Chen C.J. Physics of Solar Energy

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112  Interaction of Sunlight with Earth

Table 5.1: Monthly Clearness Index K

[%] of Selected U.S. Cities

City Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Ave

Yuma, AZ 83 87 91 94 97 98 92 91 93 93 90 83 91

Las Vegas, NV 74 77 78 81 85 91 84 86 92 84 83 75 82

Los Angeles, CA 70 69 70 67 68 69 80 81 80 76 79 72 73

Denver, CO 67 67 65 63 61 69 68 68 71 71 67 65 67

New York, NY 49 56 57 59 62 65 66 64 64 61 53 50 59

Seattle, WA 27 34 42 48 53 48 62 56 53 36 28 24 45

Hilo, HI 48 42 41 34 31 41 44 38 42 41 34 36 39

5.2.4 Beam and Diﬀuse Solar Radiation

Most solar radiation data are collected with a pyranometer to measure the global solar

irradiance from a hemisphere. A typical pyranometer is shown in Fig. 5.7. The cen-

terpiece of the instrument is a dishlike blackbody absorber (1), covered by a protective

glass dome (2). The radiation received by the absorber generates a voltage which is

proportional to the heat and is connected to a voltmeter through the cable (3). No

battery and electronics are needed. By design, the instrument receives direct (beam)

sunlight and the diﬀuse sunlight from the entire sky.

The solar radiation on a surface is always a mixture of direct (beam) sunlight

and diﬀuse sunlight. In any practical application, direct (beam) sunlight and diﬀuse

sunlight behave diﬀerently. For example, for concentrated solar applications, only direct

sunlight is used. On the other hand, for ﬂat solar thermal or photovoltaic receivers,

diﬀuse sunlight plays a signiﬁcant role. It is important to know the proportion of these

two.

In general, the cloudier the climate, the higher the proportion of diﬀuse radiation.

There should also be a quantitative correlation between the clearness index and the

ratio between beam sunlight and diﬀuse sunlight. The problem was studied in detail

by Liu and Jordan [52], and a large body of literature has been accumulated. The

central issue is to estimate the ratio of diﬀuse radiation averaged over a period of time

Figure 5.7 Pyranometer. The

center piece of the instrument is

a dishlike blackbody absorber (1),

covered by a protective glass dome

(2). It generates a voltage propor-

tional to the radiation received by

the absorber over the entire hemi-

sphere, then outputs it through the

cable (3).

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5.2 Interaction of Sunlight with Atmosphere  113

Figure 5.8 Correlation between diﬀuse radiation and clearness index. Data points are from

Liu and Jordan [52]. The thick straight lines represent the simpliﬁed model, Eqs. 5.15 – 5.17. The

curves represent third-order polynomials by Liu and Jordan [52]. Later data and analysis also support

a simple linear relation, see Ref. [54].

over the total averaged radiation H. Here we present a conceptually simple model

for a rule-of-thumb estimate. It has three components.

First, even for a perfectly clear day, diﬀuse radiation from the sky is appreciable.

Experiments show that the average proportion of diﬀuse radiation is about 15% of total

radiation. Therefore, if the clearness is greater than 85%, the ratio is 15%:

> 0.85, then

=0.15, (5.15)

where

is deﬁned in Eq. 5.14.

Second, for a heavily cloudy sky, all the radiation is diﬀuse.

≈ 0, then

=1. (5.16)

Third, for the cases in between, the simplest mathematical representation is a linear

interpolation,

If 0.85 >

> 0, then

=1−

. (5.17)

The conceptually and mathematically very simple model ﬁts the data published by

Liu and Jordan [52] reasonably well and is in agreement with several later publications.

See Fig 5.8. The thick straight lines represent the simpliﬁed model, Eqs 5.15 – 5.17.

The curves represent more complicated formulas by Liu and Jordan.

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114  Interaction of Sunlight with Earth

5.3 Penetration of Solar Energy into Earth

It is an experimental fact that in most solids the heat ﬂux q

along direction z is

proportional to the temperature gradient in the solid along that direction,

= −k



∂T

∂z



, (5.18)

where k is the thermal conductivity, a constant depending on the material nature of the

solid; and the negative sign is a result of that fact that the heat ﬂows in the direction

of decreasing temperature. In the International System of Units (SI), the unit for heat

ﬂux is watts per square meter. The unit for thermal conductivity is watts per meter

Kelvin. A table of thermal conductivities of commonly used materials in construction

is given as Table 5.2.

Consider a one-dimensional system in which the temperature and heat ﬂux are a

function of position z and time t. Practically, the system could be a bar with uniform

cross section A. The temperature and the heat ﬂux are uniform within each cross

section. If the density of the material is ρ and its heat capacitance is c

, the change in

the heat content of a slab with thickness Δz with temperature is

ΔQ = ρc

A ΔT Δz. (5.19)

Combining Eqs 5.18 and 5.19, we ﬁnd the diﬀerential equation for the temperature

distribution,

∂T

∂t

ρc

∂

∂z

. (5.20)

Let

α =

ρc

. (5.21)

Table 5.2: Thermal Property of Earth

Material Density Heat Thermal Coeﬃcient

ρ capacity c

conductivity kα

(10

kg/m

)(10

J/kg·K) (W/m·K) (10

−6

/s)

Limestone 2.18 0.91 1.5 0.626

Granite 3.0 0.79 3.5 1.47

Earth (wet) 1.7 2.1 2.5 0.70

Earth (dry) 1.26 0.795 0.25 0.25

Source : American Institute of Physics Handbook,

American Institute of Physics, New York, 3rd Ed., 1972; and Ref. [31].

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5.3 Penetration of Solar Energy into Earth  115

Equation 5.20 then becomes

∂T

∂t

= α

∂

∂z

. (5.22)

We seek a solution of Eq. 5.22 for a semi-inﬁnite space z ≥ 0 with boundary conditions

T = T

+ΔT cos ωt, z =0, (5.23)

T = T

,z= ∞, (5.24)

where ΔT is the amplitude of the temperature variation at z = 0 and ω is its circular

frequency. In the case we are considering, that is, the annual variation of temperature,

the circular frequency is

ω =

2π

(365.25 × 86400)

≈ 2 × 10

−7

−1

. (5.25)

Introducing a dimensionless temperature,

Θ=

T −T

ΔT

, (5.26)

Equation 5.22 becomes

∂Θ

∂t

= α

∂

∂z

(5.27)

with boundary conditions

Θ=cosωt, z =0, (5.28)

Θ=0,z= ∞. (5.29)

Equation 5.27 can be resolved much easier in complex numbers using the Euler relation

=cosx + i sin x,orcosx =Re[e

]. The boundary conditions now become

Θ=Re



−iωt



,z=0, (5.30)

Θ=0,z= ∞, (5.31)

and Eq. 5.27 can be resolved using the Ansatz,

Θ=Re



λz−iωt



. (5.32)

Obviously, the suggested solution (Eq. 5.32), satisﬁes the boundary condition at z =0

(Eq. 5.30). The constant λ can be determined by the diﬀerential equation Eq. 5.27.

In fact, it gives

αλ

= −iω. (5.33)

There are two solutions to Eq. 5.33,

λ = ±





2α

− i



2α



. (5.34)

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116  Interaction of Sunlight with Earth

Figure 5.9 Penetration of solar energy into Earth. On the ground, the average temperature is

, and the amplitude of annual temperature variation is ΔT . The heat penetrates into the ground

with a time delay. At a certain depth, the proﬁle of annual temperature variation is reversed.

Because of the boundary condition at z = ∞, only the negative sign is admissible.

Finally,

Θ=exp



−



2α



cos





2α

z − ωt



. (5.35)

Or, using Eq. 5.26, the solution is

T = T

+ΔT exp



−



2α



cos





2α

z − ωt



. (5.36)

For a numerical example, we use T

=15

◦

C, ΔT =12

◦

C, and limestone. From Eq. 5.25

and Table 5.2, one ﬁnds



ω/2α =



2 × 10

−7

/2 × 0.626 × 10

−6

≈ 0.33 m

−1

. Taking

the number of the month as a parameter,

T =15

◦

+12

◦

−0.33 z

cos



0.33 z −

2π × (month − 7)



. (5.37)

At a distance z where



ω/2α = π, the value of the cosine function becomes its

negative. For example, according to Eq. 5.37, when 0.33z = π,orz =9.5m,the

tempeture in the summer is the lowest and the temperature in the winter is the highest.

The proﬁle of the annual temperature variation is reversed; see Fig. 5.9. Therefore,

the stored solar energy can be eﬀectively utilized. We will discuss the details of its

implementation in chapter 6.

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Problems

5.1. Modeling the Sun as a blackbody radiator of T



= 5800 K, calculate Earth’s sur-

face temperature T

⊕

, assuming that the temperature is uniform over its entire surface

which is also a blackbody radiator.

5.2. As in the previous problem, assume the Sun as a blackbody radiator of T



5800 K. If the absorptivity of Earth is 0.65 over the entire spectral range and its entire

surface, what is Earth’s surface temperature T

⊕

5.3. By direct substitution, prove that

u(z,t)=

√

exp



−z

4αt



(5.38)

is a solution of the one-dimensional heat conduction equation

∂u

∂t

= α

∂

∂z

. (5.39)

5.4. Because the heat conduction equation is linear, if u(z, t) is a solution, then the

inﬁnite integral of u(z, t) is also a solution. Prove that

u(z, t)=A erf

√

αt

(5.40)

is a solution of Eq. 5.39, where the error function erf(x) is deﬁned as

erf(x)=



(π)



−ξ

dξ. (5.41)

5.5. If the surface temperature of Earth suddenly changes from 0

◦

CtoT

,howlong

does it take for the interior of Earth to be in equilibrium with its surface, which means

it reaches 99% of the surface temperature? Make estimates for depths of 10 m, 100 m,

and 1000 m, where the ground is made of granite.

5.6. Using the theory of solar energy penetration into Earth, determine the eﬀect of

the average daily variation of earth temperature. Assume that the amplitude of daily

temperature is ΔT =5

◦

C and the ground is made of limestone:

1. At what depth the phase of temperature proﬁle is reversed (i.e., cooler at about

3pm and warmer at about 3am)?

2. What is the ratio of the amplitude of temperature variation at the depth of

temperature proﬁle reversal versus the temperature amplitude at the surface?

5.7. The average temperature of Oklahoma City is 28

◦

CinJulyand4

◦

C in January.

The temperature proﬁle is approximately a sinusoidal curve during the entire year. If

the ground is made of granite, what is the monthly temperature variation (list the

values for each month) 10 m beneath the ground?

Problems  117

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Chapter 6

Thermodynamics of Solar Energy

Thermodynamics is a branch of physics devoted to the study of energy and its trans-

formation. It was established in the eighteenth and nineteenth centuries in attempts to

better understand the underlying principles of heat engine, which converts heat energy

to mechanical energy, such as the steam engine and the internal combustion engine. In

the middle of nineteenth century, thermodynamics evolved into a logically consistent

system starting with a few axioms, or laws, from which the entire theory could be

deduced. In the twentieth century, the theory was extended to refrigeration and heat

pumps as well as other forms of energy, such as electric, magnetic, elastic, chemical,

electrochemical, and nuclear. At the core is the ﬁrst law of thermodynamics regarding

the conservation of energy and the second law of thermodynamics regarding the conver-

sion of thermal energy to other forms of energy. In many textbooks of thermodynamics,

there is also a zeroth law and a third law. The zeroth law is a self-evident deﬁnition

of temperature. The third law has limited applications, most of which are unrelated

to the study of solar energy. The theory of thermodynamics is macroscopic in nature,

dealing directly with measurable physical quantities. The corresponding microscopic

theory is statistical physics, where the laws of thermodynamics are derived from an

atomic point of view.

In this chapter, we will present the basic concepts of thermodynamics that are essen-

tial for the understanding of solar energy. Complete presentations of thermodynamics,

including the details of the third law of thermodynamics, can be found in standard

textbooks.

6.1 Deﬁnitions

In this section, we introduce several basic deﬁnitions. The object of investigation by

thermodynamics is called a system. A typical system is a uniform body of substance

with a well-deﬁned boundary, such as a piece of solid, a volume of liquid, a package of

gas, and a surface layer. The physical objects outside the system boundary are called

the surroundings. A state of a system represents the totality of its macroscopic prop-

erties. Two types of physical quantities are present in thermodynamics: the extensive

quantity and the intensive quantity. An extensive quantity is proportional to the vol-

ume or mass of the system and is additive, such as mass, volume, energy, and entropy.

119

Physics of Solar Energy C. Julian Chen

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120  Thermodynamics of Solar Energy

Intensive quantities are independent of the volume or mass of the system and are not

additive, such as temperature, density, pressure, and the speciﬁc values of the extensive

quantities such as energy density. A thermodynamic system can be connected to the

surroundings or isolated. There is no heat or work exchange between an isolated system

and its surroundings. The state of a system can go through a process where at least

one of the physical quantities is changing over time. Two processes are of particular

interest: the isothermal process, where the temperature of the system does not change,

and the adiabatic process, where there is no heat change between the system and its

surroundings.

An inﬁnitesimal quantity of work, dW , is deﬁned as the product of the force F

acting on the boundary of the system and the length dL it moves in the direction of

the force. Because the force F is a product of pressure P and area A, it is convenient

to write work as

dW = FdL= PAdL = PdV, (6.1)

where dV is the inﬁnitesimal change of the volume of the system. The occurrence of

the negative sign is because, when the pressure is positive, the force is pointing to the

inside of the system. When the system expands under a positive pressure, it does work

to the surroundings. The total work during a process from state 1 to state 2 is then

W =



FdL=



PdV. (6.2)

Heat is deﬁned as the energy transferred to the system across the boundary without

moving the surface. If the temperature of the surroundings is higher than that of the

system, the heat is transferred to the system, which is denoted as positive. Similarly,

the total heat transferred to the system during a process from state 1 to state 2 is

Q =



dQ. (6.3)

Temperature is deﬁned by the zeroth law of thermodynamics, which asserts as fol-

lows:

When two systems are in thermodynamic equilibrium with a third system,

the two systems are in thermodynamic equilibrium with each other, and all

three systems have the same temperature.

The zeroth law tells us how to compare temperature, but it does not deﬁne the scale

of temperature. In thermodynamics, there are two independent deﬁnitions of temper-

ature. The ﬁrst one is deﬁned by Lord Kelvin based on the Carnot cycle, which bears

his name; (see Section 2.3.2). The second deﬁnition is based on the properties of ideal

gas; (see Section 2.5.1). The two deﬁnitions are equivalent within a constant multiplier.

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6.2 First Law of Thermodynamics  121

6.2 First Law of Thermodynamics

The ﬁrst law of thermodynamics asserts that energy can be converted from one form to

another but can never be created or annihilated. A succinct presentation is as follows:

It is impossible to build a perpetual motion that generates energy from noth-

ing.

The ﬁrst law of thermodynamics is by no means trivial, regarding the fact that each

year, numerous patent applications on perpetual motion are still received by the patent

oﬃces of countries all over the world, even in the era of high technology. Searching on

Google for “perpetual motion”, you would be surprised by the exotic new designs of

perpetual motion proposed by overambitious inventors.

In his autobiography, Max Planck described how his physics teacher taught him

about the concept of energy: A construction worker lifted a brick and put it at the top

of the building. His work increased the energy of the brick. But the increase of energy

was in the form of potential energy and not explicit. One day, the brick fell down from

the top. As it almost reached the ground, the brick moved fast, which had an explicit

kinetic energy. Finally, the brick hit the ground and converted the energy to heat. In

the ﬁrst step, the construction worker did the work as

W = fh = mgh, (6.4)

where m is the mass of the brick and g =9.81m/s

is the gravitational acceleration. The

gravitational force f = mg.Andh is the height, or the distance along the direction of

force the brick moves. The increase of potential energy ΔE equals the work performed

on the system,

ΔE = W = mgh. (6.5)

Just before the brick hit the ground, its velocity v is given as

v =



2gh, (6.6)

which satisﬁes the law of conservation of energy,

ΔE =

= mgh. (6.7)

The unit of energy is the product of the unit of force, the newton, and the unit

of length, the meter. The unit of energy, newton-meter, or joule, is named after En-

glish physicist James Prescott Joule, who did the ﬁrst experiment to demonstrate the

equivalence of mechanical work and heat in 1844. A schematic of Joule’s experiment is

shown in Fig. 6.1.

In Joule’s experiment, at the beginning, the weight is positioned at a predetermined

distance from its equilibrium position. The initial temperature of the water, T

,is

measured. Then, by setting the weight to move and waiting to the end of motion,

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122  Thermodynamics of Solar Energy

Figure 6.1 Joule’s experiment. A paddle-wheel 1 is placed inside an insulated barrel 2 ﬁlled with

water 3. The temperature is measured by a thermometer 4. The wheel is driven through a spindle 5

and a pulley 6 by a falling weight 7. The height of the weight is measured by the ruler 8. After setting

the paddle-wheel to move, the mechanical energy is transformed into heat, which is measured by the

thermometer.

the ﬁnal temperature of the water barrel, T

, is measured. The heat generated by

mechanical disturbance is

Q =(T

− T

)M. (6.8)

If M is the mass of water in grams, then the heat Q is in calories. The mechanical

equivalence of one calorie of heat found by Joule is 4.159 J/cal, very close to the result

of modern measurements. As a result of the equivalence of mechanical work and heat,

the increment of the energy of a system is the sum of mechanical work and heat,

ΔE = W + Q. (6.9)

Energy could transfer as heat as well. By pushing two systems with heat capacity C

and C

at temperatures T

and T

into contact, as shown in Fig. 6.5, heat transfers from

the hotter system to the cooler system. Assuming that the thermal capacities of the

two systems are constant, that is, independent of temperature within the temperature

range of interest, eventually, the temperature becomes a single value T

+ C

). (6.10)

The heat transferred from one system to another is

Q = ±

+ C

+ T

). (6.11)

In Joule’s experiment, mechanical work transforms into heat. However, there no

simple way to transform heat back to mechanical work. In the case of heat transfer,