IE 211
INTRODUCTION TO ENGINEERING THERMODYNAMICS
Chapter 7
‘’Entropy’’
Clausius Inequality
Combined system
(one normal system+ one cyclic device)
dEc= QR-Wc or E=QR-Wc
Energy balance for CLOSED SYSTEM;
Where,
Wc(total work of the combined system)=Wrev+ Wsys
If cyclic device is reversible;
(QR/ Q)rev.=TR/T
dEc = (TR/T)Q - Wc
Wc = (TR/T)Q - dEc
Suppose that the system undergoes a cycle while cyclic device undergoes integral
number of cycles;(dEc=0)
Wc = (TR/T)Q Wc= TR ∫(Q/T)
Net work for the combined cycle
Kelvin-Planck Statement of the second law; No system can produce a net
amount of work while operating in a cycle and exchanging heat with a single
thermal energy reservoir.
Wc cannot be a work output, and this cannot be a positive quantity
∫ (Q/T) ≤ 0 CLAUSIUS INEQUALITY
(Valid for all thermodynamic cycles, reversible or irreversible, including ref. cycles)
If combined system is internally reversible;
In the reversed cycle case all the quantities have the same magnitude
but the opposite sign
Since Wc cannot be positive quantity in the regular case, cannot be a
negative quantity in the reversed case;
Wc,int.rev. = 0 ∫ (Q/T)int.rev. = 0
A quantity whose cyclic integral is zero depends on the state not
the process path, thus it is property
ENTROPY: dS = (Q/T)int.rev (kJ/K)
ENTROPY CHANGE DURING A PROCESS
S= S2-S1= ∫ (Q/T)int.rev.1
2
Total entropy change may be found;
by performing integration (by writing heat in terms of
temperature)
Using tabulated data
(entropy of a substance may be assigned to zero at some arbitrarily reference point)
Total entropy change(S) between two specified states is the same no
matter what path, reversible or irreversible
1
2
reversible
irreversible
T
S(kJ/K)
The term ∫(Q/T) ; gives us entropy only if internally reversible path is followed
gives us different values for various irreversible paths and so it is not
a property for irreversible paths
Because of that , for irreversible processes integration must be performed along
imaginary internally reversible path.
A Special Case: Int. Rev. Isothermal Heat Transfer Process
T=To=const.
S= S2-S1= ∫ (Q/T)int.rev.= ∫ (Q/To)int.rev.= (1/To)∫ (Q)int.rev
S= Qint.rev./To (particularly used for the entropy changes of thermal energy reservoirs)
1
2
1
2
1
2
Entropy change of a system during an int. rev. isothermal process can be
positive or negative depending on the direction of heat transfer;
Heat transfer to the system increases the entropy of the system
Heat transfer from the system decreases the entropy of the system
THE INCREASE OF ENTROPY PRINCIPLE
A CYCLE;
∫ (Q/T) ≤ 0 CLAUSIUS INEQUALITY
Sfinal – Sinitial = S1-S2
∫ (Q/T) + ∫ (Q/T)int.rev. ≤ 0 1
2
2
1
S2-S1 ≥ ∫ (Q/T) dS ≥ Q/T1
2
Thermodynamic temperature
scale at the boundary
For Internally reversible process; For irreversible process;
dS = Q/T dS>Q/T
Entropy change (S)of a closed system during an irreversible process is greater
than the integral of Q/T
Some entropy is generated or created during an irreversible process due to
irreversibilities
ENTROPY GENERATION
Entropy change of a closed system for
irreversible processdS>Q/T S > ∫ (Q/T )
1
2
S = Sgen. + ∫ (Q/T ) S = Sgen. ≥ 01
2
For closed system (irreversible); For adiabatic closed system (irreversible);
Entropy generation;
1) Sgen. ≥ 0 (always)
2) Value depends on process, it is not a property of the system
‘’ Entropy of an isolated system during a process always increases or, in the
limiting case of reversible process, remains constant’’ (never decreases)
Increase of entropy principle
Entropy change of an isolated system is equal
to entropy generation (since no entropy transfer);
Sgen. = Stotal = Ssys. + Ssurr. 0
Sgen. = Stotal = Ssys. + Ssurr. = 0 Reversible process
Sgen. = Stotal = Ssys. + Ssurr. > 0 Irreversible process
Sgen. = Stotal = Ssys. + Ssurr. < 0 Impossible process
Entropy of universe Since no actual process is reversible, we can conclude that some entropy is generated during
a process, there the entropy of universe (isolated sys.) is continuously increasing.
entropy is a measure of the disorder in the universe
Entropy change and entropy generation
The entropy change of a system can be negative during a process, but entropy generation
cannot
SOME REMARKS ABOUT ENTROPY
(1) A process can occur in a direction only, not in any
direction(direction in which entropy increases, Sgen. ≥ 0)
(2) Entropy is nonconserved property(entropy is only conserved during reversible processes and increases during
all actual processes)
(3) Entropy generation is a measure of the magnitudes
of the irreversibilities present during a process
Sgen. α irreversibilities
Can be used as quantitative measure of irreversibilities and to establish criteria
for the performance of engineering devices
ENTROPY CHANGE OF PURE SUBSTANCES
Using a suitable reference state, the entropies of substances are evaluated
In tables entropy values are given relative to an arbitrary reference state
- In steam tables (entropy of saturated liquid sf at 0.01oC is assigned the value of zero)
- For refrigerant 134-a (sf=0 for saturated liquid at -40oC)
ISENTROPIC PROCESSES
Entropy of a fixed mass can be changed by;
(1) Heat transfer
(2) Irreversibilities
Entropy of a fixed mass doesn’t change during a process that is;
- Internally reversible and adiabatic
Isentropic process: A process during which
the entropy remains constant, (S=0)
e.g. Pumps, turbines, nozzles and diffusers(operate generally under adiabatic conditions)
All reversible adiabatic processes are isentropic but the reverse is not correct.
However, the term isentropic is used to imply an internally reversible, adiabatic
process.
PROPERTY DIAGRAMS INVOLVING ENTROPY
T-S and h-s diagrams
Area under T-S diagram
dS = (Q/T)int.rev Qint.rev = TdS
The area has no meaning for irreversible processes
(heat transfer during an internally reversible process)
ISOTHERMAL PROCESS
S= Qint.rev./To Qint.rev.= ToS or qint.rev.= Tos
Qint.rev = ∫ TdS or qint. rev.= ∫ Tds 1
2
1
2
T-s diagrams h-s diagrams(Mollier diagram)
q=0
Valuable tools for visualizing the second-
law aspects of processes and cycles
Commonly used in engineering
devices in the analysis of steady-
flow devices such as turbines,
compressors, and nozzles
Example:
THE T dS RELATIONS
CLOSED SYSTEM
-Containing a simple compressible substance
- Internally reversible process
dE = dU = Qint.rev. - Wint.rev. out
TdS PdV
TdS = dU + PdV or Tds = du + Pdv First TdS or Gibbs equations
Since h = u + Pv dh = du + Pdv + vdP
Tds
Tds = dh - vdP Second TdS equation
Both equations are valid for reversible or irreversible processes since entropy
does not depend on path followed
or
For ideal gases;
du = CV dT
dh = CP dT
Pv = RT
Tds = du + Pdv Tds = dh - vdP
ENTROPY CHANGE OF LIQUIDS AND SOLIDS
Liquids and solids can be approximated as incompressible substances
(their specific volumes remain nearly constant during a process, dv 0, CPCV C)
0
ds = du/T + Pdv/T (du = C(T)dT)
In some cases, for large temp. difference in which volume change is high
volume change term should be included.
ISENTROPIC PROCESSES OF LIQUIDS AND SOLIDS
Isentropic processes of liquids and solids are also isothermal
ENTROPY CHANGE OF IDEAL GASES
ds = du/T + Pdv/T (Pv = RT)
Cv,ave. dT/T
CP,ave. dT/T
entropy change can also be written using dh;
ds = dh/T – vdP/T
The specific heats of ideal gases, with the exception of monoatomic gases (Ar, He),
depend on temperature. However, in many cases, constant or average specific heats are
used.
1. Constant Specific Heats
When the temperature change during a process is large average or constant
specific heats cannot be used anymore in calculating the entropy change
Specific heats must be taken as a function of temperature; CP(T), CV(T)
Instead of performing the integrals each time, absolute zero (0 K) is chosen as
the reference temp. and so function is defined;
2. Variable Specific Heats (Exact Analysis)
ISENTROPIC PROCESSES OF IDEAL GASES
s = 0 = s2-s1
1. Constant Specific Heats (Approximate Analysis), IDEAL GASES;
= 0
OR
First isentropic relation
Where;
= 0
Second isentropic relation
By substituting Eq.2 into Eq. 1 third relation is obtained;
Third isentropic relation
2. Variable Specific Heats, IDEAL GASES;
=
2.a.Relative Pressure
The quantity exp(so/R) is called relative pressure, Pr
The relative pressure, Pr, depends only on temperature, since so depends
only on temperature
Pr data is tabulated
against temperature
2.b. Relative Volume
When automative engines are analysed, specific volume ratios are
given instead of pressure ratios.
The quantity T/Pr is called relative specific volume, Vr
REVERSIBLE STEADY-FLOW DEVICES
Reversible work relations for steady flow and closed systems;
Energy balance for steady flow device undergoing an internally reversible process;
When the changes in k.e and p.e. are negligible;
Reversible work output associated with an internally
reversible process in steady-flow device
To avoid negative sign in work inputs of devices such as compressors and
pumps, work term is multiplied by a minus sign;
If fluid in steady flow device is incompressible v remains constant
For steady flow devices such as NOZZLE and PIPE, the work term is zero,
wrev =0
2
called Bernoulli Equation in Fluid Mechanics
Implications of for work producing devices;
The larger v, the larger the wrev produced or consumed by the steady flow
device
During compression;
specific volume should be kept minimum to minimize the work input
During expansion;
specific volume should be maximum to maximize the work output
STEAM POWER PLANTS;
Pump handles liquid (specific volume is small)
Turbine handles vapor (high specific volume)
MINIMIZING THE COMPRESSOR WORK
Work input is minimized when the process
is executed in an internally reversible mannerwin = wrev,int
(when k.e and p.e. are negligible)
Compressor work may be minimized;
(1) By minimizing the irriversibilities (friction, turbulence, and nonquasi-eqm. comp.)
(2) Keeping the specific volume of the gas as small as possible during comp.
Redusing the work input to a compressor requires that
the gas be cooled as it is compressed
COMPARE WORK REQUIREMENTS FOR THREE KINDS OF PROCESSES;
(A)An isentropic process (involves no cooling, Q=0; PVk=const.)
(B)A polytropic process (involves some cooling, PVn=const.)
(C)An isothermal process (involves maximum cooling; PV=const.)
Work requirements;
wC,rev < wB,rev < wA,rev
Assume that all these processes are reversible and executed between the same pressure levels;
If sufficient heat is removed during compression, the value of n approaches
unity and the process becomes isothermal
ENTROPY BALANCE
The Second Law of Thermodynamics states that entropy can be created but
it cannot be destroyed;
The entropy change of a system during a process is equal to the net entropy transfer
through the system boundary and the entropy generated within the system.
ENTROPY CHANGE OF A SYSTEM
Ssys.= 0 for steady-flow devices such as nozzles, compressors, turbines, pumps, and heat exchangers
When the system properties is not uniform;
MECHANISMS of ENTROPY TRANSFER, Sin and Sout
Entropy can be transferred to or from a system by;
(1) Heat transfer
(2) Mass flow
(1) Heat Transfer
Heat is a form of disorganized energy, and some disorganization(entropy) will
flow with heat
1.a.Constant Temperature:
1.b. Temperature is not constant:
Qk : heat transfer through the boundary at
constant temperature Tk at location k.
Entropy transfer between contacting two bodies;
Heat flow from warmer body to cooler one
Entropy transfer from warmer to cooler body
No entropy is created or destroyed at the boundary
T2>T1
T2 T1
Q
Energy is transferred by both heat and work, whereas entropy is
transferred only by heat, not by work
No entropy is exchanged during a work interaction between a system
and its surroundings
(2) Mass Flow
Mass contains entropy as well as energy
Both entropy and energy are carried into or out of a system by flow of matter
When the properties of mass change with time;
ENTROPY GENERATION, Sgen
Irreversibilities such as;Friction
Mixing
Chemical reactions
Heat transfer through a finite temp. difference
Unrestrained expansion
Non-quasi eqm. compression or expansion
Increases the entropy of a system
Sgen. is a measure of the entropy created by irreversibilities
Sgen. represents the entropy generation only within the system boundary
For INTERNALLY REVERSIBLE (No irreversibilities) PROCESS;
Sgen. = 0 Entropy change of the system is equal to the entropy transfer
Entropy Balance for any system undergoing any process;
Entropy balance in rate form;
Entropy balance on a unit-mass basis;
Closed System
No mass flow
Entropy change is due to the entropy transfer accompanying heat transfer and
the entropy generation
For adiabatic closed system;
Any closed system and its surroundings can be treated as an adiabatic
system
(If temp. is const.)
Control Volumes
Heat, mass flow across the boundary
In rate form;
STEADY FLOW PROCESS
(for multiple streams)
STEADY FLOW (SINGLE STREAM) DEVICE
If steady flow single stream device is adiabatic; Q=0
Entropy of a fluid increases as it flows through an adiabatic device, Sgen0
If steady flow single stream device is both adiabatic and reversible;
se = si
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