Figure6.7bshows the more commonly used method for calculating the thrust force in con- ventional nozzles. Instead of summing all the forces from the exterior acting upon the volume of propellant contained within the thruster, the ‘internal forces method’ involves summing all the forces acting upon the surroundings from the volume of propellant within the interior of the thruster. Using this method, the net thrust force is given by:
Ft= (pst−pex)Aex−Fbl (6.5)
where Fbl (magenta → arrows) is the friction force between the boundary layer and the
wall. However,Fbl is almost always omitted when applying (6.5) to continuum regime flow
(Kn.0.01) in conventional nozzles. While this is a valid decision in those circumstances,
Fbl is crucially important in slip regime flow (0.01 . Kn . 0.1) in microthruster nozzles, where the boundary layer plays a significant role in dictating the overall flow behaviour. In PRin particular, the magnitude of Fbl is on par with that of the net thrust force Ft (Table 7.1). NeglectingFblcan thus lead to a gross overestimation of Ft to be twice or more of the actual value. Conversely, in the general thrust equation (6.4), the effects ofFbl are already imprinted in the axial velocityuz profile, and do not require separate treatment.
To explain the origin of Fbl, suppose an inviscid fluid is flowing through the discharge
chamber with a frictionless, adiabatic wall. Then the axial velocity of the flow is high and uniform across the diameter and length of the discharge chamber, and pst in the plenum is low since the gas is able to leave the discharge chamber without any resistance. In this hypothetical case, whereFbl = 0N, the total thrust is merely the pressure force difference between the internal front and rear faces of the plenum, given by(pst−pex)Aex.
Suppose that the flowing fluid was then imbued with the viscosity and friction (repres- ented by the tangential momentum accommodation coefficient αu) of a real gas, then the gas molecules incident on the wall must slow down due to the friction force, and the axial
velocity of the surrounding flow also decreases due to the viscosity of the fluid. This produces a velocity profile that is peaked in the middle of the discharge chamber, as expected with laminar pipe flow in the continuum and slip regimes. The deceleration of the boundary layer flow compared to the main flow is evidence that momentum is being transferred from the flow to the wall through friction and viscosity effects, resulting in a force Fbl acting in the
direction of flow. Since this direction is opposite to the direction of intended motion, Fbl
detracts from the total thrust.
As a direct consequence of Fbl, pst in the plenum increases as the flow through the discharge volume becomes restricted by the boundary layer effects. Additionally, if the wall was nonadiabatic, the wall material then acts as a thermal source (or sink), further increasing (or decreasing) pst accordingly. For flows on larger scales or where Kn → 0,pst
remains mostly unaffected and the magnitude of Fbl is negligible when compared with the net pressure force term in Equation (6.5). However, on miniature scales such as in PR,pstis greatly inflated and Fbl manifests as an unavoidable and significant fraction of the inflated
net pressure force. Hence, using the inflatedpst without accounting forFblinevitably results
in a overestimation of the net thrust.
In practice, quantifying Fbl by itself is greatly nontrivial. Nevertheless, Fbl can be cal- culated by equating (6.4) and (6.5) when all the other variables are known. Since Fbl is
dependent on the location and extensiveness of the boundary layer, it is ultimately depend- ent on the geometry of the microthruster. However, despite its inconvenience, information on the magnitude ofFbl can be used to optimise the surface properties or geometry of the microthruster, and thereby maximise performance and efficiency.
6.4
Chapter summary
This chapter explores modelling vacuum expansion in CFD simulations, without the use of hybrid or DSMC methods, by taking advantage of the flow velocity choking phenomenon. Flow velocity choking is a compressible flow effect, and occurs only when there is a sufficiently steep pressure gradient in the flow system. The velocity of the flow is accelerated by the pressure gradient up to the local sound speed. The location of the sonic surface depends on the geometry of the stream tube. In the cylindrical geometry of PR and MiniPR, the sonic surface is parabolic in shape. In the constricted nozzle geometry of PR-C, the sonic surface forms at the throat where the area of the stream tube is a minimum. CFD simulations performed with a vacuum downstream region are compared with those performed with a nonzero background pressure valid for fluid numerical methods, and the results are consistent.
Thrust force is calculated using the integral form of the general thrust equation, as the fluid parameter profiles are not uniform across the exit surface due to significant boundary layer effects in the slip regime. This is in contrast to conventional nozzles in the continuum regime, in which the boundary layer friction force is often ignored. The computed thrust force forMiniPRis compared against experimental measurements obtained using a pendulum thrust balance in a space simulation chamber. Discrepancies between the CFD simulation and experimental results are ascribed to experimental error, and the CFD simulation results are shown to be reasonable as the computed specific impulse is in line with theoretical pre- dictions once a justifiable correction is applied to experimental results. The CFD simulation results may be compared against future experimental results upon the integration of thePR propellant and RF power subsystems.
Discharge geometry
As discussed in Chapter6, the geometry of the discharge chamber has a significant influence on the cold gas flow behaviour inPocket Rocket. Extra factors come into play during plasma operation: in addition to flow velocity choking and boundary layer effects, the plasma is also influenced by electric potentials, plasma sheath dynamics, chemical reactions, neutral gas temperature, and neutral pumping. A comprehensive CFD-plasma model is necessary for understanding these processes, and how each of these can be controlled by shaping the physical and electrical geometry of the discharge.
This chapter presents CFD-plasma simulations of three geometrical variations ofPocket Rocket: the original PR-O (Figure 2.8) with a cylindrical discharge chamber, PR-C (Figure
6.1) with a constricted nozzle introduced in Chapter 6, and a new design prototype PR-N (Figure7.1) featuring a sculpted converging-diverging nozzle that achieves desirable plasma confinement while providing higher performance. Slight modifications to the CFD-plasma modelling technique allows the outlet boundary static pressure to be set to a lower value of
p0 = 0.1Torr, thereby producing a CFD-plasma model that gives the closest approximation of real vacuum scenarios. While numerical limitations prevent plasma modelling in a true vacuum environment, results may be obtained by extrapolating from plasma simulations per- formed in a pressurised environment, using the performance delta from cold gas simulations performed in both environments.