Introduction to High-temperature
Thermodynamics and Its Applications to
High-speed Flows
Sergei Utyuzhnikov
University of Manchester
Moscow Institute of Physics & Technology
[email protected]
2nd International School on Non-Equilibrium and High-Temperature Flows,
Saint-Petersburg, 26-28 September, 2016
OUTLINE
• Introduction
- Importance of real gas effects
- Hypersonic flow past a re-entry vehicle
- Equilibrium and nonequilibrium chemical reactions
• Gas-surface interaction
• Multicomponent diffusion
• Equilibrium high-enthalpy flows
• Elements of statistical thermodynamics
- Vibrational nonequilibrium
2
IMPORTANCE of REAL GAS EFFECTS
3
IMPORTANCE of REAL GAS EFFECTS
Shift of the centre of pressure towards top
© Griffith et al, 1983
4
IMPORTANCE of REAL GAS EFFECTS
High-temperature shock layer
for a re-entry vehicle
© Anderson, 2003
5
PERFECT & IMPERFECT GASES
Calorically perfect gas: Cp and Cv are constant
h  c pT,
e  cvT.
Calorically imperfect gas: Cp and Cv are variable
h  h(T, P),
e  e(S, ).
6
EQUILBRIUM and NONEQUILIBRIUM CHEMICAL REACTIONS
Damköhler number (Da) =
typical gas dynamic time
___________________________
typical time of a reaction
7
EQUILBRIUM and NONEQUILIBRIUM CHEMICAL REACTIONS
Equilibrium reactions:
the rate of a reaction tends to infinity
Da >> 1
Frozen reactions:
the rate of a reaction tends to zero
Da << 1
Nonequilibrium reactions:
the rate of a reaction is limited
8
High-temperature shock layer
for a re-entry vehicle
9
Hypersonic shock layer
10
© C. Johnston, NASA
11
MULTIPHYSICS PROBLEMS
• Chemical nonequilibrium
- Dissociation, ionization, …
- Internal energy excitation
• Thermal nonequilibrium
- Internal energy relaxation
 Gas-surface interaction
 Turbulence
 Radiation
 Rarefied gas effects
12
Shock Layer Temperature
Calorically perfect gas model:
TW 
2T M 2
(  1)
(  1)2
H  53 km, M   25
TW  35000K
13
Shock Layer Temperature
Comparison of real-gas temperature
against perfect gas one
© Anderson, 2003
14
REAL-GAS EFFECTS at DIFFERENT TEMPERATURES
Dissociation and ionization absorb about
75% of molecular kinetic energy of highenthalpy flow!
15
Space Shuttle
16
SPACE-SHUTTLE TRAJECTORY
Velocity-altitude map
© Tauber, Meneses 1986
17
SPACE-SHUTTLE TRAJECTORY
The influence of different chemical processes
© Sarma, 2000
18
GAS-SURFACE INTERACTION
Recombination:
O  O  A  O2  A
N  N  A  N2  A
The reactions of recombination are exothermic: Δh > 0
19
GAS-SURFACE INTERACTION
Recombination:
O  O  A  O2  A
N  N  A  N2  A
Catalytic activity:
i
Ni,rec

Ni
20
CATALYTIC ACTIVITY
i
Ni,rec

Ni
Ni is the number of atoms of species
impinging the surface per unit time.
i
Ni,rec is the number of atoms of species i
recombining at the surface.
21
CATALYTIC ACTIVITY
i
Ni,rec

Ni
γi = 1: fully catalytic surface.
γi = 0: noncatalytic surface.
22
GAS-SURFACE INTERACTION
© Barbante, 2015
23
24
CATALYTIC ACTIVITY
© VKI
25
GAS-SURFACE INTERACTION
© T. Magin, VKI
26
BINARY DIFFUSION
dci
Ii

 i 
dt
x
i is the volume source
Ii is the diffusion of an i-th species
ci  i /  is an i-th mass fraction.
Binary mixture:
 c1
I1  1U 1    D 12
is Fick's law
x
U1 is the diffusion velocity
D12 is the coefficient of binary diffusion
27
MULTICOMPONENT DIFFUSION
Ii is the diffusion of an i-th species
U i is the diffusion velocity:Ii  iU i
28
MULTICOMPONENT DIFFUSION
V
i i
i
  i(Vi  V  V )  V 
i
Consider V 
cV
U
i
i
i
i i
i
29
BINARY DIFFUSION
The conservation law of mass:
 iU i  0
i
Hence: D12  D21
30
MULTICOMPONENT DIFFUSION
dci
Ii

 i 
dt
x
i is the volume source
Ii is the diffusion of an i-th species
ci  i /  is an i-th mass fraction.
Ii
Dim
c i
   D im
is Fick's law
x
is the coefficient of multicomponent diffusion
" m" means mixture
31
BINARY DIFFUSION. HEAT FLUX
T
J  
  D 12
x
T

  D 12(h1 
x
 c1
c 2 

 h2
 h1
 
x
x 

c
h2) 1
x
dc  T

J       D 12(h1  h2) 1 
dT  x

J   eff
eff
T
x
dc1
    D12(h1  h2)
dT
32
BINARY DIFFUSION. HEAT FLUX
J   eff
T
x
eff     D12(h1  h2)
dc1
dT
Thus, in a gas mixture the efficient
heat
conductivity
necessarily
coincide
does
with
the
not
real
heat conductivity!
The
effect
of
the
multicomponent
diffusion can be significant: up to
30% of heat flux for Space Shuttle.
33
EQUILIBRIUM HIGH - ENTHALPY FLOWS
Quasi-perfect Equation of State
Perfect gas:
  const
h 
p
(  1)
Calorically imperfect gas: h = ?
35
EQUILIBRIUM AIR
air
CO2
© Lunev, 2009
36
Quasi-perfect Equation of State
Perfect gas:
  const
h 
p
(  1)
Calorically imperfect gas:
h  h(p,  )
37
Quasi-perfect Equation of State
Perfect gas:
  const
h 
p
(  1)
Calorically imperfect gas:
h
*

p
*  1
38
Efficient Ratio of Heat Capacities
© Lunev, 2009
h
*

p
*  1
39
Shock Layer Temperature
TW 
2T M 2
 *( *  1)
( *  1)2
H  53 km, M   25
  1.4 : Tw  35000K
*  1.1 : Tw  9000K
40
REAL-GAS EFFECTS at DIFFERENT TEMPERATURES
Dissociation and ionization absorb about
75% of molecular kinetic energy of highenthalpy flow!
41
ELEMENT CONTENT in AIR
Reactions:
© Barbante, 2015
42
HEAT CAPACITY
© Barbante, 2015
43
Ratio of Heat Capacities
γc, real c.
γ*, efficient c.
γf, “frozen” c.
γe, equilibrium c.
 e  a2 / p
a is the speed of sound
© Lunev, 2009
103
2*103
3*103
44
Equilibrium Air
© Lunev, 2009 1: 0.01bar; 2: 1 bar; 3: 100bar; 4: 0.1bar
45
ELEMENTS of STATISTICAL
THERMODYNAMICS
ATOM
47
MOLECULE
48
ROTATIONAL ENERGY
49
ROTATIONAL ENERGY
50
ENERGY MODES
51
ENERGY MODES
52
ENERGY MODES
53
BOLTZMANN ENERGY DISTRIBUTION
Population distribution or microstate:
 
N
i
i
i
N 
N
i
i
Translational energy:
m 

f  

 2kT 
3/2
e
 mV 2 /(2kT )
54
BOLTZMANN ENERGY DISTRIBUTION
Boltzmann distribution (equilibrium) is reached:
- after 3 collisions (on average) for translation
- after 3-4 collisions for rotation
- after 10^5 collisions for vibration
55
EQUIPARTITION of ENERGY
The equipartition theorem:
Each degree of freedom contributes the same
portion of energy:
i
kT

2
Each degree of freedom contributes the same
portion of energy per a unit mass:
ei
RT

2
56
TEMPERATURE
L is an arbitrary direction
(x,y,z) is the Cartesian coordinate system
mU y2
mU L2
mU x2
mU z2
kT




2
2
2
2
2
57
TRANSLATIONAL TEMPERATURE
The total energy of translation:
trans
3kT

2
Translational temperature:
2
2
mU
2 trans
2
s
T  mU L / k 

3 2k
3 k
58
MULTIPLE TEMPERATURES
2
2
mU
2 trans
2
s
T  mU L / k 

3 2k
3 k
Trot  rot / k
Tvib  vib / k
Te  e / k
59
ENERGY MODES
© Anderson, 2003
60
VIBRATIONAL ENERGY
High temperatures:
General case:
evib
h  / (kT )
 h  /(kT )
RT
e
 1
h is Planck’s constant:
h  6.6  1034J  s
61
HEAT CAPACITY (AIR)
e 
© Anderson, 2003
3RT
2

RT
cv

e

T |v
RT

7RT
2
62
VIBRATIONAL ENERGY
evib  evib   evib(0)  evib  et
63
GAS DYNAMIC LASER
64
GASDYNAMIC LASER
65
ENERGY MODES
© Anderson, 2003
66
GAS DYNAMIC LASER
67
SPACE-SHUTTLE TRAJECTORY
The influence of different chemical processes
© Sarma, 2000
68