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Determination of the Velocity of the Detonation Wave and the Conditions for the Appearance of Spherical Detonation during the Interaction of Hydrogen with Oxygen

By Myron Polatayko

Submitted: June 26th 2018Reviewed: October 2nd 2018Published: November 5th 2018

DOI: 10.5772/intechopen.81792

Downloaded: 112

Abstract

The well-known formula for the flat detonation wave velocity derived from the Hugoniot system of equations faces difficulties, if being applied to a spherical reactor. A similar formula has been obtained in the framework of the theory of explosion in reacting gas media with the use of a special model describing the transition of an explosive wave in the detonation. The derived formula is very simple, being also more suitable for studying the limiting processes of volume detonation. The conditions for the transition of a shock wave to a detonation wave are studied. Initial detonation conditions required for fast chemical reactions to take place at the front of a spherical explosive wave have been determined. A simple relation describing the critical detonation temperature for various pressures in the hydrogen-oxygen mixture was obtained. Using the known formulas for a shock transition, the critical temperature was coupled with the initial conditions in a static environment, such as the pressure, temperature, and hydrogen content in the mixture.

Keywords

  • detonation
  • point blast
  • spherical wave
  • Haber scheme
  • Lewis scheme
  • kinetics of chemical reactions
  • critical temperature

1. Introduction

The strong explosion in a small volume of a detonating gas mixture has been studied well in modern physics [1, 2]. The velocity of a detonation wave propagating in a spherical reactor can be calculated absolutely precisely with the use of a variety of original software programs [3]. There are also approximation formulas such as

DDn=1ArRx,E1

where ris the current radius, Rxis the critical radius, А is a constant, Dп is the velocity of a plane wave, and D is the velocity of a spherical wave. For larger charges, if the radius exceeds the critical one, the Eyring dependence

DDn=1ArE2

is used. The work is aimed at analyzing the development of the process at the time moment, when the energy of a point explosion is equal to the energy of a burned gas, r=Rx, but provided that Rx. In other words, we intended to study the initial stage of detonation in reacting gas media by determining the scalar value of detonation wave velocity. The transition to the Chapman-Jouguet regime begins at some distance Rxfrom the center [4, 5], when the energy of the system appreciably increases. The model supposes that the pressures at the front and in the explosion region are equal, which results in the appearance of a high temperature at the transition point behind the shock front, since the main part of the substance mass is concentrated in a thin layer of the blast wave.

A lot of researches were devoted to the emergence of detonation in a hydrogen-oxygen mixture [6, 7, 8]. At the same time, there are no works in the scientific literature dealing with the influence of initial conditions on the detonation process, when the blast wave propagates in the gas environment. In this paper it was possible to obtain the necessary results by studying the chain reactions [9] of the interaction of hydrogen with oxygen.

2. Determination of the detonation wave velocity in an explosive gas mixture

2.1. Explosion in a chemically inert gas mixture

Consider an explosion in a chemically inert gas mixture. Let the point explosion occur instantly in a perfect gas with density ρ0, and a shock wave propagates in the gas from the point of energy release. We intend to analyze the initial stage of the process of shock wave propagation, when the shock wave amplitude is still so high that the initial gas pressure, Р0, can be neglected. This assumption is equivalent to a neglect of the initial internal gas energy in comparison with the explosion one, i.e., we consider a strong explosion. The problem is to determine the velocity of the blast wave, when the wave front is modeled by a rigid piston compressing the volume of the gas in front (Figure 1). The main regularities of the process are well-known [10], and there is a simple approximate method to find them.

Figure 1.

Schematic diagram of a shock wave from the point explosion.

Let the total mass of a gas engaged into a blast wave be concentrated in a thin layer near the front surface. The gas density here is constant and equal to that at the front,

ρ1=γ+1γ1ρ0.E3

This formula can be derived from the formula for strong shock waves [11] in the case where the Mach number M1. To avoid a misunderstanding, note that, in this case, we mean a transformation of a medium denoted by subscript. 0 (the medium at rest before the explosion) into a medium denoted by subscript 1. The layer thickness ris determined from the condition of mass conservation,

4πR2Δrρ1=43πR3ρ0;E4

whence

Δr=Rρ03ρ1=R3γ1γ+1.E5

Since the layer is very thin, the gas velocity in it almost does not change and coincides with that at the front,

u1=2Dγ+1.E6

In the shock wave theory, a more accurate formula is considered, which couples the gas flow velocity behind the shock wave front, u1, with the front velocity D: u1=2Dγ+11b0D2, where b0 is the sound velocity in the unperturbed gas. The gas mass in the layer is finite and equal to the mass m of the gas originally contained within a sphere of radius R,

m=43πR3ρ0.E7

Let us denote the pressure at the inner layer side as Pc, and let it equal α times the pressure at the wave front, Pc=αP1. Newton’s second law for the layer rin thickness reads

ddtmu1=4πR2Pc=4πR2αP1.E8

It can be used only within the limits

0<RRx0,E9

where the quantity Rx0is determined from energy considerations. At this point, the kinetic energy of the blast wave is still high enough, and its velocity considerably exceeds that of sound in the unperturbed gas medium.

Here, we arrive at a detailed mathematical representation of formulas and equations, by using the already known relations (4) and (8), which are the conservation laws for the mass and the moment, respectively. However, the latter are not enough for the problem to be solved. One more equation is needed,

E=ET+Ek=const;E10

which is the energy conservation law. The explosion energy is constant and equal to a sum of two terms: the potential, ET, and kinetic, Ek, energies. The general system consists of three equations—Eqs. (4), (8), and (10)—appended by the condition of strong explosion, M1, when formula (6), the relations (4), (8), (10) and the condition of strong explosion

M1E11

are valid. Moreover, we have Eq. (6), and

P1=2γ+1ρ0D2,E12

where P1is the pressure at the front of the shock wave. Formula (12) follows from the relation P1P0=2γM2γ+1γ+1of work [11] in the case where M1: P1P02γM2γ+1=2γ×D2b02γ+1=2γ×D2ρ0γP0γ+1=2ρ0D2P0γ+1.

It should be noted that, in the given system of equations, relation (4) does not determine a connection between the regions separated by the shock wave front (regions 0 and 1). Instead, it couples the states before the explosion and after it. While solving this problem for the one-dimensional centrally symmetric flow, we come back to Eq. (8).

The mass itself depends on the time, so that it is the momentum mu1rather than the velocity that should be differentiated with respect to the time. The mass is subjected to the action of the force 4πR2Pcdirected from the inside, because the pressure Pcis applied to the inner side of the layer. The force acting from the outside is equal to zero, because the initial pressure of the gas is neglected. By expressing the quantities u1and P1in Eq. (8) in terms of the front velocity D=dRdtand using formulas (6) and (12), we obtain the new relation

13ddtR3D=αD2R2.E13

Bearing in mind that

ddt=ddRdRdt=DddRE14

and integrating Eq. (13), we find

D=aR31αE15

where ais the integration constant. To determine the parameters aand α, let us take the energy conservation law into account. The kinetic energy of the gas is equal to

Ek=mu122.E16

The internal energy is concentrated in a “cavity” confined by an infinitesimally thin layer. The pressure in the cavity is equal to Pc. Actually, this means that, strictly speaking, the whole mass is not contained in the layer. A small amount of the substance is included into the cavity as well. In gas dynamics, the specific internal energy of the ideal gas is calculated by the formula e=Pρ1γ1, where Pis the pressure, ρthe density, and γthe adiabatic index. Therefore, the internal energy is equal to

ET=1γ1×4πR33Pc,E17

so that

E=ET+Ek=1γ1×4πR33Pc+mu122.E18

Expressing the quantities Pcand u1once more in terms of Dand substituting D=aR31α, we obtain

E=43πρ0a22αγ21+2γ+12R361α.E19

Since the explosion energy Eis constant, the power exponent of the variable Rmust be equal to zero. This means

α=12.E20

We determine the constant afrom Eq. (19)

a=34π×γ1γ+123γ112Eρ012,E21

and substituting it together with Eq. (20) into formula (15), we arrive at the expression for the shock wave velocity in the case of point-like explosion

D=34π×γ1γ+123γ112Eρ012R32,E22

or

D=ξ0Eρ012R32,E23

where

ξ0=34π×γ1γ+123γ112=const.E24

2.2. Theory of explosion in a combustible mixture of gases

Distinctive features of the problem consist in that the exothermic chemical reactions are possible in such a medium. Therefore, it is quite reasonable to assume that the blast wave continuously transforms into the detonation one. Let us consider the following model. An explosion in the gas generates a strong shock wave, which propagates over the gas and heats it up to a state, in which burning reactions become probable. We denote the energy of explosion by E0. The energy Ureleased at the combustion of the gas is equal t

U=43πR13ρ0Q,R0<<R1E25

where Qis the specific heat released in the medium (per mass unit of the medium). The process is considered at the time moment t1, when R=R1(Figure 2). Supposing that E0>U, we determine a condition, under which the detonation energy weakly affects the gas flow [12],

R1<Rx,E26

where Rx3=3E04πQρ0. Let the charge have a finite radius R0Then, when applying the conventional theory of point explosion to the description of the motion, we have to use the estimation

R0<R<Rx.E27

Figure 2.

Scenario of the continuous transformation of a blast wave into a detonation one: R 0 is the charge radius, R x is the initial threshold, R 2 is the final threshold, where the transformation of the strong detonation mode into the Chapman-Jouguet one is possible.

It should be noticed that conditions (26) and (27) strongly restrict the scope, where the laws of point explosion in an inert gas are applicable to the flows in the detonating medium. However, if the energy E0is high, and if it is released in a small volume, the flow in the region R1<Rxwould mainly occur as it does at an ordinary point explosion. On the other hand, for the time moment t2, at which

R=R2andE0<U,E28

the combustion processes start to play a dominating role, and the gas flow will possess the main characteristics of the detonation combustion [12].

From the aforesaid, some interesting conclusions can be drawn.

  1. The theory of a point explosion is proposed to be used for a combustible mixture of gases within the limits R0<R<Rx, if the proposed model of transformation of a blast wave into a detonation one is valid for the given mixture. Another scenario is probable, when the detonation is impossible under the given physico-chemical conditions in the gas medium, and the blast wave simply fades.

  2. When RRx, the energy of the system considerably changes, Econst, increasing almost twice as much, which has to be taken into consideration while studying the gas motion at this stage.

  3. It is evident that, if RRx, the energy becomes proportional to the cube of the sphere radius, ER3.

Hence, we come to an idea of that, for the model of point explosion in a combustible mixture of gases to be valid at RRxunder our conditions, it should be either modified or extended. Look once more at formula (19) expressing the energy conservation law. In the theory of point explosion for a usual non-detonable mixture of gases, it is adopted that E=const, which results in α=12. However, in the case α=1, Eq. (19) yields ER3, which is necessary in our case. One can see that the energy conservation law allows the following set of relations:

α=1;E29
E=43πρ0a22γ21+2γ+12R3;E30
a=3γ1γ+1216πγ12Eρ012R32.E31

Substituting the new values of aand αinto formula (15), we obtain

D=a=3γ1γ+1216πγ12Eρ012R32.E32

According to the integration rules, the quantity ais a constant. Hence, a new formula for the velocity of a blast wave in the reacting gas medium is proposed:

D=3γ1γ+1216πγ12Eρ012R32=const.E33

The law of conservation of energy gives unpredictable results, but these results are quite possible, given that the energy of the system is changing.

2.3. Formula for the velocity of a spherical wave

Let us determine the shock wave velocity in the critical zone, when RRxand RR2(Figure 2). One can consider a simplified version, when the transition occurs at a distance Rxfrom the center [4], but for the formation of normal detonation it is necessary to isolate the transition interval. As an example, let us consider the detonating gas, 2H2+O2=2H2O+Q, where Q=286.5kJ/molis the thermal effect obtained at a combustion of one mole of hydrogen. Let this reaction (an initial explosion) be initiated. The energy of the system is

E=VnH2q+E0,E34

where Vis the volume of a certain ball, nH2the concentration of hydrogen molecules in it, qthe thermal effect produced by one hydrogen molecule, and E0the initial energy of a charge of radius R0(recall that R0Rx, but the current radius of the sphere RRx). The volume of the ball and the concentration of hydrogen in it are calculated using the known formulas: V=43πR3,nH2=P0KT0NAc,whereP0KT0=ρ0μ;P0, T0, and ρ0are the initial pressure, temperature, and density of the gas mixture; Kis the universal gas constant; NAis the Avogadro constant; cthe current content of hydrogen in the mixture (it is supposed that all the hydrogen burns out in the course of the reaction); and μis the molar mass of the mixture. Hence,

E=43πR3P0KT0NAcq+E0.E35

Substituting Eq. (35) into Eq. (33), we obtain

D=γ+12γ1NAqc4γμ+ξ02E0ρ0R312,

where

ξ0=3γ1γ+1216πγ12.E36

At the time moment, when RR2, where R2>RxR0, the second term in the brackets tends to zero, ξ02E0ρ0R30, whence we obtain

D=γ+12γ1Qc4γμ12,E37

taking into account that Q=NAq, where Qis the thermal energy of one hydrogen mole. The final formula (37) is suggested to describe the velocity of a detonation wave. Above the threshold R2, the charge energy E0loses its importance; further, the energy of the system is replenished only by the first term Eq. (35), which demonstrates the real wave velocity. Provided that formula (37) is valid, the examined quantity does not depend on the mixture pressure. At the initial time moment, the velocity is constant, and it is governed by the following parameters: the combustion energy per one mole of the combustible gas, Q; the fraction of the burned-out gas, c; the molar mass of the mixture, μ; and the adiabatic index for the given mixture of gases, γ.

For a plane wave, the following formula is widely known [11, 13]:

D=2γ21QE38

where Qis the ratio between the energy released by a substance to the mass flow of this substance. As a result, by comparing formulas (37) and (38), we come to a conclusion that they are very similar, although the former seems to be more adequate for the description of the spherical detonation at the beginning of the process. The results of calculations for two different gas mixtures are compared in the Table 1, where Dsis the velocity of a spherical wave calculated by the new formula (37) at the beginning of the detonation, when R=R2; Dnis the plane wave velocity at the final stage of detonation, when R, taken from work [14]; and εis the corresponding relative difference.

Gas mixtureDsm/sDnm/sε%
66.6%H2+33.3%O2255028309.9
25%C2H2+75%O22089233010.3

Table 1.

Shock wave velocities.

In this work, the ideal case of the transformation of an explosive spherical wave into the Chapman-Jouguet mode is considered. From this viewpoint, formulas (33) and (37) prove that the regime of normal spherical detonation can exist at the beginning of the process, much earlier before the curvature radius can be assumed tending to the infinity. Moreover, it demonstrates a possibility of the existence of the normal spherical detonation with a lower velocity of a shock wave in comparison with the classical one. The mathematical expression (38) is “actual” at the final stage, when the radius tends to infinity, i.e. for the plane wave. It should be noted that, in the gas dynamics researches, instead of the shock wave velocity, its ratio to the sound velocity in the unperturbed gas medium, b0, i.e. the Mach number M, is often used,

M=Db0.E39

With regard for the formula for the sound velocity,

b0=γP0ρ0=γKT0μ,E40

and expression (37), we obtain

M=γ+12γ1Qc4γ2KT012.E41

Formula (41) demonstrates the dependence of the Mach number on the adiabatic index γ, the combustion heat Q, the fraction of the burned-out gas c, and the temperature of the medium T0. By varying those quantities, it is possible to regulate the shock transition intensity.

3. Conditions for the appearance of spherical detonation in the interaction of hydrogen with oxygen

3.1. Some issues concerning the chemical reaction kinetics

The process of shock wave propagation is very fast. For instance, at the shock wave velocity D=2500m/sand the gas layer thickness r=0.005mthe shock compression of the substance lasts t=2×106s. This means that the dominant part of a compressed substance must react within such a short time interval; only in this case, we may talk about the supersonic burning as a self-supporting process [13]. Proceeding from this viewpoint, let us consider some issues of the kinetics of the chemical reaction of H2and O2.

First of all, it should be emphasized that the matter concerns chain reactions. The Haber scheme [9] and the development of a chain reaction with the Haber cycle look like

OH+H2=H2O+H,E42
H+O2+H2=H2O+OH,E43
H+O2=OH+O,E44
O+H2=OH+H,E45
OH+OH=H2O2chain break,E46
H+H=H2chain break,E47
H+wallchain break,E48
OH+wallchain break.E49

Reactions (42) and (43) correspond to the chain continuation, reactions (44) and (45) to the chain branching, and reactions (46)(49) to the chain break. For reaction (42), the corresponding activation energy is supposed to be high, with not every collision of OHand H2resulting in the reaction between them. On the contrary, reaction (43) runs at every ternary collision [9]. The cycle of reactions (42) and (43) composes a repeating chain link. According to Haber, 5–10, on the average, cycles must pass before reaction (44) occurs and there emerges a branching in the chain. Let us consider reactions (43) and (44), which compete with each other. Denoting the rate of reaction reaction (44) as W3and that of reaction (43) as W2, the probability of branching δcan be defined as the rate ratio

δ=W3W2.E50

In Semenov’s book [9], the expression for δis given as

δ=2.5×105expE3KT2H2,E51

where H2is the partial pressure of hydrogen in units of mm Hg (the numerical coefficient of 2.5×105in the nominator is multiplied by 1 mm Hg; therefore, the pressure in the denominator is expressed in terms of mm Hg units), E3is the activation energy of reaction (44), Kthe gas constant, and T2the medium temperature (in Kelvin degrees). According to Semenov’s data [9, 15], E3=16kcal/mol. Formula (51) shows that δstrongly depends on the temperature, so that the process can be substantially accelerated as the temperature grows. Moreover, it turns out that the cycle of reactions (42) and (43) with branching (44) does not describe the fastest mechanism. There may exist a case where

W3=W2,E52

or

δ=1.E53

From the physical viewpoint, this means that the probability reaches the maximum, and the branching occurs at every chain link. Then the interaction scheme changes, reaction (44) substitutes reaction (43), and a transformation to the Lewis scheme takes place. In this case, we obtain OH+H2=H2O+H, H+O2=OH+O, and so on, i.e. the temperature Tx, at which δ=1, is a critical one, when the kinetics of the interaction between hydrogen and oxygen undergoes qualitative changes. Let us write down the Lewis scheme in the complete form [9],

OH+H2=H2O+H,E54
H+O2=OH+O,E55
OH+H2=H2O+H,E56
O+H2=OH+H,E57
H+wallchain break,E58
O+wallchain break,E59
OH+wallchain break.E60

In the summarized form, the cycle reaction looks like

OH+3H2+O2=OH+2H+2H2O,E61

and just this reaction is associated with the first fastest initial chain transformations that give rise to detonation.

3.2. Medium state at the shock wave front. Critical temperature

Let a point explosion took place in a gas medium. In our case, the matter concerns the reacting gas media; therefore, the blast wave extinction may occur more slowly that usually; or it can be absent altogether, because a strong mechanism of chain reactions between hydrogen and oxygen starts to play its role. The ultimate result depends on the physico-chemical properties of the gas mixture and the initial energy of explosion. From this point of view, the most interesting is the model of a transition of the strong (overcompressed) detonation into the Chapman-Jouguet regime.

The shock wave propagates from a region with a higher pressure into a region where the pressure is lower. The gas dynamics usually considers waves that have a sharp front. The region of shock-induced transition is a discontinuous surface, the shock wave front. The unperturbed state is designated by subscript 1 and the perturbed one by subscript. 2. The density ρ, pressure P, and temperature Tchange in a jump-like manner across the front. The relations between the parameters P1T1ρ1and P2T2ρ2follow from the Hugoniot relations (the conservation laws) and the equation of ideal gas [11]. It is known that

ρ2ρ1=γ+1M22+γ1M2;E62
P2P1=2γM2γ+1γ+1;E63
T2T1=2γM2γ+12+γ1M2γ+12M2,E64

where Mach number, M, γ=CpCVis the adiabatic exponent (for a two-atom ideal gas, γ=1.4 [16]). In such a manner, when a shock wave propagates in gases, we should consider the medium near (subscript 1) and at the front (subscript 2). To characterize the latter, we must know an important parameter, the shock wave velocity or the Mach number. In our case, using expression (37) and the formula for sound velocity (40), we obtain (41).

Now, let us carry out a simple gedanken experiment. Let a spherical reactor contain a hydrogen-oxygen mixture with the initial parameters (P0,T0=293K). Let us heat up the mixture to the temperature T1<T1, where T1is the ignition temperature of the static medium. We initiate a reaction using an explosion and should observe a continuous transformation of a blast wave into the detonation of the hydrogen-oxygen mixture. At the wave front, the medium parameters are P2T2. Let

T2=Tx,E65

i.e. the critical temperature Txis attained, and the reaction develops, being driven by the chain reaction mechanism according to the Lewis scheme. In order to determine the critical temperature Tx, let us use formula (51). Taking into account that δ=1, we obtain the transcendental equation for the critical temperature Tx

2.5×105expE3KTxH2=1,E66

where

H2=cP2E67

is the partial hydrogen pressure (in mm Hg units) at the shock wave front [17], P2is the total pressure in the mixture (in mm Hg units) at the shock wave front, and cthe hydrogen content in the mixture (coefficient). With regard for Eq. (67), we obtain

2.5×105expE3KTxcP2=1.E68

Let us express P2in the denominator of Eq. (68) in terms of known quantities. Before the reaction started (the initiation of the explosion), the gas mixture pressure was P0, and its temperature was T0=293K. As the mixture is heated up to T1, its pressure increases to

P1=P0T1T0.E69

From Eq. (63), it follows that

P2=2γM2γ+1γ+1P1,E70

or, in view of Eq. (69),

P2=2γM2γ+1P0T1γ+1T0.E71

The temperature T1in formula (71) is expressed in terms of Txand the Mach number Mas follows:

TxT1=2γM2γ+12+γ1M2γ+12M2.E72

Whence, we obtain

T1=γ+12M2Tx2γM2γ+12+γ1M2,E73

or, taking Eq. (73) into account,

P2=γ+1M2TxP0T02+γ1M2.E74

The denominator in formula (68) also includes the hydrogen content, c. If we assume that all hydrogen in the gas mixture burns out, we can express cusing the Mach number (Eq. (41)) and the temperature of the gas medium T1,

c=4γ2M2KT1γ1γ+12Q,E75

or, in accordance with Eq. (73),

c=4γ2M4KTxγ12γM2γ+12+γ1M2QE76

(in this case, we impose a restriction on the gas mixture composition, 0<c0.66). From Eq. (67) and using Eqs. (76) and (74), we obtain the partial pressure of hydrogen at the shock wave front,

H2=4γ2γ+1M6KP0γ12γM2γ+12+γ1M22QT0Tx2.E77

Then, formula (66) reads

Tx2=2.5×105QT0γ12γM2γ+12+γ1M224γ2γ+1M6KP0expE3KTx.E78

Hence, we obtained the dependence which connects the initial pressure in the medium and the Mach number with the critical temperature at the shock wave front.

3.3. Results and discussion

After the substitution of the corresponding numerical values of physical parameters of the hydrogen-oxygen mixture and taking into account that γ=1.4, Q=286.5kJ/mol, K=8.31J/molK, E3=16×103×4.19J/mol, and T0=293K, Eq. (78) reads

Tx2=5.38×10102+0.4M222.8M20.4P0M6exp8067Tx.E79

Most of these quantities are well known. The values of the others are chosen for practical reasons. So, for example, T0=293K, this is the temperature at which the experimental setup operates. The most optimal for laboratory conditions is the pressure of the gas mixture P0=60mm Hg, since at P0>60mm Hg the shock wave acquires destructive energy. Using expression (79), let us calculate the critical temperature for two Mach numbers, (i) M=2.15and (ii) M=4.78, i.e. for shock waves of two types, but at the fixed initial pressure P0=60mm Hg. Experimental data indicate that detonation is not observed at M<2.15, weak shock waves become waves of compression and rarefaction. At the same time, the value M=4.78was selected as the largest one obtained from expression (41) at the following parameters: c=0.66, T1=T0=293K, and γ=1.4. In the first case (M=2.15and P0=60mm Hg)

Tx2=1.69×109exp8067Tx.E80

In the second one (M=4.78and P0=60mm Hg),

Tx2=5.93×108exp8067Tx.E81

The transcendental equations were solved with the use of the software package “Consortium Scilab (Inria, Enpc)” with the program code “Scilab-4.1.2”. After the corresponding calculations, we obtained Tx=1120Kfor the first case and Tx=1420Kfor the second one. The interval of researches can be expanded to determine the critical temperatures for Mach numbers within line segment [2; 5] with an increment of 0.2. Only real-valued roots, which have a physical sense, must be taken into consideration. The corresponding plot for the dependence TxfMat P0=60mm Hg is shown in Figure 3.

Figure 3.

Dependences of the critical temperature at the shock wave front, T x ∼ f M , and the temperature of static medium T 1 ∼ g M , at which the detonation is possible, on the Mach number at a fixed pressure P 0 = 60 mm Hg.

One can see that the critical temperature grows nonlinearly with the Mach number. This behavior is not of surprise. The stronger the shock wave, the higher is the pressure at its front, and the higher is the probability of the chain break. As a result, we obtain the critical temperature growth, because the probability of chain break can be compensated only by the probability of chain branching, which increases with the medium temperature. However, this is not the main point. Knowing the critical temperature and the Mach number, it is possible to determine the initial temperature of the gas medium required for the detonation to take place. In other words, it is possible to determine such a temperature T1in front of the shock wave front that the corresponding wave would stimulate the detonation. Using expression (73), let us plot the dependence of T1on Mat P0=60mm Hg (Figure 3). It allows us to determine the initial temperature of the medium, at which the detonation of the gas mixture becomes possible for the given Mach number. Moreover, in accordance with Eq. (41), the initial temperature and the known Mach number determine the hydrogen content. Hence, the critical temperature is unambiguously related with the Mach number and, therefore, with the initial parameters of the hydrogen-oxygen mixture.

4. Conclusions

To summarize, it should be noted that formula (37) determines the velocity of a detonation wave at the initial stage, if this wave is generated at the combustion of some “portion” (the parameter c) of a combustible gas, when RR2(see Figure 2 and the model described the transformation of a blast wave in a detonation one). This formula is valid for spherical wave, in contrast to formula (38) known from the literature, which was obtained for plane waves. Thus, provided that the shock wave velocity or the Mach number is known, the solution of one of the basic gas dynamics problems can be obtained, i.e. we can find the parameters P1T1ρ1at the wave front, if we know the set P0T0ρ0of parameters for the unperturbed medium. In particular, the determined parameters are necessary for studying the kinetics of a chemical reaction in the course of the shock transition. There is no doubt that the stoichiometric mixture of hydrogen and oxygen will generate a detonation wave. However, it is difficult to assert the same for the mixture with 12% of hydrogen. In this case, it is necessary to consider the reaction mechanism itself.

The expression (66) obtained for the critical temperature is the simplest criterion for the transformation of the blast wave into a detonation one. At this temperature, T2=Txif δ=1, i.e. the probability of branching becomes maximum for the scheme of chain reactions with the hydrogen-oxygen interaction, which was considered above. The obtained Eq. (78) allows the sought value to be determined as a function of the Mach number, provided that the initial pressure is fixed. The critical temperature is the threshold of a detonation in a gas mixture, because the supersonic burning is impossible at temperatures below it. For example, let us analyze gas mixtures with different hydrogen contents of 66.6, 60, and 50%, and the temperature of static medium T1=273K(Table 2).

Gas mixtureT1KMT2KTxK
66.6%H2+33.3%O22734.9515581427
60%H2+40%O22734.7214381420
50%H2+50%O22734.3112411390

Table 2.

Parameter changes at the shock transition (T1=273Kand P0=60mm Hg).

Comparing the T2- and Tx-values, we come to a conclusion that the process, which is of interest for us, occurs only in the first two cases. However, it is enough to raise the initial temperature T1=373Kfor the detonation to become possible at lower hydrogen concentrations (Table 3).

Gas mixtureT1KMT2KTxK
66.6%H2+33.3%O23734.2316481384
60%H2+40%O23734.0415231365
50%H2+50%O23733.7013391339

Table 3.

Parameter changes at the shock transition (T1=373Kand P0=60mm Hg).

The practical results testify that the conditions for the emergence of spherical detonation have a drastic dependence on the temperature and the mixture composition. The relation obtained in this work allows the critical values of those parameters to be determined and, in such a manner, to stimulate the regime of supersonic burning in the hydrogen-oxygen mixture. In the future, the presented results can be used to determine the range of admissible values of the parameters of the hydrogen-oxygen mixture necessary for detonation. Thus, it becomes possible to improve the performance of engines, make them more efficient.

Nomenclature

Basic designations

D

shock wave velocity, detonation velocity

Rx

critical radius

M

Mach number

P,T,ρ

pressure, temperature, density of the medium

u1

gas velocity behind the shock front

b0

speed of sound in a stationary gaseous medium

γ

adiabatic index

E0

explosion energy

U

burnt gas energy

Q

combustion energy of one mole of combustible gas

μ

molar mass

K

universal gas constant

NA

Avogadro number

c

coefficient of flammable gas content in the mixture

W

chemical reaction rate

δ

branching probability

H2

is the partial pressure of hydrogen

Tx

critical temperature

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

fx

function of the variable x

expx

exponential function

x

increment of variable x

ddx

derivative

much less

much more

aspires to

ab

line segment

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Myron Polatayko (November 5th 2018). Determination of the Velocity of the Detonation Wave and the Conditions for the Appearance of Spherical Detonation during the Interaction of Hydrogen with Oxygen [Online First], IntechOpen, DOI: 10.5772/intechopen.81792. Available from:

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