Further Analysis of Reversible Machines

P M V SubbaraoProfessor

Mechanical Engineering Department

Innovation of A New Property of A System!!!!

Performance of Reversible Machines : SSSF

Wout,C

QHC

QLC

HTR (Source)

LTR (Sink)

Carnot Engine

Wout,S

QHS

QLS

HTR (Source)

LTR (Sink)

Stirling Engine

Wout,R

QHR

QLR

HTR (Source)

LTR (Sink)

Regenerative Engine

Compound Machine using two Reversible Machines

Wout,C

QHC

QLC

HTR (Source)

Carnot Engine

Wout,S

QHS

QLS

LTR (Sink)

Stirling Engine

All Reversible Machines Working Between Same Reservoirs should have Same performance

Further Algebra of A Reversible Engine Model

H

L

H

LH

H

netCarnot T

T

Q

QQ

Q

W

1

Use a generalized indices…

HTR

LTR

HTR

LTR

T

T

Q

Q 11

HTR

LTR

HTR

LTR

T

T

Q

Q

HTR

HTR

LTR

LTR

T

Q

T

Q

0LTR

LTR

HTR

HTR

T

Q

T

Q

0LTR

LTR

HTR

HTR

T

Q

T

Q

During a time duration

01

n

i i

i

T

QFor any reversible cycle with n number heat transfer processes

•Where, i shows ith process and n is the number of process in a cycle.•But Qi is a path function and depends on process.

01

1

m

k

k

k T

Q

k is an initial state and k+1 is a final state. m is the number state Points in a cycle.Therefore for all reversible cycles.

0T

Q

•Therefore,

Cyclic integral of this new quantity is zero!Any quantity whose cyclic integral is zero is a property !

0T

Q

process. reversible afor path of indpendent is T

Q

For a reversible process an infinitesimal change in this new property is:

rev

T

QdS

Boltzman named S as Entropy of a substance.

A New cyclic Integral

f

i revif T

QSS

During a reversible process

S S (gaseous state)(gaseous state) > S > S (liquid state)(liquid state) > S > S (solid state)(solid state)

SSoo (J/K•mol) (J/K•mol)

HH22OO(liq)(liq) 69.9569.95

HH22OO(gas)(gas) 188.8 188.8

SSoo (J/K•mol) (J/K•mol)

HH22OO(liq)(liq) 69.9569.95

HH22OO(gas)(gas) 188.8 188.8

Entropy, S : A Measure of State of MatterEntropy, S : A Measure of State of Matter

For a given substance

Entropy and Order of Molecules of MatterEntropy and Order of Molecules of Matter

S˚S˚(Br2 liq)(Br2 liq) < S˚ < S˚(Br2 gas)(Br2 gas) S˚S˚(H2O solid) (H2O solid) < S˚< S˚(H2O liquid)(H2O liquid)

Increase in molecular complexity Increase in molecular complexity generally leads to increase in S.generally leads to increase in S.

Entropy, S : Molecular ComplexityEntropy, S : Molecular Complexity

Standard Molar EntropiesStandard Molar Entropies

Entropy and TemperatureEntropy and TemperatureEntropy and TemperatureEntropy and Temperature

S increases S increases slightly with Tslightly with T

S increases a S increases a large amount large amount with phase with phase changeschanges

Entropy Change during a Reversible Process

• From the definition of the entropy, it is known that Q=TdS during a reversible process.

• The total heat transfer during this process is given by Qreversible = TdS

• Therefore, it is useful to consider the T-S diagram for a reversible process involving heat transfer

• On a T-S diagram, the area under the process curve represents the heat transfer for a reversible processT

S

• A reversible adiabatic process

Carnot Cycle

• Show the Carnot cycle on a T-S diagram and identify the heat transfer at both the high and low temperatures, and the work output from the cycle.

S

TTH

TL

S1=S4 S2=S3

1 2

34

A B

• 1-2, reversible isothermal heat transferQH = TdS = TH(S2-S1) area 1-2-B-A• 2-3, reversible, adiabatic expansionisentropic process, S=constant (S2=S3)• 3-4, reversible isothermal heat transfer QL = TdS = TL(S4-S3), area 3-4-A-B• 4-1, reversible, adiabatic compressionisentropic process, S1=S4

• Net work Wnet = QH - QL, the area enclosed by 1-2-3-4, the shaded area

Nature of Reversible Machines

0 WQ For all reversible heat engines:

0

T

Q

For all reversible heat pumps and refrigerators: 0 WQ

0

T

Q

Therefore all reversible machines in this universe : 0

T

Q

Process : h-s Diagram : Mollier Diagram

• Enthalpy-entropy diagram, h-s diagram: it is valuable in analyzing steady-flow devices such as turbines, compressors, etc.

• h: change of enthalpy from energy balance (from the first law of thermodynamics)

• s: change of entropy from the second law.

• A measure of the irreversibilities during an adiabatic process.

s

h

h

s

Enthalpy Vs Entropy Diagram

Temperature- Entropy Diagram

TdS -- Equations

• For a control mass containing a pure compressible substance undergoing a reversible process (no change in KE & PE)

dU= Qrev - Wrev = TdS - pdV

TdS = dU + pdV, or Tds = du + pdv ( per unit mass) This is the famous Gibbsian equation

Eliminate du by using the definition of enthalpy h=u+pv

dh = du + pdv + vdp, thus du + pdv = dh - vdp

Tds = du + pdv, also Tds = dh - vdp

• Important: these equations relate the entropy change of a system to the changes in other properties: dh, du, dp, dv.

• Therefore, they are independent of the processes.

Entropy change of an incompressible substance

• For most liquids and all solids, the density is not changed as pressure changes, that is, dv=0.

• Gibbsian equation states that Tds=du+pdv=du, du=CdT.

• For an incompressible substance Cp=Cv=C is a function of temperature only.

T

duds

T

dTTCds

Integrating from state 1 to state 2 1

1

2

1

12

T

T T

dTTCdsss

1

1

12

T

T

avg T

dTCss

Where, Cavg is the averaged specific heat over the given temperature range.

Entropy change during change of Phase

• Consider steam is undergoing a phase transition from liquid to vapor at a constant temperature.

dvT

p

T

duds

)()(1

fgfgfgfg vvT

puu

Tsss

Determine the entropy change sfg=sg-sf using the Gibbsian equations and compare the value to that read directly from the thermodynamic table.

For a change from saturated liquid to saturated vapor

T

vp

T

usss fgfg

fgfg