Qualitative differences between detonations and deflagrations .

1. L18: Detonations and Deflagrations: C-J Analysis • Qualitative differences between detonations and deflagrations. • Derivation of the Rankine-Hugoniot relations. • Properties of the Hugoniot curve. Unburned nx,1 Burned • nx,2 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. Conservation of Mass in Two Flow Regimes • Consider a flame (term generally used for deflagration) or a combustion wave (term generally used for a detonation) in the laboratory frame of reference. Velocity of unburned gases nx,1 and velocity of burned gases = nx,2 Unburned nx,1 Burned • nx,2 r2, P2, T2, c2, Ma2 r1, P1, T1,c1, Ma1

4. Rayleigh Line: Pressure versus Specific Volume • Combine Eqs. (1) and (2) to give the Rayleigh line relation: Slope Ordinate Abscissa Constant

5. Rayleigh Line: Pressure versus Specific Volume P= 300 kPa P= 200 kPa Physically inaccessible region B P= 100 kPa Physically Inaccessible region A 0.215 0.86 m3/kg 0.43

6. Velocity difference (Momentum equation) • Now we will work with Eqn. (5) to derive some expressions that will be useful for later analysis of detonation and deflagration waves. Derive an expression for nx,2-nx,1: • From Table 16.1 from the text book we can see that for a detonation r2 > r1, while the reverse is true for a deflagration. Thus the sign of nx,2-nx,1 is negative for a detonation and positive for a deflagration.

7. Expressionfor (nx,1)2 - (nx,2)2 Momentum Eqn. • Now let’s derive an expression for (nx,1)2- (nx,2)2:

8. Derivation of the Rankine- Hugoniot RelationsEnergy Equation and Ideal Gas Law

9. Derivation of the Rankine- Hugoniot RelationsCombineEnergy Equation and Momentum Equation • Now substitute Momentum Eqn. (7) into the energy equation, Eqn. (3): • Eqn. (8) is referred to as the Hugoniot relation.

10. Derivation of the Rankine- Hugoniot EquationHugoniot Relation, Perfect Gas, Constant MW, cP

11. Derivation of the Rankine- Hugoniot EquationHugoniot Relation, Perfect Gas, Constant MW, cP • For an ideal gas: • Rearrange Eqn. (9), and then use Eqn. (10) and the ideal gas equation Eqn. (4):

12. • Substitute into the Hugoniot Relation: Eqn. (8) results in the Rankine - Hugoniot Equation: Left Hand Side: Chemical Energy from the fuel-oxidizer reaction Right Hand Side Term 1: Sensible Energy change expressed in the form of pressure and density using deal gas law and specific heat relationships. Right Hand Side Term 2: Momentum balance dictates the kinetic energy change expressed in terms of pressure change times a term representing the effective change in volume. Since it is clear that conservation of mass, momentum, energy, and equation of state (zeroth and first law of thermodynamics) are already being used. What about the second law of thermodynamics? One way second law of Thermodynamics