Lecture 4 - Deflagations Detonations › sites › cefrc › files › Files...Lecture 4 Deﬂagrations and Detonations 1 heat conduction species diﬀusion viscous dissipation chemical

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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. 1

Lecture 4 Deflagrations and Detonations

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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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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

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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

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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

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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

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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

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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

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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

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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.

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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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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


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).

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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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Lecture 4 - Deflagations Detonations › sites › cefrc › files › Files...Lecture 4 Deﬂagrations and Detonations 1 heat conduction species diﬀusion viscous dissipation chemical