Chen C.J. Physics of Solar Energy

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6.3 Second Law of Thermodynamics  123

heat can spontaneously transfer from a system at a higher temperature to a system at

a lower temperature. But heat can never spontaneously transfer from a system at a

lower temperature to a system at a higher temperature without expending mechanical

work. Such observations lead to the second law of thermodynamics.

6.3 Second Law of Thermodynamics

There are many ways to state the second law of thermodynamics. It can be shown that

all those incarnations are equivalent. A succinct formulation, similar to that of Kelvin

and Planck, is as follows:

It is impossible to build a machine that converts heat to mechanical work

from a single source of heat.

Because the heat in the ocean is unlimited, if perpetual motion of the second type

could be built, mankind would never have to worry having energy. Another formulation

of the second law of thermodynamics, due to Clausius, is as follows:

It is impossible to transfer heat from a reservoir at a lower temperature to

a reservoir at a higher temperature without spending mechanical work.

In fact, if a machine to transfer heat from a cold reservoir to a hot reservoir without

expending external mechanical energy could be built, everybody on Earth would be

able to enjoy free heating and free air conditioning.

6.3.1 Carnot Cycle

The spirit of the second law of thermodynamics can be best understood using the

Carnot cycle, proposed by Sadi Carnot in 1824 in the quest for the ultimate eﬃciency

of heat engines [1]. A schematic is shown in Fig. 6.2. The engine consists of two

heat reservoirs and a cylinder with a piston ﬁlled with a volume of working gas as the

thermodynamic system.

The Carnot cycle is an idealization of a heat engine that generates mechanical work

by transferring heat from a hot reservoir at temperature T

to a cold reservoir at tem-

perature T

. A complete cycle consists of four processes. First, the system is in contact

with the hot reservoir, undergoing an isothermal expansion process. The system, al-

ways at temperature T

, gains heat Q

from the hot reservoir. Second, the system

is isolated from the reservoir and is undergoing an adiabatic expansion process. With

no heat transfer, the temperature of the system is reduced to T

. Third, the system

is in contact with the cold reservoir, undergoing an isothermal compression process.

The system, always at temperature T

, releases heat Q

to the cold reservoir. Fourth,

the system is isolated from the reservoir and is undergoing an adiabatic compression

process. With no heat transfer, the temperature of the system is raised to T

.Anet

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124  Thermodynamics of Solar Energy

work W is performed to the surroundings. The ﬁrst law of thermodynamics requires

that

= Q

+ W. (6.12)

The eﬃciency η of a heat engine is deﬁned as the ratio of mechanical work W over

the heat energy from the hot reservoir, Q

η ≡

=1−

. (6.13)

An essential assumption of the Carnot cycle is that the processes are reversible.

The Carnot cycle in Fig. 6.2 can be operated as a refrigerator or a heat pump; see Fig.

6.3. The thermodynamic system — a body of gas conﬁned in a cylinder with a piston

— transfers heat from a cold reservoir to a hot reservoir with a cost of mechanical

work. The four processes are as follows: First, the system is in contact with the

cold reservoir, undergoing an isothermal expansion process. The system, always at

temperature T

, gains heat Q

from the cold reservoir. Second, the system is isolated

from the reservoir and is undergoing an adiabatic compression process. With no heat

transfer, the temperature of the system is raised to T

. Third, the system is in contact

with the hot reservoir, undergoing an isothermal compression process. The system,

always at temperature T

, releases heat Q

to the hot reservoir. Fourth, the system

is isolated from the reservoir and is undergoing an adiabatic expansion process. With

no heat transfer, the temperature of the system is reduced to T

Figure 6.2 Carnot cycle. The thermodynamic system is a quantity of gas conﬁned in a cylinder with

a piston. It consists of four reversible processes: (1) an isothermal expansion process at temperature

, acquiring a heat Q

from the hot reservoir; (2) an adiabatic expansion process to reduce the

temperature to T

; (3) an isothermal compression process at temperature T

, releasing a heat Q

to the hot reservoir; (4) an adiabatic compression process to raise the temperature to T

.Anet

mechanical work W is done to the surroundings.

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6.3 Second Law of Thermodynamics  125

Figure 6.3 Reverse Carnot cycle. An idealized representation of a refrigerator or a heat pump,

it consists of four processes: (1) an isothermal expansion process at temperature T

, acquiring a heat

from the cold reservoir; (2) an adiabatic compression process to raise the temperature to T

;. (3)

an isothermal compression process at temperature T

, releasing a heat Q

to the hot reservoir; (4) an

adiabatic expansion process to reduce the temperature to T

. A net mechanical work W is required

to make the transfer.

Because the Carnot cycle is reversible, the same Carnot machine can function as a

heat engine or a heat pump (or equivalently, refrigerator). That fact has a far-reaching

consequence: The eﬃciency of all Carnot cycles depends only on the temperatures of

the two heat reservoirs,

η =1−

= f(T

). (6.14)

The ingenious proof, given by Sadi Carnot in 1824, is based on a logical argument

as follows: If two Carnot machines have diﬀerent eﬃciencies, one can use the Carnot

machine with a higher eﬃciency as the heat engine and the one with a lower eﬃciency

as the heat pump. The combined machine would contradict the second law of thermo-

dynamics. There are two alternative proofs with regard to the two formulations of the

second law of thermodynamics.

The ﬁrst proof is as follows: With the same amount of heat from the hot reservoir,

that heat engine can generate more work than is required to pump heat from the cold

reservoir to recover the input heat back to the hot reservoir. Therefore, the combined

machine can convert heat into mechanical work from a single heat reservoir, which is

a perpetual motion of the second type. It contradicts the Kelvin–Planck formulation of

the second law of thermodynamics.

An alternative proof is as follows: By using all the mechanical work generated with

the heat Q

from the hot reservoir using the more eﬃcient machine, a heat greater

than Q

can be generated to put back into the hot reservoir. The combined machine is

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126  Thermodynamics of Solar Energy

capable of transferring heat from a cold reservoir to a hot reservoir without requiring

any external mechanical work. It contradicts the Clausius formulation of the second

law of thermodynamics. The conclusions of the analysis based on the Carnot cycle

is simple but far-reaching: The eﬃciency of a reversible Carnot cycle, regardless of

the nature of the process and the nature of the substance, is uniquely determined by

the two temperatures — the temperature of the hot reservoir and the temperature of

the cold reservoir. The universality of the Carnot cycle motivated William Thompson

(Lord Kelvin) to deﬁne the scale of thermodynamic temperature.

6.3.2 Thermodynamic Temperature

In Section 2.1, the condition of equality of temperature was deﬁned. But the scale of

temperature is yet to be deﬁned. Based on the theory of the Carnot cycle, William

Thomson (Lord Kelvin) deﬁned the thermodynamic temperature, also known as the

absolute temperature or the Kelvin temperature scale [2], abbreviated K.

For reversible Carnot cycles, the eﬃciency is deﬁned as

η ≡

=1−

. (6.15)

Now look at the ratio of Q

and Q

. If the temperature of the hot reservoir equals

the temperature of the cold reservoir, then no mechanical work can be generated.

Therefore, Q

= Q

. In other words,

=1, then

=1. (6.16)

Obviously, the temperature scale should also satisfy the following:

< 1, then

< 1. (6.17)

The simplest temperature scale satisfying these criteria, proposed by Lord Kelvin

in 1848, is [2]

. (6.18)

Lord Kelvin showed that this absolute temperature scale is equivalent to the tem-

perature scale based on ideal gas properties [2]. The Kelvin temperature is also iden-

tical to the temperature in statistical physics created by Maxwell and Boltzmann; see

Appendix D.

In terms of the Kelvin temperature scale, the eﬃciency of a reversible Carnot cycle

(Eq. 6.15), is

=1−

, (6.19)

which is the maximum eﬃciency any heat engine can achieve, often referred to as

the Carnot eﬃciency. As shown, the lower the temperature T

, the greater the eﬃ-

ciency. The eﬃciency can be artiﬁcially close to 1 if T

is suﬃciently low. However,

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6.4 Thermodynamic Functions  127

the eﬃciency of any heat engine cannot equal to 1; otherwise the second law of ther-

modynamics is violated. Although 0 K could never be reached, we can formally deﬁne

absolute zero temperature by

If Q

=0, or W = Q

, then T

=0. (6.20)

The Celsius temperature scale is deﬁned as a shifted Kelvin scale, with the triple

point of water (the state in which the solid, liquid, and vapor phases exist together

in equilibrium) deﬁned as 0.01

◦

C. On the Celsius scale, the boiling point of water is

found experimentally to be 100.00

◦

C. This deﬁnition also ﬁxes the constant factor in

the Kelvin scale. The relation between the Kelvin scale and the Celsius scale is

◦

C + 273.15. (6.21)

6.3.3 Entropy

Equation 6.18 can be rewritten as

, (6.22)

which has a signiﬁcant consequence. Actually, Eq. 2.22 can be generalized to any

number of steps in a cycle, where in each step an inﬁnitesimal heat δQ is transferred.

For a reversible cycle, the cyclic integral is zero,



δQ

=0. (6.23)

Therefore, a single-valued function of the state can be deﬁned,

− S

≡



δQ

. (6.24)

This function, which plays a central role is thermodynamics, is called entropy.How-

ever, heat Q is not a function of the state, because the cyclic integral of δQ is, in general,

not zero. Each state of the system can have multiple values of the same quantity. How-

ever, an inﬁnitesimal amount of heat can be expressed in terms of the state function

entropy as

δQ = TdS. (6.25)

6.4 Thermodynamic Functions

As discussed in Section 2.1, for the ﬁxed state of a system, the state functions are ﬁxed.

Examples of state functions include volume V , mass m, pressure P , temperature T ,

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128  Thermodynamics of Solar Energy

etc. Heat and the mechanical work, on the other hand, are not state functions. We

denote the diﬀerentials of heat and mechanical work as δQ and δW.

According to the ﬁrst law of thermodynamics, the total energy of a system is also

a function of the state. In fact, a mathematical representation of the ﬁrst law of

thermodynamics is that the cyclic integral of the sum of mechanical work and heat is

zero,



(δQ + δW)=0. (6.26)

The diﬀerence of energy between state 1 and state 2 is deﬁned as

− U



(δQ + δW). (6.27)

Because mechanical work is related to pressure P and volume V as δW = −PdV,

combining eq. 6.27 with Eq. 6.25, the diﬀerential of energy is

dU = TdS−PdV. (6.28)

6.4.1 Free Energy

The total energy of a system U can be deﬁned as the capability of doing work to the

surroundings or the capability of transferring heat to the surroundings. In the analysis

of actual problems, sometimes it is necessary to emphasize the portion of energy related

to the capability of doing mechanical work. The deﬁnition of free energy satisﬁes this

purpose:

F = U − TS. (6.29)

Intuitively speaking, the deﬁnition of free energy seems to eliminate the heat component

of the total energy and retain the mechanical part. Similar to Eq. 6.25, the diﬀerential

of free energy is

dF = −SdT − PdV. (6.30)

Therefore, more precisely speaking, free energy is a measure of the capability of a state

of the system to do mechanical work at a constant temperature.

6.4.2 Enthalpy

Sometimes it is useful to know the portion of the energy of a system related to the

capability of delivering heat. The deﬁnition of enthalpy satisﬁes that purpose,

H = U + PV. (6.31)

Intuitively speaking, the deﬁnition of enthalpy seems to eliminate the mechanical

work component of the total energy and retain the heat content. Similar to Eq. 6.25,

the diﬀerential of enthalpy is

dH = TdS+ VdP. (6.32)

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6.4 Thermodynamic Functions  129

More precisely speaking, enthalpy is a measure of the capability of a state of the

system to deliver heat at a constant pressure.

When a liquid evaporates under constant pressure, the volume of the gas increases.

The heat required for evaporation is the sum of the diﬀerence of internal energy U and

the work needed to expand the volume, PV. The heat transferred during the phase

change is the sum of U and PV,thatis,theenthalpy,

ΔH =Δ(U + PV) . (6.33)

6.4.3 Gibbs Free Energy

The fourth state function, introduced by Willard Gibbs, is

G = U + PV − TS. (6.34)

Similarly, the diﬀerential of the Gibbs free energy is

dG = −SdT + VdP. (6.35)

From Eq. 6.35, it is clear that if the temperature and pressure of a system are kept

constant and the quantity of the substance is constant, then the Gibbs free energy is

a constant. This property is important in the analysis of chemical reactions, because

most chemical reactions occurs under conditions of constant temperature and constant

pressure.

First, consider a single-component system in which the quantity of the substance is

a variable which by convention is expressed as the number of moles of the substance,

N. When the temperature and the pressure of the system are kept constant, the Gibbs

free energy can be expressed as

dG = −SdT + VdP+ μdN. (6.36)

The quantity μ is the Gibbs free energy per mole of the substance.

6.4.4 Chemical Potential

For a system with more than one component, the expression of the Gibbs free energy

(Eq. 6.36), can be generalized to

dG = −SdT + VdP+



, (6.37)

where i is the index for the ith component. If there is a chemical reaction in the

system, then the composition of the system, represented by the set of molar values

, will change. Under constant temperature and constant pressure, the equilibrium

condition of the system is

dG =



=0. (6.38)

The quantity μ

is known as the chemical potential of the ith component of the

system.

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130  Thermodynamics of Solar Energy

6.5 Ideal Gas

Experimentally, it is found that within a large range of temperature and pressure many

commonly encountered gases satisfy a universal relation

PV = NRT, (6.39)

where P is the pressure in pascals, V is the volume in cubic meters, N is the number

of moles of the gas, T is absolute temperature in kelvins; and R is a universal gas

constant,

R =8.3144

mol · K

. (6.40)

In solar energy storage systems, for the gases commonly used, such as nitrogen, oxy-

gen, argon, and methane, at temperatures of interest the ideal gas relation is accurate

up to a pressure of 10 MPa, or 100 standard atmospheres pressure.

In the following, we will derive all the thermodynamic functions for the ideal gases.

First, we will show that the energy U of an ideal gas depends only on temperature

but not on volume. In fact, assuming that U = U (T,V ), using Eq. 6.25, the internal

energy varies with volume as



∂U

∂V



= T



∂S

∂V



− P. (6.41)

Using the equation of state 6.39, Eq. 6.41 becomes



∂U

∂V



= T



∂P

∂T



− P = T

− P =0. (6.42)

Deﬁning the constant-volume speciﬁc heat per mole as

≡



∂U

∂V



, (6.43)

the internal energy as a function of temperature is

− U



dT. (6.44)

If in the temperature interval of interest C

is a constant,

− U

= NC

− T

). (6.45)

To obtain an explicit expression of entropy, we use Eqs. 6.25, 6.39, and 6.43,

dS =

PdV

= NC

+ NR

. (6.46)

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6.5 Ideal Gas  131

Again, assuming that the speciﬁc heat C

is a constant,

− S

= N





+ R



= N



log

+ R log



. (6.47)

Another important quantity is the constant-pressure speciﬁc heat C

. For ideal

gases, there is a simple relation between C

and C

.ToheatN moles of gas while

keeping the pressure constant, work P ΔV is done to the surroundings. According to

the ﬁrst law of thermodynamics, an additional energy P ΔV must be supplied. Because

for an ideal gas P ΔV = NRΔT , the total energy required to raise the temperature is

ΔQ = NC

ΔT + NRΔT = N(C

+ R)ΔT. (6.48)

Therefore,

= C

+ R. (6.49)

The ratio of constant-pressure speciﬁc heat and constant-volume speciﬁc heat is an

important parameter in the study of the adiabatic process, often denoted as γ:

γ ≡

=1+

. (6.50)

During an adiabatic process, the heat transfer is zero. Therefore,

dT = −PdV. (6.51)

On the other hand, from the equation of state for the ideal gas,

NRdT = VdP+ PdV. (6.52)

Combining Eqs 6.51 and 6.52, we obtain





PdV + VdP=0. (6.53)

The solution of the above diﬀerential equation is

=const, (6.54)

which is the state equation of an adiabatic process.

In the following, we show that the eﬃciency of a reversible Carnot cycle of ideal

gas is η =1− (T

), and thus the thermodynamic temperature scale is identical

to the temperature scale based on the ideal gas; see Fig. 6.4. In the ﬁrst process,

an isothermal expansion process at temperature T

from state 1 to state 2, the gas

medium receives heat from the hot reservoir,



PdV = NRT



NRT

log

, (6.55)

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132  Thermodynamics of Solar Energy

Figure 6.4 Carnot cycle with ideal gas as the system. Curve 1–2 is an isothermal expansion

process at T

. Curve 2–3 is an adiabatic expansion process. Curve 3–4 is an isothermal contraction

process at T

. Curve 4–1 is an adiabatic contraction process. The gas does work W to the surroundings.

where V

and V

are the volume of states 1 and 2, respectively. In the third process,

an adiabatic contraction process from state 3 to state 4, the gas medium releases heat

to the cold reservoir,



PdV = NRT



NRT

log

, (6.56)

where V

and V

are the volumes of state 3 and 4, respectively.

Because processes 1–2 and 3–4 are isothermal, according to the equation of state,

= P

(6.57)

On the other hand, processes 2–3 and 4–1 are adiabatic. According to Eq. 6.54,

= P

(6.58)

Combining Eqs 6.57 and 6.58, we have

. (6.59)

Applying this relation to Eqs. 6.55 and 6.56, we conclude that the thermodynamic

temperature scale and the ideal gas temperature scale are identical,

. (6.60)