Although CFD-ACE+ has an inbuilt physical properties database, the default entries for the solid materials and fluid species are not sufficiently detailed for the CFD-plasma studies presented in this thesis. In particular, the inbuilt database lists constant values for fluid parameters such as dynamic viscosityµ and thermal conductivity k, when they are in fact
functions of temperature. This is especially important in the CFD-plasma simulations ofPR as the fluid species are subject to a wide range of temperatures from ∼100K to∼1000K. Consequently, the physical properties database has to be populated with the appropriate temperature dependent data for each and every solid material and fluid species used in the CFD-plasma simulations. The comprehensive use of temperature dependent physical properties represent a significant improvement of the CFD-plasma modelling technique over previous works [15,60,66] which used only constant values.
2.3.1 Solid materials
The solid materials considered are Al2O3, Al, Cu, Macor, and zirconia (ZrO2). The solid phase specific heatcp is calculated using the Shomate equation:
with coefficients sourced from [67] within the stated nominal temperature range at a resolu- tion of∆T = 25K. The other physical properties include mass density ρ, thermal conduct-
ivity k, electrical resistivity ρel, and relative permittivity εr. These data are sourced from
[68–70], and typically have sparse data points. The data is left as is for approximately linear curves, but fitting is required for some parameters to improve smoothness. Figures 2.12
and 2.13 plot the more important physical properties for Al2O3 and Cu respectively. For example,k(T) of Al2O3 (Figure 2.12, red line) needs to be fitted at a higher resolution of∆T = 25K, but k(T) of Cu (Figure 2.13,red line) is sufficiently well described using the regular piecewise linear interpolation forT ≥300K.
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Figure 2.12: Temperature dependent physical properties of Al2O3: mass densityρ(T)(black line), solid phase specific heat cp(T) (blue line), and thermal conductivity k(T) (red line), all fitted at ∆T = 25K resolution.
The electrical resistivity ρel for electrically conductive materials such as Al and Cu uses
the relation:
ρel(T) =ρel(300K)·[1 +αel(T −300K)] (2.7)
where αel is the temperature coefficient for electrical resistivity, fitted to the data points
for calculating the temperature dependent ρel(T) from the reference value at T = 300K. Dielectric materials such as Al2O3, Macor, and ZrO2 are considered as electrical insulators, and therefore the relative permittivityεr applies instead ofρel.
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Figure 2.13: Temperature dependent physical properties of Cu: solid phase specific heat
cp(T) (blue line, fitted at ∆T = 25K resolution), thermal conductivityk(red•markers) with linear interpolation (red line), and electrical resistivity ρel (black • markers) with linear fitρel(T) (black line).
2.3.2 Fluid species
The fluid species considered are Ar, Xe, and N2. As an example, Figure 2.14 plots the dynamic viscosity µ(blue line) in units of[×10−5Pa s]and thermal conductivity k(red
line) in units of[×10−2W m−1K−1]for Ar gas atp= 1Torr over the range84K≤T ≤
700K. These fluid parameters vary considerably with temperature, and thus using default constant values may introduce nonideal deviations from the accurate result. Temperature dependent data for µ, k, and the specific heat at constant pressure cp are sourced from
[67]. As these parameters do not vary significantly with pressure, the isobaric dataset at
p= 1Torr and∆T = 1K resolution is selected, and is adequate for the0Torr≤p≤10Torr range that the PR CFD-plasma simulations are conducted in. CFD-ACE+ uses piecewise linear interpolation to calculate parameter values between data points.
Figure2.14also plots the mass diffusivityD(T)(black line) in units of[×10−2m2s−1]. Temperature dependent data for D is sourced from both theoretical calculations of inter-
molecular forces [71] and potential energy functions [72], as well as diffusion cell experiments with gases doped with tracer isotope species [73–78]. The value ofD is usually quoted for p= 1atm. Thus, it is necessary to convertDatmto Dat the nominalp= 1Torr regime via:
D= ρatm
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Figure 2.14: Temperature dependent physical properties of Ar at 1Torr: mass diffusivity
D(T) (black line, fifth order polynomial fit), dynamic viscosity µ (blue line), and
thermal conductivityk(red line), at ∆T = 1K resolution.
whereρatmandρare the temperature dependent mass density of the species atp= 1atm and
p= 1Torr respectively. Since theDatm measurements are discrete, a fifth order polynomial in T is fitted to the experimental data. The conversion to D is then calculated at finer
intervals of ∆T = 1K with ρatm and ρ sourced from [67], and a fifth order polynomial in T (Figure 2.14) is fitted to the calculatedD values. D(T) is preferred to using a constant Schmidt number Sc in:
D= µ
ρ·Sc (2.9)
as Sc also tends to vary withT.
The first approximation of the Chapman-Enskog theory [79, 80] for calculating D12 in
[m2s−1]of two species with molecular massesm1 andm2 is: D12= 1.41×10 −4 pσ122 Ω ·T 3 2 1 m1 + 1 m2 12 (2.10) wherepis the static pressure in [Torr],T is the temperature in [K], andσ12= 1/2 (σ1+σ2) is the average Lennard-Jones collision diameter of the two species in [Å]. Ω is a dimen- sionless quantity of order one, characteristic of the integration of the interaction between the two species [80]. Note that for the referenced experiments, the self-diffusion coefficient
D = D11 is obtained by putting m2 = m1 even though different isotope species are used. This assumption is generally valid as the mass dependent term in (2.10) is insensitive to
species with molecular massm&20u [75]. Overall, (2.10) is accurate to ∼10 %[80] when calculatingD11, and is also approximately valid for electronically excited species.
Temperature independent parameters for the fluid species include the following. The molecular massm for each fluid species is a constant, sourced from [81]. For CFD or CFD-
plasma simulations involving multiple fluid or neutral plasma species, the Lennard-Jones characteristic energy σ and collision diameter σ [82] are used for calculating the Lennard-
Jones potential: Vσ(r) = 4σ σ r 12 −σ r 6 (2.11) which describes the interaction between a pair of neutral molecules separated by a distance
r. For CFD-plasma simulations, additional parameters such as the electrical resistivityρel,
relative permittivity εr, charge exchange cross section σq, and polarisability volumeα0 [83]
are required and loaded into the physical properties database.