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State 2 for Detonation: The upper Chapman- Jouguet point Increase in pressure, decrease in velocity to sonic speed acr

L19: Detonation Waves and Velocities. State 2 for Detonation: The upper Chapman- Jouguet point Increase in pressure, decrease in velocity to sonic speed across a detonation wave. Detonation velocities Structure of Detonation Waves: ZND Model. Unburned n x,1. Burned. n x,2 = c 2 =.

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State 2 for Detonation: The upper Chapman- Jouguet point Increase in pressure, decrease in velocity to sonic speed acr

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  1. L19: Detonation Waves and Velocities • State 2 for Detonation: The upper Chapman-Jouguetpoint • Increase in pressure, decrease in velocity to sonic speed across a detonation wave. • Detonation velocities • Structure of Detonation Waves: ZND Model Unburned nx,1 Burned • nx,2 = c2 = r2, P2, T2, c2, Ma2 r1, P1, T1,c1, Ma1

  2. 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 normal shock. 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.

  3. 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 By definition:and velocity of burned gases =nx,2 Burned Unburned nx,1 • nx,2 = c2 = r2, P2, T2, c2, Ma2 r1, P1, T1,c1, Ma1

  4. Shock Wave: Density Ratio- Specific Heat Ratio

  5. Shock Wave: Energy Equation

  6. Shock Wave: Energy Equation: KE in terms of Props. Also see variable specific heat based shock relations: 16.26, 16.27, 16.28

  7. 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. • 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 Raleigh line. AU and AL are both Raleigh lines one corresponding to a detonation and the other corresponding to a deflagration. P2 1/r2

  9. • Applying the conservation of mass relation and the conservation of momentum relation to Region V gives us imaginary values for

  10. • 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 show that the Mach number is unity at point U. Begin with the Hugoniot relation: • Differentiate with respect to 1/r2for fixed q, P1, 1/r1:

  11. • Rearranging and solving for :

  12. • At the Chapman-Jouget points U and L, the slope is also given by the Rayleigh line which is tangent to the RH curve. P2

  13. • Equating the two expressions for : • And simplifying

  14. • But we already showed that:

  15. P2

  16. • 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 also obtain 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:

  17. Multiplying both the LHS and RHS by g/(gr1P1) we obtain:

  18. • Consider now the velocity of the burned gases V2 in the laboratory frame. Velocity of unburned gases V1 = 0, and velocity of the combustion wave Vw = V1. In the diagram below the velocities Vw and V2 are positive in the direction shown: Unburned Vw Burned V2 r1, P1, T1 r2, P2, T2

  19. • Let’s develop a relation between velocity difference and density difference across the combustion wave:

  20. • At the upper Chapman-Jouguet point we have: • For a detonation, burned gases follow the combustion wave. P2 1/r2

  21. • For the deflagration wave: • For a deflagration, burned gases move away from the combustion wave. P2 1/r2

  22. • Zeldovich, von Neumann, and Döring in the early 1940's independently formulated similar theories of the structure of detonation waves. The structure is shown in the diagram below: Induction Zone Normal Shock Reaction Zone 20 P/P1 10 T/T1 r /r1 1 1 1' 2 1"

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