of a blackbody. The actual radiation from a surface as a fraction

of blackbody radiation at a given wavelength is called its emissivity E(λ). It is less

than 1, except it equals exactly 1 for a blackbody. Based on classical thermodynam-

ics, Kirchhoﬀ showed that the emissivity of a surface at a given wavelength must equal

its absorptivity. At thermal equilibrium, the radiation energy emitted must equal the

radiation energy absorbed. Otherwise heat can transfer from a cold reservoir to a hot

reservoir, which violates the second law of thermodynamics (see chapter 6). There-

fore, the absorptivity at a given wavelength must equal the emissivity at the same

wavelength,

E(λ)=A(λ). (5.3)

Figure 5.2 Emissivity and absorptivity.

When heated, matter radiates electromagnetic

waves. The maximum radiation power spec-

trum follows Planck’s law. For the general

case, the radiation from a surface as a fraction

of blackbody radiation at a given wavelength is

called its emissivity, E(λ). At thermal equilib-

rium, the radiation energy emitted must equal

the radiation energy absorbed. Consequently,

E(λ)=A(λ).

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5.1 Interaction of Radiation with Matter  107

5.1.3 Bouguer–Lambert–Beer’s Law

An empirical relationship between the absorption of light and the property of the

absorbing medium was discovered by Pierre Bouguer before 1729, then was formulated

by Johann Heinrich Lambert in 1760 in his monograph “Photometria.” It states that

the light intensity depends exponentially on the thickness of the optical path z,

(z)=I

(0) e

−A(λ) z

, (5.4)

where A(λ) is the absorption coeﬃcient of the medium at wavelength λ, which has a

dimension of inverse length and z is the optical path.

The empirical relation was further developed by August Beer in 1852 to correlate

the absorption coeﬃcient with the concentration of absorbing particles, known as the

Beer’s law or Bouguer–Lambert–Beer’s law,

(z)=I

(0) e

−Nσ(λ) z

, (5.5)

where N is the number of absorbing particles per unit volume and σ(λ)istheabsorption

cross section of the absorbent at wavelength λ. An intuitive proof is shown in Fig. 5.3.

For a thin slice in the absorption path of thickness dz and cross sectional area S,the

fractional area dS occupied by the absorbing particles is

dS = Nσ(λ) Sdz. (5.6)

Obviously, a proportion dS/S of radiation energy is blocked by the absorbing particles,

(z)

= −

= −Nσ(λ) dz. (5.7)

Integrating over z and using the initial condition at z = 0, one obtains

(z)=I

(0) e

−Nσ(λ) z

. (5.8)

Figure 5.3 Bouguer–

Lambert–Beer’s law. Variation

of light intensity with the concen-

tration of the absorbing particles

and the length of the optical

path plays an important role in

the study of the eﬀect of the

atmosphere on sunlight.

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108  Interaction of Sunlight with Earth

Figure 5.4 Attenuation of sun-

light at azimuth θ. Variation of

light intensity with the concentration

of the absorbing particles and the

length of the optical path plays an

important role in the study of the ef-

fect of the atmosphere on sunlight.

Comparing with Eq. 5.4, one ﬁnds A(λ)=Nσ(λ). If the concentration of particles is

not uniform over the optical path, which can be described by a concentration distribu-

tion N(z), the above relation can be extended to

(z)=I

(0) exp



−



σ(λ) N (z) dz



. (5.9)

Now we look into a situation relevant to the attenuation of sunlight by the atmo-

sphere; see Fig. 5.4. In the atmosphere, to a good approximation, the distribution of

molecules and particles is a function of height z. If the Sun is at the zenith, according

to Eq. 5.9, the absorbance over the entire atmosphere is

α(λ, 0) =

(0)

(∞)

=exp



−



∞

σ(λ) N (z) dz.



. (5.10)

If the azimuth of the Sun is θ — notice that dz = ds cos θ (see Fig. 5.4) — the absorbance

becomes

α(λ, θ)=exp



−



∞

σ(λ) N (z) ds



cos θ

exp



−



∞

σ(λ) N (z) dz



α(λ, 0)

cos θ

(5.11)

5.2 Interaction of Sunlight with Atmosphere

The interaction of sunlight with the atmosphere has been studied extensively by climate

scientists. Here is a summary. Approximately, 30% of solar radiation is reﬂected or

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5.2 Interaction of Sunlight with Atmosphere  109

Figure 5.5 Interaction of sunlight with atmosphere. Approximately 30% of solar radiation is

reﬂected or scattered back to space immediately, 20% is absorbed by the atmosphere and cloud, and

50% is absorbed by Earth. (See Ref. [66], p. 94).

scattered back to space, see Fig. 5.5. Six percent is scattered by air; 20% is reﬂected by

clouds; 4% is reﬂected by the surface of Earth; and 20% is absorbed by the atmosphere:

16% is absorbed by water vapor, dust, and O

. Another 4% is absorbed by the cloud.

The solar radiation thus absorbed heats up the atmosphere. Fifty percent is absorbed

by the solid surface of Earth. The total radiation energy received by the atmosphere

and the solid Earth is about 70%.

Earth should be in thermal equilibrium with the surroundings. Indeed, the 70% of

solar energy entering the atmosphere and Earth is transferred back to space as heat

radiation.

5.2.1 AM1.5 Reference Solar Spectral Irradiance

According to several decades of careful measurements, the power density of solar radia-

tion outside the atmosphere is 1366 W/m

. On the surface of Earth, due to scattering

and absorption, even under a perfectly clear sky, when the Sun is right at the zenith,

solar radiation is reduced by about 22%. Because, on average, the Sun should have an

azimuth angle with the horizon, the reduction should be on average more than 22%.

To standardize the measurement of solar energy applications, in 1982, the American

Society for Testing and Materials (ASTM) started to promulgate Standard Tables of

Reference Solar Spectral Irradiance at Air Mass 1.5. The standard was revised in

2003 as ASTM G173-03. A separate standard for zero air mass was promulgated in

2006 to become ASTM E490-06. An extension to diﬀerent tilt angles was promulgated

in 2008 as ASTM G197-08. In both ASTM G173-03 and G197-08, atmospheric and

climatic conditions are identical. These standards represent the solar radiation under

reasonable cloudless atmospheric conditions favorable for the computerized simulation,

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110  Interaction of Sunlight with Earth

Figure 5.6 AM0 and AM1.5 solar radiation spectra. The AM0 spectrum is the solar radiation

outside the atmosphere. It is approximately a blackbody radiation at 5800 K. The integrated power

density is 1.366 kW/m

. The AM1.5 spectrum shows a number of atmospheric eﬀects. On the blue

side, there is a broad-band reduction of power density due to Rayleigh scattering from molecules and

dust particles. On the infrared side, water vapor contributes the most absorption, followed by carbon

dioxide.

comparative rating, or experimental testing of fenestration systems. The U.S. standard

was adapted by the International Organization for Standardization (ISO) to become

ISO 9845-1, 1992.

The meaning of air mass (AM) is as follows. When the Sun is at the zenith under

reasonable cloudless atmospheric conditions, the absorption due to the atmosphere is

deﬁned as air mass 1. In most cases, the zenith angle of the Sun is not zero. ASTM

chose as the standard the condition when the absorption is 1.5 times the normal air

mass, abbreviated as AM1.5. According to Eq. 5.11, the standard zenith angle is

θ = arccos

1.5

=48.19

◦

. (5.12)

Also see Fig. 5.4. The integrated power density of the AM1.5 solar radiation is 1

kW/m

. The quantity 1 kW/m

is deﬁned as a unit of radiation, called one sun.We

shall use this unit throughout the book.

The spectral irradiance or the radiation power spectrum for AM0 and AM1.5 solar

radiation is shown in Fig. 5.6. The AM0 spectrum is the solar radiation outside the

atmosphere. It is approximately a blackbody radiation at 5800 K. The integrated power

density is 1.366 kW/m

. The AM1.5 spectrum shows a number of atmospheric eﬀects.

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5.2 Interaction of Sunlight with Atmosphere  111

On the blue side, there is a broad-band reduction of power density due to Rayleigh

scattering from molecules and dust particles. The probability of Rayleigh scattering is

proportional to the inverse fourth power of the wavelength of the radiation, and thus

the short-wavelength radiation is reduced heavily. On the infrared side, water vapor

contributes the most absorption followed by carbon dioxide. Plate 1 is a colored verion

of Fig. 5.6. A table of the data is shown in Appendix E.

5.2.2 Annual Insolation Map

In Chapter 4, we discussed the variation of direct solar radiation as a function of time

(day in a year and time in a day) and location (latitude and longitude). Because of the

interaction of solar radiation with the atmosphere, the actual solar radiation received

at the surface is always less, and the percentage of reduction depends on the location.

A frequently used representation is the annual insolation map; see Plates 3–5. There

are two conventions:

1. Annual radiation energy in kilowatt-hours per square meter. The standard solar

radiation is deﬁned as one sun, or 1 kW/m

. Therefore, the insolation is often expressed

in hours per year. The number ranges from more than 2000 h/year (Sahara desert,

part of Outback of Australia, part of South Africa) to less than 600 h/year (Greenland,

northern parts of Siberia, Finland, and Canada).

2. Average diurnal radiation energy in kilowatt-hours per square meter over a year.

Similarly, it is often expressed in h/day. The number ranges from more than 6 h/day

to less than 2 h/day.

There is an obvious relation between those two conventions:

Annual insolation = 365.2422 × average diurnal insolation. (5.13)

Plate 3 is an average diurnal insolation map of the world, Plate 4 is an average diurnal

insolation map of the United States, and Plate 5 is an annual insolation map of Europe.

5.2.3 Clearness Index

In most cases the insolation shown on the maps is smaller than the solar radiation

derived in Chapter 4. The ratio is the clearness index,

, (5.14)

where

is the average diurnal insolation under cloudless conditions from Chapter 4;

and

H is the observed average diurnal insolation, typically over a year or a month.

From a practical point of view, monthly averaged diurnal insolation is the most

useful data for the application of solar energy and is also widely measured at many

solar observation stations in the world. The clearness index

, in percent, is also

collected for many cities in the world. Table 5.1 shows the clearness index at several

U.S. cities, from the sunniest city on Earth, Yuma, Arizona, to one of the wettest cities,

Hilo, Hawaii. The last column is the annual average clearness index.