Turbomachinery Aero-Thermodynamics Introduction

Turbomachinery Aero-Thermodynamics Introduction – Thermodynamics

Alexis. Giauque1

1Laboratoire de M´ecaniquedes Fluides et Acoustique Ecole 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. Introduction Some history Turbomachinery now and in the near future 2. Compressible flows: A refresher crash course Isentropic flow relations 3. Dimensionless quantities and similitude laws Dimensionless numbers Similitude laws 4. Thermodynamics Energies Effective work Kinetic energy / Work of internal forces Internal energy / mechanical dissipation Entropy / Gibbs equation Summary

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 a fluid. 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 and ressource 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 (Societ´eAnonyme des Turbomoteurs - 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 (Neuchatel Switzerland) – 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 in aeronautics - 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 in aeronautics - 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 eﬃciency of 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 production Alexis 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 progress Propulsion – Hybrid plane

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 21 / 70 And now what are the stakes and technologies?

Sustainable progress Propulsion – 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 progress Electricity 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 progress Electricity 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 20th century...

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 poor kind of thing. Anyone who expects a source of power from the transformation of these atoms is talking moonshine.” –Ernest Rutherford, circa 1930. ”As far as sinking a ship with a bomb is concerned, it just can’t be done.” –Rear Admiral Clark Woodward, 1939, US Navy. ”That is the biggest fool thing we have ever done?. The atomic bomb 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 signiﬁcant market share.” – Steve Ballmer, Microsoft CEO (April 2007) ”In ﬁve years I don’t think there’ll be a reason to have a tablet anymore.” – Thorsten Heins, BlackBerry CEO (2013)

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 26 / 70 Table of Contents

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 27 / 70 Some physical phenomena in which compressibility cannot be 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 an reversible 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 and the nozzle Isentropic transformation : – the entropy is constant during the transformation1 Since in this nozzle, there is also no work (no moving parts) exchanged with the ﬂuid, the following relations hold:

1 h0 = CpT0. The total enthalpy per unit mass is constant along the ﬂow2 γ h P ρ i 2 ∆s = Cv ln 1 = 0 regardless of the reference state ”1”. P1 ρ

1The term entropy actually refers in statistical mechanics to the volume of the phase space. To know more about entropy have a look at this video 2All quantities are considered per unit mass Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 30 / 70 Isentropic ﬂow relations

Obtaining

γ − 1 T = T 1 + M2 0 2

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 31 / 70 Isentropic ﬂow relations

γ − 1 T = T 1 + M2 0 2

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 fonction of the Mach number only.

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 32 / 70 Isentropic ﬂow relations

Obtaining Isentropic Flow relations

T γ − 1 0 = 1 + M2 T 2 γ P γ − 1 γ−1 0 = 1 + M2 P 2 1 ρ γ − 1 γ−1 0 = 1 + M2 ρ 2

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 33 / 70 Critical section and mass ﬂow rate

Obtaining

− γ+1 γ 1/2 γ − 1 2(γ−1) m˙ = P T −1/2 M 1 + M2 A 0 0 r 2

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 34 / 70 Critical section and mass ﬂow rate

Obtaining

− γ+1 γ 1/2 γ − 1 2(γ−1) m˙ = P T −1/2 M 1 + M2 A 0 0 r 2 | {z } γ+1 − γ−1 2(γ−1) ? (1+ 2 ) A

? A is fixed at the design stage√ (geometry) and the fluid (usually air) is also fixed so that the quantitym ˙ T0 enables to compare turbomachines P0 regardless of the external conditions (P0, ρ0, T0). Standard mass flow rate

The standard√ mass ﬂow rate, largely used in turbomachinery, is deﬁned as T0 Pst0 m˙ st =m ˙ √ P0 T0st

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 35 / 70 Table of Contents

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 the parameters 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-Buckingham theorem tells us that there are 7-3=4 dimensionless numbers describing the system. It also tells us that the initial relation f (ρ0, µ, U, D, Q, (∆p0), P) = 0 can be recast as f (Π1, Π2, Π3, Π4) = 0 in which the Πs are deﬁned as:

a1 a2 a4 a5 a6 a7 Π1 = ρ0 ∗ µ ∗ U ∗ D ∗ Q ∗ (∆p0) ∗ P b1 b2 b3 b4 b6 b7 Π2 = ρ0 ∗ µ ∗ U ∗ D ∗ Q ∗ (∆p0) ∗ P c1 c2 c3 c4 c5 c7 Π3 = ρ0 ∗ µ ∗ U ∗ D ∗ Q ∗ (∆p0) ∗ P d1 d2 d3 d4 d5 d6 Π4 = ρ0 ∗ µ ∗ U ∗ D ∗ Q ∗ (∆p0) ∗ P

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 38 / 70 Using Vashy-Buckingham Theorem

Obtaining

−1 Π1 = ρ0 ∗ µ ∗ U ∗ D −1 −2 Π2 = U ∗ D ∗ Q −1 −2 Π3 = ρ0 ∗ U ∗ (∆p0) −1 −3 −2 Π4 = ρ0 ∗ U ∗ D ∗ P

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 39 / 70 Using Vashy-Buckingham Theorem

Let’s define those dimensionless numbers ρUD the Reynolds Number (Re = µ ). It assesses the nature of the flow (laminar/turbulent) Q the flow coefficient (φ = UD2 ). It provides a comparison of the output velocity with the reference velocity. ∆p0 the load coefficient (Ψ = 2 ). It compares the change in pressure ρ0U to the available dynamic pressure. ¯ P the power coefficient (P = ρU3D2 ). 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

U the Mach number (Ma = c ). It assesses the importance of

compressibility effects NQ0.5 the specific speed, Ns = 3/4 . This coefficient is a normalization (∆h0) of the rotation speed. 1/4 D(∆h0) the specific diameter, Ds = Q0.5 . This is a normalization of the 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 so that the previous list can be revisited. The following coefficients are denoted the ”Rateau” coefficients. Q the flow coefficient, δ = ND3 , gHm the manometric coefficient, µ = N2D2 , where Hm is the water height, P the power coefficient, τ = ρN3D5 , NQ0.5 the specific rotation speed, Ns = 3/4 , (gHm) C the torque coefficient, γ = 3 , ρgHmD Q the opening coefficient, Φ = 0.5 2 . (gHm) D

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: piston vs 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 ﬂow rate (Qv ) and how much work you can aﬀord (∆h0). The following chart lets 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 taken into account Guide the design of representative prototypes for the system (for example a 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 the real one but in which most of the dimensionless numbers are kept constant.

It is important to remember that if all dimensionless numbers are kept constant, the physical problem is the same for the prototype and the real machine.

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 46 / 70 Unfortunately, it is quite difficult so that many scenarios are possible depending on the problem at hand. Similitude laws can be: Geometrical. In this case dimensions in different directions are all scaled by the same factor. Kinematic. The flow coefficient is kept constant. Velocity angles are also conserved. Dynamic. The load coefficient is conserved. The ratio of forces applied to the blades are the same as for the real machine. Energetic. The power coefficient is kept constant. The energy ratio are conserved.

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 47 / 70 Table of Contents

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 V 2 energy . Following the ﬁrst principle of thermodynamics, we have: 2 De D(V 2/2) Dq Dw + = + e Dt Dt Dt Dt |{z} | {z } heat exchange work of external forces

Total energy balance

De Dq Dw 0 = + e Dt Dt Dt |{z} | {z } heat exchange 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 power writes P = R ρ~f .V~ dv ev Dm The surface forces due to the stress tensor and denoted σ.¯ ~n. Their power can be written as: Z Pes = σ.¯ V~ dS ∂Dm

where σ¯ = −p¯I + τ¯. We therefore have: Z Z Z Pes = div(τ¯V~ )dv − pdiv(V~ )dv − V~ .grad~ (p)dv Dm Dm Dm | {z } | {z } Compressibility Transport

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 51 / 70 External forces work

So that

Dw 1 p V~ e = ~f .V~ + div(τ¯V~ ) − div(V~ ) − .grad~ (p) Dt ρ ρ ρ Obtaining

Dw 1 D(p/ρ) 1 ∂p e = ~f .V~ + div(τ¯V~ ) − + Dt ρ Dt ρ ∂t

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 52 / 70 Total enthalpy balance

Total enthalpy balance

Dh Dq Dw 0 = + u Dt Dt Dt |{z} | {z } heat exchange eﬀective work

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 53 / 70 Eﬀective work

Obtaining Dw 1 1 ∂p u = ~f .V~ + div(τ¯V~ ) + Dt ρ ρ ∂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 Z Dwu Pu = ρ dv Dm Dt Replacing the eﬀective work by our previous ﬁndings we have Z Z Z Z Dwu ∂p Pu = ρ dv = ρ~f .V~ dv + div(τ¯V~ )dv + dv Dm Dt Dm Dm Dm ∂t

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 55 / 70 Eﬀective power

The following hypothesis can be made that simplify the previous expression 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: Z Pu ≈ ρ~f .V~ dv Dm showing that the effective power is indeed equal to the power exchanged between the flow and the machine.

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 56 / 70 Link between mechanical eﬀective powers

Obtaining Z Pu − Pe = pV~ .~nds ∂D1 U ∂D2 | {z } Flowtransferpower

This term represents the power necessary to impose a given ﬂow rate between 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 Z Dwu Pu = ρ dv Dm Dt Z Z ∂ρwu Pu = dv + ρwuV~ .~nds Dm ∂t ∂Dm Assuming the system is in a steady state, the eﬀective work is constant at the inlet and outlet, and since the velocity is zero on solid boundaries, we have Z Pu = ρwuV~ .~nds ∂D1 U ∂D2 Pu = −mw˙ u1 +mw ˙ u2 =m ˙ ∆wu

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 58 / 70 Kinetic energy / work of internal forces

DV 2/2 Obtaining from ρ Dt Dw DV 2/2 Dw i = − e Dt Dt Dt Dw 1 i = − σ¯ : D¯ Dt ρ

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 kinetic energy and use the fact that σ¯ = τ¯ − p¯I

DV 2/2 1 1 V~ = ~f .V~ + div~ (τ.¯ V~ ) − τ¯ : D¯ − .grad~ (p) Dt ρ ρ ρ

~f .V~ represents the work done by the volume forces (the machine) 1 ~ ¯ ~ ρ div(τ.¯ V ) represents the work done by the viscous forces 1 ¯ ¯ − ρ τ¯ : D is the dissipation of kinetic energy due to the viscosity V~ ~ ρ .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 for the total energy, one obtains the conservation equation for the internal energy which writes: De Dq Dw = − i Dt Dt Dt |{z} |{z} heat exchange work of internal forces TRICKY De Dq 1 D(1/ρ) = + τ¯ : D¯ − p Dt Dt ρ Dt |{z} | {z } | {z } heat exchange mechanical dissipation compression work

mechanical/viscous dissipation We see that the mechanical dissipation decreases kinetic energy (slows down the ﬂuid) and increases the internal energy (heats up the ﬂow). This term 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 D(p/ρ) systems), we add Dt on both sides of the previous equation Dh Dq 1 1 Dp = + τ¯ : D¯ + Dt Dt ρ ρ Dt |{z} | {z } heat exchange mechanical dissipation

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-states all corresponding to the same macro-state. Sb = kblnΩ Following Gibbs, the entropy is a state function that can be computed knowing the thermodynamic state of the system. They both refer to the same quantity but it has only been deﬁnitely proven in 1965. Second principle of thermodynamics The second principle of thermodynamics states that entropy of a closed system 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 supposed to be a closed system). The universe can be split between the system of interest and its surrounding. In this case we have

dSsystem + dSsurrounding >= 0

So that if the entropy of the system decreases, the entropy of the surrounding must have increased by a larger quantity. Let’s now come back to Clausius/Gibbs deﬁnition of entropy. It is deﬁned as follows: Gibbs Equation

1 Tds = dh − dp ρ

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 64 / 70 Physical interpretation

From the previous equation one can write that:

Ds De D1/ρ T = + p Dt Dt Dt Using the conservation equation for the internal energy, one can write that Ds Dq 1 T = + τ¯ : D¯ Dt Dt ρ |{z} | {z } heat exchange 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 be radiation, convection or conduction), the mechanical dissipation

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 65 / 70 Physical interpretation

Obtaining

 2 Ds 2µ X X 2 2λ X ∂Vj T = Dij +   ≥ 0 Dt ρ ρ ∂xj i j j

which shows that the entropy can only grow in a ﬂuid that ﬂows without exchanging 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

Ds0 Dh0 1 Dp0 T0 = − Dt Dt ρ0 Dt

Stagnation entropy is equal to static entropy Because one goes from the static to the stagnation state by an isentropic deceleration, stagnation entropy and static entropy are equal.

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 67 / 70 Thermodynamics – Summary – energy/enthalpy

De Dq Dw 0 = + e Dt Dt Dt |{z} | {z } heat exchange work of external forces Dw 1 p V~ e = ~f .V~ + div(τ¯V~ ) − div(V~ ) − .grad~ (p) Dt ρ ρ ρ Dh Dq Dw 0 = + u Dt Dt Dt |{z} | {z } heat exchange eﬀective work Dw 1 1 ∂p u = ~f .V~ + div(τ¯V~ ) + Dt ρ ρ ∂t Z Z Dwu Dwe Pu − Pe = ρ − dv = pV~ .~nds D Dt Dt ∂D1∪∂D2 | {z } Flow transfer power

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 68 / 70 Thermodynamics – Summary – works

DV 2/2 Dw = t Dt Dt |{z} work of all forces Dw 1 1 V~ t = ~f .V~ + div~ (τ.¯ V~ ) − τ¯ : D¯ − .grad~ (p) Dt ρ ρ ρ Dw Dw Dw t = e + i Dt Dt Dt |{z} | {z } |{z} work of all forces work of external forces work of internal forces Dw 1 i = − σ¯ : D¯ Dt ρ

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 69 / 70 Thermodynamics – Summary – entropy

Ds Dh 1 Dp T = − Dt Dt ρ Dt Ds Dh0 1 Dp0 T0 = − Dt Dt ρ0 Dt Ds Dq 1 T = + τ¯ : D¯ Dt Dt ρ |{z} | {z } heat exchange mechanical dissipation

Alexis Giauque (LMFA/ECL) Turbomachinery Aero-Thermodynamics I Ecole Centrale Paris 70 / 70