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

#### **Figure 5.**

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

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

*Dependences of the critical temperature Tx and the detonation temperature in the motionless medium, T*<sup>1</sup>*, on*

**Figure 4** also exhibits the dependence of the detonation temperature *T*<sup>1</sup> on the

*M*<sup>2</sup> *Tx*

Mach number *M*, which can easily be obtained [4] by substituting the critical

From Eqs. (25) and (26), it follows that the detonation temperature for a

*M* 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 *Tx*, [*K*] 1130 1166 1201 1233 1260 1286 1309 1329 1349 1365

, [*K*] 609 572 537 503 470 440 412 385 360 337 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 1381 1396 1408 1420 1431 1441 1450 1458 1466 1473 1479 316 297 279 262 247 233 219 207 196 186 176

*Values of critical temperature Tx and detonation temperature T*<sup>1</sup> *depending on the Mach number M*

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

1479 *K* ≥*Tx* ≥1130 *K:* (26)

<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> *:* (27)

Mach number *M* accepts values from the following interval:

temperature *Tx* into relation (7):

**Figure 4.**

*T*1

**Table 1.**

**156**

*( P*<sup>0</sup> ¼ 60 *mm Hg).*

motionless medium falls within the interval

*the Mach number M at the fixed pressure P* ¼ 60 *mm Hg.*

*Recent Advances in Numerical Simulations*


In **Figure 6**, using expression (32) and **Table 3**, we plotted the dependences *Tx*ð Þ*c* at *P*<sup>0</sup> ¼ *const* and *T*<sup>1</sup> ¼ *const* . By the form, they are similar to the previous

> *M*<sup>2</sup> *<sup>K</sup>* <sup>∗</sup> *<sup>T</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> � � !*:* (33)

ð Þ*c* , at *P*<sup>0</sup> ¼ *const* and *T*<sup>1</sup> ¼ *const* . It

More interesting is the dependence of the detonation temperature in the

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

*Dependences of the critical temperature Tx on the hydrogen content c in the gas mixture (P* ¼ 60 *mm Hg) for*

173°K *Tx* 1040 1176 1256 1317 1354 1382 1405 1423 1436 1450 1460 1470 1480 273°K 980 1080 1160 1220 1270 1305 1338 1358 1376 1390 1408 1420 1427 373°K 1000 1080 1155 1205 1240 1276 1300 1327 1343 1364 1375 1384 473°K 1040 1113 1166 1200 1230 1254 1280 1300 1318 1330 1344 573°K 1008 1070 1116 1152 1184 1216 1240 1259 1282 1300 1316 673°K 1030 1076 1118 1148 1180 1204 1223 1244 1260 1280 773°K 1050 1085 1122 1150 1174 1195 1215 1236 1253 800°K 1078 1112 1142 1164 1187 1205 1226 1246

*Data showing the functional dependence of the critical temperature Tx on the specific content of hydrogen c at*

*c* 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.66

plots (**Figure 5**) and confirm the conclusions made for points 1 to 5.

can be determined from relation (5) with regard for Eqs. (7) and (4):

<sup>4</sup>*γ*2*<sup>K</sup>* <sup>∗</sup> *<sup>P</sup>*<sup>0</sup> <sup>2</sup>*γM*<sup>2</sup> � *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> � �*M*<sup>2</sup>

� exp � *<sup>E</sup>*2ð Þ *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup>

*QT*0ð Þ *<sup>γ</sup>* � <sup>1</sup> ð Þ *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>3</sup>

motionless medium on the hydrogen content, *T*<sup>1</sup>

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

*<sup>T</sup>*<sup>1</sup> � �<sup>2</sup> <sup>¼</sup> <sup>2</sup>*:*<sup>5</sup> � <sup>10</sup><sup>5</sup>

**Figure 6.**

**Table 3.**

**159**

*T*<sup>1</sup> ¼ *const, P*<sup>0</sup> ¼ 60 *mm Hg.*

*various temperatures in the motionless medium, T*1*.*

**Table 2.**

*Data of the Mach number M, depending on the specific content of hydrogen c at T*<sup>1</sup> ¼ *const, P*<sup>0</sup> ¼ 60 *mm Hg .*

ones confine the region of allowable Mach numbers corresponding to the normal spherical detonation. The first line corresponds to the minimum *Mmin* ¼ 2*:*2, and the second one to the maximum *Mmax* ¼ 6*:*2. 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 *Mmin* ¼ 2*:*2, 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 *T*<sup>1</sup> ¼ 173 *K*, the detonation is possible if *M* ¼ *Mmax* ¼ 6*:*2 (point 5 in **Figure 5**). In other words, for the chosen temperature, 800 *K* ≥*T*<sup>1</sup> ≥173 *K*, and hydrogen content, 0.66 ≥*c*≥0*:*075, 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 *M c*ð Þ , *T*<sup>1</sup> , this segment is curvilinear.

Let us derive the functional dependence *Tx*ð Þ*c* by substituting the *M c*ð Þ dependence (Eq. (24)) into Eq. (5). To make transformations simpler, let us rewrite Eq. (24) in a slightly different form,

$$M = \left[\eta c\right]^{\frac{1}{2}},\tag{29}$$

where

$$\eta = \frac{(\chi + \mathbf{1})^2 (\chi - \mathbf{1}) Q}{4 \eta^2 K^\* \, T\_1}. \tag{30}$$

Then we obtain the following transcendental equation for the critical temperature *Tx*:

$$T\_x^2 = \frac{2.5 \times 10^5 Q T\_0 (\chi - 1)(2\eta\wp - \chi + 1)(2 + (\chi - 1)\eta c)^2}{4\eta^2 (\chi + 1)K^\* P\_0 \eta^3 c^3} \times \exp\left(-\frac{E\_2}{K^\* T\_x}\right) \tag{31}$$

or, taking Eq. (30) into account,

$$T\_x^2 = \frac{2.5 \times 10^5 T\_0 T\_1 (2\eta\eta c - \chi + 1)(2 + (\chi - 1)\eta c)^2}{(\chi + 1)^3 P\_0 \eta^2 c^3} \exp\left(-\frac{E\_2}{K^\* T\_x}\right). \tag{32}$$

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

In **Figure 6**, using expression (32) and **Table 3**, we plotted the dependences *Tx*ð Þ*c* at *P*<sup>0</sup> ¼ *const* and *T*<sup>1</sup> ¼ *const* . By the form, they are similar to the previous plots (**Figure 5**) and confirm the conclusions made for points 1 to 5.

More interesting is the dependence of the detonation temperature in the motionless medium on the hydrogen content, *T*<sup>1</sup> ð Þ*c* , at *P*<sup>0</sup> ¼ *const* and *T*<sup>1</sup> ¼ *const* . It can be determined from relation (5) with regard for Eqs. (7) and (4):

$$\begin{split} \left( \left( T^{1} \right)^{2} = \frac{2.5 \times 10^{5} Q T o \left( \chi - 1 \right) \left( \chi + 1 \right)^{3}}{4 \chi^{2} K^{\*} P\_{0} \left( 2 \gamma M^{2} - \chi + 1 \right) M^{2}} \\ \times \exp \left( - \frac{E\_{2} \left( \chi + 1 \right)^{2} M^{2}}{K^{\*} T^{1} \left( 2 \gamma M^{2} - \chi + 1 \right) \left( 2 + \left( \chi - 1 \right) M^{2} \right)} \right) . \end{split} \tag{33}$$

#### **Figure 6.**

ones confine the region of allowable Mach numbers corresponding to the normal spherical detonation. The first line corresponds to the minimum *Mmin* ¼ 2*:*2, and the second one to the maximum *Mmax* ¼ 6*:*2. 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.

*Data of the Mach number M, depending on the specific content of hydrogen c at T*<sup>1</sup> ¼ *const, P*<sup>0</sup> ¼ 60 *mm Hg .*

*c* 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.66

173°K *M* 1.71 2.42 3 3.42 3.85 4.19 4.51 4.84 5.1 5.41 5.7 5.93 6.2 273°K 1.36 1.93 2.32 2.72 3.05 3.34 3.6 3.85 4.1 4.31 4.5 4.72 4.95 373°K 1.17 1.65 2 2.33 2.6 2.85 3.1 3.3 3.5 3.69 3.9 4.04 4.23 473°K 1.03 1.46 1.8 2.07 2.32 2.53 2.72 2.93 3.1 3.27 3.42 3.59 3.76 573°K 0.94 1.33 1.62 1.88 2.12 2.3 2.5 2.66 2.81 2.97 3.1 3.26 3.42 673°K 0.87 1.23 1.5 1.74 1.93 2.13 2.3 2.45 2.6 2.74 2.86 3.01 3.15 773°K 0.81 1.14 1.39 1.62 1.8 1.98 2.16 2.29 2.43 2.56 2.7 2.81 2.94 800°K 0.8 1.11 1.34 1.59 1.75 1.95 2.11 2.25 2.38 2.52 2.64 2.76 2.89

Let us consider points 1 to 5 in **Figure 5** separately. Segment 1–2 corresponds to the lower limit of the shock wave velocity *Mmin* ¼ 2*:*2, 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 *T*<sup>1</sup> ¼ 173 *K*, the detonation is possible if *M* ¼ *Mmax* ¼ 6*:*2 (point 5 in **Figure 5**). In other words, for the chosen temperature, 800 *K* ≥*T*<sup>1</sup> ≥173 *K*, and hydrogen content, 0.66 ≥*c*≥0*:*075, 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

Let us derive the functional dependence *Tx*ð Þ*c* by substituting the *M c*ð Þ dependence (Eq. (24)) into Eq. (5). To make transformations simpler, let us rewrite

1

ð Þ *γ* � 1 *Q* 4*γ*<sup>2</sup>*K* <sup>∗</sup> *T*<sup>1</sup>

<sup>4</sup>*γ*<sup>2</sup>ð Þ *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> *<sup>K</sup>* <sup>∗</sup> *<sup>P</sup>*0*η*<sup>3</sup>*c*<sup>3</sup> � exp � *<sup>E</sup>*<sup>2</sup>

2, (29)

2

2

*<sup>P</sup>*0*η*<sup>2</sup>*c*<sup>3</sup> exp � *<sup>E</sup>*<sup>2</sup>

*:* (30)

*K* <sup>∗</sup> *Tx* 

*K* <sup>∗</sup> *Tx* 

(31)

*:* (32)

*M* ¼ ½ � *ηc*

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

Then we obtain the following transcendental equation for the critical

*QT*0ð Þ *γ* � 1 ð Þ 2*γηc* � *γ* þ 1 ð Þ 2 þ ð Þ *γ* � 1 *ηc*

*T*0*T*1ð Þ 2*γηc* � *γ* þ 1 ð Þ 2 þ ð Þ *γ* � 1 *ηc*

ð Þ *<sup>γ</sup>* <sup>þ</sup> <sup>1</sup> <sup>3</sup>

*M c*ð Þ , *T*<sup>1</sup> , this segment is curvilinear.

*Recent Advances in Numerical Simulations*

Eq. (24) in a slightly different form,

or, taking Eq. (30) into account,

where

**Table 2.**

temperature *Tx*:

*T*2

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

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

*T*2

**158**

*Dependences of the critical temperature Tx on the hydrogen content c in the gas mixture (P* ¼ 60 *mm Hg) for various temperatures in the motionless medium, T*1*.*


#### **Table 3.**

*Data showing the functional dependence of the critical temperature Tx on the specific content of hydrogen c at T*<sup>1</sup> ¼ *const, P*<sup>0</sup> ¼ 60 *mm Hg.*

Making allowance for Eqs. (29) and (30), relation (33) can be simplified to the following form:

*T*1

medium (*T*<sup>1</sup> <sup>¼</sup> *<sup>T</sup>*<sup>1</sup>

following.

ð Þ*c* . (ii) As the hydrogen content in the mixture grows, the temperature of the detonation *T*<sup>1</sup> 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 *<sup>T</sup>*<sup>1</sup> <sup>¼</sup> <sup>173</sup> *<sup>K</sup>*). At the point of its intersection with the curve corresponding to the same temperature of the motionless medium (in our case, this is *T*<sup>1</sup> ¼ 173 *K*), the detonation condition *T*<sup>2</sup> ¼ *Tx* (point 5) is satisfied. Detonation becomes probable, because the current temperature of motionless medium reaches the detonation temperature for this

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

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**

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

1.As was indicated above, a temperature lower than *T*<sup>1</sup> ≈173÷176 *K* 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 *Mmax* ¼ 6*:*2.

Eq. (34) and **Figure 7**), the detonation is possible if the hydrogen content in

3.Physical restrictions imposed by the minimum Mach number *Mmin* ¼ 2*:*2 bring about the existence of the upper limit for the detonation temperature, *<sup>T</sup>*<sup>1</sup> <sup>¼</sup> 609 *K* . 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 *T*<sup>2</sup> ≥*Tx* is satisfied at *T*<sup>1</sup> ≥609 *K* . Whence it follows that *c* ¼ 0*:*27 is the minimum hydrogen content in the mixture, below which the detonation is mpossible even at very high

4.Experimental results testify that, if the temperature of a gas mixture is higher than *T*<sup>1</sup> ¼ 800 *K*, 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

From the reasons given above, it follows that the region of spherical supersonic

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

burning is bounded by segments 2–3, 3–4, 4–5, and 5–2 in **Figure 7**. For the

corresponds to the condition *T*<sup>2</sup> ≥*Tx* for the whole family of curves.

2.According to the plot of the functional dependence *T*<sup>1</sup>

the mixture is not lower than 0.57.

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

hydrogen-oxygen mixture can be heated.

illustrative purpose, it was hatched.

temperatures.

**5. Conclusions**

**161**

). (iv) The intersection points of any horizontal line (see item iii)

ð Þ*c* at *T*<sup>1</sup> ¼ 273 *K* (see

$$\left(T^{1}\right)^{2} = \frac{2.5 \times 10^{5} T\_{0} T\_{1} (\chi + 1)}{c P\_{0} (2\eta\eta c - \chi + 1)} \times \exp\left(-\frac{E\_{2} (\chi + 1)^{2} \eta c}{K^{\*} \, T^{1} (2\eta\eta c - \chi + 1)(2 + (\chi - 1)\eta c)}\right). \tag{34}$$

The corresponding family of curves is shown in **Figure 7**, according to **Table 4**. While analyzing the plots, the attention should be drawn to the following facts. (i) Every temperature *T*<sup>1</sup> of the gas mixture is associated with a specific dependence

#### **Figure 7.**

*Diagrams of the dependence of the detonation temperature T*<sup>1</sup> *on the hydrogen content c in an explosive gas mixture H*<sup>2</sup> þ *O*<sup>2</sup> *(P* ¼ 60 *mm Hg) for different temperatures in a stationary environment, T*1*.*


#### **Table 4.**

*Values of the detonation temperature T*<sup>1</sup>*, depending on the specific content of hydrogen c at T*<sup>1</sup> <sup>¼</sup> *const, 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*

*T*1 ð Þ*c* . (ii) As the hydrogen content in the mixture grows, the temperature of the detonation *T*<sup>1</sup> 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 *<sup>T</sup>*<sup>1</sup> <sup>¼</sup> <sup>173</sup> *<sup>K</sup>*). At the point of its intersection with the curve corresponding to the same temperature of the motionless medium (in our case, this is *T*<sup>1</sup> ¼ 173 *K*), the detonation condition *T*<sup>2</sup> ¼ *Tx* (point 5) is satisfied. Detonation becomes probable, because the current temperature of motionless medium reaches the detonation temperature for this medium (*T*<sup>1</sup> <sup>¼</sup> *<sup>T</sup>*<sup>1</sup> ). (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 *T*<sup>2</sup> ≥*Tx* 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.


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.
