Values of critical temperature and detonation temperature depending on the Mach number ().
Using elements of the theory of classical detonation and previously obtained relations for spherical waves, the author tried to establish the range of admissible values of temperature, Mach numbers, and specific hydrogen content in the gas mixture of the possible existence of normal spherical detonation. The work took into account the critical values of the parameters associated with the kinetics of the chemical reaction at the front of the blast wave and the parameters that determine the intensity of the shock transition (minimum and maximum Mach number) for a given reacting medium. Using the example of the interaction of hydrogen and oxygen in a hydrogen-oxygen mixture, it was possible to graphically determine the range of values of the main physical quantities and parameters - the critical temperature, the detonation temperature of the quiescent medium, and the specific hydrogen content in the mixture required for spherical detonation. Mathematical modeling of the process was carried out at a fixed value of the pressure of the gaseous medium.
- detonation wave
- critical temperature
- range of permissible values
- process modeling
Explosions are widely used in many areas of science and engineering, and their models are applied to elucidate various physical phenomena. Moreover, the unexpected explosions in industry and everyday life often result in catastrophes with numerous human losses, which invokes the intensive study of a supersonic burning nowadays. Those researches are carried out using both analytical methods  and numerical simulations [2, 3]. This work aims at studying the range of parameters needed for a normal spherical detonation in a gas mixture to take place. It is the kind of detonation that precedes the plane (classical) detonation, but emerges at lower shock wave velocities . The spherical wave produced by a strong point explosion corresponds to the initial stage of the whole detonation process and transforms gradually into the classical variant. In gaseous explosive mixtures, the detonation regime of the explosive transformation is possible only at certain concentrations of the combustible gas, depending on the chemical composition of the mixture, pressure and temperature. A decrease in pressure leads to the appearance of a pulsating detonation front, and subsequently to the formation of the so-called spin detonation, in which the three-shock wave configurations arising at the detonation wave front rotate along a helical line. With a further decrease in pressure, the supersonic combustion process dies out. At present, the reasons for the onset and existence of pulsating detonation  have not been fully investigated. It is hoped that in the near future this issue will be resolved after a detailed study of spherical detonation waves  and volumetric detonation.
In the earlier work , the model for the transition of an explosion spherical wave to the Chapman--Jouguet regime was proposed. In the other work , the concept of the critical temperature at the wave front was introduced as a basic criterion for the transformation of a shock wave to the detonation one. In this work, using the example of a gaseous hydrogen-oxygen mixture, an attempt is made to graphically determine the ranges of physical parameters and quantities at which spherical detonation is probable.
2. Critical values of parameters related to the chemical reaction kinetics
The classical theory considers detonation waves with sharp front edge. In its framework, the chemical transformations are assumed to begin right after a jump-like increase of the pressure. Actually, the process develops somewhat differently . The temperature and pressure profiles behind the shock front of a detonation wave are schematically shown in Figure 1. After the shock transition (1–2), the vibrational and rotational degrees of freedom of gas molecules become excited (2–3), which is accompanied by a temperature reduction.
Then the induction period (3–4) takes place, the duration of which can be equal to more than 90% of the whole chemical reaction time (3–5), if the activation energy of the process is sufficiently high (E = 20 ÷ 40 kcal/mol). In the stationary detonation regime (the Chapman--Jouguet regime), profile 1–5 does not change in time. The reaction zone adjoins the region of non-stationary flow, rarefaction wave (5–6), the profile of which can change.
It turns out that, in the case of a hydrogen-oxygen mixture compressed by a shock wave, a lot of free radicals emerge in section 2–4 [10, 11], with their concentration reaching . Rapid chain transformations start just from those initial centers  and run following the Lewis scheme. In this case, we have
Note also that the temperature , at which the branching probability equals unity,
is critical: the process becomes considerably accelerated, and the rapid chain reaction takes place. Under the indicated conditions, according to the results of work , the equality
plays the role of a criterion for qualitative variations in the kinetics of the interaction between hydrogen and oxygen. The Lewis scheme ignites the detonation mechanism, although the process itself can run in a certain different way, following a different scenario, in which the reaction rate is higher by an order of magnitude. From the chemical viewpoint, we have already stated the fact that, in order to obtain the supersonic burning at the shock wave front, it is necessary to reach the temperature in the medium, at which the branching probability equals unity. How can be determined? In work , the relation between the key parameters of a chemical reaction at the shock wave front, on the one hand, and the physical quantities that characterize the process of shock transition, on the other hand, was obtained,
where is the Mach number (it reflects the shock transition intensity); the initial, before the explosion ignition (at 293 K), pressure of the gas mixture reckoned in mm Hg units; the activation energy of the branching reaction (2); the gas constant; the combustion energy per gas mole; and the adiabatic index for the given gas mixture. For the hydrogen-oxygen mixture, the corresponding numerical values are : , kJ/mol, J/mol/K, J/mol, and K. Then Eq. (5) reads
It is evident that the temperature is different for different Mach numbers. Formula (6) describes the functional dependence of the critical temperature on the Mach number for the given initial pressure and allows one to compare its value with the real temperature
where is the temperature of the medium in front of the wave front. Hence, in our case, the important criterion,
has to be satisfied for the detonation to take place as a real process.
3. Elements of the hydrodynamic theory of detonation: limiting parameters dependent on the Mach number minimum and maximum
The detonation process of explosive materials is considered as a cumulative action of the shock wave and the chemical reaction, when the shock compression initiates the reaction, and the reaction energy maintains the detonation process afterward. The hydrodynamic theory  enables one to evaluate the size of a chemical reaction zone and the values of medium parameters in the chemical reaction zone (at the interface with the detonation products). The classical theory considers a plane detonation front,
where is the chemical reaction zone width, the reaction duration, the shock wave velocity, and the gas velocity behind the reaction front (the Jouguet point). To be exact, in a real situation (Figure 2), there exists some shock transition interval (1–2) before the temperature is achieved, which is not taken into account in this case. One can see in Figure 2 that front (3–3) separates the chemical reaction zone from detonation products. This means that the substance being suddenly compressed by the shock wave burns out completely within the time inter .
The theory is based on two important postulates: (1) the whole substance compressed by the shock wave burns out, and (2) the combustion energy is enough to maintain the shock wave velocity to be constant (). According to the theory, the pressure and the density in the chemical reaction zone at the interface with detonation products (the Jouguet point), are connected with each other by the following relations :
Where is the pressure at front (3–3) separating the reaction zone from the reaction products, the pressure in front of the shock wave front, the gas density in front of the wave front, the wave velocity, the adiabatic index, the medium density at the wave front (3–3), and the sound velocity in the motionless medium in front of the front. From the Mendeleev--Clapeyron equation
where . Here, we took into account that
when , and that
where is the molar mass. It is evident that if we consider the detonation and the support of a chemical reaction by the shock wave, the following condition has to be satisfied:
or, in a wider sense (Figure 3),
Let us analyze inequality (16) in detail. From the theory of shock waves , it is known that the temperature at point 2 in Figure 3 is determined by relation (7). Therefore, inequality (16) can be transformed as follows:
since and . By solving the equation
with respect to (keeping in mind that ), we obtain
or, substituting the corresponding -value,
The positive roots of this equation are and . For shock transitions, the most interesting is the first root, . On the basis of inequality (18), we may assert that the detonation process is not possible for all shock waves, but only for those with the Mach number . Owing to hydrodynamic reasons, there is no detonation for waves with . While analyzing Eq. (20), it is expedient to admit that the temperature equality 
describes the lower temperature limit of the detonation (Figure 3). In so doing, we took into account that the rate of the chemical reaction decreases together with the temperature in the chemical reaction zone. At the same time, according to the hydrodynamic theory, the amount of the substance that was compressed by the shock wave and interacted under its action has to remain at the previous level. This circumstance inevitably results in the time growth for the active reaction phase, and, as a consequence of the process continuity, gives rise to a considerable reduction of the induction period (interval 3–4 in Figure 1). In this connection, there emerges a possibility for the detonation wave to create a gas layer with an approximately identical temperature, and the Mach number should be considered as the lower detonation limit.
In order to determine the upper limit of the detonation wave emergence by initiating an explosion in reacting gas media, let us use the model describing the continuous transition of a spherical explosion wave into the Chapman--Jouguet regime . For the normal spherical detonation, it can be determined from the formula
derived in work . All quantities in this formula are known, except for the parameter , the specific content of the burned out gas (hydrogen). The intensity of a detonation wave can be controlled by changing, mainly, two parameters: (in the numerator) and (in the denominator). In our case, all hydrogen compressed by the shock wave burns out. The values of coefficient are confined within the interval . Let we have the stoichiometric mixture of hydrogen with oxygen (), and the medium temperature . We suppose that a further temperature decrease will result in changes of the adiabatic index and the physical properties of the reacting mixture [15, 16], i.e. let formula (24) be valid for real gases at . In this case, we obtain a rough estimate for the Mach number maximum, . Note again that a strong explosion takes place in a cooled down medium. In this case, . Hence, we estimated the interval of possible Mach numbers for the normal spherical detonation of the hydrogen-oxygen gas mixture under real conditions:
In view of relation (6), let us plot the dependence of the critical temperature on the Mach number, (M) (Figure 4). Since the Mach number range was found, we will calculate the critical temperature for every from the indicated interval with an increment of 0.2 and the fixed initial pressure (see Table 1). Transcendental equations were solved using the “Consortium Scilab (Inria, Enpc)” software package with the “Scilab-4.1.2” code. When solving equations, only roots with real values that have physical meaning should be taken into account (the procedure was applied in ).
The larger the Mach number, the higher is the critical temperature. However, at , the critical temperature growth becomes a little slower. At the lower limit , , and, at the upper limit , . Hence, in a hydrogen-oxygen mixture, the critical temperature for the allowable values of Mach number accepts values from the following interval:
This is the minimum temperature in front of the shock wave that makes the detonation possible.
4. Results of calculations and their discussion
Using the intervals obtained above for some physical quantities, let us graphically determine the region of existence for the normal spherical detonation. The upper limit of the hydrogen content in the mixture is confined in our case by the value . Above this value, the chemical reactions resulting from the interaction between hydrogen and oxygen in the mixture, which were considered in work , become more complicated, and this circumstance may result in different values of critical temperature. Further researches of this issue are required. Below, the choice for the upper limit of the medium temperature is explained in detail.
With regard for dependence (24) of the Mach number on the temperature of a motionless medium and the hydrogen content , let us plot the dependences , , and at fixed and . We proceed from the plots of the dependence at exhibited in Figure 5 for -temperatures in the interval (see Table 2). The lower curve corresponds to the gas mixture temperature , and the upper one to . According to expression (24), this family of curves has a power dependence on the hydrogen content in the mixture, , with a power exponent of 0.5. Let us fix the maximum content of burned out hydrogen, , which corresponds to the stoichiometric composition of hydrogen-oxygen mixture, and draw a vertical line. The Mach number corresponding to its intersection with the mentioned family of curves changes from at point 4 to at point 5. Another important detail should be emphasized. Four dashed lines are drawn in Figure 5. Two horizontal ones confine the region of allowable Mach numbers corresponding to the normal spherical detonation. The first line corresponds to the minimum , and the second one to the maximum . The third dashed vertical line corresponds to the stoichiometric composition of the hydrogen-oxygen mixture and is the optimal variant for the detonation. The fourth line will be discussed below.
Let us consider points 1 to 5 in Figure 5 separately. Segment 1–2 corresponds to the lower limit of the shock wave velocity , but the medium temperature for the segment points turns out lower than the detonation one (Figure 4). Therefore, the detonation is impossible in this case. The region of the probable detonation for this Mach number is restricted to segment 2–3, because the temperature of motionless medium reaches the detonation temperature values here. On the basis of Figure 4, it is also possible to draw a conclusion that, for the medium temperature , the detonation is possible if (point 5 in Figure 5). In other words, for the chosen temperature, , and hydrogen content, 0.66 , intervals, the region of the probable detonation is bounded by segments 2–3, 3–4, 4–5, and 5–2. Segment 5–2 is presented in Figure 5 schematically by a straight line. In the general case, in view of the nonlinear dependence , this segment is curvilinear.
Then we obtain the following transcendental equation for the critical temperature :
or, taking Eq. (30) into account,
While analyzing the plots, the attention should be drawn to the following facts. (i) Every temperature of the gas mixture is associated with a specific dependence . (ii) As the hydrogen content in the mixture grows, the temperature of the detonation in the motionless medium drastically decreases, which is especially appreciable at low temperatures. (iii) Let us draw a horizontal line that intersects the family of curves (for example, let it be the dashed line ). At the point of its intersection with the curve corresponding to the same temperature of the motionless medium (in our case, this is ), the detonation condition (point 5) is satisfied. Detonation becomes probable, because the current temperature of motionless medium reaches the detonation temperature for this medium (). (iv) The intersection points of any horizontal line (see item iii) correspond to the critical hydrogen contents in the mixture, below which the detonation is impossible. The dashed line connecting points 2 and 5 in Figure 7 corresponds to the condition for the whole family of curves.
Let the hydrogen content in the mixture change from 0.075 (point 1) to 0.66 (point 4). Then, on the basis of the plots shown in Figure 7, one may assert the following.
As was indicated above, a temperature lower than can give rise to a variation in the physical properties of the reacting mixture. Then the proposed formulas will produce erroneous results. The horizontal dashed line that passes through point 5 corresponds to this temperature, and point 5 testifies to the explosion with the maximum Mach number .
Physical restrictions imposed by the minimum Mach number bring about the existence of the upper limit for the detonation temperature, . Points of both segments 1–2 and 2–3 correspond to the allowable values of Mach number. However, the detonation is possible only for the points on segment 2–3, because the main condition is satisfied at . Whence it follows that is the minimum hydrogen content in the mixture, below which the detonation is mpossible even at very high temperatures.
Experimental results testify that, if the temperature of a gas mixture is higher than , the spontaneous ignition takes place, which can transform into the detonation, if the hydrogen content in the mixture is not lower than 0.37. Therefore, this temperature is a kind of upper limit, to which the hydrogen-oxygen mixture can be heated.
From the reasons given above, it follows that the region of spherical supersonic burning is bounded by segments 2–3, 3–4, 4–5, and 5–2 in Figure 7. For the illustrative purpose, it was hatched.
The dependences between the temperature, Mach number, the hydrogen content in the hydrogen-oxygen mixture as the main parameters characterizing the process of transformation of a shock wave into a detonation one and affecting the chemical reactions between reacting components are studied. On the basis of relations obtained earlier , the conditions are found, under which the probability of a chain branching reaches unity (), and a fast chain reaction is started. The existence of the critical temperature at the front of a shock wave, above which the detonation takes place, is substantiated, as well as the functional dependence (5) of the critical temperature on the Mach number. In author’s opinion, the latter should be taken as a basis, while studying the processes of spherical detonation. Summarizing the results of work , the ondition is found, which connects the kinetics of a chemical reaction with the detonation in a gas mixture. On the basis of the relations of the hydrodynamic theory of detonation, the region of possible values for the temperatures at the shock wave front, , and in the chemical reaction zone, , is determined. The equality which couples them, corresponds to the lower limit, at which the detonation is possible. The minimum and maximum values of Mach number in reacting gas media are also determined, which enables the process of supersonic burning to be analyzed in more details and the region of physical parameters and quantities (the critical emperature, the temperature of detonation in the motionless medium, and the hydrogen content in the mixture), at which the spherical detonation is probable, to be indicated. The latter is llustrated, by using the hydrogen-oxygen mixture as an example.
To summarize, it should be noted that this paper is final in a cycle of works devoted to the study of the whole process of normal spherical detonation.
The author expresses his sincere gratitude to Yu.L. Birkovoi, the former head of the technological department of microelectronics of the known, in the past, production association “Rodon”, as well as to the whole team of this department, for a long-term fruitful cooperation.
compound of an oxygen atom with a hydrogen atom
temperature at the shock front
shock wave velocity, detonation velocity
pressure, temperature, density of the medium
chemical reaction zone
the activation energy of the branching reaction
the chemical reaction zone width
the reaction duration
the gas velocity behind the reaction front
the sound velocity in the motionless medium in front of the front
temperature at the Jouguet point
coefficient of flammable gas content in the mixture
chemical reaction rate
maximum Mach number
minimum Mach number
function of the variable
increment of variable x