Effect of heat transfer and geometry on micro-thruster performance

International Journal of Thermal Sciences 146 (2019) 106063

Contents lists available at ScienceDirect

International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

Effect of heat transfer and geometry on micro-thruster performance a

a,∗

K.M. Muhammed Rafi , M. Deepu , G. Rajesh a b

T

b

Dept. of Aerospace Engineering, Indian Institute of Space Science and Technology, 695547, India Dept. of Aerospace Engineering, Indian Institute of Technology Madras, 600036, India

A R T I C LE I N FO

A B S T R A C T

Keywords: Micro-nozzle NS-DSMC solver Heat transfer in rarefied flows Continuum breakdown Nozzle cluster

Coupled Navier-Stokes and Direct Simulation Monte Carlo (NS-DSMC) simulations of gas flow in micro-nozzles for various wall thermal conditions and geometrical aspects are presented. Micro-thrusters employed in miniature spacecraft and microsatellites experience substantial changes in wall thermal conditions. This can influence the internal boundary layer development and the exit plume structure of a micro-nozzle. These changes in flow physics differ with the nozzle divergence angle and the proximity of a similar nozzle in the cluster. Continuum solvers often fail to analyze the micro-nozzles operating in vacuum conditions as the flow in micronozzles experiences continuum, transitional, and rarefied regimes. Coupled NS-DSMC solver is an effective alternative that can simulate the non-equilibrium effects in a micro-nozzle flow field. A steady solution of the entire flow field has been obtained using a non-linear Harten-Lax-van Leer-Contact (HLLC) scheme based finite volume solver with a higher order slip boundary condition. Continuum breakdown regions are identified based on the gradient-length local (GLL) Knudsen number condition. This initial steady solution on the flow transition boundary is implemented in the DSMC solver as a Dirichlet boundary condition. The present computations are useful in calibrating micro-propulsion controllers to adapt to the substantial momentum changes associated with various nozzle wall thermal conditions and the proximity of similar nozzles in the cluster.

1. Introduction Micro-satellite is an emerging preference for the cost-effective space research. Miniature thrusters with the precise control over operating characteristics are essential for its attitude control, orbit maintenance, and station keeping. Micro-thrusters are designed to develop thrusts of the order of a few milli-Newtons by means of a cold gas or a chemical reaction [1]. Micro-thrusters and associated micro-propulsion technology [1–5] have been emerging as a matured research field. Cold-gas micro-thruster is a suitable propulsive device in micro-propulsion systems [6] owing to its higher specific impulse, lower mass, and lower power consumption. Micro-nozzles operating in vacuum experience flow regimes ranging from the continuum, slip-flow, transition, and free molecular conditions as the flow is evolved from thrust chamber to the ambient vacuum conditions. Numerical methods are useful alternatives for more involved experimental methods for the exploration of hypersonic rarefied gas dynamics. However, the deviation from continuum assumption necessitates the use of the advanced kinetic theory of gases to describe the rarefied gas behavior. The probabilistic particle simulation approach, known as Direct Simulation Monte Carlo (DSMC) method [7] is used in the present study to describe the micro-nozzle flow field. DSMC method are finding application in resolving rarefied ∗

gas flow behavior in various size affected domains and can accurately simulate the fluid path in higher Knudsen number flows that can influence heat transfer effects [8]. Flow characteristics of the micro-nozzles differ considerably from that of a conventional nozzle due to its large surface-to-volume ratio and low Reynolds number of the flow. Dominant viscous effects lead to the pronounced growth of boundary layers downstream of the throat of the micro-nozzle. Boundary layer growth brings down the effective nozzle-area, adversely affecting the designed area ratio, the mass flow rate, and the performance of the nozzle [9]. Moreover, micro-nozzles employed in spacecraft are often subjected to the thermal radiation from the sun which is periodic in nature [10]. Heat from the thermal radiation is transferred through the micro-nozzle wall to the working fluid increases the fluid viscosity leading to the enhancement of the boundary layer growth and flow acceleration within the nozzle. Hence these effects pose a combined Fanno-Rayleigh flow problem with an area change in the miniature flow passage. Earlier numerical studies [10–12] used continuum approach to resolve this phenomenon which has inherent limitations in describing the rarefaction effects of dilute gas flows in micro-nozzle under actual operating conditions. Louisos and Hitt [12] used a continuum model to perform a numerical study on 2-D planar micro-nozzles to investigate the effect of divergence angle

Corresponding author. E-mail address: [email protected] (M. Deepu).

https://doi.org/10.1016/j.ijthermalsci.2019.106063 Received 26 July 2018; Received in revised form 23 August 2019; Accepted 23 August 2019 1290-0729/ © 2019 Elsevier Masson SAS. All rights reserved.

International Journal of Thermal Sciences 146 (2019) 106063

K.M.M. Rafi, et al.

Nomenclature

U u vr x

Letters c cr f F G g h Isp keff L m m· n P Q″ Rt ts T To

Mean thermal speed (m/s) Relative velocity (m/s) Distribution density Flux vector in axial direction Flux vector in radial direction Acceleration due to gravity (m/s2) Heat transfer coefficient (W/m2K) Specific Impulse (s−1) Effective thermal conductivity (W/mK) Axis-to-axis distance (m) mass of a molecule (kg) mass flow rate (kg/s) Number density (m−3) Static pressure (Pa) Heat flux (W/m2) Throat radius (m) Sampling time (s) Static temperature (K) Stagnation temperature (K)

Vector of conservation variables Axial component of velocity (m/s) radial component of velocity (m/s) Axial distance (m)

Greeks γ μ λ κ ρ σ σv σT θ

Ratio of specific heat Dynamic viscosity (N-s/m2) Mean free path (m) Thermal conductivity Density (kg/m3) Collision cross-section (m2) Momentum accommodation coefficient Thermal accommodation coefficient Divergence half-angle

Subscripts Adia s T

Adiabatic Slip Thrust

Multiscale Method (HMM) to simulate non-isothermal microscale rarefied flows. Lou et al. [20] compared results of kinetic flux-vector splitting schemes for Navier–Stokes equations with DSMC method to resolve the Knudsen layers. The Knudsen layers in an impulsively started piston, a flat plate, and a lid-driven cavity were resolved using this approach. La Torre et al. [21] studied an axisymmetric micronozzle flow using a static, one-way, state-based coupled NS-DSMC solver. This method could produce comparable results with that of complete DSMC solution with a significant reduction in computational time. Espinoza et al. [22] coupled DSMC method with a two-temperature continuum solver for solving atmospheric re-entry problems involving high thermal non-equilibrium [22]. From all the above studies, it is known that a coupled NS-DSMC solution procedure is the most suitable method for resolving the compressible flow of a dilute gas through a micro-nozzle, which experiences all degrees of flow rarefaction. Often the arrays/clusters of cold gas micro-thrusters are employed in spacecraft. Ketsdever et al. [23] performed experimental and numerical studies on the effect of multiple jet interaction on the thruster performance. They used orifices of 1 mm diameter and varied their axisto-axis distance to study the thruster array flow field. Holz et al. [24] performed an experimental study on the plume interaction of two adjacent cold-flow micro-nozzles operating in near-vacuum condition. Knudsen number varies considerably along the miniature flow passage with heat transferring boundary. Akhlaghi [25] used an iterative technique for specifying wall heat flux in DSMC method for the simulation of Micro/Nano systems wherein Knudsen number varies along the flow path. Survey on existing literature on micro-nozzle performance analysis shows that studies addressing the only independent influence of various parameters of the nozzle are available. Combined influence of the parameters such as divergence angle of individual micro-nozzles, the presence of plume from similar neighboring micro-nozzles, and real wall heat transfer conditions needs to be assessed to explore the actual behavior of conical micro-nozzle arrays operating in near-vacuum condition. Aforementioned geometrical and operating conditions have a significant influence on the flow characteristics and performance micro-nozzle arrays in rarefied conditions. Nature and extent of the boundary layer thickening due to heat transfer effect differ as divergence angle of the micro-nozzle changes. This, in turn, influences the

and the throat Reynolds number on the thruster performance. It was observed that the direct scaling of the divergence angle of a macro-scale nozzle is not logical to realize the same effect in a micro-nozzle. Xu and Zhao [13] performed continuum simulations with the slip wall boundary conditions to study the shock structure in a typical over-expanded micro nozzle flow filed. The breakdown of continuum assumptions calls for the use of Boltzmann equation [7] for simulating the flow through a micro-nozzle operating vacuum. A complete numerical solution of the Boltzmann equation to simulate such flow fields is computationally intensive. Stochastic particle simulation approach can describe the molecular motions, interactions, and collisions to obtain the solution of the flow field. Park and Baek [14] simulated unsteady semi-confined micro-flow using DSMC method and observed that the thermal accommodation is influenced by wall heat flux and wall pressure. Boyd et al. [15] performed experiments with the micro-nozzles and could establish a good agreement with their DSMC computations. Studies focusing on the effects of temperature field on rarefied gases are also available in the literature. Louisos and Hitt [10] performed a numerical study on the effect of isothermal wall conditions and consequent subsonic layer growth in planar micro-nozzles. Finite Volume Method (FVM) based N–S equation solver with a first order slip model was used for this study as the flow was well within the continuum regime. They could identify a tradeoff between viscous losses and angular expansion losses. Alexeenko et al. [16] carried out a numerical study on the transient performance of Micro Electro Mechanical Systems (MEMS) thruster using DSMC method to explore the effect of viscous boundary layer growth with the increase in temperature. Hameed et al. [17] analyzed the effect of heating and cooling on the performance of low Reynolds number micro-thrusters. They observed that the wall cooling resulted in considerable reduction of subsonic boundary layer thickness and viscous losses. Particle-based numerical models are computationally inefficient when applied to near continuum flows [7]. Therefore, coupling continuum solution with the stochastic particle simulation is useful for the accurate description of a flow field with all degrees of rarefaction. There are several coupling methods proposed in the literature. Roveda et al. [18] proposed a hybrid Euler-DSMC solver for the simultaneous simulation of a continuum and rarefied flow field. Docherty et al. [19] used a point-wise coupling approach known as the Heterogeneous 2

International Journal of Thermal Sciences 146 (2019) 106063

K.M.M. Rafi, et al.

diffused under-expanded nozzle plume structure also, due to the fact that the viscous subsonic layer exited from the nozzle encompasses the under-expanded jet at the nozzle exit. The boundary layer formed inside the nozzle is significantly influenced by the wall heat transfer effects. Thus an internal flow in the micro-nozzle with the presence of heat transfer needs to be resolved for various divergence angles in order to precisely account for the boundary layer thickening phenomenon. Moreover, the flow regimes inside the nozzle are transitional in nature. Hence a coupled NS-DSMC approach is used in the present study to resolve the miniature nozzle flow fields operating under high vacuum for various flow and operating parameters. A basic experiment has also been performed to validate the numerical results. Extensive simulations have been carried out further to analyze the effects of wall heat transfer, divergence half-angle, and the presence of identical thrusters in proximity using the coupled NS-DSMC approach.

2 − σT ⎞ ⎛ 2 ⎞ ⎛ ∂T ⎞ ⎜ ⎟ λT Ts = Twall + ⎛ ⎝ σT ⎠ ⎝ γ + 1 ⎠ ⎝ ∂n ⎠wall

2. Numerical model and validation

⎯→ ⎯ ∂ (nf ) ∂ (nf ) ∂ (nf ) +→ c⋅ → + F ⋅ → = ∂t ∂c ∂r

⎜

2.2. DSMC simulation Rarefied regimes in the micro-nozzle flow field are resolved using the direct statistical simulation of the molecular processes based on the kinetic theory. The Boltzmann equation for a simple dilute gas [7,31] is ∞ 4π

∫ ∫ n2 [f ∗ f1∗ − ff1 ] cr σdΩd→c1 −∞ 0

(4)

→ where n is the number density, f is the velocity distribution function, c ⎯→ ⎯ is the molecular velocity, F is the external force acting on the molecule, cr is the relative velocity of the collision pair, σ is the collision → cross-section, dΩ is the infinitesimal velocity space solid angle, and d c1 is the infinitesimal velocity of field molecules. Direct simulation Monte Carlo method [32] is a numerical tool based on the probabilistic particle simulation technique. DSMC is used to resolve the non-continuum regimes of the flow field in the present study. DSMC method simulates the molecular motion and wall collisions deterministically and the intermolecular interactions stochastically for the solution of dilute gas flows [7]. The conventional Bird's DSMC procedure available in the open source framework, known as dsmcFOAM [33] is used to simulate the rarefied regimes of micronozzle flow field. This method makes use of a fixed-grid, fixed-timestep, collision sampling with no-time-counter (NTC) method [34]. The non-continuum regime of the micro-nozzle flow field is discretized such that the size of the control volume is well within the mean free path and sufficient DSMC particles are created in it. The variable hard-sphere (VHS) intermolecular collision model is used for simulating binary collisions. The mean free path in VHS model is computed based on the temperature coefficient of viscosity (ω) as

Continuum regime of the flow field is resolved using the two-dimensional compressible axisymmetric form of the Navier-Stokes equations (1)

here U is the vector of the conservation variables, F and G are the fluxes along axial and radial directions. A non-linear Harten-Lax-van Leer-Contact (HLLC) scheme [26] based finitevolume method (FVM) numerical solver available in a commercial package (METACOMP CFD++ [27]) is used to simulate the continuum flow. This point-implicit solver uses an automatic Courant-number adjustment procedure (ACAP) in time integration to obtain the steady-state solution. A continuous type TVD flux-limiter could improve the convergence of solution procedure. Computational domain is initially discretized into 28,100 control volumes using a block-structured 2-D grid. Subsequently, the grid is refined to 56, 200, and 112,782 finitevolumes till an invariance for the axial pressure distribution is achieved. Air is chosen as the working fluid and is modeled as a real gas. The specific heat capacity is computed as a function of temperature using the model suggested by Ehlers et al. [28]. Viscosity is computed using the Sutherland law [29]. The finite velocity-slip in the Knudsen layer in the slip-flow regime is implemented using a second order model proposed by Beskok et al. [30], given by ⎜

2 u

2κ

temperature, λT = ρmc . The accommodation coefficients σv and σT = 1 v are invoked to model perfect diffuse reflection wall with complete thermal accommodation.

2.1. Continuum simulation

2 − σv ⎞ Kn ∂u ⎛ ⎞⎛ ⎞ us = u w + ⎛ ⎝ σv ⎠ ⎝ 1 − bKn ⎠ ⎝ ∂n ⎠wall

(3)

where, Ts is the dimensionless temperature jump, Tw is the dimensionless wall temperature, σT is the thermal accommodation coef′ 1 ficient, γ is the specific heat ratio, b = u ″ and the mean free path for

Micro-nozzle operation in vacuum ambient experiences all regimes of rarefaction. Appropriate numerical methods are essential to resolve each flow regime. Mechanism of heat transfer between air flowing inside the micronozzle and its wall is assumed to be only by molecular interaction as diatomic molecules in the present rarefied conditions are not participating in radiation heat exchange. The present computational procedure involves high fidelity continuum simulation using a robust non-linear Harten-Lax-van Leer-Contact HLLC solver with higher order slip conditions on boundaries, detection of rarefaction with gradient-length local (GLL) Knudsen number condition, followed by a DSMC solution.

∂U ∂F 1 ∂ (rG ) =0 + + ∂t ∂x r ∂r

⎟

2(5 − 2ω)(7 − 2ω) ⎞ ⎛ μ ⎞ m ⎞ λ=⎛ ⎜ ⎟⎛ 15 ⎠ ⎝ ρ ⎠ ⎝ 2πkT ⎠ ⎝

(5)

The phenomenological model by Larsen and Borgnakke [35] is used to model the energy redistribution subsequent to an inelastic collision. The time-step for numerical integration is well within the mean collision time of gas molecules, such that,

Δt =

λξ c

(6)

where, ξ is a fraction used to limit time step within the mean collision time and c is the mean thermal speed of the gas molecule. 2.3. Coupling of NS-DSMC solutions A coupling-interface for the continuum and particle simulation domains has been identified based on the continuum breakdown condition proposed by Wang and Boyd [36]. This continuum breakdown criterion is based on the Gradient Length Local Knudsen number (KnGLL), given by

⎟

(2)

where, us is the dimensionless slip wall velocity, uw is the dimensionless no-slip wall velocity, σv is the tangential momentum accommodation coefficient. The temperature-jump in the Knudsen layer in the slip-flow regime is given by

Br = max{KnGLL − ρ, KnGLL − T , KnGLL − U } = 0.02

(7)

The coupling interface is a plane perpendicular to the micro-nozzle axis, wherein the breakdown occurs. A Dirichlet-type state-based oneway information transfer of number density, velocity vector and 3

International Journal of Thermal Sciences 146 (2019) 106063

K.M.M. Rafi, et al.

deviate from experimental pressure data near the core of expanding rarefied flow. Rothe [37] measured gas density and rotational temperatures of nitrogen flow through a miniature conical nozzle (throat diameter of 5 mm and divergence half-angle of 20°) using electron-beam based measurements. Nitrogen gas was maintained at an inlet stagnation pressure and temperature of 474Pa and 300 K respectively. The nozzle exit ambient was maintained at a back pressure of 1Pa. NS-DSMC predicted Mach contours of the plume from the present numerical solution (Fig. 3 (a)) indicate the presence of a supersonic bubble embedded within the nozzle along with a shock-free viscous transition to lower Mach number flow. Gas temperature data along the nozzle axis obtained from this experiment has been used for validating the present coupled NS-DSMC approach. Computed values of gas temperature along nozzle axis agree well with that of the experiment as shown in Fig. 3 (b). Authors performed an experimental study [9] to validate the numerical data on a micro-nozzle operating in near-vacuum conditions with air as the operating fluid. A micro-nozzle with 500 μm throat-radius, area-ratio of 54 and divergence half-angle of 12° was maintained at an inlet stagnation pressure and temperature of 4400Pa and 300 K respectively. A back-pressure of 2Pa was maintained at the nozzle exit. The predicted results show a sharp transition of the subsonic boundary layer from slip-flow to the free-molecular regime at the nozzle lip in the Mach contours shown in Fig. 4 (a). This effect creates significant backflow at the nozzle exit. The axial plume static pressure measurement was done using a Sankovich probe. Fig. 4(b) shows the comparison of the measured axial static pressure with that obtained using the

temperature is implemented for transferring the data across the coupling interface between the solvers. The state-based one-way spatialcoupled NS-DSMC solution process is elaborated as a flow chart (Fig. 1).

2.4. Validations of the numerical results The aforementioned solution procedure is validated based on the experimental data of three appropriate test cases for both interior and exterior of micro-nozzles. Boyd et al. [15] performed an experiment to obtain pressure distribution in an expanding nitrogen flow issuing out of a miniature conical nozzle (throat diameter of 3.18 mm and an area ratio of 100). Stagnation pressure of 6400Pa and stagnation temperature of 699 K were maintained at the inlet. A back-pressure of 0.01Pa was maintained throughout the experiment at the exit. A stagnation pitot pressure probe was used to survey the plume pressure within an error band of ± 5Pa. NS-DSMC predicted Mach number contours of the plume are shown in Fig. 2 (a). The interface for coupling the continuum and rarefied regime and the trajectory of the Pitot probe for surveying the pressure data are also shown. Salient features of the under-expanded micro-nozzle flow field such as acceleration of the subsonic boundary layer flow and subsequent turning around the lip of the nozzle with sonic line termination are also captured. A comparison of computed pressure data with that of the experiment is given in Fig. 2 (b). The pressure profile predicted by the coupled solver agrees fairly well with that of the experiments for region away from the nozzle axis. Diffuse-reflecting wall conditions are used in the present computations to model the gas-surface interactions. This model could capture the near wall effects with higher accuracy. Whereas, computed results

Fig. 1. Flowchart of the computational procedure. 4

International Journal of Thermal Sciences 146 (2019) 106063

K.M.M. Rafi, et al.

Fig. 2. (a) Mach contours (b) Comparison of pressure profiles from NS-DSMC computation and experiment [15].

Fig. 3. (a) Mach contours (b) Comparison of computed gas temperature along the nozzle axis with the experimental data [37].

adiabatic case. The computational domains used for the continuum simulations are shown in Fig. 5 (a). Dirichlet-type state-based one-way coupling is implemented at the continuum breakdown region and is indicated as the inlet for DSMC simulation (Fig. 5 (b)). Ideally, an infinite rectangular cluster of micro-nozzles needs to be modeled for accounting the effect of proximity of other similar nozzles. This has been simulated by making use of the quarter-symmetry of the conical micro-nozzle. Schematic of the micro-nozzle cluster and computational domain used for DSMC simulations are given in Fig. 6. The effect of flow interactions on the performance of micro-thrusters with respect to the change in distance between the neighboring thrusters is analyzed by varying the shortest distance between the geometric axes (L) of two adjacent micro-thrusters.

present NS-DSMC solver.

2.5. Computational domain and boundary conditions The influence of nozzle geometry, thermal conditions on the wall, and proximity of other similar nozzles on the flow characteristics of micro-nozzles are analyzed using the coupled NS-DSMC approach deliberated before. The cold-flow operation of a conical micro-nozzle with throat radius of 500 μm and an area-ratio of 54 has been considered for the present study. Air is chosen as the operating fluid with an inlet stagnation pressure and temperature of 5000Pa and 300 K respectively. An isothermal wall condition is imposed on the diverging side of the CD micro-nozzle in order to study the effect of wall heat transfer on the flow characteristics. The wall temperature is varied from 50 K to 1000 K on the diverging nozzle wall in order to consider all possible thermal conditions in a practical small-scale satellite configuration [38]. Vacuum far-field static pressure is chosen as 1Pa, correspond to the typical vacuum experienced by a micro-thruster of a spacecraft in Earth's orbit. The effect of divergence half-angle is also analyzed by simulating flows through micro-nozzles having identical area ratio with half-angles of 10° and 12° for various wall temperature conditions including an

3. Results and discussion 3.1. Effect of wall thermal conditions Non-equilibrium in Knudsen layer for the diverging part of the nozzle is analyzed in detail for various thermal and geometric conditions. There exists a significant velocity difference between the nozzle 5

International Journal of Thermal Sciences 146 (2019) 106063

K.M.M. Rafi, et al.

Fig. 4. a) Mach contours (b Comparison of axial static pressure with experiment [9].

Fig. 5. Computational domain (a) Continuum (b) DSMC.

the wall of the micro-nozzles downstream of the throat. The variation of slip velocities obtained for both the nozzle wall divergences are similar in nature. The temperature profiles in flow closer to the wall region for both nozzles for various thermal conditions are also compared and are given in Fig. 7(b). A substantial difference in fluid temperature is observed near the throat region between the two nozzle geometries. This is attributed to the changing radial gradients in the developing thermal boundary layers for various divergence angles. Effect of the nozzle wall thermal conditions on mean slip velocity and temperature-jump in near-wall region is analyzed and the results are summarized in Table 1. It can be observed that the mean velocity-slip increases with the wall temperature. The heat addition enhances the flow rarefaction and results in larger slip velocity. This highlights the effect of wall thermal condition on the transport effects in Knudsen layer. A higher amount of slip corresponding to the heated surface condition can be observed from the mean slip velocity values. Heat transfer effects are significantly influenced by the nature of boundary layer growth inside the micronozzle. The higher temperature of wall leads to elevated levels of rarefaction near the wall which in turn reduces the gas-surface interaction. Pronounced temperature jump is evident at higher surface temperature conditions. The near wall rarefaction adversely affects the momentum transfer and heat transfer across the solid-fluid interface. Thus changes in wall thermal conditions bring significant changes in flow characteristics which affect the performance of the nozzle.

Fig. 6. Schematic of a micro-thruster array.

boundary surface and the adjoining fluid flow as the flow behavior deviates from the continuum. This difference in velocity is termed as slip velocity. Computed velocity profiles of flow near the wall region for the nozzle with 12° divergence with various thermal conditions are shown in Fig. 7(a). A significant amount of velocity-slip is observed at

3.2. Heat transfer rates The difference between the fluid temperature and the ambient 6

International Journal of Thermal Sciences 146 (2019) 106063

K.M.M. Rafi, et al.

Fig. 7. (a) Velocity profile near micro-nozzle wall (b) temperature profile near micro-nozzle wall.

(8)) in the divergent part of the micronozzle for various wall temperature conditions. Heat is transferred from the expanding gas stream to the nozzle wall when the wall is maintained at lower temperature (up to 300 K). Whereas, heat transfer occur from wall to gas for higher when the wall is maintained at higher temperature (above 300 K). Rayleigh effect is significant in this scenario, heat removal from the expanding gas realizes enhanced acceleration. Heat flux is maximum near to the throat and decreases towards the exit as observed in the conventional supersonic nozzle. Temperature jump is more pronounced near to the nozzle exit to higher extend of rarefaction that adversely affect the heat transfer effects.

Table 1 Effect of wall temperature on velocity-slip and temperature-jump. Wall Temperature

Adiabatic 50 K 100 K 200 K 300 K 400 K 500 K 600 K 700 K 800 K 900 K 1000 K

Mean slip velocity (m/s)

Avg. Temperature-jump, (Tw - Tf) (K)

θ = 10°

θ = 12°

θ = 10°

θ = 12°

64.8016 36.7306 45.6397 57.0022 66.2145 73.6675 80.5950 88.1890 95.1161 102.0062 109.2506 115.8488

84.4187 29.6824 40.4225 54.1627 63.8399 72.6744 80.3993 87.4832 95.2362 102.1190 109.1408 115.8433

– −9.8613 −8.3125 −3.0725 2.8415 8.7039 13.9133 19.0042 23.5629 27.5389 31.5251 35.2294

– −8.3277 −7.9689 −3.6263 2.2191 8.3289 14.3339 19.5822 24.8242 29.5020 33.9851 38.5420

3.3. Subsonic layer development A subsonic layer is found to develop within the diverging section of the micro-nozzle. The subsonic layer thickness increases with a rise in Twall. The possible reasons are summarized from well-known theories.

temperature initiates the heat transfer across the nozzle wall boundary. It is already established that there exist significant velocity-slip and temperature-jump on the nozzle wall. There exists comparatively thicker developing boundary layer emanating from the throat and it has a significant effect on the transfer phenomena involved in the domain. The heat transfer rate is computed from the microscopic data obtained from DSMSC solution by sampling the difference of energy fluxes [39] as follows N

qw =

• The heat transfer to the fluid increases the fluid viscosity, which results in the growth of subsonic layer. • Heat added to a supersonic flow will decelerate it according to the Rayleigh flow theory. • The surface-to-volume ratio of the micro-nozzle is much larger than •

N

s ⎤ Fnum ⎡ s 1 2 1 ∑ ⎛ mc j ⎞ − ∑ ⎛ 2 mc j2 ⎞ ⎥ ts⋅ΔA ⎢ j = 1 ⎝ 2 ⎠i j = 1 ⎝ ⎠r ⎦ ⎣

its macro-scale counterpart. This will enhance both heat and momentum transfer effects leading to a higher hydrodynamic and thermal boundary layer growth than that in a conventional nozzle. The fluid temperature in near-wall region increases due to heat addition and reduces the local Mach number.

(8) The aforementioned effects often conglomerate and augment the growth of subsonic layer within the diverging side of the micro-nozzle. Louisos and Hitt [14] also have observed similar features in a micronozzles when it is operated in a slip-flow regime with a backpressure of 1000Pa. Authors also observed in an experimental study [9] that the increased viscous boundary layer within the micro-nozzle changes the

where Ns is the number of molecules interacting with the surface; i and r denote the incident and reflected molecules respectively and Fnum is the number of real molecules represented by a DSMC particle. Heat flux (Fig. 8) is estimated using the microscopic data obtained from DSMSC solution by sampling the difference of energy fluxes (Eqn 7

International Journal of Thermal Sciences 146 (2019) 106063

K.M.M. Rafi, et al.

Fig. 8. Heat flux estimates in the divergent part of the micronozzle (θ = 12°).

the present study. This is due to the tradeoff between the divergence and viscous losses. Developing boundary layer along the diverging side of the micro-nozzle becomes thicker as it grows along the length. Boundary layer growth reduces the effective area ratio and it adversely affects the delivered thrust. The effect of viscous losses will be more pronounced for the 10° expander micro-nozzle as thicker boundary layer is developed along a comparatively longer length. Whereas, the nozzle with large expander half-angle has a greater divergence loss. Substantial augmentation in the micro-nozzle performance is brought about by the effect of a pronounced surface area to volume ratio that influences heat and momentum transfer processes.

effective nozzle-area ratios. The thermal effects which lead to the growth of viscous boundary layer alter the effective area-ratio of the micro-thrusters. Further, this effect results in significant changes in the performance and flow features downstream of the nozzle exit. The subsonic layer boundary is identified by locating the sonic line from the DSMC computations. The extent of subsonic boundary layer within the nozzle for 10° and 12° (of identical area ratio) half-divergence angles are compared in Fig. 9. This clearly brings about the geometry and wall thermal dependency of boundary layer growth in micro-nozzles. It can be observed that the subsonic layer merges at the nozzle axis, downstream of the nozzle throat for wall temperatures above 700 K for a 10° half-angle divergence. Whereas the same happens only at 800 K for the 12° case. Hence, the supersonic flow expansion commences only after some distance downstream of the throat.

3.5. Analysis of the plume structure Plume structures of micro-nozzles differ considerably as a consequence of the pronounced boundary layer growth within the nozzle. The exhaust plumes of micro-nozzles operating in near-vacuum condition are analyzed for various wall heat transfer conditions and divergence angles. The effect of wall heat transfer and divergence half-angle on plume structure are analyzed in detail. A comparison of computed Mach number contours of 10° micronozzle obtained for the extreme wall temperatures (50 K and 1000 K) are shown in Fig. 11. The viscous layer emerging out from the nozzle is found to act as an outer jet boundary. This outer layer of the plume prevents its further expansion by offering an inward deflection for the flow. This effect becomes more prominent when the degree of

3.4. Performance analysis of the micro-thrusters The wall heat-transfer and its consequent effects on flow physics have a significant impact on the performance of conical micro-thruster operating in near-vacuum conditions. This is systematically analyzed here by assessing the changes in performance parameters such as net thrust and the specific impulse for various wall thermal conditions. The thrust produced by the micro-thrusters is calculated using a control volume approach, by integrating the momentum change and pressure forces acting at the nozzle exit as,

FThrust =

∫ ρu (u⋅n) dA + ∫ Aexit

Aexit

(Pexit − P∞) dA (9)

Specific impulse produced by the micro-thruster is evaluated as.

Isp =

FThrust ˙ mg

(10)

Performance prediction of coupled NS-DSMC is compared in Fig. 10 with that using a continuum solver for micro-nozzles of identical area ratios with half-angles 10° and 12°. Considerable deviation in predictions can be observed among both solution methods. Analysis of the coupled NS-DSMC computations shows that there is a significant improvement in thrust production and specific impulse when the wall temperature is increased. The thrust produced by the 12° expander is marginally higher than that of 10° expander micro-nozzle when the wall is maintained above 80 K. 10° expander nozzle is found to perform slightly better below ∼80 K for the operating conditions considered in

Fig. 9. A comparison of the extent of subsonic layer in 10° and 12°half-angle nozzle. 8

International Journal of Thermal Sciences 146 (2019) 106063

K.M.M. Rafi, et al.

Fig. 10. Micro-thruster performance (a) Thrust (b) Specific impulse.

rarefaction is higher due to the wall heating. A schematic of the underexpanded plume structure and progressive changes in shock cell characteristics due to the effect of wall conditions are shown in Fig. 12. This schematic is based on the computations performed for a conical microthruster with an area-ratio of 54, throat radius 500 μm, and a divergence half-angle of 10°. The nozzle is operated with a pressure ratio of 5000 by maintaining a back-pressure of 1Pa for various wall thermal conditions. The nature of flow transitions during the wall heating is indicated using arrow marks. The boundary of the under-expanded rarefied jet deflects outward when the wall temperature is raised. This allows the shock cell boundary to move towards the nozzle exit. Significant differences in the primary shock cell can be observed from the Mach number contours of 10° micro-nozzle obtained for the extreme wall temperatures. The expansion of the plume to near-vacuum ambient is quite gradual towards the downstream of micro-nozzle exitplane when the heat is extracted from the fluid (For lower wall temperature). Here the Mach number attains a maximum value of 5.4 forming a strong primary shock-cell, followed by a weaker one. It resembles a typical slightly diffused under-expanded nozzle plume structure. Whereas, the expansion culminates quickly with a larger plume diameter when the heat is transferred towards the fluid (For higher wall temperature). Knudsen number contours for 10° micronozzle with the same wall temperature levels are compared in Fig. 13. A significant rarefaction is developed within the nozzle. No traces of a secondary shock-cell are formed as the flow transits to the free-molecular regime. This is due to the insufficient intermolecular collisions to

Fig. 12. Schematic of the rarefied plume structure transition during wall heat addition.

form a thin discontinuity under these rarefied flow conditions. The plume is highly rarefied for all the cases, from Twall = 50 K to Twall = 1000 K. The flow is either in transition or in free-molecular

Fig. 11. Mach contour for the 10° nozzle for (a) Wall temperature = 50 K (b) Wall temperature = 1000 K 9

International Journal of Thermal Sciences 146 (2019) 106063

K.M.M. Rafi, et al.

Fig. 13. Knudsen number contours for the 10° nozzle for (a) Wall temperature = 50 K (b) Wall temperature = 1000 K

regime, which forms shocks with finite thickness and larger viscous layers. The gradients are observed to be more diffused as the flow enters the free-molecular regime. A comparison of the axial static pressures inside the two nozzles for various wall temperatures is given in Fig. 14. The axial static pressure distribution in the vicinity of the nozzle exit plane is highlighted to indicate the formation primary shock cell. It can be inferred from the axial plume static pressure distribution that the shock formed in the primary shock-cell is weak. No shock formation is observed when higher wall temperature is maintained. This is due to a higher degree of rarefaction in the plume. Here the flow is getting decelerated in the supersonic regime due to the heat addition and viscous effects. This peculiar effect was also observed in the experiments performed by Rothe in high-vacuum conditions [33] and similar experiments performed by the authors [9]. Static pressure at the nozzle exit is found to be less when lower wall temperature is maintained and this increases when the wall temperature is raised. An analysis of the effect of the nozzle divergence angle on plume structure has been carried out when the nozzle walls are maintained at extreme temperature conditions. Plume static pressures within the nozzle for various wall conditions for both nozzles are compared in Fig. 15. Higher plume pressure is experienced when a higher temperature is maintained on wall boundary. The extent of expansion is found to be higher in 12° half-angle micro-nozzle. The nature of subsonic boundary layer growth with the rise in wall temperature (as

indicated by the arrow mark) differs for micro-nozzles when the nozzle divergence angle is varied. Thus the larger viscous layer acts like a flexible-wall which adjusts the effective flow area according to the flow conditions. The developing boundary layer inside the nozzle significantly affects the plume structure at the nozzle exit when it is getting ejected out. Hence the wall temperature conditions significantly influence the overall plume structure of the micro-nozzle.

3.6. The cluster of micro-thrusters It is established in the preceding discussions that the nature of boundary layer development inside micro-nozzles influences the underexpanded plume structure. Plumes emanating from neighboring nozzles interact when they are operated in a cluster. The under-expanded plume from a micro-nozzle significantly brings down the pressure in the immediate vicinity which can influence the expansion of neighboring nozzle plumes. In addition to this effect, low-speed fluid layer emanating from nozzles interact and prevent expansion of neighboring plumes. The nature of interaction depends on the spacing between them. Hence, a typical cold-flow operation of micro-nozzle clusters is analyzed by varying the axis-to-axis distance (L) between the nozzles. The most common configuration in spacecraft is a rectangular cluster, in which the thrusters are arranged in a rectangular array as shown in Fig. 6. Micro-nozzle of divergence half-angle 12° with an inlet stagnation pressure and temperature of 5000Pa and 300 K respectively is

Fig. 14. Axial static pressure comparison for various wall temperatures (a) 10° divergence (b) 12° divergence. 10

International Journal of Thermal Sciences 146 (2019) 106063

K.M.M. Rafi, et al.

Fig. 15. Comparison of axial plume static pressures within the nozzle for various wall conditions (a) Micro-nozzle with θ = 10° (b) Micro-nozzle with θ = 12°.

Fig. 16. Mach number contours in plumes of 12° micro-nozzle cluster for (a) L = 12 mm and (b) L = 30 mm. 11

International Journal of Thermal Sciences 146 (2019) 106063

K.M.M. Rafi, et al.

chosen for the present study as well. The distances between the axes of nozzles are progressively varied as 12 mm, 20 mm, and 30 mm. An infinite rectangular cluster configuration is analyzed by simulating the one-quarter of the flow domain as shown in Fig. 6. Flow physics of micro-nozzles, when they are operated in isolation, are established in section 3.5. The maximum flow Mach number occurs inside the primary shock-cell (Fig. 11). Mach contour plots of the exhaust plume of the cold-gas micro-thruster arrays, operated at same operating conditions, for various axis-to-axis distance (L) are compared in Fig. 16. Significant changes in thruster flow features can be observed when their axial separation is varied. The gas expansion inside this primary shock-cell is more when the adjacent micro-thrusters are more separated. The single micro-thruster has a maximum flow Mach number around 5, whereas the maximum flow Mach numbers obtained for various cluster configurations are Mmax = 2.8 for L = 12 mm, Mmax = 3.7 for L = 20 mm and Mmax = 4.5 for L = 30 mm. This happens due to the merging of the subsonic, viscous low-speed fluid layers ejected from adjacent micro-nozzles which prevent the expansion of each under-expanded plume. These viscous layers are oriented parallel to the micro-nozzle axis, thereby preventing the gas expansion outside the cluster. The plume pressure distributions for the clusters of micro-thrusters for various axis-to-axis distance are calculated from the solution of DSMC simulations. The axial plume static pressure distributions of the various cluster configurations are compared with that obtained from an isolated micro-thruster (Fig. 17 (a)). Though the ambient at the exit is maintained at a pressure of 1Pa, a considerable increase in pressure can be observed in regions downstream of the nozzle. Higher axial pressure is obtained when the spacing between nozzles in the array is minimum. This is due to the significant interaction between nozzle plumes which prevents expansion in the region downstream of the micro-nozzle exit. Pressure rise due to the plume interaction is found to affect the upstream flow characteristics within the nozzle as well. Therefore, a considerable variation in performance of the micro-thruster clusters is expected when the axis-to-axis distance between them is varied. Needless to say that the performance of an isolated micro-thruster cannot be mapped to analyze a micro-thruster cluster. The radial distribution of static pressure at two axial positions (at the exit, and 4 mm from the exit) of various cluster patterns are compared with that

Fig. 18. Net thrust and specific impulse of a micro-nozzle array.

obtained from an isolated micro-thruster (Fig. 17 (b)). Over-pressurization of ambient at the nozzle exit is observed for close cluster arrays in this analysis also. The performance aspects such as net thrust and specific impulse of micro-nozzle array of various axis-to-axis distance (L) are estimated and compared (Fig. 18). The net thrust is found to decrease slightly when the nozzle separation is increased. Thereafter the thrust increases considerably with the nozzle separation. This is due to the tradeoff among various components that contribute to the thrust when the axisto-axis distance is varied. The contribution of pressure thrust component towards overall thrust is affected drastically when the microthrusters are operated with minimum separation. Whereas, the momentum component is higher for cluster configuration with smaller spacing. Specific impulse variations with various spacings of the nozzles also show a similar trend as considerable variation in mass flow rate is not expected with the increase in separation. 4. Summary and conclusions The effect of nozzle divergence and wall heat transfer on its flow

Fig. 17. Comparison of plume pressure distribution of clusters. 12

International Journal of Thermal Sciences 146 (2019) 106063

K.M.M. Rafi, et al.

features and performance is numerically studied using an efficient coupled NS-DSMC approach. Various flow and configuration parameters such the divergence angle, wall temperature, and proximity of other nozzles have been varied to predict the flow characteristics and the performance of the nozzles. The results of the coupled NS-DSMC model are validated with available experimental data. Following are the salient observations and conclusions from the present study.

[11] J. Kujawa, D.L. Hitt, G. Cretu, Numerical Simulation of Supersonic Flow in MEMS Nozzle Geometries with Heat Loss, AIAA Paper 2003-3585, 2003, https://doi.org/ 10.2514/6.2003-3585. [12] W. Louisos, D. Hitt, Optimal Expansion Angle for Viscous Supersonic Flow in 2-D Micro-nozzles, AIAA Paper 2005-5032, 2005, https://doi.org/10.2514/6.20055032. [13] J. Xu, C. Zhao, Two-dimensional numerical simulations of shock waves in micro convergent–divergent nozzles, Int. J. Heat Mass Transf. 50 (11–12) (2007) 2434–2438. [14] J.H. Park, S.W. Baek, Investigation of influence of thermal accommodation on oscillating micro-flow, Int. J. Heat Mass Transf. 47 (6–7) (2004) 1313–1323, https:// doi.org/10.1016/j.ijheatmasstransfer.2003.08.028. [15] I.D. Boyd, P.F. Penko, D.L. Meissner, K.J. DeWitt, Experimental and numerical investigations of low-density nozzle and plume flows of nitrogen, AIAA J. 30 (10) (1992) 2453–2461, https://doi.org/10.2514/3.11247. [16] A.A. Alexeenko, D.A. Fedosov, S.F. Gimelshein, D.A. Levin, R.J. Collins, Transient heat transfer and gas flow in a MEMS-based thruster, J. Microelectromech. Syst. 15 (1) (2006) 181–194, https://doi.org/10.1109/JMEMS.2005.859203. [17] A.H. Hameed, R. Kafafy, W. Asrar, M. Idres, Two-dimensional flow properties of micronozzle under varied isothermal wall conditions, Int. J. Eng. Syst. Model Simul. 5 (4) (2013) 174–180, https://doi.org/10.1504/IJESMS.2013.056692. [18] R. Roveda, D.B. Goldstein, P.L. Varghese, Hybrid Euler/particle approach for continuum/rarefied flows, J. Spacecr. Rocket. 35 (3) (1998) 258–265, https://doi.org/ 10.2514/2.3349. [19] S.Y. Docherty, M.K. Borg, D.A. Lockerby, J.M. Reese, Coupling heterogeneous continuum-particle fields to simulate non-isothermal microscale gas flows, Int. J. Heat Mass Transf. 98 (2016) 712–727. [20] T. Lou, D.C. Dahlby, D. Baganoff, A numerical study comparing kinetic flux–vector splitting for the Navier–Stokes equations with a particle method, J. Comput. Phys. 145 (2) (1998) 489–510, https://doi.org/10.1006/jcph.1998.6040. [21] F. La Torre, S. Kenjereš, J.L. Moerel, C.R. Kleijn, Hybrid simulations of rarefied supersonic gas flows in micro-nozzles, Comput. Fluid. 49 (1) (2011) 312–322, https://doi.org/10.1016/j.compfluid.2011.06.008. [22] D.E.R. Espinoza, V. Casseau, T.J. Scanlon, R.E. Brown, An open-source hybrid CFDDSMC solver for high speed flows, AIP Conf. Proc. 1786 (1) (2016) 050007, , https://doi.org/10.1063/1.4967557. [23] N. Selden, A. Ketsdever, Plume Interactions of Multiple Jets Expanding into Vacuum: Experimental Investigation, AIAA Paper 2004-1348, 2004, https://doi. org/10.2514/6.2004-1348. [24] A. Holz, G. Dettleff, K. Hannemann, S. Ziegenhagen, Experimental Investigation of Two Interacting Thruster-Plumes Downstream of the Nozzles, DLR Institute for Aerodynamic, Gottingen, 2012https://elib.dlr.de/75691/1/SP2012_2394020.pdf. [25] H. Akhlaghi, E. Roohi, S. Stefanov, A new iterative wall heat flux specifying technique in DSMC for heating/cooling simulations of MEMS/NEMS, Int. J. Therm. Sci. 59 (2012) 111–125, https://doi.org/10.1016/j.ijthermalsci.2012.04.002. [26] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: a Practical Introduction, Springer Science & Business Media, 2013. [27] S. Chakravarthy, O. Peroomian, U. Goldberg, S. Palaniswamy, The CFD++ Computational Fluid Dynamics Software Suite, SAE Technical Paper 98-5564, 1998, https://doi.org/10.2514/6.1998-5564. [28] J.G. Ehlers, S. Gordon, S. Heimel, B.J. Mc Bride, Thermodynamic Properties to 6000 Deg K for 210 Substances Involving the First 18 Elements, NASA, 1963, https://ntrs. nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19630013835.pdf. [29] W. Sutherland, LII. The viscosity of gases and molecular force, London Edinburgh Dublin Phil. Magaz. J. Sci. 36 (223) (1893) 507–531, https://doi.org/10.1080/ 14786449308620508. [30] A. Beskok, G.E. Karniadakis, W. Trimmer, Rarefaction and compressibility effects in gas microflows, J. Fluids Eng. 118 (3) (1996) 448–456, https://doi.org/10.1115/1. 2817779. [31] C. Cercignani, The Boltzmann Equation, the Boltzmann Equation and its Applications, (1988), pp. 40–103, https://doi.org/10.1007/978-1-4612-1039-9_2. [32] C. Shen, Rarefied Gas Dynamics: Fundamentals, Simulations and Micro Flows, Springer Science & Business Media, 2006. [33] T.J. Scanlon, E. Roohi, C. White, M. Darbandi, J.M. Reese, An open source, parallel DSMC code for rarefied gas flows in arbitrary geometries, Comput. Fluid. 39 (10) (2010) 2078–2089, https://doi.org/10.1016/j.compfluid.2010.07.014. [34] G.A. Bird, Perception of numerical methods in rarefied gas dynamics, Rarefied gas dynamics: Theoretical and computational techniques, (1989), pp. 374–395, https:// doi.org/10.2514/5.9781600865923.0211.0226. [35] C. Borgnakke, P.S. Larsen, Statistical collision model for Monte Carlo simulation of polyatomic gas mixture, J. Comput. Phys. 18 (4) (1975) 405–420, https://doi.org/ 10.1016/0021-9991(75)90094-7. [36] W.-L. Wang, I. Boyd, Continuum Breakdown in Hypersonic Viscous Flows, AIAA Paper 2002-651, 2002, https://doi.org/10.2514/6.2002-651. [37] D.E. Rothe, Electron-beam studies of viscous flow in supersonic nozzles, AIAA J. 9 (5) (1971) 804–811, https://doi.org/10.2514/3.6279. [38] W. Louisos, D.L. Hitt, Influence of wall heat transfer on supersonic micronozzle performance, J. Spacecr. Rocket. 49 (3) (2012) 450–460, https://doi.org/10.2514/ 1.A32206. [39] W.W. Liou, Y. Fang, Microfluid Mechanics: Principles and Modeling, McGraw Hill Nano Science and Technology Series, 2006.

• The state-based one-way spatial-coupled continuum-DSMC solver is • •

• •

•

found to be a suitable numerical procedure for analyzing a micronozzle operated in high vacuum conditions, which experiences all degrees of rarefaction. The flow within the diverging section of the nozzle is found to be in the rarefied regime, due to the prominent presence of finite velocityslip and temperature jump at the solid-fluid interface. The effect of rarefaction near the nozzle wall is large when the more heat flux is applied to the fluid. The subsonic layer present within the micro-nozzle is found to grow with the rise in wall temperature and reduces the effective nozzle area-ratio. The growth of subsonic layer is attributed by the combined effects of the larger surface-to-volume ratio in micro-nozzles, the temperature dependence of fluid viscosity, and deceleration due to the Rayleigh-flow effect. The performance of the micro-thrusters is found to be rising with the increase in wall temperature. Thrust production is found to be more for the 12° micro-thruster due to the dominance of viscous losses over the divergence-loss. It is observed from the Knudsen number profiles that the plume of various micro-nozzles considered in the present study is either in transition or in free-molecular regime. The expansion of flow to the vacuum ambient is found to be more rapid and has larger plume diameter when the wall temperature is kept higher. A feeble secondary shock-cell structure is observed when the wall temperature is minimum (Twall ∼ 50 K). Effect of the presence of a similar thruster in the vicinity of a microthruster is studied by modeling an infinite rectangular cluster. The performance of the cluster worsens when the nozzles are brought closer (L∼12 mm) due to the development of a higher pressure region downstream of the cluster.

References [1] J. Köhler, J. Bejhed, H. Kratz, F. Bruhn, U. Lindberg, K. Hjort, L. Stenmark, A hybrid cold gas microthruster system for spacecraft, Sens. Actuators A Phys. 97 (4) (2002) 587–598, https://doi.org/10.1016/S0924-4247(01)00805-6. [2] N. Chigier, T. Gemci, A Review of Micro Propulsion Technology, AIAA Paper 2003670, 2003, https://doi.org/10.2514/6.2003-670. [3] S.K. Chou, W.M. Yang, K.J. Chua, J. Li, K.L. Zhang, Development of micro power generators–A review, Appl. Energy 88 (1) (2011) 1–16, https://doi.org/10.1016/j. apenergy.2010.07.010. [4] M.M. Micci (Ed.), Micropropulsion for Small Spacecraft, AIAA, New York, 2000. [5] A. Ketsdever, J. Mueller, Systems Considerations and Design Options for Microspacecraft Propulsion Systems, AIAA paper 1999-2723, 1999, https://doi. org/10.2514/6.1999-2723. [6] A. Eriksson, L. Stenmark, Cold Gas Micro Thrusters-Final Edition, AIAA Paper 20025765, 2002, https://doi.org/10.2514/6.2002-5765. [7] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon, Oxford, 1994. [8] E. Roohi, V. Shahabi, A. Bagherzadeh, On the vortical characteristics and cold-tohot transfer of rarefied gas flow in a lid driven isosceles orthogonal triangular cavity with isothermal walls, Int. J. Therm. Sci. 125 (2018) 381–394, https://doi.org/10. 1016/j.ijthermalsci.2017.12.005. [9] K.M.M. Rafi, B.A.H. Fahd, M. Deepu, G. Rajesh, Experimental and numerical studies on the plume structure of micro-nozzles operating at high-vacuum conditions, International Symposium on Shock Waves, Springer, 2017, pp. 927–936, , https:// doi.org/10.1007/978-3-319-91020-8_111. [10] W. Louisos, D. Hitt, Heat Transfer & Viscous Effects in 2D & 3D Supersonic Micronozzle Flows, AIAA Paper 2007-3987, 2007, https://doi.org/10.2514/6.20073987.

13