Analysis of the Limiting Values of Thermodynamic Parameters for Jouguet Detonation

Abstract

An analytical study of the interaction of an ideal gas flow with a detonation wave was performed with account for the activation energy of chemical processes. Based on the modified Rankine-Hugoniot conditions, the effect of heat release on the limiting characteristics of detonation was analyzed. A dependence of the limiting value of the exponent Arrhenius number on the Mach number before the shock wave has been obtained. As the Mach number increases, the limiting value of the Arrhenius number decreases. An equation has been derived for determining the limiting value of the compression ratio in the shock wave. The effect of heat release intensity on the limiting compression ratio in a shock wave was elucidated. Also studied were effects of the Mach number and the Arrhenius number on the limiting compression ratio in a detonation wave. A condition for determining the critical value of the Arrhenius number necessary for the onset of detonation was obtained. Effects of the Mach number and the exponent of the Arrhenius number ${\mathrm{Ar}}_E$ on the critical value of the amplitude Arrhenius number ${\mathrm{Ar}}_A$ were discussed. The symmetry analysis of the gas flow parameters when passing through a detonation wave was performed.

1. Introduction

Knowledge of the limiting values of the parameters of the detonation process is necessary for assessing the consequences of possible explosions on the safety conditions in industry [1,2,3], severe accidents in nuclear power [4], and managing thermodynamic processes of detonation engines [5,6,7]. Detonation parameters are fundamental for designing microstructure technological equipment [8].

The processes of occurrence and propagation of detonation have been actively studied for more than a century. In the middle of the last century, Zel’dovich Y.B. [9], John Von Neumann [10], and W. Döring [11] independently developed a detonation model that takes into account the time of chemical reactions as a result of heating the substance by a shock wave. This model allowed to significantly advance in understanding the phenomenon of detonation, studying the structure of the zone of chemical reactions, and the propagation of explosion products. A great contribution to the theory of rates of chemical reactions and flame propagation was made by J. O. Hirschfelder [12]. Some of his works on the theory of combustion and detonation are still relevant today. Modern experimental and theoretical studies of the detonation process continue [13,14,15]. They are aimed at studying the details of the process, clarifying the physical parameters that characterize the conditions of occurrence and mechanisms of propagation of detonation in different environments.

Studies [16,17,18] focused on the limiting parameters of detonation in gas mixtures. The influence of the initial gas composition on the efficiency of detonation suppression is analyzed.

The results of determining the range of physical parameter values for the existence of normal spherical detonation are presented in the studies [19,20]. Numerical modeling methods establish the range of permissible values for temperature, Mach numbers, and specific hydrogen content in the gas mixture for spherical detonation [20]. Critical values of parameters related to the kinetics of chemical reactions at the explosion wave front are taken into account.

The studies [21,22,23] present results of numerical modeling to determine the detonation parameters of some civilian and dual-use explosives. The critical detonation parameters of synthetic fuels for detonation engines are investigated in [24]. The role of fuel chemical composition in detonation occurrence was shown.

The study [25] presents the results of numerical modeling of the interaction between detonation and turbulence in a mixture of hydrogen, oxygen, and argon. The influence of turbulence on process parameters was demonstrated. The effect of heat diffusion and turbulence on detonation development in internal combustion engines was investigated in [26]. The impact of heat diffusion on the detonation process of hydrogen-air mixtures was assessed.

The influence of plasma discharge on detonation initiation in a pulse detonation engine was studied in [27]. The formation process of a detonation wave in a hydrogen-air mixture under the action of a plasma pulse was numerically investigated. The influence of the ignition scheme on the parameters of stable detonation wave formation was demonstrated.

The heat release in a rotating detonation combustion chamber was experimentally investigated in the studies [28]. The impact of hydrogen-air mixture detonation parameters on heat transfer characteristics in the combustion chamber was shown. The combustion characteristics of fuel in rotating detonation engines were also explored in the studies [29,30].

The studies [31,32,33,34] present results of analytical modelling of the conditions of the propagation of shock waves in ideal and non-ideal gases. These studies demonstrated the influence of parameters of non-ideal gases, such as van der Waals gas parameters [31], gas flow turbulence [32,34], and magnetic field intensity [33], etc., on detonation parameters.

A large number of theoretical and experimental studies conducted in recent years contribute to a better understanding of the physics of detonation initiation and propagation and its limiting characteristics. However, the features of the detonation process in the presence of heat release in the gas flow have been insufficiently studied.

Therefore, the objective of the present research is to develop a mathematical model that allows determining the limiting values of gas physical parameters necessary for detonation occurrence under heat supply conditions.

A potential application area of the results of this study is the development of regulatory documents in the field of industrial safety in the case of work with explosive materials. Data on the effect of thermal load on the limiting parameters of detonation will be useful for improving the design of detonation engines, creating a reliable system for their cooling. Knowledge of the criteria for the boundaries of detonation modes in a hydrogen-steam mixture is important for assessing reactor safety in severe accidents at nuclear power plants.

2. Mathematical Model

Schematic representation of the problem of the process of detonation of an ideal gas, which passes through a plane detonation wave considered in the present study, is outlined in Figure 1

A steady-state gas flow is described by the following system of mass, momentum, and energy conservation equations [35,36]

$${\rho }_1{V}_1={\rho }_2{V}_2,$$

(1)

$$p_1+{\rho }_1{V}_1^2=p_2+{\rho }_2{V}_2^2,$$

(2)

$${\rho }_1{V}_1\left[\left(h_1+{\displaystyle \frac{V_1^2}{2}}\right)+q\right]= {\rho }_2{V}_2\left(h_2+{\displaystyle \frac{V_2^2}{2}}\right),$$

(3)

Here V is velocity, ρ is gas density, p is pressure, h is enthalpy, q is the heat supply, i.e., heat released per unit mass by the corresponding process undergoing across the shock, where the subscript “1” denotes the parameters before the shock wave, the subscript “2” denotes the parameters after the shock wave.

The heat supply is determined by the following equation [37]

$$q=A \exp \left(-{\displaystyle \frac{E}{\Re T}}\right),$$

(4)

where A is the pre-exponential factor, E is the activation energy of the chemical interaction, T is the temperature, and $\Re$ is the individual (specific) gas constant. The pre-exponential factor does not depend on temperature and concentration and is equal to the number of so-called successful collisions per unit time. The physical meaning of the pre-exponential factor A is that it represents the reaction rate as the temperature approaches infinity.

System (1)–(3) is a system of Rankine–Hugoniot equations [38,39,40]. The Rankine–Hugoniot conditions (or jump conditions), also called the Rankine–Hugoniot relations, show the relation between the states before and after a shock wave or a combustion wave (deflagration or detonation) in a one-dimensional fluid flow or under one-dimensional deformation in solids.

System (1)–(3) does not take into account diffusion and dissipation effects, just as in the classical Jouguet theory of detonation.

System (1)–(3) is closed by the equation of state for an ideal gas

$$p=\rho \Re T$$

(5)

together with Mayer’s relation

$$\Re ={\mathrm{c}}_p-{\mathrm{c}}_{\upsilon }={\mathrm{c}}_p\left(1-{\displaystyle \frac{1}{k}}\right)={\mathrm{c}}_p{\displaystyle \frac{k-1}{k}}.$$

(6)

Here $k=c_p/c_{\upsilon }$ is the specific heat ratio, c_p is the specific heat capacity at constant pressure, and ${\mathrm{c}}_{\upsilon }$ is the specific isochoric heat capacity.

Let us transform Equation (4) as follows

$$q=A \exp \left(-{\displaystyle \frac{E}{\Re T}}\right)=A\exp \left(-{\displaystyle \frac{E}{\Re \left(T-T_{⁎}+T_{⁎}\right)}}\right)=A\exp \left(-{\displaystyle \frac{E}{\Re T_{⁎}\left(1+\frac{\Delta T}{T_{⁎}}\right)}}\right),$$

(7)

where

$$\Delta T=T-T_{⁎},$$

(8)

where $T_{⁎}$ is the reference temperature value. Obviously, this can be the ignition temperature, which corresponds to the critical temperature, i.e., the temperature of the flow with critical flow velocity. This velocity equals the local velocity of sound $a_{⁎}$ (critical velocity of sound). Additionally, we assume that $\Delta T/T_{⁎}<<1$. Then we can obtain

$$\begin{array}{l}q=A\exp \left(-{\displaystyle \frac{E}{\Re T_{⁎}\left(1+\frac{\Delta T}{T_{⁎}}\right)}}\right)=A\exp \left(-{\displaystyle \frac{E}{\Re T_{⁎}}}\left(1-{\displaystyle \frac{\Delta T}{T_{⁎}}}\right)\right)=A\exp \left(-{\displaystyle \frac{E}{\Re T_{⁎}}}+{\displaystyle \frac{E\Delta T}{\Re T_{⁎}^2}}\right)=\\ A\exp \left(-{\displaystyle \frac{E}{\Re T_{⁎}}}\right)\exp \left({\displaystyle \frac{E\Delta T}{\Re T_{⁎}^2}}\right)=A\exp \left(-{\displaystyle \frac{E}{\Re T_{⁎}}}\right)\left(1+{\displaystyle \frac{E\Delta T}{\Re T_{⁎}^2}}\right)=A\exp \left(-{\mathrm{Ar}}_E\right)\left(1+{\mathrm{Ar}}_E\left({\displaystyle \frac{T}{T_{⁎}}}-1\right)\right)\end{array},$$

(9)

where

$${\mathrm{Ar}}_E={\displaystyle \frac{E}{\Re T_{⁎}}}$$

(10)

is the exponent Arrhenius number [41].

Since the critical temperature is taken as $T_{⁎}$ and the condition $\Delta T/T_{⁎}<<1$ is fulfilled, the calculation error will not exceed 10%.

Equation (6) can be rewritten taking into account Equation (9)

$$c_p{T}_1+{\displaystyle \frac{V_1^2}{2}}+A\exp \left(-{\mathrm{Ar}}_E\right)\left(1+{\mathrm{Ar}}_E\left({\displaystyle \frac{T_1}{T_{⁎}}}-1\right)\right)=c_p{T}_2+{\displaystyle \frac{V_2^2}{2}}.$$

(11)

Here, we used the following equation for enthalpy

$$h=c_{p}T.$$

(12)

Taking into account Equations (5) and (6), we can recast Equation (11) as follows

$${\displaystyle \frac{k}{k-1}}{\displaystyle \frac{p_1}{{\rho }_1}}+{\displaystyle \frac{V_1^2}{2}}+A\exp \left(-{\mathrm{Ar}}_E\right)\left(1+{\mathrm{Ar}}_E\left({\displaystyle \frac{T_1}{T_{⁎}}}-1\right)\right)={\displaystyle \frac{k}{k-1}}{\displaystyle \frac{p_2}{{\rho }_2}}+{\displaystyle \frac{V_2^2}{2}}.$$

(13)

3. Symmetry Analysis

The symmetries of Equations (1)–(3) can be described by an infinitesimal generator [42]

$$gen=\xi {\displaystyle \frac{\partial }{\partial x}}+{\eta }_{\rho }{\displaystyle \frac{\partial }{\partial \rho }}+{\eta }_p{\displaystyle \frac{\partial }{\partial p}}+{\eta }_V{\displaystyle \frac{\partial }{\partial V}},$$

(14)

Here, the unknown coefficients $\xi , {\eta }_{\rho }, {\eta }_p,{\eta }_V$ are determined based on the methodology of the work [42]. Based on this methodology, we will analyze the symmetries of the system (1)–(3). In this case, we exclude temperature and enthalpy based on Equations (5) and (12). As a result, we find that one of the Lie subalgebras for this system has the following form

$$gen=x{\displaystyle \frac{\partial }{\partial x}}-\rho {\displaystyle \frac{\partial }{\partial \rho }}-p{\displaystyle \frac{\partial }{\partial p}},$$

(15)

This generator allows for determining symmetry transformations based on the defining differential equations (Lie equations)

$${\displaystyle \frac{{dx}^{⁎}}{d\epsilon }}=x^{⁎},$$

(16)

$${\displaystyle \frac{{d\rho }^{⁎}}{d\epsilon }}=-{\rho }^{⁎},$$

(17)

$${\displaystyle \frac{{dp}^{⁎}}{d\epsilon }}=-p^{⁎}.$$

(18)

Here ε is the group transformation parameter, the asterisk refers to the new transformed variables.

This system is solved under the following initial conditions

$$x^{⁎}=x\left(0\right), {\rho }^{⁎}=\rho \left(0\right), p^{⁎}=p\left(0\right),$$

(19)

where variables without an asterisk are “old” variables.

The solution of system (16)–(18) under initial conditions (19) is

$$x^{⁎}=x\exp \left(\epsilon \right),$$

(20)

$${\rho }^{⁎}=\exp \left(-\epsilon \right)\rho \left(x\right)=\exp \left(-\epsilon \right)\rho \left(x^{⁎}\exp \left(-\epsilon \right)\right),$$

(21)

$$p^{⁎}=\exp \left(-\epsilon \right)p\left(x\right)=\exp \left(-\epsilon \right)p\left(x^{⁎}\exp \left(-\epsilon \right)\right),$$

(22)

$$V^{⁎}=V\left(x\right)=V\left(x^{⁎}\exp \left(-\epsilon \right)\right),$$

(23)

It is easy to ascertain by substituting Equations (20)–(23) into Equations (1)–(3) that Equations (20)–(23) are indeed symmetry transformations for system (1)–(3).

Mathematical transformations of symmetry (20)–(23) enable obtaining a physical interpretation of the flow parameters variation when passing through a detonation wave. It follows from Equations (20)–(23) that the pressure and density demonstrate the same trend of variation when the flow passes through a discontinuity. When passing through a shock wave, the pressure and density increase; when passing through a rarefaction wave, the pressure and density decrease. However, as follows from Equation (23), there exist conditions when the flow velocity remains an invariant. As will be shown below (Section 5), this case corresponds to Jouguet detonation.

4. Modified Hugoniot Equation

Keeping Equation (1), we can recast Equation (2) as

$$p_1-p_2={\rho }_1{V}_1\left(V_2-V_1\right)={\rho }_1{V}_1\left(V_2-V_1\right)={\rho }_1{V}_1\left(V_2-V_1\right)={\rho }_1{V}_1\left(V_2-V_1\right).$$

(24)

Multiplying Equation (24) by the factor

$${\displaystyle \frac{V_2+V_1}{{\rho }_1{V}_1}}={\displaystyle \frac{1}{{\rho }_1}}+{\displaystyle \frac{1}{{\rho }_2}}$$

(25)

one can obtain

$$\left(p_1-p_2\right)\left({\displaystyle \frac{1}{{\rho }_1}}+{\displaystyle \frac{1}{{\rho }_2}}\right)=V_2^2-V_1^2.$$

(26)

It follows from Equation (13) that

$$\begin{array}{l}{V}_2^2-V_1^2={\displaystyle \frac{2k}{k-1}}\left({\displaystyle \frac{p_1}{{\rho }_1}}-{\displaystyle \frac{p_2}{{\rho }_2}}\right)+A\exp \left(-{\mathrm{Ar}}_E\right)\left(1+{\mathrm{Ar}}_E\left({\displaystyle \frac{T_1}{T_{⁎}}}-1\right)\right)=\\ {\displaystyle \frac{2k}{k-1}}\left({\displaystyle \frac{p_1}{{\rho }_1}}-{\displaystyle \frac{p_2}{{\rho }_2}}\right)+A\exp \left(-{\mathrm{Ar}}_E\right)\left(1+{\mathrm{Ar}}_E\left({\displaystyle \frac{k\Re T_1}{k\Re T_{⁎}}}-1\right)\right)\end{array},$$

(27)

here

$${\displaystyle \frac{k\Re T_1}{k\Re T_{⁎}}}={\left({\displaystyle \frac{a_1}{a_{⁎}}}\right)}^2={\displaystyle \frac{k+1}{2\left(k-1\right)}}-{\displaystyle \frac{{\lambda }_1^2}{2}},$$

(28)

where a₁ is the speed of sound in the flow in front of the shock wave,

$${\lambda }_1={\displaystyle \frac{V_1}{a_{⁎}}}$$

(29)

is the speed coefficient before the shock wave.

A comparison of Equations (26) and (27) yields

$$\begin{gathered} {\displaystyle \frac{p_2}{{\rho }_2}}\left({\displaystyle \frac{p_1}{p_2}}-1\right)\left({\displaystyle \frac{{\rho }_2}{{\rho }_1}}+1\right)={\displaystyle \frac{p_2}{{\rho }_2}}{\displaystyle \frac{2k}{k-1}}\left({\displaystyle \frac{p_1{\rho }_2}{{\rho }_1{p}_2}}-1\right)+A\exp \left(-{\mathrm{Ar}}_E\right)\left(1+{\mathrm{Ar}}_E\left({\left({\displaystyle \frac{a_1}{a_{⁎}}}\right)}^2-1\right)\right) \\ \left({\displaystyle \frac{p_1}{p_2}}-1\right)\left({\displaystyle \frac{{\rho }_2}{{\rho }_1}}+1\right)={\displaystyle \frac{2k}{k-1}}\left({\displaystyle \frac{p_1}{p_2}}{\displaystyle \frac{{\rho }_2}{{\rho }_1}}-1\right)+{\displaystyle \frac{p_1}{p_2}}{\displaystyle \frac{{\rho }_2}{{\rho }_1}}{\mathrm{Ar}}_A\exp \left(-{\mathrm{Ar}}_E\right)\left(1-{\displaystyle \frac{{\mathrm{Ar}}_E}{2}}\left(k-1\right)\left({\lambda }_1^2-1\right)\right), \end{gathered}$$

(30)

where

$${\mathrm{Ar}}_A={\displaystyle \frac{A}{\Re T_1}}={\displaystyle \frac{A}{\Re T_{⁎}}}{\displaystyle \frac{T_{⁎}}{T_1}}={\displaystyle \frac{2A}{\Re T_{⁎}\left(k+1-{\lambda }_1^2\left(k-1\right)\right)}}$$

(31)

is the amlitude Arrhenius number.

Equation (30) can be transformed using the Mach number ${\mathrm{Ma}}_1$. We obtain as a result

$$\left({\displaystyle \frac{p_1}{p_2}}-1\right)\left({\displaystyle \frac{{\rho }_2}{{\rho }_1}}+1\right)={\displaystyle \frac{2k}{k-1}}\left({\displaystyle \frac{p_1{\rho }_2}{{\rho }_1{p}_2}}-1\right)+{\displaystyle \frac{p_1}{p_2}}{\displaystyle \frac{{\rho }_2}{{\rho }_1}}{\mathrm{Ar}}_A\exp \left(-{\mathrm{Ar}}_E\right)\left(1-{\displaystyle \frac{{\mathrm{Ar}}_E}{2}}\left(1-{\displaystyle \frac{k+1}{2+\left(k-1\right){\mathrm{Ma}}_1^2}}\right)\right),$$

(32)

$${\mathrm{Ar}}_A={\displaystyle \frac{A}{\Re T_{⁎}}}{\displaystyle \frac{1+\frac{k-1}{2}{\mathrm{Ma}}_1^2}{\frac{k+1}{2}}}, {\mathrm{Ma}}_1={\displaystyle \frac{V_1}{a_1}}$$

(33)

Solving Equation (30) or (32) with respect to p₂/p₁, one can derive

$${\displaystyle \frac{p_2}{p_1}}={\displaystyle \frac{\frac{k+1}{k-1}-\frac{{\rho }_1}{{\rho }_2}+Z}{\frac{k+1}{k-1}\frac{{\rho }_1}{{\rho }_2}-1}},$$

(34)

where

$$Z={\mathrm{Ar}}_A\exp \left(-{\mathrm{Ar}}_E\right)\left(1-{\displaystyle \frac{{\mathrm{Ar}}_E}{2}}\left(k-1\right)\left({\lambda }_1^2-1\right)\right)={\mathrm{Ar}}_A\exp \left(-{\mathrm{Ar}}_E\right)\left(1-{\displaystyle \frac{{\mathrm{Ar}}_E}{2}}\left(1-{\displaystyle \frac{k+1}{2+\left(k-1\right){\mathrm{Ma}}_1^2}}\right)\right)$$

(35)

In dimensionless form, this equation looks as

$$P={\displaystyle \frac{\gamma -R^{-1}+Z}{\gamma R^{-1}-1}}={\displaystyle \frac{\gamma -S+Z}{\gamma S-1}},$$

(36)

where

$$P={\displaystyle \frac{p_2}{p_1}}, R={\displaystyle \frac{{\rho }_2}{{\rho }_1}}=S^{-1}, \gamma ={\displaystyle \frac{k+1}{k-1}}.$$

(37)

Equations (34) and (36) describe the modified Rankine-Hugoniot equation for detonation.

5. Limiting Parameters

Equations (34) and (36) exhibit an asymptote for ρ₂/ρ₁ expressed as

$$R=S^{-1}=\gamma .$$

(38)

Relation (38) is a classical expression for the maximum degree of gas compression when passing through a shock wave. It shows that for an ideal gas, the maximum compression ratio cannot exceed six. For the condition expressed as Equation (36), the pressure jump (26) becomes infinite.

With a detonation that propagates spontaneously at a constant speed, the limiting density, at a speed tending to infinity, can be written differently. These relations are determined below.

The results of calculations according to Equation (39) are shown in Figure 2. The classical Poisson’s curve for isentropic conditions

$$P=R^k=S^{-k}$$

(39)

is also plotted in Figure 2.

It can be seen that an increase in heat release Z leads to an equidistant rise in the Hugoniot curve compared to the case of Z = 0.

From a physical standpoint, heat release must be positive Z > 0. From this, one can obtain the limiting values of the exponent Arrhenius number. Equation (35) provides the condition for the limiting values of the exponent Arrhenius number in the following form

$${\displaystyle \frac{{\mathrm{Ar}}_E}{2}}\left(1-{\displaystyle \frac{k+1}{2+\left(k-1\right){\mathrm{Ma}}_1^2}}\right)<1.$$

It follows from here that

$${\mathrm{Ar}}_E<{\mathrm{Ar}}_E^{cr}=2\left(1+{\displaystyle \frac{k+1}{\left(k-1\right)\left({\mathrm{Ma}}_1^2-1\right)}}\right).$$

(40)

This dependence is plotted in Figure 3.

It is evident that as the Mach number before the shock wave increases (with the increasing intensity of the shock wave), the limiting value of the Arrhenius exponent decreases. That is, with an increase in the Mach number before the shock wave, the activation energy required for detonation decreases. Additionally, as seen in Figure 3, for diatomic gases at k = 7/5, a higher activation energy is required. This is due to the necessity of introducing additional energy for the chemical reactions of diatomic gases.

To determine which point on the Hugoniot curve corresponds to stable normal detonation with a minimum velocity, we use the Jouguet selection rule [35,36]. This point corresponds to a point on the Hugoniot curve D through which the tangent passes, which also passes through the point $\left(P,S\right)=\left(1,1\right)$ (Figure 4).

As can be seen from Figure 4, the tangent equation is determined by the following condition:

$${\displaystyle \frac{P-1}{1-S}}={\displaystyle \frac{dP}{dS}}.$$

(41)

From (36), we find that

$${\displaystyle \frac{P-1}{1-S}}={\displaystyle \frac{dP}{dS}}={\displaystyle \frac{1+\gamma P}{1-\gamma S}}.$$

(42)

Let us determine the tangent of the slope to this curve. To do this, we can find from Equations (1) and (2) that

$$V_1=\sqrt{{\displaystyle \frac{p_2-p_1}{{\rho }_2-{\rho }_1}}{\displaystyle \frac{{\rho }_2}{{\rho }_1}}}=\sqrt{{\displaystyle \frac{p_2-p_1}{{\rho }_2-{\rho }_1}}{\displaystyle \frac{{\rho }_2}{{\rho }_1}}}=\sqrt{{\displaystyle \frac{P-1}{1-S}}{\displaystyle \frac{p_1}{{\rho }_1}}},$$

(43)

$$V_2=S\sqrt{{\displaystyle \frac{P-1}{1-S}}{\displaystyle \frac{p_1}{{\rho }_1}}}.$$

(44)

Now we transform the equations for the velocity of detonation products given by Equation (44)

$$V_2^2=S^2{\displaystyle \frac{P-1}{1-S}}{\displaystyle \frac{p_1}{{\rho }_1}}$$

(45)

It is a well-known fact that at point D, the derivatives of the Poisson’s law (39) and the Hugoniot Equation (36) are equal:

$${\displaystyle \frac{dP}{dS}}={\displaystyle \frac{P-1}{1-S}}=k{\displaystyle \frac{P}{S}}$$

(46)

Comparison of Equations (45) and (46) allows us to find the equation for the velocity of detonation products

$$V_2^2=kPS{\displaystyle \frac{p_1}{{\rho }_1}}=k{\displaystyle \frac{p_2}{{\rho }_2}}=a_2$$

(47)

or

$${\mathrm{Ma}}_2=1$$

(48)

Thus, the velocity of detonation products during Jouguet detonation is equal to the critical speed. Equation (47) confirms the validity of the statement about the invariance of speed (as shown by transformation (23)). This follows from the symmetry transformations for density and pressure (20)–(22) after substitution into (47).

From Equation (43), we find that the tangent at point D is

$$\tan \left(\alpha \right)=-{\displaystyle \frac{P_D-1}{1-S_D}}=-k{\mathrm{Ma}}_1^2$$

(49)

On the other hand, the same tangent is determined by Equation (42)

$$\tan \left(\alpha \right)={\left({\displaystyle \frac{dP}{dS}}\right)}_D={\displaystyle \frac{1+\gamma P_D}{1-\gamma S_D}}$$

(50)

The solution of the system of Equations (49) and (50) allows us to find the coordinates of the point D on the Hugoniot curve. This solution is given by the well-known Jouguet relations

$$P_D={\displaystyle \frac{1+k{\mathrm{Ma}}_1^2}{1+k}},$$

(51)

$$S_D={\displaystyle \frac{1+k{\mathrm{Ma}}_1^2}{\left(1+k\right){\mathrm{Ma}}_1^2}}.$$

(52)

If we use the derivative of the Poisson’s isotropy (50) in Equation (49), we obtain the same values of parameters $S_D$ (52) and $P_D$ (51).

The limit of the solution for $S_D$ at a speed tending to infinity (Ma₁→∞) is independent of the intensity of the heat supply and can be expressed as

$$R_D={\left({\displaystyle \frac{{\rho }_2}{{\rho }_1}}\right)}_D={\displaystyle \frac{1+k}{k}}$$

(53)

It should be pointed out that Equations (51) and (52) do not depend on the parameter Z. This is explained by the fact that the parameter P is included explicitly in Equations (42), (49) and (50). Therefore, we will apply a different approach when the parameter P is expressed through the parameter S. The first option is to use Equation (49). Substituting Equation (36) into Equation (49) gives

$${\displaystyle \frac{\frac{\gamma -S_D+Z}{\gamma S_D-1}-1}{1-S_D}}=k{\mathrm{Ma}}_1^2$$

(54)

The solution to this equation is

$$S_D^{1,2}={\displaystyle \frac{1+k{\mathrm{Ma}}_1^2\pm \sqrt{{\left({\mathrm{Ma}}_1^2-1\right)}^2-\frac{k^2-1}{k}{\mathrm{Ma}}_1^{2}Z}}{\left(1+k\right){\mathrm{Ma}}_1^2}}$$

(55)

However, both solutions have no physical meaning, since in the limit Ma₁→∞ the first solution is equal to unity, and the second is equal to zero for any values of the Arrhenius numbers. The second option is to use Equations (36) and (46) together with Equation (49). This gives

$$S_D^{1,2}={\displaystyle \frac{\left({\mathrm{Ma}}_1^2-1\right)\left(\mathrm{k}-1\right)\pm \sqrt{{\left(-1+k\right)}^2{\left({\mathrm{Ma}}_1^2-1\right)}^2+4\left(1+k\right){\mathrm{Ma}}_1^2\left(1+k+\left(-1+k\right)Z\right)}}{2\left(1+k\right){\mathrm{Ma}}_1^2}}$$

(56)

Both solutions (56) in the limit Ma₁→∞ give the value expressed by Equation (38) for any values of the Arrhenius numbers. However, the negative sign before the square root in Equation (56) does not have a physical meaning. If we use Equation (50) instead

$${\displaystyle \frac{1+\gamma \frac{\gamma -S_D+Z}{\gamma S_D-1}}{1-\gamma S_D}}=-k{\mathrm{Ma}}_1^2,$$

(57)

we can solve it as follows

$$S_D^{1,2}={\displaystyle \frac{k\left(k^2+1\right){\mathrm{Ma}}_1\pm \left(k+1\right)\sqrt{k\left(4\mathrm{k}+\left(k^2-1\right)Z\right)}}{k{\left(1+k\right)}^2{\mathrm{Ma}}_1}}$$

(58)

The limiting case of these solutions at Ma₁→∞ has also the form of Equation (38) regardless of the values of the Arrhenius numbers. As in the case of Equation (56), the minus sign before the root in (58) yields non-physical values.

Table 1. Abscissa of the point D and limiting values of the compression ratio.

System	${\mathit{S}}_{\mathit{D}}$	$\underset{{\mathbf{Ma}}_1\to \mathit{\infty }}{\mathbf{lim}}{\mathit{S}}_{\mathit{D}}$
(49), (50)	(52)	${\displaystyle \frac{k}{k+1}}$
(36), (49)	(55)	$\underset{{\mathrm{Ma}}_1\to \infty }{\lim }{S}_D^1=1, \underset{{\mathrm{Ma}}_1\to \infty }{\lim }{S}_D^2=0$
(36), (49), (46)	(56)	${\displaystyle \frac{k-1}{k+1}}$
(36), (50)	(58)	${\displaystyle \frac{k-1}{k+1}}$

summarizes all solutions analyzed above.

Functions $S_D^1$ expressed by Equations (56) and (58) are plotted in Figure 5 and Figure 6.

Function (42) is also plotted here. It is evident that as the value of the exponent Arrhenius number increases, the influence of the amplitude Arrhenius number decreases. The effect of the amplitude Arrhenius number disappears approximately under the condition

$${\mathrm{Ar}}_E>3$$

(59)

Figure 7 shows a comparison of the functions $S_D^1$ calculated using Equations (56) and (58). It is evident that the data by Equation (56) lie somewhat lower than the data by Equation (58). But all curves tend in the limit to the value predicted by Equation (38).

Since the condition $S_D<1$ is always satisfied in a shock wave, we can find the restrictions for the values of the amplitude and the Arrhenius number. Using this condition, we can find from Equation (56) the critical values of the amplitude Arrhenius number

$${\mathrm{Ar}}_A^{cr}=-{\displaystyle \frac{4\exp \left({\mathrm{Ar}}_E\right)\left(-1+{\mathrm{Ma}}_1^2\right)\left(2+\left(-1+k\right){\mathrm{Ma}}_1^2\right)}{\left(-1+k\right)\left(-4+{\mathrm{Ar}}_E-{\mathrm{Ar}}_{E}k+\left(-2+{\mathrm{Ar}}_E\right)\left(-1+k\right){\mathrm{Ma}}_1^2\right)}}$$

(60)

Then we find from Equation (58)

$${\mathrm{Ar}}_A<{\mathrm{Ar}}_A^{cr}=-{\displaystyle \frac{8\exp \left({\mathrm{Ar}}_E\right)k\left(-1+{\mathrm{Ma}}_1^2\right)\left(2+\left(-1+k\right){\mathrm{Ma}}_1^2\right)}{-\left(\left(4+{\mathrm{Ar}}_E\left(-1+k\right)\right)\left(-1+k^2\right)\right)+\left(-2+{\mathrm{Ar}}_E\right){\left(-1+k\right)}^2\left(1+k\right){\mathrm{Ma}}_1^2}}$$

(61)

The critical values of amplitude and Arrhenius number are illustrated in Figure 8.

This figure confirms the trend elucidated in Figure 5 and Figure 6.

6. Conclusions

This article theoretically investigates the interaction of an ideal gas flow with a normal detonation wave in the presence of activation energy of chemical processes.

A modified Hugoniot detonation equation has been obtained, which takes into account the effect of the intensity of heat release on the limiting characteristics of detonation.

The dependence of the limiting value of the exponent Arrhenius number ${\mathrm{Ar}}_E$ on the Mach number before the shock wave has been obtained. As the Mach number increases, the limiting value of ${\mathrm{Ar}}_E$ decreases. This means that the activation energy of the chemical processes required for the occurrence of detonation decreases.

An equation has been derived for determining the limiting value of the compression ratio in the shock wave. The influence of the Mach number before the shock wave and the Arrhenius numbers Ar_E and Ar_A on the limiting value of the compression ratio has been shown. With the growth of the exponent Arrhenius number Ar_E, the influence of the amplitude Arrhenius number Ar_A on the limiting value of the compression ratio of the shock wave decreases.

A condition for determining the critical value of the Arrhenius number Ar_A necessary for the occurrence of detonation has been obtained. The influence of the Mach number and the exponent Arrhenius number Ar_E on the critical value of the amplitude Arrhenius number Ar_A has been shown.

Analysis of the symmetries of the gas flow parameters when passing through a detonation wave showed that conditions are possible when the flow velocity remains invariant. This situation corresponds to Jouguet detonation.

Further development of this research project is planned by taking into account flow turbulence, chemical reactions, and real gas properties in the model.