3.4 Evaluating boundary conditions
3.4.2 Fluid parameter profiles
Before concluding this chapter, it is worthwhile to briefly examine the profiles of a few fluid parameters generated by the different flow boundary conditions, namely those of the static pressure p (Figure 3.5), the axial velocity uz (Figure 3.6), and the temperature T (Figure 3.7). The profiles are obtained from the PR-A CFD simulations run with m˙ = 100SCCM of Ar, and plotted along the central z-axis from the front of the plenum at z = −30mm to z = 10mm just beyond the discharge chamber exit. The no slip boundary condition is represented by magenta lines, while the inviscid boundary condition is represented by magenta lines. The slip boundary condition cases are represented by blue lines for
α = 0.9, cyan lines for α = 0.5, and red lines for α = 0. The α = 1 case is again almost identical to theα= 0.9case, and not shown for clarity.
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Figure 3.5: Static pressure palong the z-axis with different boundary conditions.
The inviscid and α = 0 cases immediately stand out as being unphysical, especially for
uz (Figure 3.6) and T (Figure 3.7). The flat uz profile in the discharge chamber indicates
that the flow is unimpeded, since a boundary layer does not form under those conditions. This results in very lowp in the plenum region, as seen in Figure 3.3, that are significantly
under the experimentally measuredpst. The no slip,α= 0.9, andα= 0.5cases on the other hand exhibit the correct properties of compressible flow, and are consistent with each other. Nonetheless, the selection of the value ofα must be made according to the gas species used,
as doing otherwise produces a small but noticeable discrepancy between the CFD simulation and experimental results.
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Figure 3.6: Axial velocity uz along thez-axis with different boundary conditions.
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Figure 3.7: Temperature T along thez-axis with different boundary conditions.
Figure3.8plots the axial velocityuz across the diameter of the discharge chamber exit in
PR-A(alongz= 0mm) for the different flow boundary condition cases. As mentioned earlier, there is no boundary layer in the inviscid and α = 0 cases, and uz is essentially constant
across the diameter of the discharge chamber. In the no slip case, uz at the discharge
chamber wall is exactly 0m s−1 by definition. uslip = 31.3m s−1 introduced by α = 0.9 produces a solution that best resembles the real behaviour in the main flow. For α = 0.5,
uslip = 47.6m s
−1 is not significantly higher, but the flow behaviour is sufficiently deviant from the ideal case. However, uslip = 94.5m s−1 in the α = 0 case significantly alters the
flow characteristics, and is not appropriate at the present operating conditions.
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Figure 3.8: Axial velocity uz across the diameter of the discharge chamber exit in PR-A with different boundary conditions. The slip velocity uslip at the discharge chamber wall
(r = 2.1mm) is indicated in each case: no slip (magenta + markers), α = 0.9 (blue
◦
markers), α = 0.5 (cyan◦
markers), inviscid (magenta × markers), and α = 0 (red◦
markers).Since the experimentally measuredαu and αT for most gases are close to unity (Tables
3.1 and 3.2), using α = 0 is not a physically accurate representation of real gas flows [87] although the solution is mathematically consistent. Similarly, the applicability of inviscid flow solutions is limited only to frictionless flows (e.g. superfluid He II [110]), and the use of inviscid boundary conditions is therefore strongly discouraged for most conventional prob- lems.
3.5
Chapter summary
This chapter covers slip regime rarefied gas dynamics from a theoretical, experimental, and modelling perspective. First, the theoretical concepts of mean free path, Knudsen number, and flow regimes are introduced, before a more extensive discussion of slip flow is explained with molecular dynamics. The slip boundary conditions are implemented using tangen- tial momentum and thermal accommodation coefficients sourced from various experiments, which introduce a fictitious slip velocity and temperature jump at the wall.
Next, the volume, boundary, and initial conditions for the PR CFD simulation domain are detailed. 54 CFD simulations are performed with two geometries PR-A andPR-Z which have discharge chambers of different diameters, and with Ar and N2 cold gas propellants. The stagnation pressure in the plenum is either overestimated or underestimated unless the correct tangential momentum and thermal accommodation coefficients are used, making it a clear indicator of the accuracy of the CFD simulation results. The fluid parameter profiles demonstrate how the pressure, axial velocity, and temperature vary along the length of the discharge chamber, while the axial velocity across the diameter of the discharge chamber exit reveals the slip velocity at the discharge chamber wall.
Plasma-induced heating
Plasmas are used in a wide variety of applications across research, technology, and industry. While the properties of plasmas vary according to each specific application, there is a ubi- quitous concern for the temperatures of the species present in the plasma. For weakly ionised plasmas in particular, the neutral gas temperature is an important parameter that can have significant influence on the characteristics of the plasma and its application. For example, microelectronics fabrication require stringent uniformity of the plasma which can be dis- rupted by gas flow, gradients and variations in gas temperature, as well as ion transport in regions with large voltage biases [111, 112]. Biomedical applications demand minimal heating for the treatment of heat-sensitive materials and soft tissues, and gas temperature affects the rate of chemical reactions as well as the proportion of desired reactive species in the plasma [113]. In electric propulsion, the aim is to maximise heating of the propellant with minimal input power so as to improve thrust performance over cold gas operation.
Many experimental techniques [114] have been used to characterise neutral gas temper- atures in plasmas. These include using atomic line profiles from Doppler, Stark, and van der Waals broadening, as well as rotational spectroscopy [49,51,55,115] and laser-induced fluor- escence [116]. However, rotational spectroscopy becomes unreliable at low pressures [115,
116], and therefore computational modelling techniques are required. Though immensely useful, there are few successful plasma simulation models [111,117] due to their complexity and the required expertise to produce accurate results. The biggest challenge lies in model- ling plasmas in the∼Torr pressure regime, where the pressure is high enough for collisional
effects to be significant but still sufficiently rarefied for thermal conductivity to be ineffective. For example, a model [118] of a microwave plasma chemical vapour deposition (MPCVD) system in COMSOL utilised a heat conduction equation with a single volumetric heat source term for electron-neutral collisions. While the simulation results matched with experiments
atp= 30Torr, the gas temperature was overestimated atp= 10Torr.
In this chapter, collisional heating of the neutral background gas is modelled in thePocket Rocket Ar discharge at ∼1Torr pressure. Ion transport and heat transfer mechanisms are examined, and two local models are developed to explain the spatially resolved temperature profile of the neutral gas. Ion-neutral charge exchange collisions are demonstrated to be the dominant heating mechanism. Heating is found to be significantly greater in the plasma bulk than the plasma sheath due to different plasma parameters and ion transport behaviours in these regions. The neutral gas maintains a peak temperature on the centralz-axis since radial
thermal conduction to the discharge chamber wall is ineffective in this rarefied laminar flow system. The following sections detail the input parameters for the CFD-plasma simulation before moving on to a discussion of the results.