UNIVERSITY OF TRINIDAD AND TOBAGO
B.A.Sc. Utilities Engineering
January 15, 2013
CCTD101B Engineering Thermodynamics (Lecture Notes 6)
2nd Law of Thermodynamics
2nd Law of Thermodynamics .............................................................................................................. 1
7.
7.1
Limitations of the First Law .................................................................................................................................. 1
7.2
Introduction to the Second Law ............................................................................................................................ 1
7.3
Uses of the Second Law.......................................................................................................................................... 3
7.4
Statements of the Second Law............................................................................................................................... 3
7.4.1
Clausius statement of the Second Law ............................................................................................................ 3
7.4.2
Kelvin-Planck statement of the Second Law ................................................................................................... 4
7.5
Reversible and Irreversible processes .................................................................................................................. 5
7.5.1 Internal and External Reversibility ........................................................................................................................... 6
7.6
Applying the Second Law to Thermodynamic Cycles ........................................................................................ 6
7.6.1
Interpreting the Kelvin Planck statement ....................................................................................................... 6
7.6.2
Power Cycles interacting with two reservoirs ................................................................................................ 7
7.6.3
Refrigeration and Heat Pump Cycles interacting with two Reservoirs .......................................................... 7
7.7
The Thermodynamic Temperature Scale ............................................................................................................ 8
7.8
Maximum Performance Measures for Cycles Operating between two Reservoirs .......................................... 9
7.8.1
Power Cycle.................................................................................................................................................... 9
7.8.2
Refrigeration and Heat Pump Cycles ........................................................................................................... 10
7.
2nd Law of Thermodynamics
7.1
Limitations of the First Law
There are energy conversion phenomena that cannot be explained using the first law alone. The first law
expresses the relationship between work and heat, and it allows us to define stored energy. But it does
not allow us to predict the extent of an energy conversion nor does it indicate whether a proposed or
conceived energy conversion process is possible. As an example, experience shows that at least one type
of desirable energy conversion – heat to work cannot be carried out completely. Furthermore, certain
processes that would in no way violate the first law cannot occur.
7.2
Introduction to the Second Law
There is a definite direction for spontaneous processes. Consider the systems illustrated in the diagram
below.
System a: An object at an elevated temperature T1 placed in contact with atmospheric air at temperature
T0 would eventually cool to the temperature of the surroundings. In conformity with the conservation of
energy principle, the decrease in internal energy of the body would appear as an increase in the internal
energy of the surroundings. The inverse process would not take place spontaneously, even though
energy would be conserved. The internal energy of the surroundings would not decease spontaneously
while the body warmed from T0 to its initial temperature.
System b. Air held at a high pressure pi in a closed tank would flow spontaneously to the lower pressure
surroundings at po if the interconnecting valve were opened. Eventually fluid motions would cease and
CCTD101B– 2nd Law of Thermodynamics
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all of the air would be at the same pressure as the surroundings. The inverse process would not take
place spontaneously even though energy would be conserved.
System c. A mass suspended by a cable at an elevation zi would fall when released. When it comes to
rest, the potential energy of the mass in its initial condition would appear as an increase in the internal
energy of the mass and its surroundings in accordance with the conservation of energy principle.
Eventually the mass also would come to the temperature of the surroundings. The inverse process
would not take place spontaneously even though energy would be conserved.
In each case considered, the initial conditions of the system can be restored but not in a spontaneous
process. Some auxiliary devices would be required. By such auxiliary means the object would be
returned to its initial temperature, the air could be returned to the tank and restored to its initial pressure,
and the mass could be returned to its initial height. Also, in each case, a fuel or electrical input would be
required for the auxiliary devices to operate, so a permanent change in the surroundings would result.
The above indicates that not every process consistent with the principle of energy conservation can
occur. Generally the energy balance alone neither enables the preferred direction to be predicted nor
permits the processes that can occur to be distinguished from those that cannot. In elementary cases such
as the ones considered, experience can be used to deduce whether particular spontaneous processes can
occur and to deduce their direction. For more complex cases, where experience is lacking or uncertain, a
guiding principle would be helpful. This is provided by the second law.
CCTD101B– 2nd Law of Thermodynamics
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Also, for the above the processes may occur rapid or slow, still satisfying the conservation of energy
principle. This alone however is insufficient to predict the final equilibrium state. Another principle is
required. This is provided by the second law.
By exploiting the spontaneous processes shown above it is possible, in principle, for work to be
developed, as equilibrium is attained. e.g. instead of permitting the body in diagram (a) to cool
spontaneously with no other result, energy could be delivered by heat transfer to a system undergoing a
power cycle that would develop a net amount of work. In summary, when an imbalance exists between
two systems, there is an opportunity for developing work that would be lost if the systems were allowed
to come into equilibrium in an uncontrolled way. Two important questions for this are:
 What is the theoretical maximum work that could be attained

What are the factors that would allow attainment of the maximum value?
The second law of thermodynamics provides means for answering these two questions.
7.3
Uses of the Second Law
The second law and its corollaries provide means for:
1. Predicting the direction of processes
2. Establishing conditions for equilibrium
3. Determining the best theoretical performance of cycles, engines and other devices.
4. Evaluating quantitatively the factors that preclude the attainment of the best theoretical
performance level.
Additional uses of the second law include its roles in
1. Defining a temperature scale independent of the properties of any thermometric property.
2. Develop means for evaluating properties such as u and h in terms of properties that are more
readily obtained experimentally.
7.4
Statements of the Second Law
The second law like the first law cannot be derived from any other law of nature. Two well known
statements of the second law are the Clausius statement and the Kelvin-Planck statement.
7.4.1
Clausius statement of the Second Law
The Clausius statement of the Second Law asserts that: It is impossible for any device to operate in
such a manner that it produces no other effect other than the transfer of heat from one body to another
body at a higher temperature. The Clausius statement is often given as “heat cannot by itself pass from
a cold to a hot body”.
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The Clausius statement does not rule out the possibility of transferring energy by heat from a cooler
body to a hotter body, for this exactly what refrigerators and heat pumps do. The words ‘sole effect’
suggest when a heat transfer from a cooler to a hotter body occurs, there must be some other effect in the
system accomplishing the heat transfer, its surroundings or both.
7.4.2
Kelvin-Planck statement of the Second Law
where the words single reservoir are added to emphasize that the system communicates thermally
with only a single reservoir as it executes the cycle.
Both statements of the second law are equivalent and it can be demonstrated that the violation of each
statement implies the violation of the other. Consider the illustration shown below.
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The system on the left transfers energy QC from cold reservoir to the hot reservoir by heat transfer
without any other effects and thus violates the Clausius statement. The system on the right operates in a
cycle while receiving QH (> than QC) from the hot reservoir, rejecting QC to the cold reservoir and
delivering work Wcycle to the surroundings.
Consider the combined system shown as a dotted line which consists of the cold reservoir and the two
devices. The combined system can be regarded as executing a cycle because one part undergoes a cycle
and the other two parts experience no net change in the conditions. Moreover the combined system
receives energy (QH – QC) by heat transfer from a single reservoir, the hot reservoir and produces an
equivalent amount of work. Accordingly the combined system violates the Kelvin-Planck statement.
Thus a violation in the Clausius statement implies a violation of the Kelvin-Planck statement.
7.5
Reversible and Irreversible processes
A process is reversible if after it has occurred, both the system and the surroundings can by any means
whatsoever be returned to their original states. Any other process is irreversible.
Reversible processes are important because they provide the maximum work from work producing
devices and the minimum work input to devices that absorb work to operate. To determine whether a
process is reversible, it is necessary to apply the second law. It is easier to show that a process is
irreversible than to show that it is reversible. We can identify reversible processes by the absence of
irreversible ones.
A system that has undergone an irreversible process can be returned to its original state. However it may
not be possible to return the surroundings to its original state making the process irreversible. The
second law can be used to determine whether a process is reversible or irreversible. A reversible process
must involve no:
 friction – sliding friction as well as friction in the flow of fluids
 heat transfer across a finite temperature difference
 free expansion of a gas to a lower pressure
 spontaneous mixing of matter at different composition or states
 spontaneous chemical reaction
 Electric current through a resistance.
 inelastic deformation
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All reversible processes must meet the following conditions:
1. a reversible process must be such that, after it has occurred, the system and the surroundings can
be made to traverse, in the reverse order, the states they passed through the original process. All
energy transformations of the original process would be reversed in direction but unchanged in
form or direction.
2. the direction of a reversible process can be changed by making infinitesimal changes in the
conditions that control it.
3. during a reversible process the system and the surroundings must each at all times be in states of
equilibrium or infinitesimally close to states of equilibrium; that is, the process must be a
quasiequilibrium one.
4. a reversible process must involve no friction, unrestrained expansion, mixing, and heat transfer
across a finite temperature difference or inelastic deformation.
All actual processes are irreversible. Reversible processes do not occur. Nevertheless, reversible
processes are extremely useful and serve as standards of comparison since they are often the limiting
cases of actual processes.
Work can always be completely converted into heat, but the extent to which heat can be converted to
work is limited. Therefore, work is the more valuable form of energy in transition. A process that uses
heat to convert it all into work is always irreversible since this is a violation of the Kelvin-Planck
statement of the first law.
7.5.1 Internal and External Reversibility
In an irreversible process, irreversibilities are present within the system, its surroundings or both. For
many analyses it may be convenient to divide the irreversibilities into two categories. Internal
irreversibilities are those that occur within the system. External irreversibilities are those that occur
within the surroundings, often the immediate surroundings.
A process is reversible if it is both internally and externally irreversible. A process is internally
reversible if there are irreversibility within the system. Irreversibility may be located within the
surroundings, however, as is the case for heat transfer across the boundary. At every intermediate stage
of an internally reversible process of a closed system, all intensive properties are uniform throughout
each phase present.
7.6
Applying the Second Law to Thermodynamic Cycles
Several important applications of the second law related to power cycles and refrigeration and heat
pump cycles are explored in this section.
7.6.1
Interpreting the Kelvin Planck statement
For a single reservoir
For a system with irreversibilities
CCTD101B– 2nd Law of Thermodynamics
𝑊𝑐𝑦𝑐𝑙𝑒 ≤ 0
𝑊𝑐𝑦𝑐𝑙𝑒 < 0
6
For a system with no irreversibilities
7.6.2
𝑊𝑐𝑦𝑐𝑙𝑒 = 0
Power Cycles interacting with two reservoirs
A significant limitation on the performance of systems undergoing power cycles can be brought out by
the Kelvin Planck statement of the second law. Consider the figure shown below which depicts a system
that executes a cycle while communicating thermally with two reservoirs, a hot and cold reservoirs, and
developing net work Wcycle. The thermal efficiency of the cycle is
𝜂=
𝑊𝑐𝑦𝑐𝑙𝑒
𝑄𝐶
= 1−
𝑄𝐻
𝑄𝐻
where QH is the amount of energy received by the system from the hot reservoir by heat transfer and QC
is the amount of energy discharged from the system to the cold reservoir by heat transfer.
If the value of QC was zero, the system (see diagram) would draw energy QH from the hot reservoir and
produce an equal amount of work, while undergoing a cycle. The thermal efficiency of such a cycle
would be 100%, violating the Kelvin Planck statement and is thus not allowed. It follows that for a
system undergoing a power cycle, while operating between two reservoirs, only a portion of the heat
transfer QH can be obtained as work and the remainder QC must be discharged by heat transfer to the
cold reservoir. That is the thermal efficiency must be less than 100%. This applies to all power cycles,
whatever the details of the operation. This may be regarded as a corollary of the second law. Other
corollaries follow.
Carnot Corollaries
1. the thermal efficiency of an irreversible power cycle is always less than the thermal efficiency of
a reversible power cycle when each operates between the same two thermal reservoirs.
2. All reversible power cycles operating between the same two thermal reservoirs have the same
thermal efficiency.
Note that a cycle is considered reversible when there are no irreversibilities within the system as it
undergoes the cycle and heat transfers between the system and reservoirs occur reversibly. Power cycles
that contain irreversibilities deliver less work and thus have smaller thermal efficiencies.
7.6.3
Refrigeration and Heat Pump Cycles interacting with two Reservoirs
The second law of thermodynamics places limits on the performance of refrigeration and heat pump
cycles as it does for power cycles. Consider the figure below which shows a system undergoing a cycle
while communicating thermally with two thermal reservoirs, a hot and a cold reservoir. In accordance
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with the conservation of energy principle, the cycle discharges energy QH by heat transfer to the hot
reservoir equal to the sum of the energy QC received by heat transfer from the cold reservoir and the net
work input. The cycle might be a refrigeration or heat pump cycle, depending on whether its function is
to remove energy QC form the cold reservoir or deliver energy QH to the hot reservoir.
For a refrigeration cycle, the coefficient of performance is
𝑄𝐶
𝑄𝐶
𝛽=
=
𝑊𝑐𝑦𝑐𝑙𝑒 𝑄𝐻 − 𝑄𝐶
The coefficient of performance for a heat pump cycle is
𝛾=
𝑄𝐻
𝑄𝐻
=
𝑊𝑐𝑦𝑐𝑙𝑒 𝑄𝐻 − 𝑄𝐶
As the net work input to the cycle Wcycle tends to zero, the coefficients of performance given by the
equations above approach a value of unity. If Wcycle were identically zero, the system would draw energy
QC from the cold reservoir and deliver energy QC to the hot reservoir, while undergoing a cycle. This
method of operation is a violation of the Clausius statement of the second law. It follows that these
coefficients of performance must be finite in value. This may be regarded as another corollary of the
second law.
Corollaries for refrigeration and heat pump cycles
:
1. The coefficient of performance of an irreversible refrigeration cycle is always less than the
coefficient of performance of a reversible refrigeration cycle when each operates between the
same two thermal reservoirs.
2. All reversible refrigeration cycles operating between the same two thermal reservoirs have the
sane coefficient of performance.
By replacing the term refrigeration with heat pump, we obtain the corollaries for heat pump cycles.
7.7
The Thermodynamic Temperature Scale
A temperature scale that is entirely independent of the physical properties of any substance can be
defined. Such a scale is called the thermodynamic temperature scale. The most commonly used
thermodynamic temperature scale is the one proposed by Kelvin and defined by the relationship
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TH QH

TL
QL
where TH and TL are the temperatures of the energy reservoirs between which a reversible engine
operates when it absorbs QH of heat from the high temperature reservoir and rejects heat QL to the
low temperature reservoir. Thus two temperatures on the Kelvin scale are in the same ratio as the values
of the heat transfers absorbed and rejected by a system underdog a reversible cycle while
communicating thermally with reservoirs at these temperatures
If a reversible power cycle were operated in the opposite direction as a refrigeration or heat pump cycle,
the magnitudes of the energy transfers QH and QC would remain the same, but the energy transfers
would be oppositely directed. Accordingly the above equation still applies to each type of cycle
considered thus far, provided the system undergoing the cycle operates between two thermal reservoirs
and the cycle is reversible.
Negative temperatures are impossible on the temperature scale. It must be remembered that the
Q
T
relationship H  H only applies to reversible engines operating between two reservoirs at TH and
TL
QL
TL. The efficiency of any externally reversible engine operating between temperatures T H and TL is
given by
T
  1 L
TH
7.8
Maximum Performance Measures for Cycles Operating between two Reservoirs
This section deals with the development of expressions for the maximum thermal efficiency of power
cycle and the maximum coefficient of performance of refrigeration and heat pump cycles in terms of
reservoir temperatures evaluated on the Kelvin scale.
The following equations are utilized:
TH QH
(1)

TL
QL
𝜂=
𝛽=
𝑄𝑖𝑛 −𝑄𝑜𝑢𝑡
𝑄𝑖𝑛
𝑄
= 1 − 𝑄 𝑖𝑛
𝑜𝑢𝑡
𝑄𝑖𝑛
𝑄𝑜𝑢𝑡 −𝑄𝑖𝑛
𝛾=𝑄
𝑄𝑜𝑢𝑡
𝑜𝑢𝑡 −𝑄𝑖𝑛
7.8.1
(Thermodynamic temperature scale)
(2)
(Efficiency of a power cycle)
(3)
(COP of a refrigeration cycle)
(4)
(COP of a heat cycle)
Power Cycle
Utilizing equations 1 and 2 above, the maximum thermal efficiency of a system undergoing a reversible
power cycle while operating between two thermal reservoirs at temperatures TH and TL is given by
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 max  1 
TL
TH
This is known as the Carnot efficiency.
Recalling the two Carnot corollaries, it should be evident that the efficiency given above is the thermal
efficiency of all reversible power cycles operating between two thermal reservoirs at temperatures TH
and TL, and the maximum efficiency any power cycle can have while operating between the two
reservoirs.
7.8.2
Refrigeration and Heat Pump Cycles
Equation 2 above is applicable to reversible refrigeration and heat pump cycles operating between two
thermal reservoirs, but for these QC represents the heat added to the cycle from the cold reservoir at
temperature TC on the Kelvin scale and QH is the heat discharged to the hot reservoir at temperature at
TH.
Using equations 2 and 3, the maximum coefficient of performance for a reversible refrigeration cycle is
𝑇𝐶
𝛽𝑚𝑎𝑥 =
𝑇𝐻 − 𝑇𝐶
Similarly, using equations 3 and 4, the maximum coefficient of performance for a reversible heat pump
cycle is
𝑇𝐻
𝛾𝑚𝑎𝑥 =
𝑇𝐻 − 𝑇𝐶
These expressions can be used as standards of comparison for actual refrigerators and heat pumps.
……..
CCTD101B– 2nd Law of Thermodynamics
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