Availability and irreversibility

Availability and Irreversibility
Dr. Rohit Singh Lather

What is the maximum work possible
from a particular device, given a
specific amount of fuel?
The answer to this question is
provided by availability analyses
Simple Combustion
Automobile Engine
Steam Power Plant
Fuel Cell

Introduction
• There are many forms in which an energy can exist. But even under ideal conditions all these
forms cannot be converted completely into work. This indicates that energy has two parts :
- Available part
- Unavailable part
• ‘Available energy’ or ‘Exergy’: is the maximum portion of energy which could be converted into
useful work by ideal processes which reduce the system to a dead state (a state in equilibrium
with the earth and its atmosphere).
- There can be only one value for maximum work which the system alone could do while descending
to its dead state, therefore 'Available energy’ is a property
• ‘Unavailable energy’ or Anergy’: is the portion of energy which could not be converted into useful
work and is rejected to the surroundings

• A system which has a pressure difference from that of surroundings, work can be obtained from
an expansion process, and if the system has a different temperature, heat can be transferred to
a cycle and work can be obtained. But when the temperature and pressure becomes equal to that
of the earth, transfer of energy ceases, and although the system contains internal energy, this
energy is unavailable
• Summarily available energy denote, the latent capability of energy to do work, and in this sense it
can be applied to energy in the system or in the surroundings.
• The theoretical maximum amount of work which can be obtained from a system at any state p1 and
T1 when operating with a reservoir at the constant pressure and temperature p0 and T0 is called
‘availability’.

• First Law of Thermodynamics (law of energy conservation) used for may analyses performed
• Second Law of Thermodynamics simply through its derived property - entropy (S)
• Other ‘Second Law’ properties my be defined to measure the maximum amounts of work
achievable from certain systems
• This section considers how the maximum amount of work available from a system, when
interacting with surroundings, can be estimated
• All the energy in a system cannot be converted to work: the Second Law stated that it is
impossible to construct a heat engine that does not reject energy to the surroundings

• For stability of any system it is necessary and sufficient that, in all possible variations of
the state of the system which do not alter its energy, the variation of entropy shall be
negative
• This can be stated mathematically as ∆S < 0
• It can be seen that the statements of equilibrium based on energy and entropy, namely ∆E > 0
and ∆S < 0

• System A, which is a general system of constant composition in which the work output, 𝛿W, can
be either shaft or displacement work, or a combination of both
• Figure b, the work output is displacement work, p 𝛿V
Helmholtz Energy (Helmholtz function)
Thermal Reservoir To
𝛿Q
𝛿Qo
pop
System A
𝛿𝑊 𝑅
System B
ER

For a specified change of state these quantities, which are changes in properties, would be
independent of the process or work done. Applying the First Law of Thermodynamics to System A
gives
𝛿W = - dE + 𝛿Q
𝛿Wnet = 𝛿W + 𝛿WR
If the heat engine (ER,) and System A are considered to constitute another system, System B, then,
applying the First Law of Thermodynamics to System B gives
$%&
'&
=
Since the heat engine is internally reversible, and the entropy flow on either side is equal, then
and the change in entropy of System A during this process, because it is reversible, is dS =
$%
'
𝛿Wnet = - 𝛿E + TodS 𝛿Wnet = - d(E – ToS)
𝛿𝑄
𝑇

• The expression E - ToS is called the Helmholtz energy or Helmholtz function.
• In the absence of motion and gravitational effects the energy, E, may be replaced by the intrinsic
internal energy, U, giving 𝛿Wnet= -d(U - ToS)
• The changes executed were considered to be reversible and 𝛿Wnet was the net work obtained
from System B (i.e. System A + heat engine ER).Thus, 𝛿Wnet must be the maximum quantity of
work that can be obtained from the combined system
• The expression for 𝛿W is called the change in the Helmholtz energy, where the Helmholtz energy
is defined as F = U - TS
- Helmholtz energy is a property which has the units of energy, and indicates the maximum work
that can be obtained from a system
- It can be seen that this is less than the internal energy, U
- Product TS is a measure of the unavailable energy
𝛿Wnet = - d(E – ToS)

• The change in Helmholtz energy is the maximum work that can be obtained from a closed system
undergoing a reversible process whilst remaining in temperature equilibrium with its surroundings
• A decrease in Helmholtz energy corresponds to an increase in entropy, hence the minimum value
of the function signifies the equilibrium condition
• A decrease in entropy corresponds to an increase in F; hence the criterion dF > 0 is that for
stability
-This criterion corresponds to work being done on the system
- For a constant volume system in which W = 0, dF = 0
• For reversible processes, F1 = F2; for all other processes there is a decrease in Helmholtz energy
• The minimum value of Helmholtz energy corresponds to the equilibrium condition

System A could change its volume by 𝛿V, and while it is doing this it must perform work on the
atmosphere equivalent to po 𝛿V, where po is the pressure of the atmosphere. This work detracts
from the work previously calculated and gives the maximum useful work, as Wu = 𝛿Wnet - PodV
if the system is in pressure equilibrium with surroundings.
𝛿Wu = -d(E- ToS) – podV
= -d(E + poV - ToS) because po= constant
Hence 𝛿Wu = -d(H - ToS)
• The quantity H - TS is called the Gibbs energy, Gibbs potential, or the Gibbs function, G
Hence G = H - TS
- Gibbs energy is a property which has the units of energy
- Indicates the maximum useful work that can be obtained from a system
- It can be seen that this is less than the enthalpy
Gibbs energy (Gibbs Function)

• The change in Gibbs energy is the maximum useful work that can be obtained from a system
undergoing a reversible process whilst remaining in pressure and temperature equilibrium with its
surroundings
• The equilibrium condition for the constraints of constant pressure and temperature can be
defined as: dG)PT < 0 Spontaneous change
dG)PT= 0 Equilibrium
AG)PT > 0 Criterion of stability
The minimum value of Gibbs energy corresponds to the equilibrium condition

• The work done by a system can be considered to be made up of two parts: that done against a
resisting force and that done against the environment.
• The pressure inside the system, p, is resisted by a force, F, and the pressure of the environment.
Hence, for System A, which is in equilibrium with the surroundings p.A = F + po.d
If the piston moves distance dx, then work done by various components p.A.dx = F.dx + po.A. dx
where P.A. dx= p dV = 𝛿W = work done by the fluid in the system
F dx = 𝛿W = work done against the resisting force
poA.dx = podV = 𝛿 W = work done against the surroundings
F
System A
pop
Displacement Work

Hence the work done by the system is not all converted into useful work, but some of it is used to do
displacement work against the surroundings, i.e.
𝛿Wsym = 𝛿Wuse + 𝛿Wsurr
which can be rearranged to give 𝛿Wuse = 𝛿Wsys - 𝛿Wsurr

Thermal Reservoir To
𝛿Q
𝛿Qo
𝛿𝑊pop
System A
𝛿𝑊 𝑅
System B
ER
• All the displacement work done by a system is available to do useful work
• This concept will now be generalized to consider all the possible work outputs from a system that is not in
thermodynamic and mechanical equilibrium with its surroundings (i.e. not at the ambient, or dead state,
conditions)
Availability for a Closed System (non-steady)
• The maximum work that can be obtained from a constant
volume, closed system
𝛿WS + 𝛿WR = - (dU – TodS)
• Hence, the maximum useful work which can be achieved from a
closed system is 𝛿 WS + 𝛿 WR = -(dU + PodV -TodS)
• This work is given the symbol dA
• Since the surroundings are at fixed pressure and temperature
(i.e. po and To are constant) dA can be integrated to give
A = U + po V - TOS

• A is called the non-flow availability function
- It is a combination of properties
- A is not itself a property because it is defined in relation to the arbitrary datum values of po and
To
- It is not possible to tabulate values of A without defining both these datum levels
- The datum levels are what differentiates A from Gibbs energy G
- The maximum useful work achievable from a system changing state from 1 to 2 is given by
Wmax = ∆A= -(A2 - Al) = Al - A2
- The specific availability, a , i.e. the availability per unit mass is a = u + pov - Tos
- If the value of a were based on unit amount of substance (i.e. kmol) it would be referred to as
the molar availability
• The change of specific (or molar) availability is
∆a = a2 - a1 = (u2 + pov2 - Tos2)- (u1 + pov1-Tos1)
= ( h2 + v2(Po-P2) - (h1+ V1(Po – P1)) -To(S2 - S1)

Availability of a Steady Flow System
• Consider a steady flow system and let it be assumed that the flowing fluid has the following
properties and characteristics; Internal energy u, specific volume v, specific enthalpy h, pressure
p, velocity c and location z
Control
Volume
Carnot Engine
Inlet Outlet
P1,V1,T1
P2,V2,T2
WEngine
Q units of heat be rejected by the system
To(S1 – S2)
System delivers a work output W units
Normally, P2 &T2 ambient or state dead condition

U1 + p1 𝑣1 +
V1
2
2
+ gz1 − Q = U2 + p2 𝑣2 +
V2
2
2
+ gz2 + Ws
U1 + p1 𝑣1 − Q = U2 + p2 𝑣2 + Ws
Neglecting the kinetic and potential energy changes
H1 − Q = H2 + Ws
Shaft work Ws = H1 − H2 − Q
• Heat Q rejected by the system may be made to run a reversible heat engine, the output from the
engine equals
Wengine = Q (1 –
'&
'C
)
= Q – To (S1 – S2)
• Maximum available useful work or net work Wnet = Ws + Wengine
= H1 − H2 − Q + Q – To (S1 – S2)

• Clearly, the availability B is a state function in the strictest mathematical sense so the maximum
(or minimum) work associated with any steady state process is also independent of the path
Availability: Yields the maximum work producing potential or the minimum work requirement of a
process
- Allows evaluation and quantitative comparison of options in a sustainability context
= H1 − H2 − Q + Q – To (S1 – S2)
= H1 − To S1 – (H2 − ToS2)
= B1 – B2 Steady flow availability function H − ToS or Darrieus function
and the Keenam function
dB = (B1 – Bo) − (B2 – Bo) = B1 – B2

Available & Unavailable Energy
• If a certain portion of energy is available then obviously another part is unavailable
- the unavailable part is that which must be thrown away
- Diagram indicates an internally reversible process from a to b
- This can be considered to be made up of an infinite number of strips 1-m-n-4-1 where the
temperature of energy transfer is essentially constant, i.e. T1 = T4 = T
The energy transfer obeys
$%
'
=
$%&
'&
Where,
𝛿Q = heat transferred to system and
𝛿Qo= heat rejected from system,
As in an engine (ER) undergoing an infinitesimal Carnot cycle
- In reality 𝛿Q0 is the minimum amount of heat that can be rejected
because processes 1 to 2 and 3 to 4 are both isentropic, i.e.
adiabatic and reversible

Hence the amount of energy that must be rejected is
𝐸 𝑢𝑛𝑎𝑣 = ∫ 𝑑𝑄 𝑜 = T ∫
L%
'
𝑅= To∆S
• Note that the quantity of energy, 𝛿Q, can be written as a definite integral because the process is an
isentropic (reversible) one
• Then E, is the energy that is unavailable and is given by cdefc
• The available energy on this diagram is given by abcda and is given by
Eav = Q - Eunav =Q – T.dS where Q is defined by the area abfea

Graphical Representation of Available Energy, and Irreversibility
Thermal Reservoir TH
∆So
∆SH
A
I
Entropy
Temperature
To
TH
QH
W
-Qo
• Consider the energy transfer from a high temperature reservoir at TH through a heat engine (not
necessarily reversible)
The available energy flow
from the hot reservoir is
EH = QH -To ∆SH
The work done by the engine is
W = QH - Qo

The total change of entropy of the universe is ∑Δ𝑆 = Δ𝑆 𝐻 − Δ𝑆 𝑜 =
%Q
'Q
=
%&
'&
The energy which is unavailable due to irreversibility is defined by
Eirrev = EH - W = QH- To .∆SH - W
= QH –To . ∆SH - (QH - Qo) = Qo - To .∆SH
= To(∆S0- ∆SH)
In the case of a reversible engine ∑Δ𝑆 = 0 because entropy flow is conserved, i.e.
%Q
'Q
=
%&
'&
• Hence the unavailable energy for a reversible engine is To ∆SH while the irreversibility is zero
• However, for all other engines it is non-zero.
• The available energy is depicted by the area marked ‘A’, while the energy ‘lost’ due to
irreversibility is denoted ‘I’ and is defined Eirrev = To(∆S0 - ∆SH)

• The entropy of a system plus its surroundings (i.e. an isolated system) can never decrease (2nd
law).
• The second law states: ΔSsystem + ΔSsurr. = 0
where, Δ = final - initial > 0 irreversible (real world)
= 0 reversible (frictionless, ideal)
• In an ideal case if Q is the heat supplied from a source at T, its availability or the maximum work
it can deliver is Q(1-T0/T1) where T0 is the temperature of the surroundings.
- Invariably it will be less than this value.
- The difference is termed as irreversibility.
- Availability = Maximum possible work - Irreversibility
Wuseful = Wrev - I
Irreversibility

• Irreversibility can also be construed as the amount of work to be done to restore the system to
the original state.
- Eg: If air at 10 bar is throttled to 1 bar, the irreversibility will be p.v ln (10) which is the work
required to get 10 bar back.
- Here p is 1 bar and v is the specific volume at this condition.
- Note that the system has been restored to the original state but not the surroundings
- Therefore increase in entropy will be R ln 10.
• Combining first & second laws
TdS ≥ Δu + δ W
- It implies that the amount of heat energy to be supplied in a real process is larger than the
thermodynamic limit

• Irreversible Processes increase the entropy of the universe
• Reversible Processes do not effect the entropy of the universe
• Impossible Processes decrease the entropy of the universe
ΔS universe = 0
• Entropy Generation in the universe is a measure of lost work
ΔSUniverse = Δ SSystem + Δ SSurroundings
• The losses will keep increasing
• The sin keeps accumulating and damage to environment keeps increasing
• When the entropy of the universe goes so high, then some one has to come and set it right
HE SAYS HE WILL COME
Every religion confirms this

Heat Transfer Through a System Finite Temperature Difference
• When heat Q is transferred from a finite source, the temperature does not remain constant and
decreases as the flow of heat to the engine starts
- Heat supplied at varying temperature
- The change in entropy (S2 – S1) is calculated by integration as the temperature varies during
the heat transfer
Eav = Q - Eunav =Q –T.dS

Heat Transfer from a Finite Source
• Consider certain quantity of heat Q is transferred from a system at constant temperature T1 to
another system at constant temperature T2 (T1 > T2)
• Initial available energy
• Final available energy
• Change in the available energy = Eav1 - Eav2
= Q (1 –
'&
'C
) - Q (1 –
'&
'S
)
Eav2 = Q (1 –
'&
'S
)
= To (
%
'S
-
%
'C
)
= To (dS1 + dS2)
= To (dS)net ∆So
∆SH
e
Entropy
Temperature
To
T1
T2
b
gc
d
f
a
Increase in unavailable energy
This total change is called
entropy of universe or
entropy production

• abcd is the power cycle when heat available at T1 and area under cd represents the unavailable
energy
• efgd is the power cycle when the heat is available at T2 and area under dg represents the
unavailable energy
• Increase in the unavailable energy due to irreversible heat transfer is then represented by the
dark area under cg and given by To ( ambient temperature) and net increase in entropy of the
interacting systems
• Loss of available energy when heat transferred through finite temperature difference
• Greater temperature difference, more increase in entropy
• The concept of available energy provides the measure of quality of energy
• Energy is degraded each time it floes through a finite temperature difference (law of energy
degradation)

• The approaches derived previously work very well when it is possible to define the changes
occurring inside the system
• However, it is not always possible to do this and it is useful to derive a method for evaluating the
change of availability from ‘external’ parameters
• If a closed system goes from state 1 to state 2 by executing a process then the changes in that
system are
The change in specific availability is given by
where q, w and u, are the values of Q, Wand u per unit mass
Availability balance for a closed system
From First Law: U2 – U1 = ∫ 𝛿𝑄 − 𝛿𝑊 = ∫ 𝛿𝑄 − 𝑊
S
C
S
C
From Second Law: S2 – S1 = ∫
$%
'
+ 𝜎
S
C
a2 – a1 = u2 – u1 – To(s2 –s1) + po(v2 –v1)
= U 1 −
𝑇 𝑜
𝑇
𝛿𝑞 − 𝑤 + 𝑝𝑜 𝑣2 − 𝑣1
− 𝑇𝑜𝜎 𝑚
S
C

A2 – A1= ∫ 1 −
'&
'
𝛿𝑄 − 𝑊 + 𝑝𝑜 𝑣2 − 𝑣1
− 𝑇𝑜𝜎
S
C
Availability transfer
accompanying
Heat Transfer
Availability
transfer
accompanying
Work
Availability
destruction due to
Irreversibilities
A2 – A1= ∫ 1 −
'&
'
𝛿𝑄 − 𝑊 + 𝑝𝑜 𝑣2 − 𝑣1
− 𝐼
S
C
𝑑𝐴
𝑑𝑡
= U 1 −
𝑇 𝑜
𝑇
𝛿𝑄̇ − 𝑊̇ + 𝑝𝑜
𝑑𝑉
𝑑𝑡
− 𝑇𝑜 𝜎̇
S
C
𝑑𝐴
𝑑𝑡
= U 1 −
𝑇 𝑜
𝑇
𝛿𝑄̇ − 𝑊̇ + 𝑝𝑜
𝑑𝑉
𝑑𝑡
− 𝑇𝑜 𝐼̇

S
C
𝑑𝑎
𝑑𝑡
= 1 −
𝑇 𝑜
𝑇
𝑞̇ − 𝑤̇ + 𝑝𝑜
𝑑𝑣
𝑑𝑡
− 𝚤̇

• The effect of change of volume on the availability of the system
• The effect of ‘combustion 'on the availability of the system

• The energy content of the universe is constant, just as its mass content is
• We are always told how to “conserve” energy
• As engineers, we know that energy is already conserved
• What is not conserved is exergy, which is the useful work potential of the energy
• Once the exergy is wasted, it can never be recovered
• When we use energy (to heat our homes, for example), we are not destroying any energy; we are merely
converting it to a less useful form, a form of less exergy
Exergy and the Dead State
• The useful work potential of a system is the amount of energy we extract as useful work
• The useful work potential of a system at the specified state is called exergy
• Exergy is a property and is associated with the state of the system and the environment
• A system that is in equilibrium with its surroundings has zero exergy and is said to be at the dead state
• The exergy of the thermal energy of thermal reservoirs is equivalent to the work output of a Carnot heat
engine operating between the reservoir and the environment

Volume
Pressure
0
Dead State and Availability
P0
1’
1
Dead
State
Isotherm at T0
Avalability
Avalability
• Dead state is the state at which a system remains in
complete equilibrium with the surrounding
• There wont be finite driving potential for change to
occur
• Deas State implies that
- System is stable and uniform in composition
- Its pressure and temperature are equal to surroundings
- The system has zero velocity and minimum potential energy

• The total useful work delivered as the system undergoes a reversible process from the given
state to the dead state (that is when a system is in thermodynamic equilibrium with the
environment), which is Work potential by definition
Work Potential = Wuseful = Wmax - P0(V0 - V1)
• The work potential of internal energy (or a closed system) is either positive or zero, never
negative
• The useful work potential of enthalpy can be expressed on a unit mass basis as:
here ho and so are the enthalpy and entropy of the fluid at the dead state
• The work potential of enthalpy can be negative at sub atmospheric pressures

Reversible Work
• Reversible work Wrev is defined as the maximum amount of useful work that can be produced (or
the minimum work that needs to be supplied) as a system undergoes a process between the
specified initial and final states
- This is the useful work output (or input) obtained when the process between the initial and final
states is executed in a totally reversible manner
Irreversibility
• The difference between the reversible work Wrev and the useful work Wu is due to the
irreversibilities present during the process and is called the irreversibility I. It is equivalent to
the exergy destroyed and is expressed as
destroyed 0 gen rev, out u, out u, in rev, inI X T S W W W W= = = − = −
where Sgen is the entropy generated during the process. For a totally reversible process, the useful
and reversible work terms are identical and thus irreversibility is zero

• Irreversibility can be viewed as the wasted work potential or the lost opportunity to do work. It represents
the energy that could have been converted to work but was not
• Exergy destroyed represents the lost work potential and is also called the wasted work or lost work

Second Law Efficiency
• When assessing a power cycle we define the first-law efficiency as the quotient of the net work
done by the cycle over the exterior and the heat input to the cycle
• As a consequence of the 1st and the 2nd laws of thermodynamics, we get 0 ≤ 𝜂 < 1
• Energetic efficiency does not behave this way for all energy uses and devices
• Heat pumps present first law efficiencies greater than 1, because the energy input does not take
into account the heat input from the environment (the cold reservoir)
- Because the domain of 𝜂 is any positive real number
- the first-law efficiency does not provide a figure of merit in each energy use, but only a
quantification of the amount of energy transferred to a given desired end relative to an input
• The second-law efficiency is the widely accepted and used figure of merit for energy use systems

Second-Law Efficiency
• The second-law efficiency is a measure of the performance of a device relative to the
performance under reversible conditions for the same end states and is given by
u
, rev
th
II
th rev
W
W
η
η
η
= =For heat engines and other work-
producing devices and
For refrigerators, heat pumps, and
other work-consuming devices
rev
u
II
rev
WCOP
COP W
η = =
• In general, the second-law efficiency is expressed as
ηII = = −
Exergy recovered
Exergy supplied
Exergy destroyed
Exergy supplied
1 𝜂II =
LabcdaL aeadfg hdijbkad
dalamijh aeadfg cjnoh

• Its name is due to a figure of merit based in the second law of thermodynamics, measuring for
each process the distance from the theoretical ideal processes that can be measured in terms of
exergy
• It presents lower values as higher exergy is destroyed in a process
• With this definition, as a consequence of the second law of thermodynamics, exergy efficiency is
bonded as 0 ≤ 𝜂II ≤ 1, even for refrigerators and heat pumps, and therefore expresses a figure of
the quality and closeness to perfection of a given energy use
• Additionally, taking advantage of the concept of exergy, this definition can be reformulated as
𝜂II =
minimum exergy intake to perform the given task
actual exergy intake to perform the same task
=
pqcj
p

• We want to measure and study the performance of energy uses throughout a country or an
economy and therefore a comparable way of measuring it becomes essential
• The comparability of the energy use performance becomes clear when one states the second-law
efficiencies for several different applications
• A power-plant converts a fraction of available energy A or Wmax to useful work
- for desired output of W, Amin = W and A = Wmax
I = Wmax - W
𝜂II =
𝜂I
𝜂carnot
𝜂II =
W
Wmax
𝜂I =
W
Q1
𝜂I =
W .Wmax
Wmax . Q1
𝜂I = 𝜂II 𝜂carnot
Wmax = Q1 (1 -
To
T )
𝜂II =
W
Q1 (1 − To
T
)
If work is involved Amin = Wdesired
If heat is involved Amin = Q1 (1 -
To
T )

• The general definition of second law efficiency of a process can be obtained in terms of change in
availability during the process:

Availability and irreversibility

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Availability and irreversibility