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