Clausius inequality and entropy

This presentation helps understanding the basic concepts of Clausius Inequality and Entropy.

1. Name:-Nishant Narvekar
Branch:-Mechanical-1
EN Roll:-170410119061

2. Clausius statement
•It is impossible to construct a device as heat
pump that operates in a cycle and produces no
effect other than transfer of heat from a lower
temperature body to higher temperature body.
•OR “No process is possible whose removes heat
from a reservoir at lower temperature and
absorbs equal amount of heat by a reservior at a
higher temperature without any external
assistant

4. The Clausius Inequality
• Expressions of inequality/equality relating to heat flow at
a fixed temperature.
• The expression is required for the derivation of an
equation for entropy – which is our next main topic.
• Derived from a “thought experiment” using Carnot
engines acting in a series.

5. ENTROPY

6. OBJECTIVES
•Apply the second law of thermodynamics to processes.
•Define a new property called entropy to quantify the
second-law effects.
•Establish the increase of entropy principle.
•Calculate the entropy changes that take place during
processes for pure substances*, incompressible substances,
and ideal gases.
•* we’re limiting our studies to pure substances (2010)
}

7. ENTROP
Y }Entropy is a somewhat abstract property and it
is difficult to give a physical description of
without considering the microscopic state of
the system
The second law of thermodynamics often leads
to expressions that involve inequalities.
An irreversible heat engine is less efficient than
a reversible one operating between the same
two thermal reservoirs
Another important inequality that has major
consequences in thermodynamics is Clausius
inequality

8. The system considered in the
development of the Clausius inequality.
Clausius inequality is expressed
as;
dQ
T
—
ò
£
0
Clausius realized he had discovered a
new thermodynamic property and
named it entropy with designation of
S. It is defined as;
dS =
æ dQ ö
ç
èT
÷
ø
int, rev
K
(
k
J
)
The entropy change of a system during
a process can be determined by
integrating between the initial and final
states
DS = S
2 - S
1 =
int, rev
K
(
k
J
)
2 æ dQ
ö
ò
1
ç
èT
÷
ø

9. The entropy change
between two specified
states is the same
whether the process is
reversible or
irreversible

10. Tè
ç
æ dQ
öø
÷—
ò
int, rev
= 0
A quantity whose cyclic
integral is zero (i.e. a property
like volume)
The entropy change between two specified states is the same whether the process is reversible or
irreversible
The net change in volume (a property) during a cycle is always zero
Entropy is an extensive property of a system

11. A system and its surrounding form an
isolated system
The entropy change of an
isolated system is the sum of
the entropy changes of its
components, and is never
less than zero.
DS
isolated
³
0
= DS
sys ³
0
gen í
ì>Irreversible
process
S
ï
=
Reversible process
îï
<Impossible
process
The increase of
entropy principle

12. The entropy of a pure substance is
determined from the tables (like other
properties).
Entropy change
DS = mDS = m(s
2 - s
1 )(
kJ
K )

13. During an internally reversible, adiabatic
(isentropic) process, the entropy remains
constant.
Isentropicprocesses
A process during which the entropy
remains constant is called an
isentropic process
The isentropic process appears
as a vertical line segment on a
T-s diagram.
Ds = 0ors
2 - s
1 kg
(
kJ
× K
)

14. ANALYSIS
a. This is a reversible adiabatic (i.e., isentropic)
process, and thus s2 = s1. From the
refrigerant tables (Tables A-11 through A-13),
An insulated cylinder is initially
filled with saturated R-134a vapor
at a specified pressure. The
refrigerant expands in a reversible
manner until the pressure drops to
a specified value. The final
temperature in the cylinder and
the work done by the refrigerant
are to be determined.
SOLUT
ION
sat.
vapor
P
1 = 0.8MPaü
þ
v= v
1 [email protected] MPa
3
= 0.02561
m
kg
ýu
1 = u
[email protected] MPa
s
1 = s
[email protected] MPa
= 246.79
kJ
kg
= 0.91835
kJ
kg ×
K
R-134a
0.8 MPa
0.05 m
3
1.The kinetic and potential energy
changes are negligible.
2.The cylinder is well-insulated and
thus heat transfer is negligible.
3.The thermal energy stored in the
cylinder itself is negligible.
4.The process is stated to be reversible.
ASSUMPTION
S

15. Propertydiagrams involving
entropy
On a T-S diagram, the area under the process curve
represents the heat transfer for internally reversible
processes.
dQ
int, rev = TdS
2
Q
int, rev = ò
1
T dS
d q
int, rev = Tds
2
q
int, rev = ò
1
T ds
Q
int, rev = T
0DS
q
int, rev = T
0Ds

16. Propertydiagrams involving
entropy
For adiabatic steady-flow
devices, the vertical
distance ∆h on an h-s
diagram is a measure of
work, and the horizontal
distance ∆s is a measure of
irreversibility's.
Mollier diagram: The h-s diagram