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Turbomachinery Aero-ThermodynamicsIntroduction – Thermodynamics
Alexis. Giauque1
1Laboratoire de Mecanique des Fluides et AcoustiqueEcole Centrale de Lyon
Ecole Centrale Paris, January-February 2015
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 1 / 70
Table of Contents
1. IntroductionSome historyTurbomachinery now and in the near future
2. Compressible flows: A refresher crash courseIsentropic flow relations
3. Dimensionless quantities and similitude lawsDimensionless numbersSimilitude laws
4. ThermodynamicsEnergiesEffective workKinetic energy / Work of internal forcesInternal energy / mechanical dissipationEntropy / Gibbs equationSummary
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 2 / 70
You said Turbomachines?
Turbomachines are machines that transfer energy between a rotor and afluid.If the energy is transferred from the fluid the turbomachine, it is a turbine.If the energy is transferred to the fluid, it is a compressor.Turbomachines
have been here for long
are almost everywhere
are key ingredients in projects that will address climate change andressource scarcity issues
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 3 / 70
Let’s have a closer look to turbomachinery components
Figure : Schematic views of axial and centrifugal rotors
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 4 / 70
How it all began...
-120 — The first turbomachinery : The aeropile (Hero of Alexandria)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 5 / 70
How it all began...
1500 – Chimney Jack (Leonardo Da Vinci)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 6 / 70
How it all began...
1629 – First centrifugal impeller (Papin)
1791 – The first concept of gas turbine cycle (John Barber)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 7 / 70
How it all began...
1827 – The first underwater hydraulic turbine (Fourneyron)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 8 / 70
How it all began...
1883–1897 The first modern steam turbines (De Laval, Rateau,Parsons, Curtis)
1897 – Demonstration of first modern steam engine boat (Parsons)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 9 / 70
How it all began...
1905 – First self-sustained gas turbine cycle (Societe Anonyme desTurbomoteurs - Paris)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 10 / 70
How it all began...
1939 – First 4 MW utility power generation gas turbine (NeuchatelSwitzerland) – Thermal efficiency 17.4%
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 11 / 70
How it all began...
1934 – First turbojet engine (von Ohain - Germany)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 12 / 70
How it all began...
1939 – First turbojet airplane (Heinkel-178)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 13 / 70
How it all began...
1947 – First Mach 1 flight (Charles ”Chuck” Yeager with the X-1)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 14 / 70
What are they used for right now: Propulsion inaeronautics - Civil Applications
(a) Pratt & Whitney 4156. Fan diameter: 2.4m.Equips A310-300, A300-600, B747-400, B767-200,MD-11
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 15 / 70
What are they used for right now: Propulsion inaeronautics - Military Applications
(b) GE F404. Turbofan with post-combustion
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 16 / 70
What are they used for right now: Electricity Production -Thermal
Figure : Gas Turbine for electricity production (43 MW with a thermal efficiencyof 33%)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 17 / 70
What are they used for right now: Electricity Production -Nuclear
Figure : Steam Turbine used in nuclear facilities for electricity productionAlexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 18 / 70
What are they used for right now: Electricity Production -Hydraulic
(a) Pelton Turbine (b) Francis Turbine (c) Kaplan Turbine
Figure : Hydraulic Turbines
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 19 / 70
What are they used for right now: Electricity Production -Wind
Figure : Wind Turbine
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 20 / 70
And now what are the stakes and technologies?
Sustainable progressPropulsion – Hybrid plane
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 21 / 70
And now what are the stakes and technologies?
Sustainable progressPropulsion – Contra-Rotative Open Rotors (CRORs)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 22 / 70
And now what are the stakes and technologies?
Sustainable progressElectricity production – Break Efficiency Barriers
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 23 / 70
And now what are the stakes and technologies?
Share progressElectricity production – Improve on existing technologies
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 24 / 70
Future technologies were hard to predict in the 20thcentury...
A few statements from eminent scientists and engineers(source: Cyrus B. Meher-Homji, ASME)
”The energy produced by the breaking down of atoms is a very poorkind of thing. Anyone who expects a source of power from thetransformation of these atoms is talking moonshine.” –ErnestRutherford, circa 1930.
”As far as sinking a ship with a bomb is concerned, it just can’t bedone.” –Rear Admiral Clark Woodward, 1939, US Navy.
”That is the biggest fool thing we have ever done?. The atomicbomb will never go off, and I speak as an expert in explosives.”–Admiral William Leahy, US Navy, to President Truman, 1945.
”Space travel is utter bilge.” –Sir Richard van der Riet Wooley,Astronomer Royal, 1956.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 25 / 70
And it goes on...
A few statements from eminent scientists and engineers
”Cellular phones will absolutely not replace local wire systems.” –Marty Cooper, Director of research at Motorola (1981)
”I predict the Internet in 1996 (will) catastrophically collapse.” –Robert Metcalfe co-inventor of Internet (1995)
”The subscription model of buying music is bankrupt.” – Steve Jobs(2003)
”There’s no chance that the iPhone is going to get any significantmarket share.” – Steve Ballmer, Microsoft CEO (April 2007)
”In five years I don’t think there’ll be a reason to have a tabletanymore.” – Thorsten Heins, BlackBerry CEO (2013)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 26 / 70
Table of Contents
1. IntroductionSome historyTurbomachinery now and in the near future
2. Compressible flows: A refresher crash courseIsentropic flow relations
3. Dimensionless quantities and similitude lawsDimensionless numbersSimilitude laws
4. ThermodynamicsEnergiesEffective workKinetic energy / Work of internal forcesInternal energy / mechanical dissipationEntropy / Gibbs equationSummary
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 27 / 70
Some physical phenomena in which compressibility cannotbe ignored
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 28 / 70
Isentropic flow relations
Let’s consider the nozzle below in which the fluid is accelerated.
In this nozzle we will consider that the compressible fluid undergoes anreversible adiabatic or isentropic transformation.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 29 / 70
Isentropic flow relations
Adiabatic transformation : – no heat is exchanged between the fluid andthe nozzleIsentropic transformation : – the entropy is constant during thetransformation1
Since in this nozzle, there is also no work (no moving parts) exchangedwith the fluid, the following relations hold:
1 h0 = CpT0. The total enthalpy per unit mass is constant along theflow2
2 ∆s = Cv ln[PP1
(ρ1ρ
)γ]= 0 regardless of the reference state ”1”.
1The term entropy actually refers in statistical mechanics to the volume of the phasespace. To know more about entropy have a look at this video
2All quantities are considered per unit massAlexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 30 / 70
Isentropic flow relations
Obtaining
T0 = T
(1 +
γ − 1
2M2
)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 31 / 70
Isentropic flow relations
T0 = T
(1 +
γ − 1
2M2
)This equation provides the relation existing between the total (stagnation)temperature and the static (actual) temperature.Providing there is no heat or work exchange with the fluid, it is a fonctionof the Mach number only.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 32 / 70
Isentropic flow relations
Obtaining
Isentropic Flow relations
(T0
T
)=
(1 +
γ − 1
2M2
)(P0
P
)=
(1 +
γ − 1
2M2
) γγ−1
(ρ0
ρ
)=
(1 +
γ − 1
2M2
) 1γ−1
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 33 / 70
Critical section and mass flow rate
Obtaining
m = P0T−1/20
(γr
)1/2M
(1 +
γ − 1
2M2
)− γ+12(γ−1)
A
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 34 / 70
Critical section and mass flow rate
Obtaining
m = P0T−1/20
(γr
)1/2M
(1 +
γ − 1
2M2
)− γ+12(γ−1)
A︸ ︷︷ ︸(1+ γ−1
2 )− γ+1
2(γ−1) A?
A? is fixed at the design stage (geometry) and the fluid (usually air) is also
fixed so that the quantity m√T0P0
enables to compare turbomachinesregardless of the external conditions (P0, ρ0, T0).
Standard mass flow rate
The standard mass flow rate, largely used in turbomachinery, is defined as
mst = m√T0P0
Pst0√T0st
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 35 / 70
Table of Contents
1. IntroductionSome historyTurbomachinery now and in the near future
2. Compressible flows: A refresher crash courseIsentropic flow relations
3. Dimensionless quantities and similitude lawsDimensionless numbersSimilitude laws
4. ThermodynamicsEnergiesEffective workKinetic energy / Work of internal forcesInternal energy / mechanical dissipationEntropy / Gibbs equationSummary
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 36 / 70
Defining the parameters...
The most important job of an engineer/scientist is to define theparameters upon which the system he/she studies depends.The following list can be proposed
ρ0 (kg.m-3), fluid density,
µ (kg.s-1.m-1), fluid viscosity,
U (m.s-1), reference velocity,
D (m), reference dimension,
Q (m3.s-1), volume flow rate,
∆p0 (kg.m-1.s-2), change in total pressure,
P (kg.m2.s-3), power.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 37 / 70
Using Vashy-Buckingham Theorem
There are 7 parameters and 3 dimensions [L,T,M], Vashy-Buckinghamtheorem tells us that there are 7-3=4 dimensionless numbers describingthe system.It also tells us that the initial relation f (ρ0, µ,U,D,Q, (∆p0),P) = 0 canbe recast as f (Π1,Π2,Π3,Π4) = 0 in which the Πs are defined as:
Π1 = ρa10 ∗ µa2 ∗ U ∗ Da4 ∗ Qa5 ∗ (∆p0)a6 ∗ Pa7
Π2 = ρb10 ∗ µb2 ∗ Ub3 ∗ Db4 ∗ Q ∗ (∆p0)b6 ∗ Pb7
Π3 = ρc10 ∗ µc2 ∗ Uc3 ∗ Dc4 ∗ Qc5 ∗ (∆p0) ∗ Pc7
Π4 = ρd10 ∗ µd2 ∗ Ud3 ∗ Dd4 ∗ Qd5 ∗ (∆p0)d6 ∗ P
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 38 / 70
Using Vashy-Buckingham Theorem
Obtaining
Π1 = ρ0 ∗ µ−1 ∗ U ∗ DΠ2 = U−1 ∗ D−2 ∗ QΠ3 = ρ−1
0 ∗ U−2 ∗ (∆p0)
Π4 = ρ−10 ∗ U−3 ∗ D−2 ∗ P
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 39 / 70
Using Vashy-Buckingham Theorem
Let’s define those dimensionless numbers
the Reynolds Number (Re = ρUDµ ). It assesses the nature of the
flow (laminar/turbulent)
the flow coefficient (φ = QUD2 ). It provides a comparison of the
output velocity with the reference velocity.
the load coefficient (Ψ = ∆p0
ρ0U2 ). It compares the change in pressureto the available dynamic pressure.
the power coefficient (P = PρU3D2 ). This coefficient is a
dimensionless form of the power output.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 40 / 70
Other numbers/coefficients can be important
the Mach number (Ma = Uc ). It assesses the importance of
compressibility effects
the specific speed, Ns = NQ0.5
(∆h0)3/4 . This coefficient is a normalization
of the rotation speed.
the specific diameter, Ds = D(∆h0)1/4
Q0.5 . This is a normalization ofthe reference dimension of the turbomachine.
The efficiency, η (We will come to that later)
The degree of reaction, Λ (We will come to that later)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 41 / 70
Specifics for Hydraulic turbines
Hydraulic turbines work with water which is practically incompressible sothat the previous list can be revisited. The following coefficients aredenoted the ”Rateau” coefficients.
the flow coefficient, δ = QND3 ,
the manometric coefficient, µ = gHm
N2D2 , where Hm is the waterheight,
the power coefficient, τ = PρN3D5 ,
the specific rotation speed, Ns = NQ0.5
(gHm)3/4 ,
the torque coefficient, γ = CρgHmD3 ,
the opening coefficient, Φ = Q(gHm)0.5D2 .
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 42 / 70
Why are these numbers important
A good reason to consider dimensionless numbers
Dimensionless numbers enable to categorize turbomachines (for ex: pistonvs axial vs centrifugal compressors)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 43 / 70
Why are these numbers important
Let’s consider a compressor you need to design. You know the volume flowrate (Qv ) and how much work you can afford (∆h0). The following chartlets you decide depending on the size (D) and the rotation speed (N)which technology should be used.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 44 / 70
Similitude laws
What you can do with dimensionless numbers and similitude laws
Determine the most important parameters of your system.Limit experimental cost by ’a priori’ limiting the number of variables takeninto accountGuide the design of representative prototypes for the system (for examplea smaller one)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 45 / 70
Similitude laws
The whole idea of similitude laws is to analyze a simpler system than thereal one but in which most of the dimensionless numbers are kept constant.
It is important to remember that if all dimensionless numbers are keptconstant, the physical problem is the same for the prototype and the realmachine.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 46 / 70
Unfortunately, it is quite difficult so that many scenarios are possibledepending on the problem at hand.Similitude laws can be:
Geometrical. In this case dimensions in different directions are allscaled by the same factor.
Kinematic. The flow coefficient is kept constant. Velocity angles arealso conserved.
Dynamic. The load coefficient is conserved. The ratio of forcesapplied to the blades are the same as for the real machine.
Energetic. The power coefficient is kept constant. The energy ratioare conserved.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 47 / 70
Table of Contents
1. IntroductionSome historyTurbomachinery now and in the near future
2. Compressible flows: A refresher crash courseIsentropic flow relations
3. Dimensionless quantities and similitude lawsDimensionless numbersSimilitude laws
4. ThermodynamicsEnergiesEffective workKinetic energy / Work of internal forcesInternal energy / mechanical dissipationEntropy / Gibbs equationSummary
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 48 / 70
System of interest
We will consider the following system
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 49 / 70
Total energy
The total energy e0 is composed of the internal energy e and the kinetic
energyV 2
2. Following the first principle of thermodynamics, we have:
De
Dt+
D(V 2/2)
Dt=
Dq
Dt︸︷︷︸heat exchange
+Dwe
Dt︸︷︷︸work of external forces
Total energy balance
De0
Dt=
Dq
Dt︸︷︷︸heat exchange
+Dwe
Dt︸︷︷︸work of external forces
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 50 / 70
External forces power
External forces that apply to a volume of fluid Dm are of two types:
The volume forces denoted ~f . When applied to Dm, their powerwrites Pev =
∫Dmρ~f .~Vdv
The surface forces due to the stress tensor and denoted ¯σ.~n. Theirpower can be written as:
Pes =
∫∂Dm
¯σ.~VdS
where ¯σ = −p¯I + ¯τ . We therefore have:
Pes =
∫Dm
div(¯τ ~V )dv −∫Dm
pdiv(~V )dv︸ ︷︷ ︸Compressibility
−∫Dm
~V . ~grad(p)dv︸ ︷︷ ︸Transport
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 51 / 70
External forces work
So that
Dwe
Dt= ~f .~V +
1
ρdiv(¯τ ~V ) − p
ρdiv(~V ) −
~V
ρ. ~grad(p)
Obtaining
Dwe
Dt= ~f .~V +
1
ρdiv(¯τ ~V ) − D(p/ρ)
Dt+
1
ρ
∂p
∂t
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 52 / 70
Total enthalpy balance
Total enthalpy balance
Dh0
Dt=
Dq
Dt︸︷︷︸heat exchange
+Dwu
Dt︸ ︷︷ ︸effective work
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 53 / 70
Effective work
Obtaining
Dwu
Dt= ~f .~V +
1
ρdiv(¯τ ~V ) +
1
ρ
∂p
∂t
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 54 / 70
Effective power
Let’s consider the effective power applied to the fluid in the following
system
Pu =
∫Dm
ρDwu
Dtdv
Replacing the effective work by our previous findings we have
Pu =
∫Dm
ρDwu
Dtdv =
∫Dm
ρ~f .~Vdv +
∫Dm
div(¯τ ~V )dv +
∫Dm
∂p
∂tdv
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 55 / 70
Effective power
The following hypothesis can be made that simplify the previousexpression
The flow has reached a steady state
The velocity is zero on the solid boundaries
The viscous stress is negligible at the inlet and outlet
Using all these assumptions, one can state that:
Pu ≈∫Dm
ρ~f .~Vdv
showing that the effective power is indeed equal to the power exchangedbetween the flow and the machine.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 56 / 70
Link between mechanical effective powers
Obtaining
Pu − Pe =
∫∂D1 U ∂D2
p~V .~nds︸ ︷︷ ︸Flowtransferpower
This term represents the power necessary to impose a given flow ratebetween the inlet to the outlet. It is called the transfer power.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 57 / 70
Link between effective power and effective work
the effective power writes
Pu =
∫Dm
ρDwu
Dtdv
Pu =
∫Dm
∂ρwu
∂tdv +
∫∂Dm
ρwu~V .~nds
Assuming the system is in a steady state, the effective work is constant atthe inlet and outlet, and since the velocity is zero on solid boundaries, wehave
Pu =
∫∂D1 U ∂D2
ρwu~V .~nds
Pu = −mwu1 + mwu2 = m∆wu
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 58 / 70
Kinetic energy / work of internal forces
Obtaining from ρDV 2/2Dt
Dwi
Dt=
DV 2/2
Dt− Dwe
DtDwi
Dt= −1
ρ¯σ : ¯D
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 59 / 70
Kinetic energy
Let’s pause a moment on the expression of the conservation of kineticenergy and use the fact that ¯σ = ¯τ − p¯I
DV 2/2
Dt= ~f .~V +
1
ρ~div(¯τ.~V ) − 1
ρ¯τ : ¯D −
~V
ρ. ~grad(p)
~f .~V represents the work done by the volume forces (the machine)1ρ~div(¯τ.~V ) represents the work done by the viscous forces
−1ρ
¯τ : ¯D is the dissipation of kinetic energy due to the viscosity
~Vρ .
~grad(p) represents the work done by the pressure force (transport)
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 60 / 70
Internal energy / mechanical dissipation
By subtracting the conservation equation of kinetic energy to the one forthe total energy, one obtains the conservation equation for the internalenergy which writes:
De
Dt=
Dq
Dt︸︷︷︸heat exchange
− Dwi
Dt︸︷︷︸work of internal forces
TRICKYDe
Dt=
Dq
Dt︸︷︷︸heat exchange
+1
ρ¯τ : ¯D︸ ︷︷ ︸
mechanical dissipation
− pD(1/ρ)
Dt︸ ︷︷ ︸compression work
mechanical/viscous dissipation
We see that the mechanical dissipation decreases kinetic energy (slowsdown the fluid) and increases the internal energy (heats up the flow). Thisterm therefore does not appear in the balance of total energy.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 61 / 70
Internal enthalpy
To obtain the internal enthalpy conservation equation (useful for open
systems), we add D(p/ρ)Dt on both sides of the previous equation
Dh
Dt=
Dq
Dt︸︷︷︸heat exchange
+1
ρ¯τ : ¯D︸ ︷︷ ︸
mechanical dissipation
+1
ρ
Dp
Dt
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 62 / 70
Entropy / Gibbs equation
Entropy has two definitions that refer to the same quantity.
Following Boltzmann, it is a measure of the number of micro-statesall corresponding to the same macro-state. Sb = kblnΩ
Following Gibbs, the entropy is a state function that can be computedknowing the thermodynamic state of the system.
They both refer to the same quantity but it has only been definitely provenin 1965.
Second principle of thermodynamics
The second principle of thermodynamics states that entropy of a closedsystem can only grow.
dSclosed system >= 0
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 63 / 70
Entropy / Gibbs equation
The previous expression can be applied to the universe (which is supposedto be a closed system). The universe can be split between the system ofinterest and its surrounding. In this case we have
dSsystem + dSsurrounding >= 0
So that if the entropy of the system decreases, the entropy of thesurrounding must have increased by a larger quantity.Let’s now come back to Clausius/Gibbs definition of entropy. It is definedas follows:
Gibbs Equation
Tds = dh − 1
ρdp
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 64 / 70
Physical interpretation
From the previous equation one can write that:
TDs
Dt=
De
Dt+ p
D1/ρ
Dt
Using the conservation equation for the internal energy, one can write that
TDs
Dt=
Dq
Dt︸︷︷︸heat exchange
+1
ρ¯τ : ¯D︸ ︷︷ ︸
mechanical dissipation
The entropy variation is therefore due to the entropy creation due to
the heat exchange of the fluid with its surrounding (It can beradiation, convection or conduction),
the mechanical dissipation
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 65 / 70
Physical interpretation
Obtaining
TDs
Dt=
2µ
ρ
∑i
∑j
D2ij +
2λ
ρ
∑j
∂Vj
∂xj
2
≥ 0
which shows that the entropy can only grow in a fluid that flows withoutexchanging heat with its surrounding.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 66 / 70
Stagnation entropy
Showing that∆s = ∆s0
so that
T0Ds0
Dt=
Dh0
Dt− 1
ρ0
Dp0
Dt
Stagnation entropy is equal to static entropy
Because one goes from the static to the stagnation state by an isentropicdeceleration, stagnation entropy and static entropy are equal.
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 67 / 70
Thermodynamics – Summary – energy/enthalpy
De0
Dt=
Dq
Dt︸︷︷︸heat exchange
+Dwe
Dt︸︷︷︸work of external forces
Dwe
Dt= ~f .~V +
1
ρdiv(¯τ ~V ) − p
ρdiv(~V ) −
~V
ρ. ~grad(p)
Dh0
Dt=
Dq
Dt︸︷︷︸heat exchange
+Dwu
Dt︸ ︷︷ ︸effective work
Dwu
Dt= ~f .~V +
1
ρdiv(¯τ ~V ) +
1
ρ
∂p
∂t
Pu − Pe =
∫Dρ
(Dwu
Dt− Dwe
Dt
)dv =
∫∂D1∪∂D2
p~V .~nds︸ ︷︷ ︸Flow transfer power
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 68 / 70
Thermodynamics – Summary – works
DV 2/2
Dt=
Dwt
Dt︸︷︷︸work of all forces
Dwt
Dt= ~f .~V +
1
ρ~div(¯τ.~V ) − 1
ρ¯τ : ¯D −
~V
ρ. ~grad(p)
Dwt
Dt︸︷︷︸work of all forces
=Dwe
Dt︸︷︷︸work of external forces
+Dwi
Dt︸︷︷︸work of internal forces
Dwi
Dt= −1
ρ¯σ : ¯D
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 69 / 70
Thermodynamics – Summary – entropy
TDs
Dt=
Dh
Dt− 1
ρ
Dp
Dt
T0Ds
Dt=
Dh0
Dt− 1
ρ0
Dp0
Dt
TDs
Dt=
Dq
Dt︸︷︷︸heat exchange
+1
ρ¯τ : ¯D︸ ︷︷ ︸
mechanical dissipation
Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 70 / 70