Exergy (Availability): A Measure of Work Potential

yahoohordeMechanics

Oct 28, 2013 (3 years and 5 months ago)

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1

Exergy (Availability): A Measure of Work Potential


The 1
st

Law tells us that energy cannot be created or destroyed. Thus there is no energy shortage in the universe
because we have as much energy now as we had 100 years ago.


However the second law also

tells us that we are loosing the amount of energy that we can convert to work.


(Kelvin
-
Planck Statement) It is not possible to for any device that operates on a cycle to receive heat from a single
reservoir and produce a net amount of work (see Figure
1).





Figure 1: Heat Reservoir 1






2


(Clausius Statement) It is not possible to construct a device that operates in a cycle and produce no effect other
than the transfer of heat from a cooler body to a hotter body.




Figure 2: Heat Reservoir 2


The 2
nd

Law says all work can be converted to heat but all heat can not be converted to work.


It also says that energy has a quality. Work is the highest quality and then heat is a lower quality. The higher
the
temperature of the source from which the heat comes the higher the quality of the energy. More precisely I should
say the more different the temperature of the heat from the ultimate sink the higher the quality of energy.


Note that all energy tends to

work its way down the ladder going from higher quality to lower quality. That is, it
degrades to heat at a temperature of the ultimate sink.


This all means that, as a society, what we need to be worried about is how we use our work potential.



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Exergy (A
vailability)

-

(
x

or
X

is the symbol) Maximum possible useful work that a system in a given state in a
specified environment can produce. Availablity is another name given to exergy

Note: Work =
f(initial state, process path, final state)

Work output is

maximized when the process between two specified states is executed in a irreversable manner.



1.1.

Given


Figure 3:
Some quantity of mass

What is the maximum work we can get?

T, P,

, V,


u, h, s

T
0
, P
0
,

0
, V
0
,

0
= 0,

u
0
, h
0
, s
0

Su
rroundings



4


SURROUNDINGS AND ENVIRONMENT


1.1.1.

Note that the air outside in the atmosphere contain
s a lot of energy but it is not energy we can convert
to work. You could get work out of it if you could connect it to outer space and use that as a reservoir.





Figure 4: Surrounding and Environment


Surroundings: Everythin
g outside system boundaries.


Immediate Surroundings: Portion of the surroundings affected by the process.


Environment: Region beyond the Immediate Surroundings.


You can only get work out of a system if is at a different state than its surroundings (ho
t boiled egg and the
environment)




5

For a heat engine you need two reservoirs at different temperatures to produce work. (or Run a heat engine between
two temperature levels, T
H

and T
L
)







DEAD STATE


Dead state

-

the state o
f a system when it is in thermodynamic equilibrium with its environment.



Note: If the system is in a state different than the dead state it can always produce more work


It is differences between the system and environment that really give us the ability

to produce work.





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Normally the dead state is taken to be our environment. For many cases this is taken as 25
o
C, 101.325 kPa.


At Dead State We have:


Zero velocity relative to the environment

At the lowest elevation in the environment
-

ground level

Des
ignate properties of system at dead state with the subscript “
o


T
o
, P
o
, u
o
, h
o
, s
o


Note:

You can change the value of your exergy by changing your environment, but this is an extremely difficult
thing to do


EXERGY


Exergy can be looked at as a property o
f a system that describes its work potential.


Note work is a path function and exergy is essentially an amount of work.


The way exergy becomes a property is by defining the path and end point. The end point is a state in equilibrium
with the environment
-

dead state.



The process is a reversible one because that is what gives us the most work.


Note:

irreversibilities are items that degrade the quality of our energy.

Exergy can be defined for control mass, control volume in SSSF or USUF




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Exergy does not

include the work done against the atmosphere by the system. If your system changes size by
necessity it must do some work that is not available for other purposes and thus is not included in the exergy.


Exergy associated with different types of energy


1.1.2.

K
inetic energy




1.1.2.1.

ke

is pure work so all of this energy is exergy


1.1.3.

Potential energy




1.1.3.1.

pe

is pure work so all of this energy is exergy


1.1.4.

Internal energy (just thermal energy portion)




Note:

This
expression comes from looking at a control mass undergoing a reversible process from state
T, P, v, u,
s

to the dead state
T
o
, P
o
, v
o
, u
o
, s
o.
The
ke

and
pe
changes are taken to be zero because we are only worried about
changes in internal energy.





8

REVER
SIBLE WORK AND IRREVERSABILITY


Figure: Actual Work


Surrounding Work, W
surr
: Work done by or against the surrounding (Environment) during a process


Useful Work, W
u
: Work done by or against the system


Actual Work, W: Sum of
Surrounding Work and Useful Work


W = W
surr

+ W
u


Exergy for a closed system


Note: To obtain exergy for a closed system from thermal energy: control mass undergoing a process from its
original state to the dead state




9


Figure:
Exergy of a fixed system

1.1.4.1.

The work done is that which is done against the surroundings and that which is extra and can be
used for what we want (Boundary work)



(A)


Reversible Work

Maximum amount of useful work that can be produced or
supplied as a system undergoes a process between initial
and final states.


Irreversibility

Any difference between reversible work (W
rev
) and useful work (W
u
) during initial and final states of a process.


I = W
rev

+ W
out





10

Note:


Obtaining exergy for a
closed system from thermal energy is a reversible process. Therefore heat must be
transported through a reversible heat engine and

is (note
Q

is taken as a magnitude
-

the derivation is just for
Q

out of the system. There are some di
fferences for
Q

into the system, but the results are the same)



Since this heat is run through a heat engine an additional amount of work is produced




(B)


The total maximum useful theoretical work that we can get

from the closed system




From the
1
st

Law on a differential basis we can get an expression for




Note: that the minus sign takes care of the direction of heat transfer from the system




11



Substituting 1
st

law into maximum work expression




Thus the total useful work or the exergy between the given state and the dead state can be found by integrating from
the given state to the dead state




Integrating from the given state to the dead state gives




On a per unit mass basis




Suppose a system had
ke

and
pe
then




Exergy of a flow stream
:



12


For a flowing fluid, additional ener
gy required for a flowing process is flow energy or flow work (
W
flow

= W
pv

=Pv
).


Note:

The exergy of flow energy is the difference between flow work and work done against the atmosphere.




For a flowing fluid:


Not
e:



therefore



or



or



1.1.5.

Enthalpy




13


Note: This can be obtained by combining internal energy and flow work expressions


Exergy for a flow stream with
ke

an
d
pe




Control Mass

Exergy of a control mass
-

what is the maximum amount of work we can get out of the control mass by running
it between its present state and the dead state



1.1.6.

If you remember back to the First Law

we broke the energy contained by the system into three types:
ke, pe

and
u
.


1.1.7.

Exergy change of a control mass as it goes between state 1 and state 2





1.1.8.

Note you can multiply this by mass to get the extensive exergy change




14



1.2.

Exergy of a flow stream
-

what is the maximum amount of work we can get out of a flow by running it
between its present state and the dead state




1.2.1.

Exergy change of a flow stream as it under goes a change between state 1 an
d state 2




1.2.2.

Note you can multiply this by mass flow rate to get the extensive exergy change for a steady flow




1.3.

Exergy transported across boundary by (heat, work, and mass)


1.3.1.

HEAT


Note: We can always produce heat
from work at a temperature above the environment temperature by transferring
heat into a heat engine that rejects the waste heat into the environment.




15



1.3.1.1.

This comes from running the heat transfer through a Carnot heat engine




1.3.1.1.1.

This works whether
T
o

is greater than or less than
T


1.3.1.1.1.1.

Note that when
T
o

is greater than
T

the
Q

is that going to the low temperature
reservoir and not that coming out of the high temperature reservoir


1.3.1.1.2.

For the heat exergy
Q

should be positive
for heat into heat engine and negative for heat out
of heat engine


1.3.1.2.

If the system temperature is changing then you need to integrate



WORK

1.3.2.

Work


Recall:
W = W
surr

+ W
u


1.3.2.1.

Boundary work



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1.3.2.2.

all other work


MASS


a.

Mass contains exergy, energy, and entropy:


b.





c.








1.3.3.

Mass Flow




1.3.3.1.

For

mass with varying properties as it crosses the system boundary an integration would have to
be performed



2.

Exergy Destruction



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2.1.

We know at this time because of the second law that irreversibilities destroy our ability to do useful work.

These irreversibilities are


2.1.1.

Friction

2.1.2.

Heat transfer through a finite temperature difference

2.1.3.

Mixing

2.1.4.

Chemical reactions

2.1.4.1.

Combustion

2.1.5.

Unrestrained expansion

2.1.6.

etc


Note: Anything that generates entropy always destroys exergy



or


The Decrease of Exergy Principle:


2.1.7.

Look at combination of first and second laws for an isolated system

2.1.7.1.

First Law:



2.1.7.1.1.

There is no energy flowing in or out of an isolated system, so



2.1.7.2.

Second Law:




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2.1.7.2.1.

There is no entropy flowing in or out of an isolated system, so





2.1.7.3.

Multiply 2
nd

Law by
T
o
:




2.1.7.3.1.

Subtract it from the 1
st

Law:




2.1.7.4.

Exergy change of isolated system:




2.1.7.4.1.

Since
V
2
=V
1

for an isolated system (there can be no volume change)




2.1.7.4.2.

Since

for an isolated system




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2.1.7.5.

And we conclude that



2.1.7.6.

From the 2
nd

Law we know that
, and thus we conclude that





2.2.

For an isolated system the lost exergy is the amount of work capability that we loose




2.2.1.

Gouy Stodola Theorem


2.3.

We can look at

as a statement of the 2
nd

Law. Note
that this is not a new law, it comes from a
combination of the 1
st

and 2
nd

Laws.


3.

Exergy Balances for processes noting that there is exergy destruction


3.1.

Exergy balance on a control mass (Closed System)




20

3.1.1.

Technically, like entropy in the Second Law, exergy

is not conserved. Therefore doing an entropy or
exergy balance might sound a little funny. The reason we can do balances is because of the
S
gen

term in the
case of the 2
nd

Law and because of the
X
destroyed

term in the case of exergy.





or



or






where


3.1.1.1.

This represents the amount of work potential we have lost


3.1.2.

For a cycle




21



3.1.2.1.

Note that there is no change in the exergy within the c
ontrol mass


3.1.2.2.

There is also no change in volume



3.2.

Exergy Balance for USUF (Uniform State Uniform Flow) Control Volume











where


3.2.1.1.

This represents the a
mount of work potential we have lost


3.3.

Exergy Balance for SSSF (Steady State Steady Flow) Control Volume




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where


3.3.1.1.

This represents the work potential
that we have lost


3.4.

For all cases



Irreversible process



Reversible process



Impossible process

4.

Irreversibility


4.1.

Irreversibility (
I
)

-

the difference between the theoretical maximum amount of

useful work and the actual
useful work for a process between a given initial and final state



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for work producing devices



for devices where work is input


4.2.

Note that


4.3.

For devices that seeming
ly produce no work the first expression should be used



4.4.

The irreversibility for different processes is



control mass


SSSF


USUF


4.4.1.

This represents our lost work potential, that is the amoun
t of work that is destroyed because of
irreversibilities

5.

Reversible Work


5.1.

Reversible work

(
W
rev
)
-

the theoretical maximum amount of useful work a system can do operating
between two given states.


5.2.

For processes that absorb work the maximum work correspond
s to the minimum work input



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5.3.

In order to get the maximum amount of work from the system the process has to be reversible


5.4.

This is different than the exergy because we are now looking at the process between two arbitrary state
points. The process is still r
eversible but there are two given state points.


5.5.

Reversible work is a function of both of these state points and therefore it is not a property (point function),
but a path function. Unlike exergy which can be considered a point function because the dead
state is defined.


5.6.

Equation for the reversible work can be obtained from those for an exergy change between state 1 and 2 by
setting the exergy destruction term equal to zero and solving for the work.


5.6.1.

Control Mass




5.6.2.

SSSF



5.6.3.

USUF






25

5.6.4.

Note that all of these equations are written for the normal system. For the most part we will be using
these equations for a system that includes the normal system and the immediate surroundings. This needs to
be done
because we do not know the boundary temperatures as a function of the process time.


5.7.

The system plus surroundings equations become


5.7.1.

Control Mass




5.7.1.1.

The volume change buried in the exergy change cancels out for the extended system


5.7.1.2.

Many
times we ignore the exergy change of the immediate surroundings


5.7.1.3.

This is the same thing we did with the Second Law for the entropy change of the immediate
surroundings


5.7.2.

SSSF


5.7.3.

USUF






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5.7.3.1.

The volume change cancels out for

the extended system

5.7.3.2.

Many times we ignore the exergy change of the surroundings

6.

Second Law Efficiency


6.1.

Example: Two Heat Engines with the same First Law efficiency







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Comparison of second law efficiencies for two h
eat engines

6.1.1.

While both these engines have the same thermal efficiency one of them is actually designed better than
the other.


6.1.2.

The one utilizing the lower high temperature reservoir is actually converting more of the available
exergy of the reservoir to wo
rk than the one operating with the higher temperature reservoir



6.2.

First law efficiencies just tell us how much of the total energy available is being converted into the energy
form that we want.


6.2.1.

These are often called conversion efficiencies


Q
H

W

HE

T
L
= 300 K

T
H
= 10
00 R

Q
L

Q
H

W

HE

T
L
= 300 K

T
H
= 5
00 R

Q
L

High Temperature Heat Engine

Low Temperature Heat Engine



28

6.2.2.

The first la
w efficiencies don’t recognize that the 2
nd

Law puts theoretical limits on how much heat
can be transformed into work



6.2.3.

1
st

Law efficiency (thermal efficiency) is not really a fair comparison because the 2
nd

Law says we can
never convert all heat to work i
n a cyclic device. A better comparison is a 2
nd

Law efficiency.


6.3.

Second Law Efficiency,

II




6.3.1.

There is some disagreement as to what the exergy supplied and the exergy recovered are in certain
cases


6.3.2.

Note that exergy supplied and recovered comes in different forms



6.3.2.1.

heat

6.3.2.2.

work

6.3.2.3.

PE

6.3.2.4.

KE

6.3.2.5.

internal energy



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6.3.2.6.

enthalpy

6.3.2.7.

flow work


6.3.3.

Note

that Second Law efficiencies are different than the process or device efficiencies described in
the review

6.3.3.1.

Process efficiencies used ideal process with different states than the actual process


6.3.3.2.

Second Law efficiencies use the exact same initial and final
states in the ideal process that are
used in the actual process


6.3.4.

Note that the Second Law efficiency’s definition holds for both cycles and processes


6.3.4.1.

This is nice because you do not have to have different definitions


6.3.4.2.

This is also nice because you now can

make comparisons between cycles and devices


6.4.

Specific Definitions of Second Law Efficiencies,

II


6.4.1.

Work producing devices:




6.4.1.1.

For a cyclic heat engine:


6.4.2.

Work Consuming Devices



30


6.4.3.




6.4.3.1.

For a cyclic
refrigerator or heat pump:

7.

Hemholtz function and Gibbs function


7.1.

In many of these equations the combination of the properties


and


have appeared.


7.1.1.

This combination of properties have s
pecial names



Hemholtz function


Gibbs function


7.1.2.

These combinations of properties show up in other situations in thermodynamics. In particular in
regards to chemical reactions