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6/26/11
Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 1
Lecture 4 Deflagrations and Detonations
1
heat conductionspecies diffusionviscous dissipationchemical reactions
p0ρ0T0
Yi0
p∞ρ∞T∞Yi∞
v0 v∞
plane, steady, one-dimensional flow involving exothermic chemical reactions inwhich the properties become uniform as |x| → ∞.
One-dimensional flow
2
coordinate attached to a wave propagating to the left at speed v0
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Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior wri@en permission of the owner, M. Matalon. 2
D
Dt=
∂
∂t+ v
∂
∂xsteady, one dimensional
3
d
dx(ρv) = 0
d
dx(ρv) = −dp
dx+
4
3µd2v
dx2
d
dx(ρvh) = v
dp
dx+
4
3µ
�dv
dx
�2
− dq
dx
d
dx(ρvYi) +
d
dx(ρYiVi) = ωi i = 1, 2, . . . , N
p = ρRT/W
d
dx(ρv) = 0
d
dx
�p+ ρv2 − 4
3µdv
dx
�= 0
d
dx
�ρv(h+
1
2v2)− 4
3µv
dv
dx+ q
�= 0
d
dx(ρvYi + ρYiVi) = ωi
ρv = const.
p+ ρv2 − 43µ
dv
dx= const.
ρv(h+ 12v
2)− 43µv
dv
dx+ q = const.
d
dx(ρvYi + ρYiVi) = ωi
4
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q = −λdT
dx+
N�
i=1
ρYihoiVi
ρYiVi = −ρDidYi
dx
h =N�
i=1
Yihoi +
� T
T o
cpdT
p = ρRT/W
[ρv]+∞−∞ = 0
�p+ ρv2 − 4
3µdv
dx
�+∞
−∞= 0
�ρv(h+ 1
2v2)− 4
3µvdv
dx+ q
�+∞
−∞= 0
�d
dx(ρvYi + ρYiVi)
�+∞
−∞= [ωi]
+∞−∞
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h∞ − h0 =N�
i=1
(Yi∞−Yi0)hoi + cp(T∞ − T0)
= −Q+ cp(T∞ − T0)
assume cpi≡ cp and Wi ≡ W for all i (or take some average value)
Q is the heat of combustion per unit mass of combustible mixture
R/W =γ − 1
γcpnote that cp − cv = R/W implies
ρ0v0 = ρ∞v∞
p0 + ρ0v20 = p∞ + ρ∞v2∞
h0 +12v
20 = 1
2h∞v2∞
ωi0= 0 , ωi∞ = 0 i = 1, 2, . . . , N
p0/ρ0T0 = p∞/ρ∞T∞ = R/W
6
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after some manipulations (and use of the equation of state to eliminate T )
let m = ρ0v0 = ρ∞v∞
they determine the relation between the end and initial states
p∞ − p01/ρ∞ − 1/ρ0
= −m2
γ
γ − 1
�p∞
ρ∞
− p0ρ0
�− 1
2
�1
ρ∞
+1
ρ0
�(p∞ − p0) = Q
Rayleigh Line
Hugoniot curve
relation between thermodynamic variables only
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µ = m2/p0ρ0 , q = Q/(p0/ρ0) = Q/(γa20)
p− 1
v − 1= −µ
p =2q + γ+1
γ−1 − vγ+1γ−1v − 1
upstream state corresponds to p = v = 1
(also the specific volume)
Dimensionless variablesp = p∞/p0 , ρ = ρ∞/ρ0 , v = v∞/v0
µ = γM20 where M0 = v0/a0 is the upstream Mach number
provide p, v (and consequently ρ, T ) for a given µ
possible downstream states are intersections of the Hugoniot and Rayleigh line
8
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p
v = 1/ρ 1
1
q
q = 0
Hugoniot curves
Hugoniot is a rectangular hyperbola that asymptotes to
v = (γ − 1)/(γ + 1) as p → ∞p = −(γ − 1)/(γ + 1) as v → ∞
solutions restricted to
γ−1γ+1 ≤ v ≤ 2q + γ+1
γ−1
0 ≤ p ≤ ∞
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p
v = 1/ρ 1
1
solutions further restricted to
γ−1γ+1 ≤ v < 1 and 1+(γ − 1)q ≤ p ≤ ∞
1+ γ−1γ q < v ≤ 2q + γ+1
γ−1 and 0 ≤ p < 1
Deton
ations
Deflagrations
Ralyleigh line always goes through (1,1) and has a negative slope
10
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p
v = 1/ρ 1
1
X
shock
the other solution, for a subsonic incident wave, would correspond to ararefaction wave, but is not acceptable because it violates the 2nd law.
there is only one solution for each supersonic incident velocityit is a shock wave, with p, ρ and T increasing behind the shock,the gas slows down (v < 1) behind the shock and is subsonic.
q = 0
11
1
p
v = 1/ρ
1
strongdetona
tion
weak
deton
ation
weak
deflagra
tion
strong
deflagration
CJ
CJ
q > 0
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Deflagrations
there is a maximum value of µ and, as a consequence, a maximum wave speedcorresponding to Chapman-Jouguet (CJ) deflagration;
for µ < µCJ there are two solutions, weak and strong deflagrations
Detonations
there is a minimum value of µ and, as a consequence, a minimum wave speedcorresponding to Chapman-Jouguet (CJ) detonation
for µ > µCJ there are two solutions, strong and weak detonations
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passing through a detonation wavev < 1 the gas slows down
ρ > 1 the gas is compressedp > 1 the pressure increases
passing through a deflagration wavev > 1 the gas speeds upρ < 1 the gas expands
p < 1 the pressure decreases
−v0←−−− u = v∞ − v0u = 0v∞−→v0 −→
as seen by an observer moving with the wave as seen in the laboratory frame
u = v + Vwave is the fluid particle velocityu < 0 for detonations and u > 0 for deflagrations
t
x
front
particle path
Unburned
t
x
front
particle path
Unburned
Detonations Deflagrations
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Deflagrations:
Expansion waves that propagate subsonically (0 < M0 < 1) The burned gas behind the front expands and accelerates away from the front In a frame attached to the wave, the downstream flow beyond CJ waves is sonic (M∞ = 1); M∞ < 1 for weak deflagrations, M∞ > 1 for strong deflagrations
Strong deflagrations are not possible (there is no structure that connects
the burned and unburned states) There is a unique solution for the (weak) deflagration, giving a definite wave speed for each gas mixture Weak deflagrations (flames) travel typically at 10 - 100 cm/s and p∞/p0 ~ 1, T∞/T0 ~ 6, ρ∞/ρ0 ~ 1/6
15
Detonations:
Compression waves that propagate supersonically (1 < M0 < ∞)
The wave retards the burned gas that compresses behind the front
In a frame attached to the wave the downstream flow beyond CJ
waves is sonic (M∞ = 1).
M∞ > 1 for weak detonations; M∞ < 1 for strong detonations
A strong detonation can be produced by driving a piston at an
appropriate speed into the mixture
There is only one wave speed at which a weak detonation can
propagate (but weak detonations are seldom observed)
Self-sustained detonations are CJ; in a hydrogen/air mixture
v0 ~ 2000 – 3000 m/s and p∞/p0 ~ 20, T∞/T0 ~ 10, ρ∞/ρ0 ~ 2
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Comment
The Rankine-Hugoniot analysis provides, for given conditions, the state of thegas downstream, but does not determine (except for the CJ waves) the wavespeed v0.
The possible existence of a final state based on this analysis does not guar-antee the existence of the solution. One needs to ensure that the initial andfinal states are connected smoothly by a solutions of the differential equationsand that this solution is stable.
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Deflagrations
S
!
heat
coldhot
• Thermal di!usivity ! ! 10"5 m2 s"1,
• Chemical time tc ! 10"3 s
• S !!
!/tc ! cm/s, " !#
!tc < 1mm, "p/p $ 1
• Slow, subsonic, nearly-isobaric, di!usion-controlled
• Power density ! 102 watt/cm2
1
S
Thermal diffusivity Dth ∼ 10−5 m2/s
Chemical time tc ∼ 10−5 s
Speed S ∼�Dth/tc
Wave thickness δ ∼√Dthtc
S ∼ 10− 100cm/sp∞/p0 ∼ 1, T∞/T0 ∼ 6
Deflagrations
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Deflagrations (weak) are subsonic waves, so that disturbances behind the wavecan propagate ahead and affect the state of the unburned gas before arrival ofthe wave (this depends, of course, on the rear end boundary conditions, if openor close, etc. . . ).
The expansion of the products can therefore cause a displacement of the re-actants so that the wave moves faster than the flame speed (burning velocityrelative s quiescent mixture).
Because of intrinsic instabilities, turbulence, obstacles, etc. . . the deflagra-tion wave may accelerate and turn into a detonation, known as Deflagration toDetonation Transition (DDT).
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DetonationsThe ZND structure
Detonations
!!rINDUCTIONFIRE LEAD SHOCK
P
D
ZND structure (shock followed by fast flame)
Propagation via autoignition caused by shock compression.
Rapid, violent, spectacular.
In high-energy condensed explosives,
• shock pressures up to 500,000 atmospheres
• detonation speeds approaching 10 km/s
• temperatures up to 5,500! K
• power approaching 20 " 109 watts per cm2
Deflagrations to heat, detonations to push.
2
D
Zel’dovich, von Neueman, Doring
shock followed by a fast flame
Ahead of the wave - gas is quiescent, insignificant reactionPassage through the lead shock the gas is compressed and its temperature risesthousands of degreesThe ensuing chemical reaction goes to completion very rapidly in a thin reactionzone (or fire) behind the shock
The ZND solution is an exact solution of the reactive Euler equations
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Steady Propagating WavesSteady Travelling Waves
Du
u
Laboratory Frame Wave-fixed Frame
u0 = 0
u0 = D
State upstream (wave-fixed frame)
p0, v0 (= 1/!0), u0 (= D), "0 (= 0)
State within
p, v (= 1/!), u, "
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p, v (= ρ−1), u, λ
State within
p0, v0 (= ρ−10 ), u0, λ0(= 0)
State ahead
Here λ is a progress variable, that varies from λ = 0 in the fresh unreacted gas
to λ = 1 when reaction is complete, and u is the particle velocity.
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Steady Travelling-Wave Profiles: Weak, Strong and Hybrid
v v0
p
p0
! = 1
! = 0
C
N
N1
A
B
S
W
0
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
p
!
D = DCJ
D > DCJ
weak
strong
A
B
12
v v0
p = p0 = −ρ20D2(v − v0)
The two extreme Hugoniot curves correspond to λ = 0 and λ = 1.
For a given D, all states within the reaction zone must lie on the corresponding
Rayleigh line.
Since the Hugoniot reaction must be satisfied, the portion of the Rayleigh line
relevant is bounded by the extreme Hugoniot corresponding to λ = 0, 1.
22
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Starting with an initial state, the state of the gas will jump to the point N1
along the shock-Hugoniot (corresponding to λ=0) as it traverse the led shock.
As the particle reacts, λ increases and the state of the particles slides down
to S along the Rayleigh line crossing Hugoniot curves with varying λ.
At the end of the reaction zone, λ=1, and the particle reaches the state S.
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Steady Travelling-Wave Profiles: Weak, Strong and Hybrid
v v0
p
p0
! = 1
! = 0
C
N
N1
A
B
S
W
0
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
p
!
D = DCJ
D > DCJ
weak
strong
A
B
12
v v0
Steady Travelling-Wave Profiles: Weak, Strong and Hybrid
v v0
p
p0
! = 1
! = 0
C
N
N1
A
B
S
W
0
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
p
!
D = DCJ
D > DCJ
weak
strong
A
B
12
v v0
The lowest possible Rayleigh line is the one tangent to the Hugoniot, corre-
sponding to λ=1. The final state in this case will be C, the CJ state and the
propagation speed in this case is DCJ.
ZND structure is not possible for weak detonations.
ZND structure does not restrict the propagation speed D for strong detona-
tions. Therefore, wave speeds depend on the experimental configuration, or on
the rear BCs.
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We denote by u∗ the particle speed at the end of the reaction zone and u∗CJ the
corresponding value for the CJ detonation.
up > u∗CJ
D
x
reaction zone
up
u∗
The detonation speed D is chosensuch that up = u∗(D) steady following flow
The same qualitative picture remainsas up is reduced towards u∗
CJ
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The rear BC can be thought to be a hypothetical piston following the wave.The question is to determine D for a given piston velocity up.
D
x
reaction zone
up
constant state
up < u∗CJ
As before, u∗ is the particle speed at the end of the reaction zone and u∗CJ the
corresponding value for the CJ detonation.
u∗CJ
rarefaction The detonation wave (includingthe reaction zone) propagates at DCJ .
The following flow must be reduced to match the BC. this corresponds to atime-dependent rarefaction wave, which is then followed by a constant state (asnecessary).
26
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The conditions across the shock follow from the RH relations. The spatial
distribution behind the shock is determined from
which in a frame attached to the shock is given by
(u+D)dλ
dx= k(1− λ)ne−R/RT
Dλ
Dt= r r = k(1− λ)ne−R/RT
and can be integrated to give
x =
� λ
0
[u(λ) +D]
k(1− λ)neR/RT dλ
and when λ = 1 we find the end state.
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