# 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 $u_0/c_0$           5-10         0.00001-0.03
  Velocity ratio $u_1/u_0$        0.4-0.7      4-6
  Pressure ratio $p_1/p_0$        13-55        0.98
  Temperature ratio $T_1/T_0$     8-21         4-16
  Density ratio $\rho_1/\rho_0$   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.

\[\[Image:1d steady flow across combustion
wave.jpg\|center\|thumb\|400px\|

```{=html}
<center>
```
Figure 1: 1-D steady flow across a combustion wave

```{=html}
<center>
```
\]\]

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:

`<big>`{=html}`<big>`{=html}$\rho_0 u_0=\rho_1 u_1$`</big>`{=html}`</big>`{=html}

Conservation of momentum:

$p_0+\rho_0 u_0^2=p_1+\rho_1 u_1^2$

Conservation of energy:

$h_0+q+\frac{u_0^2}{2}=h_1+\frac{u_1^2}{2}$

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

$(p_1 - p_0)/(\nu_0 - \nu_1)=\rho_0^2u_0^2=\rho_1^2u_1^2= \dot{m^2}$,

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

$\dot{m} = \sqrt{(p_1 - p_0)/(\nu_0 - \nu_1)}$.

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 `<big>`{=html}$y= p_1 / p_0$`</big>`{=html} and
`<big>`{=html} $x= \nu_1 / \nu_0$`</big>`{=html} then we can obtain the
following equation

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

Substituting expressions for the speed of sound of the reactants
`<small>`{=html}$c_0= \sqrt{\gamma_0p_0/\rho_0}$`</small>`{=html} and
the Mach number of the combustion wave
`<big>`{=html}$M= u_0/c_0$`</big>`{=html}, the above equation can be
re-casted as

$y= (1+\gamma_0M_0^2)- (\gamma_0M_0^2)x$

The above equation defining the thermodynamic path linking state 0 to
state 1is also known as the Rayleigh line. Isolating $u_0^2$ or $u_1^2$
from the following equation

$(p_1 - p_0)/(\nu_0 - \nu_1)=\rho_0^2u_0^2=\rho_1^2u_1^2= \dot{m^2}$

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

$h_1-(h_0+q)=\frac{1}{2} (p_1-p_0)(\nu_0-\nu_1)$.

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

$y= \frac{\frac{\gamma_0+1}{\gamma_0-1} -x+\frac{2q}{p_0\nu_o}}{\frac{\gamma_0+1}{\gamma_0-1}x -1}$.

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.

\[\[Image:CJ detonation and deflagration
points.jpg\|center\|thumb\|400px\|

```{=html}
<center>
```
Figure 2: CJ detonation and deflagration solution

```{=html}
<center>
```
\]\]

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.

\[\[Image:Variation of properties across ZND
structure.jpg\|center\|thumb\|400px\|

```{=html}
<center>
```
Figure 3: Variation of properties across a ZND detonation wave

```{=html}
<center>
```
\]\]

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.

\[\[Image:Det_front_structure.jpg\|center\|thumb\|400px\|

```{=html}
<center>
```
Figure 4: Detonation front structure

```{=html}
<center>
```
\]\]

## 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

## Other links

A Detonation
database containing
a collection of experimental data related to detonations is available
online.
