Propulsion with pulsated detonation

The driving with pulsated detonation (PDE in English: Pulsed Detonation Engine) is a system of propulsion which can be useful for the Aéronautique or the Astronautique.
This type of engine was developed in order to improve the specific impulse of the propulsion compared to the traditional rocket S chemical.

Principle

Useful detonations

The engine is simple in its principle: the PDE is a tube in which one makes explode a mixture Oxydant/Réducteur which is ejected at once at high speed by a mechanism of purging (blowdown). The interest is to have here a Combustion with constant volume behind the produced shock wave, which has a thermodynamic effectiveness larger than combustion with constant pressure used in traditional chemical combustion. The Détonation S must be repeated at a fast frequency in order to produce a high Poussée average.

A cycle PDE

The principal phenomenon is the propagation of a detonation. The various stages are recapitulated hereafter:

• Stage 1: it is the initiation which creates the detonation. There exist two kinds of initiation: the direct one which requires the contribution of a large quantity of energy and that by transition (or English DDT: Deflagration to Detonation Transition) which is simpler with obtenir.
  

• Stage 2: in the tube closed at an end, the detonation (which consists of a shock wave followed by a zone of chemical reactions) is propagated in direction of the open end. Directly behind the shock wave, the power flarings is increased moving. But, like side closed in contact with the wall, speed is null, of the waves of relaxation (waves of Taylor) must be formed in order to observe this boundary condition. With final, the phase of propagation consists of a wave train made up of a shock wave followed by a succession of waves of relaxation moving towards the open end of the tube.
  

• Stage 3: once the shock wave reached the opened end, the gases flarings are expelled and the shock wave is transmitted in the form of a semicircular wave in the external environment of the tube. With the interface between the interior and the outside of the tube, a considered wave is also created: that can be a relaxation or shock wave according to the mix design. This considered wave is propagated behind in the tube.
  

• Stage 4: The considered wave reaches finally the closed end of the tube. On its course, it involved a fall of pressure in the tube in order to equalize the internal and external pressures. At the end of this stage, the tube can be again filled and a new cycle can start.

Principal parameters of design

The sizes characteristic of a PDE are the following ones: push, the frequency of detonation, the report/ratio of mixture of fuels, initial conditions (P, T) in the tube, the diameter of the tube, mass of the engine and the thrust report/ratio.

The performance

In a first approach and by neglecting all the forces of friction to the walls, one can say that the force of push is produced only by the pressure which is exerted on the walls of the tube. This pressure is evaluated by various models (analytical or numerical). One from of then deduced the ISP by the formula:
$I=S\; \backslash \; int\_\; \{0\}\; ^\; \{T\}\; \backslash \; triangle\; P\; (T)\; dt$

where S is the section of the tube, $\backslash \; P$ triangle is the difference in pressure on the walls and T the duration of the cycle.

Calculation of the average push

It is supposed here that the tube has a constant diameter over all its length. Three parameters define its geometry: the diameter D, the length L and the frequency of the cycles of detonation. The average push is calculated by multiplying the impulse of a simple cycle by the frequency of the cycles:
$T=I\; \backslash \; cdot\; V\; \backslash \; cdot\; f=I\; \backslash \; cdot\; \backslash \; frac\; \{\backslash \; pi\; \backslash \; cdot\; L\; \backslash \; cdot\; d^2\}\; \{4\}\; \backslash \; cdot\; f$

Limiting frequency

The frequency of the cycles depends mainly on the length L of the tube. As the push is directly proportional to the frequency, it is important to consider the frequency maximum possible to which the machine can function. This maximum frequency of cycle corresponds to one duration of minimum cycle $t\_\; \{cycle\}$. The cycle of a PDE can be cut out in three phases: filling, the detonation and expulsion; from where:
$t\_\; \{cycle\}\; =t\_\; \{filling\}\; +\; t\_\; \{detonation\}\; +\; t\_\; \{expulsion\}$ Calculations show that time for the detonation and expulsion is equal to $10\; \backslash \; t\_\; \{CJ\}$ where $t\_\; \{CJ\}$ is the time put by the detonation wave to traverse the tube:
$t\_\; \{CJ\}\; =\; \backslash \; frac\; \{L\}\; \{U\_\; \{CJ\}\}$ The impulse is maximum after this duration which corresponds to the beginning of negative overpressure on the wall of the tube. The time of filling, can be to him estimated coarsely by supposing one constant duration of filling:
$t\_\; \{filling\}\; =\; \backslash \; frac\; \{L\}\; \{U\_\; \{filling\}\}$ With final, one from of deduced the maximum frequency from cycle:
$f\_\; \{max\}\; =\; \backslash \; frac\; \{1\}\; \{L\}\; \backslash \; cdot\; \backslash \; frac\; \{1\}\; \{\backslash \; frac\; \{10\}\; \{U\_\; \{CJ\}\}\; +\; \backslash \; frac\; \{1\}\; \{U\_\; \{filling\}\}\}$

The first research

The detonations applied to the propulsion were studied only in the fifty last years because of the technical difficulties related on the mixture at high-speed of fuel and the control of this detonation with a report/ratio of donné.
mixture The concept of PDE goes back to work of Hoffmann, which managed to initiate intermittent detonations with mixtures of liquid hydrocarbons (benzene) and gas (acetylene). The search for an optimum of performance for the frequency of the cycles of detonation was nevertheless unfruitful and it was necessary to await work of Nicholls with operations simple and multi-cycles on hydrogen mixtures/air acetylene/air. The experiments consisted of a simple tube with detonation, opened on a side and provided on the side closed of a system of Co-annular injection for oxidant and the reducer. The question arises today of knowing if the periodic waves measured in these experiments were really the result of detonations or if they were only forms of deflagrations at high-speed.

Initiation of the detonation

Two mechanisms exist to create a detonation:

• the direct initiation which produces the detonations more “  pures  ”, but which requires a very large quantity of energy;
• the transition deflagration-detonation or DDT: this initiation is observed in very reactive mixtures (acetylene/air, hydrogen/air). The waves of combustion accelerate thanks to the warming of unburnt gases downstream from the wave caused by a succession of compression waves related to the dilation of the gases flarings. The higher temperature of unburnt gases increases the speed of sound, thus make it possible the various waves to catch up with the initial wave. This acceleration of unburnt gases involves the development of turbulent movements which cause the distortion of the face of flame which tears little by little: several zones of reactions are formed and their relative movements create important fluctuations of temperature and concentration of the components. Cells characteristic of the detonation are formed then gradually. A constraint of the DDT is to have a sufficiently important length of tube to make it possible the transition to develop. One of the approaches most usually used currently consists in using a separated room in which the DDT is initiée.
  

The detonation

The principal mechanism of the PDE is the detonation. This is why the knowledge of all its characteristics is essential to develop an effective engine.

Models

Theory of Chapman-Jouguet

It is the simplest model of detonation. It supposes that the detonation is a simple discontinuity in the flow and that the rate of chemical reaction is infinite. From there, one can establish tools specific to the detonation waves. The conservation equations are written:
$\backslash \; rho\_3\; X\_3=\; \backslash \; rho\_4\; X\_4\; \backslash ,$Masse
$\backslash \; rho\_3\; X\_3^2\; +\; P\_3\; =\; \backslash \; rho\_4\; X\_4^2+P\_4$ Momentum
$h\_4-h3=\; (X\_3^2-X\_4^2)\; /2$ Energy

This system has a single solution which corresponds to the state of Chapman-Jouguet; it defines a speed for the detonation wave (U_CJ) which is equal to 1 as well as the composition of the products of combustion.

Model ZND

It was developed during the Second world war by Zeldovich, Neumann and Döring to take into account the speed of chemical reaction. Model ZND describes the detonation wave like a shock wave immediately followed by a zone of chemical reaction (the flame). The thickness of this zone is given by the rate of reaction. Theory ZND gives the same reaction speed and the same pressures that the Chapman-Jouguet theory but takes in more in account the thickness of the wave.

Real detonations

The detonations are in realities of the structures in three dimensions including/understanding of the shock waves and the zones of reaction. The shock waves consist of segments of curves; with each line of detachment, the shock waves interact in the form of configurations of Mach. The size of this reason in the shape of fish scales created by the intersections of shock waves is characteristic of the rate of chemical reaction of the mixture and thus defines a scale characteristic of the cell of reaction. The size of these cells is a factor limiting for the diameter of the tube: a detonation cannot be propagated in a tube of diameter lower than the size of cell divided per pi, this limit corresponding to highly unstable detonations.

Propagation of the detonation

Boundary conditions

The analysis of the dynamics of gases requires to know what occurs when the incidental detonation wave reaches the open end of the tube. This interface can be modelled in 1D like a surface of contact. When the detonation wave reaches this surface, a transmitted wave will be propagated towards the outside of the tube while a considered wave sets out again towards the interior of the tube. In the case of a detonation being propagated in air with an atmosphere, the transmitted wave will be always a compression wave. The considered wave can be of compression or relaxation. The way simplest to determine its nature is to use the Pression diagram/Speed.

Dynamics of gases analyzes

The dynamics of gases in an ideal tube (i.e. without obstacles) can be analyzed thanks to the diagram outdistances/time. This diagram x-t represents the detonation wave being propagated at the speed Chapman-Jouguet U_CJ followed by a wave of relaxation of Taylor. The first characteristic thought of the interface gas/air at exit of tube is also represented. This characteristic with an initial slope determined by the conditions with the interface which is modified by the interaction with the wave of Taylor. Once it crossed this wave of Taylor, the first characteristic is propagated at the speed of sound in the c₃ medium. The area behind this first characteristic is complex because the wave of considered relaxation also interacts with the wave of Taylor.

Two characteristic times can be defined: t₁ corresponding to the reflection of the detonation wave to the external interface and t₂ corresponding to the time necessary with the first characteristic reflected to reach the fermée.
end The difference in pressure on this surface of push involves the specific impulse of a simple cycle. It is thus interesting to be focused on the history in pressure on the closed end. When the detonation is initiated, a peak of pressure is recorded (the peak of Chapman-Jouguet) then the pressure falls in P₃ following the passage of the wave of Taylor. The pressure on the surface of push remains constant until the first considered wave reaches this wall. The wave of considered relaxation then decreases the pressure up to the external value. The peak of pressure of Chapman-Jouguet occurs during a very short time and thus does not take part in the push. The pressure remains constant then during a time t1+t3 with the P₃ pressure calculated with the equations of the flow isentropique.

Calculation of the specific impulse

By taking again the formula exposed higher: $I=S\; \backslash \; int\_\; \{0\}\; ^\; \{T\}\; \backslash \; triangle\; P\; (T)\; dt$ and by using the preceding profile of pressure, one can develop the value of the impulse:
$I=S\; \backslash \; cdot\; P\_3\; (t\_1+t\_2)\; +int\_\; \{t2\}\; ^\; \{T\}\; \backslash \; triangle\; P\; (T)\; dt$
Time T1 corresponds to the time necessary with the detonation to reach the open end of the tube length L ($t\_1=\; \backslash \; frac\; \{L\}\; \{U\_CJ\}$) and t2 is the time necessary for the first considered wave to reach the wall of push. Time t2 depends mainly on the length of the tube and speed characteristic behind the wave of Taylor (speed of sound c3). One models that with a parameter &alpha:
$t\_2=\; \backslash \; alpha\; \backslash \; cdot\; L/c\_3$ The last part of the integral of pressure between t2 and T is modelled by introducing a time T3 expressed using the parameter &beta: $t\_3=\; \backslash \; beta\; \backslash \; cdot\; \backslash \; frac\; \{L\}\; \{c\_3\}$
One can thus rewrite the impulse in the form: $I=S\; \backslash \; cdot\; \backslash \; triangle\; P\_3\; \backslash \; cdot\; \backslash \; frac\; \{L\}\; \{c\_3\}$

The parameter %alpha is determined by the interaction of the considered wave and the wave of Taylor. The parameter %beta is calculated numerically or determined by experimentation.

Purging of the engine

The phase of purging is important to determine the frequency of the engine.

Conduits

The design and the integration of conduits for PDE are delicate because of the highly non stationary nature of the propagation of the detonation waves. The current studies do not make it possible to converge on a joint conclusion when to the mechanisms of interaction between the conduit and the shock wave, in particular when the shock waves come to disturb the phase of filling. In general, a divergent conduit gives a better specific impulse but with a certain delay compared to the case without conduit, which can affect the frequency of cycle.

Structural considerations

An important aspect of the design of the engines is the optimization of the Masse. The mass of the PDE is primarily given by the choice of materials and the thickness of the walls since the length and the diameter are fixed by the performance.
The constraints which are excercent on the walls of the tube in the PDE are dynamic and circular and require the use of materials with important thermomechanical capacities such as the Inconel, the Acier, the Aluminum and SI₃N₄.

References

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