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..!?!!?!