APPLIED THERMODYNAMICS (MC 270)

Module 02

Dr. Clement A. Komolafe

cakomolafe@umat.edu.gh

Class: Tuesday (2.30 – 4.30 pm)

Thursday (6.00 – 8.00 am)
 Ground Rules
The ground rules are :
1. Come to class promptly and listen attentively
2. Come to class with a pen and paper and solve related
problems copiously
3. Obtain and consult the recommended textbooks
regularly
4. Make an acquaintance with your student handbook
and adhere strictly to all its provisions
5. Do not be negatively influenced by peer pressure
6. Read, study and meditate on your books because that is
your primary focus here

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 Lecture content
At the end of the module, students should be
able to:
Vapour power cycles

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 THE CARNOT CYCLE
 You have been taught earlier that heat engines are
cyclic devices and that the working fluid of a heat
engine returns to its initial state at the end of each
cycle.

 Work is done by the working fluid during one part of

the cycle and on the working fluid during another part.

 The difference between these two is the net work

delivered by the heat engine. The efficiency of a heat-
engine cycle greatly depends on how the individual
processes that make up the cycle are executed.

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 THE CARNOT CYCLE
 The net work, thus the cycle efficiency, can be
maximized by using processes that require the least
amount of work and deliver the most, that is, by
using reversible processes.

 Therefore, it is no surprise that the most efficient

cycles are reversible cycles, that is, cycles that consists
entirely of reversible processes.

 Reversible cycles cannot be achieved in practice

because the irreversibilities associated with each
process cannot be eliminated. However, reversible
cycles provide upper limits on the performance of real
cycles. 5
 THE CARNOT CYCLE
 Heat engines and refrigerators that work on reversible cycles
serve as models to which actual heat engines and
refrigerators can be compared. Reversible cycles also serve
as starting points in the development of actual cycles and
are modified as needed to meet certain requirements.

 Probably the best-known reversible cycle is the Carnot

cycle, first proposed in 1824 by French engineer Sadi
Carnot. The theoretical heat engine that operates on the
Carnot cycle is called the Carnot heat engine.

 The Carnot cycle is composed of four reversible processes-

two isothermal and two adiabatic (isentropic) - and it can
be executed either in a closed or a steady-flow system.

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 THE CARNOT CYCLE

 It has the maximum thermal efficiency for given temperature

limits, and it serves as a standard against which actual power
cycles can be compared.

 Since it is a reversible cycle, all four processes that comprise the

Carnot cycle can be reversed. Reversing the cycle does also
reverse the directions of any heat and work interactions.

 Consider a closed system that consists of a gas contained in an

adiabatic piston–cylinder device, as shown. The insulation of
the cylinder head is such that it may be removed to bring the
cylinder into contact with reservoirs to provide heat transfer.

 The four reversible processes that make up the Carnot cycle are
as follows:
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 THE CARNOT CYCLE

 Reversible Isothermal Expansion (process 1-2, TH is

constant). Initially (state 1), the temperature of the gas is TH
and the cylinder head is in close contact with a source at
temperature TH.

 The gas is allowed to expand slowly, doing work on the

surroundings. As the gas expands, the temperature of the
gas tends to decrease.

 But as soon as the temperature drops by an infinitesimal

amount dT, some heat is transferred from the reservoir
into the gas, raising the gas temperature to TH. Thus, the
gas temperature is kept constant at TH.

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 THE CARNOT CYCLE
 Since the temperature
difference between the gas
and the reservoir never
exceeds a differential amount
dT, this is a reversible heat
transfer process. It continues
until the piston reaches
position 2. The amount of
total heat transferred to the Fig. 1.3(a)
gas during this process is QH.

 Reversible Adiabatic Expansion (process 2-3, temperature drops from TH

to TL). At state 2, the reservoir that was in contact with the cylinder head
is removed and replaced by insulation so that the system becomes
adiabatic. The gas continues to expand slowly, doing work on the
surroundings until its temperature drops from TH to TL (state 3). The
piston is assumed to be frictionless and the process to be quasi-
equilibrium, so the process is reversible as well as adiabatic.

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 THE CARNOT CYCLE
Reversible Isothermal Compression
(process 3-4, TL constant). At state 3,
the insulation at the cylinder head is
removed, and the cylinder is brought
into contact with a sink at temperature
TL. Now the piston is pushed inward
by an external force, doing work on
the gas. As the gas is compressed, its
temperature tends to rise. But as soon
as it rises by an infinitesimal amount
dT, heat is transferred from the gas to
the sink, causing the gas temperature to
drop to TL. Thus, the gas temperature
remains constant at TL. Since the
temperature difference between the gas
and the sink never exceeds a Fig. 1.3 (b)
differential amount dT, this is a
reversible
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 THE CARNOT CYCLE
 The heat engine cycle has been discussed last semester, and it is
shown that the most efficient cycle is the Carnot cycle for given
temperatures of source and sink.

 This applies to both gases and vapour is shown in Fig. 2. A brief

summary of the essential features is as follows:

- 4 to 1: heat is supplied at constant temperature and pressure

- 1 to 2: the vapour expands isentropically from the high
pressure and temperature to the low pressure. In doing so, it
does work on the surroundings, which is the purpose of the
cycle.
- 2 to 3: the vapour which is wet at 2, has to be cooled to
state point 3 such that isentropic compression from 3 will
return the vapour to its original state at 4. From 4, the cycle is
repeated. 11
 THE CARNOT CYCLE

Carnot cycle for a

wet vapour on a
T-S diagram

Fig. 2

 The cycle described shows the different types of processes

involved in the complete cycle and the changes in the
thermodynamic properties of the vapour as it passes through
the cycle.

 The four processes are physically very different from each

other and thus they each require particular equipment.
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 THE CARNOT CYCLE
 The heat supply, 4-1 can be made in a boiler. The work
output, 1-2, can be obtained by expanding the vapour
through a turbine.

 The vapour is condensed , 2-3, in a condenser, and to raise

the pressure of the wet vapour, 3-4, requires a pump or
compressor.

 Thus, the components of the plant are defined, but before

these are discussed further, the deficiencies of the Carnot cycle
as the ideal cycle for a vapour must be considered.

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 The Carnot Heat Engine
 The hypothetical heat engine that operates on the reversible Carnot
cycle is called the Carnot heat engine. The thermal efficiency of any
heat engine, reversible or irreversible, is given by
𝑄𝐿
𝜂𝑐𝑎𝑟 = 1 - (a)
𝑄𝐻

where QH is heat transferred to the heat engine from a high-temperature

reservoir at TH, and QL is heat rejected to a low-temperature reservoir at TL.
For reversible heat engines, the heat transfer ratio in the above relation can
be replaced by the ratio of the absolute temperatures of the two reservoirs,
as given by Eq. (a). Then the efficiency of a Carnot engine, or any
reversible heat engine, becomes

(b)

Heat rejected by the engines is given by Eq. (c)

𝑇𝐿 (c)
𝑄𝐿 = 𝑄
𝑇𝐻 𝐻
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 The Carnot Heat Engine
 This relation is often referred to as the Carnot efficiency, since the
Carnot heat engine is the best known reversible engine. This is the
highest efficiency a heat engine operating between the two thermal
energy reservoirs at temperatures TL and TH can have.

 All irreversible (i.e., actual) heat engines operating between these

temperature limits (TL and TH) have lower efficiencies. An actual heat
engine cannot reach this maximum theoretical efficiency value because
it is impossible to completely eliminate all the irreversibilities associated
with the actual cycle.

 Note that TL and TH in Eq. (b) are absolute temperatures. Using °C or °F

for temperatures in this relation gives results grossly in error.

THE CARNOT PRINCIPLES

 The second law of thermodynamics puts limits on the operation of cyclic
devices as expressed by the Kelvin–Planck and Clausius statements. A
heat engine cannot operate by exchanging heat with a single reservoir, and
a refrigerator cannot operate without a net energy input from an external
source. 15
 The Carnot Heat Engine
 We can draw valuable conclusions from these statements. Two
conclusions pertain to the thermal efficiency of reversible and irreversible
(i.e., actual) heat engines and they are known as the Carnot principles
expressed as follows:
1. The efficiency of an irreversible heat engine is always less than the
efficiency of a reversible one operating between the same two reservoirs;
and
2. The efficiencies of all reversible heat engines operating between the same
two reservoirs are the same.

Example 1
A Carnot heat engine receives 500 kJ of heat per cycle from a high-
temperature source at 652°C and rejects heat to a low-temperature sink at
30°C. Determine (a) the thermal efficiency of this Carnot engine and (b) the
amount of heat rejected to the sink per cycle.

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 The Carnot Heat Engine
Solution:
The heat supplied to a Carnot heat engine is given. The thermal efficiency
and the heat rejected are to be determined.
(a) The Carnot heat engine is a reversible heat engine, and so its efficiency
can be determined from Eq. (b) to be:
𝑇𝐿
𝜂𝑐𝑎𝑟 = 1 - (b)
𝑇𝐻

30+273
𝜂𝑐𝑎𝑟 = 1 - = 0.672 = 67.2%
652+273

That is, this Carnot heat engine converts 67.2 percent of the heat it receives to
work.
(b) The amount of heat rejected QL by this reversible heat engine is easily
determined from Eq. (c) to be:

𝑄𝐿 =
𝑇𝐿
𝑄 (c)
𝑇𝐻 𝐻

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 The Carnot Heat Engine
30+273
= (500) = 164 kJ
652+273

Discussion:
Note that this Carnot heat engine rejects to a low-temperature sink 164 kJ
of the 500 kJ of heat it receives during each cycle.

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 Assignment
1. A Carnot heat engine receives 650 kJ of heat from a source
of unknown temperature and rejects 250 kJ of it to a sink at
24oC. Determine (a) the temperature of the source and (b) the
thermal efficiency of the heat engine
2. A Carnot heat engine operates between a source at 1000 K and
a sink at 300 K. If the heat engine is supplied with heat at a rate
of 800 kJ/min, determine (a) the thermal efficiency and (b) the
power output of this heat engine.

3. A heat engine receives heat from a heat source at 1200oC and

has a thermal efficiency of 40 percent. The heat engine does
maximum work equal to 500 kJ. Determine the heat supplied to
the heat engine by the heat source, the heat rejected to the heat
sink, and the temperature of the heat sink.
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