Thermodynamics of Ideal Processes

P M V SubbaraoProfessor

Mechanical Engineering Department

Entropy View of Theoretical Processes …..

Reversible Cycle & Origin of Thermodynamic Entropy

Temperature- Entropy Diagram

Temperature-Entropy Change -- 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+pvdh = du + pdv + vdp, thus du + pdv = dh - vdpTds = 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.

Tduds

T

dTTCds

Integrating from state 1 to state 2 1

1

2

112

T

T TdTTCdsss

1

1

12

T

Tavg T

dTCssWhere, 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.

dvTp

Tduds

)()(1fgfgfgfg vv

Tpuu

Tsss

For a change from saturated liquid to saturated vapor

Tvp

Tu

sss fgfgfgfg

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

dPTv

Tdhdsdv

TP

Tduds 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

Tv

TdTTC

dsdvTP

TdTTcds pv or

By Integration, the change in the entropy is

1

22

112 ln

vvR

TdTTcss v

1

22

112 ln

ppR

TdTTcss por

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

ppR

TTcss p

1

2

1

212 lnln

vvR

TTcss v

Isentropic Process by an idea gas with constant propeties

1

1

2

1

2

vv

TT

1

1

2

1

2

pp

TT

2

1

1

2

vv

pp

1

11

CvT

2

1

CTp

3Cpv or or

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