Compressible Navier-Stokes equations

To consider the continuum flow inside a thruster nozzle, the 2-D axisymmetric compressible N-S equation in Eq 1 is adopted in the present study. The governing equations include the continuity, momentum, energy, and turbulence equations for the gas phase in vector form as follows (1) where z and r are the axial and radial coordinates, respectively [11]. Q is the conservation variables of the flowfield, E and F are the inviscid flux vectors while E_v and F_v are the viscous flux vectors in the z and the r direction as presented in Eq 2. (2) Here, ρ, u, v, e, h and p are the density, axial and radial velocity components, total energy, enthalpy and pressure, respectively. k and ω are turbulent kinetic energy and specific dissipation rate. Also, τ and q denote shear stresses and heat fluxes [12]. And H is the axisymmetric source term defined as below [11].

(3)

Total energy, enthalpy and the pressure of flowfield are given by the following equations (4) (5) where Y_i and M_i are the mass fraction and the molecular mass of the product gases, and R is the universal gas constant (8314.41 J/kmol K), respectively [11]. As the physical dimension of the low thrust rocket engine is small, the thickness of the wall boundary layer cannot be disregarded. To account for the turbulence behavior of the low Reynolds number flow in the viscous sub-layer and the high velocity flow region away from the wall together, the shear-stress transport (SST) k−ω turbulence model is used in the present study.

For an efficient calculation, the governing equation of Eq 1 is discretized and integrated over each grid cell according to the finite-volume methodology as Eq 6, and then solved using the implicit method [11].

(6)

Direct simulation Monte Carlo method

Generally, the characteristics of the flow regimes can be classified by the Knudsen number (Kn) which is defined as the ratio of a mean free path and a characteristic length. The application of proper fluid models and equations is divided depending on the Knudsen number range in Fig 3 [5,6]. If the Knudsen number is near or greater than one, the mean free path of a molecule is comparable to a length scale of the physical system. Thus, the continuum assumption such as the Navier-Stokes equation does not guarantee a good approximation for the high Knudsen number flow any more. In such case, the Boltzmann equation in a nonlinear form (Eq 7) should be dealt with statistical methods for the rarefied flow regime [5,6].

(7) Here, n, , , f and v_r are number density, velocity vector, position vector, probability density function and relative velocity of molecules, respectively. Also, , , dΩ and σ are external force vector, molecules of class with velocity v₁, elementary solid angle and collision cross section [5,6]. Among various methods, the Direct Simulation Monte Carlo (DSMC) method originally developed by Bird in 1960s [5,6] can be considered as the most effective technique for solving the nonlinear Boltzmann equation for the rarefied flow modeling. The DSMC is a direct particle simulation method based on kinetic theory and it uses a statistical method in which the number of representative simulated molecules is traced in space and time which models the physics of the real gas [5,6]. Thus, although the DSMC method generally requires a longer computational time than conventional continuum flow models, the present study uses the two dimensional axisymmetric DSMC code to describe the exhaust plume gas behaviors in the vacuum condition because conventional computational fluid dynamic schemes such as FDM (Finite Difference Method) and FVM (Finite Volume Method) do not predict accurate flow behaviors reasonably for the high vacuum regime. The present study uses the two dimensional axisymmetric DSMC code to describe the expanding low thrust plume gas flow into the vacuum condition. The variable hard sphere (VHS) model [5] is used as the intermolecular-collision model and the no-time counter (NTC) method is for the collision sampling technique [5]. The Larsen-Borgnakke model [16] is employed to redistribute the translational and the internal energy exchange between the colliding molecules. Because the overall plume temperature far from the nozzle exit is not so high, only the rotational mode is considered while the chemical reaction and vibrational mode excitation are neglected.

Thermodynamic properties in thruster chamber

At first, the chemical equilibrium reactions of the monopropellant and the bipropellant in the combustion chamber are calculated individually to estimate the composition of the chemical species of the plume gas flow and its highest combustion temperature at the stagnation condition. Here, hydrazine (N₂H₄) and a combination of monomethylhydrazine (MMH, CH₃N₂H₃) and nitrogen tetroxide (NTO, N₂O₄) are chosen for the monopropellant and the bipropellant in the present study because they are typically used for space propulsion application.

In case of hydrazine, thrust is provided by the catalytic decomposition and the final combustion product gases consist of three main species, H₂, N₂, and NH₃ following overall chemical equilibrium process in Eq 8 [11,17]. (8) (9) where f represents for the extent of ammonia dissociation at given temperature [11,17]

For the bipropellant thruster, the more complicated combustion process between the fuel and the oxidizer are involved. Overall chemical equilibrium reaction of MMH and NTO can be defined in Eq 10, where α is Stoichiometric coefficient and n_pi is the mole number of each product species [18,19]. (10) To solve Eq 10, the following nonlinear equations based on the mass conservation equations of the major elements (C, O, H, and N) and the equilibrium constants of eleven elementary equilibrium equations are calculated numerically at the same time [18,19]. (11) Here, n_t is total number of product gas moles and K_pi is the equilibrium constant defined by G, h, S which are Gibbs free energy, enthalpy, and entropy under standard condition in Eqs 12 and 13, respectively.

(12)(13)(14)

The thermodynamic parameters of specific heat capacities of gases (C_p), enthalpies (h), and entropies (S) are determined based on polynomials of temperature [20].

(15)(16)(17)

When the entire mole numbers of each product species are determined, adiabatic flame temperature can be predicted by the general energy equation of the chemical equilibrium reaction in Eq 18. (18) where ΔH^o_r is the heat of reaction under a standard condition and T_ad is an adiabatic flame temperature [18,19]. To calculate mole numbers of each product and the adiabatic flame temperature, Eqs 11–18 were iterated numerically by the Newton method with an initially assumed adiabatic temperature and a chamber pressure. Then, the molecular mass (M) and specific heat at constant pressure and specific heat ratio of the mixture of the combustion product gases can be calculated by the following equations [18,19].

(19)(20)(21)

The mixture ratio by mass is calculated from the reactants of the chemical reactions as follows (22) where m is a mass and n is a mole number of the fuel and the oxidizer, respectively [18,19].

The final stagnation flow conditions of each propellant with the composition of the major chemical species of the plume gas flow and its highest combustion temperature are summarized in Table 1.

Continuum gas flowfields inside thruster nozzle

Because plume gas flowfields in the vacuum space are influenced dominantly by the continuum nozzle flow inside the thruster, accurate prediction of the plume flow properties at the nozzle exit plane is important to specify the inflow boundary condition of the plume analysis. Thus, the computational fluid dynamics (CFD) method based on the Navier–Stokes (N–S) equations was used to simulate continuum gas flowfields inside the thruster nozzle. Fig 4A shows a brief configuration of the small thruster considered in this study with hydrazine as the monopropellant and the MMH-NTO combination as the bipropellant together. It has a conical shape nozzle with an expansion ratio of 50:1 to produce roughly a five newton thrust when the stagnation chamber pressure, p_c, is 1.45 MPa. The computations were performed for a calculation gird with about 2,500 nodes along the nozzle axis and radius inside the thruster show in Fig 4B. For the boundary conditions, the stagnation flow data inside the chamber were used for the nozzle inlet condition, which are specified from previous chemical equilibrium reactions in Table 1. The product gas species are assumed to be a mixture of perfect gases with chemically frozen compositions during the expansion process through the nozzle. At the center line of the nozzle, the velocity component and the derivatives of all other properties in the radial direction are zero by the axisymmetric condition. The nozzle wall is assumed to have an adiabatic and no-slip condition. Additionally, an extrapolation condition is imposed at the outflow boundary.

Fig 5 presents the major analysis outcomes of the continuum nozzle flow inside the monopropellant and bipropellant thrusters. The monopropellant results occupy the upper portion and the bipropellant results are shown in the lower portion of the given figures. Because the adiabatic stagnation temperature of the bipropellant MMH-NTO is above 3,000 K initially in the chamber, Fig 5A shows that the whole gas flow under the expansion process still maintains much higher temperature levels over the entire nozzle region than that of the monopropellant for which the decomposition temperature is about 1,300 K. For example, the analysis predicts temperatures at the center of the nozzle exit plane to be about 690 K for the bipropellant and 260 K for the monopropellant, respectively. Additionally, it can be deduced that an even higher velocity profile will be estimated in the axial direction for the bipropellant thruster shown in Fig 5B because the exhaust velocity of the nozzle increases proportional to the chamber temperature following rocket performance theory. On the other hand, a denser gas flow is produced from the monopropellant decomposition process and spreads all over the nozzle domain shown in Fig 5C in contrast to the temperature distribution because the internal nozzle flow is assumed to obey the perfect gas law in Eq 5, which describes an inverse relation between the density and the temperature at a given pressure. Moreover, from the Mach number result in Fig 5D, it was found that a supersonic gas flow starts to develop through the nozzle throat and is accelerated above Mach number 5 as it approaches toward the nozzle exit. Although the axial velocity component of the bipropellant combustion gas is much faster, the Mach number of the monopropellant gas is predicted to be slightly higher than that of the bipropellant result at the core of the nozzle exit because the Mach number is defined as a function of the reciprocal of a square root of the gas temperature. The temperature of the monopropellant gas is estimated to be about 400 K lower than that of the bipropellant gas.

The final combustion gas properties at the nozzle exit plane are summarized in Fig 6 including the density, temperature, and two velocity components. Closer to the nozzle wall, large variations in the flow properties are observed in the given profiles across the compressible boundary layer region. For the given chamber stagnation conditions, the numerical solutions predict a thrust of 4.998 and 4.993 N for the monopropellant and the bipropellant cases, respectively.

Rarefied plume gas flow in vacuum space environment

Together with the ten gas mixture composition including Table 1, the continuum flowfield results obtained from the N-S equations were applied as inlet conditions at the nozzle exit for the DSMC simulation of the plume flow in the vacuum. Fig 7 shows the calculation domain and boundary conditions for the DSMC method used in this study. The center of the nozzle exit plane is located at the point (0,0), and the size of the calculation domain is 3 m in the forward axial direction and 1 m in the radial direction to compare the plume expansion phenomena widely in the vacuum region. Additionally, -0.5 m backward axial region was considered to investigate the plume backflow generated from the boundary layer effect around the nozzle lip. The entire calculation domain is assumed to be a vacuum condition, while the axisymmetry condition is applied to the bottom r = 0 axis. The domain consisted of about 7,100 nodes and 13,800 cells with triangular grids. Almost 250,000 particles were generated when a steady state converged solution was achieved. During the DSMC calculation, the plume flowfield was sampled every 30 time steps for 10,000 iterations.

Major differences were examined in the plume gas flowfield between the monopropellant hydrazine and bipropellant MMH-NTO combination under the vacuum condition, and Figs 8–10 show the major analysis results with the upper-half region for the monopropellant and the lower-half region for the bipropellant, respectively. Total three and fifteen gas species are considered for the monopropellant and the bipropellant plume flow, respectively. First, it is clearly observed from Fig 8 that some amount of ejected plume flow turns suddenly at an angle larger than 90° around the nozzle lip due to viscous boundary layer effects when it comes out into the vacuum condition. Both hydrazine and MMH-NTO plume gases expand in such a similar form into the thruster backflow region, which may cause various plume impingement effects directly on the spacecraft surfaces. In addition, the difference of the overall temperatures between the hydrazine and MMH-NTO plume flow gases are compared in Fig 9A. The present DSMC method predicts that the higher temperature plume gas of MMH-NTO spreads over the whole calculation domain based on the temperature profile at the nozzle exit plane in Fig 6. This indicates that the amount of thermal energy released from the chemical reactions of the propellant is dominantly dependent on the adiabatic temperature inside the thruster chamber and also it can affect the exhaust plume gas flow expanding through the nozzle. For further comparison, the overall temperature variations are plotted in Fig 9B following a radial axis direction at z = +1 m and -0.5 m, respectively. In the case of the hydrazine plume, its temperature decreases gradually about from 260 K at the center of the nozzle exit to 20 K at z = +1 m following the forward axial direction, while the temperature of the MMH-NTO plume gas varies between 690 K at the center of the nozzle exit and 83 K at z = +1 m. Especially, more severe temperature deviations are observed in the backflow region in which the MMH-NTO plume gas expands from 2,054 K at the nozzle lip and 537 K at z = -0.5 m whereas the hydrazine plume gas falls from 790 K at the nozzle lip to 185 K at z = -0.5 m. Therefore, a possibility is predicted that a more excessive thermal load may be transferred to the spacecraft surfaces when the MMH-NTO bipropellant thruster is used because it ejects hotter plume gas particles than that of the monopropellant thruster. In contrast to the overall temperature distributions, Fig 10A illustrates that the hydrazine plume gas flow is predicted to spread more densely all over the calculation domain including the backflow region than that of the MMH-NTO gas because a higher density profile is initially applied at the inflow condition of the DSMC method based on the continuum flow results inside the thruster nozzle. For further comparison, the density variations are shown in Fig 10B in the radial axis direction at z = +1 m and -0.5 m, respectively. As the plume gas flow expands gradually in the vacuum space, the density of the gas ejected from the MMH-NOT thruster falls to below 2.3E-8 kg/m³ (6.8E+17 No./m³ for the number density) at z = +1 m in the forward axial direction while it remains over 2.9E-8 kg/m³ (1.2E+18 No./m³) for the hydrazine plume gas. For the backflow region at z = -0.5 m, a higher density of the hydrazine plume gas is also estimated to be about 3.3E-10 kg/m³ (1.4E+16 No./m³) compared to the 2.0E-10 kg/m³ (5.9E+15 No./m³) of the MMH-NTO bipropellant. Consequently, the disturbance influences due to the higher collisions of the plume backflow particles on the spacecraft surfaces may be possible to be increased when the monopropellant hydrazine thruster is used instead of the MMH-NTO thruster.

As the final result, density distributions of the plume gas species were compared. Among the various species in Table 1, H₂ and N₂ were selected as the representative compositions because these are commonly included in the product gases of both propellants. Actually, the different distributions of the plume gas compositions are caused by nonequilibrium species separation due to a sudden expansion into the rarefied region. Figs 11A and 12A illustrate that the H₂ and N₂ species in the hydrazine plume gas are distributed more widely over the calculation domain than that of the bipropellant because of the higher density of the overall plume gas. Especially, large deviations are observed for H₂ at the main stream (z = +1 m) and backflow (z = -0.5 m) regions in Fig 11B between the two propellants because H₂ occupies a considerable amount of the exhaust hydrazine plume gas rather than N₂ in Fig 12B. Thus, a possibility arose that the surface contamination by the deposition of H₂ molecules onto the spacecraft may influence on the sensitive equipment when the monopropellant hydrazine is used as a propellant instead of the bipropellant MMH-NTO.