Sirri Oguz – All About Pyrotechnics
LESSON 03 / DETONATION PHYSICS
Shock Hugoniots and Rayleigh Lines
3.1 The Shock Hugoniot
A shock Hugoniot is the locus of thermodynamic states accessible through a single shock from a specified initial state, subject to the jump conditions and equation of state. It is often plotted in pressure–specific-volume coordinates.
This is the Hugoniot energy equation. It relates the internal-energy difference to the change in pressure and specific volume caused by shock compression.
3.1.1 Physical Interpretation
From a given initial density and pressure, shocks of different strengths produce different compressed states. Their collection forms the shock Hugoniot. A stronger shock generally produces greater compression and higher pressure.
3.1.2 Hugoniot Versus Isentrope
An isentrope describes states connected through constant-entropy transformations. Shock compression is irreversible and increases entropy, so its Hugoniot differs from an isentropic compression path even if the paths start at the same state.
3.1.3 Inert and Reactive Hugoniots
The unreacted Hugoniot describes material response without completed chemical transformation; a reacted or equilibrium-products Hugoniot refers to chemically transformed states. Chemical energy release changes the accessible thermodynamic states. This distinction underlies CJ analysis.
3.2 Simple Examples: Inert and Reactive Hugoniots
Example 1 — Inert Hugoniot: Shock Compression of Aluminum
Imagine a shock wave traveling through a solid aluminum plate.
Figure 3.1 illustrates the pressure–relative-volume relationship for aluminum under increasing shock compression. As shock pressure increases, the material becomes progressively compressed, resulting in higher density, lower relative volume (V/V₀), and increased internal energy and temperature. The initial unshocked state corresponds to (V/V₀ = 1). The Hugoniot represents shock-compressed thermodynamic states without chemical reaction, although physical phase transformations may occur at sufficiently high pressures.
Figure 3.1. Inert Shock Hugoniot of Aluminum. Schematic pressure–relative-volume relationship illustrating the thermodynamic states produced by increasing shock compression of aluminum. The plotted values are illustrative and do not represent experimentally validated Hugoniot data.
Example 2 — Reactive Hugoniot: Detonation of TNT
Now consider a detonation wave propagating through TNT.
Two distinct physical processes occur:
- Initial shock compression: The leading shock compresses and heats the unreacted TNT. The state immediately behind the idealized shock is associated with the unreacted Hugoniot.
- Chemical reaction: Behind the shock, TNT decomposes into reaction products, releasing chemical energy. The fully reacted equilibrium states are described by a separate products Hugoniot.
The chemical energy release changes the relationship between pressure, specific volume, and internal energy.
Schematic pressure vs specific-volume diagram in Figure 3.2 illustrates the classical detonation construction for an energetic material. Point A represents the initial unreacted explosive state. The unreacted Hugoniot describes states reached by shock compression of TNT prior to substantial chemical reaction; point C represents the shock-compressed unreacted state immediately behind the leading shock. As chemical reactions proceed behind the leading shock, the material evolves from the shock-compressed unreacted state toward chemically transformed products. The products Hugoniot represents possible fully reacted equilibrium states, rather than the detailed thermodynamic path through the reaction zone. That path is addressed more explicitly by the ZND detonation model in Lesson 05. The straight Rayleigh line connects the initial state to the detonation solution, and the Chapman–Jouguet (CJ) point at B is the tangency point on the products Hugoniot corresponding to a self-sustaining detonation. The figure is conceptual and intended to illustrate the relationship among the initial state, shock-compressed state, reacted products, and CJ condition.
Figure 3.2. Reactive Hugoniot Construction for TNT Detonation. Schematic pressure–specific-volume (P–v) diagram illustrating the unreacted Hugoniot, equilibrium-products Hugoniot, and Rayleigh line. Point A represents the initial unreacted state, point C the shock-compressed unreacted state immediately behind the leading shock, and point B the Chapman–Jouguet (CJ) equilibrium-products state, where the Rayleigh line is tangent to the products Hugoniot. The diagram is conceptual and does not represent experimentally validated TNT data.
3.3 The Rayleigh Line
Momentum and mass conservation give the Rayleigh-line relation in pressure–specific-volume coordinates:
Equivalently:
Its slope is:
The Rayleigh line passes through the initial state (v₀, P₀) and has a slope of −m² on a pressure–specific-volume diagram. Each value of mass flux corresponds to a different Rayleigh line through the same initial state.
The intersection of a Rayleigh line with a Hugoniot identifies a state satisfying both the conservation constraints and the thermodynamic energy relationship. For reactive materials, the tangency condition between the Rayleigh line and the fully reacted products Hugoniot provides the classical geometric interpretation of the Chapman–Jouguet detonation state, discussed in Lesson 04.