td chapter 3

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3. Availability Concepts: Availability, irreversibility and application of 2nd law

efficiency

3.1 Introduction

3.1.1 High Grade and Low Grade EnergyFirst law of thermodynamics deals with quantity of energy and second law with qualiry of energy. Ist law does not distinguish between the

nature of heat and work whereas second law does. 2nd law states that all heat can be converted into heat and only a part of heat can be converted

into work. But all work can be converted into heat. Therefore, work is a energy of high grade and heat is energy of low grade. The examples of

high grade energy are mechanical work, electrical work, water power, wind power, tidal power etc. The examples of low grade energy are heat or

thermal energy, heat derived from nuclear energy, heat from combustion jof fuel etc.

3.1.2 Quality of Steam1 kJ of heat at 15000C will produce maximum work of 0.82 kJ by using Carnot engine when atmospheric temperature is 400C.

kJ X X T

T T QW 82.01

2731500

4015001

1

211 =

+

−=

−== η

Similarly, it can be shown that 1 kJ jof heat at 1000C will produce 0.16 kJ of work.

It is observed that 1 kJ of heat at 15000C produces 5 times more work than same amount of heat at 1000C. Therefore, it is concluded that 1 kJ

heat at 15000C is high grade as compared with 1 kJ heat at 1000C.

3.2 Available and Unavailable Energy

2.2.1 Definition of Available and Unavailable Energy1. Available EnergyThe part of low grade energy heat which can be converted into maximum useful work in a cycle is called available work.

2. Unavailable EnergyThe part of low grade energy heat which has to be rejected to a sink to produce maximum useful work in a cycle is called unavailable energy.



3.3 Available and Unavailable Energy for Carnot EngineFig. 3.1 represents the available and unavailable parts of heat energy

revW W Q

T

T AE ==−= max

0 )1(

Available energy is equal to maximum work or reversible work for a given input Q.

The corresponding minimum energy unavailable is

UA = Q - AE

)1( 0

T

T Q −−=

QT

T 0=

3.4 Available and Unavailable Energy for a Finite Reservoir

Changing its Temperature



Fig. 3.2 shows a finite reservoir which is dropping temperature from T1 to T2 while supplying heat Q. Area under curve 1-2 is heat supplied Q.

For a heat input of Qδ ,

QT

T UAand Q

T

T AE δ δ 00 )1( =−=

Qδ = T ds, AE = Qδ - T0 ds, UE = T0 ds. On integrating, we get

)(

)(

1200

1200

s sT dsT UE

s sT QdsT Q AE

−==

−−=−=

∫

∫

AE represents the maximum output when state changes from 1 to 2. It also represents decrease in available energy of the reservoir. Corresponding

increase in unavailable energy is represented by UE.

3.5 Dead State of a SystemIf a system in equilibrium with the surroundings in respect of pressure, temperature, velocity, elevation etc, the state of the system is called the

dead state.

If a system is reduced to temperature of surroundings t0, its capacity to work utilizing the temperature difference ceases and energy available

becomes zero. Similarly, the pressure of the system is reduced to atmospheric pressure 0 p , its pressure difference ceases and its available

energy becomes zero. It is also true for other properties.



The properties in dead state are denoted by subscript zero i.e. T0, p0, h0, u0 etc.

3.6 Definition and Concept of Availability of a System

The availability of a given system is defined as the maximum useful work that can be obtained in a process in which system attains dead state.

The property availability signifies the work potential at a given state. The work will be maximum when process is reversible and the systemfinally attains the dead state. The diagram explains the concept of availability.



3.7 Expression for Availability of Closed System

Fig. 3.4 illustrates a closed system a closed system having heat flow Q from atmosphere to it and work produced the system is W. The boundary

of the system changes to this interaction and change in volume of the system is V0-V1. This change in volume of the system is resisted by the

atmosphere. Hence, the useful work that can be obtained from the system is equal to maximum work minus work done against pushing the

atmosphere.

Availability = Wuseful = Wmax – p0(V0-V1) (3.1)

Applying first law to the system, we get

Q = U0 – U1 + W (3.2)

Applying second law to the system and the surroundings, we get



)(

0

100

0

10

S S T Q

or T

QS S S uni

−≤

≥−−=∆

Equality sign holds good for a reversible process which gives maximum work. So

Q = T0 ( S0 - S1) (3.3)

Putting value of Q from Eq.(3.3) in Eq. (3.2), we obtain

Wmax = ( U1 – U0 ) - T0 (S1 – S0 ) or

Availability Ф = Wuseful = Wmax – p0(V0-V1)

Ф = ( U1 + p0 V1 – T0 S1 ) - (U0 + p0 V0 – T0 S0 )

If a system undergoes change of state from 1 to 2, the change of availability Ф1 – Ф2 will be equal to maximum useful work between the states 1

to 2. It is also shown in Fig. 3.3.



3.8 Expression for Availability of a Steady Flow System

Fig. 3.5 represents a steady flow process of a fluid through a control volume. Let Q and W be heat and work interactions between the control

volume and the surroundings. m1 is the mass entering and m2 the leaving the control volume. For steady flow process mass entering and mass

leaving are equal. From Eq. (2.14), applying first law to the control volume, we get

Q = W + m { (h2 + ve22 / 2 + gz2 ) – (h1 + ve1

2 / 2 + gz1) } ……(3.4)

)5.3()2

()2

(2

2

2

21

2

1

1z g

V hm z g

V hmQW or ++−+++=

Applying second law of thermodynamics to the control volume for a reversible process, we obtain



( ) surr S S T I sy )()(0 ∆+∆=

I = T0 (∆S)uni 0≥

It is concluded that the irreversibility of a given process is equal to the product of the temperature of the surroundings and the net change in the entropy of

the interacting systems. If there is any irreversibility in the process, ∆Suni is positive and henc e the irreversibility I is also positive. If the system

undergoes a reversible process, then ∆Suni is zero and the irreversibility of the process is zero. Irreversibility represents loss of work.

Similarly for steady flow process,

I = Wmax - Wa

={ })()

2()

2( 2102

2

221

2

11 s sT m z g

V hm z g

V hm −−++−++

-

=

{ } Q z g V

hm z g V

hm +++−++ )2

()2

(2

2

2

21

2

1

1

= T0 (S1 – S0 ) - Q

= surr S T S T sy)()(

00 ∆+∆

= ( ) surr S S T sy )()(0 ∆+∆

= T0 (∆S)uni 0≥

3.10 Second law Efficiency and its Applications3.10.1Introduction1. It is observed while studying second law of thermodynamics that work is a high grade energy and maximum work is produced when all the

processes are reversible as in case of Carnot engine.

Energy is the capacity of doing work or in general capacity to produce change in a specified environment. The change may be raising of weight called

work or heating of a space called heat or changing of state of a system called process. Ist law states that energy can either be created or destroyed.

Total energy at a state remains constant. It can change from one form to another form. First law is a quantitative statement.

In contrast, quality of energy is important also. As already discussed according to second law of thermodynamics, all work can be coverted into

heat but all heat can not be converted into work. The quality of energy is defined by availability or exergy or available energy. It provides a measure

of quality of energy. Energy is always conserved for the universe where as availability can be consumed or simply wasted out.From !st law, for system and surroundings

( )

( ) surr

system

E W Q

E W Q

121212

121212

∆=−−

∆=−

Adding above two equations, we obtain( ) ( ) ( )

surr systemuniverseE E E 121212 +∆=∆ = 0

and we have seen from second law of thermodynamics,( ) ( ) ( ) 0121212 ≥∆+∆=∆

surr systemuniverseS S S



First law says that the stores energy of the universe is conserved and second law says that the entropy of the universe always increases for the real

irreversible processes. Entropy production is work lost, loss of some thing occurs in real processes and this loss manifests itself as entropy production.

Something that is lost is not energy. Loss is reduced capacity n to produce change. In other words, there is loss of energy due to irreversibility of the process. In reversible process, exergy is conserved

3.10.2 Limitation of First Law Efficiency1. The performance of heat engines executing cycles is generally monitored and indexed by measuring its first law efficiency. First law efficiency

1η is specified as

input quired

output Deried

Re=η

2. In case of devices such as turbines, compressors, pumps, nozzles and diffusers that execute processes and deliver some output. An efficiency for any

output device undergoing a process is given by

devicetheof t Idealoutpu

devicethebyrequired output Ideal efficincyocess =Pr

For all devices, the best performance is achieved for a reversible process with no entropy generation and losses. In case of adiabatic devices, the

processes must be reversible and adiabatic or isentropic. For isothermal devices , the best performance is obtained when processes are reversible

isothermal.

3. This definition of fist law efficiency is some times quite misleading and gives efficiency higher than 100%. For a refrigerator working on vapour

compression cycle, the first law efficiency is equal to COP.

input quired

output Deried

Re=η

doneWork

extracted Heat =

= 3.5

The efficiency is 350% quite unsual.

4. First law efficiency for a heat pump is also its COP. ( )devicetheoperatetoinput Energy

sourceetemperatur higher todeliverd Heat COP HP =

( ) HP COP =W

Q1

Also in heat pump, Q1 = W + Q2

( )W

QW COP HP

2+= , here Q2 is heat drawn from low temperature sink.

So, first law efficiency is also greater than 100% which not desirable.

5. First law efficiency is ratio of work done and heat supplied or



1

1Q

W =η

It is the ratio of first grade energy W to second grade energy Q1. It is not desirable to compare two uneven type of energies.

Therefore, we conclude that first law efficiency concept has limited utility to act as index of performance for engines and other applications.

3.10.2 Second Law Efficiency, II η

1. It is the ratio of minimum amount of work required to perform a task to the amount of work actually used.

act

II

W

W

used work Actuallyrequired work Minimum

min=

=η

2.First Law Second Law Efficiency, II η , for a Work ProducingDevice, Turbine



Turbine is a work producing device. The figure shows the adiabatic flow of fluid from state 1 at pressure p1 to state 2 at pressure p2. 1-2’ shows the

isentropic expansion of the fluid from pressure p1 to pressure p2.

First law efficiency for the process is

statesend samethe for work imumor versible

produceswork Actual

maxRe1 =η

=21

12

′W

W

=21

21

′−−hhhh

Second law efficiency for a work producing machine is given by

rev

act

II

W

W

statesend samethe for work imumor versible

produced work Actual

=

=maxRe

η

=21

21

ψ −Ψ− hh

therefore s sT hh ),()(000

−−−=ψ

),()( 010011 s sT hh −−−=ψ

),()( 020022s sT hh −−−=ψ and

)()(2102121s sT hh −−−=−ψ ψ

3.First Law Second Law Efficiency, II η , for a Work Absorbing



Device, Comprewsor

Compressor is a work absorbing device which compresses a gas from a low pressure to a high pressure. Most of the compressors are designed for a

adiabatic flow and some are designed for a isothermal flow. In Fig. 3.7, 1-2 represents the isentropic process and 1-2 / the adiabatic process. The first

law efficiency for a work absorbing devic is given by

act

iso

adia I

act

adia

adia I W

W

W

W == ,, , η η

21

21 /

hh

hh I

−

−

=η

The second law efficiency II η for a work absorbing device like compressor is given by



input work Actual

stateto state fromtyavailabiliin Increase

W

W

act

rev

II

21=

=η

12

12

hh −

−

=

ψ ψ

)()(1201212 S S T hh −−−=−ψ ψ

)lnln()(1

2

1

2

01212T

T C

p

p RT T T C p p −−−=−ψ ψ

4. Second Law Efficiency of a Heat Exchanger If Q is supplied by a constant temperature source, reversible work produced is calculated by using Carnot engine. If two fluids are exchanging heatwith each other and temperatures of both fluids are changing continuously, then reversible work is calculated by using the expression of availability.

1. When Temperatures of Cold and Hot Fluids are Changing

Fig. 3.8 shows a parallel flow heat exchanger in which cold fluid is entering at state 1 and leaving at state 2 hot fluid is entering at state 3 and leavingat state 4. In case of heat exchanger, second law efficiency is defined as the ratio of increase in availability of cold fluid to decrease in availability of

hot fluid or

43

12

ψ ψ

ψ ψ η

−

−= II

2. When Temperatures of Cold and Hot Fluids are Constant



Fig. 3.9 shows heat transfer Q1 from hot fluid at a constant temperature T1. In turn cold fluid at constant temperature T2 receives heat Q2. Thedifference (Q1 - Q2) is lost to the surroundings. In the heat exchanger, heating of cold fluid is the desired process and cooling of hot fluid is the input

process. Therefore, second law efficiency is defined as the ratio of increase in availability of cold fluid to decrease in availability of hot fluid or

43

12

ψ ψ

ψ ψ η

−

−= II

Increase in availability of cold fluid = Q2

−

2

01T

T

Decrease in availability of hot fluid = Q1

−

1

01T

T



= II η

−

−

1

0

1

2

0

2

1

1

T

T Q

T

T Q

)1(

)1(

1

0

2

0

T

T

T

T

I II

−

−

= η η If 0= I η

)1(

)1(

1

0

2

0

T

T

T

T

II

−

−

=η



3.11Maximum Work Associated with Heat

Fig. 3.10 represents heat flow Q frpm surroundings to a system whose surface is at a constant temperature of Ts. A Carnot engine is introduced

between the reservoir at Ts and the surroundings at T0. The maximum or reversible work produces will be Q multiplied by the efficiency of the Carnot

engine or

)1( 0

s

revT

T QW −=

td chapter 3

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