Normal Shock Waves in Chemically Reacting Flows with Exothermic and Endothermic Reactions Under High-Temperature Conditions

Abstract

This article theoretically investigates the interaction of a normal shock wave in a flow with chemical reactions under high-temperature conditions. The main novelty of the work is that the thermal effect of chemical reactions is modeled as a function of the temperature. A modified Rankine–Hugoniot model for a shock wave in a flow with chemical reactions has been developed. It is shown that for an exothermic reaction the pressure jump increases with increasing Arrhenius numbers. This is due to the additional energy introduced into the flow as heat is released during the chemical reaction. For endothermic reactions, the opposite trend is observed. The change in the speed of the adiabatic gas flow as it passes through a normal shock wave depending on the type of chemical reaction is clarified. The study provides comparisons between the results of the analytical and numerical solutions of the modified Hugoniot adiabatic equations.

1. Introduction

Shock waves cause a quick increase in pressure and temperature in the surrounding environment. The study of shock-wave interactions is crucial for understanding the mechanisms of chemical reactions in explosives [1] and the synthesis of new materials [2]. Research into chemical reactions induced by shock interactions of hypersonic vehicles [3], comets, and meteorites with planetary atmospheres [4] is of great interest. The interaction of shock waves and chemical reactions significantly impacts processes in engine combustion chambers [5]. The development of high-altitude weapons and ammunition also requires research into the propagation patterns of shock waves in environments with negative overpressure [6].

The study [7] reviews the current state of research on the structure of various types of shock-wave interactions, including thermal and chemical interactions. Shock waves can cause ultra-fast chemical reactions and transition to detonation in energetic materials [8]. Reactive molecular dynamic simulations of energetic materials have shown that temperature and intramolecular deformation accelerate chemical kinetics. Reference [2] presents numerical modeling results of shock-induced chemical reactions in a binary energetic material mixture of aluminum and iron oxides. The mixture temperature was shown to increase with the impact speed due to the shock-induced chemical reaction. The final products of the reaction are iron and aluminum oxide. Simulation results of shock-wave propagation in reactive systems of metallic powder mixtures are presented in [9]. The development of thermal and mechanical fields in the shock wave at the particle level of the mixture is shown, and the ultra-fast chemical reactions in these systems are explained.

The effect of exothermic chemical reactions initiated by a pulsed laser in the air on the speed of induced shock waves is shown in [10]. The expansion of the resulting shock wave leads to the expansion of the thermal impact zone in the surrounding atmosphere and the formation of additional shock fronts.

Reference [11] presents results on the general characteristics of an explosive wave in the air. Solutions have been obtained that allow for comparing explosive waves from different explosions, including nuclear explosions. The impact of a shock wave on an explosive substance has been studied at the molecular level [12]. This process leads to the development of a three-dimensional structure of detonation waves observed for all explosives.

Shock tubes are used for experimental studies of high-temperature chemical reactions in strong shock waves [13,14]. Using a shock tube combined with laser diagnostics allowed the study of the rapid cascade of chemical reactions that occur during fuel combustion. This will help develop cleaner and more efficient engines [14].

Theoretical studies on thermochemical effects in shock waves are presented in [15,16,17,18,19,20]. Modified Rankine–Hugoniot jump conditions, accounting for dissociation and vibrational excitation, have been obtained [15]. It has been shown that the turbulent Reynolds number increases when a shock wave passes at hypersonic Mach numbers in the presence of dissociation. Direct numerical simulations of detonation and turbulence interaction in a stoichiometric mixture of hydrogen, oxygen, and argon have been conducted [16]. The results show that the shock wave and detonation flame in the mixture significantly deform in the turbulent flow, forming many small unburned gas pockets. The main contribution of turbulence is the creation of varying shock strengths, leading to different reaction rates.

Results show that the shock wave and detonation flame in both mixtures significantly deform in the turbulent flow, forming many small unburned gas pockets. The main contribution of turbulence is creating varying shock strengths, leading to different reaction rates.

References [17,18] investigated the impact of turbulence on shock and detonation waves, assessing the influence of heat release and turbulence on shock-wave speed. The effect of the Van der Waals gas equation on pressure and velocity changes in a planar shock wave was studied in [19,20,21]. The impact of the thermodynamic properties of real gas on shock-wave parameters was shown. The effect of nanoparticles in a gas when a shock wave passes was considered in [22]. The influence of nanoparticle concentration in the gas on shock-wave intensity was elucidated. Reference [23] investigated the impact of a magnetic field on the detonation wave.

An iterative method for modeling physical and chemical processes in reactive shock waves, providing good stability and fast convergence, has been developed [24]. The proposed theoretical method has been applied to problems related to reactive shock waves, including the stability of oblique detonation waves and shock-wave reflection in dissociated air. Reference [25] presents numerical simulation results for hypersonic thermochemical non-equilibrium flows using non-linear coupled constitutive relations, showing the importance of the vibrational energy source term for accurate high-speed flow modeling.

The propagation of one-dimensional low-amplitude shock waves in a linearly elastic mixture with binary chemical reactions is considered in [26]. The effect of the type of chemical reaction and the underlying thermochemical equilibrium state on shock wave behavior is discussed. The results of mesoscale modeling of shock-wave energy dissipation through chemical reactions are presented in [27]. It was found that such chemical reactions can attenuate shock waves. It was shown how the parameters of the chemical model influence this behavior. Modeling shows that the magnitude of the volumetric collapse and the speed at which chemistry propagates are critical for shock attenuation, whereas the energy involved in the reaction plays a secondary role.

The heat and mass transfer during fluid flow in porous media which account for chemical reactions are quite complex. Studies of these phenomena are relevant to improving the technology and equipment in chemical industry. Theoretical studies of the effects of chemical reactions on flows of nanofluids in porous media are presented in [28,29]. The effect of the activation energy of chemical reactions and porosity on the nanofluid flow along a porous plate was shown in [28]. The flow of a nanofluid between two plates filled with a porous medium under the influence of chemical reactions and an external magnetic field was considered in [29]. Heat and mass transport equations were obtained that elucidate the effects of chemical reactions on the process parameters. Convective heat and mass transfer of chemically reacting fluids in rotating systems were studied in [30,31]. Variation in the velocity and concentration in a chemically reacting rotating fluid was studied. The effects of activation energy on friction and heat transfer coefficients were also considered.

Studies on the impact of thermochemical effects on shock waves in gaseous media have shown that shock-wave front behavior depends on the mixture type, the gas-flow regime, and the type of chemical reactions (exothermic or endothermic).

The objective of this work was to study the impact of chemical reactions on the intensity of shock waves in a gas flow. This article theoretically investigates (i) the interaction of a normal shock wave in a flow with chemical reactions under high-temperature conditions and (ii) the development of a modified Rankine–Hugoniot model for a shock wave in a flow with chemical reactions and (iii) obtains the Hugoniot adiabatic equation and determines the flow velocities in front of the shock wave. The main novelty of the model to be developed in the present work is that the thermal effect of chemical reactions is modeled as a function of temperature.

2. Mathematical Model

Physically, the phenomenon at issue looks like this: in a normal shock wave, within a negligible thickness (several molecular mean free paths), the functions of pressure, temperature, and density vary almost abruptly, and the flow slows down from supersonic to subsonic velocities. This enables consideration of a shock wave as a jump in the flow parameters. This thermodynamic process is adiabatic (but not isentropic). The total enthalpy of the flow after the shock wave is equal to that before the shock wave. Surface friction is also negligible. In this case, the gas flow through a normal shock wave is schematically outlined in Figure 1.

A steady-state gas flow is described by the following system of mass, momentum, and energy conservation equations [21]:

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

(1)

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

(2)

$${\rho }_1{V}_1\left[\left(h_1+{\displaystyle \frac{V_1^2}{2}}\right)+q_1\right]=\hspace{0.17em}{\rho }_2{V}_2\left[\left(h_2+{\displaystyle \frac{V_2^2}{2}}\right)+q_2\right].$$

(3)

where V is the velocity, ρ is the density of a gas, p is the pressure, h is the enthalpy, and q is the heat supply for the chemical reaction, i.e., a measure of the energy removed (q < 0—endothermic reaction) or released (q > 0—exothermic reaction) per unit mass by the corresponding process that occurs across the shock. The subscript “1” denotes the parameters before the shock wave, and the subscript “2” denotes the parameters after the shock wave.

The main limitation of the model is the linear approximation of the thermal effect of chemical reactions.

The velocity of the heat supply of a reaction is determined by the following equation [32]:

$$q=\pm A\hspace{0.17em}\exp \left(-{\displaystyle \frac{E}{\Re T}}\right),$$

(4)

where A is a function of thermophysical properties, which include the dimension of enthalpy; E is the activation energy; T is the temperature; and $\Re$ is the individual (specific) gas constant. The plus sign corresponds to an exothermic reaction, and the minus sign corresponds to an endothermic reaction.

System (1)–(4) 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)

where $k=c_p/c_{\upsilon }$ is the specific heat ratio, c_p is the specific heat capacity at a constant pressure, and $c_{\upsilon }$ is the specific isochoric heat capacity.

On condition that $E/\left(\Re T\right)<<1$, Equation (4) can be represented as follows:

$$q\approx \pm A\left(1-{\displaystyle \frac{E}{\Re T}}\right),$$

(7)

Equation (3) can be represented by taking into account Equations (1) and (7) in the following form:

$$c_p{T}_1+{\displaystyle \frac{V_1^2}{2}}\pm A\left(1-{\displaystyle \frac{E}{\Re T_1}}\right)=c_p{T}_2+{\displaystyle \frac{V_2^2}{2}}\pm A\left(1-{\displaystyle \frac{E}{\Re T_2}}\right).$$

(8)

or

$$c_p{T}_1+{\displaystyle \frac{V_1^2}{2}}\mp {\displaystyle \frac{AE}{\Re T_1}}=c_p{T}_2+{\displaystyle \frac{V_2^2}{2}}\mp {\displaystyle \frac{AE}{\Re T_2}}.$$

(9)

Here, we used the following equation for enthalpy:

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

(10)

Taking into account (5) and (6), we can rewrite (9) in the following form:

$${\displaystyle \frac{k}{k-1}}{\displaystyle \frac{p_1}{{\rho }_1}}+{\displaystyle \frac{V_1^2}{2}}\mp AE{\displaystyle \frac{{\rho }_1}{p_1}}={\displaystyle \frac{k}{k-1}}{\displaystyle \frac{p_2}{{\rho }_2}}+{\displaystyle \frac{V_2^2}{2}}\mp AE{\displaystyle \frac{{\rho }_2}{p_2}}.$$

(11)

where the plus sign corresponds to an endothermic reaction and the minus sign corresponds to an exothermic reaction.

3. Modified Hugoniot Condition

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

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

(12)

Multiplying Equation (12) by the following factor:

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

(13)

one can obtain

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

(14)

It follows from Equation (11) that

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

(15)

A comparison of Equations (14) and (15) yields

$$\left(p_1-p_2\right)\left({\displaystyle \frac{1}{{\rho }_1}}+{\displaystyle \frac{1}{{\rho }_2}}\right)={\displaystyle \frac{2k}{k-1}}\left({\displaystyle \frac{p_1}{{\rho }_1}}-{\displaystyle \frac{p_2}{{\rho }_2}}\right)\mp AE\left({\displaystyle \frac{{\rho }_1}{p_1}}-{\displaystyle \frac{{\rho }_2}{p_2}}\right)$$

(16)

or

$$\left(1-{\displaystyle \frac{p_2}{p_1}}\right)\hspace{0.17em}\hspace{0.17em}\left(1+{\displaystyle \frac{{\rho }_1}{{\rho }_2}}\right){\displaystyle \frac{p_1}{{\rho }_1}}={\displaystyle \frac{2k}{k-1}}\left(1-{\displaystyle \frac{{\rho }_1}{{\rho }_2}}{\displaystyle \frac{p_2}{p_1}}\right){\displaystyle \frac{p_1}{{\rho }_1}}\mp AE{\displaystyle \frac{{\rho }_1}{p_1}}\left(1-{\displaystyle \frac{{\rho }_2}{p_2}}{\displaystyle \frac{p_1}{{\rho }_1}}\right)$$

(17)

In dimensionless form:

$$\left(1-{\displaystyle \frac{p_2}{p_1}}\right)\hspace{0.17em}\hspace{0.17em}\left(1+{\displaystyle \frac{{\rho }_1}{{\rho }_2}}\right)={\displaystyle \frac{2k}{k-1}}\left(1-{\displaystyle \frac{{\rho }_1}{{\rho }_2}}{\displaystyle \frac{p_2}{p_1}}\right)\mp \mathrm{Ar}\left(1-{\displaystyle \frac{{\rho }_2}{p_2}}{\displaystyle \frac{p_1}{{\rho }_1}}\right),$$

(18)

where

$$\mathrm{Ar}={\displaystyle \frac{AE}{{\left(p_1/{\rho }_1\right)}^2}}={\displaystyle \frac{AE}{{\left(\Re T_1\right)}^2}}=k^2{\displaystyle \frac{AE}{a_1^4}}$$

(19)

is the Arrhenius number and a₁ is the sound speed before the shock wave.

Solving Equation (18) with respect to p₂/p₁, one can derive

$$P={\displaystyle \frac{1+R\left(k+1\mp \mathrm{Ar}\left(k-1\right)\right)-k+\sqrt{{\left(k\left(1+R\left(\mathrm{Ar}-1\right)\right)-R\left(\mathrm{Ar}+1\right)-1\right)}^2\pm 4\mathrm{Ar}{R}^2\left(k-1\right)\left(k+1-R\left(k-1\right)\right)}}{2\left(\left(R+1\right)-k\left(R-1\right)\right)}},$$

(20)

where

$$P={\displaystyle \frac{p_2}{p_1}}, R={\displaystyle \frac{{\rho }_2}{{\rho }_1}},$$

(21)

where the upper sign in Equation (20) corresponds to the minus sign in Equation (18), i.e., an exothermic reaction, and the lower sign in Equation (20) corresponds to the plus in Equation (18), i.e., an endothermic reaction.

Equation (20) describes the modified Hugoniot condition. In case there is no chemical reaction, Equation (20) is reduced to

$${\displaystyle \frac{p_2}{p_1}}={\displaystyle \frac{\left(k+1\right)\frac{{\rho }_2}{{\rho }_1}-\left(k-1\right)}{\left(k+1\right)-\left(k-1\right)\frac{{\rho }_2}{{\rho }_1}}},$$

(22)

which is the well-established Hugoniot condition for pure ordinary gases [33,34,35].

Equation (20) exhibits an asymptote for ρ₂/ρ₁ expressed as

$$R={\displaystyle \frac{k+1}{k-1}}.$$

(23)

For the condition expressed as Equation (23), the pressure jump (20) becomes infinite. This means that for k = 1.4, the maximum degree of compressibility is equal to six.

Figure 2 shows the calculations according to Equation (20) for different Arrhenius numbers and k = 1.4.

It can be seen from Figure 2 that in an exothermic reaction the condition (P = constant), the degree of compressibility increases with increasing level of Arrhenius numbers.

Obviously, this is due to the additional energy introduced into the flow as heat is released during the chemical reaction. For endothermic reactions, the opposite trend is observed.

From Figure 2 it can be seen that in an exothermic reaction under the condition (P = constan) degree of compressibility increases with increasing level of Arrhenius numbers.

The same upward trend in the pressure jump is shown in Figure 3. It can be seen that with an increase in the Arrhenius number, the magnitude of the pressure jump changes linearly. Obviously, an increase in the density ratio (or jump) also increases the pressure jump.

A decrease in the parameter k from 1.4 to 1.33 causes a slight decrease in the P value, less than 2%.

4. Verification of the Linear Approach to Heat Supply in Chemical Reactions

The equation for the modified Hugoniot condition (20) was obtained based on the linear approximation (7) for the heat supply of chemical reactions. If the exponential Equation (4) is used, the modified Hugoniot condition takes the form of a transcendental equation.

$$\left(1-{\displaystyle \frac{p_2}{p_1}}\right)\hspace{0.17em}\hspace{0.17em}\left(1+{\displaystyle \frac{{\rho }_1}{{\rho }_2}}\right)={\displaystyle \frac{2k}{k-1}}\left(1-{\displaystyle \frac{{\rho }_1}{{\rho }_2}}{\displaystyle \frac{p_2}{p_1}}\right)\pm {\mathrm{Ar}}_A\left(\exp \left(-{\mathrm{Ar}}_E\right)-\exp \left(-{\mathrm{Ar}}_E{\displaystyle \frac{{\rho }_2}{{\rho }_1}}{\displaystyle \frac{p_1}{p_2}}\right)\right),$$

(24)

where

$${\mathrm{Ar}}_A={\displaystyle \frac{A}{\Re T_1}}, {\mathrm{Ar}}_E={\displaystyle \frac{E}{\Re T_1}}.$$

(25)

Thus,

$$\mathrm{Ar}={\mathrm{Ar}}_A{\mathrm{Ar}}_E.$$

(26)

Comparative calculations using Equations (20) and (24) are presented in

Table 1. Calculation results for the endothermic reaction.

Ar	R		1.5	2	3	4	5	5.99
		Method
0	P	Analytic (20)	1.777	2.75	5.666	11.5	29.0	3494
		Numerical (24)	1.777	2.75	5.666	11.5	29.0	3494
1 ArA = 2 ArE = 0.5	P	Analytic (20)	1.733	2.630	5.239	10.28	25.0	2896
		Numerical (24)	1.748	2.669	5.367	10.61	25.98	3024
		Δ %	0.86	1.46	2.39	3.11	3.77	4.23
3 ArA = 6 ArE = 0.5	P	Analytic (20)	1.674	2.466	4.616	8.368	18.16	1703
		Numerical (24)	1.703	2.544	4.891	9.137	20.62	2085
		Δ %	1.7	3.07	5.62	8.41	11.9	18.3
1 ArA = 1 ArE = 1	P	Analytic (20)	1.733	2.630	5.239	10.28	25.0	2896
		Numerical (24)	1.758	2.696	5.457	10.85	26.69	3116
		Δ %	1.42	2.45	3.99	5.25	6.33	7.06
3 ArA = 3 ArE = 1	P	Analytic (20)	1.674	2.466	4.616	8.368	18.16	1703
		Numerical (24)	1.726	2.606	5.104	9.730	22.26	2362
		Δ %	3.01	5.37	9.56	14.0	18.4	27.9

for an exothermic reaction and in

Table 2. Calculation results for the exothermic reaction.

Ar	R		1.5	2	3	4	5	5.99
		Method
0	P	Analytic	1.777	2.75	5.666	11.5	29.0	3494
		Numerical	1.777	2.75	5.666	11.5	29.0	3494
2 ArA = 4 ArE = 0.5	P	Analytic (20)	1.925	3.106	6.782	14.38	37.67	4690
		Numerical (24)	1.860	2.964	6.457	13.55	35.51	4435
		Δ %	3.38	4.57	4.79	5.77	5.73	5.44
5 ArA = 10 ArE = 0.5	P	Analytic (20)	2.404	4.000	9.000	19.44	51.57	6483
		Numerical (24)	2.080	3.463	7.865	17.17	46.03	5847
		Δ %	13.47	13.42	12.61	11.68	10.74	9.81
2 ArA = 2 ArE = 1	P	Analytic (20)	1.925	3.106	6.782	14.38	37.67	4690
		Numerical (24)	1.825	2.881	6.160	12.97	33.95	4250
		Δ %	5.91	7.24	9.17	9.81	9.88	9.38
5 ArA = 5 ArE = 1	P	Analytic (20)	2.404	4.000	9.000	19.44	51.57	6483
		Numerical (24)	1.931	3.157	7.105	15.55	42.00	5884
		Δ %	19.7	21.1	21.1	20.0	18.6	9.24

for an endothermic reaction. Solutions to the transcendental Equation (24) are obtained by ad hoc methods [36].

Thus, the range of the validity of the analytical model is elucidated in Table 1 and Table 2, which represent comparisons of the linear and transcendental Hugoniot equations.

Table 1 and Table 2 show the relative error between the results of the analytical and numerical solutions of the modified Hugoniot adiabatic equations. As can be seen, the error is quite acceptable. Under the condition Ar = const, the error mainly increases with the increase in the Arrhenius number (Ar_E). This is understandable, as the expansion (7) is performed using this parameter because it is under the exponent. With the increase in the density jump (R), the error increases, and for large values of the parameter Ar_A (Ar_A > 3), it can reach 28% for endothermic reactions and 10% for exothermic reactions.

Therefore, it can be concluded that the linear model is valid up to Arrhenius numbers of Ar = 3 for endothermic reactions and Ar = 5 for exothermic reactions. For larger values, it is necessary to use the transcendental Equation (24).

5. Velocity Variation

We need to determine the relation between the flow velocities before and after the shock wave. Equation (2) needs to be transformed, taking into account Equation (1).

$$V_1-V_2={\displaystyle \frac{p_2}{{\rho }_2{V}_2}}-{\displaystyle \frac{p_1}{{\rho }_1{V}_1}}.$$

(27)

Les us recast Equation (11) as follows:

$${\displaystyle \frac{a_1^2}{k-1}}+{\displaystyle \frac{V_1^2}{2}}\mp k{\displaystyle \frac{AE}{a_1^2}}=const.$$

(28)

The constant in Equation (28) can be found by writing this equation for critical conditions under which the flow velocity is equal to the speed of sound, from which one can obtain

$${\displaystyle \frac{k}{k-1}}{\displaystyle \frac{p_1}{{\rho }_1}}+{\displaystyle \frac{V_1^2}{2}}\mp {\displaystyle \frac{AE}{p_1/{\rho }_1}}={\displaystyle \frac{a_1^2}{k-1}}+{\displaystyle \frac{V_1^2}{2}}\mp k{\displaystyle \frac{AE}{a_1^2}}={\displaystyle \frac{{a^{\ast }}^2}{k-1}}+{\displaystyle \frac{{a^{\ast }}^2}{2}}\mp k{\displaystyle \frac{AE}{{a^{\ast }}^2}}={a^{\ast }}^2{\displaystyle \frac{k+1}{2\left(k-1\right)}}\mp k{\displaystyle \frac{AE}{{a^{\ast }}^2}}.$$

(29)

It follows from here that

$$\begin{array}{l}{\displaystyle \frac{p_1}{{\rho }_1}}={\displaystyle \frac{1}{4{a^{\ast }}^{2}k}}\\ \left[\left(k+1\right){a^{\ast }}^4-\left(k-1\right)\left({a^{\ast }}^2{V}_1^2+2AE\right)\pm \sqrt{{\left(\left(k+1\right){a^{\ast }}^4-\left(k-1\right)\left({a^{\ast }}^2{V}_1^2+2AE\right)\right)}^2+16k\left(k-1\right){a^{\ast }}^{4}AE}\right]\end{array}.$$

(30)

Similar to that, it can be written

$$\begin{array}{l}{\displaystyle \frac{p_2}{{\rho }_2}}={\displaystyle \frac{1}{4{a^{\ast }}^{2}k}}\\ \left[\left(k+1\right){a^{\ast }}^4-\left(k-1\right)\left({a^{\ast }}^2{V}_2^2+2AE\right)\pm \sqrt{{\left(\left(k+1\right){a^{\ast }}^4-\left(k-1\right)\left({a^{\ast }}^2{V}_2^2+2AE\right)\right)}^2+16k\left(k-1\right){a^{\ast }}^{4}AE}\right]\end{array}.$$

(31)

Next, we transform Equation (27), taking into account Equations (30) and (31), which gives

$$\begin{array}{l}{\displaystyle \frac{1}{4k}}\left(\begin{array}{l}\mp 2{\mathrm{Ar}}^{\ast }\left(-1+k\right)\left({\lambda }_1-{\lambda }_2\right)-\sqrt{\pm 16{\mathrm{Ar}}^{\ast }\left(-1+k\right)k+{\left(\left(1+k\right)\mp \left(-1+k\right)\left(2{\mathrm{Ar}}^{\ast }\pm {\lambda }_1^2\right)\right)}^2}{\lambda }_2+\left({\lambda }_1-{\lambda }_2\right)\left(\left(1+k\right)+\left(-1+k\right){\lambda }_1{\lambda }_2\right)+\\ {\lambda }_1\sqrt{\pm 16{\mathrm{Ar}}^{\ast }\left(-1+k\right)k+{\left(\left(1+k\right)\mp \left(-1+k\right)\left(2{\mathrm{Ar}}^{\ast }+{\lambda }_2^2\right)\right)}^2}\end{array}\right)-\left({\lambda }_1-{\lambda }_2\right){\lambda }_1{\lambda }_2=0\end{array}$$

(32)

where

$${\mathrm{Ar}}^{\ast }={\displaystyle \frac{AE}{{a^{\ast }}^4}}$$

(33)

$${\lambda }_1={\displaystyle \frac{V_1}{a^{\ast }}}, {\lambda }_2={\displaystyle \frac{V_2}{a^{\ast }}},$$

(34)

are speed coefficients.

The upper sign in Equation (32) corresponds to an exothermic reaction, and the lower sign corresponds to an endothermic reaction.

Equation (32) is a sixth-order polynomial equation. It does not have a general analytical solution. Therefore, its solution was found numerically as the solution of a transcendental equation.

Equation (32) is a modified Prandtl’s law for a shock wave. Under the condition ${\mathrm{Ar}}^{\ast }$ = 0, it follows from Equation (32) that

$$\left({\lambda }_1\hspace{0.17em}{\lambda }_2-1\right)\left({\lambda }_1-\hspace{0.17em}{\lambda }_2\right)\left(k+1\right)=0.$$

(35)

Since in the shock wave ${\lambda }_1>{\lambda }_2$, from (35), we obtain

$${\lambda }_1\hspace{0.17em}{\lambda }_2=1.$$

(36)

This means that when ${\mathrm{Ar}}^{\ast }$ = 0, the flow velocity is supersonic ahead of the shock and subsonic after it. The results of the calculations performed using Equation (32) are presented in

Table 3. Calculation results obtained using Equation (32).

	λ1	Chemical Reaction	2.5	2	1.5	1.1
${\mathrm{Ar}}^{\ast }$ = 0	λ2	No reaction	0.4	0.5	0.667	0.909
${\mathrm{Ar}}^{\ast }$ = 0.1	λ2	Exothermic	0.387	0.494	0.663	0.905
${\mathrm{Ar}}^{\ast }$ = 0.3	λ2	Exothermic	0.385	0.489	0.658	0.900
${\mathrm{Ar}}^{\ast }$ = 0.5	λ2	Exothermic	0.384	0.487	0.656	0.897
${\mathrm{Ar}}^{\ast }$ = 0.1	λ2	Endothermic	0.407	0.521	0.673	0.915
${\mathrm{Ar}}^{\ast }$ = 0.3	λ2	Endothermic	0.418	0.532	0.711	0.939
${\mathrm{Ar}}^{\ast }$ = 0.5	λ2	Endothermic	0.427	0.539	0.718	1.039

. It is evident that, in the case of exothermic reactions, the velocity behind the shock wave is lower than in the absence of chemical reactions. This is due to the nature of the density change as the flow passes through the shock wave. As shown earlier, at a constant pressure P = const in the conditions of exothermic reactions, the degree of compressibility increases. Therefore, as derived from Equation (1), the velocity must decrease. For endothermic reactions, the velocity behind the wave increases with the parameter value and can exceed the speed of sound.

One can pass from the speed coefficients to the Mach numbers according to the following equation:

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

(37)

6. Conclusions

A modified Rankine–Hugoniot model for a shock wave in a flow with chemical reactions has been developed. The main novelty of the work is that the thermal effect of chemical reactions is modeled as a function of temperature. The Hugoniot adiabatic equation for gas flow with exothermic and endothermic reactions has been obtained. It has been shown that for exothermic reactions, the pressure jump increases with increasing Arrhenius numbers. For endothermic reactions, the opposite trend is observed.

A comparison between the results of the analytical and numerical solutions of the modified Hugoniot adiabatic equations was conducted. With the increase in the density jump (R), the error increases.

In the case of exothermic reactions, the velocity behind the shock wave is lower than in the absence of chemical reactions. This is due to the change in density as the flow passes through the shock wave. For endothermic reactions, the velocity behind the shock wave increases with the parameter value ${\mathrm{Ar}}^{\ast }$ and can exceed the speed of sound.