**1. Introduction**

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 [1] 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 [4]. 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 [5] 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 [6] and volumetric detonation.

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 1012÷1015cm�3. Rapid chain transformations start just from those initial centers [10] and run following the Lewis scheme. In this case, we have

*Determination of Values Range of Physical Quantities and Existence Parameters of Normal…*

Note also that the temperature *T*2, at which the branching probability *δ* equals

is critical: the process becomes considerably accelerated, and the rapid chain reaction takes place. Under the indicated conditions, according to the results of

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 *Tx* in the medium, at which the branching probability *δ* equals unity. How can *Tx* be determined? In work [4], 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,

*QT*0ð Þ <sup>γ</sup> � <sup>1</sup> <sup>2</sup>γ*M*<sup>2</sup> � <sup>γ</sup> <sup>þ</sup> <sup>1</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>γ</sup> � <sup>1</sup> *<sup>M</sup>*<sup>2</sup> <sup>2</sup>

where *M* is the Mach number (it reflects the shock transition intensity); *P*<sup>0</sup> the initial, before the explosion ignition (at 293 K), pressure of the gas mixture reckoned in mm Hg units; *E*<sup>2</sup> the activation energy of the branching reaction (2); *K*<sup>∗</sup> the gas constant; *Q* the combustion energy per gas mole; and *γ* the adiabatic index for the given gas mixture. For the hydrogen-oxygen mixture, the corresponding numerical values are [12]: *<sup>γ</sup>* <sup>¼</sup> <sup>1</sup>*:*4, *<sup>Q</sup>* <sup>¼</sup> <sup>286</sup>*:*5 kJ/mol, *<sup>K</sup>*<sup>∗</sup> <sup>¼</sup> <sup>8</sup>*:*31 J/mol/K, *<sup>E</sup>*<sup>2</sup> <sup>¼</sup>

*<sup>P</sup>*0*M*<sup>6</sup> � *<sup>e</sup>*

<sup>4</sup>γ<sup>2</sup>ð Þ <sup>γ</sup> <sup>þ</sup> <sup>1</sup> *<sup>M</sup>*<sup>6</sup>*<sup>K</sup>* <sup>∗</sup> *<sup>P</sup>*<sup>0</sup>

<sup>2</sup> <sup>¼</sup> <sup>5</sup>*:*<sup>38</sup> � <sup>1010</sup> <sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*4*M*<sup>2</sup> <sup>2</sup> <sup>2</sup>*:*8*M*<sup>2</sup> � <sup>0</sup>*:*<sup>4</sup>

*<sup>T</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*γM*<sup>2</sup> � *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> <sup>þ</sup> ð Þ *<sup>γ</sup>* � <sup>1</sup> *<sup>M</sup>*<sup>2</sup> ð Þ *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup>

*Tx* (6) .

It is evident that the temperature *Tx* is different for different Mach numbers. Formula (6) describes the functional dependence of the critical temperature *Tx* on the Mach number *M* for the given initial pressure *P*<sup>0</sup> and allows one to compare its

<sup>16</sup> � <sup>10</sup><sup>3</sup> � <sup>4</sup>*:*19 J/mol, and *<sup>T</sup>*<sup>0</sup> <sup>¼</sup> 293 K. Then Eq. (5) reads

unity,

*T*2

**151**

*<sup>x</sup>* <sup>¼</sup> <sup>2</sup>*:*<sup>5</sup> � <sup>10</sup><sup>5</sup>

*Tx*

value with the real temperature

work [4], the equality

*DOI: http://dx.doi.org/10.5772/intechopen.96285*

OH þ H2 ¼ H2O þ H, (1) H þ O2 ¼ OH þ O*:* (2)

*δ* ¼ 1, (3)

*T*<sup>2</sup> ¼ *Tx*, (4)

� *<sup>e</sup>*xp � *<sup>E</sup>*<sup>2</sup>

�8067*=*

*<sup>M</sup>*<sup>2</sup> � *<sup>T</sup>*1, (7)

*K* <sup>∗</sup> *Tx* ,

(5)

In the earlier work [7], the model for the transition of an explosion spherical wave to the Chapman–Jouguet regime was proposed. In the other work [8], 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 jumplike increase of the pressure. Actually, the process develops somewhat differently [9]. 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.

#### **Figure 1.**

*Schematic diagrams of the pressure, Р, and gas temperature, Т, profiles behind the shock wave front [9] under the condition* T2 ≥Tx*, where* T2 *is the temperature at point 2.*

*Determination of Values Range of Physical Quantities and Existence Parameters of Normal… DOI: http://dx.doi.org/10.5772/intechopen.96285*

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 1012÷1015cm�3. Rapid chain transformations start just from those initial centers [10] and run following the Lewis scheme. In this case, we have

$$\rm OH + H\_2 = H\_2O + H,\tag{1}$$

$$\mathbf{H} + \mathbf{O}\_2 = \mathbf{O}\mathbf{H} + \mathbf{O}.\tag{2}$$

Note also that the temperature *T*2, at which the branching probability *δ* equals unity,

$$
\delta = \mathbf{1},
\tag{3}
$$

is critical: the process becomes considerably accelerated, and the rapid chain reaction takes place. Under the indicated conditions, according to the results of work [4], the equality

$$T\_2 = T\_\infty. \tag{4}$$

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 *Tx* in the medium, at which the branching probability *δ* equals unity. How can *Tx* be determined? In work [4], 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,

$$T\_x^2 = \frac{2.5 \times 10^5 Q T\_0 (\gamma - 1) \left(2 \eta M^2 - \gamma + 1\right) \left(2 + (\gamma - 1)M^2\right)^2}{4 \eta^2 (\gamma + 1) M^6 K^\* P\_0} \times \exp\left(-\frac{E\_2}{K^\* T\_x}\right),\tag{5}$$

where *M* is the Mach number (it reflects the shock transition intensity); *P*<sup>0</sup> the initial, before the explosion ignition (at 293 K), pressure of the gas mixture reckoned in mm Hg units; *E*<sup>2</sup> the activation energy of the branching reaction (2); *K*<sup>∗</sup> the gas constant; *Q* the combustion energy per gas mole; and *γ* the adiabatic index for the given gas mixture. For the hydrogen-oxygen mixture, the corresponding numerical values are [12]: *<sup>γ</sup>* <sup>¼</sup> <sup>1</sup>*:*4, *<sup>Q</sup>* <sup>¼</sup> <sup>286</sup>*:*5 kJ/mol, *<sup>K</sup>*<sup>∗</sup> <sup>¼</sup> <sup>8</sup>*:*31 J/mol/K, *<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>16</sup> � <sup>10</sup><sup>3</sup> � <sup>4</sup>*:*19 J/mol, and *<sup>T</sup>*<sup>0</sup> <sup>¼</sup> 293 K. Then Eq. (5) reads

$$T\_x \, ^2 = \frac{5.38 \times 10^{10} \left(2 + 0.4M^2\right)^2 \left(2.8M^2 - 0.4\right)}{P\_0 M^6} \times e^{-8067/\Gamma\_x} \tag{6}$$

It is evident that the temperature *Tx* is different for different Mach numbers. Formula (6) describes the functional dependence of the critical temperature *Tx* on the Mach number *M* for the given initial pressure *P*<sup>0</sup> and allows one to compare its value with the real temperature

$$T\_2 = \frac{\left(2\chi\mathcal{M}^2 - \chi + 1\right)\left(2 + (\chi - 1)\mathcal{M}^2\right)}{\left(\chi + 1\right)^2\mathcal{M}^2} \times T\_1,\tag{7}$$

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 [5] have not been fully investigated. It is hoped that in the near future this issue will be resolved after a detailed study of spherical

In the earlier work [7], the model for the transition of an explosion spherical wave to the Chapman–Jouguet regime was proposed. In the other work [8], 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

**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 jumplike increase of the pressure. Actually, the process develops somewhat differently [9]. 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–

Then the induction period (3–4) takes place, the duration of which can be equal

*Schematic diagrams of the pressure, Р, and gas temperature, Т, profiles behind the shock wave front [9] under*

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

detonation waves [6] and volumetric detonation.

3), which is accompanied by a temperature reduction.

(5–6), the profile of which can change.

*the condition* T2 ≥Tx*, where* T2 *is the temperature at point 2.*

**Figure 1.**

**150**

spherical detonation is probable.

*Recent Advances in Numerical Simulations*

where *T*<sup>1</sup> is the temperature of the medium in front of the wave front. Hence, in our case, the important criterion,

$$T\_2 \ge T\_x,\tag{8}$$

has to be satisfied for the detonation to take place as a real process.
