© AFM, EDP Sciences 2020

1 Introduction

The spacecraft performance optimization design can be improved by thermodynamics and aerodynamics studies, for example, minimizing the pressure and friction losses for internal and external flows may increase the vehicle range and payload. For this purpose, the propulsion systems are used to correct and maintain the satellite at its nominal position. Generally, a propulsion system is a system that accelerates a material to provide a thrust force that moves a vehicle or rotates it relative to its centre of mass. The nozzle is one of the most important components of this engine because its efficiency greatly affects the performance of the engine [1]. By minimizing the losses in the nozzle, the rocket's thrust will be increased.

Rao has developed a method that optimizes a rocket nozzle contour for a given length or expansion ratio so that the maximum thrust is reached. In a study by [2,3], three large optimized Rao steeple nozzles have been compared to a 15-degree conical nozzle. The thrust produced by the optimized nozzles is considerably greater than that obtained with the conical nozzle. In this study, the choice is based on a type called axisymmetric minimum length nozzle, which gives a uniform flow and parallel to the exit section [4–6]. All previous studies were based on using a perfect gas model (PG), in which the specific heat C_P and γ are constant and do not depend on temperature. In reality these parameters vary with temperature [7–9] have developed a mathematical model that takes account of this temperature. The new model is called the model of perfect gas at high temperature (HT). In this context, the mathematical model of the perfect gas (PG) becomes a special case of the model with high temperature (HT). The design of nozzles operating with non-ideal gases is currently an active area of research in the design of ORC turbines [10–12].

The thrust of a propulsive system primarily depends on the amount of combustion products ejected through it [1]. So, considering air for the design of supersonic nozzles is far from reality [13]. The use of gas other than air influences essentially the behaviour of supersonic flows and in particular the physical thermodynamic (P/P ₀, T/T ₀, ρ/ρ ₀,) and geometric parameters (A/A*, C_F , Mass) of the nozzle design [14]. In this reference, the author did not clearly show the influence of the addition of the concentrations of the gases resulting from a combustion on the design of the nozzles. It is clear that these gases come from a combustion of bipropellants (oxidizer/fuel). For this reason, a use of the thermochemical study of the combustion of the most used and popular liquid propellants on engines (launchers and satellites) provided us the necessary information for the equilibrium composition at assumed temperature. Four engines were studied: the main stage of the Ariane-5 (Vulcain 2), the second stage of the Ariane-5 launcher (Aestus), the first stage of the Zenit launcher (RD-170) and the upper stage Falcon 9 (Raptor). These engines respectively use the following liquid propellants: hydrogen/oxygen, mono-methyl hydrazine/Nitrogen tetroxide, kerosene/oxygen and methane/oxygen. Edwards [15] and Amri et al., [16] carried out a study of the chemical reactions of these four engines. The gases types resulting from the combustion are: triatomic (H₂O, CO₂), diatomic (H₂, OH, CO, O₂, and N₂) and monoatomic (O, H, N).

The aim of the present paper is to develop a numerical program for studying the effect of these gases triatomic and diatomic on the performance of the axisymmetric MLN supersonic nozzles, using the length, the exit Mach number, the mass and the thrust coefficient parameters.

2 Design method of the axisymmetric MLN nozzle (Case of Air)

In this study, a nozzle type which gives a uniform flow and parallel to the exit section is chosen. It is named minimum length nozzle (MLN) with centered expansion. This type of nozzle has a minimum length compared to other existing types [17]. Figure 1 illustrates the general diagram of the axisymmetric nozzle of minimum length with straight sonic line

The method of characteristics (MOC) at high temperature is undertaken to perform the nozzle design in order to:

• Find the nozzle's contour for a given exit Mach number M_E .

• Calculate the internal flow field in the axisymmetric MLN.

The relations (1) and (2) below represent the characteristics equations and compatibilities for a supersonic, irrotational, adiabatic flow of a perfect gas [17–19] and are respectively valid on the C⁻ and C⁺ characteristics as shown in Figure 2.

Following ξ (1–3):$$\{\begin{array}{l}d(v+\theta )=\frac{\sin \theta \sin \mu }{y}d\xi \hfill \\ \frac{\mathrm{d}y}{\mathrm{d}x}=tg(\theta -\mu )\hfill \end{array}$$(1)

Following η (2–3):$$\{\begin{array}{l}d(v-\theta )=\frac{\sin \theta \sin \mu }{y}d\eta \hfill \\ \frac{\mathrm{d}y}{\mathrm{d}x}=tg(\theta +\mu )\hfill \end{array}$$(2)

where ν is the Prandtl Meyer function at high temperature (HT) [19].

The use of the method of the characteristics (MOC) let us to introduce a fine mesh in order to approximate each characteristic between two points by straight line segments. The properties (x, y, T, θ, ρ, P) at a flow field point can be determined from two points connected with a point considered by the characteristic lines that comes before it. Therefore, thermodynamic parameters have been calculated for each point of the mesh, as well as the performance parameters of the axisymmetric nozzle MLN. The calculation of the flow in the nozzle is divided into two regions named as Kernel and transition regions. The numerical results obtained between the radius of the numerically calculated exit section and the ratio of the standard theoretical sections (y ^* = 1.0) presented by the following formula will be compared.

Figure 2 illustrates a result of the numerical code which represents a mesh and contour design at high temperature axisymmetric nozzle (MLN) for exit Mach number M_E = 3.00 and T₀ = 2000 K. This applies for air.

Figure 3 gives the fluid's parameters results obtained numerically for an axisymmetric MLN nozzle for Mach M_E  = 3.00 and T ₀ = 2000 K. It is clear to note that the iso-Mach contour and the iso-values match those of the references [19].

After obtaining the flow at all points of the flow field in the axisymmetric MLN, we can proceed to the study of the thermodynamics parameter evolution (Mach, flow angle deviation, pressure, temperature and density) on the wall of the nozzle (see Fig. 4). To do this, a three-dimensional mesh of the nozzle contour is realized by applying the definition of the streamlines in space. A refinement of the contour is necessary by refining of the number of longitudinal sections N_L and transversely points N_T [17,18]. Figures 5–9 show the results obtained by our calculation program.

Fig. 1 Presentation of the axisymmetric MLN nozzle.

Fig. 2 Contour nozzle and mesh of the nozzle for ME  = 3.00 and T 0 = 2000 K. (a) Large grid. (b) Moderate grid. (c) Rather fine grid. (d) Fine grid.

Fig. 3 Numerical results of the flow of air in an axisymmetric nozzle MLN for ME 3.00 and T 0 = 2000 K. (a) Iso-Mach curves (Ref. [19] and numerical results). (b) Iso-Directions curves (Ref. [19] and numerical results). (c) Iso-temperatures ratio (T/T 0) curves. (d) Iso-Pressures ratio (P/P 0) curves.

Fig. 4 Thee-dimensional mesh of the axisymmetric MLN contour for T 0 = 2000 K and ME  = 3.00.

Fig. 5 Variation of M along the wall.

Fig. 6 Variation of θ* along the wall.

Fig. 7 Variation of T/T 0 along the wall.

Fig. 8 Variation of P/P 0 along the wall.

Fig. 9 Variation of ρ/ρ 0 along the wall.

3 Study of combustion for bipropellants

The objectives here are to determine the theoretical combustion temperature and the theoretical composition of the resulting reaction products. The first principle concerns the conservation of energy. The heat created by the combustion is equal to the heat necessary to raise the resulting gases adiabatically to their final combustion temperature. The heat of reaction of the combustion ΔH_r has to equal the enthalpy change ΔH of the gases.$$\mathrm{\Delta }{H}_r={\displaystyle \sum _1^n}{n}_j{\int }_{T_{ref}}^{T_1}{C}_{P}dT={\displaystyle \sum _1^n}{n}_j\mathrm{\Delta }{h}_j{{\displaystyle |}}_{T_{ref}}^{T_1}$$(3)where ΔH_r the energy difference between the reactants and products under standard conditions (kJ).

The second principle is the conservation of mass. The mass of any of the atomic species present in the reactants before the chemical reaction must be equal to the mass of the same species in the products. The combustion of the following propellant combinations was studied: hydrogen/oxygen, hydrocarbon fuel/oxygen, hydrocarbon fuel/hydrogen peroxide, hydrazine/nitrogen tetroxide and mono-methyl hydrazine/Nitrogen tetroxide. The following five (05) chemical reactions were considered:

• Liquid hydrogen (H₂) and liquid oxygen (O₂)

$$2\alpha {\mathrm{H}}_2{\text{+O}}_2\to {\text{n}}_{{\text{H}}_2\mathrm{O}}{\text{H}}_2{\text{O+n}}_{{\text{H}}_2}{\text{H}}_2{\text{+n}}_{{\mathrm{O}}_2}{\text{O}}_2{\text{+n}}_{\mathrm{O}}{\text{O+n}}_{\text{H}}{\text{H+n}}_{\text{OH}}\text{OH}$$(4)

• Hydrocarbon fuel (C_nH_m) and liquid oxygen (O₂)

$$\alpha {\mathrm{C}}_{\mathrm{n}}{\mathrm{H}}_{\mathrm{m}}{\text{+O}}_2\to {\text{n}}_{\mathrm{C}{\mathrm{O}}_2}{\text{CO}}_2{\text{+n}}_{{\text{H}}_2\mathrm{O}}{\text{H}}_2{\text{O+n}}_{{\mathrm{O}}_2}{\text{O}}_2{\text{+n}}_{\mathrm{C}\mathrm{O}}{\text{CO++n}}_{\text{OH}}{\text{OH +n}}_{{\text{H}}_2}{\text{H}}_2{\text{+n}}_{\text{O}}{\text{O+n}}_{\text{H}}\text{H}$$(5)

• Hydrocarbon fuel (C_nH_m) and hydrogen peroxide (H₂O₂)

$$\alpha {\mathrm{C}}_{\mathrm{n}}{\mathrm{H}}_{\mathrm{m}}+{\mathrm{H}}_2{\mathrm{O}}_2\to {\mathrm{n}}_{\mathrm{C}{\mathrm{O}}_2}\mathrm{C}{\mathrm{O}}_2+{\mathrm{n}}_{{\mathrm{H}}_2\mathrm{O}}{\mathrm{H}}_2\mathrm{O}+{\mathrm{n}}_{{\mathrm{O}}_2}{\mathrm{O}}_2+{\mathrm{n}}_{\mathrm{C}\mathrm{O}}\mathrm{C}\mathrm{O}++{\mathrm{n}}_{\mathrm{O}\mathrm{H}}\mathrm{O}\mathrm{H}+{\mathrm{n}}_{{\mathrm{H}}_2}{\mathrm{H}}_2+{\mathrm{n}}_{\mathrm{O}}\mathrm{O}+{\mathrm{n}}_{\mathrm{H}}\mathrm{H}$$(6)

• Mono-methyl hydrazine (MMH) and nitrogen tetroxide (N₂O₄)

$$4\alpha \mathrm{C}{\mathrm{H}}_3{\text{-NH-NH}}_{\text{2}}{\text{+5N}}_2{\text{O}}_4\to {\text{n}}_{\mathrm{C}{\mathrm{O}}_2}{\text{CO}}_2{\text{+n}}_{{\text{H}}_2\mathrm{O}}{\text{H}}_2{\text{O+n}}_{{\mathrm{O}}_2}{\text{O}}_2{\text{+n}}_{{\mathrm{N}}_2}{\mathrm{N}}_2{\text{+n}}_{\mathrm{N}\mathrm{O}}{\text{NO+n}}_{\text{CO}}{\text{CO+n}}_{\text{OH}}{\text{OH+n}}_{{\text{H}}_2}{\text{H}}_2{\text{+n}}_{\text{O}}{\text{O+n}}_{\text{H}}\text{H}$$(7)

• Hydrazine (N₂H₄) and nitrogen tetroxide (N2O4)

$$2\alpha {\text{N}}_2{\text{H}}_{\text{4}}{\text{+N}}_2{\text{O}}_4\to {\text{n}}_{{\text{H}}_2\mathrm{O}}{\text{H}}_2{\text{O+n}}_{{\mathrm{O}}_2}{\text{O}}_2{\text{+n}}_{{\mathrm{N}}_2}{\mathrm{N}}_2{\text{+n}}_{\mathrm{N}\mathrm{O}}{\text{NO+n}}_{\text{OH}}{\text{OH+n}}_{{\text{H}}_2}{\text{H}}_2{\text{+n}}_{\text{O}}{\text{O+n}}_{\text{H}}\text{H}$$(8)

It will be noted that at each iteration, the composition of the equilibrium composition of the combustion is required at any temperature in the range of [1000, 6000]. To do this, it is necessary to solve the system of nonlinear equations for each equations (4)–(8). The Lieberstein numerical method was used to solve the problem.

Figures 10–15 give the effect of pressure on the flame temperature of the combustion for each liquid propellant when the chamber pressure is 50 bar, 100 bar and 200 bar.

Figures 16–21 show all the obtained results by the realized numerical program for each bipropellant combinations when the chamber pressure is 50 bar, 100 bar and 200 bar.

This study is based on the thermochemical aspect and the combustion of the most used bipropellants combination on launchers and satellites. It is essential to evaluate their performance and provide us with the necessary information during the design of the launcher engine. As an application, four engines were studied in the present paper (Vulcain 2, Aestus, RD-170, Raptor and RD-161P engines). Table 1 shows the input data used for the study.

Figure 22 gives the results of the four engines to the combustion products of the bipropellant combinations: hydrogen/oxygen, hydrocarbon fuel/oxygen, Mono-methyl hydrazine/nitrogen tetroxide and Methane/oxygen. The gas produced from different bipropellant combustions are: triatomic, diatomic and monoatomic types.

Table 2 gives the results in term of vacuum specific impulse, flame temperature, molecular mass and politropic parameter.

In order to obtain the nozzles contours of bipropellant engines, the method of characteristics (MOC) at high temperature can be using, provided that the thermodynamic properties of the combustion products (triatomic and diatomic) of the different combustion are known. Particular attention is paid to the variation of the specific heat at constant pressure C_P (T) as a function of temperature and to the thermodynamic constant of the gases γ(T). Figure 23 illustrates these variations.

From Figure 23, we can notice that, contrary to the γ ratio, C_P (T) of all gases increases with the temperature T ₀. Computational program requires analytical forms for the C_P (T) function of each gas. Polynomial interpolations from values of the tables was used [20]. Figure 24 gives the results in term of error for C _PCO2 with different polynomials interpolations (4th, 5th, 6th, 7th, 8th and 9th degree). The choice was oriented towards polynomial interpolations of 9th degree.$${\mathit{\text{C}}}_{\mathit{\text{P}}}(T)={\text{a}}_1+\text{T}({\text{a}}_2\text{T}({\text{a}}_3+\text{T}({\text{a}}_4+\text{T}({\text{a}}_5+\text{T}({\text{a}}_6+\text{T}({\text{a}}_7+\text{T}({\text{a}}_8+\text{T}({\text{a}}_9+\text{T}({\text{a}}_{10})))))))))$$(9)where a_i are the interpolation coefficients.

Applying the method of characteristics at high temperature (MOC) with the functions of C_P (T) for each gas, various results were obtained by computational program and are presented in what follows. Figure 25 shows the shape of the nozzle for the four engines (Vulcain 2, Aestus, RD-170 and Raptor).

In order to clearly illustrate the difference between the design parameters of the various engines, the results obtained as shown in Table 3.

In minimum length nozzles, the last part of the diverging section is substantially constant and therefore it contributes only a few percent to the total thrust. For reasons of optimization, these nozzles must be truncated in order to increase their performance. The latter consists in truncating the nozzle at a well-defined section having almost the same properties as those of the exit section. As an example in practice, the ideal truncated nozzle that equips the Viking engines of the launcher Ariane 4 are cited [21]. Usually, the truncation is performed, at an angle of inclination of the flow θ equal to 3 degree in general. In this case, the mass of the nozzle is considerably reduced. However, the thrust delivered by the nozzle decreases slightly. Table 4 shows the mass gain and the loss of thrust coefficient calculated by the computation code truncated at 30%.

Fig. 10 Pressure effect on the flame temperature of the combustion H2/O2.

Fig. 11 Pressure effect on the flame temperature of the combustion N2H4/N2O4.

Fig. 12 Pressure effect on the flame temperature of the combustion MMH/N2O4.

Fig. 13 Pressure effect on the flame temperature of the combustion RP-1/O2.

Fig. 14 Pressure Effect on the flame temperature of the combustion RP-1/N2O4.

Fig. 15 Pressure effect on the flame temperature of the combustion CH4/O2.

Fig. 16 Combustion products (%) of the bipropellants H2/O2.

Fig. 17 Combustion products (%) of the bipropellants N2H4/N2O4.

Fig. 18 Combustion products (%) of the bipropellants MMH/N2O4.

Fig. 19 Combustion products (%) of the bipropellants RP-1/O2.

Fig. 20 Combustion products (%) of the bipropellants RP-1/N2O4.

Fig. 21 Combustion products (%) of the bipropellants CH4/O2.

Fig. 22 Examples of combustion products (%) of the various engines.

Fig. 23 Variation of the specific heat function CP and γ as a function of the stagnation temperature T 0.

Fig. 24 Error calculation of CpCO2 for different interpolations polynomials.

Fig. 25 Effect of the stagnation temperature T 0 on the profile of the nozzle for different gases.

4 Effect of T₀ on the design parameters

Figures 26–29 show the design parameters (length, coefficient of mass, nozzle area ration and vacuum specific impulse) of the nozzle of the following engines: Aestus, Vulcain 2, RD-170 and Raptor, for various values of the stagnation temperature T ₀. From the numerical forms and results obtained, the effect of the combustor temperature T ₀ on the design of axisymmetric MLN nozzles has also been shown as presented in the reference studies [12,15].

Fig. 26 Variation of Isp versus T 0.

Fig. 27 Variation of CMass versus T 0.

Fig. 28 Variation of L versus T 0.

Fig. 29 Variation of AE/A* versus T 0.

5 Effect of M_E on the design parameters

Figures 30–33 show the variations in the design parameters of the MLN nozzles of the four engines (Vulcain 2, Aestus, RD-170 and Raptor) as a function of M_E. It is clear that these parameters depend on M_E. For the length (L), the coefficient mass (C_Mass) as well as the nozzle area ration (A_E/A*), it can be noted that the curves of each figure representing the numerical results of each engine are almost identical up to M_E = 2.00. But, beyond this value, they begin to be different as M_E increases. It can be concluded that the nozzle, which corresponds to Aestus with its mass, its length and its vacuum specific impulse gives better results than those of others engines. On the other hand, we observe totally the opposite in the study of the vacuum specific impulse Isp, it is the Vulcain 2 engine which delivers more thrust compared to other engines.

Fig. 30 Variation of L versus ME .

Fig. 31 Variation of Isp versus ME .

Fig. 32 Variation of C Mass versus ME .

Fig. 33 Variation of AE/A* versus ME .

6 Effect of gas in the design of the nozzle

In this section, we will study the effects of diatomic and triatomic products on the combustion (oxidant/fuel) combinations of liquid propellant engines in Figures 16–21, on the performance of MLN axisymmetric supersonic nozzles. Table 5 shows a comparison with different diatomic gases on the design parameters. From the results obtained, the difference between values is negligible for the diatomic gases H₂, O₂, OH, CO and N₂. This difference is result of the gases molar mass influence on the nozzle velocity since the latter's have different molar masses. On the other hand, it is shown, therefore, that the results of the diatomic gas nozzles are almost identical with those of the air, which is also a 99% diatomic gas.

In order to illustrate the influence of the triatomic gases (H₂O/CO₂) combustion products in the design of the nozzles, we us equation (10) in the calculation code. Figure 34 shows this results for cases where λ equal to 0%, 20%, 50%, 80% and 100%.$$C{p}_{\text{m}é\text{lange}}\left(T\right)=\left(1-\lambda \right)C{p}_{{\text{Gas}}_1}\left(T\right)+\lambda C{p}_{{\text{Gas}}_2}\left(T\right)\lambda \in \left[\mathrm{0,1}\right]\text{.}$$(10)

From the numerical forms and results obtained in Figure 34, the increase and decrease in the percentage of diatomic gases (H₂, OH, O₂, NO, N₂ and CO) have little effect on design (such as: length, exit radius, nozzle area ratio) and performance of supersonic nozzle (mass coefficient and thrust coefficient). However, the difference of the contours corresponding to H₂O gas is very large and increases considerably the performances of axisymmetric minimum length nozzle, compared to that of diatomic gases. For example, the increase of 50% of H₂O, increases the nozzle design parameters (diatomic gas including air) in the order of 25%. These errors are due to variations in the function γ(T).

Table 6 presents the differences between the results of the increase in the percentage of triatomic H₂O gas with other diatomic gases. Otherwise, the obtained results are compared to these of air gas [17]. This difference is very considerable. So, considering air for the design of supersonic nozzles is far from reality.

Figure 35 illustrates the influence of the presence of the triatomic gas CO₂ on the contours of the supersonic nozzle for diatomic gases (H₂, OH, O₂, NO, N₂ and CO). These profiles, calculated numerically by our calculation code, correspond to the case where the percentage of presence of the CO₂ gas in the mixture is 0%, 20%, 50%, 80% and 100%. The comparison shows that the presence of CO₂ gas in combustion species increases considerably the design parameters of supersonic minimum nozzle. We notice, these increases depend on the percentage of these gases in combustion (oxidizer/fuel).

Table 7 summarizes the various calculated nozzle parameters and their corresponding percentage presence of triatomic CO₂ gases. For example, an increase of 50% in the CO2 content in a combustion causes an expansion of approximately 35% for the length and the exit radius. This evaluation becomes greater for the ratio of the exit sections which is of the order of 55%. This configuration will therefore increase the weight of the nozzle.

In conclusion, the performance of a rocket engine propelled by a liquid bipropellant is maximum when the presence of CO₂ in the combustion is maximum as seen in Figures 16–21. On the other hand, an increase of H₂O automatically leads to an increase in its performances.

Fig. 34 Influence of triatomic gas H2O with different gases.

Fig. 35 Influence of triatomic gas CO2 with different gases.

7 Conclusion

The present work concerns is to investigate numerically the effect of adding the gas (diatomic/triatomic) type on the design and performance of axisymmetric MLN nozzles, as well as with the flow parameters study (pressure, mass, length, coefficients of mass and thrust) using the method of characteristics (MOC) at high temperature. A FORTRAN language programme code has been developed for this purpose. The gases used in this study were calculated after a thermochemical of the combustion of liquid propellants (the most widely used in the space propulsion domain) of four spacecrafts: Ariane 5 (Vulcain 2), Ariane-5 upper stage engine (Aestus), Zenit first stage (RD-170) and Falcon 9 upper stage (Raptor). The combinations studied are: H₂/O₂, RP-1/O₂, MMH/N₂O₄ and CH₄/O₂. These gases are: H₂O, CO₂, H₂, O₂, OH, CO and N₂. Once the profile mixture between gases previously mentioned (triatomic/diatomic) is generated, an analysis of the evolution of the thermodynamic parameters (namely length, Mach number, mass and thrust coefficient) is conducted. The determination of the equilibrium composition, was solved numerically and it is function of mixture ratio by mass of oxidizer to fuel (O/F). Furthermore, spacecraft engine performance has also been calculated. These will have a maximum value at a wealth equal to 1.

The effect of the stagnation temperature T ₀ and the exit Mach number M_E on the values of the nozzle design parameters and the numerical results obtained show that the difference between each engine is negligible are almost identical up to M_E  = 2.00.

The increase and decrease in the percentage of diatomic gases (H₂, OH, O₂, NO, N₂ and CO including air) have little effect on design (such as: length, exit radius, nozzle area ratio) and performance of supersonic nozzle (mass coefficient and thrust coefficient).

The difference of the contours corresponding to H₂O gas is very large and increases considerably the performances of axisymmetric minimum length nozzle, compared to that of diatomic gases. The increase of 50% of H₂O, increases the nozzle design parameters in the order of 25%.

The comparison shows that the presence of CO₂ gas in combustion species increases considerably the design parameters of supersonic minimum nozzle. We notice, these increases depend on the percentage of these gases in combustion (oxidizer/fuel). For example, an increase of 50% in the CO₂ content in a combustion causes an expansion of approximately 35% for the length and the exit radius. This evaluation becomes greater for the ratio of the exit sections which is of the order of 55%. This configuration will therefore increase the weight of the nozzle.

The truncation method is applied in the MLN nozzles to optimize the thrust/weight ratio. The related mass gain and C_F loss depend on the position of the truncation section with respect to the exit section, the number of exit Mach number M_E of the nozzle and the stagnation temperature T ₀. For example, a 30% decrease in the length of the nozzle with the four engines application results in a gain of 37% of the mass coefficient and a loss of 0.6% of thrust coefficient.

Nomenclature

A : Section area (m²)

a : Coefficient of the specific heat at constant pressure function

C ^‑ : Left-running characteristic

C ⁺ : Right-running characteristic

C_F : Thrust coefficient

C_Mass : Coefficient of the mass of the nozzle

C_P : Specific heat at constant pressure (J kg^‑1 K^‑1)

H : Enthalpy (KJ)

HT : Abbreviation for high temperature

I_sp : Vacuum specific impulse (s)

L : Length of the nozzle (m)

M : Mach number

Mass : Mass through the nozzle normalized by (ρ_M t_M ) (m²)

MLN : Minimum Length Nozzle

P : Pressure (bar)

PG : abbreviation for perfect gas

R : Thermodynamic constant of gas

T : Temperature (K)

x : Abscissa of a section in the nozzle

y : Radius of a section in the nozzle

γ : Specific heats ratio

ρ : Density (kg/m³)

θ: Flow angle deviation (rad)

µ: Mach angle

ν: Prandtl Meyer angle

ε : Tolerance of calculation (desired precision)

ξ : Downward Mach line

η : Right-running Mach line

Indices

0: Stagnation condition (combustion chamber)

*: Critical condition

S: Supersonic section

E : Exit section

M : Material

References

All Tables

All Figures

	Fig. 1 Presentation of the axisymmetric MLN nozzle.
In the text

	Fig. 2 Contour nozzle and mesh of the nozzle for ME  = 3.00 and T 0 = 2000 K. (a) Large grid. (b) Moderate grid. (c) Rather fine grid. (d) Fine grid.
In the text

	Fig. 3 Numerical results of the flow of air in an axisymmetric nozzle MLN for ME 3.00 and T 0 = 2000 K. (a) Iso-Mach curves (Ref. [19] and numerical results). (b) Iso-Directions curves (Ref. [19] and numerical results). (c) Iso-temperatures ratio (T/T 0) curves. (d) Iso-Pressures ratio (P/P 0) curves.
In the text

	Fig. 4 Thee-dimensional mesh of the axisymmetric MLN contour for T 0 = 2000 K and ME  = 3.00.
In the text

	Fig. 5 Variation of M along the wall.
In the text

	Fig. 6 Variation of θ* along the wall.
In the text

	Fig. 7 Variation of T/T 0 along the wall.
In the text

	Fig. 8 Variation of P/P 0 along the wall.
In the text

	Fig. 9 Variation of ρ/ρ 0 along the wall.
In the text

	Fig. 10 Pressure effect on the flame temperature of the combustion H2/O2.
In the text

	Fig. 11 Pressure effect on the flame temperature of the combustion N2H4/N2O4.
In the text

	Fig. 12 Pressure effect on the flame temperature of the combustion MMH/N2O4.
In the text

	Fig. 13 Pressure effect on the flame temperature of the combustion RP-1/O2.
In the text

	Fig. 14 Pressure Effect on the flame temperature of the combustion RP-1/N2O4.
In the text

	Fig. 15 Pressure effect on the flame temperature of the combustion CH4/O2.
In the text

	Fig. 16 Combustion products (%) of the bipropellants H2/O2.
In the text

	Fig. 17 Combustion products (%) of the bipropellants N2H4/N2O4.
In the text

	Fig. 18 Combustion products (%) of the bipropellants MMH/N2O4.
In the text

	Fig. 19 Combustion products (%) of the bipropellants RP-1/O2.
In the text

	Fig. 20 Combustion products (%) of the bipropellants RP-1/N2O4.
In the text

	Fig. 21 Combustion products (%) of the bipropellants CH4/O2.
In the text

	Fig. 22 Examples of combustion products (%) of the various engines.
In the text

	Fig. 23 Variation of the specific heat function CP and γ as a function of the stagnation temperature T 0.
In the text

	Fig. 24 Error calculation of CpCO2 for different interpolations polynomials.
In the text

	Fig. 25 Effect of the stagnation temperature T 0 on the profile of the nozzle for different gases.
In the text

	Fig. 26 Variation of Isp versus T 0.
In the text

	Fig. 27 Variation of CMass versus T 0.
In the text

	Fig. 28 Variation of L versus T 0.
In the text

	Fig. 29 Variation of AE/A* versus T 0.
In the text

	Fig. 30 Variation of L versus ME .
In the text

	Fig. 31 Variation of Isp versus ME .
In the text

	Fig. 32 Variation of C Mass versus ME .
In the text

	Fig. 33 Variation of AE/A* versus ME .
In the text

	Fig. 34 Influence of triatomic gas H2O with different gases.
In the text

	Fig. 35 Influence of triatomic gas CO2 with different gases.
In the text