detonation waves and velocities
Post on 03-Jun-2018
268 Views
Preview:
TRANSCRIPT
-
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
4/22
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
15/22
P2
21/ r
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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
-
8/12/2019 Detonation Waves and Velocities
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'
-
8/12/2019 Detonation Waves and Velocities
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
top related