**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 [13] 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,

$$d = \Delta t \left( D - v\_{\mathfrak{g}} \right), \tag{9}$$

where *d* is the chemical reaction zone width, Δ*t* the reaction duration, *D* the shock wave velocity, and *vg* 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 *T*<sup>2</sup> 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 Δ*t*.

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 (*D* ¼ *const*). According to the theory, the pressure *P*<sup>3</sup> and the density *ρ*<sup>3</sup> in the chemical reaction zone at the interface with detonation products (the Jouguet point), are connected with each other by the following relations [13]:

$$P\_3 = \frac{P\_1 + \rho\_1 D^2}{1 + \chi},\tag{10}$$

$$\frac{\rho\_3}{\rho\_1} = \frac{D^2(\chi + \mathbf{1})}{b\_1^{\chi^2} + \chi D^2},\tag{11}$$

*<sup>T</sup>*<sup>3</sup> <sup>¼</sup> *<sup>ρ</sup>*1*D*<sup>2</sup> *γ* þ 1

**Figure 2.**

when *<sup>P</sup>*<sup>3</sup>

**153**

� *μ K*<sup>∗</sup>

*<sup>γ</sup>D*<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*<sup>1</sup> <sup>2</sup>

ð Þ *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>¼</sup> *μ γD*<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*<sup>1</sup>

*front P*2*, and in the chemical reaction zone (the Jouguet point, P*3*) (b). Reaction rates (c).*

where *D* ¼ *b*1*M* . Here, we took into account that

<sup>2</sup> *<sup>K</sup>* <sup>∗</sup> ð Þ *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> <sup>¼</sup> *<sup>μ</sup>b*<sup>1</sup>

*Schematic structure of a plane detonation wave: explosive substance (ES), detonation products (DP), and chemical reaction zone (CRZ) (a). The pressure changing in time: in front of the wave front P*1*, at the wave*

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

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

*<sup>P</sup>*<sup>3</sup> <sup>≈</sup> *<sup>ρ</sup>*1*D*<sup>2</sup> *γ* þ 1

*K* <sup>∗</sup> *T*<sup>1</sup>

*b*1 <sup>2</sup> <sup>¼</sup> *<sup>γ</sup>*

2 *K*<sup>∗</sup>

*<sup>γ</sup>M*<sup>2</sup> <sup>þ</sup> <sup>1</sup>

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

, (14)

*<sup>μ</sup>* , (15)

*<sup>γ</sup>M*<sup>2</sup> <sup>þ</sup> <sup>1</sup> ð Þ *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> , (13)

*ρ*1*D*<sup>2</sup>

*<sup>P</sup>*<sup>1</sup> ≫ 1, and that

Where *P*<sup>3</sup> is the pressure at front (3–3) separating the reaction zone from the reaction products, *P*<sup>1</sup> the pressure in front of the shock wave front, *ρ*<sup>1</sup> the gas density in front of the wave front, *D* the wave velocity, *γ* the adiabatic index, *ρ*<sup>3</sup> the medium density at the wave front (3–3), and *b*<sup>1</sup> the sound velocity in the motionless medium in front of the front. From the Mendeleev–Clapeyron equation

$$PV = \frac{m}{\mu} K^\* T \Rightarrow T = \frac{P\mu}{\rho K^\*},\tag{12}$$

substituting the values of *P*<sup>3</sup> (10) and *ρ*<sup>3</sup> (11), we determine the temperature *T*<sup>3</sup> at the Jouguet point,

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

#### **Figure 2.**

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

**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 [13] 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

*d* ¼ Δ*t D* � *vg*

where *d* is the chemical reaction zone width, Δ*t* the reaction duration, *D* the shock wave velocity, and *vg* 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 *T*<sup>2</sup> 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 Δ*t*. 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 (*D* ¼ *const*). According to the theory, the pressure *P*<sup>3</sup> and the density *ρ*<sup>3</sup> in the chemical reaction zone at the interface with detonation products (the Jouguet point), are connected with each

*<sup>P</sup>*<sup>3</sup> <sup>¼</sup> *<sup>P</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ρ</sup>*1*D*<sup>2</sup>

ð Þ *γ* þ 1

*<sup>K</sup>* <sup>∗</sup> *<sup>T</sup>* ) *<sup>T</sup>* <sup>¼</sup> *<sup>P</sup><sup>μ</sup>*

substituting the values of *P*<sup>3</sup> (10) and *ρ*<sup>3</sup> (11), we determine the temperature *T*<sup>3</sup>

¼ *D*2

*b*1

Where *P*<sup>3</sup> is the pressure at front (3–3) separating the reaction zone from the reaction products, *P*<sup>1</sup> the pressure in front of the shock wave front, *ρ*<sup>1</sup> the gas density in front of the wave front, *D* the wave velocity, *γ* the adiabatic index, *ρ*<sup>3</sup> the medium density at the wave front (3–3), and *b*<sup>1</sup> the sound velocity in the motionless

*ρ*3 *ρ*1

medium in front of the front. From the Mendeleev–Clapeyron equation

*PV* <sup>¼</sup> *<sup>m</sup> μ*

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

**3. Elements of the hydrodynamic theory of detonation: limiting**

*T*<sup>2</sup> ≥*Tx*, (8)

, (9)

<sup>1</sup> <sup>þ</sup> *<sup>γ</sup>* , (10)

<sup>2</sup> <sup>þ</sup> *<sup>γ</sup>D*<sup>2</sup> , (11)

*<sup>ρ</sup>K*<sup>∗</sup> , (12)

our case, the important criterion,

*Recent Advances in Numerical Simulations*

considers a plane detonation front,

other by the following relations [13]:

at the Jouguet point,

**152**

*Schematic structure of a plane detonation wave: explosive substance (ES), detonation products (DP), and chemical reaction zone (CRZ) (a). The pressure changing in time: in front of the wave front P*1*, at the wave front P*2*, and in the chemical reaction zone (the Jouguet point, P*3*) (b). Reaction rates (c).*

$$T\_3 = \frac{\rho\_1 D^2}{\chi + 1} \times \frac{\mu}{K^\*} \frac{\left(\mathcal{y} D^2 + b\_1^{-2}\right)}{\rho\_1 D^2 \left(\mathcal{y} + 1\right)} = \frac{\mu \left(\mathcal{y} D^2 + b\_1^{-2}\right)}{K^\* \left(\mathcal{y} + 1\right)^2} = \frac{\mu b\_1^{-2}}{K^\*} \frac{\left(\mathcal{y} M^2 + 1\right)}{\left(\mathcal{y} + 1\right)^2} = T\_1 \mathcal{Y} \times \frac{\left(\mathcal{y} M^2 + 1\right)}{\left(\mathcal{y} + 1\right)^2},\tag{13}$$

where *D* ¼ *b*1*M* . Here, we took into account that

$$P\_3 \approx \frac{\rho\_1 D^2}{\gamma + 1},\tag{14}$$

when *<sup>P</sup>*<sup>3</sup> *<sup>P</sup>*<sup>1</sup> ≫ 1, and that

$$\left|b\_{1}\right|^{2} = \gamma \frac{K^\* \, T\_1}{\mu},\tag{15}$$

where *μ* 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:

$$T\_3 > T\_2;\tag{16}$$

with respect to *M* (keeping in mind that *M* >0), we obtain

Eq. (20), it is expedient to admit that the temperature equality [14]

or, substituting the corresponding *γ* -value,

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

as the lower detonation limit.

formula

**155**

*<sup>M</sup>*<sup>4</sup> <sup>2</sup>*<sup>γ</sup>* � *<sup>γ</sup>*<sup>2</sup> � � <sup>þ</sup> *<sup>M</sup>*<sup>2</sup> *<sup>γ</sup>*<sup>2</sup> � <sup>5</sup>*<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> � � <sup>þ</sup> <sup>2</sup>*<sup>γ</sup>* � <sup>2</sup> <sup>¼</sup> <sup>0</sup> (21)

The positive roots of this equation are *M*<sup>1</sup> ¼ 2*:*145 and *M*<sup>2</sup> ¼ 0*:*455. For shock transitions, the most interesting is the first root, *M*<sup>1</sup> ¼ 2*:*145≈2*:*2 . 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 *M* >2*:*2 . Owing to hydrodynamic reasons, there is no detonation for waves with *M* < 2*:*2. While analyzing

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

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 *M* ≈ 2*:*2 should be considered

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 [4]. For the normal spherical detonation, it can be determined from the

ð Þ *γ* � 1 *Qc*

2

6*:*2≥ *M* ≥2*:*2*:* (25)

, (24)

*C*≈173*K* ¼

4*γ*<sup>2</sup>*K* <sup>∗</sup> *T*<sup>1</sup> " #<sup>1</sup>

derived in work [4]. All quantities in this formula are known, except for the parameter *c*, the specific content of the burned out gas (hydrogen). The intensity of a detonation wave can be controlled by changing, mainly, two parameters: *c* (in the numerator) and *T*<sup>1</sup> (in the denominator). In our case, all hydrogen compressed by the shock wave burns out. The values of coefficient *c* are confined within the interval 0*:*66≥*c* >0. Let we have the stoichiometric mixture of hydrogen with

*min* . We suppose that a further temperature decrease will result in changes of the adiabatic index *γ* and the physical properties of the reacting mixture [15, 16], i.e. let formula (24) be valid for real gases at *T*<sup>1</sup> ≥173*K*. In this case, we obtain a rough estimate for the Mach number maximum, *Mmax* ¼ 6*:*2. Note again that a strong explosion takes place in a cooled down medium. In this case, *Mmax* ¼ 6*:*2. Hence, we estimated the interval of possible Mach numbers for the normal spherical detona-

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

oxygen (*<sup>c</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>66</sup> <sup>¼</sup> *max* ), and the medium temperature *<sup>T</sup>*<sup>1</sup> ¼ �100°

tion of the hydrogen-oxygen gas mixture under real conditions:

<sup>0</sup>*:*84*M*<sup>4</sup> � <sup>4</sup>*:*04*M*<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*<sup>8</sup> <sup>¼</sup> <sup>0</sup>*:* (22)

*T*<sup>3</sup> ≈*T*<sup>2</sup> ≈*Tx* (23)

or, in a wider sense (**Figure 3**),

$$T\_3 > T\_2 > T\_x \tag{17}$$

Let us analyze inequality (16) in detail. From the theory of shock waves [13], 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:

$$T\_1 \gamma \frac{\gamma \mathbf{M}^2 + \mathbf{1}}{\left(\gamma + \mathbf{1}\right)^2} > T\_1 \frac{\left(2\gamma \mathbf{M}^2 - \gamma + \mathbf{1}\right)\left(2 + (\gamma - \mathbf{1})\mathbf{M}^2\right)}{\left(\gamma + \mathbf{1}\right)^2 \mathbf{M}^2} \tag{18}$$

or

$$\log\left(\mathbf{y}\mathbf{M}^2+\mathbf{1}\right)-\frac{\left(2\mathbf{y}\mathbf{M}^2-\mathbf{y}+\mathbf{1}\right)\left(2+(\mathbf{y}-\mathbf{1})\mathbf{M}^2\right)}{\mathbf{M}^2} > 0,\tag{19}$$

since *T*<sup>1</sup> >0 and *γ* > 0. By solving the equation

$$\chi \left( \chi \mathbf{M}^2 + \mathbf{1} \right) - \frac{\left( 2\chi \mathbf{M}^2 - \chi + \mathbf{1} \right) \left( 2 + (\chi - \mathbf{1}) \mathbf{M}^2 \right)}{\mathbf{M}^2} = \mathbf{0} \tag{20}$$

**Figure 3.**

*Schematic diagrams for the temperature profiles behind the wave front: the general case (a) and the case where T*<sup>3</sup> ≈*T*<sup>2</sup> ≈*Tx (b) corresponding to the limiting detonation process.*

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

with respect to *M* (keeping in mind that *M* >0), we obtain

$$M^4 \left(2\gamma - \gamma^2\right) + M^2 \left(\chi^2 - 5\gamma + 1\right) + 2\gamma - 2 = 0\tag{21}$$

or, substituting the corresponding *γ* -value,

where *μ* 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

*T*<sup>3</sup> >*T*<sup>2</sup> >*Tx* (17) .

*γ γM*<sup>2</sup> <sup>þ</sup> <sup>1</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>

*γ γM*<sup>2</sup> <sup>þ</sup> <sup>1</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>

*Schematic diagrams for the temperature profiles behind the wave front: the general case (a) and the case where*

*T*<sup>3</sup> ≈*T*<sup>2</sup> ≈*Tx (b) corresponding to the limiting detonation process.*

Let us analyze inequality (16) in detail. From the theory of shock waves [13], it is known that the temperature at point 2 in **Figure 3** is determined by relation (7).

> <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>

*T*<sup>3</sup> >*T*2; (16)

*<sup>M</sup>*<sup>2</sup> (18)

*<sup>M</sup>*<sup>2</sup> <sup>&</sup>gt;0, (19)

*<sup>M</sup>*<sup>2</sup> <sup>¼</sup> 0 (20)

satisfied:

or

**Figure 3.**

**154**

or, in a wider sense (**Figure 3**),

*Recent Advances in Numerical Simulations*

*T*1*γ*

Therefore, inequality (16) can be transformed as follows:

*<sup>γ</sup>*М<sup>2</sup> <sup>þ</sup> <sup>1</sup> ð Þ *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> <sup>&</sup>gt;*T*<sup>1</sup>

since *T*<sup>1</sup> >0 and *γ* > 0. By solving the equation

$$0.84M^4 - 4.04M^2 + 0.8 = 0.\tag{22}$$

The positive roots of this equation are *M*<sup>1</sup> ¼ 2*:*145 and *M*<sup>2</sup> ¼ 0*:*455. For shock transitions, the most interesting is the first root, *M*<sup>1</sup> ¼ 2*:*145≈2*:*2 . 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 *M* >2*:*2 . Owing to hydrodynamic reasons, there is no detonation for waves with *M* < 2*:*2. While analyzing Eq. (20), it is expedient to admit that the temperature equality [14]

$$T\_3 \approx T\_2 \approx T\_x \tag{23}$$

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 *M* ≈ 2*:*2 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 [4]. For the normal spherical detonation, it can be determined from the formula

$$M = \left[\frac{(\chi+\mathbf{1})^2(\chi-\mathbf{1})Q\mathbf{c}}{4\chi^2 K^\* T\_1}\right]^{\frac{1}{2}},\tag{24}$$

derived in work [4]. All quantities in this formula are known, except for the parameter *c*, the specific content of the burned out gas (hydrogen). The intensity of a detonation wave can be controlled by changing, mainly, two parameters: *c* (in the numerator) and *T*<sup>1</sup> (in the denominator). In our case, all hydrogen compressed by the shock wave burns out. The values of coefficient *c* are confined within the interval 0*:*66≥*c* >0. Let we have the stoichiometric mixture of hydrogen with oxygen (*<sup>c</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>66</sup> <sup>¼</sup> *max* ), and the medium temperature *<sup>T</sup>*<sup>1</sup> ¼ �100° *C*≈173*K* ¼ *min* . We suppose that a further temperature decrease will result in changes of the adiabatic index *γ* and the physical properties of the reacting mixture [15, 16], i.e. let formula (24) be valid for real gases at *T*<sup>1</sup> ≥173*K*. In this case, we obtain a rough estimate for the Mach number maximum, *Mmax* ¼ 6*:*2. Note again that a strong explosion takes place in a cooled down medium. In this case, *Mmax* ¼ 6*:*2. Hence, we estimated the interval of possible Mach numbers for the normal spherical detonation of the hydrogen-oxygen gas mixture under real conditions:

$$\text{6.2} \geq \text{M} \geq \text{2.2}.\tag{25}$$

#### **Figure 4.**

*Dependences of the critical temperature Tx and the detonation temperature in the motionless medium, T*<sup>1</sup>*, on the Mach number M at the fixed pressure P* ¼ 60 *mm Hg.*

In view of relation (6), let us plot the dependence of the critical temperature on the Mach number, *Tx* (M) (**Figure 4**). Since the Mach number range was found, we will calculate the critical temperature *Tx* for every *M* from the indicated interval with an increment of 0.2 and the fixed initial pressure *P*<sup>0</sup> (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 [4]).

The larger the Mach number, the higher is the critical temperature. However, at *M* ≥5, the critical temperature growth becomes a little slower. At the lower limit *M* ¼ 2*:*2, *Tx* ¼ 1130 *K*, and, at the upper limit *M* ¼ 6*:*2, *Tx* ¼ 1479 *K* . Hence, in a hydrogen-oxygen mixture, the critical temperature *Tx* for the allowable values of Mach number *M* accepts values from the following interval:

$$1479\ K \ge T\_x \ge 1130\ K.\tag{26}$$

609 *K* ≥ *T*<sup>1</sup> ≥176 *K:* (28)

This is the minimum temperature in front of the shock wave that makes the

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

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 *c* ¼ 0*:*66. Above this value, the chemical reactions resulting from the interaction between hydrogen and oxygen in the mixture, which were considered in work [4], become more complicated, and this circumstance may result in different values of critical temperature. Further researches of this issue are required. Below, the choice *T*<sup>1</sup> ¼ 800 *K* for the upper limit of the medium temperature is explained in detail.

With regard for dependence (24) of the Mach number *M* on the temperature of a motionless medium *T*<sup>1</sup> and the hydrogen content *c*, let us plot the dependences

dependence *M c*ð Þ at *T*<sup>1</sup> ¼ *const* exhibited in **Figure 5** for *T*<sup>1</sup> -temperatures in the interval 800 *K* ≥*T*<sup>1</sup> ≥ 173 *K* (see **Table 2**). The lower curve corresponds to the gas mixture temperature *T*<sup>1</sup> ¼ 800 *K*, and the upper one to *T*<sup>1</sup> ¼ 173 *K*. According to expression (24), this family of curves has a power dependence on the hydrogen content in the mixture, *c*, with a power exponent of 0.5. Let us fix the maximum content of burned out hydrogen, *c* ¼ 0*:*66, 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 *M* ¼ 2*:*8 at point 4 to *M* ¼ 6*:*2 at point 5. Another important detail should be emphasized. Four dashed lines are drawn in **Figure 5**. Two horizontal

*Dependence of the Mach number M on the hydrogen content c in the gas mixture (P*<sup>0</sup> ¼ 60 *mm Hg) for various*

ð Þ*c* at fixed *T*<sup>1</sup> and *P*0. We proceed from the plots of the

**4. Results of calculations and their discussion**

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

detonation possible.

*M c*ð Þ, *Tx*ð Þ*<sup>c</sup>* , and *<sup>T</sup>*<sup>1</sup>

**Figure 5.**

**157**

*temperatures in the motionless medium, T*1*:*

**Figure 4** also exhibits the dependence of the detonation temperature *T*<sup>1</sup> on the Mach number *M*, which can easily be obtained [4] by substituting the critical temperature *Tx* into relation (7):

$$T^1 = \frac{(\chi + \mathbf{1})^2 \mathbf{M}^2 T\_x}{\left(2\chi \mathbf{M}^2 - \chi + \mathbf{1}\right)\left(2 + (\chi - \mathbf{1})\mathbf{M}^2\right)}.\tag{27}$$

From Eqs. (25) and (26), it follows that the detonation temperature for a motionless medium falls within the interval


**Table 1.**

*Values of critical temperature Tx and detonation temperature T*<sup>1</sup> *depending on the Mach number M ( P*<sup>0</sup> ¼ 60 *mm Hg).*

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

$$1609\,\mathrm{K} \geq T^1 \geq 176\,\mathrm{K}.\tag{28}$$

This is the minimum temperature in front of the shock wave that makes the detonation possible.
