UNIT 3 Thermodynamics
UNIT 3 Thermodynamics
UNIT 3 Thermodynamics
UNIT III
Limitations of the First Law Thermal Reservoir, Heat Engine, Heat pump, Parameters
of performance, Second Law of Thermodynamics, Kelvin-Planck and Clausius
Statements and their Equivalence / Corollaries, PMM of Second kind, Carnots principle,
Carnot cycle and its specialties, Thermodynamic scale of Temperature,
Clausius Inequality, Entropy, Principle of Entropy Increase Energy Equation, Availability
and Irreversibility Thermodynamic Potentials, Gibbs and Helmholtz Functions, Maxwell
Relations Elementary Treatment of Third Law of Thermodynamics
Limitations of first law
The first law of thermodynamics has its own limitations
in actual practice. Some situations are given below.
1. According to the first law of thermodynamics, heat and work
are mutually convertible .complete conversion is not possible in
real practice.
2. According to the first law of thermodynamics, there is no
restriction on the direction of flow of work and heat, which is
not true.
3. According to the first law of thermodynamics, in energy
cyclic process work and heat are exchangeable completely but
from experience it is not.
4. In natural way heat is not completely converted to work, but
reverse is not automatically true.
5. Heat flows from hot to cold region, but reverse is not
automatically true.
6. high pressure gas expands to low pressure but reverse is not
atomatically true.
7. from the above cases some external source of energy is
required for reverse processto occur which again violates the
1st law of thermodynamics.
8.Joules experiment amply demonstrate that energy, when supplied to a
system in the form of work, can be completely converted into heat(work
transfer -+ internal energy increase -+ heat transfer). But the complete
conversion of heat into work in a cycle is not possible. So heat and work are
not completely interchangeable forms of energy
. When work is converted into heat, we always have
but when heat is converted into work in a complete closed cycle process
ENERGY RESERVOIRS
A thermal energy reservoir (TER) is defined as a large body of infinite
heat capacity. which is capable of absorbing or rejecting an unlimited quantity
of heat without suffering appreciable changes in its thermodynamic coordinates.
The changes that do take place in the large body as heat enters or leaves
are so very slow and so very minute that all processes within it are quasi-static.
The thermal energy reservoir TERH from which heat QI is transferred to
the system operating in a heat engine cycle is called the source.
The thermal energy reservoir TERL to which heat Q2 is rejected from the
system during a cycle is the sink.
A typical source is a constant temperature furnace where fuel is
continuously burnt, and a typical sink is a river or sea or the atmosphere itself.
A mechanical energy reservoir (MER) is a large body enclosed by an
adiabatic impermeable wall capable of storing work as potential energy (such as
a raised weight or wound spring) or kinetic energy (such as a rotating flywheel).
All processes of interest within an MER are essentially quasi-static. An MER
receives and delivers mechanical energy quasi-statically,
If Q2 =0, the heat engine will produce net work in a complete cycle
by exchanging heat with only one reservoir, thus violating the Kelvin-Planck
statement fig 1 Such a heat engine is called a perpetual motion machine of the
second kind, abbreviated to PMM2. A PMM2 is impossible
.A heat engine has, therefore, to exchange heat with two thermal energy
reservoirs at two different temperatures to produce net work in a complete cycle
fig 2 .So long as there is a difference in temperature, motive power(i.e. work)
can be produced. If the bodies with which the heat engine exchange heat are of
finite heat capacities, work will be produced by the heat engine till the
temperatures of the two bodies are equalized.
FIG 1 A PMM2
If the second law were not true, it would be possible to drive a ship
across theocean by extracting heat from the ocean or to run a power plant by
extracting heat from the surrounding air. Neither of these impossibilities
violates the first law of thermodynamics. Both the ocean and the surrounding air
contain an enormous store of internal energy, which, in principle, may be
extracted in the form of a flow of beat. There is nothing in the first law to
preclude the possibility of converting this heat completely into work. The
second law is, therefore, a separate law of nature, and not a deduction of the
first law ".The first law denies the possibility of creating or destroying energy;
the second denies the possibility of utilizing energy in a particular way. The
continual operation of a machine that creates its own energy and thus violates
the first law is called the PMMI. The operation of a machine that utilizes the
internal energy of only one TER, thus violating the second law, is called the
PMM2.
Clausius Statement of the second law
..3
EQUIVALENCE
STATEMENTS :
OF
KELVIN
PLANCK
AND
CLAUSIUS
Let us assume a cyclic heat pump (P) extracting heat Q2 from a low
temperature reservoir at t2 and discharging heat to the high temperature
reservoir at t1 with the expenditure of work W equal to what the PMM2
delivers in a complete cycle. So E and P together constitute a heat pump
working in cycles and producing the sole effect of transferring heat from a
lower to a higher temperature body, thus violating the Clausius statement.
A cyclic heat engine operating on the Carnot cycle is called a Carnot heat
engine.
Fig 3 two cyclic heat engine Ea and Eb operating between the same source and
sink, of which Eb is revesible
Therefore,
Now, let Eb be reversed. Since Eb is a reversible heat engine, the magnitudes
of heat and work transfer quantities will remain the same, but their directions
will be reversed, as shown in Fig.4. Since Wa> Wb, some part of WA (equal to
(Wa) may be fed to drive the reversed heat engine
Fig 4 Eb reversed
Since Q1 A = Q1b= Q1 the heat discharged by
source may, therefore, be eliminated (Fig. 6.27). The net result is thatEA and3B
together constitute a heat engine which, operating in a cycle, produces net work
WA - W0' while exchanging heat with a single reservoir at (2' This violates the
Kelvin-Planck statement of the second law. Hence the assumption that
is
wrong.
Therefore
Therefore
Since the efficiencies of all reversible heat engines operating between the same
heat reservoirs are the same, the efficiency of a reversible engine is independent
of the nature or amount of the working substance undergoing the cycle.
Absolute Thermodynamic Temperature Scale :
The efficiency of any heat engine cycle receiving heatQI and rejecting heat
Q2 is
given by
.1
..4
The temperatures
dependsonly on t1. and t2, and is independent of t3.So t3 will drop out from the
ratio on the right in equation 4. After it has been cancelled, the numerator can be
writtenas
and the denominator as
where is another unknown
function.
Thus
Since
as proposed by Kelvin.
Then, by definition
5
can be measured. In defining the Kelvin temperature scale also, the triple point
of water is taken as the standard reference point. For a Carnot engine operating
between reservoirs at temperatures T and Tt, Tt being the triple point of water
fig 2, arbitrarily assigned the value 273.16 K,
6
If this equation is compared it is seen that in the Kelvin scale. Q plays the role
of thermometric property. The amount of heat supply Q changes with change in
temperature.just like the thermal emf in a thermocouple.
It follows from the Eq6
that the heat transferred isothermally between the given adiabatics decreases as
the temperature decreases. Conversely, the smaller the value of Q, the lower
the corresponding T. The smallest possible value of Q is zero, and the
corresponding T is absolute zero. Thus, if a system undergoes a reversible
isothermal process without transfer of heat, the temperature at which this
process takes place is called the. absolute zero. Thus, at absolute zero. an
isotherm and an adiabatic are identical.
That the absolute thermodynamic temperature scale has a definite zero
point can be shown by imagining a series of re versible engines, extending from
a source at T1 to lower temperatures fig 3
Since
fig 3
between the steam point and the ice point could be realized by a series of one
hundred Carnot engines operating as in Fig.3. Such a scale would be
independent of the working substance.
If enough engines are placed in series to make the total work output equal to
Q1, then by the first law the heat rejected from the last engine will be zero. By
the second law, however, the operation of a cyclic heat engine with zero heat
rejection cannot be achieved, although it may be approached as a limit. When
the heat rejected approaches zero, the temperature of heat rejection also
approaches zero as a limit. Thus it appears that a definite zero point exists on
the absolute
temperature scale but this point cannot be reached without a violation 0f the
second law.
ENTROPY
First law of thermodynamics was stated in teerms of cycles. First ,and it was
shown that
Q= w
When the first law was applied for thermodynamic processes, the existence of a
property, the internal energy, was found.
Similarly, the second law was also first stated in terms of cycles executed by
systems. When applied to processes, the second law also leads to the definition
ofa new property, known as entropy. If the first law is said to be the law of
internal energy, then second law may be stated to be the law of entropy. In fact,
thermodynamics is the study of three E's, namely ,energy, equilibrium and
entropy.
Two Reversible Adiabatic Paths Cannot Intersect Each Other
Fig 2
..1
Heat transferred in the process i-f is equal to the heat transferred in the
isothermal process a-b.
Thus any reversible path may be substituted by a reversible zigzag path,
between the same end states, consisting of a reversible adiabatic followed by a
reversible isotherm and then by a reversible adiabatic, such that the heat
transferred during the isothermal process is the same as that transferred during
the original process.
Let a smooth closed curve representing a reversible cycle (Fig. 3) be
considered. Let the closed cycle be divided into a large number of strips by
means of reversible adiabatics. Each strip may be closed at the top and bottom
by reversible isotherms.
The original closed cycle is thus replaced by a zigzag closed path consisting of
alternate adiabatic and isothermal processes, such that the heat transferred
during all the isothermal processes is equal to the heat transferred in the original
cycle. Thus the original cycle is replaced by a large number of Carnot cycles. If
the 'adiabatics are close to one another and the number ofCarnot cycles is large.
the saw-toothed zigzag line will coincide with the original cycle.
For the cycle abcd,
heat is
rejected reversibly at T2
Fig 3
If similar equations are written for all the elemental Carnot cycles, then for
the whole original cycle
3
as Clausiustheorem. The letter R emphasizes the fact that the equation is valid
only for a reversible cycle.
CLAUSIUS INEQUALITY:
Let us consider a cycle ABCD (Fig.1). Let AB be a general process, either
reversible or irreversible, while the other processes in the cycle are reversible.
Let the cycle be divided into a number of elementary cycles, as shown. For one
of these elementary cycles
Fig 1
Now, the efficiency of a general cycle will be equal to or less than the
efficiency of a reversible cycle.
.1
......2
Since entropy is a property and the cyclic integral of any property is zero
..3
Entropy Principle
For any infinitesimal process undergone by a system, we have Eq
for the total mass
For an isolated system which does not undergo any energy interaction with
the surroundings,
Therefore, for an isolated system
An isolated system can always be formed by including any system and its
surroundings within a single boundary (Fig. 1). Sometimes the original
system which is then only a part of the isolated system is called a 'subsystem'.
Fig 1
The system and the surroundings together (the universe or the isolated
system) include everything which is affected by the process. For all possible
processes that a system in the given surroundings can undergo
Entropy may decrease locally at some region within the isolated system, but
it must be compensated by a greater increase of entropy somewhere within the
system so that the net effect of an irreversible process is an entropy increase
ofthe whole system. The entropy increase of an isolated system is a measure of
the extent of irreversibility of the process undergone by the system.
Rudolf Clausius summarized the first and second laws of thermodynamics in
the following words:
(a) Die Energie der Welt ist Constant.
(b) Die Entropie der Welt strebt einem Maximum zu.
[(a) The energy of the world (universe) is constant.
(b) The entropy of the world tends towards a maximum.]
The entropy of an isolated system always increases and becomes a maximum
at the state of equilibrium. If the entropy of an isolated system varies with some
parameter x, then there is a certain value of xe. which maximizes the entropy
and represents the equilibrium stateThe system is .
then said to exist at the peak of the entropy hill, and dS = O. When the system is
at equilibrium. any conceivable change in entropy would be zero.
Fig 1
The rod connecting the reservoirs suffers no entropy change
. because, once in the steady state, its coordinates do not change.
Therefore, for the isolated system comprising the reservoirs and the rod, and
since entropy is an additive property
since
If
negative
is
positive, and the entropy of the system increases. When heat is removed from
the system, dQ is negative, and the entropy of the system decreases.
Heat transferred to the system of fixed mass increases the internal energy of
the system, as a result of which the molecules (of a gas) move with higher
kinetic energy and collide more frequently, and so the disorder in the system
increases. Heat is thus regarded as disorganised or disordered energy transfer
which increases molecular state. If heat Q flows reversibly
from the system to the surroundings at (To) fig ,the entropy increase of the
surroundings is
The temperature of the boundary where heat transfer occurs is the constant
temperature To. It may be said that the system has lost entropy to the
surroundings. Alternatively, one may state that the surroundings have gained
entropy from the system. Therefore, there is entropy transfer from the system
to the surroundings along with heat flow. In other words, since the heat inflow
increases the molecular disorder, there is flow of disorder along with heat. The
sign of entropy transfer is the same as the sign of heat transfer: positive, if into
the system, and negative, if out of the system.
where A and B are reversible processes and C is an irreversible process. For the
reversible cycle consisting of A and B
The equality sign holds good for a reversible process and the inequality sign
for an irreversible process.
AVAILABLE ENREGY , AVAILABILITY & IRREVERSIBILITY:
Available energy is also called exergy . Unavailable energy is also called anergy
The sources of energy can be divided into two groups, viz. high grade energy
and low grade energy.
High grade energy
Low grade energy
The conversion of high grade energy Conversion of low grade energy is
to shaft work is exempted
subject to limitations of 2nd law
from the limitations of the second law
Ex:
Ex:
Mechanical work, electrical work,
Heat or thermal energy , Heat derived
water power, wind power, kinetic
from nuclear fission or fusion, Heat
energy of a jet, tidal power
derived from combustion of fossil
fuels
The bulk of the high grade energy in the form of mechanical work or electrical
energy is obtained from sources of low grade energy, such as fuels, through the
medium of the cyclic heat engine. The complete conversion of low grade
energy,heat, into high grade energy, shaft-work, is impossible by virtue of the
second law of thermodynamics. That part of the low grade energy which is
available for conversion is referred to as available energy, while the part which,
according to the second law, must be rejected, is known as unavailable energy.
The originator of availability concept is Josiah Willard Gibbs. He indicated that
environment plays an important part in evaluating the available energy.
Available Energy referred to a Cycle
The maximum work output obtainable from a certain heat input in a cyclic heat
engine (Fig. .1) is called the available energy (A.E.), or the available' part of the
energy supplied. The minimum energy that has to be rejected to the sink by the
second law is called the unavailable energy (U.E), or the unavailable part of the
energy supplied.
Therefore,
Fig 1
fig 2
For the heat engine receiving heat for the whole process x-y, and rejecting heat
at To
1
The unavailable energy is thus the
product of the lowest temperature of heat
rejection, and the change of entropy of the system during the process of
supplying heat (fig 3)
Fig 3
Reversible Work by an Open System Exchanging Heat only with the
Surroundings
Let us consider an open system exchanging energy only with the surroundings
at
For the maximum work, the process must be entirely reversible. There is a
temperature difference between the control volume and the surroundings. To
make the heat transfer process reversible, let us assume a reversible heat engine
E operating between the two. Again, the temperature of the fluid in the control
volume may be different at different points. It is assumed that heat transfer
occurs at points of the control surface where the temperature is T. Thus in an
infinitesimal reversible process an amount of heat
is absorbed by the engine
E from the surroundings at temperature
is rejected by
the engine reversibly to the system where the temperature is T, and an amount
of work
is done by the engine. For a reversible engine,
.2
The work
flow. When
be positive.
Now, since the process is reversible, the entropy change of the system will be
equal to the net entropy transfer, and
Therefore,
3
.4
5
Substituting eq 1 for
in eq.5
is
.6
from eq.3
.7
Eqn 7 is the general expression for the maximum work of an open system which
exchanges heat only with the surroundings at
THERMODYNAMIC POTENTIAlS
The thermodynamical state of a system can be described in terms of the
basic independent coordinates p,v,T & s. These four coordinates are insufficient
to obtain complete knowledge of the system we use certain energy terms which
are easily measurable, known as thermodynamics potentials.
Thermodynamic energy potentials are energy functions that are
mathematically formed by combining basic thermodynamic coordinates p,v,T&
s in different ways .
dh= Tds+vdp .. 8
Tds =dh vdp
Helmholt Free energy (f):
Consider a real isothermal process at constant volume which is always
irreversible .The entropy increses in an irreversible process never decreases.
Therefore ds dQ/T
Tds dQ
d(Ts) dQ ( since T is constant)
d(Ts) du + pdv
d(Ts) du pdv.9
Since isothermal process is taking place at constant volume ,
Therefore d(Ts-u) 0
d(u-Ts) 0 10
u-ts is thus a physical quantity representing energy which never increses during
an Isothermal iso volumic process.
This u-Ts is called Helmholtz free energy or Helmholtz free energy function f
this f= u-Ts
df =du-(Tds+sdT)
df= -pdv-sdt
df=-pdv-sdt
Gibbs free energy:
Considering a real isothermal process at constant pressure (isobaric) which is
always irreversible , from eqn 9
d(Ts) du d(pv)
d (Ts-u-pv) 0
d(u+pv-Ts) 0
this physical quantity representing energy which never increases in a isothermal
process isobaric process is called Gibbs free energy (g)
g=u+pv-Ts
If the initial and final equilibrium states of the system are at the same pressure
and temperature of the surroundings, say
Then,
11
Then for two equilibrium states at the same pressure P and temperature T
..13
From eqs 12 &13
..14
...15
The decrease in the Gibbs function of a system sets an upper limit to the work
that can be performed, exclusive of pdV work, in any process between two
equilibrium states at the same temperature and pressure, provided the system
exchanges heat only with the environment which is at the same temperature and
pressure as the end states of the system If theprocess is irreversible, the useful
work is less than the maximum.
MAXWELLS EQUATION :
These equations are differential relations among the basic thermodynamic
coordinates from thermodynamics potentials