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# Determination of Values Range of Physical Quantities and Existence Parameters of Normal Spherical Detonation by the Method of Numerical Simulation

By Myron Polatayko

Submitted: November 23rd 2020Reviewed: January 29th 2021Published: February 25th 2021

DOI: 10.5772/intechopen.96285

## Abstract

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.

### Keywords

• detonation wave
• critical temperature
• range of permissible values
• process modeling

## 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.

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 jump-like 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.

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÷1015cm3. Rapid chain transformations start just from those initial centers [10] and run following the Lewis scheme. In this case, we have

OH+H2=H2O+H,E1
H+O2=OH+O.E2

Note also that the temperature T2, at which the branching probability δequals unity,

δ=1,E3

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

T2=Tx,E4

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 Txin the medium, at which the branching probability δequals unity. How can Txbe 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,

Tx2=2.5×105QT0γ12γM2γ+12+γ1M224γ2γ+1M6KP0×expE2KTx,E5

where Mis the Mach number (it reflects the shock transition intensity); P0the initial, before the explosion ignition (at 293 K), pressure of the gas mixture reckoned in mm Hg units; E2the activation energy of the branching reaction (2); Kthe gas constant; Qthe combustion energy per gas mole; and γthe adiabatic index for the given gas mixture. For the hydrogen-oxygen mixture, the corresponding numerical values are [12]: γ=1.4, Q=286.5kJ/mol, K=8.31J/mol/K, E2=16×103×4.19J/mol, and T0=293K. Then Eq. (5) reads

Tx2=5.38×10102+0.4M222.8M20.4P0M6×e8067Tx.E6

It is evident that the temperature Txis different for different Mach numbers. Formula (6) describes the functional dependence of the critical temperature Txon the Mach number Mfor the given initial pressure P0and allows one to compare its value with the real temperature

T2=2γM2γ+12+γ1M2γ+12M2×T1,E7

where T1is the temperature of the medium in front of the wave front. Hence, in our case, the important criterion,

T2Tx,E8

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 [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=tDvg,E9

where dis the chemical reaction zone width, tthe reaction duration, Dthe shock wave velocity, and vgthe 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 T2is 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 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=const). According to the theory, the pressure P3and the density ρ3in the chemical reaction zone at the interface with detonation products (the Jouguet point), are connected with each other by the following relations [13]:

P3=P1+ρ1D21+γ,E10
ρ3ρ1=D2γ+1b12+γD2,E11

Where P3is the pressure at front (3–3) separating the reaction zone from the reaction products, P1the pressure in front of the shock wave front, ρ1the gas density in front of the wave front, Dthe wave velocity, γthe adiabatic index, ρ3the medium density at the wave front (3–3), and b1the sound velocity in the motionless medium in front of the front. From the Mendeleev--Clapeyron equation

PV=mμKTT=ρK,E12

substituting the values of P3(10) and ρ3(11), we determine the temperature T3at the Jouguet point,

T3=ρ1D2γ+1×μKγD2+b12ρ1D2γ+1=μγD2+b12Kγ+12=μb12KγM2+1γ+12=T1γ×γM2+1γ+12,E13

where D=b1M. Here, we took into account that

P3ρ1D2γ+1,E14

when P3P11, and that

b12=γKT1μ,E15

where μis 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:

T3>T2;E16

or, in a wider sense (Figure 3),

T3>T2>Tx.E17

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:

T1γγМ2+1γ+12>T12γM2γ+12+γ1M2γ+12M2E18

or

γγM2+12γM2γ+12+γ1M2M2>0,E19

since T1>0and γ>0. By solving the equation

γγM2+12γM2γ+12+γ1M2M2=0E20

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

M42γγ2+M2γ25γ+1+2γ2=0E21

or, substituting the corresponding γ-value,

0.84M44.04M2+0.8=0.E22

The positive roots of this equation are M1=2.145and M2=0.455. For shock transitions, the most interesting is the first root, M1=2.1452.2. 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]

T3T2TxE23

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 M2.2should 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=γ+12γ1Qc4γ2KT112,E24

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 T1(in the denominator). In our case, all hydrogen compressed by the shock wave burns out. The values of coefficient care confined within the interval 0.66c>0. Let we have the stoichiometric mixture of hydrogen with oxygen (c=0.66=max), and the medium temperature T1=100°C173K=min. We suppose that a further temperature decrease will result in changes of the adiabatic index γand the physical properties of the reacting mixture [15, 16], i.e. let formula (24) be valid for real gases at T1173K. In this case, we obtain a rough estimate for the Mach number maximum, Mmax=6.2. Note again that a strong explosion takes place in a cooled down medium. In this case, Mmax=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:

6.2M2.2.E25

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 Txfor every Mfrom the indicated interval with an increment of 0.2 and the fixed initial pressure P0(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]).

 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 T1, [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

### Table 1.

Values of critical temperature Txand detonation temperature T1depending on the Mach number M(P0=60mmHg).

The larger the Mach number, the higher is the critical temperature. However, at M5, the critical temperature growth becomes a little slower. At the lower limit M=2.2, Tx=1130K, and, at the upper limit M=6.2, Tx=1479K. Hence, in a hydrogen-oxygen mixture, the critical temperature Txfor the allowable values of Mach number Maccepts values from the following interval:

1479KTx1130K.E26

Figure 4 also exhibits the dependence of the detonation temperature T1on the Mach number M, which can easily be obtained [4] by substituting the critical temperature Txinto relation (7):

T1=γ+12M2Tx2γM2γ+12+γ1M2.E27

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

609KT1176K.E28

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 c=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 T1=800Kfor the upper limit of the medium temperature is explained in detail.

With regard for dependence (24) of the Mach number Mon the temperature of a motionless medium T1and the hydrogen content c, let us plot the dependences Mc, Txc, and T1cat fixed T1and P0. We proceed from the plots of the dependence Mcat T1=constexhibited in Figure 5 for T1-temperatures in the interval 800KT1173K(see Table 2). The lower curve corresponds to the gas mixture temperature T1=800K, and the upper one to T1=173K. 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=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=2.8at point 4 to M=6.2at 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 Mmin=2.2, and the second one to the maximum Mmax=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.

 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

### Table 2.

Data of the Mach number M, depending on the specific content of hydrogen cat T1=const, P0=60mmHg.

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=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 T1=173K, the detonation is possible if M=Mmax=6.2(point 5 in Figure 5). In other words, for the chosen temperature, 800KT1173K, and hydrogen content, 0.66 c0.075, 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 McT1, this segment is curvilinear.

Let us derive the functional dependence Txcby substituting the Mcdependence (Eq. (24)) into Eq. (5). To make transformations simpler, let us rewrite Eq. (24) in a slightly different form,

M=ηc12,E29

where

η=γ+12γ1Q4γ2KT1.E30

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

Tx2=2.5×105QT0γ12γηcγ+12+γ1ηc24γ2γ+1KP0η3c3×expE2KTxE31

or, taking Eq. (30) into account,

Tx2=2.5×105T0T12γηcγ+12+γ1ηc2γ+13P0η2c3expE2KTx.E32

In Figure 6, using expression (32) and Table 3, we plotted the dependences Txcat P0=constand T1=const. By the form, they are similar to the previous plots (Figure 5) and confirm the conclusions made for points 1 to 5.

 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 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

### Table 3.

Data showing the functional dependence of the critical temperature Txon the specific content of hydrogen cat T1=const, P0=60mmHg.

More interesting is the dependence of the detonation temperature in the motionless medium on the hydrogen content, T1c, at P0=constand T1=const. It can be determined from relation (5) with regard for Eqs. (7) and (4):

T12=2.5×105QT0γ1γ+134γ2KP02γM2γ+1M2×expE2γ+12M2KT12γM2γ+12+γ1M2.E33

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

T12=2.5×105T0T1γ+1cP02γηcγ+1×expE2γ+12ηсKT12γηсγ+12+γ1ηс.E34

The corresponding family of curves is shown in Figure 7, according to Table 4.

 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 T1 680 542 456 398 346 313 280 258 232 217 200 188 173 273°K 658 568 500 442 409 372 345 320 302 280 266 250 373°K 626 576 516 477 436 412 381 360 340 328 310 473°K 620 572 535 500 467 436 416 392 378 360 573°K 660 616 580 540 508 476 458 432 418 398 673°K 652 616 576 543 512 492 470 454 438 773°K 644 609 580 548 526 504 484 470 800°K 658 625 600 570 548 523 507 493

### Table 4.

Values of the detonation temperature T1, depending on the specific content of hydrogen cat T1=const, P0=60mmHg.

While analyzing the plots, the attention should be drawn to the following facts. (i) Every temperature T1of the gas mixture is associated with a specific dependence T1c. (ii) As the hydrogen content in the mixture grows, the temperature of the detonation T1in 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 T1=173K). At the point of its intersection with the curve corresponding to the same temperature of the motionless medium (in our case, this is T1=173K), the detonation condition T2=Tx(point 5) is satisfied. Detonation becomes probable, because the current temperature of motionless medium reaches the detonation temperature for this medium (T1=T1). (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 T2Txfor 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.

1. As was indicated above, a temperature lower than T1173÷176Kcan 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=6.2.

2. According to the plot of the functional dependence T1cat T1=273K(see Eq. (34) and Figure 7), the detonation is possible if the hydrogen content in the mixture is not lower than 0.57.

3. Physical restrictions imposed by the minimum Mach number Mmin=2.2bring about the existence of the upper limit for the detonation temperature, T1=609K. 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 T2Txis satisfied at T1609K. Whence it follows that c=0.27is the minimum hydrogen content in the mixture, below which the detonation is mpossible even at very high temperatures.

4. Experimental results testify that, if the temperature of a gas mixture is higher than T1=800K, 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.

## 5. Conclusions

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 [4], the conditions are found, under which the probability of a chain branching reaches unity (δ=1), and a fast chain reaction is started. The existence of the critical temperature Txat 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 [4], the ondition T2Txis 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, T2, and in the chemical reaction zone, T3, is determined. The equality TxT2T3which 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.

## Acknowledgments

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.

## Nomenclature

Basic designations:H2

hydrogen molecule

O2

oxygen molecule

H2O

water molecule

O

oxygen atom

H

hydrogen atom

OH

compound of an oxygen atom with a hydrogen atom

Tx

critical temperature

T2

temperature at the shock front

δ

branching probability

D

shock wave velocity, detonation velocity

M

Mach number

P,T,ρ

pressure, temperature, density of the medium

ES

explosive substance

DP

detonation products

CRZ

chemical reaction zone

γ

Q

combustion energy of one mole of combustible gas

μ

molar mass

K

universal gas constant

E2

the activation energy of the branching reaction

d

the chemical reaction zone width

t

the reaction duration

vg

the gas velocity behind the reaction front

b1

the sound velocity in the motionless medium in front of the front

m

gas mass

T3

temperature at the Jouguet point

c

coefficient of flammable gas content in the mixture

W

chemical reaction rate

T1

detonation temperature

Mmax

maximum Mach number

Mmin

minimum Mach number

fx

function of the variable x

expx

exponential function

x

increment of variable x

much less

much more

almost equal

ab

line segment

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Myron Polatayko (February 25th 2021). Determination of Values Range of Physical Quantities and Existence Parameters of Normal Spherical Detonation by the Method of Numerical Simulation [Online First], IntechOpen, DOI: 10.5772/intechopen.96285. Available from: