detonation waves and velocities

8/12/2019 Detonation Waves and Velocities

1/22

State 2 for Detonation: The upper Chapman-Jouguet point

Increase in pressure, decrease in velocity to sonic speedacross a detonation wave.

Detonation velocities

Structure of Detonation Waves: ZND Model

Unburnednx,1

Burned

r1, P1, T1,c1, Ma1 r2, P2, T2, c2, Ma2

nx,2= c2=

L19: Detonation Waves and Velocities

2RT


2/22

Detonations and Deflagrations: Comparison

Typical values for detonations and deflagrations are shown above

(Turns, Table 16.1, p. 617). Ma1 is prescribed to be 5.0 for normalshock. For normal shock and deflagration for each P2/P1 a unique

normal Ma1 exists based on combined conservation of mass and

momentum. For detonation, a range exists based on the heat

release rate.

Property Normal

Shock

Detonation Deflagration

Ma1 5.0 5-10 0.001

Ma2 0.42 1.0 0.003

nx,2/nx,1 0.2 0.4-0.7 7.5

P2/P1 29 13-55 1

T2/T1 5.8 8-21 7.5

r2/r1 5.0 1.7-2.6 0.13


3/22

Definition of Detonation Velocity

The speed at which the unburned mixture enters the

detonation wave approximated as one dimensional for an

observed riding with the one dimensional detonation wave Bydefinition: and velocity of burned gases =nx,2

1 ,1 2 22 2

1 1 ,1 2 2 ,22 2

1 ,1 2 ,2

1 1 1 2 2 2

/ (1)

(2)

/ 2 / 2 (3)

; (4)

x x

x x

x x

m A m v c

P v P v

h v h v

P RT P RT

r r

r r

r r

Unburnednx,1

Burned

r1, P1, T1,c1, Ma1 r2, P2, T2, c2, Ma2

,1D xv v

nx,2= c2= 2RT


4/22


5/22

Shock Wave: Energy Equation

2 2

1 ,1 2 ,2

, ,1 2

1 1 2 2

2 2

2 1 ,1 ,2

2 2

2 1 ,1 ,2

( ) / 2 ( ) / 2

;

( ) ( ) ( ) / 2 /

/ ( ) / 2

p ref x p ref x

O O

i f i i f i

ref ref x x p p

p x x p

c T T v q c T T v

q Y h Y h

P RT P RT

T T T T v v c q c

T T q c v v c

r r

2 2

1 ,1 2 ,2/ 2 / 2 (3)x xh v h v


6/22

Shock Wave: Energy Equation: KE in terms of Props.

2 2

2 1 ,1 ,2

2

2 1 2 1 2

2

1 2

2

1

12

,1 1

/ ( ) / 2

/ (( / ) 1) / 2

1/ ( / 2 ) ( ) 1

2/

1

,

2( 1) ( / )

p x x p

p p

p p

p

D x p

T T q c v v c

T T q c RT c

T q c RT c

T q c

Finally

v v R T q c

r r

Also see variable specific heat based shock relations:16.26, 16.27, 16.28

P ti f th H i t C


7/22

Properties of the Hugoniot Curve

The Hugoniot curve is a plot of all possible values of (1/r2, P2)

for given values of q and (1/r1, P1). The point (1/r1, P1) is the

origin of the Hugoniot plot and is designated by the symbol A.

P2

1/r2


8/22

The Hugoniot curve can be divided into five regions by drawing

tangents to the curve from point A and by drawing horizontal

and vertical lines from point A. Region V can be eliminated

because it does not give us real intersections with any Raleighline. AU and AL are both Raleigh lines one corresponding to

a detonation and the other corresponding to a deflagration.

P2

1/r2


9/22

1

1 1 2 2 2 1

2

22 2 21

1 1 1 2 2 2 2 1

2

1 22

1 2

1 2 1

1 / 1 /1 0

v v v v

P v P v P v

vP P

rr r

r

rr r

r

r r

r

Applying the conservation of mass relation and the

conservation of momentum relation to Region V gives us

imaginary values for 1v


10/22

It turns out that usually the only physically realizable

conditions, as established by experiments, are the point U (M2

= 1) and region III (subsonic deflagration). Now we will showthat the Mach number is unity at point U. Begin with the

Hugoniot relation:

2 1 1 2 122 1 1 2

1 11

P Pq P P r r r r

Differentiate with respect to 1/r2for fixed q, P1, 1/r1:

22

2 2

21 2 2 1

2

101 1 1/

1 11/ 1/

2 1/ 2

dPPd

dPP P

d

r r

r rr


11/22

Rearranging and solving for :

2 1 22

2 2 1 2

2 1

2 1

2 1 2 1

2 1 1 2

2 1

1/ 2 1 1/ 1/ 1/

1 1

1 1/ 1 1/

1/ 1/ 1/ 1/

P P PdP

d

P P

P P P P

r r r r

r r

r r r r

2

21 /

dP

d r


12/22

At the Chapman-Jouget points U and L, the slope is also

given by the Rayleigh line which is tangent to the RH curve.

2 2 1

2 2 11/ 1 1

C J

dP P P

d r r r

P2

21r


13/22

Equating the two expressions for :

2 1 2 1 2 1

2 1 2 1 1 21 1 1/ 1/ 1/ 1/P P P P P P

r r r r r r

2 2/ (1/ )dP d r

And simplifying

2 1 2 1 2 1

2 1 2 1 2 1

2 2 1 2 2 1

2 2 1 2 2 1

2 1

2 2

2 1

1/

1/ 1/

1/

1/

1/

P P P P 1/

1/ 1/ P P

P 1/ 1/ P P

P 1/ 1/ P P

P PP

1/

r r

r r r r

r r r

r r r

rr r


14/22

But we already showed that:

2 2 1 2 2

2 2 2 2

2 1

2 22 2 2

2 2 2 22

2 2

2 2

1 1

P Pm v P

P P

v R T c

v c

r rr r

r

r r


15/22

P2

21/ r


16/22

At the Chapman-Jouget points U and L the speed of the

burned gases in a reference frame fixed to the combustion

wave is equal to the speed of sound (M2= 1). We can alsoobtain an expression for the Mach number of the unburned

gases in the reference frame attached to the combustion

wave. Rewrite conservation of mass and momentum in terms

of Mach number M1:

2 2 2 2 1

1 1

1 2

51 1

P Pm ur

r r

1 1 1 1 1

v

/

/p

c R T P

c c

r


17/22

Multiplying both the LHS and RHS by /(r1P1) we obtain:

2 2

2 2 21 11 1 12

1 1 1 1 1

2 12 1

1 1 1 2 1 2

2 12

1

1 2

/

1

1 1 1

1

1

v vv MP P c

P PP P

P

P PM

r

r r

r r r r r

r r


18/22

Consider now the velocity of the burned gases V2in the

laboratory frame. Velocity of unburned gases V1= 0, and

velocity of the combustion wave Vw= V1. In the diagram belowthe velocities Vwand V2are positive in the direction shown:

1

2 2

w

w

V V

V V v

Unburned

Vw

Burned

V2

r1, P1, T1 r2, P2, T2


19/22

Lets develop a relation between velocity difference and

density difference across the combustion wave:

2 2

2 2

2 2 1 1 2 1

2 1

2 1

2 1

1 1

m mv v m v m v

v v m

r rr r

r r


20/22

At the upper Chapman-Jouguet point we have:

2 1 2 1 1 2

2 1 2

1 1 0

0

v v m v v

V v v

r r

For a detonation, burned gases follow the combustion wave.

P2

1/r2


21/22

For the deflagration wave:

2 1 2 1 1 2

2 1 2

1 1 0

0

v v m v v

V v v

r r

P2

1/r2

For a deflagration, burned gases move away from the

combustion wave.

Z ld i h N d D i i th l 1940'


22/22

Zeldovich, von Neumann, and Dring in the early 1940's

independently formulated similar theories of the structure of

detonation waves. The structure is shown in the diagram below:

20

10

1

P/P1

T/T1

r /r1

Reaction ZoneNormalShock

InductionZone

1 1' 1" 2

detonation waves and velocities

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