Concepts of Thermodynamics
Prof. Suman Chakraborty Department of Mechanical Engineering
Indian Institute of Technology, Kharagpur
Lecture – 53
Exergy (Availability) Today, we will start with a new chapter which
is titled Exergy or Availability. So, these are very synonymous terms; availability
is a little bit old-fashioned term; exergy is little bit more modernized. So, before
getting into what it is, I will start with an example. Let us say that there is a chamber
in which you have air and fuel. Let us say this is completely insulated from the surroundings.
Then, you transfer heat to this system. It was originally insulated.
Now, you transfer heat to the system so that there is burning of the fuel. So, maybe you
do a sparking and all these things. So, you have still air and fuel, and then, finally,
you have products of combustion. So, you had air-fuel mixture totally isolated from the
surroundings, you removed the isolation and transferred some heat so that there is combustion,
and then you have products of combustion. These three stages we are considering.
So, in all these processes, if you include the heat transfer effect, then energy is conserved,
right? Energy conservation is not violated. But what happens is that here you had air
and fuel with tremendous potential of doing work. Why? Because you could have used this
fuel to run a power plant, you could have used this fuel to drive a car, you could do
this—use this fuel to do many tasks. In other words, the chemical energy associated
with the fuel could be used for many purposes, but now, when this is burnt, the products
of combustion they get formed, but you do not utilise any work out of that, it’s just
a closed volume chamber. No work is extracted from this, eventually, if you leave it to
interact with the surroundings, it will come to a state when it is in equilibrium with
the surroundings, and then, the potential of utilising work from the system has degraded
in this process. So, we have two things: one is energy, another
is the quality of energy. So, energy can be expressed in—or energy can be sort of explained
in two perspectives: one is the quantity of energy, another is the quality of energy.
Quality of energy gives the potential to do work. So, in terms of potential to do work,
out of these three cases which one is maximum? This has the greatest potential of doing work;
this is still less, but still, you know, the combustion is not complete so you could still
extract some work out of this. This, in equilibrium with the surroundings, is at a state where
it has lost all potential of doing work and this is called as thermodynamically dead state,
when it is in equilibrium with the surroundings. So, the quality of energy, although the quantity
of energy remains conserved by taking into account heat transfer and all this, but the
quality of energy has degraded as you go from state 1 to state 2 to state 3. High quality
of energy means high quality of extract—a capability of extracting work out of the system.
Low quality of energy means low capability of extracting work out of the system. So,
this is high work potential, and this goes towards low work potential. So, in other words,
it is a high quality energy degraded to low quality energy, because work potential is
associated with a high quality of energy. Why work is always considered to be a high
grade form of energy or high quality energy? You have—you must have noticed, that in
the second law analysis, there is heat—there is entropy transport associated with heat
transfer, but there is no entropy transport associated with work transfer, right? So,
entropy transfer or entropy transport which includes entropy generation effect, there
when you are considering the net change in entropy, heat transfer get reflected in that,
and there is also an obvious outcome of the second law that 100 percent heat in a cyclic
process you cannot convert to 100 percent work. Therefore, a particular rate of heat
transfer, an equivalent energy transferred in terms of heat doesn’t have the full potential
of getting converted into work. So, it’s energy of a low grade.
Now, this quality of energy or work potential is expressed by a quantification which is
called as exergy or availability. So, to understand this, we have to consider a benchmark. Before
establishing that benchmark, I give you an analogous example which has nothing to do
with thermodynamics, but thermodynamics is such a beautiful subject, it has analogy with
society, economics or any branch of science and humanities.
So, let us say, this is very common to our advanced education system in India. So, if
you consider premier institutes, when students enter the first year of undergraduate studies,
they are at this state. So, tremendous work potential because it’s a highly competitive
environment and students come after a whole lot of screening, going through very rigorous
entrance tests where—I mean, I can definitely say that—I mean, not many countries will
have such a high level of competitive exam through which students have to go through
to enter into, you know, very advanced institutes. So, the students entering the first year of
undergraduate study, tremendous work potential they have. Now, as they go through different
years of studies, you know, people are relaxed and they understand that it is more difficult
to enter into these institutes, but much easier to pass through these institutes. So, they
will go through a process where their entire potential as, you know, as a scholar, as a
scholarly individual, in most of the cases is not exploited, and they—eventually when
they graduate, not all of them, but in many of the cases, you find that the work potential
has become much less than the state at which they have entered.
So, this is a law of nature—I mean it’s not a question that, you know, you can blame
the system or you can blame the individual. What I am trying to impress upon you that
it is a law of nature, that you start with a high potential, and then, because you do
not have—if you do not have a conscious effort of exploiting the potential, then your
potential will automatically degrade, and that is the law of nature. How much the potential
will degrade, you have to establish a benchmark. What is that benchmark? The benchmark is a
reversible process. So, you have to see that, given a set of inputs, what could be the reversible
work that you could extract out of it, and then what is the actual work. The difference
between the reversible work and actual work is the irreversibility which is due to entropy
generation during the process. So, with this broad understanding, we will try to develop
an expression for the reversible work. So, to generalise, we will assume that there
is a control volume but you can consider the control mass system as a special case considering
that there is some inlet, there is some exit and within the control volume, there is state
change of state from 1 to 2. If there is no mass flow across i and e, this control volume
becomes a control mass system. So, control mass system is a special case of this if m
i and m e are 0, okay? So, to have a general description which considers both flow and
non-flow process, I have considered a generic arrangement like this.
What is specified? What is specified is that there is an ambient T 0; in addition to that,
there is a thermal reservoir at a temperature T H from which you are having a transfer of
heat Q H to the system. Okay? Again, what is specified? So, let us encircle what is
specified, this is very important: T H, T naught, Q H, state i, state e, state 1, state
2. So, all the states are fixed. So, given all these states are fixed, the puzzle that
we have to solve is that, given this arrangement, what is the maximum work that we could derive
out of this. That maximum work is the reversible work.
So, one missing link here is that, given that you expose this system to a surrounding—so,
there is a difference between two terminologies: one is called as immediate surrounding and
another is surrounding. So, this one is an immediate surrounding, because there is a
arrangement by which this thermal reservoir directly interacts with the system, but you
have a broader surrounding which is the ambient. So, there is a potential of heat transfer
by virtue of the temperature difference between the ambient and the system boundary. So, that
is not given. So, let us say that, with this T 0, you have
a Q 0. If it is a reversible process, let us call this Q 0 reversible, okay? This is
not something which is given; this is a spontaneous phenomenon because of the temperature difference
between the system and the surrounding, okay? So, this is not given, but this will happen
because of the temperature difference. So, now we will apply the—so, this will have
a work output which is reversible work. So, how do we quantify this reversible work? We
will—we can quantify through first law and second law.
So, first law. You have Q H plus Q 0 reversible, which is not known, plus energy associated
with state i. So, what is energy associated with state i? m i into h i plus V i square
by 2 plus g Z i. So, in short form, we are writing this in this way. So, just for the
first time, I am writing this, next time onwards I will not write this. So, this is m i into
h i plus V i square by 2 plus g Z i. This is equal to E 2 minus E 1 within the control
volume. Now, what is this E? This is for example,
is—I have not completed the equation yet, but I am writing the expression—this is
m 2, this is energy of the control volume, so, it is internal energy and not enthalpy.
Okay? Similarly, E 1 plus E i—sorry, plus E e, plus W rev. Okay? But, how do you guarantee
that this work is reversible work? You can guarantee that this work is reversible work
if there is internal and external reverse reversibility associated with it. So, that
is guaranteed by 0 entropy generation in the entropy transport.
So, second law. So, S 2 minus S 1 plus S e minus S i—these are all capital S, that
means specific entropy into mass, this is the change in entropy—is equal to Q H by
T H plus Q 0 reversible by T 0 plus there is no entropy generation, because it is reversible.
So, if you just want me to write entropy generation I can write, but you straightaway have to
set it as 0. So, what you can do is, see it is a very straight
forward exercise. This Q 0 reversible is not known, so, we eliminate that from these two
equations. So, Q 0 reversible is equal to T 0 into S 2 minus S 1 plus T 0 into S e minus
S i minus Q H into T 0 by T H. Okay? You substitute that in first law. So, Q H plus Q 0 reversible
is this expression, so, you have another term with Q H, so, you will have Q H into 1 minus
T 0 by T H plus E i minus T 0 S i minus E e minus T 0 S e is equal to E 2 minus T 0
S 2 minus E 1 minus T 0 S 1 plus W rev. Right? So, I will try to explain this equation physically,
so, you will get an expression for reversible work, so, if you are interested for that expression,
we will isolate all the terms. So, Q H into 1 minus T 0 by T H plus E i minus T 0 S i
minus E e minus T 0 S e plus E 1 minus T 0 S 1 plus E 2—sorry, minus E 2 minus T 0
S 2, right? Okay? So, see a couple of interesting things. First
of all, this is the maximum work potential of the system which you can exploit because
it’s reversible, there is no loss. So, a part of the work potential is associated with
this heat transfer Q H. And what is the maximum work that you can get? If you fit a Carnot
cycle engine that connects the thermal reservoir one with T H and another with the surrounding,
then this is actually the work done by that Carnot cycle in a cyclic process. So, with
this Q H, Q H—so, this is like Q H into one minus T l by T H, T l is T 0 here.
So, this is a reversible utilisation of this Q H for work and this is the maximum potential
governed by the initial and the final state and inlet and exit state, and you can see
that it is just like the first law expression, but energy is replaced by energy minus T 0
S. So, this, E minus T S, is considered to be a sort of a term called as free energy
which we will discuss later on. But the concept starts from here that you have an energy that
is freely available which is not the difference in E, but the difference in E minus T S, where
the T is the reference temperature of the surroundings. So, this tells you, so, this—these
are the—this totality is the work potential due to the change in state and this is the
work potential due to heat transfer. So, the total maximum work that you can extract is
this one. Now, consider the reality, so, the real process. Real process is not reversible,
right? So, I will keep the same box, but for your notes, please draw a fresh diagram. So, the same box which is the control volume,
you have state i, let me encircle it, same inputs, that is you have heat transfer Q H
to the box, it changes from 1 to 2 and the exit is E, T 0 is there, but the process is
not reversible. Because the process is not reversible, instead of this W reversible,
we will have W actual. Okay? Instead of W reversible, we will have W actual; instead
of Q 0 reversible, we will have Q 0, which is Q 0 actual.
So, then, let us write the first law and the second law. Q H plus Q 0 actual is equal to
plus E i—now I will not write what it is, we have already written—is equal to E 2
minus E 1 plus E e plus W actual. Okay? Again, Q 0 actual
you do not know, so, you have the second law from which you get an expression for this
and you eliminate in the process. So, second law: Q H by—sorry, S 2 minus S 1 plus S
e minus S i is equal to Q H by T H plus Q 0 actual by T 0; now there will be entropy
generation because of irreversibility, right? So, you can eliminate Q 0 actual as T 0 into
S 2 minus S 1 plus S T 0 into S e minus S i minus T 0 into Q H by T H minus T 0 into
entropy generation. Okay? You substitute that here. So, Q H into 1 minus T 0 by T H plus
E i minus T 0 S i is equal to E 2—sorry, then we will have minus T 0 S gen, right?—is
equal to E 2 minus T 0 S 2 minus E 1 minus T 0 S 1, okay?—plus E e—sorry, yes, so—plus
E e minus T 0 S e plus W actual. Right? So, this is one expression where you have
W reversible, this is one expression you have from which you can get W actual. The irreversibility
is defined as the difference between reversible work and
actual work. So, if you subtract these two expression for W actual from W reversible,
all the terms will cancel, you can see, except T 0 into entropy generation. So, this is equal to T 0 into entropy generation,
and this sounds meaningful. This makes sense because the irreversibility is associated
with entropy generation. So, irreversibility, which is a difference between reversible and
actual work, is nothing but the T 0 into the entropy generation. We will stop here in this
lecture; we will continue with this in the next lecture.
Thank you very much.
