Entropy Change

P M V SubbaraoProfessor

Mechanical Engineering Department

A Single Reason for Every Thing That Happens!!!

The Thermodynamics of Temperature Creation

• The Gibbsian equation,defines the change in specific entropy of any substance during any reversible process.

vdpdhpdvduTds • Consider a control mass

executing a constant volume process:

pdvduTds

constant

vs

uT

The relative change in internal energy of a control mass w.r.t. change in entropy at constant volume is called as absolute

temperature.

The Thermodynamics of Temperature Creation

vdpdhTds

• Consider a control volume executing a reversible constant pressure process:

constant

ps

hT

The relative change in enthalpy of a control volume w.r.t. change in entropy at constant pressure is called as absolute

temperature.

Entropy change of an ideal gas

• From the Gibbsian equations, the change of entropy of a substance can be expressed as

dPT

v

T

dhdsdv

T

P

T

duds or

For an ideal gas, u=u(T) and h=h(T), du=cv(T)dT and dh=cp(T)dT and Pv=RT

dP

T

v

T

dTTCdsdv

T

P

T

dTTcds pv or

By Integration, the change in the entropy is

1

22

1

12 lnv

vR

T

dTTcss v

1

22

1

12 lnp

pR

T

dTTcss p

or

Ideal Gas with constant specific heats

• When specific heats are constant (calorically perfect gas), the integration can be simplified:

• If a process is isentropic (that is adiabatic and reversible), ds=0, s1=s2,

1

2

1

212 lnln

p

pR

T

Tcss p

1

2

1

212 lnln

v

vR

T

Tcss v

Isentropic Process with idea gas

0lnln1

2

1

2

p

pR

T

Tcp0lnln

1

2

1

2

v

vR

T

Tcv

1

2

1

2 lnlnv

vR

T

Tcv

1

2

1

2 lnlnp

pR

T

Tcp

1

2

1

2 lnlnv

vcc

T

Tc pvv

1

2

1

2 lnlnp

pcc

T

Tc vpp

1

2

1

2 ln1lnv

v

c

c

T

T

v

p

1

2

1

2 ln1lnp

p

c

c

T

Tc

p

vp

1

2

1

2 ln1lnv

v

T

T

1

2

1

2 ln1

1lnp

p

T

T

1

1

2

1

2

v

v

T

T

1

1

2

1

2

p

p

T

T

1

1

2

1

1

2

p

p

v

v

1

1

2

1

2

p

p

v

v

1

1

2

1

2

p

p

v

v

1

2

1

2

v

v

p

p

2

1

1

2

v

v

p

p 1122 vpvp

Isentropic Process by an idea gas with constant propeties

1

1

2

1

2

v

v

T

T

1

1

2

1

2

p

p

T

T

2

1

1

2

v

v

p

p

1

11

Cv

T

2

1

CT

p

3Cpv or or

Are the reversible Process practicable?100% perfection is possible but may not ne practicable..!?!!?!

Practical Processes are influenced by Irreversibilities

• Fluid friction

• Solid friction

• Electrical resistance

• Thermo-chemical Reactions (Combustion)

• Unrestrained motion

• Heat Transfer with a finite temperature difference

Solid Friction is an Irreversibility

PE KE

Q

Solid Friction is an Irreversibility

PE KEQ

Solid Friction is an Irreversibility

PE KE

Q

Solid Friction is an Irreversibility

QQ

ReverseTHIS IS NOT POSSIBLE.

Q?

Solid Friction is an Irreversibility

Q

1 2

43

Irreversible and Reversible engines

LTR

QHER

QLER

ER

WnetR

Assume that an irreversibleEngine is more efficient than

the reversible engine.

QHEI

QLEI

EI

WnetI

QHE

QLE

EWnetI

revirr

HER

Rnet

HEI

Inet

Q

W

Q

W ,,

HTR

• For same Wnet, QHEI < QHER

• Implies that, |QLEI | < |QLER|

• But a reversible engine can be completely reversed and it will work as a heat pump.

Wnet,I Wnet,R>QHEI QHER

revrev

1

HPRHEIrev

irr QQ

1

Let us construct a compound machine using an irreversible engineand reversed reversible engine (reversible Heat Pump).

For same |Wnet |,

QHPR

QLPR

QHEI

EIR

WnetR

QHEI < |QHPR |

QLE

LTR (Source)

HTR (Sink)

|QLEI| < QLPR

QLPR - |QLEI |

|QHPR | - QHEI

Irreversible Machines

• The efficiency of an irreversible heat engine will always less than the efficiency of a reversible engine working between the same reservoirs.

• The COP of an irreversible heat pump will always less than the COP of a reversible heat pump working between the same reservoirs.

Further Discussions

irrrev

irrrev but, 1/revrev

irrrev (Mathematically possible but thermodynamically impossible).

• Similarly, irrrev 1/irrrev

irrrev (Mathematically possible but thermodynamically impossible).

Isotope Half-Life DecayHe-3 Stable N/A

He-4Stable N/A≈ 0.5 x 10-21 sec - 1 x 10-21 sec p or n

He-5 1 x 10-21 sec n

He-60.8 sec β-5 x 10-23 sec - 5 x 10-21 sec n

He-7 3 x 10-22 sec - 4 x 10-21 sec n

He-80.1 sec β-0.5 x 10-21 sec - 1 x 10-21 sec n/α

He-9 unknown unknown

Increase of Entropy Principle

Entropy change

Entropy Generation

•The principle states that for an isolated Or a closed adiabatic Or System + Surroundings;•A process can only take place such that Sgen 0 where Sgen = 0 for a reversible process only and Sgen can never be less than zero.

Entropy Transfer

(due to heat transfer)

Increase of Entropy Principle

Define entropy generation Sgen as,

For a general Process

Implications of Increase of Entropy Principle

• Entropy, unlike energy, is non-conservative since it is always increasing.

• The entropy of the universe is continuously increasing, in other words, it is becoming disorganized and is approaching chaotic.

• The entropy generation is due to the presence of irreversibilities.

• Therefore, the higher irreversibilities lead to the higher the entropy generation and the lower the efficiency of a device.

• The above is Engineering statement of the second law

Second Law & Entropy Balance

• Increase of Entropy Principle is another way of stating the Second Law of Thermodynamics:

• Second Law : Entropy can be created but NOT destroyed

• In contrast, the first law states: Energy is always conserved.

• Note that this does not mean that the entropy of a system cannot be reduced, it can.

• However, total entropy of a system + surroundings cannot be reduced.

Entropy of Universe

A quantity of heat Q is spontaneously transferred from the surroundings at temperature T0 to the control mass at temperature T. Let the work done during this process be W. For this process by control mass and write

For the surroundings at T0, Q is negative, and we assume a reversible heat extraction so

The total net change of entropy is therefore

Since T0 > T, the quantity [(1/T) - (1/T0)] is positive, and we conclude that

Net Change in Entropy of Universe

If T > T0, the heat transfer is from the control mass to the surroundings

It should be noted that the right-hand side of above equation represents an external entropy generation due to heat transfer through a finite temperature difference.

+

The Third Law of Thermodynamics

The entropy change of a system during a reversible isothermal process tends

towards zero when the thermodynamic temperature of the system tends towards

zero. In the neighbourhood of absolute zero, all

reactions in a liquid or solid in internal equilibrium take place with no change in

entropy. [Nernst 'principle'].

Planck’s statement of the 3rd law

• In 1911, Planck one step further and made the hypothesis that not only does the entropy difference vanish as T → 0, but that:

• Planck’s statement of the Third Law: The entropy of every solid or liquid substance in internal equilibrium at absolute zero is itself zero.

• Planck is just saying:

0lim0

ST

Engineering Relations from Second Law

Entropy as A Rate Equation

• The second law of thermodynamics was used to write the balance of entropy for a infinitesimal variation for a finite change.

• Here the equation is needed in a rate form so that a given process can be tracked in time.

• Take the incremental change and divide by t.

• We get

• For a given control mass we may have more than one source of heat transfer, each at a certain surface temperature (semi-distributed situation).

The rate of entropy change is due to the flux of entropy into the control mass from heat transfer and an increase due to

irreversible processes inside the control mass.

The Second Law Of ThermodynamicsFor A Control Volume

• The rate of change of property B of the system .

inoutCVCM smsm

dt

dS

dt

dS

• Let B = Entropy of the system, S = ms.

inoutCVCM BB

dt

dB

dt

dB

genCM S

T

Q

dt

dS

Entropy Rate Equation for CV

Rate of change in entropy of a CV = Entropy in flow rate –Entropy out flow rate + the flux of entropy into the control mass from heat

transfer + Rate of Entropy generation

The Steady State Steady Flow Process

• For the steady-state process, which has been defined before, we conclude that there is no change with time of the property (entropy) per unit mass at any point within the control volume.

• That is,

so that, for the steady-state process,

• If in a steady-state process there is only one area over which mass enters the control volume at a uniform rate and only one area over which mass leaves the control volume at a uniform rate,

• we can write

and dividing the mass flow rate out gives

Since sgen is always greater than or equal to zero, for an adiabatic process it follows that

where the equality holds for a reversible adiabatic process.

Geometry of Turbine Blades for High Efficiency

Transient Process

• For the transient process, the second law for a control volume, it can be written in the following form:

If this is integrated over the time interval t, we have

Therefore, for this period of time t, we can write the second law for the transient process as

Since in this process the temperature is uniform throughout the control volume at any instant of time, the integral on the right reduces to

and therefore the second law for the transient process can be written

Mechanical Engineering Inventions

• Carnot Cycle

• Lenoir Cycle

• Otto Cycle

• Stirling Cycle

• Atkinson Cycle

• Diesel Cycle

• Brayton cycle

• Rankine Cycle

• Vapour Compression Refrigeration Cycle

• Vapour Absorption Refrigeration Cycle