# Engineering Acoustics/Detonation

## Detonation

A detonation wave is a combustion wave propagating at supersonic speeds. It is composed of a leading shock which adiabatically compresses the reactants, followed by a reaction zone which converts the reactants into products. During this process, a significant amount of heat is released, increasing the temperature and decreasing the pressure and density. The products are expanded in the reaction zone, giving the detonation a forward thrust.

In contrast, a deflagration wave, which can be thought of as a propagating flame, is a combustion wave which propagates at subsonic speeds.A deflagration consist of a precursor shock followed by a reaction zone. A deflagration propagates via heat and mass diffusion from the reaction zone to ignite the reactants ahead of it. Being a subsonic wave, information downstream can travel upstream and change the initial thermodynamic state of the reactants.

A qualitative difference between these two combustion modes are tabulated below.

Detonation Deflagration
Mach number ${\displaystyle u_{0}/c_{0}}$ 5-10 0.00001-0.03
Velocity ratio ${\displaystyle u_{1}/u_{0}}$ 0.4-0.7 4-6
Pressure ratio ${\displaystyle p_{1}/p_{0}}$ 13-55 0.98
Temperature ratio ${\displaystyle T_{1}/T_{0}}$ 8-21 4-16
Density ratio ${\displaystyle \rho _{1}/\rho _{0}}$ 1.7-2.6 0.06-0.25

According to this table, the burned products (with respect to a stationary combustion wave) experience a deceleration across a detonation wave,and an acceleration in a deflagration wave. The pressure and density rise across the detonation which is why a detonation wave is known as a compression wave. In contrast, the pressure decreases slightly across a deflagration, hence it is considered an expansion wave.

## Rayleigh line and Hugoniot curve

Let’s first treat the detonation as a black box that brings reactants of state 0 to products of state 1.

Figure 1: 1-D steady flow across a combustion wave

The basic conservation equations are applied to relate state 0 to state 1. Considering a one dimension steady flow across the combustion wave, the basic equations are:

Conservation of mass:

${\displaystyle \rho _{0}u_{0}=\rho _{1}u_{1}}$

Conservation of momentum:

${\displaystyle p_{0}+\rho _{0}u_{0}^{2}=p_{1}+\rho _{1}u_{1}^{2}}$

Conservation of energy:

${\displaystyle h_{0}+q+{\frac {u_{0}^{2}}{2}}=h_{1}+{\frac {u_{1}^{2}}{2}}}$

where ρ, p,u,h and q are the density, pressure, velocity, enthalpy and the difference between the enthalpies of formation of the reactants and the products, respectively.

Combining the conservation of mass and momentum, the following equation is obtained

${\displaystyle (p_{1}-p_{0})/(\nu _{0}-\nu _{1})=\rho _{0}^{2}u_{0}^{2}=\rho _{1}^{2}u_{1}^{2}={\dot {m^{2}}}}$,

where ${\displaystyle \nu }$ is the specific volume and ${\displaystyle {\dot {m}}=\rho u}$ is the mass flux per unit area. The mass flux can also be written as

${\displaystyle {\dot {m}}={\sqrt {(p_{1}-p_{0})/(\nu _{0}-\nu _{1})}}}$.

Remember that the mass flux must be a real number. Therefore, if the numerator is positive so must the denominator and vice versa.

If we define ${\displaystyle y=p_{1}/p_{0}}$ and ${\displaystyle x=\nu _{1}/\nu _{0}}$ then we can obtain the following equation

${\displaystyle {\dot {m^{2}}}=(y-1)/(1-x)}$

Substituting expressions for the speed of sound of the reactants ${\displaystyle c_{0}={\sqrt {\gamma _{0}p_{0}/\rho _{0}}}}$ and the Mach number of the combustion wave ${\displaystyle M=u_{0}/c_{0}}$, the above equation can be re-casted as

${\displaystyle y=(1+\gamma _{0}M_{0}^{2})-(\gamma _{0}M_{0}^{2})x}$

The above equation defining the thermodynamic path linking state 0 to state 1is also known as the Rayleigh line. Isolating ${\displaystyle u_{0}^{2}}$ or ${\displaystyle u_{1}^{2}}$ from the following equation

${\displaystyle (p_{1}-p_{0})/(\nu _{0}-\nu _{1})=\rho _{0}^{2}u_{0}^{2}=\rho _{1}^{2}u_{1}^{2}={\dot {m^{2}}}}$

we can eliminate the velocity terms from the energy equation and obtain the equation of the Hugoniot curve,

${\displaystyle h_{1}-(h_{0}+q)={\frac {1}{2}}(p_{1}-p_{0})(\nu _{0}-\nu _{1})}$.

The Hugoniot curve represents the locus of all possible downstream states, given an upstream state. It is also possible to express the Hugoniot curve with respect to the variables x and y, similar to the Rayleigh line, as

${\displaystyle y={\frac {{\frac {\gamma _{0}+1}{\gamma _{0}-1}}-x+{\frac {2q}{p_{0}\nu _{o}}}}{{\frac {\gamma _{0}+1}{\gamma _{0}-1}}x-1}}}$.

Note that the case where q=0 corresponds to a non reacting shock wave and the Hugoniot curve will pass by the point (1,1) in the x-y plane. When the Rayleigh line is tangent to the Hugoniot curve, the tangent points are called Chapman-Jouquet (CJ) points. The upper CJ point corresponds to the CJ detonation solution whereas the lower CJ point is referred as the CJ deflagration solution.

Figure 2: CJ detonation and deflagration solution

Note that the CJ theory does not take into account the detailed structure of the detonation wave. It simply links upstream conditions to downstream conditions via steady one dimensional conservation laws.

## Detonation wave structure

### ZND model

Assuming a one dimensional steady flow, the Zel’dovich, von Neumann and Döring (ZND) model is an idealized representation of the detonation wave. The model essentially describes the detonation wave as a leading shock followed by chemical reactions. The leading shock adiabatically compresses the reactants, increasing the temperature, pressure and density across the shock. An induction zone is followed where the reactants are dissociated into radicals and free radicals are generated. The induction zone is thermally neutral in the sense that the thermodynamic properties remain relatively constant. When enough active free radicals are produced, a cascade of reactions occurs to convert the reactants into products. Chemical energy is released resulting in a rise in temperature and a drop in pressure and density. The decrease in pressure in the reaction zone is further decreased by expansion waves and creates a forward thrust that will support the leading shock front. In other words, the proposed propagation mechanism of a detonation wave is autoignition by the leading shock which is supported by the thrust provided by the expansion of the products.

The variation of thermodynamic properties is illustrated in the following sketch.

Figure 3: Variation of properties across a ZND detonation wave

Although the ZND provides a description of the structure of the detonation wave, it does not consider any boundary conditions. In reality, both the initial conditions ( thermodynamic states, mixture composition) as well as boundary conditions ( geometry, degree of confinement, nature of walls) affect the detonation velocity. Under certain conditions, initial and boundary conditions can even make the propagation of detonations impossible. To-date, no quantitative theoretical model can accurately predict the limits of detonations.

### Experimental observations

Although the ZND models the detonation in a one-dimensional frame, experimental observations indicate that the detonation front is actually three-dimensional and unstable. Instabilities are manifested in both longitudinal ( pulsating detonation) and transverse directions. The front consists of many curved shocks composed of Mach stems and incident shocks. At the intersection between these curved shocks, reflected shocks, also referred to as transverse waves, extend into the reacting mixture. The intersection of these three shocks is referred as the triple point. These transverse waves move back and forth sweeping across the entire front. The trajectories of these triple points can be recorded on a sooth covered surface as the detonation passes by. The wave spacing can be measured from smoked foils and is referred to as the cell size.

A sketch of a simplified cellular detonation structure is shown below. λ represents the cell size.

Figure 4: Detonation front structure

## How to initiate a detonation

There are a few ways to initiate a detonation. Here are some examples:

• Deflagration-to-detonation transition (DDT): assuming that a deflagration has already been formed, it needs to accelerate to a certain velocity via turbulence. When conditions permit, the deflagration abruptly transits into a detonation.Few key processes of the development of the detonation wave are summarized as follows:
• Generation of compression waves ahead of the propagating flame
• Coalescence of the compression wave to form a shock wave.
• Generation of turbulence
• Creation of blast wave from a local explosion in the reaction zone resulting into a detonation “bubble”. A detonation bubble catches up to the precursor shock and an overdriven detonation is formed. Transverse pressure waves are generated.
• Direct initiation: Method of initiating a detonation bypassing the deflagration phase. A detonation may be formed directly through the use of a powerful ignition source.
• Diffraction of a planar detonation into a larger volume to form a spherical detonation.

## Limits of detonation

Detonation limits refer to the critical set of conditions outside of which a self-sustained detonation can no longer propagate. The detonation velocity is actually affected by initial conditions of the explosive mixture (thermodynamic states, composition, dilution, etc.) and the boundary conditions (degree of confinement, geometry and dimensions of confinement, type of wall surface, etc.). For example, given a set of thermodynamic states and a particular experimental apparatus, we change the mixture composition rich or lean. At a particular fuel concentration the detonation will cease to propagate. This type of limit approach reveals the composition limit of an explosive mixture. For given initial conditions, we can also vary the dimensions of the experimental apparatus. Detonation propagating inside a tube below a critical tube diameter, no detonation can propagate. This yields the critical tube diameter. It is worth noting that it becomes more difficult to initiate a detonation as the detonation limit is approached. The critical energy required for initiation increases exponentially. The limit is not a characteristic property of the explosive mixture since it is affected by both initial and boundary conditions.

Since the steady propagation velocity of the detonation depends on initial and boundary conditions, a common observation as the detonation limit is approached is the presence of velocity deficit. Studies in round tubes reveal a velocity deficit of about 15% that of CJ velocity before failure of the detonation wave. Near the limit, longitudinal fluctuation of the velocity has also been observed. Depending on the magnitude of these fluctuations and their duration, different unstable behaviour such as stuttering and galloping can be manifested.

Another characteristic indicating the approach of the limit is the detonation cell size compared to the tube dimension. Away from the limit, the detonation cell size is small compared to the dimensions of the detonation tube. As the limit is approached, there are less transverse waves and the wave spacing increases until a single transverse wave propagates around the perimeter of the tube, indicative of a spinning detonation. The magnitude of the transverse pressure wave oscillations become larger and larger as the limit is approached.

## Prevention

Since it takes less energy to initiate a deflagration, this is the mode of combustion most likely to occur in industrial accidents. Although the key factors required for the formation of a detonation or a deflagration ( explosive mixture and ignition source) may not be eliminated in a chemical plant, some prevention mechanisms have been developed to stop the propagation of a deflagration and prevent a detonation from forming. Here are a few examples:

• Inhibition of flames : once a flame is detected, flame suppressant is injected. The suppressant will combine with active radicals. By taking away the radical necessary for the chemical reactions to take place, the flame will cease to spread.
• Venting : to avoid formation of a detonation, any pressure build up is released. However, by actually releasing the pressure, the turbulence created can accelerate the flame.
• Quenching : it is possible to quench (or suppress) the flame, even near detonation velocities.

## References

• Lee, J.H.S.,The Detonation Phenomenon, Cambridge University Press, 2008
• Kuo, K.K.,Principles of Combustion, John Wiley and Sons, Inc. 2005, 2nd ed.
• Fickett, W. and Davis, W.C.,Detonation, University of California Press, 1979