Defining Exergy
Notice: an opportunity for doing useful work exists when two systems
at different states are placed in communication, i.e.:
Assumptions:
1. when two systems are placed in communication they come to
equilibrium,
2. system “ A” may transfer heat ( Q A,B ) with environment only,
Exergy (definition No 1): exergy is the maximum useful work,
(e. g. mechanical or electrical work) obtainable when system
interacts with environment until equilibrium is reached, that
means that the process of reaching the equilibrium is performed
under the control aimed at maximizing the amount of useful
work – it is not a spontaneous process, example: transformation
of hydrogen – oxygen mixture to water in fuel cell will result in
amount if useful work being closer to exergy than the one
obtained via combustion,
1
Defining Exergy
Notice: an opportunity for doing useful work exists when two systems
at different states are placed in communication, i.e.:
Assumptions:
1. when two systems are placed in communication they come to
equilibrium,
2. system “ A” may transfer heat ( Q A,B ) with environment only,
Exergy (definition No 2):” exergy is the minimum useful work
required to form a quantity of matter from substances present
in the environment and to bring matter to a specified state, it
means that most efficient and least energy consuming
transformations are followed when producing the matter and
achieving the required state,
2
Defining Exergy
Properties of exergy:
Exergy is defined as work, i.e. it is expressed in units of energy (kJ ,
kWh ),
Exergy is an extensive property of the system (other examples of
extensive properties ? –volume, mass, energy i.e. properties which
depend on the size of the system – extensive properties are not
additive, for completeness – examples of intensive properties –
temperature, pressure, intensive properties do not depend on the
size of the system and are not additive),
Exergy is a measure of the departure of the state of the system from
the state of environment, i.e. once the environment is specified, the
value may be assigned to the exergy of the system, exergy may have
different values for another environment,
Exergy may be destroyed due to irreversibilities and generally it is
not conserved, i.e. if the system tends to equilibrium in a
spontaneous (uncontrolled) manner, then it is possible that no
useful work will be developed ( example ? – a compressed spring
which may develop useful work in controlled transformation or
may not develop any useful work in spontaneous expansion),
Exergy must not be negative, since useful work must be at least zero,
Exergy of all casual components of environment is zero,
Exergy may be transferred between systems, it is the extensive
property so it can be transferred like energy or entropy,
3
Defining Exergy
Environment – some portion of the surroundings, intensive
properties of each phase of the surroundings (e.g. pressure,
temperature, density) are uniform and do not change during
any processes occurring in the environment,
Environment – is free from irreversibilities –any irreversibilities
which exist are located within the system (internal
irreversibilities) or reside in the immediate surroundings of the
system (external irreversibilities),
Environment – is composed of common substances existing in
abundance within the Earth’s atmosphere, oceans and crust,
these substances are in their stable forms as they exist naturally
and there is no possibility of developing work from interactions
either physical or chemical – between parts of environment,
Environment – its intensive properties do not change, extensive
properties of environment can change due to interactions with
other systems,
Environment – kinetic and potential energies are evaluated relative
to coordinates in the environment, since its parts are at rest
with respect to one another, then the change in the energy of the
environment is the change of its internal energy only,
Environment – is usually modeled as a simple compressible system,
large in extent and uniform in temperature T0 and pressure p0 ,
in our analysis T0 = 25 [o C] ; pO = 1 [atm], for atmospheric air
TO will be an average air temperature, if both air and water
from natural surroundings will be used, then T0 will be specified
as the lower of average temperatures of air and water,
4
Defining Exergy
Environment – when the pressure, temperature, composition, velocity
or elevation of a system is different from the environment,
there is an opportunity to develop work,
Environment – when the pressure, temperature, composition, velocity
or elevation of a system change towards those of the
environment, the opportunity to develop work diminishes,
Environment – the opportunity to develop work ceases to exist when
the system and the environment are at rest and in equilibrium,
Environment – dead state when the conditions of mechanical,
thermal and chemical equilibrium between the system and the
environment are satisfied, i.e. pressure, temperature and
chemical potentials of the system are equal to those of
environment, furthermore the system has zero velocity and
elevation with respect to coordinates of the environment,
Dead state – under these conditions there is no possibility to develop
work, no possibility for a spontaneous change within the system,
nor is the possibility for the system and the environment to
interact,
Restricted dead state - when the conditions of mechanical and
thermal equilibrium between the system and the environment are
satisfied, i.e. a fixed quantity of matter which constitutes the
system is sealed in an impermeable envelope (no mass flow
allowed) which is at zero velocity and elevation with respect to
environment and at temperature T0 and pressure p0,
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Defining Exergy
Exergy components:
E = E KN + E PT + E PH + E CH
(3.1)
where:
E KN
E PT
E PH
E CH
- kinetic exergy
- potential exergy
- physical exergy
- chemical exergy
Thermomechanical exergy:
ETM = E KN + E PT + E PH
(3.2)
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Defining Exergy
Specific exergy:
e = e KN + e PT + e PH + e CH
(3.3)
where:
e
- total specific exergy (per unit mass or mole)
Kinetic and potential exergies are determined with respect to
coordinates of the environment, so they are equal to potential and
kinetic energies (i.e. kinetic and potential energies can be fully
converted to useful work when brought to rest ):
e
KN
1 2
= V
2
e PT = g ⋅ z
(3.3.a)
(3.3.b)
where:
V
z
- velocity relative to coordinates of environment
- elevation relative to coordinates of environment
Total specific exergy (eq. 3.3) equals:
e=e
PH
1 2
+ V + g ⋅ z + eCH
2
(3.3.c)
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Defining Exergy
When system at rest relative to environment:
e KN = e PT = 0
(3.3.d)
then total specific exergy:
e = e PH + eCH
e PH
e CH-
(3.4)
- physical exergy i.e. maximum theoretical useful work
obtainable when system passes from its initial state
characterized by temperature T and pressure p to the
restricted dead state characterized by temperature T0 and
pressure p0
chemical exergy i.e. maximum theoretical useful work
obtainable when system passes from restricted dead state to
the dead state where it is in complete equilibrium with the
environment
8
Physical Exergy
Definition: physical exergy of closed system at a given state is given
by the the formula:
E PH = (U − U 0 ) + p0 (V − V0 ) − T0 ( S − S 0 )
(4.1)
where:
U
V
S
- internal energy
- volume
at the given state
- entropy
U0
- internal energy
V0
- volume
S0
- entropy
at the restricted dead state
(U − U 0 )
p0 (V − V0 )
T0 ( S − S0 )
- change of internal energy
- work done by external forces (from I law of
thermodynamics)
- change of heat
9
Physical Exergy
Derivation of eq. (4.1):
Assumptions:
- closed system and and environment constitute the combined
system,
- control boundary of the combined system allow for transfer
of work only, heat transfer is prohibited,
- control boundary of the closed system allow for transfer of
heat and work,
- closed system is at rest relative to environment,
- volumes of closed system and environment may vary, but
total volume of combined system remains constant (i.e. it
ensures that the work developed is useful because it is not
expended for useless displacements of the surroundings by
the moving boundary of the combined system),
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Physical Exergy
Energy balance of the combined system:
∆U C = QC − WC
(4.2)
Boundary of the combined system allows for work transfer only, i.e.:
∆U C = −WC
(4.3)
where:
WC
∆U C
- work (useful) of the combined system (hence subscript C)
- change of internal energy of the combined system ( i.e.
the sum of changes of internal energy of the closed
system and environment)
Change of internal energy of the combined system:
∆U C = (U 0 − U ) + ∆U e
(4.4)
where:
U - internal energy of closed system at given state
U 0 - internal energy of closed system at restricted dead state
∆U e - change of internal energy of the environment
Notice: ( I law of thermodynamics – thermodynamic formulation)
Q1, 2 = U 2 − U1 + L1, 2
(1.2)
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Physical Exergy
Work done by external forces:
L1, 2 =
V2
∫ pdV
(1.3)
V1
Relation between heat and entropy:
S2
Q1, 2 = ∫ TdS
(2.2)
S1
Combining eqs. (1.2) (1.3) and (2.2)
dU = TdS − pdV
(4.5)
The temperature and pressure of the environment:
T0 = idem; p0 = idem
(4.6)
The change of internal energy of the environment in finite differences:
∆U e = T0 ⋅ ∆S e − p0 ⋅ ∆V e
(4.7)
(i.e. change of internal energy of environment may only be due to
changes of entropy
Se
and volume
Ve
of the environment)
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Physical Exergy
Combining eqs. (4.3 ; 4.4 and 4.7):
WC = (U − U 0 ) − (T0 ⋅ ∆S e − p0 ⋅ ∆V e )
(4.8)
Notice: total volume of the combined system is constant
The change of volume of the environment:
where:
∆V e = −(V0 − V )
V
- volume of the closed system at the given state
V0
- volume of the closed system at the restricted dead state
(4.9)
Notice: the change of volume of the environment is equal in
magnitude but opposite in sign to volume change of the closed
system,
substituting (4.9) into eq. (4.8):
WC = (U − U 0 ) + p0 (V − V0 ) − T0 ⋅ ∆S e
(4.10)
Notice: eq. (4.10) gives the work performed by the combined system
as the closed system passes from the given state i.e.
U ; V; T; S
to the restricted dead state, interacting only with environment:
U 0 ; V0 ; To ; S0
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Physical Exergy
Question: how to determine entropy change of the environment?
∆S e = ???
Entropy balance of the combined system:
∆SC = S gen
(4.11)
where:
S gen
- entropy generation within the combined system as the
closed system comes to equilibrium with the environment
Notice: no heat transfer occurs through the boundary of the combined
system (see scheme at transp.10), so the only source of
entropy change is the generation of entropy within combined
system
Entropy change of the combined system:
where:
( S0 − S )
∆SC = ( S 0 − S ) + ∆S e
(4.12)
- entropy change for the closed system
Notice: eq.(4.12) states, that entropy change of the combined system
is the sum of entropy change for the closed system and for the
environment
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Physical Exergy
Combining eqs. (4.10) ; (4.11) and (4.12):
WC = (U − U 0 ) + p0 (V − V0 ) − T0 ( S − S0 ) +
− T0 ⋅ S gen
(4.13)
Notice: first three terms
(U − U 0 ) + p0 (V − V0 ) − T0 ( S − S0 )
depend on the parameters of the given (initial) state and restricted
dead state (final state) only, they do not depend on the details of the
process
Question: what about term N0 4? determined by entropy generation?
S gen = ???
From II law of thermodynamics:
dQ
dS =
(rev)
T
;
dQ
dS ≥
(irrev) ;
T
S gen = 0
S gen > 0
(2.3)
(2.3)
Question: for which process WC from eq. (4.13) will become exergy?
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Physical Exergy
Notice: exergy is the maximum theoretical useful work, i.e. eq. (4.13)
will present exergy only when:
E PH = WCMAX
(4.14)
which is fulfilled (see eq. 4.13) when:
S gen = 0
that leads to:
WCMAX = (U − U 0 ) + p0 (V − V0 ) − T0 ( S − S 0 )
(4.15)
and proves that:
E PH = (U − U 0 ) + p0 (V − V0 ) − T0 ( S − S 0 )
(4.1)
Notice: For many practical applications there are no chemical
processes involved in thermal systems, then the thermomechanical
exergy is sufficient to determine the exergy change between two states
of the closed system (combine eqs. (3.2) and (4.1)):
E2 − E1 = (U 2 − U1 ) + p0 (V2 − V1 ) +
− T0 ( S 2 − S1 ) + ( KE2 − KE1 ) + ( PE2 − PE1 )
(4.16)
which may further be simplified with common assumption:
KE2 ≈ KE1 ≈ PE2 ≈ PE1 ≈ 0
(4.17)
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